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Hitting two BSM particles with one lepton-jet: search for a top partner
decaying to a dark photon, resulting in a lepton-jet
K. du Plessis1, M. M. Flores1,2, D. Kar1*, S. Sinha1 H. van der Schyf1,
1 School of Physics, University of Witwatersrand, Johannesburg, South Africa.
2 National Institute of Physics, University of the Philippines, Diliman,
Quezon City, Philippines.
*<EMAIL_ADDRESS>
## Abstract
A maverick top partner model, decaying to a dark photon was suggested in Ref.
[1]. The dark photon decays to two overlapping electrons for dark photon
masses of $\approx 100$ GeV, and results in a so-called lepton-jet. The event
includes a top quark as well, which results in events with two boosted
objects, one heavy and the other light. We propose a search strategy
exploiting the unique signal topology. We show that for a set of kinematic
selections, both in hadronic and leptonic decay channel of the SM top quark,
almost all background can be eliminated, leaving enough signal events up to
top partner mass of about 3 TeV for the search to be viable at the LHC.
###### Contents
1. 1 Introduction
2. 2 Maverick Top Partners
1. 2.1 Brief Survey of the Model
2. 2.2 Event Generation
3. 2.3 Current experimental bounds on VLTs
3. 3 Analysis strategy
1. 3.1 Overview
2. 3.2 Analysis with the top quark decaying hadronically
3. 3.3 Analysis with the top quark decaying leptonically
4. 4 Conclusions
## 1 Introduction
The Standard Model (SM) of particle physics [2] has by far been successful in
explaining almost all the experimentally observed microscopic phenomena.
However, it is deemed an incomplete theory due to a number of reasons, one of
which is its failure to explain dark matter (DM), whose existence is
empirically established from various astrophysical data [3, 4]. Although the
presence of dark matter is pretty much a consensus, little is known regarding
its interaction with itself and with the elementary particles of SM. One
possibility is that dark matter particles may interact through some dark force
akin to the electromagnetic force felt by ordinary particles. This leads to
the conjecture of the existence of a new gauge boson that would mediate this
dark force, analogous to the role of the photon in Quantum Electrodynamics.
For this reason, the new gauge boson is referred to in literature as the dark
photon [5] (which will be denoted in this paper as $\gamma_{d}$), although the
terms paraphoton [6] and U-boson [7] have been used.
In this paper, we propose a new search strategy for a chimera-like scenario,
where a dark photon will come not from known SM particles, but rather from
hypothetical vectorlike quarks (VLQ), $T$ (also known as top partners), dubbed
the maverick top partners [1], i.e., VLTs [8, 9, 10, 11] with unconventional
decays. The primary model is motivated by [1], in which a top-partner is
charged under both the SM and a new gauge $U(1)_{d}$ group, where the SM is
neutral under the $U(1)_{d}$.
The paper is divided into the following sections: in Section 2 we discuss the
general idea of maverick top partners and the model in concern, followed by
event generation procedure which is provided in 2.2. In Section 3, we discuss
the particle - level analysis performed for the model in question, and
estimate our ability to experimentally probe such a chimera-like scenario, and
finally present our conclusions in Section 4.
## 2 Maverick Top Partners
### 2.1 Brief Survey of the Model
There are theoretical reasons to believe that dark photons could be massless
or massive [5], and their respective features and experimental signatures are
quite distinct. In this paper we focus on the massive photon since it is more
readily accessible in the collider searches (due to the fact that it couples
to ordinary matter).
Typical production mechanisms of massive dark photons include [5, 12]:
* •
Bremsstrahlung, whereby an incoming electron scatters off a nuclei target (Z)
and emits dark photon, i.e., $e^{-}Z\rightarrow e^{-}Z\gamma_{d}$;
* •
Annihilation, whereby an electron-positron pair annihilates into a photon and
a dark photon, i.e., $e^{-}e^{+}\rightarrow\gamma\gamma_{d}$; and
* •
Drell-Yan, whereby a quark-antiquark pair annihilates into a dark photon,
which consequently decays into a lepton pair or hadrons, i.e.,
$q\bar{q}\rightarrow\gamma_{d}\rightarrow
l^{-}l^{+}\,\,\mbox{or}\,\,h^{-}h^{+}$.
Here we consider a special production mechanism where the dark photon comes
from a VLT, and thus the production chain will look like VLT$\rightarrow
t\gamma_{d}$, where $t$ is the SM top quark.
The decay modes of the massive dark photon is further divided into whether it
is visible or invisible, with the borderline requirement of visibility being
$m_{\gamma_{d}}>2m_{e}\simeq 1~{}\mbox{MeV}$ because then it can decay into SM
charged particles (electron-positron pairs being the extreme case) which leave
a signature in the detectors. Otherwise, it cannot decay into known SM charged
particles and the decay is therefore invisible. We choose to study this
extreme case of dark photon decaying into electron-positron pairs since these
final states result into the lesser studied unusual topology known as lepton-
jets. This choice constrains our massive photon to have a mass between 1 MeV
to 200 MeV, otherwise the braching ratio (BR) into electron-positron pairs is
not 100% and we get non-zero BR into muon-antimuon pairs [12, 1].
Figure 1: Feynman diagram showing the production mechanism of VLT going to a
top quark and a dark photon, which subsequently decays to two leptons.
Hence, we perform a phenomenological study on final states involving a lepton-
jet in association with the top quark, investigating both the scenarios where
the top quark decays hadronically or leptonically.
No evidence in favor of VLTs has been found at the Large Hadron Collider (LHC)
when probing via its traditional decays into SM particles. Hence,
nontraditional decays are searched for, including decay into the dark photon
which becomes dominant provided that its mass is very less compared to the SM
electroweak sector $(m_{\gamma_{d}}<<m_{Z})$. This appealing scenario provides
a probe for light dark sector by searching for heavy particles at the LHC,
thus bridging the two heavy and light mass regimes of BSM particles, VLTs and
dark photons respectively.
In order for the VLT decays into the dark photon to be dominant, we follow the
model discussed in-depth in [1], where a new Abelian gauge symmetry $U(1)_{d}$
is introduced whose gauge boson is the dark photon itself (i.e., SM is
extended to $SU(3)\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{d}$). The SM
particles are singlets under this new symmetry while the VLT has a charge
$+1$. Specifically, this VLT is a singlet under $SU(2)_{L}$ with SM
hypercharge $Y=2/3$. With this, the dark photon gauge boson kinetically mixes
with the SM $U(1)_{Y}$ field. While this opens up a decay channel of VLT into
the dark photon (plus top), its decay into fully SM final states (such as
$W/Z/h$ plus top) is still feasible. To enhance the former, we take the
$U(1)_{d}$ scale to be significantly smaller than the electroweak scale
($m_{\gamma_{d}}<<m_{Z}$) since the partial widths of VLT decay into
$\gamma_{d}$ relative to SM electroweak bosons is proportional to
$(v_{EW}/v_{d})^{2}$, where $v_{EW}=246$ GeV is the Higgs boson vacuum
expectation value (vev) and $v_{d}$ is the vev of the dark sector Higgs boson.
It should be noted that VLTs at the LHC can either be pair produced
$(pp\rightarrow T\bar{T})$, or produced singly in association with a jet
$(pp\rightarrow T/\bar{T}+\mbox{jet})$. The single production mechanism
surpasses the pair production in a wide range of $m_{VLT}$, with the latter
only outmatching the former in relatively low $M_{VLT}$ masses
($M_{VLT}\lesssim 1000$ GeV) [1, 13]. In this work, we only focus on the
single VLT production since not only is it the more favored production phase
space, the final state it produces is also relatively “cleaner” as opposed to
pair production.
With the requirement that the kinetic mixing $\varepsilon$ is small on top of
the $m_{\gamma_{d}}<<m_{Z}$ mass requirement, enhances VLT decay into dark
photons, and the $U(1)_{d}$ gauge boson inherits couplings to SM particles
through the electromagnetic current with coupling strength $\varepsilon eQ$
[1]. This in turn, kinematically opens up dark photon decays into
$e^{+}e^{-}$, for the values of $m_{\gamma_{d}}$ considered in this paper.
### 2.2 Event Generation
We generated events using MadGraph5 interfaced with Pythia8 [14] (using the
default NN23LO1 parton distribution function set [15]) from UFO [16] files
based on the maverick top partner model in [1], with the relevant parameters
above set as: $v_{d}=10$ GeV, $\varepsilon=0.001$. We vary the VLT masses from
$1$ TeV, $3$ TeV to $5$ TeV (with cross sections of $47.6$ fb, $0.41$ fb and
$0.0054$ fb respectively) to make sure that we remain at the phase space where
it is singly produced. Also, we chose $m_{\gamma_{d}}=0.1$ GeV which ensures
that the branching ratio (BR) to $e^{+}e^{-}$ is 1.
For each signal mass point, 100,000 events were generated, while for each
background processes, 1 million events were generated. All the background
processes have been generated using Pythia8 as well. While a leading order
generator cannot model all the background accurately, our aim here is to get
the general characteristics of the background processes, and suggest ways to
reduce them, for which Pythia8 is adequate. In any case, the region of
interest is the tail of reconstructed VLT mass distribution, so only general
features can be studied in a particle level analysis with limited Monte Carlo
statistics. Finally, all distributions below are normalised using an
integrated luminosity of $300$ fb-1, corresponding to projected combined
integrated luminosity after LHC Run 3.
### 2.3 Current experimental bounds on VLTs
Currently, the most stringent limits on VLT masses are set by the ATLAS and
CMS experiments [17, 18, 19, 20]. The excluded masses for VLT depend on the
branching ratios (BR) assumed; and for extreme values of 100% BR, a singlet
VLT is excluded for masses below 2 TeV.
## 3 Analysis strategy
### 3.1 Overview
The signal is characterised by an opposite sign electron pair with high
$p_{\mathrm{T}}$ and very little angular separation. They form what is known
as a lepton-jet (more specifically electron-jet in our case), where tracks
from the two electrons will overlap, and the energy deposits will be collected
in a single jet. Electron reconstruction algorithm used in LHC experiments
will mostly mis-reconstruct it as a single electron. So unlike standard
searches, electron multiplicity is a misleading observable here.
lepton-jets have been studied sparingly at the LHC [21, 22, 23], mostly to
search for SUSY and Higgs portal models with relatively high mass bosons
compared to the dark photon considered here. As mass calibration for jets in
experiments is deemed unreliable below 10 GeV, we are in no position to
exploit the lepton-jet mass. However, since the lepton-jet will be boosted,
the mass over transverse momentum ratio (subsequently referred to as
$m/$$p_{\mathrm{T}}$) of such jets can be used as a discriminating variable,
even accounting for the inherent uncertainty in the measured mass value. The
standard jets from quarks and gluons tend to have a much higher mass compared
to $\approx$ 100-200 MeV. Experimentally, the lepton-jet is expected to have a
larger than usual fraction of energy deposited in electromagnetic calorimeter,
which can be exploited as well.
The event also contain a top quark, and a at least one more jet coming from
the initial quark. A sketch of the event topology is shown in Fig. 2. We do
note this is aid visualisation, not an accurate indicator of the event
kinematics.
Figure 2: A cartoon of different objects produced in a typical signal event.
All decays shown are prompt.The large-radius jet shows the hadroniccaly
decalying top quark, the the b-tagged jet shown as shaded. In leptonic decay
mode of the top quark, it will contain a charged lepton and corresponding
neutrino, along with the b-tagged jet.
The decay mode of the top quark on the other side will determine the full
topology. We investigate the two scenarios of the top quark decaying
hadronically and leptonically separately.
We note that due to the lepton-jet being very boosted, the resulting two
electrons will have a very skewed energy (hence transverse momentum)
distribution, with one usually produced with a much higher $p_{\mathrm{T}}$.
This has crucial implications in the leptonic channel, as we will see. We also
note that the initial quark produced along with the VLT can be a bottom quark,
so we expect two b-tagged jets in some events.
In the following analysis, we will use objects as they are commonly used in
experiments. Jets will be the ones reconstructed using anti-$k_{t}$ [24]
algorithm with radius parameter of 0.4, $p_{\mathrm{T}}$$>30$ GeV and
$|\eta|<4.4$. Large radius jets are reconstructed using anti-$k_{t}$ algorithm
with radius parameter of 1.0, with $p_{\mathrm{T}}$$>150$ GeV and $|\eta|<2$.
We have not used any grooming as it is a particle level study without any
pileup, but application of standard trimming or soft-drop would not change any
conclusions. Muons or neutrinos are not used as jet inputs, and the presence
of a b-hadron inside a jet is used to determine if the jet is b-tagged or not.
Lepton are chosen with $p_{\mathrm{T}}$$>25$ GeV and $|\eta|<2.5$, dressed
with photons within a radius of $0.1$ and not originating from hadron decays.
The missing transverse momentum is calculated from the negative four-vector
sum of all visible particles. The most inclusive suggested trigger will be a
single electron trigger with appropriate threshold. Rivet analysis toolkit
[25] has been used.
### 3.2 Analysis with the top quark decaying hadronically
The signal topology in hadronic channel is two electrons being identified as
one as a part of a high $p_{\mathrm{T}}$, low mass lepton-jet, a boosted top
quark jet, and at least another jet. The event selection requirement is at
least one electron, no muons, and at least three jets, at least one of them
being b-tagged.
The largest background by far is the usual multijet processes, where a jet is
misreconstructed as an electron. Top quark pair production and all-hadronic
and (electron-channel) semileptonic decay modes will consist of the other
significant background processes. The cross-sections of other possible
background processes, like $\textrm{top}+Z+$jets. $\textrm{top}+W+$jets,
$\textrm{top}+\gamma+$jets are tiny, and therefore neglected. We required
there be at least one small radius jet and at least three large radius jets.
The lepton-jet candidate is the jet closest in $\Delta R$ to the leading
$p_{\mathrm{T}}$ electron. This is verified by checking the $\Delta R$ between
the two electrons and between the leading electron and the closest jet in Fig.
3.
Figure 3: Distributions of $\Delta R$ between the two electrons (left) and
$\Delta R$ between leading electron and closest jet (right) for three
representative signal points
As expected, the two electrons are almost on top of each other, and there is
always a jet collinear with them. In order to be consistent with the
experimental signature, we require minimum of only one electron, not two.
However, for multijet and hadronic $t\overline{t}$ backgrounds, this one
electron requirement had to be relaxed, as jets misidentified as electrons
will result in these reducible backgrounds. Here we have scaled the cross-
sections by $0.1$ to mimic the electron misidentification rate. Requiring a
tight electron identification criteria will help in reducing these
backgrounds.
The top-jet candidate is a highest mass large radius jet, satisfying a top-
mass window (chosen rather liberally to be between 125–225 GeV) requirement.
The large-radius jet is also required to contain a b-tagged jet with $\Delta
R<1$. While any of the standard top-tagging techniques [26, 27] or
requirements on jet substructure observables can be used to increase the
purity of the top-jet selection, we leave that for the experimental analysis.
Since there is no real electron in multijet and hadronic $t\overline{t}$
background events, we have considered the jets farthest from the top jet
candidate as the lepton-jet. Then the invariant mass of the top partner is
reconstructed from four-momenta of the lepton-jet and top jet candidates. This
is the key observable for the search.
Kinematic selections are used based on signal characteristics to reduce the
background contributions. Fig. 4 shows lepton-jet and top jet $p_{\mathrm{T}}$
distributions, and based on these, a 200 GeV requirement is applied on both.
Additionally based on Fig. 5, a $\Delta\phi>1.5$ requirement is applied
between the lepton-jet and top jet candidate, as they are expected to be more
back-to-back than most background processes. After all the above kinematic
requirements, we are still left with a significant background, as can be seen
in the top partner mass distribution in reconstructed VLT mass distribution on
the right of Fig. 5.
Figure 4: Distributions of lepton-jet $p_{\mathrm{T}}$ (left) and the top jet
candidate $p_{\mathrm{T}}$ (right) for three representative signal points and
dominant background processes
Figure 5: Distributions of $\Delta\phi$ between the lepton-jet and hadronic
top jet and VLT mass after all kinematic requirements for three representative
signal points and dominant background processes
Then in Fig. 6, $m/$$p_{\mathrm{T}}$ for signals and backgrounds is compared,
and requiring the ratio to be less than $0.01$ essentially keeps only the
signal, as can be seen in the right figure.
Figure 6: Distributions of lepton-jet $m/p_{T}$ ratio before (right) and
reconstructed VLT mass after the $m/p_{T}<0.01$ requirement for three
representative signal points and dominant background processes
A detailed study on the effect of these kinematic requirement on signal and
background was performed and summarised in Table 1 for the signal points.
Requirement | Signal efficiency reduction for TP mass in $\%$:
---|---
| 1 TeV | 3 Tev | 5 TeV
Lepton multiplicity | 97 | 98 | 98
Jet multiplicity | 93 | 94 | 92
Top mass window | 75 | 79 | 71
B-jet multiplicity | 68 | 75 | 68
lepton-jet $p_{\mathrm{T}}$ | 64 | 73 | 67
Top jet $p_{\mathrm{T}}$ | 64 | 73 | 67
B-jet containment | 57 | 57 | 41
Lepton and top jet separation | 56 | 56 | 40
$m/$$p_{\mathrm{T}}$ | 17 | 29 | 27
Table 1: Effect of kinematic requirements on leptonic channel signal points.
Each requirement is mentioned earlier.
While the requirements enforced decrease the signal significantly, the
$m/$$p_{\mathrm{T}}$ requirement essentially makes it a zero background
search. The different impact of b-jet containment and $m/$$p_{\mathrm{T}}$
requirement on different signals is due to how boosted the signal is. After
all selections, about a thousand signal events remain for VLT mass of 1 TeV,
and about ten for VLT mass of 3 TeV.
### 3.3 Analysis with the top quark decaying leptonically
The signal topology in leptonic channel is two electrons being identified as
one as a part of a high $p_{\mathrm{T}}$, a low mass lepton-jet, a top quark
decaying leptonically, and at least another jet. Depending on lepton flavour
from top quark decay, two scenarios are possible, termed electron and muon
channels. At least two jets, with at least one of them b-tagged is required.
In both cases, the top quark is reconstructed by using the pseudotop method
[28]. The algorithm starts with the reconstruction of the neutrino four-
momentum. While the $x$ and $y$ components of the neutrino momentum are set to
the corresponding components of the missing transverse momentum, the $z$
component is calculated by imposing the $W$ boson mass constraint on the
invariant mass of the charged-lepton–neutrino system. If the resulting
quadratic equation has two real solutions, the one with the smaller value of
$|p_{z}|$ is chosen. If the discriminant is negative, only the real part is
considered. The leptonically decaying $W$ boson is reconstructed from the
charged lepton and the neutrino. The leptonic top quark is reconstructed from
the leptonic $W$ and the b-tagged jet closest in $\Delta R$ to the charged
lepton. We will refer it to as the leptonic top candidate.
In the muon channel, the muon is always used to reconstruct the leptonic top.
In the electron channel, a possible ambiguity can arise, as there are three
electrons, two of them are expected to be overlapping, and the third from the
top quark. We cannot a priori assume that the leading electron is from the
dark photon decay, or the electron closest to the b-tagged jet is from the top
quark decay, as a large fraction of events have two b-tagged jets, the second
one from the initial quark. However, we have found that the electron pair with
the highest invariant $p_{\mathrm{T}}$ comes from the dark photon about 99.5%
of the time, so we use this pair as the single merged electron seeding the
lepton-jet, as seen by the detector. Then the remaining electron is used for
leptonic top reconstruction. Then the invariant mass of the top partner is
reconstructed from four-momenta of the lepton-jet and leptonic top candidates.
As in the hadronically decaying top quark scenario above, we need to be
careful about lepton multiplicity when considering the backgrounds. For the
muon channel, in order to be consistent with the experimental signature, we
require at least one electron and at least one muon. The largest background is
the dileptonic mixed flavour $t\overline{t}$ process, which is an irreducible
background. Since mis-reconstruction of muons as jets are relatively rare as
compared to electrons, we do not have to worry about background that does not
contain a real muon. The semileptonic $t\overline{t}$ process with a muon will
be a reducible background, where a jet from leptonic top can mimic the lepton-
jet. As before, for this case, we have loosened the electron requirement, and
applied a normalisation factor of 0.1.
For the electron channel, we start by requiring at least three electrons, and
no muons. However, then the merged lepton is used, so the effective
requirement is two. The largest background is the dileptonic $t\overline{t}$
process with both top quarks decaying to electrons, which is an irreducible
background. The semileptonic $t\overline{t}$ process with an electron will be
a reducible background. where a jet from hadronic top can mimic the lepton-jet
or the leptonic top. As before, for this case, we have loosened the electron
requirement, and applied an extra normalisation factor of 0.1.
The other significant backgrounds can be $W/Z$ boson (decaying to electrons)
with jets. As the $Z$-boson mass is much higher than the dark photon mass, the
electrons rarely end up overlapping, and the $b$-jet requirement essentially
gets rid of almost all the $W$+jets contribution. We have not explicitly
considered $\tau$ decay modes of the top quark as the hadronic decay mode will
be equivalent to the fully hadronic signature considered, and the leptonic
decay modes will be similar to the electron and muon channels.
The kinematic requirements are inline with those applied in hadronic case,
with requirements of lepton-jet $p_{\mathrm{T}}$ and leptonic top candidate
$p_{\mathrm{T}}$ of 200 GeV. Most kinematic distributions are similar to the
hadronic channel ones shown above, so we did not repeat them. The distribution
of them is shown in Fig. 7.
Figure 7: Distributions of lepton-jet $p_{\mathrm{T}}$ (top row) and the
leptonic top candidate $p_{\mathrm{T}}$ (bottom row) for muon (left) and
electron (right) channels for three representative signal points and dominant
background processes
Similarly the b-tagged jet is required to be with $\Delta R<1$ of the leptonic
top candidate, and a $\Delta\phi>2.5$ requirement is applied between the
lepton-jet and leptonic top candidate, the latter distribution is shown in
Fig. 8. A loose minimum mass requirement of 100 GeV is applied for the
leptonic top candidate. Finally the same $m/$$p_{\mathrm{T}}$ requirement as
before is applied, and it again helps to massively reduce all backgrounds, as
can be seen in Fig. 9.
Figure 8: Distributions of $\Delta\phi$ between the lepton-jet and the
leptonic top candidate for the muon channel (left) and the electron channel
(right) for three representative signal points and dominant background
processes
Figure 9: Distributions of reconstructed VLT mass before (top row) and after
the $m/p_{T}<0.01$ requirement, for muon (left) and electron (right) channels
for three representative signal points and dominant background processes
A detailed study on the effect of these kinematic requirement on signal and
background was performed and summarised in Table 2 separately for electron and
muon channels, as defined above. The initial drop is due to one of the leptons
falling below the kinematic thresholds, and while the $m/$$p_{\mathrm{T}}$
requirement does reduce the efficiencies drastically, it also eliminates
almost all of the background, as seen above. We can also consider a
requirement on missing transverse momentum, but this is always better to leave
for a detector level analysis. Similarly, a requirement of about 100 GeV on
the leading electron $p_{\mathrm{T}}$ keeps almost all of the signal and will
help in reducing backgrounds, but this is better optimised using the potential
mis-reconstructed electrons at the detector level. After all selections, about
a thousand signal events remain for VLT mass of 1 TeV, and about ten for VLT
mass of 3 TeV.
Requirement | Signal efficiency reduction for TP mass in $\%$:
---|---
| 1 TeV | 3 TeV | 5 TeV
| El | Mu | El | Mu | El | Mu
Lepton multiplicity | 70 | 82 | 88 | 92 | 92 | 94
Jet multiplicity | 62 | 72 | 84 | 88 | 88 | 92
lepton-jet $p_{\mathrm{T}}$ | 60 | 68 | 82 | 88 | 88 | 92
Top jet $p_{\mathrm{T}}$ | 58 | 66 | 82 | 88 | 88 | 92
Top mass minimum | 58 | 64 | 82 | 86 | 88 | 92
B-jet containment | 54 | 60 | 80 | 84 | 86 | 90
Lepton and top separation | 52 | 58 | 78 | 82 | 84 | 88
$m/$$p_{\mathrm{T}}$ | 16 | 18 | 38 | 40 | 54 | 56
Table 2: Effect of kinematic requirements on leptonic channel signal
points.Each requirement is mentioned earlier.
## 4 Conclusions
A maverick top partner model, decaying to a dark photon was suggested in Ref.
[1]. A phenomenological exploration of the model has been performed, and a
search strategy has been proposed exploiting the unique signal topology. We
show that for a set of kinematic selections, both in hadronic and leptonic
decay channel of the SM top quark, almost all background can be eliminated,
leaving enough signal events for the search to be viable at the LHC upto VLT
mass of 3 TeV.
## Acknowledgements
We thank the authors of the maverick top partner paper, specifically Ian M.
Lewis and Samuel D. Lane for providing us with model files to cross check
ours. We thank Peter Loch for illuminating discussion on applicability of jet
mass calibration and suggesting the use of mass over transverse momentum
ratio. DK is funded by National Research Foundation (NRF), South Africa
through Competitive Programme for Rated Researchers (CPRR), Grant No: 118515.
MMF is funded by the Department of Science and Technology (DOST) - Accelerated
Science and Technology Human Resource Development Program (ASTHRDP)
Postdoctoral Research Fellowship. SS thanks University of Witwatersrand
Research Council for Sellschop grant and NRF for grant-holder linked bursary.
KDP thanks the National Astrophysics and Space Science Programme for their
ongoing financial support. HVDS thanks SA-CERN programme for excellence
scholarship.
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|
# Real-Time Formal Verification of Autonomous Systems With An FPGA
Minh Bui, Michael Lu, Reza Hojabr, Mo Chen, Arrvindh Shriraman All the authors
are with the School of Computing Science, Simon Fraser University.
{minh_bui_3, mla233, shojabro, mochen<EMAIL_ADDRESS>
###### Abstract
Hamilton-Jacobi reachability analysis is a powerful technique used to verify
the safety of autonomous systems. This method is very good at handling non-
linear system dynamics with disturbances and flexible set representations. A
drawback to this approach is that it suffers from the curse of dimensionality,
which prevents real-time deployment on safety-critical systems. In this paper,
we show that a customized hardware design on a Field Programmable Gate Array
(FPGA) could accelerate 4D grid-based Hamilton-Jacobi (HJ) reachability
analysis up to 16 times compared to an optimized implementation and 142 times
compared to MATLAB ToolboxLS on a 16-thread CPU. Our design can overcome the
complex data access pattern while taking advantage of the parallel nature of
the HJ PDE computation. Because of this, we are able to achieve real-time
formal verification with a 4D car model by re-solving the HJ PDE at a
frequency of 5Hz on the FPGA as the environment changes. The latency of our
computation is deterministic, which is crucial for safety-critical systems.
Our approach presented here can be applied to different systems dynamics, and
moreover, potentially leveraged for higher dimensions systems. We also
demonstrate obstacle avoidance with a robot car in a changing environment.
## I Introduction
Autonomous systems are becoming more prevalent in our lives. Examples of these
systems include self-driving cars, unmanned aerial vehicles, rescue robots,
etc. One key factor that will allow wider adoption of autonomous systems is
the guaranteed safety of these systems. Despite tremendous progress in
autonomous system research in areas such as motion planning, perception, and
machine learning, deployment of these systems in environments that involve
interactions with humans and other robots remains limited due to the potential
danger these robotic systems can cause. Formal verification methods can help
autonomous robots reach their untapped potential.
Hamilton-Jacobi (HJ) reachability analysis is a formal verification approach
that provides guaranteed safety and goal satisfactions to autonomous systems
under adversarial disturbances. There are many ways to do reachability
analysis, solving the HJ PDE is one way to characterize sets of safe states
and synthesizes optimal controllers, which involves calculating Backward
Reachable Tube (BRT) that describes a set of states the system must stay out
of in order to avoid obstacles. HJ reachability analysis has been successfully
applied in practical applications such as aircraft safe landing [1], multi-
vehicle path planning, multi-player reach avoid games [2]. The appeal of this
particular method is that it’s very powerful in handling control and
disturbances, nonlinear system dynamics, and flexible set representations.
The main downside to HJ reachability is that it’s solved on a multi-
dimensional grid with the same number of dimensions as the number of state
variables and scales exponentially with the number of dimensions. This
prevents HJ formulation to be applied on real-time systems where safety is
increasingly demanded. While 3D or smaller systems could be computed quickly
with multi-core CPUs, practical systems that usually involve 4 to 5 system
components can take several minutes to hours to compute. There have been
researches that proposed decomposing high dimensional systems into smaller
tractable sub-systems that can exactly compute [3] or overapproximate the BRT
in certain cases [4]. However, that challenge of applying HJ formulation on
real-time systems remains, as some systems cannot be decomposed further than
four dimensions, and over-approximation is introduced if projection methods
are used.
In this paper, we expand the limit of the number of dimensions for which we
could directly compute the BRT in _real time_ through the use of FPGA. We
would argue that customized hardware accelerators could complement well with
those decomposition methods in making higher dimensional systems provably safe
in real-time. As general-purpose computer no longer double its performance
every two years due to the end of Moore’s law, we have seen examples of
successful hardware accelerations in other areas such as machine learning’s
training/inference [5, 6, 7], robot’s motion planning[8].
In this paper, our contributions are as follows:
* •
We prototype a customized hardware design on FPGA that accelerates HJ
reachability analysis to 16x compared to state-of-the-art implementation and
142x compared to [9] on 16-thread CPU for 4D system
* •
We demonstrate that the system could meet real-time requirement of
guaranteeing safety in changing environments by re-computing BRT at 5Hz
* •
Demonstrate obstacle avoidance with a robot car driving in an environment in
which new obstacles are introduced during run time at 5Hz.
## II PRELIMINARIES
### II-A Hamilton-Jacobi Reachability Analysis
Let $s\leq 0$ be time and $z\in\mathbb{R}^{n}$ be the state of an autonomous
system. The evolution of the system state over time is described by a system
of ordinary differential equations (ODE) below.
$\dot{z}=\derivative{z(s)}{s}=f(z(s),u(s),d(s))$ (1)
where $u(\cdot)$ and $d(\cdot)$ denote the control and disturbance function
respectively. The system dynamics $f$ are assumed to be uniformly continuous,
bounded and Lipschitz continuous in $z$ for fixed $u$ and $d$. Given
$u(\cdot)$ and $d(\cdot)$, there exists a unique trajectory that solves
equation (1).
The trajectory or solution to equation (1) is denoted as
$\zeta(s;z,t,u(\cdot),d(\cdot)):[t,0]\rightarrow\mathbb{R}^{n}$ , which starts
from state $z$ at time $t$ under control $u(\cdot)$ and disturbances
$d(\cdot)$. $\zeta$ satisfies (1) almost everywhere with initial condition
$\zeta(t;z,t,u(\cdot),d(\cdot))=z$.
In reachability analysis, we begin with a system dynamics described by an ODE
and a target set that represents unsafe states/obstacles [10]. We then solve a
HJ PDE to obtain Backward Reachable Tube (BRT), defined as follows:
$\displaystyle\bar{\mathcal{A}}=\\{z:\exists\gamma\in\Gamma,\forall
u(\cdot)\in\mathbb{U},\exists s\in[t,0],$ (2)
$\displaystyle\zeta(s;z,t,u(\cdot),d(\cdot))\in\mathcal{T}\\}$
In HJ reachability analysis, a target set
$\mathcal{T}\subseteq\mathcal{R}^{n}$ is represented by the implicit surface
function $V_{0}(z)$ as $\mathcal{T}=\\{z:V_{0}(z)\leq 0\\}$. The BRT is then
the sub-level set of a value function $V(z,s)$ defined as below.
$\displaystyle
V(z,s)=\min_{d(\cdot)}\max_{u(\cdot)\in\mathbb{U}}\min_{s\in[t,0]}V_{0}(\zeta(0;z,t,u(\cdot),d(\cdot)))$
(3)
We assume disturbance is applied with non-anticipative strategies[11]. In a
zero-sum differential game, the control input and disturbances have opposite
objectives.
The value function $V(z,s)$ can be obtained as the viscosity solution of this
HJ PDE:
$\begin{gathered}\min\\{D_{s}V(z,s)+H(z,\nabla V(z,s)),V(z,0)-V(z,s)\\}=0\\\
V(z,0)=l(z),s\in[t,0]\\\ H(z,\nabla
V(z,s))=\min_{\gamma[u](\cdot)}\max_{u(\cdot)}\nabla
V(z,s)^{T}f(z,u)\end{gathered}$ (4)
We compute the HJ PDE until it converges. Numerical toolboxes based on level
set methods such as [9] are used to obtain a solution on a multi-dimensional
grid for the above equation.
### II-B Basic Numerical Solution
Let us store the value function on a multi-dimensional grid, with the
numerical solution of the value function denoted as $V$. Let $N_{d}$ be the
grid size on the $d$th axis ($1\leq d\leq 4$). We also let $x_{d,i}$ denote
the state of grid $i$ in dimension $d$. In our approach throughout this paper,
we will adopt the central differencing scheme for approximating derivatives in
dimension $d$, which is defined as follows:
$\begin{gathered}D_{d}^{-}V(x_{d,i})=\dfrac{V(x_{d,i})-V(x_{d,i-1})}{\Delta
x_{d}},\\\ D_{d}^{+}V(x_{d,i})=\dfrac{V(x_{d,i+1})-V(x_{d,i})}{\Delta
x_{d}},\\\
D_{d}V(x_{d,i})=\dfrac{D_{d}^{+}V(x_{d,i})+D_{d}^{-}V(x_{d,i})}{2}\end{gathered}$
(5)
The two terms $D_{d}^{-}$ and $D_{d}^{+}$ are the left and right
approximations respectively. Note that for grid points at each end of each
dimension (i.e $i=N_{d}-1$, $i=0$), (5) is computed with extrapolated points.
The basic algorithm for solving (4) on-grid for 4D systems is then described
as follows:
Algorithm 1 Value function solving procedures
1:$V_{0}[N_{1}][N_{2}][N_{3}][N_{4}]\leftarrow l(z)$
2://Compute Hamiltonian term, and max, min deriv
3:for $i=0:N_{1}-1$; $j=0:N_{2}-1$; $k=0:N_{3}-1$; $l=0:N_{4}-1$ do
4: Compute (5) for $1\leq d\leq 4$
5: $minDeriv\leftarrow min(minDeriv,D_{d}V(x))$
6: $maxDeriv\leftarrow max(maxDeriv,D_{d}V(x))$
7: $\displaystyle u_{opt}\leftarrow\arg\max_{u\in\mathbb{U}}\nabla
V(z,s)^{\top}f(z,u)$
8: $\dot{x}\leftarrow f(z,u_{opt})$
9: $H_{i,j,k,l}\leftarrow\nabla V(z,s)^{\top}\dot{x}$
10:end for
11:// Compute dissipation and add to H
12:for $i=0:N_{1}-1$; $j=0:N_{2}-1$; $k=0:N_{3}-1$; $l=0:N_{4}-1$ do
13:
$\alpha_{d}(x)\leftarrow\max_{p\in[minDeriv,maxDeriv]}\absolutevalue{\dfrac{\partial
H(x,p)}{\partial p_{d}}}$
14: $H_{i,j,k,l}\leftarrow
H_{i,j,k,l}-\Sigma_{d=1}^{4}\alpha_{d}(x)\dfrac{D_{d}^{+}\check{(}x)-D_{d}^{-}\check{(}x)}{2}$
15: $\alpha_{d}^{max}\leftarrow max(\alpha_{d}^{max},\alpha_{d})$
16:end for
17://Compute stable integration time step
18:$\Delta
t\leftarrow(\Sigma_{d=1}^{4}\dfrac{\absolutevalue{\alpha_{d}^{max}}}{\Delta
x_{d}})^{-1}$
19:$V_{t+1}\leftarrow H\Delta{t}+V_{t}$
20:$V_{t+1}\leftarrow min(V_{0},V_{t+1})$
21:$\epsilon\leftarrow\absolutevalue{V_{t+1}-V_{t}}$
22:if $\epsilon<threshold$ then
23: $V_{t}\leftarrow V_{t+1}$
24: Go to line 3
25:end if
The above algorithm loops through the 4D array three times. In the first grid
iteration, the Hamiltonian terms, maximum and minimum derivative is determined
(lines 3-9). In the next grid iteration, the dissipation is computed and added
to the Hamiltonian in order to make the computation stable. At the same time,
the maximum alphas in all dimensions defined in line 13 are computed. These
$\alpha_{d}^{max}$ are used to determine the step bound $\Delta t$. In the
third grid iteration (line 19), each grid point is integrated for a length of
$\Delta t$.
Figure 1: 9 memory accesses (yellow + green colored grid) within each
iteration for computing derivatives in all 4 directions as in line 3
(algorithm 2)
In certain cases, $\alpha_{d}(x)$ in line 13 is the same as computing the
absolute value of $\dot{x}$, which has been computed in line 8. In addition,
in a lot of cases, $\alpha_{d}^{max}$ stays the same across different time
iterations. We also observed that $\Delta t$ depends only on grid
configuration and $\alpha_{d}^{max}$. So instead of re-computing $\Delta t$
every time and then loop through the 4D grid array again, we could pre-compute
$\Delta t$ and re-use it for all the time iterations. Combining these ideas
together, throughout this paper, we will use the following algorithm with one
grid looping, which is more computationally efficient:
Algorithm 2 Value function solving procedures
1:$V_{0}[N_{1}][N_{2}][N_{3}][N_{4}]\leftarrow l(z)$
2:for $i=0:N_{1}-1$; $j=0:N_{2}-1$; $k=0:N_{3}-1$; $l=0:N_{4}-1$ do
3: Compute (5) for $1\leq d\leq 4$
4: $\displaystyle u_{opt}\leftarrow\arg\max_{u\in\mathbb{U}}\nabla
V(z,s)^{\top}f(z,u)$
5: $\dot{x}\leftarrow f(z,u_{opt})$
6: $H_{i,j,k,l}\leftarrow\nabla V(z,s)^{\top}\dot{x}$
7: $H_{i,j,k,l}\leftarrow
H_{i,j,k,l}-\Sigma_{d=1}^{4}\absolutevalue{\dot{x_{d}}}\dfrac{D_{d}^{+}(x)-D_{d}^{-}(x)}{2}$
8: $V_{t+1,(i,j,k,l)}\leftarrow
H_{i,j,k,l}\Delta{t}_{precomputed}+V_{t,(i,j,k,l)}$
9: $V_{t+1,(i,j,k,l)}\leftarrow min(V_{0,(i,j,k,l)},V_{t+1,(i,j,k,l)})$
10:end for
11:$\epsilon\leftarrow\absolutevalue{V_{t+1}-V_{t}}$
12:if $\epsilon<threshold$ then
13: $V_{t}\leftarrow V_{t+1}$
14: Go to line 2
15:end if
### II-C Field Programmable Gate Arrays (FPGA)
FPGA are configurable intergrated circuits that are programmed for specific
applications using hardware description language (HDL).
Figure 2: System overview on FPGA (Right). The initial value array is first
transferred from DRAM to FPGA’s on-chip memory. The memory buffer then
distributes data to the 4 PEs to concurrently compute the new value function
at 4 consecutive grid points. The output from PE is then written back to DRAM.
Each fully pipelined PE outputs one grid point every clock cycle (Left).
Inside the PE, there are hardware components that sequentially solve algorithm
2
Computing platforms such as CPUs, GPUs, and FPGAs have a memory component and
computing cores. Compute cores must request and receive all the necessary data
from the memory component before proceeding with the computation. If the
memory component cannot provide all data accesses the application requires to
proceed at once, cores have to stall and wait, slowing down the computation.
Figure 3: Pipelining schedule of a single PE. The PE’s operation is an
assembly line where multiple grid points could be processed at the same time
at different stages. Each stage is physical hardware that computes specific
parts of algorithm 2. At a particular stage and a particular cycle, the PE is
busy computing a certain part of algorithm 2 for the grid point at the indices
shown. Note that for simplicity, the indices shown here are for a single PE
only.
Efficient systems need to have both fast computing cores and fast data
distributions from the memory. Depending on the application, the memory access
and computing pattern will vary. General-purpose CPU/GPU are often architected
towards a reasonable performance for a wide variety of applications, but
unoptimized for any particular application. FPGA chip, on the other hand,
provides programmable digital circuits to design customized computing core and
memory blocks. Thus, one can leverage knowledge about the details of the
computing workload to design an efficient system accordingly with FGPA. With
FPGA, one could control and achieve a higher degree of parallelism from the
digital hardware level at the cost of programmability.
### II-D Problem Description
A key observation of algorithm 2 is that each new grid point $V_{t+1}$ could
be computed independently with each other within one time iteration and
therefore, in parallel. We could then leverage a high degree of parallelism on
FPGA by having many cores to update as many grid points concurrently as
possible.
However, before that, two challenges must be addressed. Firstly, memory blocks
need to efficiently distribute data to compute cores. In order for a loop
computation to proceed, each of these cores needs up to 9 data inputs (Fig 2.)
and a memory design needs to satisfy this. Secondly, a four-dimensional grid
takes up tens of megabytes in memory and therefore cannot be fully fit to
FPGA’s on-chip memory for fast access.
In this paper, our goal is twofold. First, we will discuss our hardware design
that can solve the above challenges and maximize parallel computation of
algorithm 2 while efficiently making use of FPGA’s on-chip memory. Next, we
will show that this enables low latency of computation on FPGA which could be
deployed in real-time systems.
## III Solving the HJ PDE Using FPGAs
Before going into details of the design, we will introduce some terminologies
that will be relevant throughout the next section.
In digital systems, time is discretized into the unit of a _clock cycle_ ,
which is the amount of time it takes for an operation such as computing,
loading, and storing to proceed. Each clock cycle typically is a few
nanoseconds. Dynamic Random Access Memory (DRAM) is a type of memory that sits
outside of the FPGA, which has higher memory capacity but takes a lot more
clock cycles to access.
Our custom hardware comprised two main components: on-chip memory buffer, and
processing elements (PE) or computing cores (shown in Fig 2). The memory
buffer is on-chip storage, providing access to all the grid points a PE needs
to compute a new value function. Each PE is a digital circuit that takes 9
grid points from the memory buffer to compute a new value function at a
particular grid point according to algorithm 2 (line 3-10). In the following
subsections, we will go into the details of each component.
### III-A Indexed Processing Element (PE)
The PE has the following target design objectives: (1) increase compute
throughput (defined as the number of output generated per second) through
pipelining, (2) reduce the computation time of each PE, (3) and ensure the
correctness of result while minimizing data transfer between DRAM and FPGA.
In our design, we use 4 PEs (as shown in figure 2). Each PE has an index $idx$
with $0\leq idx\leq 3$ associated with it and computes the grid point
$V_{t+1}(i,j,k,l+idx)$. At the beginning of the computation of algorithm 2,
each PE takes as input a grid index $(i,j,k,l)$ and its 8 neighbours to start
computing $V_{t+1}(i,j,k,l)$ according to algorithm 2 (line 2-10).
To increase computation throughput, each PE is fully pipelined. Similar to an
assembly line, the PE operation is divided into multiple stages taking a few
clock cycles to complete (Fig. 3). Each stage within the pipeline is physical
hardware that has a computation corresponding to one of the lines in algorithm
2 (line 3-10) for a particular index $i,j,k,l$. Every clock cycle, the result
from previous stages will be loaded to the next stage, following the
sequential order of algorithm 2. At any time during operations, the processing
element is computing different intermediate components for multiple indices
concurrently (explained in Fig.3).
To ensure that the computation is correct, inside each of the PE, there are
indices counters to keep track of loop variable $i,j,k,l$, with the inner loop
variable incrementing by one every clock cycle. These indices are used to
correctly address the state vectors during system dynamics computation. To
avoid accessing external DRAM we store these 4 state/position vectors $x$ or
any fixed non-linear functions such as $\cos(\cdot)$ and $\sin(\cdot)$ of
these states as a lookup table stored in on-chip memory, as state vectors
depend only on grid configuration and do not change with the environment. Each
PE will have its own look-up table to avoid communications between PEs. Having
this data on-chip will only require a few kilobytes of memory and no need to
access DRAM throughout the computation.
Figure 4: Four lines of memory buffer supply all the grid data to the four
PEs. Each of the rectangle blocks is a FIFO queue synthesized as Block RAM
(BRAM). The overhead notation is the size of the FIFO queue with
$N_{1},N_{2},N_{3},N_{4}$ as the four grid dimensions. Note that the queue’s
size depends only on three dimensions. Every clock cycle, new grid points
streamed from DRAM start entering each buffer line (left-hand side) and grid
points at the end of the lines are discarded (right-hand side).
### III-B On-Chip Memory Buffer
The memory buffer has the following key design objectives: (1) minimizing the
amount of on-chip memory usage and external DRAM accesses while (2)
concurrently providing 9 grid points to each PE every clock cycle.
One problem of working with a high-dimensional grid is that the whole grid can
take up tens of megabytes and therefore cannot be fully fit to a state-of-the-
art FPGA’s on-chip memory. Instead of storing everything on-chip, in our
design, grid points are streamed continuously from DRAM into an on-chip memory
buffer (shown in Fig.2) and can be re-used many times for spatial derivatives
computation in 4 dimensions before being thrown away. From the grid
dimensions, we could compute the maximum reuse distance beyond which a grid
point can be safely discarded as no longer needed. This maximum reuse distance
is equal to the minimum size of on-chip memory buffer, which is dependent only
on $N-1$ dimensions [12] and can be fitted to an FPGA’s on-chip memory. Our
memory buffer structure is implemented as First In First Out (FIFO) queue data
structure. Every clock cycle, a new grid point supplied from DRAM will start
entering the FIFO queue while the grid point reaching at the end of the FIFO
queue will be discarded (shown in Fig. 4).
FPGA on-chip memory buffers are composed of standard Blocks of Random Access
Memory (BRAM). Each BRAM has two-ports and at most two reads can be requested
concurrently in the same clock cycle. If all 9 grid points (shown in Fig. 1)
are stored in the same BRAM at the same time, a PE would then have to wait for
5 clock cycles before performing the computation. One way to increase the
number of accesses per clock cycle is to duplicate the data in multiple BRAMs,
but this would not work well for multidimensional arrays since these array
copies easily exceed FPGA on-chip memory. A different technique would be
_memory banking_ , which is to partition the memory on-chip into multiple BRAM
that could concurrently deliver data to the PE, allowing the PE to start to
compute new value function for a grid point in one clock cycle.
To allow concurrent access for multiple PEs, we adopted the parallel memory
buffer microarchitecture from [12]. Corresponding to the number of PEs, our
on-chip storage structure is made of 4 line buffers. Each of these line
buffers is a sequence of BRAM connected acting in a queue fashion: a grid
point moves towards the end of the line every clock cycle. The two endpoints
of each BRAM (shown in Fig 4) provide tapping points that are connected as
inputs to the PEs. The number of PEs, therefore, is mainly limited by the DRAM
bandwidth.
We also made modifications to the execution flow in [13] to accommodate for
computing values function at the boundary. Once each of the buffer lines is
half full, all the processing elements can start computing a new value
function.
### III-C Fixed-Point Representation
Computing a new value function based on algorithm 2 involves multiple addition
operations on floating-point numbers. At the hardware level, the addition of
floating-point numbers is as computationally expensive as fixed-point
multiplication, which would take up lots of resources and chip’s area.
Instead, we use fixed-point representations for our data to reduce the burden
on the hardware. We will show in the next section that this has little impact
on the correctness of the computation if the radix point is chosen carefully
for the grid configuration.
## IV EXPERIMENT & RESULT
### IV-A Experiment setup
In this section, we demonstrate that our system can meet the real-time
requirement through an obstacle avoidance demonstration in a changing
environment.
We used a Tamiya TT02 model RC car[14] controlled by an on-board Nvidia Jetson
Nano microcontroller inside a $4$m $\times$ $6$m room. We use the following
extended Dubins car model for its dynamics:
$\begin{split}\dot{x}&=v\cos(\theta)\\\ \dot{y}&=v\sin(\theta)\\\
\dot{v}&=a\\\ \dot{\theta}&=v\frac{\tan(\delta)}{L}\end{split}$ (6)
where $a\in\left[-1.5,1.5\right]$,
$\delta\in\left[-\frac{\pi}{18},\frac{\pi}{18}\right]$, and $L=0.3$m. The
control inputs are the acceleration $a$ and the steering angle $\delta$. We
use a grid size of $60\times 60\times 20\times 36$ with resolutions of $0.1$m,
$0.067$m, $0.2$m/s and $0.17$ rad for $x$-position, $y$-position, speed and
angle, respectively.
Inside the room, we use orange cones as obstacles and a motion capture system
is used to accurately track the car’s state and the position of the obstacles.
We initialize the initial value function as follows:
$V_{0}(x,y,v,\theta)=\sqrt{(x-x_{o})^{2}+(y-y_{o})^{2}}-R$ (7)
where $x_{o}$ and $y_{o}$ are the obstacle’s positions and R is the radius of
the cone. Obstacle’s positions can be obtained from the motion capture system.
Each of the cones has a radius of $0.08$m but is set as $0.75$m to account for
the model mismatch between the car and the dynamics used.
For the experiment, we considered three different environments, with different
cone placements, set up inside the room as shown in Fig. 5. For each
environment, a user manually controls the car and tries to steer into the
cones.
$V(x,y,v,\theta)<0.15$ (8)
Given the car’s state, when (8) is satisfied, the car is near the boundary of
a BRT so optimal control computed from the value function is applied to safely
avoid the cone. The optimal control is obtained from the value function as
follows:
$u_{opt}=\arg\max_{u\in\mathbb{U}}\nabla
V(x,y,v,\theta,s)^{\top}f(x,y,v,\theta,u)$ (9)
(a) Environment 1
(b) Environment 2
(c) Environment 3
Figure 5: Different BRTs are used as the placement of cones change over time
which limits where the RC car can be as it drives around the room.
We pre-compute the BRTs with a horizontal time of $0.5$s for three
environments using optimized_dp[15] and demonstrate safety by correctly
loading the value functions as the environment changes. We choose to pre-
compute the BRTs in order to emulate having an FPGA on-board without extra
latency resulted from communication with a remote AWS instance. For all
environments, the maximum time step to make the integration stable is
$0.007497$s. Initially, the room had a single cone but changed over time to
different cone placements. The BRT of a new environment could not be used
until 200ms after the environment has changed, which is longer than the time
taken to compute the same BRT on an FPGA. A video of these experiments can be
found at
https://www.youtube.com/playlist?list=PLUBop1d3Zm2vgPL4Hxtz8JufnIPmvrwlC
### IV-B Hardware Correctness
We use fixed-point data representations for hardware computation. In
particular, we use 32 bits with 5 bits to represent the integer part
(including sign) and 27 bits for the decimal part. With this choice, the
precision of our computation is $2^{-27}=7.45\times 10^{-9}$ and the range of
our computation is from $-16$ to $16$. The area we use for the experiment is
$4$m $\times$ $6$m, hence the largest absolute distance is the diagonal of
$7.2$m. Therefore, the number of integer bits is enough to represent all
possible values in the solution $V$, which has the physical interpretation of
minimum distance to collision over time, given (3) and the choice of $V_{0}$
in (7).
We choose to synthesize and implement our design on AWS F1 FPGA because of its
flexibility and availability. To correctly input data to the FPGA, we first
generate an initial value array based on the obstacles’ positions and radius
described by (7). Then this value array is converted from floating-point to
fixed-point number correctly based on the bit choice discussed above.
Afterward, the value array is passed to the FPGA for the HJ PDE solving
procedure to start.
For all three experiment, we verified the correctness of BRT generated by our
hardware with the toolbox at [15] by comparing the maximum error between
corresponding grid points. The toolbox uses 32-bit floating-point numbers. The
numerical error resulting from the different representations is shown in table
below for the three environments in table I.
TABLE I: ERROR COMPARISON
| Env. 1 | Env. 2 | Env. 3
---|---|---|---
Error | $1.68\times 10^{-6}$ | $1.78\times 10^{-6}$ | $1.37\times 10^{-6}$
These negligible errors are due to precision difference between fixed-point
and floating point number. Even though the computation is repeated for many
iterations, the maximum error does not grow dramatically over time. We believe
that is because of the convergence property of BRT. As time grows, the rate of
changes in the grid values also slows down leading to stable discrepancy
between the correct floating point and fixed-point values.
### IV-C Computational speed and Resources Usage
To measure the speed up for all three environments, we compare the computation
time on AWS FPGA running at 250MHz against [15] and [9] running on a 16-thread
Intel(R) Core(TM) i9-9900K CPU at 3.60GHz. The latency here is the time it
takes to compute the BRT. For FPGA, latency can be computed by multiplying the
clock cycles with the clock period. The result is summarized in the table
below.
TABLE II: FPGA
| Clock cycles | Period | Iterations | Latency
---|---|---|---|---
Env. 1 | 44155209 | 4 ns | 67 | 0.176s
Env. 2 | 44155209 | 4 ns | 67 | 0.176s
Env. 3 | 44155209 | 4 ns | 67 | 0.176s
TABLE III: optimized_dp[15]
| Latency | Iterations | FGPA speed up
---|---|---|---
Env. 1 | 3.35 s | 67 | $\times$18.9
Env. 2 | 2.99 s | 67 | $\times$17
Env. 3 | 3.42 s | 67 | $\times$19.4
TABLE IV: ToolboxLS[9]
| Latency | Iterations | FPGA speed up
---|---|---|---
Env. 1 | 25.11 s | 70 | $\times$142
Env. 2 | 25.14 s | 70 | $\times$142
Env. 3 | 25.18 s | 70 | $\times$142
It can be observed that the latency of computation on FPGA is fixed and
deterministic for all three environments while the latency on CPUs varies even
though the computation remains the same. With the lower latency of $0.176$s,
we are able to update the value function at a frequency of $5.68$Hz. The
resources usage of our design for 4 PEs is shown in the table below.
| LUT | BRAM | DSP
---|---|---|---
Used | 26319 | 519 | 598
Available | 111900 | 1680 | 5640
Utilization | 14.03% | 30.89% | 10.6%
On an FPGA, arithmetic operations on numbers are implemented using Digital
Signal Processing (DSP) hardware or Look Up Table (LUT) that perform logical
functions. Our design does not significantly consume most of the available
resources and could be scaled up to a larger grid size.
## V CONCLUSION
This paper introduces a novel customized hardware design on FPGA that allows
HJ reachability analysis to be computed $16$x faster than state-of-the-art
implementation on a 16-thread CPU. Because of that, we are able to solve the
HJ PDE at a frequency of 5Hz. The latency of our computation on FPGA is
deterministic for all computation iterations, which is crucial for safety-
critical systems. Our design approach presented here can be applied to system
dynamics and potentially higher dimensional systems. Finally, we demonstrate
that at 5Hz, a robot car can safely avoid obstacles and guarantee safety.
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|
# High-resolution Elemental Abundance Measurements of Cool JWST Planet Hosts
Using AutoSpecFit: An Application to the Sub-Neptune K2-18b’s Host M dwarf
Neda Hejazi Department of Physics and Astronomy, University of Kansas,
Lawrence, KS 66045, USA Department of Physics and Astronomy, Georgia State
University, Atlanta, GA 30303, USA<EMAIL_ADDRESS>Ian J. M. Crossfield
Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045,
USA Diogo Souto Departamento de Física, Universidade Federal de Sergipe, Av.
Marcelo Deda Chagas, S/N Cep 49.107-230, São Cristóvão, SE, Brazil Jonathan
Brande Department of Physics and Astronomy, University of Kansas, Lawrence, KS
66045, USA Thomas Nordlander Research School of Astronomy & Astrophysics,
Australian National University, Canberra, ACT 2611, Australia The ARC Centre
of Excellence for All Sky Astrophysics in 3 Dimensions, Canberra, ACT 2611,
Australia Emilio Marfil Departamento de Física de la Tierra y Astrofísica and
IPARCOS-UCM (Unidad de Física de Partículas y del Cosmos de la UCM), Facultad
de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg,
Germany Katia Cunha Department of Astronomy and Steward Observatory,
University of Arizona, Tucson, AZ 85721, USA David R. Coria Department of
Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA Zachary
G. Maas Department of Astronomy, Indiana University, Bloomington, IN 47405,
USA Alex S. Polanski Department of Physics and Astronomy, University of
Kansas, Lawrence, KS 66045, USA Natalie R. Hinkel Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA Joseph E.
Hand Department of Physics and Astronomy, University of Kansas, Lawrence, KS
66045, USA
###### Abstract
We present an in-depth, high-resolution spectroscopic analysis of the M dwarf
K2-18 that hosts a sub-Neptune exoplanet in its habitable zone. We show our
technique to accurately normalize the observed spectrum, which is crucial for
a proper spectral fitting. We also introduce a new automatic, line-by-line
model-fitting code, AutoSpecFit, that performs an iterative ${\chi}^{2}$
minimization process to measure individual elemental abundances of cool
dwarfs. We apply this code to the star K2-18, and measure the abundance of 10
elements - C, O, Na, Mg, Al, K, Ca, Sc, Ti, and Fe. We find these abundances
moderately supersolar, except for Fe with a slightly subsolar abundance. The
accuracy of the inferred abundances is limited by the systematic errors due to
uncertain stellar parameters. We also derive the abundance ratios associated
with several planet-building elements such as Al/Mg, Ca/Mg, Fe/Mg, and (a
solar-like) C/O=0.568 $\pm$ 0.026, which can be used to constrain the chemical
composition and the formation location of the exoplanet. On the other hand,
the planet K2-18 b has attracted considerable interest, given the JWST
measurements of its atmospheric composition. Early JWST studies reveal an
unusual chemistry for the atmosphere of this planet, which is unlikely to be
driven by formation in a disk of unusual composition. The comparison between
the chemical abundances of K2-18 b from future JWST analyses and those of the
host star can provide fundamental insights into the formation of this
planetary system.
Cool dwarfs — Planet-host stars — Elemental abundances — Model atmospheres —
Spectral synthesis — Planet formation
## 1 Introduction
Cool dwarfs ($M{\lesssim}0.75M_{\sun}$) are optimal and primary targets for
transit and radial velocity surveys of planets beyond our Solar system, since
their lower mass, radius, and luminosity make planetary signatures easier to
detect compared to those exoplanets orbiting more massive dwarfs. M dwarfs
($M{\lesssim}0.6M_{\sun}$) are particularly the most abundant stars in the
Galaxy ($70\%$ by number, Reid & Gizis 1997; Henry et al. 2006), and there is
likely at least one planet orbiting around these stars (e.g. Dressing &
Charbonneau, 2013, 2015; Tuomi et al., 2014; Hardegree-Ullman et al., 2019). M
dwarfs therefore dominate the general occurrence rates of planets around main
sequence stars. The presence and properties of planets are believed to be
linked to the chemical composition of their host stars (e.g. Santos et al.,
2004; Fische & Valenti, 2005; Beauge & Nesvorny, 2013). Accordingly, M dwarfs
provide ideal sites to probe the formation mechanisms of planetary systems.
Planets are formed in a protoplanetary disk around a new star, which are all
embedded in a larger molecular cloud. As a result, there is a mutual
interaction between planets and their host stars, which can alter the
properties of the two components over their lifetimes. In particular, the
accretion of material from protoplanetary disk into the star as well as post-
formation events, such as planet engulfment, may imprint planetary signatures
in stellar chemical abundances (e.g. Pinsonneault et al., 2001; Oh et al.,
2018; Nagar et al., 2020; Spina et al., 2021). The detailed abundance
measurements of host stars are, therefore, of vital importance to
characterizing planetary systems and can provide fundamental insights into
planetary formation, evolution, and composition.
Although significant progress has been made in understanding star-planet
chemical connections, most studies have been focused on more massive FGK-type
dwarfs rather than M dwarfs. The spectra of cool M dwarfs are dominated by
millions of molecular lines in both the optical (e.g., TiO and CaH) and near-
infrared (NIR, e.g., H2O) regions, which are blended with each other and many
atomic lines. This causes a significant flux depression and, in turn, makes
identifying the continuum level in many wavelength areas challenging. Combined
with the substantial line crowding, established methodologies for FGK-type
dwarfs and giant stars, such as equivalent width measurements, are therefore
inappropriate for M dwarfs. As a result, most spectroscopic studies of M
dwarfs rely on the spectral synthesis and model fitting (e.g. Rajpurohit et
al., 2014; Lindgren et al., 2016; Souto et al., 2022).
Recently, high-resolution NIR spectroscopy has opened the way for detailed
elemental abundance measurements of M dwarfs with a reasonable accuracy
($\lesssim$ 0.2 dex). Modern spectrographs, along with methods to correct
spectra for telluric contamination, have made it possible to detect and
analyze various atomic and molecular lines and scrutinize the effect of
physical parameters on their shape. Parallel advances in modeling the
atmospheres of low-mass M dwarfs and calculating atomic and molecular line
lists are of great importance in measuring the parameters and chemical
abundances of these stars. Various previous studies have attempted to model
M-dwarf atmospheres assuming one-dimensional radiative-convective equilibrium
(e.g. Allard & Hauschildt, 1995; Tsuji et al., 1996; Gustafsson et al., 2008;
Kurucz, 2011; Husser et al., 2013). However, the synthetic spectra associated
with the same set of physical parameters and elemental abundances using
different model atmospheres and spectral synthesis methods show discrepancies
over many wavelength ranges. These are likely due to differences in model
assumptions and opacity calculations as well as atomic and molecular line
lists incorporated in synthesizing spectra (Iyer et al. 2023, specifically see
Figure 1). All these complications motivate more profound and detailed studies
to understand any missing source of line and continuum opacity and better
characterize the atmosphere of M dwarfs. Nevertheless, significant progress
has been made in determining the physical parameters of M dwarfs using high-
resolution NIR spectroscopy with synthetic model fitting (e.g. Lindgren et
al., 2016; Lindgren & Heiter, 2017; Rajpurohit et al., 2018; Passegger et al.,
2018, 2019; López-Valdivia et al., 2019; Souto et al., 2017, 2020, 2021, 2022;
Marfil et al., 2021; Wanderley et al., 2023) using different methods and
various combinations of model atmospheres and line data. Although these
studies have shown agreement between their own observed and best-fit model
spectra, the consistency in parameter values among different analyses is still
under debate (e.g. Olander et al., 2021; Passegger et al., 2022).
In contrast to the numerous efforts aimed at the determination of M-dwarf
physical parameters, measuring the individual elemental abundances of these
cool dwarfs using line-by-line model fitting, particularly in high-resolution
NIR spectra, is still in the early stage (e.g. Souto et al., 2017; Abia et
al., 2020; Shan et al., 2021; Souto et al., 2022). The accuracy of inferred
elemental abundances from such methods highly depends on model atmospheres and
atomic and molecular line lists used in spectral synthesis, as well as the
continuum/pseudo-continuum normalization of observed spectra. Souto et al.
(2017, 2022) derived the abundances of 13-14 elements for a few M-dwarf
samples by synthesizing spectra using the widely-used radiative transfer code
Turbospectrum (Alvarez & Plez, 1998; Plez, 2012) along with one-dimensional
(1D) plane-parallel MARCS model atmospheres (Gustafsson et al., 2008), and
then performing a ${\chi}^{2}$ minimization approach for each single selected
spectral line. In our previous work (Hejazi et al. 2023, hereafter Paper I),
we further extended this method by carrying out an iterative ${\chi}^{2}$
minimization process, where after each iteration, a new grid of synthetic
spectra for each element was generated based on the new inferred abundances,
which were then used in the next iteration. This procedure was repeated until
the abundance of all elements converged to their final values. In Paper I, the
transition from one iteration to the next was implemented manually, but we
have developed a model fitting code, AutoSpecFit, that automatically allows
Turbospectrum to produce the synthetic spectra required for each iteration “on
the fly” without interrupting the run. In this paper, we apply this automatic
code to the planet-hosting M dwarf K2-18 to obtain its elemental abundances.
The sub-Neptune K2-18b has been targeted by several James Webb Space Telescope
(JWST) observing programs, and the comparison between the composition of this
planet and its host star can shed light on its formation history.
This paper is outlined as follows. In Section 2, we describe the properties of
the exoplanet K2-18 b that has been observed by both the Hubble Space
Telescope (HST) and JWST. We summarize the spectroscopic observations of the
host star K2-18, the data reduction method, and the pre-processing needed to
prepare the spectra for the analysis in Section 3. The spectral synthesis and
the line lists used in this study are described in Section 4. The process of
line selection and continuum/pseudocontinuum normalization are presented in
Section 5. Physical parameters of the target K2-18 determined from other
independent methods, except for microturbulent velocity that is obtained from
this spectroscopic analysis, are shown in Section 6. All steps of AutoSpecFit
for measuring elemental abundances are detailed in Section 7. In Section 8, we
utilize our abundance technique to derive the abundances of 10 elements as
well as the abundance ratios associated with several planet-building elements
for K2-18. The error analysis of the inferred abundances and abundance ratios
is also demonstrated in this section. The summary and conclusion of this
study, particularly in the context of the star-planet connection, are
presented in Section 9.
## 2 Exoplanet K2-18 b
The K2-18 system is host to two planets, one of which (K2-18 b) is a
transiting super-Earth/sub-Neptune (2.61 $\pm$ 0.09 $R_{\oplus}$,
$9.51^{+1.57}_{-1.89}$ $M_{\oplus}$111The earlier study of Cloutier et al.
(2017) inferred a planet mass of 8.63 $\pm$ 1.35 $M_{\oplus}$, which is
consistent with the planet mass from Radica et al. (2022) within the
uncertainties.; Benneke et al. 2019; Radica et al. 2022) in the star’s
habitable zone (Montet et al., 2015; Crossfield et al., 2016), and the other
(K2-18 c) is a non-transiting planet of similar mass ($6.92^{+0.96}_{-0.99}$
$M_{\oplus}\sin i$; Radica et al., 2022). Given K2-18 b’s amenability to
transit spectroscopy and its temperate instellation, it has been a high-
priority target for observational and theoretical characterization.
Initial HST/WFC3 transmission spectroscopy revealed a clear atmosphere for the
planet, as well as the presence of water vapor (Benneke et al., 2019; Tsiaras
et al., 2019). The prospect of water vapor on a habitable zone world spurred a
flurry of further modeling to explain the observed data and more thoroughly
model the planet’s upper atmosphere and deeper interior. Madhusudhan et al.
(2020) modeled the interior structure of the planet and how varying interior
compositions would affect the planet’s observed spectrum, finding that, while
their modeling efforts are consistent with rocky planets, classical gas
dwarfs, and water-dominated ocean worlds, K2-18 b is likely to have a small
($\lesssim 6\%$) H/He fraction, and the planet could still support habitable
conditions. Bézard et al. (2022) noted that methane has strong absorption
features that overlap with water vapor in the HST/WFC3 near-IR bandpass and
found that, after reanalyzing the data, methane is a much more likely absorber
given self-consistent radiative-equilibrium models of K2-18 b’s atmosphere.
This predicted methane signal was confirmed with JWST observations of the
planet, clearly detecting methane and carbon dioxide (and not detecting water
vapor) at wavelengths without contaminating features from other absorbers
(Madhusudhan et al., 2023). Again, many more theoretical investigations
followed this reversal of the previous observational results, focusing on the
potential for K2-18 b to be a “Hycean” (water ocean under a hydrogen
atmosphere) planet compared to a more typical Neptune-like gas-dwarf. By
modeling the convective processes on K2-18 b, Leconte et al. (2024) predict
that the planet may not be Hycean, as its clear atmosphere would allow too
much incident radiation to maintain a liquid water ocean, while Shorttle et
al. (2024) show that a magma ocean interior could also reproduce the current
observed JWST spectrum.
Finally, Wogan et al. (2024) model the planet and favor a typical Neptune-like
composition over Hycean compositions as Hycean planets may not be able to
produce sufficient methane through photochemical processes to match the
observed methane abundance in the JWST data. Other similar exoplanets have
also been observed in the same mass/radius/temperature range as K2-18 b, such
as TOI-270 d, another habitable-zone sub-Neptune with methane and carbon
dioxide, but also water vapor (Benneke et al., 2024; Holmberg & Madhusudhan,
2024). The persistent uncertainties around K2-18 b’s true nature and the
infancy of panchromatic, high-precision studies of these temperate worlds both
motivate deeper studies of the system itself, especially this work.
## 3 Spectroscopic Observations and Pre-processing
We observed K2-18 with the IGRINS high-resolution (R$\sim$45,000) spectrograph
(Yuk et al., 2010; Park et al., 2014) at the Gemini-South Observatory as part
of program GS-2023A-Q-203 (PI: Ian Crossfield). The star was observed on
2023-01-20 with a single ABBA nod sequence; each frame had an exposure time of
245 s. For telluric corrections, the facility observers selected the nearby
A0V star HIP 61628 and observed a single ABBA sequence with 50 s integration
times. The data were processed in the same way as described in Paper I. In
brief, the raw 2D echelleograms were processed and reduced by the standard
IGRINS Pipeline Package (Lee et al., 2017), with the order-by-order 1D spectra
provided through the Raw & Reduced IGRINS Spectral Archive (Sawczynec et al.,
2022). We then further processed the spectra by running the spectra of K2-18
and its A0V calibrator through the SpeXTool pipeline’s xtellcor_general
routine (Cushing et al., 2004) to account for any small wavelength offset
between the spectra of K2-18 and the A0V star, and then through xmerge_orders
to combine the individual echelle orders into a single, 1D spectrum. The final
spectrum spans a wavelength range of 1.45-2.45 µm covering both H- and K-
bands, with a median S/N of 270 per pixel, which is higher than the minimum
median S/N ($\sim$200) required for detailed abundance measurements of cool
dwarfs at the resolution provided by IGRINS spectra (or even at the lower
resolution of APOGEE spectra, i.e., $\sim$22,500).
In order to flatten the observed spectrum, we divide the spectrum into smaller
parts, typically ranging between 50 and 150 Å, and fit a low-order polynomial
to the data points of each part. We then exclude those wavelengths whose
fluxes are less than the respective values of the polynomial. We further fit a
new low-order polynomial to the remaining data points and again exclude those
wavelengths with fluxes less than the relevant polynomial values. This
procedure is repeated until a final polynomial, that passes only through the
spectral peaks and does not cross any absorption line, is reached. Lastly, we
divide the spectrum of each part to the corresponding final polynomial to
obtain the flattened spectrum normalized to unity, and then combine all the
flattened parts back together. It should be noted that the resulting flattened
spectrum does not present a continuum-normalized spectrum as the continuum
level of M-dwarf spectra cannot be identified in many spectral regions.
## 4 Spectral Synthesis and Line Data
We generate the synthetic, continuum-normalized spectra222The synthetic
continuum-normalized spectra are calculated by dividing the absolute flux line
spectrum by the absolute flux continuum spectrum. The continuum is calculated
in the same way as the line spectrum, but instead of line opacities, only
continuous opacities are used. This approach is a standard practice in high-
resolution spectroscopic analyses, as the continuum generally exhibits smooth
variations on scales longer than the width of a spectral order. The continuum
is calculated on a more coarse wavelength scale than the line spectrum, and
then interpolated onto the exact same wavelengths. (hereafter, “synthetic
models/spectra” or “model spectra”, for simplicity) required for our analysis
by employing Turbospectrum (version v15.1) assuming local thermodynamic
equilibrium (LTE)333The non-LTE version of Turbospectrum (Gerber et al., 2023)
has also been publicly available., which can consistently handle very large
line lists including millions of spectral lines related to all atoms and tens
of molecules. We use 1D hydrostatic MARCS model atmospheres that were computed
in LTE and solves the radiative transfer equation in plane-parallel geometry
for dwarf stars. The MARCS model grid is based on the solar abundances from
Grevesse et al. (2007), but the abundances of $\alpha$-elements are enhanced
for models with subsolar metallicities ([M/H]$<$0) following the typical
trends of [$\alpha$/Fe] as a function of metallicity for stars in the Solar
neighborhood. To synthesize model spectra, we also use a set of atomic and
molecular line lists that are described in Paper I, but with some
improvements, as shown below.
To examine our selected atomic line list (and also to choose the best spectral
lines and perform the pseudo-continuum normalization process, Section 5), we
need to compare our observed spectrum to an initial guess of best-fit model.
To this end, We use the interpolation routine developed by Thomas
Masseron444https://marcs.astro.uu.se/software.php to interpolate the MARCS
model associated with the star’s physical parameters (see Section 6.1). Using
the interpolated model, we then produce the synthetic spectrum with the star’s
parameters, assuming microturbulence $\xi$=1.0 km/s, and the absolute
abundances equal to the solar values plus the overall metallicity, i.e.,
A(X)=A(X)☉+[M/H], or equivalently, the relative abundances555In this paper, we
use several abundance notations that are defined as follows:
“absolute abundance” $\rm{A(X)=log({N_{X}}/{N_{H}})_{star}+12}$,
“abundance”
$\rm{[X/H]=log({N_{X}}/{N_{H}})_{star}-log({N_{X}}/{N_{H}})_{\sun}}$,
or $\rm{[X/H]=A(X)_{star}-A(X)_{\sun}}$,
“relative abundance” [X/Fe]=[X/H]$-$[M/H],
“abundance ratio” $\rm{X/Y=10^{(A(X)-A(Y))}=N_{X}/N_{Y}}$,
where X and Y indicate the elements X and Y, $\rm{N_{X}}$ means the number
density of element X, and [M/H] shows the overall metallicity. equal to the
solar values, i.e., [X/Fe]=0, where X denotes the element X. These solar
relative abundances are the default values when using Turbospectrum without
any abundance customization. Although we first assume a microturbulence value
of $\xi$=1.0 km/s based on previous studies of M dwarfs (e.g. Souto et al.,
2017), we later find this value as the best-fit parameter for K2-18 (see
Section 6.2). This synthesized spectrum represents a first-order approximation
of the star’s best-fit model (hereafter “(Model)${}_{\textrm{approx}}$”).
For Radial velocity correction, we compare this model with the observed
spectrum that is Doppler shifted to obtain the star’s radial velocity. We
first visually examine different radial velocities (RVs) with a large step of
10.0 km/s over several spectral regions, and after finding a rough estimate of
RV, we determine the best-fit RV value by fine tuning using small steps
between 0.5 and 1.0 km/s. However, smaller RV adjustments which can be as
small as $\pm$ 0.1 km/s) may still be needed for some spectral lines before
synthetic model fitting. This slight radial velocity offset may be due to the
uncertainty of the best-fit value, the inaccuracy of the line lists, or the
insufficiency of the wavelength calibration in data reduction. The observed
wavelengths are shifted according to the inferred radial velocity, which are
used in the following steps of our analysis whenever the observed and model
spectra are compared together.
On occasion, the line parameters such as oscillator strength log($gf$) drawn
from some line databases are not accurate enough or updated using more recent
studies to well reproduce some specific observed spectral lines. To inspect
the atomic line list, we have compared the log($gf$) of all identified atomic
lines in the spectra of our target from the Vienna Atomic Line Database
(VALD3, Ryabchikova et al. 2015; Pakhomov et al. 2017, 2019) with those from
Linemake666https://github.com/vmplacco/linemake (an open-source atomic and
molecular line list generator, Placco et al. 2021). We have found 11 lines
that have different values of log($gf$) in these two line databases: Ti I
(15334.85 Å), Ti I (15602.84 Å), Ti I (15715.57 Å), Mg I (15748.99 Å), Ca I
(16197.07 Å), Ca I (19853.09 Å), Ca I (19933.73 Å), Sc I (22052.14 Å), Sc I
(22065.23 Å), Sc I (22266.73 Å), and Ca I (22651.18 Å). We have noted that
only the three Ti I lines show better consistency between the observed
spectrum and (Model)${}_{\textrm{approx}}$ if log($gf$) values from Linemake,
rather than from the VALD3, are used. We have accordingly updated the
log($gf$) of these three lines in the VALD3 line list using the values from
Linemake that are originally from Lawler et al. (2013). We have also replaced
the FeH line list in our previous set used in Paper I (Dulick et al., 2003)
with the more recent one from Hargreaves et al. (2010). This new line list
reproduces synthetic models that are in significantly better agreement with
observed spectra over regions dominated by FeH lines.
## 5 Spectral Line Selection and Continuum/Pseudocontinuum Normalization
For our spectral fitting analysis, the ideal atomic and molecular lines are
those that show consistency between the observed spectrum and the best-fit
model. Since the best-fit model is undetermined before performing the fitting
code, we compare the observed spectrum with (Model)${}_{\textrm{approx}}$ over
the spectral line candidates and select the best lines for the analysis.
However, for a reasonable comparison, the observed spectrum needs to be
locally continuum-normalized, or pseudo continuum-normalized for most regions
where the flux level is lower than unity due to a large number of H2O lines
and the continuum cannot be identified. The reliability of inferred abundances
therefore strongly depends on the propriety of spectral line selection, which
relies on the accuracy of the normalization process.
Prior to normalizing the observed spectrum, the synthetic spectra are smoothed
at the observed spectral resolution using a Gaussian kernel and then
interpolated at the shifted observed wavelengths. The
continuum/pseudocontinuum normalization is performed using the method
described in Paper I, which is based on the general concept presented in
Santos-Peral et al. (2020), but with some modifications. The most appropriate
data points on the continuum/pseudocontinuum around the analyzed spectral
lines are chosen using a routine that implements a linear fit to the residuals
R=O/S, where O is the observed flux and S is the synthetic flux, both at
shifted, observed wavelengths, followed by two or three iterative
$\sigma$-clippings. The value of the clippings changes from the first to the
third iteration as 2$\sigma$, 1.5$\sigma$, and 1$\sigma$. For the cases where
the three $\sigma$-clippings do not end up with enough number of normalizing
data points, only the first two $\sigma$-clippings are performed. The
normalized spectrum is obtained after dividing the observed spectrum by the
linear fit to the residuals of the final data points.
We identify the well-defined and almost isolated spectral lines that have a
proper shape (e.g., not deformed by noise or bad pixels) and that are strong
enough to be distinguished from the prevalent background molecular opacities
(while these lines might still be weakly blended with molecular lines). We
then look for the continuum/pseudocontinuum regions on both sides around these
line candidates. Often, a few lines are close together, and common normalizing
regions around, and in some cases, also between these lines, are determined.
We test different pairs of continuum/pseudocontinuum ranges for each studied
line (or a few lines if they are close together) and then normalize the
observed spectrum using the process described above. We choose the pair of
continuum/pseudocontinuum regions that lead to a normalized spectrum
consistent with (Model)${}_{\textrm{approx}}$ within those regions. It should
be noted that at least two normalizing data points (one each side) are
required to perform the final linear fit and normalize the observed spectrum.
We further test the selected pairs of ranges by changing the corresponding
elemental abundances and checking whether the normalizing points still remain
on the continuum/pseudocontinuum. This inspection is important because, in the
model fitting procedure, the observed spectrum is normalized relative to a
number of model spectra with varying abundances before calculating
${\chi}^{2}$ values (see Section 7). As the abundance of an element varies
while the physical parameters remain unchanged, the flux may be slightly
changed over the neighboring regions around or even beyond of the respective
spectral lines. This flux redistribution could reshape the pseudocontinuum
levels around some spectral lines and even cause some weak absorption lines to
appear. For example, after increasing the abundance of oxygen (that is linked
to the abundance of H2O), some H2O absorption lines may arise within
pseudocontinuum regions. In this case, the determined normalizing data points
may show up inside the absorption lines that emerge after changing elemental
abundances.
We illustrate the normalizing regions and normalizing data points for a few
spectral lines using the spectrum of our target in Figures 1 and 2. Figure 1
shows the synthetic flux of the normalizing data points (black circles at the
edges of the panels) within the selected pseudocontinuum ranges on both sides
of the K I 15168.38 Å line (left panels) and the OH 16526.25 Å (right panels),
which are separated from the inner spectral regions by green dashed-lines. The
observed spectrum (red lines and circles) is normalized relative to
(Model)${}_{\textrm{approx}}$ (blue lines) as shown in the middle panels. The
top and bottom left panels present the observed spectrum that is normalized
relative to the model spectra similar to (Model)${}_{\textrm{approx}}$, but
with the relative abundance of potassium equal to [K/Fe]=$-$0.20 dex and
[K/Fe]=+0.20 dex (or [K/H]=$-$0.03 dex and [K/H]=+0.37 dex, following the
equation [X/H]=[X/Fe]+[M/H]), respectively. In the same way, the top and
bottom right panels demonstrate the observed spectrum normalized with respect
to the synthetic spectra similar to (Model)${}_{\textrm{approx}}$, except for
the relative oxygen abundance that is equal to [O/Fe]=$-$0.20 dex and
[O/Fe]=+0.20 dex (or [O/H]=$-$0.03 dex and [O/H]=+0.37 dex), respectively. For
both lines, although there is a slight change in the overall flux level with
abundance variation, the shape of the pseudocontinuum in the selected ranges
does not change, and the already chosen data points remain the most suitable
normalizing points. If this was not the case, we would explore other ranges on
the pseudocontinuum to find those that would meet the above condition.
Figure 2 shows the same plots as Figure 1, but for two neighboring spectral
lines, Sc I 22266.73 Å and Ti I 22274.02 Å. Clearly, there is no observed
pseudocontinuum region between these two lines that is in agreement with the
synthetic models, and we, therefore, determine common normalizing ranges
around both sides of the lines. The middle panels again display the observed
spectrum normalized relative to (Model)${}_{\textrm{approx}}$ (the two middle
panels are repeated for better comparison between different abundances of each
line, from top to bottom). In the top and bottom left panels, the relative
abundance of Sc changes in the same way as in Figure 1, while the relative
abundance of Ti is fixed, equal to the solar value. Similarly, in the top and
bottom right panels, the relative abundance of Ti varies in the same manner as
above, but the relative abundance of Sc is constant, equal to the solar value.
For the two cases, the chosen normalizing ranges and data points remain on the
pseudocontinuum level and continue to be the best for the analysis.
We examine the atomic and molecular (CO, OH, and FeH) line candidates and
identify their best normalizing ranges, while adjusting for radial velocity if
needed. We then normalize the observed spectrum according to the synthetic
spectra with different relative abundances spanning from [X/Fe]=$-$0.30 dex to
[X/Fe]=+0.30 dex with steps of 0.10 dex777We find this range of abundances
sufficiently broad to examine all the studied lines and determine the best-fit
model. for the lines associated with the element X. We visually compare the
resulting normalized observed spectrum with the respective models, which gives
us an early understanding of how consistently the synthetic models can
reproduce the observed spectral lines assuming 1D LTE. For example, the two
alkali lines, K I 15163.07 Å and Na I 22056.4 Å, show no adequate agreement
with model spectra, which may be due to the insufficiency in the line lists,
the NLTE effect (Olander et al., 2021), or other factors. For this reason, we
removed these two lines from our analysis.
After a careful examination, we selected 148 spectral lines for 10 different
species, CO, OH, Na, Mg, Al, K, Ca, Sc, Ti, and FeH, as listed in Table 1. We
then manually determine a fitting or ${\chi}^{2}$ window for the selected
lines (the third column of Table 1), mainly around the cores and far from the
outermost part of the wings (that are more influenced by spectral noise), to
perform the ${\chi}^{2}$ minimization. For some adjoining doublet lines of the
same species (e.g., some OH lines), a single common ${\chi}^{2}$ window is
defined. We use the adopted normalizing ranges and ${\chi}^{2}$ windows of the
lines as input to run AutoSpecFit in the next step.
As presented in the last column of Table 1, four atomic lines in our line set
(Na I 22083.66 Å, Al I 16718.96 Å, Al I 16750.56 Å, and Sc I 22266.73 Å) are a
combination of multiple lines, including lines from hyperfine structure (HFS)
splitting. HFS data have been included to the VALD3 database (Pakhomov et al.,
2017, 2019), and have shown to properly model the HFS lines in M dwarfs (Shan
et al., 2021). In addition, several atomic lines (Mg I 15765.84 Å, Ti I
15334.85 Å, Ti I 15715.57 Å, Ti I 21782.94 Å, Ti I 22211.24 Å, Ti I 23441.48
Å) are blended with a few other lines associated with the same elements, most
of which are too weak to influence the shape of the main (central) lines
(Table 1). It should be pointed out that every single line is included in the
line list and modeled by our spectral synthesis.
## 6 Physical Parameters of Planet-Host M dwarf K2-18
### 6.1 Effective Temperature, Metallicity, and Surface Gravity
K2-18, an M2.5V dwarf star (Schweitzer et al. 2019), resides in the Solar
vicinity at a distance of 38 pc (Gaia Collaboration et al. 2021). Due to its
proximity, numerous studies in the literature have determined its stellar
parameters. These studies report an effective temperature of approximately
3500 K (Montet et al. 2015, Stassun et al. 2019, Martinez et al. 2017,
Schweitzer et al. 2019, Zhang et al. 2020, Reiners et al. 2022), surface
gravity around 4.70 (Stassun et al. 2019, Schweitzer et al. 2019, Shan et al.
2021, Queiroz et al. 2023), and metallicity varying notably across different
works, from $-$0.30 (Ding et al. 2022) to +0.26 dex (Hardegree-Ullman et al.
2020).
We use our $H-$band spectra to determine the atmospheric parameters of K2-18
($T_{\rm eff}$ and log $g$) using the methodology of Souto et al. (2020). To
summarize, we derive oxygen abundances from two species (H2O and OH lines) for
a set of effective temperatures ranging from 3200 to 3800 K in steps of 100 K.
Because the H2O and OH lines display different sensitivity to changes in
$T_{\rm eff}$, there will be a unique solution yielding the oxygen abundance
to a $T_{\rm eff}$ value. To derive the surface gravity, we employ the same
methodology but determine the oxygen abundance for a set of log($g$) from 4.50
to 5.00 in steps of 0.10 dex. The abundances are inferred from the best-fit
synthetic models compared to the observed spectrum that was generated in the
same way as described in this study, i.e., employing the Turbospectrum code in
conjunction with MARCS models, as well as using the APOGEE line list (Smith et
al. 2021). To derive the uncertainties in the atmospheric parameters ($T_{\rm
eff}$ and log $g$), we propagate the oxygen abundance uncertainty into the
atmospheric parameter determination methodology. For K2-18, we obtain $T_{\rm
eff}$ = 3547 $\pm$ 85 and log $g$ = 4.9 $\pm$ 0.10. Another product from this
analysis is the stellar overall metallicity, which is determined from the Fe I
and FeH lines available in the $H-$band (see Souto et al. 2020, Souto et al.
2021). We obtain that K2-18 is metal-rich, where [Fe/H] = +0.17 $\pm$ 0.10
dex. We adopt the uncertainty of metallicity from Melo et al. (2024).
It is important to emphasize that all steps of the parameter determination
procedure are completely different from our abundance analysis described in
this paper. Nevertheless, we find very good agreement between
(Model)${}_{\textrm{approx}}$ associated with the derived physical parameters
and the observed spectrum, which assures the reliability of the abundances
based on these parameters.
### 6.2 Microturbulent Velocity
Low-mass stars generally have microturbulence between 0 and 2 km/s (e.g. Reid
& Hawley, 2005; Bean et al., 2006; Tsuji & Nakajima, 2014; Pavlenko, 2017;
Souto et al., 2017; Olander et al., 2021; Recio-Blanco et al., 2023). We
determine the microturbulence $\xi$ using the approach described in Paper I.
We start with the three species having the most abundant selected lines, i.e.,
CO (whose abundance is an indicator of carbon abundance), OH (whose abundance
is an indicator of oxygen abundance), and FeH (whose abundance is an indicator
of iron abundance). We use the molecular FeH lines to measure iron abundance
because they are significantly more numerous than atomic Fe I lines. It is
important to note that there is a difference between the iron abundance
inferred from the methodology of Souto et al. 2020 and Souto et al. 2021
(Section 6.1) and that derived from this analysis (Section 8), though
consistent within the errors. However, the former abundance has shown to be a
very good estimate of the overall metallicity of the target and has been used
to measure the abundances of the analyzed elements.
For any of these three species, we generate a grid of models, all associated
with the target’s parameters but having different values of $\xi$ ranging from
0 to 2 km/s with steps of 0.1 km/s, and different relative abundances spanning
from [X/Fe]=-0.30 dex to [X/Fe]=+0.30 dex with steps of 0.01 dex, where X
denotes C, O, or Fe, while assuming the solar relative abundance for all other
elements Y (([Y/Fe]=0), leading to 1281 synthetic spectra in total for each
species. We then perform a single ${\chi}^{2}$ minimization routine (Section
7) over all the spectral lines corresponding to each of the three
aforementioned molecules individually. To this end, the observed lines are
normalized relative to the model spectra that correspond to a specific value
of $\xi$ and varied abundances of the respective element and then compared to
those models to obtain the best-fit abundance. We calculate the average and
the standard deviation of abundances for each species and each $\xi$ value. We
find the CO lines the most sensitive to $\xi$, as the average of CO abundances
shows the largest variation as a function of $\xi$. The standard deviation of
CO abundances is minimum when $\xi$=1 km/s, and we therefore adopt $\xi$=1.0
$\pm$ 0.1 km/s for K2-18 in our analysis. As we see in Section 8, the CO
spectral lines are indeed the most sensitive to microturbulent velocity as
compared to the lines of all other studied species (Table 2). It should be
noted that the effect of rotational velocity and magnetic field on the
target’s spectrum is negligible, and we do not include these two physical
parameters in our study.
### 6.3 Mass, Radius, and Age
Since M-dwarf seismology is infeasible (Rodriguez-Lopez, 2019), the mass and
age of M dwarfs are determined using some indirect techniques. For example,
Guinan & Engle (2019) estimated an age of $\sim$2.4$\pm$0.6 Gyr for K2-18
using the Prot-Age relation for M2.5–6.0 stars (Engle & Guinan, 2018), as
K2-18 has a well-determined rotation-period Prot = 39.6 $\pm$ 0.9 days (Sarkis
et al., 2018). One may consider a theoretical method using stellar isochrones
and evolution tracks to infer M-dwarf age. However, M dwarfs evolve very
slowly once they reach the main sequence and there is no age dependence
associated with these methods.
M-dwarf Mass can be estimated using mass-luminosity relation (MLR, e.g.,
Benedict et al. 2016; Mann et al. 2019) that connect the luminosity of a
lower-main-sequence star to its mass. Cloutier et al. (2019) derived a mass of
0.495 $\pm$ 0.004 M☉ for the host star K2-18 using the MLR from Benedict et
al. (2016) based on absolute K-band magnitudes (which is favored over the
V-band whose dispersion about the relation is twice that in the K-band.)
Interferometry can be used to accurately measure the angular diameter, which
together with a well-measured bolometric flux can yield an accurate $T_{\rm
eff}$ measurement. However, this technique is expensive in terms of time and
analysis, and limited to stars that are sufficiently large ($\gtrsim$0.3 mas)
and bright ($\gtrsim$8 mag). Empirical relations are therefore more
appropriate to derive M-dwarf radius. For instance, Cloutier et al. (2019)
estimated a radius of 0.469 $\pm$ 0.010 R☉ for our target K2-18 using the
mass-radius relationship for M dwarfs from Boyajian et al. (2012).
The above inferred mass and radius of K2-18 from empirical relations are not
accurate enough to improve our inferred values of $T_{\rm eff}$ and log $g$
using high-resolution spectroscopy. The advantage of our method is that these
two parameters can be derived consistently from the same spectra using the
same diagnostic features. This is possible, thanks to the excellent quality of
our IGRINS spectra, which allows deriving consistent parameters for similar
stars observed with the same instrument.
## 7 AutoSpecFit: An Automatic Model Fitting Code for Elemental Abundance
Measurements
We present the AutoSpecFit code that carries out an automatic line-by-line
${\chi}^{2}$ minimization in an iterative manner and allows Turbospectrum to
generate the required synthetic spectra for each iteration without
interrupting the run. The abundances of the selected lines are determined
separately and modified in each iteration until the final abundances are
reached. The selected normalizing ranges and ${\chi}^{2}$ windows are used as
input for running the code. Physical parameters, i.e., effective temperature
$\rm{T_{eff}}$, metallicity [M/H], surface gravity log($g$), and
microturbulent velocity $\xi$) are also required in advance to execute
AutoSpecFit, and these parameters are not changed during the run. We find the
spectral lines sensitive to variations in physical parameters, and as a
result, these parameter can be degenerate with chemical abundances, causing
significant uncertainties in inferred abundance values. We accordingly use the
derived parameters from other independent methods (see Section 6.1 and also
Section 6.2 for the microturbulence parameter inferred from an examination
independent of AutoSpecFit) and keep them fixed with no further adjustment
when measuring elemental abundances.
The pipeline first generates a number of synthetic spectra for each studied
element X associated with the physical parameters of the star ($\rm{T_{eff}}$,
[M/H], log($g$), and $\xi$), but varied relative abundances of that particular
element usually ranging from [X/Fe]=$-$0.30 dex to [X/Fe]=+0.30 dex in steps
of 0.01 dex (61 models for each element) that are needed for a detailed
abundance analysis, and the solar relative abundances ([Y/Fe]=0) for all other
elements Y. These spectra are used in the first iteration of ${\chi}^{2}$
minimization as follows. The observed spectrum is normalized relative to all
the synthetic models over each spectral line. We perform the normalization
process during each iteration, i.e., normalizing the observed spectrum with
respect to each model under examination before calculating ${\chi}^{2}$. This
is in contrast with some other studies in which the observed spectrum is
normalized relative to a first-guess model spectrum and then used in the
${\chi}^{2}$ minimization routine without any further change (e.g. Kirby et
al., 2010; Sarmento et al., 2021; Recio-Blanco et al., 2023). However, it is
important to note that the variation of abundances generally results in a
change in the flux level of model spectra. For example, the right panels of
Figure 1 show a noticeable shift in the overall flux level of the models
around the OH line by changing the relative abundance of oxygen from
[O/Fe]=$-$0.20 dex to [O/Fe]=+0.20 dex. Since the observed spectrum is
normalized relative to each of these three models, it is also scaled in the
same way as the models, and a proper comparison can thus be made between the
observed spectrum and the models for different abundances. This is the reason
why we prefer to normalize the observed spectrum relative to all the models
used in each minimization to have a meaningful comparison.
The observed flux errors are also normalized with the same linear fit used to
normalize the observed spectrum. These normalized errors are then included in
the ${\chi}^{2}$ formula as below:
${\chi}^{2}=\sum_{i}\frac{\rm{(O_{i}-S_{i})^{2}}}{\rm{{(Oerr)_{i}}}^{2}}$ (1)
where $\rm{O_{i}}$ is the continuum/pseudocontinuum-normalized, observed flux,
$\rm{S_{i}}$ is the continuum-normalized, synthetic flux, and $\rm{Oerr_{i}}$
is the normalized, observed flux error (as described above), all at the
observed, shifted wavelength “i”. The ${\chi}^{2}$ value is calculated within
the defined ${\chi}^{2}$ window of each spectral line (Section 5). Using the
generated models, the ${\chi}^{2}$ related to each model within the chosen
${\chi}^{2}$ or fitting window of any selected spectral line is calculated,
and a polynomial fit is implemented to the resulting ${\chi}^{2}$ values as a
function of abundances. The abundance that minimizes the polynomial function
is recorded as the best-fit abundance of each particular line. For those
elements that have more than one spectral line, we calculate the average
abundance following the approach described in Adibekyan et al. (2015). We use
a weighted mean with the inverse of the distance from the median abundance as
a weight, where the distance is expressed in terms of the standard deviation
(SD) of the abundances. Since the weights corresponding to the lines with
abundances close to the median abundance are very high, we bin the distances
with a size of 0.1$\times$SD. In this way, a weight of 1/(0.1$\times$SD) is
given to the lines with abundances that are between 0 and 0.1$\times$SD away
from the median abundance, a weight of 1/(0.2$\times$SD) is given to the lines
with abundances that are between 0.1$\times$SD and 0.2$\times$SD away from the
median abundance, and so on. We prefer this method, which reduces the impact
of outlier lines without removing them. Adibekyan et al. (2015) argue that the
detection of real outliers is a difficult task, and the commonly-used outlier-
removal methods (e.g. Tukey, 1977; Shiffler, 1988; Iglewicz & Hoaglin, 1993;
Carling, 2000) are dependent on the models and the applied thresholds, and
also are not based on a clear prescription or a theoretical foundation. The
authors, therefore, recommend the use of a weighted mean instead of any
outlier-removal technique.
The abundance of elements with a single line or the average abundance of
elements with multiple lines inferred from the first iteration is used for the
second iteration. A number of model spectra are generated for each element X,
again associated with the target’s parameters and varied relative abundances
of that specific element ranging from [X/Fe]=$-$0.30 dex to [X/Fe]=+0.30 dex
in steps of 0.01 dex, but with relative abundances of all the other studied
elements Y inferred from the first iteration888It should be noted that the
relative abundance of the non-studied elements remain to be the solar values,
which are the default abundances when running Turbospectrum without any
abundance customization.. These new synthetic spectra are used in the model
fitting process exactly in the same way as the first iteration, and an average
abundance for each element of interest is derived using the procedure as
outlined above. The algorithm is repeated, and every time a series of model
spectra are generated that are optimized by the abundances obtained from the
previous iteration and employed in the next one until the abundances converge
to their final values, i.e., the difference in inferred abundance between two
consecutive iterations is less than 0.01 dex. When this condition is met for
all the studied elements simultaneously, the abundances are recorded as the
final best-fit values, and the code stops. Figure 3 shows a flowchart of the
performance of AutoSpecFit.
AutoSpecFit allows Turbospectrum to automatically produce the model spectra
required for each iteration “on the fly”. This is an advantage over
traditional methods in which the models with all possible combinations of
elemental abundances need to be generated in advance because the abundances
obtained from each iteration are unknown prior to running the fitting code.
However, for a detailed abundance measurement, this would lead to an extremely
large number of model spectra. For example, in this study, the combinations of
the 61 abundances for 10 elements would require $61^{10}\simeq 7\times
10^{17}$ spectra with traditional grid sampling. The generation of this number
of synthetic spectra would be computationally intensive and exceedingly time-
consuming, even using high-performance computing systems, which is practically
impossible. Instead, our pipeline produces $61\times 10=610$ models for each
iteration, and for instance, an analysis with 15 iterations (which is more
than enough for a typical abundance measurement, see Section 8) would require
9150 models in total, which is computationally manageable to generate999We
make use of a high-performance computing system which enables us to produce
610 model spectra within around 6 hours through 10 parallel jobs
(corresponding to 10 elements). With an additional (less than) one hour for
the fitting process (given that our original code is in MATLAB), each
iteration takes around 7 hours, on average. For a typical analysis with 8
iterations, the total time to perform the AutoSpecFit is $\sim$56 hours or
$\sim$2.3 days..
In addition, AutoSpecFit enables us to take into account the complex impact of
the abundance variation of different elements on each other. A change in the
abundance of an element (while the physical parameters are kept constant) may
cause a slight flux redistribution over different regions, which can be
reflected in the abundance measurements of other elements. That is why we use
an iterative spectral fitting routine to account for this effect, which can be
perceived by the abundance change of an element from one iteration to another
(Figures 4 and A.1-A.8). The code proceeds until all elements reach their
final abundances that are globally consistent.
## 8 Application of AutoSpecFit to the Planet-Host M dwarf K2-18
### 8.1 Chemical Abundances
We apply our technique to the planet-host M dwarf K2-18 to measure the
abundances of 10 elements: C (using CO lines), O (using OH lines), Na, Mg, Al,
K, Ca, Sc, Ti, and Fe (using FeH lines), as listed in the first column of
Table 2. The number of the lines corresponding to each species, N, is
presented in the second column of this table. As already mentioned, the star’s
physical parameters, i.e., $T_{\rm eff}$ = 3547 $\pm$ 85 K, [M/H] = 0.17 $\pm$
0.10 dex, log($g$) = 4.90 $\pm$ 0.10 dex, and $\xi$ = 1.0 $\pm$ 0.1 km/s, as
well as the selected normalizing ranges and ${\chi}^{2}$ windows are used as
input to run the AutoSpecFit. The fitting process converges after five
iterations. Figure 4 shows how the elemental abundances change from one
iteration to another until reaching their final best values, which clearly
indicates the correlation between the abundances of different elements.
The number of lines corresponding to each element is shown in the second
column, and the resulting abundances ([X/H]) are shown in the third column of
Table 2. We obtain a carbon-to-oxygen ratio for our target C/O=0.568 (for
reference, the solar ratio is (C/O)☉=0.540 using the solar abundances from
Grevesse et al. (2007)). We also determined the abundance ratios associated
with several planet-building elements such as Al/Mg=0.080, Ca/Mg=0.065, and
Fe/Mg=0.698. Figure 5 compares the normalized observed spectrum (red lines and
circles) and the final best-fit model (blue lines) that corresponds to the
target’s parameters and the derived abundances over 10 spectral lines related
to the 10 analyzed elements.
### 8.2 Abundance Errors
To determine the parameter sensitivity and the systematic uncertainties of the
derived abundances, we deviate the physical parameters by their errors
(Sections 6.1 and 6.2), in both positive and negative direction one at a time,
i.e., $T_{\rm eff}$ \+ 85 = 3632 K, $T_{\rm eff}$ $-$ 85 = 3462 K, [M/H] +
0.10 = 0.27 dex, [M/H] $-$ 0.10 = 0.07 dex, log($g$) + 0.10 = 5.00 dex,
log($g$) - 0.10 = 4.80 dex, $\xi$ \+ 0.10 = 1.10 km/s, and $\xi$ \- 0.10 =
0.90 km/s. We then perform the AutoSpecFit code eight times, in each of which
only one parameter is deviated while the other parameters remain the same as
the target’s parameter values, and the abundances of the analyzed elements are
obtained from each run. Using the synthetic models associated with the
targets’ parameters but only one parameter departed by its error, we visually
inspect the normalizing ranges over the selected spectral lines and find these
regions are still appropriate for normalizing observed spectrum even with
abundance variation. This assures us, for our future studies, that once we
determine the best normalizing ranges relative to the models with the target’s
parameters, they can also be used for models with parameters that are deviated
by their errors. Large departures beyond typical parameter uncertainties would
definitely require a new set of normalizing ranges.
Figures A.1-A.8 in the Appendix display the abundance of the 10 studied
elements as a function of iteration number for eight AutoSpecFit runs using
different input parameters, as shown in the captions. The number of iterations
required for performing the AutoSpecFit using the deviated parameters is
generally equal or more than that required for running the code using the
target’s parameters (Figure 4). For each case, the abundances change more
significantly in the first few iterations, and then smoothly converge towards
their final values.
In Table 1, the columns 4-11 show the abundance variation due to the deviated
parameters relative to the abundances obtained from the models with the star’s
parameters. The abundance variation of each element depends on the deviated
parameter, as elemental abundances show different sensitivities to different
parameters as well as the direction these parameters change. In addition, the
abundance variation differs from one element to another for the same parameter
change. For example, the abundance of the elements Ca, Al, and Mg are most
sensitive to $T_{\rm eff}$, while the abundance of the light element C (from
CO lines) is least sensitive to $T_{\rm eff}$. The abundance of the element Na
shows the highest sensitivity to [M/H], but the abundance of the elements C,
O, K, and Sc shows no significant sensitivity to [M/H]. The abundances of all
the 10 studied elements are rather sensitive to log($g$), with the elements Al
and Ca having the highest and K having the lowest sensitivity. The variation
of microturbulence velocity $\xi$ generally has a weaker influence on the
elemental abundances compared to other parameters (e.g. Souto et al., 2022;
Hejazi et al., 2023), with the abundance of element C showing the highest
sensitivity to $\xi$.
We take an average of the absolute values of the two abundance variations
related to the change of each parameter in two directions (i.e., negative and
positive). We then calculate the quadrature sum of these four averages for
each element as the systematic abundance error, $\rm{\sigma_{sys}}$, which is
shown in the column 12 of Table 2. We also obtain the random (statistical)
abundance error of the four species, CO, OH, Ti, and FeH that have a
statistically large number of lines, i.e., N $\geq$ 10, using the standard
error of the mean, i.e., $\rm{\sigma_{ran}}$=std/$\rm{\sqrt{N}}$, where std is
the standard deviation of the abundances from different lines of each species,
as shown in the column 13 of Table 2. The last column of the table presents
the quadrature sum of the systematic and random (if applicable) errors, as the
total error of the derived abundances. It should be noted that random errors
are too small to significantly contribute to the total errors. For those
elements with no random error, the total error may be slightly underestimated.
Figure 6 presents the abundances of the 10 analyzed elements as a function of
their atomic number, and their total abundance errors are shown as vertical
error bars. Using the abundance errors, we obtain the uncertainty of the
abundance ratios: C/O=0.568 $\pm$ 0.026, Al/Mg=0.080 $\pm$ 0.011, Ca/Mg=0.065
$\pm$ 0.010, and Fe/Mg=0.698 $\pm$ 0.178. We recall that the abundance ratio
of two elements depends on the subtraction of their absolute abundances, i.e.,
$\rm{X/Y=10^{(A(X)-A(Y))}}$, and as a result, their systematic uncertainties
related to the variation of different stellar parameters largely cancel. In
addition, the (uncorrelated) random uncertainties (if applicable) are very
small (see Table 2). All these have led to relatively small uncertainties of
abundance ratios, other than Fe/Mg for which the rather large difference
between the systematic errors of the two elements Fe and Mg associated with
effective temperature has resulted in a significantly larger uncertainty.
It is important to note that abundance errors highly depend on the uncertainty
of input physical parameters. Smaller deviations of parameters, in particular
effective temperature, would give rise to smaller abundance errors (Melo et
al., 2024). To derive more accurate elemental abundances, we need to have more
accurate input stellar parameters, which requires more reliable model
atmospheres and line lists as well as more robust techniques for parameter
determination.
Figure 1: Comparison between the normalized observed spectrum (red lines and
circles) of K2-18 and the model spectra (blue lines) associated with the
target’s parameters but varying abundances of the element K (left panels) and
the element O (right panels), while assuming solar relative abundances for all
other elements. The black circles (at the edges of the panels) show the
normalizing points within the selected continuum/pseudocontinuum normalizing
ranges that are separated from the inner spectral regions by green dashed
lines.
Figure 2: Comparison between the normalized observed spectrum (red lines and circles) of K2-18 and the model spectra (blue lines) associated with the target’s parameters but varying abundances of the element Sc (left panels) and the element Ti (right panels), while assuming solar relative abundances for all other elements. The black circles (at the edges of the panels) show the normalizing points within the selected continuum/pseudocontinuum normalizing ranges that are separated from the inner spectral regions by green dashed lines. Table 1: 148 atomic and molecular lines selected for this analysis Species | Central wavelength (Å) | ${\chi}^{2}$ window (Å) | Comments
---|---|---|---
CO | 23006.89 | 23006.25-23007.40 |
CO | 23015.00 | 23014.50-23015.50 |
CO | 23023.52 | 23023.00-23024.10 |
CO | 23032.43 | 23032.00-23033.15 |
CO | 23061.59 | 23061.05-23062.10 |
CO | 23083.04 | 23082.60-23083.50 |
CO | 23094.37 | 23093.95-23094.80 |
CO | 23118.23 | 23117.75-23118.75 |
CO | 23170.81 | 23170.35-23171.40 |
CO | 23184.97 | 23184.50-23185.45 |
CO | 23341.22 | 23340.70-23341.95 |
CO | 23351.41 | 23350.95-23352.05 |
CO | 23421.19 | 23420.77-23421.70 |
CO | 23426.30 | 23425.78-23426.70 |
CO | 23447.76 | 23447.40-23448.25 |
CO | 23461.67 | 23461.20-23462.10 |
CO | 23476.00 | 23475.60-23476.40 |
CO | 23505.90 | 23505.40-23506.55 |
CO | 23637.61 | 23637.20-23638.00 |
CO | 23658.53 | 23658.15-23658.95 |
CO | 23661.26 | 23660.78-23661.73 |
CO | 23724.24 | 23723.73-23724.75 |
CO | 23745.10 | 23744.65-23745.60 |
CO | 23759.17 | 23758.70-23759.70 |
CO | 24009.23 | 24008.50-24009.75 |
CO | 24023.59 | 24023.10-24024.00 |
CO | 24128.68 | 24128.20-24129.15 |
CO | 24198.13 | 24197.60-24198.70 |
OH | 15002.15 | 15001.85-15003.45 |
OH | 15003.12 | 15001.85-15003.45 |
OH | 15145.77 | 15145.50-15146.10 |
OH | 15147.94 | 15147.60-15148.30 |
OH | 15264.60 | 15264.30-15264.90 |
OH | 15266.17 | 15265.90-15266.45 |
OH | 15278.52 | 15278.16-15278.85 |
OH | 15281.05 | 15280.70-15281.41 |
OH | 15409.17 | 15408.90-15409.40 |
OH | 15419.46 | 15419.10-15419.72 |
OH | 15422.37 | 15421.97-15422.70 |
OH | 15558.02 | 15557.73-15558.37 |
OH | 15560.24 | 15559.90-15560.55 |
OH | 15568.78 | 15568.45-15569.11 |
OH | 15572.08 | 15571.72-15572.40 |
OH | 15626.70 | 15626.42-15627.70 |
OH | 15627.41 | 15626.42-15627.70 |
OH | 15651.90 | 15651.55-15652.20 |
OH | 15653.48 | 15653.20-15653.75 |
OH | 15719.70 | 15719.30-15720.10 |
OH | 15726.72 | 15726.44-15727.00 |
OH | 15755.52 | 15755.27-15755.77 |
OH | 15756.53 | 15756.20-15756.85 |
OH | 15884.90 | 15884.50-15885.30 |
OH | 15892.13 | 15891.80-15892.50 |
OH | 15893.53 | 15893.15-15893.80 |
OH | 15897.70 | 15897.30-15898.10 |
OH | 15910.42 | 15910.05-15910.80 |
OH | 15912.73 | 15912.33-15913.10 |
Table 1: Continued
Species | Central wavelength (Å) | ${\chi}^{2}$ window (Å) | Comments
---|---|---|---
OH | 16036.89 | 16036.43-16037.20 |
OH | 16038.54 | 16038.20-16038.85 |
OH | 16052.76 | 16052.43-16053.10 |
OH | 16055.46 | 16055.10-16055.78 |
OH | 16065.05 | 16064.80-16065.40 |
OH | 16069.52 | 16069.17-16069.90 |
OH | 16190.13 | 16189.80-16190.50 |
OH | 16192.13 | 16191.80-16192.40 |
OH | 16207.19 | 16206.70-16207.50 |
OH | 16247.88 | 16247.53-16248.27 |
OH | 16260.15 | 16259.74-16260.56 |
OH | 16346.18 | 16345.81-16346.57 |
OH | 16352.22 | 16351.75-16352.65 |
OH | 16354.58 | 16354.22-16354.96 |
OH | 16364.59 | 16364.20-16364.95 |
OH | 16368.13 | 16367.78-16368.53 |
OH | 16448.05 | 16447.70-16448.50 |
OH | 16450.37 | 16449.98-16450.80 |
OH | 16456.04 | 16455.70-16456.40 |
OH | 16471.15 | 16470.82-16471.50 |
OH | 16523.50 | 16523.15-16523.80 |
OH | 16526.25 | 16525.90-16526.60 |
OH | 16534.58 | 16534.28-16534.93 |
OH | 16538.59 | 16538.10-16538.88 |
OH | 16581.27 | 16580.95-16581.70 |
OH | 16582.32 | 16581.98-16582.60 |
OH | 16649.95 | 16649.60-16650.40 |
OH | 16654.65 | 16654.32-16654.98 |
OH | 16655.99 | 16655.65-16656.37 |
OH | 16662.20 | 16661.87-16662.55 |
OH | 16704.36 | 16703.95-16704.90 |
OH | 16866.69 | 16866.30-16867.05 |
OH | 16879.09 | 16878.70-16879.52 |
OH | 16895.18 | 16894.68-16895.64 |
OH | 16902.73 | 16902.32-16903.17 |
OH | 16904.28 | 16903.90-16904.75 |
OH | 16905.63 | 16905.25-16905.95 |
OH | 16909.29 | 16908.90-16909.75 |
OH | 17052.20 | 17051.85-17052.60 |
OH | 17066.13 | 17065.77-17066.50 |
OH | 17069.48 | 17069.15-17069.78 |
OH | 17094.52 | 17094.20-17094.95 |
OH | 17096.39 | 17095.97-17096.80 |
OH | 17239.72 | 17239.45-17240.00 |
OH | 17767.06 | 17766.75-17767.35 |
Na I | 22083.66 | 22082.35-22085.00 | The combination of four Na I lines:
| | | 22083.617, 22083.627*, 22083.684*, 22083.694*,
| | | including three HFS lines
Mg I | 15040.25 | 15039.80-15040.65 |
Mg I | 15047.71 | 15047.20-15048.10 |
Mg I | 15765.84 | 15765.30-15766.32 | Blended with two Mg I lines:
| | | 15765.645, 15765.747,
| | | significantly weaker than the main, central line (i.e., 15765.84),
| | | the three blended lines have different J values of the upper levels
Mg I | 17108.63 | 17108.10-17109.05 |
Al I | 16718.96 | 16718.10-16719.70 | The combination of six Al I lines:
| | | 16718.911, 16718.925*, 16718.943*,
| | | 16718.945*, 16718.963*, 16718.990*,
| | | including five HFS lines
Al I | 16750.56 | 16750.00-16751.10 | The combination of 12 Al I lines:
| | | 16750.455, 16750.539*, 16750.550*, 16750.608*,
| | | 16750.616*, 16750.627*, 16750.660*, 16750.665*,
| | | 16750.673*, 16750.698*, 16750.703*, 16750.717*
| | | including 11 HFS lines
Note. — * denotes a line resulted from hyperfine structure (HFS) splitting.
Table 1: Continued
Species | Central wavelength (Å) | ${\chi}^{2}$ window (Å) | Comments
---|---|---|---
K I | 15168.38 | 15167.95-15168.80 |
Ca I | 19853.09 | 19852.57-19853.70 |
Ca I | 19933.73 | 19933.20-19934.30 |
Ca I | 22607.94 | 22607.20-22608.65 |
Sc I | 22266.73 | 22266.25-22267.25 | The combination of six Sc I lines:
| | | 22266.533, 22266.637*, 22266.715*,
| | | 22266.739*, 22266.871*, 22266.975*,
| | | including five HFS lines
Ti I | 15334.85 | 15334.47-15335.20 | Blended with three weak Ti I lines:
| | | 15334.139, 15335.039, 15335.458,
| | | too weak to influence the shape of the main, central line (i.e., 15334.85),
| | | the four blended lines have different J values of the lower and/or upper levels
Ti I | 15715.57 | 15715.10-15716.20 | Blended with four weak Ti I:
| | | 15715.758, 15715.887, 15716.008, 15716.484,
| | | too weak to influence the shape of the main, central line (i.e., 15715.57),
| | | the five blended lines have different J values of the lower and/or upper levels
Ti I | 21782.94 | 21782.20-21783.75 | Blended with three Ti I lines:
| | | 21782.555, 21782.560, 21782.996,
| | | too weak to influence the shape of the main, central line (i.e., 21782.94)
| | | the four blended lines have different J values of the lower and/or upper levels
Ti I | 21897.39 | 21896.75-21898.15 |
Ti I | 22004.51 | 22004.00-22004.95 |
Ti I | 22211.24 | 22210.55-22211.95 | Blended with one Ti I line:
| | | 22210.631,
| | | too weak to influence the shape of the main, central line (i.e., 22211.24),
| | | the two blended lines have different J value of the lower levels
Ti I | 22232.86 | 22232.20-22233.50 |
Ti I | 22274.02 | 22273.45-22274.55 |
Ti I | 22963.33 | 22962.67-22963.94 |
Ti I | 23441.48 | 23441.15-23441.95 | Blended with two Ti I:
| | | 23440.630, 23441.669,
| | | too weak to influence the shape of the main line, central (i.e., 23441.48),
| | | the three blended lines have different J values of the lower and/or upper levels
FeH | 15872.67 | 15872.31-15873.00 |
FeH | 15915.94 | 15915.70-15916.22 |
FeH | 15945.71 | 15945.39-15946.00 |
FeH | 15993.22 | 15992.93-15993.60 |
FeH | 16058.56 | 16058.27-16058.89 |
FeH | 16067.85 | 16067.60-16068.20 |
FeH | 16172.62 | 16172.35-16173.00 |
FeH | 16182.95 | 16182.70-16183.25 |
FeH | 16184.38 | 16184.10-16184.80 |
FeH | 16249.70 | 16249.30-16249.98 |
FeH | 16319.36 | 16319.08-16319.70 |
FeH | 16330.67 | 16330.20-16330.93 |
FeH | 16361.74 | 16361.45-16362.08 |
FeH | 16466.93 | 16466.45-16467.20 |
FeH | 16682.00 | 16681.70-16682.30 |
FeH | 16735.42 | 16735.15-16735.65 |
FeH | 16738.29 | 16737.97-16738.58 |
FeH | 16796.38 | 16796.05-16796.68 |
FeH | 16862.14 | 16861.77-16862.42 |
FeH | 16922.75 | 16922.40-16923.00 |
FeH | 17068.40 | 17068.05-17068.75 |
FeH | 17277.76 | 17277.40-17278.10 |
FeH | 17293.38 | 17292.90-17293.70 |
FeH | 17544.47 | 17544.12-17544.75 |
Note. — * denotes a line resulted from hyperfine structure (HFS) splitting.
Table 2: The chemical abundances and their corresponding uncertainties for the ten studied elements Species | N | [X/H] | ${\Delta}T_{\rm eff}$ | $\Delta$[M/H] | $\Delta$log($g$) | $\Delta\xi$ | $\sigma_{\rm{sys}}$ | $\sigma_{\rm{ran}}=\rm{std}/\sqrt{N}$ | ${\sigma}{\rm[X/H]}_{\rm tot}$
---|---|---|---|---|---|---|---|---|---
| | | $-$85 | +85 | $-$0.10 | +0.10 | $-$0.10 | +0.10 | $-$0.10 | +0.10 | | |
| | (dex) | (K) | (K) | (dex) | (dex) | (dex) | (dex) | km/s | km/s | (dex) | (dex) | (dex)
C (CO) | 28 | +0.104 | $-$0.003 | $-$0.004 | +0.004 | $-$0.006 | $-$0.084 | +0.081 | +0.027 | $-$0.029 | 0.088 | 0.011 | 0.089
O (OH) | 74 | +0.080 | +0.014 | $-$0.019 | $-$0.005 | +0.007 | $-$0.079 | +0.077 | +0.018 | $-$0.021 | 0.083 | 0.002 | 0.083
Na | 1 | +0.066 | +0.064 | $-$0.076 | +0.073 | $-$0.085 | $-$0.080 | +0.076 | +0.010 | $-$0.012 | 0.132 | – | 0.132
Mg | 4 | +0.043 | +0.174 | $-$0.142 | +0.005 | +0.023 | $-$0.075 | +0.096 | +0.005 | $-$0.004 | 0.181 | – | 0.181
Al | 2 | +0.105 | +0.177 | $-$0.156 | +0.056 | $-$0.038 | $-$0.130 | +0.133 | +0.003 | $-$0.007 | 0.218 | – | 0.218
K | 1 | +0.040 | $-$0.019 | +0.025 | +0.002 | $-$0.007 | $-$0.026 | +0.025 | +0.002 | $-$0.003 | 0.035 | – | 0.035
Ca | 3 | +0.074 | +0.183 | $-$0.176 | +0.018 | $-$0.007 | $-$0.138 | +0.122 | +0.012 | $-$0.002 | 0.223 | – | 0.223
Sc | 1 | +0.134 | +0.039 | $-$0.028 | $-$0.002 | +0.001 | $-$0.083 | +0.083 | +0.003 | $-$0.006 | 0.090 | – | 0.090
Ti | 10 | +0.088 | +0.105 | $-$0.091 | +0.025 | $-$0.028 | $-$0.103 | +0.103 | +0.011 | $-$0.016 | 0.145 | 0.016 | 0.146
Fe (FeH) | 24 | $-$0.033 | +0.051 | $-$0.048 | +0.053 | +0.007 | $-$0.082 | +0.100 | +0.012 | $-$0.023 | 0.110 | 0.012 | 0.111
Figure 3: The flowchart of the AutoSpecFit performance from step 1 to step 7.
The first two steps are run only in the first iteration. The pipeline returns
back to step 3 to start the next iteration. Figure 4: The abundance of the 10
analyzed elements as a function of the iteration number. The abundances are
inferred using the models associated with the target’s parameters, i.e.,
$T_{\rm eff}$ = 3547 K, [M/H] = 0.17 dex, log($g$) = 4.90 dex, and $\xi$ = 1.0
km/s. The total number of iterations is 5.
Figure 5: Comparison between the normalized observed spectrum (red lines and
circles) and the final best-fit model (blue lines) over 10 spectral lines
corresponding to the 10 analyzed elements. Figure 6: The final inferred
abundances of the 10 analyzed elements versus their corresponding atomic
number. The error bars show the uncertainty of the abundances (as presented in
the last column of Table 2). The blue dashed line shows the zero abundance
level.
## 9 Summary and Conclusion
### 9.1 High-resolution Spectroscopic Analysis of K2-18
We introduce AutoSpecFit, a new automatic line-by-line synthetic model fitting
code, to measure the chemical abundances of cool dwarfs. The code performs a
series of iterative ${\chi}^{2}$ minimization processes and allows
Turbospectrum to generate the synthetic spectra required for each iteration,
which are optimized using the abundances inferred from the previous iteration.
We illustrate how the abundances of different elements are dependent on each
other and pass through multiple iterations to reach their final abundances
that are globally consistent. Our abundance analysis offers a technique that
carefully takes into account the complex dependency between different elements
when varying their abundances in a timely manner. In addition, we present our
method for continuum/pseudocontinuum normalization to make a meaningful
comparison between the observed and model spectrum in the ${\chi}^{2}$
minimization. Since the continuum level cannot be identified in many spectral
regions of cool dwarfs, we normalize the observed spectrum relative to
synthetic, continuum-normalized spectra using several wavelength data points
around the spectral lines of interest.
We apply our technique to the high-resolution IGRINS H- and K-band spectra of
the sub-Neptune K2-18’s host M dwarf and measure the abundances of 10
elements, C (using CO lines), O (using OH lines), Na, Mg, Al, K, Ca, Sc, Ti,
and Fe (using FeH lines), along with their detailed error analysis. We find
near-solar abundances and carbon-to-oxygen ratio, C/O=0.568 $\pm$ 0.026. We
also obtain the abundance ratios of some key planet-building elements, such as
Al/Mg, Ca/Mg, and Fe/Mg. We emphasize that the accuracy of inferred abundances
depends on the accuracy of the input physical parameters as well as the
normalization procedure. In particular, more accurate parameters, especially
effective temperature, would lead to more accurate elemental abundances.
### 9.2 Star-Planet Connection
The exoplanet K2-18 b has been targeted by several JWST programs, and its
atmosphere is being characterized more accurately than from previous studies,
for example, using HST observations. Historically, exoplanet abundances have
been derived assuming Solar abundances, however, it is the stellar abundances
that are the relevant benchmark (Turrini et al., 2021; Pacetti et al., 2022).
The assumption of Solar vs. stellar abundances can significantly affect the
inferred planetary abundances, leading to abundance errors larger than the
expected JWST atmospheric measurement precision (Greene et al., 2016). The
detailed elemental abundances of the host star k2-18 will be beneficial for
future JWST analyses to accurately measure the chemical composition of the
exoplanet K2-18 b.
The abundance ratios of volatile elements such as C/O play an important role
in the location of planet formation within the protoplanetary disk (Öberg et
al., 2011). A planet with a sub-stellar C/O ratio is likely to have a water-
rich atmosphere (Madhusudhan, 2012; Teske et al., 2014) with a formation
location within the H2O ice line. On the other hand, a planet with a super-
stellar C/O ratio is likely to be rich in carbonaceous compounds and have a
formation location beyond the H2O ice line, which has then experienced an
inward migration to its current place (e.g. Reggiani et al., 2022).
Furthermore, an overabundance of alkali metals, Na and K, has been found in
the atmospheres of some hot gas giants relative to their host stars (Hands &
Helled, 2022). Such an enhancement of alkali species is thought to be a result
of planet formation exterior to the H2O ice line followed by inward migration.
However, due to the uncertainties on K2-18 b’s internal structure, its C/O
ratio has not yet been confidently measured. For example, the observed carbon-
bearing species combined with no observed water vapor would imply a relatively
high C/O ratio, but this only holds for classical gas-dominated models. If
instead, the observed atmosphere is blanketing a planetary ocean, we wouldn’t
observe any of the water present in the planet and would erroneously infer a
high C/O ratio. Madhusudhan et al. (2023) did not present a C/O ratio in their
atmospheric observations, and Wogan et al. (2024) assumed a solar C/O ratio in
their planetary atmosphere models. As of now, we are unable to measure K2-18
b’s C/O ratio with confidence, but hopefully our understanding of the planet
and its interior structure will improve with future observations and modeling
efforts. This, together with stellar C/O ratio measured in this study, will
help us to better understand the formation pathway of the planet.
For our follow-up research, we will attempt to develop an alternative method
to determine stellar parameters by performing a deep analysis of parameter
sensitivity and the correlation between parameters and elemental abundances.
The degeneracy effect is one of the major issues in the spectroscopic
determination of stellar parameters, in particular for cool dwarfs. Many
spectroscopic studies use inferred values of one or two parameters from
empirical photometric relations and take them out of synthetic spectral
fitting. However, current photometric calibrations may result in unreliable
parameter values for some stars, causing large uncertainties in determining
the free parameters. One way to overcome this problem is to find the spectral
regions/features that are mostly sensitive to only one parameter. Utilizing
such collected wavelength intervals will isolate the contribution of each
parameter to the respective spectral lines and features during model fitting.
This may enable us to determine the input parameters with higher accuracy,
which can yield more accurate elemental abundances.
In our future work, We will also apply our abundance measurement technique to
other observed cool JWST host stars and measure their chemical abundances,
which can then be used to determine the properties of their exoplanet in
upcoming JWST analyses.
We wish to thank the anonymous referee for their helpful comments and
suggestions, which improved our manuscript. We extend our thanks to Justin
Cantrell for his technical support with the high-performance computing system
of the physics and astronomy department, Georgia State University, which was
used for this study. N.H. and I.J.M.C. acknowledge support from NSF AAG grant
No. 2108686 and from NASA ICAR grant No. NNH19ZDA001N. D.S. thanks the
National Council for Scientific and Technological Development – CNPq. T.N.
acknowledges support from the Australian Research Council Centre of Excellence
for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project No.
CE170100013. D.S. thanks the National Council for Scientific and Technological
DevelopmentCNPq. E.M. acknowledges financial support through a “Margarita
Salas” postdoctoral fellowship from Universidad Complutense de Madrid
(CT18/22), funded by the Spanish Ministerio de Universidades with
NextGeneration EU funds.
## Appendix A Figures
Figure A.1: Identical to Figure 3, except that abundances are inferred using
the models associated with the deviated effective temperature by +85 K, i.e.,
$T_{\rm eff}$ = 3632 K, [M/H] = 0.17 dex, log($g$) = 4.90 dex, and $\xi$ = 1.0
km/s. The total number of iterations is 5. Figure A.2: Identical to Figure 3,
except that abundances are inferred using the models associated with the
deviated overall metallicity by +0.10 dex, i.e., $T_{\rm eff}$ = 3547 K, [M/H]
= 0.27 dex, log($g$) = 4.90 dex, and $\xi$ = 1.0 km/s. The total number of
iterations is 6. Figure A.3: Identical to Figure 3, except that abundances are
inferred using the models associated with the deviated surface gravity by
+0.10 dex, i.e., $T_{\rm eff}$ = 3547 K, [M/H] = 0.17 dex, log($g$) = 5.00
dex, and $\xi$ = 1.0 km/s. The total number of iterations is 5. Figure A.4:
Identical to Figure 3, except that abundances are inferred using the models
associated with the deviated microturbulence by +0.10 km/s, i.e., $T_{\rm
eff}$ = 3547 K, [M/H] = 0.17 dex, log($g$) = 4.90 dex, and $\xi$ = 1.1 km/s.
The total number of iterations is 6. Figure A.5: Identical to Figure 3, except
that abundances are inferred using the models associated with the deviated
effective temperature by $-$85 K, i.e., $T_{\rm eff}$ = 3462 K, [M/H] = 0.17
dex, log($g$) = 4.90 dex, and $\xi$ = 1.0 km/s. The total number of iterations
is 5. Figure A.6: Identical to Figure 3, except that abundances are inferred
using the models associated with the deviated overall metallicity by $-$0.10
dex, i.e., $T_{\rm eff}$ = 3547 K, [M/H] = 0.07 dex, log($g$) = 4.90 dex, and
$\xi$ = 1.0 km/s. The total number of iterations is 5. Figure A.7: Identical
to Figure 3, except that abundances are inferred using the models associated
with the deviated surface gravity by $-$0.10 dex, i.e., $T_{\rm eff}$ = 3547
K, [M/H] = 0.17 dex, log($g$) = 4.80 dex, and $\xi$ = 1.0 km/s. The total
number of iterations is 8. Figure A.8: Identical to Figure 3, except that
abundances are inferred using the models associated with the deviated
microturbulence by $-$0.10 km/s, i.e., $T_{\rm eff}$ = 3547 K, [M/H] = 0.17
dex, log($g$) = 4.90 dex, and $\xi$ = 0.9 km/s. The total number of iterations
is 6.
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|
††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work
# Constructing higher-order topological states in higher dimension
Yao Wang Center for Integrated Quantum Information Technologies, School of
Physics and Astronomy, State Key Laboratory of Advanced Optical Communication
Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Yongguan Ke Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing & School of Physics and Astronomy,
Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China Nonlinear
Physics Centre, Research School of Physics and Engineering,
Australian National University, Canberra ACT 2601, Australia Yi-Jun Chang
Center for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China Yong-Heng Lu
Center for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China Jun Gao Center
for Integrated Quantum Information Technologies, School of Physics and
Astronomy, State Key Laboratory of Advanced Optical Communication Systems and
Networks, Shanghai Jiao Tong University, Shanghai 200240, China Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China Chaohong Lee
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum
Metrology and Sensing & School of Physics and Astronomy,
Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China State Key
Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, China Xian-Min Jin
<EMAIL_ADDRESS>Center for Integrated Quantum Information
Technologies, School of Physics and Astronomy, State Key Laboratory of
Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong
University, Shanghai 200240, China Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China
Higher-order topological phase as a generalization of Berry phase attracts an
enormous amount of research. The current theoretical models supporting higher-
order topological phases, however, cannot give the connection between lower
and higher-order topological phases when extending the lattice from lower to
higher dimensions. Here, we theoretically propose and experimentally
demonstrate a topological corner state constructed from the edge states in one
dimensional lattice. The two-dimensional square lattice owns independent
spatial modulation of coupling in each direction, and the combination of edge
states in each direction come up to the higher-order topological corner state
in two-dimensional lattice, revealing the connection of topological phase in
lower and higher dimensional lattices. Moreover, the topological corner states
in two-dimensional lattice can also be viewed as the dimension-reduction from
a four-dimensional topological phase characterized by vector Chern number,
considering two modulation phases as synthetic dimensions in Aubry-André-
Harper model discussed as example here. Our work deeps the understanding to
topological phases breaking through the lattice dimension, and provides a
promising tool constructing higher topological phases in higher dimensional
structures.
## I Introduction
The higher-order topological phases introduced in higher-dimensional lattices
recently extend the conventional understanding on the topological nontrivial
materials, where the $d$-dimensional lattice owns not only the first-order
($d-1$)-dimensional edge states but also the $n$-order ($d-n$)-dimensional
edge states T20171 ; T20172 ; T20181 ; T20190 ; T20191 ; T20192 ; T20201 ;
T20202 . The second-order corner states in two-dimensional (2D) lattices are
widely investigated since 2019 in sonic Es1 ; Es2 ; Es3 ; Es4 ; Es5 ; Es6 ,
ring resonator Er , waveguide Ew1 ; Ew2 ; Ew3 , cavity Ec1 ; Ec2 , and cold
atom Ecold systems. Recently, the higher-order topological states in three-
dimensional lattices are also reported E3D1 ; E3D2 . The investigations on
higher-order topological phases in both theories and experiments promote and
extend the development of topological photonics Topo_review_1 ; Topo_review_2
.
The current principles to seek high-order topological states are mainly based
on analyzing spatial or (and) nonspatial symmetries T20171 ; T20172 ; TPRL118
; T20181 ; Langbehn2017 ; Song2017 ; Linhu2018 ; Max2018 . In spatial-
symmetric (such as inversion- or rotational-symmetric) systems, high-order
topological states may originate from quantized dipole polarization TPRL118 ;
T20181 or multipole moments Langbehn2017 ; Song2017 . In nonspatial-symmetric
(such as chiral-symmetric) systems, corner states may arise due to nontrivial
edge winding numbers Linhu2018 . By combining nonspatial and spatial
symmetries, second-order topological insulator and superconductors have been
partially classified Max2018 . The existing schemes requiring delicate designs
of overall symmetry, as a top-to-bottom approach, cannot provide insight of
the connection between lower-order and higher-order topological states. Since
lower-order edge states are well-known, we may wonder whether it is possible
to use lower-order topological states as building blocks assembling to higher-
order topological states. If possible, what are their topological
correspondence to the bulk?
Here, we theoretically propose and experimentally demonstrate a bottom-to-top
scheme for constructing topological corner states by using topological edge
states as building blocks. In each direction the topological edge states are
snapshot states in a topological pumping by means of a changing one-
dimensional dipole moment which is related to Chern number. Such scheme
naturally extends Chern number to vector Chern number with individual
components separately defined in each direction, from lower- to higher-
dimensional lattices. The hierarchical relation between two-dimensional Zak
phase TPRL118 ; T20181 and vector Chern number can be understood as a varying
of two-dimensional dipole polarization generates quantized charge pumping in
two directions. The fact that corner states are guaranteed by nontrivial
vector Chern number can be termed as _bulk-corner correspondence_ , and they
inherit the topological origin of edge states as a dimension reduction of
quantum Hall phase. We have to emphasize that such corner states do not
require any fine tuning of spatial or nonspatial symmetries.
Taking the off-diagonal Aubry-André-Harper (AAH) model for example, we
theoretically analyze the topological origin when extending the lattice from
one dimension to two dimensions. We construct the two-dimensional photonic
topological lattice and successfully observe the higher-order topological
corner states predicted in theory. Our model gives an intuitive understanding
on the raising of higher-order topological phases in higher dimensional
lattice, which connects the topological phases in different dimensional
lattices and provides a convenient tool for constructing higher-order
topological phases in higher dimensional lattices.
Figure 1: Schematic of constructing corner states. (a) Three types of edge
states in one-dimensional lattices. The edge states in one-dimensional lattice
can be regarded as the building block for the higher-order topological states.
(b-c) Corner states in two- and three-dimensional lattices built by edge
states in one-dimensional lattices. We can find the connection between the
topological states in different dimensional lattice by projecting the higher-
dimensional lattice into one-dimensional lattice in direction of x, y and z
axis respectively.
## II Vector Chern number and corner states
We explore a systematical method to construct corner states in a two-
dimensional square lattice. Consider that the position of a particle is
denoted by $(i,j)$, where $i$ and $j$ are the site indices of $x$ and $y$
directions respectively. The coupling between $(i,j)$th and $(k,l)$th lattice
sites has the form, $H_{x}(i,k)\delta_{j,l}+\delta_{i,k}H_{y}(j,l)$, where
$H_{x}(i,k)$ is the coupling matrix along $x$ direction irrelevant to
positions in $y$ direction, and vice versa. The motion of a particle hopping
in such lattice is governed by the Hamiltonian,
$H=H_{x}\otimes I_{y}+I_{x}\otimes H_{y},$ (1)
where $I_{s}$ is an identical matrix in $s$ direction with $s\in x,\ y$. Once
we obtain the eigenvalues $E_{p}^{s}$ and eigenstates $|\psi_{p}^{s}\rangle$
corresponding to $H_{s}$ where $p$ are quantum numbers, we can immediately
prove that
$H|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle=(E_{m}^{x}+E_{n}^{y})|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle,$
(2)
that is, $E_{m}^{x}+E_{n}^{y}$ and
$|\psi_{m}^{x}\rangle\otimes|\psi_{n}^{y}\rangle$ are the eigenvalues and
eigenstates corresponding to $H$, respectively. If $|\psi_{m}^{x}\rangle$ and
$|\psi_{n}^{y}\rangle$ are topological edge states, the product of these edge
states becomes a topological corner state in two dimensions. Hence the seeking
for topological corner states is transformed to separately design the coupling
matrix in each direction that supports topological edge states.
Consider that the coupling matrix $H_{s}$ is controlled by two parameters
$\phi^{s}$ and $\theta^{s}$, which satisfied
$H_{s}(\phi^{s},\theta^{s})=H_{s}(\phi^{s}+2\pi,\theta^{s}+2\pi)$. In
practice, $\phi^{s}$ is a modulated phase, and $\theta^{s}$ is a twisted phase
when a particle hopping across the boundary in $s$ direction by imposing
twisted boundary condition. To characterize the bulk topology, we can define
vector Chern number $(C_{\mathbf{u}}^{x};C_{\mathbf{v}}^{y})$ with individual
components given by
$C_{\mathbf{w}}^{s}=\frac{1}{{2\pi
i}}\int_{BZ}d\theta^{s}{d\phi^{s}\det[\mathcal{F}({\phi^{s},\theta^{s}})]}.$
(3)
Here,
$[\mathcal{F}({\phi^{s},\theta^{s}})]^{m,n}=\partial_{\phi^{s}}A_{\theta^{s}}^{m,n}-\partial_{\theta^{s}}A_{\phi^{s}}^{m,n}+i[A_{\phi^{s}},A_{\theta^{s}}]^{m,n}$
are elements of the non-Abelian Berry curvature with elements of Berry
connection
$(A_{\mu})^{m,n}=\langle\psi_{m}^{s}({\phi^{s},\theta^{s}})|\nabla_{\mu}|\psi_{n}^{s}({\phi^{s},\theta^{s}})\rangle$.
$m,n\in\mathbf{w}$ which is a subset of near degenerate bands. If both the two
components of the vector Chern number
$(C_{\mathbf{u}}^{x};C_{\mathbf{v}}^{y})$ are integer, in the open boundary
condition there exists at least one topological corner state at some fixed
modulated phases $(\phi^{x},\phi^{y})$. We term this relation as _bulk-corner
correspondence_ , which gives a clear topological origin of corner states,
i.e., a dimensional reduction of a four-dimensional topological space
characterized by vector Chern number.
Figure 2: The density of states for (a) one-dimensional AAH lattice and (b)
two-dimensional AAH lattices. The finite energy gaps between corner states and
extended states inherit from those between edge states and extended states.
The corner states are constructed from the edge states, the former has twice
the energy of the latter. The local density of states are shown in the insets,
the bulk state and edge state in one-dimensional AAH lattice are shown in (i)
and (ii) respectively, the edge states and corner states in two-dimensional
AAH lattice are shown in (iii, v) and (iv, vi) respectively. The parameters in
simulation are adopted as: $t_{x(y)}=0.5$, $\lambda_{x(y)}=0.95$,
$b_{x(y)}=(\sqrt{5}+1)/2$, and there are 15 sites for one-dimensional lattice
and 15$\times$15 sites for two-dimensional lattice. DOS: density of states.
Figure 3: Corner states. (a) Schematic of the fabricated photonic
quasicrystal. Nine input sites are set in the lattice, four waveguides marked
as C1, C2, C3 and C4 are designed to observe the corner states. Four
waveguides marked as B1, B2, B3 and B4 are designed to observe the edge
states. The waveguide marked as T is designed to observe bulk state. (b)
Spectrum of one-dimensional off-diagonal AAH lattice. For 16-sited lattice,
two boundary modes (green lines) cross in the band gap, for 15-sited lattice,
only one boundary mode connects the bands separated by the gap. The red dash
lines give the $\phi$ adopted in experiment. (c-d) Measured corner states. The
photons are confined in the corners when we excite the lattice corner sites
(c). The white dotted squares point out the ranges of corner states. The
quantified results confirm the theoretically predicted topological corner
states arising by extending topological lattice with edge state from one
dimension to two dimensions. Figure 4: Edge states. (a) The measured result of
edge states. We inject the photons into the lattices from the input sites B1,
B2, B3, and B4 respectively, the photons are localized in the edges of the
lattices. (b) The trivial cases. The photon cannot be confined in the edges of
the lattices. The white dotted squares point out the ranges of edge states.
(c) The quantified results. The edge states can be conveniently extended from
lower dimension to higher dimension.
For explicitness, we focus on a two-dimensional off-diagonal AAH lattice with
hopping strengths varying in space,
$\displaystyle H=$ $\displaystyle\sum_{i,j}t_{x}[1+\lambda_{x}\cos(2\pi
b_{x}i+\phi^{x})]\hat{a}_{i,j}\hat{a}_{i+1,j}^{\dagger}$
$\displaystyle+t_{y}[1+\lambda_{y}\cos(2\pi
b_{y}j+\phi^{y})]\hat{a}_{i,j}\hat{a}_{i,j+1}^{\dagger}+H.c.,$ (4)
where $\hat{a}_{i,j}^{\dagger}$ ($\hat{a}_{i,j}$) is the creation
(annihilation) operator at site ($i,j$), $t_{x(y)}$ is the average coupling
strengths, $\lambda_{x(y)}$, $b_{x(y)}$, and $\phi^{x(y)}$ are modulated
strengths, frequencies and phases, respectively. In numerical calculations of
vector Chern number, we choose $t_{x(y)}=0.5$, $\lambda_{x(y)}=0.95$,
$b_{x(y)}=(\sqrt{5}+1)/2$, and the total sites are $15\times 15$ for two-
dimensional lattice. There are three subsets of bands in each direction, and
the vector Chern number takes values of $(1,-2,1;1,-2,1)$, indicating the
existence of topological corner states.
In each direction, there are three types of one-dimensional topological edge
states, one localized at both edges, one localized at the left edge and the
last one localized at the right edge; see Fig. 1(a). These edge states
constitute basic building blocks to construct topological corner states in
higher dimension. As shown in Fig. 1(b), we can construct corner states by
using the edge states in both $x$ and $y$ directions (see Supplemental
Materials for more details SM ). Since the couplings along $x$ and $y$
directions are independent, the robustness of corner states inherits that of
one-dimensional edge states in each dimension. Taking edge states in $x$
direction for example, there are energy gaps between edge states and extended
states; see Fig. 2(a). These topological edge states are robust to
considerable disorder, perturbation and long-range couplings that mix the two
dimensions provided energy gaps keep open. Hence, the constructing corner
states also share similar topological protection to the one dimensional edge
states, where there are finite energy gaps between corner states and extended
states; see Fig. 2(b). Apart from corner states, products of edge state in one
direction and extended state in other direction form a continuum subset which
is also separated from extended states. Such approach can be naturally
generalized to construct corner in three dimensional periodically-modulated
lattice, see Fig. 1 (c), along with hinge and surface states (see Supplemental
Materials for more details SM ).
What’s more, when $b_{x}=b_{y}=1/2$, the above model reduces to a two-
dimensional Su-Schrieffer-Heeger (SSH) lattice, where coupling matrices are
changed in a stagger way in both $x$ and $y$ directions TPRL118 . Indeed,
topological corner states are predicted with an alternative theory of two-
dimensional polarization T20201 ; T20181 and observed in experiment of
photonic crystal slabs Es1 ; Ew2 ; Ew3 . The varying of two-dimensional
polarization TPRL118 could give rise to topologial charge pumping in two
dimensions which is characterized by vector Chern number. However, on one
hand, these corner states can be more easily and naturally understood in our
theoretical framework, that is, they are the product of two edge states in
both $x$ and $y$ directions. On the other hand, in contrast to two-dimensional
polarization which relies on spatial symmetries, our theory of vector Chern
number can also predict corner states without requiring any fine-tuning of
symmetries.
## III Experimental implement
In experiment, we first realize the Hamiltonian (II) in a two-dimensional
array of waveguides with modulated spacing. The site number is 15 or 16 in
both $x$ and $y$ directions, the average coupling strength $t_{x}=t_{y}=t$ is
adopted as 0.3 for photon with wavelength of 810 nm, the modulating amplitude
$\lambda_{x}=\lambda_{y}=\lambda$ is set as 0.5, the periodic parameter
$b_{x}=b_{y}=b$ is $(\sqrt{5}+1)/2$, and the initial phases $\phi^{x}$ and
$\phi^{y}$ are set as the same value. We fabricate the photonic waveguide
lattices according the Hamiltonian using the femtosecond laser direct-writing
technique fabri_1 ; fabri_2 ; fabri_3 ; fabri_4 ; PIT_Gap . As shown in Fig.
3(a), the propagation direction of the waveguide lattice maps the evolution
time, hence the fabricated tree dimensional waveguide lattice realizes the
designed two-dimensional off-diagonal AAH lattice. We further set nine sites
for injecting the photons, including four corner sites labeled as from C1 to
C4, four boundary sites labeled as from B1 to B4, and one site in the lattice
center labeled as T.
According to the prediction in theory, the corner state will appear when we
extend the one-dimensional topological lattice with edge state to a two-
dimensional lattice, and the corresponding topological origin is also extended
to higher dimensions. As shown in Fig. 3(b), there are edge states in both two
ends of lattice for 16-sited one-dimensional off-diagonal AAH lattice with
initial phase $\phi$ = 0.14$\pi$. We fabricate the two-dimensional off-
diagonal AAH lattice with initial phase $\phi^{x}=\phi^{y}$ = 0.14$\pi$ to
demonstrate the predicted corner states. We inject the photons with wavelength
of 810 nm in to the lattice from four corners respectively, the photons will
be confined in the excited lattice corners if there are higher-order corner
states in theoretical prediction. As shown in Fig. 3(c), the photon output
distributions after 40 mm evolution distance are localized in the white dotted
squares, which give the ranges of corner states.
In Fig. 3(d), we give the quantified results for measured corner states, which
is calculated by
$\xi_{p,q}=\sum_{i,j}\hat{a}_{i,j}^{\dagger}\hat{a}_{i,j}\quad(|i-p|\leq
l,|j-q|\leq l),$ (5)
where $(p,q)$ presents the excited site indices, and $l$ describes the range
of corner states adopted as 3. Compared with the 16$\times$16-sited lattice
with $\phi=0.14\pi$, there is no corner state for the case of $\phi=0.75\pi$,
and the photons flow out of the corner state range. This is because there is
no edge state in the one-dimensional AAH lattice for the case of
$\phi=0.75\pi$. Furthermore, we fabricate three two-dimensional
15$\times$15-sited lattices with phase $\phi=0.14\pi$, 0.75$\pi$, and
1.25$\pi$ respectively. There is only left (right) edge state for one-
dimensional lattice with $\phi=0.14\pi$ (1.25$\pi$), therefore the corner
state can only be observed by exciting the lattice from input C2 (C4). Similar
to 16$\times$16-sited lattice, there is no corner state for the case of
$\phi=$ 0.75$\pi$. We excite the lattices from input C2 and C4 respectively
and measure the photon output distributions, the quantified results, together
with the results of 16$\times$16-sited lattices, confirm theoretical
predictions on topological corner states arising by extending topological
lattice with edge state from one dimension to two dimension.
The corner states appearing in two-dimensional lattice require combination of
the one-dimensional lattices owning edge states in both the $x$ and $y$
directions. Differently, The edge states in higher dimensional lattices, as a
product of edge state in one direction and extended state in other direction,
can be naturally extended from the edge states in lower dimensional lattices.
As shown in Fig. 4(a), we inject the photons into the 16$\times$16-sited
lattice with $\phi$ = 0.14$\pi$ from the input B1, B2, B3 and B4 respectively,
the photons are confined in the boundaries. For the case of $\phi$ =
0.75$\pi$, there is no edge state that can be extended, so we can find that
the photons flow out the boundary ranges, as shown in Fig. 4(b) taking the
cases of B2 and B3 for example. The quantified results in Fig. 4(c) show the
observed edge states extended from one-dimensional lattices. As intuitive
understanding, edge states are extended from dots to lines when the
topological lattices are extended from one dimension to two dimensions.
In conclusion, we present a theoretical explanation on topological origin of
higher-order topological phase in higher dimensional lattices, which is
connected to the topological phase in lower dimensional lattices. We
experimentally observe the theoretically predicted higher-order topological
corner states in two-dimensional off-diagonal AAH lattices. Our model
intuitively explains the connection of topological phases in different
dimensional lattices, which is universal to various models and is promising to
be a convenient and practical tool for constructing higher-order topological
phases in higher-order lattices.
###### Acknowledgements.
The authors thank Yidong Chong and Jian-Wei Pan for helpful discussions.
X.-M.J. is supported by National Key R&D Program of China (2019YFA0308700,
2017YFA0303700), National Natural Science Foundation of China (11761141014,
61734005, 11690033), Science and Technology Commission of Shanghai
Municipality (17JC1400403), Shanghai Municipal Education Commission
(2017-01-07-00-02-E00049). X.-M.J. acknowledges additional support from a
Shanghai talent program. C. Lee is supported by the Key-Area Research and
Development Program of GuangDong Province (2019B030330001), the National
Natural Science Foundation of China (11874434, 11574405), and the Science and
Technology Program of Guangzhou (201904020024). Y. K. is partially supported
by the Office of China Postdoctoral Council (20180052), the National Natural
Science Foundation of China (11904419), and the Australian Research Council
(DP200101168).
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## IV Supplementary Materials:
Constructing higher-order topological states in higher dimension
### IV.1 Constructing corner states with off-diagonal AAH model
Figure S1: Appearance of edge modes and their topological origin. (a) Energy
bands in ($k,\phi^{s}$) space. The corresponding Chern numbers for three bands
are $(-1,2,-1)$. (b) Energy spectrum as a function of $\phi^{s}$ under open
boundary condition. (c) Asymmetry edge state and (d) Symmetry edge state for
$\phi^{s}=0(2\pi)$. The parameters are chosen as $t_{s}=1$, $\lambda_{s}=0.5$,
$b_{s}=1/3$, and the system size is $30$.
The Hamiltonian (4) in the main text can be rewritten as $H=H_{x}\otimes
I_{y}+I_{x}\otimes H_{y}$ with
$H_{s}=\sum_{j}\\{t_{s}[1+\lambda_{s}\cos(2\pi
b_{s}j+\phi_{s})]\hat{a}_{j}\hat{a}_{j+1}^{\dagger}+H.c.\\}.$ (S1)
Before proceeding to corner states, we first study the topological properties
of the Hamiltonian (S1) with rational modulated frequency $b_{s}=\mu/\nu$,
where $\mu$ and $\nu$ are coprime numbers. In this case, we can calculate the
vector Chern number in an alternative way. When the twisted angle is $0$
(i.e., in periodic boundary condition), the system is invariant when
translating a particle from the $j$th to $(j+\nu)$th sites, that is, the
system has translational symmetry. According to Bloch theorem, the eigenstates
are Bloch functions with quasi-momentum $k$, and the eigenvalues form $\nu$
Bloch bands. Here, the quasi-momentum is a good quantum number and provides an
intrinsic parameter to define topological invariant. Instead of integral over
twisted angle, each component of vector Chern number can be defined in the
space formed by quasi-momentum and modulated phase,
${C_{n}^{s}}=\frac{1}{{2\pi
i}}\int_{BZ}dk{d\phi^{s}\left({\langle{\partial_{\phi^{s}}}\psi_{n}^{s}|{\partial_{k}}\psi_{n}^{s}\rangle}\right)-\langle{\partial_{k}}\psi_{n}^{s}|{\partial_{\phi^{s}}}\psi_{n}^{s}\rangle}.$
(S2)
Such definition is consistent with Eq. (3) in the main text. According to
bulk-boundary correspondence, when the sum of all the Chern numbers below a
gap is non-zero, edge modes should appear in this gap. To show this, in the
periodic boundary condition we calculate the energy bands in the
($k,\phi^{s}$) parameter space and their Chern numbers, and in the open
boundary condition we calculate energy spectrum as a function of modulated
phase $\phi^{s}$; see Fig. S1. The parameters are chosen as $t_{s}=1$,
$\lambda_{s}=0.5$, $b_{s}=1/3$, and the system size is $30$. There are three
bands in each direction and the corresponding vector Chern number is
$(-1,2,-1;-1,2,-1)$. Indeed, in each direction, there are isolated edge modes
in each energy gap. When $\phi^{s}=0(2\pi)$, the two edge states in the first
gap become degenerate and their wave-functions are asymmetric [Fig. S1(c)] and
symmetric [Fig. S1(d)]. Here, we have clearly shown that the appearance of the
edge states results from dimension reduction of topological phases in the two-
dimensional parameter space $(k,\phi^{s})$.
Figure S2: Constructing corner states with edge states in $x$ and $y$
directions. (a) and (b): Energy spectral for off-diagonal AAH models in $y$
and $x$ directions, respectively. (c) corner states obtained by combining
symmetric and asymmetric edge states in $x$ and $y$ directions. (d) The corner
states obtained by diagonalizing Hamiltonian (4) in the main text with the
same energies as those in the top panel. The parameters are chosen as
$t_{x}=t_{y}=1$, $\lambda_{x}=0.4$, $\lambda_{y}=0.5$, $b_{x}=b_{y}=1/3$,
$\phi^{x}=\phi^{y}=0$ and the numbers of lattices in both $x$ and $y$
directions are $30$.
With the edge states at hand, we can construct the corner states with the
method depicted in the main text. For a set of degenerate eigenstates, the
superposition of these eigenstates is also the eigenstates. To avoid
unnecessary degeneracy, we consider the parameters for off-diagonal AAH model
in the $x$ direction are slightly different from those in the $y$ dimension.
The parameters are chosen as $t_{x}=t_{y}=1$, $\lambda_{x}=0.4$,
$\lambda_{y}=0.5$, $b_{x}=b_{y}=1/3$, $\phi^{x}=\phi^{y}=0$ and the numbers of
lattices in both $y$ and $x$ directions are $30$. The eigenvalues of the off-
diagonal AAH models in $y$ and $x$ directions are shown in Fig. S2(a) and (b),
respectively. Since the parameters are slightly different, the energy spectral
in $x$ and $y$ directions are quite similar. The symmetric and asymmetric edge
modes are marked by ‘S’ and ‘A’ in the spectral. With the edge modes in the
first gap, we can construct four corner states (namely, $SS$, $SA$, $AS$ and
$AA$) by any combination of the symmetric and asymmetric edge states in two
dimensions; see Fig. S2(c) respectively. For comparison, we show the corner
states obtained by directly diagonalizing Hamiltonian (4) in the main text;
see Fig. S2(d). The eigenvalues and spatial distributions of the corner states
in the bottom panel are exactly the same as those in the top panel. It means
that the constructed corner states are indeed the eigenstates of the two-
dimensional off-diagonal AAH model.
For irrational modulated frequency, translational symmetry is broken and
quasi-momentum is no-longer a good quantum number to define topological
invariant. However, vector Chern number for the irrational off-diagonal AAH
model can still be defined by introducing twisted angles, as shown in the main
text. The vector Chern number well characterizes the topology of such
irrational off-diagonal AAH model. Topological corner states also appear in
the energy gap as a consequence of such nontrival vector Chern number. We show
experimental observation of corner states based on the irrational off-diagonal
AAH model in the main text.
### IV.2 Generally constructing corner states in two-dimensional lattice
Figure S3: Constructing corner states in two-dimensional lattice. The second-
order states can be assembled using the edge states in one-dimensional
lattices.
As we have shown in main text and last section by taking the examples, one is
able to freely construct the higher-order topological phase in two-dimensional
lattice by assembling edge states in one-dimensional lattices. In this way,
the higher-order states can be assembled using the edge states in lower-
dimensional lattices, similar to playing LEGO game. In Fig. S3, we show the
combination way to constructing corner states in two-dimensional lattice using
edge states in $x$ and $y$ directional one-dimensional lattices. For example,
we can choose the modulation of the lattice owning both left and right edge
states in $x$ direction and just owning left edge state in $y$ direction to
construct the corner states in top left and top right corners.
### IV.3 Constructing corner states in three dimensional lattice
Here we extend our approach to a three-dimensional cubic lattice model. The
position of a particle is denoted by $(i,j,k)$ which are site indices of $x$,
$y$ and $z$ directions respectively. The coupling between $(i,j,k)$th and
$(l,m,n)$th lattice sites has the form,
$H_{x}(i,l)\delta_{j,m}\delta_{k,n}+\delta_{i,l}H_{y}(j,m)\delta_{k,n}+\delta_{i,l}\delta_{j,m}H_{z}(k,n)$,
where $H_{x}(i,k)$ is the coupling matrix along $x$ direction irrelevant to
positions in $y$ and $z$ directions, and vice versa. The motion of a particle
hopping in such lattice is governed by the Hamiltonian,
$H=H_{x}\otimes I_{y}\otimes I_{z}+I_{x}\otimes H_{y}\otimes
I_{z}+I_{x}\otimes I_{y}\otimes H_{z},$ (S3)
where $I_{\\{x,y,z\\}}$ are identical matrices in $\\{x,y,z\\}$ directions.
Solving the eigenproblems,
$\displaystyle H_{x}|\psi_{p}^{x}\rangle$ $\displaystyle=$ $\displaystyle
E_{p}^{x}|\psi_{p}^{x}\rangle,$ $\displaystyle H_{y}|\psi_{q}^{y}\rangle$
$\displaystyle=$ $\displaystyle E_{q}^{y}|\psi_{q}^{y}\rangle,$ $\displaystyle
H_{z}|\psi_{l}^{z}\rangle$ $\displaystyle=$ $\displaystyle
E_{l}^{z}|\psi_{l}^{z}\rangle,$ (S4)
we can obtain the eigenvalues $\\{E_{p}^{x},E_{q}^{y},E_{l}^{z}\\}$ and
eigenstates
$\\{|\psi_{p}^{x}\rangle,|\psi_{q}^{y}\rangle,|\psi_{l}^{z}\rangle\\}$
corresponding to $\\{H_{x},H_{y},H_{z}\\}$. We can immediately prove that
$\displaystyle
H|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle$
(S5) $\displaystyle=$
$\displaystyle(E_{p}^{x}+E_{q}^{y}+E_{l}^{z})|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle,$
that is, $E_{p}^{x}+E_{q}^{y}+E_{l}^{z}$ and
$|\psi_{p}^{x}\rangle\otimes|\psi_{q}^{y}\rangle\otimes|\psi_{l}^{z}\rangle$
are the eigenvalues and eigenstates corresponding to $H$, respectively. If
$\\{|\psi_{p}^{x}\rangle,|\psi_{q}^{y}\rangle,|\psi_{l}^{z}\rangle\\}$ are the
edge states, the product of the three edge states becomes a corner state in
three dimensions.
Similarly, we can also construct hinge state by selecting edge states in any
two dimensions and extended state in the other dimension, and the surface
states by constructing extended states in any two dimensions and edge state in
the other dimension.
Figure S4: Constructing (a) corner state, (b) Hinge state and (c) Surface
state in a three-dimensional lattice. The parameters are chosen as
$t_{x}=t_{y}=t_{z}=1$, $\lambda_{x}=\lambda_{y}=\lambda_{z}=0.5$,
$b_{x}=b_{y}=b_{z}=1/3$, $\phi^{x}=\phi^{y}=\phi_{z}=0$ and the numbers of
lattices in $x$, $y$ and $z$ directions are $30$. Figure S5: Constructing
higher-order topological states in three-dimensional lattice. Higher-order
topological states are observed in quasi-crystal lattice (a) and SSH lattice
(b) in simulation. The line width and color depth of markers show the output
probabilities. The red arrows point out the exciting position in simulation.
In Fig. S4, we show the construction of corner, hinge and surface states in a
three-dimensional lattice with $H_{s}$ given by Eq. (S1). The parameters are
chosen as $t_{x}=t_{y}=t_{z}=1$, $\lambda_{x}=\lambda_{y}=\lambda_{z}=0.5$,
$b_{x}=b_{y}=b_{z}=1/3$, $\phi^{x}=\phi^{y}=\phi_{z}=0$ and the numbers of
lattices in $x$, $y$ and $z$ directions are $30$. The corner state is the
product of the first symmetric edge states [as shown in Fig. S2(d)] in $x$,
$y$ and $z$ directions. The Hinge state is the product of the extended state
with the highest energy of the first band in $z$ direction and the first
symmetric edge states in $x$ and $y$ directions. The Surface state is the
product of the extended states with the highest energy of the first band in
$x$ and $z$ directions and the first symmetric edge state in $y$ direction.
To simulate the observation of higher-order topological states in three-
dimensional lattices, in the Fig. S5, we show the evolution results of photons
in assembled higher-order topological three-dimensional lattices. In the first
example, the parameters are chosen as $t_{x(y,z)}=0.5$,
$\lambda_{x(y,z)}=0.5$, $\phi_{x(y,z)}=0.15$, $b_{x(y,z)}=(\sqrt{5}+1)/2$, the
site numbers of lattice in both $x$ and $y$ directions are $16$, and the site
number of lattice in $z$ direction is $15$. This is a three-dimensional quasi-
crystal lattice based on AAH model, and the simulated results is shown in Fig.
S5(a). In the second example, the parameters are chosen as $t_{x(y,z)}=0.5$,
$\lambda_{x(y,z)}=0.5$, $\phi_{x(y,z)}=0$, $b_{x(y,z)}=1/2$, the site numbers
of lattice in both $x$ and $y$ directions are $16$, and the site number of
lattice in $z$ direction is $15$. This is a three-dimensional SSH lattice, and
the simulated results is shown in Fig. S5(b).
|
# Online Fair Allocation of Perishable Resources††A preliminary version of
this paper appeared as an extended abstract in ACM SIGMETRICS 2023.
Siddhartha Banerjee School of Operations Research and Information
Engineering, Cornell University Chamsi Hssaine University of Southern
California, Marshall School of Business Sean R. Sinclair Laboratory for
Information and Decision Sciences, Massachusetts Institute of Technology
###### Abstract
We consider a practically motivated variant of the canonical online fair
allocation problem: a decision-maker has a budget of perishable resources to
allocate over a fixed number of rounds. Each round sees a random number of
arrivals, and the decision-maker must commit to an allocation for these
individuals before moving on to the next round. The goal is to construct a
sequence of allocations that is envy-free and efficient. Our work makes two
important contributions toward this problem: we first derive strong lower
bounds on the optimal envy-efficiency trade-off that demonstrate that a
decision-maker is fundamentally limited in what she can hope to achieve
relative to the no-perishing setting; we then design an algorithm achieving
these lower bounds which takes as input $(i)$ a prediction of the perishing
order, and $(ii)$ a desired bound on envy. Given the remaining budget in each
period, the algorithm uses forecasts of future demand and perishing to
adaptively choose one of two carefully constructed guardrail quantities. We
demonstrate our algorithm’s strong numerical performance — and state-of-the-
art, perishing-agnostic algorithms’ inefficacy — on simulations calibrated to
a real-world dataset.
###### Contents
1. 1 Introduction
1. 1.1 Our contributions
2. 1.2 Related work
2. 2 Preliminaries
1. 2.1 Basic setup
2. 2.2 Notions of fairness and efficiency
3. 3 Limits of perishability
1. 3.1 Restricting the aggressiveness of the perishing process
2. 3.2 Unavoidable loss due to errors in allocation schedule
4. 4 The Algorithm
1. 4.1 Performance guarantee
2. 4.2 On $\delta$-offset expiry
3. 4.3 Analysis
5. 5 Numerical experiments
1. 5.1 Geometric perishing
2. 5.2 Non-i.i.d. perishing
6. 6 Conclusion
7. A Table of notation
8. B Omitted proofs
1. B.1 Section 3 omitted proofs
2. B.2 Tightness of bounds
3. B.3 Section 4.1 omitted proofs
4. B.4 Section 4.2 omitted proofs
5. B.5 Section 4.3 omitted proofs
6. B.6 Section 5 omitted proofs
7. B.7 Proof of Proposition 5.2
9. C Useful lemmas
10. D Simulation details
## 1 Introduction
Despite a consistent decline in food insecurity in the United States over the
past decade, 2022 saw a marked increase in individuals struggling to access
enough food to fulfill basic needs. A recent report by the U.S. Department of
Agriculture found that over 44 million individuals faced some form of hunger
in 2022 — 45% more than the previous year (Godoy, , 2023). Due in part to
rising food prices and the rolling back of pandemic-era social security
measures, this disturbing statistic has further underscored the important role
of local food banks; for instance, the Feeding America network of food banks,
food agencies, and local food programs distributed over 5.2 billion meals that
same year (Feeding America, , 2022).
In distributing food throughout their operating horizon, food banks have two
competing objectives: distributing as much food as possible to communities in
need, and ensuring equitable access to donations. This tension has attracted
much attention in the operations literature, with recent work characterizing
the fundamental trade-offs between fairness and overall utility in sequential
allocation problems (Bertsimas et al., , 2011; Donahue and Kleinberg, , 2020;
Lien et al., , 2014; Manshadi et al., , 2023; Sinclair et al., , 2022).
Understanding such trade-offs in theory is useful, as they allow a system
designer to recognize and choose their desired operating point, balancing the
loss in efficiency and equity. Despite the useful insights derived from prior
work, to the best of our knowledge an important reality of food bank
operations remains overlooked: the existence of perishable goods, which
constitute a substantial portion of food donations. The Los Angeles Regional
Food Bank, for instance, distributed over 26 million pounds of produce in 2022
alone (Los Angeles Regional Food Bank, 2022a, ). Perishable goods pose
significant challenges for these organizations, who frequently find themselves
needing to throw out spoiled goods (Los Angeles Regional Food Bank, 2022b, ).
Indeed, the equity-efficiency trade-off is exacerbated in the presence of
perishables: while equity requires a decision-maker to allocate conservatively
across arriving streams of demand (Sinclair et al., , 2022), perishability
starts a “race against time.” As goods perish due to a slow allocation rate,
not only is efficiency further harmed, but so may be equity, as spoilage runs
the risk of a decision-maker running out of items, with nothing left to give
out by the end of the operating horizon.
Thus motivated, this paper seeks to answer the following questions:
Do established equity-efficiency trade-offs in dynamic environments persist in
the presence of perishable goods? If not, what limits do they impose on fair
and efficient allocations? Can we design policies that perform well under
these limits?
Before detailing our contributions, we highlight that, though this work is
motivated by food bank distribution systems, the interplay between fairness
and perishability is an important consideration in several other settings,
e.g., vaccine distribution (Manshadi et al., , 2023), electric vehicle
charging (Gerding et al., , 2019), and federated cloud computing (Aristotle
Cloud Federation Project, , 2022; Ghodsi et al., , 2011).
### 1.1 Our contributions
We consider a model in which a decision-maker has a fixed budget of items
(also referred to as goods, or resources) to be allocated over $T$ discrete
rounds. An a priori unknown number of individuals arrives in each period, each
seeking a share of goods. Each agent is characterized by an observable type
(drawn from a known, potentially time-varying distribution), associated with a
linear utility function over the received allocation. Moreover, each unit of
good has a stochastic perishing time (similarly drawn from a known
distribution, independent of incoming arrivals), after which the good spoils
and can no longer be allocated. The decision-maker allocates goods according
to a fixed ordering (also referred to as perishing prediction, or allocation
schedule), e.g., in increasing order of expected perishing time. The goal is
to find a policy that trades off between three ex-post metrics:
1. 1.
_Hindsight Envy_ (Envy) – Maximum difference in utility obtained by any two
agents.
2. 2.
_Counterfactual Envy_ $(\Delta_{\text{\it EF}})$ – Maximum difference between
the utility obtained by any agent, and their utility under the static,
proportional allocation (optimal in the no-perishing setting).
3. 3.
_Inefficiency_ $(\Delta_{\text{\it efficiency}})$ – Amount of unallocated
(including spoiled) goods at the end of the horizon.
For this setting, we first characterize the fundamental limits of
perishability placed on any online algorithm. We argue that — contrary to the
setting without perishable resources — envy ceases to be a meaningful metric
for a large class of perishing processes. To see this, consider an extreme
scenario in which all items perish at the end of the first round. Clearly,
there is no hope of achieving low envy in such a setting since future demand
can never be satisfied. Our first main contribution identifies a necessary and
sufficient condition on the joint perishing and arrival distribution for low
counterfactual envy to be an achievable desideratum (Theorem 3.2). From a
managerial perspective, this characterization — which at a high level states
that the cumulative perishing must lag behind cumulative arrivals — can be
viewed as guidance on the composition of perishables in the initial budget. It
moreover underscores the importance of leveraging joint information over
perishing and demand: if demand is back-loaded, perishing times should be
concentrated late in the horizon; if most demand arrives early, however, early
perishing times are acceptable.
$\Delta_{\text{\it efficiency}}$$\Delta_{\text{\it EF}}$Envy-Efficiency
Tradeoff$T^{-1/2}$Achievable with Perishing-GuardrailImpossible due to Demand
UncertaintyImpossible due to Perishing
Uncertainty$T\mathcal{L}^{\textsf{perish}}$
(a) Envy-Efficiency Pareto Frontier
$L_{T}$$\mathcal{L}^{\textsf{perish}}$$T^{-1/2}$$T^{-1/2}$High-envy,low-
perishingLow-envyHigh-envy,high-perishing
(b) Dependence of Algorithm 1 performance on envy-perishing regime
Figure 1: Graphical representation of Theorems 4.2 and 3.7. Fig. 1(a)
illustrates the envy-efficiency trade-off ($\Delta_{\text{\it efficiency}}$
vs. $\Delta_{\text{\it EF}}$) achieved by Perishing-Guardrail (Algorithm 1).
The dotted lines represent the impossibility results due to either demand or
perishing uncertainty. The region below the solid line represents the
impossibility due to the envy-efficiency trade-off; the green region is the
achievable region for Perishing-Guardrail. Fig. 1(b) illustrates the phase
transition between the performance of Perishing-Guardrail depending on the
spoilage loss $\mathcal{L}^{\textsf{perish}}$ ($x$-axis) and envy parameter
$L_{T}$ ($y$-axis).
For this class of processes, which we term offset-expiring, zero spoilage
occurs if the perishing prediction is perfect. We show, however, that
inaccuracies in the allocation schedule pose an insurmountable barrier to any
algorithm’s performance by characterizing an unavoidable loss in equity and
efficiency due to these errors (Theorem 3.7). In contrast to the no-perishing
setting, in which the only source of loss is exogenous uncertainty in demand,
in our setting the loss due to spoilage is endogenous: it crucially depends on
the rate at which the algorithm allocates items. This endogeneity poses
significant challenges in the analysis of any adaptive algorithm; designing a
tractable approach to analyzing ex-post spoilage is the main technical
contribution of our work. Additionally, the lower bounds we derive give rise
to the key insight that, contrary to the “race against time” intuition under
which a decision-maker must increase the allocation rate to prevent avoidable
spoilage, achieving low hindsight envy in the presence of unavoidable spoilage
requires a decision-maker to potentially allocate significantly less than the
proportional allocation. Hence, perishability throws a wrench into the well-
studied envy-efficiency trade-off: while hindsight and counterfactual envy are
aligned in the no-perishing setting, these two may be at odds when goods
spoil, since only high counterfactual-envy solutions may yield low hindsight-
envy. The tension between efficiency and equity is exacerbated for the same
reason, relative to the classical setting.
In our final technical contribution, we leverage these insights to construct
an adaptive threshold algorithm (Algorithm 1) that achieves these lower bounds
(Theorem 4.2). Our algorithm takes as input $(i)$ the fixed allocation
schedule, $(ii)$ a desired upper bound on hindsight envy $L_{T}$, and $(iii)$
a high-probability parameter $\delta$. Given these inputs, it computes a high-
probability lower bound on a budget-respecting zero-hindsight-envy solution,
and an “aggressive” efficiency-improving allocation that is $L_{T}$ away. In
each round the algorithm chooses which of the two quantities to allocate to
each individual, cautiously doing so by constructing pessimistic forecasts of
future arrivals and perishing. While this algorithm is similar in flavor to
state-of-the-art algorithms for the no-perishing setting (Sinclair et al., ,
2022), the main challenge it contends with is forecasting (endogenous) future
spoilage. Here, we leverage the bounding technique used to construct our lower
bounds, which relies on the analysis of a knife-edge, “slow” consumption
process that tractably decouples past allocations from future perishing. Our
algorithm’s bounds give rise to three salient regimes (depicted graphically in
Figure 1(b)), as a function of hindsight envy tolerance $L_{T}$ and the
unavoidable loss due to spoilage per period, denoted by
$\mathcal{L}^{\textsf{perish}}$:
1. 1.
Low-envy ($L_{T}\lesssim 1/\sqrt{T}$): there are no efficiency or
counterfactual envy gains from deviating from the equitable solution
$L_{T}=0$.
2. 2.
High-envy, high-perishing ($L_{T}\gtrsim
1/\sqrt{T},\mathcal{L}^{\textsf{perish}}\gtrsim 1/\sqrt{T}$): inefficiency is
invariant to $L_{T}$; setting $L_{T}\sim\mathcal{L}^{\textsf{perish}}$ is
optimal with respect to counterfactual envy.
3. 3.
High-envy, low-perishing ($L_{T}\gtrsim
1/\sqrt{T},\mathcal{L}^{\textsf{perish}}\lesssim 1/\sqrt{T}$): counterfactual
envy increases as $L_{T}$, and inefficiency decreases as $1/L_{T}$, until
reaching the unavoidable cumulative spoilage loss of
$T\mathcal{L}^{\textsf{perish}}$.
These results further highlight the extent to which a decision-maker is
restricted in leveraging inequity to improve efficiency (and vice versa).
We complement our theoretical bounds in Section 5 by testing the practical
performance of our algorithm on both synthetic and real-world datasets. Our
experiments show that the unfairness required to achieve these efficiency
gains is order-wise larger than in settings without perishable resources.
Additionally, they underscore the weakness of perishing-agnostic online
algorithms. We observe that these latter algorithms are incapable of
leveraging unfairness to improve efficiency across a variety of perishing
regimes. In contrast to these, our algorithm’s construction of a perishing-
aware baseline allocation $\underline{X}$ is necessary to mitigate stockouts
across all — rather than simply worst-case — instances. These include
instances where offset-expiry fails to hold with high probability, as is the
case in the real-world dataset we use to calibrate our experiments (Keskin et
al., , 2022). Perhaps most surprisingly, despite our baseline allocation being
significantly lower than that of algorithms that don’t take into account
perishability, our algorithm is more efficient than these more aggressive
algorithms, in addition to being more fair. This observation contradicts the
“race against time” intuition that aggressive allocations are necessarily more
efficient than cautious ones. Finally, we numerically explore the question of
setting practical allocation schedules that perform well along all metrics.
Our main managerial insight is that ordering items in increasing order of a
high-probability lower bound on their perishing time robustly trades off
between the natural ordering that allocates items in increasing order of
expected perishing time, and the inherent variability in the perishing
process.
#### Paper organization.
We next survey the related literature. We present the model in Section 2, and
formalize the limits of perishability in Section 3. In Section 4 we design and
analyze an algorithm that achieves the derived lower bounds; we conclude with
numerical experiments in Section 5.
### 1.2 Related work
Fairness in resource allocation has a long history in the economics and
computation literature, beginning with Varian’s seminal work (Varian, , 1974,
1976). More recently, there has been ongoing work studying the intersection of
fairness and operations, including assortment planning (Chen et al., , 2022),
pricing (Cohen et al., , 2022; den Boer et al., , 2022), incentive design
(Freund and Hssaine, , 2021), algorithmic hiring (Salem and Gupta, , 2023),
and societal systems more generally (Gupta and Kamble, , 2021). We highlight
the most closely related works below, especially as they relate to online fair
allocation; see Aleksandrov and Walsh, 2019b for a survey.
##### Fair allocation without perishable resources.
There exists a long line of work in which the non-perishable resource becomes
available to the decision-maker online, whereas agents are fixed (Benade et
al., , 2018; Aleksandrov et al., , 2015; Mattei et al., , 2017, 2018;
Aleksandrov and Walsh, 2019a, ; Banerjee et al., , 2020; Bansal et al., ,
2020; Bogomolnaia et al., , 2022; He et al., , 2019; Aziz et al., , 2016; Zeng
and Psomas, , 2020). These models lie in contrast to the one we consider,
wherein resources are fixed and individuals arrive online. Papers that
consider this latter setting include Kalinowski et al., (2013), which
considers maximizing utilitarian welfare with indivisible goods; Gerding et
al., (2019) considers a scheduling setting wherein agents have fixed and
known arrival and departure times, as well as demand for the resource;
Hassanzadeh et al., (2023) allows individuals to arrive in multiple
timesteps. A series of papers also consider the problem of fair division with
minimal disruptions relative to previous allocations, as measured by a
fairness ratio, a competitive ratio analog of counterfactual envy in our
setting (Friedman et al., , 2017; Cole et al., , 2013; Friedman et al., ,
2015). Other works design algorithms with attractive competitive ratios with
respect to Nash Social Welfare (Azar et al., , 2010; Banerjee et al., , 2020),
or the max-min objective (Lien et al., , 2014; Manshadi et al., , 2023).
The above papers situate themselves within the adversarial, or worst-case,
tradition. A separate line of work considers fair resource allocation in
stochastic settings (Donahue and Kleinberg, , 2020; Elzayn et al., , 2019;
Freund and Hssaine, , 2021), as we do. The algorithms developed in these
papers, however, are non-adaptive: they decide on the entire allocation
upfront, before observing any of the realized demand. In contrast, we consider
a model where the decision-maker makes the allocation decision in each round
after observing the number of arrivals. Freeman et al., (2017) considers a
problem in which agents’ utilities are realized from an unknown distribution,
and the budget resets in each round. They present algorithms for Nash social
welfare maximization and discuss some of their properties. Our work is most
closely related to (and indeed, builds upon) Sinclair et al., (2022), which
first introduced the envy-freeness and efficiency tradeoff we are interested
in. Our work considers the more challenging setting of perishable goods, which
none of the aforementioned works consider.
##### Perishable resources.
Though dynamic resource allocation of perishable goods has a long history in
the operations research literature (see, e.g., Nahmias, (2011) for a
comprehensive survey of earlier literature), to the best of our knowledge, the
question of fairly allocating perishable goods has attracted relatively little
attention. We highlight the few relevant papers below. Perry, (1999) and
Hanukov et al., (2020) analyze FIFO-style policies for efficiency
maximization in inventory models with Poisson demand and deterministic or
Poisson perishing times. Motivated by the problem of electric vehicle
charging, Gerding et al., (2019) consider an online scheduling problem where
agents arrive and compete for a perishable resource that spoils at the end of
every period, and as a result must be allocated at every time step. They
consider a range of objectives, including: maximum total resource allocated,
maximum number of satisfied agents, as well as envy-freeness. Bateni et al.,
(2022) similarly consider a setting wherein an arriving stream of goods perish
immediately. Recent empirical work by Sengul Orgut and Lodree, (2023)
considers the problem of a food bank equitably and efficiently allocating
perishable goods under complete information. Their case study on data from a
partnering food bank numerically validates our theoretical results: in low-
budget settings, there is little or no benefit to increasing inequity to
counteract the inefficiency due (in part) to spoilage. In contrast to these
latter papers, the model we consider locates itself within the smaller
category of inventory models in which products have random lifetimes. The
majority of these assume that items have exponentially distributed or
geometric lifetimes (Bakker et al., , 2012).
## 2 Preliminaries
We consider a decision-maker who, over $T$ rounds, must allocate $B$ divisible
units (also referred to as items) of a single type of resource among a
population of individuals. Let $\mathcal{B}$ denote the set of these $B$
units.
### 2.1 Basic setup
##### Demand model.
At the start of each round $t\in[T]$, a random number of individuals arrives,
each requesting a share of units. Each individual is characterized by her type
$\theta\in\Theta$, with $|\Theta|<\infty$. Each type $\theta$ is associated
with a known utility function $u_{\theta}(x)=w_{\theta}\cdot x$ for a given
allocation $x\in\mathbb{R}_{+}$ of the resource, with $w_{\theta}>0$. We let
$N_{t,\theta}$ denote the number of type $\theta$ arrivals in round $t$;
$N_{t,\theta}$ is drawn independently from a known distribution, with
$N_{t,\theta}\geq 1$ almost surely for all $t\in[T],\theta\in\Theta$. This
latter assumption is for ease of exposition; our results continue to hold (up
to constants) as long as $\mathbb{P}(N_{t,\theta}=0)$ does not scale with $T$.
For ease of notation we define $N_{t}=\sum_{\theta\in\Theta}N_{t,\theta}$ and
$N=\sum_{t\in[T],\theta\in\Theta}N_{t,\theta}$. We assume
$\mathbb{E}[N]=\Theta(T)$, and define $\beta_{avg}=B/\mathbb{E}[N]$ to be the
average number of units per individual, with $\beta_{avg}=\Theta(1)$.
##### Perishing model.
Each unit of resource $b\in\mathcal{B}$ is associated with a perishing time
$T_{b}\in\mathbb{N}^{+}$ drawn from a known distribution. Items’ perishing
times are independent of one another and of the arrival process, and perishing
occurs at the end of each round, after items have been allocated to
individuals. For $t\in[T]$, we let
$P_{t}=\sum_{b\in\mathcal{B}}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}=t}\right\\}$
denote the number of units of resource perishing in period $t$.
The decision-maker has access to a predicted ordering according to which items
perish; we will often refer to this ordering as the allocation schedule. We
use $\sigma:\mathcal{B}\rightarrow[B]$ to denote this ordering, i.e.,
$\sigma(b)$ is rank of $b$ in this ordering. For $b\in[B]$, $\sigma^{-1}(b)$
is used to denote the identity of the $b$th ranked item in $\sigma$, with
$\sigma^{-1}(1)$ being the item that comes first in the allocation schedule.
While our results allow $\sigma$ to be arbitrary, in Section 5 we investigate
natural choices of $\sigma$, such as increasing order of
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]$.
###### Remark 2.1.
In this paper we restrict our attention to static (rather than time-varying
and sample path-dependent) allocation schedules, given their practical
relevance to the motivating real-world applications described in Section 1. We
leave the nonstationary extension to future work.
##### Additional notation.
For any time-dependent quantity $Y_{t}$, we define $Y_{\leq
t}=\sum_{t^{\prime}\leq t}Y_{t^{\prime}}$, $Y_{\geq t}=\sum_{t^{\prime}\geq
t}Y_{t^{\prime}}$, along with their strict analogs. We let
$w_{max}=\max_{\theta}w_{\theta}$,
$\sigma_{t,\theta}^{2}=\text{Var}(N_{t,\theta})<\infty$, and assume
$\rho_{t,\theta}=|N_{t,\theta}-\mathbb{E}[N_{t,\theta}]|<\infty$ almost
surely. Finally, let
$\mu_{\max}=\max_{t}\mathbb{E}\mathopen{}\mathclose{{}\left[N_{t}}\right],\sigma^{2}_{\min}=\min_{t,\theta}\sigma^{2}_{t,\theta},\sigma^{2}_{\max}=\max_{t,\theta}\sigma^{2}_{t,\theta}$,
and $\rho_{\max}=\max_{t,\theta}\rho_{t,\theta}$. We use $\lesssim$ and
$\gtrsim$ to denote the fact that inequalities hold up to polynomial factors
of
$\beta_{avg},|\Theta|,w_{max},\mu_{\max},\sigma^{2}_{\min},\sigma^{2}_{\max}$,
$\log T$, and $\log(1/\delta)$. We summarize all notation in Table 5.
### 2.2 Notions of fairness and efficiency
The goal is to design a fair and efficient online algorithm that determines
the amount to allocate to all $N_{t,\theta}$ individuals in each round $t$,
for all $\theta\in\Theta$, given the remaining budget in each round. We assume
this amount is allocated uniformly across all $N_{t,\theta}$ individuals of
type $\theta$. We use $X_{t,\theta}^{alg}\in\mathbb{R}$ to denote the per-
individual amount distributed in period $t$, with
$X^{alg}=(X_{t,\theta}^{alg})_{t\in[T]}$.
Our notions of online fairness and efficiency are motivated by the offline
notion of Varian Fairness (Varian, , 1974), and are the same as those
considered in past works (Sinclair et al., , 2022).
###### Definition 2.2 (Counterfactual Envy, Hindsight Envy, and Efficiency).
Given budget $B$, realized demands $(N_{t,\theta})_{t\in[T],\theta\in\Theta}$,
perishing times $(T_{b})_{b\in[B]}$, and allocation schedule $\sigma$, for any
online allocation defined by $X^{alg}$ we define:
* •
_Counterfactual Envy_ :
$\displaystyle\Delta_{\text{\it
EF}}\triangleq\max_{t\in[T],\theta\in\Theta}\mathopen{}\mathclose{{}\left|w_{\theta}\mathopen{}\mathclose{{}\left(X_{t,\theta}^{alg}-\frac{B}{N}}\right)}\right|.$
(1)
* •
_Hindsight Envy_ :
$\displaystyle\textsc{Envy}\triangleq\max_{t,t^{\prime}\in[T]^{2},\theta,\theta^{\prime}\in\Theta^{2}}w_{\theta}(X_{t^{\prime},\theta^{\prime}}^{alg}-X_{t,\theta}^{alg}).$
(2)
* •
_Inefficiency_ :
$\displaystyle\Delta_{\text{\it efficiency}}\triangleq
B-\sum_{t\in[T],\theta\in\Theta}N_{t,\theta}X_{t,\theta}^{alg}.$ (3)
In the offline setting without perishable goods, Varian, (1974) established
that $X^{opt}_{t,\theta}=B/N$ (referred to as the proportional allocation) is
the optimal fair and efficient per-individual allocation. Hence,
counterfactual envy $\Delta_{EF}$ can be interpreted as a form of regret with
respect to this strong no-perishing benchmark, and can be used to characterize
the impact of perishability on our algorithm’s performance. Hindsight envy, on
the other hand, measures how differently the online algorithm treats any two
individuals across time. Finally, the efficiency of the online algorithm,
$\Delta_{\text{\it efficiency}}$, measures how wasteful the algorithm was in
hindsight. This could happen in two ways: either through spoilage, or because
the decision-maker allocated too conservatively throughout the horizon, thus
leaving a large number of unspoiled goods unallocated by $T$.
Even in simple settings without perishability, it is known that these metrics
are at odds with each other in online settings. To see this, consider the
following two scenarios. On the one hand, an algorithm can trivially achieve a
hindsight envy of zero by allocating nothing to individuals in any round;
this, however, would result in both high counterfactual envy, in addition to
maximal inefficiency. On the other hand, a distance to efficiency of zero can
trivially be satisfied by exhausting the budget in the first round, at a cost
of maximal hindsight envy as individuals arriving at later rounds receive
nothing. Sinclair et al., (2022) formalized this tension for the additive
utility setting without perishable resources via the following lower bounds.
###### Theorem 2.3 (Theorems 1 and 2, Sinclair et al., (2022)).
Under any arrival distribution satisfying mild regularity conditions, there
exists a problem instance without perishing, such that any algorithm must
incur $\Delta_{\text{\it EF}}\gtrsim\frac{1}{\sqrt{T}}$, where $\gtrsim$ drops
poly-logarithmic factors of $T$, $\log(1/\delta)$, $o(1)$ terms, and absolute
constants. Moreover, any algorithm that achieves $\Delta_{\text{\it
EF}}=L_{T}=o(1)$ or $\textsc{Envy}=L_{T}=o(1)$ must also incur waste
$\Delta_{\text{\it efficiency}}\gtrsim\min\\{\sqrt{T},1/L_{T}\\}.$
Since settings without perishable resources are a special case of our setting
(e.g., a perishing process with $T_{b}>T$ a.s., for all $b\in\mathcal{B}$),
this lower bound holds in our case; the goal then is to design algorithms that
achieve this lower bound with high-probability. However, as we will see in the
following section, perishing stochasticity is fundamentally distinct from, and
more challenging than, demand stochasticity. This difference is particularly
salient in regards to the envy-efficiency trade-off.
## 3 Limits of perishability
In the presence of perishable resources, a decision-maker must contend with
two obstacles: $(i)$ the “aggressiveness” of the perishing process, and $(ii)$
errors in the perishing prediction $\sigma$. In this section we formalize the
impact of these two challenges. Namely, we identify classes of perishing
processes for which there is no hope of achieving the optimal fair allocation,
and derive lower bounds on any algorithm’s performance, as a function of the
quality of the prediction $\sigma$.
In the remainder of the section, we say that an online algorithm is feasible
over a sample path if it does not run out of budget. Note that, if an
algorithm is infeasible over a sample path, it necessarily achieves
$\Delta_{\text{\it EF}}=\Theta(1)$.
### 3.1 Restricting the aggressiveness of the perishing process
We first argue that the proportional allocation $X^{opt}_{t,\theta}=B/N$ is
unachievable unless one places restrictions on the rate at which items perish,
even under full information over perishing times and demand.
To see this, consider an instance where all items perish at the end of the
first round. There is no hope of achieving low envy in this setting, since
there are no items left for arrivals from $t=2$ onwards. The following result
establishes that the only perishing time realizations for which $B/N$ is a
meaningful benchmark are ones in which the fraction of perished items “lags
behind” the proportion of arrivals in each period. We formalize this via the
notion of offset-expiry, defined below.
###### Definition 3.1 (Offset-expiring process).
A perishing process $(T_{b})_{t\in[T]}$ is offset-expiring if:
$\frac{P_{<t}}{B}\leq\frac{N_{<t}}{N}\quad\forall\ t\geq 2.$
Theorem 3.2 establishes that offset-expiry exactly captures the trajectories
whereby $B/N$ is a feasible allocation when units are allocated in increasing
order of $T_{b}$. We refer to this latter ordering as the hindsight optimal
ordering. We defer its proof to Section B.1.
###### Theorem 3.2.
$X_{t,\theta}=B/N$ for all $t,\theta$ is feasible under the hindsight optimal
ordering if and only if the perishing process if offset-expiring.
Thus motivated, for our main algorithmic result we restrict our attention to
processes that satisfy the offset-expiry condition with high probability (via
a relaxed notion of $\delta$-offset-expiry, see Definition 4.1). We moreover
provide verifiable necessary and sufficient conditions for this condition to
hold in Section 4.2. From a managerial perspective, the (high-probability)
offset-expiry condition provides decision-makers with guidance on the
selection of perishable goods to stock at the beginning of the horizon. Within
the context of food banks, for instance, it highlights that rejecting
perishable goods outright is too severe a policy, and that the reasonable rule
of thumb that says “don’t have more spoilage than what you want to give out”
is the only correct rule of thumb. It moreover underscores the importance of
jointly considering demand and perishing processes in this selection.
Finally, we note that though the remainder of our work is focused on high-
probability offset-expiring processes, an interesting future research
direction is the question of fairly allocating non-offset-expiring goods under
more meaningful notions of envy and inefficiency that don’t penalize for
aggressive perishing.
### 3.2 Unavoidable loss due to errors in allocation schedule
The previous section established that, even under full information about the
perishing process, there exist restrictions on the aggressiveness of the
process for the optimal fair allocation to be achievable. We next show that,
even under offset-expiry, the quality of the allocation schedule $\sigma$ is
crucial in determining what any online algorithm can achieve. To see this,
consider an instance where $B=T$, $|\Theta|=1$ (with $w_{\theta}=1)$, and
$N_{t}=1$ for all $t$. Suppose moreover that exactly one unit perishes in each
period (i.e., the perishing process is offset-expiring), but $\sigma$ reverses
the true perishing order. In this case, allocating $B/N=1$ to each arrival
under $\sigma$ is infeasible, since after $T/2$ rounds the algorithm will have
run out of items.
Motivated by this example, our key insight is that, for any static allocation
$X$, there exists a worst-case loss due to errors in $\sigma$, denoted by
$\overline{\Delta}(X)$, that the algorithm incurs. As a result, rather than
having a budget of $B$ items, the algorithm has an effective budget of
$B-\overline{\Delta}(X)$ items. Under this effective budget, any feasible
stationary allocation must set $X$ such that
$\overline{N}X\leq{B-\overline{\Delta}(X)}$, where $\overline{N}$ is a high-
probability upper bound on $N$. Noting that $X=0$ is always a feasible
solution to this inequality, a natural choice is to set:
$\displaystyle\underline{X}=\sup\mathopen{}\mathclose{{}\left\\{X\ \mid\
X\leq\frac{B-\overline{\Delta}(X)}{\overline{N}}}\right\\},$ (4)
if this supremum is achieved. When $\overline{\Delta}(X)=0$ for all $X$ (i.e.,
either items don’t perish or predictions are perfect),
$\underline{X}=B/\overline{N}$, and we recover the conservative allocation of
the no-perishing setting (Sinclair et al., , 2022). The notion of
$\sigma$-induced loss plays a central role in our results.
###### Definition 3.3 ($\sigma$-induced loss).
We define $\mathcal{L}^{\textsf{perish}}=\frac{B}{\overline{N}}-\underline{X}$
to be any algorithm’s $\sigma$-induced loss. We moreover term
$T\mathcal{L}^{\textsf{perish}}$ to be the cumulative $\sigma$-induced loss.
By (4), given the worst-case perishing loss $\overline{\Delta}(X)$ for any
allocation $X$, one can compute $\underline{X}$ via line search. Obtaining
tight bounds on this quantity, however, is the challenging piece. To see this,
note that for any algorithm, the quantity of goods that perished by the end of
the horizon is:
$\displaystyle\sum_{t\in[T]}\sum_{b\in\mathcal{B}_{t}^{alg}}(B_{t}^{alg}(b)-X_{t}^{alg}(b))^{+}\cdot\mathds{1}\\{T_{b}=t\\},$
(5)
where $\mathcal{B}_{t}^{alg}$ denotes the set of remaining items at the
beginning of period $t$, $B_{t}^{alg}(b)$ is the quantity of item $b$
remaining at the beginning of period $t$, and $X_{t}^{alg}(b)$ is the quantity
of item $b$ given out in period $t$. Since $X_{t}^{alg}(b)$ depends on the
perishing realizations of previous rounds, computing this quantity requires
the ability to simulate sufficiently many replications of the static
allocation process under $X$, for all $X\in[0,B/\overline{N}]$, and for each
of these replications to compute the number of unallocated goods that perished
by the end of the horizon under this allocation, an approach which fails to
scale.
To tackle this difficulty, it will be useful for us to consider a “slow”
consumption process, in which $\underline{N}_{\leq t}$ — a high-probability
lower bound on $N_{\leq t}$ — individuals arrive before $t+1$,
$\underline{N}_{\leq t}{X}$ items are allocated up to period $t\in[T]$, and no
items perish. For $b\in\mathcal{B}$, we let ${\tau}_{b}(1\mid X,\sigma)$ be
the period in which $b$ would have been entirely allocated under this slow
consumption process. Formally,
$\displaystyle{\tau}_{b}(1\mid
X,\sigma)=\inf\mathopen{}\mathclose{{}\left\\{t\geq 1:\underline{N}_{\leq
t}X\geq\sigma(b)}\right\\}.$ (6)
${\tau}_{b}(1\mid X,\ \sigma)$ represents an upper bound on the time an
algorithm using static allocation $X$ would allocate $b$, since items ranked
higher than $b$ may have perished, thus decreasing the time at which $b$ is
allocated. We define
$\mu(X)=\sum_{b\in\mathcal{B}}\mathbb{P}\mathopen{}\mathclose{{}\left(T_{b}<\min\mathopen{}\mathclose{{}\left\\{T,{\tau}_{b}(1\mid
X,\sigma)}\right\\}}\right)$ and let
$\displaystyle\overline{\Delta}({X})=\min\mathopen{}\mathclose{{}\left\\{B,\mu(X)+\textsc{Conf}^{P}_{1}(\mu(X))}\right\\},$
(7)
where $\textsc{Conf}^{P}_{1}(\mu(X))$ is an appropriately chosen confidence
bound, to be specified later.
Figure 2: Illustrating the $\underline{X}$ construction (4) for the toy
instance in Example 3.6. The dashed line corresponds to the line $Y=X$, and
the solid line to $(B-\overline{\Delta}(X))/\overline{N}$. Here,
$\underline{X}$ is represented by the red star, the point at which the solid
and dashed lines intersect.
###### Remark 3.4.
We henceforth assume for simplicity that the supremum on the right-hand side
of (4) is attained. This is without loss to our results, since our bounds
depend on the $\sigma$-induced loss $\mathcal{L}^{\textsf{perish}}$. Hence, if
the supremum fails to be attained, one can define
$\underline{X}=X^{*}-\epsilon$, where $X^{*}$ is the point of discontinuity
and $\epsilon=o(1)$ guarantees feasibility of $\underline{X}$.
###### Remark 3.5.
Since $N_{\leq t}\geq t$ for all $t\in[T]$ one can similarly define
${\tau}_{b}(1\mid X,\sigma)=\inf\\{t\geq
1:tX\geq\sigma(b)\\}=\lceil\frac{\sigma(b)}{X}\rceil$ as an upper bound on the
latest possible perishing time. This quantity can then be interpreted as the
“effective rank” of item $b$. This interpretable simplification comes at the
cost of our algorithm’s practical performance, but does not affect our
subsequent theoretical bounds.
Example 3.6 illustrates the $\underline{X}$ construction for a toy instance.
###### Example 3.6.
Consider a setting where $B=T=4$, $|\Theta|=1$, and $N_{t}=1$ for all
$t\in[T]$, with the following perishing time distributions for each item:
$\displaystyle T_{1}=\begin{cases}1\quad\text{w.p. }1/2\\\ 2\quad\text{w.p.
}1/2\end{cases}\quad T_{2}=\begin{cases}1\quad\text{w.p. }1/2\\\
4\quad\text{w.p. }1/2\end{cases}\quad T_{3}=\begin{cases}2\quad\text{w.p.
}1/2\\\ 3\quad\text{w.p. }1/2\end{cases}\quad
T_{4}=\begin{cases}3\quad\text{w.p. }1/2\\\ 4\quad\text{w.p. }1/2\end{cases}.$
Let $\sigma(b)=b$ for all $b$ (i.e., items are allocated in increasing order
of expected perishing time, with ties broken in favor of earliest possible
perishing time). Fig. 2 illustrates the solution to (4) for this instance,
with $\textsc{Conf}_{1}^{P}(\mu(X))=0$ for all $X$. Observe that
$\underline{X}=0.25$, whereas the proportional allocation $B/N=1$.
Observe that the perishing process described in Example 3.6 is not almost
surely offset-expiring, since items 1 and 2 both perish at the end of period 1
with probability 1/4. However, lack of offset-expiry is not the reason that
any online algorithm incurs an additional perishing-related loss. Theorem 3.7
establishes that any online algorithm’s performance necessarily scales with
the $\sigma$-induced loss, even under offset-expiry.
###### Theorem 3.7.
There exists an offset-expiring instance such that, for any online algorithm
that is feasible with probability at least $\alpha$, the following holds with
probability at least $\alpha$:
$\Delta_{\text{\it efficiency}}\geq
T\mathcal{L}^{\textsf{perish}}\quad\quad\Delta_{\text{\it
EF}}\geq\mathcal{L}^{\textsf{perish}}.$
###### Proof.
Consider an instance with $B=T$, $|\Theta|=1$ and $N_{t}=1$ for all $t$.
Suppose moreover that resources have deterministic perishing times, with
$T_{b}=b$ for all $b$, and $\sigma(b)=T+1-b$. ($T_{b}=b$ implies that the
process is offset-expiring.) Since the perishing and demand processes are
deterministic, we let $\textsc{Conf}_{1}^{P}(\mu(X))=0$ for all $X$, and
$\overline{N}=N$. For ease of notation, we omit the dependence of all
quantities on $\theta$ in the remainder of the proof. The following lemma
states that under this flipped ordering, any online algorithm is severely
limited in its total allocation.
###### Lemma 3.8.
Any feasible algorithm must have $\sum_{t}X_{t}^{alg}\leq 1$.
###### Proof.
For any feasible algorithm, there must exist an available unit in period $T$.
Since the only unit that has not perished by $t=T$ is $b=1$, it must be that
this unit is available in period $T$. Thus, we must have
$\sum_{t}X_{t}^{alg}\leq 1$ (else, $b=1$ will have been allocated before $T$).
∎
Lemma 3.8 implies that any feasible stationary algorithm, which allocates a
fixed amount $X_{t}^{alg}=X$ for all $t$, must have $X\leq\frac{1}{N}$. We use
this fact to bound $\overline{\Delta}(X)$, for any feasible stationary
allocation.
###### Lemma 3.9.
For any $0<X\leq\frac{1}{N}$, $\overline{\Delta}(X)\geq T-1$.
###### Proof.
By definition:
$\displaystyle{\tau}_{b}(1\mid X,\sigma)=\inf\\{t>0:N_{\leq
t}X\geq\sigma(b)\\}=\inf\\{t>0:tX\geq\sigma(b)\\}=\lceil\frac{\sigma(b)}{X}\rceil=\lceil\frac{T+1-b}{X}\rceil$
$\displaystyle\geq T(T+1-b),$
where the final inequality follows from the fact that
$X\leq\frac{1}{N}=\frac{1}{T}$. Hence,
$\displaystyle\overline{\Delta}(X)=\sum_{b}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}<\min\\{T,{\tau}_{b}(1\mid
X,\sigma)\\}}\right\\}\geq\sum_{b}\mathds{1}\mathopen{}\mathclose{{}\left\\{b<\min\\{T,T({T+1-b})\\}}\right\\}=\sum_{b}\mathds{1}\mathopen{}\mathclose{{}\left\\{b<T}\right\\}$
$\displaystyle=T-1,$
where the inequality uses the lower bound on ${\tau}_{b}(1\mid X,\sigma)$ in
addition to the assumption that $T_{b}=b$. ∎
Putting Lemma 3.8 and Lemma 3.9 together, we have:
$\underline{X}:=\sup\\{X:X\leq\frac{B-\overline{\Delta}(X)}{N}\\}\leq\sup\\{X:X\leq\frac{T-(T-1)}{T}\\}=\frac{1}{T}\\\
\implies\mathcal{L}^{\textsf{perish}}\geq 1-1/T.$
We now show the lower bounds on $\Delta_{\text{\it EF}}$ and
$\Delta_{\text{\it efficiency}}$. By Lemma 3.8, $\sum_{t}X_{t}^{alg}\leq 1$,
which implies that $\min_{t}X_{t}^{alg}\leq\frac{1}{T}$. Hence,
$\Delta_{\text{\it EF}}=\max_{t}|1-X_{t}^{alg}|\geq
1-\frac{1}{T}=\mathcal{L}^{\textsf{perish}}$. Moreover, $\Delta_{\text{\it
efficiency}}=T-\sum_{t}X_{t}^{alg}\geq T-1=T\mathcal{L}^{\textsf{perish}}$. ∎
## 4 The Algorithm
Our algorithm, Perishing-Guardrail, takes as input $(i)$ a desired bound on
envy $L_{T}$, and $(ii)$ a high-probability parameter $\delta$. The
algorithmic approach is tackled in three steps:
1. 1.
Constructing a static allocation (also referred to as baseline allocation or
lower guardrail), $\underline{X}$, under which the algorithm doesn’t run out
of budget with high probability. Motivated by Section 3.2, we let
$\underline{X}=\sup\mathopen{}\mathclose{{}\left\\{X\mid
X\leq\frac{B-\overline{\Delta}(X)}{\overline{N}}}\right\\},$
with $\overline{N}=\mathbb{E}[N]+\textsc{Conf}_{0,T}^{N}$, for an
appropriately defined high-probability confidence term
$\textsc{Conf}_{0,T}^{N}$.
2. 2.
Setting an “aggressive” allocation $\overline{X}=\underline{X}+L_{T}$ to
improve efficiency.
3. 3.
Determining an appropriate threshold condition that indicates when to allocate
$\overline{X}$.
Though the above approach is similar to the guardrail algorithm proposed by
Sinclair et al., (2022) for the no-perishing setting, we emphasize that
identifying the appropriate static allocation and threshold condition under
perishing uncertainty poses significant challenges that do not exist in the
classical setting. In this latter setting, the natural static allocation that
guarantees budget-feasibility is the proportional allocation in a high-demand
regime, i.e., $\underline{X}=B/\overline{N}$. Part of the reason this is
easily handled is the fact that arrival uncertainty is exogenous, i.e. it is
invariant to the decisions made by the algorithm. On the other hand,
uncertainty around perishing is endogenous: as discussed in Section 3, though
the distribution around perishing times is fixed, how many — and which — items
perish depends heavily on the rate at which items are being allocated, which
itself depends on the rate at which items perish. The threshold condition we
next describe must contend with this knife-edge effect.
##### Determining the threshold condition.
Recall, the “aggressive” allocation $\overline{X}$ will be used to improve our
algorithm’s efficiency, at the cost of higher envy. In each period, our
algorithm checks whether there is enough budget remaining to accommodate $(i)$
an aggressive allocation in the current period, $(ii)$ a conservative
allocation in all future periods, under high demand, and $(iii)$ perishing
that may occur under future conservative allocations. The main challenge here
lies in estimating $(iii)$: over-optimism runs an increased risk of exhausting
the budget early, due to the same phenomenon as that described in Section 3.2,
whereas over-pessimism fails to take advantage of efficiency gains from
aggressively allocating.
For $t\in[T]$, we let $\overline{P}_{t}$ denote our algorithm’s worst-case
perishing prediction. As above, the “slow” consumption process will allow us
to obtain a closed-form characterization of $\overline{P}_{t}$. In particular,
for $t\in[T]$, $b\in\mathcal{B}$, we consider the notion of “latest allocation
time” after $t$:
$\displaystyle{\tau}_{b}(t\mid\underline{X},\sigma)=\inf\mathopen{}\mathclose{{}\left\\{t^{\prime}\geq
t:\underline{N}_{<t}\underline{X}+\underline{N}_{[t,t^{\prime}]}\underline{X}\geq\sigma(b)}\right\\},$
(8)
where
$\underline{N}_{[t,t^{\prime}]}=\mathbb{E}\mathopen{}\mathclose{{}\left[N_{[t,t^{\prime}]}}\right]-\textsc{Conf}_{t,t^{\prime}}^{N}$
for an appropriately defined high-probability confidence term
$\textsc{Conf}_{t,t^{\prime}}^{N}$. In words,
$\underline{N}_{<t}\underline{X}+\underline{N}_{[t,t^{\prime}]}\underline{X}$
corresponds to the least amount the algorithm could have consumed by
$t^{\prime}$ (either by allocating or via perishing). Hence, if $b$ is in the
set of remaining items at the beginning of period $t$, with high probability
it will be allocated before ${\tau}_{b}(t\mid\underline{X},\sigma)$. Via
similar logic, we let
$\overline{\mathcal{B}}_{t}=\\{\sigma^{-1}(\lceil\underline{N}_{<t}\underline{X}\rceil),\ldots,\sigma^{-1}(B)\\}$
be the set of items remaining under this slow process, and define the expected
number of items that perish from $t$ onwards as:
$\displaystyle\eta_{t}=\sum_{b\in\overline{\mathcal{B}}_{t}}\mathbb{P}\mathopen{}\mathclose{{}\left(t\leq
T_{b}<\min\\{T,{\tau}_{b}(t\mid\underline{X},\sigma)\\}}\right).$ (9)
The pessimistic forecast of future spoilage is then defined as:
$\displaystyle\begin{cases}\overline{P}_{t}&=\min\mathopen{}\mathclose{{}\left\\{\overline{P}_{t-1},\eta_{t}+\textsc{Conf}_{t}^{P}(\eta_{t})}\right\\}\quad\forall\
t\in[T],\\\ \overline{P}_{0}&=B,\end{cases}$ (10)
for an appropriately defined confidence term $\textsc{Conf}_{t}^{P}(\cdot)$.
Note that $\overline{P}_{1}=\overline{\Delta}(\underline{X})$ (see Eq. 7).
We present our algorithm, Perishing-Guardrail, in Algorithm 1. For $t\in[T]$,
$\mathcal{B}^{alg}_{t}$ is used to denote the set of remaining resources at
the beginning of time $t$, and $B_{t}^{alg}$ the quantity of remaining
resources at the beginning of the period. Moreover, let $\mathcal{A}_{t}$ be
the set of items allocated in round $t$, and $\textsc{PUA}_{t}^{alg}$ the
quantity of unallocated items that perished at the end of round $t$.
Input: Budget $B=B_{1}^{alg}$, allocation schedule $\sigma$, envy parameter
$L_{T}$, arrival confidence terms
$(\textsc{Conf}_{t,t^{\prime}}^{N})_{t,t^{\prime}\in\\{0,\ldots,T\\}}$,
perishing confidence terms
$(\textsc{Conf}_{t}^{P}(\cdot))_{t\in\\{1,\ldots,T\\}}$, and perishing inputs
$(\eta_{t})_{t\in[T]}$ given by (9)
Output: An allocation $X^{alg}\in\mathbb{R}^{T\times|\Theta|}$
Compute $\underline{X}=\sup\mathopen{}\mathclose{{}\left\\{X\mid
X\leq\frac{B-\overline{\Delta}(X)}{\overline{N}}}\right\\}$ and set
$\overline{X}=\underline{X}+L_{T}$.
for _$t=1,\ldots,T$_ do
Compute
$\overline{P}_{t}=\min\mathopen{}\mathclose{{}\left\\{\overline{P}_{t-1},\eta_{t}+\textsc{Conf}_{t}^{P}(\eta_{t})}\right\\}$
// Compute ‘‘worst-case’’ future perishing
if _$B_{t}^{alg} <N_{t}\underline{X}$_ then // insufficient budget to
allocate lower guardrail
Set $X_{t,\theta}^{alg}=\frac{B_{t}^{alg}}{N_{t}}$ for each $\theta\in\Theta$.
Allocate items $b\in\mathcal{B}^{alg}_{t}$ according to $\sigma$.
else if
_$B_{t}^{alg}-N_{t}\overline{X}\geq\underline{X}(\mathbb{E}\mathopen{}\mathclose{{}\left[N_{
>t}}\right]+\textsc{Conf}_{t,T}^{N})+\overline{P}_{t}$_ then// use upper
guardrail
Set $X_{t,\theta}^{alg}=\overline{X}$ for each $\theta\in\Theta$. Allocate
items $b\in\mathcal{B}^{alg}_{t}$ according to $\sigma$.
else // use lower guardrail
Set $X_{t,\theta}^{alg}=\underline{X}$ for each $\theta\in\Theta$. Allocate
items $b\in\mathcal{B}^{alg}_{t}$ according to $\sigma$.
Update $B_{t+1}^{alg}=B_{t}^{alg}-N_{t}X_{t}^{alg}-\textsc{PUA}^{alg}_{t}$
end for
return _$X^{alg}$_
ALGORITHM 1 Perishing-Guardrail
### 4.1 Performance guarantee
###### Definition 4.1 ($\delta$-Offset-Expiry.).
A perishing process is _$\delta$ -offset-expiring_ if:
$\mathbb{P}\mathopen{}\mathclose{{}\left(\frac{P_{<t}}{B}\leq\frac{N_{<t}}{N}\quad\forall\,t\geq
2}\right)\geq 1-\delta$
###### Theorem 4.2.
For $t^{\prime}>t$, define the confidence terms:
* •
$\textsc{Conf}^{N}_{t,t^{\prime}}=\sqrt{2(t^{\prime}-t)|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)}$
* •
$\textsc{Conf}^{P}_{t}(\eta_{t})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\log(3t\log
T/\delta)+\sqrt{\log^{2}(3t\log T/\delta)+8\eta_{t}\log(3t\log
T/\delta)}}\right)$
Then, for any $\delta$-offset-expiring perishing process, with probability at
least $1-3\delta$, Algorithm 1 achieves:
$\displaystyle\Delta_{\text{\it
EF}}\lesssim\max\\{L_{T},\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}\\}$
$\displaystyle\Delta_{\text{\it
efficiency}}\lesssim\min\mathopen{}\mathclose{{}\left\\{\sqrt{T},L_{T}^{-1}+\sqrt{TL_{T}^{-1}\mathcal{L}^{\textsf{perish}}}}\right\\}+T\mathcal{L}^{\textsf{perish}}$
$\displaystyle\textsc{Envy}\lesssim L_{T}$
where $\lesssim$ drops poly-logarithmic factors of $T$, $\log(1/\delta)$,
$o(1)$ terms, and absolute constants.
In Section B.2, we show that these bounds are indeed tight, relative to the
lower bounds in Theorem 2.3 and Theorem 3.7. We dedicate the remainder of this
section to further parsing the bounds on counterfactual envy and efficiency —
graphically represented in Fig. 1 — given the scalings of $L_{T}$ and
$\mathcal{L}^{\textsf{perish}}$. In Section 4.2 we derive necessary and
sufficient conditions on the perishing process for $\delta$-offset-expiry to
hold. We finally prove Theorem 4.2 in Section 4.3.
Note that our result is a strict generalization of Sinclair et al., (2022):
in the simplest setting where $\mathcal{L}^{\textsf{perish}}=0$ (i.e., no
perishing, or perishing with perfect predictions), we recover the trade-off
they identified. The following corollary simplifies our bounds in the “low-
envy” setting where $L_{T}\lesssim 1/\sqrt{T}$.
###### Corollary 4.3 (Low-Envy).
Suppose $L_{T}\lesssim 1/\sqrt{T}$. Then, Perishing-Guardrail achieves with
probability at least $1-3\delta$:
$\displaystyle\Delta_{\text{\it
EF}}\lesssim\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}$
$\displaystyle\Delta_{\text{\it
efficiency}}\lesssim\sqrt{T}+T\mathcal{L}^{\textsf{perish}}.$
Corollary 4.3 implies that there is no efficiency benefit to increasing
$L_{T}$ as long as $L_{T}\lesssim 1/\sqrt{T}$. When
$\mathcal{L}^{\textsf{perish}}\lesssim 1/\sqrt{T}$, our algorithm incurs
$\widetilde{O}(\sqrt{T})$ envy and inefficiency. In this case, these
quantities are driven by the exogenous uncertainty in demand. When
$\mathcal{L}^{\textsf{perish}}\gtrsim 1/\sqrt{T}$, on the other hand, envy and
inefficiency are driven by unavoidable perishing due to prediction errors.
Corollary 4.4 next considers the “high-envy, high-perishing” setting, on the
other extreme of the spectrum.
###### Corollary 4.4 (High-Envy, High-Perishing).
Suppose $L_{T}\gtrsim 1/\sqrt{T}$ and $\mathcal{L}^{\textsf{perish}}\gtrsim
1/\sqrt{T}$. Then, Perishing-Guardrail achieves with probability $1-3\delta$:
$\displaystyle\Delta_{\text{\it
EF}}\lesssim\max\\{L_{T},\mathcal{L}^{\textsf{perish}}\\}$
$\displaystyle\Delta_{\text{\it efficiency}}\lesssim
T\mathcal{L}^{\textsf{perish}}.$
Similarly in this regime, increasing $L_{T}$ doesn’t guarantee arbitrary gains
in efficiency; thus, setting $L_{T}\sim\mathcal{L}^{\textsf{perish}}$ is
optimal. We conclude by exploring the more nuanced “high-envy, low-perishing”
regime. In this setting, our algorithm’s guarantees depend on whether the
efficiency gain from increasing envy, $L_{T}^{-1}$, exceeds
$T\mathcal{L}^{\textsf{perish}}$. We defer its proof to Section B.3.
###### Corollary 4.5 (High-Envy, Low-Perishing).
Suppose $L_{T}\gtrsim 1/\sqrt{T}$, $\mathcal{L}^{\textsf{perish}}\lesssim
1/\sqrt{T}$, and $T\mathcal{L}^{\textsf{perish}}\lesssim L_{T}^{-1}$. Then,
Perishing-Guardrail achieves with probability at least $1-3\delta$:
$\displaystyle\Delta_{\text{\it EF}}\lesssim L_{T}$
$\displaystyle\Delta_{\text{\it efficiency}}\lesssim L_{T}^{-1}.$
On the other hand, if $T\mathcal{L}^{\textsf{perish}}\gtrsim L_{T}^{-1}$,
Perishing-Guardrail achieves with probability at least $1-3\delta$:
$\displaystyle\Delta_{\text{\it EF}}\lesssim L_{T}$
$\displaystyle\Delta_{\text{\it efficiency}}\lesssim
T\mathcal{L}^{\textsf{perish}}.$
Thus, in “high-envy, low-perishing” regimes, if the cumulative
$\sigma$-induced loss is order-wise dominated by the efficiency gains from
envy (otherwise phrased, our allocation schedule $\sigma$ is high-enough
quality that perishing is low), increasing $L_{T}$ allows Perishing-Guardrail
to achieve inversely proportional gains in efficiency. One can do this until
moving into the regime where $L_{T}^{-1}\lesssim
T\mathcal{L}^{\textsf{perish}}$ (i.e., the cumulative $\sigma$-induced loss
dominates efficiency gains from envy). At this point, further increasing
$L_{T}$ hurts envy, and has no order-wise impact on efficiency.
Section 4.2 next provides conditions on the perishing distribution to satisfy
$\delta$-offset expiry.
### 4.2 On $\delta$-offset expiry
In order to highlight the salient parameters of the perishing process that
dictate offset-expiry, in this section we assume $B=N=T$ almost surely, with
$N_{t}=1$ for all $t$. At the cost of cumbersome algebra, one can relax this
assumption and derive entirely analogous results. In this case,
$\delta$-offset expiry (Definition 4.1) reduces to
$\mathbb{P}\mathopen{}\mathclose{{}\left(P_{<t}\leq t-1\ \forall\,t\geq
2}\right)\geq 1-\delta.$
For $t\in[T]$, let
$\mathcal{B}^{rand}_{<t}=\\{b:\mathbb{P}\mathopen{}\mathclose{{}\left(T_{b}<t}\right)\in(0,1)\\}$,
and
$\mathcal{B}^{det}_{<t}=\\{b:\mathbb{P}\mathopen{}\mathclose{{}\left(T_{b}<t}\right)=1\\}$.
Proposition 4.6 states that, in expectation, no more that $t-1$ items can
perish for $\delta$-offset expiry to hold for non-trivial values of $\delta$.
We defer all proofs in this section to Appendix B.4.
###### Proposition 4.6.
Suppose there exists $t\geq 2$ such that $\mathbb{E}[P_{<t}]>t-1$. If
$\mathcal{B}^{rand}_{<t}=\emptyset$, $\delta$-offset-expiry cannot be
satisfied for any value of $\delta\in(0,1)$. Else, $\delta$-offset-expiry
cannot be satisfied for
$\delta<\frac{1}{2}-{\mathrm{Std}\mathopen{}\mathclose{{}\left[P_{<t}}\right]^{-3}}\cdot
T$.
Note that this necessary condition fails to hold for one of the most standard
models of perishing: geometrically distributed perishing with parameter $1/T$
(that is, a constant fraction of items perish in each period). This highlights
that one of the most popular models in the literature is, in a sense, far too
pessimistic; for this setting, there is no hope of achieving low envy and
efficiency with high probability. Proposition 4.7 next establishes a
sufficient condition for $\delta$-offset expiry to hold.
###### Proposition 4.7.
Suppose that $\mathbb{E}[P_{<t}]\leq t-1$ for all $t\geq 2$. Then, the
perishing process is $\delta$-offset-expiring for any
$\delta\geq\sum_{t=2}^{T}\min\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\frac{\mathrm{Std}\mathopen{}\mathclose{{}\left[P_{<t}}\right]}{t-\mathbb{E}[P_{<t}]}}\right)^{2},\exp\mathopen{}\mathclose{{}\left(-\frac{{2}(t-\mathbb{E}[P_{<t}])^{2}}{|\mathcal{B}^{rand}_{<t}|}}\right)}\right\\}\mathds{1}\\{|\mathcal{B}^{rand}_{<t}|>0\\}.$
Proposition 4.7 states that either the expected lag $t-\mathbb{E}[P_{<t}]$
must be large, or the coefficient of variation with respect to the random lag
process $t-P_{<t}$, must be small. This then reduces to a bound on the
variability of $P_{<t}$ in settings where perishing closely tracks demand. In
our numerical experiments (Section 5) we instantiate the above bounds for
common distributions.
### 4.3 Analysis
The proof of Theorem 4.2 is based on three main building blocks:
* •
Defining and bounding the “good event”: (Section 4.3.1): We first show that,
with probability at least $1-\delta$, the realizations of future arrivals and
perishing are no worse than our algorithm’s pessimistic predictions. As a
result, it suffices to condition over such “good” sample paths.
* •
Establishing feasibility of $\underline{X}$ (Section 4.3.2): For this good
event, we show that the static allocation $\underline{X}$ computed at the
start of the horizon will never exhaust the budget, despite incurring
$\sigma$-induced loss.
* •
Improving efficiency via $\overline{X}$ (Section 4.3.3): We next show that the
threshold condition guarantees that the algorithm allocates aggressively
enough throughout the horizon to ensure high efficiency.
We use these building blocks for our final bounds on envy and efficiency in
Section 4.3.4.
#### 4.3.1 Defining and bounding the “good event”
We analyze the performance of our algorithm under a so-called “good event”
$\mathcal{E}$, the intersection of the following three events:
1. 1.
$\mathcal{E}_{N}=\mathopen{}\mathclose{{}\left\\{|N_{(t,t^{\prime}]}-\mathbb{E}\mathopen{}\mathclose{{}\left[N_{(t,t^{\prime}]}}\right]|\leq\textsc{Conf}^{N}_{t,t^{\prime}}\
\forall\ t,t^{\prime}>t\,\,}\right\\},$
2. 2.
$\mathcal{E}_{\overline{P}}=\mathopen{}\mathclose{{}\left\\{\overline{P}_{t}\geq\textsc{PUA}^{alg}_{\geq
t}\,\forall\ t\in\\{0,1,\ldots,T\\}}\right\\}$, where
$\textsc{PUA}^{alg}_{\geq t}$ denotes the quantity of unallocated items that
perished between the end of round $t$ and the end of round $T-1$,
3. 3.
$\mathcal{E}_{oe}=\mathopen{}\mathclose{{}\left\\{\frac{P_{<t}}{B}\leq\frac{N_{<t}}{{N}}\quad\forall\,t\geq
2}\right\\}$.
$\mathcal{E}$ represents the event that the arrival process falls close to its
mean, that $\overline{P}_{t}$ is indeed a pessimistic estimate of the
unallocated goods that perish in the future, and that the process is offset-
expiring.
Since the process is $\delta$-offset expiring, we have that
$\mathbb{P}(\mathcal{E}_{oe})\geq 1-\delta$ by assumption. The following lemma
provides the high-probability bound on $\mathbb{P}(\mathcal{E}_{N})$. We defer
its proof — which follows from a standard application of Hoeffding’s
inequality — to Appendix B.5.
###### Lemma 4.8.
$\mathcal{E}_{N}$ holds with probability at least $1-\delta$.
The main challenge in analyzing $\mathcal{E}$ lies in showing that
$\overline{P}_{t}$ is indeed a pessimistic estimate of our algorithm’s future
perishing. To upper bound the amount of unallocated resources that perished
between $t$ and $T-1$, we must account for both the uncertainty in arrivals
and the realized order in which resources perished, and relate these two
sources of uncertainty to the time at which the algorithm intended to allocate
these resources. Establishing this upper bound hinges upon the careful
construction of the “slow” consumption process, which decouples future
perishing from future allocations to compute $\overline{P}_{t}$. We formalize
these ideas in Lemma 4.9.
###### Lemma 4.9.
Given $\mathcal{E}_{N}$, $\mathcal{E}_{\overline{P}}$ holds with probability
at least $1-\delta$.
###### Proof of Lemma 4.9.
We prove the claim by induction.
Base case: $t=0$. Since $\overline{P}_{0}=B$ by definition, we have
$\overline{P}_{0}\geq\textsc{PUA}^{alg}_{\geq 0}$ trivially.
Inductive step: $t-1\rightarrow t.$ Since $\textsc{PUA}^{alg}_{\geq t}$
represents the amount of unallocated goods that perished between $t$ and
$T-1$, we have:
$\displaystyle\textsc{PUA}^{alg}_{\geq t}$
$\displaystyle{=}\sum_{\tau=t}^{T-1}\sum_{b\in\mathcal{B}^{alg}_{\tau}}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}=\tau,b\not\in\mathcal{A}_{\tau}}\right\\}=\sum_{b\in\mathcal{B}^{alg}_{t}}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}\geq
t,b\text{ not allocated before }T_{b}}\right\\}$ (11)
Recall, ${\tau}_{b}(t\mid\underline{X},\sigma)$ (8) is an upper bound on the
latest possible time the algorithm would have allocated $b$. This follows from
the fact that the least the algorithm could have allocated before $t$ under
$\mathcal{E}_{N}$ is $\underline{N}_{<t}\underline{X}$. Similarly, the least
amount of goods that the algorithm can allocate between $t$ and
$t^{\prime}\geq t$ is $\underline{N}_{[t,t^{\prime}]}\underline{X}$. Hence, if
$\underline{N}_{<t}\underline{X}+\underline{N}_{[t,t^{\prime}]}\underline{X}\geq\sigma(b)$
and $b$ did not perish before $t^{\prime}$, it must be that the algorithm
allocated $b$. Applying this logic to (11), we have:
$\displaystyle\textsc{PUA}_{\geq t}^{alg}$
$\displaystyle\leq\sum_{b\in\mathcal{B}^{alg}_{t}}\mathds{1}\mathopen{}\mathclose{{}\left\\{t\leq
T_{b}<\min\\{T,{\tau}_{b}(t\mid\underline{X},\sigma)\\}}\right\\},$ (12)
since it could have been that an item $b^{\prime}$ such that
$\sigma(b^{\prime})<\sigma(b)$ perished early, resulting in an earlier
allocation time for $b$. A similar argument gives us that
$\mathcal{B}^{alg}_{t}\subseteq\overline{\mathcal{B}}_{t}$. Plugging this into
(12):
$\displaystyle\textsc{PUA}^{alg}_{\geq t}$
$\displaystyle\leq\sum_{b\in\overline{\mathcal{B}}_{t}}\mathds{1}\mathopen{}\mathclose{{}\left\\{t\leq
T_{b}<\min\\{T,{\tau}_{b}(t\mid\underline{X},\sigma)\\}}\right\\}.$ (13)
Recall, $\eta_{t}=\sum_{b\in\overline{\mathcal{B}}_{t}}\mathbb{P}(t\leq
T_{b}<\min\\{T,{\tau}_{b}(t\mid\underline{X},\sigma)\\})$. Applying a Chernoff
bound (see Corollary C.3) to the right-hand side of (13), we obtain that, with
probability at least $1-\delta/(3t\log(T))$:
$\displaystyle\textsc{PUA}^{alg}_{\geq t}$
$\displaystyle\leq\eta_{t}+\frac{1}{2}\mathopen{}\mathclose{{}\left(\log(3t\log(T)/\delta)+\sqrt{\log^{2}(3t\log(T)/\delta)+8\eta_{t}\log(3t\log(T)/\delta)}}\right)$
$\displaystyle=\eta_{t}+\textsc{Conf}_{t}^{P}(\eta_{t}).$
Moreover, $\textsc{PUA}^{alg}_{\geq
t}\leq\textsc{PUA}^{alg}_{\geq(t-1)}\leq\overline{P}_{t-1}$, where the second
inequality follows from the inductive hypothesis. Putting these two facts
together, we obtain:
$\displaystyle\textsc{PUA}^{alg}_{\geq
t}\leq\min\mathopen{}\mathclose{{}\left\\{\overline{P}_{t-1},\eta_{t}+\textsc{Conf}_{t}^{P}(\eta_{t})}\right\\}=\overline{P}_{t}$
with probability at least $1-\delta/(3t\log(T))$. A union bound over $t$
completes the proof of the result. ∎
Lemma 4.10 follows from these high-probability bounds. We defer its proof to
Section B.5.
###### Lemma 4.10.
Let
$\mathcal{E}=\mathcal{E}_{N}\cap\mathcal{E}_{\overline{P}}\cap\mathcal{E}_{oe}$.
Then, $\mathbb{P}(\mathcal{E})\geq 1-3\delta$.
In the remainder of the proof, it suffices to restrict our attention to
$\mathcal{E}$.
#### 4.3.2 Feasibility of $\underline{X}$
We now show that, given $\mathcal{E}$, our algorithm never runs out of budget,
and as a result always allocates
$X_{t,\theta}^{alg}\in\\{\underline{X},\overline{X}\\}$. Since
$X_{t,\theta}^{alg}=X_{t,\theta^{\prime}}^{alg}$ for all
$\theta,\theta^{\prime}$, for ease of notation in the remainder of the proof
we omit the dependence of $X_{t,\theta}^{alg}$ on $\theta$.
###### Lemma 4.11.
Under event $\mathcal{E}$, $B_{t}^{alg}\geq N_{\geq t}\underline{X}$ for all
$t\in[T]$.
###### Proof of Lemma 4.11.
By induction on $t$.
Base Case: $t=1$. By definition:
$\displaystyle\underline{X}\leq\frac{B-\overline{\Delta}(\underline{X})}{\overline{N}}\implies
B\geq\overline{N}\underline{X}+\overline{\Delta}(\underline{X})\geq
N\underline{X},$
where the final inequality follows from $\overline{N}\geq N$ under
$\mathcal{E}$, and $\overline{\Delta}(\underline{X})\geq 0$.
Step Case: $t-1\rightarrow t$. We condition our analysis on
$(X_{\tau}^{alg})_{\tau<t}$, the algorithm’s previous allocations.
Case 1: $X_{\tau}^{alg}=\underline{X}$ for all $\tau<t$. By the recursive
budget update, $B_{t}^{alg}=B-N_{<t}\underline{X}-\textsc{PUA}^{alg}_{<t}$,
where $\textsc{PUA}^{alg}_{<t}$ denotes the quantity of unallocated goods that
perished before the end of round $t$. To show that $B_{t}^{alg}\geq N_{\geq
t}\underline{X}$, it then suffices to show that $B-\textsc{PUA}^{alg}_{<t}\geq
N\underline{X}$. We have:
$\displaystyle\textsc{PUA}^{alg}_{<t}\leq\textsc{PUA}^{alg}_{\geq
1}\leq\overline{P}_{1}=\overline{\Delta}(\underline{X}),$
where the final inequality follows from Lemma 4.9. Under $\mathcal{E}$, then,
as in the base case:
$\displaystyle B-\textsc{PUA}^{alg}_{<t}\geq
B-\overline{\Delta}(\underline{X})\geq N\underline{X}.$
Case 2: There exists $\tau<t$ such that $X_{\tau}^{alg}=\overline{X}$. Let
$t^{*}=\sup\\{\tau<t:X_{\tau}^{alg}=\overline{X}\\}$ be the most recent time
the algorithm allocated $\overline{X}$. Again, by the recursive budget update:
$B_{t}^{alg}=B_{t^{*}}^{alg}-N_{t^{*}}\overline{X}-N_{(t^{*},t)}\underline{X}-\textsc{PUA}^{alg}_{[t^{*},t)}.$
Since $\overline{X}$ was allocated at $t^{*}$, it must have been that
$B_{t^{*}}^{alg}\geq
N_{t^{*}}\overline{X}+\overline{N}_{>t^{*}}\underline{X}+\overline{P}_{t^{*}}$.
Plugging this into the above and simplifying:
$\displaystyle B_{t}^{alg}$ $\displaystyle\geq
N_{t^{*}}\overline{X}+\overline{N}_{>t^{*}}\underline{X}+\overline{P}_{t^{*}}-N_{t^{*}}\overline{X}-N_{(t^{*},t)}\underline{X}-\textsc{PUA}^{alg}_{[t^{*},t)}$
$\displaystyle=\overline{N}_{>t^{*}}\underline{X}-N_{(t^{*},t)}\underline{X}+\overline{P}_{t^{*}}-\textsc{PUA}^{alg}_{[t^{*},t)}$
$\displaystyle\geq N_{\geq
t}\underline{X}+\overline{P}_{t^{*}}-\textsc{PUA}^{alg}_{[t^{*},t)},$
where the second inequality follows from the fact that
$\overline{N}_{>t^{*}}\geq N_{>t^{*}}$ under $\mathcal{E}$. Thus, it suffices
to show that $\overline{P}_{t^{*}}\geq\textsc{PUA}^{alg}_{[t^{*},t)}$. This
holds since $\textsc{PUA}^{alg}_{[t^{*},t)}\leq\textsc{PUA}^{alg}_{\geq
t^{*}}\leq\overline{P}_{t^{*}}$ by Lemma 4.9. ∎
#### 4.3.3 Improving efficiency via $\overline{X}$
Having established that the algorithm never runs out of budget, it remains to
investigate the gains from allocating $\overline{X}$. By the threshold
condition, whenever the algorithm allocates $\overline{X}$ it must be that
there is enough budget remaining to allocate $\overline{X}$ in the current
period, and $\underline{X}$ in all future periods, under high demand and high
perishing. Thus, at a high level, $\overline{X}$ being allocated is an
indication that the algorithm has been inefficient up until round $t$. The
following lemma provides a lower bound on the last time the algorithm
allocates $\underline{X}$. This lower bound will later on allow us to
establish that, for most of the time horizon, the remaining budget is low
relative to future demand, ensuring high efficiency.
###### Lemma 4.12.
Given $\mathcal{E}$, let $t_{0}=\sup\\{t:X_{t}^{alg}=\underline{X}\\}$ be the
last time that $X_{t}^{alg}=\underline{X}$ (or else $0$ if the algorithm
always allocates according to $\overline{X}$). Then, for some
$\tilde{c}=\widetilde{\Theta}(1)$,
$t_{0}>T-\tilde{c}\mathopen{}\mathclose{{}\left(\frac{1}{L_{T}}+\sqrt{\frac{T\mathcal{L}^{\textsf{perish}}}{L_{T}}}}\right)^{2}.$
We defer the proof of Lemma 4.12 to Appendix B.5. Observe that, as
$\mathcal{L}^{\textsf{perish}}$ increases, our algorithm stops allocating
$\underline{X}$ earlier on. We will next see that this loss propagates to the
our final efficiency bound.
#### 4.3.4 Putting it all together
With these building blocks in hand, we prove our main result.
###### Proof of Theorem 4.2.
By Lemma 4.11, the algorithm never runs out of budget under event
$\mathcal{E}$, which occurs with probability at least $1-3\delta$. As a result
$X_{t,\theta}^{alg}\in\\{\underline{X},\overline{X}\\}$ for all $t\in[T]$,
$\theta\in\Theta$. We use this to bound envy and efficiency.
Counterfactual Envy: Recall, $\Delta_{\text{\it
EF}}=\max_{t,\theta}|w_{\theta}(X_{t,\theta}^{alg}-\frac{B}{N})|\leq
w_{\max}\cdot|X_{t,\theta}-\frac{B}{N}|$. We consider two cases.
Case 1: $\underline{X}\leq\overline{X}\leq\frac{B}{N}$. By definition:
$\displaystyle\frac{B}{N}-\underline{X}=\frac{B}{N}-\frac{B}{\overline{N}}+\frac{B}{\overline{N}}-\underline{X}\leq\frac{B}{\underline{N}}-\frac{B}{\overline{N}}+\mathcal{L}^{\textsf{perish}},$
where the inequality follows from the fact that $\underline{N}\leq N$ under
$\mathcal{E}$, and
$\mathcal{L}^{\textsf{perish}}=B/\overline{N}-\underline{X}$ by definition. We
turn our attention to the first two terms:
$\displaystyle\frac{B}{\underline{N}}-\frac{B}{\overline{N}}$
$\displaystyle\leq\frac{B}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]-\textsc{Conf}_{0,T}^{N}}-\frac{B}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]+\textsc{Conf}_{0,T}^{N}}$
$\displaystyle=\frac{B}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}\mathopen{}\mathclose{{}\left(\frac{1}{1-\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}-\frac{1}{1+\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}}\right)=\beta_{avg}\mathopen{}\mathclose{{}\left(\frac{1}{1-\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}-\frac{1}{1+\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}}\right).$
Using the fact that
$\textsc{Conf}_{0,T}^{N}=\sqrt{2T|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)}$
and $\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]=\Theta(T)$, there
exists $c_{1},c_{2}=\widetilde{\Theta}(1)$ such that, for large enough $T$,
$\mathopen{}\mathclose{{}\left(1-\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}\right)^{-1}\leq\mathopen{}\mathclose{{}\left(1-c_{1}/\sqrt{T}}\right)^{-1}\leq
1+2c_{1}/\sqrt{T}$ and
$\mathopen{}\mathclose{{}\left(1+\frac{\textsc{Conf}_{0,T}^{N}}{\mathbb{E}\mathopen{}\mathclose{{}\left[N}\right]}}\right)^{-1}\geq\mathopen{}\mathclose{{}\left(1+c_{2}/\sqrt{T}}\right)^{-1}\geq
1-c_{2}/\sqrt{T}$. Plugging this into the above:
$\displaystyle\frac{B}{\underline{N}}-\frac{B}{\overline{N}}$
$\displaystyle\leq\beta_{avg}\mathopen{}\mathclose{{}\left(1+2c_{1}/\sqrt{T}-(1-c_{2}/\sqrt{T})}\right)\leq\beta_{avg}(2c_{1}+c_{2})/\sqrt{T}\lesssim
1/\sqrt{T}.$
Thus, we obtain $|X_{t,\theta}^{alg}-B/N|\lesssim
1/\sqrt{T}+\mathcal{L}^{\textsf{perish}}$.
Case 2: $\underline{X}\leq\frac{B}{N}\leq\overline{X}$. We have:
$|X_{t,\theta}^{alg}-B/N|=\max\mathopen{}\mathclose{{}\left\\{\frac{B}{N}-\underline{X},\overline{X}-\frac{B}{N}}\right\\}\leq\overline{X}-\underline{X}=L_{T}.$
Combining these two cases, we obtain $\Delta_{\text{\it
EF}}\lesssim\max\\{1/\sqrt{T}+\mathcal{L}^{\textsf{perish}},L_{T}\\}$.
Hindsight Envy: Envy is trivially bounded above by $w_{max}\cdot L_{T}\lesssim
L_{T}$ since, for any $t,t^{\prime}$:
$w_{\theta}(X_{t^{\prime},\theta}-X_{t,\theta})\leq
w_{max}(\overline{X}-\underline{X})=w_{max}L_{T}.$
Efficiency: Let $t_{0}=\sup\\{t:X_{t}^{alg}=\underline{X}\\}$. Then:
$\displaystyle\Delta_{\text{\it efficiency}}$
$\displaystyle=B-\sum_{t,\theta}N_{t,\theta}X_{t,\theta}=B-\sum_{t}N_{t}X_{t}^{alg}$
$\displaystyle=B_{t_{0}}+\sum_{t<t_{0}}N_{t}X_{t}^{alg}+\textsc{PUA}^{alg}_{<t_{0}}-\sum_{t}N_{t}X_{t}^{alg}$
$\displaystyle=B_{t_{0}}-\sum_{t\geq
t_{0}}N_{t}X_{t}^{alg}+\textsc{PUA}^{alg}_{<t_{0}}$
$\displaystyle<N_{t_{0}}\overline{X}+\overline{N}_{>t_{0}}\underline{X}+\overline{P}_{t_{0}}-N_{t_{0}}\underline{X}-N_{>t_{0}}\overline{X}+\textsc{PUA}^{alg}_{<t_{0}}$
$\displaystyle=\underline{X}(\overline{N}_{>t_{0}}-N_{>t_{0}})-(\overline{X}-\underline{X})(N_{>t_{0}}-N_{t_{0}})+\overline{P}_{t_{0}}+\textsc{PUA}^{alg}_{<t_{0}},$
where the inequality follows from $X_{t_{0}}^{alg}=\underline{X}$, and the
threshold condition for allocating $\overline{X}$.
Noting that $\underline{X}\leq\beta_{avg}$ and
$\overline{N}_{>t_{0}}-N_{>t_{0}}\leq 2\textsc{Conf}_{t_{0},T}^{N}$, we have:
$\displaystyle\underline{X}(\overline{N}_{>t_{0}}-N_{>t_{0}})$
$\displaystyle\leq\beta_{avg}\cdot
2\sqrt{2({T}-{t_{0}})|\Theta|\rho_{max}^{2}\log(2T^{2}/\delta)}$
$\displaystyle\leq
2\beta_{avg}\sqrt{2\tilde{c}|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)}\min\mathopen{}\mathclose{{}\left\\{\sqrt{T},L_{T}^{-1}+\sqrt{TL_{T}^{-1}\mathcal{L}^{\textsf{perish}}}}\right\\},$
(14)
where the second inequality follows from Lemma 4.12.
We loosely upper bound the second term by:
$\displaystyle-(\overline{X}-\underline{X})(N_{>t_{0}}-N_{t_{0}})\leq(\overline{X}-\underline{X})N_{t_{0}}\leq
L_{T}|\Theta|(\mu_{max}+\rho_{\max}).$ (15)
Finally, consider $\overline{P}_{t_{0}}+\textsc{PUA}_{<t_{0}}^{alg}$. By
construction,
$\overline{P}_{t_{0}}\leq\overline{P}_{1}=\overline{\Delta}(\underline{X})$.
To upper bound $\textsc{PUA}_{<t_{0}}^{alg}$, we consider the process that
allocates $\underline{X}$ in each period to all arrivals. Let
$B_{t}(\underline{X})$ denote the quantity of remaining items under this
process, and $\mathcal{B}_{t}(\underline{X})$ the set of remaining items. We
use $\textsc{PUA}_{t}(\underline{X})$ to denote the quantity of unallocated
items that perish at the end of period $t$ under this process, and
$\textsc{PUA}_{<t}(\underline{X})$ those that perished before the end of
period $t$. The following lemma allows us to tractably bound
$\textsc{PUA}_{<t_{0}}^{alg}$ via this process. We defer its proof to Appendix
B.5.4.
###### Lemma 4.13.
For all $t\in[T]$,
1. 1.
$\mathcal{B}^{alg}_{t}\subseteq\mathcal{B}_{t}(\underline{X})$
2. 2.
$\textsc{PUA}_{t}^{alg}\leq\textsc{PUA}_{t}(\underline{X})$.
Using these two facts, we have:
$\textsc{PUA}_{<t_{0}}^{alg}\leq\textsc{PUA}_{<t_{0}}(\underline{X})\leq\textsc{PUA}_{\geq
1}(\underline{X})\leq\overline{P}_{1}=\overline{\Delta}(\underline{X}).$
Hence,
$\displaystyle\overline{P}_{t_{0}}+\textsc{PUA}_{<t_{0}}\leq
2\overline{\Delta}(\underline{X})\leq
2\overline{N}\mathcal{L}^{\textsf{perish}}\leq
2\mu_{max}(1+\sqrt{2|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)})T\mathcal{L}^{\textsf{perish}},$
(16)
where the second inequality follows from $\overline{\Delta}(\underline{X})\leq
B-\overline{N}\underline{X}=B-\overline{N}\mathopen{}\mathclose{{}\left(\frac{B}{\overline{X}}-\mathcal{L}^{\textsf{perish}}}\right)=\overline{N}\mathcal{L}^{\textsf{perish}}$,
and the last inequality uses the definition of $\overline{N}$. Putting bounds
(4.3.4), (15) and (16) together, we obtain:
$\displaystyle\Delta_{\text{\it efficiency}}$ $\displaystyle\leq
2\beta_{avg}\sqrt{2\tilde{c}|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)}\min\mathopen{}\mathclose{{}\left\\{\sqrt{T},L_{T}^{-1}+\sqrt{TL_{T}^{-1}\mathcal{L}^{\textsf{perish}}}}\right\\}+L_{T}|\Theta|(\mu_{max}+\rho_{\max})$
$\displaystyle\qquad+2\mu_{max}(1+\sqrt{2|\Theta|\rho_{\max}^{2}\log(2T^{2}/\delta)})T\mathcal{L}^{\textsf{perish}}.$
Using the fact that $L_{T}=o(1)$, we obtain the final bound on efficiency. ∎
## 5 Numerical experiments
In this section we study the practical performance of Perishing-Guardrail via
an extensive set of numerical experiments. We first consider one of the most
popular (and aggressive) models of perishing: geometrically distributed
perishing times. For this tractable perishing process, we establish
distribution-dependent bounds on the $\sigma$-induced loss
$\mathcal{L}^{\textsf{perish}}$, and empirically explore the dependence of the
envy-efficiency trade-off on the perishing rate. We leverage these empirical
trade-off curves to provide guidance on how to select the envy parameter
$L_{T}$, and compare the performance of Perishing-Guardrail to its perishing-
agnostic counterpart (Sinclair et al., , 2022). We moreover demonstrate the
robustness of our algorithm on a real-world dataset on ginger perishability
(Keskin et al., , 2022). We conclude our numerical study by considering the
non-i.i.d. perishing setting to gain insights into the choice of allocation
schedule. ††Code available at https://github.com/seanrsinclair/Online-
Resource-Allocation
### 5.1 Geometric perishing
Consider the setting in which each available unit perishes independently with
probability $p$ in each period, i.e., $T_{b}\sim\text{Geometric}(p)$, for all
$b\in\mathcal{B}$. Throughout the section, we assume $|\Theta|=1$, as our
insights are invariant to the number of types. Since perishing times are
identically distributed, the allocation order $\sigma$ does not have any
impact on the performance of the algorithm; hence, in the remainder of this
section we assume $\sigma$ is the identity ordering.
#### 5.1.1 Quantifying the unavoidable perishing loss
We first investigate the impact of perishability by characterizing the lower
guardrail $\underline{X}$ as a function of the perishing rate $p.$ As in
Section 4.2, we instantiate our theoretical bounds assuming $B=T,N_{t}=1$ for
all $t\in[T]$. In this case, Proposition 4.6 implies that $p\leq\frac{1}{T}$
is necessary to guarantee $\delta$-offset-expiry, for nontrivial values of
$\delta$. Proposition 5.1 below provides a lower bound on $\underline{X}$ for
this setting. We defer its proof to Section B.6.
###### Proposition 5.1.
Suppose $T_{b}\sim\emph{Geometric}(p)$ for all $b\in\mathcal{B}$, with $p\leq
1/T$. Then, the perishing process is $\delta$-offset-expiring for any
$\delta\geq 2\log T\cdot\frac{Tp}{(1-Tp)^{2}}$. Moreover, $\underline{X}\geq
1-3Tp-\frac{\log(3\log(T)/\delta)}{T}$ for any ordering $\sigma$.
Figure 3: Maximum feasible allocation $\underline{X}$ vs. $\alpha$, for
$T_{b}\sim\emph{Geometric}(T^{-(1+\alpha)})$, $B=200$, $T=150$, $N_{t}$ drawn
from a truncated normal distribution $\mathcal{N}(2,0.25)$, and $\delta=1/T$.
Here, $\underline{X}$ was calculated via line search, with Monte Carlo
simulation used to estimate $\overline{\Delta}(X)$ for each value of $X$. The
dashed line represents the “naive” allocation $B/\overline{N}=0.89$ which
ignores possible perishing, and the green line is the curve of best fit to
$\underline{X}$.
Proposition 5.1 establishes that, in the worst case, $\underline{X}$ decays
linearly in the rate $p$ at which goods spoil. This highlights the extent to
which the commonly used (and practically pessimistic) geometric model limits
both the kinds of perishable goods selected before allocation, as well as the
rate at which a decision-maker can allocate goods. Letting
$p=T^{-(1+\alpha)}$, $\alpha\in(0,1)$, Proposition 5.1 implies that
$\mathcal{L}^{\textsf{perish}}$ is on the order of $T^{-\alpha}$.
Alternatively, if a decision-maker wants no more than $T^{-\alpha}$ loss
relative to the proportional allocation, Proposition 5.1 provides an upper
bound of $T^{-(1+\alpha)}$ on the (exogenous) rate at which goods spoil.
We validate the scaling of $\underline{X}$ numerically in Fig. 3, for an
instance where the decision-maker also faces demand uncertainty. We observe
that $\underline{X}$ is concave increasing in $\alpha$, and that our lower
bound on $\underline{X}$ in the idealized setting provides a good fit for
$\underline{X}$, even under demand uncertainty. For $\alpha$ close to 0 (i.e.,
$p\sim 1/T$), $\underline{X}\approx 0.4$, less than half of the “naive” no-
perishing allocation, $B/\overline{N}$. For $\alpha=1$, $\underline{X}\approx
B/\overline{N}$. Note that, for $\alpha>1$, $\underline{X}$ is limited by the
confidence bound $\log(3\log T/\delta)/T$ in Proposition 5.1. Plugging the
lower bound on $\delta$ into this term, this implies that, even under no
demand uncertainty, $\underline{X}$ incurs a loss on the order of $1/T$
relative to the proportional allocation.
(a)
$T_{b}\sim\emph{Geometric}(T^{-1.1}):\underline{X}=0.70,\mathcal{L}^{\textsf{perish}}=0.4,\mathbb{P}(\mathcal{E}_{OE})=0.89$.
(b)
$T_{b}\sim\emph{Geometric}(T^{-1.2}):\underline{X}=0.84,\mathcal{L}^{\textsf{perish}}=0.26,\mathbb{P}(\mathcal{E}_{OE})=0.97$
(c)
$T_{b}\sim\emph{Geometric}(T^{-1.25}):\underline{X}=0.89,\mathcal{L}^{\textsf{perish}}=0.21,\mathbb{P}(\mathcal{E}_{OE})=0.99$
(d)
$T_{b}\sim\emph{Geometric}(T^{-1.3}):\underline{X}=0.93,\mathcal{L}^{\textsf{perish}}=0.17,\mathbb{P}(\mathcal{E}_{OE})=0.99$
Figure 4: Empirical trade-off between
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]$ and
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$. The
points on the trade-off curve correspond to increasing values of $L_{T}$, from
left to right. Static-$\frac{B}{\overline{N}}$ and Static-$\underline{X}$
respectively correspond to Vanilla-Guardrail and Perishing-Guardrail for
$L_{T}=0$.
#### 5.1.2 Numerical performance of Perishing-Guardrail
##### Empirical Trade-off Curves.
We numerically investigate the impact of the perishing rate on the envy-
efficiency frontier, and use the empirical trade-off curve to provide guidance
on how decision-makers should select $L_{T}$ in the setting of geometric
i.i.d. perishing. The demands $N_{t}$ are drawn from a truncated normal
distribution $\mathcal{N}(2,0.25)$. We let $T=100$, $B=200$, and vary
$\alpha\in\\{0.1,0.2,0.25,0.3\\}$ (see Appendix D, Fig. 7 for additional
values of $\alpha$). For these instances, we compare Perishing-Guardrail and
Vanilla-Guardrail (Sinclair et al., , 2022), a guardrail-based algorithm
designed for settings without perishable resources, with $L_{T}=T^{-\beta}$,
$\beta\in\\{0,0.05,0.1,\ldots,1\\}$. All results are averaged over 150
replications.
The empirical trade-off curves can be found in Fig. 4. For
$\alpha\in\\{0.1,0.2\\}$, Vanilla-Guardrail makes close to no gains in
efficiency for any value of $L_{T}$. In these high-perishing settings, then,
setting $L_{T}=0$ is optimal (there is no trade-off), in stark contrast to the
classic setting without perishability. As $\alpha$ increases, Vanilla-
Guardrail attains small gains in efficiency, but plateaus very quickly. This
yields the important insight that perishing-agnostic algorithms are not able
to leverage unfairness to improve efficiency in the presence of perishable
resources. Perishing-Guardrail, on the other hand, sees extremely large gains
in efficiency for a very small increase in envy, across all values of
$\alpha$. Even for this algorithm, however, larger values of $L_{T}$ do not
provide marginal gains in efficiency; the horizontal asymptote observed across
all values of $\alpha$ is precisely the cumulative $\sigma$-induced loss,
$T\mathcal{L}^{\textsf{perish}}$. Moreover, the vertical asymptote across all
plots corresponds to the unavoidable loss due to demand uncertainty (Theorem
3.7).
Note that, for small values of $\alpha$, the Perishing-Guardrail empirical
trade-off curve lies to the left of that of Vanilla-Guardrail, i.e., it
achieves lower counterfactual envy across all values of $L_{T}$. As we will
see below, this is due to the fact that Vanilla-Guardrail achieves extremely
high stockout rates. This effect is diminished as $\alpha$ increases (i.e.,
the perishing rate decreases). As this happens, both curves move down and to
the left (and closer) as they achieve lower counterfactual envy and
inefficiency due to spoilage. When the perishing rate is negligible (largest
value of $\alpha$), the empirical trade-off curve of Vanilla-Guardrail is
slightly to the left of that of Perishing-Guardrail; this is due to the loss
incurred by our modified $\underline{X}$ construction, which always allocates
less than $B/\overline{N}$ as a baseline. However, even when perishing is
negligible Perishing-Guardrail is slightly more efficient that Vanilla-
Guardrail, despite its baseline allocation being lower. This runs counter to
the intuition that Vanilla-Guardrail should be more efficient since it has a
higher baseline allocation. The reason for this is the difference in the two
algorithms’ threshold allocation decisions. Our algorithm, Perishing-
Guardrail, allocates $\overline{X}=\underline{X}+L_{T}$ if it forecasts that
it has enough budget remaining to allocate $\overline{X}$ in period $t$, and
$\underline{X}$ onwards. On the other hand, Vanilla-Guardrail allocates
$B/\overline{N}+L_{T}$ if it has enough budget remaining to allocate this high
amount in period $t$, and $B/\overline{N}$ in all future periods. Since
$B/\overline{N}>\underline{X}$, Vanilla-Guardrail depletes its budget faster
than Perishing-Guardrail whenever they both allocate the lower guardrail.
Hence, Perishing-Guardrail is able to allocate aggressively more frequently
than Vanilla-Guardrail, which results in improved efficiency and decreased
spoilage.
These empirical trade-off curves help to provide guidance on the choice of
$L_{T}$. In particular, across all experiments, the cusp of the trade-off
curve lies at $L_{T}\sim T^{-0.35}$. This value of $L_{T}$ is larger than no-
perishing cusp of $L_{T}\sim T^{-1/2}$ (Sinclair et al., , 2022). This is due
to the fact that our baseline allocation is significantly lower to avoid
perishing-induced stockouts; hence, in order to recover from this
inefficiency, $L_{T}$ must be higher. We use this observation in the following
experiments, comparing the performance of Perishing-Guardrail and Vanilla-
Guardrail for $L_{T}\sim T^{-0.35}$.
##### Perishing-Guardrail performance: synthetic experiments.
We compare the performance of Perishing-Guardrail to three benchmark
algorithms:
* •
Vanilla-Guardrail (Sinclair et al., (2022));
* •
Static-$\underline{X}$, the algorithm which allocates
$X_{t,\theta}=\underline{X}$ for all $t,\theta$, until it runs out of
resources;
* •
Static-$\frac{B}{\overline{N}}$, the algorithm which allocates
$X_{t,\theta}=\frac{B}{\overline{N}}$ for all $t,\theta$, until it runs out of
resources.
We are interested in five metrics: $(i)$ expected counterfactual envy
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$,
$(ii)$ expected hindsight envy
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$, $(iii)$
expected inefficiency
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]$, $(iv)$ expected spoilage
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{PUA}_{\geq 1}}\right]$, and
$(v)$ the stockout probability
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Stockout}}\right]$, i.e. the
proportion of replications for which the algorithm runs out of resources
before the end of the time horizon. We use the same simulation setup as above,
with $B=2T$ (see Appendix D, Fig. 8 for additional values of $\alpha$). Fig. 5
illustrates the performance of each algorithm across the first four metrics of
interest. We identify three regimes:
(a) $T_{b}\sim\emph{Geometric}(T^{-1.1})$
(b) $T_{b}\sim\emph{Geometric}(T^{-1.2})$
(c) $T_{b}\sim\emph{Geometric}(T^{-1.3})$
Figure 5: Algorithm comparison across $\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right],\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it efficiency}}}\right],\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$, and $\mathbb{E}\mathopen{}\mathclose{{}\left[\text{Spoilage}}\right]$, for $\alpha\in\\{0.1,0.2,0.25,0.3\\}$. $\alpha$ | Static-$\frac{B}{\overline{N}}$ | Static-$\underline{X}$ | Vanilla-Guardrail | Perishing-Guardrail
---|---|---|---|---
$.1$ | $0.99\pm 0.020$ | $0.00\pm 0.0$ | $1.00\pm 0.0$ | $0.11\pm 0.06$
$.2$ | $0.63\pm 0.095$ | $0.00\pm 0.0$ | $0.68\pm 0.091$ | $0.03\pm 0.03$
$.3$ | $0.03\pm 0.037$ | $0.00\pm 0.0$ | $0.06\pm 0.046$ | $0.00\pm 0.0$
Table 1: Comparison of stockout probabilities, for $T=150$,
$T_{b}\sim\text{Geometric}(T^{-(1+\alpha})$. The second number in each cell
corresponds to $95\%$ confidence intervals.
* •
High Perishing ($\alpha=0.1$): While unfairness (as measured by
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$ and
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$) is decreasing
in $T$ under Perishing-Guardrail and Static-$\underline{X}$, Vanilla-Guardrail
and Static-$\frac{B}{\overline{N}}$ perform remarkably poorly along these two
metrics. This is due to the fact that these latter algorithms fail to account
for the unavoidable perishing loss, resulting in an extremely high stockout
probability, as illustrated in Table 1, for $T=150$. In contrast, the two
perishing-aware algorithms rarely run out of resources. This underscores the
importance of modifying the baseline guardrail $\underline{X}$, which was
specifically constructed to avoid stockouts due to unavoidable perishing.
Comparing Static-$\underline{X}$ to Perishing-Guardrail, our results also
demonstrate that, in this high-perishing regime, the strategy of cautiously
over-allocating by $L_{T}$ comes at a significant reduction in inefficiency
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]$, at close to no increase in counterfactual envy
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$.
* •
Medium Perishing ($\alpha=0.2$): Though we observe similar trends as when
$\alpha=0.1$, all algorithms perform better across the board. Still, in Table
1 we see that the perishing-agnostic algorithms run out of resources in over
50% of replications. As observed in Fig. 3, Perishing-Guardrail exhibits both
higher efficiency and lower spoilage than its perishing-agnostic counterpart
in this regime, since it satisfies the threshold condition more frequently, as
described above.
* •
Low Perishing ($\alpha=0.3$): For this smaller perishing rate, Vanilla-
Guardrail stocks out significantly less frequently. Putting this together with
the fact that $B/\overline{N}>\underline{X}$, this explains the fact that it
has lower counterfactual envy than Perishing-Guardrail. However, along all
other metrics Perishing-Guardrail improves upon Vanilla-Guardrail. The
improvements in efficiency and spoilage are due to the same effects as
described above; moreover, our algorithm improves upon Vanilla-Guardrail on
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$ since it never
stocks out.
Overall, our results highlight the robustness of Perishing-Guardrail to
perishability, as it is able to achieve similar if not improved performance as
Vanilla-Guardrail in settings where there is limited perishing, with vastly
superior performance in high-perishing settings.
Algorithm | $\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$ | $\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$ | $\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Spoilage}}\right]$ | $\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Stockout}}\right]$ | $\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it efficiency}}}\right]$
---|---|---|---|---|---
Static-$\frac{B}{\overline{N}}$ | $1.18\pm 0.01$ | $1.03\pm 0.0$ | $346.4\pm 2.6$ | $1.0\pm 0.0$ | $444.6\pm 2.7$
Static-$\underline{X}$ | $0.60\pm 0.01$ | $0.0\pm 0.0$ | $475.9\pm 2.7$ | $0\pm 0.0$ | $605.5\pm 3.0$
Vanilla-Guardrail | $1.17\pm 0.01$ | $1.44\pm 0.0$ | $341.4\pm 3.0$ | $1.0\pm 0.0$ | $343.5\pm 2.9$
Perishing-Guardrail | $0.78\pm 0.04$ | $0.42\pm 0.05$ | $372.2\pm 3.2$ | $0.39\pm 0.09$ | $372.7\pm 3.2$
Table 2: Performance of the different algorithms (for $L_{T}=T^{-0.35}$) on
the “ginger” dataset in Keskin et al., (2022). The second number in each cell
corresponds to 95% confidence intervals.
##### Real-World Instance.
We next investigate the performance of our algorithm using the “ginger”
dataset provided by Keskin et al., (2022), which tracks demand,
replenishments, and perishing of ginger across $T=365$ days. We treat the time
between each replenishment as an independent sample (102 samples in total),
and fit a geometric distribution to the quantity of goods that perish in each
sample, obtaining $p=0.00224$. We similarly fit a truncated normal
distribution to the dataset, with $\mathcal{N}(3.2,1.85)$. Finally, we let
$B=365\cdot 3.2=1168$. For these inputs, $B/\overline{N}=0.89$ and
$\underline{X}=0.46$. Under these parameters the offset-expiry condition is
only satisfied 65.2% of the time; given this aggressive perishing, perishing-
agnostic algorithms are expected to perform particularly poorly.
In Table 2 we compare the performance of the different algorithms. We observe
the following:
* •
As conjectured, Static-$\frac{B}{\overline{N}}$ and Vanilla-Guardrail stock
out on 100% of replications since they fail to account for endogenous
perishing. The high stockout probabilities of Static-$\frac{B}{\overline{N}}$
and Vanilla-Guardrail lead to high unfairness (vis-à-vis hindsight and
counterfactual envy), since later arrivals receive allocations of zero. In
contrast, Static-$\underline{X}$ never stocks out. Perishing-Guardrail
achieves a higher stockout rate of 40%, likely due to the fact that threshold
condition does not account for non-offset-expiring trajectories. Still, our
algorithm’s counterfactual envy and hindsight envy are over 30% and 70% lower,
respectively, than that of Vanilla-Guardrail.
* •
Perishing-Guardrail allocates approximately 10% fewer goods than Vanilla-
Guardrail. It is notable, however, that it is more efficient than
Static-$\frac{B}{\overline{N}}$; this highlights that naively allocating more
aggressively need not always generate gains.
Overall, we see that even when offset-expiry holds with much lower
probability, for small losses in efficiency our algorithm makes major gains in
fairness relative to perishing-agnostic algorithms.
### 5.2 Non-i.i.d. perishing
As seen in Section 3, the performance of any algorithm is a function of
$\mathcal{L}^{\textsf{perish}}$, which depends on the allocation order
$\sigma$ in non-i.i.d. settings. A natural question that arises, then, is how
a decision-maker should choose $\sigma$. In this section, we investigate the
impact of three practical allocation orders on the numerical performance of
Perishing-Guardrail.
Given Theorem 3.7, a reasonable allocation order to choose would be
$\sigma^{*}\in\arg\min_{\sigma}\mathcal{L}^{\textsf{perish}}(\sigma),$ where
we emphasize the dependence of $\mathcal{L}^{\textsf{perish}}$ on the order
$\sigma$. Computing such a $\sigma^{*}$, however, is infeasible given the
space of $N!$ orderings. In lieu of this, we identify sufficient conditions on
the perishing process and allocation order that guarantee that
$\mathcal{L}^{\textsf{perish}}$ remain low (equivalently, that $\underline{X}$
remain high), and use these conditions to identify practical and easily
interpretable allocation schedules. Proposition 5.2 below provides these
sufficient conditions, for $B=T,N_{t}=1$ for all $t\in[T]$. We defer the proof
to Section B.7.
###### Proposition 5.2.
Suppose there exists $\alpha\in(0,1)$ such that:
1. 1.
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]>\min\mathopen{}\mathclose{{}\left\\{T,\lceil\frac{\sigma(b)}{1-T^{-\alpha}}\rceil}\right\\}$
2. 2.
$\sum_{b}\frac{\mathrm{Var}\mathopen{}\mathclose{{}\left[T_{b}}\right]}{\mathopen{}\mathclose{{}\left(\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]-\min\\{T,\lceil\frac{\sigma(b)}{1-T^{-\alpha}}\rceil\\}}\right)^{2}}\leq\frac{1}{2}T^{1-\alpha}$
Then, for any $\delta\geq 3\log(T)e^{-\frac{1}{8}T^{1-\alpha}}$, the process
is $\delta$-offset-expiring, and $\underline{X}\geq 1-T^{-\alpha}$.
Proposition 5.2 highlights the two key components that determine the baseline
allocation: the distance between the expected perishing time
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]$ and the expected
allocation time $\min\\{T,\lceil\frac{\sigma(b)}{1-T^{-\alpha}}\rceil\\}$
(which we colloquially refer to as “room to breathe”), and the variance of the
perishing time. Specifically, Condition 1 implies that, if
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]$ is low, it must be
that the item is allocated early on in the horizon (i.e., $\sigma(b)$ is low).
This encodes the “race against time” intuition that is typically held around
perishing. Condition 2 can be viewed as an upper bound on the cumulative
adjusted coefficient of variation (CV) of the perishing process. High-variance
perishing times and smaller “room to breathe” push $\alpha$ down, resulting in
a lower allocation rate. Hence, to guarantee a high allocation rate, the
perishing process needs to satisfy one of two conditions: (1) low variability,
or (2) high room to breathe. Having identified these driving factors, we
compare the following candidate allocation orders in experiments:
* •
Increasing Mean: Increasing order of
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]$;
* •
Decreasing Coefficient of Variation (CV): Decreasing order of
$\mathrm{Std}\mathopen{}\mathclose{{}\left[T_{b}}\right]/\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]$.
For fixed expected perishing time, this schedule allocates high-variance units
earlier on. Conversely, for fixed variance, it allocates items according to
the Increasing Mean schedule;
* •
Increasing Lower Confidence Bound (LCB): Increasing order of
$\mathbb{E}\mathopen{}\mathclose{{}\left[T_{b}}\right]-1.96\mathrm{Std}\mathopen{}\mathclose{{}\left[T_{b}}\right]$.
This ordering allocates items according to the lower bound of the 95%
confidence interval of the normal approximation to its perishing time. This
lower bound is expected to be small if either the expected perishing time is
small or the variance is large.
We break ties randomly in all cases.
As in Section 5.1, we draw the demands $N_{t}$ from a truncated normal
distribution, $\mathcal{N}(2,0.25)$; we moreover let $T=50$, $B=100$,
$\delta=\frac{1}{T}$, and $L_{T}={T}^{-0.35}$. Finally, we consider two sets
of perishing distributions:
* •
Instance 1: Front-loaded variability
$\displaystyle T_{b}=\begin{cases}\text{Uniform}(T/2-b/2,T/2+b/2)&\quad b\leq
T\\\ T&\quad b>T\end{cases}$ (17)
* •
Instance 2: Back-loaded variability
$\displaystyle T_{b}=\begin{cases}b+1&\quad b\leq T\\\
\text{Uniform}(b+1,T)&\quad b>T\end{cases}$ (18)
It can easily be verified that both instances are $\delta$-offset-expiring,
for $\delta=0.05$. Tables 3 and 4 show the performance of our algorithm over
these instances.
Order
$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{L}^{\textsf{perish}}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Spoilage}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Stockout}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]$ Increasing Mean $0.10\pm 0.004$ $0.36\pm 0.02$ $0.51\pm
0.02$ $5.78\pm 0.4$ $0.04\pm 0.04$ $6.28\pm 0.4$ Decreasing CV / Increasing
LCB $0.0\pm 0.0$ $0.44\pm 0.02$ $0.48\pm 0.02$ $1.24\pm 0.1$ $0.06\pm 0.04$
$1.79\pm 0.1$
Table 3: Performance of Perishing-Guardrail for $L_{T}=T^{-0.35}$ on the
distributions given in Eq. 17.
Order
$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{L}^{\textsf{perish}}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it EF}}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Envy}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Spoilage}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\textsc{Stockout}}\right]$
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]$ Increasing Mean / LCB $0.0\pm 0.0$ $0.41\pm 0.02$
$0.46\pm 0.02$ $0.0\pm 0.0$ $0.1\pm 0.05$ $0.56\pm 0.07$ Decreasing CV
$0.47\pm 0.0$ $0.51\pm 0.05$ $0.48\pm 0.02$ $48.3\pm 0.08$ $0.01\pm 0.0$
$48.7\pm 0.07$
Table 4: Performance of Perishing-Guardrail for $L_{T}=T^{-0.35}$ on the
distributions given in Eq. 18.
For the first instance, the Increasing Mean schedule allocates the first $T$
items uniformly at random, ignoring the fact that, for $b\leq T$, as $b$
increases the item is more likely to perish earlier on in the horizon. The
Decreasing CV / Increasing LCB schedules, on the other hand, are identical:
they allocate the first $T$ resources in decreasing order of $b$, and allocate
the remaining uniformly at random. Notably, the Decreasing CV / Increasing LCB
order achieves
$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{L}^{\textsf{perish}}}\right]=0$,
i.e., $\underline{X}=B/\overline{N}$, as in the no-perishing setting. (Note
that $\mathcal{L}^{\textsf{perish}}=0$ implies that this is an optimal
ordering.) Since its baseline allocation is higher it results in 78% less
spoilage than the Increasing Mean order, and a 71% decrease in inefficiency.
However, this order performs slightly worse with respect to counterfactual
envy and stockouts: this is again due to the more aggressive allocations.
For the second instance, the Increasing Mean and Increasing LCB schedules are
identical: they allocate items lexicographically. The Decreasing CV schedule,
on the other hand, allocates the last $T$ items (in increasing order of $b$)
before the first $T$ resources, since
$\mathrm{Std}\mathopen{}\mathclose{{}\left[T_{b}}\right]=0$ for all $b\leq T$.
In this setting, the first schedule is optimal with respect to
$\sigma$-induced loss, with
$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{L}^{\textsf{perish}}}\right]=0$.
This more aggressive allocation results in a 10% stockout rate (versus 1% for
the Decreasing CV schedule), but outperforms the Decreasing CV order across
all other metrics. This is intuitive as the number of errors in this latter,
clearly bad order results in
$\mathbb{E}\mathopen{}\mathclose{{}\left[\mathcal{L}^{\textsf{perish}}}\right]=0.47$,
approximately $50\%$ of the baseline allocation $B/\overline{N}$. The
algorithm then incurs both high inefficiency and spoilage.
These results indicate that the Increasing LCB schedule is both a practical
and robust candidate allocation order as it hedges against the inherent
variability of the perishing process.
## 6 Conclusion
This paper considers a practically motivated variant of the canonical problem
of online fair allocation wherein a decision-maker has a budget of perishable
resources to allocate fairly and efficiently over a fixed time horizon. Our
main insight is that perishability fundamentally impacts the envy-efficiency
trade-off derived for the no-perishing setting: while a decision-maker can
arbitrarily sacrifice on envy in favor of efficiency in this latter setting,
this is no longer the case when there is uncertainty around items’ perishing
times. We derive strong lower bounds to formalize this insight, which are a
function of both the quality of the decision-maker’s prediction over perishing
times, as well as the inherent aggressiveness of the perishing process. We
moreover design an algorithm that achieves these lower bounds; this algorithm
relies on the construction of a baseline allocation that accounts for the
unavoidable spoilage incurred by any online algorithm. From a technical
perspective, the main challenge that the perishing setting presents is that
the uncertainty around the quantity of resources that spoil in the future is
endogenous, in contrast to the exogenous uncertainty on the number of arrivals
in the classical setting. Deriving tight bounds on spoilage (both for our
lower bounds as well as in the design of our algorithm) relied on the “slow
allocation” construction, which rendered the highly coupled process amenable
to tractable analysis. Finally, our numerical experiments demonstrate our
algorithm’s strong performance against state-of-the-art perishing-agnostic
benchmarks.
In terms of future directions, our work identified offset-expiry as a
necessary condition for which the classical notion of envy-freeness is even
meaningful. While our algorithm performs well numerically in “low-probability”
offset-expiring settings, relaxing this assumption in theory remains an
interesting open question. Though we conjecture that the slow allocation
construction will remain a useful tool in more aggressive perishing settings,
the philosophical question of how to define more appropriate notions of envy
is likely the more important one. In addition to this, our model assumes that
the decision-maker allocates items according to a fixed allocation schedule.
Though our results do not require that the perishing distribution be
memoryless, allowing for time-varying / adaptive allocation schedules, though
less practical, would improve our algorithm’s performance in non-memoryless
settings. This relates back to the question of deriving theoretical insights
into the structure of optimal allocation schedules. Finally, though this paper
considered exogenous depletion of the budget, a natural practical extension is
one wherein $B$ evolves stochastically, accounting for external donations
independent of the allocations made by the algorithm.
## Acknowledgments
The authors would like to thank Connor Lawless for insightful conversations
about this work. Part of this work was done while Sean Sinclair and Sid
Banerjee were visiting the Simons Institute for the Theory of Computing for
the semester on Data-Driven Decision Processes. We gratefully acknowledge
funding from the National Science Foundation under grants ECCS-1847393,
DMS-1839346, CCF-1948256, CNS-195599, and CNS-1955997, the Air Force Office of
Scientific Research under grant FA9550-23-1-0068, and the Army Research
Laboratory under grants W911NF-19-1-0217 and W911NF-17-1-0094.
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## Appendix A Table of notation
Symbol | Definition
---|---
Problem setting specifications
$T$ | Total number of rounds
$B,\mathcal{B}$ | Number of resources and set of resources available
$\Theta,\theta$ | Set of types for individuals, and specification for individual’s type
$w_{\theta},w_{max}$ | Preference for the resource of individuals of type $\theta$ and $w_{max}=\max_{\theta}w_{\theta}$
$N_{t,\theta}$ | Number of individuals of type $\theta$ in round $t$
$N_{t}$ | $\sum_{\theta\in\Theta}N_{t,\theta}$
$N_{\geq t}$ | $\sum_{t^{\prime}\geq t}N_{t^{\prime}}$
$\sigma_{t,\theta}^{2},\rho_{t,\theta},\mu_{t,\theta}$ | $\mathrm{Var}\mathopen{}\mathclose{{}\left[N_{t,\theta}}\right]$, bound on $|N_{t,\theta}-\mathbb{E}\mathopen{}\mathclose{{}\left[N_{t,\theta}}\right]|$, and $\mathbb{E}\mathopen{}\mathclose{{}\left[N_{t,\theta}}\right]$
$\sigma_{min}^{2},\sigma_{max}^{2}$ | The respective maximum and minimum value of each quantity
$T_{b},P_{t}$ | Perishing time for resource $b\in[B]$ and $P_{t}=\sum_{b}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}=t}\right\\}$
$\beta_{avg}$ | $B/\sum_{\theta\in\Theta}\mathbb{E}\mathopen{}\mathclose{{}\left[N_{\theta}}\right]$
$X^{opt},X^{alg}$ | Optimal fair allocation in hindsight $X^{opt}_{t}=B/N$ and allocation by algorithm
$\Delta_{\text{\it EF}}$ | $\max_{t\in[T],\theta\in\Theta}|w_{\theta}X_{t,\theta}-w_{\theta}\frac{B}{N}|$
Envy | $\max_{t,t^{\prime}\in[T]^{2},\theta,\theta^{\prime}\in\Theta^{2}}w_{\theta}X_{t^{\prime},\theta^{\prime}}^{alg}-w_{\theta}X_{t,\theta}^{alg}$
$\Delta_{\text{\it efficiency}}$ | $B-\sum_{t,\theta}N_{t,\theta}X_{t,\theta}^{alg}$
$\overline{Y},\underline{Y}$ | High probability upper bound and lower bound of a random variable $Y$
$\sigma$ | $\sigma:\mathcal{B}\rightarrow[B]$ the allocation schedule
$\underline{X}$ | Maximum feasible allocation subject to endogenous perishing
$\mathcal{L}^{\textsf{perish}}$ | $\frac{B}{\overline{N}}-\underline{X}$
Algorithm specification
$L_{T}$ | Desired bound on $\Delta_{\text{\it EF}},\textsc{Envy}$
$\delta$ | High probability constant
$\textsc{Conf}_{t}$ | Confidence bound on $N_{\geq t}$ and $\textsc{PUA}_{\geq}t$, indicated by superscript
$B_{t}^{alg}$ | Budget available to the algorithm at start of round $t$
${\tau}_{b}(t\mid X,\sigma)$ | $\inf\\{t^{\prime}\geq t\mid\underline{N}_{<t}\underline{X}+\underline{N}_{[t,t^{\prime}]}\underline{X}\geq\sigma(b)\\}$
$\overline{\Delta}(X)$ | $\sum_{b}\mathbb{P}(T_{b}<\min\\{T,{\tau}_{b}(1\mid X,\sigma)\\})+\textsc{Conf}_{1}^{P}$
$\mathcal{A}_{t}$ | Set of resources allocated by algorithm at time $t$
$\textsc{PUA}_{\geq t}^{alg}$ | $\sum_{\tau=t}^{T-1}\sum_{B\in\mathcal{B}_{\tau}^{alg}}\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}=\tau,b\not\in\mathcal{A}_{\tau}}\right\\}$, perished and unallocated resources after $t$
$\overline{P}_{t}$ | Upper bound on $\textsc{PUA}_{\geq t}$
Additional notation
$\Phi(\cdot)$ | Standard normal CDF
Table 5: Common notation
## Appendix B Omitted proofs
### B.1 Section 3 omitted proofs
#### B.1.1 Proof of Theorem 3.2
###### Proof.
We first argue that offset-expiry implies feasibility of $B/N$. Consider the
allocation schedule which allocates goods in increasing order of perishing
time (breaking ties arbitrarily), and is such that $X_{t,\theta}=B/N$ for all
$t,\theta$, as long as there are resources remaining. Noting that
$(B/N)N_{<t}$ is precisely the cumulative allocation at the beginning of round
$t$, this implies that we allocate (weakly) more than the number of goods with
perishing time before round $t$ (i.e. $P_{<t}$). Since we allocate goods in
increasing order of perishing time, this also implies that no unit ever
perishes under this sequence of allocations. Thus, the total allocation by the
end of the horizon is $\frac{B}{N}\cdot N=B$, implying that $B/N$ is feasible.
We now argue that offset-expiry is necessary for $B/N$ to be feasible. To see
this, consider the first period $t\geq 2$ for which $P_{<t}/B>N_{<t}/N$ (i.e.,
by the end of period $t-1$, there existed some unallocated goods that had
perished). Then, the remaining budget at the start of period $t$ for any
algorithm, denoted by $B_{t}^{alg}$, is:
$\displaystyle B_{t}^{alg}\leq B-P_{<t}<B-N_{<t}\cdot\frac{B}{N}=N_{\geq
t}\cdot\frac{B}{N},$
which implies that the remaining budget does not suffice to allocate $B/N$ to
all arrivals from $t$ onwards. Hence, $B/N$ is not feasible. ∎
### B.2 Tightness of bounds
Consider the random problem instance which achieves the lower bounds of
Theorem 2.3 with probability 1/2, and the lower bounds of Theorem 3.7 with
probability 1/2. Putting these two bounds together, we have:
$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
EF}}}\right]\gtrsim\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}.$
By Theorem 4.2, our algorithm achieves
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
EF}}}\right]\lesssim\max\\{L_{T},\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}\\}.$
Letting $L_{T}\lesssim\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}$ then, our
algorithm achieves this lower bound. We now argue that our algorithm is tight
with respect to efficiency in this regime. Suppose $L_{T}=0$. By Theorem 2.3
and Theorem 3.7, any online algorithm incurs:
$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\gtrsim T\mathcal{L}^{\textsf{perish}}+\sqrt{T},$
which is achieved by our algorithm.
Consider now the regime in which $\Delta_{\text{\it EF}}=L_{T}$, i.e.,
$L_{T}\gtrsim\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}$. Again, randomizing
between the two lower bounds, we have:
$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\gtrsim
T\mathcal{L}^{\textsf{perish}}+\min\\{\sqrt{T},L_{T}^{-1}\\}.$ (19)
Case 1:
$L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}\gtrsim\sqrt{T}$.
Here, our algorithm achieves
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\lesssim\sqrt{T}+T\mathcal{L}^{\textsf{perish}}.$
If $L_{T}^{-1}\gtrsim\sqrt{T}$, we achieve the bound in (19). Suppose now that
$L_{T}^{-1}=o(\sqrt{T})$. Then, (19) implies that
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\gtrsim T\mathcal{L}^{\textsf{perish}}+L_{T}^{-1}.$ We
argue that, if $L_{T}^{-1}=o(\sqrt{T})$, then in this case
$T\mathcal{L}^{\textsf{perish}}\gtrsim\sqrt{T}$.
$T\mathcal{L}^{\textsf{perish}}$ then dominates both the lower bound in (19),
as well as our upper bound, which gives us tightness.
Case 2:
$L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}\lesssim\sqrt{T}$.
Here, our algorithm achieves
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\lesssim
L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}+T\mathcal{L}^{\textsf{perish}}.$
Since $L_{T}^{-1}\lesssim\sqrt{T}$, (19) reduces to
$\mathbb{E}\mathopen{}\mathclose{{}\left[\Delta_{\text{\it
efficiency}}}\right]\gtrsim T\mathcal{L}^{\textsf{perish}}+L_{T}^{-1}.$ It is
easy to check that
$\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}\lesssim\max\\{L_{T}^{-1},T\mathcal{L}^{\textsf{perish}}\\}$,
which completes the tightness argument.
### B.3 Section 4.1 omitted proofs
#### B.3.1 Proof of Corollary 4.5
###### Proof.
Consider first the case where $T\mathcal{L}^{\textsf{perish}}\lesssim
L_{T}^{-1}$. Then:
$\displaystyle
L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}\lesssim
L_{T}^{-1}\lesssim\sqrt{T},$
since $L_{T}\gtrsim 1/\sqrt{T}$ by assumption. Thus,
$\displaystyle\Delta_{\text{\it
efficiency}}\lesssim\min\mathopen{}\mathclose{{}\left\\{\sqrt{T},L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}}\right\\}+T\mathcal{L}^{\textsf{perish}}\lesssim
L_{T}^{-1},$
where again we’ve used the assumption that
$T\mathcal{L}^{\textsf{perish}}\lesssim L_{T}^{-1}.$
For the bound on $\Delta_{\text{\it EF}}$, we use the facts that $L_{T}\gtrsim
1/\sqrt{T}$ and $\mathcal{L}^{\textsf{perish}}\lesssim 1/\sqrt{T}$ to obtain:
$\Delta_{\text{\it
EF}}\lesssim\max\\{L_{T},\mathcal{L}^{\textsf{perish}}+1/\sqrt{T}\\}\lesssim
L_{T}.$
Suppose now $T\mathcal{L}^{\textsf{perish}}\gtrsim L_{T}^{-1}$. In this case:
$\displaystyle
L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}\lesssim
T\mathcal{L}^{\textsf{perish}}.$
Using the fact that $\mathcal{L}^{\textsf{perish}}\lesssim 1/\sqrt{T}$, we
obtain:
$\displaystyle\Delta_{\text{\it
efficiency}}\lesssim\min\mathopen{}\mathclose{{}\left\\{\sqrt{T},L_{T}^{-1}+\sqrt{T\mathcal{L}^{\textsf{perish}}L_{T}^{-1}}}\right\\}+T\mathcal{L}^{\textsf{perish}}\lesssim
T\mathcal{L}^{\textsf{perish}}.$
For the bound on $\Delta_{\text{\it EF}}$, we similarly have
$\Delta_{\text{\it EF}}\lesssim L_{T}$, since
$\mathcal{L}^{\textsf{perish}}\lesssim 1/\sqrt{T}\lesssim L_{T}$, by
assumption. ∎
### B.4 Section 4.2 omitted proofs
For ease of notation, we let $\nu_{t}=\mathbb{E}[P_{<t}]$ for all
$t\in\\{2,\ldots,T\\}$.
#### B.4.1 Proof of Proposition 4.6
###### Proof.
Let $t\in[T]$ be such that $\mathbb{E}[P_{<t}]>t-1$. Then:
$\displaystyle\mathbb{P}(P_{<t}\leq t-1\ \forall\ t\geq 2)$
$\displaystyle\leq\mathbb{P}(P_{<t}\leq
t-1)=\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}}\mathds{1}\\{T_{b}<t\\}\leq
t-1}\right),$ (20)
where the second equality is by definition.
Consider first the case where $\mathcal{B}^{rand}_{<t}=\emptyset$. In this
case, if $b$ perishes before $t$ with strictly positive probability, it must
be that $b\in\mathcal{B}^{det}_{<t}$. Then:
$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}}\mathds{1}\\{T_{b}<t\\}\leq
t-1}\right)$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{det}_{<t}}\mathds{1}\\{T_{b}<t\\}\leq
t-1}\right)=\mathbb{P}\mathopen{}\mathclose{{}\left(|\mathcal{B}^{det}_{<t}|\leq
t-1}\right),$ (21)
where the second equality follows from the fact that items in
$\mathcal{B}^{det}_{<t}$ perish before $t$ with probability 1. By the same
reasoning:
$\displaystyle
t-1<\mathbb{E}[P_{<t}]=\sum_{b\in\mathcal{B}}\mathbb{P}(T_{b}<t)=\sum_{b\in\mathcal{B}^{det}_{<t}}\mathbb{P}(T_{b}<t)=|\mathcal{B}^{det}_{<t}|\implies\mathbb{P}(|\mathcal{B}^{det}_{<t}|\leq
t-1)=0.$
Plugging this back into (20), we obtain $\mathbb{P}(P_{<t}\leq t-1\ \forall\
t\geq 2)=0$.
Consider now the case where $\mathcal{B}^{rand}_{<t}\neq\emptyset$. The goal
is to show the existence of $\epsilon$ such that $\mathbb{P}(P_{<t}\leq t-1\
\forall t\geq 2)\leq\epsilon$. Define the random variable:
$Y_{b}=\mathds{1}\mathopen{}\mathclose{{}\left\\{T_{b}<t}\right\\}-\mathbb{P}(T_{b}<t),\quad
b\in\mathcal{B}^{rand}_{<t}.$
By construction, $\mathbb{E}[Y_{b}]=0,0<\mathbb{E}[Y_{b}^{2}]\leq 1$, and
$\mathbb{E}[|Y_{b}|^{3}]\leq 1$. We have:
$\displaystyle\mathbb{P}(P_{<t}\leq t-1)$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathds{1}\\{T_{b}<t\\}\leq
t-1-|\mathcal{B}^{det}_{<t}|}\right)=\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{rand}_{<t}}Y_{b}\leq
t-1-|\mathcal{B}^{det}_{<t}|-\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{P}(T_{b}<t)}\right).$
By assumption,
$\mathbb{E}[P_{<t}]=|\mathcal{B}^{det}_{<t}|+\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{P}(T_{b}<t)>t-1$.
Hence,
$\displaystyle\mathbb{P}(P_{<t}\leq t-1)$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{rand}_{<t}}Y_{b}\leq
0}\right)=\mathbb{P}\mathopen{}\mathclose{{}\left(\frac{\sum_{b\in\mathcal{B}^{rand}_{<t}}Y_{b}}{\sqrt{\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[Y_{b}^{2}]}}\leq
0}\right).$
Let $\Phi(\cdot)$ denote the cdf of the standard normal distribution. By the
Berry-Esseen Theorem,
$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\frac{\sum_{b\in\mathcal{B}^{rand}_{<t}}Y_{b}}{\sqrt{\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[Y_{b}^{2}]}}\leq
0}\right)$
$\displaystyle\leq\Phi(0)+\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[Y_{b}^{2}]}\right)^{-3/2}\cdot\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[|Y_{b}|^{3}]$
$\displaystyle=\frac{1}{2}+\mathopen{}\mathclose{{}\left(\sum_{b\in\mathcal{B}^{rand}_{<t}}\text{Var}[\mathds{1}\\{T_{b}<t\\}]}\right)^{-3/2}\cdot\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[|Y_{b}|^{3}]$
$\displaystyle=\frac{1}{2}+\mathopen{}\mathclose{{}\left(\text{Var}\mathopen{}\mathclose{{}\left[\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathds{1}\\{T_{b}<t\\}}\right]}\right)^{-3/2}\cdot\sum_{b\in\mathcal{B}^{rand}_{<t}}\mathbb{E}[|Y_{b}|^{3}]$
$\displaystyle\leq\frac{1}{2}+\text{Var}[P_{<t}]^{-3/2}\cdot T$
$\displaystyle=\frac{1}{2}+{\mathrm{Std}\mathopen{}\mathclose{{}\left[P_{<t}}\right]^{-3}}\cdot
T.$
Putting this all together, we obtain: |
# PentestGPT: Evaluating and Harnessing Large Language Models for Automated
Penetration Testing
Gelei Deng1 Yi Liu1 Víctor Mayoral-Vilches23 Peng Liu4 Yuekang Li5 Yuan Xu1
Tianwei Zhang1 Yang Liu1 Martin Pinzger3 Stefan Rass6
1Nanyang Technological University 2Alias Robotics 3Alpen-Adria-Universität
Klagenfurt 4Institute for Infocomm Research ($I^{2}R$), A*STAR, Singapore
5University of New South Wales 6Johannes Kepler University Linz
###### Abstract
Penetration testing, a crucial industrial practice for ensuring system
security, has traditionally resisted automation due to the extensive expertise
required by human professionals. Large Language Models (LLMs) have shown
significant advancements in various domains, and their emergent abilities
suggest their potential to revolutionize industries. In this work, we
establish a comprehensive benchmark using real-world penetration testing
targets and further use it to explore the capabilities of LLMs in this domain.
Our findings reveal that while LLMs demonstrate proficiency in specific sub-
tasks within the penetration testing process, such as using testing tools,
interpreting outputs, and proposing subsequent actions, they also encounter
difficulties maintaining a whole context of the overall testing scenario.
Based on these insights, we introduce PentestGPT, an LLM-empowered automated
penetration testing framework that leverages the abundant domain knowledge
inherent in LLMs. PentestGPT is meticulously designed with three self-
interacting modules, each addressing individual sub-tasks of penetration
testing, to mitigate the challenges related to context loss. Our evaluation
shows that PentestGPT not only outperforms LLMs with a task-completion
increase of 228.6% compared to the GPT-3.5 model among the benchmark targets,
but also proves effective in tackling real-world penetration testing targets
and CTF challenges. Having been open-sourced on GitHub, PentestGPT has
garnered over 6,200 stars in 9 months and fostered active community
engagement, attesting to its value and impact in both the academic and
industrial spheres.
## 1 Introduction
Securing a system presents a formidable challenge. Offensive security methods
like penetration testing (pen-testing) and red teaming are now essential in
the security lifecycle. As explained by Applebaum [1], these approaches
involve security teams attempting breaches to reveal vulnerabilities,
providing advantages over traditional defenses, which rely on incomplete
system knowledge and modeling. This study, guided by the principle _“the best
defense is a good offense”_ , focuses on offensive strategies, specifically
penetration testing.
Penetration testing is a proactive offensive technique for identifying,
assessing, and mitigating security vulnerabilities [2]. It involves targeted
attacks to confirm flaws, yielding a comprehensive inventory of
vulnerabilities with actionable recommendations. This widely-used practice
empowers organizations to detect and neutralize network and system
vulnerabilities before malicious exploitation. However, it typically relies on
manual effort and specialized knowledge [3], resulting in a labor-intensive
process, creating a gap in meeting the growing demand for efficient security
evaluations.
Large Language Models (LLMs) have demonstrated profound capabilities,
showcasing intricate comprehension of human-like text and achieving remarkable
results across a multitude of tasks [4, 5]. An outstanding characteristic of
LLMs is their emergent abilities [6], cultivated during training, which
empower them to undertake intricate tasks such as reasoning, summarization,
and domain-specific problem-solving without task-specific fine-tuning. This
versatility posits LLMs as potential game-changers in various fields, notably
cybersecurity. Although recent works [7, 8, 9] posit the potential of LLMs to
reshape cybersecurity practices, including the context of penetration testing,
there is an absence of a systematic, quantitative assessment of their aptitude
in this regard. Consequently, an imperative question presents: To what extend
can LLMs automate penetration testing?
Motivated by this question, we set out to explore the capability boundary of
LLMs on real-world penetration testing tasks. Unfortunately, the current
benchmarks for penetration testing [10, 11] are not comprehensive and fail to
assess progressive accomplishments fairly during the process. To address this
limitation, we construct a robust benchmark that includes test machines from
HackTheBox [12] and VulnHub [13]—two leading platforms for penetration testing
challenges. Comprising 13 targets with 182 sub-tasks, our benchmark
encompasses all vulnerabilities appearing in OWASP’s top 10 vulnerability list
[14] and 18 Common Weakness Enumeration (CWE) items [15]. The benchmark offers
a more detailed evaluation of the tester’s performance by monitoring the
completion status for each sub-task.
With this benchmark, we perform an exploratory study using GPT-3.5 [16], GPT-4
[17], and Bard [18] as representative LLMs. Our test strategy is interactive
and iterative. We craft tailored prompts to guide the LLMs through penetration
testing. Each LLM, presented with prompts and target machine information,
generates step-by-step penetration testing operations. We then execute the
suggested operations in a controlled environment, document the results, and
feed them back to the LLM to inform and refine its next steps. This cycle
(prompting, executing, and feedback) is repeated until the LLM completes the
entire penetration testing process autonomously. To evaluate LLMs, we compare
their results against baseline solutions from official walkthroughs and
certified penetration testers. By analyzing similarities and differences in
their problem-solving approaches, we aim to better understand LLMs’
capabilities in penetration testing and how their strategies differ from human
experts.
Our investigation yields intriguing insights into the capabilities and
limitations of LLMs in penetration testing. We discover that LLMs demonstrate
proficiency in managing specific sub-tasks within the testing process, such as
utilizing testing tools, interpreting their outputs, and suggesting subsequent
actions. Compared to human experts, LLMs are especially adept at executing
complex commands and options with testing tools, while models like GPT-4 excel
in comprehending source code and pinpointing vulnerabilities. Furthermore,
LLMs can craft appropriate test commands and accurately describe graphical
user-interface operations needed for specific tasks. Leveraging their vast
knowledge base, they can design inventive testing procedures to unveil
potential vulnerabilities in real-world systems and CTF challenges. However,
we also note that LLMs have difficulty in maintaining a coherent grasp of the
overarching testing scenario, a vital aspect for attaining the testing goal.
As the dialogue advances, they may lose sight of earlier discoveries and
struggle to apply their reasoning consistently toward the final objective.
Additionally, LLMs overemphasize recent tasks in the conversation history,
regardless of their vulnerability status. As a result, they tend to neglect
other potential attack surfaces exposed in prior tests and fail to complete
the penetration testing task.
Building on our insights into LLMs’ capabilities in penetration testing, we
present PentestGPT111PentestGPT is King Arthur’s legendary sword, known for
its exceptional cutting power and the ability to pierce armor., an interactive
system designed to enhance the application of LLMs in this domain. Drawing
inspiration from the collaborative dynamics commonly observed in real-world
human penetration testing teams, PentestGPT is particularly tailored to manage
large and intricate projects. It features a tripartite architecture comprising
Reasoning, Generation, and Parsing Modules, each reflecting specific roles
within penetration testing teams. The Reasoning Module emulates the function
of a lead tester, focusing on maintaining a high-level overview of the
penetration testing status. We introduce a novel representation, the
Pentesting Task Tree (PTT), based on the cybersecurity attack tree [19]. This
structure encodes the testing process’s ongoing status and steers subsequent
actions. Uniquely, this representation can be translated into natural language
and interpreted by the LLM, thereby comprehended by the Generation Module and
directing the testing procedure. The Generation Module, mirroring a junior
tester’s role, is responsible for constructing detailed procedures for
specific sub-tasks. Translating these into exact testing operations augments
the generation process’s accuracy. Meanwhile, the Parsing Module deals with
diverse text data encountered during penetration testing, such as tool
outputs, source codes, and HTTP web pages. It condenses and emphasizes these
texts, extracting essential information. Collectively, these modules function
as an integrated system. PentestGPT completes complex penetration testing
tasks by bridging high-level strategies with precise execution and intelligent
data interpretation, thereby maintaining a coherent and effective testing
process.
We assessed PentestGPT across diverse testing scenarios to validate its
effectiveness and breadth. In our custom benchmarks, PentestGPT significantly
outperformed direct applications of GPT-3.5 and GPT-4, showing increases in
sub-task completion rates of 228.6% and 58.6%, respectively. Furthermore, when
applied to real-world challenges such as the HackTheBox active machine
penetration tests [20] and picoMini [21] CTF competition, PentestGPT
demonstrated its practical utility. It successfully resolved 4 out of 10
penetration testing challenges, incurring a total cost of 131.5 US Dollars for
the OpenAI API usage. In the CTF competition, PentestGPT achieved a score of
1500 out of a possible 4200, placing 24th among 248 participating teams. This
evaluation underscores PentestGPT’s practical value in enhancing penetration
testing tasks’ efficiency and precision. The solution has been made publicly
available on GitHub222The project is at:
https://github.com/GreyDGL/PentestGPT., receiving widespread acclaim with over
6,200 stars to the date of writing, active community engagement, and ongoing
collaboration with multiple industrial partners.
User1\. ExploitFlow2\. PentestGPT3\.
PentestPerfTargetexploitflowgraphadaptersmodelsstateparsingreasoninggenerationprogramatically
in Pythongoal description in textexchange exploit treeBenchmarks an exploit
flow4\. Malism2\. PentestGPTExternal entityOther future papersThis paperInner
Component Figure 1: Architecture of our framework to develop a fully automated
penetration testing tools, Malism. Figure depicts the various interaction
flows that an arbitrary User could follow using Malism to pentest a given
Target. 1. Corresponds with ExploitFlow, a modular library to produce security
exploitation routes (_exploit flows_) that caputures the state of the system
being tested in a flow after every discrete action. 2\. (this paper)
Corresponds with PentestGPT, a testing tool that leverages the power of LLMs
to produce testing guidance (heuristics) for every given discrete state. 3.
PentestPerf is a comprehensive penetration testing benchmark to evaluate the
performances of penetration testers and automated tools across a wide array of
testing targets. 4. captures Malism, our framework to develop fully automated
penetration testing tools which we name _cybersecurity cognitive engines_.
As a long term research goal, we aim to contribute to unlocking the potential
of modern machine learning approaches and develop a fully automated
penetration testing framework that helps produce cybersecurity cognitive
engines. Our overall architecture is depicted in Figure 1, showing our current
work and future planned contributions. Our proposed framework, Malism, is
designed to enable a user without in-depth security domain knowledge to
produce its cybersecurity cognitive engine that helps conduct penetration
testing over an extensive range of targets. This framework comprises three
primary components:
1. 1.
ExploitFlow [22]: A modular library to produce cyber security exploitation
routes (_exploit flows_). ExploitFlow aims to combine and compose exploits
from different sources and frameworks, capturing the state of the system being
tested in a flow after every discrete action, which allows learning attack
trees that affect a given system. ExploitFlow’s main motivation is to
facilitate and empower Game Theory and Artificial Intelligence (AI) research
in cyber security. It uniquely represents the exploitation process that
encodes every facet within it. Its representation can be effectively
integrated with various penetration testing tools and scripts, such as
Metasploit [23] to perform end-to-end penetration testing. Such representation
can be further visualized to guide the human experts to reproduce the testing
process.
2. 2.
PentestGPT (this paper): An automated penetration testing system that
leverages the power of LLMs to produce testing guidance and intuition at every
given discrete state. It functions as the core component of the Malism
framework, guiding the LLMs to utilize their domain knowledge in real-world
testing scenarios efficiently.
3. 3.
PentestPerf: A comprehensive penetration testing benchmark developed to
evaluate the performances of penetration testers and automated tools across a
wide array of testing targets. It offers a fair and robust platform for
performance comparison.
The harmonious integration of these three components forms an automated, self-
evolving penetration testing framework capable of executing penetration tests
over various targets, Malism. This framework to develop fully automated
penetration testing tools, which we name _cybersecurity cognitive engines_ ,
aims to revolutionize the field of penetration testing by significantly
reducing the need for domain expertise and enabling more comprehensive and
reliable testing.
In summary, we make the following contributions:
* •
Development of a Comprehensive Penetration Testing Benchmark. We craft a
robust and representative penetration testing benchmark, encompassing a
multitude of test machines from leading platforms such as HackTheBox and
VulnHub. This benchmark includes 182 sub-tasks covering OWASP’s top 10
vulnerabilities, offering fair and comprehensive evaluation of penetration
testing. To the best of our knowledge, this is the first benchmark in the
field that can provide progressive accomplishments assessments and
comparisons.
* •
Comprehensive Evaluation of LLMs for Penetration Testing Tasks. By employing
models like GPT-3.5, GPT-4, and Bard, our exploratory study rigorously
investigates the strengths and limitations of LLMs in penetration testing. To
the best of our knowledge, this is the first systematic and quantitative study
for the capability of LLMs in performing automated penetration testing. The
insights gleaned from this study shed valuable light on the capabilities and
challenges faced by LLMs, enriching our understanding of their applicability
in this specialized domain.
* •
Development of an Innovative LLM-powered Penetration Testing System. We
engineer PentestGPT, a novel interactive system that leverages the strengths
of LLMs to carry out penetration testing tasks automatically. Drawing
inspiration from real-world human penetration testing teams, PentestGPT
integrates a tripartite design that mirrors the collaborative dynamics between
senior and junior testers. This architecture optimizes LLMs’ usage,
significantly enhancing the efficiency and effectiveness of automated
penetration testing. We have open-sourced PentestGPT and it has received over
6,500 stars on GitHub, active community contributions, and industry partners
including AWS, Huawei, and ByteDance to collaborate.
## 2 Background & Related Work
### 2.1 Penetration Testing
Penetration testing, or “pentesting”, is a critical practice to enhance
organizational systems’ security. In a typical penetration test, security
professionals, known as penetration testers, analyze the target system, often
leveraging automated tools. The standard process is divided into five key
phases [24]: Reconnaissance, Scanning, Vulnerability Assessment, Exploitation,
and Post Exploitation (including reporting). These phases enable testers to
understand the target system, identify vulnerabilities, and exploit them to
gain access.
Despite significant advancements [25, 26, 11], a fully automated penetration
testing system remains out of reach. This gap results from the need for deep
vulnerability understanding and a strategic action plan. Typically, testers
combine depth-first and breadth-first search techniques [24]. They first grasp
the target environment’s scope, then drill down into specific vulnerabilities.
This method ensures comprehensive analysis, leaning on expertise and
experience. The multitude of specialized tools further complicate the
automation. Thus, even with artificial intelligence, achieving a seamless
automated penetration testing solution is a daunting task.
### 2.2 Large Language Models
Large Language Models (LLMs), including OpenAI’s GPT-3.5 and GPT-4, are
prominent tools with applications extending to various cybersecurity-related
fields, such as code analysis [27] and vulnerability repairment [28]. These
models are equipped with wide-ranging general knowledge and the capacity for
elementary reasoning. They can comprehend, infer, and produce text resembling
human communication, aided by a training corpus encompassing diverse domains
like computer science and cybersecurity. Their ability to interpret context
and recognize patterns enables them to adapt knowledge to new scenarios. This
adaptability, coupled with their proficiency in interacting with systems in a
human-like way, positions them as valuable assets in enhancing penetration
testing processes. Despite inherent limitations, LLMs offer distinct
attributes that can substantially aid in the automation and improvement of
penetration testing tasks. The realization of this potential, however,
requires the creation and application of a specialized and rigorous benchmark.
## 3 Penetration Testing Benchmark
### 3.1 Motivation
The comprehensive evaluation of LLMs in penetration testing necessitates a
robust and representative benchmark. Existing benchmarks in this domain [10,
11] have several limitations. First, they are often restricted in scope,
focusing on a narrow range of potential vulnerabilities, and thus fail to
capture the complexity and diversity of real-world cyber threats. For
instance, the OWASP juiceshop project [29] is the most widely adopted
benchmark for web vulnerability evaluation. However, it does not include
privilege escalation vulnerabilities, which is an essential aspect of
penetration testing. Second, existing benchmarks may not recognize the
cumulative value of progress through the different stages of penetration
testing, as they tend to evaluate only the final exploitation success. This
approach overlooks the nuanced value each step contributes to the overall
process, resulting in metrics that might not accurately represent actual
performance in real-world scenarios.
To address these concerns, we propose the construction of a comprehensive
penetration testing benchmark that meets the following criteria:
Task Variety. The benchmark must encompass diverse tasks, reflecting various
operating systems and emulating the diversity of scenarios encountered in
real-world penetration testing.
Challenge Levels. To ensure broad applicability, the benchmark must include
tasks of varying difficulty levels suitable for challenging novice and expert
testers.
Progress Tracking. Beyond mere success or failure metrics, the benchmark must
facilitate tracking of incremental progress, thereby recognizing and scoring
the value added at each stage of the penetration testing process.
### 3.2 Benchmark Design
Following the criteria outlined previously, we develop a comprehensive
benchmark that closely reflects real-world penetration testing tasks. The
design process progresses through several stages.
Task Selection. We begin by selecting tasks from HackTheBox [12] and VulnHub
[13], two leading penetration testing training platforms. Our selection
criteria are designed to ensure that our benchmark accurately reflects the
challenges encountered in practical penetration testing environments. We
meticulously review the latest machines available on both platforms, aiming to
identify and select a subset that comprehensively covers all vulnerabilities
listed in the OWASP [14] Top 10 Project. Additionally, we choose machines that
represent a mix of difficulties, classified according to traditional standards
in the penetration testing domain into easy, medium, and hard categories. This
process guarantees that our benchmark spans the full spectrum of
vulnerabilities and difficulties. Note that our benchmark does not include
benign targets to assess false positives. In penetration testing, benign
targets are sometimes explored. Our main objective remains identifying true
vulnerabilities.
Task Decomposition. We further parse the testing process of each target into a
series of sub-tasks, following the standard solution commonly referred to as
the “walkthrough” in penetration testing. Each sub-task corresponds to a
unique step in the overall process. We decompose sub-tasks following NIST
800-115 [30], the Technical Guide to Security Testing. Each sub-task is one
step declared in the Guide (e.g., network discovery, password cracking), or an
operation that exploits a unique vulnerability categorised in the Common
Weakness Enumeration (CWE) [15] (e.g., exploiting SQL injection - CWE-89
[31]). In the end, we formulate an exhaustive list of sub-tasks for every
benchmark target. We provide the complete list of the decomposed sub-tasks in
Appendix Table 7.
Benchmark Validation. The final stage of our benchmark development involves
rigorous validation, which ensures the reproducibility of these benchmark
machines. To do this, three certified penetration testers independently
attempt the penetration testing targets and write their walkthrough. We then
adjust our task decomposition accordingly because some targets may have
multiple valid solutions.
Ultimately, we have compiled a benchmark that effectively encompasses all
types of vulnerabilities listed in the OWASP [14] Top 10 Project. It comprises
13 penetration testing targets, each at varying levels of difficulty. These
targets are broken down into 182 sub-tasks across 26 categories, covering 18
distinct CWE items. This number of targets is deemed sufficient to represent a
broad spectrum of vulnerabilities, difficulty levels, and varieties essential
for comprehensive penetration testing training. Detailed information about the
included categories can be found in the Appendix Section 7. To foster
community development, we have made this benchmark publicly available online
at our anonymous project website [32].
## 4 Exploratory Study
We conduct an exploratory study to assess the capabilities of LLMs in
penetration testing, with the primary objective of determining how well LLMs
can adapt to the real-world complexities and challenges in this task.
Specifically, we aim to address the following two research questions:
RQ1 (Capability): To what extent can LLMs perform penetration testing tasks?
RQ2 (Comparative Analysis): How do the problem-solving strategies of human
penetration testers and LLMs differ?
We utilize the benchmark described in Section 3 to evaluate the performance of
LLMs on penetration testing tasks. In the following, we first delineate our
testing strategy for this study. Subsequently, we present the testing results
and an analytical discussion to address the above research questions.
### 4.1 Testing Strategy
LLMs are text-based and cannot independently perform penetration testing
operations. To address this, we develop a human-in-the-loop testing strategy,
serving as an intermediary method to accurately assess LLMs’ capabilities.
This strategy features an interactive loop where a human expert executes the
LLM’s penetration testing directives. Importantly, the human expert functions
purely as an executor, strictly following the LLM’s instructions without
adding any expert insights or making independent decisions.
Figure 2 decipits the testing strategy with the following steps: ❶ We initiate
the looped testing procedure by presenting the target specifics to the LLM,
seeking its guidance on potential penetration testing steps. ❷ The human
expert strictly follows the LLM’s recommendations and conducts the suggested
actions in the penetration testing environment. ❸ Outcomes of the testing
actions are collected and summarized: direct text outputs such as terminal
outputs or source code are documented; non-textual results, such as graphical
representations, are translated by the human expert into succinct textual
summaries. The data is then fed back to the LLM, setting the stage for its
subsequent recommendations. ❹ This iterative process persists either until a
conclusive solution is identified or an deadlock is reached. We then compile a
record of the testing procedures, encompassing successful sub-tasks,
ineffective actions, and any reasons for failure, if applicable. For a more
tangible grasp of this strategy, we offer illustrative examples of prompts and
corresponding outputs from GPT-4 related to one of our benchmark targets in
the Appendix Section A.
To ensure the evaluation’s fairness and accuracy, we employ several
strategies. First, we involve expert-level penetration testers333We selected
Offensive Security Certified Professionals (OSCP) testers. as the human
testers. With their deep pentesting knowledge, these testers can precisely
comprehend and execute LLM-generated operations, thus accurately assessing
LLMs’ true capabilities. Second, we instruct the penetration testers to
strictly execute the commands given by the LLMs, without altering any content
or information, even upon identifying clear errors. They are also instructed
to faithfully report the testing results back to the LLM without any
additional commentary. Third, for managing UI-based operations and graphical
results, we have adopted specific measures. Initially, we instruct the LLMs to
minimize the use of GUI-based tools. For indispensable tools that cannot be
avoided (e.g., BurpSuite), we propose a result-oriented approach: upon
receiving a GUI operation instruction, the testers first execute the operation
based on their expert knowledge. Subsequently, they are required to provide
detailed, step-by-step textual descriptions of their actions and the observed
responses at each step, which are then communicated back to the LLM. Should
the LLM express any objections or comments concerning a particular step, the
operation is to be repeated. This protocol ensures the integrity of the
feedback loop, guaranteeing that the LLM obtains a comprehensive understanding
of the testing results.
Figure 2: Overview of strategy to use LLMs for penetration testing. Table 1:
Overall performance of LLMs on Penetration Testing Benchmark.
| Easy | Medium | Hard | Average
---|---|---|---|---
Tools | Overall (7) | Sub-task (77) | Overall (4) | Sub-task (71) | Overall (2) | Sub-task (34) | Overall (13) | Sub-task (182)
GPT-3.5 | 1 (14.29%) | 24 (31.17%) | 0 (0.00%) | 13 (18.31%) | 0 (0.00%) | 5 (14.71%) | 1 (7.69%) | 42 (23.07%)
GPT-4 | 4 (57.14%) | 55 (71.43%) | 1 (25.00%) | 30 (42.25%) | 0 (0.00%) | 10 (29.41%) | 5 (38.46%) | 95 (52.20%)
Bard | 2 (28.57%) | 29 (37.66%) | 0 (0.00%) | 16 (22.54%) | 0 (0.00%) | 5 (14.71%) | 2 (15.38%) | 50 (27.47%)
Average | 2.3 (33.33%) | 36 (46.75%) | 0.33 (8.33%) | 19.7 (27.70%) | 0 (0.00%) | 6.7 (19.61%) | 2.7 (20.5%) | 62.3 (34.25%)
### 4.2 Evaluation Settings
We proceed to assess the performances of various LLMs in penetration testing
tasks using the strategy mentioned above.
Model Selection. Our study focuses on three cutting-edge LLMs that are
currently accessible: GPT-3.5 with 8k token limit, GPT-4 with 32k token limit
from OpenAI, and LaMDA [33] from Google. These models are selected based on
their prominence in the research community and consistent availability. To
interact with the LLMs mentioned above, we utilize chatbot services provided
by OpenAI and Google, namely ChatGPT [34] and Bard [18]. For this paper, the
terms GPT-3.5, GPT-4, and Bard will represent these three LLMs.
Experimental Setup. Our experiments occur in a local setting with both target
and testing machines on the same private network. The testing machine runs on
Kali Linux [35], version 2023.1.
Tool Usage. Our study aims to assess the innate capabilities of LLMs on
penetration testing, without reliance on end-to-end automated vulnerability
scanners such as Nexus [36] and OpenVAS [37]. Consequently, we explicitly
instruct the LLMs to refrain from using these tools. We follow the LLMs’
recommendations for utilizing other tools designed to validate specific
vulnerability types (e.g., sqlmap [38] for SQL injections). Occasionally,
versioning discrepancies may lead the LLMs to provide incorrect instructions
for tool usage. In such instances, our penetration testing experts evaluate
whether the instructions would have been valid for a previous version of the
tool. They then make any necessary adjustments to ensure the tool’s correct
operation.
### 4.3 Capability Evaluation (RQ1)
To address RQ1, we evaluate the performance of three leading LLMs: GPT-4,
Bard, and GPT-3.5. We summarize these findings in Table 1. Each LLM
successfully completes at least one end-to-end penetration test, highlighting
their versatility in simpler environments. Of these, GPT-4 excels, achieving
success on 4 easy and 1 medium difficulty targets. Bard and GPT-3.5 follow
with success on 2 and 1 easy targets, respectively. In sub-tasks, GPT-4
completes 55 out of 77 on easy targets and 30 out of 71 on medium. Bard and
GPT-3.5 also show potential, finishing 16 (22.54%) and 13 (18.31%) of medium
difficulty sub-tasks, respectively. However, on hard targets, all models’
performance declines. Though they can initiate the reconnaissance phase, they
struggle to exploit identified vulnerabilities. This is anticipated since hard
targets are designed to be especially challenging. They often feature
seemingly vulnerable services that are non-exploitable, known as rabbit holes
[39]. The pathways to exploit these machines are unique and unpredictable,
resisting automated tool replication. For example, the target Falafel has
specialized SQL injection vulnerabilities resistant to sqlmap. Current LLMs
cannot tackle these without human expert input.
Finding 1: Large Language Models (LLMs) have shown proficiency in conducting
end-to-end penetration testing tasks but struggle to overcome challenges
presented by more difficult targets.
Table 2: Top 10 Types of Sub-tasks completed by each tool.
Sub-Tasks | WT | GPT-3.5 | GPT-4 | Bard
---|---|---|---|---
Web Enumeration | 18 | 4 (22.2%) | 8 (44.4%) | 4 (22.2%)
Code Analysis | 18 | 4 (22.2%) | 5 (27.2%) | 4 (22.2%)
Port Scanning | 12 | 9 (75.0%) | 9 (75.0%) | 9 (75.0%)
Shell Construction | 11 | 3 (27.3%) | 8 (72.7%) | 4 (36.4%)
File Enumeration | 11 | 1 (9.1%) | 7 (63.6%) | 1 (9.1%)
Configuration Enumeration | 8 | 2 (25.0%) | 4 (50.0%) | 3 (37.5%)
Cryptanalysis | 8 | 2 (25.0%) | 3 (37.5%) | 1 (12.5%)
Network Enumeration | 7 | 1 (14.3%) | 3 (42.9%) | 2 (28.6%)
Command Injection | 6 | 1 (16.7%) | 4 (66.7%) | 2 (33.3%)
Known Exploits | 6 | 2 (33.3%) | 3 (50.0%) | 1 (16.7%)
We further examine the detailed sub-task completion performances of the three
LLMs compared to the walkthrough (WT), as presented in Table 2. Analyzing the
completion status, we identify several areas where LLMs excel. First, they
adeptly utilize common penetration testing tools to interpret the
corresponding outputs, especially in enumeration tasks correctly. For example,
all three evaluated LLMs successfully perform nine Port Scanning sub-tasks.
They can configure the widely-used port scanning tool, nmap [40], comprehend
the scan results, and formulate subsequent actions. Second, the LLMs reveal a
deep understanding of prevalent vulnerability types, connecting them to the
services on the target system. This understanding is evidenced by the
successful completion of sub-tasks related to various vulnerability types.
Finally, LLMs demonstrate their effectiveness in code analysis and generation,
particularly in the tasks of Code Analysis and Shell Construction. These tasks
require the models to read and generate codes in different programming
languages. This often culminates in identifying potential vulnerabilities from
code snippets and crafting the corresponding exploits. Notably, GPT-4
outperforms the other two models regarding code interpretation and generation,
marking it the most suitable candidate for penetration testing tasks.
Finding 2: LLMs can efficiently use penetration testing tools, identify common
vulnerabilities, and interpret source codes to identify vulnerabilities.
### 4.4 Comparative Analysis (RQ2)
Table 3: Top Unnecessary Operations Prompted by LLMs on the Benchmark Targets
Unnecessary Operations | GPT-3.5 | GPT-4 | Bard | Total
---|---|---|---|---
Brute-Force | 75 | 92 | 68 | 235
Exploit Known Vulnerabilities (CVEs) | 29 | 24 | 28 | 81
SQL Injection | 14 | 21 | 16 | 51
Command Injection | 18 | 7 | 12 | 37
To address RQ2, we examine the problem-solving strategies that LLMs employ,
contrasting them with human penetration testers. In each penetration testing
trial, we concentrate on two main aspects: (1) Identifying the unnecessary
operations that LLMs prompt, which are not conducive to successful penetration
testing, as compared to a standard walkthrough; and (2) Understanding the
specific factors that prevent LLMs from successfully executing penetration
tests.
We analyze the unnecessary operations prompted by LLMs by breaking down the
recorded testing procedures into sub-tasks. We employ the same method to
formulate benchmark sub-tasks, as Section 3 outlines. By comparing this to a
standard walkthrough, we identify the primary sub-task trials that fall
outside the standard walkthrough and are thus irrelevant to the penetration
testing process. The results are summarized in Table 3. We find that the most
prevalent unnecessary operation prompted by LLMs is brute force. For all
services requiring password authentication, LLMs typically advise brute-
forcing it. This is an ineffective strategy in penetration testing. We surmise
that many hacking incidents in enterprises involve password cracking and brute
force. LLMs learn these reports from accident reports and are consequently
considered viable solutions. Besides brute force, LLMs suggest that testers
engage in CVE studies, SQL injections, and command injections. These
recommendations are common, as real-world penetration testers often prioritize
these techniques, even though they may not always provide the exact solution.
Table 4: Top causes for failed penetration testing trials
Failure Reasons | GPT3.5 | GPT4 | Bard | Total
---|---|---|---|---
Session context lost | 25 | 18 | 31 | 74
False Command Generation | 23 | 12 | 20 | 55
Deadlock operations | 19 | 10 | 16 | 45
False Scanning Output Interpretation | 13 | 9 | 18 | 40
False Source Code Interpretation | 16 | 11 | 10 | 37
Cannot craft valid exploit | 11 | 15 | 8 | 34
To understand penetration testing trial failures, we categorize the reasons
for the 195 trials, as shown in Table 4. The primary failure cause is loss of
session context. This means models often lose awareness of previous test
outcomes, missing essential past results. This issue arises from LLMs’
challenge in handling conversation context. Each LLM has a fixed token window,
such as GPT-4 with a capacity of 8,000 tokens [41]. If critical information
for a complex task exceeds this limit, trimming it causes the loss of
important details. This is problematic in intricate tests where identifying
vulnerabilities across services and forming a cohesive exploit strategy is
vital. This design flaw impacts the model’s efficacy in dealing with layered,
detailed tasks.
Finding 3: LLMs struggle to maintain long-term memory, which is vital to link
vulnerabilities and develop exploitation strategies effectively.
Secondly, LLMs strongly prefer the most recent tasks, adhering rigorously to a
depth-first search approach. They tend to immerse deeply into resolving the
issues mentioned in the most recent conversation, seldom branching out to new
targets until the ongoing path is exhaustively explored. This behavior aligns
with the studies [42, 43] that LLMs primarily concentrate their attention at
the prompt’s beginning and end. In contrast, seasoned penetration testers
adopt a more holistic approach, strategically plotting moves that promise the
highest potential outcomes. When coupled with the aforementioned session
context loss, this proclivity drives LLMs to become excessively anchored to
one specific service. As the testing advances, the models often neglect prior
discoveries, leading to an impasse.
Finding 4: LLMs strongly prefer recent tasks and a depth-first search
approach, often resulting in an over-focus on one service and forgetting
previous findings.
Lastly, LLMs have inaccurate result generation and hallucination issues, as
noted in [44]. This phenomenon ranks as the second most frequent cause of
failures and is characterized by the generation of false commands. In our
study, we observe that LLMs frequently identify the appropriate tool for the
task but stumble in configuring the tools with the correct settings. In some
cases, they even concoct non-existent testing tools or tool modules.
Finding 5: LLMs may generate inaccurate operations or commands, often stemming
from inherent inaccuracies and hallucinations.
Our exploratory study on three LLMs in penetration testing highlights their
capability to complete sub-tasks. However, they face issues with long-term
memory retention, reliance on a depth-first strategy, and ensuring operation
accuracy. In the subsequent section, we detail our approach to mitigate these
challenges and describe the design of our LLM-based penetration testing tool.
## 5 Methodology
### 5.1 Overview
In light of the challenges identified in the preceding section, we present our
proposed solution, PentestGPT, which leverages the synergistic interplay of
three LLM-powered modules. As illustrated in Figure 3, PentestGPT incorporates
three core modules: the Reasoning Module, the Generation Module, and the
Parsing Module. Each module reserves one LLM session with its conversation and
context. The user interacts seamlessly with PentestGPT, where distinct modules
process different types of messages. This interaction culminates in a final
decision, suggesting the subsequent step of the penetration testing process
that the user should undertake. In the following sections, we elucidate our
design reasoning and provide a detailed breakdown of the engineering processes
behind PentestGPT.
Figure 3: Overview of PentestGPT.
### 5.2 Design Rationale
Our central design considerations emerged from the three challenges observed
in the previous Exploratory Study (Section 4): The first challenge (Finding 3)
pertains to the issue of penetration testing context loss due to memory
retention. LLMs in their original form struggle to maintain such long-term
memory due to token size limits. The second obstacle (Finding 4) arises from
the LLM chatbots’ tendency to emphasize recent conversation content. In
penetration testing tasks, this focuses on optimizing the immediate task. This
approach falls short in the complex, interconnected task environment of
penetration testing. The third obstacle (Finding 5) is tied to the inaccurate
results generation by LLMs. When tasked to produce specific operations for a
step in penetration testing directly, the outputs are often imprecise,
sometimes even leading to false directions.
PentestGPT has been engineered to address these challenges, rendering it more
apt for penetration testing tasks. We draw inspiration from the methodologies
employed by real-world penetration testing teams, where directors plan
overarching procedures, subdividing them into subtasks for individual testers.
Each tester independently performs their task, reporting results without an
exhaustive understanding of the broader context. The director then determines
the following steps, possibly redefining tasks, and triggers the subsequent
round of testing. Essentially, the director manages the overall strategy
without becoming entrenched in the minutiae of the tests. This approach is
mirrored in PentestGPT’s functionality, enhancing its efficiency and
adaptability in conducting penetration tests. Our strategy divides penetration
testing into two processes: identifying the next task and generating the
concrete operation to complete the task. Each process is powered by one LLM
session. In this setup, the LLM session responsible for task identification
retains the complete context of the ongoing penetration testing status. At the
same time, the generation of detailed operations and parsing of information is
managed by other sessions. This division of responsibilities fosters effective
task execution while preserving the overarching context.
To assist LLMs in effectively carrying out penetration testing tasks, we
design a series of prompts that align with user inputs. We utilize the Chain-
of-Thought (CoT) [45] methodology during this process. As CoT reveals, LLMs’
performance and reasoning capabilities can be significantly enhanced using the
input, chain-of-thought, output prompting format. Here, the chain-of-thought
represents a series of intermediate natural language reasoning steps leading
to the outcome. We dissect the penetration testing tasks into micro-steps and
design prompts with examples to guide LLMs through processing penetration
testing information step-by-step, ultimately leading to the desired outcomes.
The complete prompts are available at our anonymized open-source project[32].
### 5.3 Reasoning Module
The Reasoning Module plays a pivotal role in our system, analogous to a team
lead overseeing the penetration testing task from a macro perspective. It
obtains testing results or intentions from the user and prepares the testing
strategy for the next step. This testing strategy is passed to the generation
module for further planning.
To effectively supervise the penetration testing process and provide precise
guidance, it is crucial to translate the testing procedures and outcomes into
a natural language format. Drawing inspiration from the concept of an attack
tree [46], which is often used to outline penetration testing procedures, we
introduce the notion of a pentesting task tree (PTT). This novel approach to
testing status representation is rooted in the concept of an attributed tree
[47]:
###### Definition 1 (Attributed Tree)
A attributed tree is an edge-labeled, attributed polytree
$G=(V,E,\lambda,\mu)$ where $V$ is a set of nodes (or vertices), $E$ is a set
of directed edges, $\lambda:E\to\Sigma$ is an edge labeling function assigning
a label from the alphabet $\Sigma$ to each edge and $\mu:(V\cup E)\times K\to
S$ is a function assigning key(from K)-value(from S) pairs of properties to
the edges and nodes.
Given the definition of attributed tree, PTT is defined as follows:
###### Definition 2 (Pentesting Task Tree)
A PTT $T$ is a pair $(N,A)$, where: (1) $N$ is a set of nodes organized in a
tree structure. Each node has a unique identifier, and there is a special node
called the root that has no parent. Each node, other than the root, has
exactly one parent and zero or more children. (2) $A$ is a function that
assigns to each node $n\in N$ a set of attributes $A(n)$. Each attribute is a
pair $(a,v)$, where $a$ is the attribute name and $v$ is the attribute value.
The set of attributes can be different for each node.
Figure 4: Pentesting Task Tree in a) visualized tree format, and b) natural
language format encoded in LLM.
As outlined in Figure 3, the Reasoning Module’s operation unfolds over four
key steps operating over the PTT. ❶ The module begins by interpreting the
user’s objectives to create an initial PTT, formatted in natural language.
This involves instructing the LLM with designed prompts that contain the above
PTT definition and real-world examples. The outputs from the LLM are parsed to
ensure that the tree structure is correctly represented, which can be
formatted in natural language through layered bullets, as shown in Figure 4.
The Reasoning Module effectively overcomes the memory-loss issue by
maintaining a task tree that encompasses the entire penetration testing
process. ❷ After updating the tree information, a verification step is
conducted on the newly updated PTT to ascertain its correctness. This process
checks explicitly that only the leaf nodes of the PTT have been modified,
aligning with the principle that atomic operations in the penetration testing
process should only influence the status of the lowest-level sub-tasks. This
step confirms the correctness of the reasoning process, safeguarding against
any potential alterations to the overall tree structure due to hallucination
by the LLM. If discrepancies arise, the information is reverted to the LLM for
correction and regeneration. ❸ With the updated PTT, the Reasoning Module
evaluates the current tree state and pinpoints viable sub-tasks that can serve
as candidate steps for further testing. ❹ Finally, the module evaluates the
likelihood of these sub-tasks leading to successful penetration testing
outcomes. It then recommends the top task as the output. The expected results
of this task are subsequently forwarded to the Generation Module for an in-
depth analysis. This is feasible, as demonstrated in the exploratory study,
since LLMs, particularly GPT-4, can identify potential vulnerabilities when
provided with system status information. This procedural approach enables the
Reasoning Module to address one of the inherent limitations of LLMs, precisely
their tendency to concentrate solely on the most recent task. Note that in
cases where the tester identifies that the correct task is incorrect or not
completed in a preferred way, he could also manually revise the PTT through
the interactive handle further discussed in Section 5.6.
We devise four sets of prompts to sequentially guide the Reasoning Module
through the completion of each stage. To bolster the reproducibility of our
results, we optimize these prompts further with a technique known as hint
generation [48]. From our practical experience, we observe that LLMs are adept
at interpreting the tree-structured information pertinent to penetration
testing and can update it accurately in response to test outputs.
Figure 5: A demonstration of the task-tree update process on the testing
target HTB-Carrier
### 5.4 Generation Module
The Generation Module translates specific sub-tasks from the Reasoning Module
into concrete commands or instructions. Each time a new sub-task is received,
a fresh session is initiated in the Generation Module. This strategy
effectively isolates the context of the overarching penetration task from the
immediate task under execution, enabling the LLM to focus entirely on
generating specific commands.
Instead of directly transforming the received sub-task into specific
operations, our design employs the CoT strategy [45] to partition this process
into two sequential steps. This design decision directly addresses the
challenges associated with model inaccuracy and hallucination by enhancing the
model’s reasoning capability. In particular, ❺ upon the receipt of a concise
sub-task from the Reasoning Module, the Generation Module begins by expanding
it into a sequence of detailed steps. Notably, the prompt associated with this
sub-task requires the LLM to consider the possible tools and operations
available within the testing environment. ❻ Subsequently, the Generation
Module transforms each of these expanded steps into precise terminal commands
ready for execution or into detailed descriptions of specific Graphical User
Interface (GUI) operations to be carried out. This stage-by-stage translation
eliminates potential ambiguities, enabling testers to follow the instructions
directly and seamlessly. Implementing this two-step process effectively
precludes the LLM from generating operations that may not be feasible in real-
world scenarios, thereby improving the success rate of the penetration testing
procedure.
By acting as a bridge between the strategic insights provided by the Reasoning
Module and the actionable steps required for conducting a penetration test,
the Generation Module ensures that high-level plans are converted into precise
and actionable steps. This transformation process significantly bolsters the
overall efficiency of the penetration testing procedure, and also provides
human-readable outputs of the complete testing process. We present a detailed
PTT generation process for a complete penetration testing target in Appendix
Figure 9, accompanied by an illustrative example to aid understanding.
An Illustrative Example. We utilize a real-world running example to illuminate
how the Reasoning Module and the Generation Module collaboratively operate to
complete penetration testing tasks. Figure 5 illustrates a single iteration of
PentestGPT working on the HackTheBox machine Carrier [49], a medium-difficulty
target. As depicted in a-1), the PTT, in natural language format, encodes the
testing status, revealing the open ports (21, 22, 80) with running services.
The Reasoning Module is subsequently instructed to identify the available
tasks. As highlighted in red, service scanning is the only available task on
the leaf node of the PTT. This task is therefore chosen and forwarded to the
Generation Module for command generation. The generated command is executed in
the testing environment, and the execution result is conveyed to the Reasoning
Module to update the PTT. In a-2), the Reasoning Module integrates the
previous scanning result into the PTT, cross-referencing it with the earlier
PTT to update only the leaf nodes. It then looks for the available tasks to
execute. In this case, two tasks emerge: scanning the web service on port 80
and checking the SSH service for known vulnerabilities. The LLM evaluates
which task is more promising and chooses to investigate the web service, often
seen as more vulnerable. This task is passed to the Generation Module. The
Generation Module turns this general task into a detailed process, employing
nikto [50], a commonly used web scanning script. The iterative process
continues until the tester completes the penetration testing task.
### 5.5 Parsing Module
The Parsing Module operates as a supportive interface, enabling effective
processing of the natural language information exchanged between the user and
the other two core modules. Two needs can primarily justify the existence of
this module. First, security testing tool outputs are typically verbose, laden
with extraneous details, making it computationally expensive and unnecessarily
redundant to feed these extended outputs directly into the LLMs. Second, users
without specialized knowledge in the security domain may struggle to extract
key insights from security testing outputs, presenting challenges in
summarizing crucial testing information. Consequently, the Parsing Module is
essential in streamlining and condensing this information.
In PentestGPT, the Parsing Module is devised to handle four distinct types of
information: (1) user intentions, which are directives provided by the user to
dictate the next course of action, (2) security testing tool outputs, which
represent the raw outputs generated by an array of security testing tools, (3)
raw HTTP web information, which encompasses all raw information derived from
HTTP web interfaces, and (4) source codes extracted during the penetration
testing process. Users must specify the category of the information they
provide, and each category is paired with a set of carefully designed prompts.
For source code analysis, we integrate the GPT-4 code interpreter [51] to
execute the task.
### 5.6 Active Feedback
While LLMs can produce insightful outputs, their outcomes sometimes require
revisions. To facilitate this, we introduce an interactive handle in
PentestGPT, known as active feedback, which allows the user to interact
directly with the Reasoning Module. A vital feature of this process is that it
does not alter the context within the Reasoning Module unless the user
explicitly desires to update some information. The reasoning context,
including the PTT, is stored as a fixed chunk of tokens. This chunk of tokens
is provided to a new LLM session during an active feedback interaction, and
users can pose questions regarding them. This ensures that the original
session remains unaffected, and users can always query the reasoning context
without making unnecessary changes. If the user believes it necessary to
update the PTT, they can explicitly instruct the model to update the reasoning
context history accordingly. This provides a robust and flexible framework for
the user to participate in the decision-making process actively.
### 5.7 Discussion
We explore various design alternatives for PentestGPT to tackle the challenges
identified in Exploratory Study. We have experimented with different designs,
and here we discuss some key decisions.
Addressing Context Loss with Token Size: a straightforward solution to
alleviate context loss is the employment of LLM models with an extended token
size. For instance, GPT-4 provides versions with 8k and 32k token size limits.
This approach, however, confronts two substantial challenges. First, even a
32k token size might be inadequate for penetration testing scenarios, as the
output of a single testing tool like dirbuster [52] may comprise thousands of
tokens. Consequently, GPT-4 with a 32k limit cannot retain the entire testing
context. Second, even when the entire conversation history fits within the 32k
token boundary, the API may still skew towards recent content, focusing on
local tasks and overlooking broader context. These issues guided us in
formulating the design for the Reasoning Module and the Parsing Module.
Vector Database to Improve Context Length: Another technique to enhance the
context length of LLMs involves a vector database [53, 54]. By transmuting
data into vector embeddings, LLMs can efficiently store and retrieve
information, practically creating long-term memory. Theoretically, penetration
testing tool outputs could be archived in the vector database. In practice,
though, we observe that many results closely resemble and vary in only nuanced
ways. This similarity often leads to confused information retrieval. Solely
relying on a vector database fails to overcome context loss in penetration
testing tasks. Integrating the vector database into the design of PentestGPT
is an avenue for future research.
Precision in Information Extraction: Precise information extraction is crucial
for conserving token usage and avoiding verbosity in LLMs [55, 56]. Rule-based
methods are commonly employed to extract diverse information. However, rule-
based techniques are engineeringly expensive given natural language’s inherent
complexity and the variety of information types in penetration testing. We
devise the Parsing Module to manage several general input information types, a
strategy found to be both feasible and efficient.
Limitations of LLMs: LLMs are not an all-encompassing solution. Present LLMs
exhibit flaws, including hallucination [57, 58] and outdated knowledge. Our
mitigation efforts, such as implementing task tree verification to ward off
hallucination, might not completely prevent the Reasoning Module from
producing erroneous outcomes. Thus, a human-in-the-loop strategy becomes
vital, facilitating the input of necessary expertise and guidance to steer
LLMs effectively.
## 6 Evaluation
In this section, we assess the performance of PentestGPT, focusing on the
following four research questions:
RQ3 (Performance): How does the performance of PentestGPT compare with that of
native LLM models and human experts?
RQ4 (Strategy): Does PentestGPT employ different problem-solving strategies
compared to those utilized by LLMs or human experts?
RQ5 (Ablation): How does each module within PentestGPT contribute to the
overall penetration testing performance?
RQ6 (Practicality): Is PentestGPT practical and effective in real-world
penetration testing tasks?
### 6.1 Evaluation Settings
We implement PentestGPT with 1,900 lines of Python3 code and 740 lines of
prompts, available at our anonymized project website [32]. We evaluate its
performance over the benchmark constructed in Section 3, and additional real-
world penetration testing machines (Section 6.5). In this evaluation, we
integrate PentestGPT with GPT-3.5 and GPT-4 to form two working versions:
PentestGPT-GPT-3.5 and PentestGPT-GPT-4. Due to the lack of API access, we do
not select other LLM models, such as Bard. In line with our previous
experiments, we use the same experiment environment setting and instruct
PentestGPT to only use the non-automated penetration testing tools.
### 6.2 Performance Evaluation (RQ3)
The overall task completion status of PentestGPT-GPT-3.5, PentestGPT-GPT-4,
and the naive usage of LLMs is illustrated in Figure 6(a). As the Figure
shows, our solutions powered by LLMs demonstrate superior penetration testing
capabilities compared to the naive application of LLMs. Specifically,
PentestGPT-GPT-4 surpasses the other three solutions, successfully solving 6
out of 7 easy difficulty targets and 2 out of 4 medium difficulty targets.
This performance indicates that PentestGPT-GPT-4 can handle penetration
testing targets ranging from easy to medium difficulty levels. Meanwhile,
PentestGPT-GPT-3.5 manages to solve only two challenges of easy difficulty, a
discrepancy that can be attributed to GPT-3.5 lacking the knowledge related to
penetration testing found in GPT-4.
The sub-task completion status of PentestGPT-GPT-3.5, PentestGPT-GPT-4, and
the naive usage of LLM is shown in Figure 6(b). As the Figure illustrates,
both PentestGPT-GPT-3.5 and PentestGPT-GPT-4 perform better than the standard
utilization of LLMs. It is noteworthy that PentestGPT-GPT-4 not only solves
one more medium difficulty target compared to naive GPT-4 but also
accomplishes 111% more sub-tasks (57 vs. 27). This highlights that our design
effectively addresses context loss challenges and leads to more promising
testing results. Nevertheless, all the solutions struggle with hard difficulty
testing targets. As elaborated in Section 4, hard difficulty targets typically
demand a deep understanding from the penetration tester. To reach testing
objectives, they may require modifications to existing penetration testing
tools or scripts. Our design does not expand the LLMs’ knowledge of
vulnerabilities, so it does not notably enhance performance on these more
complex targets.
(a) Overall completion status.
(b) Subtask completion status.
Figure 6: The performance of GPT-3.5, GPT-4, PentestGPT-GPT-3.5, and
PentestGPT-GPT-4 on overall target completion and sub-task completion.
### 6.3 Strategy Evaluation (RQ4)
We analyze PentestGPT’s problem-solving methods, comparing them with LLMs and
human experts. Through manual examination, we identify PentestGPT’s approach
to penetration testing. Notably, PentestGPT breaks down tasks similarly to
human experts and prioritizes effectively. Rather than just addressing the
latest identified task, PentestGPT identifies key sub-tasks that can result in
success.
Figure 7 contrasts the strategies of GPT-4 and PentestGPT on the VulnHub
machine, Hackable II [59]. This machine features two vulnerabilities: an FTP
service for file uploads and a web service to view FTP files. A valid exploit
requires both services. The figure shows GPT-4 starting with the FTP service
and identifying the upload vulnerability (❶-❸). Yet, it does not link this to
the web service, causing an incomplete exploit. In contrast, PentestGPT shifts
between the FTP and web services. It first explores both services (❶-❷), then
focuses on the FTP (❸-❹), realizing the FTP and web files are identical. With
this insight, PentestGPT instructs the tester to upload a shell (❺), achieving
a successful reverse shell (❻). This matches the solution guide and
underscores PentestGPT’s adeptness at integrating various testing aspects.
Figure 7: Penetration testing strategy comparison between GPT-3.5 and
PentestGPT on VulnHub-Hackable II.
Our second observation is that although PentestGPT behaves more similarly to
human experts, it still exhibits some strategies that humans will not apply.
For instance, PentestGPT still prioritizes brute-force attacks before
vulnerability scanning. This is obvious in cases where PentestGPT always tries
to brute-force the SSH service on target machines.
We analyze cases where penetration testing with PentestGPT failed, identifying
three primary limitations. First, PentestGPT struggles with image
interpretation. LLMs are unable to process images, which are crucial in
certain penetration testing scenarios. Addressing this limitation may require
the development of advanced multimodal models that can interpret both text and
visual data. Second, PentestGPT lacks the ability to employ certain social
engineering techniques and to detect subtle cues. For example, while a human
tester might generate a brute-force wordlist from information extracted from a
target service, PentestGPT can retrieve names from a web service but fails to
guide the usage of tools needed to create a wordlist from these names. Third,
the models struggle with accurate exploitation code construction within a
limited number of trials. Despite some proficiency in code comprehension and
generation, the LLM falls short in producing detailed exploitation scripts,
particularly with low-level bytecode operations. These limitations underline
the necessity for improvement in areas where human insight and intricate
reasoning are still more proficient than automated solutions.
### 6.4 Ablation Study (RQ5)
We perform an ablation study on how the three modules: Reasoning Module,
Generation Module, and Parsing Module, contribute to the performance of
PentestGPT. We implement three variants:
1. 1.
PentestGPT-no-Parsing: the Parsing Module is deactivated, causing all data to
be directly fed into the system.
2. 2.
PentestGPT-no-Generation: the Generation Module is deactivated, leading to the
completion of task generation within the Reasoning Module itself. The prompts
for task generation remain consistent.
3. 3.
PentestGPT-no-Reasoning: the Reasoning Module is disabled. Instead of PTT,
this variant adopts the same methodology utilized with LLMs for penetration
testing, as delineated in the Exploratory Study.
All the variants are integrated with GPT-4 API for testing.
(a) Overall completion status
(b) Sub-task completion status
Figure 8: The performance of PentestGPT, PentestGPT-No-Annotation, PentestGPT-
Operation-Only, and PentestGPT-Parameter-Only on both normalized average code
coverage ($\mu LOC$) and bug detection.
Figure 8 presents the outcomes of three tested variants on our benchmarks.
Among these, PentestGPT consistently outperforms the ablation baselines in
both target and sub-task completion. Our primary observations include: (1)
Without its Parsing Module, PentestGPT-no-Parsing sees only a slight drop in
performance for task and sub-task completion. Though parsing aids in
penetration testing, the 32k token limit generally covers diverse outputs. The
Reasoning Module’s design, which retains the full testing context, compensates
for the absence of the Parsing Module, ensuring minimal performance reduction.
(2) PentestGPT-no-Reasoning has the lowest success, achieving just 53.6% of
the sub-tasks of the full variant. This is even lower than the basic GPT-4
setup. The Generation Module’s added sub-tasks distort the LLM context. The
mismatched prompts and extended generation output cloud the original context,
causing the test’s failure. (3) PentestGPT-no-Generation slightly surpasses
the basic GPT-4. Without the Generation Module, the process mirrors standard
LLM usage. The module’s main role is guiding precise testing operations.
Without it, testers might require additional information to use essential
tools or scripts.
### 6.5 Practicality Study (RQ6)
Table 5: PentestGPT performance over the active HackTheBox Challenges.
Machine | Difficulty | Completions | Completed Users | Cost (USD)
---|---|---|---|---
Sau | Easy | 5/5 (✓) | 4798 | 15.2
Pilgramage | Easy | 3/5 (✓) | 5474 | 12.6
Topology | Easy | 0/5 (✗) | 4500 | 8.3
PC | Easy | 4/5 (✓) | 6061 | 16.1
MonitorsTwo | Easy | 3/5 (✓) | 8684 | 9.2
Authority | Medium | 0/5 (✗) | 1209 | 11.5
Sandworm | Medium | 0/5 (✗) | 2106 | 10.2
Jupiter | Medium | 0/5 (✗) | 1494 | 6.6
Agile | Medium | 2/5 (✓) | 4395 | 22.5
OnlyForYou | Medium | 0/5 (✗) | 2296 | 19.3
Total | - | 17/50 (6) | - | 131.5
Table 6: PentestGPT performance over picoMini CTF.
Challenge | Category | Score | Completions
---|---|---|---
login | web | 100 | 5/5 (✓)
advance-potion-making | forensics | 100 | 3/5 (✓)
spelling-quiz | crypto | 100 | 4/5 (✓)
caas | web | 150 | 2/5 (✓)
XtrOrdinary | crypto | 150 | 5/5 (✓)
tripplesecure | crypto | 150 | 3/5 (✓)
clutteroverflow | binary | 150 | 1/5 (✓)
not crypto | reverse | 150 | 0/5 (✗)
scrambled-bytes | forensics | 200 | 0/5 (✗)
breadth | reverse | 200 | 0/5 (✗)
notepad | web | 250 | 1/5 (✓)
college-rowing-team | crypto | 250 | 2/5 (✓)
fermat-strings | binary | 250 | 0/5 (✗)
corrupt-key-1 | crypto | 350 | 0/5 (✗)
SaaS | binary | 350 | 0/5 (✗)
riscy business | reverse | 350 | 0/5 (✗)
homework | binary | 400 | 0/5 (✗)
lockdown-horses | binary | 450 | 0/5 (✗)
corrupt-key-2 | crypto | 500 | 0/5 (✗)
vr-school | binary | 500 | 0/5 (✗)
MATRIX | reverse | 500 | 0/5 (✗)
We demonstrate PentestGPT’s applicability in real-world penetration testing
scenarios, extending beyond standardized benchmarks. For this analysis, we
deploy PentestGPT in two distinct challenge formats: (1) HackTheBox (HTB)
active machine challenges, which present a series of real-world penetration
testing scenarios accessible to a global audience. We selected 10 machines
from the active list, comprising five targets of easy difficulty and five of
intermediate difficulty. (2) picoMini [21], a jeopardy-style Capture The Flag
(CTF) competition organized by Carnegie Mellon University and redpwn [60]. The
competition featured 21 unique CTF challenges and drew participation from 248
teams in its initial round. These challenges are now freely accessible online
for practice and reattempts. Our evaluation employed PentestGPT in conjunction
with the GPT-4 32k token length API, defining the capture of the root flag as
the metric for a successful trial. We conduct five trials on each target and
documented the number of successful captures. Note that we consider single
successful capture out of five trials as successful attempt over the target.
This criterion reflects the iterative nature of real-world penetration testing
and CTF challenges, where multiple attempts are allowed, and success is
ultimately determined by achieving the objective at least once.
Tables 5 presents PentestGPT’s performance across both sets of challenges. In
the HackTheBox challenges, PentestGPT successfully completed four easy and one
medium difficulty challenges, incurring a total cost of 131.5 USD—an average
of 21.9 USD per target. This performance indicates PentestGPT’s effectiveness
in tackling easy to intermediate-level penetration tests at a reasonable cost.
Table 6 demonstrates the performance of PentestGPT in the picoMini CTF. In
particular, PentestGPT managed to solve 9 out of 21 challenges, with the
average cost per attempt being 5.1 USD. Ultimately, PentestGPT accumulated a
total of 1400 points444Each challenge’s points were assigned based on its
difficulty level and ranked 24th out of 248 teams with valid submissions [61].
These outcomes suggest a promising performance of PentestGPT on real-world
penetration testing tasks among various types of challenges.
## 7 Discussion
It is possible that LLMs used by PentestGPT were trained on walkthroughs of
the benchmark machines, which could invalidate evaluation results. To counter
this, we employ two methods. First, We ensure the LLM lacks prior knowledge of
the target machine. We ascertain this by querying LLMs about the tested
machine’s familiarity. Secondly, our benchmark comprises machines launched
post-2021, ensuring they are beyond OpenAI models’ training data. Our study on
recent HackTheBox challenges confirms PentestGPT’s ability to solve without
pre-existing target knowledge.
While we aim for universally applicable prompts, certain LLMs avoid producing
specific hacking content. For instance, OpenAI has implement model alignments
[62] to ensure the GPT model outputs do not violate usage policies, including
generating malicious exploitation contents. We incorporate jailbreak
techniques [63, 64, 65, 66, 67, 68, 69] to coax LLMs into producing relevant
data. Improving reproducibility of PentestGPT remains a focus area.
LLMs occasionally "hallucinate" [57], producing outputs deviating from
training data. This impacts our tool’s dependability. To combat this, we’re
researching methods [70] to minimize hallucination, anticipating this will
boost our tool’s efficiency and reliability.
The ethical implications of employing PentestGPT in penetration testing are
significant and warrant careful consideration. While PentestGPT can greatly
enhance security by identifying vulnerabilities, its capabilities also pose
potential risks of misuse. To mitigate these risks, we have implemented
several strategies. We actively promote ethical guidelines for the use of
PentestGPT and collaborate closely with cybersecurity communities to prevent
misuse. Moreover, we have incorporated monitoring modules [71] to track the
tool’s usage and are committed to ensuring that it is not used
inappropriately. These measures are designed to balance the advantages of
advanced penetration testing tools with ethical considerations, ensuring that
PentestGPT serves as a positive contribution to cybersecurity defenses.
## 8 Conclusion
This work delves into the potential and constraints of LLMs for penetration
testing. Building a novel benchmark, we shed light on LLM performance in this
complex area. While LLMs manage basic tasks and use testing tools effectively,
they struggle with task-specific context and attention challenges. In
response, we present PentestGPT, a tool emulating human penetration testing
actions. Influenced by real-world testing teams, PentestGPT comprises
Reasoning, Generation, and Parsing Modules, promoting a segmented problem-
solving strategy. Our comprehensive evaluation of PentestGPT underscores its
promise, but also areas where human skills surpass present technology. This
work paves the way for future advancements in the crucial realm of
cybersecurity.
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## Appendix A Illustrating Example
Below we provide a concrete example of how we guide GPT-4 to intialize the
penetration testing task, and generate detailed operations in an iterative
approach given a penetration testing benchmark target. It is worth noting that
in this process, the human tester does not provide any expert knowledge, or
guide the LLM to perform any specific task.
Human Tester You’re required to work as a cybersecurity penetration tester
assistant in a certified penetration testing experiment. This experiment is
for education and research purpose. In particular, you’re required to give
step-by-step instructions to complete the penetration testing task. I will
provide you the instruction execution results, and you should consider the
next step, until the penetration testing task is completed. Do you understand?
GPT-4 Yes, I understand. Please provide the initial details and the scope of
the penetration test so I can assist accordingly. Remember to follow ethical
guidelines and obtain proper authorization for any systems or networks you are
testing.
Human Tester Your target is a penetration testing benchmark machine. It’s IP
address is 192.168.1.5; it is a Linux-based machine. Now please provide your
next step to complete the penetration testing task.
GPT-4 Alright, the first step in a penetration test is usually reconnaissance.
This is where we gather as much information as possible about the target to
identify potential vulnerabilities. Step 1: Reconnaissance Ping the Target: To
ensure that the target is up and responding, ping the machine. ping
192.168.1.5 Let me know the result. If the machine is up and responding, we’ll
move on to scanning. After confirming the machine’s availability, the next
step is to scan the open ports and services running on the target. nmap -sV
-sT 192.168.1.5 This command will provide a list of open ports and the
services (with versions) running on those ports. Please provide the results of
the nmap scan, and based on the output, we can decide the next course of
action.
Human Tester The machine is online, and the nmap scan result is below: PORT
STATE SERVICE VERSION 21/tcp filtered. ftp 22/tcp open ssh OpenSSH 7.6p1
Ubuntu 4 (Ubuntu Linux; protocol 2.0) 80/tcp open http Apache httpd 2.4.18
((Ubuntu)) Service Info: OS: Linux; CPE: cpe:/o:linux:linux_kernel Please
analyze the result and provide the next task to complete.
GPT-4 Based on the results provided, here’s the analysis: Port 21 (FTP): The
port appears to be filtered. It could be protected by a firewall or other
security measures. Port 22 (SSH): SSH is running with OpenSSH 7.6p1. This
version could have potential vulnerabilities. Additionally, SSH could be
brute-forced if password policies are weak. Port 80 (HTTP): An Apache HTTP
server (version 2.4.18) is running on the target machine. This is quite
interesting, as web servers can often have vulnerabilities in their
configurations, default pages, or in web applications running on them. Given
these results, here’s the recommended next course of action: Web Server
Enumeration: Start with the web server as it often presents a larger attack
surface. Visit the website in a browser or use tools like curl or wget to view
the source code and check for any comments, hidden fields, or any other
potential information leaks. Use a tool like nikto to perform a deeper scan of
the web server: nikto -h http://192.168.1.5
This will check for potential misconfigurations, default files, or any known
vulnerabilities in the web server. Also, use a tool like dirb or dirbuster to
perform directory brute-forcing, which might reveal hidden directories or
files. dirb http://192.168.1.5
Table 7: Summarized 26 types of sub-tasks in the proposed penetration testing
benchmark.
Phase | Technique | Description | Related CWEs
---|---|---|---
Reconnaissance | Port Scanning | Identify the open ports and related information on the target machine. | CWE-668
Web Enumeration | Gather detailed information about the target’s web applications.
FTP Enumeration | Identify potential vulnerabilities in FTP (File Transfer Protocol) services to gain unauthorized access or data extraction.
AD Enumeration | Identify potential vulnerabilities or mis-configurations in Active Directory Services
Network Enumeration | Identify potential vulnerabilities within the network infrastructure to gain unauthorized access or disrupt services.
Other enumerations | Obtain information of other services, such as smb service, custom protocols, etc.
Exploitation | Command Injection | Inject arbitrary commands to be run on a host machine, often leading to unauthorized system control. | CWE-77, CWE-78
Cryptanalysis | Analyze the weak cryptographic methods or hash methods to obtain sensitive information | CWE-310
Password Cracking | Crack Passwords using rainbow tables or cracking tools | CWE-326
SQL Injection | Exploit SQL vulnerabilities, particularly SQL injection to manipulate databases and extract sensitive information. | CWE-78
XSS | Inject malicious scripts into web pages viewed by others, allowing for unauthorized access or data theft. | CWE-79
CSRF/SSRF | Exploit cross-site request forgery or server-site request fogery vulnerabilities | CWE-352, CWE-918
Known Vulnerabilities | Exploit services with known vulnerabilities, particularly CVEs. | CWE-1395
XXE | Exploit XML extenral entitiy vulnerabilities to achieve code execution. | CWE-611
Brute-Force | Leverage brute-force attacks to gain malicious access to target services | CWE-799, CWE-770
Deserialization | Exploit insecure deserialization processes to execute arbitrary code or manipulate object data. | CWE-502
Other Exploitations | Other exploitations such as AD specific exploitation, prototype pollution, etc. |
Privilege Escalation | File Analysis | Enumerate system/service files to gain malicious information for privilege escalation | CWE-200, CWE-538
System Configuration Analysis | Enumerate system/service configurations to gain malicious information for privilege escalation | CWE-15, CWE-16
Cronjob Analysis | Analyze and manipulate scheduled tasks (cron jobs) to execute unauthorized commands or disrupt normal operations. | CWE-250
User Access Exploitation | Exploit the improper settings of user access in combination with system properties to conduct privilege escalation | CWE-284
Other techniques | Other general techniques, such as exploiting running processes with known vulnerabilities |
General Techniques | Code Analysis | Analyze source codes for potential vulnerabilities |
Shell Construction | Craft and utilize shell codes to manipulate the target system, often enabling control or extraction of data. |
Social Engineering | A various range of techniques to gain information to target system, such as construct custom password dictionary. |
Others | Other techniques |
## Appendix B PTT Generation Process
To demonstrate the PTT Generation Process in its entirety, we deploy
PentestGPT on the benchmark system Hackable II. Figure 9 illustrates the
complete PTT. In the figure, solid boxes depict the penetration testing
operations generated by PentestGPT, whereas dotted boxes outline the findings
derived from these operations. Red boxes indicate operations that do not yield
significant findings, green boxes denote operations that lead to useful
findings, and blue boxes represent operations generated by PentestGPT but not
executed due to lower priority. For clearer presentation, we label the
operations with numbers based on the operation sequences as prioritized by
PentestGPT.
As depicted in Figure 9, PentestGPT emulates the strategic approach typically
employed by human penetration testers, encompassing four steps including
enumeration, web user access via reverse shell, and privilege escalation to
both normal user and root levels on this particular benchmark machine.
Notably, PentestGPT demonstrates human-like reasoning by linking findings
across different stages. During the Web User Access phase, it connects a
vulnerability in the FTP service with earlier findings from the web service to
facilitate an attack by uploading and triggering a reverse shell via FTP.
Similarly, in the Privilege Escalation to Normal User phase, PentestGPT
identifies a user named "shrek" on the system, which it then exploits to crack
the password and escalate privileges. These instances illustrate PentestGPT’s
capability to integrate and leverage disparate pieces of information,
mirroring the cognitive processes of human testers.
Figure 9: A complete PTT example on the testing target Vulnhub-Hackable II
|
it may have been more difficult to form multiple giant planets nearby, because
the inner one may have migrated inward and away from outer-growing
protoplanets. Ida et al. (2013) did not take account of secular perturbations
between giant planets in their Monte Carlo method for planet-planet scattering
either, which can also trigger orbital instabilities and is of course
automatically included in our N-body simulations. Furthermore, Ida et al.
(2013) explored a relatively narrow range of metallicities of
${\rm[Fe/H]}=[-0.2,\,0.2]$ as opposed to our ${\rm[Fe/H]}=[-0.5,\,0.5]$. Since
the number of giant planets per system is more sensitive to disc masses for
lower metallicities (e.g. see cases with ${\rm[Fe/H]}\lesssim 0.0$ in Figure
7), they may have generated a lower number of giant planets per system on the
average. Although, this effect may have been mitigated to an extent since they
covered a wide range of disc masses (spanning two orders of magnitude), and
the low metallicities can be compensated for by more massive discs. Also, it
is possible that the eccentricity damping effect was too strong in their
simulations, which prevented the occurrence of dynamical instabilities (see
next sub-section as well as Bitsch et al., 2020).
### 4.7 Comparison with Bitsch et al. (2019)
Here, we briefly compare our work with a similar N-body work on giant planet
formation by Bitsch et al. (2019). A direct comparison is not possible since
the details of their models are different from ours (e.g. equation of motion,
pebble and gas accretion models, pebble isolation mass). However, we list
differences in disc parameters below and compare their results with ours for
similar disc parameters. We find that there are both similarities and
differences.
Their gas accretion rate follows Hartmann et al. (1998) and Bitsch et al.
(2015), and it is the same as the formula adopted by Matsumura et al. (2017):
$\log\left(\frac{\dot{M}_{*,\,B19}}{M_{\odot}\,{\rm
yr}^{-1}}\right)=-8-\frac{7}{5}\log\left(\frac{t+10^{5}\,{\rm
yr}}{10^{6}\,{\rm yr}}\right).$ (52)
Although we did not use the formula for this paper, their accretion rate very
closely follows Disc 2 of our model (which corresponds to
$M_{d}=0.06\,M_{\odot}$ with $t_{{\rm diff}}=0.1\,$Myr and thus $\alpha_{{\rm
acc}}\sim 7.4\times 10^{-2}$). Since they evolve the disc from 2 Myr to 5 Myr,
their mass accretion changes from $\dot{M}_{*}\sim 3.8\times
10^{-9}\,M_{\odot}$ to $\sim 1\times 10^{-9}\,M_{\odot}$. They also adopted
$\alpha_{{\rm acc}}=5.4\times 10^{-3}$ for disc accretion and $\alpha_{{\rm
turb}}=5.4\times 10^{-4}$ and $10^{-4}$ for disc turbulence. Although the mass
evolution is similar to our Disc 2, their disc could be close to our Disc 6
(i.e. $M_{d}=0.02\,M_{\odot}$ with $t_{{\rm diff}}=1\,$Myr and thus
$\alpha_{{\rm acc}}\sim 7.4\times 10^{-3}$) since they started with an evolved
disc and assumed $\alpha_{{\rm acc}}=5.4\times 10^{-3}$. In our simulations,
both produce similar types of planets (see Figure 9 and a discussion below).
On the other hand, their pebble mass flux is
$\dot{M}_{F,\,B19}\propto Z^{7/3}\alpha^{-1}\dot{M}_{*,\,B19},$ (53)
where $Z=1\%$ is the metallicity. Although they assumed the solar metallicity
for their simulations, they scaled the pebble mass flux by the factor of
$S_{{\rm peb}}=1-10$, which corresponds to exploring the metallicities of
${\rm[Fe/H]}=0.0$ to $0.43$. Since our pebble mass flux is $\dot{M}_{F}\propto
Z^{2}\alpha^{-1}\dot{M}_{*}$, the dependence on each parameter is similar.
One of the conclusions by Bitsch et al. (2019) was that formation of CJs was
possible for cores originating from 10 au with $\alpha_{{\rm turb}}=5.4\times
10^{-4}$ and those from 5 au with $\alpha_{{\rm turb}}=10^{-4}$. From their
Figures $4-6$, the formation of giant planets up to $\sim 300\,M_{\oplus}$ was
possible for both $\alpha_{{\rm turb}}$ cases above $S_{{\rm peb}}=2.5$ (which
corresponds to ${\rm[Fe/H]}\gtrsim 0.17$), while no giant planets formed for
the solar metallicity case. Our simulations show similar trends. In Disc 6
with $\alpha_{{\rm turb}}=10^{-4}$, giant planets primarily become low-mass
CJs with $\sim 100-400\,M_{\oplus}$ for metallicities ${\rm[Fe/H]}=0.3$ and
$0.5,$ while no giant planets form for ${\rm[Fe/H]}\leq 0.0$ (see Figure 9).
In Disc 2, the outcomes are similar except that planetary masses are lower
$\sim 100-300\,M_{\oplus,}$ and very low-mass giant planets can be formed with
${\rm[Fe/H]}=0.0$ (but not for lower metallicities). Out of formed giant
planets, those starting from $3-5\,$au typically become the closest-in CJs
with orbital radius beyond $\sim 1\,$au. For higher (lower) $\alpha_{{\rm
turb}}$, planets tend to migrate further (less) (see Section 3.2.1 and Figure
12) as indicated by Bitsch et al. (2019).
One major difference between Bitsch et al. (2019) and our work is that their
simulations are left with a number of giant planets with low eccentricities,
while we have successfully reproduced the eccentricity distribution of giant
planets (see Figure 5). The result may be surprising at a glance because the
number of giant planets that formed and survived in simulations by Bitsch et
al. (2019) is higher than in our simulations, and typically $\sim 5$ or more
from their Figures 6 and 7. We note that the higher number of surviving
planets is not likely due to the difference in the initial number of cores (60
for their simulations as opposed to 10 for ours). Jurić & Tremaine (2008)
showed that, even when there were 50 giant planets, the final number of
planets would be 2-3, as long as the dynamical instabilities occur. The
eccentricity distribution from such simulations with an initially high number
of giant planets agrees well with observations, and also with dynamical
instability simulations with lower number of giant planets (e.g. Jurić &
Tremaine, 2008; Chatterjee et al., 2008). Thus, the large number of surviving
giant planets in Bitsch et al. (2019) indicates that the dynamical instability
among giant planets was rare in their simulations. Indeed, even their long-
term evolution of 100 Myr did not lead to a dramatic increase in orbital
eccentricities (Bitsch et al., 2019).
While this manuscript was under revision, we became aware of the work by
Bitsch et al. (2020),in which they studied the effects of eccentricity and
inclination damping efficiencies on the eccentricity distribution of giant
planets. They parameterised the eccentricity and inclination damping
timescales as $\tau_{a}=K\,\tau_{e}=K\,\tau_{i}$ with $K=5,\,50,\,500$, and
$5000$, and found that the observed eccentricity distribution of giant planets
can be recovered for slow damping with $K\sim 5-50$. Since Bitsch et al.
(2019) adopted a faster damping with $K=100$, it may be the reason why they
did not obtain eccentric giants.
As seen in Figure 1, we also have $K\sim 100,$ though we managed to reproduce
the eccentricity distribution of giants. Since our eccentricity and
inclination damping prescriptions are similar to those by Bitsch et al.
(2019), it is possible that subtle differences in disc conditions changed the
dynamical outcomes of simulations. For example, the choice of $\alpha_{{\rm
acc}}=5.4\times 10^{-3}$ in Bitsch et al. (2019) may be inconsistent with the
time evolution of the stellar mass accretion rate they adopted (Equation 52).
The fit to this observed mass accretion rate requires a rather high viscosity
$\alpha_{{\rm acc}}\sim 0.01$ for the disc size of $10-100\,$au (Hartmann et
al., 1998) and even higher $\alpha_{{\rm acc}}$ for a larger disc. By assuming
the lower $\alpha_{{\rm acc}}$ for the same accretion rate, the estimated
surface mass density and thus the disc mass becomes higher, which leads to
more efficient eccentricity and inclination damping. In fact, we observed a
similar lack of dynamical instabilities in the no-migration simulations of
Matsumura et al. (2017) — we adopted the same stellar mass accretion equation
as Equation 52 and used $\alpha\leq 5\times 10^{-3}$ when $3-5$ giant planets
were formed (except for one case with 3 giant planets with $\alpha=0.01$). For
completeness, Matsumura et al. (2017) had $K\sim 100$ in the type I regime and
$K\sim 10$ in the type II regime, which should favour more eccentric systems.
Moreover, we had twice as many cores in a much narrower range of disc radii
($0.3-5\,$au) compared to the current runs. It is possible, however, since
these simulations did not include migration, that planets were separated too
far from one another to invoke dynamical instabilities within simulation
times. In that case, convergent migration may also play an important role in
determining the eccentricity and inclination distributions of planetary
systems.
## 5 Conclusion
For this paper, we studied the formation of planetary systems via pebble
accretion by using N-body simulations, and we investigated the effects of disc
parameters such as masses, dissipation timescales, and metallicities. This is
a continuation of Matsumura et al. (2017), in which we modified the N-body
code SyMBA (Duncan et al., 1998) to incorporate the pebble accretion model by
Ida et al. (2016), gas accretion, type I and type II migration, and
eccentricity and inclination damping. In this work, we updated the code as
detailed in Section 2 to take account of the recent development of the field,
and we also adopted a two-$\alpha$ disc model, where mass accretion and disc
turbulence have different drivers.
We find that the disc masses, dissipation timescales, and stellar
metallicities all affect the efficiency of planet formation. The effects of
each parameter can be summarised as follows (see Section 4.6):
* •
Disc metallicities ${\rm[Fe/H]}$ affect the formation timescales of
protoplanetary cores, but they do not strongly affect the final planetary
masses.
* •
Initial disc masses $M_{D,0}$ affect both core formation and gas accretion
timescales, and thus the final planetary masses.
* •
Disc diffusion timescales $t_{{\rm diff}}$ set time limits on planet formation
in the disc, and thus affect the final planetary masses.
We identified two types of giant planet formation trends, depending on whether
planet formation is fast compared to the disc’s dispersal or not. When a
disc’s dissipation timescales are long in comparison to typical planet
formation timescales (Discs 4, 5, 7, and 8 in our simulations, formation-
dominated case), giant planets are massive ($\sim M_{J}$ or higher) and
distributed over a wide range of orbital radii (from $\sim 0.1\,$au to over
$100\,$au). On the other hand, when a disc’s dissipation timescales are
comparable to planet formation timescales (Discs 1, 2, 3, and 6, disc-
dissipation-limited case), giant planets tend to be low-mass ($\sim M_{J}$ or
lower) and CJs (with $a\gtrsim 1\,$au). The formation timescale depends both
on stellar metallicities and disc masses — the timescale is shorter for more
massive, more metal-rich discs. Therefore, protoplanetary cores tend to
migrate significantly before accreting gas to become giant planets in low
metallicity discs, while giant planets can form in situ in the outer part of
high-metallicity, massive discs. For low-mass, low-metallicity discs, giant
planet formation is difficult.
Our main findings are the following:
* •
Differently from Matsumura et al. (2017), we successfully reproduced the
overall distribution trends of semi-major axes $a$, eccentricities $e$, and
planetary masses $M_{p}$ of extrasolar giant planets (see Section 3.1.1 and
Figure 5), though we tend to overproduce CJs compared to HJs and WJs. The
success of reproducing the $a-M_{p}$ distribution, especially for CJs, is
largely due to the new type II migration formula and the two-$\alpha$ disc
model, as proposed by Ida et al. (2018). The success in reproducing the $e$
distribution is likely due to a more self-consistent disc model, a higher
number of giant planets formed per system compared to Matsumura et al. (2017),
and not too efficient eccentricity and inclination damping in the disc (see
Section 4.7).
* •
The overall occurrence rates of giant planets as a function of orbital periods
agree well with observed trends (see Section 4.1). The occurrence rates
increase with periods in the inner region, decrease in the outer region, and
peak at $\sim 1-10\,$au. The most abundant giant planets are CJs ($>50\%$),
and thus more than half the giant planets in our simulations stay near their
formation region.
* •
As discussed in Section 4.2, our simulations naturally explain why HJs tend to
be alone (e.g. Wright et al., 2009; Steffen et al., 2012; Huang et al., 2016),
and also why the eccentricities of HJs are low around low-metallicity stars
and vary more widely around high-metallicity ones (e.g. Dawson & Murray-Clay,
2013; Shabram et al., 2016; Buchhave et al., 2018). The same trend is expected
for stellar obliquities of their host stars, and the current observations
support that (see Section 3.1.2).
* –
In low-metallicity discs, HJs tend to form in situ: protoplanetary cores
migrate to the inner disc and accrete gas there. This is because planet
formation is slower in the low-metallicity discs, which leads to greater
migration of a protoplanetary core before it reaches the PIM and starts
accreting a significant gas to become a giant planet. Since the low-
metallicity discs tend to form just 1-2 giant planets, HJs tend to be alone
and on nearly circular and coplanar orbits.
* –
In high-metallicity discs, HJs can be formed either via tidal circularisation
of highly eccentric orbits or via a migration scenario (including in situ
formation; see Section 3.1.2). The higher metallicity discs tend to produce a
number of giant planets that are prone to dynamical instabilities. A HJ could
be formed from a WJ/CJ as its eccentric orbit is circularised. Alternatively,
HJs could be first formed in situ (i.e. via core migration followed by gas
accretion) or via migration, along with WJs and CJs. The dynamical
instabilities in such systems often remove either HJs and/or WJs, leaving
either (i) only CJs, or (ii) HJs with CJs. HJs formed in high-metallicity
discs have a wider variety of eccentricities and inclinations and also tend to
be alone.
* •
If an SCJ is formed, as a giant planet grows within $\sim 20\,$au and then
gets scattered outward, we expect that such an SCJ (1) was born in a high-
metallicity disc ($\left[{\rm Fe/H}\right]\gtrsim 0.0$), (2) has an eccentric
orbit, and (3) tends to be accompanied by another giant planet ( $\sim
80\,\%$) (see Section 3.1.3).
* •
Most warm Jupiters ($0.1\,{\rm au}\lesssim a\lesssim 1\,$au) are formed in the
formation-dominated discs (i.e. Discs 4, 5, 7, and 8 in our simulations). In
other words, in our simulations, it is difficult to form WJs in rapidly
dissipating, low-mass and/or low-metallicity discs.
* •
CJs tend to be formed in high-mass and/or high-metallicity discs, where the
planet formation timescale is comparable to or shorter than the disc
dissipation timescale.
Finally, there are still several issues that need to be resolved/explored in
our work. Most importantly, type I migration is still too fast and we tend to
lose SEs. For example, type I migration can be slowed in the inner disc region
if we fully adopt the wind-driven disc, as in Ogihara et al. (2018). Resolving
the migration issue is also important when choosing a more appropriate gas
accretion formula, which would provide more accurate planetary compositions
(see Section 4.5). Furthermore, when $\alpha_{{\rm turb}}\ll\alpha_{{\rm
acc}}$ as we assumed, the gap depth may also be affected by the wind-driven
accretion.
###### Acknowledgements.
We thank Man Hoi Lee and Eduard Vorobyov for useful discussions and an
anonymous referee for detailed comments. SM is grateful to Aurora Sicilia-
Aguilar for valuable discussions and also for kindly sharing her data from
Sicilia-Aguilar et al. (2010). SM would also like to thank the Earth-Life
Science Institute at Tokyo Institute of Technology for its hospitality, where
part of this work has been done. SM is partly supported by the STFC grant
number ST/S000399/1 (The Planet-Disk Connection: Accretion, Disk Structure,
and Planet Formation). RB acknowledges financial assistance from the Japan
Society for the Promotion of Science (JSPS) Shingakujutsu Kobo (JP19H05071).
SI is supported by MEXT Kakenhi 18H05438.
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# Variants of the Selberg sieve, and almost prime $k$-tuples
Paweł Lewulis Supported by NCN Sonatina 3, 2019/32/C/ST1/00341.
###### Abstract
Let $k\geq 2$ and $\mathcal{P}(n)=(A_{1}n+B_{1})\cdots(A_{k}n+B_{k})$ where
all the $A_{i},B_{i}$ are integers. Suppose that $\mathcal{P}(n)$ has no fixed
prime divisors. For each choice of $k$ it is known that there exists an
integer $\varrho_{k}$ such that $\mathcal{P}(n)$ has at most $\varrho_{k}$
prime factors infinitely often. We used a new weighted sieve set-up combined
with a device called an $\varepsilon$-trick to improve the possible values of
$\varrho_{k}$ for $k\geq 7$. As a by-product of our approach, we improve the
conditional possible values of $\varrho_{k}$ for $k\geq 4$, assuming the
generalized Elliott–Halberstam conjecture.
## 1 Introduction
### State of the art
Let us begin with recalling the following notion.
###### Definition (Admissible tuples).
Fix a positive integer $k$. For each $i=1,\dots,k$ fix integers $A_{i}$,
$B_{i}$, such that $A_{i}>0$, and let
$L_{i}\colon\mathbf{Z}^{+}\rightarrow\mathbf{Z}$ be a function given by the
formula $L_{i}(n):=A_{i}n+B_{i}$. For each positive integer $n$ put
$\mathcal{P}(n):=\prod_{i=1}^{k}L_{i}(n).$
We call $\mathcal{H}:=\\{L_{1},\dots,L_{k}\\}$ an admissible $k$–tuple, if for
every prime $p$ there is an integer $n_{p}$ such that none of the
$L_{i}(n_{p})$ is a multiple of $p$.
We are interested in the following problem being a vast generalization of the
twin primes conjecture.
###### Conjecture 1 (Dickson–Hardy–Littlewood).
Fix a positive integer $k$. Let $\\{L_{1},\dots,L_{k}\\}$ be an admissible
$k$–tuple. Then,
$\liminf_{n\rightarrow\infty}\Omega(\mathcal{P}(n))=k.$ (1)
One may reformulate the statement above into a general question about the
total number of prime factors contained within $\mathcal{P}(n)$. This creates
a way to ‘measure’ of how far we are from proving Conjecture 1.
###### Problem 2 ($DHL_{\Omega}$).
Fix positive integers $k$ and $\varrho_{k}\geq k$. Let
$\\{L_{1},\dots,L_{k}\\}$ be an admissible $k$–tuple. The task is to prove
that
$\liminf_{n\rightarrow\infty}\Omega(\mathcal{P}(n))\leq\varrho_{k}.$ (2)
From this point on, if inequality (2) is true for some precise choice of $k$
and $\varrho_{k}$, then we say that $DHL_{\Omega}[k;\varrho_{k}]$ holds. In
the case $k=1$, the classical Dirichlet’s theorem is equivalent to
$DHL_{\Omega}[1;1]$. This is also the only instance where the optimal possible
value of $\varrho_{k}$ is already known. For $k=2$ we have $DHL_{\Omega}[2;3]$
by Chen’s theorem proven in [Chen]. If $k\geq 3$, then the state of the art
and recent progress are described below.
Table A. State of the art – obtained values $\varrho_{k}$ for which
$DHL_{\Omega}[k;\varrho_{k}]$ holds.
Unconditional case.
$k$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---
Halberstam, Richert [Halberstam-Richert] | 10 | 15 | 19 | 24 | 29 | 34 | 39 | 45 |
Porter [Porter] | 8 | | | | | | | |
Diamond, Halberstam [DH] | | 12 | 16 | 20 | 25 | 29 | 34 | 39 |
Ho, Tsang [HT] | | | | | 24 | 28 | 33 | 38 |
Maynard [3-tuples, MaynardK] | 7 | 11 | 15 | 18 | 22 | 26 | 30 | 34 |
Lewulis [Lewulis] | | | 14 | | | | | |
This work | | | | | 21 | 25 | 29 | 33 |
The $GEH$ case.
$k$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---
Sono [Sono] | 6 | | | | | | | |
Lewulis [Lewulis] | | 10 | 13 | 17 | 20 | 24 | 28 | 32 |
This work | | 8 | 11 | 14 | 17 | 21 | 24 | 27 |
### Notation
The letter $p$ with possible indices always denotes a prime number and
${\log}$ denotes the natural logarithm. We use the notation
$\mathbf{N}=\\{1,2,3,\dots\\}$. We also use the following definitions listed
below:
* •
$\varphi(n):=\\#\left(\mathbf{Z}/n\mathbf{Z}\right)^{\times}$ denotes Euler
totient function;
* •
$\tau(n):=\sum_{d|n}1$ denotes the divisor function;
* •
$\Omega(n)$ denotes the number of prime factors of $n$;
* •
$\pi(x):=\\#\left\\{n\in\mathbf{N}:n\leq x,\leavevmode\nobreak\ n\text{ is
prime}\right\\}$;
* •
$\pi(x;q,a):=\\#\left\\{n\in\mathbf{N}:n\leq x,\leavevmode\nobreak\ n\equiv
a\bmod q,\leavevmode\nobreak\ n\text{ is prime}\right\\}$;
* •
$\log_{y}x:=\frac{\log x}{\log y}$ for $x,y>0$ and $y\not=1$;
* •
By $(a,b)$ and $[a,b]$ we denote the greatest common divisor and the lowest
common multiple, respectively;
* •
For a logical formula $\phi$ we define the indicator function
$\mathbf{1}_{\phi(x)}$ that equals $1$ when $\phi(x)$ is true and $0$
otherwise;
* •
For a set $A$ we define the indicator function $\mathbf{1}_{A}$ that equals
$1$ when the argument belongs to $A$ and $0$ otherwise;
* •
By $\text{gpf}(n)$ and $\text{lpf}(n)$ we denote the greatest and the lowest
prime divisor of $n$ respetively;
* •
The condition $n\sim x$ means that $x<n\leq 2x$;
* •
For a function $F$ being a map between some two abelian groups we define the
difference operator $\partial_{y}F(x):=F(x+y)-F(x)$;
* •
We define an analogous operator for a function $F$ with $m$ variables, namely
$\partial_{y}^{(i)}F(x_{1},\dots,x_{m}):=F(x_{1},\dots,x_{i-1},x_{i}+y,x_{i+1},\dots,x_{m})-F(x_{1},\dots,x_{m})$;
* •
For every compactly supported function
$F\colon[0,+\infty)\rightarrow\mathbf{R}$ we define
$S(F):=\sup\left(\\{x\in\mathbf{R}\colon F(x)\not=0\\}\cup\\{0\\}\right);$
* •
We define a normalizing expression $B:=\frac{\varphi(W)\log x}{W}$ (cf. next
subsection);
* •
Symmetric polynomials of degree $m$ and $k$ variables
$\sum_{j=1}^{k}t_{j}^{m}$ are denoted as $P_{m}$;
* •
For a finitely supported arithmetic function
$f\colon\mathbf{N}\rightarrow\mathbf{C}$ we define a discrepancy
$\Delta\left(f;a\bmod q\right):=\sum_{n\equiv a\bmod
q}f(n)-\frac{1}{\varphi(q)}\sum_{(n,q)=1}f(n)\,;$
* •
For any $f\colon\mathbf{R}\rightarrow\mathbf{R}$ we define a function related
to Selberg weights
$\lambda_{f}(n):=\sum_{d|n}\mu(d)f(\log_{x}d);$
* •
We also make use of the ‘big $O$’, the ‘small $o$’, and the ‘$\ll$’ notation
in a standard way.
### The general set-up
Let us fix $k\in\mathbf{Z}^{+}$ and consider the expression
$\mathcal{S}:=\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\left(\varrho_{k}-\Omega(\mathcal{P}(n))\right)\nu(n),$ (3)
where $\nu$ is some arbitrarily chosen sequence of non-negative weights. Put
$W:=\prod_{p<D_{0}}p$
for $D_{0}:=\log\log\log x$ and take some integer $b$ coprime to $W$. We
choose some residue class $b$ such that $\mathcal{P}(b)$ is coprime to $W$ and
then, we restrict our attention to $n\equiv b\bmod W$. This way we discard all
irregularities caused by very small prime numbers. Put
$A:=4\max\left\\{|A_{1}|,|B_{1}|,\dots,|A_{k}|,|B_{k}|\right\\}$. Assume that
$x>10\,000$ and $D_{0}>A$. Thus, our goal is to show that
$\mathcal{S}=\varrho_{k}\mathcal{S}_{0}-\mathcal{S}_{\Omega}>0,$ (4)
where
$\begin{split}\mathcal{S}_{0}&:=\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv
b\bmod W\end{subarray}}\nu(n),\\\
\mathcal{S}_{\Omega}&:=\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\Omega(\mathcal{P}(n))\nu(n).\\\ \end{split}$ (5)
The main difficulty is to calculate $\mathcal{S}_{\Omega}$ with sufficient
accuracy. One possible method and a good source of inspiration for new tools
is the following identity valid for square-free $n\leqslant x$:
$\Omega(n)=\sum_{p|n}1=\mathbf{1}_{\text{gpf}(n)>U}+\sum_{\begin{subarray}{c}p|n\\\
p\leq U\end{subarray}}1,$ (6)
where $U>x^{1/2}$ (usually, $U=x^{1/2+\epsilon}$ for some small $\epsilon>0$
has been considered). For instance, one can exploit the simple inequality
$\Omega(\mathcal{P}(n))\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{i=1}^{k}\mathbf{1}_{\text{gpf}(L_{i}(n))>U}\leavevmode\nobreak\
+\sum_{\begin{subarray}{c}p|\mathcal{P}(n)\\\ p\leq
U\end{subarray}}1\leavevmode\nobreak\ \leq\leavevmode\nobreak\
k\leavevmode\nobreak\ +\sum_{\begin{subarray}{c}p|\mathcal{P}(n)\\\ p\leq
U\end{subarray}}1$ (7)
under the previous assumptions. This reasoning leads to results that are
nontrivial, but weaker than already existing in literature. However, the
interesting observation about this identity is that one does not need to rely
on any distributional claims about primes in arithmetic progressions in order
to exploit it.
In [MaynardK] and [Lewulis] the authors applied the following identity valid
for all square-free $n\sim x$:
$\Omega(n)=\frac{\log n}{\log T}+\sum_{p|n}\left(1-\frac{\log p}{\log
T}\right),$ (8)
where $T:=x^{l}$ for some exponent $l\in(0,1]$. This approach combined with
(6) gives some flexibility, because the expression in the parentheses in (8)
is negative for $p>T$. In such case, we can transform the task of seeking for
upper bounds for $\mathcal{S}_{\Omega}$ into problem of establishing lower
bounds. The idea was to apply the following partition of unity:
$1=\sum_{r}\mathbf{1}_{\Omega(n)=r}\geq\sum_{r\leq
H}\mathbf{1}_{\Omega(n)=r},$ (9)
valid for any $H>0$, and then, to calculate the contribution of $S_{\Omega}$
via (8) and (9), usually for $H=3,4$, depending on specific cases.
In this work we propose a different approach and we establish the asymptotic
behaviour of $\mathcal{S}_{\Omega}$. Such a result is sufficient to improve
the currently known values of $\varrho_{k}$ in the conditional case, that is
when $GEH$ (cf. Section ‘Preparing the sieve’ for definitions) is true. It
also greatly simplifies the unconditional results from [Lewulis] and explains
Conjecture 4.2 formulated there, which turns out to be slightly incorrect.
To tackle the uncondtitional case, we need to expand the sieve support beyond
the domain offered by the standard claims regarding primes in arithemetic
progressions (Theorem 5, in particular). Hence, we incorporate a device
invented in [Polymath8] called an $\varepsilon$-trick. In order to do so, we
have to apply (8). The reason for this is that the $\varepsilon$-trick is all
about bounding the sieve weights from below. In the same time, we wish to
apply this tool to $\mathcal{S}_{\Omega}$, which has to be estimated from
above. As we noticed before, (8) enables us to partially convert upper bounds
into lower bounds, at least until the prime factors are sufficiently large. On
the other hand, if they are small, we do not need to rely on any
distributional claim on primes in arithmetic progressions at all, so in this
case we can expand the sieve support almost freely.
To summarize, we propose a general set-up that is flexible enough to cover all
applications appearing in this work. We have the following criterion for our
main problem.
###### Lemma 3.
Let $k\geq 2$ and $\varrho\geq k$ be fixed integers. Suppose that for each
fixed admissible $k$–tuple $\\{L_{1},\dots,L_{k}\\}$ and each residue class
$b\bmod W$ such that $(L_{i}(b),W)=1$ for all $i=1,\dots,k$, one can find a
non-negative weight function $\nu\colon\mathbf{N}\rightarrow\mathbf{R}^{+}$
and fixed quantities $\alpha>0$, and $\beta_{1},\dots\beta_{k}\geq 0$, such
that one has the asymptotic lower bound
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\nu(n)\geq\left(\alpha-o(1)\right)B^{-k}\frac{x}{W},$ (10)
and the asymptotic upper bounds
$\displaystyle\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\sum_{p|L_{i}(n)}\left(1-\ell\log_{x}p\right)\nu(n)$
$\displaystyle\leq(\beta_{i}+o(1))B^{-k}\frac{x}{W},$ (11)
$\displaystyle\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ not sq-
free}\end{subarray}}\tau(\mathcal{P}(n))\left|\nu(n)\right|$
$\displaystyle\leq o(1)\times B^{-k}\frac{x}{W}$ (12)
for all $i=1,\dots,k$, and the key inequality
$\varrho>\frac{\beta_{1}+\dots+\beta_{k}}{\alpha}+\ell k.$
Then, $DHL_{\Omega}[k;\varrho]$ holds. Moreover, if one replaces inequalities
(10–11) with equalities, then the right-hand side of the key inequality above
is constant with respect to the $\ell$ variable.
###### Proof.
We have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\left(\varrho-\Omega(\mathcal{P}(n)\right)\nu(n)=\varrho\left(\sum_{\begin{subarray}{c}n\sim
x\\\ n\equiv b\bmod
W\end{subarray}}\nu(n)\right)-\left(\sum_{\begin{subarray}{c}n\sim x\\\
n\equiv b\bmod W\\\ \mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\Omega(\mathcal{P}(n))\nu(n)\right)\\\
+O\left(\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ not sq-
free}\end{subarray}}\tau(\mathcal{P}(n))\nu(n)\right).$ (13)
We also observe that
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\Omega(\mathcal{P}(n))\nu(n)=\left(\sum_{i=1}^{k}\sum_{\begin{subarray}{c}n\sim
x\\\ n\equiv b\bmod W\\\ \mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\sum_{p|L_{i}(n)}\left(1-\ell\log_{x}p\right)\nu(n)\right)\\\
+(\ell k+o(1))\left(\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\nu(n)\right).$ (14)
Combining (13–14) with the assumptions we arrive at
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\left(\varrho-\Omega(\mathcal{P}(n)\right)\nu(n)\geq\left((\varrho+\ell
k)\,\alpha-\sum_{i=1}^{k}\beta_{i}-o(1)\right)B^{-k}\frac{x}{W}.$ (15)
Note that (15) becomes an equality, if one replaces inequalities (10–11) with
equalities – in such a case the left-hand side of (15) obviously does not
depend on the $\ell$ variable, so the same has to be true for the right-hand
side of (15). We conclude that the left-hand side of (15) is asymptotically
greater than $0$ if
$\varrho>\frac{\beta_{1}+\dots+\beta_{k}}{\alpha}+\ell k.$ (16)
∎
As mentioned in Table A, the main goal of this work is to prove the following
result.
###### Theorem 4 (Main Theorem).
$DHL_{\Omega}[k,\varrho_{k}]$ holds with the values $\varrho_{k}$ given in a
table below __
Table B.
$k$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---
Unconditionally | 4 | 7 | 11 | 14 | 18 | 21 | 25 | 29 | 33 |
Assuming $GEH$ | 3 | 6 | 8 | 11 | 14 | 17 | 21 | 24 | 27 |
_(bolded text indicates the novelties in the field).___
### Preparing the sieve
In this subsection we are focused on motivating our future choice of sieve
weights $\nu(n)$, so this discussion will be slightly informal. Our task is to
make the sum (3) greater than 0 for some fixed $\varrho_{k}$. That would be
sufficient to prove that $DHL_{\Omega}[k;\varrho_{k}]$ holds. Hence, the
weight $\nu$ has to be sensitive to almost prime $k$-tuples. We observe that
the von Mangoldt function satisfies
$\Lambda(n)=\left(\mu*\log\right)(n)=-\sum_{d|n}\mu(d)\log d,$
which for square-free $n\sim x$ gives
$\mathbf{1}_{n\text{ is
prime}}\approx\sum_{d|n}\mu(d)\left(1-\log_{x}d\right).$ (17)
That motivates the following construction of the Selberg sieve:
$\mathbf{1}_{n\text{ is prime}}\lessapprox
f(0)\left(\sum_{\begin{subarray}{c}d|n\end{subarray}}\mu(d)f(\log_{x}d)\right)^{2},$
(18)
where $f\colon[0,+\infty)\rightarrow\mathbf{R}$ is piecewise smooth and
supported on $[0,1)$. The problem is that the Bombieri–Vinogradov theorem
usually forces us to assume that $\mbox{supp}(f)\subset[0,\theta)$ for some
fixed positive $\theta$. The usual choice here is $\theta$ somewhat close to
$1/4$, or greater, if one assumes the Elliott–Halberstam conjecture.
In the multidimensional setting we have
$\mathbf{1}_{L_{1}(n),\dots,L_{k}(n)\text{ are all primes}}\lessapprox
f(0,\dots,0)\left(\sum_{\begin{subarray}{c}d_{1},\dots,d_{k}\\\ \forall
i\leavevmode\nobreak\
d_{i}|L_{i}(n)\end{subarray}}\left(\prod_{i=1}^{k}\mu(d_{i})\right)f\left(\log_{x}d_{1},\dots,\log_{x}d_{k}\right)\right)^{2}$
(19)
for some $f\colon[0,+\infty)^{k}\rightarrow\mathbf{R}$ being piecewise smooth
and compactly supported. In certain cases this approach can be more efficient
than (18), as was shown in [Maynard], where it was introduced. Dealing with
multivariate summations may be tedious at times, so we would like to transform
the right-hand side of $(\ref{MSS})$ a bit by replacing the function $f$ with
tensor products
$f_{1}(\log_{x}d_{1})\cdots f_{k}(\log_{x}d_{k}),$ (20)
where $f_{1},\dots,f_{k}\colon\mathbf{[}0,+\infty)\rightarrow\mathbf{R}$. By
the Stone–Weierstrass theorem we can approximate $f$ by a linear combination
of functions of such form, so essentially we lose nothing here. Our more
convenient sieve weights look as follows:
$\left(\sum_{j=1}^{J}c_{j}\prod_{i=1}^{k}\lambda_{f_{j,i}}(L_{i}(n))\right)^{2}$
(21)
with some real coefficients $c_{j}$, some smooth and compactly supported
functions $f_{i,j}$. Recall that
$\lambda_{f}(n):=\sum_{d|n}\mu(d)f(\log_{x}d).$
It is clear that such a weight can be decomposed into linear combination of
functions of the form
$n\mapsto\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n)).$
(22)
In fact, (21) is exactly our choice in Section 4.
### Distributional claims concerning primes
In this work we refer to the generalised Elliott–Halberstam conjecture,
labeled further as $GEH[\vartheta]$ for some $0<\vartheta<1$. This broad
generalisation first appeared in [GEH]. Its precise formulation can be found
for example in [Polymath8]. The best known result in this direction is
currently proven by Motohashi [Motohashi].
###### Theorem 5.
$GEH[\vartheta]$ holds for every $\vartheta\in(0,1/2)$.
In this work we actually need only one specific corollary of $GEH$, which can
be perceived as an ‘Elliott–Halberstam conjecture for almost primes’.
###### Theorem 6.
Assume $GEH[\vartheta]$. Let $r\geq 1$, $\epsilon>0$, and $A\geq 1$ be fixed.
Let
$\Delta_{r,\epsilon}=\\{(t_{1},\dots,t_{r})\in[\epsilon,1]^{r}\colon\leavevmode\nobreak\
t_{1}\leq\dots\leq t_{r};\leavevmode\nobreak\ t_{1}+\dots+t_{r}=1\\},$
and let $F\colon\Delta_{r,\epsilon}\rightarrow{\bf R}$ be a fixed smooth
function. Let $f\colon{\bf N}\rightarrow{\bf R}$ be the function defined by
setting
$\displaystyle f(n)=F\left(\log_{n}p_{1},\dots,\log_{n}p_{r}\right)$
whenever $n=p_{1}\dots p_{k}$ is the product of $r$ distinct primes
$p_{1}<\dots<p_{r}$ with $p_{1}\geq x^{\epsilon}$ for some fixed $\epsilon>0$,
and $f(n)=0$ otherwise. Then for every $Q\ll x^{\vartheta}$, we have
$\displaystyle\sum_{q\leq
Q}\max_{\begin{subarray}{c}(a,q)=1\end{subarray}}\left|\Delta\left(\mathbf{1}_{[1,x]}f;a\bmod
q\right)\right|\ll x\log^{-A}x.$
## 2 Outline of the key ingredients
Let us start from presenting a minor variation of [Polymath8, Theorem 3.6].
The only change we impose is replacing the linear forms of the shape $n+h_{i}$
by slightly more general $L_{i}(n)$. This, however, does not affect the proof
in any way.
###### Proposition 7 (Non-$\Omega$ sums).
Let $k\geq 1$ be fixed, let $\\{L_{1},\dots,L_{k}\\}$ be a fixed admissible
k-tuple, and let $b\bmod W$ be such that $(L_{i}(b),W)=1$ for each
$i=1,\dots,k$. For each fixed $1\leq i\leq k$, let
$F_{i},\,G_{i}\colon[0,+\infty)\rightarrow\mathbf{R}$ be fixed smooth
compactly supported functions. Assume one of the following hypotheses:
1. 1.
(Trivial case) One has
$\sum_{i=1}^{k}(S(F_{i})+S(G_{i}))<1.$
2. 2.
(Generalized Elliott–Halberstam) There exist a fixed $0<\vartheta<1$ and
$i_{0}\in\\{1,\dots,k\\}$ such that $GEH[\vartheta]$ holds, and
$\sum_{\begin{subarray}{c}1\leq i\leq k\\\
i\not=i_{0}\end{subarray}}(S(F_{i})+S(G_{i}))<\vartheta.$
Then, we have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))=(c+o(1))B^{-k}\frac{x}{W},$
where
$c:=\prod_{i=1}^{k}\left(\int\limits_{0}^{1}F_{i}^{\prime}(t_{i})\,G_{i}^{\prime}(t_{i})\,dt_{i}\right).$
The next result is a crucial component of this work and is a novelty in the
topic. Together with Proposition 7 it creates a way to transform
$\mathcal{S}_{0}$ and $\mathcal{S}_{\Omega}$ into integrals, effectively
converting the main task of finding almost primes into an optimization
problem.
###### Proposition 8 (Sums containing $\Omega$ function).
Let $k\geq 1$ and $i_{0}\in\\{1,\dots,k\\}$ be fixed, let
$\\{L_{1},\dots,L_{k}\\}$ be a fixed admissible k-tuple, and let $b\bmod W$ be
such that $(L_{i}(b),W)=1$ for each $i=1,\dots,k$. For each fixed $1\leq i\leq
k$, let $F_{i},\,G_{i},\colon[0,+\infty)\rightarrow\mathbf{R}$ be fixed smooth
compactly supported functions, and let
$\Upsilon\colon[0,+\infty)\rightarrow\mathbf{R}$ be a bounded Riemann
integrable function continuous at $1$. Assume that there exist
$\vartheta,\,\vartheta_{0}\in(0,1)$ such that one of the following hypoteses
holds:
1. 1.
(Trivial case) One has
$\sum_{i=1}^{k}(S(F_{i})+S(G_{i}))<1-\vartheta_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
and\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ S(\Upsilon)<\vartheta_{0}.$
2. 2.
(Generalized Elliott–Halberstam) Assume that $GEH[\vartheta]$ holds, and
$\sum_{\begin{subarray}{c}1\leq i\leq k\\\
i\not=i_{0}\end{subarray}}(S(F_{i})+S(G_{i}))<\vartheta.$
Then, we have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\leavevmode\nobreak\ \textup{sq-
free}\end{subarray}}\left(\sum_{p|L_{i_{0}}(n)}\Upsilon(\log_{x}p)\right)\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))=(c+o(1))B^{-k}\frac{x}{W},$
(23)
where
$\begin{split}c:=\left(\Upsilon(1)\,F_{i_{0}}(0)\,G_{i_{0}}(0)\leavevmode\nobreak\
+\leavevmode\nobreak\
\int\limits_{0}^{1}\frac{\Upsilon(y)}{y}\int\limits_{0}^{1-y}\partial_{y}F^{\prime}_{i_{0}}(t_{i_{0}})\,\partial_{y}G^{\prime}_{i_{0}}(t_{i_{0}})\,dt_{i_{0}}\,dy\right)\prod_{\begin{subarray}{c}1\leq
i\leq k\\\
i\not=i_{0}\end{subarray}}\left(\int\limits_{0}^{1}F_{i}^{\prime}(t_{i})\,G_{i}^{\prime}(t_{i})\,dt_{i}\right).\end{split}$
The first case of Proposition 8 is strongly related to [MaynardK, Proposition
5.1] and [Lewulis, Proposition 1.13]. It is worth mentioning that the
conditional results in the latter of these two cited papers relied only on
$GEH[2/3]$. It was not possible to invoke the full power of $GEH$ by methods
studied there due to certain technical obstacles. The second case of
Proposition 8 is strong enough to overcome them. It also paves a way to
conveniently apply a device called an $\varepsilon$-trick in the unconditional
setting.
The role of the last proposition in this section is to deal with the
contribution from $n$ such that $\mathcal{P}(n)$ is not square-free.
###### Proposition 9 (Sums with double prime factors).
Let $k\geq 1$ be fixed, let $\\{L_{1},\dots,L_{k}\\}$ be a fixed admissible
k-tuple, and let $b\bmod W$ be such that $(L_{i}(b),W)=1$ for each
$i=1,\dots,k$. For each fixed $1\leq i\leq k$, let
$F_{i},\,G_{i}\colon[0,+\infty)\rightarrow\mathbf{R}$ be fixed smooth
compactly supported functions. Then, we have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\ \mathcal{P}(n)\emph{
not sq-
free}\end{subarray}}\tau(\mathcal{P}(n))\left|\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))\right|=o(1)\times
B^{-k}\frac{x}{W}.$
Now, we combine Propositions 7–9 to obtain Theorems 10, 12, and 13 giving us
criteria for the $DHL_{\Omega}$ problem. Theorem 10 refers to sieving on
standard simplex $\mathcal{R}_{k}$, which can be considered as a default range
for the multidimensional Selberg sieve. The next one, Theorem 12, deals with
the extended simplex $\mathcal{R}_{k}^{\prime}$, which was applied in
[Lewulis], where $DHL_{\Omega}[5;14]$ was proven. We also prove Theorem 13
being the most general of these three. It describes sieving on the epsilon-
enlarged simplex. In fact, Theorems 10 and 12 are corollaries from Theorem 13,
as noted in Remark 2.
###### Theorem 10 (Sieving on a standard simplex).
Suppose that there is an arbitrarily chosen fixed real parameter $\ell$ and a
fixed $\theta\in(0,\frac{1}{2})$ such that $GEH[2\theta]$ holds. Let $k\geq 2$
and $m\geq 1$ be fixed integers. For any fixed compactly supported square-
integrable function $F\colon[0,+\infty)^{k}\rightarrow\mathbf{R}$, define the
functionals
$\begin{split}I(F):=&\int_{[0,+\infty)^{k}}F(t_{1},\dots,t_{k})^{2}\,dt_{1}\dots
dt_{k},\\\ Q_{i}(F):=&\int_{0}^{\frac{1}{\theta}}\frac{1-\ell\theta
y}{y}\int_{[0,+\infty)^{k-1}}\left(\int_{0}^{\frac{1}{\theta}-y}\left(\partial_{y}^{(i)}F(t_{1},\dots,t_{k})\right)^{2}dt_{i}\right)\,dt_{1}\dots
dt_{i-1}\,dt_{i+1}\dots dt_{k}\,dy,\\\
J_{i}(F):=&\int_{[0,+\infty)^{k-1}}\left(\int_{0}^{\infty}F(t_{1},\dots,t_{k})\,dt_{i}\right)^{2}dt_{1}\dots
dt_{i-1}\,dt_{i+1}\dots dt_{k},\end{split}$ (24)
and let $\Omega_{k}$ be the infimum
$\Omega_{k}:=\inf_{F}\left(\frac{\sum_{i=1}^{k}\left(Q_{i}(F)+\theta(1-\ell)J_{i}(F)\right)}{I(F)}+\ell
k\right),$ (25)
over all square integrable functions $F$ that are supported on the simplex
$\mathcal{R}_{k}:=\\{(t_{1},\dots,t_{k})\in[0,+\infty)^{k}\colon
t_{1}+\dots+t_{k}\leq 1\\},$
and are not identically zero up to almost everywhere equivalence. If
$m>\Omega_{k},$
then $DHL_{\Omega}[k;m-1]$ holds.
###### Remark 1.
Due to the continuity of $\Omega_{k}$ we can replace the condition that
$GEH[2\theta]$ holds by a weaker one that $GEH[2\theta^{\prime}]$ holds for
all $\theta^{\prime}<\theta$. Therefore, we are also permitted to take
$\theta=1/4$ unconditionally and $\theta=1/2$ assuming $GEH$. The same remark
also applies to Theorems 11, 12, and 13.
The choice of parameter $\ell$ does not affect the value of $\Omega_{k}$.
Substituting
$F(t_{1},\dots,t_{k})=f(t_{1}+\dots+t_{k})$
for some $f\colon[0,+\infty)\rightarrow\mathbf{R}$ and fixing $\ell=1$ we get
the following result.
###### Theorem 11 (One-dimensional sieving).
Suppose that there is a fixed $\theta\in(0,\frac{1}{2})$ such that
$GEH[2\theta]$ holds. Let $k\geq 2$ and $m\geq 1$ be fixed integers. For any
fixed and locally square-integrable function
$f\colon[0,+\infty)\rightarrow\mathbf{R}$, define the functionals
$\begin{split}\bar{I}(f):=&\int\limits_{0}^{1}f(t)^{2}\,t^{k-1}\,dt,\\\
\bar{Q}^{(1)}(f):=&\int\limits_{0}^{1}\frac{1-\theta
y}{y}\int\limits_{0}^{1-y}\left(f(t)-f(t+y)\right)^{2}t^{k-1}\,dt\,dy,\\\
\bar{Q}^{(2)}(f):=&\left(\int\limits_{0}^{1}\int\limits_{1-y}^{1}\,+\,\int\limits_{1}^{\frac{1}{\theta}-1}\int\limits_{0}^{1}\,+\,\int\limits_{\frac{1}{\theta}-1}^{\frac{1}{\theta}}\int\limits_{0}^{\frac{1}{\theta}-y}\,\right)\frac{1-\theta
y}{y}\,f(t)^{2}\,t^{k-1}\,dt\,dy,\\\
\bar{Q}^{(3)}(f):=&\int\limits_{\frac{1}{\theta}-1}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{\frac{1}{\theta}-y}^{1}f(t)^{2}\,\left(t^{k-1}-\left(t+y-\frac{1}{\theta}\right)^{k-1}\right)dt\,dy,\end{split}$
(26)
and let $\bar{\Omega}_{k}$ be the infimum
$\bar{\Omega}_{k}:=\inf_{f}\left(\frac{\sum_{i=1}^{3}\bar{Q}^{(i)}(f)}{\bar{I}(f)}+1\right)\cdot
k,$
over all square integrable functions $f$ that are not identically zero up to
almost everywhere equivalence. If
$m>\bar{\Omega}_{k},$
then $DHL_{\Omega}[k;m-1]$ holds.
We obviously have $\bar{\Omega}_{k}\geq\Omega_{k}$ for every possible choice
of $k$. We may apply Theorem 11 to get some non-trivial improvements over the
current state of the art in the $GEH$ case. We perform optimization over
polynomials of the form $f(x)=a+b(1-x)+c(1-x)^{2}+d(1-x)^{3}$ for
$-1<a,b,c,d<1$. This choice transforms the functionals (26) into quadratic
forms depending on the parameters $a,b,c,d$. Details including close to
optimal polynomials (up to a constant factor) for each $k$ are covered in the
table below.
Table C. Upper bounds for $\Omega_{k}$.
$k$ | $\theta=1/4$ | $\theta=1/2$ | $f(1-x)$ |
---|---|---|---|---
$2$ | 5.03947 | 3.84763 | $3+25x-x^{2}+x^{3}$ |
$3$ | 8.15176 | 6.31954 | $1+12x-2x^{2}+9x^{3}$ |
$4$ | 11.49211 | 9.00542 | $1+15x-x^{2}+19x^{3}$ |
$5$ | 15.01292 | 11.86400 | $1+16x+5x^{2}+32x^{3}$ |
$6$ | 18.68514 | 14.86781 | $1+26x-8x^{2}+86x^{3}$ |
$7$ | 22.48318 | 17.99402 | $1+24x+6x^{2}+110x^{3}$ |
$8$ | 26.39648 | 21.23219 | $1+30x+x^{2}+200x^{3}$ |
$9$ | 30.40952 | 24.56817 | $1+30x+3x^{2}+260x^{3}$ |
$10$ | 34.51469 | 27.99372 | $1+36x-x^{2}+400x^{3}$ |
It turns out that close to optimal choices in the unconditional setting are
also close to optimal under $GEH$. These results are sufficient to prove the
conditional part of Theorem 4 in every case except for $k=4$. Unfortunately,
by this method we cannot provide any unconditional improvement over what is
already obtained in [MaynardK], as presented in Table C. Therefore, let us try
to expand the sieve support a bit.
###### Theorem 12 (Sieving on an extended simplex).
Suppose that there is a fixed $\theta\in(0,\frac{1}{2})$ such that
$GEH[2\theta]$ holds and an arbitrarily chosen fixed real parameter $\ell$.
Let $k\geq 2$ and $m\geq 1$ be fixed integers. Let $\Omega_{k}^{\emph{ext}}$
be defined as in (25), but where the supremum now ranges over all square-
integrable and non-zero up to almost everywhere equivalence $F$ supported on
the extended simplex
$\mathcal{R}^{\prime}_{k}:=\\{(t_{1},\dots,t_{k})\in[0,+\infty)^{k}\colon\forall_{i\in\\{1,\dots,k\\}}\leavevmode\nobreak\
t_{1}+\dots+t_{i-1}+t_{i+1}+\dots+t_{k}\leq 1\\}.$
If
$m>\Omega^{\emph{ext}}_{k},$
then $DHL_{\Omega}[k;m-1]$ holds.
It is difficult to propose a one-dimensional variation of Theorem 12 in a
compact form, because the precise shape of functionals analogous to (26)
varies depending on $k$. We deal with this problem in Subsection 5.2. Given
that, we apply Theorem 12 directly and perform optimization over polynomials
of the form $F(t_{1},\dots,t_{k})=a+b(1-P_{1})+c(1-P_{1})^{2}=:f(P_{1})$ for
$-1<a,b,c<1$. Our choice is motivated by the fact that the values of symmetric
polynomials generated only by $P_{1}$ depend only on the sum
$t_{1}+\dots+t_{k}$, so they behave ’one-dimensionally’, which makes all
necessary calculations much easier. Moreover, our numerical experiments
suggest that including $P_{2}$ does not provide much extra contribution. Some
good choices of polynomials (again, up to a constant factor) and the bounds
they produce are listed below.
Table D. Upper bounds for $\Omega^{\text{ext}}_{k}$.
$k$ | $\theta=1/4$ | $\theta=1/2$ | $f(1-x)$ |
---|---|---|---|---
$2$ | 4.49560 | 3.35492 | $6+8x+3x^{2}$ |
$3$ | 7.84666 | 6.03889 | $2+7x+7x^{2}$ |
$4$ | 11.27711 | 8.80441 | $1+6x+9x^{2}$ |
$5$ | 14.84534 | 11.70582 | $1+7x+15x^{2}$ |
$6$ | 18.55409 | 14.74036 | $1+9x+32x^{2}$ |
$7$ | 22.38208 | 17.89601 | $1+10x+46x^{2}$ |
$8$ | 26.32546 | 21.16260 | $1+10x+65x^{2}$ |
$9$ | 30.37012 | 24.52806 | $1+10x+90x^{2}$ |
$10$ | 34.50669 | 27.98326 | $1+11x+121x^{2}$ |
The results from the $\theta=1/4$ column in Table D predict the limitations of
methods developed in [Lewulis]. In the conditional case we also get a strong
enhancement over what is achievable by sieving on the standard simplex in the
$k=4$ case. In the $k=2$ case we observe a standard phenomenon that passing
through the constant $3$ seems impossible, most probably because of the parity
obstruction as mentioned in [Polymath8]. In this work we do not make any
attempt to break this notorious barrier, so we do not expect to outdo the
result of Chen – even assuming very strong distributional claims like $GEH$.
In order to push our results even more, we would like to apply a device called
an $\varepsilon$-trick, which made its debut in [Polymath8]. The idea is to
expand the sieve support even further than before, but at a cost of turning
certain asymptotics into lower bounds. This is also the place where the $\ell$
parameter starts to behave non-trivially.
###### Theorem 13 (Sieving on an epsilon-enlarged simplex).
Suppose that there is a fixed $\theta\in(0,\frac{1}{2})$ such that
$GEH[2\theta]$ holds, and arbitrarily chosen fixed real parameters $\ell>1$,
$\varepsilon\in[0,1)$, and $\eta\geq 1+\varepsilon$ subject to the constraint
$2\theta\eta+\frac{1}{\ell}\leq 1.$ (27)
Let $k\geq 2$ and $m\geq 1$ be fixed integers. For any fixed compactly
supported square-integrable function
$F\colon[0,+\infty)^{k}\rightarrow\mathbf{R}$, define the functionals
$\begin{split}J_{i,\varepsilon}(F):=&\int_{(1-\varepsilon)\cdot\mathcal{R}_{k-1}}\left(\int_{0}^{\infty}F(t_{1},\dots,t_{k})\,dt_{i}\right)^{2}dt_{1}\dots
dt_{i-1}\,dt_{i+1}\dots dt_{k},\\\
Q_{i,\varepsilon}(F):=&\int_{0}^{\frac{1}{\theta}}\frac{1-\ell\theta
y}{y}\int_{\Phi(y)\cdot\mathcal{R}_{k-1}}\left(\,\int_{0}^{\frac{1}{\theta}-y}\left(\partial_{y}^{(i)}F(t_{1},\dots,t_{k})\right)^{2}dt_{i}\right)dt_{1}\dots
dt_{i-1}\,dt_{i+1}\dots dt_{k}\,dy,\end{split}$ (28)
where $\Phi\colon[0,+\infty)\rightarrow\mathbf{R}$ is a function given by the
formula
$\Phi(y):=\begin{cases}1+\varepsilon,&\emph{for
}y\in\left[0,\frac{1}{\ell\theta}\right),\\\ 1-\varepsilon,&\emph{for
}y\in\left[\frac{1}{\ell\theta},\frac{1}{\theta}\right],\\\
0,&\emph{otherwise.}\end{cases}$ (29)
Let $\Omega_{k,\varepsilon}$ be the infimum
$\Omega_{k,\varepsilon}:=\inf_{\eta,F}\left(\frac{\sum_{i=1}^{k}\left(Q_{i,\varepsilon}(F)-\theta(\ell-1)J_{i,\varepsilon}(F)\right)}{I(F)}+\ell
k\right),$ (30)
over all square integrable functions $F$ that are supported on the region
$(1+\varepsilon)\cdot\mathcal{R}_{k}^{\prime}\,\cap\,\eta\cdot\mathcal{R}_{k},$
and are not identically zero up to almost everywhere equivalence. If
$m>\Omega_{k,\varepsilon},$
then $DHL_{\Omega}[k;m-1]$ holds. Moreover, if $\varepsilon=0$, then
constraint (27) can be discarded and the functional inside the parentheses in
(30) is constant with respect to the $\ell$ variable.
###### Remark 2.
Observe that Theorems 10 and 12 follow easily from Theorem 13. In the first
case we just consider $\varepsilon=0$ and $\eta=1$. To prove the latter, we
take the same $\varepsilon$ and any $\eta\geq k/(k-1)$.
Constraint (27) refers to the hypotheses mentioned in the ‘trivial case’ from
Proposition 8. Notice that we do not have to restrict the support of the
$Q_{i,\varepsilon}$ integrals for $y\in\left[0,\frac{1}{\ell\theta}\right)$,
because we do not apply any $EH$-like theorem/conjecture in this interval.
Below we present some upper bounds for $\Omega_{k,\varepsilon}$ obtained via
considering $\eta=1+\varepsilon$ and optimizing over polynomials of the form
$a+b(1-P_{1})+c(1-P_{1})^{2}$ for $-1<a,b,c<1$ supported on the simplex
$(1+\varepsilon)\cdot\mathcal{R}_{k}$:
Table E. Upper bounds for $\Omega_{k,\varepsilon}$.
$k$ | $\varepsilon$ | $\theta=1/4$ |
---|---|---|---
$2$ | 1/3 | 4.69949 |
$3$ | 1/4 | 7.75780 |
$4$ | 1/5 | 11.05320 |
$5$ | 1/6 | 14.54134 |
$6$ | 1/7 | 18.19060 |
$7$ | 1/9 | 21.99368 |
$8$ | 1/10 | 25.90287 |
$9$ | 1/10 | 29.90565 |
$10$ | 2/21 | 34.01755 |
We are also able to obtain the bound $33.93473$ for $k=10$ and the same
$\varepsilon$, if one optimizes over polynomials of the form
$a+b(1-P_{1})+c(1-P_{1})^{2}+d(1-P_{1})^{3}$ for $-1<a,b,c,d<1$. We observe
that results provided by the $\varepsilon$-trick are considerably stronger
than those listed in Table D for every $k\geq 3$. They surpass the currently
known value of $\varrho_{k}$ in (2) for $7\leq k\leq 10$. Let us also notice
that the bigger $k$ we take, the better improvement over Tables C and D we
obtain. The reason for this is that the region $\mathcal{R}^{\prime}_{k}$ is
much larger than simplex $\mathcal{R}_{k}$ for small $k$, but the difference
in size is far less spectacular for bigger values of $k$. In the same time,
the epsilon-enlarged simplex $(1+\varepsilon)\cdot\mathcal{R}_{k}$ does not
share this weakness.
###### Remark 3.
It is possible to consider other choices of $\eta$ than $1+\varepsilon$. One
of them is $(1+\varepsilon)k/(k-1)$, which gives an access to a larger domain
$(1+\varepsilon)\cdot\mathcal{R}_{k}^{\prime}$. However, expanding the sieve
support so far makes the constaint (27) more restrictive. As for this moment,
numerical experiments suggest that one loses more than wins by implementing
such a manouver. The author also tried excluding the fragment
$\\{(t_{1},\dots,t_{k})\in[0,+\infty)^{k}\colon\forall_{i\in\\{1,\dots,k\\}}\leavevmode\nobreak\
t_{1}+\dots+t_{i-1}+t_{i+1}+\dots+t_{k}>1-\varepsilon\\},$
motivated by the fact, that it contibutes neither to $J_{i,\varepsilon}(F)$,
nor the negative part of $Q_{i,\varepsilon}(F)$, and in the same time it
contributes to $I(F)$. Unfortunately, this technique did not generate any
substancial advantage.
## Lemmata
We have the following lemma enabling us to convert certain sums into
integrals.
###### Lemma 14.
Let $m\geq 1$ be a fixed integer and let
$f\colon(0,+\infty)^{m}\rightarrow\bf{C}$ be a fixed compactly supported,
Riemann integrable function. Then for $x>1$ we have
$\sum_{\begin{subarray}{c}p_{1},\dots,p_{m}\\\ p_{1}\cdots p_{m}\sim
x\end{subarray}}\,f\left(\log_{x}p_{1},\dots,\log_{x}p_{m}\right)=\left(c_{f}+o(1)\right)\frac{x}{\log
x},$
where
$c_{f}:=\int_{\begin{subarray}{c}t_{1}+\dots+t_{m}=1\end{subarray}}f(t_{1},\dots,t_{m})\frac{dt_{1}\dots
dt_{m-1}}{t_{1}\cdots t_{m}},$
where we lift Lebesgue measure $dt_{1}\dots dt_{m-1}$ up to the hyperplane
$t_{1}+\cdots+t_{m}=1$.
###### Proof.
Follows from prime number theorem combined with elementary properties of the
Riemann integral. ∎
We introduce an another useful lemma which helps us with discarding those
$n\sim x$ having low prime factors.
###### Lemma 15 (Almost primality).
Let $k\geq 1$ be fixed, let $(L_{1},\dots,L_{k})$ be a fixed admissible
$k$–tuple, and let $b\bmod W$ be such that $(L_{i}(b),W)=1$ for each
$i=1,\dots,k$. Let further
$F_{1},\dots,F_{k}\colon[0,+\infty)\rightarrow\mathbf{R}$ be fixed smooth
compactly supported functions, and let $m_{1},\dots,m_{k}\geq 0$ and
$a_{1},\dots,a_{k}\geq 1$ be fixed natural numbers. Then,
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\prod_{j=1}^{k}\left(\left|\lambda_{F_{j}}(L_{j}(n))\right|^{a_{j}}\tau(L_{j}(n))^{m_{j}}\right)\ll
B^{-k}\frac{x}{W}.$
Furthermore, if $1\leq j_{0}\leq k$ is fixed and $p_{0}$ is a prime with
$p_{0}\leq x^{1/10k}$, then we have the variant
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\prod_{j=1}^{k}\left(\left|\lambda_{F_{j}}(L_{j}(n))\right|^{a_{j}}\tau(L_{j}(n))^{m_{j}}\right)\mathbf{1}_{p_{0}|L_{j_{0}}(n)}\ll\frac{\log_{x}p_{0}}{p_{0}}B^{-k}\frac{x}{W}.$
As a consequence, we have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\prod_{j=1}^{k}\left(\left|\lambda_{F_{j}}(L_{j}(n))\right|^{a_{j}}\tau(L_{j}(n))^{m_{j}}\right)\mathbf{1}_{\textup{lpf}(L_{j_{0}}(n))\leq
x^{\epsilon}}\ll\epsilon B^{-k}\frac{x}{W},$
for any $\epsilon>0$.
###### Proof.
This is a trivial modification of [Polymath8, Proposition 4.2]. ∎
## 3 Proof of Propositions 8 and 9
Contraty to the numerical ordering, we tackle Proposition 9 first, because it
is going to be needed throughout the rest of this section.
### Propositon 9
###### Proof.
It suffices to show that
$\sum_{p}\,\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
p^{2}|\mathcal{P}(n)\end{subarray}}\tau(\mathcal{P}(n))\left|\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))\,\right|\,=\,o(1)\times
B^{-k}\frac{x}{W}.$ (31)
Choose an $\epsilon>0$. We decompose the outer sum in (31) as follows:
$\sum_{p}\leavevmode\nobreak\ =\leavevmode\nobreak\ \sum_{p\leq
x^{\epsilon}}\leavevmode\nobreak\ +\leavevmode\nobreak\
\sum_{p>x^{\epsilon}}.$ (32)
We apply the divisor bound $\tau(n)\ll n^{o(1)}$, valid for all
$n\in\mathbf{N}$, to conclude that the second sum from the right-hand side of
(32) is
$\ll x^{o(1)}\sum_{p>x^{\epsilon}}\sum_{\begin{subarray}{c}n\sim x\\\
p^{2}|\mathcal{P}(n)\end{subarray}}1\ll x^{1-\epsilon+o(1)}.$ (33)
The first sum, by the third part of Lemma 15, can be easily estimated as being
$\ll\leavevmode\nobreak\ \epsilon B^{-k}\frac{x}{W}.$
To this end, we only have to send $\epsilon\rightarrow 0$ sufficently slowly.
∎
### The trivial case of Proposition 8
###### Proof.
We shall take $i_{0}=k$, as the other cases can be proven exactly the same
way. Proposition 9 implies that our task is equivalent to showing that
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\sum_{p|L_{k}(n)}\Upsilon(\log_{x}p)\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))=(c+o(1))B^{-k}\frac{x}{W}.$
(34)
Interchanging the order of summation, we get that the left-hand side of (34)
equals
$\sum_{p}\Upsilon(\log_{x}p)\sum_{\begin{subarray}{c}d_{1},\dots,d_{k}\\\
e_{1},\dots,e_{k}\end{subarray}}\left(\prod_{i=1}^{k}\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})\right)S_{p}(d_{1},\dots,d_{k},e_{1},\dots,e_{k}),$
(35)
where
$S_{p}(d_{1},\dots,d_{k},e_{1},\dots,e_{k}):=\sum_{\begin{subarray}{c}n\sim
x\\\ n\equiv b\bmod W\\\ \forall_{i}\,[d_{i},e_{i}]|L_{i}(n)\\\
p|L_{k}(n)\end{subarray}}1.$ (36)
By hypotheses, all the $L_{i}(n)$ are coprime to $W$. We also assumed that for
all distinct $i,\,j$ we have $|A_{i}B_{j}-A_{j}B_{i}|<D_{0}$. On the other
hand, if there exists a prime $p_{0}$ dividing both $[d_{i},e_{i}]$ and
$[d_{j},e_{j}]$, then $A_{i}B_{j}-A_{j}B_{i}\equiv 0\bmod p_{0}$, which forces
$p_{0}\leq D_{0}$. By this contradiction, we may further assume in this
subsection that $W,\,[d_{1},e_{1}],\dots,[d_{k},e_{k}]$ are pairwise coprime,
because otherwise $S_{p}$ vanishes. We mark this extra constraint by the ′
sign next to the sum (see (41) for an example). Under these assumptions, we
can can merge the congruences appearing under the sum in (36) into one:
$n\equiv a\bmod q,$ (37)
where
$q:=W\,[d_{k},e_{k},p]\prod_{i=1}^{k-1}[d_{i},e_{i}]$ (38)
and $(a,q)=1$. This gives
$S_{p}(d_{1},\dots,d_{k},e_{1},\dots,e_{k})=\sum_{\begin{subarray}{c}n\sim
x\\\ n\equiv a\bmod q\end{subarray}}1\,=\,\frac{x}{q}+O(1).$ (39)
The net contribution of the $O(1)$ error term to (35) is at most
$\ll\,\left(\sum_{d,e\leq
x}\frac{1}{[d,e]}\right)^{k-1}\sum_{\begin{subarray}{c}d,e,p\leq
x\end{subarray}}\frac{1}{[d,e,p]}\ll\left(\sum_{r\leq
x}\frac{\tau(r)^{O(1)}}{r}\right)^{k}\leq x^{o(1)}.$ (40)
Therefore, it suffices to show that
$\sum_{p}\frac{\Upsilon(\log_{x}p)}{p}\left(\prod_{i=1}^{k}\sideset{}{{}^{\prime}}{\sum}_{d_{i},e_{i}}\frac{\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})}{\psi_{i}([d_{i},e_{i}])}\right)=(c+o(1))B^{-k},$
(41)
where
$\psi_{i}(n):=\begin{cases}n,&\text{for }i\in\\{1,\dots,k-1\\},\\\
[n,p]/p,&\text{for }i=k.\end{cases}$ (42)
By [Lewulis, Lemma 2.2 and Lemma 2.6] and the polarization argument we get
$\prod_{i=1}^{k}\sideset{}{{}^{\prime}}{\sum}_{d_{i},e_{i}}\frac{\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})}{\psi_{i}([d_{i},e_{i}])}=(c^{\prime}c^{\prime\prime}+o(1))B^{-k},$
(43)
with
$\displaystyle c^{\prime}$
$\displaystyle:=\prod_{i=1}^{k-1}\int_{0}^{1}F^{\prime}_{i}(t)G^{\prime}_{i}(t)\,dt,$
(44) $\displaystyle c^{\prime\prime}$
$\displaystyle:=\int_{0}^{1-\log_{x}p}\partial_{y}F^{\prime}_{k}(t)\,\partial_{y}G^{\prime}_{k}(t)\,dt\,dy.$
(45)
###### Remark 4.
To justify this application, we need to consider (under the notation used
within the cited work)
$\lambda_{d_{1},\dots,d_{k}}:=\prod_{i=1}^{k}\mu(d_{i})\widetilde{F}_{i}(\log_{x}d_{i})$
in one case and
$\lambda_{d_{1},\dots,d_{k}}:=\prod_{i=1}^{k}\mu(d_{i})\widetilde{G}_{i}(\log_{x}d_{i})$
in the other – we are permitted to choose these weights arbitrarily due to
[Lewulis, Lemma 1.12]. The key relationship in that paper between
$\lambda_{d_{1},\dots,d_{k}}$ and $y_{r_{1},\dots r_{k}}$ may be established
via [Lewulis, (1.20) and Lemma 2.6]. Then, from a simple formula
$\widetilde{F}^{2}-\widetilde{G}^{2}=(\widetilde{F}-\widetilde{G})(\widetilde{F}+\widetilde{G})$
we deduce that after defining $\widetilde{F}$, $\widetilde{G}$ in such a way
that $F=\widetilde{F}-\widetilde{G}$ and $G=\widetilde{F}+\widetilde{G}$, and
comparing the two mentioned choices of $\lambda_{d_{1},\dots,d_{k}}$, our
argument is completed.
The expression $1-\log_{x}p$ in the upper limit of the integral may seem a bit
artificial. Its role is to unify this part of Proposition 8 with the second
one. Now, it suffices to show that
$\sum_{p}\frac{\Upsilon(\log_{x}p)}{p}\int_{0}^{1-\log_{x}p}\partial_{y}F^{\prime}_{k}(t)\,\partial_{y}G^{\prime}_{k}(t)\,dt=\int_{0}^{1}\frac{\Upsilon(y)}{y}\int_{0}^{1-y}\partial_{y}F^{\prime}_{k}(t_{k})\,\partial_{y}G^{\prime}_{k}(t_{k})\,dt_{k}\,dy.$
(46)
This is a direct application of Lemma 14. ∎
### The Elliott–Halberstam case of Proposition 8
###### Proof.
As in the previous subsection, we can take $i_{0}=k$ without loss of
generality. Again, by Proposition 9 we have to prove that
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\sum_{p|L_{k}(n)}\Upsilon(\log_{x}p)\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))=(c+o(1))B^{-k}\frac{x}{W}.$
(47)
Take some $\epsilon>0$. We decompose the studied sum as follows:
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}=\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\text{lpf}(L_{k}(n))\leq
x^{\epsilon}\end{subarray}}+\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\\\ \text{lpf}(L_{k}(n))>x^{\epsilon}\end{subarray}}.$ (48)
We show that the contribution of the first sum from the right-hand side of
(48) is $\ll\epsilon B^{-k}xW^{-1}$. To do so we bound
$\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))\leq\frac{1}{2}\left(\lambda_{F_{i}}(L_{i}(n))^{2}+\lambda_{G_{i}}(L_{i}(n))^{2}\right)$
(49)
for each $i=1,\dots,k$. We also recall the trivial inequality
$\sum_{p|L_{k}(n)}\Upsilon(\log_{x}p)\ll\tau(L_{k}(n)).$ (50)
By (49) and (50) we can present the first sum from the right-hand side of (48)
as a linear combination of sums that can be threated straightforwardly by
Lemma 15.
Let us define a function
$\Omega^{\flat}(n):=\sum_{\begin{subarray}{c}p|n\\\
p>x^{\epsilon}\end{subarray}}\Upsilon(\log_{x}p)$
Now, it sufficies to show that for any $\epsilon>0$ we have
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\text{lpf}(L_{k}(n))>x^{\epsilon}\end{subarray}}\Omega^{\flat}(L_{k}(n))\prod_{i=1}^{k}\lambda_{F_{i}}(L_{i}(n))\lambda_{G_{i}}(L_{i}(n))=(c_{\epsilon}+o(1))B^{-k}\frac{x}{W},$
(51)
where $c_{\epsilon}\rightarrow c$ when $\epsilon\rightarrow 0$. After
expanding the $\lambda_{F_{i}},\,\lambda_{G_{i}}$ we conclude that the left-
hand side of (51) equals
$\sum_{\begin{subarray}{c}d_{1},\dots,d_{k-1}\\\
e_{1},\dots,e_{k-1}\end{subarray}}\left(\prod_{i=1}^{k-1}\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})\right)S_{\epsilon}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1}),$
(52)
where
$S_{\epsilon}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1}):=\sum_{\begin{subarray}{c}n\sim
x\\\ n\equiv b\bmod W\\\ \text{lpf}(L_{k}(n))>x^{\epsilon}\\\
\forall_{i\not=k}\,[d_{i},e_{i}]|L_{i}(n)\end{subarray}}\Omega^{\flat}(L_{k}(n))\,\lambda_{F_{k}}(L_{k}(n))\,\lambda_{G_{k}}(L_{k}(n)).$
(53)
Notice that $n\equiv b\bmod W$ implies that all of the $L_{i}(n)$ are coprime
to $W$. We also assumed that for all distinct $i,\,j$ we have
$|A_{i}B_{j}-A_{j}B_{i}|<D_{0}$, so if there exists a prime $p_{0}$ dividing
both $[d_{i},e_{i}]$ and $[d_{j},e_{j}]$, then $A_{i}B_{j}-A_{j}B_{i}\equiv
0\bmod p_{0},$ which forces $p_{0}\leq D_{0}$. That is a contradiction.
Therefore, we may further assume in this subsection that
$W,\,[d_{1},e_{1}],\dots,[d_{k},e_{k}]$ are pairwise coprime and that
$\text{lpf}\,([d_{k},e_{k}])>x^{\epsilon}$, because otherwise $S_{\epsilon}$
vanishes. Under these assumptions we can merge all the congruences under the
sum (53) into two:
$n\equiv a\bmod q\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ L_{k}(n)\equiv
0\bmod[d_{k},e_{k},p],$ (54)
where we redefine $q$ and $a$ as
$q:=W\prod_{i=1}^{k-1}[d_{i},e_{i}],$ (55)
and $a$ being some residue class coprime to its modulus such that
$(L_{i}(a),W)=1$ for each possible choice of index $i$. This gives
$S_{\epsilon}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})=\sum_{\begin{subarray}{c}n\sim
x\\\ \text{lpf}(L_{k}(n))>x^{\epsilon}\\\ n\equiv a\bmod
q\end{subarray}}\Omega^{\flat}(L_{k}(n))\,\lambda_{F_{k}}(L_{k}(n))\,\lambda_{G_{k}}(L_{k}(n)).$
(56)
We would like to perform a substitution $m:=L_{k}(n)$ in the sum from (56), so
we have to transform the congruence $n\equiv a\bmod q$ appropriately. In order
to do so, we split it into two: $n\equiv a\bmod[A_{k},q]/A_{k}$ and $n\equiv
a\bmod\mbox{rad}\,A_{k}$, where $\mbox{rad}\,A_{k}$ denotes the square-free
part of $A_{k}$. The former congruence is simply equivalent to $m\equiv
L_{k}(a)\bmod[A_{k},q]/A_{k}$. The latter is equivalent to $m\equiv
L_{k}(a)\bmod A_{k}\,\text{rad}\,A_{k}$ and it also implies $m\equiv
B_{k}\bmod A_{k}$, which has to be satisfied by our substitution. Note that
$(L_{k}(a),[A_{k},q]/A_{k})=(L_{k}(a),A_{k}\,\mbox{rad}A_{k})=1,$ (57)
so we can combine the two considered congruences into one $m\equiv
a^{\prime}\bmod[A_{k},q]\,\mbox{rad}A_{k}$. Hence,
$S_{\epsilon}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})=\sum_{\begin{subarray}{c}A_{k}x+B_{k}<m\leq
2A_{k}x+B_{k}\\\ \text{lpf}(m)>x^{\epsilon}\\\ m\equiv a^{\prime}\bmod
q^{\prime}\end{subarray}}\Omega^{\flat}(m)\,\lambda_{F_{k}}(m)\,\lambda_{G_{k}}(m),$
(58)
where $q^{\prime}:=[A_{k},q]\,\mbox{rad}\,A_{k}=qA_{k}$ and $a^{\prime}$ is a
residue class $\bmod\,q$ coprime to its modulus. Thus, we have
$S_{\epsilon}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})=\frac{1}{\varphi(q^{\prime})}\sum_{\begin{subarray}{c}A_{k}x+B_{k}<m\leq
2A_{k}x+B_{k}\\\
(m,q^{\prime})=1\end{subarray}}\Omega^{\flat}(m)\lambda_{F_{k}}(m)\lambda_{G_{k}}(m)\mathbf{1}_{\text{lpf}(m)>x^{\epsilon}}\,\\\
+\Delta\left(\Omega^{\flat}\lambda_{F_{k}}\lambda_{G_{k}}\mathbf{1}_{\text{lpf}(\cdot)>x^{\epsilon}}\mathbf{1}_{[A_{k}x+B_{k},2A_{k}x+B_{k}]};a^{\prime}\bmod
q^{\prime}\right).$ (59)
We split
$\sum_{p}S_{\epsilon}=S_{1}-S_{2}+S_{3},$ (60)
where
$\begin{split}S_{1}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})&=\frac{1}{\varphi(q^{\prime})}\sum_{p}\Upsilon(\log_{x}p)\sum_{\begin{subarray}{c}A_{k}x+B_{k}<m\leq
2A_{k}x+B_{k}\\\
p|m\end{subarray}}\lambda_{F_{k}}(m)\lambda_{G_{k}}(m)\mathbf{1}_{\text{lpf}(m)>x^{\epsilon}},\\\
S_{2}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})&=\frac{1}{\varphi(q^{\prime})}\sum_{\begin{subarray}{c}A_{k}x+B_{k}<m\leq
2A_{k}x+B_{k}\\\
(m,q^{\prime})>1\end{subarray}}\Omega^{\flat}(m)\lambda_{F_{k}}(m)\lambda_{G_{k}}(m)\mathbf{1}_{\text{lpf}(m)>x^{\epsilon}},\\\
S_{3}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1})&=\Delta\left(\Omega^{\flat}\lambda_{F_{k}}\lambda_{G_{k}}\mathbf{1}_{\text{lpf}(\cdot)>x^{\epsilon}}\mathbf{1}_{[A_{k}x+B_{k},2A_{k}x+B_{k}]};a^{\prime}\bmod
q^{\prime}\right).\end{split}$ (61)
For $j\in\\{1,2,3\\}$ we put
$\Sigma_{j}=\sum_{\begin{subarray}{c}d_{1},\dots,d_{k-1}\\\
e_{1},\dots,e_{k-1}\end{subarray}}\left(\prod_{i=1}^{k-1}\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})\right)S_{j}(d_{1},\dots,d_{k-1},e_{1},\dots,e_{k-1}).$
(62)
Therefore, it suffices to derive the main term estimate
$\Sigma_{1}=(c_{\epsilon}+o(1))B^{-k}\frac{x}{W},\\\ $ (63)
the ‘correction’ error term estimate
$\Sigma_{2}\ll x^{1-\epsilon+o(1)},\\\ $ (64)
and the ‘GEH-type’ error term estimate
$\Sigma_{3}\ll x\log^{-A}x$ (65)
for any fixed $A>0$.
Let us begin with (64). We observe that since $\text{lpf}(m)>x^{\epsilon}$,
there exists a prime $x^{\epsilon}<p\leq x$ dividing both $m$ and one of
$d_{1},e_{1},\dots,d_{k-1},e_{k-1}$ (if $k=1$, then $\Sigma_{2}$ vanishes; we
also claim that $\epsilon$ tends to 0 slowly enough to ensure that
$D_{0}<x^{\epsilon}$). Thus, we may safely assume that $p|d_{1}$, for the
remaining $2k-3$ cases are analogous. Hence, we get
$\Sigma_{2}\ll x^{o(1)}\sum_{x^{\epsilon}<p\leq
x}\sum_{\begin{subarray}{c}d_{1},\dots,d_{k-1}\leq x\\\
e_{1},\dots,e_{k-1}\leq x\\\
p|d_{1}\end{subarray}}\prod_{i=1}^{k-1}\frac{1}{\varphi([d_{i},e_{i}])}\leavevmode\nobreak\
\sum_{\begin{subarray}{c}n\ll x\\\ p|n\end{subarray}}1\ll
x^{1+o(1)}\sum_{x^{\epsilon}<p\leq x}\frac{1}{p^{2}}\ll x^{1-\epsilon+o(1)}.$
(66)
To deal with (65) we just repeat the reasoning from [Polymath8, Subsection
‘The generalized Elliott-Halberstam case’, Eq (62)] combined with
$\Omega^{\flat}(m)=O(1/\epsilon)$.
Let us move to (63). We have
$\varphi(q^{\prime})=A_{k}\varphi\left(W\prod_{i=1}^{k-1}[d_{i},e_{i}]\right),$
so again by [Lewulis, Lemma 2.6] (or [Polymath8, Lemma 4.1] for an even more
direct application) we get
$\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}d_{1},\dots,d_{k-1}\\\
e_{1},\dots,e_{k-1}\end{subarray}}\frac{\prod_{i=1}^{k-1}\mu(d_{i})\mu(e_{i})F_{i}(\log_{x}d_{i})G_{i}(\log_{x}e_{i})}{\varphi\left(q^{\prime}\right)}=\frac{A_{k}^{-1}}{\varphi(W)}(c^{\prime}+o(1))B^{1-k},$
(67)
where
$c^{\prime}:=\prod_{i=1}^{k-1}\int_{0}^{1}F^{\prime}_{i}(t)G^{\prime}_{i}(t)\,dt.$
By (61) it suffices to show that
$\sum_{p}\,\Upsilon(\log_{x}p)\sum_{\begin{subarray}{c}A_{k}x+B_{k}<m\leq
2A_{k}x+B_{k}\\\
p|m\end{subarray}}\lambda_{F_{k}}(m)\lambda_{G_{k}}(m)\mathbf{1}_{\text{lpf}(m)>x^{\epsilon}}=\left(c_{\epsilon}^{\prime\prime}+o(1)\right)\frac{A_{k}x}{\log
x},$ (68)
where $c_{\epsilon}^{\prime\prime}$ satisfies
$\lim_{\epsilon\rightarrow
0}c_{\epsilon}^{\prime\prime}=\Upsilon(1)\,F_{k}(0)\,G_{k}(0)\leavevmode\nobreak\
+\leavevmode\nobreak\
\int_{0}^{1}\frac{\Upsilon(y)}{y}\int_{0}^{1-y}\partial_{y}F^{\prime}_{k}(t)\,\partial_{y}G^{\prime}_{k}(t)\,dt\,dy.$
(69)
We simplify the restriction $A_{k}x+B_{k}<m\leq 2A_{k}x+B_{k}$ into $m\sim
A_{k}x$ at the cost of introducing to the left-hand side of (68) an error term
of size not greater than $x^{o(1)}$. We factorize $m=p_{1}\cdots p_{r}p$ for
some $x^{\epsilon}\leq p_{1}\leq\dots\leq p_{r}\leq 2A_{k}x$, $p\geq
x^{\epsilon}$, and $0\leq r\leq\frac{1}{\epsilon}$. The contribution of those
$m$ having repeated prime factors is readily $\ll x^{1-\epsilon}$, so we can
safely assume that $m$ is square-free. In such a case, we get
$\lambda_{F_{k}}(m)=(-1)^{r}\partial_{\,\log p_{1}}\dots\partial_{\,\log
p_{r}}(\partial_{\,\log p}F_{k}(0))$ (70)
and an analogous equation for $\lambda_{G_{k}}(m)$. Therefore, the left-hand
side of (68) equals
$\sum_{0\leq
r\leq\frac{1}{\epsilon}}\,\sum_{p}\Upsilon(\log_{x}p)\sum_{\begin{subarray}{c}x^{\epsilon}<p_{1}<\dots<p_{r}\\\
p_{1}\dots p_{r}p\,\sim A_{k}x\end{subarray}}\partial_{\,\log
p_{1}}\dots\partial_{\,\log p_{r}}(\partial_{\,\log
p}F_{k}(0))\,\cdot\,\partial_{\,\log p_{1}}\dots\partial_{\,\log
p_{r}}(\partial_{\,\log p}G_{k}(0)).$ (71)
Note that for the index $r=0$ the summand above equals
$(\Upsilon(1)+o(1))\sum_{p\sim A_{k}x}\,F_{k}(0)\,G_{k}(0).$ (72)
We apply Lemma 14 to (71–72) and obtain an asymptotic (68) with
$\begin{split}c_{\epsilon}^{\prime\prime}=\sum_{1\leq
r\leq\frac{1}{\epsilon}}\int_{0}^{1}\Upsilon(y)\int_{\begin{subarray}{c}\phantom{2}\\\
t_{1}+\dots+t_{r}=1-y\\\
\epsilon<t_{1}<\dots<t_{r}\end{subarray}}\partial_{t_{1}}\dots\partial_{t_{r}}(\partial_{y}F_{k}(0))\cdot\partial_{t_{1}}\dots\partial_{t_{r}}(\partial_{y}G_{k}(0))\frac{dy\,dt_{1}\dots
dt_{r-1}}{y\,t_{1}\cdots t_{r}}\\\ +\leavevmode\nobreak\
\Upsilon(1)\,F_{k}(0)\,G_{k}(0).\end{split}$ (73)
The first part of Lemma 15 gives us $c_{\epsilon}^{\prime\prime}\ll 1$ when
$\epsilon\rightarrow 0^{+}$. Now, consider any sequence of positive numbers
$(\epsilon_{1},\epsilon_{2},\dots)$ satisfying $\epsilon_{n}\rightarrow 0$ as
$n\rightarrow\infty$. In view of (68) and the last part of Lemma 15, we
conclude that
$\left(c_{\epsilon_{1}}^{\prime\prime},c_{\epsilon_{2}}^{\prime\prime}\dots\right)$
forms a Cauchy sequence, and hence it has a limit. Thus, by dominated
convergence theorem it suffices to establish for each $y\in[0,1]$ the
following equality
$\sum_{r\geq 1}\int_{\begin{subarray}{c}\phantom{2}\\\
t_{1}+\dots+t_{r}=1-y\\\
0<t_{1}<\dots<t_{r}\end{subarray}}\partial_{t_{1}}\dots\partial_{t_{r}}(\partial_{y}F_{k}(0))\cdot\partial_{t_{1}}\dots\partial_{t_{r}}(\partial_{y}G_{k}(0))\frac{dt_{1}\dots
dt_{r-1}}{t_{1}\cdots t_{r}}\\\
=\int_{0}^{1-y}\partial_{y}F_{k}^{\prime}(t)\,\partial_{y}G_{k}^{\prime}(t)\,dt.$
(74)
By depolarization argument it suffices to show that for each $y\in[0,1]$, we
have
$\sum_{r\geq 1}\int_{\begin{subarray}{c}\phantom{2}\\\
t_{1}+\dots+t_{r}=1-y\\\
0<t_{1}<\dots<t_{r}\end{subarray}}\left|\partial_{t_{1}}\dots\partial_{t_{r}}(\partial_{y}F(0))\right|^{2}\frac{\,dt_{1}\dots
dt_{r-1}}{t_{1}\cdots
t_{r}}=\int_{0}^{1-y}\left|\partial_{y}F^{\prime}(t)\right|^{2}\,dt$ (75)
for any smooth $F\colon[0,\infty)\rightarrow\mathbf{R}$. For the sake of
clarity, we relabel $\partial_{y}F(x)$ as $H(x)$. We substitute $u:=t/(1-y)$
and $u_{i}:=t_{i}/(1-y)$ for all possible choices of $i$. With these settings
(75) is equivalent to
$\sum_{r\geq 1}\int_{\begin{subarray}{c}\phantom{2}\\\ u_{1}+\dots+u_{r}=1\\\
0<u_{1}<\dots<u_{r}\end{subarray}}\left|\partial_{(1-y)u_{1}}\dots\partial_{(1-y)u_{r}}H(0)\right|^{2}\frac{\,du_{1}\dots
du_{r-1}}{u_{1}\cdots
u_{r}}=(1-y)^{2}\int_{0}^{1}\left|H^{\prime}(u(1-y))\right|^{2}\,du.$ (76)
Note that one of the $(1-y)$ appeared from transforming $t_{r}\mapsto u_{r}$.
Put $\widetilde{H}(x):=H(x(1-y))$. We get
$\partial_{(1-y)u_{1}}\dots\partial_{(1-y)u_{r}}H(0)=\partial_{u_{1}}\dots\partial_{u_{r}}\widetilde{H}(0),$
and $\widetilde{H}^{\prime}(x)=(1-y)H^{\prime}(x(1-y))$ by the chain rule.
Thus, it suffices to show that
$\sum_{r\geq 1}\int_{\begin{subarray}{c}\phantom{2}\\\ u_{1}+\dots+u_{r}=1\\\
0<u_{1}<\dots<u_{r}\end{subarray}}\left|\partial_{u_{1}}\dots\partial_{u_{r}}\widetilde{H}(0)\right|^{2}\frac{\,du_{1}\dots
du_{r-1}}{u_{1}\cdots
u_{r}}=\int_{0}^{1}\left|\widetilde{H}^{\prime}(u)\right|^{2}\,du.$ (77)
To this end, we apply the key combinatorial identity [Polymath8, (67)]. ∎
## 4 Proof of Theorem 13
###### Proof.
Let $k,m,\varepsilon,\theta,\ell$ be as in Theorem 13. Let us assume that we
have a non-zero square integrable function
$F\colon[0,+\infty)^{k}\rightarrow\mathbf{R}$ supported on
$(1+\varepsilon)\cdot\mathcal{R}_{k}^{\prime}\cap\,\eta\cdot\mathcal{R}_{k}$
and satisfying
$\frac{\sum_{i=1}^{k}\left(Q_{i,\varepsilon}(F)-\theta(\ell-1)J_{i,\varepsilon}(F)\right)}{I(F)}+\ell
k<m.$ (78)
Now, we perform an analogous sequence of simplifications as in [Polymath8,
(72–84)] and eventually arrive at a non-zero smooth function
$f\colon\mathbf{R}^{k}\rightarrow\mathbf{R}$ being the linear combination of
tensor products – namely
$f(t_{1},\dots,t_{k})=\sum_{j=1}^{J}c_{j}f_{1,j}(t_{1})\cdots f_{k,j}(t_{j})$
(79)
with $J$, $c_{j}$, $f_{i,j}$ fixed, for which all the components
$f_{1,j}(t_{1}),\dots,f_{k,j}(t_{k})$ are supported on the region
$\left\\{(t_{1},\dots,t_{k})\in\mathbf{R}^{k}\colon\sum_{i=1}^{k}\max\left(t_{i},\delta\right)\leq\theta\eta-\delta\right\\}\\\
\cap\left\\{(t_{1},\dots,t_{k})\in\mathbf{R}^{k}\colon\forall_{1\leq i_{0}\leq
k}\sum_{\begin{subarray}{c}1\leq i\leq k\\\
i\not=i_{0}\end{subarray}}\max\left(t_{i},\delta\right)\leq(1+\varepsilon)\theta-\delta\right\\}$
(80)
for some sufficently small $\delta>0$ – that obeys
$\frac{\sum_{i=1}^{k}\left(\widetilde{Q}_{i,\varepsilon}(f)-(\ell-1)\widetilde{J}_{i,\varepsilon}(f)\right)}{\widetilde{I}(f)}+\ell
k<m,$ (81)
where
$\displaystyle\widetilde{I}(f):=$
$\displaystyle\int\limits_{[0,+\infty)^{k}}\left|\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f(t_{1},\dots,t_{k})\right|^{2}dt_{1}\dots dt_{k},$
(82) $\displaystyle\widetilde{J}_{i,\varepsilon}(f):=$
$\displaystyle\int\limits_{(1-\varepsilon)\theta\cdot\mathcal{R}_{k-1}}\left|\frac{\partial^{k-1}}{\partial
t_{1}\dots\partial t_{i-1}\partial t_{i+1}\dots\partial
t_{k}}f(t_{1},\dots,t_{i-1},0,t_{i+1},\dots,t_{k})\right|^{2}dt_{1}\dots
dt_{i-1}dt_{i+1}\dots dt_{k},$
$\displaystyle\widetilde{Q}_{i,\varepsilon}(f):=$
$\displaystyle\int\limits_{0}^{1}\frac{1-\ell
y}{y}\int\limits_{{\Psi}(y)\cdot\mathcal{R}_{k-1}}\left(\int\limits_{0}^{1-y}\left|\partial_{y}^{(i)}\frac{\partial^{k}}{\partial
t_{1}\dots\partial
t_{k}}f(t_{1},\dots,t_{k})\right|^{2}dt_{i}\right)dt_{1}\dots
dt_{i-1}\,dt_{i+1}\dots dt_{k}\,dy,$
with $\Psi\colon[0,+\infty)\rightarrow\mathbf{R}$ being a function given as
$\Psi(y):=\begin{cases}1+\varepsilon,&\text{for
}y\in\left[0,\frac{1}{\ell}\right),\\\ 1-\varepsilon,&\text{for
}y\in\left[\frac{1}{\ell},1\right],\\\ 0,&\text{otherwise.}\end{cases}$ (83)
We construct a non-negative sieve weight
$\nu\colon\mathbf{N}\rightarrow\mathbf{Z}$ by the formula
$\nu(n):=\left(\sum_{j=1}^{J}c_{j}\lambda_{f_{1,j}}(L_{1}(n))\cdots\lambda_{f_{k,j}}(L_{k}(n))\right)^{2}.$
(84)
Notice that if $\varepsilon>0$, then for any $1\leq j,j^{\prime}\leq J$ we
have
$\sum_{i=1}^{k}(S(f_{i,j})+S(f_{i,j^{\prime}}))<2\theta\eta<1$ (85)
from the $2\theta\eta+\frac{1}{\ell}\leq 1$ assertion. On the flip side, if
$\varepsilon=0$, then $\text{supp}(F)\subset\mathcal{R}_{k}^{\prime}$ and
consequently for every $1\leq i_{0}\leq k$ we have
$\sum_{\begin{subarray}{c}1\leq i\leq k\\\
i\not=i_{0}\end{subarray}}(S(f_{i,j})+S(f_{i,j^{\prime}}))<2\theta.$ (86)
Applying results from [Polymath8, Subsection ‘Proof of Theorem 3.12’], we get
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod
W\end{subarray}}\nu(n)=\left(\alpha+o(1)\right)B^{-k}\frac{x}{W},$ (87)
where
$\alpha=\widetilde{I}(f).$
Now, let us consider the sum
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\nu(n)\sum_{p|L_{k}(n)}\left(1-\ell\log_{x}p\right).$ (88)
We can expand the sum above as a linear combination of expressions
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\sum_{p|L_{k}(n)}\left(1-\ell\log_{x}p\right)\prod_{i=1}^{k}\lambda_{f_{i,j}}(L_{i}(n))\lambda_{f_{i,j^{\prime}}}(L_{i}(n))$
(89)
for various $1\leq j,j^{\prime}\leq J$. We seek for the upper bound of the sum
(89). We can achieve this goal by applying Proposition 8. We also observe that
the first part of this result should be more effective for smaller values of
$p$, and the second part for larger values of $p$. Therefore, we perform a
decomposition of the expression (89) as follows:
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\sum_{p|L_{k}(n)}=\sum_{\begin{subarray}{c}n\sim x\\\
n\equiv b\bmod W\\\ \mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\left(\sum_{\begin{subarray}{c}p|L_{k}(n)\\\ p\leq
x^{1/\ell}\end{subarray}}+\sum_{\begin{subarray}{c}p|L_{k}(n)\\\
p>x^{1/\ell}\end{subarray}}\right).$ (90)
For the $p\leq x^{\ell}$ sum we apply the trivial case of Proposition 8 with
$\vartheta_{0}=1/\ell$ and
$\Upsilon(y)=(1-\ell y)\mathbf{1}_{y\leq 1/\ell}.$
Under these assumptions we have
$\displaystyle\sum_{i=1}^{k}\left(S(f_{i,j})+S(f_{i,j^{\prime}})\right)$
$\displaystyle<2\theta(1+\varepsilon)\leq 1-\frac{1}{\ell},$ (91)
$\displaystyle S(\Upsilon)$ $\displaystyle\leq\frac{1}{\ell},$ (92)
so the necessary hypotheses from the ‘trivial case’ of Proposition 8 are
indeed satisfied. Observe that under $\varepsilon=0$ the inequality (91)
satisfies the second case of Proposition 8, so in these circumstances we do
not have to rely on the constraint (27) any longer. Thus, we get
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\nu(n)\sum_{\begin{subarray}{c}p|L_{i}(n)\\\ p\leq
x^{1/\ell}\end{subarray}}\left(1-\ell\log_{x}p\right)=\left(\beta_{k}^{(1)}+\,o(1)\right)B^{-k}\frac{x}{W},$
(93)
where
$\begin{split}\beta_{k}^{(1)}=\sum_{j,j^{\prime}=1}^{J}c_{j}c_{j^{\prime}}\left(\int\limits_{0}^{1}\frac{\Upsilon(y)}{y}\int\limits_{0}^{1-y}\partial_{y}f_{k,j}^{\prime}(t_{k})\,\partial_{y}f_{k,j^{\prime}}^{\prime}(t_{k})\,dt_{k}\,dy\right)\prod_{i=1}^{k-1}\left(\int\limits_{0}^{1}f_{k,j}^{\prime}(t_{i})\,f_{k,j^{\prime}}^{\prime}(t_{i})\,dt_{i}\right).\end{split}$
(94)
From (84) we see that $\beta_{k}^{(1)}$ factorizes as
$\beta_{k}^{(1)}=\int\limits_{0}^{1/\ell}\frac{1-\ell
y}{y}\int\limits_{(1+\varepsilon)\theta\cdot\mathcal{R}^{k-1}}\int\limits_{0}^{1-y}\left|\partial_{y}^{(k)}\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f(t_{1},\dots,t_{k})\right|^{2}dt_{i}\,dt_{1}\dots
dt_{k-1}\,dy.$ (95)
Now we deal with the $p>x^{\ell}$ case. We apply the $GEH$ case of Proposition
8 with $\vartheta=1/2$ and
$\Upsilon(y)=(1-\ell y)\mathbf{1}_{y>1/\ell}.$
We decompose $\\{1,\dots,J\\}$ into $\mathcal{J}_{1}\cup\mathcal{J}_{2}$,
where $\mathcal{J}_{1}$ consists of those indices $j\in\\{1,\dots,J\\}$
satisfying
$\sum_{i=1}^{k-1}S(f_{i,j})<(1-\varepsilon)\theta,$ (96)
and $\mathcal{J}_{2}$ is the complement. As in [Polymath8] we apply the
elementary inequality
$(x_{1}+x_{2})^{2}\geq(x_{1}+2x_{2})x_{1}$
to obtain the pointwise lower bound
$\begin{split}\nu(n)\geq\left(\left(\sum_{j\in\mathcal{J}_{1}}+\leavevmode\nobreak\
2\sum_{j\in\mathcal{J}_{2}}\right)c_{j}\lambda_{f_{1,j}}(L_{1}(n))\cdots\lambda_{f_{k,j}}(L_{k}(n))\right)\left(\sum_{j^{\prime}\in\mathcal{J}_{1}}c_{j^{\prime}}\lambda_{f_{1,j^{\prime}}}(L_{1}(n))\cdots\lambda_{f_{k,j^{\prime}}}(L_{k}(n))\right).\end{split}$
(97)
Therefore, if $j\in\mathcal{J}_{1}\cup\mathcal{J}_{2}$ and
$j^{\prime}\in\mathcal{J}_{1}$, then from (96) one has
$\sum_{i=1}^{k-1}\left(S(f_{i,j})+S(f_{i,j^{\prime}})\right)<2\theta,$
so the hypothesis from the ‘Generalised Elliott–Halberstam’ case of
Proposition 8 is indeed satisfied. Thus, by Proposition 8 and (97) we get
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\nu(n)\sum_{\begin{subarray}{c}p|L_{i}(n)\\\
p>x^{1/\ell}\end{subarray}}\left(1-\ell\log_{x}p\right)\leq\left(\beta_{k}^{(2)}+\,o(1)\right)B^{-k}\frac{x}{W},$
(98)
where
$\begin{split}\beta_{k}^{(2)}=\left(\sum_{j\in\mathcal{J}_{1}}+\leavevmode\nobreak\
2\sum_{j\in\mathcal{J}_{2}}\right)\sum_{j^{\prime}\in\mathcal{J}_{1}}c_{j}c_{j^{\prime}}\left(\Upsilon(1)\,f_{k,j}(0)\,f_{k,j^{\prime}}(0)+\int\limits_{0}^{1}\frac{\Upsilon(y)}{y}\int\limits_{0}^{1-y}\partial_{y}f_{k,j}^{\prime}(t_{k})\,\partial_{y}f_{k,j^{\prime}}^{\prime}(t_{k})\,dt_{k}\,dy\right)\\\
\times\,\prod_{i=1}^{k-1}\left(\int\limits_{0}^{1}f_{k,j}^{\prime}(t_{i})\,f_{k,j^{\prime}}^{\prime}(t_{i})\,dt_{i}\right).\end{split}$
(99)
For $s=1,2$ let us define
$f_{s}(t_{1},\dots,t_{k}):=\sum_{j\in\mathcal{J}_{s}}c_{j}f_{1,j}(t_{1})\cdots
f_{k,j}(t_{k}).$
From (84) we observe that $\beta_{k}^{(2)}$ can be factorized as
$\beta_{k}^{(2)}=\beta_{k}^{(2,1)}+\beta_{k}^{(2,2)},$ (100)
where
$\beta_{k}^{(2,1)}:=\int\limits_{1/\ell}^{1}\frac{1-\ell
y}{y}\int\limits_{(1-\varepsilon)\theta\cdot\mathcal{R}^{k-1}}\int\limits_{0}^{1-y}\left(\partial_{y}^{(k)}\frac{\partial^{k}}{\partial
t_{1}\dots\partial
t_{k}}f_{1}(t_{1},\dots,t_{k})+2\partial_{y}^{(k)}\frac{\partial^{k}}{\partial
t_{1}\dots\partial t_{k}}f_{2}(t_{1},\dots,t_{k})\right)\\\
\times\partial_{y}^{(k)}\frac{\partial^{k}}{\partial t_{1}\dots\partial
t_{k}}f_{1}(t_{1},\dots,t_{k})\,dt_{i}\,dt_{1}\dots dt_{k-1}\,dy$
and
$\beta_{k}^{(2,2)}:=(1-\ell)\int\limits_{(1-\varepsilon)\theta\cdot\mathcal{R}_{k-1}}\left(\frac{\partial^{k-1}}{\partial
t_{1}\dots\partial
t_{k-1}}f_{1}(t_{1},\dots,t_{k-1},0)+2\frac{\partial^{k-1}}{\partial
t_{1}\dots\partial t_{k-1}}f_{2}(t_{1},\dots,t_{k-1},0)\right)\\\
\times\frac{\partial^{k-1}}{\partial t_{1}\dots\partial
t_{k-1}}f_{1}(t_{1},\dots,t_{k-1},0)\,dt_{1}\dots dt_{k-1}.$
Let $\delta_{1}>0$ be a sufficiently small fixed quantity. By a smooth
partitioning, we may assume without loss of generality that all of the
$f_{i,j}$ are supported on intervals of length at most $\delta_{1}$, while
keeping the sum
$\sum_{j=1}^{J}|c_{j}||f_{1,j}(t_{1})|\cdots|f_{k,j}(t_{k})|$
bounded uniformly in $t_{1},\dots,t_{k}$ and in $\delta_{1}$. Therefore, the
supports of $f_{1}$ and $f_{2}$ overlap only on some set of measure at most
$O(\delta_{1})$. Hence, we conclude that
$\beta_{k}:=\beta_{k}^{(1)}+\beta_{k}^{(2)}=\,\widetilde{J}_{k,\varepsilon}(f)+\widetilde{Q}_{k,\varepsilon}(f)+O(\delta_{1}),$
(101)
which implies
$\sum_{\begin{subarray}{c}n\sim x\\\ n\equiv b\bmod W\\\
\mathcal{P}(n)\textup{ sq-
free}\end{subarray}}\nu(n)\sum_{p|L_{k}(n)}\left(1-\ell\log_{x}p\right)\leq\left(\beta_{k}+o(1)\right)B^{-k}\frac{x}{W}.$
(102)
A similar argument provides results analogous to (102) for all remaining
indices $1\leq i\leq k-1$. If we set $\delta_{1}$ to be small enough, then the
claim $DHL_{\Omega}[k;\varrho_{k}]$ follows from Lemma 3 and (81). We also
note that if $\varepsilon=0$, then (102) becomes an equality, because in this
case we have $\mathcal{J}_{2}=\emptyset$. ∎
## 5 Solving variational problems
In this Section we focus on applying Theorems 10, 12, and 13 to prove Theorem
4.
### 5.1 Proof of Theorem 11
###### Proof.
This is a direct application of Theorem 10. We choose
$F(t_{1},\dots,t_{k})=\bar{f}(t_{1}+\dots+t_{k})$ for a function
$\bar{f}\colon[0,+\infty)\rightarrow\mathbf{R}$ defined as
$\bar{f}(x):=\begin{cases}f(x),&\text{for }x\in[0,1],\\\
0,&\text{otherwise.}\end{cases}$ (103)
We also set $\ell=1$, so the contribution from $J_{i}(F)$ vanishes for each
possible choice of index $i$.
First, we calculate $I(F)$. We substitute $t_{1}+\dots+t_{k}\mapsto t$ and
leave $t_{j}$ the same for $j=2,\dots,k$. We get
$I(F)=\int\limits_{0}^{1}f(t)^{2}\left(\int\limits_{t\cdot\mathcal{R}_{k-1}}dt_{2}\dots
dt_{k}\right)dt\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{(k-1)!}\,\int\limits_{0}^{1}f(t)^{2}\,t^{k-1}dt\leavevmode\nobreak\
=\leavevmode\nobreak\ \bar{I}(f).$ (104)
Let us move on to the $Q_{i}(F)$ integral. For the sake of convenience let us
choose $i=k$. By the same substitution as before we arrive at
$Q_{k}(F)=\int\limits_{0}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{0}^{1}\left(\bar{f}(t)-\bar{f}(t+y)\right)^{2}\int\limits_{t\cdot\mathcal{R}_{k-1}}\mathbf{1}_{t_{k}\leq\frac{1}{\theta}-y}\,dt_{2}\dots
dt_{k}\,dt\,dy.$ (105)
We wish to replace $\bar{f}$ with $f$ and discard the indicator function. The
latter can be simply performed by calculating the inner integral. Note that it
may be geometrically intepreted as a volume of a ‘bitten’ simplex. We define
$H_{y,t}:=\left\\{(t_{2},\dots,t_{k})\in\mathbf{R}^{k-1}\colon
t_{2}+\dots+t_{k}\leq t\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
t_{k}>1/\theta-y\right\\}.$
Observe that $H_{y,t}$ is just a translated simplex
$(t-1/\theta+y)\cdot\mathcal{R}_{k-1}$ for $1/\theta-t<y\leq 1/\theta$ and an
empty set for $y\leq 1/\theta-t$ . Thus, we obtain
$\frac{1}{(k-1)!}\int\limits_{t\cdot\mathcal{R}_{k-1}}\mathbf{1}_{t_{k}\leq\frac{1}{\theta}-y}\,dt_{2}\dots
dt_{k}\\\
=\text{Vol}(t\cdot\mathcal{R}_{k-1})-\text{Vol}(H_{y,t})=\begin{cases}t^{k-1},&\text{for
}y\in[0,\frac{1}{\theta}-t],\\\ t^{k-1}-(t-1/\theta+y)^{k-1},&\text{for
}y\in(\frac{1}{\theta}-t,\frac{1}{\theta}].\end{cases}$ (106)
For $0\leq y\leq 1$ we also have
$\bar{f}(t)-\bar{f}(t+y)=\begin{cases}f(t)-f(t+y),&\text{for }t\in[0,1-y],\\\
f(t),&\text{for }t\in(1-y,1],\end{cases}$ (107)
and simply $\bar{f}(t)-\bar{f}(t+y)=\bar{f}(t)$ for greater $y$. We decompose
the domain of integration
$D:=\\{(y,t)\in\mathbf{R}^{2}\colon 0<t<1\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 0<y<1/\theta\\}$
into
$D=D_{1}\cup D_{2}\cup D_{3}\cup D_{4}\cup D_{5}\cup\left(\text{some set of
Lebesgue measure 0}\right),$ (108)
where
$\displaystyle D_{1}$ $\displaystyle:=\\{(y,t)\in\mathbf{R}^{2}\colon
0<y<1\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 0<t<1-y\\},$
$\displaystyle D_{2}$ $\displaystyle:=\\{(y,t)\in\mathbf{R}^{2}\colon
0<y<1\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 1-y<t<1\\},$
$\displaystyle D_{3}$ $\displaystyle:=\\{(y,t)\in\mathbf{R}^{2}\colon
1<y<1/\theta-1\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 0<t<1\\},$
$\displaystyle D_{4}$ $\displaystyle:=\\{(y,t)\in\mathbf{R}^{2}\colon
1/\theta-1<y<1/\theta\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
0<t<1/\theta-y\\},$ $\displaystyle D_{5}$
$\displaystyle:=\\{(y,t)\in\mathbf{R}^{2}\colon
1/\theta-1<y<1/\theta\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
1/\theta-y<t<1\\}.$
Therefore, from (105–108) we get
$Q_{k}(F)=\iint\limits_{D_{1}}\frac{1-\theta
y}{y}\,(f(t)-f(t+y))^{2}\,t^{k-1}\,dt\,dy\leavevmode\nobreak\ \\\
+\iint\limits_{D_{2}\cup D_{3}\cup D_{4}}\frac{1-\theta
y}{y}\,f(t)^{2}\,t^{k-1}\,dt\,dy\leavevmode\nobreak\ +\leavevmode\nobreak\
\iint\limits_{D_{5}}\frac{1-\theta
y}{y}\,f(t)^{2}\,\left(t^{k-1}-\left(t+y-1/\theta\right)^{k-1}\right)\,dt\,dy.$
(109)
The same reasoning applies to $Q_{i}(F)$ for $i=1,\dots,k-1$.
### 5.2 Collapse of Theorem 12 into one dimension and bounds for
$\Omega_{k}^{\textmd{ext}}$
We wish to transform Theorem 12 into its one-dimensional analogue in a similar
manner as we did in Subsection 5.1. For the sake of convenience, let us assume
in this subsection that $k\geq 3$. In the $k=2$ case, Theorem 12 can be
applied directly without any intermediate simplifications – it also does not
provide anything beyond what is already known anyway, as presented in Table D.
We take
$F(t_{1},\dots,t_{k})=f(t_{1}+\dots+t_{k})\mathbf{1}_{(t_{1},\dots,t_{k})\in\mathcal{R}_{k}^{\prime}},$
where $f\colon[0,+\infty)\rightarrow\mathbf{R}$ is some locally square-
integrable function. We also put $\ell=1$, so the contribution from $J_{i}(F)$
vanishes for each possible choice of index $i$. Let us begin with $I(F)$
integral. This time we substitute
$\begin{cases}t_{1}+\dots+t_{k}&\longmapsto\leavevmode\nobreak\
\leavevmode\nobreak\ x,\\\ t_{1}+\dots+t_{k-1}&\longmapsto\leavevmode\nobreak\
\leavevmode\nobreak\ t,\\\ t_{1}&\longmapsto\leavevmode\nobreak\
\leavevmode\nobreak\ t_{1},\\\ &\vdots\\\
t_{k-2}&\longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\
t_{k-2}.\end{cases}$ (110)
We also relabel $t_{k-1}$ as $s$. It is calculated in [Lewulis, Subsubsection
‘Calculating J’] that
$I(F)=\int\limits_{\mathcal{R}_{k}^{\prime}}F(t_{1},\dots,t_{k})^{2}\,dt_{1}\dots
dt_{k}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{(k-3)!}\int\limits_{0}^{1}\int\limits_{0}^{t}\int\limits_{t}^{1+\frac{s}{k-1}}f(x)^{2}\,(t-s)^{k-3}\,dx\,ds\,dt.$
(111)
Let us focus on the $Q_{i}(F)$ integral. Again, for the sake of convenience we
choose $i=k$. We have
$Q_{k}(F)=\int\limits_{0}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{\mathcal{R}_{k-1}}\left(\int\limits_{0}^{\rho(t_{1},\dots,t_{k-1})}\left(\partial_{y}\bar{f}(t_{1}+\dots+t_{k})\right)^{2}\mathbf{1}_{t_{k}\leq\frac{1}{\theta}-y}\,dt_{k}\right)\,dt_{1}\dots
dt_{k-1}\,dy,$ (112)
where
$\rho(t_{1},\dots,t_{k}):=\sup\\{t_{k}\in\mathbf{R}\colon(t_{1},\dots,t_{k})\in\mathcal{R}_{k}^{\prime}\\}.$
We observe that any permutation of the variables $t_{1},\dots,t_{k-1}$ does
not change the integrand. We also notice that if we consider an extra
assertion $0<t_{1}<\dots<t_{k-1}$, then
$\rho(t_{1},\dots,t_{k-1})=1-t_{2}-\dots-t_{k-1}.$
Therefore, $Q_{k}(F)$ equals
$(k-1)!\,\int\limits_{0}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{\begin{subarray}{c}\mathcal{R}_{k-1}\\\
0<t_{1}<\dots<t_{k-1}\end{subarray}}\left(\int\limits_{0}^{1-t_{2}-\dots-
t_{k-1}}\left(\partial_{y}\bar{f}(t_{1}+\dots+t_{k})\right)^{2}\mathbf{1}_{t_{k}\leq\frac{1}{\theta}-y}\,dt_{k}\right)\,dt_{1}\dots
dt_{k-1}\,dy,$ (113)
In order to calculate the inner integral, we perform the same substitution as
described (110). This way we obtain
$\int\limits_{\begin{subarray}{c}\mathcal{R}_{k-1}\\\
0<t_{1}<\dots<t_{k-1}\end{subarray}}\left(\int\limits_{0}^{1-t_{2}-\dots-
t_{k-1}}\left(\partial_{y}\bar{f}(t_{1}+\dots+t_{k})\right)^{2}\mathbf{1}_{t_{k}\leq\frac{1}{\theta}-y}\,dt_{k}\right)\,dt_{1}\dots
dt_{k-1}\\\ =\leavevmode\nobreak\
\int\limits_{0}^{1}\int\limits_{\begin{subarray}{c}0<t_{1}<\dots<t_{k-2}<t-\sum_{i=1}^{k-2}t_{i}\end{subarray}}\left(\int\limits_{t}^{1+t_{1}}\left(\partial_{y}\bar{f}(x)\right)^{2}\mathbf{1}_{x-t\leq\frac{1}{\theta}-y}\,dx\right)\,dt_{1}\dots
dt_{k-2}\,dt$ (114)
For the sake of clarity, we relabel $t_{1}$ as $s$. Thus, the expression from
(114) equals
$\int\limits_{0}^{1}\int\limits_{0}^{\frac{t}{k-1}}\left(\int\limits_{t}^{1+s}\left(\partial_{y}\bar{f}(x)\right)^{2}\mathbf{1}_{x-t\leq\frac{1}{\theta}-y}\,dx\right)\left(\int\limits_{s}^{\frac{t-s}{k-2}}\int\limits_{t_{2}}^{\frac{t-s-
t_{2}}{k-3}}\cdots\int\limits_{t_{k-3}}^{\frac{t-s-t_{2}-\dots-
t_{k-3}}{2}}\,dt_{k-2}\dots dt_{2}\right)\,ds\,dt.$ (115)
If $k=3$, then the inner integral simplifies to $1$. For $0\leq s\leq t$ let
us define
$\mathscr{L}(k;t,s):=\int\limits_{s}^{\frac{t-s}{k-2}}\int\limits_{t_{2}}^{\frac{t-s-
t_{2}}{k-3}}\cdots\int\limits_{t_{k-3}}^{\frac{t-s-t_{2}-\dots-
t_{k-3}}{2}}\,dt_{k-2}\dots dt_{2}.$
We apply the induction over $k$ to show that
$\mathscr{L}(k;t,s)=\frac{(t-(k-1)s)^{k-3}}{(k-2)!(k-3)!}.$ (116)
Our claim is obviously true for $k=3$. For every $k\geq 3$ we observe the
identity
$\mathscr{L}(k+1;t,s)=\int\limits_{s}^{\frac{t-s}{k-1}}\mathscr{L}(k;t-s,u)\,du.$
(117)
To finish the proof of the claim one has to put (116) into (117) and
substitute
$t-s-(k-1)u\mapsto z.$
Combining (113–115) with the claim discussed above we conclude that $Q_{k}(F)$
equals
$\frac{(k-1)!}{(k-2)!(k-3)!}\,\int\limits_{0}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{0}^{1}\int\limits_{0}^{\frac{t}{k-1}}\left(\int\limits_{t}^{1+s}\left(\partial_{y}\bar{f}(x)\right)^{2}\mathbf{1}_{x-t\leq\frac{1}{\theta}-y}\,dx\right)(t-(k-1)s)^{k-3}\,ds\,dt\,dy.$
(118)
Let us relabel $s$ as $s/(k-1)$ to simplify the expression above. We arrive at
$\displaystyle Q_{k}(F)$
$\displaystyle=\frac{1}{(k-3)!}\,\int\limits_{0}^{\frac{1}{\theta}}\frac{1-\theta
y}{y}\int\limits_{0}^{1}\int\limits_{0}^{t}\left(\int\limits_{t}^{1+\frac{s}{k-1}}\left(\partial_{y}\bar{f}(x)\right)^{2}\mathbf{1}_{x-t\leq\frac{1}{\theta}-y}\,dx\right)(t-s)^{k-3}\,ds\,dt\,dy$
$\displaystyle=\frac{1}{(k-3)!}\int\limits_{E}\frac{1-\theta
y}{y}\left(\partial_{y}\bar{f}(x)\right)^{2}(t-s)^{k-3}\,dx\,dt\,ds\,dy,$
(119)
where
$E:=\left\\{(y,s,t,x)\in\mathbf{R}^{4}\colon
0<y<\frac{1}{\theta},\leavevmode\nobreak\ 0<t<1,\leavevmode\nobreak\
0<s<t,\leavevmode\nobreak\ t<x<1+\frac{s}{k-1},\leavevmode\nobreak\
x-t<\frac{1}{\theta}-y\right\\}.$
We wish to drop the bar from $\bar{f}$. Hence, we decompose
$E=E_{1}\cup E_{2},$
where
$\displaystyle E_{1}$ $\displaystyle:=\left\\{(y,s,t,x)\in E\colon x+y\leq
1+\frac{s}{k-1}\right\\},$ $\displaystyle E_{2}$
$\displaystyle:=\left\\{(y,s,t,x)\in E\colon x+y>1+\frac{s}{k-1}\right\\}.$
From (5.2) we have that $Q_{k}(F)$ equals $1/(k-3)!$ times
$\int\limits_{E_{1}}\frac{1-\theta
y}{y}\left(f(x)-f(x+y)\right)^{2}(t-s)^{k-3}\,dx\,dt\,ds\,dy\leavevmode\nobreak\
+\leavevmode\nobreak\ \int\limits_{E_{2}}\frac{1-\theta
y}{y}f(x)^{2}(t-s)^{k-3}\,dx\,dt\,ds\,dy.$ (120)
Now, we would like to convert two integrals above into a finite sum of
integrals with explicitely given limits, just like in (109). If we choose the
order of integration
$y\rightarrow x\rightarrow t\rightarrow s,$
then we get
$\displaystyle\int\limits_{E_{1}}\boxtimes\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\int\limits_{0}^{1}\int\limits_{s}^{1}\int\limits_{t}^{1+\frac{s}{k-1}}\leavevmode\nobreak\
\int\limits_{0}^{1+\frac{s}{k-1}-x}\boxtimes\leavevmode\nobreak\
dy\,dx\,dt\,ds,$ (121)
$\displaystyle\int\limits_{E_{2}}\boxtimes\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\int\limits_{0}^{1}\int\limits_{s}^{1}\int\limits_{t}^{1+\frac{s}{k-1}}\int\limits_{1+\frac{s}{k-1}-x}^{\frac{1}{\theta}+t-x}\boxtimes\leavevmode\nobreak\
dy\,dx\,dt\,ds,$ (122)
where $\boxtimes$ denotes an arbitrary integrable function.
###### Remark 5.
From the computational point of view, the variable $y$ should be integrated in
the last order, because it engages a non-polynomial function. The author found
the following order of integration as the most computationally convenient:
$x\rightarrow t\rightarrow s\rightarrow y.$
Unfortunately, in this case there is no decompsition of $E$ similar to (108)
that is common for all possible choices of $k$ and $\theta$. In the
$k=4,\,\theta=1/2$ case, which accordingly to Tables C and D is the only one,
where we can expect a qualitative improvement over Theorem 11, we are able to
convert the integral over $E$ into 15 integrals with explicitely given limits.
Such a conversion is a straightforward operation (quite complicated to perform
without a computer program, though). We do not present the precise shape of
these integrals here.
Let us set $k=4$, $\theta=1/2$, and
$f(x)=12+63x+100x^{2}.$
Combining (111) with (121–122), and performing the calculations on a computer,
we get
$\displaystyle I(F)=\leavevmode\nobreak\ $
$\displaystyle\frac{2977019}{51030}>58.3386047422.$ $\displaystyle
Q_{k}(F)=\leavevmode\nobreak\ $
$\displaystyle\frac{132461570733345\log\frac{5}{3}-997242435\log
3-49178701703144}{4629441600}$
$\displaystyle+\frac{6144554}{105}\log\frac{6}{5}-\frac{15996989}{280}\,\text{arcoth}\,4<70.0214943902.$
This combined with Theorem 12 gives
$\Omega_{4}^{\text{ext}}\left(\theta=\frac{1}{2}\right)<8.80105,$ (123)
which proves the $k=4$ case of the conditional part of Theorem 4.
### 5.3 bounds for $\Omega_{k,\varepsilon}$
We apply Theorem 13 with $\eta=1+\varepsilon$ and some $\varepsilon$, $\ell$
satisfying
$2\theta(1+\varepsilon)+\frac{1}{\ell}=1.$
We choose $F(t_{1},\dots,t_{k})=\bar{f}(t_{1}+\dots+t_{k})$ for a function
$\bar{f}\colon[0,+\infty)\rightarrow\mathbf{R}$ satisfying
$\bar{f}(x):=\begin{cases}f(x),&\text{for }x\in[0,1+\varepsilon],\\\
0,&\text{otherwise.}\end{cases}$ (124)
First, we calculate $I(F)$. We proceed just like in (104) and get
$I(F)\,=\
\int\limits_{0}^{1+\varepsilon}f(t)^{2}\left(\,\int\limits_{t\cdot\mathcal{R}_{k-1}}dt_{2}\dots
dt_{k}\right)dt\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{(k-1)!}\,\int\limits_{0}^{1+\varepsilon}f(t)^{2}\,t^{k-1}\,dt.$ (125)
Next, let us consider $J_{i,\varepsilon}(F)$. As before, let us put $i=k$. We
have
$J_{k,\varepsilon}(F)\,=\,\int\limits_{(1-\varepsilon)\cdot\mathcal{R}_{k-1}}\left(\int\limits_{0}^{1+\varepsilon-
t_{1}-\dots-t_{k-1}}f(t_{1}+\dots+t_{k})\,dt_{k}\right)^{2}dt_{1}\dots
dt_{k-1}.$ (126)
We perform the same substitution as in (104). We get that
$J_{k,\varepsilon}(F)$ equals
$\begin{gathered}\,\int\limits_{0}^{1-\varepsilon}\left(\int\limits_{0}^{1+\varepsilon-t}f(t+t_{k})\,dt_{k}\right)^{2}\int\limits_{t\cdot\mathcal{R}_{k-1}}\,dt_{1}\dots
dt_{k-2}\,dt\,=\,\int\limits_{0}^{1-\varepsilon}\left(\int\limits_{t}^{1+\varepsilon}f(x)\,dx\right)^{2}\frac{t^{k-2}}{(k-2)!}\,dt.\end{gathered}$
(127)
We perform analogous calculations for $i=1,\dots,k-1$.
Let us move to $Q_{i,\varepsilon}(F)$. Put
$\begin{cases}t_{1}+\dots+t_{k-1}&\longmapsto\leavevmode\nobreak\
\leavevmode\nobreak\ t,\\\ t_{2}&\longmapsto\leavevmode\nobreak\
\leavevmode\nobreak\ t_{2},\\\ &\vdots\\\
t_{k}&\longmapsto\leavevmode\nobreak\ \leavevmode\nobreak\ t_{k}.\end{cases}$
(128)
and split
$Q_{k,\varepsilon}(F)=Q_{(1)}(f)+Q_{(2)}(f),$ (129)
where
$\displaystyle Q_{(1)}(f)$
$\displaystyle:=\frac{1}{(k-2)!}\int\limits_{0}^{\frac{1}{\ell\theta}}\frac{1-\ell\theta
y}{y}\int\limits_{0}^{1+\varepsilon}\left(\,\int\limits_{0}^{\frac{1}{\theta}-y}\left(\bar{f}(t+t_{k})-\bar{f}(t+t_{k}+y)\right)^{2}\,dt_{k}\right)t^{k-2}\,dt\,dy,$
$\displaystyle Q_{(2)}(f)$
$\displaystyle:=\frac{1}{(k-2)!}\int\limits_{\frac{1}{\ell\theta}}^{\frac{1}{\theta}}\frac{1-\ell\theta
y}{y}\int\limits_{0}^{1-\varepsilon}\left(\,\int\limits_{0}^{\frac{1}{\theta}-y}\left(\bar{f}(t+t_{k})-\bar{f}(t+t_{k}+y)\right)^{2}\,dt_{k}\right)t^{k-2}\,dt\,dy,$
(130)
Therefore, we put $t_{k}+t\mapsto x$ and decompose
$(k-2)!\left(Q_{(1)}(f)+Q_{(2)}(f)\right)=\\\\[4.30554pt]
\int\limits_{H_{1}\cup H_{3}}\frac{1-\ell\theta
y}{y}f(x)^{2}\,t^{k-2}\,dx\,dt\,dy\,+\,\int\limits_{H_{2}\cup
H_{4}}\frac{1-\ell\theta
y}{y}\left(f(x)-f(x+y)\right)^{2}t^{k-2}\,dx\,dt\,dy,$ (131)
where
$H:=\\{(y,t,x)\in\mathbf{R}^{3}\colon 0<y<1/\theta,\leavevmode\nobreak\
0<t<x<1+\varepsilon,\leavevmode\nobreak\ x-t<1/\theta-y\\},$
and
$\displaystyle H_{1}$ $\displaystyle:=\\{(y,t,x)\in H\colon 0<y\leq
1/(\ell\theta)\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
x+y<1+\varepsilon\\},$ $\displaystyle H_{2}$ $\displaystyle:=\\{(y,t,x)\in
H\colon 0<y\leq 1/(\ell\theta)\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ x+y>1+\varepsilon\\},$ $\displaystyle H_{3}$
$\displaystyle:=\\{(y,t,x)\in H\colon
1/(\ell\theta)<y<1/\theta\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
0<t<1-\varepsilon\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
x+y<1+\varepsilon\\},$ $\displaystyle H_{4}$ $\displaystyle:=\\{(y,t,x)\in
H\colon 1/(\ell\theta)<y<1/\theta\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 0<t<1-\varepsilon\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ x+y>1+\varepsilon\\},$
Unfortunately, with varying $k$, $\varepsilon$, $\theta$ there is no uniform
way to decompose $H_{1},\dots,H_{4}$ further into integrals with explicitely
given limits. In the unconditional setting, namely with $\theta=1/4$ fixed,
every choice of parameters described in Table E provides less than 10
different integrals to calculate. For these choices we present close to
optimal polynomials minimalizing the $\Omega_{k,\varepsilon}$ functional.
Table G. Upper bounds for $\Omega_{k,\varepsilon}$.
$k$ | $\varepsilon$ | $f(1+\varepsilon-x)$ | bounds for $\Omega_{k}$ |
---|---|---|---|---
$2$ | 1/3 | $1+5x+3x^{2}$ | 4.6997 |
$3$ | 1/4 | $1+7x+10x^{2}$ | 7.7584 |
$4$ | 1/5 | $1+7x+19x^{2}$ | 11.0533 |
$5$ | 1/6 | $1+7x+33x^{2}$ | 14.5415 |
$6$ | 1/7 | $1+7x+51x^{2}$ | 18.1907 |
$7$ | 1/9 | $1+8x+70x^{2}$ | 21.9939 |
$8$ | 1/10 | $1+8x+102x^{2}$ | 25.9038 |
$9$ | 1/10 | $1+5x+132x^{2}$ | 29.9059 |
$10$ | 2/21 | $1+35x+30x^{2}+470x^{3}$ | 33.9384 |
These bounds are sufficient to prove the unconditional part of Theorem 4.
∎
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|
11institutetext: Shanghai Institute for Advanced Communication and Data
Science, Shanghai JiaoTong University, Shanghai, 200240, China
11email<EMAIL_ADDRESS>
# Cost of Dietary Data Acquisition with Smart Group Catering
Jiapeng Dong Pengju Wang Weiqiang Sun
###### Abstract
The need for dietary data management is growing with public awareness of food
intakes. As a result, there are increasing deployments of smart canteens where
dietary data is collected through either Radio Frequency Identification (RFID)
or Computer Vision(CV)-based solutions. As human labor is involved in both
cases, manpower allocation is critical to data quality. Where manpower
requirements are underestimated, data quality is compromised. This paper has
studied the relation between the quality of dietary data and the manpower
invested, using numerical simulations based on real data collected from
multiple smart canteens. We found that in both RFID and CV-based systems, the
long-term cost of dietary data acquisition is dominated by manpower. Our study
provides a comprehensive understanding of the cost composition for dietary
data acquisition and useful insights toward future cost effective systems.
###### keywords:
CV systems, data accuracy, dietary data acquisition, dietary management,
health management, RFID systems, Smart group catering
## 1 Introduction
The emerging public concern with health has led to the proliferation of health
management applications, individual health monitoring[1] and nutritional
assessments[2, 3] in which dietary data is often an important component. Given
the diversity of food and unpredictable dining locations, however, recording a
person s regular intake has never been an easy task. Since 2017, smart group
catering (SGC) systems targeted at canteens of different sizes, and featuring
automatic billing and data acquisition, have become popular. Onsite
experiences with SGC systems indicate that even though dietary data
acquisition technologies seem to be readily available, data quality may vary
drastically from one canteen to another. One important reason behind this is a
widespread under-estimation of the necessary manpower needed for accurate data
acquisition. Thus, it is important to understand this cost and its
relationship to other factors.
There are two types of widely used SGC systems implemented for dietary data
acquisition Radio Frequency Identification (RFID)- and Computer Vision
(CV)-based solutions. The data acquisition workflows of the two types of
solutions are shown in Fig. 1. In RFID-based systems, special dishes with
embedded tags are used when food is served. The food information is read when
customers checkout. In CV-based systems, cameras are used at checkout counters
to recognize the dish . Fig. 1 shows the basic workflow of a traditional
canteen and the extra procedural intrusion of the two systems.
Figure 1: Workflow of Traditional Canteen and Intrusion of Two Systems
RFID is a mature technology and has countless applications, [4] e.g. RFID
systems were first proposed for SGC by Yao.X et al.[5] in 2011. There were
subsequent implementations by Y.H.Liang et al.[6], Pai-Hsun.C et al.[7] and
E.B.Kossonon et al.[8], which were mainly prototypes and somewhat different
from the systems deployed in canteens, as shown in Fig. 1. With the advantage
of mature technology and simple software, RFID systems currently account for
over 80% of the market.
CV systems are less mature. The food detection algorithm used in CV systems
has become popular in recent years. Lead by Bossard.L et al.[9] in 2014, some
researchers put up new datasets[10], while others focused on a food
recognition algorithm. A group of studies tried to solve image segmentation
before further recognition[11, 12]. Those that directly tackle the entire food
detection mission[13, 14] also achieve good performance. Among all these
studies, the sequential works[15, 16] are the milestone for the canteen
scenarios, which also provides solid references for our study.
In this paper, we aimed to study the relation between dietary data quality and
invested manpower. We conducted our study by means of numerical simulations,
with parameters taken from real-life canteens operating SGC systems. Our
contributions are:
* •
Information flow-centric modeling of dietary data acquisition for RFID and CV-
based SGC systems.
* •
A comprehensive numerical analysis of the cost of dietary data acquisition
with the two types of technologies.
* •
Comparative analyses of RFID and CV-based systems’ application scenarios and
limitations based on the dynamic relationships between cost, data accuracy and
other relevant factors of deployment.
* •
Future directions and evolutionary trends for dietary data collection using
SGC systems.
The rest of the paper is organized as follows: Chapter 2 builds the cost
accuracy model based on information flows and key procedure properties.
Chapter 3 describes our data set and basic settings of the experiments.
Chapter 4 progressively analyzes the cost of dietary data harvesting and its
major influence factors. Other accessory factors are discussed in Chapter 5.
Finally, the conclusions and future outlooks are presented.
## 2 Models
In this section, we modeled different types of costs for both systems and the
relationships between relevant ones and data accuracy. We included the cost of
the system itself and the corresponding extra costs necessary to maintain
normal operation and obtain accurate data.
### 2.1 Cost Composition: Key Procedures and Cost Groups
In order to excavate the essential mechanism of dietary data harvesting in
both RFID and CV systems, we reviewed its general workflow, as illustrated in
Fig. 1. We considered the nature of data harvesting as an information flow.
Thus, we ruled out all the factors that were not concerned with the
information flow. Afterwards, we defined and located the key procedures for
the necessary transformation or transmission of dietary information. The
rearranged and expanded workflow chart is presented as an information flow
chart in Fig. 2.
Figure 2: Information Flow of Two Systems: Step blocks in white background are
the intermediate formation or carrier of dietary information and those in gray
represent sources and targets.
From Fig. 2, the data quality of RFID systems is much more dependent on staff
operation at three key sequential procedures, namely inputting, setting and
labeling. The flow of CV systems, however, is primarily determined by its
algorithm performance and secondarily by sampling procedures. The information
flow of RFID systems looks simpler and more direct while the flow of CV
systems contains fewer procedures requiring staff operation. Both systems have
to correct errors at checkout to ensure normal billing.
All costs are incurred during the procedure of information transformation and
transmission. In order to discriminate between different cost items, we
divided them into two groups according to whether they are staff operation
related costs (SORC) or not staff operation related cost (NSORC). For each
type of system:
$SORC^{RFID}=C_{input}+C_{set}+C_{label}+C_{correct}$ (1) $NSORC^{RFID}=\sum
C^{RFID}_{devices}+C^{RFID}_{software}+C_{plate-loss}$ (2)
$SORC^{CV}=C_{sample}+C_{correct}$ (3) $NSORC^{CV}=\sum
C^{CV}_{devices}+C^{CV}_{software}$ (4)
, where $C_{plate-loss}$ represents the RFID-embedded plates malfunctioning
during the usage, determined by the plate number used per meal and the
statistical loss rate. Assuming five years of lifespan, devices and their
converted per meal cost for each system are presented in Table. 1, where $m$
depends on the ratio of canteen total throughput and unit checkout velocity.
Table 1: NSORC Items Comprised in Two Systems System | NSORC Item | Number | Value (RMB)
---|---|---|---
RFID | RFID Writer | $T$ | $1.37\times 10^{-1}$ |
RFID Reader | $m$ | $5.48\times 10^{-1}$ |
Control Terminal | $1$ | $5.48\times 10^{-1}$ |
Checkout Terminal | $m$ | $5.48\times 10^{-1}$ |
Peripheral Network | $1$ | $2.74\times 10^{-1}$ |
Server | $1$ | $8.22\times 10^{-1}$ |
RFID software | $1$ | $5.48$ |
RFID plate | $n*rate$ | $3.5$ |
CV | Embedded Camera | $m$ | $8.22\times 10^{-2}$ |
Checkout Terminal | $m$ | $8.22\times 10^{-1}$ |
Server (extra training) | $1$ | $1.37$ |
CV software | $1$ | $10.96$ |
a$T$ for dish types and $n$ for total plate number. |
### 2.2 Factors of Accuracy: Staff Operation and Sample Accumulation
From the information flow in Fig. 2, we concluded that the data accuracy
mechanism differed between the two systems. The RFID system contained three
staff-operated procedures and did not generate any false data when all three
procedures were operated without any failure. Although the CV system required
sampling and marking procedures, the number of executions was much smaller
compared to the scale of the data to be collected. Meanwhile, the data was
determined by the deduction results, which made the performance of the CV
system mainly dependent on the CV dish recognition model applied and the
number of samples in the training set.
man-hours was adopted as the measure of staff work in different procedures.
Moreover, we expanded the man-hour concept into equivalent man-hours (EMH)
which was defined to equivalently measure the extra cost invested to harvest
dietary data across distinct procedures. With regards to the staff’s non-
standard operation and corresponding accuracy, based on our on-site knowledge,
the following assumptions were raised:
* •
The accuracy of a key procedure carried out by staff once was proportionate to
the extra EMH that the staff was provided.
* •
The accuracy of a key procedure always reached one hundred percent with
sufficient extra EMH.
* •
As the provided extra EMH increased, the marginal accuracy growth continuously
decreased toward the endpoint of accuracy.
* •
The pattern of the marginal accuracy growth variation differed per procedure
according to their attributes.
Since the power function is the simplest function fitting all these
assumptions, it was used to construct our EMH Accuracy (EMH-A) model:
$Accuracy(h)=(\frac{h}{S})^{\alpha},\quad 0\leq\alpha\leq 1$ (5)
, where S is the standard EMH needed when accuracy reaches one hundred
percent, and alpha is the procedure distinction coefficient representing
procedure features’ effects on the marginal accuracy growth patterns. The
specific $S$ values of inputting, setting, labeling and correction procedures
are based on the average time taken in a real canteen environment. In
addition, the knowledge and skills required by the procedures are also taken
into account. The baseline values are listed in Table.2
Table 2: Parameters of EMH Accuracy Model Procedure | $S$ (hour) | $\alpha$
---|---|---
Inputting | $1.7\times 10^{-1}$ | $0.6$ |
Setting | $6.7\times 10^{-2}$ | $0.4$ |
Labeling | $1.39\times 10^{-3}$ | $0.1$ |
Correction | $1.1\times 10^{-2}(1.1\times 10^{-3})$ | $0.15$ |
The value of $\alpha$ depended on procedure features. The staff-related
procedures in dietary data harvesting scenarios usually have three features:
automation degree, throughput pressure, and internal complexity. A procedure
with more automation is more accurate, and thus the value should be bigger. It
is more difficult for a pressured procedure to reach high accuracy, which
leads to a small value. Higher accuracy can be easier to achieve with lower
EMH if the procedure is comparatively simple, so the value should be small. A
binding system like RFID usually has high pressured labeling and correction
procedures, a less pressured but more complex and automatic setting procedure
and a more complex but less pressured inputting procedure. Therefore, using
the values of $\alpha$ shown in Table. 2, the curves of each procedure are
drawn in Fig. 3, where the EMH-axis offset of the correction procedure is
determined by the fixed cost of total price correction, which is crucial for
normal billing.
Figure 3: Accuracy by Unit EMH Cost of Four Procedures’ EMH-A Models
As in the previous discussion, the algorithm used in the CV model is the major
internal factor for data accuracy. The procedure of information inputting
including sample dish preparation, sampling (photo taking), and marking, with
time reserved for training, was also taken into consideration. The cost of
this procedure is very high, about 0.33 h of EMH. This forces the on-site
manager to minimize the sample number and to use the served dish for new
samples at checkout as much as possible. Based on the general characteristics
of the CV learning model, three assumptions were raised to model the
relationship between deduction accuracy and sample size used in training:
* •
The performance of deduction always has a less-than-one upper bound decided by
the algorithm the model applies.
* •
When the accuracy is low, the increment caused by sample number increase is
prominent.
* •
As the accuracy approximates its upper bound, its marginal growth drops
increasingly rapidly.
We chose the sigmoid function as our prototype to approximate the actual
learning process of the dish recognition model. The Sample Number Accuracy
(SNA) model is as follows:
$Accuracy(n_{sample})=U*sigmoid(\beta*n_{sample})$ (6)
, where $n_{sample}$ represents the number of samples, $U$ the upper bound,
and $\beta$ is the transmission coefficient representing the algorithm’s
feature extraction efficiency. Based on the algorithm performance in [16] and
actual deployment situations, the value of $U$ was set to 0.85.
## 3 Experiments
### 3.1 Data Sets and Baseline Parameters
Our data set was collected from thirteen canteens throughout mainland China.
The types of canteens included government departments, primary schools,
colleges, private and state enterprises. The data content contained the SGC
system deployment profile of each canteen, menu update records of over four
hundred dish types and over a million dish transaction details over a time
span of more than half a year. The data set established a numerical foundation
for our simulations.
According to previous procedure and information flow analyses, four main
canteen features which have major impact on data acquisition cost were
extracted. These four canteen parameters are listed in Table. 3, with their
definitions and units. The values here were based on the profile of our most
familiar canteen, and were used as the baseline in the experiments. In
addition, the product of $T$ and $N$ represents the canteen’s scale, i.e., the
customer capacity, while $F$ and $R$ show the canteen’s service quality.
Table 3: Canteen Feature Parameters Parameter | Definition | Unit | Value
---|---|---|---
T | per meal dish types | types / meal | 20 |
N | dish number of each type | dishes / type | 70 |
F | frequency of adding new dish type | types / meal | 0.3 |
R | rotation of old dish type | types / meal | 6 |
### 3.2 Basic System Characteristics and Experimental Settings
Here we briefly look into the basic characteristics of the two systems in
order to make preliminary settings for experiments.
#### 3.2.1 Cost Allocation of RFID Systems
RFID systems comprise of three sequential key procedures among which cost can
be allocated in various proportions. The function between a set of EMH costs,
i.e., $(H_{input},H_{set},H_{label})$ and the corresponding accuracy, cannot
be clearly depicted in graphs. After changing the $F$ of our canteen baseline
to zero for simplification and better demonstration, we were able to draw the
accuracy contour by the summed EMH cost of setting and labeling in Fig. 4a.
In this simplified condition, we can prove that for each convex accuracy
contour, there is a point where the total cost of the two procedures is
optimal. These points, as are shown in Fig.4a, form an optimal path joined
first by the tangent points of the auxiliary total cost line with the contour,
and then by the intersections of the line with the upper boundary of setting.
The segmentation of optimal path like this is common since the cost of
labeling is major in most conditions.
In summary, there is always an optimal path for EMH cost allocation and the
path can be approximated as a broken line with several fixed proportions.
Therefore, the path is adopted wherever relevant throughout this paper.
#### 3.2.2 Cold Start Problem of CV Systems
CV systems are stateful and their accuracy depends on the size of the training
set. Accuracy is attainable, but a ramping-up process is necessary considering
the high cost of sampling which the canteen usually tends to skip or cut down.
Thus, as demonstrated in Fig. 4b, the cold start problem of CV systems results
in the ramping-up period (RP) of accuracy which zooms at the very beginning
and ends when the differential accuracy increment (3-meal-long window) drops
below $1\times 10^{-5}$. The rest part is defined as the stable period (SP)
because the accuracy only fluctuates in a comparatively small range. The cost
decreases sharply during the RP, especially between the first two meals (about
a hundredth), and in SP peaks in accordance with the frequency of new dish
addition, with an overall low average afterwards.
(a) RFID Systems: Accuracy Contour and Optimal Allocation Path
(b) CV Systems: Accuracy and Total EMH of First Hundred Meals
Figure 4: Basic Characteristics
Extra EMH in RP, which can be multiple times that of SP, is essential and
requires preparation in advance. Fortunately, the cold start problem only
happens when a great many types of new dishes need to be sampled, and in most
conditions only once. The infrequent occurrence of the problem and the large
EMH gap between the two periods make it possible and reasonable to ignore the
RP in a long term study. We will regard the EMH of the CV system to be
specialized in SP unless otherwise mentioned.
## 4 Results
In order to conduct a thorough study into the cost of dietary data harvesting
with RFID and CV systems, we started our baseline from the general cost and
its composition. Then we focused on the SORC dynamics of the two systems under
variable accuracy targets. After another set of experiments conducted under
various canteen conditions, four typical canteens were specified and used as
examples for straightforward understanding.
### 4.1 General Cost Composition
Simulations were carried out with the two separate systems to evaluate the
total cost of dietary data harvesting of a single meal with target accuracy of
1 in the baseline canteen. For a better comparison of NSORC in canteens of
different scales, another set of experiments on an enlarged canteen were also
appended. For experiment settings, NSORC items’ statistical prices and
baseline values of model parameters were adopted as listed in TABLE. 1. The
enlarged canteen was set with twice the customers (900 people) and more dish
types (50 types). Furthermore, the values of EMH were transformed into RMB by
current average hour wage level. In this way, we had four groups of total cost
with composition as shown in Fig. 5.
Figure 5: Mealy Cost of 100% Target Accuracy by Cost Group, System and Canteen
Scale: SORC(left) and NSORC(right) for each system and canteen.
NSORC accounted for around 10% to 15%, much less than SORC does in both
systems and both canteens. The sum of NSORC of both systems in the baseline
canteen differs little (about 1% of total), but as the scale of the canteen
increases, the NSORC of RFID systems increases faster than CV systems, due to
its extra plate loss (about 1.9% of total) and RFID writing devices bonded to
dish type numbers (about 2.2% of total). Meantime, the impact on CV systems
from device addition appears minimal with barely a 0.7% increment of the
total.
Compared to the insignificant increase of NSORC, SORC ascends synchronously
with the canteen scale both in sum and ratio. Based on the invariance and
minority role played by NSORC in data accuracy, SORC is more worthy of further
study.
### 4.2 SORC by Target Accuracy
In this section we explore the relationship between the SORC and data
accuracy, in a specific baseline canteen scenario. As was illustrated above,
RFID systems’ accuracy could be assigned arbitrarily between 0 and 1 while
that of CV systems was fixed at the average level of the stable period without
manual data correction at checkouts. The higher accuracy of CV systems
required manual data correction at checkout. The same correction could also be
available for RFID systems but will be discussed later.
Experiments were performed conforming with the baseline canteen and with three
accuracy targets, as is shown in Fig. 6a. The diagram describes the
distinction of the two systems in general: With base SP accuracy, the cost of
the labeling procedure accounts for about 33% of RFID systems, while in CV
systems the sampling procedure dominates. When the target accuracy rises,
despite the correction cost decrement, the labeling procedure takes up almost
the entire cost increment (about 25.7% and 52.3% of total) while setting and
inputting procedures remain nearly the same. When it comes to CV systems, with
the invariant sampling cost, the correction cost increment is minor (about
3.1% of total) when the accuracy grows to 90%, but it then increases to full
accuracy. Therefore, the target accuracy was expanded to the whole range. The
results of more detailed experiments are demonstrated in Fig. 6b. The cost
curve when data is manually harvested is also provided for comparison.
(a) Cost Composition
(b) Total Cost Comparison
Figure 6: SORC by Target Accuracy
In the baseline scenario, the RFID system out-performed the CV system only
when accuracy was above 0.96. The curve of the RFID system in Fig. 6b showed
convexity to some degree, with a minimal cost around accuracy of 0.6, because
of the increased total price correction cost at low accuracy and the high cost
required for high accuracy. Without data correction, the accuracy of CV
systems stabilized at 0.84 and increased to 0.92. This efficiency was lost
when the power law of the correction cost dominated and the cost surpassed the
level of RFID systems (around 0.96).
To summarize, compared to the manual means of data harvesting at checkout, the
SGC systems take the lead beginning from 0.63 accuracy by CV systems and then
after 0.95 accuracy by RFID systems. Deploying an SGC system can save over 80%
of the cost of dietary data harvesting with an accuracy greater than 0.8.
### 4.3 SORC by Canteen Features
To extend our analysis to more canteens in real circumstances, experiments
were designed to determine how the four features of canteens, i.e., $T$, $N$,
$F$ and $R$ as listed in TABLE. 3, influenced the accuracy of harvested data
and the cost. $T$ and $N$ were grouped together as were $F$ and $R$. Since a
dynamic analysis was required and the effect of target accuracy has already
been studied, we concentrated on the average cost and accuracy of CV systems’
stable period and the cost across five different target accuracies of RFID
systems, that is, $0.6,0.7,0.8,0.9,1$.
#### 4.3.1 Dish Type Numbers and Dish Number of Each Type
The canteen scale is consistent with the product of a canteen’s dish type
number ($T$) and dish number of each type ($N$). The range of $T$ and $N$ are
expanded in both directions simultaneously from our baseline values. The
results are shown in Fig. 7.
(a) by $T$ of RFID
(b) by $N$ of RFID
(c) by $T$ of CV
(d) by $N$ of CV
Figure 7: SR Cost by T
Similarities in the influence patterns of both parameters were apparent in the
diagrams. In RFID systems, both $T$ and $N$ increased with approximate
linearity as their value ascended,. The speed of increase was relatively
small, about $5.56\times 10^{-3}$ h/type and $2.78\times 10^{-3}$ h/dish,
under low target accuracy (0.8 and lower). When the target accuracy rose
beyond 0.8, however, the speed was intensified, up to $1\times 10^{-1}$ h/type
and $2.78\times 10^{-2}$ h/dish at accuracy of 1. For CV systems, both
influences on the cost fluctuated only slightly, without recognizable
patterns. The increment of the two features caused weak growth of average
accuracy (no more than 0.02). Furthermore, the accuracy deviation reduction
against boosting dish types was caused by the decreasing proportion of fixed
new dish frequency($F$).
#### 4.3.2 New Dish Frequency
The frequency of appending new dishes ($F$) can differ significantly among
canteens, from only one new dish in weeks to several new dishes every meal.
Therefore, we used the logarithmic scale for the F value. The results of both
systems are shown in Fig. 8a and Fig. 8b.
(a) RFID Systems
(b) CV Systems
Figure 8: SR Cost by F
The results revealed the distinct effect of $F$ on the costs of the two
systems: The range of $F$ was segmented by 1 type/meal for RFID systems. The
cost grew slowly in logarithm when $F$ was smaller than 1, i.e., new dishes
were not added at each meal. As $F$ passed 1, the curve was computed to be
approximately linear and the cost grew much faster than that of a smaller-
than-one $F$. With accuracy of 1, the rate was about $2.08\times 10^{-1}$
h/type$*$meal. However, $F$’s effect on CV systems shows uniformity, i.e.,
linearity along the whole range (about 1.67 h/type$*$meal). Another
significant point is the reverse impact on accuracy, an approximately linear
decrement with a ratio of about 0.011 /type$*$meal.
The distinction proves the far bigger influence of $F$’s value on CV systems
than on RFID systems.
#### 4.3.3 Dish Rotation Number
CV systems are free of the costs of dish rotation because the features of the
old dishes have been extracted and stored inside the CV model. RFID systems,
however, have no way to deal with the problem except by updating the settings
before or during the meal. Experiments were arranged with expanded $R$ range
from the baseline, as is shown in Fig. 9.
Figure 9: SR Cost by R of RFID Systems
The result illustrated that with different target accuracies, the cost of RFID
systems changed linearly against the value of $R$ at an approximately marginal
rate, about $5.56\times 10^{-2}$ h/type$*$meal. The higher the target accuracy
was, the more sparsely the curves were distributed. This proved the $R$
parameter affected the cost independently of data accuracy.
#### 4.3.4 Comprehensive analysis
After four canteen features were studied, all the results were organized into
Table. 4 for comparison.
Table 4: Marginal Cost (h) by Systems and Canteen Feature Effects Parameter | RFID-80% | RFID-100% | CV SR Cost | CV Accuracy
---|---|---|---|---
T | $+1\times 10^{-1}$ | $+5.56\times 10^{-3}$ | - | $+4\times 10^{-5}\quad(T<45)$ |
N | $+2.78\times 10^{-2}$ | $+2.78\times 10^{-3}$ | - | $+4\times 10^{-5}\quad(N<80)$ |
F | $+2.08\times 10^{-1}$ | $+1.94\times 10^{-1}$ | $+1.67$ | $-1.1\times 10^{-2}$ |
R | $+5.56\times 10^{-2}$ | $+6.39\times 10^{-2}$ | - | - |
By analyzing the table, we could make the following conclusions. Firstly, it
was expensive for both systems to add new dish types, but the degree was
comparatively smaller for RFID systems at about one eighth of that of CV.
Secondly, the cost of CV systems was directly affected by $F$, along with
collateral damage to data accuracy. Finally, four features affect the cost of
RFID systems to similar degrees but when target accuracy escalates, the effect
from $T$ and $N$ multiplies and thus dominates.
### 4.4 Cost in Typical Canteen Scenarios
The values of the four features we studied have different distributions based
on our dataset from real canteen situations. Four major combinations of
feature values were extracted to construct four typical canteens as listed in
Table. 5. We noticed that the value of $N$ was relatively stable in real
circumstances and the scale of the canteen was mainly indicated by $T$.
Moreover, a canteen usually tended to choose either a high $F$ or $R$ to
ensure menu variation and reasonable cost.
Table 5: Typical Canteens Based on Statistics | $T$ | $N$ | $F$ | $R$ | Customer number
---|---|---|---|---|---
TYPE I | 20 | 70 | 0.3 | 6 | 450 |
TYPE II | 20 | 70 | 3 | 12 | 450 |
TYPE III | 50 | 70 | 0.75 | 15 | 1200 |
TYPE IV | 50 | 70 | 7.5 | 30 | 1200 |
With these four canteens, we demonstrated the cost in a more specific way.
Apart from the general results explained, there were new insights as depicted
in Fig. 10: For a canteen with a high old dish rotation like types I and III,
if data accuracy was no more than 0.95, the CV system was recommended and
could save up to 50%-75% of costs. The bigger the canteen, the larger the cost
saving. As the accuracy increased, the cost difference declined and the RFID
system became better for ultimate accuracy (higher than 0.95). Meanwhile, for
a canteen with a high new dish frequency, like types II and IV, the RFID
system was always the better choice. Costs could be saved up to 50% for
moderate, and 70% for large scale canteens. The cost difference decreased a
little as the accuracy increased from 0.8 to 0.95.
(a) Type I
(b) Type II
(c) Type III
(d) Type IV
Figure 10: Total Cost by Typical Canteens
## 5 Discussions
This section discusses supplemental factors, including model parameter
sensitivity, correction features and balancing, and standardization of dishes,
in order to generalize our research and facilitate applications.
### 5.1 Model Parameter Sensitivity: A Long Term Perspective
The values of parameters listed in the Model section are based on our dataset
and on-site measurements, which may have some deviations on a real occasion.
Moreover, the change of social labor prices and technology improvements in the
long term can also cause the variation of these values. To make our work more
practical, extra experiments were performed to identify the extent of the
influences caused by these deviations. Experiments were conducted in the
baseline environment.
#### 5.1.1 The $S$ and $\alpha$ in EMH-A Models
The EMH-Accuracy model was most widely used throughout our research. The basic
key procedures of RFID systems, namely, inputting, setting and labeling, plus
correction procedures contained in both systems, all applied to this model to
describe the corresponding relationship between manpower cost and data
accuracy. Thus, we selected labeling, as it was found to be the dominant
procedure in RFID systems, to experiment on the two parameters’ value
deviation effects on the total cost. The results are drawn in Fig. 11.
Further calculation based on the diagrams showed that the variation degree of
total cost was proportionate to the deviation degree of $S$ by 1 accuracy, 50%
$S$ increment vs. 45% cost increment in Fig. 11a. The effect decreased as the
target accuracy dropped and became minor when the accuracy dropped beneath
0.8. Since the cost of labeling was dominant, the effect of $S$ deviation was
smaller when it came to other procedures like inputting, setting and
correcting. This also applied to $S$ in the sampling procedure in CV systems.
In terms of the $\alpha$ deviation, it mainly affected the total cost in a
range of accuracy between 0.7 to 0.93, about 50% $\alpha$ increment for 20%
total cost increment. The effect increased slowly as the $\alpha$ value
decreased. In addition, the $\alpha$ decrement also led to relatively more
difficulty to achieve a higher accuracy.
(a) SR Cost by $S$
(b) Unit EMH by $S$
(c) SR Cost by $\alpha$
(d) Unit EMH by $\alpha$
Figure 11: SR Cost by $S$ and $\alpha$ Deviation
#### 5.1.2 The $\beta$ in SNA Model
For the SNA model in CV systems, it is predictable that experimenting on the
$U$ parameter will not generate anything substantial, because of its direct
bond with SP accuracy. Hence, the transmission coefficient $\beta$, also the
expansion degree of our adopted sigmoid curve, was selected. Experiments were
carried out in a similar way to Section 5.1.1, with results as shown in Fig.
12.
Figure 12: SP Cost and Accuracy by $\beta$ Deviation
The results showed that there was no obvious bond between the $\beta$ value
and the total cost. However, the $\beta$ value growth caused a small accuracy
decrement around 0.002 for 50% deviation.
Finally, the long term change is also worth mentioning. As a global trend, the
inevitable manpower price increases will cause $S$ to increment and thus the
total cost will increase; The enhancement of systems’ degree of automation
will lead to a bigger $\alpha$ value which will make it more economical to
pursue accurate data; The development of the CV algorithm will trigger a
smaller $\beta$ and thereby higher accuracy and lower cost for CV systems.
### 5.2 Correction Features: Balancing in Local Conditions
As stated in Section 4.2, data correction at checkout is the only measure for
CV systems to improve their data quality. Further, correction is optional for
RFID systems. The cost of correction is decided by the number of errors and
expected accuracy improvement. Since CV systems have firm requirements, we
experimented on canteens with different customer capacities which is
proportionate with the product of $T$ and $N$, to explore its impact on
correction cost. The results are depicted in Fig. 13a, where the dotted line
is located by points with the same marginal cost growth right before the
abrupt rise. Considering the efficiency of the accuracy improvement by
correction, there is a dynamic limit for each canteen scale. In addition, the
space for improvement lessens by about 0.022 as the canteen capacity grows
from 190 to 700 customers.
(a) by Customer Capacity of CV Systems
(b) by Improvement Degree of RFID Systems
Figure 13: Correction Cost by Customer Capacity and Improvement Degree
For the RFID system, we constrained the accuracy improvement within 0.2 and
activated the correction procedure. The results are illustrated in Fig. 13b.
The results proved that it costs much more to depend on correction when
perfect accuracy is demanded since humans are more apt to make mistakes
especially in a high-pressured condition like checkout. Correction can save
the cost, about 10% maximum, when the target accuracy is between 0.7 and 0.97.
Overall, because of the elevated cost along improvement degree, correction can
only bring about small cost savings in a small accuracy range and thus cannot
be relied on. For CV systems, there is an efficient improvement limit which is
comparatively constant. The cost of accuracy above the limit multiplies and is
not economical.
### 5.3 Standardization of Dish Supply: An Inevitable Trend
The frequently changed menus and recipes in modern canteens have always been
the biggest liability to dietary data harvesting, and even more so for Chinese
foods. Except for the extra cost paid for information inputting and on-site
setting, the irregularity of the menus and the recipes also costs more in
sections outside the SGC systems. Inconsistent material ordering, temporary
dish interference and other expenditures can be saved if a relatively unified
menu and recipes are coordinated. To determine the cost saving of dish
standardization, a complementary experiment was carried out and the results
are drawn in Fig. 14.
Figure 14: Cost in Standardized Canteen: Baseline canteen is used for
comparison.
So, as is shown in Fig. 14, a canteen without dish addition and rotation can
lead to cost savings in dietary data harvesting of up to 46.7% for RFID
systems and 55.1% for CV systems. Against the background of emerging self-
awareness of health concerns and dietary management, it seems possible for
customers to compromise their preferences if effective dietary management is
provided. Moreover, the dish standardization may be more economical where
dietary management is of greater urgency, for example in hospitals, rehab
facilities and so on. We believe that instead of passively waiting for
technology developments, it would be wiser to embrace the trend of dietary
management and make self-adjustments to our own habits.
## Conclusions
In this paper, we conducted an in-depth analysis of the essential mechanisms
and the fundamental distinctions between RFID- and CV-based SGC systems. We
analyzed the manpower costs required for dietary data acquisition, which is
often overlooked. The tag binding of RFID systems leads to its long pipelined
manual operations and the upper bound of CV systems’ recognition models
constrains system reliability, which results in unacceptable costs to maximum
data accuracy. Two models based on the characteristics of staff operation in
specific procedures(the EMH-A model) and sample accumulation (the SNA model)
were proposed. Datasets collected from real canteens were used as input data.
In relation to the cost of dietary data acquisition, we have developed major
numerical conclusions as follows:
* •
In order for accurate dietary data acquisition, a large amount of extra costs
are required, of which staff operation related costs account for up to 90%.
* •
When deploying RFID systems, the cost of labeling procedures accounts for 80%
of the staff related costs. The labeling procedure is also the bottleneck to
achieve perfect accuracy.
* •
The accuracy of CV systems can be improved by around 0.08 to 0.92 through
efficient checkout corrections, but it is unrealistic to go on forcing the
accuracy to 1, since the total cost will multiply about 8 times.
* •
For both types of systems, the marginal costs continued to rise when higher
data accuracy was demanded. CV systems had a much higher rising rate, making
its total costs bypass those of RFID systems after 0.95 accuracy.
* •
It is expensive to continually introduce new dishes although RFID systems were
comparatively more suitable for a moderate-sized canteen with frequent new
dish additions.
* •
CV systems were vulnerable to new dish additions, which increased costs by
about 1.67 h / type and jeopardized the accuracy by about 0.01 / type, while
there were advantages for a large-scaled.
Based on our analysis, the current advantages of RFID systems will diminish if
no improvements on labeling procedures occur. CV systems can benefit from
improving recognition algorithms, and higher levels of automation in sampling.
They may also benefit from publicly available dietary datasets.
## Acknowledgment
We acknowledge the support of the National Natural Science Foundation of China
under grant 61433009, and the National Key Research and Development Project of
China under grant 2019YFC1709800.
—- Bibliography —-
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|
# IMAGE DENOISING WITH LESS ARTEFACTS:
NOVEL NON-LINEAR FILTERING ON FAST PATCH REORDERINGS
###### Abstract
Leading denoising methods such as 3D block matching (BM3D) are patch-based.
However, they can suffer from frequency domain artefacts and require to
specify explicit noise models. We present a patch-based method that avoids
these drawbacks. It combines a simple and fast patch reordering with a non-
linear smoothing. The smoothing rewards both patch and pixel similarities in a
multiplicative way. We perform experiments on real world images with additive
white Gaussian noise (AWGN), and on electron microscopy data with a more
general additive noise model. Our filter outperforms BM3D in 77% of the
experiments, with improvements of up to 29% with respect to the mean squared
error.
Index Terms— non-local patch-based methods, diffusion methods, image
denoising, additive white Gaussian noise
## 1 Introduction
Non-local patch-based methods [1, 2, 3, 4, 5, 6] have been producing superior
image denoising results since quite a few years now. These models offer two
main advantages: Firstly, the assumption that similar pixels have similar
neighbourhoods around them, is quite robust in a noisy scenario. Secondly, one
can access similar data from distant regions in the image.
The non-local Bayes (NLB) [7, 2] and BM3D [3, 8] approaches produce state-of-
the-art results. NLB uses Bayesian modelling while BM3D employs Fourier domain
filtering. However, it is well known that when the assumptions about noise-
free images are violated, one can see artefacts in the denoised images. The
idea that we can find completely similar patches within an image in general
need not be satisfied. Thus, there is a risk that data from dissimilar regions
can diffuse into each other, which leads to the above mentioned artefacts.
This observation is well documented for both NLB and BM3D [6].
One remedy to eliminate these artefacts is to use an additional post-
processing step [6]. Another possibility which was proposed by Ram et al. [4,
5] is to employ a smooth reordering of patches for subsequent filtering.
However, the underlying reason why the latter method is better than BM3D is
not well understood. It appears plausible to us that it minimises information
exchange between dissimilar regions with the help of patch reordering, thus
reducing the artefacts and consequently leading to better results. However,
this comes at a computationally very expensive reordering step, which
basically requires to solve a travelling salesman problem.
Our Contribution. We introduce a new method to solve the above artifact
problem without the need of an additional post-processing step and also within
a relatively low computational time. In contrast to the methods of Ram et al.
[4, 5], we use a simpler and much faster patch reordering, and combine it with
a more sophisticated non-linear filtering. Hence, we call our method non-
linear filtering on fast patch reorderings (NFPR). In particular, we employ a
filtering technique which is a novel combination of weights that reward both
patch and pixel similarities. Moreover, we always use disc-shaped windows,
thus leading to a rotationally invariant model. In contrast to NLB and BM3D,
we avoid an explicit AWGN assumption and hence are more robust with respect to
the noise type.
Paper Structure. In Section 2, we introduce our proposed NFPR framework for
noise elimination along with proper motivations. In Section 3, we showcase a
comparative evaluation of NFPR with NLB, BM3D and the method of Ram et al.
[4], for both real-world test images and electron microscopy data. Finally, in
Section 4, we conclude with a summary of our contribution and an outlook to
future work.
Image | $\sigma$ | $\lambda$ | $k_{\textrm{max}}$ | NFPR | NLB | BM3D
---|---|---|---|---|---|---
L40 | 150 | 11.5 | 16 | 74.00 | 69.30 | 68.27
L60 | 160 | 15.5 | 16 | 104.58 | 109.3 | 104.83
L80 | 175 | 20.0 | 14 | 139.37 | 154.40 | 143.23
L100 | 175 | 23.5 | 16 | 164.85 | 198.89 | 183.78
L120 | 190 | 27.0 | 15 | 196.96 | 254.90 | 228.39
L140 | 195 | 31.5 | 15 | 231.48 | 312.95 | 273.09
B40 | 130 | 15.0 | 13 | 254.87 | 233.36 | 233.34
B60 | 160 | 20.5 | 8 | 333.37 | 333.22 | 315.37
B80 | 165 | 26.0 | 9 | 400.71 | 425.61 | 391.02
B100 | 180 | 30.5 | 9 | 457.70 | 486.65 | 453.37
B120 | 180 | 34.5 | 10 | 505.86 | 551.94 | 512.38
B140 | 190 | 36.5 | 11 | 556.54 | 616.51 | 572.44
H40 | 140 | 10.5 | 27 | 58.77 | 62.16 | 56.01
H60 | 160 | 12.0 | 27 | 81.24 | 104.85 | 92.15
H80 | 180 | 15.0 | 23 | 116.63 | 156.88 | 130.64
H100 | 185 | 17.5 | 17 | 153.27 | 218.91 | 187.60
H120 | 205 | 22.5 | 17 | 192.11 | 289.12 | 235.53
H140 | 200 | 25.0 | 19 | 233.62 | 356.59 | 300.88
P40 | 155 | 11.5 | 16 | 57.73 | 60.03 | 58.91
P60 | 160 | 16.0 | 17 | 83.73 | 95.22 | 91.16
P80 | 185 | 18.5 | 15 | 112.39 | 128.95 | 124.08
P100 | 195 | 21.0 | 15 | 139.05 | 171.88 | 160.88
P120 | 205 | 25.0 | 15 | 167.70 | 216.02 | 200.58
P140 | 205 | 29.5 | 15 | 198.83 | 266.79 | 241.20
Table 1: MSE values of denoised images including NFPR parameters. L40 stands
for Lena image with $\sigma_{\textrm{noise}}$ = 40. B, H, P denote Bridge,
House and Peppers respectively.
## 2 Modelling of our denoising algorithm
Our NFPR technique consists of two parts: The goal of the first step is to
achieve a fast patch-based reordering of the pixels. In the second step, we
employ a non-linear smoothing on the reordered pixels which yields the
denoised image. In the following we describe these steps in detail.
Step 1 : Fast Patch Reordering. In order to compute a smooth reordering of
pixels, we employ a patch-based similarity approach: We first consider a disc-
shaped search region $B_{\textrm{search}}$ of radius $\rho_{\textrm{search}}$
around every pixel $u_{i}$ in the 2D image domain. We then compute the $L_{2}$
norm $d_{ij}$ between disc-shaped patches of radius $\rho_{\textrm{sim}}$,
centered around $u_{i}$ and $u_{j},$ for all $j\in B_{\textrm{search}}$. This
is followed by constructing a set $P_{i}$ of $N$ pixels within
$B_{\textrm{search}}$ which have the least distance from $u_{i}$ according to
$d_{ij}$. This set characterises the desired smooth reordering of pixels. In
contrast to Ram et al. [4, 5] who solve instances of the NP-hard travelling
salesman problem, we compute the reordering using just a simple sort
operation. Ideally, when we average noisy versions of the same greyvalue, we
should not introduce artefacts. However, we have noisy versions of
approximately equal grey values in the set $P_{i}$. Moreover, the above simple
reordering is achieved at the cost of some disordered pixels in $P_{i}$, that
come from areas of dissimilar greyvalues. In the second step of the algorithm,
we employ a very robust non-linear smoothing technique, to deal with both
problems.
Step 2 : Non-linear Smoothing. The goal of this step is to optimally combine
the set of pixels $P_{i}$, obtained from the first step, and compute a final
denoised image. To this end, we apply a non-linear smoothing process on this
set. This can be thought of as diffusing information between pixels in a space
defined by the neighbourhood similarity distances $d_{ij}$ instead of the
generally used spatial distances. We utilise two assumptions which form the
core of our structure preserving smoothing technique: Firstly, similar pixels
have relatively smaller absolute tonal differences $|u_{j}-u_{i}|$. Secondly,
they also have similar neighbourhoods around them. Although we have already
used such an idea for patch reordering, we will be re-using the distances
$d_{ij}$ through a multiplicative combination of both assumptions. This gives
us an advantage in scenarios where one of the assumptions might be violated in
the presence of noise. The discrete evolution equation for our smoothing
process is given by
$\displaystyle\frac{u_{i}^{k+1}-u_{i}^{k}}{\tau}=a_{i}^{k}\cdot\left(\sum_{\begin{subarray}{c}j\in
P_{i}^{k}\end{subarray}}g\left({u^{k}_{\sigma j}}-{u^{k}_{\sigma
i}}\right)h\left(d_{ij}^{k}\right)\left(u_{j}^{k}-u_{i}^{k}\right)\right.$
$\displaystyle+\left.\sum_{\begin{subarray}{c}j\in
P^{\textrm{add},k}_{i}\end{subarray}}g\left({u^{k}_{\sigma j}}-{u^{k}_{\sigma
i}}\right)h\left(d_{ij}^{k}\right)\left(u_{j}^{k}-u_{i}^{k}\right)\right).$
(1)
This equation has two terms on the right hand side, which model two types of
information exchange: Remember that $P_{i}$ denotes the set of pixels which
are closest to pixel $u_{i}$ according to the distance $d_{ij}$. Every pixel
in the image will have its own reordered set. Thus, the pixel $u_{i}$ could
also be part of sets other than $P_{i}$. The symbol $P^{\textrm{add}}_{i}$
denotes an additional set of pixels in whose corresponding reordered sets,
$u_{i}$ is present. The two terms mentioned in the above equation represent
interactions with these two sets of pixels $P_{i}$ and $P^{\textrm{add}}_{i}$,
respectively. This can also be seen as collaborative filtering similar to BM3D
and NLB.
Image | $\sigma$ | $\lambda$ | $k_{\textrm{max}}$ | NFPR | REC
---|---|---|---|---|---
L50 | 150 | 14.5 | 17 | 91.36 | 82.90
L75 | 170 | 19.0 | 15 | 126.90 | 123.75
L100 | 175 | 23.5 | 16 | 164.85 | 163.05
H50 | 160 | 11.0 | 23 | 70.94 | 74.45
H75 | 165 | 15.5 | 23 | 108.71 | 120.41
H100 | 185 | 17.5 | 17 | 153.27 | 167.52
Table 2: MSE values of denoised images including NFPR parameters.
Abbreviations as in Table 1, REC - Ram et al. [4].
noisy NLB BM3D NFPR original
Fig. 1: Top to Bottom: Zoom into Lena, Bridge, House and Peppers images
($\sigma_{\mathrm{noise}}=80$).
noisy NFPR FRC plot
Fig. 2: Left: Zoom into ribosome image of a yeast cell, with original size 256
$\times$ 256\. Right: Zoom into the [0.4-1.0] correlation range of the
corresponding FRC plot calculated for 64 frequency levels. NFPR parameters
used:- $\sigma=170,\lambda=2.5,k_{\textrm{max}}=35$. For NLB and BM3D we have
optimised the unknown $\sigma_{\textrm{noise}}$ with respect to FRC.
Let us now discuss the details of the above two individual terms: The
functions $g$ and $h$ model the above mentioned tonal and neighbourhood
similarity assumptions, respectively. However, if we look closely at the
argument of $g$, we have ${u_{\sigma j}}-{u_{\sigma i}}$ instead of
${u_{j}}-{u_{i}}$ which are the real tonal differences. This idea of
calculating the tonal differences on a denoised image $\bm{u}_{\sigma}$, for a
robust performance, is inspired from diffusion-based methods [9]. We have
chosen a collaborative version of the non-local means approach [1] for this
initial denoising process:
$\displaystyle u^{k}_{\sigma
i}=b_{i}^{k}\cdot\left(\sum_{\begin{subarray}{c}j\in
P_{i}^{k}\end{subarray}}h\left(d_{ij}^{k}\right)u_{j}^{k}+\sum_{\begin{subarray}{c}j\in
P^{\textrm{add},k}_{i}\end{subarray}}h\left(d_{ij}^{k}\right)u_{j}^{k}\right).$
(2)
The symbols $a_{i}$ and $b_{i}$ in (2) and (2), respectively, are the
normalisation constants. The functions $g$ [10] and $h$ in (2) are chosen as
$\displaystyle
g\left(s\right)=1-\text{exp}\left(\frac{-3.31488}{\left(\frac{s}{\lambda}\right)^{8}}\right),$
$\displaystyle h(s)=\text{exp}\left(\frac{-s^{2}}{2\sigma^{2}}\right).$
The time step size $\tau$ in (2) is selected such that the maximum-minimum
principle is not violated. This means that the dynamic range of the denoised
image does not exceed that of the initial noisy image. We can achieve this by
choosing $a_{i}$ = $b_{i}/M_{i}$, where $M_{i}$ is the sum of number of
elements in $P_{i}$ and $P_{i}^{\textrm{add}}$, and $\tau\leq 1$. Finally, we
iterate NFPR for $k_{\textrm{max}}$ times. We initialise the non-linear
smoothing with the initial noisy image $\bm{f}$ and the patch reordering using
a Gaussian smoothed version of $\bm{f}$ with standard deviation $\sigma_{G}$.
## 3 Experiments and Discussion
In order to test the robustness of our method with respect to the noise type,
we have performed denoising experiments on both real-world test images and
electron microscopy data. For saving time, we have restricted the usage of our
patch reordering step to just two iterations. This has negligible effect on
the denoising output. We have fixed the following parameters:
$\rho_{\textrm{search}}=10$, $\rho_{\textrm{sim}}=10$, $\sigma_{G}=2.5$,
$\tau=0.95$ and the number of elements in the reordered set $N=35$. In order
to have a correspondence for the parameter $\sigma$ between real-world and
electron microscopy data, we have performed an affine rescaling of the
distances $d_{ij}$ within the set $P_{i}$, to [0, 255]. Thus, we just optimise
the parameters $\sigma$, $\lambda$ and $k_{\textrm{max}}$. As already
mentioned, our denoising experiments employ NLB, BM3D, Ram et al. [4] for
comparison purposes with available implementations and detailed parameter
studies in [7, 8, 11, 4].
We first present our results on the real-world images Lena, Bridge, House and
Peppers111http://sipi.usc.edu/database/, which have been corrupted with AWGN.
We use the mean squared error (MSE) for both measuring the quality of the
denoised images and optimising the parameters. Figure 1 and Table 1 show the
comparison with NLB and BM3D. Visual advantages in terms of less artefacts and
more pleasant edges are larger than MSE advantages would suggest. From Table
2, we can conclude that our method is competetive with the approach of Ram et
al. [4].
Experiments on a GPU222NVIDIA GeForce GTX 970 graphics card using C++ and CUDA
show that our method takes just 2 and 6.5 seconds for denoising the $256\times
256$ sized House and $512\times 512$ sized Lena images
($\sigma_{\textrm{noise}}=100$), respectively. This implementation could
further be improved with pre-computing the weighting functions and also
through faster implementations of patch-similarity computations. The available
non-parallel CPU333Intel(R) Core(TM) i7-6700 CPU 3.4 GHz machine using
MATLAB/C implementation of [4] takes 1128 and 7021 seconds in the above
scenarios. This is very expensive, even if we take into account the technical
differences in the implementations.
We have also considered ribosomal data in yeast cells acquired using an
electron microscope. For measuring the quality of the denoised images, we use
a popular frequency domain measure in electron microscopy called Fourier ring
correlation (FRC). It computes a cross-correlation coefficient between the
denoised versions of two different images of the same scene at different
frequency levels [12, 13]. Figure 2 shows the corresponding results. We
observe higher correlation coefficients for NFPR in the FRC curves. This
indicates that it does a better job in preserving image structures during the
denoising process.
All the above results can be attributed to the previously mentioned modelling
advantages of NFPR. In contrast to NLB and BM3D, it also benefits by avoiding
an explicit AWGN noise approach. This leads to even better NFPR results for
electron microscopy data, as such kind of data is generally approximated with
a more general additive noise model (see Chapter 11 of [14]). On the other
hand, there are certainly some cases when NLB, BM3D and Ram et al. [4] are
better than NFPR for real-world images like Lena and Bridge which have some
amount of texture.
We observe that an important message from the above results is in accordance
with the conclusion in our multi-frame denoising research [11]: The process of
choosing the combination of pixels that undergo non-linear smoothing is as
important as choosing the type of non-linearity itself. In BM3D and NLB, we
filter a group of similar patches altogether. In NFPR, we utilise a carefully
chosen set of pixels and subsequently filter them using a robust procedure.
This is superior although we do not use any explicit spatial context in the
filtering process. Unlike NLB and BM3D, we apply it only for patch reordering.
In [11], we observed that a linear temporal filter can outperform a non-linear
one, but only if the pixels that undergo the denoising process are chosen
carefully.
## 4 Conclusions and Outlook
Although most people would agree that artifact avoidance is desirable in
denoising, this goal has hardly been addressed in practice. It appears that
smooth patch reordering is helpful in that aspect, but its computational
workload is usually very high. The message of our paper is that this patch
reordering can be fairly simplistic and therefore fast, provided that one
comes up with more sophisticated non-linear filters that reward pixel and
patch similarities in a multiplicative manner. Moreover, refraining from
explicit noise models can be beneficial in real-world applications that may
deviate from ideal noise assumptions. We believe that these are fairly general
design principles that deserve further exploration in the future. This is also
part of our ongoing work.
## References
* [1] A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Diego, CA, June 2005, vol. 2, pp. 60–65.
* [2] M. Lebrun, A. Buades, and J.M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM Journal on Imaging Sciences, vol. 6, no. 3, pp. 1665–1688, Sept. 2013.
* [3] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Transactions on Image Processing, vol. 16, no. 8, pp. 2080–2095, Aug. 2007.
* [4] I. Ram, M. Elad, and I. Cohen, “Image processing using smooth ordering of its patches,” IEEE Transactions on Image Processing, vol. 22, no. 7, pp. 2764–2774, July 2013.
* [5] I. Ram, M. Elad, and I. Cohen, “Image denoising using NL-means via smooth patch ordering,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Vancouver, BC, Canada, May 2013, pp. 1350–1354.
* [6] N. Pierazzo, M. E. Rais, J. M. Morel, and G. Facciolo, “DA3D: Fast and data adaptive dual domain denoising,” in Proc. IEEE International Conference on Image Processing (ICIP), Quebec City, Canada, Sept. 2015, pp. 432–436.
* [7] M. Lebrun, A. Buades, and J.M. Morel, “Implementation of the Non-Local Bayes (NL-Bayes) image denoising algorithm,” Image Processing On Line, vol. 3, pp. 1–42, June 2013.
* [8] M. Lebrun, “An analysis and implementation of the BM3D image denoising method,” Image Processing On Line, vol. 2, pp. 175–213, Aug. 2012.
* [9] F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 182–193, Mar. 1991.
* [10] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 1998.
* [11] K. Bodduna and J. Weickert, “Removing Multi-frame Gaussian Noise by Combining Patch-based Filters with Optical Flow,” arXiv:2001.08058 [eess.IV], Jan. 2020.
* [12] W. O. Saxton and W. Baumeister, “The correlation averaging of a regularly arranged bacterial cell envelope protein,” Journal of Microscopy, vol. 127, no. 2, pp. 127–138, Aug. 1982\.
* [13] P. A. Penczek, “Resolution measures in molecular electron microscopy,” Methods in Enzymology, vol. 482, pp. 73–100, 2010.
* [14] J. Frank, Electron Tomography: Methods for Three-dimensional Visualisation for Structures in the Cell, Springer, New York, second edition, 2008.
|
# Vanishing of Drude weight in interacting fermions on $\hbox{\msytw Z}^{d}$
with quasi-periodic disorder
Vieri Mastropietro University of Milano, Department of Mathematics “F.
Enriquez”, Via C. Saldini 50, 20133 Milano, Italy
###### Abstract
We consider a fermionic many body system in $\hbox{\msytw Z}^{d}$ with a short
range interaction and quasi-periodic disorder. In the strong disorder regime
and assuming a Diophantine condition on the frequencies and on the chemical
potential, we prove at $T=0$ the exponential decay of the correlations and the
vanishing of the Drude weight, signaling Anderson localization in the ground
state. The proof combines Ward Identities, Renormalization Group and KAM
Lindstedt series methods.
## 1 Introduction
The conductivity properties in fermionic systems, describing electrons in
metals, are strongly affected by the presence of disorder, which breaks the
perfect periodicity of an ideal lattice and is unavoidable in real systems.
Disorder can be represented either by a random variable or by a quasi-periodic
potential; the first description is more suitable for impurities in solids
while the second appears naturally in quasi-crystals or cold atoms
experiments. In absence of many body interaction disorder produces the
phenomenon of Anderson localization [1], consisting in an exponential decay of
all eigenstates and in an insulating behavior with vanishing conductivity.
Such a phenomenon relies on the properties of the single particle Schroedinger
equation and it has been the subject of a deep mathematical investigation.
With random disorder Anderson localization was established for strong disorder
in any dimension [2], [3] and in one dimension with any disorder. In the case
of quasi-periodic disorder localization in one dimension is present only for
large disorder [4], [5], while for weak disorder is absent; in higher
dimensions localization was proved for strong disorder in $d=2$ [6], [7] and
for any $d$ in [8].
The interplay between disorder and interaction has been deeply analyzed in the
physical literature soon after [1]. The presence of many body interaction
induces new processes which can indeed destroy localization. At zero
temperature $T=0$ with random disorder qualitative scaling arguments gave
evidence of persistence of localization in $d=3$ [9], [10] for short range
weak interaction; in $d=1$ a second order Renormalization Group analysis was
shown to produce a complex phase diagram [11]. The case of quasi-random
disorder has been less studied, with the exception of [12], [13] focusing on
the extended weak disorder regime at $T=0$. In more recent times the
properties at $T>0$ were analyzed in [14], where perturbative arguments for
the vanishing of conductivity up to a certain critical $T$ in any dimension
were given (many body localized phase). Subsequently numerical simulations
found localization in certain systems in all the spectrum and vanishing of
conductivity for any $T$, a phenomenon called many body localization, see [15]
for random and [16] for quasi-periodic disorder. If all states are localized
one expects, in a non-equilibrium setting, that interaction is unable to
produce thermalization in an isolated quantum system, a phenomenon that in
classical mechanics is due to closeness to an integrable system. Interacting
quantum systems with quasi-periodic disorder have been realized in cold atoms
experiments [17], [18],[19] ; quasi-periodic disorder with many body
interaction has been extensively numerically analyzed [20]-[28].
While the above works suggest that localization persists in presence of
interaction, results based on numerical or perturbative analysis cannot be
conclusive. In particular the presence of small divisors has the effect that
physical informations are difficult to be extracted by lower order analysis
but are typically encoded in convergence or divergence of the whole series.
This is a well known phenomenon in classical mechanics; the Birkoff series for
prime integrals in Hamiltonian systems are generically diverging while
Lindsdtet series for Kolomogorov-Arnold-Moser (KAM) tori converge, even if
both series are order by order finite and present similar small divisors.
Therefore, even if perturbative analysis in [14] or [29] get localization at
finite temperature and in any dimension, one cannot exclude that the series
are divergent and localization eventually disappear (this would say that
thermalization in experiments is eventually reached, even if at long times). A
non-perturbative proof of many body localization for all eigenstates has been
indeed finally obtained in $d=1$ with random disorder in [30] but the result
is based on a certain unproven assumption. A complete proof have been obtained
only with vanishing densities [31], [32]. Arguments for breaking of many body
localization in $d>1$ have been indeed presented in [33].
In order to get rigorous results as benchmark for conjectures and
approximations, a natural starting point is the zero temperature case in the
thermodynamic limit. Our approach is to compute thermodynamical correlations;
they not only provide physical observables at equilibrium but give also
information on the spectrum (so their computation is of interest even for
situation where equilibrium is not reached). In particular at zero temperature
they provide information of correlations over the ground state, while the
vanishing of conductivity at any temperature is a signal of many body
localization in all the spectrum. It has been proven in [34],[35],[36] for one
dimensional interacting fermions with strong quasi-periodic disorder the $T=0$
exponential decay of 2-point correlations, indicating persistence of
localization in the ground state. Aim of this paper is twofold. The first is
to investigate the $d>1$ case. We consider a disorder of the form
$f(\vec{\omega}\vec{x})$ with $f$ periodic, as the one considered in [6] for
the single particle Schroedinger equation ; more general forms of disorder are
however possible, as $f(\vec{\omega}_{1}\vec{x},\vec{\omega}_{2}\vec{x})$
considered in [6]. The second aim is to compute the $T=0$ conductivity
expressed by Kubo formula, whose properties can be analyzed via a combination
of information provided by Ward Identities with regularity properties of the
current correlations. The thermodynamical quantities are expressed by a series
expansion showing a peculiar combinations of properties appearing in classical
and quantum physics; they show a small divisor problem, as in the Lindstedt
series for KAM [37], but loop graphs appear in the expansion, a signature of
quantum physics totally absent in classical mechanics. In order to achieve
convergence and exclude non perturbative effects one has from one side to show
that divisors can be controlled by number theoretical conditions on
frequencies, and from the other that the huge number of loop graphs is
compensated by cancellations from the fermionic anticommutative nature of the
problem.
The paper is organized in the following way. In §2 the model is presented and
in §3 the main results, together with open problems, are presented. In §4 we
discuss the implications of Ward Identities and regularity bounds. In §5 we
introduce the Grassmann representation and in §6 we introduce the multiscale
analysis. In §7 we prove the convergence of series expansion and in §8 we get
the asymptotic decay of correlations.
## 2 Interacting fermions with quasi-periodic disorder
We introduce the Fock space $\mathcal{F}_{L}=\bigoplus_{N\geq
0}\mathfrak{h}_{L}^{\wedge N}$ where the $N$ particle Hilbert space
$\mathfrak{h}_{L}^{\wedge N}$ is the set of the totally antisymmetric square
integrable functions in
$\Lambda_{L}:=\\{\vec{x}\in\mathbb{Z}^{d}\mid\vec{x}=n_{1}\vec{e}_{1}+n_{2}\vec{e}_{2}+...\;,\quad-L/2\leq
n_{i}\leq L/2\;,\quad i=1,2,..,d\\}$ where $\vec{e}_{i}$ are unit vectors. The
$a^{\pm}_{\vec{x}}$ are fermionic creation or annihilation operators sending
an element of $\mathfrak{h}_{L}^{\wedge N}$ in $\mathfrak{h}_{L}^{\wedge N+1}$
(creation) or $\mathfrak{h}_{L}^{\wedge N-1}$ (annihilation) and
$\\{a^{+}_{\vec{x}}\,,a^{-}_{\vec{y}}\\}=\delta_{\vec{x},\vec{y}}$,
$\\{a^{+}_{\vec{x}}\,,a^{+}_{\vec{y}}\\}=\\{a^{-}_{\vec{x}}\,,a^{-}_{\vec{y}}\\}=0$.
The Hamiltonian is
$H=-{\varepsilon\over
2}\sum_{\vec{x}}\sum_{i=1}^{d}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}+a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})+u\sum_{\vec{x}}\phi_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}+\lambda\sum_{\vec{x}}\sum_{i=1}^{d}a^{+}_{\vec{x}}a^{-}_{\vec{x}}a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}+\vec{e}_{i}}$
(1)
where $a^{+}_{\vec{x}}$ must be interpreted as zero for
$\vec{x}\not\in\Lambda_{L}$ and
$\phi_{\vec{x}}=\bar{\phi}(\vec{\omega}\vec{x})$ with
$\bar{\phi}(t):\hbox{\msytw T}\rightarrow\hbox{\msytw R}$ periodic of period
$1$. In order to describe a quasi-periodic disorder we impose that
$\vec{\omega}$ is rationally independent and ”badly” approximated by rationals
(Diophantine condition). The first term in (1) represents the kinetic energy
of the fermions hopping on a lattice, the second represents the interaction
with a quasi-periodic potential and the last term represents a 2 body
interaction.
There are several interesting limits; $\lambda=0$ is the non interacting
limit; $\lambda=u=0$ is the integrable limit;ùù $\lambda=\varepsilon=0$ is the
anti-integrable limit (the therminology was introduced in [38] ). We consider
the case in which $\lambda,\varepsilon$ are small with respect to $u$, and we
set $u=1$ for definiteness; that is we consider a perturbation of the anti-
integrable limit.
If $N=\sum_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}$ we define
$\langle\cdot\rangle_{\beta,L}=\frac{\mathrm{Tr}_{\mathcal{F}_{L}}\cdot
e^{-\beta(H-\mu
N)}}{\mathcal{Z}_{\beta,L}}\;,\qquad\mathcal{Z}_{\beta,L}=\mathrm{Tr}_{\mathcal{F}_{L}}e^{-\beta(H-\mu
N)}$ (2)
where $\mu$ is the chemical potential, which is fixed by the density in the
Grand-Canonical ensamble, and $\mathcal{Z}_{\beta,L}$ is the partition
function. In the limit $\beta\rightarrow\infty$ they provide information on
the ground states. We define
$\langle\cdot\rangle=\lim_{\beta\rightarrow\infty}\lim_{L\rightarrow\infty}\langle\cdot\rangle_{\beta,L}$
(3)
The imaginary-time (or Euclidean) evolution of the fermionic operators is
$a^{\pm}_{{\bf x}}=e^{x_{0}(H-\mu N)}a^{\pm}_{\vec{x}}e^{-x_{0}(H-\mu N)}$ (4)
with ${\bf x}=(x_{0},\vec{x})\quad\text{with}\quad x_{0}\in[0,\beta)$, The
2-point function is given by
$S_{\beta,L}({\bf x},{\bf y})=\left\langle Ta^{-}_{\bf x}a^{+}_{\bf
y}\right\rangle_{\beta,L}$ (5)
and $T$ is the time order product. We also consider the truncated expectations
$\left\langle TA;B\right\rangle_{\beta,L}=\left\langle
TAB\right\rangle_{\beta,L}-\left\langle TA\right\rangle_{\beta,L}\left\langle
TB\right\rangle_{\beta,L}$. The density and the current are given by
$\rho_{\vec{x}}=a^{+}_{\vec{x}}a^{-}_{\vec{x}}\quad\quad
j^{i}_{\vec{x}}={\varepsilon\over
2i}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}-a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})$
(6)
The (Euclidean) conductivity density in the zero temperature limit is defined
by Kubo formula
$\sigma^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}{1\over
p_{0}}\lim_{\beta\rightarrow\infty}\lim_{L\rightarrow\infty}[\sum_{\vec{x}\in\Lambda_{L}}\int_{0}^{\beta}dx_{0}e^{ip_{0}x_{0}}\left\langle
Tj^{i}_{\vec{x},x_{0}};j^{i}_{\vec{y},0}\right\rangle_{\beta,L}+<\tau^{i}_{\vec{y}}>_{\beta,L}]$
(7)
where
$\tau^{i}_{\vec{y}}=-{\varepsilon\over
2}(a^{+}_{\vec{y}+\vec{e}_{i}}a^{-}_{\vec{y}}+a^{+}_{\vec{y}}a^{-}_{\vec{y}+\vec{e}_{i}})$
(8)
The conductivity can be equivalently expressed in terms of the Fourier
transform which is, in the $\beta\rightarrow\infty,L\rightarrow\infty$ limit ,
$i=1,,d$
$\widehat{H}_{ii}({\bf p},\vec{y})=\sum_{\vec{x}\in\Lambda}\int_{\hbox{\msytw
R}}dx_{0}e^{i{\bf p}{\bf x}}<Tj^{i}_{\vec{x},x_{0}0};j^{i}_{\vec{y},0}>$ (9)
and similarly we define $\widehat{H}_{\mu\nu}({\bf p},\vec{y})$, with
$\mu=0,1,...d$ ($\mu=0$ is the density and $\mu=1,...,d$ the current
component). We can rewrite (7) as
$\sigma^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow
0}{1\over p_{0}}[\widehat{H}_{ii}({\bf p},\vec{y})+<\tau^{i}_{\vec{y}}>]$ (10)
Finally the (zero temperature) Drude weight, see eg [39], [40] , is defined as
$D^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow
0}[\widehat{H}_{ii}({\bf p},\vec{y})+<\tau^{i}_{\vec{y}}>]$ (11)
In a perfect metal at equilibrium the Drude weight is non-vanishing implying
that the conductivity is infinite; a vanishing Drude weight signals a non-
metallic behavior.
In the above definitions of conductivity the order in which the limits are
taken is essential; already in the integrable limit $u=\lambda=0$ reversing
the order of the limits one obtains a zero result, while the Drude weight is
indeed non vanishing as a consequence of the non-continuity of the Fourier
transform of the current correlation.
## 3 Main result
In the anti-integrable limit $\lambda=\varepsilon=0$ the eigenvalues of the
Hamiltonian are, $\vec{x}\in\Lambda_{L}$
$H_{0}=\sum_{\vec{x}\in\Lambda_{L}}\bar{\phi}(\vec{\omega}\vec{x})n_{\vec{x}}\quad\quad
n_{\vec{x}}=0,1$ (12)
and the single particle eigenfunctions have the form of
$\delta_{\vec{x},\vec{y}}$. The 2-point function is given by
$g({\bf x},{\bf
y})=\delta_{\vec{x},\vec{y}}e^{(\phi_{\vec{x}}-\mu)(x_{0}-y_{0})}[\theta(x_{0}-y_{0}){1\over
1+e^{\beta(\phi_{\vec{x}}-\mu)}}-\theta(y_{0}-x_{0}){e^{\beta(\phi_{\vec{x}}-\mu)}\over
1+e^{\beta(\phi_{\vec{x}}-\mu)}}]$ (13)
which can be equivalently written as
$g({\bf x},{\bf
y})=\delta_{\vec{x},\vec{y}}{1\over\beta}\sum_{k_{0}={2\pi\over\beta}(n_{0}+{1\over
2})}e^{-ik_{0}(x_{0}-y_{0})}\widehat{g}(\vec{x},k_{0})=\delta_{\vec{x},\vec{y}}\bar{g}(\vec{x};x_{0}-y_{0})$
(14)
with
$\widehat{g}(\vec{x},k_{0})={1\over-ik_{0}+\phi_{\vec{x}}-\mu}$ (15)
We define
$\mu=\bar{\phi}(\alpha)$ (16)
and the occupation number on the ground state is
$\theta(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))$; the choice of
$\mu$ fixes the averaged density. The conductivity is exactly vanishing as the
is proportional to $\varepsilon$. The density correlation is
$<\rho_{\bf x};\rho_{\bf
y}>=\delta_{\vec{x},\vec{y}}\bar{g}(\vec{x};x_{0}-y_{0})\bar{g}(\vec{x};y_{0}-x_{0})$
(17)
We want to investigate what happens when we consider a non-vanishing hopping
$\varepsilon\not=0$ and interaction $\lambda\not=0$. As usual in small divisor
problems, we need to impose a Diophantine condition on the frequencies
$\vec{\omega}$ of the quasi-periodic disorder that is
$||(\vec{\omega}\vec{x})||_{\hbox{\msytw T}}\geq
C_{0}|\vec{x}|^{-\tau}\quad\quad\vec{x}\in\hbox{\msytw Z}^{d}/\vec{0}$ (18)
$||.||$ being the norm on the one dimensional torus with period $1$; we
require also a Diophantine condition on the chemical potential, that is
$||(\vec{\omega}\vec{x})\pm 2\alpha||_{\hbox{\msytw T}}\geq
C_{0}|\vec{x}|^{-\tau}\quad\quad\vec{x}\in\hbox{\msytw Z}^{d}/\vec{0}$ (19)
The complementary of the set of numbers $\omega,\alpha$ verifying the
diophantine conditions for some $C_{0}$ has measure $O(C_{0})$, see eg [41].
In general the value of the chemical potential is modified by the interaction;
in order to fix the interacting chemical potential to the value
$\bar{\phi}(\alpha)$ we choose the bare one to $\mu=\bar{\phi}(\alpha)+\nu$
with $\nu$ chosen properly.
Our main result is the following
###### Theorem 3.1.
Assume that $\mu=\bar{\phi}(\alpha)+\nu$ and
$\phi_{x}=\bar{\phi}(\vec{\omega}\vec{x})$ with
$\bar{\phi}:\mathbf{T}\rightarrow\hbox{\msytw R}$, even, differentiable and
such that $v_{0}=\partial\bar{\phi}(\alpha)\not=0$: in addition $\vec{\omega}$
verifies (18) and $\alpha$ verifies (19). There exists $\varepsilon_{0}$ and a
suitable choice of $\nu=O(\varepsilon_{0})$ such that, for
$|\lambda|\leq|\varepsilon|\leq\varepsilon_{0}$ in the zero temperature and
infinite volume limit
1. 1.
The 2-point correlation verifies, for any $N$
$|S({\bf x},{\bf y})|\leq|\log\Delta_{\vec{x},\vec{y}}|C_{N}{e^{-{1\over
4}|\log|\varepsilon|||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}-y_{0}|)^{N}}$ (20)
with
$\Delta_{\vec{x},\vec{y}}=(1+\min(|\vec{x}|,|\vec{y}|))^{-\tau}$ (21)
2. 2.
The density and current correlations verify
$|H_{\mu,\nu}({\bf x},{\bf
y})|\leq\Delta_{\vec{x},\vec{y}}^{-4}C_{N}{e^{-{1\over
4}|\log|\varepsilon|||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}-y_{0}|)^{N}}$ (22)
3. 3.
The Drude weight is vanishing
$D^{i}_{\vec{x}}=0$ (23)
The above result says that there is exponential decay in the coordinate
difference in the fermionic and current correlations, signaling localization
in the ground state with quasi periodic potential of the form
$\bar{\phi}(\vec{\omega}\vec{x})$ in any dimension. Moreover the Drude weight
at $T=0$ is vanishing, implying a non-metallic behavior. This result is
obtained assuming a Diophantine condition on the frequencies and on the
chemical potential (or equivalently on the densities), see (19). As the
estimate of the radius of convergence $\varepsilon_{0}$ is proportional to
$C_{0}$ to some power, with fixed $\varepsilon,\lambda$ we get a large measure
set of densities for which localization is present (but not on an interval).
Information on the conductivity are obtained by combining the Ward Identities
following from the conservation of the current with regularity properties of
the Fourier transform of the correlations, which are related to the decay in
the coordinate space. In the case of non-interacting fermions, or for $1d$
interacting fermions without disorder, the slow power law decay of
correlations implies a non vanishing Drude weight, see [42]. In the present
case, the decay in space is exponentially fast but the decay in the imaginary
time has rate not uniform in $\vec{x},\vec{y}$, due to the lack of translation
invariance. As a consequence, we can deduce the vanishing of the Drude weight
but not of the conductivity.
The analysis is based on an extension of the Lindstedt series approach to KAM
tori with exact Renormalization Group methods for fermions. The correlations
are expressed by a series expansion showing a small divisor problem, as in the
Lindstedt series for KAM, in graphs with loops, which are a peculiarity of
quantum physics. Small divisors are controlled by the Diophantine conditions
and the huge number of loop graphs is compensated by cancellations due to
anticommutativity.
While we have proved here the vanishing of the Drude weight, it would be
interesting to understand if also the conductivity is vanishing or if a zero
result is found only by a suitable averaging over the phase, as is done in
numerical simulations [27].
The effective interaction is irrelevant in the Renormalization Group sense, as
consequence of Diophantine conditions and by cancellations due to
anticommutativity. The presence of spin [43] and an anisotropic hopping [44]
produce extra marginal couplings. They can in principle destroy the
convergence result of the present paper, and it is interesting to observe that
numerical [45] or cold atoms experiments [19] have found evidence of
delocalization is such cases. Another important point would be to extend the
analysis to a more general kind of disorder like
$f(\vec{\omega}_{1}\vec{x},\vec{\omega}_{2}\vec{x})$. The condition of strong
disorder is non technical; in the case of weak quasiperiodic disorder there is
no localization; in particular, this is the case of the interacting Aubry-
Andre’ model [46], of the bidimensional Hofstadter model [47] or of three
dimensional Weyl semimetals [48]. Finally, we stress that a rigorous
understanding of $T=0$ properties of interacting fermions with finite density
and random disorder is still unknown.
The main open problem if of course to extend the above result on transport
coefficients to finite temperature to get information on localization beyond
the ground state. While an extension of [39] allows to pass from Euclidean to
real time conductivity at $T=0$, this is expected to be a major difficulty for
$T>0$. Another difficulty is due to the fact that we do not get ground state
localization in an interval of densities, but only in a large measure set. The
absence of thermalization in the classical case is considered related to KAM
theorem; it is interesting to note that the persistence of localization in a
quantum system, which is considered an obstruction to thermalization, is also
obtained via the generalization of KAM methods in a quantum context.
## 4 Vanishing of Drude weight
We show that the vanishing of Drude weight (23) is consequence of the bound
(22) combined with Ward Identities. Note first that the Fourier transform in
the infinite volume limit is continuous as
$\displaystyle|\widehat{H}_{\mu,\nu}({\bf p},\vec{y})|\leq\sum_{\vec{x}}\int
dx_{0}|H_{\mu,\nu}({\bf x},{\bf y})|\leq\sum_{\vec{x}}\int
dx_{0}\Delta_{\vec{x},\vec{y}}^{-4}C_{N}{e^{-{1\over
4}|\log|\varepsilon||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}|)^{N}}\leq$ (24) $\displaystyle
C_{1}\sum_{\vec{x}}(|\vec{x}+\vec{y}|^{5\tau}+|\vec{y}|^{5\tau})e^{-{1\over
4}|\log|\varepsilon||\vec{x}||}\leq C_{2}\sum_{\vec{x}}e^{-{1\over
4}|\log|\varepsilon||\vec{x}||}(|\vec{x}|^{5\tau}+2|\vec{y}|^{5\tau})\leq
C_{3}|\vec{y}|^{5\tau}/(|\log|\varepsilon||)^{d+5\tau}$
Ward identities can be deduced from the continuity equation,
$\partial_{0}\rho_{\bf x}=[H,\rho_{\bf x}]=-i\sum_{i}(j^{i}_{\bf
x}-j^{i}_{{\bf x}-e_{i}})$ (25)
we get, setting $\partial_{i}j_{\bf x}\equiv j_{\bf x}-j_{{\bf x}-{\bf
e}_{i}}$ , $i=1,...,d$, ${\bf e}_{i}=(0,\vec{e}_{i})$
$\displaystyle\partial_{0}<T\rho_{\bf x};\rho_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj_{\bf x}^{i};\rho_{\bf
y}>+\delta(x_{0}-y_{0})<[\rho_{\bf x},\rho_{\bf y}]>$
$\displaystyle\partial_{0}<T\rho_{\bf x};j^{j}_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};j^{j}_{\bf
y}>+\delta(x_{0}-y_{0})<[\rho_{\bf x},j^{j}_{\bf y}]>$ (26)
Note that $[\rho_{\vec{x},x_{0}},\rho_{\vec{y},x_{0}}]=0$ while
$[\rho_{\vec{x},x_{0}},j^{j}_{\vec{y},x_{0}}]=-i\delta_{\vec{x},\vec{y}}\tau^{j}_{\vec{x}}+i\delta_{\vec{x}-\vec{e}_{j},\vec{y}}\tau^{j}_{\vec{y}}$
(27)
so that, in the $L,\beta\rightarrow\infty$ limit
$\displaystyle\partial_{0}<T\rho_{\bf x};\rho_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};\rho_{\bf y}>$ (28)
$\displaystyle\partial_{0}<T\rho_{\bf x};j^{j}_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};j^{j}_{\bf
y}>-i\delta(x_{0}-y_{0})(-\delta_{\vec{x},\vec{y}}<\tau^{j}_{\vec{y}}>+\delta_{\vec{x}-\vec{e}_{j},\vec{y}}<\tau^{j}_{\vec{y}}>)$
Taking the Fourier transform in ${\bf x}$ we get, using translation invariance
in time and setting $y_{0}=0$
$\sum_{\vec{x}}\int dx_{0}e^{i{\bf p}{\bf x}}(\partial_{0}<T\rho_{{\bf
x}};j^{j}_{\vec{y}}>+i\sum_{i}\partial_{i}<Tj^{i}_{\bf
x};j^{j}_{\vec{y}}>+i\delta(x_{0})(-\delta_{\vec{x},\vec{y}}<\tau^{j}_{\vec{y}}>+\delta_{\vec{x}-\vec{e}_{j},\vec{y}}<\tau^{j}_{\vec{y}}>)=0$
(29)
with $p_{0}\in\hbox{\msytw R}$ and $\vec{p}\in[-\pi,\pi)^{d}$ so that
$-ip_{0}\widehat{H}_{0,j}({\bf
p},\vec{y})+i\sum_{i}(1-e^{-ip_{i}})(\widehat{H}_{i,j}({\bf
p},\vec{y})+e^{-i\vec{p}\vec{y}}<\tau^{j}_{y,0}>)=0$ (30)
Setting $j=1$ for definiteness, we set $\bar{\vec{p}}=(p_{1},0,0)$ so that
$-ip_{0}\widehat{H}_{0,1}(\bar{\bf
p},\vec{y})+i(1-e^{-ip_{1}})(\widehat{H}_{1,1}(\bar{\bf
p},\vec{y})+e^{-ip_{1}y_{1}}<\tau^{1}_{y,y_{0}}>)=0$ (31)
so that
$\lim_{p_{1}\rightarrow
0}(\widehat{H}_{1,1}(0,p_{1},\vec{y})+e^{-ip_{1}y_{1}}<\tau^{1}_{y,y_{0}}>)=0$
(32)
but $\lim_{p_{1}\rightarrow 0}(e^{-ip_{1}y_{1}}-1)=0$. In conclusion
$\lim_{p_{1}\rightarrow
0}(\widehat{H}_{1,1}(0,p_{1},\vec{y})+<\tau^{1}_{y,y_{0}}>)=0$ (33)
Due to (4) $\widehat{H}_{1,1}({\bf p},\vec{y})$ is continuous in ${\bf p}$ so
that we can exchange the limits
$\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow 0}(\widehat{H}_{1,1}({\bf
p},\vec{y})+<\tau^{1}_{y,y_{0}}>)=D^{1}_{\vec{x}}=0$ (34)
and this shows that the Drude weight is vanishing. Note the crucial role
played by continuity of the Fourier transform, following by the fast decay of
the correlations; without quasi-periodic disorder the Fourier transform is not
continuous due to its slow decay and the Drude weight is non vanishing.
## 5 Perturbation theory and Grassmann representation
The starting point of the analysis consists in expanding around the anti-
integrable limit (12); defining
$\displaystyle H-\mu N=H_{0}+V$ (35) $\displaystyle
H_{0}=\sum_{\vec{x}}(\phi_{\vec{x}}-\bar{\phi}(\alpha))a^{+}_{\vec{x}}a^{-}_{\vec{x}}$
$\displaystyle
V=\varepsilon\sum_{\vec{x},i}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}+a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})+\lambda\sum_{\vec{x},i}a^{+}_{\vec{x}}a^{-}_{\vec{x}}a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}+\vec{e}_{i}}+\nu\sum_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}$
(36)
and using the Trotter formula one can write the partition function and the
correlations as a power series expansion in $\lambda,\varepsilon$.
${\bf x}\pm{\bf e}_{i}$
${\bf x}\pm{\bf e}_{i}$
${\bf x}$
${\bf x}$
${\bf x}$
${\bf x}\pm{\bf e}_{i}$
${\bf x}$
${\bf x}$
$\nu$
$\varepsilon$
$\lambda$
Figure 1: Graphical representation of the three terms in ${\cal V}(\psi)$
eq.(38)
The correlations can be equivalently written in terms of Grassmann integrals.
We can write
$e^{W(\eta,J)}=\int P(d\psi)e^{-{\cal V}(\psi)-{\cal B}(\psi,J,\eta)}$ (37)
with ${\bf e_{i}}=(0,\vec{e}_{i})$
${\cal V}(\psi)=\varepsilon\sum_{i}\int d{\bf x}(\psi^{+}_{{\bf x}+{\bf
e_{i}}}\psi^{-}_{{\bf x}}+\psi^{+}_{{\bf x}-{\bf e_{i}}}\psi^{-}_{{\bf
x}})+\lambda\int d{\bf x}\sum_{i}\psi^{+}_{{\bf x}}\psi^{-}_{{\bf
x}}\psi^{+}_{{\bf x}+{\bf e}_{i}}\psi^{-}_{{\bf x}+{\bf e_{i}}}+\nu\int d{\bf
x}\psi^{+}_{{\bf x}}\psi^{-}_{{\bf x}}$ (38)
where $\int d{\bf x}=\sum_{x\in\Lambda_{L}}\int_{-{\beta\over 2}}^{\beta\over
2}dx_{0}$ and $\psi^{\pm}_{\bf x}$ is vanishing outside $\Lambda_{L}$;
moreover
${\cal B}(\psi,J,\eta)=\int d{\bf x}[\eta^{+}_{{\bf x}}\psi^{-}_{{\bf
x}}+\psi^{+}_{{\bf x}}\eta^{-}_{{\bf x}}+\sum_{\mu=0}^{d}J_{\mu}({\bf
x})j_{\mu}({\bf x})]$ (39)
with
$\displaystyle j_{0}({\bf x})=\psi^{+}_{{\bf x}}\psi^{-}_{\bf x}\quad\quad
j_{i}({\bf x})=\varepsilon(\psi^{+}_{{\bf x}+{\bf e}_{i}}\psi^{-}_{\bf
x}-\psi^{+}_{{\bf x}}\psi^{-}_{{\bf x}+{\bf e_{i}}})$ (40)
The 2-point and the current correlations are given by
$S_{2}^{L,\beta}({\bf x},{\bf y})={\partial^{2}\over\partial\eta^{+}_{{\bf
x}}\partial\eta^{-}_{{\bf y}}}W(\eta,J)|_{0,0}\quad\quad H_{\mu,\nu}({\bf
x},{\bf y})={\partial^{2}\over\partial J_{\mu,{\bf x}}\partial J_{\nu,{\bf
y}}}W(\eta,J)|_{0,0}$ (41)
By expanding in $\lambda,\varepsilon,\nu$ one can write the correlations as a
series expansion, which can be expressed in terms of Feynman graphs obtained
contracting the half lines of vertices, see Fig. 1, and associating to each
line the propagator $g({\bf x},{\bf y})$. There is a basic difference between
the perturbative expansion in the non interacting case $\lambda=0$ and the
interacting case $\lambda\not=0$. In the first case there are only chain
graphs, while in the second there are also loops, producing further
combinatorial problems. One can verify that the perturbative expansions
obtained by Trotter formula for (2) and by the Grassmann generating functions
are the same (this is true up to the so called ”tadpoles” which can be easily
taken into account, see §1 D in [35]). The identity between (2) and (37) is
true in a rigorous sense provided that the Grassmann integral representation
is analytic in a disk uniformly in $L,\beta$, as proven in the following
sections. Indeed at finite $L,\beta$ the partition function in (2) is entire
and it coincides order by order with the Grassmann representation, which is
analytic in a disk independent on the volume, so they coincide. As the
denominator of the correlations is non vanishing in this finite disk and the
numerator is entire at finite $\beta,L$, also the correlations (2) is analytic
and coincide with the Grassmann representation, and the identity holds also in
the limit.
## 6 Multiscale decomposition and renormalization
The difficulty in controlling the perturbative expansion is due to a ”small
divisor problem” related to the size of the propagator; the denominator of
$\widehat{g}(\vec{x},k_{0})$ can be arbitrarily small if $\vec{\omega}\vec{x}$
is close to $\pm\alpha$, a fact which can produce in principle $O(n!)$-terms
which could destroy convergence. The starting point of the analysis is to
separate the propagator in two terms, one containing the quasi-singularity and
a regular part; we write
$g({\bf x},{\bf y})=g^{(1)}({\bf x},{\bf y})+\sum_{\rho=\pm}g_{\rho}^{(\leq
0)}({\bf x},{\bf y})$ (42)
where
$\displaystyle g^{(1)}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(1)}(\vec{\omega}\vec{x},k_{0}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(1)}(\vec{x},x_{0}-y_{0})$
$\displaystyle g^{(\leq 0)}_{\rho}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(0)}_{\rho}(\vec{\omega}\vec{x},k_{0}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})$ (43)
with
$\chi^{(0)}_{\rho}(\vec{\omega}\vec{x},k_{0})={\widetilde{\theta}}_{\rho}(\vec{\omega}\vec{x})\bar{\chi}_{0}(\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})$
with ${\widetilde{\theta}}_{\rho}$ is the periodic theta function
(${\widetilde{\theta}}_{\pm}=1$ if $\vec{\omega}\vec{x}$ mod. $1$ is
positive/negative and zero otherwise) and $\bar{\chi}_{0}$ such that
$C^{\infty}(\hbox{\msytw R}^{+})\rightarrow\hbox{\msytw R}$ such that
$\bar{\chi}_{0}(t)=1$ with $t\leq 1$ and $\bar{\chi}_{0}(t)=0$ for
$t\geq\gamma>1$; moreover $\chi^{(1)}+\sum_{\rho=\pm}\chi_{\rho}=1$. The
”infrared” propagator $g^{(\leq 0)}({\bf x},{\bf y})$ has denominator
arbitrarily small. We can further decompose the infrared propagator as sum of
propagators with smaller and smaller denominators
$g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})=\sum_{h=-\infty}^{0}g^{(h)}_{\rho}(\vec{x},x_{0}-y_{0})$
(44)
with $g^{(h)}_{\rho}$ similar $g^{(\leq 0)}_{\rho}$ witrh $f^{h}$ replacing
$\bar{\chi}_{0}$ with
$f^{h}=\bar{\chi}_{0}(\gamma^{h}\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})-\bar{\chi}_{0}(\gamma^{h-1}\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})$
(45)
For any integer $N$ one has
$|g^{(h)}_{\rho}(\vec{x},x_{0}-y_{0})|\leq{C_{N}\over
1+(\gamma^{h}|x_{0}-y_{0}|)^{N}}$ (46)
if $C_{N}$ is a suitable constant.
The integration of (37) is done iteratively by using two crucial properties of
Grassmann integrations. If $P(d\psi^{(1)})$ and $P(d\psi^{(\leq 0)})$ are
gaussian Grassmann integrations with propagators $g^{(1)}$ and $g^{(\leq 0)}$,
we can write $P(d\psi)=P(d\psi^{(1)})P(d\psi^{(\leq 0)})$ so that
$\displaystyle e^{W(\eta,J)}=\int P(d\psi^{(1)})P(d\psi^{(\leq 0)})e^{-{\cal
V}(\psi^{(1)}+\sum_{\rho=\pm}\psi^{(\leq 0)}_{\rho})-{\cal
B}(\psi^{(1)}+\sum_{\rho=\pm}\psi^{(\leq 0)}_{\rho},\eta,J)}=$
$\displaystyle\int P(d\psi^{(\leq 0)})e^{-{\cal V}^{(0)}(\psi^{(\leq
0)}_{\rho},\eta,J)}$ (47)
with
${\cal V}^{(0)}(\psi^{(\leq 0)}_{\rho},\eta,J)=\sum_{n=0}^{\infty}{1\over
n!}{\cal E}^{T}_{1}({\cal V}+{\cal B};n)$ (48)
and ${\cal E}^{T}_{1}$ are fermionic truncated expectations with propagator
$g^{(1)}$. By integrating $\psi^{(0)},\psi^{(-1)},..,\psi^{(h+1)}$ one obtains
a sequence of effective potentials ${\cal V}^{(h)}$, $h=0,-1,-2,..$. The way
in which we define the integration is dictated by the scaling dimension which
is, as we will see below, $D=1$; that is all terms are relevant in the
Renormalization Group sense. Remark Note that after the integration of
$\psi^{1}$ one gets a theory defined in terms of two fields
$\psi_{+},\psi_{-}$. This is due to the fact that
$\bar{\phi}(t)=\bar{\phi}(\alpha)$ in correspondence of two points
$\pm\alpha$. If we consider more general forms of quasi periodic disorder,
like $\bar{\phi}(t_{1},t_{2})$ as the one in [7] , then
$\bar{\phi}(t_{1},t_{2})-\mu=0$ in a set corresponding to a surface. In this
case one gets a description in terms of a field $\psi_{\rho}$, with $\rho$ a
parameter parametrizing this curve, a situation somewhat analogue to what
happens in interacting fermions with extended Fermi surface.
The multiscale integration is described iteratively in the following way.
Assume that we have already integrated the fields
$\psi^{(0)},\psi^{(-1)},..,\psi^{(h+1)}$ obtaining (we set $\eta=0$ for the
moment)
$e^{W(0,J)}=\int P(d\psi^{(\leq h)})e^{-{\cal V}^{(h)}(\psi^{(\leq h)},J)}$
(49)
where $P(d\psi^{(\leq h)}$ has propagator
$g^{(\leq h)}_{\rho}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(h)}_{\rho}(k_{0},\vec{\omega}\vec{x}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})$ (50)
and
${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{l\geq 0,m\geq
0}\sum_{\underline{\varepsilon},\underline{\rho}}\int d{\bf x}_{1}...d{\bf
x}_{l}d{\bf y}_{1}...d{\bf y}_{m}H^{h}_{l,m}(\underline{{\bf
x}},\underline{{\bf y}})\prod_{i=1}^{l}\psi^{\varepsilon_{i}(\leq
h)}_{\rho_{i},{\bf x}_{i}}\prod_{i=l}^{m}J_{{\bf y}_{i}}$ (51)
If there is a subset of $\psi^{\varepsilon_{i}}_{\rho_{i},{\bf x}_{i}}$ with
the same $\varepsilon,\rho$ and $\vec{x}_{i}$, by the anticommuting properties
of Grassmann variables we can write, if $l>1$
$\prod_{i=1}^{l}\psi^{\varepsilon}_{\vec{x},x_{0,i}}=\psi^{\varepsilon}_{\vec{x},x_{0,1}}\prod_{i=2}^{l}D^{\varepsilon}_{\vec{x},x_{0,i},x_{0,1}}\quad\quad\quad
D^{\varepsilon}_{\vec{x},x_{0,i},x_{0,1}}=\psi^{\varepsilon}_{\vec{x},x_{0,i}}-\psi^{\varepsilon}_{\vec{x},x_{0,1}}$
(52)
We can therefore rewrite that effective potential in the following way
${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{l\geq 0,m\geq
0}\sum_{\underline{\varepsilon},\underline{\rho}}\int d{\bf x}_{1}...d{\bf
x}_{l}d{\bf y}_{1}...d{\bf y}_{m}H^{h}_{l,m}(\underline{{\bf
x}},\underline{{\bf
y}})\prod_{i=1}^{l}d^{\sigma_{i}}\psi^{\varepsilon_{i}}_{\rho_{i},{\bf
x}_{i}}\prod_{i=l}^{m}J_{{\bf y}_{i}}$ (53)
with $\sigma=0,1$ and $d^{0}\psi=\psi$ and $d^{1}\psi=D$.
We define resonant the terms with fields with the same coordinate $\vec{x}$,
that is ${\bf x}_{i}=(x_{0,i},\vec{x})$. Note that all the resonant terms with
$l\geq 4$ are such that there are at least two $D$ fields; the fields have the
same $\rho$ index as have the same $\vec{\omega}\vec{x}$.
We define a renormalization operation ${\cal R}$ in the following way
1. 1.
If $l=2$, $m=0$
${\cal R}\sum_{\vec{x}}\int dx_{0,1}dx_{0,2}H_{2,0}^{(h)}\psi^{+(\leq
h)}_{\vec{x},x_{0,1},\rho}\psi^{-(\leq
h)}_{\vec{x},x_{0,2},\rho}=\sum_{\vec{x}}\int
dx_{0,1}dx_{0,2}H_{2,0}^{(h)}\psi^{+(\leq h)}_{\vec{x},x_{0,1},\rho}T^{-(\leq
h)}_{\vec{x},x_{0,1},x_{0,2}\rho}$ (54)
with
$T^{-(\leq h)}_{\vec{x},x_{0,1},x_{0,2}\rho}=\psi^{-(\leq
h)}_{\vec{x},x_{0,2},\rho}-\psi^{-(\leq
h)}_{\vec{x},x_{0,1},\rho}-(x_{0,1}-x_{0,2})\partial\psi^{-(\leq
h)}_{\vec{x},x_{0,1},\rho}$ (55)
2. 2.
${\cal R}=0$ otherwise
We define ${\cal R}=1-{\cal L}$ and by definition ${\cal L}{\cal V}^{(h)}$ is
given by the following expression
${\cal L}{\cal
V}^{(h)}=\gamma^{h}F^{(h)}_{\nu}+F^{(h)}_{\zeta}+F^{(h)}_{\alpha}$ (56)
where, if
$H_{2,0}^{(h)}(\vec{x},x_{0}-y_{0})\equiv\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0}-y_{0})$
one has
$\nu_{h}=\int
dx_{0}\bar{H}_{2,0}^{(h)}(\rho\alpha,x_{0})\quad\xi_{h}(\vec{x})=\int
dx_{0}{\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0})-\bar{H}_{2,0}^{(h)}(\rho\alpha,x_{0})\over\vec{\omega}\vec{x}-\rho\alpha}$
(57)
and $\alpha_{h}(\vec{x})=\int
dx_{0}x_{0}\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0})$; moreover
$\displaystyle F^{(h)}_{\nu}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}\nu_{h}\psi^{+(\leq h)}_{{\bf x},\rho}\psi^{-(\leq h)}_{{\bf
x},\rho}\quad\quad F^{(h)}_{\zeta}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}((\vec{\omega}\vec{x})-\rho\alpha)\zeta_{h,\rho}(\vec{x})\psi^{+(\leq
h)}_{{\bf x},\rho}\psi^{-(\leq h)}_{{\bf x},\rho}$ $\displaystyle
F^{(h)}_{\alpha}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}\alpha_{h,\rho}(\vec{x})\psi^{+(\leq h)}_{{\bf
x},\rho}\partial_{0}\psi^{-(\leq h)}_{{\bf x},\rho}\quad\quad$ (58)
The running coupling constants $\vec{v}_{h}=(\nu_{h},\alpha_{h},\xi_{h})$ are
independent from $\rho$, as (37) is invariant under parity
$\vec{x}\rightarrow-\vec{x}$. Note also that
$(\widehat{g}^{(k)})^{*}(\vec{x},k_{0})=\widehat{g}^{(k)}(\vec{x},-k_{0})$ so
that
$(\widehat{H}^{(h)}_{2,\rho}(\vec{x},k_{0}))^{*}=\widehat{H}^{(h)}_{2,\rho}(\vec{x},-k_{0})$,
and this implies that $\nu_{h}$ is real.
Remark The ${\cal R}$ operation is defined in order to act non trivially on
the resonant terms with two fields and no $J$ fields; they are the only
resonant terms with no $D$ fields. This fact would be not true of there is the
spin or an extra degree of freedom, as in the case of lattice Weyl semimetals
[48]. In that case the local part of the effective potential would contain
also effective interactions. With the above definitions we can write (49)
$e^{W(0,J)}=\int P(d\psi^{(\leq h-1)})\int P(d\psi^{(h)})e^{-{\cal L}{\cal
V}^{(h)}(\psi^{(\leq h)},J)-{\cal R}{\cal V}^{(h)}(\psi^{(\leq h)},J)}=\int
P(d\psi^{(\leq h-1)})\ e^{-{\cal L}{\cal V}^{(h)}(\psi^{(\leq h-1)},J)}$ (59)
and the procedure can be iterated.
## 7 Convergence of series expansion
The effective potential can be written as a sum over Gallavotti trees $\tau$,
see Fig.2
${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}_{h,n}}V^{(h)}(\tau,\psi^{(\leq h)})$ (60)
where $\tau$ are trees constructed adding labels to the unlabeled trees,
obtained by joining a point, the root, with an ordered set of $n\geq 1$
points, the endpoints, so that the root is not a branching point.
$v_{0}$
$v$
$v^{\prime}$
$h_{v}$
$1$
$2$
Figure 2: A labeled tree
The set of labeled trees ${\cal T}_{h,n}$ is defined associating a label
$h\leq 0$ with the root and introducing a family of vertical lines, labeled by
an integer taking values in $[h,2]$ intersecting all the non-trivial vertices,
the endpoints and other points called trivial vertices.To a vertex $v$ is
associated $h_{v}$ and, if $v_{1}$ and $v_{2}$ are two vertices and
$v_{1}<v_{2}$, then $h_{v_{1}}<h_{v_{2}}$. Moreover, there is only one vertex
immediately following the root, which will be denoted $v_{0}$ and can not be
an endpoint; its scale is $h+1$. To the end-points are associated ${\cal
V}+{\cal B}$ , and in such a case the scale is $2$; or ${\cal L}{\cal
V}^{h_{v}-1}(\psi^{(\leq h_{v}-1)},J)$ and in this case the scale is
$h_{v}\leq 1$ and there is the constraint that $h_{v}=h_{\bar{v}}+1$, if
$\bar{v}$ is the first non trivial vertex immediately preceding $v$. The tree
structure induces a jerarchy of end-points which can be represented by
clusters, see Fig.3.
$1$
$2$
$3$
$4$
$5$
$\Longleftrightarrow$
$1$
$2$
$3$
$4$
$5$
Figure 3: A tree of order 5 and the corresponding clusters.
If $v_{0}$ is the first vertex of $\tau$ and $\tau_{1},..,\tau_{s}$
($s=s_{v_{0}}$) are the subtrees of $\tau$ with root $v_{0}$,
$V^{(h)}(\tau,\psi^{(\leq h)})$ is defined inductively by the relation
$V^{(h)}(\tau,\psi)={(-1)^{s+1}\over s!}{\cal
E}^{T}_{h+1}[\bar{V}^{(h+1)}(\tau_{1},\psi^{(\leq
h+1)});..;\bar{V}^{(h+1)}(\tau_{s},\psi^{(\leq h+1)})]$ (61)
where $\bar{V}^{(h+1)}(\tau_{i},\psi^{(\leq h+1)})$ it is equal to ${\cal
R}{\cal V}^{(h+1)}(\tau_{i},\psi^{(\leq h+1)})$ if the subtree $\tau_{i}$ is
non trivial;if $\tau_{i}$ is trivial, it is equal to ${\cal L}{\cal
V}^{(h+1)}$. By iterating (61) we get a jerarchy of truncated expectations,
with a certain subset of fields contracted in each expectations. We can
therefore write $V^{(h)}(\tau,\psi^{(\leq h)})$ as sum over sets defined in
the following way. We call $I_{v}$ the set of $\psi$ associated to the end-
points following $v$ and $P_{v}$ is a subset of $I_{v}$ denoting the external
$\psi$. We denote by $Q_{v_{i}}$ the intersection of $P_{v}$ and $P_{v_{i}}$;
they are such that $P_{v}=\cup_{i}Q_{v_{i}}$ and the union ${\cal I}_{v}$ of
the subsets $P_{v_{i}}\setminus Q_{v_{i}}$ is, by definition, the set of the
internal fields of $v$, and is non empty if $S_{v}>1$. The effective potential
can be therefore written as
${\cal V}^{(h)}(\tau,\psi^{(\leq h)})=\sum_{{\bf P}\in{\cal P}_{\tau}}{\cal
V}^{(h)}(\tau,{\bf P})\quad\bar{\cal V}^{(h)}(\tau,{\bf P})=\int d{\bf
x}_{v_{0}}\widetilde{\psi}^{(\leq h)}(P_{v_{0}})K_{\tau,{\bf P}}^{(h+1)}({\bf
x}_{v_{0}})\;,$ (62)
where $\widetilde{\psi}^{(\leq h)}(P)=\prod_{f\in P}\psi_{{\bf x}(f)}$. If we
expand the truncated expectations by the Wick rule we get a sum of Feynman
graphs with an associated cluster structure; an example is in Fig.4.
Figure 4: An example of graph with $\lambda$ and $\varepsilon$ vertices and
the associated cluster structure; the propagator in the cluster, represented
as a circle, has scale $h$ smaller than the scales of the propagators external
to the cluster.
The truncated expectations can be written by the Brydges-Battle-Federbush
formula
${\cal
E}^{T}_{h_{v}}({\widetilde{\psi}}^{(h_{v})}(P_{1}/Q_{1}),\cdots,{\widetilde{\psi}}^{(h_{v})}(P_{s}/Q_{s})))=\sum_{T_{v}}\prod_{l\in
T_{v}}\big{[}\delta_{\vec{x}_{l},\vec{y}_{l}}\bar{g}^{(h_{v})}(\vec{x}_{l},x_{0,l}-y_{0,l})\big{]}\,\int
dP_{T}({\bf t})\;{\rm det}\,G^{h_{v},T}({\bf t})\;,$ (63)
where $T_{v}$ is a set of lines forming an anchored tree graph between the
clusters of points ${\bf x}^{(i)}\cup{\bf y}^{(i)}$, that is $T_{v}$ is a set
of lines, which becomes a tree graph if one identifies all the points in the
same cluster. Moreover ${\bf t}=\\{t_{ii^{\prime}}\in[0,1],1\leq
i,i^{\prime}\leq s\\}$, $dP_{T_{v}}({\bf t})$ is a probability measure with
support on a set of ${\bf t}$ such that $t_{ii^{\prime}}={\bf u}_{i}\cdot{\bf
u}_{i^{\prime}}$ for some family of vectors ${\bf u}_{i}\in\hbox{\msytw
R}^{s}$ of unit norm.
$G^{h,T}_{ij,i^{\prime}j^{\prime}}=t_{ii^{\prime}}\delta_{\vec{x}_{ij},\vec{y}_{i^{\prime}j^{\prime}}}\bar{g}^{(h)}(\vec{x}_{ij},x_{0,ij}-y_{0,i^{\prime}j^{\prime}})\;,$
(64)
We define $\bar{T}_{v}=\bigcup_{w\geq v}T_{w}$ starting from $T_{v}$ and
attaching to it the trees $T_{v_{1}},..,T_{v_{S_{v}}}$ associated to the
vertices $v_{1},..,v_{S_{v}}$ following $v$ in $\tau$, and repeating this
operation until the end-points of $\tau$ are reached.
$w_{1}$
$w_{a}$
$w_{b}$
$w_{c}$
$w_{2}$
Figure 5: A tree $\bar{T}_{v}$ with attached wiggly lines representing the
external lines $P_{v}$; the lines represent propagators with scale $\geq
h_{v}$ connecting $w_{1},w_{a},w_{b},w_{c},w_{2}$, representing the end-points
following $v$ in $\tau$.
The tree $\bar{T}_{v}$ connects the end-points $w$ of the tree $\tau$. To each
end-point $w$ we associate a factor $\vec{\delta}_{w}^{i_{w}}$, and a)
$\vec{\delta}^{i}_{w}=0$ if $w$ corresponds to a
$\nu_{h},\alpha_{h},\zeta_{h}$ end-point; b) $\vec{\delta}_{w}^{i}$ one among
$\pm\vec{e}_{i}$, $i=1,2,3$ if it corresponds to an $\varepsilon$ end-point;
c) $\delta^{i}_{w}$ one among $0,\pm\vec{e}_{i}$, $i=1,2,3$ if it corresponds
to a $\lambda$ end-point. If $\vec{x}_{w_{1}}$ and $\vec{x}_{w_{2}}$ are
coordinates of the external fields ${\widetilde{\psi}}(P_{v})$ we have, see
Fig.5
$\vec{x}_{w_{1}}-\vec{x}_{w_{2}}=\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}$ (65)
where $c_{w_{1},w_{2}}$ is the set of endpoints in the path in $\bar{T}$
connecting $w_{1}$ and $w_{2}$. The above relation implies, in particular,
that the coordinates of the external fields ${\widetilde{\psi}}(P_{v})$ are
determined once that the choice of a single one of them and of
$\tau,\bar{T}_{v}$ and ${\bf P}$ is done. We can therefore write the effective
potential as sum over trees $T$, setting the Kronecker deltas in the
propagators in $l\in T$ equal to $1$
${\cal V}^{(h)}(\tau,\psi^{(\leq h)})=\sum_{{\bf P}\in{\cal
P}_{\tau}}\sum_{T}{\cal V}^{(h)}(\tau,{\bf P},T)\quad\bar{\cal
V}^{(h)}(\tau,{\bf P},T)=\sum_{\vec{x}}\int
dx_{0,v_{0}}\widetilde{\psi}^{(\leq h)}(P_{v_{0}})K_{\tau,{\bf
P},T}^{(h+1)}({\bf x}_{v_{0}})\;,$ (66)
where in $K_{\tau,{\bf P},T}^{(h+1)}$ the propagators in $T$ are
$g^{(h)}(\vec{x},x_{0}-y_{0})$ and the determinants are product of determinats
involving propagators with the same $\vec{x}$. We can bound the propagators in
$T$ by
$\int dx_{0}|g^{(h)}(\vec{x},x_{0}-y_{0})|\leq C\gamma^{-h}$ (67)
Moreover the determinants in the BFF formula can be bounded by the Gram-
Hadamard inequality . We introduce an Hilbert space ${\cal H}=\hbox{\msytw
R}^{s}\otimes L^{2}(\hbox{\msytw R}^{1})$ so that
${\widetilde{G}}^{h,T}_{ij,i^{\prime}j^{\prime}}=\Big{(}{\bf u}_{i}\otimes
A(x_{0,ij}-,x_{ij})\;,\ {\bf u}_{i^{\prime}}\otimes
B(y_{0,i^{\prime}j^{\prime}}-,x_{ij})\Big{)}\;,$ (68)
where ${\bf u}\in\hbox{\msytw R}^{s}$ are unit vectors
$(u_{i},u_{i})=t_{ii^{\prime}}$, and $A,B$
$(A,B)=\int dz_{0}A(\vec{x},x_{0}-z_{0})B^{*}(\vec{x},z_{0}-y_{0})$ (69)
given by
$A(\vec{x},x_{0}-z_{0})={1\over\beta}\sum_{k_{0}}e^{-ik_{0}(x_{0}-z_{0})}\sqrt{f_{h}}\quad\quad
B(\vec{x},y_{0}-z_{0})={1\over\beta}\sum_{k_{0}}{e^{-ik_{0}(y_{0}-z_{0})}\sqrt{f_{h}}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}$
Moreover $||A_{h}||^{2}=\int dz_{0}|A_{h}(x^{\prime},z_{0})|^{2}\leq
C\gamma^{h}$ and $||B_{h}||^{2}\leq C\gamma^{-h}$ so that By Gram-Hadamard
inequality we get:
$|{\rm det}{\widetilde{G}}^{h_{v},T_{v}}({\bf t}_{v})|\leq
C^{\sum_{i=1}^{S_{v}}|P_{v_{i}}|-|P_{v}|-2(S_{v}-1)}\;.$ (70)
One get therefore the bound, for $|\lambda|,|\vec{v}_{h}|\leq\varepsilon_{0}$,
$|K_{\tau,{\bf P},T}^{(h+1)}({\bf x}_{v_{0}})|\leq
C^{n}\varepsilon_{0}^{n}\prod_{v}{1\over S_{v}!}\gamma^{-h_{v}(S_{v}-1)}$ (71)
which is not suitable for summing over $\tau$ and $P$. In order to improve the
above bound we need to implement in the bounds some constraints which have
been neglected in the derivation of (71), and to take into account the effect
of the presence of the $D$ fields.
We define $V_{\chi}$ the set of non trivial vertices or the trivial ones with
non zero internal lines; we define $v^{\prime}$ the first vertex in $V_{\chi}$
following $v$. We say that $v$ is a non-resonant vertex if in
${\widetilde{\psi}}(P_{v})$ there are at least two different coordinates, and
a resonant vertex when all coordinates are equal. We define
$S_{v}=S^{L}_{v}+S^{H}_{v}$ where $S^{L}_{v}$ is the number of non resonant
subtrees (including trivial ones) and $S^{H}_{v}$ the number of resonant ones
(inluding trivial ones). We also call $H$ the set of $v\in V_{\chi}$ which are
resonant and $L$ the $v\in V_{\chi}$ which are non resonant. Consider a non
resonant vertex $v$ so that there are at least two fields in $P_{v}$ with
different spatial coordinates $\vec{x}$, say
$\vec{x}_{w_{1}}\not=\vec{x}_{w_{2}}$. The fields ${\widetilde{\psi}}^{(\leq
h_{v})}(P_{v})$ have scale $\leq\gamma^{h_{v^{\prime}}}$, $v^{\prime}\in
V_{\chi}$ the first vertex belonging to $V_{\chi}$ after $v$ so that
$||(\vec{\omega}\vec{x}_{w_{1}})-\rho_{1}\alpha||_{\hbox{\msytw T}}\leq
cv_{0}^{-1}\gamma^{h_{v^{\prime}}-1}\quad\quad||(\vec{\omega}\vec{x}_{w_{2}})-\rho_{2}\alpha||_{\hbox{\msytw
T}}\leq cv_{0}^{-1}\gamma^{h_{v^{\prime}}-1}$ (72)
so that
$2cv_{0}^{-1}\gamma^{h_{v^{\prime}}}\geq||(\vec{\omega}\vec{x}_{w_{1}})-\rho_{1}\alpha||_{\hbox{\msytw
T}}+||(\vec{\omega}\vec{x}_{w_{2}})-\rho_{2}\alpha||_{\hbox{\msytw
T}}\geq||\vec{\omega}(\vec{x}_{w_{1}}-\vec{x}_{w_{2}})-(\rho_{1}-\rho_{2})\alpha||_{\hbox{\msytw
T}}$ (73)
and by (65)
$2cv_{0}^{-1}\gamma^{h_{v^{\prime}}}\geq||\vec{\omega}(\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}})+(\rho_{1}-\rho_{2})\alpha||_{\hbox{\msytw
T}}\geq{C_{0}\over|\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}|^{\tau}}$ (74)
where the Diophantine conditions have been used. Therefore
$\sum_{w\in c_{w_{1},w_{2}}}|\vec{\delta}_{w}^{i_{w}}|\geq|\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}|\geq C\gamma^{-h_{v^{\prime}}/\tau}$
(75)
and, if $N_{v}$ is the number of end-points following $v$ in $\tau$
$\sum_{w\in c_{w_{1},w_{2}}}|\vec{\delta}_{w}^{i_{w}}|\leq N_{v}$ (76)
as $|\vec{\delta}_{w}^{i_{w}}|=0,1$ so that
$N_{v}\geq C\gamma^{-h_{v^{\prime}}/\tau}$ (77)
Note that to each endpoint is associated a small factor $\varepsilon_{0}$ and
the fact that $N_{v}$ is large by (77) produces a gain for the $v$ with the
fields with different $\vec{x}$. Of course there can be several $\bar{T}_{v}$
with different $v$ passing through the same end-points. Therefore, given a
constant $c<1$, we can multiply the contribution to each tree $\tau$ with
$n$-endpoints by $c^{-n}c^{n}$ (the factor $c^{-n}$ is of course armless); we
can then write
$c=\prod_{h=-\infty}^{0}c^{2^{h-1}}$ (78)
and associate to each $v$ a factor $c^{N_{v}2^{h-1}}$. If there are two fields
in $P_{v}$ (that is external to the cluster $v$) with different $\vec{x}$ we
get in the bounds, by assuming $\gamma^{1\over\tau}/2\equiv\gamma^{\eta}>1$
than, for any $N$
$c^{A\gamma^{-h\over\tau}2^{h}}=e^{-|\log c|A\gamma^{-\eta
h}}\leq\gamma^{N\eta h}{N\over[|\log|c||A]^{N}e^{N}}$ (79)
as $e^{-\alpha x}x^{N}\leq[{N\over\alpha}]^{N}e^{-N}$, and we can choose
$N=3/\eta$; therefore given a couple of fields external to a vertex $v$ with
different $\vec{x}$, we can associate a factor $\gamma^{2h_{v^{\prime}}}$ in
the bounds.
On the other hand if there is a $D$ field we get in the bound an extra
$\gamma^{h_{v^{\prime}}-h_{v}}$ from the expression
$\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},x_{0,1}-z_{0})-\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},x_{0,2}-z_{0})=(x_{0,1}-x_{0,2})\int_{0}^{1}dt\partial\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},\widehat{x}_{0,1,2}(t)-z_{0})$
(80)
where $\widehat{x}_{0,1,2}(t)=x_{0,1}+t(x_{0,2}-x_{0,1})$. In conclusion
1. 1.
To each non-resonant $v$ we associate a factor (79) so that we get in the
bound an extra factor $\prod_{v\in V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}$
2. 2.
There is a factor $\prod^{*}_{v}\gamma^{h_{v^{\prime}}}$ where $v$ are the
endpoints $\nu,\alpha,\xi$ (it comes from the definition of $\nu$ and the
presence $(x_{0}-y_{0})$ or $(\vec{\omega}\vec{x}-\rho\alpha$).
3. 3.
In the resonant $v$ with $l\geq 2$ fields there is a factor $\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}$. For $l=2$ this it is due to the ${\cal
R}$ definition, for $l\geq 4$ by anticommutativity.
4. 4.
In the terms with $|P_{v}|\geq 8$ we can consider the fields
$\psi^{\varepsilon}_{x}$ whose number is maximal; we can group them in couples
connected by path in $\bar{T}$ non overlapping, and or have different
$\vec{x}$, hence there is a path in $\bar{T}$ connecting them giving an extra
$\gamma^{2h_{v^{\prime}}}$, or they have the same $\vec{x}$ so that there is
an extra $\gamma^{2(h_{v^{\prime}}-h_{v})}$. This produces an extra
$\gamma^{-\alpha|P_{v}|}$, see §F in [36].
We bound first the effective potential ($J=0$). If $\tau\in{\cal T}_{h,n}$,
the set of trees with $n$ end-points and defining
$||K_{\tau,{\bf P},T}^{(h+1)}||={1\over\beta L^{d}}\sum_{\vec{x}}\int
dx_{0,v_{0}}|K_{\tau,{\bf P},T}^{(h+1)}|$ (81)
we get
$||K_{\tau,{\bf P},T}^{(h+1)}||\leq C^{n}\varepsilon_{0}^{n}\prod_{v}{1\over
S_{v}!}\gamma^{-h_{v}(S_{v}-1)}\prod_{v\in
V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (82)
If the first vertex $v_{0}\in V_{\chi}$ is non resonant we get
$\prod_{v\in
V_{\chi}}\gamma^{-h_{v}S_{v}}\prod_{v}\gamma^{h_{v}S^{L}_{v}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H,v\not=v_{0}}\gamma^{h_{v^{\prime}}}=1\quad\quad\prod_{v\in
V_{\chi}}\gamma^{h_{v}}\prod_{v\in
H,v\not=v_{0}}\gamma^{-h_{v}}\leq\gamma^{h_{v_{0}}}$ (83)
We use that $S_{v}=S_{v}^{L}+S_{v}^{H}$,
$\prod_{v}\gamma^{h_{v}S^{L}_{v}}=\prod_{v\in
L}\gamma^{h_{v^{\prime}}}\prod^{**}_{v}\gamma^{h_{v}}$, with $\prod^{**}_{v}$
is over the first vertex $v\in V_{\chi}$ after the $\varepsilon,\lambda$
endpoints, and that $\prod_{v\in L}\gamma^{h_{v^{\prime}}}\leq\prod_{v\in
L}\gamma^{h_{v^{\prime}}-h_{v}}$
$\displaystyle||K_{\tau,{\bf P},T}^{(h+1)}||\leq
C^{n}\varepsilon_{0}^{n}\gamma^{h_{v_{0}}}\prod_{v}{1\over S_{v}!}\prod_{v\in
V_{\chi}}\gamma^{(h_{v^{\prime}}-h_{v})}\prod^{**}_{v}\gamma^{h_{v}}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (84)
where $\prod^{**}_{v}$ is over the vertices $v\in V_{\chi}$ following from the
end-points associated to $\varepsilon,\lambda$. Note that $\sum_{\bf
P}[\prod_{v\in V_{\chi}}\gamma^{-{1\over 8}|P_{v}|}]\leq C^{n}$; moreover
$\sum_{\bf T}[\prod_{v}{1\over S_{v}!}]\leq C^{n}$. The sum over the trees
$\tau$ is done performing the sum of unlabeled trees and the sum over scales.
The unlabeled trees can be bounded by $4^{n}$ by Caley formula, and the sum
over the scales reduces to the sum over $h_{v}$, with $v\in V_{\chi}$, as
given a tree with such scales assigned, the others are of course determined.
Let us consider now the case in which the first vertex $v_{0}$ is resonant; we
can distinguish two cases. If we are considering the contribution to the beta
function then there is no ${\cal R}$ applied in $v_{0}$ so that the same bound
as above is found with $h_{v_{0}}=h+1$. Instead if ${\cal R}$ is applied we
get instead of (83), as there is an extra
$\gamma^{h_{v^{\prime}_{0}}-h_{v_{0}}}$
$\prod_{v\in
V_{\chi}}\gamma^{-h_{v}S_{v}}\prod_{v}\gamma^{h_{v}S^{L}_{v}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H}\gamma^{h_{v^{\prime}}}=\gamma^{h_{v^{\prime}_{0}}}\quad\quad\prod_{v\in
V_{\chi}}\gamma^{h_{v}}\prod_{v\in H}\gamma^{-h_{v}}\leq 1$ (85)
and the same bound is found, as $h_{v^{\prime}_{0}}=h+1$. In conclusion we get
$\sum_{\tau\in{\cal T}_{h,n}}\sum_{{\bf P},T}||K_{\tau,{\bf
P},T}^{(h+1)}||\leq C^{n}\varepsilon_{0}^{n}\gamma^{h}$ (86)
The running coupling constant $\alpha_{h},\xi_{h}$ verify
$\alpha_{h-1}=\alpha_{h}+O(\varepsilon_{0}^{2}\gamma^{h\over
2})\quad\quad\xi_{h-1}=\xi_{h}+O(\varepsilon_{0}^{2}\gamma^{h\over 2})$ (87)
where the factor $\gamma^{h\over 2}$ is due to the fact that the trees have at
least an $\varepsilon,\lambda$ endpoint, from the factor
$\prod^{**}_{v}\gamma^{h_{v}}$ in (84) (short memory property). The flow of
$z_{h},\alpha_{h}$ is therefore summable; in addition one can choose $\nu$ so
that $\nu_{h}$ is bounded, by proceeding as in Lemma 2.7 of citeM3.
## 8 Decay of correlations
We consider now the current correlations, which can be written as
$H_{\mu,\nu}({\bf x},{\bf y})=\sum_{h,n}\sum_{\tau\in{\cal
T}_{h,n+2}}\sum_{{\bf P},T}G_{\tau,{\bf P},T}({\bf x},{\bf y})$ (88)
where ${\cal T}_{h,n+2}$ is the set of trees with $n+2$ end-points, two of
them associated to the $J$ end-points. In the trees $\tau$ we can identify a
vertex $v_{x}$ for the end-point corresponding to $J_{\bf x}$, and $v_{y}$ for
the end-point corresponding to $J_{\bf y}$ with $h_{v_{x}}=h_{v_{y}}=+2$; we
call $\widehat{v}$, with scale $\widehat{h}$, the first vertex $v\in V_{\chi}$
such that $v_{x},v_{y}$ follows $\widehat{v}$, and $v_{0}$ the first vertex
$\in V_{\chi}$, with scale $h$. There are several constraints.
1. 1.
By (65) and using that $\vec{x}-\vec{y}=\sum_{w\in
C_{v_{x},v_{y}}}\vec{\delta}_{w}^{i_{w}}$ we get $n\geq\sum_{w\in
C_{v_{x},v_{y}}}|\vec{\delta}_{w}^{i_{w}}|\geq|\vec{x}-\vec{y}|$
2. 2.
$h\geq\bar{h}(n)$ with, if $|\vec{z}|=1+\min(|\vec{x}|,|\vec{y}|)$
$\gamma^{-\bar{h}}\leq\sup_{\vec{q}=\sum_{i=1}^{n}\vec{e}_{i}}{1\over||{\vec{\omega}(\vec{x}+\vec{q})-\rho\alpha}||}\leq
C(|\vec{z}|+n)^{\tau}$ (89)
With respect to the bound for the $J=0$ case there are the following
differences. If $T_{\widehat{v}}$ is the tree connecting the 2 $J$ endpoints,
we have an extra $\gamma^{\widehat{h}}$ due to the fact that we do not
integrate over the coordinates of the $J$ fields, and we can extract from the
the propagators in $\prod_{l\in\bar{T}_{\widehat{v}}}g^{(h_{l})}$,
$h_{l}\geq\widehat{h}$ a decay factor
${1\over 1+(\gamma^{\widehat{h}}|x_{0}-y_{0}|)^{N}}$ (90)
Moreover there is no ${\cal R}$ in the resonant terms with one or two external
$J$ lines. We can multiply and divide by
$\gamma^{-4\bar{h}}\gamma^{4\bar{h}}$: we can select two paths in $\tau$
$v_{0}<v_{1}<..v_{x}$ and $v_{0}<v^{\prime}_{1}<..v_{y}$, writing
$\gamma^{2\bar{h}}=\gamma^{2(\bar{h}-h_{v_{1}})}...\gamma^{2h_{v^{\prime}_{x}}}\quad\quad\gamma^{2\bar{h}}=\gamma^{2(\bar{h}-h_{v^{\prime}_{1}})}...\gamma^{2h_{v^{\prime}_{y}}}$
(91)
where $v^{\prime}_{x}$, $v^{\prime}_{y}$ are the first vertex $\in V_{\chi}$
after $v_{x}$, $v_{y}$. We get therefore the following bound
$|G_{\tau,{\bf P},T}({\bf x},{\bf
y})|\leq\gamma^{-4\bar{h}}{C^{n}|\varepsilon|^{n}\gamma^{\widehat{h}}\over(\gamma^{\widehat{h}}|x_{0}-y_{0}|)^{N}}\prod_{v}{1\over
S_{v}!}\gamma^{-h_{v}(S_{v}-1)}\prod_{v\in
V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}\prod^{*}_{v}\gamma^{h_{v}}\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (92)
where $H$ now includes also resonant terms with one or two $J$ fields.
Proceeding as in §7 and for $|x_{0}-y_{0}|>1$, if ${\cal T}_{n}$ are the trees
with $n$ end-points
$\sum_{\tau\in{\cal T}_{h,n}}\sum_{{\bf P},T}|G_{\tau,{\bf P},T}({\bf x},{\bf
y})|\leq\gamma^{-3\bar{h}}{C^{n}|\varepsilon|^{n}\over
1+(\gamma^{\bar{h}}|x_{0}-y_{0}|)^{N}}\leq
C^{n}|\varepsilon|^{n}{|\vec{z}|^{3\tau}\over(|\vec{z}|^{-3\tau}|x_{0}-y_{0}|)^{N}}(1+{n\over|\vec{z}|})^{(N+3)\tau}$
(93)
The sum over $h\geq\bar{h}$ can be bounded by an an extra $\gamma^{-\bar{h}}$.
As $|\vec{z}|\geq 1$ and $n/|\vec{z}|\leq n$; we can sum over $n$ obtaining,
remembering the constraint $n\geq|\vec{x}-\vec{y}|$
$|H_{\mu,\nu}({\bf x},{\bf y})|\leq
C{|\vec{z}|^{4\tau}\over(|\vec{z}|^{-3\tau}|x_{0}-y_{0}|)^{N}}|\varepsilon|^{|\vec{x}-\vec{y}|/4}$
(94)
The analysis of the 2-point function is done in a similar way; there are 2
endpoints associated with the externl fields, so with respect to the bound for
the effective potential there is an extra factor $\gamma^{-2\bar{h}}$ and an
extra $\gamma^{\bar{h}}$ from the lack of integration; the sum over the scales
produces an extra $|\bar{h}|$. Acknowledgements. This work has been supported
by MIUR, PRIN 2017 project MaQuMA, PRIN201719VMAST01.
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|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-078 LHCb-PAPER-2013-017 June 19, 2013
Differential branching fraction
and angular analysis of
the decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The determination of the differential branching fraction and the first angular
analysis of the decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ are
presented using data, corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$, collected by the LHCb experiment at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The differential branching fraction
is determined in bins of $q^{2}$, the invariant dimuon mass squared.
Integration over the full $q^{2}$ range yields a total branching fraction of
${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})=\left(7.07\,^{+0.64}_{-0.59}\pm
0.17\pm 0.71\right)\times 10^{-7}$, where the first uncertainty is
statistical, the second systematic, and the third originates from the
branching fraction of the normalisation channel. An angular analysis is
performed to determine the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$,
and $A_{9}$. The observables are consistent with Standard Model expectations.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso59, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S.
Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van
Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M.
Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58, V. Bocci24, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V.
Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J.
Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51,
I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K. Ciba37, X. Cid
Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11,
F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A.
Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34,
A. Dziurda25, A. Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S.
Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L.
Eklund50,37, I. El Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C.
Färber11, G. Fardell49, C. Farinelli40, S. Farry51, V. Fave38, D. Ferguson49,
V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32,
M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A.
Golutvin52,30,37, A. Gomes2, H. Gordon54, M. Grabalosa Gándara5, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening54, S. Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E.
Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C.
Haen37, S.C. Haines46, S. Hall52, T. Hampson45, S. Hansmann-Menzemer11, N.
Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, V. Heijne40,
K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, C. Hombach53, P.
Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D.
Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, B. Mc Skelly51, J. McCarthy44, A. McNab53, R. McNulty12, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J.
Morello22,s, R. Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28,
B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54,
A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S.
Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora
Goicochea2, P. Owen52, A. Oyanguren35,o, B.K. Pal58, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35,
B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F.
Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10,
B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51,
C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, V. Salustino Guimaraes2, C. Salzmann39, B. Sanmartin Sedes36,
M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k,
M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D.
Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K.
Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34,
I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L.
Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva
Coutinho47, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, M. Smith53, M.D.
Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B.
Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K.
Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M. Szczekowski27, P.
Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36,
S. Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37,
B. Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D.
Volyanskyy10, D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10,
R. Waldi60, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wishahi9, M. Witek25, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, F. Xing54, Z. Xing58,
Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ ($\phi\\!\rightarrow
K^{+}K^{-}$) decay111The inclusion of charge conjugated processes is implied
throughout this paper. involves a $b\rightarrow s$ quark transition and
therefore constitutes a flavour changing neutral current (FCNC) process. Since
FCNC processes are forbidden at tree level in the Standard Model (SM), the
decay is mediated by higher order (box and penguin) diagrams. In scenarios
beyond the SM new particles can affect both the branching fraction of the
decay and the angular distributions of the decay products.
The angular configuration of the $K^{+}K^{-}\mu^{+}\mu^{-}$ system is defined
by the decay angles $\theta_{K}$, $\theta_{\ell}$, and $\Phi$. Here,
$\theta_{K}$ ($\theta_{\ell}$) denotes the angle of the $K^{-}$ ($\mu^{-}$)
with respect to the direction of flight of the $B^{0}_{s}$ meson in the
$K^{+}K^{-}$ ($\mu^{+}\mu^{-}$) centre-of-mass frame, and $\Phi$ denotes the
relative angle of the $\mu^{+}\mu^{-}$ and the $K^{+}K^{-}$ decay planes in
the $B^{0}_{s}$ meson centre-of-mass frame [1]. In contrast to the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$, the final state of the decay
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is not flavour specific. The
differential decay rate, depending on the decay angles and the invariant mass
squared of the dimuon system is given by
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{4}\Gamma}{{\rm d}q^{2}{\rm d}\cos{\theta_{\ell}}{\rm
d}\cos{\theta_{K}}{\rm d}\Phi}=\frac{9}{32\pi}$
$\displaystyle\bigl{[}S_{1}^{s}\sin^{2}\theta_{K}+S_{1}^{c}\cos^{2}\theta_{K}$
$\displaystyle+S_{2}^{s}\sin^{2}\theta_{K}\cos
2\theta_{\ell}+S_{2}^{c}\cos^{2}\theta_{K}\cos 2\theta_{\ell}$
$\displaystyle+{S_{3}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\Phi+{S_{4}}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\Phi$
$\displaystyle+{A_{5}}\sin
2\theta_{K}\sin\theta_{\ell}\cos\Phi+{A_{6}}\sin^{2}\theta_{K}\cos\theta_{\ell}$
$\displaystyle+{S_{7}}\sin 2\theta_{K}\sin\theta_{\ell}\sin\Phi+{A_{8}}\sin
2\theta_{K}\sin 2\theta_{\ell}\sin\Phi$
$\displaystyle+{A_{9}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\Phi\bigr{]},$ (1)
where equal numbers of produced $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are assumed [2]. The
$q^{2}$-dependent angular observables $S_{i}^{(s,c)}$ and $A_{i}$ correspond
to $C\\!P$ averages and $C\\!P$ asymmetries, respectively. Integrating Eq. 1
over two angles, under the assumption of massless leptons, results in three
distributions, each depending on one decay angle
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{K}}$
$\displaystyle=\frac{3}{4}(1-{F_{\rm
L}})(1-\cos^{2}\theta_{K})+\frac{3}{2}{F_{\rm L}}\cos^{2}\theta_{K},$ (2)
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{\ell}}$
$\displaystyle=\frac{3}{8}(1-{F_{\rm
L}})(1+\cos^{2}\theta_{\ell})+\frac{3}{4}{F_{\rm
L}}(1-\cos^{2}\theta_{\ell})+\frac{3}{4}{A_{6}}\cos\theta_{\ell},$ (3)
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\Phi}$
$\displaystyle=\frac{1}{2\pi}+\frac{1}{2\pi}{S_{3}}\cos
2\Phi+\frac{1}{2\pi}{A_{9}}\sin 2\Phi,$ (4)
which retain sensitivity to the angular observables $F_{\rm
L}(=S_{1}^{c}=-S_{2}^{c})$, $S_{3}$, $A_{6}$, and $A_{9}$. Of particular
interest is the $T$-odd asymmetry $A_{9}$ where possible large
$C\\!P$-violating phases from contributions beyond the SM would not be
suppressed by small strong phases [1].
This paper presents a measurement of the differential branching fraction and
the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ in six bins
of $q^{2}$. In addition, the total branching fraction is determined. The data
used in the analysis were recorded by the LHCb experiment in 2011 in $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and correspond to an
integrated luminosity of $1.0\mbox{\,fb}^{-1}$.
## 2 The LHCb detector
The LHCb detector [3] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The LHCb trigger system [4] consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage which
applies a full event reconstruction.
Simulated signal event samples are generated to determine the trigger,
reconstruction and selection efficiencies. Exclusive samples are analysed to
estimate possible backgrounds. The simulation generates $pp$ collisions using
Pythia 6.4 [5] with a specific LHCb configuration [6]. Decays of hadronic
particles are described by EvtGen [7] in which final state radiation is
generated using Photos [8]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [9,
*Agostinelli:2002hh] as described in Ref. [11]. Data driven corrections are
applied to the simulated events to account for differences between data and
simulation. These include the IP resolution, tracking efficiency, and particle
identification performance. In addition, simulated events are reweighted
depending on the transverse momentum ($p_{\rm T}$) of the $B^{0}_{s}$ meson,
the vertex fit quality, and the track multiplicity to match distributions of
control samples from data.
## 3 Selection of signal candidates
Signal candidates are accepted if they are triggered by particles of the
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ ($\phi\\!\rightarrow K^{+}K^{-}$)
final state. The hardware trigger requires either a high transverse momentum
muon or muon pair, or a high transverse energy ($E_{\rm T}$) hadron. The first
stage of the software trigger selects events containing a muon (or hadron)
with $\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
($\mbox{$E_{\rm T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and a minimum
IP with respect to all primary interaction vertices in the event of
$80\,\upmu\rm m$ ($125\,\upmu\rm m$). In the second stage of the software
trigger the tracks of two or more final state particles are required to form a
vertex that is significantly displaced from all primary vertices (PVs) in the
event.
Candidates are selected if they pass a loose preselection that requires the
kaon and muon tracks to have a large $\chi^{2}_{\rm IP}$ ($>9$) with respect
to the PV. The $\chi^{2}_{\rm IP}$ is defined as the difference between the
$\chi^{2}$ of the PV reconstructed with and without the considered particle.
The four tracks forming a $B^{0}_{s}$ candidate are fit to a common vertex,
which is required to be of good quality ($\chi^{2}_{\rm vtx}<30$) and well
separated from the PV ($\chi^{2}_{\rm FD}>121$, where FD denotes the flight
distance). The angle between the $B^{0}_{s}$ momentum vector and the vector
connecting the PV with the $B^{0}_{s}$ decay vertex is required to be small.
Furthermore, $B^{0}_{s}$ candidates are required to have a small IP with
respect to the PV ($\chi^{2}_{\rm IP}<16$). The invariant mass of the
$K^{+}K^{-}$ system is required to be within
$12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\phi$ mass [12].
To further reject combinatorial background events, a boosted decision tree
(BDT) [13] using the AdaBoost algorithm [14] is applied. The BDT training uses
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
$({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-})$
candidates as proxy for the signal, and candidates in the
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ mass sidebands
($5100<m(K^{+}K^{-}\mu^{+}\mu^{-})<5166{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and
$5566<m(K^{+}K^{-}\mu^{+}\mu^{-})<5800{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
as background. The input variables of the BDT are the $\chi^{2}_{\rm IP}$ of
all final state tracks and of the $B^{0}_{s}$ candidate, the angle between the
$B^{0}_{s}$ momentum vector and the vector between PV and $B^{0}_{s}$ decay
vertex, the vertex fit $\chi^{2}$, the flight distance significance and
transverse momentum of the $B^{0}_{s}$ candidate, and particle identification
information of the muons and kaons in the final state.
Several types of $b$-hadron decays can mimic the final state of the signal
decay and constitute potential sources of peaking background. The resonant
decays $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ and $B^{0}_{s}\\!\rightarrow\psi{(2S)}\phi$ with
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ are rejected by applying vetoes on
the dimuon mass regions around the charmonium resonances,
$2946<m(\mu^{+}\mu^{-})<3176{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$3592<m(\mu^{+}\mu^{-})<3766{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To
account for the radiative tails of the charmonium resonances the vetoes are
enlarged by $200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to lower
$m(\mu^{+}\mu^{-})$ for reconstructed $B^{0}_{s}$ masses below
$5316{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the region
$5416<m(B^{0}_{s})<5566{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ the vetoes
are extended by $50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to higher
$m(\mu^{+}\mu^{-})$ to reject a small fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ decays that
are misreconstructed at higher masses. The decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ ($K^{*0}\\!\rightarrow K^{+}\pi^{-}$) can be
reconstructed as signal if the pion is misidentified as a kaon. This
background is strongly suppressed by particle identification criteria. In the
narrow $\phi$ mass window, $2.4\pm 0.5$ misidentified $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ candidates are expected within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass of
$5366{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [12]. The resonant decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ can
also constitute a source of peaking background if the $K^{+}$ ($K^{-}$) is
misidentified as $\mu^{+}$ ($\mu^{-}$) and vice versa. Similarly, the decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
($K^{*0}\rightarrow K^{+}\pi^{-}$) where the $\pi^{-}$ ($\mu^{-}$) is
misidentified as $\mu^{-}$ ($K^{-}$) can mimic the signal decay. These
backgrounds are rejected by requiring that the invariant mass of the
$K^{+}\mu^{-}$ ($K^{-}\mu^{+}$) system, with kaons reconstructed under the
muon mass hypothesis, is not within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass of
$3096{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [12], unless both the kaon and
the muon pass stringent particle identification criteria. The expected number
of background events from double misidentification in the $B^{0}_{s}$ signal
mass region is $0.9\pm 0.5$. All other backgrounds studied, including
semileptonic $b\rightarrow c\,\mu^{-}\bar{\nu}_{\mu}(c\rightarrow
s\,\mu^{+}\nu_{\mu})$ cascades, hadronic double misidentification from
$B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}(D^{-}_{s}\rightarrow\phi\pi^{-})$, and
the decay $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax(1520)\,\mu^{+}\mu^{-}$, have been found to be negligible.
## 4 Differential branching fraction
Figure 1 shows the $\mu^{+}\mu^{-}$ versus the $K^{+}K^{-}\mu^{+}\mu^{-}$
invariant mass of the selected candidates. The signal decay
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is clearly visible in the
$B^{0}_{s}$ signal region. The determination of the differential branching
fraction is performed in six bins of $q^{2}$, given in Table 1, and
corresponds to the binning chosen for the analysis of the decay
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ [15]. Figure 2 shows the
$K^{+}K^{-}\mu^{+}\mu^{-}$ mass distribution in the six $q^{2}$ bins. The
signal yields are determined by extended unbinned maximum likelihood fits to
the reconstructed $B^{0}_{s}$ mass distributions. The signal component is
modeled by a double Gaussian function. The resolution parameters are obtained
from the resonant
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decay. A $q^{2}$-dependent scaling factor, determined with simulated
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ events, is introduced to account
for the observed $q^{2}$ dependence of the mass resolution. The combinatorial
background is described by a single exponential function. The veto of the
radiative tails of the charmonium resonances is accounted for by using a scale
factor. The resulting signal yields are given in Table 1. Fitting for the
signal yield over the full $q^{2}$ region, $174\pm 15$ signal candidates are
found. A fit of the normalisation mode
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ yields
$(20.36\pm 0.14)\times 10^{3}$ candidates.
Figure 1: Invariant $\mu^{+}\mu^{-}$ versus $K^{+}K^{-}\mu^{+}\mu^{-}$ mass.
The charmonium vetoes are indicated by the solid lines. The vertical dashed
lines indicate the signal region of $\pm
50{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ around the known $B^{0}_{s}$ mass
in which the signal decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is
visible.
Figure 2: Invariant mass of $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ candidates in six bins of invariant dimuon mass squared. The fitted signal component is denoted by the light blue shaded area, the combinatorial background component by the dark red shaded area. The solid line indicates the sum of the signal and background components. Table 1: Signal yield and differential branching fraction ${\rm d}{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$ in six bins of $q^{2}$. Results are also quoted for the region $1<q^{2}<6{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ where theoretical predictions are most reliable. The first uncertainty is statistical, the second systematic, and the third from the branching fraction of the normalisation channel. $q^{2}$ bin $({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})$ | $N_{\rm sig}$ | ${\rm d}{\cal B}/{\rm d}q^{2}~{}(10^{-8}\mathrm{\,Ge\kern-1.00006ptV}^{-2}c^{4})$
---|---|---
$0.10<q^{2}<\parbox{0.0pt}{2.00}$ | $25.0\,^{+5.8}_{-5.2}$ | $4.72\,^{+1.09}_{-0.98}\pm 0.20\pm 0.47$
$2.00<q^{2}<\parbox{0.0pt}{4.30}$ | $14.3\,^{+4.9}_{-4.3}$ | $2.30\,^{+0.79}_{-0.69}\pm 0.11\pm 0.23$
$4.30<q^{2}<\parbox{0.0pt}{8.68}$ | $41.2\,^{+7.5}_{-7.0}$ | $3.15\,^{+0.58}_{-0.53}\pm 0.12\pm 0.31$
$10.09<q^{2}<\parbox{0.0pt}{12.90}$ | $40.7\,^{+7.7}_{-7.2}$ | $4.26\,^{+0.81}_{-0.75}\pm 0.26\pm 0.43$
$14.18<q^{2}<\parbox{0.0pt}{16.00}$ | $23.8\,^{+5.9}_{-5.3}$ | $4.17\,^{+1.04}_{-0.93}\pm 0.24\pm 0.42$
$16.00<q^{2}<\parbox{0.0pt}{19.00}$ | $26.6\,^{+5.7}_{-5.3}$ | $3.52\,^{+0.76}_{-0.70}\pm 0.20\pm 0.35$
$1.00<q^{2}<\parbox{0.0pt}{6.00}$ | $31.4\,^{+7.0}_{-6.3}$ | $2.27\,^{+0.50}_{-0.46}\pm 0.11\pm 0.23$
The differential branching fraction of the signal decay in the $q^{2}$
interval spanning from $q^{2}_{\rm min}$ to $q^{2}_{\rm max}$ is calculated
according to
$\displaystyle\frac{{\rm d}{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})}{{\rm d}q^{2}}$
$\displaystyle=\frac{1}{q^{2}_{\rm max}-q^{2}_{\rm min}}\frac{N_{\rm
sig}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}}\frac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}}{\epsilon_{\phi\mu^{+}\mu^{-}}}{\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi){\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}),~{}$ (5)
where $N_{\rm sig}$ and $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$
denote the yields of the $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
candidates and $\epsilon_{\phi\mu^{+}\mu^{-}}$ and
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$ denote their
respective efficiencies. Since the reconstruction and selection efficiency of
the signal decay depends on $q^{2}$, a separate efficiency ratio
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}/\epsilon_{\phi\mu^{+}\mu^{-}}$ is determined for every $q^{2}$
bin. The branching fractions used in Eq. 5 are given by ${\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(10.50\pm 1.05\right)\times 10^{-4}$ [16] and ${\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})=\left(5.93\pm 0.06\right)\times 10^{-2}$
[12]. The resulting $q^{2}$-dependent differential branching fraction ${\rm
d}{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$ is shown
in Fig. 3. Possible contributions from $B^{0}_{s}$ decays to
$K^{+}K^{-}\mu^{+}\mu^{-}$, with the $K^{+}K^{-}$ pair in an S-wave
configuration, are neglected in this analysis. The S-wave fraction is expected
to be small, for the decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
it is measured to be $(1.1\pm 0.1\,^{+0.2}_{-0.1})\%$ [16] for the
$K^{+}K^{-}$ mass window used in this analysis.
The total branching fraction is determined by summing the differential
branching fractions in the six $q^{2}$ bins. Using the form factor
calculations described in Ref. [17] the signal fraction rejected by the
charmonium vetoes is determined to be $17.7\%$. This number is confirmed by a
different form factor calculation detailed in Ref. [18]. No uncertainty is
assigned to the vetoed signal fraction. Correcting for the charmonium vetoes,
the branching fraction ratio ${\cal
B}\left(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}\right)/{\cal
B}\left(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi\right)$ is measured to be
$\displaystyle\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$
$\displaystyle=\left(6.74\,^{+0.61}_{-0.56}\pm 0.16\right)\times 10^{-4}.$
The systematic uncertainties will be discussed in detail in Sec. 4.1. Using
the known branching fraction of the normalisation channel the total branching
fraction is
$\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})$
$\displaystyle=\left(7.07\,^{+0.64}_{-0.59}\pm 0.17\pm 0.71\right)\times
10^{-7},$
where the first uncertainty is statistical, the second systematic and the
third from the uncertainty on the branching fraction of the normalisation
channel.
Figure 3: Differential branching fraction ${\rm d}{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$. Error bars
include both statistical and systematic uncertainties added in quadrature.
Shaded areas indicate the vetoed regions containing the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonances.
The solid curve shows the leading order SM prediction, scaled to the fitted
total branching fraction. The prediction uses the SM Wilson coefficients and
leading order amplitudes given in Ref. [2], as well as the form factor
calculations in Ref. [17]. $B^{0}_{s}$ mixing is included as described in Ref.
[1]. No error band is given for the theory prediction. The dashed curve
denotes the leading order prediction scaled to a total branching fraction of
$16\times 10^{-7}$ [19].
### 4.1 Systematic uncertainties on the differential branching fraction
The dominant source of systematic uncertainty on the differential branching
fraction arises from the uncertainty on the branching fraction of the
normalisation channel
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$),
which is known to an accuracy of $10\%$ [16]. This uncertainty is fully
correlated between all $q^{2}$ bins.
Many of the systematic uncertainties affect the relative efficiencies
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}/\epsilon_{\phi\mu^{+}\mu^{-}}$ that are determined using
simulation. The limited size of the simulated samples causes an uncertainty of
$\sim 1\%$ on the ratio in each bin. Simulated events are corrected for known
discrepancies between simulation and data. The systematic uncertainties
associated with these corrections (e.g. tracking efficiency and performance of
the particle identification) are typically of the order of $1\text{--}2\%$.
The correction procedure for the impact parameter resolution has an effect of
up to $5\%$. Averaging the relative efficiency within the $q^{2}$ bins leads
to a systematic uncertainty of $1\text{--}2\%$. Other systematic uncertainties
of the same magnitude include the trigger efficiency and the uncertainties of
the angular distributions of the signal decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$. The influence of the signal mass
shape is found to be $0.5\%$. The background shape has an effect of up to
$5\%$, which is evaluated by using a linear function to describe the mass
distribution of the background instead of the nominal exponential shape.
Peaking backgrounds cause a systematic uncertainty of $1\text{--}2\%$ on the
differential branching fraction. The size of the systematics uncertainties on
the differential branching fraction, added in quadrature, ranges from
$4\text{--}6\%$. This is small compared to the dominant systematic uncertainty
of $10\%$ due to the branching fraction of the normalisation channel, which is
given separately in Table 1, and the statistical uncertainty.
## 5 Angular analysis
The angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ are
determined using unbinned maximum likelihood fits to the distributions of
$\cos{\theta_{K}}$, $\cos{\theta_{\ell}}$, $\Phi$, and the invariant mass of
the $K^{+}K^{-}\mu^{+}\mu^{-}$ system. The detector acceptance and the
reconstruction and selection of the signal decay distort the angular
distributions given in Eqs. 2–4. To account for this angular acceptance
effect, an angle-dependent efficiency is introduced that factorises in
$\cos{\theta_{K}}$ and $\cos{\theta_{\ell}}$, and is independent of the angle
$\Phi$, i.e.
$\epsilon(\cos{\theta_{K}},\cos{\theta_{\ell}},\Phi)=\epsilon_{K}(\cos{\theta_{K}})\cdot\epsilon_{\ell}(\cos{\theta_{\ell}})$.
The factors $\epsilon_{K}(\cos{\theta_{K}})$ and
$\epsilon_{\ell}(\cos{\theta_{\ell}})$ are determined from fits to simulated
events. Even Chebyshev polynomial functions of up to fourth order are used to
parametrise $\epsilon_{K}(\cos{\theta_{K}})$ and
$\epsilon_{\ell}(\cos{\theta_{\ell}})$ for each bin of $q^{2}$. The point-to-
point dissimilarity method described in Ref. [20] confirms that the angular
acceptance effect is well described by the acceptance model.
Taking the acceptances into account and integrating Eq. 1 over two angles,
results in
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{K}}$
$\displaystyle=\epsilon_{K}(\cos{\theta_{K}})\biggl{[}\frac{3}{4}(1-{F_{\rm
L}})(1-\cos^{2}\theta_{K})\,\xi_{1}+\frac{3}{2}{F_{\rm
L}}\cos^{2}\theta_{K}\,\xi_{2}\biggr{]},$ (6) $\displaystyle\frac{1}{{\rm
d}\Gamma/{\rm d}q^{2}}\frac{{\rm d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm
d}\cos\theta_{\ell}}$
$\displaystyle=\parbox{0.0pt}{$\epsilon_{\ell}(\cos{\theta_{\ell}})$}\biggl{[}\frac{3}{8}(1-{F_{\rm
L}})(1+\cos^{2}\theta_{\ell})\,\xi_{3}+\frac{3}{4}{F_{\rm
L}}(1-\cos^{2}\theta_{\ell})\,\xi_{4}$
$\displaystyle\hphantom{=\epsilon_{K}(\cos{\theta_{K}})}+\frac{3}{4}{A_{6}}\cos\theta_{\ell}\,\xi_{3}\biggr{]},$
(7) $\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\Phi}$
$\displaystyle=\biggl{[}\frac{1}{2\pi}\xi_{1}\xi_{3}+\frac{1}{2\pi}F_{\rm
L}(\xi_{2}\xi_{4}-\xi_{1}\xi_{3})$
$\displaystyle\hphantom{={}}+\frac{1}{2\pi}{S_{3}}\cos
2\Phi\,\xi_{2}\xi_{3}+\frac{1}{2\pi}{A_{9}}\sin
2\Phi\,\xi_{2}\xi_{3}\biggr{]}.$ (8)
The terms $\xi_{i}$ are correction factors with respect to Eqs. 2–4 and are
given by the angular integrals
$\displaystyle\xi_{1}$
$\displaystyle=\frac{3}{8}\int_{-1}^{+1}(1+\cos^{2}\theta_{\ell})\epsilon_{\ell}(\cos{\theta_{\ell}}){\rm
d}\cos{\theta_{\ell}},$ $\displaystyle\xi_{2}$
$\displaystyle=\frac{3}{4}\int_{-1}^{+1}(1-\cos^{2}\theta_{\ell})\epsilon_{\ell}(\cos{\theta_{\ell}}){\rm
d}\cos{\theta_{\ell}},$ $\displaystyle\xi_{3}$
$\displaystyle=\frac{3}{4}\int_{-1}^{+1}(1-\cos^{2}\theta_{K})\epsilon_{K}(\cos{\theta_{K}}){\rm
d}\cos{\theta_{K}},$ $\displaystyle\xi_{4}$
$\displaystyle=\frac{3}{2}\int_{-1}^{+1}\cos^{2}\theta_{K}\epsilon_{K}(\cos{\theta_{K}}){\rm
d}\cos{\theta_{K}}.$ (9)
Three two-dimensional maximum likelihood fits in the decay angles and the
reconstructed $B^{0}_{s}$ mass are performed for each $q^{2}$ bin to determine
the angular observables. The observable $F_{\rm L}$ is determined in the fit
to the $\cos{\theta_{K}}$ distribution described by Eq. 6. The
$\cos{\theta_{\ell}}$ distribution given by Eq. 7 is used to determine
$A_{6}$. Both $S_{3}$ and $A_{9}$ are measured from the $\Phi$ distribution,
as described by Eq. 8. In the fit of the $\Phi$ distribution a Gaussian
constraint is applied to the parameter $F_{\rm L}$ using the value of $F_{\rm
L}$ determined from the $\cos{\theta_{K}}$ distribution. The constraint on
$F_{\rm L}$ has negligible influence on the values of $S_{3}$ and $A_{9}$. The
angular distribution of the background events is fit using Chebyshev
polynomial functions of second order. The mass shapes of the signal and
background are described by the sum of two Gaussian distributions with a
common mean, and an exponential function, respectively. The effect of the veto
of the radiative tails on the combinatorial background is accounted for by
using an appropriate scale factor.
The measured angular observables are presented in Fig. 4 and Table 2. The
$68\%$ confidence intervals are determined using the Feldman-Cousins method
[21] and the nuisance parameters are included using the plug-in method [22].
Figure 4: a) Longitudinal polarisation fraction $F_{\rm L}$, b) $S_{3}$, c) $A_{6}$, and d) $A_{9}$ in six bins of $q^{2}$. Error bars include statistical and systematic uncertainties added in quadrature. The solid curves are the leading order SM predictions, using the Wilson coefficients and leading order amplitudes given in Ref. [2], as well as the form factor calculations in Ref. [17]. $B^{0}_{s}$ mixing is included as described in Ref. [1]. No error band is given for the theory predictions. Table 2: Results for the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ in bins of $q^{2}$. The first uncertainty is statistical, the second systematic. $q^{2}$ bin $({\mathrm{\,Ge\kern-0.80005ptV^{2}\\!/}c^{4}})$ | $F_{\rm L}$ | $S_{3}$ | $A_{6}$ | $A_{9}$
---|---|---|---|---
$0.10<q^{2}<\parbox{0.0pt}{2.00}$ | $0.37\,^{+0.19}_{-0.17}\pm 0.07$ | $-0.11\,^{+0.28}_{-0.25}\pm 0.05$ | $0.04\,^{+0.27}_{-0.32}\pm 0.12$ | $-0.16\,^{+0.30}_{-0.27}\pm 0.09$
$2.00<q^{2}<\parbox{0.0pt}{4.30}$ | $0.53\,^{+0.25}_{-0.23}\pm 0.10$ | $-0.97\,^{+0.53}_{-0.03}\pm 0.17$ | $0.47\,^{+0.39}_{-0.42}\pm 0.14$ | $-0.40\,^{+0.52}_{-0.35}\pm 0.11$
$4.30<q^{2}<\parbox{0.0pt}{8.68}$ | $0.81\,^{+0.11}_{-0.13}\pm 0.05$ | $0.25\,^{+0.21}_{-0.24}\pm 0.05$ | $-0.02\,^{+0.20}_{-0.21}\pm 0.10$ | $-0.13\,^{+0.27}_{-0.26}\pm 0.10$
$10.09<q^{2}<\parbox{0.0pt}{12.90}$ | $0.33\,^{+0.14}_{-0.12}\pm 0.06$ | $0.24\,^{+0.27}_{-0.25}\pm 0.06$ | $-0.06\,^{+0.20}_{-0.20}\pm 0.08$ | $0.29\,^{+0.25}_{-0.26}\pm 0.10$
$14.18<q^{2}<\parbox{0.0pt}{16.00}$ | $0.34\,^{+0.18}_{-0.17}\pm 0.07$ | $-0.03\,^{+0.29}_{-0.31}\pm 0.06$ | $-0.06\,^{+0.30}_{-0.30}\pm 0.08$ | $0.24\,^{+0.36}_{-0.35}\pm 0.12$
$16.00<q^{2}<\parbox{0.0pt}{19.00}$ | $0.16\,^{+0.17}_{-0.10}\pm 0.07$ | $0.19\,^{+0.30}_{-0.31}\pm 0.05$ | $0.26\,^{+0.22}_{-0.24}\pm 0.08$ | $0.27\,^{+0.31}_{-0.28}\pm 0.11$
$1.00<q^{2}<\parbox{0.0pt}{6.00}$ | $0.56\,^{+0.17}_{-0.16}\pm 0.09$ | $-0.21\,^{+0.24}_{-0.22}\pm 0.08$ | $0.20\,^{+0.29}_{-0.27}\pm 0.07$ | $-0.30\,^{+0.30}_{-0.29}\pm 0.11$
### 5.1 Systematic uncertainties on the angular observables
The dominant systematic uncertainty on the angular observables is due to the
angular acceptance model. Using the point-to-point dissimilarity method
detailed in Ref. [20], the acceptance model is shown to describe the angular
acceptance effect for simulated events at the level of $10\%$. A cross-check
of the angular acceptance using the normalisation channel
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ shows
good agreement of the angular observables with the values determined in Refs.
[23] and [24]. For the determination of the systematic uncertainty due to the
angular acceptance model, variations of the acceptance curves are used that
have the largest impact on the angular observables. The resulting systematic
uncertainty is of the order of $0.05\text{--}0.10$, depending on the $q^{2}$
bin.
The limited amount of simulated events accounts for a systematic uncertainty
of up to $0.02$. The simulation correction procedure (for tracking efficiency,
impact parameter resolution, and particle identification performance) has
negligible effect on the angular observables. The description of the signal
mass shape leads to a negligible systematic uncertainty. The background mass
model causes an uncertainty of less than $0.02$. The model of the angular
distribution of the background can have a large effect since the statistical
precision of the background sample is limited. To estimate the effect, the
parameters describing the background angular distribution are determined in
the high $B^{0}_{s}$ mass sideband
($5416<m(K^{+}K^{-}\mu^{+}\mu^{-})<5566{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
using a relaxed requirement on the $\phi$ mass. The effect is typically
$0.05\text{--}0.10$. Peaking backgrounds cause systematic deviations of the
order of $0.01\text{--}0.02$. Due to the sizeable lifetime difference in the
$B^{0}_{s}$ system [24] a decay time dependent acceptance can in principle
affect the angular observables. The deviation of the observables due to this
effect is studied and found to be negligible. The total systematic
uncertainties, evaluated by adding all components in quadrature, are small
compared to the statistical uncertainties.
## 6 Conclusions
The differential branching fraction of the FCNC decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ has been determined. The results are
summarised in Fig. 3 and in Table 1. Using the form factor calculations in
Ref. [17] to determine the fraction of events removed by the charmonium
vetoes, the relative branching fraction ${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)$ is
determined to be
$\displaystyle\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$
$\displaystyle=$ $\displaystyle\left(6.74\,^{+0.61}_{-0.56}\pm
0.16\right)\times 10^{-4}.$
This value is compatible with a previous measurement by the CDF collaboration
of ${\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(11.3\pm 1.9\pm 0.7\right)\times 10^{-4}$ [25] and a recent
preliminary result which yields ${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(9.0\pm 1.4\pm 0.7\right)\times 10^{-4}$ [26]. Using the
branching fraction of the normalisation channel, ${\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(10.50\pm 1.05\right)\times 10^{-4}$ [16], the total
branching fraction of the decay is determined to be
$\displaystyle{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle\left(7.07\,^{+0.64}_{-0.59}\pm 0.17\pm
0.71\right)\times 10^{-7},$
where the first uncertainty is statistical, the second systematic, and the
third from the uncertainty of the branching fraction of the normalisation
channel. This measurement constitutes the most precise determination of the
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ branching fraction to date. The
measured value is lower than the SM theory predictions that range from
$14.5\times 10^{-7}$ to $19.2\times 10^{-7}$ [19, 27, 28, 29]. The
uncertainties on these predictions originating from the form factor
calculations are typically of the order of $20\text{--}30\%$.
In addition, the first angular analysis of the decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ has been performed. The angular
observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ are determined in bins
of $q^{2}$, using the distributions of $\cos{\theta_{K}}$,
$\cos{\theta_{\ell}}$, and $\Phi$. The results are summarised in Fig. 4, and
the numerical values are given in Table 2. All measured angular observables
are consistent with the leading order SM expectation.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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# Interpretable Hyperspectral AI: When Non-Convex Modeling meets Hyperspectral
Remote Sensing
Danfeng Hong, Wei He, Naoto Yokoya, Jing Yao, Lianru Gao, Liangpei Zhang,
Jocelyn Chanussot, and Xiao Xiang Zhu D. Hong is with the Remote Sensing
Technology Institute (IMF), German Aerospace Center (DLR), 82234 Wessling,
Germany, and also with the Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-
lab, 38000 Grenoble, France. (e-mail: danfeng.hong@dlr.de)W. He is with the
Geoinformatics Unit, RIKEN Center for Advanced Intelligence Project (AIP),
RIKEN, 103-0027 Tokyo, Japan. (e-mail: wei.he@riken.jp)N. Yokoya is with
Graduate School of Frontier Sciences, the University of Tokyo, 277-8561 Chiba,
Japan, and also with the Geoinformatics Unit, RIKEN Center for Advanced
Intelligence Project (AIP), RIKEN, 103-0027 Tokyo, Japan. (e-mail:
naoto.yokoya@riken.jp)L. Gao and J. Yao are with the Key Laboratory of Digital
Earth Science, Aerospace Information Research Institute, Chinese Academy of
Sciences, Beijing 100094, China. (e-mail: gaolr@aircas.ac.cn)L. Zhang is with
the State Key Laboratory of Information Engineering in Surveying, Mapping and
Remote Sensing, Wuhan University, Wuhan 430072, China.
(e-mail:zlp62@whu.edu.cn)J. Chanussot is with the Univ. Grenoble Alpes, INRIA,
CNRS, Grenoble INP, LJK, 38000 Grenoble, France, also with the Aerospace
Information Research Institute, Chinese Academy of Sciences, 100094 Beijing,
China. (e-mail: jocelyn@hi.is)X. Zhu is with the Remote Sensing Technology
Institute (IMF), German Aerospace Center (DLR), 82234 Wessling, Germany, and
Data Science in Earth Observation (SiPEO), Technical University of Munich
(TUM), 80333 Munich, Germany. (e-mail<EMAIL_ADDRESS>
###### Abstract
This is the pre-acceptance version, to read the final version please go to
IEEE Geoscience and Remote Sensing Magazine on IEEE Xplore. Hyperspectral
imaging, also known as image spectrometry, is a landmark technique in
geoscience and remote sensing (RS). In the past decade, enormous efforts have
been made to process and analyze these hyperspectral (HS) products mainly by
means of seasoned experts. However, with the ever-growing volume of data, the
bulk of costs in manpower and material resources poses new challenges on
reducing the burden of manual labor and improving efficiency. For this reason,
it is, therefore, urgent to develop more intelligent and automatic approaches
for various HS RS applications. Machine learning (ML) tools with convex
optimization have successfully undertaken the tasks of numerous artificial
intelligence (AI)-related applications. However, their ability in handling
complex practical problems remains limited, particularly for HS data, due to
the effects of various spectral variabilities in the process of HS imaging and
the complexity and redundancy of higher dimensional HS signals. Compared to
the convex models, non-convex modeling, which is capable of characterizing
more complex real scenes and providing the model interpretability technically
and theoretically, has been proven to be a feasible solution to reduce the gap
between challenging HS vision tasks and currently advanced intelligent data
processing models.
This article mainly presents an advanced and cutting-edge technical survey for
non-convex modeling towards interpretable AI models covering a board scope in
the following topics of HS RS:
* •
HS image restoration,
* •
dimensionality reduction,
* •
data fusion and enhancement,
* •
spectral unmixing,
* •
cross-modality learning for large-scale land cover mapping.
Around these topics, we will showcase the significance of non-convex
techniques to bridge the gap between HS RS and interpretable AI models with a
brief introduction on the research background and motivation, an emphasis on
the resulting methodological foundations and solution, and an intuitive
clarification of illustrative examples. At the end of each topic, we also pose
the remaining challenges on how to completely model the issues of complex
spectral vision from the perspective of intelligent ML combined with physical
priors and numerical non-convex modeling, and accordingly point out future
research directions.
This paper aims to create a good entry point to the advanced literature for
experienced researchers, Ph.D. students, and engineers who already have some
background knowledge in HS RS, ML, and optimization. This can further help
them launch new investigations on the basis of the above topics and
interpretable AI techniques for their focused fields.
###### Index Terms:
Artificial intelligence, image processing, interpretability, hyperspectral,
machine learning, modeling, non-convex, remote sensing, signal processing.
## I Introduction
### I-A Background and Significance of Hyperspectral Remote Sensing
Imaging spectroscopy, which was first-ever to be conceptualized by Goetz1 et
al. in 1980’s [1], is a seminal hyperspectral (HS) imaging technique of truly
achieving the integration of the 1-D spectrum and the 2-D image for earth
remote sensing (RS). Imaging spectroscopy is a typical “passive” RS technique,
which assembles spectroscopy and digital photography into a unified system.
Fig. 1 shows the data acquisition process of two different imaging patterns:
“active” RS and “passive” RS [2]. The resulting HS image collects hundreds of
2-D images finely sampled from the approximately contiguous wavelength across
the whole electromagnetic spectrum [3] (see Fig. 2). This enables the
recognition and identification of the materials, particularly for those that
have extremely similar spectral signatures in visual cues (e.g., RGB) [4], at
a more accurate and finer level. As a result, HS RS has been significantly
advanced and widely applied in many challenging tasks of earth observation
[5], such as fine-grained land cover classification, mineral mapping, water
quality assessment, precious farming, urban planning and monitoring, disaster
management and prediction, and concealed target detection.
Figure 1: An illustration to clarify the data acquisition process of two
different imaging patterns, i.e., “active” RS and “passive” RS. Figure 2: A
showcase to clarify the electromagnetic spectrum: the order from low to high
frequency is Long-waves, Radio-waves, Micro-waves, Infrared, Visible,
Ultraviolet, X-rays, and Gamma-rays, where four widely-concerned intervals,
e.g., Radio-waves, Infrared, Visible, and X-rays, are finely partitioned.
More specifically, characterized by the distinctive 3-D signaling structure,
the advantages of the HS image over conventional 1-D or 2-D signal products
can be summarized as
* •
compared to the common 1-D signal system, the HS 2-D spatial pattern provides
us more structured information, enabling the discrimination of underlying
objects of interest at a more semantically meaningful level;
* •
Beyond the 2-D natural images, the rich spectral information in HS images is
capable of detecting materials through tiny discrepancies in the spectral
domain, since the HS imaging instruments exploit the sensors to collect
hundreds or thousands of wavelength channels with an approximately continuous
spectral sampling at a subtle interval (e.g., 10nm).
Furthermore, the significance of HS images compared to other RS imaging
techniques with a lower spectral resolution, e.g., multispectral (MS) or RGB
imaging, mainly embodies in the following three aspects:
* 1)
HS images are capable of finely discriminating the different classes that
belong to the same category, such as Bitumen and Asphalt, Stressed Grass and
Synthetic Grass, Alunite and Kaolin. For those optical broadband imaging
products (e.g., MS imagery), they can only identify certain materials with the
observable differences in the spectral signatures, e.g., Water, Trees, Soil.
* 2)
The higher spectral resolution creates the possibilities to some challenging
applications that can be hardly achieved by only depending on formerly imaging
techniques, e.g., parameter extraction of biophysics and biochemistry,
biodiversity conservation, monitoring and management of the ecosystem,
automatic detection of food safety, which provides new insight into RS and
geoscience fields.
* 3)
Due to the limitations of image resolution either in spectral or spatial
domains, physically and chemically atmospheric effects, and environmental
conditions (e.g., the interference of soil background, illumination, the
uncontrolled shadow caused by clouds or building occlusion, topography change,
complex noises), those traditional RS imaging techniques were to a great
extent dominated by qualitative analysis. As HS RS arises, quantitative or
semi-quantitative analysis becomes increasingly plausible in many practical
cases.
### I-B An Ever-Growing Relation between Non-convex Modeling and
Interpretable AI in Hyperspectral Remote Sensing
In recent years, a vast number of HS RS missions (e.g., MODIS, HypSEO, DESIS,
Gaofen-5, EnMap, HyspIRI, etc.) have been launched to enhance our
understanding and capabilities to the Earth and environment, contributing to
the rapid and better development in a wide range of relevant applications,
such as land cover land use classification, spectral unmixing, data fusion,
image restoration, and multimodal data analysis. With the ever-growing
availability of RS data sources from both satellite and airborne sensors on a
large scale and even global scale, expert system-centric data processing and
analysis mode has run into bottlenecks and can not meet the demand of the big
data era. For this reason, data-driven signal and image processing, machine
learning (ML), AI models have been garnering growing interest and attention
from the researchers in the RS community.
Supported by well-established theory and numerical optimization, convex models
have been proven to be effective to model a variety of HS tasks under highly
idealized assumptions. However, there exist unknown, uncertain, and
unpredictable factors in the complex real scenes. Due to these factors that
lead to the lack of sound understanding and modeling capability to the scene,
convex models fail to work properly. The specific reasons could be two-fold.
On the one hand, integrating the benefits of 1-D and 2-D signals, the 3-D
structurized HS images offer greater potential and better solutions (compared
to natural images) to deal with the varying situation, but simultaneously
increase the model’s complexity and uncertainty to some extent. On the other
hand, due to unprecedented spatial, spectral, and temporal resolutions of HS
images in remotely sensed HS imaging, the difficulties and challenges in the
sophisticated HS vision approaches are mainly associated with the volume of
the HS data, complex material (spectral) mixing behavior, and uncontrolled
degradation mechanisms in data acquisition caused by illumination, noise, and
atmospheric effects.
The aforementioned factors, to a great extent, limit convex models to be
intelligent approaches for fully understanding and interpreting the real-life
scenario. Therefore, this naturally motivates us to investigate the
possibility of processing and analyzing the HS data in a non-convex modeling
fashion. In the following, we briefly make a qualitative comparison between
convex and non-convex models to clarify that non-convex modeling might be an
optimally feasible solution towards interpretable AI models in HS RS.
* •
Convex models are theoretically guaranteed to converge to the global optimal
solution, yet most tasks related to HS RS are complex in reality and hardly
simplified to an equivalent and perfect convex formulation. This to some
extent makes convex models inapplicable to practical tasks, due to the lack of
interpretability and completeness for problem modeling.
* •
Rather, non-convex models are capable of characterizing the complex studied
scene in HS RS more finely and completely, thereby more tending to achieve
automatization and intelligentization in the real world. Moreover, by
excavating the intrinsic properties from the HS data effectively to yield
physically meaningful priors, the solution space of non-convex models can be
shrunk to a “good” region bit by bit.
* •
Although non-convex models are complex by considering more complicated prior
knowledge, possibly leading to the lack of stable generalization ability, they
hold the higher potential that convex models do not have, particularly in
explaining models, understanding scenes, and achieving intelligent HS image
processing and analysis. Furthermore, this might be able to provide
researchers with a broader range of HS vision related topics, making it
applicable for more real cases in a variety of HS RS-related tasks.
### I-C Contributions
With the advent of the big data era, an ever-increasing data bulk and
diversity brings rare opportunities and challenges for the development of HS
RS in earth observation. Data-driven AI approaches, e.g., ML-based, deep
learning (DL)-based, have occupied a prominent place in manifold HS RS
applications. Nevertheless, how to open the “model” and give them
interpretability remains unknown yet. In this article, we raise a bold and
understandable standpoint, that is, non-convex modeling might be an effective
means to bridge the gap between interpretable AI models and HS RS. To support
the opinion, this article provides a detailed and systematic overview by
reviewing advanced and latest literature with an emphasis on non-convex
modeling in terms of five classic and burgeoning topics related to HS RS. More
specifically,
* •
We present a comprehensive discussion and analysis related to non-convex
modeling in five well-noticed and promising HS RS-related applications, such
as HS image restoration, dimensionality reduction and classification, spectral
unmixing, data fusion and enhancement, and cross-modality learning.
* •
For each topic, a few representative works are emphatically and detailedly
introduced by the attempts to make a connection between non-convex modeling
and intelligent/interpretable models. Moreover, the example experiments
(qualitatively or quantitatively) are subsequently performed after the
detailed method description. Those selected methods that engage in the
comparative experiments are accompanied by available code and data links for
the sake of reproducibility. Finally, the remaining challenges are highlighted
to further clarify the gap between interpretable ML/AI models and practical HS
RS applications.
* •
Regarding the three aspects of non-convex modeling, interpretable AI, and HS
RS, the end of this article concludes with some remarks, makes summary
analysis, and hints at plausible future research work.
We need to point out and emphasize, however, that this paper features food for
thoughts for advanced readers, and it is not an introduction for beginners
entering the field. The goal of this paper is to provide a cutting-edge survey
rather than a real tutorial. As a result, readers are expected to have some
prior knowledge across multidisciplinary, such as HS RS, convex optimization,
non-convex modeling, ML, and AI, where some basic principle, definitions, and
deductions need to be mastered. For beginners who are willing to start with
new researches on non-convex modeling for HS RS applications, we recommend
reading and learning following materials and references to get to know, for
example,
* •
what is the HS imaging or HS RS (e.g., principles, superiority,
practicability) and its relevant applications (topics) [5];
* •
convex optimization and its solutions (including a detailed description of
physical meaningful priors, e.g., low-rank, sparsity, graph regularization,
non-negativity, sum-to-one, etc.) as well as its relationship with non-convex
modeling [6];
* •
a general guideline on why and how to build non-convex models and how to solve
non-convex minimization problems [7];
* •
classic and advanced ML algorithms, including essential ideas, designing
thought, implementation process [8];
* •
a big picture about what is the AI and how to build the basic AI models [9].
Moreover, we hope that this paper can be also regarded as a good starting
point and evolve many novels, interesting, and noteworthy research issues
around the fields of non-convex modeling, interpretable ML/AI, and HS RS,
serving to more application cases in reality.
Figure 3: Illustration of five promising topics in HS RS, including image
restoration, dimensionality reduction and classification, data fusion and
enhancement, spectral unmixing, and cross-modality learning.
## II Outline and Basic Preparation
This paper starts with a brief introduction to a general non-convex
minimization problem in signal or image processing, and then specifies the
non-convex modeling with each topic in HS RS from an ML perspective. According
to the characteristics of HS imaging, these different issues may bring fresh
challenges to researches on non-convex modeling and optimization, which
contributes to boosting the development of both HS RS and ML for intelligent
data processing and analysis. Very recently, some successful showcases have
garnered growing attention from researchers who engage in HS data processing
and analysis, ML, or statistical optimization, leading to many newly-developed
non-convex models and their solutions. They can be roughly categorized into
the following several groups in a wide range of real applications related to
HS RS. Fig. 3 gives the illustration for each topic.
### II-A A Brief Introduction from Convex to Non-convex Models
As the name suggests, in the convex model, the shape of the area represented
by the function (or model) is convex. Accordingly, let the function
$f:\mathbb{R}_{n}\rightarrow\mathbb{R}$ be convex, if the domain of $f$ is a
convex set, and for the variable $\theta\in[0,1]$, any two points (e.g., $x$
and $y$) meet the following condition:
$\displaystyle f(y)\geq\theta f(x)+(1-\theta)f(y),$ (1)
then the function $f$ is convex, whose necessary and sufficient condition is
$f(y)\geq f(x)+\bigtriangledown f(x)(x-y)$.
Within the domain, the globally optimal solution of the convex model can be
obtained by common convex optimization techniques, such as linear programming
[10], quadratic programming [11], second order cone programming [12].
Informally, although convex methods have been widely used to model various
tasks, owing to the existence and uniqueness of solutions and the relatively
low model complexity, the real scenes are complex and changeable, inevitably
leading to many uncertainties and difficulties in the modeling process. For
this reason, non-convex modeling is capable of providing stronger modeling
power to the algorithm designer and can fully meet the demand of
characterizing the real complex scenes well. This naturally motivates us to
shift our emphases on some key issues related to non-convex modeling.
A general non-convex model usually consists of a smooth objective function
(e.g., Euclidean loss, negative log-likelihood) with Lipschitz gradient
$f(\mathbf{X})$ and the non-convex constraints $\mathcal{C}$, which can be
generalized by optimizing the following minimization problem:
$\displaystyle\mathop{\min}_{\mathbf{X}}\frac{1}{2}f(\mathbf{X})\;\;{\rm
s.t.}\;\mathbf{X}\in\mathcal{C},$ (2)
where $\mathbf{X}$ is the to-be-estimated variable, which can be defined as a
vector (1-D signal), a matrix (2-D image), or an unfolded matrix (3-D HS
image). The constraints $C$ could be sparsity-promoting variants, low-rank,
TV, and others, which need to be determined by the specific tasks.
Unlike convex models, there exist many local minimums in non-convex models.
This poses a big challenge on finding globally optimal solutions. For possible
solutions of non-convex methods, one strategy is to relax the non-convex
problem to an approximately convex model [13]. Another can break the non-
convex problems down into several convex subproblems and solve them in
parallel by the means of convex ways [14].
In the light of the different research goals, the general model in Eq. (2) can
be extended to the task-driven variants covering a broad scope within the HS
RS, including image restoration, dimensionality reduction, data fusion and
enhancement, spectral unmixing, and cross-modality learning. It should be
noted that in the following sections, some representative methods will be
introduced with a focus on non-convex modeling, while the alternating
direction method of multipliers (ADMM) optimization framework is recommended
for use as a general solver to solve these non-convex models. For specific
solutions of these models in each topic, please refer to the cited references.
### II-B Main Abbreviation
AI: | artificial intelligence.
---|---
ANN: | artificial neural network.
CML: | cross-modality learning.
DL: | deep learning.
DR: | dimensionality reduction.
GSD: | ground sampling distance.
HS: | hyperspectral.
KNN: | $k$ nearest neighbors.
LDA: | linear discriminant analysis.
LMM: | linear mixing model.
MA: | manifold alignment.
ML: | machine learning.
MML: | multimodality learning.
MS: | multispectral.
NMF: | non-negative matrix factorization.
RS: | remote sensing.
SAR: | synthetic aperture radar.
SOTA: | state-of-the-art.
SSL: | shared subspace learning.
SU: | spectral unmixing.
1-D: | one-dimensional.
2-D: | two-dimensional.
3-D: | three-dimensional.
### II-C Nomenclature
$\boldsymbol{\mathcal{X}}$: | to-be-estimated 3-D HS image.
---|---
$\boldsymbol{\mathcal{Y}}$: | observed 3-D HS image.
$\boldsymbol{\mathcal{N}}_{G}$: | 3-D Gaussian noise.
$\boldsymbol{\mathcal{N}}_{S}$: | 3-D sparse noise.
$\boldsymbol{\mathcal{O}}$: | core tensor.
$r$: | rank of matrix.
$\mathbf{X}$: | unfolded 2-D matrix of $\boldsymbol{\mathcal{X}}$.
$\mathbf{x}_{i}$: | the $i$-th pixel (1-D vector) of $\mathbf{X}$.
$\mathbf{Y}$: | unfolded 2-D matrix of $\boldsymbol{\mathcal{Y}}$.
$\mathbf{N}_{S}$: | unfolded 2-D matrix of $\boldsymbol{\mathcal{N}}_{S}$.
$\mathbf{H}$: | first-order difference matrix.
$\mathcal{C}$: | model constraint set.
$\Phi$: | to-be-estimated variable set.
$f$: | transformation functions.
$\mathbf{Q}$: | combination coefficients of NMF.
$\mathbf{W}$: | graph or manifold structure.
$\mathbf{L}$: | Laplacian matrix.
$\mathbf{D}$: | degree matrix.
$\mathbf{U}$: | subspace projection matrix.
$\mathbf{I}$: | identity matrix.
$\mathbf{d}$: | distance or similarity matrix.
$\sigma$: | standard derivation.
$C_{k}$: | sample set of the $k$-th class.
$\mathbf{M}$: | one-hot encoded matrix.
$\mathbf{P}$: | regression matrix.
$\mathbf{E}$: | endmember matrix.
$\mathbf{E}_{0}$: | reference endmember matrix.
$\mathbf{A}$: | abundance matrix.
$\mathbf{S}$: | scaling factors (matrix).
$\mathbf{V}$: | spectral variability dictionary (matrix).
$\mathbf{J}$: | coefficients corresponding to $\mathbf{V}$.
$\mathbf{R}$: | spatial degradation function.
$\mathbf{G}$: | spectral response function.
$\mathbf{N}_{H}$: | HS noise.
$\mathbf{N}_{H}$: | MS noise.
$\mathbf{Z}$: | high spatial resolution MS image.
$m$: | the number of the considered modality.
$c$: | scaling constant.
### II-D Notation
$\lVert\mathbf{X}\rVert_{\operatorname{F}}$ | Forbenius norm of $\mathbf{X}$, obtained by $\sqrt{\sum_{i,j}\mathbf{X}_{i,j}^{2}}$
---|---
$\lVert\mathbf{X}\rVert_{1,1}$ | $\ell_{1}$ norm of $\mathbf{X}$, obtained by $\sum_{i,j}|\mathbf{X}_{i,j}|$
$\lVert\mathbf{X}\rVert_{2,1}$ | $\ell_{2,1}$ norm of $\mathbf{X}$, obtained by $\sum_{i}|\sqrt{\sum_{j}\mathbf{X}_{i,j}^{2}}|$
$\lVert\mathbf{X}\rVert_{1/2}$ | $\ell_{1/2}$ norm of $\mathbf{X}$, obtained by $\sum_{i,j}\mathbf{x}_{j}(i)^{1/2}$
$\lVert\mathbf{X}\rVert_{q}$ | $\ell_{q}$ norm of $\mathbf{X}$, obtained by $\sum_{i,j}\mathbf{x}_{j}(i)^{q}$
$\lVert\mathbf{X}\rVert_{{\rm TV}}$ | ${\rm TV}$ norm of $\mathbf{X}$, obtained by $\lVert\mathbf{H}_{h}\mathbf{X}+\mathbf{H}_{v}\mathbf{X}\rVert_{2,1}$
$\lVert\mathbf{X}\rVert_{0}$ | $\ell_{0}$ norm of $\mathbf{X}$, obtained by $\lim_{p\rightarrow 0}\sum_{i,j}|\mathbf{X}_{i,j}|^{p}$
$\operatorname{tr}(\mathbf{X})$ | trace of $\mathbf{X}$, obtained by $\sum_{i}\mathbf{X}_{i,i}$
$\lVert\mathbf{X}\rVert_{*}$ | nuclear norm of $\mathbf{X}$, obtained by $\operatorname{tr}(\sqrt{\mathbf{X}^{\top}\mathbf{X}})$
$\lVert\mathbf{x}\rVert_{2}$ | $\ell_{2}$ norm of $\mathbf{x}$, obtained by $\sqrt{\sum_{j}\mathbf{x}_{j}^{2}}$
$\odot$ | the element-wise multiplication operator
$\phi_{i}$ | the neighbouring pixels of the target pixel $i$
## III Hyperspectral Image Restoration
Owing to the wealthy spectral information, the HS image has been widely used
in different kinds of applications, including urban planning, agriculture,
forestry, target detection, and so on. However, due to the limitation of
hyperspectral imaging systems and the weather conditions, HS images are always
suffering from the pollution of various noise. Fig. (4) illustrates different
noise types of HS images observed from airborne and spaceborne sensors.
Therefore, HS image denoising and restoration is a necessary pre-processing
for the subsequent applications to assist the noise.
Figure 4: Examples for different noise types of HS Images observed from
airborne and spaceborne sensors.
The statistical distribution of hyperspectral noise is complicated. For
instance, the readout noise, which is assumed to obey the Gaussian
distribution, is produced by the imaging device (Charge-coupled Device) during
the conversation from electrons to the final image [15]. The stripes are
generated in the hyperspectral data collected by the pushroom hyperspectral
sensors [16]. Due to the weather environment, the obtained HS data are always
suffering from the cloud and cloud shadow [17]. Besides, the HS images are
also suffering from the signal-dependent noise [18], multiplicative noise,
impulsive noise, Laplace noise and so on. Furthermore, In this paper, we
follow the mainstream and focus on the Gaussian noise removal [19, 20] and
mixed noise removal problem [21, 22].
Since 1980s, researchers have paid attention to the HS noise analysis. For
example, maximum noise fraction (MNF) transformation [23] noise-adjusted
principal component analysis [19] are utilized to extract high-quality
components for the subsequent classification and reject the low-quality
components. Following the mainstream of gray/color image denoising in computer
vision society, various state-of-the-art technologies, such as wavelets [24],
sparse representation [25, 26], TV [20, 27], non-local means processing [28,
29], low-rank matrix representation [21, 30, 31, 22], tensor representation
[32, 33, 34], DL [35, 36] and so on.
The non-convex regularized methods have also been developed for HS image
restoration. From [21] the HS images are assumed to be corrupted by the
additive noise, including Gaussian noise, stripes, deadlines, pixel missing,
impulse noise and so on. The observation model is formulated as:
$\boldsymbol{\mathcal{Y}}=\boldsymbol{\mathcal{X}}+\boldsymbol{\mathcal{N}}_{G}+\boldsymbol{\mathcal{N}}_{S},$
(3)
where $\boldsymbol{\mathcal{Y}}$ represents the observed noisy image,
$\boldsymbol{\mathcal{X}}$ stands for the latent clean image,
$\boldsymbol{\mathcal{N}}_{G}$ is the Gaussian noise, and
$\boldsymbol{\mathcal{N}}_{S}$ is the sparse noise, including stripes,
deadlines, pixel missing and impulse noise. Typically, when sparse noise
$\boldsymbol{\mathcal{N}}_{S}$ is omitted, the model (3) is degraded to the
Gaussian noise removal problem. From [21], the low-rank property exploration
of clean image $\boldsymbol{\mathcal{X}}$ has attracted much attention and
achieved state-of-the-art HS image denoising performance [29, 34]. Generally
speaking, the mainstreams are from two points. Firstly, how to explore the
low-rank property of clean image $\boldsymbol{\mathcal{X}}$. Until now,
spectral low-rank property [21, 30], spatial low-rank property [37, 33] and
non-local low-rank property [32, 29] have been well studied. Furthermore, how
to balance the contribution of different low-rank properties is also a key
problem [38, 29]. Secondly, the low-rank constraint of
$\boldsymbol{\mathcal{X}}$ formulates a non-convex optimization problem.
Therein, how to efficiently solve the rank constraint non-convex problem is
another key problem. In the next subsection, we review several outstanding
works and illustrate how these works formulate the low-rank modeling, and how
to solve the non-convex problem.
TABLE I: Prior properties of the selected methods. One, two, and three $\bullet$ denote the low, medium, and high intensity of prior information, respectively. Methods | Low-rankness | Local smoothness
---|---|---
spatial | spectral | non-local | spatial | spectral
LRTA | $\bullet\bullet$ | $\bullet\bullet$ | | |
NAILRMA | | $\bullet\bullet$ | | |
TDL | | $\bullet$ | $\bullet\bullet\bullet$ | |
FastHyDe | | $\bullet\bullet$ | $\bullet$ | |
NGmeet | | $\bullet\bullet$ | $\bullet\bullet$ | |
LRMR | | $\bullet\bullet$ | | |
LRTV | | $\bullet\bullet$ | | $\bullet\bullet$ |
LRTDTV | $\bullet$ | $\bullet\bullet$ | | $\bullet\bullet$ | $\bullet\bullet$
LRTDGS | $\bullet$ | $\bullet\bullet\bullet$ | | $\bullet\bullet\bullet$ |
LRTF-FS | | $\bullet\bullet\bullet$ | | $\bullet\bullet\bullet$ | $\bullet\bullet$
TABLE II: The restoration results of different selected methods on Gaussian noise and mixed noised, respectively. Index | Gaussian noise removal | Mixed noise removal
---|---|---
LRTA | NAILRMA | TDL | FastHyDe | NGmeet | LRMR | LRTV | LRTDTV | LRTDGS | LRTF-FS
PSNR | 25.99 | 32.81 | 32.11 | 33.51 | 33.82 | 32.22 | 33.05 | 32.34 | 33.33 | 33.26
SSIM | 0.7095 | 0.9519 | 0.9443 | 0.9601 | 0.9607 | 0.9401 | 0.9459 | 0.9335 | 0.9596 | 0.9549
MSA | 11.35 | 4.75 | 4.72 | 4.42 | 4.38 | 4.95 | 5.06 | 4.09 | 4.24 | 4.44
### III-A Gaussian Noise Removal
In this section, five methods are selected to represent the state-of-the-art
HS image Gaussian noise removal approaches. These methods utilize different
low-rank matrix/tensor decomposition models to exploit the spatial, spectral,
or non-local low-rank properties of the original clean HS image. The
properties of these five methods are summarized in Table I. The five methods
are briefly described in the following.
#### III-A1 LRTA
111https://www.sandia.gov/tgkolda/TensorToolbox/
On the basis of the observation model (3) with ignoring sparse noise $S$, low-
rank tensor approximation (LRMA) [39] tries to restore the HS image from the
following objective function
$\min_{\boldsymbol{\mathcal{X}}}\|\boldsymbol{\mathcal{Y}}-\boldsymbol{\mathcal{X}}\|_{\operatorname{F}}^{2}\;\;{\rm
s.t.}\;\boldsymbol{\mathcal{X}}=\boldsymbol{\mathcal{O}}\times_{1}\mathbf{A}\times_{2}\mathbf{B}\times_{3}\mathbf{C},$
(4)
where
$\boldsymbol{\mathcal{X}}=\boldsymbol{\mathcal{O}}\times_{1}\mathbf{A}\times_{2}\mathbf{B}\times_{3}\mathbf{C},$
is the Tucker decomposition,
$\boldsymbol{\mathcal{O}}\in\mathbb{R}^{r_{1}\times r_{2}\times r_{3}}$ stands
for the core tensor, and $\mathbf{A}\in\mathbb{R}^{M\times
r_{1}},\mathbf{B}\in\mathbb{R}^{N\times
r_{2}},\mathbf{C}\in\mathbb{R}^{B\times r_{3}}$ are the factors related to
different dimensions. With the rank $(r_{1},r_{2},r_{3})$ of Tucker
decomposition set in advance, the LRTA model (4) can simultaneously capture
the global spatial and spectral low-rank properties. (4) provides a simple
general model for different kinds of low-rank matrix/tensor decomposition
based HS image denoising methods, that’s to say, we can change the Tucker
decomposition constraint of $\boldsymbol{\mathcal{X}}$ to different kinds of
matrix/tensor decomposition, such as canonical polyadic (CP) decomposition,
tensor train decomposition, tensor ring decomposition and so on.
#### III-A2 NAILRMA
222https://sites.google.com/site/rshewei/home
Noise-adjusted iterative LRMA (NAILRMA) [30] method assumes that the spectral
low-rank property is more important than that of spatial ones, and therefore
simply spectral low-rank regularizer is utilized to restrict the original
spectral image $\boldsymbol{\mathcal{X}}$. From other works [40], it also
indicates that the spatial TV regularizer is more important than that of the
spatial low-rank regularizer. In HS images, the similar signatures
representing the same class also appear in the nearby spatial location. To
enhance the spectral low-rank property, NAILRMA segmented the HS image into
spatial overlapping patches and process each patch individually. The noise
intensity in different bands of the HS image is different, which is a big
challenge appearing in the HS image Gaussian noise removal, is mitigated by
the noise-adjusted iterative strategy [30]. At last, the randomized singular
value decomposition (RSVD) is utilized to solve the non-convex low-rank
approximation problem.
#### III-A3 TDL
333http://gr.xjtu.edu.cn/web/dymeng/
Tensor dictionary learning (TDL) combines the non-local regularization and
low-rank tensor approximation. The noisy HS image is firstly segmented into
spatial overlapping patches, and the similar patches are clustered together to
formulate a higher-order tensor. In this way, the non-local spatial
information is collected. Then the higher-order tensors are denoised as the
same of (4), and finally the denoised higher-order tensors are utilized to
formulate the final denoised HS image. TDL represents the first method to
exploit non-local low-rank property, and the subsequent methods LLRT [38], KBR
[32], and NLTR [34] also achieve remarkable HS image Gaussian noise removal
results.
#### III-A4 FastHyDe
444https://github.com/LinaZhuang/FastHyDe_FastHyIn
The main difference between HS images and color/multispectral images is the
number of spectral bands. To eliminate this difference and utilize well
developed color/multispectral denoising methods for HS image denoising, Zhuang
et.al proposed the fast HS denoising (FastHyDe) [41] method by translating the
HS image to low-dimensional reduced image via SVD. By this translation,
various state-of-the-art color/multispectral image denoising methods, such as
wavelets [42] and BM3D [41] are used to denoise the reduced image. Finally,
the denoised reduced image is translated back to the denoised HS image via
inverse SVD. Generally speaking, under the framework of FastHyDe, the HS image
noise removal task is linked to the development of color/multispectral image
noise removal tasks.
#### III-A5 NGmeet
555https://github.com/quanmingyao/NGMeet
Spatial non-local low-rank regularizer can produce state-of-the-art HS image
noise removal performance. However, as the increase of spectral number, the
time cost of non-local related methods also increase incredibly [38, 32]. Non-
local meets global (NGmeet) method also tries to translate the HS image to the
reduced image, and utilizes a non-local low-rank method to denoise the reduced
image. Different from FastHyDe, NGmeet tries to perfect the framework by
iteratively eliminating the error caused by SVD on the noisy HS image, and
automatically estimating the spectral rank of the reduced image.
### III-B Mixed Noise Removal
In this section, we select five representative methods for the HS image mixed
noise removal. These methods are on the basis of the observation model (3). We
focus on the non-convex low-rank regularizer of original image
$\boldsymbol{\mathcal{X}}$. The properties of these five methods are
summarized in Table I.
#### III-B1 LRMR
Zhang et al. firstly introduced the observation model (3) to analysis of
complex HS noise [21]. LRMR tries to restore the original clean image
$\boldsymbol{\mathcal{X}}$ from the noisy image via low-rank and sparse
decomposition model as follows:
$\min_{\mathbf{X}}\lVert\mathbf{Y}-\mathbf{X}-\mathbf{N}_{S}\rVert_{\operatorname{F}}^{2}+\lambda_{1}rank(\mathbf{X})+\lambda_{2}card(\mathbf{N}_{S}),$
(5)
where $\mathbf{Y},\mathbf{X},\mathbf{N}_{S}$ are the reshaped matrices of
$\boldsymbol{\mathcal{Y}},\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{N}}_{S}$
along the spectral dimension, respectively, $\lambda_{1}$ and $\lambda_{2}$
are the parameters to trade-off the contributions of $rank(\mathbf{X})$ and
non-zero elements $card(\mathbf{N}_{S})$. LRMR utilizes “GoDec” algorithm [43]
to alternatively update non-convex constraint $\mathbf{X}$ and
$\mathbf{N}_{S}$, and finally obtains the restored image. To improve the
efficiency of the optimization to (5), several non-convex substitutions, such
as reweighted nuclear norm [31], $\gamma$-norm [44], smooth rank approximation
[45] and normalized $\epsilon$-Penalty [46], are further developed to exploit
the spectral low-rank property of $\mathbf{X}$.
#### III-B2 LRTV
Low-rank total variation (LRTV) claimed that spectral low-rank property is not
enough to describe the property and HS images, and therein introduced TV to
explore the spatial smoothness via TV. Generally, low-rank regularization and
TV are the two most studied regularizers, and the combination of them to
produce the state-of-the-art HS image mixed noise removal is becoming popular.
Most of the following works try to either improve the low-rank modeling [47,
48] or the smoothness modeling [49, 33, 50] of the HS image to further improve
the restoration accuracy. To further combine low-rank and TV, the low-rank
exploration of the HS difference image is also developed [51, 52].
#### III-B3 LRTDTV
Low-rank tensor decomposition with TV (LRTDTV) [48] tries to improve LRTV by
utilizing low-rank tensor decomposition to exploit the low-rank property of HS
images, meanwhile spatial-spectral TV (SSTV) to explore the spatial and
spectral smoothness simultaneously. Although LRTDTV achieved better mixed
noise removal results as reported in [48], the spatial rank utilized in LRTDTV
is much larger than that of spectral rank. This is mainly because the spatial
low-rank property of HS images is not so important compared to the spectral
low-rank property. From another side, the spatial non-local low-rank
regularization is proved to be more efficient [40] than spatial low-rank
property for HS restoration problem.
#### III-B4 LRTDGS
666https://chenyong1993.github.io/yongchen.github.io/
Low-rank tensor decomposition with group sparse regularization (LRTDGS) [33]
also utilizes low-rank rank tensor decomposition to exploit the low-rank
property of HS images. Differently, LRTDGS explores the group sparsity of the
difference image instead of SSTV in LRTDTV. From the mathematical modeling,
LRTDGS utilizes weighted $ell_{2,1}$ norm regularization to fulfill the row-
group sparsity of the difference image.
#### III-B5 LRTF-FR
777https://yubangzheng.github.io/homepage/
Following the idea of NGmeet [29], factor-regularized low-rank tensor
factorization (LRTF-FR) [53] also utilizes matrix decomposition to decouple
the spatial and spectral priors. From one side, the spectral signatures of the
HS image are assumed to be of smooth structure. From another side, the reduced
image is assumed to have a group sparse structure in the difference domain.
The optimization model of LRTF-FR is
$\displaystyle\min_{\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{N}}_{S}}$
$\displaystyle\lVert\boldsymbol{\mathcal{Y}}-\boldsymbol{\mathcal{X}}-\boldsymbol{\mathcal{N}}_{S}\rVert_{\operatorname{F}}^{2}+\lambda_{1}\lVert\boldsymbol{\mathcal{X}}\times_{3}\mathbf{H}_{3}\rVert_{2,1}$
(6)
$\displaystyle+\lambda_{2}\sum_{k=1}^{2}\lVert\boldsymbol{\mathcal{X}}\times_{k}\mathbf{H}_{k}\rVert_{\operatorname{F}}^{2}+\lambda_{3}\lVert\boldsymbol{\mathcal{N}}_{S}\rVert_{1,1},$
where $\mathbf{H}_{k},k=1,2,3$ are the first-order difference matrices.
Furthermore, in the optimization to (6), the reweighted strategy is utilized
to update $\ell_{2,1}$ norm and $L\ell_{1}$ norm to further improve the
restored results.
### III-C Experimental Study
We choose the HS image from DLR Earth Sensing Imaging Spectrometer (DESIS)
installed on the International Space Station (ISS)[54] for the experimental
study to compare different methods on the Gaussian and mixed noise removal
tasks. We remove the noisy bands and select a sub-image of size $400\times
400\times 200$ as the clean reference image, which is normalized to $[0,1]$.
Firstly, we add Gaussian noise of noise variance $0.1569$ to simulate the
Gaussian noisy image, and apply different Gaussian noise removal methods to
remove the Gaussian noise. Furthermore, we add salt & pepper noise and stripes
to simulate the mixed noisy image, and apply mixed noise removal methods to
remove the mixed noise. As similar in [33], we choose the mean of peak signal-
to-noise rate (PSNR) over all bands, the mean of structural similarity (SSIM)
over all bands, and the mean of spectral angle mapping (MSA) overall spectral
vectors to evaluate the restored results.
Figure 5: The illustration of different methods on the noise removal results.
The first row shows the results of different methods on the Guassian noise
remove experiments (R:70, G:100, B:36). The second row displays the results of
different methods on the mixed noise remove experiments (R:31, G:80, B:7).
Table II presents the evaluation values of different methods on Gaussian noise
and mixed noise removal results, respectively. For the Gaussian noise removal
task, NGmeet achieves the best values of three evaluation indices. However,
the gap between NGmeet and FastHyDe is limited. For the mixed noise removal
task, LRTDGS achieves the best accuracy in PSNR and SSIM values, meanwhile
LRTDTV achieved the best MSA value. Combining Tables I and II, we can conclude
that, firstly, the spectral low-rank prior information is important for HS
restoration. Secondly, the contribution of spatial low-rank prior information
for HS restoration is limited. Thirdly, on the basis of spectral low-rank
regularization, spatial and spectral smoothness prior can further improve the
final HS restoration results.
### III-D Remaining Challenges
Up to date, many non-convex regularized methods have been proposed to develop
the low-rank priors and local smoothness priors, and achieve remarkable HS
restoration results for Gaussian and mixed noise removal. However, these
methods still face several challenges for further work. We summary these
challenges as the following.
* •
Efficiency. Although low-rank related methods have achieved state-of-the-art
restoration results, they are time-consuming. For instance, NGmeet and LRTVGS
speed more than 10 minutes to process the HS image of size $400\times
400\times 200$. Furthermore, the related state-of-the-art restoration methods
always exploit multiple priors of the HS image, resulting in the confusion of
the parameter chosen. Therein, how to reduce the model complexity and improve
the optimization efficiency of the HS image restoration is the key challenge.
* •
Scalability. Previous non-convex related methods always focus on the small HS
image processing. However, HS images are used to observe the earth and the
spatial size of one scene is usually very large. How to improve the
scalability of the restoration approaches is the key challenge. DL provides
the possibility for fast and large scale processing of HS images. Whereas DL
approaches always rely on the quality of training samples, and the applicable
scope of the test data is always limited. To improve the scalability, how to
embed well studied non-convex regularizers to the DL architecture should also
be further analyzed.
* •
Real Application. Until now, most HS image restoration methods are evaluated
on the simulated experiments. However, in most cases, the evaluation indices
fail to predict the accuracy of the real HI image restoration results. From
another side, the noise distribution in the real noisy HS images is complex.
How to testify the related methods on the real HS images should be also
further analyzed. From another side, the training samples in the real
application are always limited. The blind and unsupervised approaches will
become the mainstream of future real HS image restoration.
## IV Dimensionality Reduction
The HS dimensionality reduction (DR) and feature extraction have long been a
fundamental but challenging research topic in HS RS [55, 56]. The main reasons
mainly lie in the following aspects. Due to the highly-correlated
characteristic between spectral bands, the HS images are subjected to
information redundancy, which could hurt the ability to discriminate the
materials under the certain extremely-conditioned cases (curse of
dimensionality). Plus, as the HS dimension gradually increases along with the
spectral domain, large storage capability and high-performance computing are
needed. Furthermore, these dimension-reduced features are usually applied for
the high-level classification or detection task [57, 58]. Recently, many works
based on non-convex modeling have shown to be effective for automatically
extracting dimension-reduced features of HS images. Linking with Eq. (2), the
DR task can be generalized to the following optimization problem:
$\displaystyle\mathop{\min}_{f_{\Phi},\mathbf{X}}\frac{1}{2}\lVert
f_{\Phi}(\mathbf{Y})-\mathbf{X}\rVert_{\operatorname{F}}^{2}\;\;{\rm
s.t.}\;\mathbf{X},f_{\Phi}\in\mathcal{C},$ (7)
where $f_{\Phi}(\bullet)$ denotes the transformation from the original HS
space to dimension-reduced subspaces with the respect to the variable set
$\Phi$, and $\mathbf{X}$ is the low-dimensional representations of
$\mathbf{Y}$. Revolving around the general form in Eq. (7), we review
currently advanced DR methods from three different aspects: unsupervised,
supervised, and semi-supervised models.
Figure 6: An illustration for supervised DR models in HS images with two
different groups: discriminant analysis based DR and regression-induced DR.
### IV-A Unsupervised Model
Non-negative matrix factorization (NMF) [59] is a common unsupervised learning
tool, which has been widely applied in HS DR. These works can be well
explained by Eq. (7), the NMF-based DR problem can be then formulated as
$\displaystyle\mathop{\min}_{\mathbf{Q}\geq\mathbf{0},\mathbf{X}\geq\mathbf{0}}\frac{1}{2}\lVert\mathbf{Y}-\mathbf{X}\mathbf{Q}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{X})+\Omega(\mathbf{Q}),$
(8)
where $\mathbf{Q}$ denotes the combination coefficients, $\Phi(\mathbf{X})$
and $\Omega(\mathbf{Q})$ are the potential regularization terms for the
variables $\mathbf{X}$ and $\mathbf{Q}$, respectively. Until the current,
there have been some advanced NMF-based works in HS DR. Gillis et al. [60]
used sparse NMF under approximations for HS data analysis. Yan et al. [61]
proposed a graph-regularized orthogonal NMF (GONMF) model with the application
to spatial-spectral DR of HS images. Wen et al. [62] further extended the
GONMF with combining multiple features for HS DR. Rasti et al. [63] designed
an orthogonal total variation component analysis (OTVCA) approach for HS
feature extraction. Moreover, the HS data are directly regarded as a high-
dimensional tensor structure in [64], where the low-rank attribute is fully
considered in the process of low-dimensional embedding. In detail, we
summarize the regularization and constraints of the above methods as follows:
* •
Sparsity [60]: $\Omega(\mathbf{Q})=\lVert\mathbf{Q}\rVert_{0}$;
* •
Graph Regularization [61]:
$\Psi(\mathbf{X})=\operatorname{tr}(\mathbf{X}\mathbf{L}\mathbf{X}^{\top}),\;\rm{s.t.}\;\mathbf{X}\mathbf{X}^{\top}=\mathbf{I}$;
* •
Multi-graph Regularization [62]:
$\Psi(\mathbf{X})=\sum_{i=1}^{s}\operatorname{tr}(\mathbf{X}\mathbf{L}^{i}\mathbf{X}^{\top}),\;\rm{s.t.}\;\mathbf{X}\mathbf{X}^{\top}=\mathbf{I}$;
* •
Total Variation [63]:
$\Psi(\mathbf{X})=\sum_{i=1}^{r}\lVert\sqrt{(\mathbf{H}_{h}\mathbf{X}_{i})^{2}+(\mathbf{H}_{v}\mathbf{X}_{i})^{2}}\rVert_{1},\\\
\rm{s.t.}\;\mathbf{Q}\mathbf{Q}^{\top}=\mathbf{I}$;
* •
Low-rank Graph [64]:
$\Psi(\mathbf{X})=\lVert\mathbf{X}\rVert_{*}+\operatorname{tr}(\mathbf{X}\mathbf{L}\mathbf{X}^{\top})$.
$\mathbf{L}=\mathbf{D}-\mathbf{W}$ is the Laplacian matrix, where
$\mathbf{D}_{i,i}=\sum_{j}\mathbf{W}_{i,j}$ is the degree matrix and
$\mathbf{W}$ is the graph (or manifold) structure of $\mathbf{X}$ [65].
$\lVert\bullet\rVert_{0}$, $\lVert\bullet\rVert_{2,1}$, and
$\lVert\bullet\rVert_{*}$ denote the $\ell_{0}$-norm [66], $\ell_{2,1}$-norm
[67], and nuclear norm [68], respectively.
Another type of unsupervised DR approaches is graph embedding, also known as
manifold learning, which can be also grouped into Eq. (7) well (according to
[69]):
$\displaystyle\mathop{\min}_{\mathbf{U},\mathbf{X}}\frac{1}{2}\lVert\mathbf{X}-\mathbf{U}\mathbf{Y}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{X})+\Omega(\mathbf{U})\;{\rm
s.t.}\;\mathbf{X}\mathbf{X}^{\top}=\mathbf{I},$ (9)
where $\mathbf{U}$ denotes the to-be-estimated projection matrix that bridges
the high-dimensional data $\mathbf{Y}$ with the low-dimensional embedding
$\mathbf{X}$. The regularization term for the variable $\mathbf{U}$ can be
usually expressed as
$\displaystyle\Omega(\mathbf{U})=\operatorname{tr}(\mathbf{U}\mathbf{Y}\mathbf{L}\mathbf{Y}^{\top}\mathbf{U}^{\top})+\lVert\mathbf{U}\rVert_{\operatorname{F}}^{2},$
(10)
while the regularizer with respect to $\mathbf{X}$ can be given by
$\displaystyle\Psi(\mathbf{X})=\operatorname{tr}(\mathbf{X}\mathbf{L}\mathbf{X}^{\top}).$
(11)
The main difference between these manifold learning-based DR approaches lies
in the graph construction, i.e., $\mathbf{W}$. Ma et al. [70] integrated the
KNN classifier with several representative manifold learning algorithms, i.e.,
locally linear embedding [71], Laplacian eigenmaps [65], and local tangent
space alignment [72], for HS image classification. Huang et al. [73] embedded
the sparse graph structure, which is performed by solving a $\ell_{1}$-norm
optimization problem, for the DR of HS images. He et al. [74] extended the
work of [73] by generating a weighted sparse graph. Hong et al. [75] developed
a new spatial-spectral graph for the DR of HS images, called RLMR, by jointly
taking neighbouring pixels of a target pixel in spatial and spectral domains
into account. An et al. [76] attempted to learn the low-dimensional tensorized
HS representations on a sparse and low-rank graph. To sum up, core graphs of
the aforementioned methods can be obtained by
* •
Sparse Graph [73]:
$\min_{\mathbf{W}}\lVert\mathbf{W}\rVert_{1,1},\;\rm{s.t.}\;\mathbf{Y}=\mathbf{Y}\mathbf{W}$;
* •
Weighted Sparse Graph [74]:
$\min_{\mathbf{W}}\lVert\mathbf{d}\odot\mathbf{W}\rVert_{1,1},\;\rm{s.t.}\;\mathbf{Y}=\mathbf{Y}\mathbf{W},$
where $\mathbf{d}$ denotes a weighted matrix on $\mathbf{W}$ and $\odot$ is
the element-wise multiplication operator;
* •
Spatial-spectral Graph [75]:
$\mathop{\min}_{\mathbf{w}_{i,0}}\sum_{j\in\phi_{i}^{spa}}\lVert\mathbf{y}_{i,j}-\sum_{k\in\phi_{i}^{spe}}\mathbf{y}_{i,k}w_{i,k,j}\rVert_{2}^{2}\\\
{\rm
s.t.}\;\lVert\sum_{k\in\phi_{i}^{spe}}\mathbf{y}_{i,k}(4w_{i,k,0}-\sum_{k=1}^{4}w_{i,k,j})\rVert_{2}^{2}\leq\eta,\\\
\qquad\mathbf{w}_{i,j}^{\operatorname{T}}\mathbf{w}_{i,j}=1,$
where $\phi_{i}^{spa}$ and $\phi_{i}^{spe}$ denote the neighbouring pixels in
spatial and spectral spaces, respectively;
* •
Sparse and Low-rank Graph [76]:
$\min_{\mathbf{W}}\lVert\mathbf{W}\rVert_{1,1}+\lVert\mathbf{W}\rVert_{*},\;\rm{s.t.}\;\mathbf{Y}=\mathbf{Y}\mathbf{W}$.
### IV-B Supervised Model
Unlike unsupervised DR that relies on embedding various priors to reduce the
dimension of HS data, supervised models are capable of better learning class-
separable low-dimensional representations via the use of label information.
The supervised DR models can be described from two different categories in
this subsection, as shown in Fig. 6. A typical group is the discriminant
analysis [55] closely related to graph embedding and manifold learning.
Intuitively speaking, these methods belong to a special case of unsupervised
graph embedding, which means they can be well explained by Eq. (9). The main
difference lies in that the labels are used for constructing the graph
structure, i.e., $\mathbf{W}$, thereby yielding a more discriminative
subspace.
In the supervised DR, a direct graph structure is written as
${\bf W}_{ij}=\begin{cases}\begin{aligned} 1,\;\;&\text{if ${\bf y}_{i}$ and
${\bf y}_{j}\in C_{k}$;}\\\ 0,\;\;&\text{otherwise,}\end{aligned}\end{cases}$
(12)
where $C_{k}$ means the sample set of the $k$-th class. Furthermore, more
advanced supervised graphs have been developed to better represent the HS data
in a low-dimensional subspace, such as sparse graph discriminant analysis
[77], collaborative graph discriminant analysis [78], feature space
discriminant analysis (FSDA) [79], spatial-spectral local discriminant
embedding [80]. These approaches sought to construct a soft graph instead of a
hard graph in Eq. (12). That is, the graph is built by using radial basis
function (RBF) to measure the similarity between samples belonging to the same
class [81]:
${\bf W}_{ij}=\begin{cases}\begin{aligned}
\exp\frac{-\lVert\mathbf{y}_{i}-\mathbf{y}_{j}\rVert_{2}^{2}}{2\sigma^{2}},\;\;&\text{if
${\bf y}_{i}$ and ${\bf y}_{j}\in C_{k}$;}\\\
0,\;\;&\text{otherwise,}\end{aligned}\end{cases}$ (13)
or by solving $\ell_{1}$-norm or $\ell_{2}$-norm optimization functions in the
same class set, e.g., [77], [78].
The DR behavior can be also modeled from a regression perspective by directly
connecting samples and labels [82], which provides a new insight into the
research of the supervised HS DR. A general form for the regression-based
supervised DR model can be formulated as
$\displaystyle\mathop{\min}_{\mathbf{P},\mathbf{U}}$
$\displaystyle\frac{1}{2}\lVert\mathbf{M}-\mathbf{P}\mathbf{X}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{P})+\Omega(\mathbf{U})$
(14) $\displaystyle{\rm
s.t.}\;\mathbf{X}=\mathbf{U}\mathbf{Y},\;\mathbf{U}\mathbf{U}^{\top}=\mathbf{I},$
where the variable $\mathbf{P}$ denotes the regression coefficients or basis
signals, and $\mathbf{M}$ is the one-hot encoded matrix obtained by labels.
Eq. (14) can be, to some extent, regarded as an interpretable linearized
artificial neural network (ANN) mode (shallow network). Ji et al. [83] jointly
performed DR and classification, which is a good fit for Eq. (14) with
$\Psi(\mathbf{P})=\lVert\mathbf{P}\rVert_{\operatorname{F}}^{2}$. To enhance
the spectrally discriminative ability, Hong et al. [84] employed a LDA-like
graph on the basis of [83] to regularize the low-dimensional representations
in a Laplacian matrix form, i.e.,
$\Omega(\mathbf{U})=\operatorname{tr}(\mathbf{U}\mathbf{Y}\mathbf{L}\mathbf{Y}^{\top}\mathbf{U}^{\top})$.
In the same work [84], Hong et al. further extended their model to a deep
version, called JPlay, with a $k$-layered linear regression:
$\displaystyle\mathop{\min}_{\mathbf{P},\\{\mathbf{U}_{i}\\}_{i=1}^{k}}\frac{1}{2}\lVert\mathbf{M}-\mathbf{P}\mathbf{X}_{i}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{P})+\Omega(\\{\mathbf{U}_{i}\\}_{i=1}^{k})$
(15) $\displaystyle{\rm
s.t.}\;\mathbf{X}_{i}=\mathbf{U}_{i}\mathbf{X}_{i-1},\;\mathbf{X}_{1}=\mathbf{U}_{1}\mathbf{Y},\;\mathbf{X}_{i}\geq\mathbf{0},\;\lVert\mathbf{x}_{i}\rVert_{2}\leq
1,$
with $\Psi(\mathbf{P})=\lVert\mathbf{P}\rVert_{\operatorname{F}}^{2}$ and
$\displaystyle\Omega(\\{\mathbf{U}_{i}\\}_{i=1}^{k})$
$\displaystyle=\sum_{i=1}^{k}\operatorname{tr}(\mathbf{U}_{i}\mathbf{X}_{i-1}\mathbf{L}\mathbf{X}_{i-1}^{\top}\mathbf{U}_{i}^{\top})$
$\displaystyle+\sum_{i=1}^{k}\lVert\mathbf{X}_{i-1}-\mathbf{U}_{i}^{\top}\mathbf{U}_{i}\mathbf{X}_{i-1}\rVert_{\operatorname{F}}^{2}.$
The J-Play attempts to open the “black box” of deep networks in an explainable
way by multi-layered linearized modeling. With explicit mappings and
physically meaningful priors, the non-convex J-Play takes a big step towards
the interpretable AI model.
### IV-C Semi-supervised Model
Due to the fact that labeling samples is extremely expensive, particularly for
RS images covering a large geographic region, the joint use of labeled and
unlabeled information then becomes crucial in DR and classification.
A simple and feasible strategy for semi-supervised learning is to integrate
supervised and unsupervised techniques, e.g., LDA and locality preserving
projections [85]. By simultaneously constructing graphs of labeled and
unlabeled samples (e.g., using Eqs. (12) and (13), respectively), Eq. (9) can
be easily extended to a semi-supervised version, leading to semi-supervised
discriminant analysis (SSDA) [86]. Zhao et al. [87] further improved the SSDA
performance by using “soft” (or “pseudo”) labels predicted by label
propagation instead of directly using unsupervised similarities between
unlabeled samples. Similarly, Wu et al. [88] generated pseudo-labels using the
Dirichlet process mixing model and achieved a novel SSDA approach to learn the
low-dimensional HS embedding. These above-mentioned methods are performed
surrounding various hand-crafted graph structures ($\mathbf{W}$).
Figure 7: An example to clarify the graph structure of JPSA method, where
$\mathbf{W}^{p}$ and $\mathbf{W}^{sp}$ denote the pixel-wise and superpixel-
wise subgraphs as well as $\mathbf{W}^{a}$ is the aligned graph between pixels
and superpixels.
A different idea is to simulate the brain-like or human-like behaviors in the
semi-supervised DR task. It is well known that the feedback reward is a key
component that forms the intelligent information processing system. Inspired
by it, [89] developed an iterative multitask learning (IMR) framework by
adaptively learning the label propagation (LP) on graphs to simulate the
feedback mechanism, thereby achieving the HS DR process more effectively and
efficiently. The IMR is a semi-supervised extension of Eq. (14) with graph
learning, which can be generally modeled as
$\displaystyle\mathop{\min}_{\mathbf{P},\mathbf{U},\mathbf{L}}$
$\displaystyle\sum_{j=1}^{2}\lVert\mathbf{M}_{j}-\mathbf{P}\mathbf{U}\mathbf{Y}_{j}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{P})+\Omega(\mathbf{U})$
(16)
$\displaystyle\mathrm{s.t.}\;\mathbf{U}\mathbf{U}^{\top}=\mathbf{I},\;\mathbf{L}\in\mathcal{C},$
where $\mathbf{Y}_{1}$ and $\mathbf{Y}_{2}$ denote the labeled and unlabeled
samples from $\mathbf{Y}$, respectively.
$\Psi(\mathbf{P})=\lVert\mathbf{P}\rVert_{\operatorname{F}}^{2}$ and
$\Omega(\mathbf{U})=\operatorname{tr}(\mathbf{U}\mathbf{Y}\mathbf{L}\mathbf{Y}^{\top}\mathbf{U}^{\top})$.
The non-convex constraint $\mathcal{C}$ with the respect to the variable
$\mathbf{L}$ can be summarized as
$\displaystyle\mathcal{C}:=\\{\mathbf{L}=\mathbf{L}^{\top},\;\mathbf{L}_{p,q,p\neq
q}\preceq 0,\;\mathbf{L}_{p,q,p=q}\succeq
0,\;\operatorname{tr}(\mathbf{L})=c\\},$
where $c>0$ is a scaling constant. Eq. (16) is a typical data-driven graph
learning model, which is capable of automatically learning the graph structure
from the data without any hand-crafted priors. By using the iterative strategy
to simulate the feedback system, $\mathbf{M}_{2}^{(t+1)}$ in the
$t$$+$$1$-step can be updated by the graph-based LP on the learned graph of
the $t$-step $\mathbf{W}^{(t)}$:
$\displaystyle\cdots\cdots\mathbf{M}_{2}^{(t+1)}\leftarrow\mathbf{W}^{(t)}\leftarrow\mathbf{M}_{2}^{(t)}\cdots\cdots.$
(17)
Besides, another intelligent feature extraction algorithm, named JPSA, which
is extended from [84], was presented in [90] by the attempts to align pixels
and superpixels for spatial-spectral semi-supervised HS DR. JPSA basically
follows the JPlay framework and the major difference is the graph structure
$\mathbf{W}$. The graph in JPSA consists of not only pixel-wise and
superpixel-wise similarities but also aligned components between pixels and
superpixels. Fig. 7 gives an example to clarify the graph structure of JPSA.
Note that the JPSA’s graph can be seen as a full data-driven structure, which
can, to a great extent, self-express the intrinsic properties of HS data and
further achieves intelligent information extraction and DR.
TABLE III: Quantitative comparison of different DR algorithms in terms of OA,
AA, and $\kappa$ using the NN classifier on the Indian Pines dataset. The best
one is shown in bold.
Methods | dimension | OA (%) | AA (%) | $\kappa$
---|---|---|---|---
OSF | 220 | 65.89 | 75.71 | 0.6148
OTVCA [63] | 16 | 74.18 | 77.61 | 0.7228
RLMR [75] | 20 | 83.75 | 86.90 | 0.8147
FSDA [79] | 15 | 64.14 | 74.52 | 0.5964
JPlay [84] | 20 | 83.92 | 89.35 | 0.8169
IMR [89] | 20 | 82.80 | 86.27 | 0.8033
JPSA [90] | 20 | 92.98 | 95.40 | 0.9197
### IV-D Experimental Study
Classification is explored as a potential application to evaluate the
performance of state-of-the-art (SOTA) DR algorithms, including original
spectral features (OSF), OTVCA888https://github.com/danfenghong/HyFTech [63],
RLMR999https://github.com/danfenghong/IEEE_JSTARS_RLML [75], FSDA [79],
JPlay101010https://github.com/danfenghong/ECCV2018_J-Play [84], IMR [89], and
JPSA [90]. Experiments are performed on the Indian Pine data using the nearest
neighbor (NN) classifier in terms of three indices: Overall Accuracy (OA),
Average Accuracy (AA), and Kappa Coefficient ($\kappa$). The scene consists of
$145\times 145$ pixels and $220$ spectral bands ranging from $0.4\mu m$ to
$2.5\mu m$. More details regarding training and test samples can be found in
[91].
Table III lists the quantitative results of different DR methods. Overall, OSF
without feature extraction or DR yields the worst classification performance,
compared to those SOTA DR methods. This, to a great extend, demonstrates the
effectiveness and necessity of DR in the HS image classification task. It is
worth noting that the approaches with spatial-spectral modeling, e.g., OTVCA,
RLMR, JPSA, tend to obtain better classification results. The performance of
RLMR is superior to that of OTVCA, owing to the full consideration of the
neighboring information in a graph form rather than the smoothing operation
only modeled by the TV regularization. As a linearized deep model, supervised
JPlay obviously performs better than others, especially FSDA that is also a
supervised DR model. More importantly, the JPSA with a semi-supervised
learning strategy dramatically outperforms other competitors, since it can
jointly learn richer representations from both pixels and superpixels by means
of spatial-spectral manifold alignment and deep regressive regression.
### IV-E Remaining Challenges
Although extensive SOTA methods have recently shown the effectiveness and
superiority in the HS DR and classification, there is still a long way to go
towards the AI-guided intelligent information processing. We herein summarize
the potential remaining challenges briefly.
* •
Optimal Subspace Dimension. Subspace dimension is a crucial parameter in DR,
which is determined experimentally and empirically in most of existing
methods. Despite some parameter estimation algorithms, e.g., intrinsic
dimension [92], subspace identification [93], they fail to avoid the pre-
survey of various prior knowledge and human intervention in the dimension
estimation process.
* •
Effects of Noises. HS images usually suffer from noise degradation in remotely
sensed imaging. These noises are complex and closely associated with spectral
signatures. Therefore, separating noises from HS data effectively and reducing
the noise sensitivity (or preserving spectral discrimination) in the DR
process remains challenging.
* •
Robustness and Generalization. Robust estimation and advanced non-convex
regularizers have been widely applied to model the DR behavior, yet the
complex noise type, the limited training samples, and noisy labels hinder the
robustness and generalization ability to be further improved. For this reason,
more robust and intelligent models should be developed in either theory or
practice emphatically in the next generation DR technique.
Figure 8: A showcase in a real HS scene (Pavia City Centre) to have a quick
look at the 3-D HS cube, spectral signals, and material mixture as well as
pure pixels (i.e., endmember) and mixed pixels. In the studied scene, the pure
pixels correspond to two spectral reflectance curves of vegetation and water,
respectively, while the examples of mixed pixels explain the case of spectral
mixing, e.g., the two mixed pixels comprise of three pure components
(endmembers) in varying proportions. In addition, the figure located in the
right upper gives two toy examples to illustrate the material miscibility.
## V Spectral Unmixing
Spectral unmixing (SU) can be usually seen as a special case of blind source
separation (BSS) problem in ML, referring to a procedure that decomposes the
observed pixel spectrum of the HS image into a series of constituent spectral
signals (or endmembers) of pure materials and a set of corresponding abundance
fractions (or abundance maps) [94]. Due to the meter-level ground sampling
distance (GSD) of HS imaging, the spectral signatures for most pixels in HS
images are acquired in the form of a complex mixture that consists of at least
two types of materials. Fig. 8 gives a showcase to visualize the HS cube,
spectral signatures, and material mixing process as well as pure and mixed
pixels. Different from general signals, e.g., digital signals, speech signals,
there are specific absorption properties in the spectrum signals of different
materials. Plus, HS images suffer from miscellaneous unknown degradation,
either physically or chemically, in the remotely sensed imaging, inevitably
bringing many uncertainties in SU. Therefore, SU plays a unique role in HS RS,
yielding many challenging researchable tasks compared to BSS in ML.
Ideally, a linear mixing model (LMM) can be used to accurately describe the SU
process [95], which is modeled as the following constrained optimization
problem:
$\displaystyle\mathop{\min}_{\mathbf{E},\mathbf{A}}\frac{1}{2}\lVert\mathbf{Y}-\mathbf{E}\mathbf{A}\rVert_{\operatorname{F}}^{2}\;\;\rm{s.t.}\;\mathbf{E},\mathbf{A}\in\mathcal{C}.$
(18)
The variables $\mathbf{E}$ and $\mathbf{A}$ in Eq. (18) stand for the
endmembers and abundance maps in the SU issue, respectively. According to the
endmembers ($\mathbf{E}$) that are available (given) or not in the process of
SU, existing SU models can be loosely divided into blind SU and endmember-
guided SU.
### V-A Blind Spectral Unmixing
NMF is a baseline model in a wide range of applications, and the same is true
in SU. Up to the present, NMF-based interpretable models have been developed
extensively for pursing the intelligent SU with the consideration of
physically meaningful priors with respect to $\mathbf{E}$ and $\mathbf{A}$,
e.g., the abundance non-negative constraint (ANC), the abundance sum-to-one
constraint (ASC). The resulting basic blind SU model can be written as
$\displaystyle\mathop{\min}_{\mathbf{E},\mathbf{A}}\frac{1}{2}\lVert\mathbf{Y}-\mathbf{E}\mathbf{A}\rVert_{\operatorname{F}}^{2}+\Phi(\mathbf{E})+\Omega(\mathbf{A})\;\;\rm{s.t.}\;\mathbf{E},\mathbf{A}\in\mathcal{C},$
(19)
where the constraint $\mathcal{C}$ is
$\displaystyle\mathcal{C}:=\\{\mathbf{E}\geq\mathbf{0},\;\mathbf{A}\geq\mathbf{0},\;\mathbf{1}^{\top}\mathbf{A}=\mathbf{1}\\}.$
On the basis of the model (19), Yang et al. [96] proposed sparse NMF for SU
with a well-designed S-measure sparseness. Qian et al. [97] imposed the
sparsity constraint on abundances and used $\ell_{1/2}$-regularized NMF for
blind SU, which has shown to be more effective than $\ell_{0}$\- and
$\ell_{1}$-norm terms. In [98], Sigurdsson et al. relaxed $\ell_{1/2}$-norm to
$\ell_{q}$-norm ($0\leq q\leq 1$) for a better estimation of abundances.
Thouvenin et al. [99] developed an improved LMM, called perturbed LMM (PLMM),
by the attempts to model spectral variabilities as perturbed information that
simply meets the Gaussian distribution. A similar work is presented in [100],
where the scaling factor, as a major spectral variability (SV), is modeled
into LMM to yield an extended LMM (ELMM) for the blind SU task. He et al.
[101] employed total variation (TV) and weighted $\ell_{1}$-norm terms to
further enhance the smoothness and sparseness of abundances. Yao et al. [102]
sought to explain the NMF-based SU model by simulating human observations on
HS images, e.g., sparsity, non-local, smooth properties, in a non-convex
modeling fashion. Another type of interesting SU strategy is to embed the
graph or topological structure of the HS data. The local neighboring relation
is introduced into the NMF model, showing robust SU results [103]. Similarly,
Lu et al. [104] enforced the abundances to follow the manifold structure of
spectral signatures in the form of Laplacian regularization form for HS
unmixing. Wang et al. [105] used a structuralized hypergraph regularization in
sparse NMF to better depict the underlying manifolds of the HS data. Very
recently, Qin et al. [106] proposed a novel graph TV regularization to
estimate endmembers and abundances more effectively. There are still other
variants that directly unmix the 3-D HS tensor by preserving spatial structure
information as much as possible. For that, Qian et al. [107] proposed a
matrix-vector non-negative tensor factorization framework for blind SU.
Imbiriba et al. [108] modeled the low-rank properties in the HS tensor to
address the SV for robust SU. A further modified work based on [108] is
proposed via weighted non-local low-rank tensor decomposition for sparse HS
unmixing.
Broadly, these key non-convex priors of the above models can be briefly
summarized as follows:
* •
$\ell_{1/2}$-NMF [97]:
$\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{1/2}=\sum_{k,n=1}^{K,N}\mathbf{a}_{n}(k)^{1/2}$;
* •
$\ell_{q}$-NMF [98]:
$\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{q}=\sum_{k,n=1}^{K,N}\mathbf{a}_{n}(k)^{q}$;
* •
PLMM [99]:
$\Phi(\mathbf{E})=\frac{1}{2}\lVert\mathbf{E}-\mathbf{E}_{0}\rVert_{\operatorname{F}}^{2}$,
$\Omega(\mathbf{A})=\frac{1}{2}\lVert\mathbf{A}\mathbf{H}\rVert_{\operatorname{F}}^{2}$,
$\Psi(\Delta)=\frac{1}{2}\sum_{n=1}^{N}\lVert\Delta_{n}\rVert_{\operatorname{F}}^{2}$,
where $\mathbf{E}_{0}$, $\mathbf{H}$, and $\Delta$ denote the reference
endmembers, the matrix differences in spatial four nearest neighbors, and
pixel-wise perturbed information, respectively.
* •
ELMM [100]:
$\Phi(\mathbf{E})=\sum_{n=1}^{N}\lVert\mathbf{E}_{n}-\mathbf{E}_{0}\mathbf{S}_{n}\rVert_{\operatorname{F}}^{2}$,
$\Omega(\mathbf{A})=\lVert\mathbf{H}_{h}(\mathbf{A})\rVert_{2,1}+\lVert\mathbf{H}_{v}(\mathbf{A})\rVert_{2,1}$,
$\Psi(\mathbf{S})=\lVert\mathbf{H}_{h}(\mathbf{S})\rVert_{\operatorname{F}}^{2}+\lVert\mathbf{H}_{v}(\mathbf{S})\rVert_{\operatorname{F}}^{2}$,
where $\mathbf{H}_{h}$ and $\mathbf{H}_{v}$ are the horizontal and vertical
gradients;
* •
TV-RSNMF [101]:
$\Omega(\mathbf{A})=\lVert\mathbf{d}\odot\mathbf{A}\rVert_{1,1}+\lVert\mathbf{A}\rVert_{\rm
TV}$;
* •
NLHTV [102]:
$\Omega(\mathbf{A})=\sum_{n=1}^{N}\lVert
J_{w}\mathbf{a}_{n}\rVert_{\mathcal{S}_{1}}+\sum_{i,j}\log(|x_{i,j}|+\epsilon)$,
where $J_{w}$ and $\lVert\bullet\rVert_{\mathcal{S}_{1}}$ are defined as non-
local Jacobian operator and the Schatten-1 norm, respectively.
* •
Graph $\ell_{1/2}$-NMF [104]:
$\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{1/2}+\operatorname{tr}(\mathbf{A}\mathbf{L}\mathbf{A}^{\top})$;
* •
Graph TV [106]: $\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{\rm
TV}+\operatorname{tr}(\mathbf{A}\mathbf{L}\mathbf{A}^{\top})$.
Owing to the powerful data fitting ability, DL-based SU approaches have
recently been paid increasing attention and achieved better unmixing results
[109, 110, 111, 112]. Although these methods still suffer from the effects of
“black box”, i.e., the lack of model interpretability, yet their performances
have preliminary shown the effectiveness and feasibility in unmixing the HS
data more accurately.
### V-B Endmember-Guided Spectral Unmixing
A mass of blind SU methods has been developed and shown to be effective to
simultaneously obtain endmembers and abundance maps. However, these blind
methods tend to extract physical meaningless endmembers, e.g., noisy signals,
spectral signatures corresponding to non-existent materials, due to the lack
of certain interpretable model guidance or prior knowledge. A straightforward
solution is to provide nearly real endmembers extracted from HS images. This
naturally leads to the researches on endmember-guided SU. As the name
suggests, the SU process is performed with given reference endmembers or the
guidance of extracted endmembers from the HS image. That is, the endmembers
$\mathbf{E}$ in Eq. (19) are known. Accordingly, the endmember-guided SU can
be implemented in a three-stage way.
* •
Firstly, the number of endmembers can be estimated by subspace estimation
algorithms, e.g., HySime [93];
* •
Secondly, the endmembers can be extracted based on geometric observations of
HS data structure. Several well-known methods are vertex component analysis
(VCA) [113], pixel purity index (PPI) [114], and fast autonomous endmember
extraction (N-FINDER) [115].
* •
Lastly, the abundances of materials are estimated using regression-based
methods, which can generally written as
$\displaystyle\mathop{\min}_{\mathbf{A}}\frac{1}{2}\lVert\mathbf{Y}-\mathbf{E}\mathbf{A}\rVert_{\operatorname{F}}^{2}+\Omega(\mathbf{A})\;\;{\rm
s.t.}\;\mathbf{A}\in\mathcal{C}.$ (20)
Following the three steps, many well-working non-convex models have been
successfully developed to estimate the abundance maps of different materials
at a more accurate level. Heinz et al. [116] thoroughly analyzed the spectral
mixture in the SU issue, yielding a fully constrained least-squares unmixing
(FCLSU) algorithm. Due to the hard ASC, the abundances can not be fully
represented in a simplex. For this reason, a partial constraint least-squares
unmixing (PCLSU) [117] model emerges as required without ASC. Bioucas-Dias et
al. [118] relaxed the strong $\ell_{0}$-norm to the solvable $\ell_{1}$-norm
in the sparse HS unmixing model and designed a fast and generic optimization
algorithm based on the ADMM framework [14], called sparse unmixing by variable
splitting and augmented Lagrangian (SUnSAL). In [119], a TV spatial
regularization is considered to further enhance the unmixing performance.
Iordache et al. [120] extended the sparse regression model to the
collaborative version regularized by $\ell_{2,1}$-norm for SU. Fu et al. [121]
proposed a semi-blind HS unmixing model by correcting the mismatches between
estimated endmembers and pure spectral signatures from the library. Huang et
al. [122] jointly imposed sparsity and low-rank properties on the abundances
for better estimating abundance maps. Hong et al. [123] devised an interesting
and effective subspace-based abundance estimation model. The model neatly
sidesteps to directly decompose the HS data in the complex high-dimensional
space instead of projecting the HS data into a more robust subspace, where the
SV tends to be removed in a more generalized way with low-rank attribute
embedding. Beyond the current framework, Hong et al. [124] further augmented
the basic LMM by fully modeling SVs, e.g., principal scaling factors and other
SVs that should be incoherent or low-coherent with endmembers, in order to
yield an interpretable and more intelligent SU model, called augmented LMM
(ALMM).
Figure 9: A visual example to clarify SVs in a real HS scene (Pavia City
Centre). An image patch cropped from the scene is select to show the spectral
bundles involving spectral variations of trees in (a). (b) shows a pure
spectral signature (i.e., endmember) of trees acquired from the laboratory.
(c) represents the differences between (a) and (b), which is seen as SVs.
Figure 10: Visualization of abundance maps estimated by different SOTA SU
algorithms on the Urban data, where SAM is computed to generate the
classification-like maps regarded as the GT to measure the shape similarity of
abundance maps obtained by different SU methods.
The non-convexity of these methods on priors, constraints, or modeling can be
summarized as follows:
* •
FCLSU [116]: $\mathbf{A}\geq\mathbf{0}$,
$\mathbf{1}^{\top}\mathbf{A}=\mathbf{1}$;
* •
PCLSU [117]: $\mathbf{A}\geq\mathbf{0}$;
* •
SUnSAL [118]: $\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{1,1}$,
$\mathbf{A}\geq\mathbf{0}$, $\mathbf{1}^{\top}\mathbf{A}=\mathbf{1}$;
* •
SUnSAL-TV [119]:
$\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{1,1}+\lVert\mathbf{A}\rVert_{\rm
TV}$, $\mathbf{A}\geq\mathbf{0}$;
* •
CSR [120]:
$\Omega(\mathbf{A})=\lVert\mathbf{a}\rVert_{2,1}=\sum_{n=1}^{N}\lVert\mathbf{a}_{n}\rVert_{2}$,
$\mathbf{A}\geq\mathbf{0}$;
* •
DANSER [121]:
$\Omega(\mathbf{A})=\sum_{n=1}^{N}(\lVert\mathbf{a}_{n}\rVert_{2}^{2}+\tau)^{p/2}$,
$\mathbf{A}\geq\mathbf{0}$,
$\Phi(\mathbf{E})=\lVert\mathbf{E}-\mathbf{E}_{0}\rVert_{\operatorname{F}}^{2}$;
* •
SULoRA [123]:
$\Psi(\mathbf{U})=\lVert\mathbf{Y}-\mathbf{U}\mathbf{Y}\rVert_{\operatorname{F}}^{2}+\lVert\mathbf{U}\rVert_{*}$,
$\Omega(\mathbf{A})=\lVert\mathbf{A}\rVert_{1,1}$, $\mathbf{A}\geq\mathbf{0}$,
where $\mathbf{U}$ denotes the subspace projection and
$\lVert\bullet\rVert_{*}$ is the nuclear norm that approximates the rank
property of the matrix $\bullet$;
* •
ALMM [124]: $\Phi(\mathbf{A})=\lVert\mathbf{A}\rVert_{1,1}$,
$\mathbf{A}\geq\mathbf{0}$,
$\Gamma(\mathbf{J})=\lVert\mathbf{J}\rVert_{\operatorname{F}}^{2}$,
$\Psi(\mathbf{V})=\lVert\mathbf{A}^{\top}\mathbf{V}\rVert_{\operatorname{F}}^{2}+\lVert\mathbf{V}^{\top}\mathbf{V}-\mathbf{I}\rVert_{\operatorname{F}}^{2}$,
where $\mathbf{V}$ and $\mathbf{J}$ denote the SV dictionary and corresponding
coefficients, respectively.
### V-C Experimental Study
A real urban HS data acquired by the HYDICE over the urban area, Texas, USA,
in 2015 (the latest version111111http://www.tec.army.mil/Hypercube) is used to
evaluate the performance of several selected SOTA unmixing methods
qualitatively, including $\ell_{1/2}$-NMF [97],
PLMM121212https://pthouvenin.github.io/unmixing-plmm/ [99],
ELMM131313https://openremotesensing.net/knowledgebase/spectral-variability-
and-extended-linear-mixing-model/ [100], NLHTV [102], FCLSU [116],
SUnSAL141414http://www.lx.it.pt/~bioucas/ [118],
SULoRA151515https://github.com/danfenghong/IEEE_JSTSP_SULoRA [123], and
ALMM161616https://github.com/danfenghong/ALMM_TIP [124]. The HS image consists
of $307\times 307$ pixels and 162 spectral bands after removing noisy bands in
the wavelength range of $0.4\mu m$ to $2.5\mu m$ at a $2m$ GSD. Moreover, four
main materials (or endmembers) are investigated in the studied scene, i.e.,
asphalt, grass, trees, and roof. Furthermore, HySime [93] and VCA [113]
algorithms are adopted to determine the number of endmembers and extract
endmembers from the HS image (as the initialization for blind SU methods) for
all compared algorithms, respectively.
Fig. 10 shows the visual comparison between different SOTA unmixing algorithms
in terms of abundance maps. Owing to the consideration of real endmembers
extracted from the HS scene, the last four endmember-guided SU methods perform
evidently better than the blind SU ones. ELMM models the scaling factors,
tending to better capture the distributions of different materials. The
embedding of non-local spatial information makes the NLHTV method obtain a
more similar shape of abundance maps to the GT, yielding comparable unmixing
performance with ELMM. Remarkably, the unmixing results with regard to
abundance maps of SULoRA and ALMM algorithms are superior to those of other
methods, since the SVs can be fully considered by robustly embedding low-rank
attributes in a latent subspace using SULoRA and characterizing complex real
scenes more finely using ALMM.
### V-D Remaining Challenges
SU has long been a challenging and widely concerned topic in HS RS. Over the
past decades, tons of SU works have been proposed by the attempts to unmix
these mixed spectral pixels more effectively. Yet, some key and essential
issues and challenges still remain to be solved.
* •
Benchmark Data. Unlike classification, recognition, and detection tasks, the
ground truth of material abundances is able to be hardly collected, due to the
immeasurability of abundance values in reality. On the other hand, spectral
signatures (i.e., endmembers) of pure materials are often acquired in the lab.
This usually leads to uncertain mismatches between real endmembers and lab
ones. It turns to be urgent to establish benchmark datasets for SU by drawing
support from more advanced imaging techniques or developing interpretable
ground truth generation models or processing chain.
* •
Evaluation Criteria. Reconstruction errors (RE) or spectral angle mapper (SAM)
are the two commonly used evaluation indices in SU. It should be noted,
however, that the results of RE or SAM are not equivalent to those of
unmixing. Linking to the issue of benchmark data, the measurement between real
results and estimated ones is the optimal choice, if we have the ground truth
for abundances and endmembers. If not, developing meaningful and reasonable
evaluation indices (e.g., classification accuracy) should give the top
priority in future work.
* •
Spectral Variability. Spectral signatures inevitably suffer from various SVs
caused by illumination and topography change, noise effects from external
conditions and internal equipment, atmospheric interference, and complex
mixing of materials in the process of imaging. Fig. 9 shows a visual example
to specify the SVs (e.g., trees) in a real HS scene. Considerable
uncertainties brought by these factors have a big negative impact on accurate
estimation of abundances and endmembers in SU.
* •
Nonlinearity. The complex interactions (e.g., intimate mixing, multilayered
mixing [94]) between multiple materials, also known as nonlinearity,
inevitably occur in the process of HS imaging. The nonlinearity in SU is a
longstanding and pending challenge. Most of existing nonlinear unmixing models
only attempt to consider certain special cases [125], e.g., bilinear mixing,
intimate mixtures, etc. Consequently, there is still lack of a general and
powerful model that can robustly address various nonlinearities in SU.
* •
Model Explainability. The non-negativity and the sum-to-one constraint
considered in LMM are the basic priors for spectral signals in HS images.
However, only the two constraints fail to model the complex unmixing process
in an explainable fashion. To further enhance the explainability, new spectral
mixture models should be developed beyond the classic LMM by fully excavating
the intrinsic attribute knowledge that lies in the HS image.
## VI Data Fusion and Enhancement
The high spectral resolution of HS images enables the identification and
discrimination of materials, meanwhile the high spatial resolution can provide
the possibility of the derivation of surface parameters [126]. However, due to
the equipment limitation, there is usually a trade-off between the spatial and
spectral resolutions, and the HS images obtained by the spaceborne imaging
spectrometers are usually with a moderate ground sampling distance [126]. To
enhance the spatial resolution, one popular way is to fuse the HS images with
high spatial MS images to generate new high spatial-spectral HS (HrHS) images.
In particular, enormous effects have been recently made to enhance the spatial
or spectral resolutions of HS images by means of ML techniques. Fig. 11
illustrates the fusion process of HS-MS images to generate the HrHS image.
Suppose we have the low-spatial resolution HS image
$\boldsymbol{\mathcal{Y}}\in\mathbb{R}^{m\times n\times B}$ and high-spatial
resolution MS image $\boldsymbol{\mathcal{Z}}\in\mathbb{R}^{M\times N\times
b}$ with $M\gg m$, $N\gg n$ and $B\gg b$, the fusion purpose is to generate
the high-spatial resolution HS image
$\boldsymbol{\mathcal{X}}\in\mathbb{R}^{M\times N\times B}$. The degradation
models from $\boldsymbol{\mathcal{X}}$ to $\boldsymbol{\mathcal{Y}}$ and
$\boldsymbol{\mathcal{Z}}$ are formulated as
$\displaystyle\mathbf{Y}=\mathbf{X}\mathbf{R}+\mathbf{N}_{H}$ (21)
$\displaystyle\mathbf{Z}=\mathbf{G}\mathbf{X}+\mathbf{N}_{M}$ (22)
where $\mathbf{X},\mathbf{Y},\mathbf{Z}$ are the reshaped matrices along the
spectral dimension of
$\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{Y}},\boldsymbol{\mathcal{Z}}$,
respectively, $\mathbf{R}$ is the mixed cyclic convolution and downsampling
operator, $\mathbf{G}$ is the spectral response function (SRF) of the MS image
sensor, $\mathbf{N}_{H}$ and $\mathbf{N}_{M}$ are the corresponding MS-HS
noise. To unify different observation models [127, 128, 129, 130, 126, 131],
$\mathbf{N}_{H}$ and $\mathbf{N}_{M}$ are assumed to be the independent
identically distributed Gaussian noise. Via the maximum a posteriori (MAP)
estimation method and Bayes rule [127, 129, 130], the following non-convex
optimization model is obtained
$\displaystyle\min_{\mathbf{X}}\|\mathbf{Y}-\mathbf{X}\mathbf{R}\|_{\operatorname{F}}^{2}+\|\mathbf{Z}-\mathbf{G}\mathbf{X}\|_{\operatorname{F}}^{2},$
(23)
where $\mathbf{R}$ and $\mathbf{G}$ are assumed to be known (in [129, 126],
$\mathbf{R}$ and $\mathbf{G}$ are estimated in advance of the optimization).
and As mentioned in [129, 130], the optimization of (23) is a NP-hard problem,
and over-estimation of $\mathbf{Z}$ will result in the unstable fusion
results. Therein, additional property of $\mathbf{X}$ and prior regularizers
should be exploited in the optimization model (23). It should be noted,
however, that the two functions $\mathbf{R}$ and $\mathbf{G}$ can be given
according to known sensors and also can be learned or automatically estimated
from the data itself.
Figure 11: Illustration of MS-HS image fusion to generate the HrHS image.
HS pansharpening is a heuristic way to perform the HS-MS fusion [132], which
has been widely applied in the HS image enhancement task. Component
substitution (CS) and multiresolution analysis (MRA) are the two main types of
pansharpening techniques. The former one aims to inject detailed information
of MS images into the low-resolution HS image, thereby generating the high-
resolution HS product. The latter one is to pansharpen the HS image by
linearly combining MS bands to synthesize a high-resolution HS band using
regression techniques. Another group for the HS-MS fusion task is the
subspace-based model, which roughly consists of Bayesian and unmixing based
methods (see [126]). Different from pansharpening, the subspace-based
approaches project the to-be-fused MS and HS images to a new space where the
dimension is generally smaller than that of the unknown high-resolution HS
image, by the means of the probability-driven Bayesian estimation (Bayesian-
based methods) or SU-guided matrix joint factorization (unmixing-based
methods).
In the following, we focus on the subspace methods, and review the related HS-
MS image fusion methods from the non-convex modeling perspective. A more
detailed review can be referred to [126, 132].
Figure 12: The fusion results of different methods with Chikusei image. The
color is composed of bands 70, 100, 36. An enlarged region is framed in green
and the corresponding residual image between the fused image and MS-GT is
framed in red.
### VI-A Unmixing based methods
Hyperspectral unmixing (HU) [5, 101] assumes that the mixed class of an HS
image can be decomposed to the collection of constitute spectra (endmembers),
and their corresponding proportions (abundances). With LMM assumption, the
different endmembers do not interfere with each other [5]. By embedding the
LMM model into (23), we can obtain the following general unmixing based
approaches
$\displaystyle\min_{\mathbf{X},\mathbf{E},\mathbf{A}}\|\mathbf{Y}-\mathbf{X}\mathbf{R}\|_{\operatorname{F}}^{2}+\|\mathbf{Z}-\mathbf{G}\mathbf{X}\|_{\operatorname{F}}^{2},$
(24) $\displaystyle{\rm
s.t.}\;\;\mathbf{X}=\mathbf{E}\mathbf{A},\;\mathbf{E},\mathbf{A}\geq
0,\;\mathbf{1}_{R}^{\top}\mathbf{A}=\mathbf{1}_{MN},$
where $\mathbf{E}$, $\mathbf{A}$ are the endmember matrix and abundance
matrix, which are assumed to obey the non-negative and abundance sum-to-one
constraints. Generally, nonlinear unmixing models [5] can be also utilized for
the fusion task of HS-MS images. However, due to the generality of the LMM
model, we focus on the review of LMM based fusion approaches.
Eismann et al. proposed a maximum a posteriori estimation method to deduce the
cost objective function, and introduced a stochastic mixing model (MAP-SMM) to
embed LMM into the cost function [127]. MAP-SMM method tries to estimate the
prior probabilities for all the mixture classes, including the mean vectors
and covariance matrices of the endmember classes. The learned prior
probabilities are passed to the cost function to help the reconstruction of
the final HrHS image $\mathbf{X}$.
Yokoya et al., regarded (24) as the coupled NMF (CNMF) problem [128] and
introduced the multiplicative update rules to optimize (24). Firstly, CNMF
utilizes
$\|\mathbf{Y}-\mathbf{E}\mathbf{A}\mathbf{R}\|_{\operatorname{F}}^{2}$ as the
cost function to update $\mathbf{E}$ and $\mathbf{A}_{h}$ with the endmember
matrix $\mathbf{E}$ initialized by vertex component analysis (VCA). Here,
$\mathbf{A}_{h}=\mathbf{A}\mathbf{R}$ is the abundance matrix from HS images.
Secondly, by initializing $\mathbf{E}_{m}=\mathbf{G}\mathbf{E}$ which is the
endmember matrix from the MS image, CNMF again utilizes the multiplicative
update rules to update $\mathbf{E}$ from the cost function
$\|\mathbf{Z}-\mathbf{G}\mathbf{E}\mathbf{A}\|_{\operatorname{F}}^{2}$.
Finally, the HrHS image $\mathbf{X}$ is reconstructed from
$\mathbf{E}\mathbf{A}$. The following works [133, 134, 135] also utilize the
CNMF framework to fuse HS-MS images. Differently, [133, 135] introduced a non-
negative dictionary learning strategy, while [134] proposed the proximal
alternating linearized minimisation algorithm to update $\mathbf{E}$ and
$\mathbf{A}$.
On the basis of (24), Wang et al., further regularized $\mathbf{X}$ with non-
local low-rank Tucker decomposition [136]. The improved non-local Tucker
decomposition regularized CNMF model [136] was solved by the multi-block ADMM,
and achieved remarkable fusion results. It indicates that additional
regularizers on $\mathbf{X}$ can further improve the fusion accuracy. From
another side, it is necessary to make a trade-off between the complex models
with higher accuracy and the computation efficiency for real large scale HS-MS
image fusion task.
### VI-B Orthogonal subspace based methods
Another common assumption in HS-MS fusion is that the spectral information of
$\mathbf{X}$ underlies a orthogonal subspace, whose dimension is much smaller
than the number of bands $B$ [129, 101], i.e.,
$\mathbf{X}=\mathbf{E}\mathbf{A}$ with $\mathbf{E}\in\mathbb{R}^{B\times k}$,
$\mathbf{A}\in\mathbb{R}^{k\times MN}$, and $k\ll B$. $\mathbf{E}^{\top}$ is
an orthogonal matrix with $\mathbf{E}^{\top}\mathbf{E}=\mathbf{I}_{k}$.
Therefore, the subspace based model is formulated as
$\displaystyle\min_{\mathbf{X},\mathbf{E},\mathbf{A}}\|\mathbf{Y}-\mathbf{X}\mathbf{R}\|_{\operatorname{F}}^{2}+\|\mathbf{Z}-\mathbf{G}\mathbf{X}\|_{\operatorname{F}}^{2},$
(25) $\displaystyle{\rm
s.t.}\;\;\mathbf{X}=\mathbf{E}\mathbf{A},\;\mathbf{E}^{\top}\mathbf{E}=\mathbf{I}_{k}.$
Although additional spectral subspace prior is exploited, the optimization of
(25) still faces several challenges. Firstly, if $k\gg b$, that’s to say, the
dimension number of the subspace is larger than the bands’ number of the MS
image, the optimization of (25) is the under-estimate problem. Therefore, to
ensure a reasonable solution, prior information of coefficient $\mathbf{A}$
need to be exploited. [129] pre-trains a dictionary to represent $\mathbf{A}$,
and updates $\mathbf{A}$ via sparse representation. Hyperspectral super-
resolution (HySure) in [130] assumes that $\mathbf{A}$ appears the spatial
smoothness structure and regularize $\mathbf{A}$ with band-by-band TV. [137]
translates the optimization of $\mathbf{A}$ to a Sylvester equation and
proposes a fast fusion method for (25) (FUSE).
Secondly, the optimization of orthogonal matrix $\mathbf{E}^{\top}$ is another
challenge due to the non-convex of (25). One appearing approach [129, 130,
137] is to pre-estimate $\mathbf{E}$ from $\mathbf{Y}$ in advance, and fix the
variable $\mathbf{E}$ during the optimization of (25). Specially, FUSE [137]
adopted principal component analysis (PCA), meanwhile, HySure utilized VCA to
extract $\mathbf{E}$ from $\mathbf{Y}$. Another strategy is to regularize the
update of $\mathbf{E}$ and $\mathbf{A}$ as a coupled matrix factorization
problem, and blind dictionary learning strategy is utilized to update
$\mathbf{E}$ [138]. A hybrid inexact block coordinate descent [139] is
introduced to exactly estimate $\mathbf{E}$.
### VI-C Tensor based methods
The above subspace based methods utilize low-rank matrix decomposition to
exploit the low-rank property of the reshaped high-spatial resolution HS image
$\mathbf{X}$. However, the original HS image is a 3-D tensor, and therein, the
researchers introduce the tensor decomposition to simultaneously capture the
spatial and spectral low-rank property. The coupled sparse tensor
factorization (CSTF) approach [140] utilized Tucker decomposition, presented
as follows:
$\displaystyle\boldsymbol{\mathcal{X}}=\boldsymbol{\mathcal{O}}\times_{1}\mathbf{E}_{1}\times_{2}\mathbf{E}_{2}\times_{3}\mathbf{E}_{3},$
(26) $\displaystyle{\rm
subject\;to}\;\;\mathbf{E}_{i}^{\top}\mathbf{E}_{i}=\mathbf{I},\;\|\boldsymbol{\mathcal{O}}\|_{0}\leq\mathcal{C},$
to regularize the high-spatial resolution HS image $\boldsymbol{\mathcal{X}}$.
In (26), the core tensor $\boldsymbol{\mathcal{O}}$ is assumed to obey the
sparse property, and $\mathbf{E}_{i}$ is the orthogonal matrix of the $i$-th
dimension. Subsequently, CP decomposition [141], tensor train decomposition
[142], tensor ring decomposition [143, 131], and so on, are utilized to
regularize $\boldsymbol{\mathcal{X}}$. Furthermore, non-local LRTD is also
investigated for the fusion task [144, 145, 146].
It is worth noting that unmixing, orthogonal subspace, and tensor based
methods share the common idea that spectral space of $\mathbf{X}$ should lie
in the low-dimensional space. Unmixing based approaches interpret the low-rank
property as the endmembers and abundances, which are assumed to be non-
negative, meanwhile Orthogonal subspace and tensor based methods ignore the
non-negative restrict. Unmixing based approaches are interpretable from the
physical meaning, but suffering from the unstable convergence in the
optimization. Orthogonal subspace and tensor based methods lose physical
meaning, but can be optimized more elegantly.
Very recently, there are some preliminary works to perform the fusion task by
means of DL-based methods [147, 148, 149, 150, 151, 152, 153] and show the
effective and competitive fusion performance. A similar problem existed in
these methods is the model interpretability and rationality. Clearly
explaining the intrinsic meaning in each layer of deep networks would
contribute to better modeling the fusion task and further obtaining higher-
quality products.
TABLE IV: Quantitative comparison of different algorithms on the HS-MS image
fusion experiments. The best one is shown in bold.
Methods | RMSE | ERGAS | SA | SSIM
---|---|---|---|---
CNMF [128] | 6.404 | 0.715 | 4.89 | 0.8857
ICCV’15 [154] | 5.203 | 0.589 | 4.64 | 0.9139
HySure [130] | 8.537 | 0.812 | 9.45 | 0.8527
FUSE [137] | 8.652 | 0.869 | 9.51 | 0.8401
CSTF [140] | 8.32 | 0.841 | 8.34 | 0.8419
STEREO [141] | 9.4425 | 0.891 | 9.78 | 0.8231
NLSTF [144] | 8.254 | 0.819 | 8.36 | 0.8424
(a) Training for MML and CML (b) Testing for MML (c) Testing for CML
Figure 13: An illustration for the model’s training and testing in MML- and
CML-based classification tasks (take the bi-modality as an example). (a) They
share the same training process, i.e., two modalities are used for model
training. The main difference lies in the testing phase, (b) MML still needs
the input of two modalities, (c) while one modality is absent in CML.
### VI-D Experimental Study
In this section, we select unmixing based methods:
CNMF171717http://naotoyokoya.com/Download.html [128],
ICCV’15181818https://github.com/lanha/SupResPALM [154]; subspace based
methods: HySure191919https://github.com/alfaiate [130],
FUSE202020https://github.com/qw245/BlindFuse [137]; tensor decomposition
regularized methods: STEREO212121https://sites.google.com/site/harikanats/
[141], CSTF [140]; finally non-local tensor decomposition regularized method:
NLSTF222222https://sites.google.com/view/renweidian/ [144] for the comparison
and analysis. We use evaluation indices, including the root mean square error
(RMSE), relative dimensional global error in synthesis (ERGAS) [155], MSA, and
SSIM [156] as evaluation criteria for the fusion results of different methods.
The selected dataset for the experiment is the Chikusei dataset obtained at
Chikusei, Ibaraki, Japan, on 29 July 2014 [126]. The selected high-spatial
resolution HS image is of size $448\times 448\times 128$, and the simulated
HS-MS images are of size $448\times 448\times 3$ and $14\times 14\times 128$,
respectively. Tab. (IV) presents the quantitative comparison results of
different algorithms on the HS-MS image fusion, meanwhile Fig. (12) presents
the visual illustration. From the results, it can be observed that even the HS
image is spatially degraded by 32 times, the fusion methods can efficiently
reconstruct the spatial details with the help of a 3 band MS image. On this
tested toy dataset, ICCV’15 performed the best. However, different datasets
need different kinds of regularizers. The fusion of HS-MS images for efficient
and large scale applications is still a challenge for further research.
### VI-E Remaining Challenges
Subspace based non-convex methods for the fusion of HS-MS images have been
well developed. However, most remarkable results are achieved on the simulated
experiments. For the real applications with HS-MS images from two different
satellite sensors, there still remain several challenges.
* •
Blind. Most fusion methods assume the linear spatial and spectral downsampling
from HR-HS image to HS-MS images. However, in real applications, the
degradation is complex and unknown in advance. how to blindly reconstruct the
HR-HS image is a challenge in future research.
* •
Regularizer. We reviewed the subspace based fusion methods from unmixing,
orthogonal subspace, and tensor decomposition perspectives. Different
assumptions are suitable for the exploited for the different structure of the
HS image. How to mine the essence of HS images and develop efficient
regularizers for large scale processing still remains a challenge.
* •
Evaluation. In the real cases, the enhanced HrHS images from HS and MS images
are not existed as the reference images in the real scenario. How to evaluate
the final enhanced HrHS images is also a key problem for the future fusion
approach development of HS-MS.
## VII Cross-modality Learning for Large-scale Land Cover Mapping
With the ever-growing availability of diverse RS data sources from both
satellite and airborne sensors, multimodal data processing and analysis in RS
[157, 158] can provide potential possibilities to break the performance
bottleneck in many high-level applications, e.g., land cover classification.
HS data are featured by rich spectral information, which enables the high
discrimination ability for material recognition at a more accurate and fine
level. It should be noted, however, that the HS image coverage from space is
much narrow compared to MS imaging due to the limitations of imaging
principles and devices. That means the HS-dominated multimodal learning (MML)
fails to identify the materials on a large geographic coverage and even global
scale [159]. But fortunately, large-scale MS or synthetic aperture radar (SAR)
images are openly available from e.g., Sentinel-1, Sentinel-2, Landsat-8.
This, therefore, drives us to ponder over a problem: can HS images acquired
only in a limited area improve the land cover mapping performance using a
larger area covered by the MS or SAR images? This is a typical issue of cross-
modality learning (CML) from a ML’s point of view.
Take the bi-modality as an example, CML for simplicity refers to that training
a model using two modalities and one modality is absent in the testing phase,
or _vice versa_ (only one modality is available for training and bi-modality
for testing) [160]. Such a CML problem that exists widely in a variety of RS
tasks is more applicable to real-world cases. Fig. 13 illustrates the
differences between MML and CML in terms of training and testing process. The
core idea of CML is to find a new data space, where the information can be
exchanged effectively across different modalities. Thereupon, we formulate
this process in a general way as follows:
$\displaystyle\mathop{\min}_{\mathbf{X},\\{\mathbf{U}_{s}\\}_{s=1}^{m}}\sum_{s=1}^{m}\frac{1}{2}\lVert\mathbf{X}-\mathbf{U}_{s}\mathbf{Y}_{s}\rVert_{\operatorname{F}}^{2}\;\;{\rm
s.t.}\;\mathbf{X},\\{\mathbf{U}_{s}\\}_{s=1}^{m}\in\mathcal{C},$ (27)
where $m$ is the number of input modality. For simplicity, we only consider
the bi-modality case in this topic, i.e., $m=2$. According to different
learning strategies on modalities, CML can be roughly categorized into two
groups: manifold alignment (MA) and shared subspace learning (SSL). The
differences between the two types of approaches mainly lie in
* •
MA learns the low-dimensional embedding by preserving the aligned manifold (or
graph) structure between different modalities. In the process of graph
construction, the similarities between samples (unsupervised MA) and indirect
label information (supervised or semi-supervised MA) are used. Despite the
competitive performance obtained by MA-based approaches for the CML task, the
discrimination ability of learned features remains limited due to the lack of
directly bridging low-dimensional features with label information.
* •
SSL, as the name suggests, aims to find a latent shared subspace, where the
features of different modalities are linked via a manifold alignment
regularizer. Also, the learned features are further connected with label
information. The two steps are jointly optimized in a SSL model, tending to
yield more discriminative feature representations.
More specifically, we will briefly review and detail some representative
approaches belonging to the aforementioned two groups as follows.
### VII-A Manifold Alignment based Approach
As the name suggests, MA is capable of aligning multiple modalities on
manifolds into a latent subspace, achieving a highly effective knowledge
transfer [161]. Due to the interactive learning ability, MA has a good fit for
large-scale RS image classification. In [162], the domain adaptation was
investigated to reduce the gap between the source and target domains of HS
data for land cover classification. By simultaneously considering labeled and
unlabeled samples, Tuia et al. [163] used semi-supervised MA (SSMA) techniques
[164] to align the multi-view RS images onto the manifold space by the
attempts to eliminate the effects of image variants caused by different views.
Matasci et al. [165] modified the classic transfer component analysis [166],
making it applicable to land cover classification of RS images. Moreover, a
kernelized MA approach presented in [167] projected the multimodal RS data to
a higher dimensional space and aligned them in a nonlinear way. Hu et al.
[168] deeply reviewed the semi-supervised MA methods with respect to the
fusion classification of HS and polarimetric SAR images. Based on the work in
[168], the same investigators made full use of topological data analysis and
designed a new graph structure for optical (e.g., HS) and SAR data fusion
[169].
Mathematically, the MA idea can be implemented by solving the following non-
convex model:
$\displaystyle\mathop{\min}_{\\{\mathbf{U}\\}_{s=1}^{m}}\frac{A+C}{B},$ (28)
where $A$, $B$, and $C$ are
$\displaystyle
A=\frac{1}{2}\sum_{p=1}^{m}\sum_{q=1}^{m}\sum_{i=1}^{n}\sum_{j=1}^{n}\lVert\mathbf{U}_{p}\mathbf{y}_{p}^{i}-\mathbf{U}_{q}\mathbf{y}_{q}^{j}\rVert_{2}^{2}\mathbf{W}_{sim}^{i,j},$
$\displaystyle
B=\frac{1}{2}\sum\limits_{p=1}^{m}\sum\limits_{q=1}^{m}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\lVert\mathbf{U}_{p}\mathbf{y}_{p}^{i}-\mathbf{U}_{q}\mathbf{y}_{q}^{j}\rVert_{2}^{2}\mathbf{W}_{dis}^{i,j},$
$\displaystyle
C=\frac{1}{2}\sum_{t=1}^{m}\sum_{i=1}^{n}\sum_{j=1}^{n}\lVert\mathbf{U}_{t}\mathbf{y}_{t}^{i}-\mathbf{U}_{t}\mathbf{y}_{t}^{j}\rVert_{2}^{2}\mathbf{W}_{t}^{i,j}.$
By minimizing the problem (28), the $\\{\mathbf{U}\\}_{s=1}^{m}$ can be
estimated via generalized eigenvalues decomposition. We then have
$\mathbf{X}=\mathbf{U}_{s}\mathbf{Y}_{s}$. Three different graphs need to be
pre-computed in Eq. (28), including the similarity graph, i.e.,
$\mathbf{W}_{sim}$:
$\displaystyle\mathbf{W}_{sim}=\left[\begin{matrix}\mathbf{W}_{sim}^{1,1}&\mathbf{W}_{sim}^{1,2}&\cdots&\mathbf{W}_{sim}^{1,m}\\\
\mathbf{W}_{sim}^{2,1}&\mathbf{W}_{sim}^{2,2}&\cdots&\mathbf{W}_{sim}^{2,m}\\\
\vdots&\vdots&\ddots&\vdots\\\
\mathbf{W}_{sim}^{m,1}&\mathbf{W}_{sim}^{m,2}&\cdots&\mathbf{W}_{sim}^{m,m}\\\
\end{matrix}\right],$ (29)
the dissimilarity matrix, i.e., $\mathbf{W}_{dis}$:
$\displaystyle\mathbf{W}_{dis}=\left[\begin{matrix}\mathbf{W}_{dis}^{1,1}&\mathbf{W}_{dis}^{1,2}&\cdots&\mathbf{W}_{dis}^{1,m}\\\
\mathbf{W}_{dis}^{2,1}&\mathbf{W}_{dis}^{2,2}&\cdots&\mathbf{W}_{dis}^{2,m}\\\
\vdots&\vdots&\ddots&\vdots\\\
\mathbf{W}_{dis}^{m,1}&\mathbf{W}_{dis}^{m,2}&\cdots&\mathbf{W}_{dis}^{m,m}\\\
\end{matrix}\right],$ (30)
and the topology structure for each single modality obtained by $knn$ graph,
i.e., $\mathbf{W}_{t}$:
$\displaystyle\mathbf{W}_{t}=\left[\begin{matrix}\mathbf{W}_{t}^{1,1}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{W}_{d}^{2,2}&\cdots&\mathbf{0}\\\
\vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\cdots&\mathbf{W}_{t}^{m,m}\\\ \end{matrix}\right].$
(31)
In Eqs. (29)$-$(31), $\mathbf{W}_{sim}^{i,j}$, $\mathbf{W}_{dis}^{i,j}$, and
$\mathbf{W}_{t}^{i,j}$ are given, respectively, by
$\mathbf{W}_{sim}^{i,j}=\begin{cases}\begin{aligned} 1,\;\;&\text{if
$\mathbf{y}_{p}^{i}$ and $\mathbf{y}_{q}^{j}\in C_{k}$}\\\
0,\;\;&\text{otherwise,}\end{aligned}\end{cases}$
$\mathbf{W}_{dis}^{i,j}=\begin{cases}\begin{aligned} 1,\;\;&\text{if
$\mathbf{y}_{p}^{i}$ and $\mathbf{y}_{q}^{j}\notin C_{k}$ }\\\
0,\;\;&\text{otherwise,}\end{aligned}\end{cases}$
$\mathbf{W}_{t}^{i,j}=\begin{cases}\begin{aligned}
\exp\frac{\lVert\mathbf{y}^{i}-\mathbf{y}^{j}\rVert_{2}^{2}}{2\sigma^{2}},\;\;&\text{if
$\mathbf{y}_{p}^{i}\in\phi_{k}(\mathbf{y}_{q}^{j})$;}\\\
0,\;\;&\text{otherwise,}\end{aligned}\end{cases}$
where $\phi_{k}(\bullet)$ denotes the $k$ nearest neighbors of $\bullet$.
### VII-B Shared Subspace Learning based Approach
Due to the lack of the direct relational modeling between the learned features
and label information, MA-based approaches fail to activate the connections
across modalities effectively [160], thereby yielding the relatively weak
transferability between different modalities, particularly heterogeneous data.
There have been some tentative works in recent years, providing potential
solutions to overcome the aforementioned challenges. For example, Hong et al.
[170] for the first time proposed a supervised CoSpace model to learn a latent
discriminative subspace from HS-MS correspondences for the CML-related
classification problem. Beyond it, the same authors [171] fully tapped the
potential of the CoSpace by learning the data-driven graph structure from both
labeled and unlabeled samples, yielding a learnable manifold alignment (LeMA)
approach. Moreover, [172] deeply investigated and analyzed different
regression techniques, i.e., $\ell_{2}$-norm ridge regression, $\ell_{1}$-norm
sparse regression, in CoSpace. In [173], a semi-supervised graph-induced
aligned learning (GiAL) was developed by jointly regressing labels and pseudo-
labels.
Accordingly, these methods can be generalized to be a unified model [170] to
address the CML’s problem in a regression-based fashion:
$\displaystyle\mathop{\min}_{\mathbf{P},\\{\mathbf{U}_{s}\\}_{s=1}^{m}}$
$\displaystyle\frac{1}{2}\lVert\mathbf{M}-\mathbf{P}\mathbf{U}_{s}\mathbf{Y}_{s}\rVert_{\operatorname{F}}^{2}+\Psi(\mathbf{P})+\Omega(\\{\mathbf{U}_{s}\\}_{s=1}^{m})$
(32) $\displaystyle{\rm
s.t.}\;\;\mathbf{U}_{s}\mathbf{U}_{s}^{\top}=\mathbf{I},\;s=1,\cdots,m,$
where $\\{\mathbf{U}_{s}\\}_{s=1}^{m}$ denote the projections linking to the
shared features for different modalities. To avoid the over-fitting of the
model and stabilize the learning process, $\mathbf{P}$ can be regularized by
the Frobenius-norm [170] or $\ell_{1,1}$-norm [172]:
$\displaystyle\Psi(\mathbf{P})=\lVert\mathbf{P}\rVert_{\operatorname{F}}^{2},\;\text{or}\;\lVert\mathbf{P}\rVert_{1,1},$
(33)
and $\Omega(\\{\mathbf{U}_{s}\\}_{s=1}^{m})$ is specified as a manifold
alignment term on the multimodal data, which is written as
$\displaystyle\Omega(\\{\mathbf{U}_{s}\\}_{s=1}^{m})=\operatorname{tr}(\mathbf{U}\mathbf{Y}\mathbf{L}\mathbf{Y}^{\top}\mathbf{U}^{\top}),$
(34)
where $\mathbf{U}=[\mathbf{U}_{1},\mathbf{U}_{2},\cdots,\mathbf{U}_{m}]$ and
$\displaystyle\mathbf{Y}=\left[\begin{matrix}\mathbf{Y}_{1}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{Y}_{2}&\cdots&\mathbf{0}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\cdots&\mathbf{Y}_{m}\\\ \end{matrix}\right].$
Similar to Fig. 7, $\mathbf{L}$ is a joint Laplacian matrix.
Using the general model in Eq. (32),
* •
[170] considers the HS-MS correspondences that exist in an overlapped region
as the model input. The learned shared representations (e.g.,
$\mathbf{X}=\mathbf{U}_{s}\mathbf{Y}_{s}$) can be then used for classification
on a larger area, even though only MS data are available in the inference
phase;
* •
Differently, [171] inputs not only the labeled HS-MS pairs but also unlabeled
MS data in large quantity. With the graph learning, i.e., the variable
$\mathbf{L}$ is to be learned from the data rather than fixed by a given RBF,
the unlabeled information can be made use of to find a better decision
boundary. According to the equivalent form of Eq. (34), we then have
$\displaystyle\operatorname{tr}(\mathbf{U}\mathbf{Y}\mathbf{L}\mathbf{Y}^{\top}\mathbf{U}^{\top})=\frac{1}{2}\operatorname{tr}(\mathbf{W}\mathbf{d})=\frac{1}{2}\lVert\mathbf{W}\odot\mathbf{d}\rVert_{1,1},$
(35)
where $\mathbf{d}_{i,j}=\lVert\mathbf{x}_{i}-\mathbf{x}_{j}\rVert_{2}^{2}$
denotes the pair-wise distance in Euclidean space. Using Eq. (35), the
resulting optimization problem with respect to the variable $\mathbf{W}$ is
$\displaystyle\frac{1}{2}$
$\displaystyle\lVert\mathbf{W}\odot\mathbf{d}\rVert_{1,1}$ (36)
$\displaystyle{\rm s.t.}\;\mathbf{W}=\mathbf{W}^{\top},\;\mathbf{W}_{i,j}\geq
0,\;\lVert\mathbf{W}\rVert_{1,1}=c.$
* •
Inspired by the brain-like feedback mechanism presented in [89], a more
intelligent CML model was proposed [173]. With the joint use of labels and
pseudo-labels updated by the graph feedback in each iteration, more
representative features can be also learned (even if a certain modality is
absent, i.e., the CML case).
TABLE V: Quantitative comparison of SOTA algorithms related to the CML’s issue
in terms of OA, AA, and $\kappa$ using the NN classifier on the Houston2013
datasets. The best one is shown in bold.
Methods | OA (%) | AA (%) | $\kappa$
---|---|---|---
O-Baseline | 62.12 | 65.97 | 0.5889
USMA [85] | 65.54 | 68.81 | 0.6251
SMA [161] | 68.01 | 70.50 | 0.6520
SSMA [164] | 69.29 | 72.00 | 0.6659
CoSpace [170] | 69.38 | 71.69 | 0.6672
LeMA [171] | 73.42 | 74.76 | 0.7110
GiAL [173] | 80.66 | 81.31 | 0.7896
### VII-C Experimental Study
We evaluate the performance of several SOTA algorithms related to the CML’s
issue both quantitatively and qualitatively. They are O-Baseline (i.e., using
original image features), unsupervised MA (USMA) [85], supervised MA
(SMA)232323https://sites.google.com/site/changwangnk/home/ma-html [161], SSMA
[164], CoSpace242424https://github.com/danfenghong/IEEE_TGRS_CoSpace [170],
LeMA252525https://github.com/danfenghong/ISPRS_LeMA [171], and GiAL [173].
Three common indices, e.g., OA, AA, and $\kappa$, are adopted to quantify the
classification performance using the SVM classifier on the Houston2013 HS-MS
datasets that have been widely used in many researches [170, 171, 172, 173].
Table V gives the quantitative comparison between the above-mentioned methods
for the CML-related classification, while Fig. 14 visualizes a region of
interest (ROI) of classification maps. By and large, the classification
accuracy of O-Baseline, i.e., only using MS data, is much lower than other
methods. By aligning multimodal data on manifolds, MA-based approaches perform
better than O-Baseline with the approximated increase of $3\%$ OA in USMA,
$6\%$ OA in SMA, and $7\%$ OA in SSMA. As expected, the classification
performance of SSL-based models, e.g., CoSpace, LeMA, and GiAL, is obviously
superior to that of MA-based ones. In particular, GiAL dramatically
outperforms other competitors, owing to the use of the brain-like feedback
mechanism and graph-driven pseudo-label learning. Visually, shared learning
methods tend to capture more robust spectral properties and achieve more
realistic classification results. As can be seen from Fig. 14, the shadow
region covered by clouds can be finely classified by CoSpace, LeMA, and GiAL,
while MA-based models fail to identify the materials well in the region.
Figure 14: ROI visualization of classification maps using different SOTA
methods related to the CML’s issue.
### VII-D Remaining Challenges
CML has drawn growing interest from researchers in computer vision and ML, yet
it is rarely investigated in the RS community. In other words, CML is a
relatively emerging topic in RS, which means there are lots of difficulties
(or challenges) to be overcome. In detail,
* •
Data Preparation. Since multimodal data are acquired by different contexts,
sensors, resolutions, etc., this inevitably poses a great challenge to data
collection and processing. For example, the errors caused by interpolation of
different resolutions, registration methods of geographical coordinates,
pixel-wise biases of different sensors, and uncertainties of image degradation
in the imaging process easily generate unregistered multimodal data to a great
extent.
* •
Model Transferability. Due to different imaging mechanisms and principles, the
viscosity between pixels from the same modality is much stronger than from the
different modalities. This might lead to difficulties in fusing multimodal
information at a deep level, particularly heterogeneous data (e.g., HS and SAR
data), further limiting the model’s transferability.
* •
Labeling. Unlike natural images or street view images that are relatively easy
and accurate to be labeled manually, labeling RS scenes (field trips needed)
is extremely expensive and time-consuming. Consequently, a limited number of
labeled samples are available for training and even worse, there are many
noisy labels in these samples. These problems will be noteworthy and to-be-
solved key points of the next generation interpretable AI models in the RS-
related CML task.
## VIII Conclusion and Future Prospect
Characterized by the nearly continuous spectral profile that is capable of
sampling and representing the whole electromagnetic spectrum, HS images play
an important role in both promoting developments of new techniques and
accelerating the practical applications, not only limiting to the fields of
RS, geoscience, signal and image processing, numerical optimization and
modeling, ML, and AI. However, there still exist severe difficulties and
challenges that need to be carefully considered in the development and
application of HS RS techniques. One sign reveals that HS data analysis
methods dominated by expert systems have been unable to meet the demand of an
ever-growing volume of HS data whether in performance gain or in processing
efficiency. Another sign is that despite the currently unprecedented progress
made on computer vision, ML, and AI techniques, the model compatibility and
interpretability for HS RS applications remain limited.
Due to the SVs of HS data caused by various degradation mechanisms (e.g.,
environmental condition, atmospheric effects, spectral nonlinear mixing,
etc.), the redundancy of high-dimensional HS signals, and the complex
practical cases underlying in the HS products (e.g., low spatial resolution,
narrow imaging range, instrumental noises), convex models under ideal
circumstances usually fail to extract useful and diagnostic information from
HS images (especially those products that are corrupted seriously) and thereby
understand our environment. Considering that non-convex modeling is capable of
characterizing more complex real scenes and better providing the model
interpretability, in this article we present a comprehensive and technical
survey over five promising and representative research topics related to HS RS
with a focus on non-convex modeling, such as HS image restoration,
dimensionality reduction and classification, data fusion and enhancement,
spectral unmixing, and cross-modality learning. Among these topics, we review
the current state-of-the-art methods with illustrations, show the significance
and superiority of non-convex modeling to bridge the gap between HS RS and
interpretable AI, and point out remaining challenges and future research
directions.
It is well-known that the HS image processing and analysis chain is wide-
ranging. Apart from the five topics covered in this paper, we are not able to
detailedly report all the important and promising applications related to HS
RS missions. There are several very noteworthy and active fields that should
be paid more attention in future work, including target/change detection,
time-series analysis, multitemporal fusion/classification, physical parameter
inversion, image quality assessment, and various practical applications (e.g.,
precious farming, disaster management and response). Moreover, some crucial
steps for the HS image pre-processing algorithms are also missing, such as
atmospheric and geometric corrections, geographic coordinate registration,
etc. Furthermore, the methodologies summarized and reported in this article
mainly focus on the survey of shallow non-convex models. Undeniably, deep
models, e.g., DL-based methods, are capable of excavating deeper and intrinsic
properties of HS data. There is, therefore, room for improvement in the
development of more intelligent DL-related non-convex modeling with
application to HS RS. For example, embedding more physically meaningful priors
and devising advanced and novel deep unfolding [174] or unrolling [175]
strategies to closely integrate data-driven DL and theoretically-guaranteed
optimization technique is to open and interpret the so-called “black box” in
DL models.
Finally, we have to admit that non-convex modeling and optimization is a
powerful tool across multidisciplinary and the relevant studies along the
direction have been made tremendous progress theoretically and technically.
This provides the possibility of creating new methodologies and implementing
interpretable AI for various HS RS applications. In this paper, we attempt to
“intellectualize” these models by introducing more interpretable and
physically meaningful knowledge to meet the actual needs in a non-convex
modeling fashion. In other words, we hope that non-convex modeling can play
the role as a bridge to connect interpretable AI models and various research
topics in HS RS. Our efforts in this paper are made to foster curiosity and
create a good starting point for post-graduate, Ph.D. students, and senior
researchers working in the HS-related fields, thereby further looking for new
and advanced research directions in the interdisciplinary involving signal and
image processing, ML, AI, and RS.
## Acknowledgement
The authors would like to thank Prof. D. Landgrebe from Purdue University for
providing the AVIRIS Indian Pines data, Prof. P. Gamba from the University of
Pavia for providing the ROSIS-3 Pavia University and Centre data, the
Hyperspectral Image Analysis group at the University of Houston for providing
the CASI University of Houston dataset used in the IEEE GRSS DFC2013 and
DFC2018, and the Hyperspectral Digital Imagery Collection Experiment (HYDICE)
for sharing the urban dataset free of charge.
This work from the D. Hong and X. Zhu sides is jointly supported by the German
Research Foundation (DFG) under grant ZH 498/7-2, by the Helmholtz Association
through the framework of Helmholtz Artificial Intelligence (HAICU) - Local
Unit “Munich Unit @Aeronautics, Space and Transport (MASTr)” and Helmholtz
Excellent Professorship “Data Science in Earth Observation - Big Data Fusion
for Urban Research”, by the German Federal Ministry of Education and Research
(BMBF) in the framework of the international AI future lab “AI4EO – Artificial
Intelligence for Earth Observation: Reasoning, Uncertainties, Ethics and
Beyond”. This work from the L. Gao side is supported by the National Natural
Science Foundation of China under Grant 42030111 and Grant 41722108. This work
from the W. He and N. Yokoya sides is supported by the Japan Society for the
Promotion of Science under KAKENHI 19K20308 and KAKENHI 18K18067. This work
from the J. Chanussot side has been partially supported by MIAI@Grenoble
Alpes, (ANR-19-P3IA-0003) and by the AXA Research Fund.
The corresponding authors of this paper are Dr. Wei He and Prof. Lianru Gao.
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11institutetext: Department of Physics, Anyang Normal University, Anyang,
455000, China, 11email<EMAIL_ADDRESS>
# Pairwise Entanglement and Geometric Phase in High Dimensional Free-Fermion
Lattice Systems
H. T. Cui Y. F. Zhang
(Received: / Revised version: )
###### Abstract
The pairwise entanglement, measured by concurrence and geometric phase in high
dimensional free-fermion lattice systems have been studied in this paper. When
the system stays at the ground state, their derivatives with the external
parameter show the singularity closed to the phase transition points, and can
be used to detect the phase transition in this model. Furthermore our studies
show for the free-fermion model that both concurrence and geometric phase show
the intimate connection with the correlation functions. The possible
connection between concurrence and geometric phase has been also discussed.
###### pacs:
03.65.Vf Phases: geometric; dynamic or topological; 03.65.Ud Entanglement and
quantum nonlocality; and 05.70.Fh Phase transitions: general studies
## 1 introduction
The understanding of quantum many-body effects based on the fundamentals of
quantum mechanics, has been raising greatly because of the rapid development
in quantum information theoryafov07 . Encouraged by the suggestion of
Preskillpreskill , the connection between the quantum entanglement and quantum
phase transition has been demonstrated first in 1D spin-$1/2$ $XY$ modeoo02 ,
and then was extended to more other spin-chain systems and fermion systems
(see Ref afov07 for a review). Furthermore the decoherence of a simple
quantum systems coupled with the quantum critical environment has been shown
the significant features closed to the critical points ycw06 ; quan .
Regarding these findings, the fidelity between the states across the
transition point has also been introduced to mark the happening of the phase
transitions zanardi . These intricate connections between quantum entanglement
and phase transition in many-body systems have sponsored great effort devoted
to the understanding of many-body effects from quantum information pointafov07
. In general quantum entanglement as a special correlation, is believed to
play an essential role for the many-body effects since it is well accepted
that the non-trivial correlation is at the root of many-body effects. Although
the ambiguity existsyang , quantum entanglement provides us a brand-new
perspective into quantum many-body effects. However the exact physical meaning
of quantum entanglement in many body systems remains unclearvedral07 .
Although the entanglement witnesses has been constructed in some many-body
systemswvb05 , a general and physical understanding of quantum entanglement in
many-body systems is still absent.
On the other hand, the geometric phase, which was first studied systemically
by Berryberry and had been researched extensively in the past 20 yearsgp ,
recently has also been shown the intimate connection to quantum phase
transitionscp05 ; zhu ; hamma ; cui06 ; plc06 ; cui08 ; hkh08 (or see a recent
review Ref.zhu08 ). This general relation roots at the topological property of
the geometric phase, which depicts the curvature of the Hilbert space, and
especially has direct relation to the property of the degeneracy in quantum
systems. The degeneracy in the many-body systems is critical in our
understanding of the quantum phase transition sachdev . Thus the geometric
phase is another powerful tool for detecting the quantum phase transitions.
Moreover recently geometric phase has been utilized to distinguish different
topological phases in quantum Hall systemsshen , in which the traditional
phase transition theory based on the symmetry-broken theory is not in
functionSenthil .
Hence it is very interesting to discuss the possible connection between
entanglement and geometric phase, since both issues show the similar
sensitivity to the happening of quantum phase transition. Recently the
connection between the entanglement entropy and geometric phase has first been
discussed with a special model in strongly correlated systems; the geometric
phase induced by the twist operator imposed on the filled Fermi sphere, was
shown to present a lower bound for the entanglement entropyrh06 . This
interesting result implies the important relation between quantum entanglement
and geometric phase, and provides an possible understanding of entanglement
from the topological structure of the systems. In another way the two-particle
entanglement was also importantoo02 . Especially in spin-chain systems two-
particle entanglement is more popular and general because of the interaction
between spins, and furthermore the quantum information transferring based on
spin systems are generally dependent on the entanglement between two
particlesss05 . So it is a tempting issue to extend this discussion to the
universal two-particle entanglement situation.
For this purpose the pairwise entanglement and geometric phase are studied
systemically in this paper. Our discussion focuses on nearest-neighbor
entanglement in the ground state in free-Fermion lattice systems because of
the availability of the exact results. By our own knowledge, this paper first
presents the exact results of entanglement and geometric phase in higher
dimensional systems. In Sec.2 the model will be provided, and the entanglement
measured by Wootter’s concurrence is calculated by introducing pseudospin
operators. Furthermore the geometric phase is obtained by imposing a globe
rotation, and its relation with concurrence are also discussed generally. In
Sec.3, we discussed respectively the concurrence and geometric phase in 2D and
3D cases. Finally, the conclusion is presented in Sec.4.
## 2 Model
The Hamiltonian for spinless fermions in lattice systems reads
$H=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{\mathbf{ij}}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}+\text{h.c.}),$
(1)
in which $c_{\mathbf{i}}^{(\dagger)}$ is fermion annihilation(creation)
operator and $L$ is the total number of lattice sites. The hermitity of $H$
imposes that matrix $A$ is Hermit and $B$ is an anti-symmetry matrix. The
configuration of lattice does not matter for Eq. (1) since our discussion
focuses on the general case and available exact results. This model obviously
is solvable exactly and can be transformed into the free Bogoliubov fermionic
model. So it is also called free-fermion model. By Jordan-Wigner
transformationjw28 one can convert the spin-chain systems into spinless
fermions systems, in which the physical properties can be readily determined.
Therefore an alternative approach is necessary by which one can treat solvable
fermion systems of arbitrary size. The model Eq. (1) serves this purpose.
Without the loss of generality we assume $A$ and $B$ to be reallsm61 . An
important property of Eq. (1) is
$[H,\prod_{\mathbf{i}}^{L}(1-2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}})]=0.$ (2)
This symmetry would greatly simplify the consequent calculation of the reduced
density matrix for two fermions. One can diagonalize Eq. (1) by introducing
linear transformation with real $g_{\mathbf{ki}}$ and $h_{\mathbf{ki}}$lsm61
$\eta_{\mathbf{k}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{i}}^{L}g_{\mathbf{ki}}c_{\mathbf{i}}+h_{\mathbf{ki}}c_{\mathbf{i}}^{\dagger},$
(3)
in which the normalization factor $1/\sqrt{L}$ have been included to ensure
the convergency under the thermodynamic limit. After some algebra, the
Hamiltonian Eq. (1) becomes
$H=\sum_{\mathbf{k}}\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$
(4)
in which $\Lambda_{\mathbf{k}}^{2}$ is the common eigenvalue of the matrices
$(A-B)(A+B)$ and $(A+B)(A-B)$ with the corresponding eigenvectors
$\phi_{\mathbf{ki}}=g_{\mathbf{ki}}+h_{\mathbf{ki}}$ and
$\psi_{\mathbf{ki}}=g_{\mathbf{ki}}-h_{\mathbf{ki}}$ respectively (see
Ref.lsm61 for details). The ground state is defined as $|g\rangle$, which
satisfies the relation
$\eta_{\mathbf{k}}|g\rangle=0$ (5)
With respect to fermi operator $\eta_{\mathbf{k}}$, one has relations
$\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}$
$\displaystyle=$ $\displaystyle\delta^{(3)}_{\mathbf{k^{\prime}k}}$
$\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}$
$\displaystyle=$ $\displaystyle 0$ (6)
Furthermore the requirement that $\\{\phi_{k},\forall k\\}$ and
$\\{\psi_{k},\forall k\\}$ be normalized and complete, reinforce the relations
lsm61
$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}g_{\mathbf{kj}}+h_{\mathbf{ki}}h_{\mathbf{kj}}$
$\displaystyle=$ $\displaystyle\delta_{\mathbf{ij}}$
$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}h_{\mathbf{kj}}+h_{\mathbf{ki}}g_{\mathbf{kj}}$
$\displaystyle=$ $\displaystyle 0$ (7)
With the help of these formula above, one obtains
$c_{\mathbf{i}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{k}}g_{\mathbf{ki}}\eta_{\mathbf{k}}+h_{\mathbf{ki}}\eta_{\mathbf{k}}^{\dagger},$
(8)
which would benefit our calculation for the correlation functions.
### 2.1 Concurrence
The concurrence, first introduced by Wootterswootters for the measure of two-
qubit entanglement, is defined as
$c=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (9)
in which $\lambda_{i}(i=1,2,3,4)$ are the square roots of eigenvalues of
matrix $R=\rho(\sigma^{y}\otimes\sigma^{y})\rho(\sigma^{y}\otimes\sigma^{y})$
with decreasing order. Then the critical step is to determine the two-body
reduced density operator $\rho$. The reduced density operator
$\rho_{\mathbf{ij}}$ for two spin-half particles labeled $\mathbf{i,j}$ can be
written generally as,
$\rho_{\mathbf{ij}}=\text{tr}_{\mathbf{ij}}\rho=\frac{1}{4}\sum_{\alpha,\beta=0}^{4}p_{\alpha,\beta}\sigma^{\alpha}_{\mathbf{i}}\otimes\sigma^{\beta}_{\mathbf{j}},$
(10)
in which $\rho$ is the density matrix for the whole system and $\sigma^{0}$ is
the $2\times 2$ unity matrix and $\sigma^{\alpha}(\alpha=1,2,3)$ are the Pauli
operators $\sigma^{x},\sigma^{y},\sigma^{z}$, which also the generators of
$SU(2)$ group.
$p_{\alpha\beta}=\text{tr}[\sigma^{\alpha}_{\mathbf{i}}\sigma^{\beta}_{\mathbf{j}}\rho_{\mathbf{ij}}]=\langle\sigma^{\alpha}_{\mathbf{j}}\sigma^{\beta}_{\mathbf{j}}\rangle$
is the correlation function. With the symmetry Eq. (2), one can verify that
only $p_{00},p_{03},p_{30},p_{11},p_{22},p_{33},p_{12},p_{21}$ are not
vanishing. After some efforts, one obtain
$c=\max\\{0,c_{I},c_{II}\\},$ (11)
in which
$\displaystyle c_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\sqrt{(p_{11}+p_{22})^{2}+(p_{12}-p_{21})^{2}}$
$\displaystyle-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]$ $\displaystyle
c_{II}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[|p_{11}-p_{22}|-\sqrt{(1-p_{33})^{2}-(p_{30}-p_{03})^{2}}].$
(12)
In order to obtain the reduced density operator for two fermions, it is
crucial to construct $SU(2)$ algebra for the fermions in lattice systems. In
1D case, the Jordan-Wigner (JW) transformation is availablejw28 ; cp05 ; zhu ;
lsm61 . For higher dimension cases the JW-like transformation has been
constructed by different methodsjw . However the transformation is very
complex and the calculation is difficult. Hence instead of a general
calculation, we focus on the nearest neighbor two lattices in this paper. In
this situation, the $SU(2)$ algebra can be readily constructed
$\displaystyle\sigma_{\mathbf{i}}^{+}=(\sigma_{\mathbf{i}}^{x}+i\sigma_{\mathbf{i}}^{y})/2=c^{\dagger}_{\mathbf{i}}$
$\displaystyle\sigma_{\mathbf{i}}^{-}=(\sigma_{\mathbf{i}}^{x}-i\sigma_{\mathbf{i}}^{y})/2=c_{\mathbf{i}}$
$\displaystyle\sigma_{\mathbf{i}}^{z}=2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1$
$\displaystyle\sigma_{\mathbf{i}+1}^{+}=(\sigma_{\mathbf{i}+1}^{x}+i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c^{\dagger}_{\mathbf{i}+1}$
$\displaystyle\sigma_{\mathbf{i}+1}^{-}=(\sigma_{\mathbf{i}+1}^{x}-i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c_{\mathbf{i}+1}$
$\displaystyle\sigma_{\mathbf{i}+1}^{z}=2c_{\mathbf{i}+1}^{\dagger}c_{\mathbf{i}+1}-1$
(13)
in which $\mathbf{i}+1$ denotes the nearest neighbor lattice for site
$\mathbf{i}$. This point can be explained as the following. The difficulty for
the JW transformation in higher dimension case comes from the absence of a
natural ordering of particles. However when one focuses on the nearest
neighbored particle, this difficulty does not appear since for a definite
direction the nearest neighbor particle is unique (for non-nearest neighbored
case one have to consider the effect from the other particles). Then the
correlation functions for the ground state are in this case
$\displaystyle p_{00}$ $\displaystyle=$ $\displaystyle
1,p_{30}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2};p_{03}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{k(i+1)}}^{2};$
$\displaystyle p_{11}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}-g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}+g_{\mathbf{k(i+1)}});$
$\displaystyle p_{22}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}+g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}-g_{\mathbf{k(i+1)}})$
$\displaystyle p_{33}$ $\displaystyle=$
$\displaystyle(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{ki}})(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{k(i+1)}})$
$\displaystyle+\frac{4}{L^{2}}\sum_{\mathbf{k,k^{\prime}}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}g_{\mathbf{k^{\prime}i}}g_{\mathbf{k^{\prime}(i+1)}}-h_{\mathbf{ki}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}(i+1)}}g_{\mathbf{k^{\prime}(i+1)}}$
$\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$ (14)
### 2.2 Geometric Phase
Following the method in Refs.cp05 ; zhu , one can introduce a globe rotation
$R(\phi)=\exp[i\phi\sum_{i}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}]$ to obtain
the geometric phase(GP). Then we have Hamiltonian with parameter $\phi$
$H(\phi)=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{ij}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}e^{2i\phi}+\text{h.c.}),$
(15)
and the ground state becomes $|g(\phi)\rangle=R(\phi)|g\rangle$. GP is defined
as berry
$\displaystyle\gamma_{g}$ $\displaystyle=$ $\displaystyle-i\int d\phi\langle
g(\phi)|\frac{\partial}{\partial\phi}|(\phi)\rangle$ (16) $\displaystyle=$
$\displaystyle\frac{\phi}{L}\sum_{\mathbf{i}}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$
Regarding to Eq.(15), one only require $\phi=\pi$ for a cycle evolution. Hence
one has
$\gamma_{g}=\frac{\pi}{L}\sum_{i}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}=\frac{1}{L}\sum_{\mathbf{i}}\gamma_{g\mathbf{i}}$.
### 2.3 GP vs. Concurrence
At a glance of Eq.(2.1) and Eq.(16), GP and concurrence both are related
directly to correlation functions. Hence it is tempting to find the relation
between the two quantities, which would benefit to the understanding of the
physical meaning of concurrence.
According to Eqs.(2.1) and (2.1), the following inequality can be obtained
(see Appendix for details of calculations)
$\displaystyle c_{I}$ $\displaystyle\leq$
$\displaystyle\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}$
$\displaystyle c_{II}$ $\displaystyle\leq$ $\displaystyle
1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$
(17)
For the first inequality, a much tighter bound is difficult to find. While if
the average of $c_{II}$ over all site $\mathbf{i}$ is considered, $c_{II}\leq
1-\frac{1}{2L^{3}\pi^{2}}\sum_{i}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$.
Fortunately in the following examples $c_{I}$ is always negative. Although the
existence of this defect, in our own points, the relation between GP and
concurrence have been displayed genuinely from the inequality above.
## 3 GP and Concurrence in Higher Dimensional $XY$ model
The previous section presents the general discussion of GP and concurrence in
free fermion lattice system Eq.(1). In this section a concrete model would be
checked explicitly, of which the Hamiltonian is
$H=\sum_{\langle\mathbf{i,j}\rangle}[c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}-\gamma(c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}^{\dagger}+\text{h.c.})]-2\lambda\sum_{\mathbf{i}}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}},$
(18)
in which $\langle\mathbf{i,j}\rangle$ denotes the nearest-neighbor lattice
sites and $c_{\mathbf{i}}$ is fermion operator. This Hamiltonian, first
introduced in Ref.li06 , depicts the hopping and pairing between nearest-
neighbor sites in hypercubic lattice systems, in which $\lambda$ is the
chemical potential and $\gamma$ is the pairing potential. Eq.(18) could be
considered as a $d$-dimensional generalization of 1D XY model. However for
$d>1$ case, this model shows different phase features li06 .
The Hamiltonian can be diagonalized by introducing the $d$-dimensional Fourier
transformation with periodic boundary condition in momentum space li06
$H=\sum_{\mathbf{k}}2t_{\mathbf{k}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}-i\Delta_{\mathbf{k}}(c_{\mathbf{k}}^{\dagger}c_{-\mathbf{k}}^{\dagger}-\text{h.c.}),$
(19)
in which $t_{\mathbf{k}}=\sum_{\alpha=1}^{d}\cos k_{\alpha}-\lambda$ and
$\Delta_{\mathbf{k}}=\gamma\sum_{\alpha=1}^{d}\sin k_{\alpha}$. With the help
of Bogoliubov transformation, one obtains
$H=\sum_{\mathbf{k}}2\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$
(20)
in which
$\Lambda_{\mathbf{k}}=\sqrt{t_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$.
Based on the degeneracy of the eigenenergy $\Lambda_{\mathbf{k}}=0$, the phase
diagram can be determined clearlyli06 ; When $d=2$, the phases diagram should
be identified as two different situations; for $\gamma=0$, the degeneracy of
the ground state occurs when $\lambda\in[0,2]$, whereas the gap above the
ground state is non-vanishing for $\lambda>2$. However for $\gamma\neq 0$
three different phases can be identified as $\lambda=0$, $\lambda\in(0,2]$ and
$\lambda>2$. The first two phases correspond to case that the energy gap above
the ground state vanishes, whereas not for $\lambda>2$. One should note that
$\lambda=0$ means a well-defined Fermi surface with $k_{x}=k_{y}\pm\pi$, whose
symmetry is lowered by the presence of $\lambda$ term. For $d=3$ two phases
can be identified as $\lambda\in[0,3]$ with the vanishing energy gap above the
ground state and $\lambda>3$ with a non-vanishing energy gap above ground
state. In a word the critical points can be identified as
$\lambda_{c}=d(d=1,2,3)$ for any anisotropy of $\gamma$, and $\lambda=0$ for
$d=2$ with $\gamma\neq 0$. One should note that since the $\gamma^{2}$
dependence of $\Lambda_{\mathbf{k}}$, the sign of $\gamma$ does not matter.
Hence the plots below are only for positive $\gamma$.
The correlation functions between nearest-neighbor lattice sites would play a
dominant role in the transition between different phases because of the
nearest-neighbor interaction, similar to the case in XY model oo02 . Then it
is expected that the pairwise entanglement is significant in this model. In
the following, concurrence for the nearest-neighbor sites of ground state is
calculated for $d=2,3$ respectively. The geometric phase of ground state is
also calculated by imposing a globe rotation $R(\phi)$. our calculation shows
that both quantities show interesting singularity closed to the boundary of
different phases.
### 3.1 Concurrence
For $d>1$ case, the nearest-neighbor lattice sites appear in different
directions. In order to eliminate the dependence of orientations, the
calculation of correlation functions Eqs.(2.1) is implemented by averaging in
all directions. With the transformation Eq.(2.1), one can determine under the
thermodynamic limit
$\displaystyle p_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin
k_{\alpha}-t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$
$\displaystyle p_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin
k_{\alpha}+t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$
$\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$
$\displaystyle p_{03}$ $\displaystyle=$ $\displaystyle
p_{30}=p_{3}=\frac{1}{(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}\frac{t_{k}}{\Lambda_{k}}$
$\displaystyle p_{33}$ $\displaystyle=$ $\displaystyle
p_{3}^{2}-(\frac{p_{11}+p_{22}}{2})^{2}+(\frac{p_{11}-p_{22}}{2})^{2}$ (21)
$d=2$ Our calculation shows that $c_{I}$ is negative. So in Fig. 1, only
$c_{II}$ and its derivative with $\lambda$ are numerically illustrated. In
order to avoid the ambiguity because of the cutoff in the definition of
concurrence, the derivative of $c_{II}$ with $\lambda$ is depicted in all
region whether $c_{II}$ positive or notyang . Obviously the singularity for
$\partial c_{II}/\partial\lambda$ can be found at the point $\lambda=0,2$
respectively, which are consistent with our knowledge about phase transitions.
$d=3$ Similar to the case of $d=2$, our calculation shows $c_{I}<0$. Only
$c_{II}$ and its derivative with $\lambda$ are numerically displayed in Fig.2.
Different from the case of $d=2$, no singularity of the first derivative of
$c_{II}$ with $\lambda$ is found at $\lambda=3$. While a cusp appears at
$\lambda=1$. A further calculation demonstrates that the second derivative of
$c_{II}$ is divergent genuinely at exact $\lambda=3$, as shown in Figs.2(c).
which means the phase transition at this points. Furthermore our numerical
calculations show that $\partial^{2}c_{II}/\partial\lambda^{2}$ is finite at
$\lambda=1$, as shown in Figs.2(b). Hence one cannot attribute this feature to
the phase transition. The similar feature has been found in the previous
studies oo02 ; yang ; gu . However the underlying physical reason is unclear
in general. But this special feature is not unique for concurrence; van Hove
singularity in solid state physics displays the similar feature, which is
because of the vanishing of the moment-gradient of the energy. Although we
cannot established the direct relation between these two issues because of the
bad definition of the moment-gradient of the energy when degeneracy happening,
we affirm that this feature is not an accident and the underlying physical
reason is still to be found.
In a word the discussion above first demonstrates the exact connection between
concurrence and quantum phase transitions in high-dimensional many body
systems. However a question is still open; what the physical interpretation of
concurrence is in many-body systems. In this study, we includes the negative
part of $c_{II}$ to identify the phase diagram in free-fermion systems. In
general, it is believed that the negative $c_{II}$ means no entanglement
between two particles and then include no any useful information about state.
But from the discussion one can note that the omission of the negative part of
$c_{II}$ would lead to incorrect results. Moreover, for $\gamma=0$, our
calculations show that $c_{I},c_{II}$ always are zero, and so one cannot
obtain any the phase transition information from pair wise entanglement in
this case. Further discussions will be presented in the final part of this
paper.
### 3.2 Geometric Phase
Geometric phase manifests the structure of Hilbert in the system and has
intimate relation to the degeneracy. GP, defined in Eq. (16) by imposing a
globe rotation $R(\phi)$ on ground state $|g\rangle$ is calculated in this
section. After some algebra, one obtains
$\gamma_{g}=\frac{\pi}{2(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha=1}^{d}dk_{\alpha}(1-\frac{t_{k}}{\Lambda_{k}}).$
(22)
$d=2$ In Fig.3, $\gamma_{g}$ and its derivative with $\lambda$ are displayed
explicitly. Obviously one notes that $\partial\gamma_{g}/\partial\lambda$
shows the singularity closed to $\lambda=0,2$, which are exactly the phase
transition points of Hamiltonian Eq.(18). An interesting observation is that
closed to these points, both GP and concurrence $c_{II}$ show the similar
behaviors.
$d=3$ GP and its derivative are plotted explicitly in Fig.(4). One should note
that there is a platform below $\lambda=1$ for
$\partial\gamma_{g}/\partial\lambda$, as shown in Fig.4(a), but a further
calculation shows that $\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is
continued (Fig.4(b)) and $\partial\gamma_{g}/\partial\lambda$ has no
divergency at this point. This phenomena is very similar to the case of
concurrence (see Fig.2(b, c)). As expected,
$\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is divergent at exact
$\lambda=3$, which means a phase transition happens at this point. Together
with respect of the case of $d=2$, it makes us a suspect that GP and
concurrence in our model have the same physical origination.
Furthermore for $\gamma=0$, GP fails to mark the phase transition too. This is
similar to the case of concurrence, but has different physical reason. The
further discussion is presented in the next section.
## 4 Discussion and Conclusions
The pairwise entanglement and geometric phase for ground state in
$d$-dimensional ($d=2,3$) free-fermion lattice systems are discussed in this
paper. By imposing the transformation Eq.(2.1), the reduce two-body density
matrix for the nearest neighbor particles can be determined exactly for any
dimension, and the concurrence is also calculated explicitly. Furthermore
geometric phase for ground state, obtained by introducing a globe rotation
$R(\phi)$, has also been calculated. Given the known results for XY model oo02
; cp05 ; zhu , our calculations show again that both GP and concurrence
display intimate connection with the phase transitions. Moreover an inequality
relation between concurrence and geometric phase is also presented in Eq.
(2.3). The similar scaling behaviors at the transition point $\lambda=3$ has
also been shown in Figs. 5. These facts strongly mean the intimate connection
between the two items. This point can be understand by noting that both of
them are connected to the correlation functions, as shown in Eqs. (2.1) and
(16).
An interesting point in our study is that in order to obtain all information
of phase diagram in model Eq.(18), the negative part of $c_{II}$ has to be
included to avoiding the confusion because of the mathematical cutoff in the
definition of concurrenceyang . In general, it is well accepted that the
negative part of $c_{II}$ gives no any information of quantum pairwise
entanglement, and then is considered to be meaningless. However, in our
calculation, the negative part of $c_{II}$ appears as an indispensable
consideration to obtain the correct phase information. This point means that
the pairwise entanglement does not provide the all information about the
system since the two-body reduced density operator throw away much
information.
As for the geometric phase, defined in Eq. (16), it is obvious that
$\gamma_{g}$ can tell us the happening of phase transition at the point, where
$\gamma_{g}$ display some kinds of singularity. However it cannot
distinguished the degenerate region from the nondegenerate, as shown in Figs.
3 and 4. Recently GP imposing by the twist operator in many-body systems is
introduced as an order parameter to distinguish the phases cui08 ; hkh08 . For
the free-fermion lattice system, this GP have also calculated and shows the
intimate connection with the vanishing of energy gap above the ground state.
However the boundary between the two different phases becomes obscure with the
increment of dimensionality in that discussion cui08 , and moreover it cannot
distinguish the phase transition not come from the degeneracy of the ground-
state energy. While the geometric phase imposing by the globe rotation
$R(\phi)$ clearly demonstrate the existence of this kind of phase transition,
as shown in Fig.3, whether originated from the degeneracy or not. In fact this
point can be understood by noting the intimate relation between $\gamma_{g}$
and correlation functions. It maybe hint that one has to find different
methods for different many-body systems to identify the phase diagram.
Although the intimate relationship of concurrence and GP with phase
transitions in the model Eq.(18), a exceptional happens when $\gamma=0$, in
which $c_{I},c_{II}$ are zero and GP is a constant independent of $\lambda$.
From Eq.(18), $\gamma=0$ means the hopping of particles is dominant, and the
position of particle becomes meaningless. Since the calculation of concurrence
depend on the relative position of lattice site, the pairwise entanglement is
disappearing. However one could introduce the spatial entanglement to detect
the phase transition in this casehav07 . For GP, $\gamma=0$ means the
emergency of new symmetry. One can find
$[\sum_{\mathbf{i}}c^{\dagger}_{\mathbf{i}}c_{\mathbf{i}},H]=0$ in this case,
which leads to the failure of $R(\phi)$ for construction of nontrivial GP.
Finally we try to transfer two viewpoints in this paper. One is that
concurrence and geometric phase can be used to mark the phase transition in
many-body systems since both of them are intimately connected to the
correlation functions. The other is that concurrence and the geometric phase
are connected directly by the inequality Eq. (2.3). Then it is interesting to
extend this relation to multipartite entanglement in the future works, which
would be helpful to establish the physical understanding of entanglement.
###### Acknowledgements.
The author (Cui) would appreciate the help from Dr. Kai Niu (DLUT) and Dr.
Chengwu Zhang (NJU) in the numerical calculations and permission of the usage
of their powerful computers. We also thank greatly the enlightening discussion
with Dr. Chong Li (DLUT). Especially we thank the first referee for his/her
important hint for the van Hove singularity. This work is supported by the
Special Foundation of Theoretical Physics of NSF in China, Grant No.
10747159\.
## APPENDIX
For the first inequality, one should note
$\displaystyle|p_{11}+p_{22}|$ (23) $\displaystyle=$
$\displaystyle\frac{4}{L}|\sum_{\mathbf{k}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}|\leq\frac{4}{L}\sum_{\mathbf{k}}|h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}|$
$\displaystyle\leq$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+h^{2}_{\mathbf{k(i+1)}})=\frac{2}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})}).$
From inequality $\sqrt{x^{2}-y^{2}}\geq|x|-|y|(|x|>|y|)$, one reduces
$\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]\geq|1+p_{33}|-|p_{30}+p_{03}|.$
(24)
Then one obtains
$c_{I}\leq\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})+\frac{1}{2}(|p_{30}+p_{03}|-|1+p_{33}|).$
(25)
However a much tighter bound is difficult to decide because of the complexity
of $p_{33}$.
For the second inequality, it can be obtained easily by observing
$\displaystyle p_{33}\leq
1-\frac{1}{L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}^{2}-\gamma_{g(\mathbf{i+1})}^{2}),$
(26)
in which we have used the relation $2ab\leq a^{2}+b^{2}$. Then $1-p_{33}$ is
non-negative and
$\displaystyle c_{II}$ $\displaystyle=$
$\displaystyle\frac{2}{L}\sum_{\mathbf{k}}|h_{\mathbf{ki}}g_{\mathbf{k(i+1)}}|+\frac{p_{33}-1}{2}$
(27) $\displaystyle\leq$
$\displaystyle\frac{1}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+g^{2}_{\mathbf{k(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$
$\displaystyle\leq$ $\displaystyle
1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$
in which
$1/L\sum_{\mathbf{k}}g_{\mathbf{ki}}^{2}=1-1/L\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$
is used.
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\begin{overpic}[width=170.71652pt]{1.eps} \put(-5.0,40.0){\large$c_{II}$}
\put(50.0,0.0){\large$\lambda$}\put(97.0,50.0){\large\begin{rotate}{-90.0}$\partial
c_{II}/\partial\lambda$\end{rotate}} \end{overpic} Figure 1: $c_{II}$
($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) vs. $\lambda$
when $d=2$. We have chosen $\gamma=1$ for this plot.
\begin{overpic}[width=170.71652pt]{2a.eps}
\put(15.0,60.0){(a)}\put(-5.0,40.0){\large$c_{II}$}
\put(45.0,0.0){\large$\lambda$}
\put(94.0,50.0){\large\begin{rotate}{-90.0}$\partial
c_{II}/\partial\lambda$\end{rotate}} \end{overpic}
\begin{overpic}[width=128.0374pt]{2b.eps} \put(18.0,78.0){(b)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}/\partial\lambda^{2}$\end{rotate}}
\put(40.0,0.0){\large$\lambda$}
\end{overpic}\begin{overpic}[width=136.5733pt]{2c.eps}
\put(22.0,78.0){(c)}\put(40.0,0.0){\large$\lambda$} \end{overpic}
Figure 2: $c_{II}$ ($\bigcirc$) and its derivative with $\lambda$
($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this
plot. The second derivative of $c_{II}$ with $\lambda$ are also displayed in
this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$ (c).
\begin{overpic}[width=170.71652pt]{3.eps}
\put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}}
\put(50.0,0.0){\large$\lambda$}\put(98.0,53.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}}
\end{overpic} Figure 3: $\gamma_{g}$ ($\bigcirc$) and its derivative with
$\lambda$ ($\triangle$) vs. $\lambda$ when $d=2$. We have chosen $\gamma=1$
for this plot.
\begin{overpic}[width=170.71652pt]{4a.eps}
\put(13.0,60.0){(a)}\put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}}
\put(50.0,0.0){\large$\lambda$}
\put(93.0,52.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}}
\end{overpic}\begin{overpic}[width=113.81102pt]{4b.eps} \put(13.0,77.0){(b)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}}
\put(35.0,0.0){\large$\lambda$}
\end{overpic}\begin{overpic}[width=130.88284pt]{4c.eps}
\put(22.0,78.0){(c)}\put(42.0,0.0){\large$\lambda$} \end{overpic}
Figure 4: $\gamma_{g}$ ($\bigcirc$) and its derivative with $\lambda$
($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this
plot. The second derivative‘ of $\gamma_{g}$ with $\lambda$ are also displayed
in this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$
(c).
\begin{overpic}[width=125.19194pt]{5a.eps} \put(55.0,78.0){(a)}
\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}\partial\lambda^{2}$\end{rotate}}
\put(50.0,0.0){\large$\log|\lambda-\lambda_{c}|/\lambda_{c}$}
\end{overpic}\begin{overpic}[width=130.88284pt]{5b.eps}
\put(57.0,78.0){(b)}\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}}
\end{overpic}
Figure 5: The scaling of GP and concurrence for 3D case closed to the critical
point $\lambda_{c}=3$. We have chosen $\gamma=1$ for this plot.
|
# Graph-Based Optimization for Technology Pathway Analysis:
A Case Study in Decarbonization of University Campuses
Blake Lopez†, Jiaze Ma†, Victor M. Zavala†
†Department of Chemical and Biological Engineering
University of Wisconsin - Madison, 1415 Engineering Dr, Madison, WI 53706, USA
Corresponding Author<EMAIL_ADDRESS>
###### Abstract
Industrial sectors such as urban centers, chemical companies, manufacturing
facilities, and microgrids are actively exploring strategies to help reduce
their carbon footprint. For instance, university campuses are complex urban
districts (involving collections of buildings and utility systems) that are
seeking to reduce carbon footprints that originate from diverse activities
(e.g., transportation operations and production of heating, cooling, and power
utilities). This work presents an optimization framework to identify
technology pathways that enable decarbonization of complex industrial sectors.
The framework uses a graph abstraction that compactly captures
interdependencies between diverse products and technologies as well as diverse
externalities (e.g., market, policy, and carbon prices). Duality analysis
reveals that the formulation can be interpreted as an economy, market, or
value chain that uses technologies to generate economic value (wealth) by
transforming basic products into higher value products. This interpretation
also reveals that the formulation identifies pathways that maximize the profit
of stakeholders, helps reveal the inherent value (prices) of intermediate
products, and helps analyze the impact of externalities and technology
specifications on product values. Our developments are illustrated via a case
study involving a prototypical university campus that seeks to identify
pathways that reduce its carbon footprint (e.g., via electrification and
deployment of hydrogen technologies). We use the framework to determine carbon
tax values, technology specifications, and investment budgets that activate
different technology pathways and that achieve different levels of
decarbonization.
Keywords: optimization; technology pathways; graph theory; decarbonization
## 1 Introduction
The global mean surface temperature reached 1℃ above pre-industrial times in
2017 and is projected to reach 1.5℃ as early as 2030 [31]. Significant efforts
need to be made to decrease greenhouse gas emissions to limit warming. The
most notable effort to accomplish this is the Paris agreement, which seeks to
limit warming to at most 2℃ by providing technological and financial support
as well as capacity building between the 192 countries in the agreement [34].
To satisfy the goals of global agreements, industrial sectors such as chemical
companies, manufacturing facilities, farms, microgrids, and urban centers need
to shift their operations so as to minimize CO2 emissions. For instance,
ammonia production facilities are exploring the implementation of electrolysis
technologies that harness renewable power to replace hydrogen sourced from
natural gas, and utility companies are relying on falling renewable costs and
energy storage technologies to mitigate carbon emissions [29, 43, 42].
An important and representative set of industrial systems that is actively
seeking to decarbonize operations are university campuses. To give some
perspective into the magnitude of these systems, we note that there are over
16.5 million students in the US reside on university campuses [2]. Campuses
involve large collections of buildings that consume heating, cooling, power,
and transportation services. Many university campuses own and operate utility
systems that provide steam, hot water, and chilled water to buildings and
research facilities. Currently, most of these systems are powered using
combined heat and power (CHP) systems that run on natural gas [39]. Electrical
power is also used for diverse essential services (e.g., ventilation,
lighting, circulation, lectures) and for resource-intensive equipment such as
computational facilities (e.g., computing clusters and data centers). The
generation and distribution of all these utilities generate a massive carbon
footprint. Moreover, universities also typically operate transportation
services, which further contribute to the carbon footprint. In essence,
university campuses are representative systems in that they combine elements
of urban centers and manufacturing facilities. It has been estimated that
university campuses in the US emit 7.7 MTCO2e per student and are responsible
for 2% of the 5,222 million MTCO2e of the US greenhouse gas emissions [39,
44]. Notably, these emissions are higher than those of ammonia production.
Technologies such as renewable power generation, heat pumps, electric and
hydrogen-based transportation vehicles, and energy storage systems are
actively being studied as means to help decrease carbon footprints [19, 28,
45, 37, 27]. The number of new technologies that can be used is steadily
increasing and complicates decision-making processes, because it is important
to understand how new technologies will interact with existing infrastructure
and operations. In the context of university campuses, cost-minimization
models have been proposed to determine the energy mix, capacities, and storage
requirements for a university campus (model is cast as a mixed integer
nonlinear program) [40]. This modeling approach requires detailed data for
investment costs, operating costs, operating lifetimes, and scaling factors
for the technologies proposed and time-dependent demand data to determine a
minimized cost scenario with no emissions. Other modeling approaches have used
multiobjective optimization to determine technology operations that can help
reach emissions reduction targets, determine lowest cost for maximum renewable
use, and determine cost of reducing emissions [35, 33, 25]. Existing
optimization models are typically intended to provide long-term investment
plans to achieve decarbonization. Detailed models are ought to be used to make
important investment decisions but these often hindered by the fact that there
might not be sufficient data to evaluate emerging technologies; in addition,
the use of detailed models might require significant implementation and
computational effort. As such, there is also a need for simple models that can
help screen options prior to investing resources in model detailed studies.
In this work, we provide an optimization framework for analyzing technology
pathways that can provide desired product demand targets while minimizing
supply and technology costs. The framework uses a compact, graph/network
representation that aims to facilitate the analysis of complex
interdependencies that arise between products and technologies. In the
proposed representation, we consider the supply of basic products to the
system (e.g., natural gas and electricity) and technologies that transform
these basic products to higher-value intermediate and final products (e.g.,
hydrogen and cooling water) that are delivered to consumers. The formulation
operates at a high level of abstraction, which enables capturing diverse types
of products (e.g., material, energy, labor, services) and byproducts (e.g.,
carbon emissions and waste) in a unified manner. The proposed formulation can
be viewed as a superstructure network that interconnects different
stakeholders in a system and helps determine how externalities (e.g., policy
and technology specifications) impact the system. Moreover, duality analysis
reveals that the formulation has a natural economic/market interpretation;
this reveals that the formulation aims to identify technology pathways that
maximize total economic value (by using technologies for transforming products
into higher-value products) and explains how externalities can help
activate/deactive pathways. The market interpretation also allows us to
discover inherent values (prices) for key intermediate products and helps
evaluate the impact of externalities and technology specifications on such
values. Moreover, the framework provides insights into how expanding the
domain of the system (e.g., by considering alternative products) can activate
technology pathways. The modeling framework uses minimal data on products and
technologies, which facilitates analysis and enables fast screening of diverse
technologies and externalities. We provide a case study that analyzes
decarbonization pathways for a prototypical university campus to illustrate
the developments.
## 2 Graph-Based Optimization Framework
We consider a system comprised of a set of basic, intermediate, and final
products as well as a set of technologies that transform basic products into
intermediate and final products. The goal is to determine technology pathways
that can maximize the value of served product demands, while minimizing supply
and technology costs.
A couple of optimization models are introduced to determine optimal technology
pathways that conduct desired goals. A management model will be used to
understand how suppliers, technologies, and demands interact and to determine
optimal allocations for all these stakeholders. The model uses a graph
abstraction that captures the topology/connectivity that arises from
interactions between products and technologies. The model can be interpreted
as a value chain that aims to identify pathways that maximize profit for all
stakeholders involved; this is done by using technologies to create wealth
(transforming lower-value products into higher-value products). This
interpretation will also allows us to determine the inherent value (prices) of
products, which is particularly useful in attributing value for intermediate
products and to understand how externalities (e.g., carbon prices or disposal
costs for waste products) and technology costs/efficiencies propagate through
the technology pathways and influence product prices. An investment model will
be used to prioritize the selection of pathways under constrained investment
budgets and to trade-off profits with investment costs.
Our models aim to use minimum specification data for technologies (e.g.,
efficiencies, capacities, operating costs, investment costs), so as to provide
$``$high-level$"$ picture of the technology landscape and on potential factors
that influence their economic viability. This is also, in part, motivated by
the fact that there is often limited data available for existing technologies.
As expected, any simplification to the representation of technologies will
come with a downside of inaccuracy. Our high-level abstraction aims to provide
an intuitive approach that helps navigate complex interdependencies between
products, technologies, and externalities.
The proposed model can be interpreted as a value chain in which there is no
transportation and spatial/geographical context; specifically, we note that
our model is a simplification of the multi-product supply chain model
presented in [38, 41]. This observation is important, as the model proposed
here can provide valuable preliminary insights that can inform the development
of more sophisticated models; for instance, it can help determine technology
pathways that will or will not be selected in supply chain designs.
### 2.1 Management Model
We define a system comprised of a set of products $\mathcal{P}$, set of
suppliers $\mathcal{S}$ that offer products, technologies $\mathcal{T}$ that
transform products, and consumers $\mathcal{D}$ that request products. These
elements are interconnected via a graph that captures product-technology
dependencies. We can think of this system as an economy, market, or value
chain that aims to generate economic value (wealth) by transforming basic
products into higher value products. In other words, we think about this
system as an economy in which elements (suppliers, consumers, technologies)
are stakeholders that provide/request services and that aim to generate
wealth. Factors that affect the ability of this economy from generating wealth
include costs of basic products, technology factors (e.g., costs, capacities,
and efficiencies), and externalities (e.g., policy, external markets, and
taxes).
Each supplier $i\in\mathcal{S}$ has an offered value
$\alpha_{i}\in\mathbb{R}$, associated allocation $s_{i}\in\mathbb{R}$ (flow of
product provided to the system), an available capacity
$\bar{s}_{i}\in\mathbb{R}$ (maximum product amount that it can provide), and a
supplied product $p(i)\in\mathcal{P}$. Each consumer $j\in\mathcal{D}$ has an
associated offered value $\alpha_{j}\in\mathbb{R}$, allocation
$d_{j}\in\mathbb{R}$ (flow of product extracted from the system), an available
capacity $\bar{d}_{j}\in\mathbb{R}$ (maximum product amount that it can take),
and a requested product $p(j)\in\mathcal{P}$. We denote the set of suppliers
or consumers that provide/request a given product $p\in\mathcal{P}$ as
$\mathcal{S}_{p}\subseteq\mathcal{S}$ and
$\mathcal{D}_{p}\subseteq\mathcal{D}$, respectively.
Each technology $k\in\mathcal{T}$ has an offered value
$\alpha_{k}\in\mathbb{R}$ (service cost), has an allocation flow
$t_{k}\in\mathbb{R}$ (flow processed by the technology); this flow corresponds
to a reference (input) product $p(k)\in\mathcal{P}$ and an available
processing capacity $\bar{t}_{k}\in\mathbb{R}$ (maximum amount of reference
product that it can process). Technologies are key assets, as they generate
wealth by conducting transformation of products; this is captured using the
transformation factors
$\gamma_{k,p}\in\mathbb{R},\;k\in\mathcal{T},p\in\mathcal{P}$. A positive
transformation factor ($\gamma_{k,p}>0$) indicates that product $p$ is
produced by technology $k$, a negative factor ($\gamma_{k,p}<0$) indicated
that the product is consumed by the technology, and a zero factor
$\gamma_{k,p}=0$ indicates that the product is neither produced nor consumed
(does not participate in the technology). The set of technologies that
generate or consume a specific product $p\in\mathcal{P}$ are denoted as
$\mathcal{T}_{p}\subseteq\mathcal{T}$. Transformation factors play a key role
in dictating the behavior of the system, as they provide measures of
efficiency for technologies; moreover, these factors capture product-
technology interdependence (connectivity) and capture the generation of
desirable (e.g., value-added) products and undesirable (e.g., waste)
byproducts. In other words, transformation of products that occur in
technologies define the topology of the system graph; this topology captures
diverse pathways that might exist to obtain specific products and encodes key
information that explains how diverse externalities propagate through the
system.
The data for suppliers, consumers, and technologies is summarized into a graph
(a superstructure) that captures all product-technology interdependencies. The
goal is to determine pathways in this superstructure that generate maximum
wealth, while filtering out those that do not generate wealth. Figure 1
provides a simple illustration on how elements are connected in the graph and
how competing technology pathways emerge. Moreover, this aims to illustrate
how externalities (emissions costs/taxes) can favor some pathways over others.
A more complex graph showing diverse pathways to decarbonize a university
campus is presented in Figure 2. This illustrates how complexity quickly
arises due to the presence of diverse pathways and product-technologies
connectivity; as such, it is necessary to use systematic techniques to
identify optimal pathways.
Figure 1: Illustration of a simple graph in which supplies of basic products
(gasoline and solar radiation) are used by technologies (gas-powered vehicles,
electricity vehicles, and solar panels) to satisfy demand of higher value
products (travel distance) and that generate undesired products (CO2
emissions). In this case, electrical power generated by the photovoltaic
panels is an intermediate product that is used by the electric vehicle to
generate a value-added and final product (distance). If the cost of CO2
emissions is not taken into account (and/or gasoline is inexpensive), the gas
vehicle pathway will be preferred. On the other hand, when emissions costs are
taken into account and/or renewable power is inexpensive, the electric vehicle
pathway will be preferred. Figure 2: Graph showing diverse suppliers,
technologies, and consumers that participate in a university campus. Complex
interdependencies/connectivity arises between products and technologies and it
is thus nonobvious what are the best pathways. See Table 1 for technology
descriptions.
We now propose an optimization formulation that helps identify optimal
technology pathways from the graph. The formulation is given by:
$\displaystyle\max_{(s,d,t)}$
$\displaystyle\quad\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}$
(1a) s.t
$\displaystyle\quad\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}=0,\quad
p\in\mathcal{P},\quad(\pi_{p})$ (1b) $\displaystyle\quad 0\leq
s_{i}\leq\bar{s}_{i},\quad i\in\mathcal{S}$ (1c) $\displaystyle\quad 0\leq
d_{j}\leq\bar{d}_{j},\quad j\in\mathcal{D}$ (1d) $\displaystyle\quad 0\leq
t_{k}\leq\bar{t}_{k},\quad k\in\mathcal{T}$ (1e)
This $``$management$"$ model aims to identify product allocations $(s,d,t)$
for all system elements that maximize the total surplus (1b). The total
surplus balances the value of demand served (to be maximized) and the costs of
all supplies and technologies (to be minimized). We will see (via duality
analysis) that the total surplus is equivalent to the total system profit
(revenue minus cost) and captures the profit of all system elements.
The constraints (1b) encode the graph that captures the connectivity between
products and technologies. In this graph, products are interpreted as nodes
and we enforce conservation of product at such nodes. Specifically, for each
product $p\in\mathcal{P}$, we must have that all input by suppliers,
generation/consumption by technologies, and extraction by consumers must be
balanced.
The dual variables $\pi_{p},p\in\mathcal{P}$ of the balance constraints (1b)
can be interpreted as inherent values (prices) for the products. Specifically,
the dual variables capture how valuable a given product is in the economy. For
instance, a high-value product can be critical because it enables diverse
technology pathways and generate wealth via sales of final products;
conversely, a low-value product can be of low priority (or even irrelevant)
for the generation of wealth in the system. In fact, we will see later that
prices can have negative values, indicating that certain products (e.g.,
waste) can be detrimental to the creation of wealth in the economy. However,
products with negative values might be necessary to achieve alternate goals in
the system, such as mitigation of social and environmental impacts by the
economy. Determining prices for intermediate products is particularly
important, as such products typically do not have external markets (they are
means to an end).
We can gain important insights into the economic properties of the
optimization problem by analyzing the (partial) Lagrangian dual function:
$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}+\sum_{p\in\mathcal{P}}\pi_{p}\left({\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}}\right).$
(2)
If we now consider the following identities:
$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{i\in\mathcal{S}_{p}}{s_{i}}=\sum_{i\in\mathcal{S}}{\pi_{i}s_{i}}$
(3a)
$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}=\sum_{k\in\mathcal{T}}{\pi_{k}t_{k}}$
(3b)
$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{j\in\mathcal{D}_{p}}{d_{j}}=\sum_{j\in\mathcal{D}}{\pi_{j}d_{j}},$
(3c)
we can see that the Lagrange function can be rewritten as:
$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{(\alpha_{j}^{\mathcal{D}}-\pi_{j})d_{j}}+\sum_{i\in\mathcal{S}}{(\pi_{i}-\alpha_{i}^{\mathcal{S}})s_{i}}+\sum_{k\in\mathcal{T}}{(\pi_{k}-\alpha^{\mathcal{T}}_{k})t_{k}}.$
(4)
Here, we use the short-hand notation $\pi_{i}:=\pi_{p(i)}$ to denote the price
of the product that supplier $i\in\mathcal{S}$ provides. We also define the
prices of the products for the consumers as $\pi_{j}:=\pi_{p(j)}$ and we
define $\pi_{k}:=\sum_{p\in\mathcal{P}}\gamma_{k,p}\pi_{p}$ as the technology
price/value (this is the weighted sum of its input and output products,
weighted by the transformation factors). If we now define the profit
functions:
$\displaystyle\phi_{i}$ $\displaystyle:=(\pi_{i}-\alpha_{i})s_{i},\quad
i\in\mathcal{S}$ (5a) $\displaystyle\phi_{k}$
$\displaystyle:=(\pi_{t}-\alpha_{k})t_{k},\quad k\in\mathcal{T}$ (5b)
$\displaystyle\phi_{j}$ $\displaystyle:=(\alpha_{j}-\pi_{j})d_{j},\quad
j\in\mathcal{D},$ (5c)
we can see that the Lagrange function can be rewritten as:
$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{\phi_{j}}+\sum_{i\in\mathcal{S}}{\phi_{i}}+\sum_{k\in\mathcal{T}}{\phi_{k}}.$
(6)
Because the optimization problem is a linear program, we can assume that
strong duality holds and thus its solution can be determined by solving the
Lagrangian dual problem:
$\displaystyle\max_{\pi}\max_{(s,d,t)\in\mathcal{C}}\mathcal{L}(s,d,t,\pi)$
(7)
Here, the set $\mathcal{C}$ captures the capacity constraints (1c),(1d), and
(1e). From these observations, we can see that the optimization model is
seeking to maximize the total profit of all system elements (suppliers,
consumers, and technologies). Moreover, because strong duality holds, we have
that the optimal allocations and prices are such that total surplus equals the
total profit:
$\displaystyle\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}=\sum_{j\in\mathcal{D}}{\phi_{j}}+\sum_{i\in\mathcal{S}}{\phi_{i}}+\sum_{k\in\mathcal{T}}{\phi_{k}}.$
(8)
The total surplus can thus be interpreted as the system-wide profit; this
profit in turn can be interpreted as the total wealth created by the system
(i.e., the system can be interpreted as a market or an economy).
Lagrangian duality analysis also reveals the role of the dual variables in
remunerating elements. Specifically, the profit of the supplier is given by
difference between the actual payment ($\pi_{i}s_{i}$) and the expected cost
($\alpha_{i}s_{i}$); as such, the supplier desires that
$\pi_{i}\geq\alpha_{i}$ (this guarantees non-negative profit) and that
$\pi_{i}$ is as large as possible (to maximize its profit). For the consumers,
we have that the profit is given by the difference between the expected cost
($\alpha_{j}d_{j}$) and the actual payment ($\pi_{j}d_{j}$); as such, we the
consumer desired that $\pi_{j}\leq\alpha_{j}$ (this guarantees non-negative
profit) and that $\pi_{j}$ is as low as possible (to maximize its profit). For
the technologies we have a more interesting case; specifically, for the profit
to be non-negative, we require that $\pi_{k}\geq\alpha_{k}$; by definition, we
have that $\pi_{k}=\sum_{p\in\mathcal{P}}\gamma_{k,p}\pi_{p}$. As such, it is
important to observe that we need that the value created by a technology needs
to be higher than its operating cost $\alpha_{k}$. This can only be achieved
if the generated output products have a higher value than the consumed input
products (weighted by the transformation factors). This insight is important,
as it clearly shows how technologies are wealth creators (generate higher-
value products from lower-value products). Moreover, any technology that
cannot achieve wealth creation (due to mismatches in input/output costs or
technologies inefficiencies) will simply not participate in the economy.
Additional insight into the prices $\pi_{p},\;p\in\mathcal{P}$ generated by
the optimization problem can be determined by posing its dual problem:
$\displaystyle\min_{\pi,\lambda}\quad\sum_{i\in\mathcal{S}}{\bar{s}_{j}\lambda_{i}}+\sum_{j\in\mathcal{D}}{\bar{d}_{j}\lambda_{j}}+\sum_{k\in\mathcal{T}}{\bar{t}_{k}\lambda_{k}}$
(9a) $\displaystyle\text{s.t}\qquad\pi_{i}-\lambda_{i}\geq\alpha_{i},\quad
i\in\mathcal{S},\quad(s_{i})$ (9b)
$\displaystyle\qquad\qquad\pi_{j}-\lambda_{j}\leq\alpha_{j},\quad
j\in\mathcal{D},\quad(d_{j})$ (9c)
$\displaystyle\qquad\qquad\pi_{k}-\lambda_{k}\geq\alpha_{k},\quad
k\in\mathcal{T},\quad(t_{k})$ (9d)
This formulation reveals that the prices must be bounded by the offered
values. Here, we have that $\lambda_{i},i\in\mathcal{S}$,
$\lambda_{j},j\in\mathcal{D}$, and $\lambda_{k},k\in\mathcal{T}$ are the dual
variables assigned for the upper bound constraints of the suppliers,
consumers, and technologies. These dual variables must be positive; as such,
they indicate that the supplier and technology prices must be higher than
their offering values, while the prices for the consumers must be lower than
their offering costs. This again indicates that prices generated by the model
are such that all system elements generate a profit (or at least break even).
This also indicates that the system-wide profit must be non-negative;
specifically, the entire system must generate wealth (or at least break even);
in other words, it makes no sense to have an economy that does not generate
value.
Note also that an outcome of the optimization model is that all allocations
are zero $(s,d,t)=(0,0,0)$. This can occur, for instance, if the supplied
products and technology services are too expensive compared to values of
requested products (and thus the system cannot generate wealth). As such, the
system-wide profit can be seen as a measure of economic efficiency (a total
profit of zero indicates the economy/system/value-chain is not economically
efficient). Another possible outcome of the model is that the allocations of
some specific technology pathways are zero; this would indicate that such
pathways do not generate wealth. This also indicates that the model has the
goal of identifying pathways that generate the most value (and eliminate the
ones that do not generate value); as such, the model can be used to quickly
identify promising pathways. From all these observations, we can see that the
proposed model makes economic sense; this is important, as it facilitates
analysis and interpretation of model outcomes. For instance, it is possible to
use the framework to explore what values of technology efficiencies can help
activate pathways and generate wealth.
The primal formulation (1) of the optimization provides the physical
allocations of the products, while the dual variables
$\pi_{p},\,p\in\mathcal{P}$ from the dual formulation (1) provides economic
allocations of the products. These formulations together make up the
optimization model which takes in information from system elements, maximizes
the total profit, and returns the physical and economic allocations to the
elements as described by Figure 3.
Figure 3: Schematic representation of optimization model, indicating elements
involved as well as model inputs/outputs.
The model proposed treats products $p\in\mathcal{P}$ as abstract objects; as
such, these can represent any type of asset that is exchanged in the system
(e.g., material, energy, labor, mileage, computing service). For instance,
vehicles could take in fuel ($gal~{}gas$) to produce distance ($km$) and
emissions ($kg~{}CO_{2}$). The abstract product objects can also be used to
represent a discrete spectrum of products (e.g., green, blue, and brown
hydrogen or water of different types).
The model also treats values for products as abstract objects and we can use
these to capture a range of behaviors. For instance, offered values for
suppliers ($\alpha_{i}$) and consumers ($\alpha_{j}$) are typically positive
or non-negative. The price bounds obtained from duality indicate that, in such
a case, the product prices will be positive. However, the offering values can
can also be allowed to be negative; this provides flexibility to capture
policy and taxes or negative impact of byproducts from the system (e.g., waste
generation). For instance, one of the pathways shown in Figure 1 produces a
waste (CO2) as a byproduct of transportation. This waste CO2 has to be taken
by the environment; as such, we can interpret the environment as a consumer
that is willing to take the waste product but at a negative cost (the negative
cost can be interpreted as a carbon tax). This is analogous to how landfills
and wastewater treatment facilities operate (they charge a tipping fee to take
on waste). We can also envision a supplier that has a product that they want
to get rid of; as such, they can offer the product at a negative cost (this is
common when dealing with undesired waste). For instance, we are willing to pay
a fee for garbage to be taken away form our homes (as garbage is an undesired
product). Accounting for undesired outcomes of the economy (via negative
prices) is important in capturing social or environmental impacts of the
system.
We highlight that the optimization model does not enforces satisfaction of
demands explicitly ($d_{j}=\bar{d}_{j}$); instead, the model will only decide
to satisfy demands if this makes economic sense (i.e., if it maximizes the
total profit). For instance, it might be possible that forcing the
satisfaction of a demand of a non-valuable product will incur significant
costs that render the system non-profitable. It is also worth highlighting
that the model will select pathways between basic products and final products
that make the most economic sense; as such, the model is fully driven by
economic incentives. It is possible to modify the model to enforce demand
satisfaction in a couple of ways. For instance, we can set a large offering
value $\alpha_{j}^{\mathcal{D}}$ to demands that need to be satisfied; this is
analogous to how electricity markets operate (a high value to electricity
demands is set to ensure delivery). It is also possible to enforce strict
constraints on satisfaction of demands, but such constraints can be shown to
introduce economic inefficiencies [41], because this can force the selection
of pathways that are non-profitable.
The proposed framework is useful as a screening tool because typical
applications contain a large number of technology pathways. For instance, a
superstructure of all possible pathways for a specific application is shown in
Figure 2. This graph contains the current infrastructure and potential new
pathways to reach decarbonization for a university campus (as we will discuss
in our case study). For this example, electrolysis, fuel cells, electric
vehicles, hydrogen vehicles, hydrogen generators, and combined heat and power
systems can be added as potential technologies. This could become even more
complex as new technologies are developed or new supplies are discovered or
demands shift towards new products (thus expanding the boundary of the
system). The bottom right demand is for CO2 which has to be accounted for
analysis, as this is an undesired byproduct of the economy/system. The
proposed optimization model can be used to screen technology pathways that
make most sense under different market conditions (e.g., under different
carbon tax scenarios).
### 2.2 Investment Model
It is often of interest to restrict the selection of technology pathways based
on investment costs or budgets that might be available. We can easily extend
the proposed model to account for this; we will use this formulation to study
how available investment budgets can constraint certain technology pathways,
as well as to understand how we should prioritize technologies. We define a
binary variable $y_{k}\in\\{0,1\\},~{}k\in\mathcal{T}$ to represent if a
technology is selected with an investment cost of $\beta_{k}\in\mathbb{R}$.
This gives the formulation:
$\displaystyle\max_{s,d,t,y}$
$\displaystyle\quad\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha}_{k}t_{k}$
(10a) s.t
$\displaystyle\quad\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}=0,\quad
p\in\mathcal{P}$ (10b) $\displaystyle\quad 0\leq s_{i}\leq\bar{s}_{i},\quad
i\in\mathcal{S}$ (10c) $\displaystyle\quad 0\leq d_{j}\leq\bar{d}_{j},\quad
j\in\mathcal{D}$ (10d) $\displaystyle\quad 0\leq t_{k}\leq\bar{t}_{k}\cdot
y_{k},\quad k\in\mathcal{T}$ (10e)
$\displaystyle\quad\sum_{k\in\mathcal{T}}{\beta_{k}\cdot
y_{k}}\leq\bar{\beta}$ (10f)
The constraint (10f) enforces that the total investment cost is less than or
equal to a given budget $\mathcal{\beta}\in\mathbb{R_{+}}$. The maximum
technology input constraint (10e) is augmented such that if the technology is
not built ($y_{k}=0$) then the corresponding input value will be zero
($t_{k}=0$). For technologies that already in place, we simply set the
corresponding binary values to one. In our modeling approach, we can capture
economies of scale by defining technologies of different capacities. The
budget constraint will enable studies to determine what budget is required to
transition to different pathways. We note that it is also possible to
incorporate the investment costs into the objective and let the model decide
what investments are needed to maximize the total surplus (without any
constraints in budget). In this context, it is important to ensure that
operating costs and investment costs are on the same time basis (e.g.,
annualized costs). The investment model reduces to a management model if the
binary variables are fixed. We can thus think of the investment model as a
framework that can be used to study how strategic deployment of technologies
can be used to activate an economy and generate wealth.
## 3 Case Study: Decarbonizing a University Campus
We consider a case study for a simulated university campus to illustrate the
modeling capabilities of the framework and the types of insights that it can
provide. The model was developed using basic data and assumptions that aim to
capture general features of a prototypical, large campus (inspired by the UW-
Madison campus); however, the model is not a real and validated representation
of the actual system. As such, results and conclusions reached in this study
are only intended to illustrate the capabilities of the model, and should not
be used as actual recommendations.
We consider a campus that serves 48,000 students and 24,000 staff members on a
900-acre campus with 25 million square feet of buildings [4, 22]. Under
existing operations (and associated technology pathways), the university
outputs 500,000 tonnes of CO2 annually to meet demands of four key products
(see Figure 4) [22]. The carbon footprint is equivalent to the annual
emissions of 34,000 average citizens in the US [10] (thus comparable in
magnitude to the campus population). To decrease the CO2 footprint, we must
identify new technology pathways that can satisfy the product demands (campus
services) while reducing CO2 emissions.
Figure 4: Sankey diagram summarizing the annual CO2 emissions and their
sources for a university campus. Generation of electrical power and heating
are the dominant sources.
### 3.1 Existing Technologies
We assume that the campus purchases electricity from the local utility with a
special rate scheme due to the large demand of the university of 400,000 MWh
per year [22]. The university has some small renewable energy projects to
generate electricity, but these total to only 30 MWh per year. These are
primarily rooftop solar photo-voltaic solar panels. The rest of the power
comes from the power grid. The largest power sources for electricity in the
state are coal and natural gas at 42% and 34%, respectively [16]. There is one
large nuclear plant supplying 15% of electricity and hydroelectric supplies
4%; the remaining 5% is composed of non-hydroelectric renewables. This leaves
the grid electricity with an estimated CO2 output of 560 kg CO2/MWh [16]. Grid
electricity is represented by GP in Table 4 and 5, which takes in MWh grid and
converts it into MWh and kg CO2. The grid is represented as a technology, to
account for generated CO2 emissions associated with the use of grid
electricity. Operating costs for the technologies are estimated and are
represented relative to the reference product.
The campus has an annual average highest outdoor temperature of 35℃ and lowest
temperature of -27℃, so both heating and cooling is required to keep buildings
comfortable for teaching and research [17]. To meet these heating and cooling
demands, the campus uses utility network that delivers steam and chilled water
to buildings [26]. These services are supplied by a couple of heating and
cooling plants and one combined heat and power (CHP) plant. These plants burn
natural gas to produce steam, which is used to heat the buildings and are
represented by SGNG. The plants also use refrigeration systems to chill water
to cool the buildings, represented by WC. These plants, along with the CHP
plant, meet the annual demand for the university of 1 million tonnes of steam
and 24 million tonnes of chilled water to maintain buildings at a comfortable
temperature as shown in Table 2.
The natural gas-fired CHP plant has a rated electricity output of 150 MW, can
produce 220 tonne/hr of steam, and 27,000 tonne/hr of chilled water [1]. For
this plant, the water chillers are powered by electricity and will be
represented separately by WC. The steam and power output will be represented
by SGNG.
The campus has a fleet of 912 vehicles to maintain the campus and meet travel
demands (measured as distance) [22]. A total of 865 of the vehicles are
exclusively gasoline powered with the rest being hybrid vehicles and just 2
fully electric vehicles. To compare different types of vehicles, there is a
demand of distance (km) to meet demands rather than a demand for fuel for
transportation purposes. These vehicles satisfy the 6 million km the
university requires each year at a rate estimated by the university fleet rate
[9]. These vehicles are represented by GV with operating costs based on
average values for internal combustion vehicles.
These technologies make up the current pathways that meet the campus demands
and are displayed in Figure 5. Supplies of natural gas, gasoline, and grid
electricity are added with pricing details found in Table 3. When implemented
in the model, the quantities for these supplies are set to a sufficiently
large value such that the technologies are not limited by the available
supply. Demand data can be found in Table 2.
Figure 5: Superstructure graph showing currently-available suppliers,
technologies, and demands for the university campus. See Table 1 for
technology descriptions.
### 3.2 Potential New Technologies
Electricity generation on campus produces the most CO2; to decarbonize
generation, renewable electricity generation is required. To do this, the
university would have to either wait for the power grid to use more renewable
power or install their own renewable generators. However, renewable capacity
depends on location and cannot be placed wherever is convenient. Further, the
times that renewable power generation occurs does always match the demand,
meaning some energy would have to be stored for later use. Based on this
constraint, the power produced from renewable power will be treated as a
separate product ($MWh~{}renew$) that cannot be used directly to satisfy
electricity demands. For the purposes of this case study, solar photovoltaic
and wind power will be considered. Solar power will be represented by SP and
wind power will be represented by WP to enable the use in the investment model
and capture the operational costs of the two technologies [13]. These
technologies are modular, meaning that additional capacity can be added in
small increments for the same cost. As such, the investment cost and capacity
reflect these aspects and, when implemented in the model, will have a set
number of units available. These technologies take into consideration the
required land usage, and an additional supply of land is added to account for
this. This means that the technologies will be competing for a supply of land
(in m2). In other words, our model captures land as a resource that needs to
be strategically used to various uses. This illustrates how the abstract
products can be used to capture diverse assets.
Hydrogen-based technologies have been proposed as an energy carrier with
similar applications as natural gas. As such, hydrogen-based technologies
could be critical to continue to meet the demands of the university. To be
decarbonized, hydrogen can be produced through electrolysis using renewable
power to create a CO2 free energy carrier. As such, electrolysis will be
represented by WS which takes in renewable power ($MWh~{}renew$) and produces
hydrogen. The hydrogen could then be used by a fuel cell for electricity by
FC, which is a modular technology as well [30, 32].
The produced hydrogen can be used by the existing CHP plant by blending
hydrogen and natural gas as represented by CHPB. This option has an investment
cost of just 1% of the initial investment cost of the turbines, because many
natural gas powered turbines can handle hydrogen blends up to a volume
fraction [49]. This study will use a blend of 20% hydrogen by volume.
Alternatively, the university could replace the existing turbines to run
entirely off hydrogen with an associated investment cost of 25% of the initial
cost of the turbines of the CHP plant as represented by CHPH2 [49].
Alternatively, hydrogen could be used by a fuel cell CHP plant which uses the
high temperatures a fuel cell operates at to generate heat that could be used
by the university as well as electricity. The fuel cell CHP will be
represented by CHPFC. Hydrogen could also be used to decarbonize the heating
demand of the university by implementing hydrogen steam generators as
described by SGH2. This can be implemented without having to make any
additional upgrades, unlike the CHP plant [47].
To reduce the CO2 emissions of the university travel demand, alternative
vehicles must be considered. Hybrid vehicles have 50% higher fuel efficiency
than traditional gasoline vehicles and will be represented by HV [3]. However,
even the most fuel efficient gasoline vehicle still emits CO2. Electric
vehicles could be a CO2 free alternative if the electricity is sourced from
renewable power such as wind and solar and will be represented by EV. The
shift to electric vehicles has already been observed as more companies release
new vehicles [5]. Hydrogen fuel-cell vehicles are another alternative,
although they are less commercially available than electric vehicles. However,
they have faster refueling times compared to the charging times of electric
vehicles [6, 7]. The hydrogen vehicles in this study will be described by H2V.
For all the alternative vehicle options, they will also be represented in a
modular way meaning the purchase of one of these vehicles provides the
capacity for 8000 km and the purchase of multiple vehicles will be required.
Modular technologies, such as vehicles and electrolysis, will have an
associated number of units that are available in Table 7. While the single
items like switching to a blend of hydrogen and natural gas feed for CHP are a
single unit. This will allow the model to see what technology pathways are
chosen at budgets that would not allow for a complete transition from, for
example, gasoline vehicles to electric vehicles. All of the new technologies
are shown in Figure 6.
All of the possible pathways from supply through technologies to demands
associated with potential technologies are displayed in Figure 2. While all of
the pathways are displayed here, a subset of them will be chosen by the model
as being the optimal pathway. By varying some key attributes such as a carbon
tax or the budget, we can determine what technologies and products will have
the most impact under those conditions.
Figure 6: Superstructure graph of potential technology pathways that could be
chosen by the university campus to decarbonize the system (current
technologies have been blurred out to facilitate comparison).
### 3.3 Results
#### 3.3.1 Management Model
The purpose of the management model is to determine optimal technology
pathways for the campus to meet its needs, without any restrictions on the
budget needed to build such technologies. These optimal technologies will vary
based on external conditions of the system (e.g., policy, markets, technology
efficiencies). For instance, a key external condition is the cost of CO2.
There is work to determine the externality of releasing CO2 that places a cost
of 113 USD per tonne CO2, although estimates vary [46]. However, our work will
aim to ask the question: What CO2 cost/tax would incentivize the university to
reduce or stop emitting CO2? The tax does not necessarily have to be
interpreted as an externality (e.g., government-imposed), but can also be
interpreted as an implicit value that the university is placing to its carbon
footprint (which can potentially trigger negative public perception).
The carbon emissions are modeled as a demand for CO2 that is charged at a
negative bid value (it is a waste product). This can be interpreted in
different ways; for instance, this can be interpreted as a tax that the
government is introducing for any emissions generator by the system.
Alternatively, this can be interpreted as a “tipping fee” that the environment
is charging the system to take its CO2 waste. The CO2 cost can also be
interpreted as an internal (inherent) value that campus is placing on
emissions; this is analogous to how companies that are currently seeking to
decarbonize think about this waste (e.g., they are internally placing a
negative value to CO2 that might be implicitly connected to branding or public
perception). This illustrates how the proposed modeling framework can help
capture diverse scenarios of interest.
To establish a benchmark, the pathway model with all potential and current
technologies will first be optimized with no carbon tax. This leads to the
technology pathway as shown in Figure 7. This pathway, however, still emits
CO2 because it is using the natural gas-powered CHP plant as much as possible.
Meaning that without an incentive, the university has no need to fully
decarbonize. The pathway, however, makes use of solar power to produce
hydrogen to be used to supplement the CHP electricity for use in electric
vehicles, to operate the water chiller, and meet the electricity demands of
the university.
Figure 7: Optimal technology pathways under no CO2 tax showing the
technologies used satisfy the demands of university campus; pathways that are
not selected are blurred out. Product prices are displayed in the legend,
flows going in and/or out of each stakeholder are reported in the units of the
legend, and the number of technology units required are reported above each
technology.
To study the impact of the CO2 tax ($\alpha_{kg~{}CO_{2}}$), it will involve
discretizing a range of values and solving the management model for each
value. This is computationally cheap as the optimization problem is a linear
program and therefore quick to solve. The demand bid for CO2 will be negative
because it is being modeled as a tax; meaning the university will have to pay
the environment/government to take the CO2 it emits. The impact of this tax on
the utility cost of the university was also determined; utility cost is the
cost to operate the technologies, cost of supplies, and disposal cost of waste
products (CO2), and is given by:
$\displaystyle\sum_{k\in\mathcal{T}}{\alpha}_{k}t_{k}+\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\alpha_{kg~{}CO_{2}}d_{kg~{}CO_{2}}$
(11)
Figure 8 shows the impact of varying the CO2 tax on the CO2 emissions and
utility cost of the university. This shows three different technology pathways
depending on the CO2 tax. Below 45 USD per tonne, the pathway chosen is the
same as in Figure 7. While the CO2 output stays constant for each pathway, the
utility cost increases when there is still CO2 emissions because the
university has to pay for the environment to take the emissions.
Figure 8: Dependence of CO2 output and utility cost of the university on the
CO2 tax (when all possible technologies can be used). Results indicate that
there are three different pathways that are activated at different tax levels.
Between 45 USD and 130 USD per tonne is the pathway shown in Figure 9. This
pathway is the one that would be chosen if the estimated externality of CO2
was implemented as a tax. The pathway has a 77% lower CO2 output than in
Figure 7 but also has a higher utility cost. This pathway has fully
decarbonized the demand for electricity, cooling, and transportation but still
relies on natural gas for steam.
Figure 9: Optimal technology pathways for a CO2 tax between 45 USD and 130 USD
per tonne showing the technologies used satisfy the demands of university
campus; pathways that are not selected are blurred out. Product prices are
displayed in the legend for a CO2 0.075 USD per tonne, flows going in and/or
out of each stakeholder are reported in the units of the legend, and the
number of technology units required are reported above each technology. This
pathway has decarbonized all of the university’s demands besides the one for
steam which still relies on natural gas fired steam generation.
For CO2 taxes above 130 USD per tonne, the optimal pathway is shown in Figure
10. This pathway has no CO2 emissions because it has switched to using
hydrogen fuel cell CHP to meet the demands for steam and some of the
electricity. Fuel cells are used to supplement the electricity for use by the
electric vehicles for distance, the water chiller for cooling, and the
universities demand for electricity. This has the highest utility cost at 55%
higher than when there is no CO2 tax. This pathway also has the highest steam
price as the fuel cell CHP plant is the most expensive to operate of the
options for CHP, while all other prices remained constant between the three
different pathways.
Figure 10: Optimal technology pathways for a CO2 taxes above 130 USD per tonne
showing the technologies used satisfy the demands of university campus;
pathways that are not selected are blurred out. Product prices are displayed
in the legend, flows going in and/or out of each stakeholder are reported in
the units of the legend, and the number of technology units required are
reported above each technology. This pathway has decarbonized all of the
university’s demands through the use of solar power and hydrogen fuel cell CHP
and hydrogen fuel cells.
The management model results identify that meeting the demand for steam to
heat the buildings seems to be a major factor that drives technology
selection. Additionally, the choice of using electric vehicles was consistent
for the three different pathways because they have the lowest cost to operate
per distance and the low cost of power from renewables. Solar power is chosen
over wind power for three pathways because it has a lower operating cost per
MWh produced than wind. With the use of solar power in each pathway, the
hydrogen price remained constant at 0.97 USD/kg H2 for the three different
pathways. The electricity price also remained constant at 62.3 USD/MWh as it
was set by the solar power to electrolysis to fuel cell pathway operating
cost. This made the price for distance (km) and chilled water to remain
constant.
Steam price did not remain constant across the three different pathways. Under
no CO2 tax (Figure 7) the steam price is -0.007 USD/kg. The negative value
means that the natural gas fired CHP would be willing to pay for the steam to
be taken because it already profits from electricity price being set by the
fuel cell pathway. When the steam is instead produced by the natural gas fired
steam generator (Figure 9) the steam price is no longer tied to the
electricity price. The operating cost of the steam generator and the tax on
the CO2 released must be covered by the steam demand and a positive price of
0.016 USD/kg steam is observed at a CO2 tax of 0.075 USD/kg CO2. When the
steam is produced by the hydrogen fuel cell CHP and there are no CO2 emissions
(Figure 9) the steam price is 0.022 USD/kg steam. For this pathway, the
positive price is a result of the revenue required from the steam to offset
the higher operating cost of the hydrogen fuel cell CHP than the hydrogen fuel
cell.
#### 3.3.2 Investment Model
The investment model allows for the further consideration of a budget on
choosing the technology pathways under different conditions. The key condition
to incentives reducing CO2 emissions will be a CO2 tax, and analysis will
involve solving the model for each value in a range. However, the impact of
the budget has to also be considered. This means for each CO2 tax, the
investment model will have to optimize for each budget at each CO2 tax. This
is more computationally expensive than the management model because of the
binary variables, but it is still readily solvable for each CO2 tax and budget
combination.
Figure 11 shows the CO2 emissions (a) and utility cost of the university (b)
at varying CO2 taxes and at varying investment budgets. When the budget is
low, little can be done to shift away from the existing infrastructure so as
the CO2 tax increases, the emissions stay the same and the utility cost
increases. This can be observed for all CO2 taxes when the budget is less than
80 million USD. While at sufficiently large budgets ($\geq$2.7 billion USD),
the potential pathways match the results of the management model and the same
dependence on the CO2 tax is observed. This occurs because the investment
model has a high enough budget that it can choose any technology it desires,
essentially eliminating the budget constraint.
Figure 11: Dependence of the annual (a) CO2 output and (b) utility cost of the
university on the CO2 tax and budget when all possible technologies are
available for purchase and use.
As the budget increases above 80 million USD, technologies are chosen to
reduce the impact of the CO2 tax. Figure 12, shows the pathway at a budget of
100 million USD and a CO2 of 200 USD per tonne. This pathway is at a point
where the budget is insufficient to allow for a complete switch to technology
alternatives. Based on the management results, the optimal pathway at this CO2
tax would be the one observed in Figure 10; however, the budget is not high
enough for this pathway to be possible. Instead, the pathway chosen uses the
existing gasoline vehicles to meet the distance demand, the natural gas
powered CHP, and grid power which all produce CO2. There is some use of solar
power for production of hydrogen which is then used by a single fuel cell CHP
plant. The number of solar power units here are less than in the management
model. The inability to offset the CO2 tax results in a 2.4 times larger price
for the power in this pathway than what the price would be without any budget
constraint.
Figure 12: Investment model results at a CO2 tax of 200 USD per tonne and a
budget of 100 million USD showing the technologies used satisfy the demands of
university campus; pathways that are not selected are blurred out. Product
prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
makes use of some solar power to hydrogen for a fuel cell CHP unit however not
demand is completely decarbonized.
Another point where the budget limits the pathway options is at a budget of 1
billion USD and a CO2 tax of 75 USD per tonne. At this CO2 tax, the optimal
pathway without any budget constraint would be the one displayed in Figure 9
which had decarbonized everything besides the demand for steam. However, the
budget limits the pathway to continue using the natural gas CHP plant and the
gasoline vehicles. There is more use of solar power than in Figure 12 due to
the higher budget and more fuel cell CHP units, but it is still insufficient
to meet the full demand for steam and electricity. Fuel cell units are used in
this pathway to supplement the electricity from the natural gas and fuel cell
CHP as well. This pathway has higher prices than in the budget unconstrained
case with a 35% higher power price. This is a result of the electricity price
being impacted by the operating cost of the natural gas fired CHP and the tax
on the CO2 it emits.
Figure 13: Investment model results at a CO2 tax of 75 USD per tonne and a
budget of 1 billion USD showing the technologies used satisfy the demands of
university campus; pathways that are not selected are blurred out. Product
prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
uses no grid power but still uses natural gas fired CHP and steam generators.
Solar power is used to produce hydrogen to be used by the fuel cell CHP and
fuel cells.
The impact of the CO2 and budget constraint on the hydrogen production and
hydrogen price is reported in Figure 14. Hydrogen production is highest in the
pathway described in Figure 9 because it relies on fuel cells to meet the
entire electricity use and demand. Hydrogen production is lower when CHP is
used, because they are more efficient. The hydrogen price is 0.97 USD per kg;
however, it spikes to as high as 3.30 USD per kg when the budget is 810
million USD and the CO2 tax is 240 USD per tonne. This pathway is displayed in
Figure 15. The high hydrogen price is caused by the use of a hydrogen vehicle.
By participating in the same demand, the hydrogen price becomes tied to the
cost of using the most expensive technology used. For this pathway and
condition, gasoline vehicles are the most expensive because of the high CO2
tax and lower efficiency than the hybrid vehicles. This effect on hydrogen
price demonstrates the importance of understanding how intermediate products
are priced as a result of being used in competing pathways; specifically, the
inherent value of intermediate products is inherently tied to the final uses.
Figure 14: Dependence of (a) hydrogen production and (b) hydrogen prices on
the CO2 tax and budget when all possible technologies are available for
purchase and use. Figure 15: Investment model results at a CO2 tax of 240 USD
per tonne and a budget of 810 million showing the technologies used satisfy
the demands of the campus; pathways that are not selected are blurred out.
Product prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
makes uses a mix of gasoline, hybrid, and a single hydrogen fuel cell vehicle
to meet the distance (km) demand of the university resulting in a high
hydrogen price.
The investment model results demonstrate the impact of the budget constraint
on technology selection under different budget and CO2 tax conditions. These
technology selections impact the utility cost for the university and the
prices of products. When the budget is low, little can be done to compensate
for the CO2 tax leading to high utility costs and prices. As the budget is
increased, technology selections favor the use of solar power production for
hydrogen production due to the low operating cost and greatest impact
offsetting the CO2 emissions associated with the use of grid power.
Additionally, the investment model results indicate that there are some
budgets where potential technologies (e.g., hybrid vehicles) are chosen
despite them not being used in the corresponding budget unconstrained case.
This suggests that, periodic investments of 100 million USD per year are made
for 10 years, may result in a different technology pathway compared to a
single investment of 1 billion USD.
## 4 Conclusions and Future Work
We have presented an optimization framework for conducting technology pathway
analysis. The proposed framework uses a graph abstraction that captures
interdependencies between products and technologies and diverse factors that
are of interest for high-level analysis such as techno-economic factors,
economies of scale, efficiencies, and externalities (e.g., policy). Duality
analysis also reveals that the optimization formulation has a natural economic
interpretation that reveals how stakeholders participating in the system can
be remunerated and that also reveals in inherent value of intermediate
products. We have illustrated the use of the framework for studying technology
pathways that enable decarbonization of a representative university campus.
Here, we show how the framework can model diverse types of products,
technologies, and externalities. This analysis identified key technologies and
products and demonstrated the importance of understanding product pricing
dependencies that can arise as decarbonization occurs. As part of future work,
we will explore the use of the proposed framework to decarbonize other systems
(such as plastics manufacturing) and to understand the impact of additional
technologies (such as ammonia production) and connections with other sectors
(such as agriculture).
## Acknowledgments
We acknowledge funding from NSF CAREER award CBET-1748516.
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## Appendix
Table 1: Acronyms for technologies considered in University campus study. Acronym | Meaning
---|---
CHPNG | Combined Heat and Power Plant (using natural gas)
SGNG | Steam Generator (using natural gas)
WC | Water Chiller
GP | Grid Power
GV | Gasoline Vehicle
SP | Solar Power (Photovoltaic)
WP | Wind Power
WS | Water Splitting (Electrolysis)
CHPB | Combined Heat and Power Plant (using blend of natural gas and hydrogen)
CHPH2 | Combined Heat and Power Plant (using hydrogen)
CHPFC | Combined Heat and Power Plant (using Hydrogen Fuel Cell)
SGH2 | Steam Generator (using Hydrogen)
FC | Hydrogen Fuel Cell
H2V | Hydrogen Fuel Cell Vehicle
EV | Electric Vehicle
HV | Hybrid Gasoline Vehicle
Table 2: University Campus Annual Demands (dashed value(s) represent variable or not constrained) Product | Quantity | Current Price | Reference
---|---|---|---
Electricity | $4.03\times 10^{5}~{}MWh$ | $99.8~{}USD/MWh$ | [22, 11]
Steam | $1.04\times 10^{9}~{}kg~{}steam$ | $0.01~{}USD/kg~{}steam$ | [22]
Chilled Water | $2.43\times 10^{1}0~{}kg~{}chilled$ | $0.00023~{}USD/kg~{}chilled$ | [22]
Distance | $7.30\times 10^{6}~{}km$ | $0.10~{}USD/km$ | [9, 22]
CO2 | $-~{}kg~{}CO_{2}$ | $0~{}USD/kg~{}CO_{2}$ |
Table 3: University Campus Annual Available Supplies Product | Quantity | Price | Reference
---|---|---|---
Grid Power | $1\times 10^{10}~{}MWh~{}grid$ | $99.8~{}USD/MWh~{}grid$ | [11]
Natural Gas | $1\times 10^{10}~{}kg~{}CH_{4}$ | $0.14~{}USD/kg~{}CH_{4}$ | [14]
Gasoline | $1\times 10^{10}~{}gal~{}gas$ | $3.10~{}USD/gal~{}gas$ | [15]
Solar Power | $1\times 10^{10}~{}MWh~{}solar$ | $0~{}USD/MWh~{}solar$ |
Wind Power | $1\times 10^{10}~{}MWh~{}wind$ | $0~{}USD/MWh~{}wind$ |
Land | $1\times 10^{10}~{}m^{2}$ | $0~{}USD/m^{2}$ |
Table 4: Inputs and Outputs of the Current University Campus Technologies Supply | Technology | Equation
---|---|---
CHPNG | Natural Gas CHP | $253~{}kg~{}CH_{4}\rightarrow 1500~{}kg~{}steam+1~{}MWh+760~{}kg~{}CO_{2}$
SGNG | Natural Gas SG | $1~{}kg~{}CH_{4}\rightarrow 23~{}kg~{}steam+2.8~{}kg~{}CO_{2}$
WC | Water Chiller | $0.024~{}MWh\rightarrow 1000~{}kg~{}chilled$
GP | Grid Power | $1~{}MWh~{}grid\rightarrow 1~{}MWh+560~{}kg~{}CO_{2}$
GV | Gasoline Vehicle | $1~{}gal~{}gas\rightarrow 40~{}km+8~{}kg~{}CO_{2}$
Table 5: Economic and Capacity Properties of the Current University Campus Technologies (dashed values are treated as unconstrained) Acronym | Technology | Operating Costs | Annual Capacity | Reference
---|---|---|---|---
CHPNG | Natural Gas CHP | $0.063~{}USD/kg~{}CH_{4}$ | $3.3\times 10^{8}~{}kg~{}CH_{4}$ | [20, 1]
SGNG | Natural Gas SG | $0.03~{}USD/kg~{}CH_{4}$ | $2.9\times 10^{8}~{}kg~{}CH_{4}$ | [26, 23]
WC | Water Chiller | $1~{}USD/MWh$ | $-~{}MWh$ | [26]
GP | Grid Power | $0~{}USD/MWh~{}grid$ | $-~{}MWh~{}grid$ | [16]
GV | Gasoline Vehicle | $1.52~{}USD/gal~{}gas$ | $2.9\times 10^{5}~{}gal~{}gas$ | [3, 24]
Table 6: Inputs and Outputs of Potential Technologies Acronym | Technology | Equation
---|---|---
SP | Solar Power | $1~{}MWh~{}solar\rightarrow 1MWh~{}renew$
WP | Wind Power | $1~{}MWh~{}wind\rightarrow 1~{}MWh~{}renew$
WS | Electrolysis | $0.043~{}MWh~{}renew\rightarrow 1~{}kg~{}H_{2}$
CHPB | Blend CHP | $236~{}kg~{}CH_{4}+7.3~{}kg~{}H_{2}\rightarrow 1500~{}kg~{}steam+1~{}MWh+707~{}kg~{}CO_{2}$
CHPH2 | Hydrogen CHP | $107~{}kg~{}H_{2}\rightarrow 1500~{}kg~{}steam+1~{}MWh$
CHPFC | Hydrogen Fuel Cell CHP | $60~{}kg~{}H_{2}\rightarrow 1414~{}kg~{}steam+1~{}MWh$
SGH2 | Hydrogen SG | $1~{}kg~{}H_{2}\rightarrow 46~{}kg~{}steam$
FC | Fuel Cell | $1~{}kg~{}H_{2}\rightarrow 0.02~{}MWh$
H2V | Hydrogen Vehicle | $1~{}kg~{}H_{2}\rightarrow 100~{}km$
EV | Electric Vehicle | $1~{}MWh\rightarrow 100~{}km$
HV | Hybrid Vehicle | $1~{}gal~{}gas\rightarrow 60~{}km+8~{}kg~{}CO_{2}$
Table 7: Economic and Capacity Properties of Potential Technologies Acronym | Operating Costs | Annual Capacity | Investment Cost | Units | Reference
---|---|---|---|---|---
SP | $8.54~{}USD/MWh~{}solar$ | $5000~{}MWh~{}solar$ | $234,650~{}USD$ | $750$ | [13, 36, 18]
WP | $10.21~{}USD/MWh~{}wind$ | $5000~{}MWh~{}wind$ | $893,550~{}USD$ | $750$ | [13, 21, 18]
WS | $0.018~{}USD/MWh~{}renew$ | $5000~{}MWh~{}renew$ | $58,250~{}USD$ | $700$ | [32]
CHPB | $0.066~{}USD/kg~{}CH_{4}$ | $3.1\times 10^{8}~{}kg~{}CH_{4}$ | $763,000~{}USD$ | $1$ | [1, 49, 12]
CHPH2 | $0.15~{}USD/kg~{}H_{2}$ | $1.4\times 10^{8}~{}kg~{}H_{2}$ | $20,000,000~{}USD$ | $1$ | [1, 49, 12]
CHPFC | $0.60~{}USD/kg~{}H_{2}$ | $1.5\times 10^{6}~{}kg~{}H_{2}$ | $12,880,000~{}USD$ | $17$ | [8]
SGH2 | $0.07~{}USD/kg~{}H_{2}$ | $4.4\times 10^{7}~{}kg~{}H_{2}$ | $0~{}USD$ | $1$ | [47]
FC | $0.28~{}USD/kg~{}H_{2}$ | $2.5\times 10^{5}~{}kg~{}H_{2}$ | $12,500,000~{}USD$ | $100$ | [32]
H2V | $3.11~{}USD/kg~{}H_{2}$ | $80~{}kg~{}H_{2}$ | $50,000~{}USD$ | $913$ | [3, 24, 48]
EV | $66.4~{}USD/MWh$ | $2.3~{}MWh$ | $40,000~{}USD$ | $913$ | [3, 24]
HV | $2.05~{}USD/gal~{}gas$ | $133~{}gal~{}gas$ | $30,000~{}USD$ | $913$ | [3, 24]
|
# Deep neural network Grad-Shafranov solver constrained with measured magnetic
signals
Semin Joung<EMAIL_ADDRESS>Department of Nuclear and Quantum
Engineering, KAIST, Daejeon 34141, Republic of Korea Jaewook Kim Department
of Nuclear and Quantum Engineering, KAIST, Daejeon 34141, Republic of Korea
Sehyun Kwak Department of Nuclear and Quantum Engineering, KAIST, Daejeon
34141, Republic of Korea Max-Planck-Institut f$\ddot{u}$r Plasmaphysik,
Teilinstitut Greifswald, D-17491 Greifswald, Germany J.G. Bak National
Fusion Research Institute, Daejeon 34133, Republic of Korea S.G. Lee
National Fusion Research Institute, Daejeon 34133, Republic of Korea H.S. Han
National Fusion Research Institute, Daejeon 34133, Republic of Korea H.S. Kim
National Fusion Research Institute, Daejeon 34133, Republic of Korea Geunho
Lee Mobiis Co., Ltd., Seongnam-si, Gyeonggi-do 13486, Republic of Korea
Daeho Kwon Department of Nuclear and Quantum Engineering, KAIST, Daejeon
34141, Republic of Korea Mobiis Co., Ltd., Seongnam-si, Gyeonggi-do 13486,
Republic of Korea Y.-c. Ghim<EMAIL_ADDRESS>Department of Nuclear and
Quantum Engineering, KAIST, Daejeon 34141, Republic of Korea
###### Abstract
A neural network solving Grad-Shafranov equation constrained with measured
magnetic signals to reconstruct magnetic equilibria in real time is developed.
Database created to optimize the neural network’s free parameters contain off-
line EFIT results as the output of the network from $1,118$ KSTAR experimental
discharges of two different campaigns. Input data to the network constitute
magnetic signals measured by a Rogowski coil (plasma current), magnetic pick-
up coils (normal and tangential components of magnetic fields) and flux loops
(poloidal magnetic fluxes). The developed neural networks fully reconstruct
not only the poloidal flux function $\psi\left(R,Z\right)$ but also the
toroidal current density function $j_{\phi}\left(R,Z\right)$ with the off-line
EFIT quality. To preserve robustness of the networks against a few missing
input data, an imputation scheme is utilized to eliminate the required
additional training sets with large number of possible combinations of the
missing inputs.
Keywords: Neural network, Grad-Shafranov equation, EFIT, poloidal flux,
toroidal current, imputation, KSTAR
## I Introduction
Magnetic equilibrium is one of the most important information to understand
the basic behavior of plasmas in magnetically confined plasmas, and the off-
line EFIT Lao _et al._ (1985) code has been extensively used to reconstruct
such equilibria in tokamaks. Its fundamentals are basically finding a solution
to an ideal magnetohydrodynamic equilibrium with toroidal axisymmetry, known
as the Grad-Shafranov (GS) equation Freidberg (1987):
$\begin{split}\Delta^{*}\psi&\equiv\left(R\frac{\partial}{\partial
R}\frac{1}{R}\frac{\partial}{\partial R}+\frac{\partial^{2}}{\partial
Z^{2}}\right)\psi\\\ &=-\mu_{0}Rj_{\phi}\\\
&=-\mu_{0}R^{2}\frac{dp(\psi)}{d\psi}-F(\psi)\frac{dF(\psi)}{d\psi},\end{split}$
(1)
where $\psi=\psi\left(R,Z\right)$ is the poloidal flux function,
$j_{\phi}=j_{\phi}\left(R,Z\right)$ the toroidal current density function,
$p(\psi)$ the plasma pressure. $F(\psi)$ is related to the net poloidal
current. Here, $R$, $\phi$ and $Z$ denote the usual cylindrical coordinate
system. As the $\Delta^{*}$ is a two-dimensional nonlinear partial
differential operator, the off-line EFIT Lao _et al._ (1985) finds a solution
with many numerical iterations and has been implemented in many tokamaks such
as Diii@-D Lao _et al._ (2005), JET O’Brien _et al._ (1992), NSTX Sabbagh
_et al._ (2001), EAST Jinping _et al._ (2009) and KSTAR Park _et al._ (2011)
to name some as examples.
With an aim of real-time control of tokamak plasmas, real-time EFIT (rt-EFIT)
Ferron _et al._ (1998) code is developed to provide a magnetic equilibrium
fast enough whose results are different from the off-line EFIT results. As
pulse lengths of tokamak discharges become longer Van Houtte and SUPRA (1993);
Ekedahl _et al._ (2010); Itoh _et al._ (1999); Zushi _et al._ (2003);
Saoutic (2002); Park _et al._ (2019); Wan _et al._ (2019), demand on more
elaborate plasma control is ever increased. Furthermore, some of the ITER
relevant issues such as ELM (edge localized mode) suppression with RMP
(resonant magnetic perturbation) coils Park _et al._ (2018) and the detached
plasma scenarios Reimold _et al._ (2015); Jaervinen _et al._ (2016) require
sophisticated plasma controls, meaning that the more accurate magnetic
equilibria we have in real time, the better performance we can achieve.
There has been an attempt to satisfy such a requirement of acquiring a more
accurate, i.e., closer to the off-line EFIT results compared to the rt-EFIT
results, magnetic equilibrium in real-time using graphics processing units
(GPUs) Yue, X N _et al._ (2013) by parallelizing equilibrium reconstruction
algorithms. The GPU based EFIT (P-EFIT) Yue, X N _et al._ (2013) enabled one
to calculate a well-converged equilibrium in much less time; however, the
benchmark test showed similar results to the rt-EFIT rather than the off-line
results Huang, Yao _et al._ (2016).
Thus, we propose a reconstruction algorithm based on a neural network that
satisfies the GS equation as well as the measured magnetic signals to obtain
accurate magnetic equilibrium in real time. We note that usage of neural
networks in fusion community is increasing rapidly, and examples are radiated
power estimation Barana _et al._ (2002), identifying instabilities Murari, A
_et al._ (2013), estimating neutral beam effects Boyer _et al._ (2019),
classifying confinement regimes Murari _et al._ (2012), determination of
scaling laws Murari _et al._ (2010); Gaudio _et al._ (2014), disruption
prediction Kates-Harbeck, Julian _et al._ (2019); Cannas _et al._ (2010);
Pau _et al._ (2019), turbulent transport modelling Meneghini, O _et al._
(2014); Meneghini _et al._ (2017); Citrin, J _et al._ (2015); Felici _et
al._ (2018), plasma tomography with the bolometer system Matos, Francisco A
_et al._ (2017); Ferreira _et al._ (2018), coil current prediction with the
heat load pattern in W7-X Böckenhoff _et al._ (2018), filament detection on
MAST-U Cannas _et al._ (2019), electron temperature profile estimation via
SXR with Thomson scattering Clayton, D J _et al._ (2013) and equilibrium
reconstruction Lister and Schnurrenberger (1991); Coccorese _et al._ (1994);
Bishop _et al._ (1994); Cacciola, Matteo _et al._ (2006); Jeon _et al._
(2001); Wang _et al._ (2016) together with an equilibrium solver van Milligen
_et al._ (1995). Most of previous works on the equilibrium reconstruction with
neural networks have paid attention to finding the poloidal beta, the plasma
elongation, positions of the X-points and plasma boundaries, i.e., last closed
flux surface, and gaps between plasmas and plasma facing components, rather
than reconstructing the whole internal magnetic structures we present in this
work.
The inputs to our developed neural networks consist of plasma current measured
by a Rogowski coil, normal and tangential components of magnetic fields by
magnetic pick-up coils, poloidal magnetic fluxes by flux loops and a position
in $\left(R,Z\right)$ coordinate system, where $R$ is the major radius, and
$Z$ is the height as shown in Figure 1. The output of the neural networks is a
value of poloidal flux $\psi$ at the specified $\left(R,Z\right)$ position. To
train and validate the neural networks, we have collected a total of $1,118$
KSTAR discharges from two consecutive campaigns, i.e., $2017$ and $2018$
campaigns. We, in fact, generate three separate neural networks which are
NN${}_{\textrm{2017}}$, NN${}_{\textrm{2018}}$ and NN${}_{\textrm{2017,
2018}}$ where subscripts indicate the year(s) of KSTAR campaign(s) that the
training data sets are obtained from. Additional $163$ KSTAR discharges (from
the same two campaigns) are collected to test the performance of the developed
neural networks.
We train the neural networks with the KSTAR off-line EFIT results, and let
them be accurate magnetic equilibria. Note that disputing on whether the off-
line EFIT results we use to train the networks are accurate or not is beyond
the scope of this work. If we find more accurate EFIT results, e.g.,
MSE(Motional Stark Effect)-constrained EFIT or more sophisticated equilibrium
reconstruction algorithms that can cope with current-hole configurations
(current reversal in the core) Rodrigues and Bizarro (2005, 2007); Ludwig _et
al._ (2013), then we can always re-train the networks with new sets of data as
long as the networks follow the trained EFIT data with larger similarity than
the rt-EFIT results do. This is because supervised neural networks are limited
to follow the training data. Hence, as a part of the training sets we use the
KSTAR off-line EFIT results as possible examples of accurate magnetic
equilibria to corroborate our developed neural networks.
To calculate the output data a typical neural network requires the same set of
input data as it has been trained. Therefore, even a single missing input (out
of input data set) can result in a flawed output van Lint _et al._ (2005).
Such a case can be circumvented by training the network with possible
combinations of missing inputs. As a part of input data, we have $32$ normal
and $36$ tangential magnetic fields measured by the magnetic pick-up coils. If
we wish to cover a case with one missing input data, then we will need to
repeat the whole training procedure with $68$ ($32+36$) different cases. If we
wish to cover a case with two or three missing input data, then we will need
additional $2,278$ and $50,116$ different cases to be trained on,
respectively. This number becomes large rapidly, and it becomes formidable, if
not impossible, to train the networks with reasonable computational resources.
Since the magnetic pick-up coils are susceptible to damages, we have developed
our networks to be capable of inferring a few missing signals of the magnetic
pick-up coils in real-time by invoking an imputation scheme Joung _et al._
(2018) based on Bayesian probability Sivia and Skilling (2006) and Gaussian
processes Rasmussen and Williams (2006).
In addition to reconstructing accurate magnetic equilibria in real-time, the
expected improvements with our neural networks compared to the previous
studies are at least fourfold: (1) the network is capable of providing whole
internal magnetic topology, not limited to boundaries and locations of
X-points and/or magnetic axis; (2) spatial resolution of reconstructed
equilibria is arbitrarily adjustable within the first wall of KSTAR since
$\left(R,Z\right)$ position is a part of the input data; (3) the required
training time and computational resources for the networks are reduced by
generating a coarse grid points also owing to $\left(R,Z\right)$ position
being an input, and (4) the networks can handle a few missing signals of the
magnetic pick-up coils using the imputation method.
We, first, present how the data are collected to train the neural networks and
briefly discuss real-time preprocessing of the measured magnetic signals in
Section II. For the readers who are interested in thorough description of the
real-time preprocessing, Appendix A provides the details. Then, we explain the
structure of our neural networks and how we train them in Section III. In
Section IV, we present the results of the developed neural network EFIT (nn-
EFIT) in four aspects. First, we discuss how well the NN${}_{\textrm{2017,
2018}}$ network reproduces the off-line EFIT results. Then, we make
comparisons among the three networks, NN${}_{\textrm{2017}}$,
NN${}_{\textrm{2018}}$ and NN${}_{\textrm{2017, 2018}}$, by examining in-
campaign and cross-campaign performance. Once the absolute performance
qualities of the networks are established, we compare relative performance
qualities between nn-EFIT and rt-EFIT. Finally, we show how the imputation
method support the networks when there exist missing inputs. Our conclusions
are presented in Section V.
## II Collection and real-time preprocessing of data
Figure 1: A poloidal cross-section of KSTAR with the first wall (blue dotted
line). Green dotted line indicates a Rogowski coil measuring the plasma
current ($I_{\textrm{p}}$). Green open circles and crosses depict locations of
the magnetic pick-up coils measuring $32$ normal ($B_{\textrm{n}}$) and $36$
tangential ($B_{\textrm{t}}$) magnetic fields, respectively, whereas green
triangles represent $22$ flux loops measuring poloidal magnetic fluxes
($\Psi_{\textrm{FL}}$). Black asterisks ($22\times 13$ spatial positions) show
locations where we obtain the values of $\psi$ from the off-line EFIT results.
Figure 1 shows locations where we obtain the input and the output data with
the first wall (blue dotted line) on a poloidal cross-section of KSTAR. The
green dotted line indicates a Rogowski coil measuring the plasma current
($I_{\textrm{p}}$). The green open circles and crosses show locations of the
magnetic pick-up coils measuring $32$ normal ($B_{\textrm{n}}$) and $36$
tangential ($B_{\textrm{t}}$) components of magnetic fields, respectively,
whereas the green triangles show $22$ flux loops measuring the poloidal
magnetic fluxes ($\Psi_{\textrm{FL}}$). These magnetic signals are selectively
chosen out of all the magnetic sensors in KSTAR Lee _et al._ (2008) whose
performance has been demonstrated for many years, i.e., less susceptible to
damages.
Figure 2: Before (blue) and after (red) the magnetic signal adjustments for
(a) normal and (b) tangential components of magnetic fields measured by the
magnetic pick-up coils, and (c) poloidal magnetic flux measured by one of the
flux loops. The signals return closer to zeros after the adjustment when all
the external magnetic coils (except the toroidal field coils) are turned off
at around $30$ sec in this KSTAR discharge. See Appendix A for detailed
description.
Although KSTAR calibrates the magnetic sensors (magnetic pick-up coils and
flux loops) regularly during a campaign to remove drifts in the magnetic
signals, it does not guarantee to fully eliminate such drifts. Thus, we
preprocess the signals to adjust the drifts. Figure 2 shows examples of before
(blue) and after (red) the drift adjustment for (a) normal and (b) tangential
components of magnetic fields measured by the magnetic pick-up coils and (c)
poloidal magnetic flux measured by one of the flux loops. Here, a KSTAR
discharge is sustained until about $20$ sec, and all the external magnetic
coils (except the toroidal field coils) are turned off at about $30$ sec.
Therefore, we expect all the magnetic signals to return to zeros at around
$30$ sec. If not, we envisage that there has been residual drifts. This means
that we need to be able to preprocess the magnetic signals in real-time so
that the input signal characteristics for predictions are similar to the
trained ones. Appendix A describes in detail how we preprocess the magnetic
signals in real-time.
The black asterisks in Figure 1 show the $22\times 13$ grid points where we
obtain the values of $\psi$ from the off-line EFIT results as outputs of the
networks. We note that the original off-line EFIT provides the values of
$\psi$ with $65\times 65$ grid points. The $22\times 13$ grid points are
selected such that the distances between the neighboring channels in $R$ and
$Z$ directions are as similar as possible while covering whole region within
the first wall. By generating such coarse grid points we can decrease the
number of samples to train the network, thus consuming less amount of
computational resources. Nevertheless, we do not lose the spatial resolution
since $\left(R,Z\right)$ position is an input, i.e., the network can obtain
the value of $\psi$ at any position within the first wall (see Section IV).
Table 1: Summary of the data samples to train and validate the networks
Parameter | Definition | Data size | No. of samples
---|---|---|---
$I_{\textrm{p}}$ | Plasma current | 1 |
| (Rogowski coil) | |
$B_{\textrm{n}}$ | Normal magnetic field | 32 |
| (Magnetic pick-up coils) | |
| | | 217,820
$B_{\textrm{t}}$ | Tangential magnetic field | 36 | (time slices)
| (Magnetic pick-up coils) | |
$\Psi_{\textrm{FL}}$ | Poloidal magnetic flux | 22 |
| (Flux loops) | |
$R$ | Position in major radius | 1 | 286
| | | ($22\times 13$ grids)
$Z$ | Position in height | 1 |
Network Input size | | 93 (+1 for bias) |
Total no. of samples | | | 62,296,520
With an additional input for the spatial position $R$ and $Z$, each data
sample contains $93$ inputs (and yet another input for bias) and one output
which is a value of $\psi$ at the specified $\left(R,Z\right)$ location. We
randomly collect a total of $1,118$ KSTAR discharges from $2017$ and $2018$
campaigns. Since each discharge can be further broken into many time slices,
i.e., every $50$ msec following the temporal resolution of the off-line EFIT,
we obtain $217,820$ time slices. With a total of $286$ value of $\psi$ from
$22\times 13$ spatial points, we have a total of
$62,296,520\left(=217,820\times 286\right)$ samples to train and validate the
networks. $90$% of the samples are used to train the networks, while the other
$10$% are used to validate the networks to avoid overfitting problems. Note
that an overfitting problem can occur if a network is overly well trained to
the training data following the very details of them. This inhibits
generalization of the trained network to predict unseen data, and such a
problem can be minimized with the validation data set. All the inputs except
$R$ and $Z$ are normalized such that the maximum and minimum values within the
whole samples become $1$ and $-1$, respectively. We use the actual values of
$R$ and $Z$ in the unit of meters.
Table 1 summarizes the training and validation samples discussed in this
section. Additionally, we also have randomly collected another $163$ KSTAR
discharges in the same way discussed here which are different from the $1,118$
KSTAR discharges to test the performance of the networks.
## III Neural network model and training
### III.1 Neural network model
We develop the neural networks that not only output a value of $\psi$ but also
satisfies Equation (1), the GS equation. With the total of $94$ input nodes
($91$ for a plasma current and magnetic signals, two for $R$ and $Z$ position,
one for the bias) and one output node for a value of $\psi$, each network has
three fully connected hidden layers with an additional bias node at each
hidden layer. Each layer contains $61$ nodes including the bias node. The
structure of our networks is selected by examining several different
structures by error and trials.
Denoting the value of $\psi$ calculated by the networks as
$\psi^{\textrm{NN}}$, we have
$\begin{split}&\psi^{\textrm{NN}}\\!=\\!s_{0}+\sum_{l=1}^{60}\\!s_{l}\\\
&\times
f\\!\\!\left(\\!u_{l0}\\!+\sum_{k=1}^{60}\\!u_{lk}f\\!\\!\left(\\!v_{k0}\\!+\\!\sum_{j=1}^{60}\\!v_{kj}f\\!\\!\left(\\!w_{j0}\\!+\\!\sum_{i=1}^{93}\\!w_{ji}x_{i}\\!\right)\\!\\!\\!\right)\\!\\!\\!\right),\end{split}$
(2)
where $x_{i}$ is the $i^{\textrm{th}}$ input value with $i=1,\dots,93$, i.e.,
$91$ measured values with the various magnetic diagnostics and two for $R$ and
$Z$ positions. $w_{ji}$ is an element in a $61\times 94$ matrix, whereas
$v_{kj}$ and $u_{lk}$ are elements in $61\times 61$ matrices. $s_{l}$ connects
the $l^{\textrm{th}}$ node of the third (last) hidden layer to the output
node. $w,v,u$ and $s$ are the weighting factors that need to be trained to
achieve our goal of obtaining accurate $\psi$. $w_{j0},v_{k0},u_{l0}$ and
$s_{0}$ are the weighting factors connecting the biases, where values of all
the biases are fixed to be unity. We use a hyperbolic tangent function as the
activation function $f$ giving the network non-linearity Haykin (2008):
$f\\!\left(t\right)=\tanh(t)=\frac{2}{1+e^{-2t}}-1.$ (3)
The weighting factors are initialized as described in Glorot and Bengio (2010)
so that a good training can be achieved. They are randomly selected from a
normal distribution whose mean is zero with the variance set to be an inverse
of total number of connecting nodes. For instance, our weighting factor $w$
connects the input layer (94 nodes with bias) and the first hidden layer (61
nodes with bias), therefore the variance is set to be $1/(94+61)$. Likewise,
the variances for $v$, $u$ and $s$ are $1/(61+61)$, $1/(61+61)$ and
$1/(61+1)$, respectively.
### III.2 Training
With the aforementioned network structure, training (or optimizing) the
weighting factors to predict the correct value of $\psi$ highly depends on a
choice of the cost function. A typical choice of such cost function would be:
$\epsilon=\frac{1}{N}\sum_{i=1}^{N}\left(\psi_{i}^{\textrm{NN}}-\psi_{i}^{\textrm{Target}}\right)^{2},$
(4)
where $\psi^{\textrm{Target}}$ is the target value, i.e., the value of $\psi$
from the off-line EFIT results in our case, and $N$ the number of data sets.
As will be shown shortly, minimizing the cost function $\epsilon$ does not
guarantee to satisfy the GS equation (Equation (1)) even if
$\psi^{\textrm{NN}}$ and $\psi^{\textrm{Target}}$ matches well, i.e., the
network is well trained with the given optimization rule. Since
$\Delta^{*}\psi$ provides information on the toroidal current density
directly, it is important that $\Delta^{*}\psi^{\textrm{NN}}$ matches
$\Delta^{*}\psi^{\textrm{Target}}$ as well. We have an analytic form
representing $\psi^{\textrm{NN}}$ as in Equation (2); therefore, we can
analytically differentiate $\psi^{\textrm{NN}}$ with respect to $R$ and $Z$,
meaning that we can calculate $\Delta^{*}\psi^{\textrm{NN}}$ during the
training stage. Thus, we introduce another cost function:
$\begin{split}\epsilon^{\textrm{new}}&=\frac{1}{N}\sum_{i=1}^{N}\left(\psi_{i}^{\textrm{NN}}-\psi_{i}^{\textrm{Target}}\right)^{2}\\\
&+\frac{1}{N}\sum_{i=1}^{N}\left(\Delta^{*}\psi_{i}^{\textrm{NN}}-\Delta^{*}\psi_{i}^{\textrm{Target}}\right)^{2},\end{split}$
(5)
where we obtain the value of $\Delta^{*}\psi^{\textrm{Target}}$ from the off-
line EFIT results as well.
Figure 3: An example of the two networks’ results trained with the cost
function (a) $\epsilon$ and (b) $\epsilon^{\textrm{new}}$ for KSTAR shot#
$17939$ at $0.950$ sec. Both networks (red dashed line) reproduce the
$\psi^{\textrm{Target}}$ (black line) well (left panels), but only the network
trained with $\epsilon^{\textrm{new}}$ reproduces
$\Delta^{*}\psi^{\textrm{Target}}$ (right panels).
To acknowledge difference between the two cost functions $\epsilon$ and
$\epsilon^{\textrm{new}}$, we first discuss the results. Figure 3 shows the
outputs of the two trained networks with the cost function (a) $\epsilon$ and
(b) $\epsilon^{\textrm{new}}$. It is evident that in both cases the network
output $\psi^{\textrm{NN}}$ (red dashed line) reproduces the off-line EFIT
$\psi^{\textrm{Target}}$ (black line). However, only the network trained with
the cost function $\epsilon^{\textrm{new}}$ reproduces the off-line EFIT
$\Delta^{*}\psi^{\textrm{Target}}$. Both networks are trained well, but the
network with the cost function $\epsilon$ does not achieve our goal, that is
correctly predicting $\psi^{\textrm{Target}}$ and
$\Delta^{*}\psi^{\textrm{Target}}$.
Since our goal is to develop a neural network that solves the GS equation, we
choose the cost function to be $\epsilon^{\textrm{new}}$ to train the
networks. We optimize the weighting factors by minimizing
$\epsilon^{\textrm{new}}$ with the Adam Kingma and Ba (2014) which is one of
the gradient-based optimization algorithms. With $90$% and $10$% of the total
data samples for training and validation of the networks, respectively, we
stop training the networks with a fixed number of iterations that is large
enough but not too large such that the validation errors do not increase,
i.e., to avoid overfitting problems. The whole workflow is carried out with
Python and Tensorflow Abadi _et al._ (2015).
With the selected cost function we create three different networks that differ
only by the training data sets. NN${}_{\textrm{2017}}$, NN${}_{\textrm{2018}}$
and NN${}_{\textrm{2017, 2018}}$ refer to the three networks trained with the
data sets from only $2017$ ($744$ discharges), from only $2018$ ($374$
discharges) and from both $2017$ and $2018$ ($744+374$ discharges) campaigns,
respectively.
Figure 4: The descending feature of training (blue line) and validation (red
dashed line) errors as a function of iterations. Shaded areas represent
standard deviation of the errors.
The descending feature of the cost function $\epsilon^{\textrm{new}}$ as a
function of the training iteration for NN2017,2018 network is shown in Figure
4. Both the training errors (blue line) and validation errors (red dashed
line) decrease together with similar values which means that the network is
well generalized. Furthermore, since the validation errors do not increase,
the network does not have an overfitting problem. Note that fluctuations in
the errors, i.e., standard deviation of the errors, are represented as shaded
areas.
Small undulations repeated over the iterations in Figure 4 are due to the
mini-batch learning. Contrary to the batch learning, i.e., optimizing the
network with the entire training set in one iteration, the mini-batch learning
divides the training set into some number of small subsets ($1,000$ subsets
for our case) to optimize the networks sequentially. One cycle that goes
through all the subsets once is called an epoch. The mini-batch learning helps
to escape from local minima in the weighting factor space Ge _et al._ (2015)
via the stochastic gradient descent scheme Bottou (2010).
## IV Performance of the developed neural networks: Benchmark tests
In this section, we present how well the developed networks perform. Main
figures of merit we use are peak signal-to-noise ratio (PSNR) and mean
structural similarity (MSSIM) as have been used perviously Matos, Francisco A
_et al._ (2017) in addition to the usual statistical quantity R2, coefficient
of determination. We note that obtaining full flux surface information
$\psi\left(R,Z\right)$ on $22\times 13$ or $65\times 65$ spatial grids with
our networks takes less than $1$ msec on a typical personal computer.
First, we discuss the benchmark results of the NN2017,2018 network. Then, we
compare the performance of NN2017, NN2018 and NN2017,2018 networks. Here, we
also investigate cross-year performance, for instance, applying the NN2017
network to predict the discharges obtained from 2018 campaign and vice versa.
Then, we evaluate the performance of the networks against the rt-EFIT results
to examine possibility of supplementing or even replacing the rt-EFIT with the
networks. Finally, we show how the imputation scheme supports the networks’
performance. Here, all the tests are performed with the unseen (to all three
networks, i.e., NN2017, NN2018 and NN2017,2018) KSTAR discharges which are
$88$ and $75$ KSTAR discharges from 2017 and 2018 campaigns, respectively.
### IV.1 Benchmark results of the NN2017,2018 network
Figure 5: Performance tests of the NN2017,2018 network on the unseen KSTAR
discharges from (a)(b) 2017 campaign and (c)(d) 2018 campaign. The values of
R2 and histograms of (a)(c) $\psi^{\textrm{NN}}$ vs. $\psi^{\textrm{Target}}$
and (b)(d) $\Delta^{*}\psi^{\textrm{NN}}$ vs.
$\Delta^{*}\psi^{\textrm{Target}}$ with colors representing number of counts
manifest goodness of the NN2017,2018 network. Red dashed line is the $y=x$
line. Figure 6: The actual reconstruction results for the KSTAR shot#18057,
comparing the network results and off-line EFIT reconstructions for ramp-up
((b) and (c)), flat-top ((d) and (e)), ramp-down ((f) and (g)) phases
following (a) the plasma current evolution. Black lines indicate the flux
surfaces from the off-line EFIT, overlaid with the red dotted lines which
stand for the NN reconstructions. As a figure of merit, magnitudes of PSNR
metric are written on each figure.
Figure 5 show the benchmark results of the NN2017,2018 network, i.e., network
trained with the data sets from both 2017 and 2018 campaigns. (a) and (b) show
the results with the test discharges from 2017 campaign; while (c) and (d)
present the results with the test discharges from 2018 campaign. Histograms of
(a)(c) $\psi^{\textrm{NN}}$ vs. $\psi^{\textrm{Target}}$ and (b)(d)
$\Delta^{*}\psi^{\textrm{NN}}$ vs. $\Delta^{*}\psi^{\textrm{Target}}$ are
shown with colors representing the number of counts. For instance, there is a
yellow colored point in Figure 5(a) around $(-0.1,-0.1)\pm\varepsilon$, where
$\varepsilon$ is a bin size for the histogram. Since yellow represents about
$2\times 10^{5}$ counts, there are approximately $2\times 10^{5}$ data whose
neural network values and EFIT values are $-0.1\pm\varepsilon$ simultaneously
within our test data set. Note that each KSTAR discharge contains numerous
time slices whose number depends on the actual pulse length of a discharge,
and each time slice generates the total of $22\times 13=286$ data points. The
values of $\psi^{\textrm{Target}}$ and $\Delta^{*}\psi^{\textrm{Target}}$ are
obtained from the off-line EFIT results. It is clear that the network predicts
the target values well.
As a figure of merit, we introduce the R2 metric (coefficient of
determination) defined as
$\textrm{R}^{2}=1-\frac{\sum_{i=1}^{L}\left(y_{i}^{\textrm{Target}}-y_{i}^{\textrm{NN}}\right)^{2}}{\sum_{i=1}^{L}\left(y_{i}^{\textrm{Target}}-\frac{1}{L}\sum_{j=1}^{L}y_{j}^{\textrm{Target}}\right)^{2}},$
(6)
where $y$ takes either $\psi$ or $\Delta^{*}\psi$, and $L$ is the number of
test data sets. The calculated values are written in Figure 5, and they are
indeed close to unity, implying the existence of very strong linear
correlations between the predicted (from the network) and target (from the
off-line EFIT) values. Note that R${}^{2}=1$ means the perfect prediction. The
red dashed lines on the figures are the $y=x$ lines.
Figure 6 is an example of reconstructed magnetic equilibria using KSTAR shot
#18057 from 2017 campaign. (a) shows the evolution of the plasma current. The
vertical dashed lines indicate the time points where we show and compare the
equilibria obtained from the network (red) and the off-line EFIT (black) which
is our target. (b) and (c) are taken during the ramp-up phase, (d) and (e)
during the flat-top phase, and (f) and (g) during the ramp-down phase. In each
sub-figure from (b) to (g), left panels compare $\psi$, and right panels are
for $\Delta^{*}\psi$. We mention that the equilibria in Figure 6 are
reconstructed with $65\times 65$ grid points even though the network is
trained with $22\times 13$ grid points demonstrating how spatial resolution is
flexible in our networks.
For a quantitative assessment of the network, we use an image relevant figure
of merit that is peak signal-to-noise ratio (PSNR) Huynh-Thu and Ghanbari
(2008) (see Appendix B) originally developed to estimate a degree of artifacts
due to an image compression compared to an original image. Typical PSNR range
for the JPEG image, which preserves the original quality with a reasonable
degree, is generally in 30–50 dB Matos, Francisco A _et al._ (2017); Ebr
(2004). For our case, the networks errors relative to the off-line EFIT
results can be treated as artifacts. As listed on Figure 6(b)-(g), PSNR for
$\psi$ is very good, while we achieve acceptable values for $\Delta^{*}\psi$.
### IV.2 The NN2017, NN2018 and NN2017,2018 networks
Figure 7: Same as Figure 5 for the the NN2017 network, i.e., trained with the
data sets from 2017 campaign. Figure 8: Same as Figure 5 for the the NN2018
network, i.e., trained with the data sets from 2018 campaign.
Similar to shown in Figure 5, we show the benchmark results of the NN2017
(trained with the data sets from 2017 campaign) and the NN2018 (trained with
the data sets from 2018 campaign) in Figures 7 and 8, respectively. R2 metric
is also provided on the figures. Again, overall performance of the networks
are good.
The NN2017 and NN2018 networks are trained with only in-campaign data sets,
e.g., NN2018 with the data sets from only 2018 campaign, and we find slightly
worse results, but still good, on predicting cross-campaign magnetic
equilibria, e.g. NN2018 predicting equilibria for 2017 campaign. Notice that
the NN2017 seems to predict cross-campaign equilibria better than in-campaign
ones by comparing Figure 7(a) and (c) which contradicts our intuition.
Although the histogram in Figure 7(c) seems tightly aligned with the $y=x$
line (red dashed line), close inspection reveals that the NN2017 network, in
general, underestimates the off-line EFIT results from 2018 campaign
marginally. This will be evident when we compare image qualities.
Mean structural similarity (MSSIM) Wang _et al._ (2004) (see Appendix B) is
another image relevant figure of merit used to estimate perceptual similarity
(or perceived differences) between the true and reproduced images based on
inter-dependence of adjacent spatial pixels in the images. MSSIM ranges from
zero to one, where the closer to unity the better the reproduced image is.
Figure 9: Histograms of MSSIM (left panel) and PSNR (right panel) for (a)
NN2017, (b) NN2018 and (c) NN2017,2018. Red (green) line indicates the test
results on the data sets from 2017 (2018) campaign. In each sub-figure, top
(bottom) panel show the results for $\psi$ ($\Delta^{*}\psi$). The off-line
EFIT results are used as reference.
Together with PSNR, Figure 9 shows MSSIM for (a) NN2017, (b) NN2018 and (c)
NN2017,2018 where the off-line EFIT results are used as reference. Notice that
counts in all the histograms of MSSIM and PSNR in this work correspond to the
number of reconstructed magnetic equilibria (or a number of time slices) since
we obtain a single value of MSSIM and PSNR from one equilibrium; whereas
counts in Figures 5, 7 and 8 are much bigger since $286(=22\times 13)$ data
points are generated from each time slice. Red (green) line indicates the test
results on the data sets from 2017 (2018) campaign. In general, whether the
test data sets are in-campaign or cross-campaign, image reproducibility of all
three networks, i.e., predicting the off-line EFIT results, is good as
attested by the fact that MSSIM is quite close to unity and PSNR for $\psi$
($\Delta^{*}\psi$) ranges approximately $40$ to $60$ ($20$ to $40$). It is
easily discernible that in-campaign results are better for both NN2017 and
NN2018 unlike what we noted in Figure 7(a) and (c). Not necessarily
guaranteed, we find that the NN2017,2018 network works equally well for both
campaigns as shown in Figure 9(c).
### IV.3 Comparisons among nn-EFIT, rt-EFIT and off-line EFIT
Figure 10: An example of reconstructed $\psi\left(R,Z\right)$ (left panel) and
$\Delta^{*}\psi\left(R,Z\right)$ (right panel) for KSTAR shot #$17975$ at
$0.7$ sec comparing (a) rt-EFIT (green) and off-line EFIT (black) and (b) nn-
EFIT (NN2017,2018) (red) and off-line EFIT (black).
It is widely recognized that rt-EFIT results and off-line results are
different from each other. If we allow the off-line EFIT results used to train
the networks to be accurate ones, then the reconstruction of equilibria with
the neural networks (nn-EFIT) must satisfy the following criterion: nn-EFIT
results must be more similar to the off-line EFIT results than rt-EFIT results
are to the off-line EFIT as mentioned in Section I. Once this criterion is
satisfied, then we can always improve the nn-EFIT as genuinely more accurate
EFIT results are collected. For this reason, we make comparisons among the nn-
EFIT, rt-EFIT and off-line EFIT results.
Figure 10 shows an example of reconstructed magnetic equilibria for (a) rt-
EFIT vs. off-line EFIT and (b) nn-EFIT (the NN2017,2018 network) vs. off-line
EFIT for KSTAR shot #$17975$ at $0.7$ sec with $\psi$ (left panel) and
$\Delta^{*}\psi$ (right panel). Green, red and black lines indicate rt-EFIT,
nn-EFIT and off-line EFIT results, respectively. This simple example shows
that the nn-EFIT is more similar to the off-line EFIT than the rt-EFIT is to
the off-line EFIT, satisfying the aforementioned criterion.
Figure 11: Histograms of MSSIM (left panel) and PSNR (right panel) of $\psi$
(top) and $\Delta^{*}\psi$ (bottom) calculated by the nn-EFIT (black) and the
rt-EFIT (green), where the nn-EFIT is the NN2017,2018. For both the nn-EFIT
and the rt-EFIT, the off-line EFIT is treated as reference.
To validate the criterion statistically, we generate histograms of MSSIM and
PSNR for the nn-EFIT and the rt-EFIT with reference to the off-line EFIT. This
is shown in Figure 11 as histograms, where MSSIM (left panel) and PSNR (right
panel) of $\psi$ (top) and $\Delta^{*}\psi$ (bottom) are compared between the
nn-EFIT (black) and the rt-EFIT (green). Here, the nn-EFIT results are
obtained with the NN2017,2018 network on the test data sets. We confirm that
the criterion is satisfied with the NN2017,2018 network as the histograms in
Figure 11 are in favour of the nn-EFIT, i.e., larger MSSIM and PSNR are
obtained by the nn-EFIT. This is more conspicuous for $\Delta^{*}\psi$ than
$\psi$.
Figure 12: Same as Figure 11 with the NN2017 as the nn-EFIT where the test
data sets are obtained from (a) 2017 campaign and (b) 2018 campaign. Figure
13: Same as Figure 11 with the NN2018 as the nn-EFIT where the test data sets
are obtained from (a) 2017 campaign and (b) 2018 campaign.
We perform the similar statistical analyses for the other two networks, NN2017
and NN2018, which are shown in Figures 12 and 13. Since these two networks are
trained with the data sets from only one campaign, we show the results where
the test data sets are prepared from (a) 2017 campaign or (b) 2018 campaign so
that in-campaign and cross-campaign effects can be assessed separately. We
find that whether in- or cross-campaign, the criterion is fulfilled for both
$\psi$ and $\Delta^{*}\psi$.
### IV.4 The NN2017,2018 network with the imputation scheme
If one or a few magnetic pick-up coils which are a part of the inputs to the
nn-EFIT are impaired, then we will have to re-train the network without the
damaged ones or hope that the network will reconstruct equilibria correctly by
padding a fixed value, e.g., zero-padding, to the broken ones. Of course, one
can anticipate training the network by considering possible combinations of
impaired magnetic pick-up coils. With the total number of $68$ signals from
the magnetic pick-up coils being inputs to the network in our case, we
immediately find that the number of possible combinations increases too
quickly to consider it as a solution.
Since inferring the missing values is better than the null replacement van
Lint _et al._ (2005), we resolve the issue by using the recently proposed
imputation method Joung _et al._ (2018) based on Gaussian processes (GP)
Rasmussen and Williams (2006) and Bayesian inference Sivia and Skilling
(2006), where the likelihood is constructed based on Maxwell’s equations. The
imputation method infers the missing values fast enough, i.e., less than $1$
msec to infer at least up to nine missing values on a typical personal
computer; thus, we can apply the method during a plasma discharge by replacing
the missing values with the real-time inferred values.
Figure 14: Measured (blue open circles) and inferred with the imputation
method Joung _et al._ (2018) (red crosses with their uncertainties) values
for (a) $B_{\textrm{n}}$ and (b) $B_{\textrm{t}}$. Probe # on the horizontal
axis is used as an identification index of magnetic pick-up coils. Inferred
probes are Probe #3, 4, 6, 14, 18, 24, 30, 35, 37 for $B_{\textrm{n}}$ and
Probe #4, 6, 8, 11, 17, 29, 30, 32, 35 for $B_{\textrm{t}}$. Table 2: The
imputation results shown in Figure 14 with KSTAR shot #20341 at 2.1 sec.
$B{{}_{n}}$ [T] $\times 10^{-2}$ | $B{{}_{t}}$ [T] $\times 10^{-2}$
---|---
No. | Measured | Inferred | No. | Measured | Inferred
3 | -1.45 | -1.88$\pm$0.22 | 4 | -14.69 | -13.97$\pm$0.47
4 | -1.72 | -2.31$\pm$0.24 | 6 | -12.38 | -11.42$\pm$0.97
6 | 4.62 | 4.45$\pm$0.65 | 8 | -7.82 | -7.88$\pm$0.67
14 | 6.13 | 6.36$\pm$0.27 | 11 | -3.15 | -3.22$\pm$0.65
18 | -8.27 | -8.11$\pm$0.48 | 17 | 0.10 | 0.30$\pm$0.52
24 | 1.86 | 1.65$\pm$0.30 | 29 | 3.84 | 2.65$\pm$0.64
30 | -7.52 | -7.19$\pm$0.18 | 30 | 1.15 | 0.49$\pm$0.61
35 | -7.93 | -7.08$\pm$0.65 | 32 | -2.65 | -2.11$\pm$0.62
37 | -4.27 | -1.41$\pm$0.93 | 35 | -8.07 | -8.87$\pm$0.55
Figure 15: Top panel: nn-EFIT (NN2017,2018 network) reconstructed equilibria
without any missing values (black line), and with two missing values replaced
with the inferred values using the imputation method (green line) or with the
zeros using the zero-padding method (pink dashed line), where the missing
values are (a) $B_{\textrm{n}}$ Probe #$14$ and $30$ (left panel) and (b)
$B_{\textrm{t}}$ Probe #$4$ and $8$ (right panel). Bottom panels: histograms
of MSSIM and PSNR using the imputation method (green) and the zero-padding
method (pink) for all the equilibria obtained from KSTAR shot #20341, where
the reference values are those obtained using nn-EFIT without any missing
values. Note that there are many more counts less than $0.9$ for MSSIM with
the zero-padding method.
We have applied the imputation method to KSTAR shot #$20341$ at $2.1$ sec for
the normal ($B_{\textrm{n}}$) and tangential ($B_{\textrm{t}}$) components of
the magnetic pick-up coils as an example. We have randomly chosen nine signals
from the $32$ $B_{\textrm{n}}$ measurements and another nine from the $36$
$B_{\textrm{t}}$ measurements and pretended that all of them ($9+9$) are
missing simultaneously. Figure 14 shows the measured (blue open circles) and
the inferred (red crosses with their uncertainties) values for (a)
$B_{\textrm{n}}$ and (b) $B_{\textrm{t}}$. Probe # on the horizontal axis is
used as an identification index of the magnetic pick-up coils. Table 2
provides the actual values of the measured and inferred ones for better
comparisons. We find that the imputation method infers the correct (measured)
values very well except Probe #$37$ of $B_{\textrm{n}}$. Inferred (missing)
probes are Probe #3, 4, 6, 14, 18, 24, 30, 35, 37 for $B_{\textrm{n}}$ and
Probe #4, 6, 8, 11, 17, 29, 30, 32, 35 for $B_{\textrm{t}}$. Here, we provide
all the Probe #’s used for the neural network: $B_{\textrm{n}}$ Probe #[2,
$\dots$, 6, 8, 9, 11, $\dots$, 15, 17, $\dots$, 20, 23, $\dots$, 26, 28,
$\dots$, 32, 34, 35, 37, $\dots$, 41] (a total of $32$) and $B_{\textrm{t}}$
Probe #[2, $\dots$, 6, 8, 9, 11, $\dots$, 32, 34, 35, 37, $\dots$, 41] (a
total of $36$).
Comparisons between the nn-EFIT without any missing values, which we treat as
reference values, and the nn-EFIT with the imputation method or with the zero-
padding method are made. Here, nn-EFIT results are obtained using the
NN2017,2018 network. Top panel of Figure 15 shows $\psi\left(R,Z\right)$
obtained from the nn-EFIT without any missing values (black line) and from the
nn-EFIT with the two missing values replaced with the inferred values (green
line), i.e., imputation method, or with zeros (pink dashed line), i.e., zero-
padding method for (a) $B_{\textrm{n}}$ (left panel) and (b) $B_{\textrm{t}}$
(right panel) at $2.1$ sec of KSTAR shot #20341. Probe #14 and 30 for
$B_{\textrm{n}}$ and Probe #4 and 8 for $B_{\textrm{t}}$ are treated as the
missing ones. Bottom panels compare histograms of MSSIM and PSNR using the
imputation method (green) and the zero-padding method (pink) for all the
equilibria obtained from KSTAR shot #20341.
It is clear that nn-EFIT with the imputation method (green line) is not only
much better than that with the zero-padding method (pink dashed line) but it
also reconstructs the equilibrium close to the reference (black). In fact, the
zero-padding method is too far off from the reference (black line) to be
relied on for plasma controls.
Figure 16: Same color code as in Figure 15. Missing values are (a) eight
$B_{\textrm{t}}$ (except only Probe #$6$) and (b) all nine $B_{\textrm{t}}$.
Figure 17: Same color code as in Figure 15. Missing values are (a) eight
$B_{\textrm{n}}$ (except only Probe #$37)$, (b) all nine $B_{\textrm{n}}$.
Motivated by such a successful result of the nn-EFIT with the imputation
method on the two missing values, we have increased number of missing values
as shown in Figures 16 and 17 for the same KSTAR discharge, i.e., KSTAR shot
#20341. Let us first discuss Figure 16 which are with (a) the eight (except
only Probe #6) and (b) nine (all) missing values of $B_{\textrm{t}}$. Color
codes are same as in Figure 15, i.e., the reference is black, and nn-EFIT with
the imputation method green or with the zero-padding method pink. It is
evident that the nn-EFIT with the imputation method performs well at least up
to nine missing values. Such a result is, in fact, expected since the
imputation method has inferred the missing values well as shown in Figure
14(b) in addition to the fact that a well-trained neural network typically has
a reasonable degree of resistance on noises. Again, the nn-EFIT with the zero-
padding method is not reliable.
Figure 17 (a) and (b) are results with the eight (except only Probe #37) and
nine (all) missing values of $B_{\textrm{n}}$, respectively. Color codes are
same as in Figure 15. We find that the nn-EFIT with the eight missing values
reconstructs the equilibrium similar to the reference one, while the
reconstruction quality becomes notably worse for nine missing values. This is
caused mostly due to poor inference of Probe #$37$ by the imputation method
(see Figure 14(a)). Nevertheless, the result is still better than the zero-
padding method. Figure 18 shows the reconstruction results with the same color
codes as in Figure 15 when we have (a) $4+4$ and (b) $9+9$ combinations of
$B_{\textrm{n}}$ and $B_{\textrm{t}}$ missing values simultaneously.
All these results suggest that the nn-EFIT with the imputation method
reconstructs equilibria reasonably well except when the imputation infers the
true value poorly, e.g., $B_{\textrm{n}}$ Probe #37 in Figure 14(a) and Table
2. In fact, the suggested imputation method Joung _et al._ (2018) infers the
missing values based on the neighboring intact values (using Gaussian
processes) while satisfying the Maxwell’s equations (using Bayesian
probability theory). Consequently, such a method becomes less accurate if (1)
the neighboring channels are also missing AND (2) the true values change fast
from the neighboring values. In fact, $B_{\textrm{n}}$ Probe #37 happens to
satisfy these two conditions, i.e., Probe #35 is also missing, and the true
values of Probe #35, #37 and #38 are changing fast as one can discern from
Figure 14(a).
Figure 18: Same color code as in Figure 15. Combinations of missing
$B_{\textrm{n}}$ and $B_{\textrm{t}}$ are examined: (a) four missing
$B_{\textrm{n}}$ and four mssing $B_{\textrm{t}}$ case and (b) nine missing
$B_{\textrm{n}}$ and nine missing $B_{\textrm{t}}$ case.
## V Conclusions
We have developed and presented the neural network based Grad-Shafranov solver
constrained with the measured magnetic signals. The networks take the plasma
current from a Rogowski coil, 32 normal and 36 tangential components of the
magnetic fields from the magnetic pick-up coils, 22 poloidal fluxes from the
flux loops, and $\left(R,Z\right)$ position of the interest as inputs. With
three fully connected hidden layers consisting of 61 nodes each layer, the
network outputs a value of poloidal flux $\psi$. We set the cost function used
to train the networks to be a function of not only the poloidal flux $\psi$
but also the Grad-Shafranov equation $\Delta^{*}\psi$ itself. The networks are
trained and validated with $1,118$ KSTAR discharges from 2017 and 2018
campaigns.
Treating the off-line EFIT results as accurate magnetic equilibria to train
the networks, our networks fully reconstruct magnetic equilibria, not limited
to obtaining selected information such as positions of magnetic axis, X-points
or plasma boundaries, more similar to the off-line EFIT results than the rt-
EFIT is to the off-line EFIT. Owing to the fact that $\left(R,Z\right)$
position is a part of the input, our networks have adjustable spatial
resolution within the first wall. The imputation method supports the networks
to obtain the nn-EFIT results even if there exist a few missing inputs.
As all the necessary computation time is approximately $1$ msec, the networks
have potential to be used for real-time plasma control. In addition, the
networks can be used to provide large number of automated EFIT results fast
for many other data analyses requiring magnetic equilibria.
## Acknowledgement
This research is supported by National R&D Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT
(grant numbers NRF-2017M1A7A1A01015892 and NRF-2017R1C1B2006248) and the
KUSTAR-KAIST Institute, KAIST, Korea.
## Appendix A Real-time preprocess on magnetic signals
As shown in Figure 2 and discussed in Section II, normal ($B_{\textrm{n}}$)
and tangential ($B_{\textrm{t}}$) components of magnetic fields measured by
the magnetic pick-up coils and poloidal magnetic fluxes ($\Psi_{\textrm{FL}}$)
measured by the flux loops tend to have residual drifts after calibrating the
magnetic diagnostics (MDs). We train the neural networks with preprocessed,
i.e., drift adjusted, magnetic signals. Therefore, we must be able to
preprocess the signals in real time as well. Here, we introduce how we
preprocess the magnetic signals in detail. The same preprocess is applied to
all the training, validation and test data sets. Note that we do not claim
that how we adjust the magnetic signals corrects the drifts completely.
### A.1 Real-time drift adjustment with information obtained during the
initial magnetization stage
To adjust the signal drifts, we deem a priori that the signals drift linearly
in time Strait _et al._ (1997); Xia, Yu-Jun _et al._ (2015); Ka, E M _et
al._ (2008). Of course, non-linear drift may well exist in the signals.
However, we need to come up with a very simple and fast solution to adjust the
drifts in real time with the limited amount of information. One can consider
such linearization in time as taking up to the first order of Taylor expanded
drifting signals. Therefore, we take the drifting components of the signals
($y_{i}^{m}$) from various types (the magnetic pick-up coils or the flux
loops) of MDs to follow:
$y_{i}^{m}=a_{i}^{m}t+b_{i}^{m},$ (7)
where $t$ is the time. $a_{i}^{m}$ and $b_{i}^{m}$ are the slope and the
offset, respectively, of a drift signal for the $i^{\textrm{th}}$ magnetic
sensor of a type $m$ (magnetic pick-up coils or flux loops). Then, our goal
simply becomes finding $a_{i}^{m}$ and $b_{i}^{m}$ for all $i$ and $m$ of
interests before a plasma starts or the blip time ($t=0$) so that $y_{i}^{m}$
can be subtracted from the measured magnetic signals in real-time, i.e.,
preprocessing the magnetic signals for the neural networks.
Figure 19: An example of temporal evolutions of (a) currents in the PF coils,
(b) normal and (c) tangential components of magnetic fields measured by the
magnetic pick-up coils, respectively, and (d) poloidal flux measured by one of
the flux loops during the initial magnetization stage, i.e., $t<0$, for a
typical KSTAR discharge. Information from the time interval d1 (d2) is used to
estimate $a_{i}^{m}$ ($b_{i}^{m}$).
We use two different time intervals during the initial magnetization stage,
i.e., before the blip time, for every plasma discharge to find $a_{i}^{m}$ and
$b_{i}^{m}$, sequentially. Figure 19 shows an example of temporal evolutions
of currents in the poloidal field (PF) coils, $B_{\textrm{n}}$ and
$B_{\textrm{t}}$ and poloidal magnetic flux up to the blip time ($t=0$) of a
typical KSTAR discharge.
During the time interval d1 in Figure 19, all the magnetic signals must be
constant in time because there are no changes in currents of all the PF coils
as well as there are no plasmas yet that can change the magnetic signals.
Therefore, any temporal changes in a magnetic signal during d1 can be regarded
as due to a non-zero $a_{i}^{m}$. With the knowledge of $a_{i}^{m}$ from d1
time interval, we obtain the value of $b_{i}^{m}$ using the fact that all the
magnetic signals must be zeros during the time interval d2 because there are
no sources of magnetic fields, i.e., all the currents in the PF coils are
zeros.
Summarizing our procedure, (1) we first obtain the slopes $a_{i}^{m}$ based on
the fact that all the magnetic signals must be constant in time during d1 time
interval, and then (2) find the offsets $b_{i}^{m}$ based on the fact that all
the magnetic signals, after the linear drifts in time are removed based on the
knowledge of $a_{i}^{m}$, must be zeros during d2 time interval.
### A.2 Bayesian inference
Bayesian probability theory Sivia and Skilling (2006) has a general form of
$p\left(\mathcal{W}|\mathcal{D}\right)=\frac{p\left(\mathcal{D}|\mathcal{W}\right)p\left(\mathcal{W}\right)}{p\left(\mathcal{D}\right)},$
(8)
where $\mathcal{W}$ is a (set of) parameter(s) we wish to infer, i.e.,
$a_{i}^{m}$ and $b_{i}^{m}$ for our case, and $\mathcal{D}$ is the measured
data, i.e., measured magnetic signals during the time intervals of d1 and d2
in Fig. 19. The posterior $p\left(\mathcal{W}|\mathcal{D}\right)$ provides us
probability of having a certain value for $\mathcal{W}$ given the measured
data $\mathcal{D}$ which is proportional to a product of likelihood
$p\left(\mathcal{D}|\mathcal{W}\right)$ and prior $p\left(\mathcal{W}\right)$.
Then, we use the maximum a posterior (MAP) to select the value of
$\mathcal{W}$. The evidence $p\left(\mathcal{D}\right)$ (or marginalized
likelihood) is typically used for a model selection and is irrelevant here as
we are only interested in estimating the parameter $\mathcal{W}$, i.e.,
$a_{i}^{m}$ and $b_{i}^{m}$.
We estimate values of the slope $a_{i}^{m}$ and the offset $b_{i}^{m}$ based
on Equation (8) in two steps as described above:
$\text{Step (1)}\>:\>p(a_{i}^{m}|\mathcal{\vec{D}}_{i,d1}^{m})\propto
p(\vec{D}_{i,d1}^{m}|a_{i}^{m})p(a_{i}^{m}),$ (9) $\text{Step
(2)}\>:\>p(b_{i}^{m}|\mathcal{\vec{D}}_{i,d2}^{m},a_{i}^{m*})\propto
p(\vec{D}_{i,d2}^{m}|b_{i}^{m},a_{i}^{m*})p(b_{i}^{m}),$ (10)
where $\mathcal{\vec{D}}_{i,d1}^{m}$ ($\mathcal{\vec{D}}_{i,d2}^{m}$) are the
time series data from the $i^{\text{th}}$ magnetic sensor of a type $m$
(magnetic pick-up coils or flux loops) during the time intervals of d1 (d2) as
shown in Fig. 19. $a_{i}^{m*}$ is the MAP, i.e., the value of $a_{i}^{m}$
maximizing the posterior $p(a_{i}^{m}|\mathcal{\vec{D}}_{i,d1}^{m})$. Since we
have no prior knowledge on $a_{i}^{m}$ and $b_{i}^{m}$, we take priors,
$p(a_{i}^{m})$ and $p(b_{i}^{m})$, to be uniform allowing all the real
numbers. Note that a correct $p(a_{i}^{m})$ would be equal to
$1/\left[\pi\left(1+\left(a_{i}^{m}\right)^{2}\right)\right]$ von Toussaint
(2011), but we sacrifice rigor to obtain a fast solution. Furthermore, the
posterior for $b_{i}^{m}$ should, rigorously speaking, be obtained by
marginalizing over all possible $a_{i}^{m}$, i.e.,
$p(b_{i}^{m}|\mathcal{\vec{D}}_{i,d2}^{m})=\int
p(b_{i}^{m}|\mathcal{\vec{D}}_{i,d2}^{m},a_{i}^{m})p(a_{i}^{m}|\mathcal{\vec{D}}_{i,d1}^{m})da_{i}^{m}$.
Again, as we are interested in real-time application, such a step is
simplified just to use $a_{i}^{m*}$.
With Equation (7), we model likelihoods, $p(\vec{D}_{i,d1}^{m}|a_{i}^{m})$ and
$p(\vec{D}_{i,d2}^{m}|b_{i}^{m},a_{i}^{m*})$, as Gaussian:
$\displaystyle
p(\vec{D}_{i,d1}^{m}|a_{i}^{m})=\frac{1}{\sqrt{(2\pi)^{L}}|\sigma_{i,d1}^{m}|}$
(11) $\displaystyle\\!\times\\!\exp\\!\left(\\!-\frac{\sum\limits_{t_{l}\in
d1}^{L}\left[a_{i}^{m}(t_{l}-t_{0})-\left(D_{i,d1}^{m}(t_{l})-\left<D_{i,d1}^{m}(t_{0})\right>\right)\right]^{2}}{2(\sigma_{i,d1}^{m})^{2}})\\!\right),$
(12) $\displaystyle p(\vec{D}_{i,d2}^{m}|b_{i}^{m},$ $\displaystyle
a_{i}^{m*})=\frac{1}{\sqrt{(2\pi)^{K}}|\sigma_{i,d2}^{m}|}$ (14)
$\displaystyle\times\\!\exp\\!\left(\\!-\frac{\sum\limits_{t_{k}\in
d2}^{K}\left[b_{i}^{m}-\left(D_{i,d2}^{m}(t_{k})-a_{i}^{m*}t_{k}\right)\right]^{2}}{2(\sigma_{i,d2}^{m})^{2}}\\!\right),$
(15)
which simply state that noises in the measured signals follow Gaussian
distributions. Here, $\sigma_{i,d1}^{m}$ and $\sigma_{i,d2}^{m}$ are the
experimentally obtained noise levels for the $i^{\text{th}}$ magnetic sensor
of a type $m$ (magnetic pick-up coils and flux loops) during the time
intervals of d1 and d2 in Figure 19, respectively. $t_{l}$ and $t_{k}$ define
the actual time intervals of d1 and d2, i.e., $t_{l}\in[-6,-1]$ sec and
$t_{k}\in[-14,-13]$ sec with $L$ and $K$ being the numbers of the data points
in each time interval, respectively. $t_{0}$ can be any value within the d1
time interval, and we set $t_{0}=-2$ sec in this work.
$\left<D_{i,d1}^{m}(t_{0})\right>$, removing the offset effect to obtain only
the slope, is the time averaged value of $D_{i,d1}^{m}(t)$ for
$t\in[t_{0}-0.5,t_{0}+0.5]$ sec. We use the time averaged value to minimize
the effect of the noise in $D_{i,d1}^{m}(t)$ at $t=t_{0}$.
With our choice of uniform distributions for priors in Equations (9) and (10),
MAPs for $a_{i}^{m}$ and $b_{i}^{m}$, which we denote them as $a_{i}^{m*}$ and
$b_{i}^{m*}$, coincide with the maximum likelihoods which can be analytically
obtained by maximizing Equations (LABEL:eq:like-slope) and (LABEL:eq:like-
offset) with respect to $a_{i}^{m}$ and $b_{i}^{m}$, respectively:
$a_{i}^{m*}=\frac{\sum\limits_{t_{l}\in
d1}^{L}\left[\left(D_{i,d1}^{m}(t_{l})-\left<D_{i,d1}^{m}(t_{0})\right>\right)\left(t_{l}-t_{0}\right)\right]}{\sum\limits_{t_{l}\in
d1}^{L}\left[t_{l}-t_{0}\right]^{2}},$ (17)
$b_{i}^{m*}=\frac{1}{K}\sum\limits_{t_{k}\in
d2}^{K}\left[D_{i,d2}^{m}(t_{k})-a_{i}^{m*}t_{k}\right].$ (18)
Now, we have attained simple algebraic equations based on Bayesian probability
theory which can provide us values of the slope $a_{i}^{m}$ and the offset
$b_{i}^{m}$ before the blip time, i.e., before $t=0$.
Since the required information ($a_{i}^{m}$ and $b_{i}^{m}$) to adjust drifts
in the magnetic signals is obtained before every discharge starts, we can
preprocess the magnetic signals in real time. This is how we have adjusted the
drift signals shown in Figure 2.
## Appendix B Image relevant figures of merit - PSNR and MSSIM
In Section IV, we used two image relevant figures of merit, namely PSNR (peak
signal-to-noise ratio) Huynh-Thu and Ghanbari (2008); Ebr (2004) and MSSIM
(mean structural similarity) Wang _et al._ (2004), to examine performance of
the developed neural networks. Although these figures of merit are widely used
and well known, we present short descriptions of PSNR and MSSIM for the sake
of readers’ convenience. Notice that we treat a reconstructed magnetic
equilibrium as an image whose dimension (a number of pixels) is set by the
spatial grid points.
### B.1 Peak signal-to-noise ratio (PSNR)
PSNR is calculated as
$\textrm{PSNR}=10\times\log_{10}\left[\frac{\max\left(y^{\textrm{Target}}\right)^{2}}{\frac{1}{M}\sum_{i=1}^{M}\left(y_{i}^{\textrm{Target}}-y_{i}^{\star}\right)^{2}}\right],$
(19)
where $y_{i}$ is the value of either $\psi$ or $\Delta^{*}\psi$ at the
$i^{\textrm{th}}$ position of the spatial grid (analogous to a pixel value of
an image), and $M$ for the total number of the grid points, i.e., either
$286(=22\times 13)$ or $4225(=65\times 65)$ depending on our choice for
reconstructing an equilibrium. $\max(\cdot)$ operator selects the maximum
value of an argument, and $y^{\textrm{Target}}$ is an array containing ‘pixel’
values of a reference EFIT ‘image’, that is a reconstructed magnetic
equilibrium. $y^{\star}$ is also an array, and depending on whether we wish to
compare the off-line EFIT result with either rt-EFIT result or nn-EFIT result,
we select the corresponding values.
### B.2 Mean structural similarity (MSSIM)
MSSIM is calculated as
$\textrm{MSSIM}=\frac{\left(2\mu_{y^{\textrm{Target}}}\mu_{y^{\star}}+C_{1}\right)\left(2\sigma_{y^{\textrm{Target}}y^{\star}}+C_{2}\right)}{\left(\mu^{2}_{y^{\textrm{Target}}}+\mu^{2}_{y^{\star}}+C_{1}\right)\left(\sigma^{2}_{y^{\textrm{Target}}}+\sigma^{2}_{y^{\star}}+C_{2}\right)},$
(20)
where $\mu_{y^{\textrm{Target}}}$ and $\mu_{y^{\star}}$ are the mean values of
$y^{\textrm{Target}}$ and $y^{\star}$, respectively. Here,
$y^{\textrm{Target}}$ and $y^{\star}$ mean the same as in Section B.1.
$\sigma^{2}_{y^{\textrm{Target}}}$ and $\sigma^{2}_{y^{\star}}$ are the
variances of $y^{\textrm{Target}}$ and $y^{\star}$, respectively; while
$\sigma_{y^{\textrm{Target}}y^{\star}}$ is the covariance between
$y^{\textrm{Target}}$ and $y^{\star}$. $C_{1}$ and $C_{2}$ are used to prevent
a possible numerical instability, i.e., denominator being zero, and set to be
small numbers. Following Wang _et al._ (2004), we have $C_{1}=10^{-4}$ and
$C_{2}=9\times 10^{-4}$.
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|
# Generating Harder Cross-document Event Coreference Resolution Datasets using
Metaphoric Paraphrasing
Shafiuddin Rehan Ahmed1 Zhiyong Eric Wang2 George Arthur Baker1
Kevin Stowe3 James H. Martin1
Departments of 1Computer Science & 2CLASIC, University of Colorado, Boulder,
USA
{shah7567<EMAIL_ADDRESS>
3Education Testing Service (ETS)
###### Abstract
The most widely used Cross-Document Event Coreference Resolution (CDEC)
datasets fail to convey the true difficulty of the task, due to the lack of
lexical diversity between coreferring event triggers (words or phrases that
refer to an event). Furthermore, there is a dearth of event datasets for
figurative language, limiting a crucial avenue of research in event
comprehension. We address these two issues by introducing ECB+META, a
lexically rich variant of Event Coref Bank Plus (ECB+) for CDEC on figurative
and metaphoric language. We use GPT-4 as a tool for the metaphoric
transformation of sentences in the documents of ECB+, then tag the original
event triggers in the transformed sentences in a semi-automated manner. In
this way, we avoid the re-annotation of expensive coreference links. We
present results that show existing methods that work well on ECB+ struggle
with ECB+META, thereby paving the way for CDEC research on a much more
challenging dataset.111 Code/data: github.com/ahmeshaf/llms_coref
## 1 Introduction
Cross-Document Event Coreference Resolution (CDEC) involves identifying
mentions of the same event within and across documents. An issue with CDEC is
that the widely used dataset, Event Coref Bank plus (ECB+; Cybulska and Vossen
(2014)), is biased towards lexical similarities, both for triggers and
associated event arguments, and therefore has a very strong baseline Cybulska
and Vossen (2015); Kenyon-Dean et al. (2018); Ahmed et al. (2023a). To see
this, consider the excerpts from ECB+ shown in Figure 1(a). This consists of
three killing events selected from separate articles sharing a common trigger.
An algorithm capable of matching the triggers and tokens within the sentences,
such as "Vancouver" and "office," can readily discern that Event 2 is
coreferent with Event 3, and not Event 1. This leads to the question of
whether the state-of-the-art methods using this corpus Held et al. (2021)
learn the semantics of event coreference, or are merely exploiting surface
triggers.
Figurative language, encompassing metaphors, similes, idioms, and other non-
literal expressions, is an effective tool for assessing comprehension across
cognitive, linguistic, and social dimensions Lakoff and Johnson (1980); Winner
(1988); Gibbs (1994); Palmer and Brooks (2004); Palmer et al. (2006).
Figurative language, by its nature, draws on a wide array of cultural,
contextual, and imaginative resources to convey meanings in nuanced and often
novel ways. Consequently, it employs a broader vocabulary and more unique word
combinations than literal language Stefanowitsch (2006). Most recent work on
metaphors has been focused on generation Stowe et al. (2020, 2021b);
Chakrabarty et al. (2021a), interpretation Chakrabarty et al. (2022, 2023),
and detection Li et al. (2023); Joseph et al. (2023); Wachowiak and Gromann
(2023). Yet, there is a dearth of event datasets for figurative language which
limits an important research direction of event comprehension.
Figure 1: Using GPT-4 to Generate ECB+META from ECB+Corpus. Event 2 & Event 3
are coreferent, while Event 1 is not. ECB+META has metaphorically transformed
triggers, e.g., killing -> silencing the life. The triggers are hand-corrected
by an annotator. ECB+META challenges previous work—Held et al. (2021) & Ahmed
et al. (2023a).
In this paper, we address these two challenges by leveraging GPT-4 in
constrained metaphoric paraphrasing of ECB+documents. We introduce a novel
dataset named ECB+META , which we generate using a semi-automatic approach.
This involves applying metaphoric transformations to the event triggers within
ECB+ and then hand-correcting the tagged triggers in the new corpus. As
depicted in Figure 1(b), the trigger word killing in Events 2 and 3 of ECB+
become slaying and snuffing out the flame of life of in ECB+META,
respectively.
This approach preserves the coreference annotations from ECB+, thereby
avoiding an expensive coreference re-annotation task. Thus, we create several
versions of “tougher” CDEC benchmark datasets with enhanced lexical diversity
with varying levels of metaphoricity. We present baseline results using
previous methods—Held et al. (2021) and Ahmed et al. (2023a) (described in
§3.2), and show the limitation of these approaches on this dataset. Finally,
we correlate lexical diversity and text complexity with CDEC and test the
hypothesis that CDEC gets more difficult as the lexical diversity/complexity
of the corpus increases.
## 2 Related Work
### 2.1 CDEC Datasets
ECB+222Corpus detailed in §A is the most widely used dataset for CDEC, yet it
has limited utility in realistic applications because of how simple the
dataset is. The Gun Violence Corpus (GVC; Vossen et al. (2018)), for instance,
was introduced as a way of adding ambiguity to the task. Yet, both these
datasets lack lexical diversity in terms of coreferent event triggers.
Ravenscroft et al. (2021) is one such work that addresses the diversity
question through cross-domain coreference, however, a dataset focusing CDEC on
figurative language does not exist to our best knowledge.
Even with the use of modern annotation tools Klie et al. (2018); Ahmed et al.
(2023b), annotating CDEC datasets is expensive. Works such as Bugert and
Gurevych (2021); Eirew et al. (2021) use Wikipedia as a way of bootstrapping
ECR annotations automatically. In a similar vein, we bootstrap CDEC
annotations for figurative language in a synthetic way using GPT-4.
### 2.2 Metaphoric Paraphrasing
The task of metaphoric paraphrasing has been explored through a variety of
methods. A primary theme is sentential paraphrasing by replacing literal words
with metaphors Stowe et al. (2021a, b); Chakrabarty et al. (2021b). These
approaches fine tune language models with control codes to indicate metaphors,
exploiting available metaphoric data to facilitate transformations from
literal language to metaphoric. However, they rely on extensive data, and
there is evidence that modern large language models excel at metaphor
generation Chakrabarty et al. (2023) and paraphrasing Kojima et al. (2023);
OpenAI (2023). For this reason, we leverage GPT-4 via ChatGPT functionality
for our experiments.
### 2.3 CDEC Methods
Non-filtering Methods: Previous works Meged et al. (2020); Zeng et al. (2020);
Cattan et al. (2021); Allaway et al. (2021); Caciularu et al. (2021); Yu et
al. (2022) in CDEC have been successful using pairwise mention representation
learning models, a method popularly known as cross-encoding. These methods use
distributed and contextually-enriched “non-static” vector representations of
mentions from Transformer-based Vaswani et al. (2017) language models like
various BERT-variants Devlin et al. (2019); Beltagy et al. (2020) to calculate
supervised pairwise scores for those event mentions. While these methods
demonstrate SoTA performance, their applicability is hindered by their
quadratic complexity at inference.
Filtering Methods: Keeping usability and tractability in mind, we experiment
only with the recent work that adds a low-compute mention pair filtering step
before crossencoding. These approaches aid in the removal of numerous
irrelevant mention pairs, thereby directing focus toward the most pertinent
pairs with resource-intensive models. For instance, in their work, Held et al.
(2021) propose a retrieval, vector-based K-nearest neighbor method, that helps
find and focus only on the hard negatives in the corpus. In contrast, Ahmed et
al. (2023a) employ simplified lexical similarity metrics to filter out a
substantial number of truly non-coreferent pairs in the corpus.
## 3 Methodology
We first synthetically create ECB+META by employing metaphoric paraphrasing of
the original corpus. Then we tag the event triggers of the original corpus in
ECB+META in a semi-automated manner. Finally, we adopt two existing CDEC
methods to test this new dataset. We describe each of these steps:
### 3.1 Metaphoric Paraphrasing using GPT-4
We paraphrase ECB+’s sentences in a constrained manner in which we convert
only the event triggers in a sentence into metaphors. We first extract the
event mentions from each sentence of the documents in the corpus, then prompt
GPT-4 to convert only the trigger words in the sentence to metaphors. We adopt
a chain of thought prompting approach Kojima et al. (2022), where we provide
the steps that need to be followed in the conversion (see §B).
To enhance diversity and sample appropriate metaphors, we generate five
metaphors for every trigger word in the sentence and then task GPT-4 to select
the most coherent one from the list. We diversify metaphoricity levels by
using both single-word and multi-word metaphors. As illustrated in Figure 3,
the conversion of "killing" into a single-word metaphor is "slaying," while
its transformation into a multi-word phrasal metaphor is "extinguishing the
candle of life." We develop two versions of ECB+META, designated as ECB+META1
for single-word transformations and ECB+METAm for multi-word transformations,
respectively.
Using the generated conversions, we first automatically tag the original
events in the transformed sentences. Then, we hand-correct cases where the
conversion is ambiguous. In the end, we are left with two versions of the
validation and the test sets of ECB+META preserving the original coreference
annotations of ECB+.
### 3.2 CDEC Methods
#### Filtering Step for CDEC:
The BiEncoder K-NN ($\mathtt{KNN}$) approach, introduced by Held et al. (2021)
involves a novel approach to mention pair retrieval before doing CDEC. This
method focuses on selecting mentions that are most similar to a given target
mention using their static vector representations and a Vector Store (like
FAISS Johnson et al. (2019)). To achieve this, they fine-tune the RoBERTa-
Large Liu et al. (2019) pre-trained model using a contrastive Categorical Loss
function, with categories corresponding to event clusters within the corpus.
This fine-tuning process utilizes token embeddings generated by the language
model and trains on the centroid representations of gold standard event
clusters. Due to computation constraints, we use RoBERTa-Base instead of
RoBERTa-Large in this work. For the same reason, we use triplet-loss with
mention pairs instead of the centroid of clusters.
The Lemma Heuristic (LH; Ahmed et al. (2023a)) leverages lexical features to
pre-filter non-coreferent pairs before CDEC. This way, they eliminate the need
for an additional fine-tuning step as required in the $\mathtt{KNN}$ approach.
LH focuses on creating a balanced set of coreferent and non-coreferent pairs
while minimizing the inadvertent exclusion of coreferent pairs (false
negatives) by the heuristic. It accomplishes this by first generating a set of
synonymous lemma pairs from the training corpus and then applying a sentence-
level word overlap ratio to prune pairs that don’t meet the threshold or lack
synonymy. In this work, we use the LH method for filtering and also as a
baseline lexical method following Ahmed et al. (2023a).
#### Cross-encoder333Described in more detail in §C:
The Cross-Encoder (CE) functions within CDEC as a pairwise classifier,
leveraging joint representations of a mention pair $(e_{i},e_{j})$. First, it
combines the two event mentions with their respective contexts into a single
unified string to facilitate cross-attention. Next, it derives the token-level
representations of each mention after encoding this unified string. Finally,
the joint representation is the concatenation of the context-enhanced token
representations $(v_{e_{i}},v_{e_{j}})$ along with their element-wise product,
as illustrated below:
$v_{(e_{i},e_{j})}=[v_{e_{i}},v_{e_{j}},v_{e_{i}}\odot v_{e_{j}}]$ (1)
The resulting vector $v_{(e_{i},e_{j})}$ is then refined through a binary
cross-entropy loss function using logistic regression that learns coreference.
In our work, we use the learned weights of the
$\texttt{CE}_{\texttt{LH}}$444Provided by the authors. For the $\mathtt{KNN}$
cross-encoder ($\texttt{CE}_{\tt KNN}$), we trained the weights of RoBERTA-
Base using the $\mathtt{KNN}$ to generate focused mention pairs. We carry out
our experiments in a transfer learning format where we train the crossencoders
only on the training set of ECB+ and use the test sets of ECB+META. This is
motivated by the work of Ortony et al. (1978), which argues the human
processes required for comprehension of figurative and literal uses of
language are essentially similar.
#### GPT-4 as Pairwise Classifier:
Yang et al. (2022) demonstrated the viability of a prompt-based binary
coreference classifier using GPT-2, though the results were sub-par. Building
on their work, we employ a similar prompting technique with GPT-4 to develop
an enhanced classifier. This classifier determines whether a pair of events,
identified by marked triggers in sentences, are coreferent by responding with
“Yes” or “No”. Similar to CE, we vary this method by incorporating the two
fitering techniques ($\texttt{GPT}_{\texttt{LH}}$, $\texttt{GPT}_{\tt KNN}$)
## 4 Results
### 4.1 Metaphor Quality Control
To assess the quality of the generated metaphors, an annotator familiar with
the events in the ECB+ dataset manually examines the $\tt{Dev}_{small}$ sets.
We chose a familiarized annotator because metaphors often abstract away many
of the details that make coreference obvious, and we are interested in whether
or not the generated paraphrases would (by any stretch of the imagination)
reasonably be interpreted as referring to the original event.
The annotator examines each of the original event mentions alongside their
paraphrased versions and makes a binary judgment as to whether the two can be
reasonably interpreted as referring to the same event. We estimate based on
the results that approximately 99% of ECB+META1 and 95% of ECB+METAm could be
reasonably interpreted by a human as being coreferent to the original event
mentions from which they are derived.
### 4.2 Coreference & Lexical Diversity
We use $\textsc{B}^{3}$ Bagga and Baldwin (1998) and CoNLL Denis and Baldridge
(2009); Pradhan et al. (2012) clustering metrics, in which we use the
$\textsc{B}^{3}_{\textsc{R}}$ for estimating recall, CoNLL as the overall
metric (evaluated using CoVal Moosavi et al. (2019)). For the methods that use
LH as the filtering step, we follow Ahmed et al. (2023a)’s clustering with
connected components. For $\mathtt{KNN}$ as the filtering step, we use Held et
al. (2021)’s greedy agglomeration.
#### Filtering Scores:
Following previous work, we first assess the $\textsc{B}^{3}_{\textsc{R}}$
score on oracle results. This tests how well the filtering methods perform in
minimizing false negatives (coreferent pairs that are eliminated
inadvertently). From Table 1 we observe a substantial difference in the recall
measures of ECB+ and ECB+META versions. The LH approach particularly takes a
toll because it relies on synonymous lemma pairs from the train set.
Interestingly, $\mathtt{KNN}$ does well on the ECB+META versions, with only a
minor drop in recall for ECB+META1 and about 10% drop for ECB+METAm. Between
ECB+META1 and ECB+METAm, as expected, the recall drops more in ECB+METAm as
more complex metaphors are used here.
| Method | | Dev | $\tt{Dev}_{small}$ | Test |
---|---|---|---|---|---|---
ECB+ | | LH | | 76.3 | 87.9 | 81.5 |
| $\mathtt{KNN}$ | | 95.7 | 95.3 | 94.9 |
ECB+ META1 | | LH | | 45.8 | 64.6 | 58.2 |
| $\mathtt{KNN}$ | | 91.8 | 93.7 | 91.4 |
ECB+ METAm | | LH | | 38.4 | 59.4 | 51.3 |
| $\mathtt{KNN}$ | | 84.4 | 86.5 | 85.6 |
Table 1: $\textsc{B}^{3}_{\textsc{R}}$ Oracle Results on Dev, $\tt{Dev}_{small}$ and Test sets of ECB+, ECB+META1, and ECB+METAm. Method | ECB+ | ECB+ META1 | ECB+ METAm |
---|---|---|---|---
LH | 74.1 | 49.8 | 54.0 |
$\texttt{CE}_{\texttt{LH}}$ | 78.1 | 60.9 | 50.6 |
$\texttt{CE}_{\tt KNN}$ | 78 | 71.4 | 54.8 |
$\texttt{GPT}_{\texttt{LH}}$ | 78.23 | 62.5 | 55.6 |
$\texttt{GPT}_{\tt KNN}$ | 67.73 | 60.15 | 55.5 |
Table 2: CoNLL F1 Baseline and Cross-encoder results on ECB+, ECB+META1 and
ECB+METAm Test sets.
#### CDEC Scores:
We present the overall CoNLL F1 scores in Table 2 for the baseline (LH), the
two fine-tuned cross-encoders ($\texttt{CE}_{\texttt{LH}}$, $\texttt{CE}_{\tt
KNN}$), and the methods that use GPT-4 ($\texttt{GPT}_{\texttt{LH}}$,
$\texttt{GPT}_{\tt KNN}$). From the table, it is evident that LH is no longer
a strong baseline for ECB+META versions with a drop in 20% score. Both
$\texttt{CE}_{\texttt{LH}}$ and $\texttt{CE}_{\tt KNN}$ show a pattern of
reducing score from ECB+META1 to ECB+METAm, with $\texttt{CE}_{\texttt{LH}}$
performing considerably worse. Interestingly, the drop in scores for
$\texttt{CE}_{\tt KNN}$ is not substantial for ECB+META1 but there is a
dramatic drop of 20% for ECB+METAm. $\texttt{GPT}_{\texttt{LH}}$ achieves the
highest scores on ECB+ and ECB+METAm, demonstrating that GPT-4’s performance
aligns with the state-of-the-art, unlike its predecessor GPT-2. However, the
financial implications of using $\texttt{GPT}_{\texttt{LH}}$ and
$\texttt{GPT}_{\tt KNN}$ are noteworthy; running CDEC with these methods
incurred approximately $75 in API costs to OpenAI.
From these results, we can conclude three things: a) ECB+ is an easy dataset,
b) datasets with complex metaphors are harder benchmarks, and c) GPT-4 is only
as good as the CE methods with a significant amount of added costs.
#### Lexical Diversity:
We estimate the lexical diversity (MLTD; McCarthy and Jarvis (2010)) of the
mention triggers of event clusters. We first eliminate singleton clusters.
Then we calculate a weighted average (by cluster size) of the MLTD score for
each cluster. The scores we achieved for the test sets of each version of ECB+
are as follows: ECB+: 7.33. ECB+META1: 11.92, ECB+METAm: 26.48. From the lower
CDEC scores from Table 2 and the increasing diversity scores of the more
complex corpus, we can establish a negative correlation between CDEC scores
and MLTD.
Overall, the results confirm our hypothesis that when a dataset a) moves away
from strong lexical overlap and b) has figurative language usage, the CDEC
scores drop.
## 5 Analysis
### 5.1 Coreference Resolution Difficulty
We evaluate whether the paraphrased versions are more difficult for humans to
determine as coreferent. On the $\tt{Dev}_{small}$ splits of ECB+META1,
ECB+METAm, and ECB+, a human annotator reaches the same coreference verdict
regardless of the degree of figurative language approximately 98% of the time.
Cases in which the human annotator did not reach the same verdict generally
involved convergent metaphorical language, for example:
Event a: The Indian navy unfurled the words that it had ensnared 23 pirates in
the law’s net who cast ominous shadows over a merchant vessel in the Gulf of
Aden on Saturday, the latest in a series of recent violent ballets with Somali
pirates. Event b: Indian Naval Ship throws a net over three pirate vessels in
a single orchestrated symphony .
were incorrectly identified as coreferent; in actuality the former refers to
the arrest of the pirates but the latter refers to the interception of their
ships. This analysis supports the findings of Ortony et al. (1978): that, for
humans, figurative language use and literal language do not substantially
affect comprehension.
### 5.2 Qualitative Error Analysis
We examined the coreference predictions of $\texttt{CE}_{\tt KNN}$ on 142
common mention pairs between ECB+, ECB+META1, and ECB+METAm, as
$\texttt{CE}_{\tt KNN}$ achieved the best overall performance. For mention
pairs that $\texttt{CE}_{\tt KNN}$ correctly predicted as coreferent across
all versions, we noticed a pattern: the same event trigger was shared in each
(see Figure 4).
In cases where $\texttt{CE}_{\tt KNN}$ got the prediction right on ECB+ but
wrong on the META versions, the event triggers in ECB+ were changed to
different ones in the META versions (see Figure 5). When $\texttt{CE}_{\tt
KNN}$ incorrectly predicted coreference on ECB+ but correctly predicted it in
the META versions, it was because the same triggers in ECB+ were altered to
different ones (see Figure 6). This further affirms that the model heavily
relies on surface triggers for making coreference decisions.
## 6 Future Work
Future research could explore applying more recent CDEC techniques on
ECB+META. These techniques could include symbolic grounding, as discussed in
Ahmed et al. (2024b, a), and event type categorical cross-encoding, as
proposed by Otmazgin et al. (2023). Another outcome of this research is to use
CDEC as a text complexity metric Hale (2016) of a corpus. We argue that a
corpus is more complex if a CDEC algorithm is not able to identify that
different explanations of the same event are the same. An interesting line of
future work would be to automatically generate an optimally complex CDEC
corpus, i.e., a corpus that yields the lowest coreference score.
In this work, we rely on the GPT-4’s metaphor list and substitution choice.
The only control we have is to make a coherent choice, however, we find
ourselves subjected to the unpredictable outputs, colloquially referred to as
“hallucinations”, generated by GPT-4. In the future, we aim to integrate human
feedback into the process of metaphor selection and to employ annotated
metaphor databases from studies such as Joseph et al. (2023).
## 7 Conclusion
In this paper, we introduced ECB+META a lexically rich variant of ECB+ using
constrained metaphoric paraphrasing of the original corpus. We provide hand-
corrected event trigger annotations of two versions of ECB+META differing in
the kind of metaphoric transformation using either single words or phrases. We
finally provide baseline results using existing SoTA methods on this dataset
and show their limitations when there is substantial lexical diversity in the
corpus. Through the provided data and methodology, we lay a path forward for
future research in Cross-Document Event Coreference Resolution on more
challenging datasets.
## Limitations
The study faced several limitations, including its focus on a single language-
English. Some experiments were conducted within a small sample space,
especially for $\tt{Dev}_{small}$, potentially leading to biased results and
limiting the generalizability of the findings. Finally, while the study
utilized variations within a single dataset, the reliance on this sole dataset
could introduce inherent biases, affecting the broader applicability of the
research outcomes.
Reproducibility Concern: All the coreferencing experiments are reproducible,
but the generation of ECB+META is not. So we may have vastly different results
if a new version of ECB+META is created with the methodology. However, we
released all the generated text that came out of our work and the code to run
the experiments.
LLMs on ECB+. Contamination Concern The GPT-4 has likely been contaminated by
the test sets of ECB+, i.e., GPT-4 has been pretained on this benchmark. With
the recent work involving GPT and ECB+ Yang et al. (2022); Ravi et al. (2023a,
b), it seems likely the test set is also been used in the instruction fine-
tuning of GPT-4. But we stress the synthesizing of datasets to battle
contamination as we do in our work.
## Ethics Statement
AI-generated text should always be thoroughly scrutinized before being used
for any application. In our work, we provide methods to synthesize new
versions of the same real articles. This can have unintentional usage in the
propagation of disinformation. This work is only intended to be applied to
research in broadening the field of event comprehension. Our work carries with
it the inherent biases in news articles of ECB+ corpus and has the potential
of exaggerating it with the use of GPT-4, which in itself has its own set of
risks and biases.
## Acknowledgements
We thank the anonymous reviewers for their helpful suggestions that improved
our paper. We are also grateful to Susan Brown, Alexis Palmer, and Martha
Palmer from the BoulderNLP group for their valuable feedback before
submission. Thanks also to William Held and Vilém Zouhar for their insightful
comments. We gratefully acknowledge the support of DARPA FA8750-18-2-0016-AIDA
– RAMFIS: Representations of vectors and Abstract Meanings for Information
Synthesis and a sub-award from RPI on DARPA KAIROS Program No.
FA8750-19-2-1004. Any opinions, findings, conclusions, or recommendations
expressed in this material are those of the authors and do not necessarily
reflect the views of DARPA or the U.S. government.
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## Appendix A ECB+ Corpus
Train | Dev* | $\tt{Dev}_{small}$* | Test |
---|---|---|---|---
Topics | 25 | 8 | 8 | 10 |
Documents | 594 | 156 | 40 | 206 |
Mentions | 3808 | 968 | 277 | 1780 |
Table 3: Corpus statistics for event mentions in ECB+
The ECB+ corpus Cybulska and Vossen (2014) is a popular English corpus used to
train and evaluate systems for event coreference resolution. It extends the
Event Coref Bank corpus (ECB; Bejan and Harabagiu (2010)), with annotations
from around 500 additional documents. The corpus includes annotations of text
spans that represent events, as well as information about how those events are
related through coreference. We divide the documents from topics 1 to 35 into
the training and validation sets555Validation set includes documents from the
topics 2, 5, 12, 18, 21, 34, and 35, and those from 36 to 45 into the test
set, following the approach of Cybulska and Vossen (2015). We further break
the documents of the validation set into two subsets: Dev and
$\tt{Dev}_{small}$ for our error analysis. Full corpus statistics can be found
in Table 3.
Metaphoric Paraphrasing You are a metaphor expert. Your task is to transform
specific words in a given sentence into metaphors. These metaphors can only be
single-word/multi-word replacements. Here are the detailed steps you need to
follow: Read the Sentence Provided: Focus on understanding the context and
meaning of the sentence. Review the Word List: This list contains the words
you need to transform into metaphors. Generate Metaphors: Create 5 distinct
single-word/multi-word metaphors for each word in the list. Compose a New
Sentence: Replace the original words with your chosen metaphors randomly.
Ensure the new sentence maintains logical and grammatical coherence. Sentence
to Transform: """{{sentence}}""" Word List to Convert into Metaphors:
"""{{trigger_list}}""" Output Requirements: Provide your final output in JSON
format, including: The "Original Sentence". The "Original Word List". The
"Metaphoric Word List" (with your chosen metaphors). The "Metaphoric Sentence"
(the sentence with metaphors incorporated). Remember, the goal is to use
metaphors to convey the original sentence’s meaning in a more nuanced or
impactful way without altering the core information. Figure 2: Metaphoric
Paraphrasing Prompt following Chain of Thought Reasoning. We provide the steps
in this prompt to follow.
Original Sentence A Vancouver man has been charged with first-degree murder
after a killing at an office party.
Metaphoric Paraphrasing Single-word Metaphors: A Vancouver man has been
implicated with first-degree murder after a slaying at an office soirée.
Multi-word Phrasal Metaphors: A Vancouver man has been ensnared in the web of
the law with first-degree murder after extinguishing the candle of life at a
conclave of festive hearts.
Figure 3: Metaphoric Paraphrasing: Transforming a Sentence with Figurative
Language. Event triggers, indicated in italics, undergo modification in
paraphrased versions, annotated by GPT-4 with two variations.
## Appendix B Metaphoric Paraphrase Prompt
We present the prompt used with GPT-4 in Figure 2 for generating the
Metaphoric Paraphrasing of ECB+ documents. We use two separate prompts for
generating single-word metaphors and multi-word metaphors. We ran this prompt
on the validation and test sets of ECB+ using GPT-4 as the LLM and a
temperature value of 0.7. We force GPT-4 to produce JSON-style output to avoid
parsing issues. It costs about $16 to generate ECB+META1 and $18 to generate
ECB+METAm with GPT-4 API calls. In the future, we plan to provide this
conversion of the training set of ECB+ as well.
## Appendix C Experiment Setup
LH details: we set the sentence-level word overlap ratio threshold at 0.005.
We employ spaCy 3.7.4 as the lemmatizer to extract the root forms of words.
$\mathtt{KNN}$ details: we adopt the RoBERTa-Base model, enhanced with a
triplet loss function calculated by F.triplet_margin_loss with a 10 margin, L2
norm ($p=2$), and $\epsilon=1e-6$ for stability, without swapping and mean
reduction. Our optimization uses AdamW, targeting bi-encoder parameters with a
$1\times 10^{-5}$ learning rate across 20 iterations and batches of 4.
$\texttt{CE}_{\texttt{LH}}$ details: We utilize the RoBERTa-Base model with
the AdamW optimizer. Learning rates are set to $1\times 10^{-5}$ for BERT
class parameters and $1\times 10^{-4}$ for the classifier. The model is
trained over 20 epochs, using the sentences in which the two mentions occur as
context, and mention pairs generated by LH.
$\texttt{CE}_{\tt KNN}$ details: It mirrors the
$\texttt{CE}_{\texttt{LH}}$configuration but it is trained on mention pairs
from $\mathtt{KNN}$exclusively.
All Non-GPT experiments are conducted on a single NVIDIA RTX 3090 with 24GB of
VRAM. For generating the META datasets, we utilized GPT-4 (model version:
gpt4-0613), setting the temperature parameter to 0.7.
## Appendix D ECB+METAm Complete Results
We provide the baseline results for validation sets of ECB+METAm. As shown in
Table 4, the results are consistent even for the development sets, where we
see significantly low coreference scores with the used methods. Interestingly,
LH performs better than the cross-encoder methods on these splits.
Split | Method | $\textsc{B}^{3}_{\textsc{R}}$ | $\textsc{B}^{3}_{\textsc{P}}$ | $\textsc{B}^{3}_{\textsc{F1}}$ | CoNLL |
---|---|---|---|---|---|---
| LH | 51.8 | 64.5 | 57.4 | 56.3 |
| $\texttt{CE}_{\texttt{LH}}$ | 47.2 | 77.3 | 58.6 | 55.3 |
Dev | $\texttt{CE}_{\tt KNN}$ | 42.4 | 86.2 | 56.8 | 49.2 |
| LH | 68.4 | 78.3 | 73.1 | 62.0 |
| $\texttt{CE}_{\texttt{LH}}$ | 64.8 | 84.7 | 73.4 | 59.0 |
$\tt{Dev}_{small}$ | $\texttt{CE}_{\tt KNN}$ | 62.4 | 91.6 | 74.2 | 55.5 |
Table 4: Baseline and Cross-encoder results on ECB+METAm Dev and
$\tt{Dev}_{small}$sets.
## Appendix E Error Analysis
ECB+METAm Event a: On Saturday, Cheeks was shown the door as head coach of the
Philadelphia 76ers. Event b: Maurice Cheeks was shown the exit door Saturday
as coach of the Philadelphia 76ers, who are hitting a rough patch at 9-14 a
year after making it to the high stakes showdown.
ECB+META1 Event a: On Saturday , Cheeks was ousted as head coach of the
Philadelphia 76ers . Event b: Maurice Cheeks was ousted Saturday as coach of
the Philadelphia 76ers , who are stumbling at 9-14 a year after entering the
duel .
ECB+ Event a: On Saturday , Cheeks was fired as head coach of the Philadelphia
76ers . Event b: Maurice Cheeks was fired Saturday as coach of the
Philadelphia 76ers , who are slumping at 9-14 a year after making the playoffs
.
Figure 4: Correct prediction of coreferent mention pair across all datasets
with $\texttt{CE}_{\tt KNN}$. Pairs have the same event trigger in each case.
ECB+METAm Event a: The Indian Navy proclaimed Saturday it had reeled in 23
pirates as they struggled to scale the ship of an Ethiopian-flagged vessel in
the Gulf of Aden . Event b: An Indian warship , INS Mysore anchored in
position in the Gulf of Aden unleashed fury upon two boats of pirates after
harvesting signals from a ship that the pirates were grappling to usurp the
helm of .
ECB+META1 Event a: "The Indian Navy proclaimed Saturday it had ensnared 23
pirates as they struggled to invade an Ethiopian-flagged vessel in the Gulf of
Aden. Event b: An Indian warship , INS Mysore anchored in the Gulf of Aden
pounced on two boats of pirates after intercepting signals from a ship that
the pirates were struggling to seize."
ECB+ Event a: "The Indian Navy said Saturday it had captured 23 pirates as
they tried to board an Ethiopian-flagged vessel in the Gulf of Aden . Event b:
An Indian warship , INS Mysore deployed in the Gulf of Aden attacked two boats
of pirates after receiving signals from a ship that the pirates were trying to
hijack ."
Figure 5: Correct coreference prediction in ECB+ but not in the META versions,
simply because the triggers got changed.
ECB+METAm Event a: "Chargers defensive tackle Jamal Williams was ensnared in
the net under the cloud of doubt of maneuvering in a state of intoxication,
the team’s second such ensnarement in less than a month. Event b: Chargers
wide receiver Vincent Jackson was ensnared by the law’s clutches early
yesterday under the shadow of doubt of the reckless dance with intoxication."
ECB+META1 Event a: "Chargers defensive tackle Jamal Williams was captured
under speculation of spirited steering, the team’s second such ensnarement in
less than a month. Event b: Chargers wide receiver Vincent Jackson was hooked
early yesterday on doubt of booze-cruising."
ECB+ Event a: "Chargers defensive tackle Jamal Williams was arrested on
suspicion of drunken driving , the team’s second such arrest in less than a
month . Event b: Chargers wide receiver Vincent Jackson was arrested early
yesterday on suspicion of drunken driving ."
Figure 6: Correct non-coreference prediction in ECB+META but not in ECB+,
simply because the META versions’ event triggers were changed.
For more examples, please checkout the provided excel file in data repository.
|
# A Survey on Optimal Transport for Machine Learning: Theory and Applications
Luis Caicedo Torres
Department of Mathematics and Statistics
Florida International University
Miami, FL, 33199
<EMAIL_ADDRESS>
& Luiz Manella Pereira
Knight Foundation School of Computing and Information Sciences
Florida International University, solid lab
Miami, FL,33199
<EMAIL_ADDRESS>
& M. Hadi Amini
Knight Foundation School of Computing and Information Sciences
Sustainability, Optimization, and Learning for InterDependent networks
laboratory (solid lab)
Florida International University
Miami, FL, 33199
<EMAIL_ADDRESS>
###### Abstract
Optimal Transport (OT) theory has seen an increasing amount of attention from
the computer science community due to its potency and relevance in modeling
and machine learning. It introduces means that serve as powerful ways to
compare probability distributions with each other, as well as producing
optimal mappings to minimize cost functions. Therefor, it has been deployed in
computer vision, improving image retrieval, image interpolation, and semantic
correspondence algorithms, as well as other fields such as domain adaptation,
natural language processing, and variational inference. In this survey, we
propose to convey the emerging promises of the optimal transport methods
across various fields, as well as future directions of study for OT in machine
learning. We will begin by looking at the history of optimal transport and
introducing the founders of this field. We then give a brief glance into the
algorithms related to OT. Then, we will follow up with a mathematical
formulation and the prerequisites to understand OT, these include Kantorovich
duality, entropic regularization, KL Divergence, and Wassertein barycenters.
Since OT is a computationally expensive problem, we then introduce the
entropy-regularized version of computing optimal mappings, which allowed OT
problems to become applicable in a wide range of machine learning problems. In
fact, the methods generated from OT theory are competitive with the current
state-of-the-art methods. The last portion of this survey will analyze papers
that focus on the application of OT within the context of machine learning. We
first cover computer vision problems; these include GANs, semantic
correspondence, and convolutional Wasserstein distances. Furthermore, we
follow this up by breaking down research papers that focus on graph learning,
neural architecture search, document representation, and domain adaptation. We
close the paper with a small section on future research. Of the
recommendations presented, three main problems are fundamental to allow OT to
become widely applicable but rely strongly on its mathematical formulation and
thus are hardest to answer. Since OT is a novel method, there is plenty of
space for new research, and with more and more competitive methods (either on
an accuracy level or computational speed level) being created, the future of
applied optimal transport is bright as it has become pervasive in machine
learning.
_Keywords_ Optimal Transport $\cdot$ Machine Learning $\cdot$ Computer Vision
$\cdot$ Wasserstein distance
## 1 Introduction
The Optimal Transport problem sits at the intersection of various fields,
including probability theory, PDEs, geometry, and optimization theory. It has
seen a natural progression in its theory from when Monge first posed the
problem in 1781 [24]. Now, it serves as a powerful tool due to its natural
formulation in various contexts. It has recently seen a wide range of
applications in computer science–most notably in computer vision, but also in
natural language processing and other areas. Different elements such as the
Convolutional Wasserstein Distance [36] and the Minibatch Energy Distance [4]
have made significant improvements on image interpolation, heat maps, and
GANs. These are examples of some problems in machine learning that are being
recast using Optimal Transport elements, such as Wasserstein distance being
used as an error measure for comparing different probability distributions. We
note the effectiveness with which optimal transport deals with both discrete
and continuous problems and the easy transition between the two classes of
problems. The powerful tools from convex geometry and optimization theory have
made optimal transport more viable in applications. To that extent, we note
the remarkable implementation of Sinkhorn’s algorithm to significantly speed
up computation of Wasserstein distances [10].
Although the theory is well-developed [42], much work is being made in
determining the state-of-the-art algorithms for computing optimal transport
plans under various conditions. In this survey, we explore the main tools from
the theory and summarize some of the major advancements in its application.
While it is not all-encompassing, we aim to provide an application-focused
summary.
The rest of this paper is organized as follows: Section 2 provides an overview
of algorithms from different applications and major breakthroughs in
computation. Section 3 presents a brief history of the topic. Section 4
details some mathematical formalism. Section 5 reviews ways to overcome the
computational challenges. Section 6 and on then explores applications of OT to
different fields, most notably in GANs and general image processing. We then
conclude with remarks and proposed directions and close with open problems.
The interested reader can dive deeper into the rich OT material using some
superb books such as [42], [41], [26], [33].
## 2 OT Algorithms at a Glance
Table 1: OT Algorithms in Machine Learning Presented Application | Publication | Metric Employed | Year
---|---|---|---
Computations | Sinkhorn Entropy-Reg OT [10] | Ent-Reg W-Distance | 2013
Computations | 2-W Barycenters [11] | Ent-Reg W-Distance | 2014
Comp. Vision | Conv-W Dist [36] | Conv-W | 2015
Comp. Vision | WGANs [4] | EMD | 2017
Comp. Vision | OT-GAN [31] | MED | 2018
Graphs | GWL [18] | Gromov - W Dist | 2018
Domain Adaptation | GCG [12] | Ent-Reg W-Distance | 2016
An Overview of the algorithms presented in detail. Abbreviations used: Entropy
Regularized Wasserstein Distance (Ent-Reg W-Distance), Minibatch Energy
Distance (MED), Convolutional Wasserstein Distance (Conv-W), Gromov
Wasserstein Distance (Gromov-W Dist) Earth Mover Distance (EMD), Domain
Adaptation (Dom. Adap.), 2-Wasserstein (2-W), Gromov-Wasserstein Learning
(GWL), Generalized Conditional Gradient (GCG)
## 3 History
The central idea of Optimal Transport (OT) can be found in the work by French
geometer Gaspard Monge. In his paper, Mémoire sur la théorie des déblais et
des remblais, published in 1781, Monge asked the question: How do I move a
pile of earth (some natural resource) to a target location with the least
amount of effort, or cost [24]? The idea was to find a better way of
optimizing such cost that was not simply iterating through every possible
permutation of supplier vs. receiver and choosing the one with the lowest
cost. One of the major breakthroughs following Monge’s work was by Russian
mathematician Leonid Vitaliyevich Kantorovich who was the founder of linear
programming. His research in optimal resource allocation, which earned him his
Nobel Prize in Economics, led him to study optimal coupling and duality,
thereby recasting some parts of the OT problem into a linear programming
problem. Kantorovich’s work led to the renaming of optimal coupling between
two probability measures as the Monge-Kantorovich problem.
After Kantorovich, the field of OT gained traction and its applications
expanded to several fields. For example, while John Mather worked on
Lagrangian dynamical systems, he developed the theory of action-minimizing
stationary measures in phase space, which led to the solution of certain
Monge-Kantorovich problems [22]. Although he did not make the connection
between his work and OT, Buffoni and Bernard in their paper Optimal mass
transportation and Mather theory showed the existence of an optimal transport
map while studying the "Monge transportation problem when the cost is the
action associated to a Lagrangian function on a compact manifold [6]."
Several other names helped expand the field OT. For example, Yann Brenier
introduced optimal coupling to his research in incompressible fluid mechanics,
thus linking the two fields. Mike Cullen introduced OT in meteorology while
working on semi-geostrophic equations. Both Brenier’s and Cullen’s work
brought forth the notion that there is a connection, previously not expected,
between OT and PDEs. Fields medalist Cédric Villani also contributed much to
the field in connection with his work in statistical mechanics and the
Boltzmann equation.
Recently, OT is being applied in several fields, including Machine Learning
(ML). It started with image processing by utilizing color histograms of images
(or gray images) and Wasserstein’s distance to compute the similarity between
images. Then, it was followed by shape recognition [25, 13, 2]. For example,
in A Metric for Distributions with Applications to Image Databases, Rubner et
al. introduced a new distance between two distributions, called Earth Mover’s
Distance (EMD), which reflects the minimal amount of work that must be
performed to transform one distribution into the other by moving "distribution
mass" around [29, 30]. Next, Haker et al. introduced a method for computing
elastic registration and warping maps based on the Monge-Kantorovich theory of
OT [16, 17].
Due to the important role of matrix factorization in ML, it was a natural
progression to use OT as the divergence component of Nonnegative Matrix
Factorization (NMF) [32]. In 2014, Solomon et al., looked at the applications
of OT in semi supervised learning in their paper Wasserstein Propagation for
Semi-Supervised Learning [38]. Other applications have been utilizing OT in
mappings between distributions; more specifically, a recent paper was
published on using Wasserstein’s metric in variational inference, which lies
at the heart of ML [3].
More recently, researchers have made advancements in the theory of OT with
Marco Cuturi proposing methods to solve approximations of the OT problems by
introducing a regularization term [10]. The field is now more active than
ever, with researchers extending the theories that work for low-dimensional ML
problems into high-dimensional problems, bringing forth several complex
theoretical and algorithmic questions [34].
## 4 Mathematical Formalism
### 4.1 Problem Statement
Given a connected compact Riemannian manifold $M$, Optimal Transport Plans (OT
plans) offer a way to mathematically formulate the mapping of one probability
measure $\mu_{0}$ _onto_ another probability measure $\mu_{1}$. These plans
$\pi$ are couplings that obey mass conservation laws and therefore belong to
the set
$\Pi(\mu_{0},\mu_{1})=\\{\pi\in\text{Prob}(M\times
M)|\pi(\cdot,M)=\mu_{0},\pi(M,\cdot)=\mu_{1}\\}$
Here, $\Pi$ is meant to be the set of all joint probabilities that exhibit
$\mu_{0}$ and $\mu_{1}$ as marginal distributions. The OT plan $\pi(x,y)$
seeks to transport mass from point $x$ to point $y$. This formulation allows
for _mass-splitting_ which is to say that the optimal transport map can take
portions of the mass at point $x$ to multiple points $y_{i}$. Kantorovich
sought to rephrase the Monge question into a minimization of a linear
functional
$\pi\rightarrow\text{inf}\int_{M\times M}c(x,y)d\pi(x,y)$ (1)
on the nonempty and convex $\Pi$ and appropriate cost function $c$. We note
that some formulations accommodating multiple cost have also been proposed,
e.g. [35]. Alternatively, these OT plans will minimize the distance between
two measures denoted formally as the _2-Wasserstein Distance_ , where $d$ is a
metric:
$W^{d}_{2}(\mu_{0},\mu_{1})=\inf_{\pi\in\prod(\mu_{0},\mu_{1})}\bigg{(}\int_{M\times
M}d(x,y)^{2}d\pi(x,y)\bigg{)}^{1/2}$ (2)
This distance defines a metric111Here, we mean a metric in the mathematics
sense, i.e. a function $d(\cdot,\cdot):M\times M\to\mathbb{R}_{+}$ that is
positive definite, symmetric, and subadditive on a metrizable space $M$. See
Appendix A for more details. as shown in Villani’s book [42]. This distance
metric will be integral to applications as we will see that it offers a new
way to define loss functions. The goal is to find, or approximate, the optimal
transport plan, $\pi$.
### 4.2 Kantorovich Duality
Duality arguments are central to both the theoretical and numerical arguments
in the OT framework. Kantorovich noticed that the minimization of the linear
functional problem emits a dual problem. Here, let $c$ denote a lower
semicontinuous cost function, $\mu_{0}$ and $\mu_{1}$ denote marginal
probabilities, and $\Pi$ be the set of all probability measures on $M\times M$
which emit $\mu_{0}$ and $\mu_{1}$ as marginals. Then, for continuous
$\phi(x),\psi(y)$, we have that
$\inf\limits_{\Pi(\mu_{0},\mu_{1})}\int_{M\times
M}c(x,y)d\pi(x,y)=\sup\limits_{\phi,\psi}\int_{M}\phi(x)d\mu_{0}+\int_{M}\psi(y)d\mu_{1}$
(3)
The right-hand side of the equation is known as the _dual problem_ of the
minimization problem and is a very useful tool in proving consequences
regarding optimal transport maps. A proof of this result, along with further
discussion, can be found in [42].
### 4.3 Entropic Regularization
We can define the entropy of a coupling on $M\times M$ by the negative energy
functional coming from information theory:
$H(\pi)=-\int\int_{M\times M}\pi(x,y)\ln(\pi(x,y))dxdy$ (4)
This entropy essentially tracks information loss of a given estimate versus
the true value as it proves a lower bound for the square loss error. Then, we
can consider the entropy-regularized Wasserstein distance:
$W^{2}_{2,\gamma}(\mu_{0},\mu_{1})=\inf_{\pi\in\Pi(\mu_{0},\mu_{1})}\bigg{[}\int\int_{M\times
M}d(x,y)^{2}d\pi(x,y)-\gamma H(\pi)\bigg{]}$ (5)
Cuturi proved this regularized distance offers a transport plan that is more
spread out and also offers much faster computational convergence convergence
[10]. This computational breakthrough will be pivotal in the tractability of
Wasserstein-distance dependent algorithms.
### 4.4 KL Divergence
A lot of results for optimal transport maps can be related to the familiar KL
divergence. If we define $p(x)$ and $q(x)$ as probability distributions given
a random variable x over a manifold of distributions, then we define the KL
divergence as:
$D_{KL}(p(x)|q(x))\coloneqq\int p(x)\bigg{(}\ln\frac{p(x)}{q(x)}\bigg{)}dx$
(6)
### 4.5 Wasserstein barycenters
The barycenter problem is central to the interpolation of points in Euclidean
space. Agueh and Carlier present the analog in Wasserstein space, proving its
existence, uniqueness, and providing characterizations [1]. The analog is
presented as the solution to the minimzation of a convex combination problem
$\inf_{\mu}\sum\limits_{i=1}^{p}\alpha_{i}W^{2}_{2}(\mu_{i},\mu)$ (7)
where $\mu_{i}$ are probability measures and the $\alpha_{i}$’s, known as
barycentric coordinates, are nonnegative and sum to unity. These conclusions
are derived from considering the problem dual to the problem and desirable
properties of the Lengendre-Fenchel transform as well as conclusions from
convex geometry. These barycenters are also uniquely characterized in relation
to Brennier maps which offers direct formulation as a push forward operator.
Barycenteres will play a major role in applications such as the interpolation
of images under transport maps as in [36]. Computing these barycenters is
discussed in the Computational Challenges section.
## 5 Computational Challenges
One the biggest challenges in the implementation of optimal transport has been
its computational cost. One widely used implementation of Sinkhorn’s algorithm
was formulated by Cuturi significantly decreased computation cost [10]. In the
following, $KL$ denotes the Kullback-Leibler divergence, $U$ denotes the
transport polytope of transport plans $P$ that emit $r$ and $c$ as marginal
distributions: $U(r,c)=\\{P\in\mathbb{R}^{d\times
d}_{+}|P\mathbbm{1}_{d}=r,P^{T}\mathbbm{1}_{d}=c\\}$; and
$U_{\alpha}(r,c)=\\{P\in U(r,c)|KL(P|rc^{T})\leq\alpha\\}$. We present the
discrete version as opposed to the continuous analog presented in equation
(5). Define the Sinkhorn distance as
$d_{M,\alpha}\coloneqq\min\limits_{P\in U_{\alpha}(r,c)}\langle P,M\rangle$
(8)
Then we can introduce an entropy regularization argument stated in a
Lagrangian for $\lambda>0$:
$\displaystyle d_{M}^{\lambda}(r,c)\coloneqq$ $\displaystyle\langle
P^{\lambda},M\rangle,\quad$ (9) $\displaystyle\text{where}\quad
P^{\lambda}=\text{argmin}_{P\in U(r,c)}$ $\displaystyle\langle
P,M\rangle-\frac{1}{\lambda}h(P)$
where $h(P)=-\sum\limits_{i,j=1}^{d}p_{ij}\log(p_{ij})$ is the entropy of $P$.
Then, Sinkhorn’s famed algorithm for finding the the minimum, which we know
from the general theory will be found on one of the vertices of the polytope,
will serve as a proper approximation tool as seen in Algorithm 1. Here, a main
result proved by Cuturi is used which states that the solution $P^{\lambda}$
is unique and, moreover, has the particular form of
$P^{\lambda}=\textit{diag}(u)K\textit{diag}(v)$, where $u,v$ are two
nonnegative vectors that are unique up to constants and $K=e^{-\lambda M}$
denotes the matrix exponential of $-\lambda M$. This result is pivotal in
further speeding up the computation of (9). This type of result is also
commonly used as in, for example, [38] which is explored in (6.1.3).
Input: $M,\lambda,r,C=[c_{1},...,c_{N}]$
$I=(r>0);r=r(I);M=M(I,:);K=exp(-\lambda M)$
$u=ones(length(r),N)/length(r)$
$\hat{K}=diag(1./r)K$
While $u$ changes or any stopping criterion Do
$\quad u=1./(\hat{K}(C./(K^{T}u)))$
end while
$v=C./(K^{T}u)$
$d=sum(u.*((K.*M)v))$
$d=[d^{\lambda}_{M}(r,c_{1}),d^{\lambda}_{M}(r,c_{2}),...,d^{\lambda}_{M}(r,c_{N})]$
Algorithm 1 Computation of Entropy-Regularized
$d=[d^{\lambda}_{M}(r,c_{1}),d^{\lambda}_{M}(r,c_{2}),...,d^{\lambda}_{M}(r,c_{N})]$
The implementation of Sinkhorn’s algorithm to find optimal transport maps has
improved the general tractability OT algorithms. We note the improvement on
the problem of computing barycenters in Wasserstein space made by Cuturi and
Doucet in [11] where they prove the polyhedral convexity of a function that is
like a discrete version of (7)
$f(r,X)=\frac{1}{N}\sum\limits_{i=1}^{N}d(r,c_{i},M_{XY_{i}})$
where $d(\cdot,\cdot)$ is the previously defined Sinkhorn distance (8), and
$r,c$ are the marginal probabilities. $M_{XY}$ is the pairwise distance
matrix. Here, the problem of the optimal p is phrased using the dual linear
programming from known as the dual optimal transport problem:
$d(r,c,M)=\max_{(\alpha,\beta)\in C_{M}}\alpha^{T}r+\beta^{T}c$
where $C_{M}$ is the polyhedron of dual variables
$C_{M}=\\{(\alpha,\beta)\in\mathbb{R}^{n+m}|\alpha_{i}+\beta_{j}\leq
m_{ij}\\}$
This problem then has a solution and the computation of the barycenters
centers around this.
While the theoretical groundwork for optimal transport has been laid,
efficient algorithms are still needed for it to be implemented in large scale.
Genevay, _et al._ formulate stochastic descent methods for large scale
computations, making use of the duality arguments previously presented along
with entropic regularization for various cases in [15]. Then, Sinkhorn’s
algorithm will play an important role in the discrete case while the
continuous case is very elegantly dealt with using reproducing kernel Hilbert
spaces.
For a complete discussion of the numerical methods associated with the OT
problem as well as other relevant algorithms, see [40, 23].
## 6 Applications
Here, we hope to bring light to some of the many applications of OT within a
machine learning setting.
### 6.1 Computer Vision
OT finds a natural formulation within the context of computer vision. The
common method is to make a probability measure out of color histograms
relating to the image. Then, one can find a dissimilarity measure between the
images using the Wasserstein distance. An early formulations of OT in computer
visions can be in [30] and those relating to the Earth Mover’s Distance (EMD)
which acts as a slighty different discrete version of the 1-Wasserstein
distance. A formulation of the EMD on discrete surfaces can be found in [37].
In the forthcoming, we note a use of OT in improving GANs and the
Convolutional Wasserstein Distances which serve well for image interpolation.
#### 6.1.1 OT Meets GANs
Multiple attempts have been made to improve GANs using optimal transport.
Arjovski _et al._ recast the GANs problem into an OT theory problem [4]. OT
lends itself well to the GANs problem of learning models that generate data
like images or text with a distribution that is similar to that of training
data. Here in WGANs, we can take two probability measures $\mu_{0},\mu_{1}\in
M$ with $\mu_{1}$ being the distribution of a locally Lipschitz
$g_{\theta}(Z)$ acting as a neural network with _nice_ convergence properties
and with $Z$ a random variable with density $\rho$ and $g_{\theta}$ and the
Kantorovich-Rubinstein duality gives
$W(\mu_{0},\mu_{1})=\sup\limits_{||f||\leq
1}\mathbb{E}_{x\sim\mu_{0}}[f(x)]-\mathbb{E}_{x\sim\mu_{1}}[f(x)]$
with supremum taken over all Lipschitz continuous functions
$f:M\to\mathbb{R}$. It is shown here that there is a solution to this problem
with relevant gradient
$\nabla_{\theta}W(\mu_{0},\mu_{1})=-\mathbb{E}_{z\sim\rho}[\nabla_{\theta}f(g_{\theta}(z))]$
wherever both are well-defined. This formulation poses an alternative to the
classical GANs and it is found to be more stable, specially when dealing with
lower dimensional data, than its counterparts.
We also note the progress made by Salimans _et al._ [31] where they improve
upon the idea of mini-batches [14] and using energy functionals [5] to
introduce an OT variant using the W-distance named the Minibatch Energy
Distance:
$\displaystyle
D^{2}_{MED}(\mu_{0},\mu_{1})=2\mathbb{E}[W_{c}(X,Y)]-\mathbb{E}[W_{c}(X,X^{\prime})]-\mathbb{E}[W_{c}(Y,Y^{\prime})]$
where $X,X^{\prime}$ are sampled mini-batches from $\mu_{0}$ and
$Y,Y^{\prime}$ are sampled mini-batches from $\mu_{1}$ and $c$ is the optimal
transport function that is learned adversarially through the alternating
gradient descent common to GANs. These algorithms are seeing a greater
statistical consistency.
#### 6.1.2 Semantic Correspondence
OT is one of the few, if not the only, method that deals with mass-splitting
phenomenon which commonly occurs in establishing dense correspondence across
semantically similar images. This occurrence is in the form of a many-to-one
matching in the assignment of pixels from a source of pixels to a target pixel
as well as a one-to-many matching of the same type. The one-to-one matching
problem can be recast as an OT problem as done in [21]. Liu _et al._ replace
it with maximizing a total correlation where the optimal matching probability
is denoted as
$P^{*}=\text{argmax}_{P}\sum\limits_{i,j}P_{ij}C_{ij}$
where $P\in\mathbb{R}^{n\times
m}_{+},P\mathbbm{1}_{n}=r,P^{T}\mathbbm{1}_{m}=c$ and $r,c$ are marginals in
the same vein as in the section on computational challenges. Then, we can call
$M=1-C$ to be the cost matrix. Then, the problem becomes the optimal transport
problem
$P^{*}=\text{argmin}_{P}\sum_{i,j}P{ij}M_{ij}$
where $P\in\mathbb{R}^{n\times
m}_{+},P\mathbbm{1}_{n}=r,P^{T}\mathbbm{1}_{m}=c$. This problem can then be
solved using known algorithms, like those proposed in the computation
challenges section. Using the percentage of correct keypoints (PCK) evaluation
metric, their proposed algorithm outperformed state-of -the-art algorithms by
7.4 (or 26%), making it a huge improvement over other methods.
#### 6.1.3 Convolutional Wasserstein Distances
In [36], Solomon _et al._ propose an algorithm for approximating optimal
transport distances across geometric domains. Here, they make use of the
entropy-regularized Wasserstein distance given by (5) for its computational
advantages discussed in the Computational Challenges section:
$\displaystyle
W^{2}_{2,\gamma}(\mu_{0},\mu_{1})=\inf\limits_{\pi\in\Pi(\mu_{0},\mu_{1})}\bigg{[}\int_{M\times
M}d(x,y)^{2}d\pi(x,y)-\gamma H(\pi)\bigg{]}$ (10)
They use Varadhan’s formula [39] to approximate the distance $d(x,y)$ by
transferring heat from $x$ to $y$ over a short time interval:
$d(x,y)^{2}=\lim\limits_{t\to 0}[-2t\ln H_{t}(x,y)]$
where $H_{t}$ is the heat kernel associated to the geodesic distance $d(x,y)$.
Then, we can use this value in a kernel defined by
$K_{\gamma}(x,y)=e^{-\frac{d(x,y)^{2}}{\gamma}}$. We can conclude through
algebraic manipulations that
$W_{2,\gamma}^{2}(\mu_{0},\mu_{1})=\gamma[1+\min\limits_{\pi\in\Pi}KL(\pi|K_{\gamma})]$
where $KL$ denotes the K-L divergence (6). Then, in order to compute the
convolutional distances, we can discretize the domain $M$ with function and
density vectors $\mathbf{f}\in\mathbb{R}^{n}$. Then, define area weights
vector $\mathbf{a}\in\mathbb{R}^{n}_{+}$ with $\mathbf{a}^{T}\mathbbm{1}=1$
and a symmetric matrix $\mathbf{H}_{t}$ discretizing $H_{t}$ such that
$\int_{M}f(x)dx\approx\mathbf{a}^{T}\mathbf{f}\quad\text{and}\quad\int_{M}f(y)H_{t}(\cdot,y)dy\approx\mathbf{H}_{t}(\mathbf{a}\otimes\mathbf{f})$
Thus we are ready to compute the convolutional Wasserstein distance as in
Algorithm 2.
Input: $\mu_{0},\mu_{1},H_{t},a,\gamma$
Sinkhorn Iterations:
$\mathbf{v},\mathbf{w}\leftarrow 1$
for $i=1,2,3,...$
$\mathbf{v}\leftarrow\mu_{0}./\mathbf{H}_{t}(\mathbf{a}.*\mathbf{w})$
$\mathbf{w}\leftarrow\mu_{1}./\mathbf{H}_{t}(\mathbf{a}.*\mathbf{v})$
KL Divergence:
Return
$\gamma\mathbf{a}^{t}[(\mu_{0}.*\ln(\mathbf{v}))+(\mu_{1}.*\ln(\mathbf{w})]$
Algorithm 2 Convolutional Wasserstein Distance
We note the authors’ use of the Convolution Wasserstein Distance along with
barycenters in the Wasserstein space to implement an image interpolation
algorithm.
### 6.2 Graphs
The OT problem also lends itself to the formulation of dissimilarity measures
within different contexts. In [19], authors developed a fast framework,
referred to as WEGL (Wasserstein Embedding for Graph Learning), to embed
graphs in a vector space.
We find that analogs of the dissimilarity measures can be defined on graphs
and manifolds where the source manifold and target manifolds need not be the
same. In [43], Xu et al. propose a new method to solve the joint problem of
learning embeddings for associated graph nodes and graph matching. This is
done using a regularized Gromov-Wasserstein discrepancy when computing the
levels of dissimilarity between graphs. The computed distance allows us to
study the topology each of the spaces. The Gromov-Wasserstein discrepancy was
proposed by Peyre as a succession to the Gromov-Wasserstein distance which is
defined as follows:
Definition: Let $(X,d_{X},\mu_{X})$ and $(Y,d_{Y},\mu_{Y})$ be two metric
measure spaces, where $(X,d_{X})$ is a compact metric space and $\mu_{X}$ is a
probability measure on X (with $(Y,d_{Y},\mu_{Y})$ defined in the same way).
The Gromov Wasserstein distance $d_{GW}(\mu_{X},\mu_{Y})$ is defined as
$\inf\limits_{\pi\in\Pi(\mu_{X},\mu_{Y})}\int\limits_{X\times
Y}\int\limits_{X\times
Y}L(x,y,x^{\prime},y^{\prime})d\pi(x,y)d\pi(x^{\prime},y^{\prime}),$
where $L(x,y,x^{\prime},y^{\prime})=|d_{X}(x,x^{\prime})-d_{Y}(y,y^{\prime})|$
is the loss function and $\Pi(\mu_{X},\mu_{Y})$ is the set of all probability
measures on X x Y with $\mu_{X}$ and $\mu_{Y}$ as marginals. We note that the
loss function could be continuous depending on the topology the metric space
$X$ is endowed with. At the very least, we would want it to be
$\pi$-measurable.
When $d_{x}$ and $d_{y}$ are replaced with dissimilarity measurements rather
than strict distance metrics and the loss function _L_ is defined more
flexibly, the GW distance can be relaxed to the _discrepancy_. From graph
theory, a graph is represented by its vertices and edges, _G(V,E)_. If we let
a metric-measure space be defined by the pair _(C, $\mu$)_, then we can define
the Gromov-Wasserstein discrepancy between two spaces, _( $C_{s},\mu_{s}$)_
and _( $C_{t},\mu_{t}$)_, as:
$\displaystyle d_{GW}(\mu_{s},\mu_{t})$
$\displaystyle=\min_{T\in\pi(\mu_{s},\mu_{t})}\sum_{i,j,i^{\prime},j^{\prime}}L(c^{s}_{ij},c^{t}_{i^{\prime}j^{\prime}})Tii^{\prime}Tjj^{\prime}$
$\displaystyle=\min_{T\in\pi(\mu_{s},\mu_{t})}\langle
L(C_{s},C_{t},T),T\rangle$
In order to learn the mapping that includes the correspondence between graphs
and also the node embeddings, Xu et al. proposed the regularized GW
discrepancy:
$\displaystyle\min\limits_{X_{s},X_{t}}\min\limits_{T\in\Pi(\mu_{s},\mu_{t})}\langle
L(C_{s}(X_{s}),C_{t}(X_{t}),T),T\rangle+\alpha\langle
K(X_{s},X_{t}),T\rangle+\beta R(X_{s},X_{t})$
To solve this problem, the authors present Algorithm 3.
Input: $\\{C_{s},C_{t}\\}$, $\\{\mu_{s}$,$\mu_{t}\\}$, $\beta$, $\gamma$, the
dimension D, the number of outer/inner iterations $\\{M,N\\}$.
Output: $X_{s}$, $X_{t}$, and $\hat{T}$
Initialize $X_{s}^{(0)}$, $X_{t}^{(0)}$ randomly,
$\hat{T}^{(0)}=\mu_{s}\mu_{t}^{T}$.
For $m=0:M-1$:
Set $\alpha_{m}=\frac{m}{M}$.
For $n=0:N-1$
Update optimal transport $\hat{T}^{(m+1)}$
Obtain $X_{s}^{(m+1)}$, $X_{t}^{(m+1)}$
$X_{s}=X_{s}^{(M)},X_{t}=X_{t}^{(M)}$ and $\hat{T}=\hat{T}^{(M)}.$
Graph matching:
Initialize correspondence set $P=\emptyset$
For $v_{i}\in V_{s}$
$j=\mathrm{argmax}_{j}\hat{T}_{ij}.P=P\bigcup\\{(v_{i}\in V_{s},v_{j}\in
V_{t})\\}$.
Algorithm 3 Gromov-Wasserstein Learning (GWL)
The proposed methodology produced matching results that are better than all
other comparable methods and opens the opportunity for the improvement of
well-known systems (i.e. recommendation systems).
We note that the Gromov-Wasserstein discrepancy can also be used to improve
GANs, as is done in [7]. Here, Bunne, et al., adapt the generative model to
use the Gromov-Wasserstein discrepancy to perform GANs across different types
of data.
### 6.3 Neural Architecture Search
In this section we will look at the following paper: _Neural Architecture
Search with Bayesian Optimisation and Optimal Transport_ [18]. Bayesian
Optimization (BO) refers to a set of methods used for optimization of a
function $f$, thus making it perfect for solving the _model selection_ problem
over the space of neural architectures. The difficulty posed in BO when
dealing with network architecture is figuring out how to quantify
_(dis)similarity_ between any two networks. To do this, the authors developed
what they call a (pseudo-)distance for neural network architectures, called
OTMANN (Optimal Transport Metrics for Architectures of Neural Networks). Then,
to perform BO over neural network architectures, they created NASBOT, or
Neural Architecture Search with Bayesian Optimization and Optimal Transport.
To understand their formulation, we first look at the following definitions
and terms. First, a Gaussian process is a random process characterized by an
expectation function (mean function) $\mu:\chi\rightarrow\mathbb{R}$ and a
covariance (kernel) $\kappa=\chi^{2}\rightarrow\mathbb{R}$. In the context of
architecture search, having a large $\kappa(x,x^{\prime})$, where
$x,x^{\prime}\in\chi$ and $\kappa(x,x^{\prime})$ is the measure of similarity
so that $f(x)$ and $f(x^{\prime})$ are highly correlated; implying the GP
imposes a smoothness condition on $f:\chi\rightarrow\mathbb{R}$. Next, the
authors view a neural network (NN) as a graph whose vertices are the layers of
the network $G=(L,E)$, where $L$ is a set of layers and $E$ the directed
edges. Edges are denoted by a pair of layers, $(u,v)\in E$. A layer $u\in L$
is equipped with a layer label $ll(u)$, which denotes the type of operations
performed at layer $u$ (i.e. $ll(1)=conv3$ means 3x3 convolutions). Then, the
attribute $lu$ denotes the number of computational units in a layer.
Furthermore, each network has _decision layers_ , which are used to obtain the
predictions of the network. When networks have more than one decision layer,
one considers the average of the output given by each layer. Lastly, each
network has an input and output layer, $u_{in}$ and $u_{op}$ respectively; any
other layer is denoted as a _processing layer_.
Using the definitions above, the authors describe the distance for neural
architectures as $d:\chi^{2}\rightarrow\mathbb{R}_{+}$; with the goal of
obtaining a kernel for the GP where $\kappa(x,x^{\prime})=exp(-\beta
d(x,x^{\prime})^{p})$, given that $\beta,p\in\mathbb{R}_{+}$. We first look at
the OTMANN distance. OTMANN is defined as the minimum of a matching scheme
which attempts to match the computation at the layers of one network to the
layers of another, where penalties occur given that different types of
operations appear in matched layers. The OTMANN distance is that which
minimizes said penalties. Given two networks $G_{1}(L_{1},E_{1})$ and
$G_{2}(L_{2},E_{2})$ with $n_{1},n_{2}$ layers respectively, the OTMANN
distance is computed by solving the following optimization problem:
$\displaystyle\underset{Z}{\text{minimize}}\hskip
8.0pt\phi_{lmm}(Z)+\phi_{nas}(Z)+\nu_{str}\phi_{str}(Z)$
$\displaystyle\text{subject to}\sum\limits_{j\in L_{2}}Z_{ij}\leq
lm(i),\sum\limits_{i\in L_{1}}Z_{ij}\leq lm(j),\forall i,j$
In the above equation, $\phi_{lmm}$ is the label mismatch penalty,
$\phi_{str}$ is the structural term penalty, $\phi_{nas}$ is the non-assigment
penalty, $Z\in\mathbb{R}^{n_{1}xn_{2}}$ denoting hte maount of mass matched
between layer $i\in G_{1}$ and $j\in G_{2}$,
$l_{m}:L\rightarrow\mathbb{R}_{+}$ is a layer mass, and lastly $\nu_{str}>0$
determines the trade-off between the structural term and other terms. This
problem can be formulated as an Optimal Transport problem and is proved in the
appendix of the paper.
Next, we look at NASBOT. The goal here is to use the kernel $\kappa$, as
previously mentioned, to define the neural architectures and to find a method
to optimize the acquisition function:
$\displaystyle\phi_{t}(x)=\mathbb{E}[\max$
$\displaystyle\\{0,f(x)-\tau_{t-1}\\}|\\{(x_{i},y_{i})\\}_{i=1}^{t-1}],$
$\displaystyle\tau_{t-1}$ $\displaystyle=\underset{i\leq
t-1}{\text{argmax}}\hskip 4.0ptf(x_{i})$
The authors solve this optimization problem using an evolutionary algorithm,
whose solution leads to the creation of NASBOT. Detailed explanations on the
algorithm and the methodology onto which the optimization was solved can be
found in the appendix of the original paper. After running an experiment to
compare NASBOT against known methods, the authors show that NASBOT
consistently had the smallest cross validation mean squared error. For the
interested reader, there are illustrations for the best architectures found
for the problem posed in the experiment proposed.
### 6.4 Document Representation
In this section we will look at the following paper: _Hierarchical Optimal
Transport for Document Representation_ [45]. In this paper, Yurochkin, _et
al._ combine hierarchical latent structures from topic models with geometry
from word embeddings. _Hierarchical_ optimal topic transport document
distances, referred to as HOTT, this method combines language information (via
word embeddings) with topic distributions from latent Dirichlet allocation
(LDA) to measure the similarities between documents. Given documents $d^{1}$
and $d^{2}$, HOTT is defined as:
$HOTT(d^{1},d^{2})=W_{1}(\sum_{k=1}^{|T|}\bar{d}_{k}^{1}\delta_{t_{k}},\sum_{k=1}^{|T|}\bar{d}_{k}^{2}\delta_{t_{k}})$
(11)
Here, $\bar{d}^{i}$ represents document distributions over topics and the
Dirac delta $\delta_{t_{k}}$ is a probability distribution supported on the
corresponding topic $t_{k}$ and $W_{1}(d^{1},d^{2})=WMD(d^{1},d^{2})$ (_WMD_
being the Word Movers Distance). By truncating topics, the authors were able
to reduce the computational time and make HOTT a competitive model against
common methods. Their experiments show that although there is no uniformly
best method, HOTT has on average the smallest error with respect to nBOW
(normalized bag of words). More importantly, what was shown was that the
process of truncating topics to improve computational time does not hinder the
goal of obtaining high-quality distances. Interested readers will find in the
paper more detailed reports about the setup and results of the experiments
run.
### 6.5 Domain Adaptation
In this section we will cover _Optimal Transport for Domain Adaptation_ [12].
In their paper, Flamary, _et al._ , propose a regularized unsupervised optimal
transportation model to perform an alignment of the representations in the
source and target domains. By learning a transportation plan that matches the
source and target PDFs, they constrained labeled samples of the same class
during the transport. This helps solve the discrepancies (known as drift) in
data distributions.
In real world problems, the drift that occurs between the source and target
domains generally implies a change in marginal and conditional distributions.
In this paper, the authors assume the domain drift is due to “an unknown,
possibly nonlinear transformation of the input space
$T:\Omega_{s}\rightarrow\Omega_{t}$ (omega is a measurable space, s is source,
t is target). Because searching for T is an intractable problem and requires
restrictions to become approximated. Here, the authors consider the problem of
finding T the same as choosing a T such that one minimizes the transportation
cost C(T):
$C(T)=\int_{\Omega_{s}}c(x,T(x))d\mu(x)$ (12)
where $c:\Omega_{s}\mathrm{x}\Omega_{t}\rightarrow\mathbb{R}^{+}$ and $\mu(x)$
is a probability mass (or measure from x to T(x).)
This is precisely the optimal transport problem. Then, to further improve the
computational aspect of the model, a regularization component that preserves
label information and sample neighborhood during the transportation is
introduced. Now, the problem is as follows:
$\min\limits_{\pi\in\Pi}\langle\pi,C\rangle_{F}+\lambda\Omega_{s}(\pi)+\eta\Omega_{c}(\pi)$
(13)
where $\lambda\in\mathbb{R}$, $\eta\geq 0$, $\Omega_{c}(\cdot)$ is a class-
based regularization term, and
$\Omega_{s}(\pi)=\sum_{i,j}\mathrm{\pi(i,j)log(i,j)}$
This problem is solved using Algorithm 4:
Initialize: $k=0$, and $\pi^{0}\in P$
repeat
With $G\in\nabla f(\pi^{k})$, solve $\pi^{*}=\underset{\pi\in
B}{\mathrm{argmin}}\langle\pi,G\rangle_{F}+g(\pi)$
Find the optimal step $\alpha^{k}$, $\alpha^{k}=\underset{0\leq\alpha\leq
1}{\mathrm{argmin}}f(\pi^{k}+\alpha\Delta\pi)+g(\pi^{k}+\alpha\Delta\pi)$,
with $\Delta\pi=\pi^{*}-\pi^{k}$
$\pi^{k+1}\leftarrow\pi^{k}+\alpha^{k}\Delta\pi$, set $k\leftarrow k+1$
until Convergence
Algorithm 4 Generalized Conditional Gradient
In the algorithm above, $f(\pi)=\langle\pi,C\rangle_{F}+\eta\Omega_{c}(\pi)$
and $g(\pi)=\lambda\Omega_{s}(\pi)$. Using the assumption that $\Omega_{c}$ is
differentiable, step 3 of the algorithm becomes
$\pi^{*}=\underset{\pi\in
Pi}{\mathrm{argmin}}\langle\pi,C+\eta\nabla\Omega_{c}(\pi^{k})\rangle_{F}+\lambda\Omega_{s}(\pi)$
By using a constrained optimal transport method, the overall performance was
better than other state-of-the-art methods. Readers can find detailed reports
on Table 1 in [12].
For readers interested in domain adaptation, a varying approach to study
heterogeneous domain adaptation problems using OT can be found in [44].
## 7 Future Research
Further research will allow OT to be implemented in more areas and become more
widely acceptable. The main problem with optimal transport is scaling onto
higher dimensions. The optimal mappings that need to be solved are currently
intractable in high-dimensions, which is where most of the current problems
today lie. For example, Google’s NLP model has roughly 1 trillion parameters.
This type of problem is currently outside the scope of OT. Another interesting
research topic is the use of optimal transport in approximating intractable
distributions. This would compete with current known methods like KL-
divergence and open up interesting opportunities when working with variational
inference and/or expectation propagation. Another fundamental area to explore
lies with the choice of using the Wasserstein distance. As shown throughout
the paper, it is the most commonly used metric, but as one can see in Appendix
1, there are various others metrics, or distances, that may be used to replace
W-distance. Interested readers can read more about them in Villani’s Book,
_Optimal transport: old and new_ [41]. For further research from an applied
perspective, one possibility is the use of the GWL framework explained in
section 6.2 to improve on recommendation systems. On the other hand, all of
the papers we have referenced above are quite novel in their applications and
thus they all provide space for continuation or extension into more specific
sub-fields within their respective context.
## 8 Concluding Remarks
Throughout this survey, we have shown that Optimal Transport is seeing growing
attention within the machine learning community due to its applicability in
different areas. Although OT is becoming widely accepted in the machine
learning world, it is deeply rooted in mathematics and so we extracted the
most important topics so that interested readers can access only what is
needed to have a high-level understanding of what is happening. These excerpts
explain Kantorovich duality, entropic regularization, KL divergence, and
Wasserstein barycenters. Although the applications of OT span a wide range, it
is limited by computational challenges. Within this section we explored how
using an entropic regularization term allowed for the formation of an
algorithm that made OT problems computationally feasible and thus applicable.
This takes us to the last section of this survey, the applications of optimal
transport in machine learning. We began with computer vision, as it was one of
the first applications of OT in ML. First, OT has been used to improve GANs by
providing better statistical stability in low-dimensional data. Furthermore,
since OT is one of the few methods that deal with the mass-splitting
phenomenon, it allowed for many-to-one matching in pixel assignments which
yielded a new approach to semantic correspondence with a 26% performance
improvement over state-of-the-art methods. The last application we covered
with respect to computer vision was the use of W-distance to create a novel
method for image interpolation called Convolutional Wasserstein Distance.
Next, with respect to graphs, OT has allowed for the creation of the Gromov-
Wasserstein Learning (GWL) algorithm which have also been shown to improve
GANs. Other interesting areas that OT has shown promising results include
neural architecture search, document representation, and domain adaptation.
All of the papers we have analyzed and summarized will show that in some form
(computational/accuracy) the use of OT has yielded better results than
traditional methods. Although the computational inefficiencies are prevalent,
the future for optimal transport in machine learning looks promising as more
researchers become aware of this new intersection of areas.
## Appendix A
The implementation of the conclusions of OT in machine learning rely mostly on
the implementation of the various metrics that can be used as error measures
in model tuning. The most notable ones arise from the reformulation or
approximation of metrics into convex functionals that can be optimized by
drawing on the many beautiful conclusions of convex geometry. Here, we recall
a metric, many times called a distance, as a function $d(\cdot,\cdot):X\times
X\to\mathbf{R}_{+}$, where $X$ is a metrizable space, that satisfies
* •
Positive Definite: $d(x,y)\geq 0$ with $d(x,y)=0$ if, and only if, $x=y$
* •
Symmetric: $d(x,y)=d(y,x)$ for all $x,y\in X$
* •
Subadditive: $d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in X$
Here, we want to note some different error estimates that come up in the OT
literature as well as some that are traditionally used to compare probability
distributions. The most notable comparison of probability measures in the OT
literature is the p-Wasserstein Distance
$W^{d}_{p}(\mu_{0},\mu_{1})=\inf_{\pi\in\Pi(\mu_{0},\mu_{1})}\bigg{(}\int_{M\times
M}d(x,y)^{p}d\pi(x,y)\bigg{)}^{1/p}$ (14)
In 14, d is a metric. We see from definition that it should very much act like
a minimal $L^{p}$ distance on the space of probability measures. The most
relevant choice of parameter p is $p=1,2$. This distance was formulated in the
most general sense possible and it has a natural discrete formulation for
discrete measures. Therefore, it allows for different contexts. For example,
we saw the analog in the context of graphs as the Gromov-Wasserstein distance
as:
Let $(X,d_{X},\mu_{X})$ and $(Y,d_{Y},\mu_{Y})$ be two metric measure spaces,
where $(X,d_{X})$ is a compact metric space and $\mu_{X}$ is a probability
measure on X (with $(Y,d_{Y},\mu_{Y})$ defined in the same way). The Gromov-
Wasserstein distance $d_{GW}(\mu_{X},\mu_{Y})$ is defined as
$\inf\limits_{\pi\in\Pi(\mu_{X},\mu_{Y})}\int\limits_{X\times
Y}\int\limits_{X\times
Y}L(x,y,x^{\prime},y^{\prime})d\pi(x,y)d\pi(x^{\prime},y^{\prime}),$ (15)
where
$L(x,y,x^{\prime},y^{\prime})=|d_{X}(x,x^{\prime})-d_{Y}(y,y^{\prime})|$.
Here, we see that the formulas look naturally similar. The Gromov-Wasserstein
distance would be a particular choice of the 1-Wasserstein distance to a
general metric space which can then be relaxed to be able to work with graphs
as we saw before.
The novelty in using OT in applications is principally the different error
estimates. We recall some of the well-known distances that are traditionally
used to compare probability measures:
* •
KL Divergence:
$\displaystyle KL(\pi|\kappa)\coloneqq$ $\displaystyle\int\int_{M\times
M}\pi(x,y)\bigg{[}\ln\frac{\pi(x,y)}{\kappa(x,y)}-1\bigg{]}dxdy$
* •
Hellinger distance:
$H^{2}(\mu_{0},\mu_{1})=\frac{1}{2}\int(\sqrt{\frac{d\mu_{0}}{d\lambda}}-\sqrt{\frac{d\mu_{1}}{d\lambda}})^{2}d\lambda$,
where $\mu_{0},\mu_{1}$ are absolutely continuous with respect to $\lambda$
and $\frac{d\mu_{0}}{d\lambda},\frac{d\mu_{1}}{d\lambda}$ denote the Radon-
Nykodym derivatives, respectively.
* •
Lèvy-Prokhorov distance: $d_{P}(\mu_{0},\mu_{1})=\inf\\{\epsilon>0;\exists
X,Y;\inf\mathbb{P}[d(X,Y)>\epsilon]\leq\epsilon\\}$
* •
Bounded Lipschitz distance (or Fortet-Mourier distance):
$d_{b}L(\mu_{0},\mu_{1})=\sup\\{\int\phi d\mu_{0}-\int\phi
d\mu_{1};||\phi||_{\infty}+||\phi||_{\text{Lip}}\leq 1\\}$
* •
(in the case of nodes) Euclidean distance: $d(x,y)=\sqrt{(x-y)^{2}}$
We note that the Lèvy-Prokhorov and bounded Lipschitz distances can work in
much the same way that the Wasserstein distance does. At the present, the
Wasserstein distance proves useful because of it’s capabilities in dealing
with large distances and its convenient formulation in many problems such as
the ones presented in this paper as well as others coming from partial
differential equations. It’s definition using infimum makes it easy to
majorate. Its duality properties are useful–particularly in the case when
$p=1$ as we see with the Kantorovich-Rubinstein distance where it is defined
as an equivalence to its dual:
$W_{1}(\mu_{0},\mu_{1})=\sup_{||\phi||_{\text{Lip}}\leq
1}\bigg{\\{}\int_{X}\phi d\mu_{0}-\int_{x}\phi d\mu_{1}\bigg{\\}}$ (16)
The interested reader can read more about the different distances in [27, 41]
As we presently see in this paper, we notice that much of the work on the
optimal transport in machine learning is in the reformulation of the
algorithms, which classically used the traditional distances, into new
versions that use the Wasserstein distance. Then, a lot of the work is done in
dealing with the computational inefficiency of the Wasserstein distance.
Moving forward, the authors think that many machine learning algorithms will
implement some of the "deeper" features of the optimal transport theory to
improve such algorithms after their best formulation becomes abundantly clear.
## Appendix B
For the readers interested in papers that apply OT in machine learning, here
are a few more references to be considered. First we have OT in GANS:
* •
A geometric view of optimal transportation and generative model [20]
Next, for semantic correspondence and NLP we have:
* •
Improving sequence-to-sequence learning via optimal transport [8]
Lastly, on domain adaptation we have:
* •
Joint distribution optimal transportation for domain adaptation [9]
* •
Theoretical analysis of domain adaptation with optimal transport [28]
## Acknowledgement
This material is based upon Luiz Manella Pereira’s work supported by the U.S.
Department of Homeland Security under Grant Award Number, 2017-ST-062-000002.
The views and conclusions contained in this document are those of the authors
and should not be interpreted as necessarily representing the official
policies, either expressed or implied, of the U.S. Department of Homeland
Security.
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|
# Maximizing Social Welfare in Score-Based
Social Distance Games
Robert Ganian TU Wien
Vienna, Austria<EMAIL_ADDRESS>Utrecht University
Utrecht, NetherlandsCzech Technical University in Prague
Prague, Czech RepublicPenn State University
Pennsylvania, USACzech Technical University in Prague
Prague, Czech RepublicCzech Technical University in Prague
Prague, Czech Republic Thekla Hamm Utrecht University
Utrecht, Netherlands<EMAIL_ADDRESS>Czech Technical University in
Prague
Prague, Czech RepublicPenn State University
Pennsylvania, USACzech Technical University in Prague
Prague, Czech RepublicCzech Technical University in Prague
Prague, Czech Republic Dušan Knop Czech Technical University in Prague
Prague, Czech Republic<EMAIL_ADDRESS>Penn State University
Pennsylvania, USACzech Technical University in Prague
Prague, Czech RepublicCzech Technical University in Prague
Prague, Czech Republic Sanjukta Roy Penn State University
Pennsylvania, USA<EMAIL_ADDRESS>Czech Technical University in Prague
Prague, Czech RepublicCzech Technical University in Prague
Prague, Czech Republic Šimon Schierreich Czech Technical University in Prague
Prague, Czech Republic<EMAIL_ADDRESS>Czech Technical University in
Prague
Prague, Czech Republic Ondřej Suchý Czech Technical University in Prague
Prague, Czech Republic<EMAIL_ADDRESS>
###### Abstract
Social distance games have been extensively studied as a coalition formation
model where the utilities of agents in each coalition were captured using a
utility function $\operatorname{u}$ that took into account distances in a
given social network. In this paper, we consider a non-normalized score-based
definition of social distance games where the utility function
$u^{\vec{\operatorname{s}}}$ depends on a generic scoring vector
$\vec{\operatorname{s}}$, which may be customized to match the specifics of
each individual application scenario.
As our main technical contribution, we establish the tractability of computing
a welfare-maximizing partitioning of the agents into coalitions on tree-like
networks, for every score-based function $u^{\vec{\operatorname{s}}}$. We
provide more efficient algorithms when dealing with specific choices of
$u^{\vec{\operatorname{s}}}$ or simpler networks, and also extend all of these
results to computing coalitions that are Nash stable or individually rational.
We view these results as a further strong indication of the usefulness of the
proposed score-based utility function: even on very simple networks, the
problem of computing a welfare-maximizing partitioning into coalitions remains
open for the originally considered canonical function $\operatorname{u}$.
## 1 Introduction
Coalition formation is a central research direction within the fields of
algorithmic game theory and computational social choice. While there are many
different scenarios where agents aggregate into coalitions, a pervasive
property of such coalitions is that the participating agents exhibit
_homophily_ , meaning that they prefer to be in coalitions with other agents
which are similar to them. It was this observation that motivated Brânzei and
Larson to introduce the notion of _social distance games_ (SDG) as a basic
model capturing the homophilic behavior of agents in a social network [15].
Brânzei and Larson’s SDG model consisted of a graph $G=(V,E)$ representing the
social network, with $V$ being the agents and $E$ representing direct
relationships or connections between the agents. To capture the utility of an
agent $v$ in a coalition $C\subseteq V$, the model considered a single
function: $u(v,C)=\frac{1}{|C|}\cdot\sum\nolimits_{w\in
C\setminus\\{v\\}}\frac{1}{d_{C}(v,w)}$ where $d_{C}(v,w)$ is the distance
between $v$ and $w$ inside $C$.
Social distance games with the aforementioned utility function
$\operatorname{u}$ have been the focus of extensive study to date, with a
number of research papers specifically targeting algorithmic and complexity-
theoretic aspects of forming coalitions with maximum social welfare [2, 3, 4,
29]. Very recently, Flammini et al. [22, 23] considered a generalization of
$\operatorname{u}$ via an adaptive real-valued scoring vector which weights
the contributions to an agent’s utility according to the distances of other
agents in the coalition, and studied the price of anarchy and stability for
non-negative scoring vectors. However, research to date has not revealed any
polynomially tractable fragments for the problem of computing coalition
structures with maximum social welfare (with or without stability-based
restrictions on the behavior of individual agents), except for the trivial
cases of complete (bipartite) graphs [15] and trees [36].
#### Our Contribution.
The undisputable appeal of having an adaptive scoring vector—as opposed to
using a single canonical utility function $\operatorname{u}$—lies in the fact
that it allows us to capture many different scenarios with different dynamics
of coalition formation. However, it would also be useful for such a model to
be able to assign negative scores to agents at certain (larger) distances in a
coalition. For instance, guests at a gala event may be keen to accept the
presence of friends-of-friends (i.e., agents at distance $2$) at a table,
while friends-of-friends may be less welcome in private user groups on social
networks, and the presence of complete strangers in some scenarios may even be
socially unacceptable.
Here, we propose the study of social distance games with a family of highly
generic non-normalized score-based utility functions. Our aim here is twofold.
First of all, these should allow us to better capture situations where agents
at larger distances are unwelcome or even unacceptable for other agents. At
the same time, we also want to obtain algorithms capable of computing welfare-
maximizing coalition structures in such general settings, at least on well-
structured networks.
Our model considers a graph $G$ accompanied with an integer-valued, fixed but
adaptive _scoring vector_ $\vec{\operatorname{s}}$ which captures how
accepting agents are towards other agents based on their pairwise
distance.111Formal definitions are provided in the Preliminaries. The utility
function $u^{\vec{\operatorname{s}}}(v,C)$ for an agent $v$ in coalition $C$
is then simply defined as $u^{\vec{\operatorname{s}}}(v,C)=\sum\nolimits_{w\in
C\setminus\\{v\\}}\vec{\operatorname{s}}(d_{C}(v,w))$; we explicitly remark
that, unlike previous models, this is not normalized with respect to the
coalition size. As one possible example, a scoring vector of $(1,0,-1)$ could
be used in scenarios where agents are welcoming towards friends, indifferent
to friends-of-friends, slightly unhappy about friends-of-friends-of-friends
(i.e., agents at distance $3$), and unwilling to group up with agents who are
at distance greater than $3$ in $G$. A concrete example which also illustrates
the differences to previous SDG models is provided in Figure 1.
Figure 1: A social network illustrating the difference of maximising social
welfare in our model compared to previous SDG models. (1) In Brânzei and
Larson’s SDG model, the welfare-maximum outcome is the grand coalition. (2) A
welfare-maximum outcome in the normalized model of Flammini et al. with a
scoring vector of $(1,0,0,0)$ is marked with dashed lines, while the same
scoring vector in our non-normalized model produces the grand coalition. (3) A
scoring vector of $\vec{\operatorname{s}}=(1,0,-1)$ in our model produces the
welfare-maximizing outcome marked with bold lines, with a welfare of $18$. (4)
A ‘less welcoming’ scoring vector of $\vec{\operatorname{s}}=(1,-3)$ leads to
the welfare maximizing dash-circled partition with a welfare of $14$ (compared
to only $12$ for the bold-circled one).
While non-normalized scoring functions have not previously been considered for
social distance games, we view them a natural way of modeling agent utilities;
in fact, similar ideas have been successfully used in models for a variety of
other phenomena including, e.g., committee voting [21], resource allocation
[14, 13] and Bayesian network structure learning [25, 37]. Crucially, it is
not difficult to observe that many of the properties originally established by
Brânzei and Larson for SDGs also hold for our non-normalized score-based model
with every choice of $\vec{\operatorname{s}}$, such as the small-world
property [15, 28] and the property that adding an agent with a close (distant)
connection to a coalition positively (negatively) impacts the utilities of
agents [15]. In addition, the proposed model can also directly capture the
notion of _enemy aversion_ with symmetric preferences [5, 35] by setting
$\vec{\operatorname{s}}=(1)$.
Aside from the above, a notable benefit of the proposed model lies on the
complexity-theoretic side of things. Indeed, a natural question that arises in
the context of SDG is whether we can compute an outcome—a partitioning of the
agents into coalitions—which maximizes the social welfare (defined as the sum
of the utilities of all agents in the network). This question has been studied
in several contexts, and depending on the setting one may also require the
resulting coalitions to be stable under _individual rationality_ (meaning that
agents will not remain in coalitions if they have negative utility) or _Nash
stability_ (meaning that agents may leave to join a different coalition if it
would improve their utility). But in spite of the significant advances in
algorithmic aspects of other coalition formation problems in recent years [10,
11, 17, 24], we lack any efficient algorithm capable of producing such a
welfare-optimal partitioning when using the utility function
$\operatorname{u}$ even for the simplest types of networks.
To be more precise, when viewed through the refined lens of _parameterized
complexity_ [18, 20] that has recently become a go-to paradigm for such
complexity-theoretic analysis, no tractable fragments of the problem are
known. More precisely, the problem of computing a welfare-maximizing outcome
under any of the previously considered models is not even known to admit an XP
algorithm when parameterized by the minimum size of a vertex cover in the
social network $G$—implying a significant gap towards potential fixed-
parameter tractability. This means that the complexity of welfare-maximization
under previous models remains wide open even under the strongest non-
trivializing restriction of the network.
As our main technical contribution, we show that non-normalized score-based
utility functions do not suffer from this drawback and can in fact be computed
efficiently under fairly mild restrictions on $G$. Indeed, as our first
algorithmic result we obtain an XP algorithm that computes a welfare-
maximizing partitioning of the agents into coalitions parameterized by the
treewidth of $G$, and we strengthen this algorithm to also handle additional
restrictions on the coalitions in terms of individual rationality or Nash
stability. As with numerous treewidth-based algorithms, we achieve this result
via leaf-to-root dynamic programming along a tree-decomposition. However, the
records we keep during the dynamic program are highly non-trivial and require
an advanced branching step to correctly pre-computed the distances in the
stored records. We remark that considering networks of small treewidth is
motivated not only by the fundamental nature of this structural graph measure,
but also by the fact that many real-world networks exhibit bounded treewidth
[34].
In the next part of our investigation, we show that when dealing with simple
scoring functions or bounded-degree networks, these results can be improved to
fixed-parameter algorithms for welfare-maximization (including the cases where
we require the coalitions to be individually rational or Nash stable). This is
achieved by combining structural insights into the behavior of such coalitions
with a different dynamic programming approach. Furthermore, we also use an
entirely different technique based on quadratic programming to establish the
fixed-parameter tractability of all 3 problems under consideration w.r.t. the
minimum size of a vertex cover in $G$. Finally, we conclude with some
interesting generalizations and special cases of our model and provide some
preliminary results in these directions.
## 2 Preliminaries
We use $\mathbb{N}$ to denote the set of natural numbers, i.e., positive
integers, and $\mathbb{Z}$ for the set of integers. For $i\in\mathbb{N}$, we
let $[i]=\\{1,\ldots,i\\}$ and ${[i]_{0}=[i]\cup\\{0\\}}$. We assume basic
familiarity with graph-theoretic terminology [19].
#### Social Distance Games.
A _social distance game_ (SDG) consists of a set $N=\\{1,\ldots,n\\}$ of
_agents_ , a simple undirected graph $G=(N,E)$ over the set of agents called a
_social network_ , and a non-increasing _scoring vector_
$\vec{\operatorname{s}}=(s_{1},\dots,s_{{\delta}})$ where a) for each
$a\in[{\delta}]$, $s_{a}\in\mathbb{Z}$ and b) for each $a\in[{\delta}-1]$,
$s_{a+1}\leq s_{a}$.
In some cases, it will be useful to treat $\vec{\operatorname{s}}$ as a
function from $\mathbb{N}$ rather than a vector; to this end, we set
$\vec{\operatorname{s}}(a)=s_{a}$ for each $a\leq{\delta}$ and
$\vec{\operatorname{s}}(a)=-\infty$ when $a>{\delta}$. The value “$-\infty$”
here represents an inadmissible outcome, and formally we set
$-\infty+z=-\infty$ and $-\infty<z$ for each $z\in\mathbb{Z}$.
A _coalition_ is a subset $C\subseteq N$, and an outcome is a partitioning
$\Pi=(C_{1},\dots,C_{\ell})$ of $N$ into coalitions; formally,
$\bigcup_{i=1}^{\ell}C_{i}=N$, every $C_{i}\in\Pi$ is a coalition, and all
coalitions in $\Pi$ are pairwise disjoint. We use $\Pi_{i}$ to denote the
coalition the agent $i\in N$ is part of in the outcome $\Pi$. The _utility_ of
an agent $i\in N$ for a coalition $\Pi_{i}\in\Pi$ is
$\operatorname{u}^{\vec{\operatorname{s}}}(i,\Pi_{i})=\sum_{j\in\Pi_{i}\setminus\\{i\\}}\vec{\operatorname{s}}(\operatorname{dist}_{\Pi_{i}}(i,j)),$
where $\operatorname{dist}_{\Pi_{i}}(i,j)$ is the length of a shortest path
between $i$ and $j$ in the graph $G[\Pi_{i}]$, i.e., the subgraph of $G$
induced on the agents of $\Pi_{i}$. We explicitly note that if $\Pi_{i}$ is a
singleton coalition then
$\operatorname{u}^{\vec{\operatorname{s}}}(i,\Pi_{i})=0$. Moreover, in line
with previous work [15] we set $\operatorname{dist}_{\Pi_{i}}(i,j):=+\infty$
if there is no $i$-$j$ path in $G[\Pi_{i}]$, meaning that
$\operatorname{u}^{\vec{\operatorname{s}}}(i,\Pi_{i})=-\infty$ whenever
$G[\Pi_{i}]$ is not connected.
For brevity, we drop the superscript from $u^{\vec{\operatorname{s}}}$
whenever the scoring vector $\vec{\operatorname{s}}$ is clear from the
context. To measure the satisfaction of the agents with a given outcome, we
use the well-known notation of _social welfare_ , which is the total utility
of all agents for an outcome $\Pi$, that is,
$\operatorname{SW}^{\vec{\operatorname{s}}}(\Pi)=\sum_{i\in
N}\operatorname{u}^{\vec{\operatorname{s}}}(i,\Pi_{i}).$
Here, too, we drop the superscript specifying the scoring vector whenever it
is clear from the context.
We assume that all our agents are selfish, behave strategically, and their aim
is to maximize their utility. To do so, they can perform _deviations_ from the
current outcome $\Pi$. We say that $\Pi$ admits an _IR-deviation_ if there is
an agent $i\in N$ such that $\operatorname{u}(i,C)<0$; in other words, agent
$i$ prefers to be in a singleton coalition over its current coalition. If no
agent admits an IR-deviation, the outcome is called _individually rational_
(IR). We say that $\Pi$ admits an _NS-deviation_ if there is an agent $i$ and
a coalition $C\in\Pi\cup\\{\emptyset\\}$ such that
$\operatorname{u}(i,C\cup\\{i\\})>\operatorname{u}(i,\Pi_{i})$. $\Pi$ is
called _Nash stable_ (NS) if no agent admits an NS-deviation. We remark that
other notions of stability exist in the literature [14, Chapter 15], but Nash
stability and individual rationality are the most basic notions used for
stability based on individual choice [30, 39].
Having described all the components in our score-based SDG model, we are now
ready to formalize the three classes of problems considered in this paper. We
note that even though these are stated as decision problems for complexity-
theoretic reasons, each of our algorithms for these problems can also output a
suitable outcome as a witness. For an arbitrary fixed scoring vector
$\vec{\operatorname{s}}$, we define:
$\vec{\operatorname{s}}$-SDG-WF Input: A social network $G=(N,E)$, desired
welfare $b\in\mathbb{N}$. Question: Does the distance game given by $G$ and
$\vec{\operatorname{s}}$ admit an outcome with social welfare at least $b$?
$\vec{\operatorname{s}}$-SDG-WF-IR and $\vec{\operatorname{s}}$-SDG-WF-Nash
are then defined analogously, but with the additional condition that the
outcome must be individually rational or Nash stable, respectively.
We remark that for each of the three problems, one may assume w.l.o.g. that
$s_{1}>0$; otherwise the trivial outcome consisting of $|N|$ singleton
coalitions is both welfare-optimal and stable. Moreover, without loss of
generality we assume $G$ to be connected since an optimal outcome for a
disconnected graph $G$ can be obtained as a union of optimal outcomes in each
connected component of $G$.
The last remark we provide to the definition of our model is that it trivially
also supports the well-known _small world_ property [28] that has been
extensively studied on social networks. In their original work on SDGs,
Brânzei and Larson showed that their model exhibits the small world property
by establishing a diameter bound of $14$ in each coalition in a so-called
_core partition_ [15]. Here, we observe that for each choice of
$\vec{\operatorname{s}}$, a welfare-maximizing coalition will always have
diameter at most ${\delta}$.
#### Parameterized Complexity.
The _parameterized complexity_ framework [18, 20] provides the ideal tools for
the fine-grained analysis of computational problems which are NP-hard and
hence intractable from the perspective of classical complexity theory. Within
this framework, we analyze the running times of algorithms not only with
respect to the input size $n$, but also with respect to a numerical parameter
$k\in\mathbb{N}$ that describes a well-defined structural property of the
instance; the central question is then whether the superpolynomial component
of the running time can be confined by a function of this parameter alone.
The most favorable complexity class in this respect is FPT (short for “fixed-
parameter tractable”) and contains all problems solvable in $f(k)\cdot
n^{\mathcal{O}(1)}$ time, where $f$ is a computable function. Algorithms with
this running time are called _fixed-parameter algorithms_. A less favorable,
but still positive, outcome is an algorithm with running time of the form
$n^{f(k)}$; problems admitting algorithms with such running times belong to
the class XP.
#### Structural Parameters.
Let $G=(V,E)$ be a graph. A set $U\subseteq V$ is a _vertex cover_ if for
every edge $e\in E$ it holds that $U\cap e\not=\emptyset$. The _vertex cover
number_ of $G$, denoted $\operatorname{vc}(G)$, is the minimum size of a
vertex cover of $G$. A _nice tree-decomposition_ of $G$ is a pair
$(\mathcal{T},\beta)$, where $\mathcal{T}$ is a tree rooted at a node $r\in
V(\mathcal{T})$, $\beta\colon V(\mathcal{T})\to 2^{V}$ is a function assigning
each node $x$ of $\mathcal{T}$ its _bag_ , and the following conditions hold:
* •
for every edge $\\{u,v\\}\in E(G)$ there is a node $x\in V(\mathcal{T})$ such
that $u,v\in\beta(x)$,
* •
for every vertex $v\in V$, the set of nodes $x$ with $v\in\beta(x)$ induces a
connected subtree of $\mathcal{T}$,
* •
$|\beta(r)|=|\beta(x)|=0$ for every _leaf_ $x\in V(\mathcal{T})$, and
* •
there are only tree kinds of internal nodes in $\mathcal{T}$:
* –
$x$ is an _introduce node_ if it has exactly one child $y$ such that
$\beta(x)=\beta(y)\cup\\{v\\}$ for some ${v\notin\beta(y)}$,
* –
$x$ is a _join node_ if it has exactly two children $y$ and $z$ such that
$\beta(x)=\beta(y)=\beta(z)$, or
* –
$x$ is a _forget node_ if it has exactly one child $y$ such that
$\beta(x)=\beta(y)\setminus\\{v\\}$ for some $v\in\beta(y)$.
The _width_ of a nice tree-decomposition $(\mathcal{T},\beta)$ is $\max_{x\in
V(\mathcal{T})}|\beta(x)|-1$, and the treewidth $\operatorname{tw}(G)$ of a
graph $G$ is the minimum width of a nice tree-decomposition of $G$. Given a
nice tree-decomposition and a node $x$, we denote by $G^{x}$ the subgraph
induced by the set $V^{x}=\bigcup_{y\text{ is a descendant of }x}\beta(y)$,
where we suppose that $x$ is a descendant of itself. It is well-known that
optimal nice tree-decompositions can be computed efficiently [8, 31, 32].
Integer Quadratic Programming. Integer Quadratic Programming (IQP) over $d$
dimensions can be formalized as the task of computing
$\max\left\\{x^{T}Qx\mid Ax\leq b,\,x\geq 0,\,x\in\mathbb{Z}^{d}\right\\}\,,$
(IQP)
where $Q\in\mathbb{Z}^{d\times d}$, $A\in\mathbb{Z}^{m\times d}$,
$b\in\mathbb{Z}^{m}$. That is, IQP asks for an integral vector
$x\in\mathbb{Z}^{d}$ which maximizes the value of a quadratic form subject to
satisfying a set of linear constraints.
###### Proposition 1 ([33, 40], see also [26]).
Integer Quadratic Programming is fixed-parameter tractable when parameterized
by $d+\|A\|_{\infty}+\|Q\|_{\infty}$.
## 3 Structural Properties of Outcomes
As our first set of contributions, we establish some basic properties of our
model and the associated problems that are studied within this paper. We begin
by showcasing that the imposition of individual rationality or Nash stability
as additional constraints on our outcomes does in fact have an impact on the
maximum welfare that can be achieved (and hence it is indeed necessary to
consider three distinct problems). We do not consider this to be obvious at
first glance: intuitively, an agent $i$’s own contribution to the social
welfare can only improve if they perform an IR- or NS-deviation, and the fact
that the distance function $\operatorname{dist}_{\Pi_{i}}$ is symmetric would
seem to suggest that this can only increase the total social welfare.
Social Network from Lemma 2.[.48] $x$ $P$ $K$ Social Network from Lemma
3.[.48] $x$ $y$ $P$ $K$
###### Lemma 2.
There is a scoring vector $\vec{\operatorname{s}}$ and a social network $G$
such that the single outcome achieving the maximum social welfare is not
individually rational.
###### Proof.
Consider a scoring function $\vec{\operatorname{s}}$ such that
$\vec{\operatorname{s}}=(1,1,-1,-1,-1,-1)$. Consider the social network $G$ in
Section 3 formed from a path $P$ on $5$ vertices and a clique $K$ on $5$
vertices by connecting the endpoints of $P$ to all vertices of $K$. Let $x$ be
the central agent of $P$. Let $C$ be the grand coalition in $G$. The graph can
be viewed as a $6$-cycle with $K$ forming one “bold” agent. All vertices on
the cycle contribute positively to the agent’s utility, except for the one
that is exactly opposite on the cycle. Hence, $\operatorname{u}(x,C)=4-5=-1$,
while utility of all other agents is $8-1=7$ in $C$. This gives total social
welfare of $62$ for the grand coalition.
However, if $x$ leaves the coalition to form its own one, their utility will
improve from $-1$ to $0$, whereas the total social welfare drops. Indeed, in
$C\setminus\\{x\\}$ there are 2 agents with utility $6-2=4$, 2 agents with
utility $7-1=6$ and 5 agents with utility $8-0$, giving total social welfare
of $60$. If any $y\neq x$ was to be excluded from $C$ to form outcome
$\\{y\\},C\setminus\\{y\\}$, then $y$ joining $C$ improves social welfare,
proving that it was not optimal. Finally, if the outcome consists of several
coalitions with the largest one of size 8, then the welfare is at most $8\cdot
7+2\cdot 1=56$, if the largest size is 7, then we get at most $7\cdot 6+3\cdot
2=48$, for 6 it is $6\cdot 5+4\cdot 3=42$ and for 5 it is $5\cdot 4+5\cdot
4=40$.
Hence the grand coalition $C$ is the only outcome with maximal social welfare,
but it is not individually rational (and therefore not Nash stable), as
$\operatorname{u}(x,C)=-1$. ∎
###### Lemma 3.
There is a scoring vector $\vec{\operatorname{s}}$ and a social network $G$
such that the single individually rational outcome achieving the maximum
social welfare among such outcomes is not Nash stable.
###### Proof.
Consider again the scoring function
$\vec{\operatorname{s}}=(1,1,-1,-1,-1,-1)$. Similarly to previous lemma,
consider the social network $G$ in Section 3 formed from a path $P$ on $5$
vertices and a clique $K$ on $4$ vertices by connecting the endpoints of $P$
to all vertices of $K$ and adding a agent $y$ only connected to the central
agent of $P$ which we call $x$. Let $C$ be the coalition containing all
vertices of $G$ except for $y$. As in the previous lemma, $G[C]$ can be viewed
as a $6$-cycle with $K$ forming one “bold” agent. Hence,
$\operatorname{u}_{x}(C)=4-4=0$, while utility of other agents in $C$ is
$7-1=6$. Trivially $\operatorname{u}_{y}(\\{y\\})=0$, hence the outcome
$(\\{y\\},C)$ is individually rational. It has total social welfare of $48$.
However, it is not Nash stable, as $x$ wants to deviate to $\\{x,y\\}$ giving
them utility $1$.
However, the outcome $(\\{x,y\\},C\setminus\\{x\\})$, which is Nash stable,
has total social welfare only $46$. Note that
$\operatorname{u}_{z}(C\setminus\\{x\\})\geq 3$ for every agent $z\in
C\setminus\\{x\\}$, so any outcome $(\\{x,y,z\\},C\setminus\\{x,z\\})$ cannot
be Nash stable. While the total social welfare of the grand coalition is $46$,
the utility of $y$ is $3-6=-3$ in this coalition, so this outcome is not even
individually rational. From the computations in the previous lemma, it
follows, that to attain the social welfare of $48$, the largest coalition in
the outcome must be of size at least $7$. Moreover, if it is of size exactly
$7$, then these $7$ vertices must be at mutual distance at most $2$. However,
there are no $7$ vertices in mutual distance at most $2$ in $G$. Hence, in any
outcome with social welfare $48$ the largest coalition must be of size at
least $8$. Agent $y$ has only $3$ agents in distance at most $2$ in $G$.
Hence, for $y$ to get a positive utility from some coalition, the coalition
must be of size at most $7$, i.e., $y$ cannot be part of the largest coalition
in any outcome with social welfare at least $48$. However, for every $z\in C$,
$z$ joining the coalition $C\setminus\\{z\\}$ improves the social welfare of
the outcome, proving that it was not optimal.
Hence the outcome $(\\{y\\},C)$ is the only individually rational outcome with
maximal social welfare, but it is not Nash stable. ∎
It should be noted that Lemmas 2 and 3 also contrast many other models where
outputs maximizing social welfare are stable for symmetric utilities [12, 7,
16].
As our next two structural results, we prove that on certain SDGs it is
possible to bound not only the diameter but also the size of each coalition in
a welfare-maximum outcome. Notably, we establish such bounds for SDGs on
bounded-degree networks and SDGs which have a simple scoring vector on a tree-
like network. While arguably interesting in their own right, these properties
will be important for establishing the fixed-parameter tractability of
computing welfare-optimal outcomes in the next section.
###### Lemma 4.
For every scoring vector $\vec{\operatorname{s}}=(s_{1},\ldots,s_{\delta})$,
if $G$ is a graph of maximum degree $\Delta(G)$ and $C$ is a coalition of size
more than $(s_{1}+1)\cdot\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}$, then for
every $i\in C$ we have $\operatorname{u}(i,C)<0$.
###### Proof.
Let $i\in C$. There are at most $\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}$
agents in distance at most ${\delta}$ from $i$. Each of these agents
contributes at most $s_{1}$ to $\operatorname{u}(i,C)$. Every other agent
contributes at most $-1$. Hence, if there are more than
$(s_{1}+1)\cdot\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}$ agents in $C$, then
more than $s_{1}\cdot\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}$ of them have a
negative contribution to $\operatorname{u}(i,C)$ and
$\operatorname{u}(i,C)<s_{1}\cdot\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}-1\cdot
s_{1}\cdot\Delta(G)\cdot(\Delta(G)-1)^{{\delta}-1}=0.\qed$
###### Lemma 5.
Let $\vec{\operatorname{s}}=(s_{1},\ldots,s_{\delta})$ be such that $s_{2}<0$.
If $G$ is a graph of treewidth $\operatorname{tw}$ and $C$ is a coalition of
size more than $2(s_{1}+1)\cdot\operatorname{tw}+1$, then $\sum_{i\in
C}\operatorname{u}(i,C)<0$.
###### Proof.
Each agent adjacent to $i$ contributes $s_{1}$ to $\operatorname{u}(i,C)$,
whereas all the other agents contribute at most $-1$. Since a graph of
treewidth $\operatorname{tw}$ is $\operatorname{tw}$-degenerate, there are
$|E(G[C])|\leq|C|\cdot\operatorname{tw}$ pairs of adjacent agents and
$\binom{|C|}{2}-|E(G[C])|$ pairs of non-adjacent agents. We have
$\displaystyle\sum_{i\in C}\operatorname{u}(i,C)$ $\displaystyle=\sum_{i,j\in
C;i\neq j}\vec{\operatorname{s}}\left(\operatorname{dist}(i,j)\right)$
$\displaystyle\leq
2\left(s_{1}\cdot\left|E\left(G[C]\right)\right|-\left(\binom{|C|}{2}-\left|E\left(G[C]\right)\right|\right)\right)$
$\displaystyle=2\left((s_{1}+1)\cdot\left|E\left(G[C]\right)\right|-\binom{|C|}{2}\right)$
$\displaystyle\leq 2(s_{1}+1)\cdot|C|\cdot\operatorname{tw}-|C|(|C|-1)$
$\displaystyle=|C|\left(2(s_{1}+1)\cdot\operatorname{tw}-(|C|-1)\right)$
$\displaystyle<|C|\left(2(s_{1}+1)\cdot\operatorname{tw}-\left(2(s_{1}+1)\cdot\operatorname{tw}+1-1\right)\right)=0.\qed$
## 4 Computing Optimal Outcomes
### 4.1 Intractability
As our first step towards an understanding of the complexity of computing a
welfare-optimal outcome in an SDG, we establish the NP-hardness of
$\vec{\operatorname{s}}$-SDG-WF, $\vec{\operatorname{s}}$-SDG-WF-IR and
$\vec{\operatorname{s}}$-SDG-WF-Nash even for a very simple choice of
$\vec{\operatorname{s}}$.
###### Theorem 6.
Let $\vec{\operatorname{s}}=(s_{1})$ for any $s_{1}>0$. Then
$\vec{\operatorname{s}}$-SDG-WF, $\vec{\operatorname{s}}$-SDG-WF-IR and
$\vec{\operatorname{s}}$-SDG-WF-Nash are NP-hard.
###### Proof Sketch.
As our first step, we prove the NP-hardness of the intermediate problem called
3-Coloring Triangle Covered Graph (3CTCG) via an adaptation of a known
reduction from NotAllEqual-3-SAT [38, Theorem 9.8]:
3-Coloring Triangle Covered Graph (3CTCG)
Input: An undirected graph $G=(V,E)$ with $|V|=3n$ vertices such that $G$
contains a collection of $n$ mutually vertex disjoint triangles. Question:
Does $G$ have a 3-coloring?
Next, we reduce 3CTCG to our three problems via a single construction. Let $G$
be an instance of 3CTCG with $3n$ vertices and $T_{1},\ldots,T_{n}$ the
corresponding collection of triangles. Let $\overline{G}$ be a complement of
$G$, let $s_{1}=s_{1}(\vec{\operatorname{s}})$ and let $b=3ns_{1}\cdot(n-1)$.
To establish the NP-hardness of $\vec{\operatorname{s}}$-SDG-WF, it suffices
to show that $G$ is a Yes-instance of 3CTCG if and only if $\overline{G}$
admits an outcome with social welfare at least $b$; for the remaining two
problems, we additionally show that such an outcome will furthermore be
individually rational and Nash stable. ∎
### 4.2 An Algorithm for Tree-Like Networks
We complement Theorem 6 by establishing that all three problems under
consideration can be solved in polynomial time on networks of bounded
treewidth—in other words, we show that they are XP-tractable w.r.t. treewidth.
We first describe the “baseline” algorithm for solving
$\vec{\operatorname{s}}$-SDG-WF, and then prove that this may be adapted to
also solve the other two problems by expanding on its records and procedures
(see the appendix).
###### Theorem 7.
For every fixed scoring vector $\vec{\operatorname{s}}$, the
$\vec{\operatorname{s}}$-SDG-WF, $\vec{\operatorname{s}}$-SDG-WF-IR, and
$\vec{\operatorname{s}}$-SDG-WF-Nash problems are in XP when parameterized by
the treewidth of the social network $G$.
###### Proof Sketch.
Our algorithm is based on leaf-to-root dynamic programming along a nice tree-
decomposition of the input social network with rather complicated structure.
In each node $x$ of the tree-decomposition, we store a set $\mathcal{R}_{x}$
of partial solutions called _records_. Each record realizes a single
_signature_ which is a triple $(C,S,T)$, where
* •
$C$ is a partition of bag agents into parts of coalitions; there are at most
$\operatorname{tw}+1$ different coalitions intersecting $\beta(x)$ and, thus,
at most ${tw^{\mathcal{O}(\operatorname{tw})}}$ possible partitions of
$\beta(x)$.
* •
$S$ is a function assigning each pair of agents that are part of the same
coalition according to $C$ the shortest intra-coalitional path; recall that
for fixed $\vec{\operatorname{s}}$, the diameter of every coalition is bounded
by a constant ${\delta}$ and, therefore, there are
${n^{\mathcal{O}({\delta})}=n^{\mathcal{O}(1)}}$ possible paths for each pair
of agents which gives us ${n^{\mathcal{O}(\operatorname{tw}^{2})}}$
combinations in total.
* •
$T$ is a table storing for every coalition $P$ and every possible vector of
distances to bag agents that are in $P$ the number of agents from $P$ that
were already forgotten in some node of the tree-decomposition; the number of
possible coalitions is at most $\operatorname{tw}+1$, the number of potential
distance vectors is
${\delta}^{\operatorname{tw}+1}=2^{\mathcal{O}(\operatorname{tw})}$, and there
are at most $n$ values for every combination of coalition and distance vector
which leads to at most ${n^{2^{\mathcal{O}(\operatorname{tw})}}}$ different
tables $T$.
The value of every record is a pair $(\pi,w)$, where $\pi$ is a partition of
$V^{x}$ such that $\operatorname{SW}(\pi)=w$ and $\pi$ witnesses that there is
a partition of $V^{x}$ corresponding to the signature of the record, as
described above. We store only one record for every signature – the one with
the highest social welfare. Therefore, in every node $x$, there are at most
$n^{2^{\mathcal{O}(\operatorname{tw})}}$ different records.
Once the computation ends, we check the record in the root node $r$ and based
on the value of $w$, we return the answer; Yes if $w\geq b$ and No otherwise.
Moreover, as $G^{r}=G$, the partition $\pi$ is also an outcome admitting
social-welfare $w$. ∎
### 4.3 Fixed-Parameter Tractability
A natural follow-up question to Theorem 7 is whether one can improve these
results to fixed-parameter algorithms. As our final contribution, we show that
this is possible at least when dealing with simple scoring vectors, or on
networks with stronger structural restrictions. To obtain both of these
results, we first show that to obtain fixed-parameter tractability it suffices
to have a bound on the size of the largest coalition in a solution (i.e., a
welfare-optimal outcome).
###### Theorem 8.
For every fixed scoring vector $\vec{\operatorname{s}}$, the variants of
$\vec{\operatorname{s}}$-SDG-WF, $\vec{\operatorname{s}}$-SDG-WF-IR,
$\vec{\operatorname{s}}$-SDG-WF-Nash where we only consider outcomes
consisting of coalitions of at most a prescribed size are FPT parameterized by
the treewidth of the network and the maximum coalition size combined.
###### Proof Sketch.
Similar to the previous ones, we design a dynamic programming (DP) on a nice
tree decomposition, albeit the procedure and records are completely different.
Given a subset of agents $X\subseteq N$, let
$\Pi=(\pi_{1},\pi_{2},\dots,\pi_{\ell})$ be a partition of a set containing
$X$ and some “anonymous” agents. We use _$\mathsf{T}(\Pi)$_ to denote a set of
graph topologies on $\pi_{1},\pi_{2},\dots,\pi_{\ell}$ given $X$. That is,
$\mathsf{T}(\Pi)=\\{\mathsf{T}(\pi_{1}),\dots,\mathsf{T}(\pi_{\ell})\\}$ where
$\mathsf{T}(\pi_{i})$ is some graph on $|\pi_{i}|$ agents, namely $\pi_{i}\cap
X$ and $|\pi_{i}\setminus X|$ “anonymous” agents, for each $i\in[\ell]$. The
maximum coalition size of any welfare maximizing partition is denoted by
$\operatorname{sz}$. Table, M, contains an entry M$[x,C,\mathsf{T}(\Pi)]$ for
every node $x$ of the tree decomposition, each partition $C$ of $\beta(x)$,
and each set of graph topologies $\mathsf{T}(\Pi)$ given $\beta(x)$ where
$\Pi$ is a partition of at most $\operatorname{sz}\cdot\operatorname{tw}$
agents. An entry of M stores the maximum welfare in $G^{x}$ under the
condition that the partition into coalitions satisfies the following
properties. Recall that for a partition $P$ of agents and an agent $a$, we use
$P_{a}$ to denote the coalition agent $a$ is part of in $P$.
1. 1.
_$C$ and $\Pi$ are consistent_, i.e., the partition of the bag agents
$\beta(x)$ in $G^{x}$ is denoted by $C$ and $C_{a}=\Pi_{a}\cap\beta(x)$ for
each agent $a\in\beta(x)$.
2. 2.
The coalition of agent $a\\!\in\\!\beta(x)$ in the graph $G^{x}$ is $\Pi_{a}$.
3. 3.
_$\mathsf{T}(\Pi)$ is consistent with $G^{x}$_ i.e., the subgraph of $G^{x}$
induced on the agents in coalition of $a$ is $\mathsf{T}(\Pi_{a})$, i.e.,
$G^{x}[\Pi_{a}]=\mathsf{T}(\Pi_{a})$.
Observe that we do not store $\Pi$. We only store the topology of $\Pi$ which
is a graph on at most $\operatorname{sz}\cdot\operatorname{tw}$ agents.
We say an entry of M$[x,C,\mathsf{T}(\Pi)]$ is _valid_ if it holds that
1. 1.
_$C$ and $\Pi$ are consistent_, i.e., $C_{a}=\Pi_{a}\cap\beta(x)$ for each
agent $a\in\beta(x)$,
2. 2.
Either $C_{a}=C_{b}$, or $C_{a}\cap C_{b}=\emptyset$ for each pair of agents
$a,b\in\beta(x)$,
3. 3.
_$\mathsf{T}(\Pi)$ is consistent with $G^{x}$ in $\beta(x)$_, i.e., for each
pair of agents $a,b\in\beta(x)$ such that $\Pi_{a}=\Pi_{b}$, there is an edge
$(a,b)\in\mathsf{T}(\Pi_{a})$ if and only if $(a,b)$ is an edge in $G^{x}$.
Once the table is computed correctly, the solution is given by the value
stored in M$[r,C,\mathsf{T}(\Pi)]$ where $C$ is empty partition and
$\mathsf{T}(\Pi)$ is empty. Roughly speaking, the basis corresponds to leaves
(whose bags are empty), and are initialized to store $0$. For each entry that
is not valid we store $-\infty$. To complete the proof, it now suffices to
describe the computation of the records at each of the three non-trivial types
of nodes in the decomposition and prove correctness. ∎
Similarly to Theorem 7, we design a dynamic programming on a nice tree
decomposition, albeit the procedure and records are completely different.
From Lemma 5 it follows that if $s_{2}<0$ and $\operatorname{tw}(G)$ is
bounded, then the maximum coalition size of a welfare maximizing outcome is
bounded. Hence, using Theorem 8 we get the following.
###### Corollary 9.
$\vec{\operatorname{s}}$-SDG-WF-Nash, $\vec{\operatorname{s}}$-SDG-WF-IR, and
$\vec{\operatorname{s}}$-SDG-WF are fixed-parameter tractable parameterized by
the treewidth $\operatorname{tw}(G)$ if $s_{2}<0$.
Turning back to general scoring vectors, we recall that Lemma 4 provided a
bound on the size of the coalitions in a welfare-optimal outcome in terms of
the maximum degree $\Delta(G)$ of the network $G$. Applying Theorem 8 again
yields:
###### Corollary 10.
$\vec{\operatorname{s}}$-SDG-WF-Nash, $\vec{\operatorname{s}}$-SDG-WF-IR, and
$\vec{\operatorname{s}}$-SDG-WF are fixed-parameter tractable parameterized by
the treewidth $\operatorname{tw}(G)$ and the maximum degree $\Delta(G)$ of the
social network.
As our final contribution, we provide fixed-parameter algorithms for computing
welfare-optimal outcomes that can also deal with networks containing high-
degree agents. To do so, we exploit a different structural parameter than the
treewidth—namely the vertex cover number of $G$ ($\operatorname{vc}(G)$). We
note that while the vertex cover number is a significantly more “restrictive”
graph parameter than treewidth, it has found numerous applications in the
design of efficient algorithms in coalition formation, including for other
types of coalition games [6, 9, 27].
###### Theorem 11.
$\vec{\operatorname{s}}$-SDG-WF-Nash, $\vec{\operatorname{s}}$-SDG-WF-IR, and
$\vec{\operatorname{s}}$-SDG-WF are fixed-parameter tractable parameterized by
the vertex cover number $\operatorname{vc}(G)$ of the social network.
###### Proof Sketch.
Let $k=\operatorname{vc}(G)$ and let $U$ be a vertex cover for $G$ of size
$k$. Observe that in each solution there are at most $k$ non-singleton
coalitions, since $G$ has a vertex cover of size $k$ and each coalition must
be connected. Furthermore, the vertices of $G-U$ can be partitioned into at
most $2^{k}$ groups according to their neighborhood in the set $U$. That is,
there are $n_{W}$ vertices in $G-U$ such that their neighborhood is $W$ for
some $W\subseteq U$; denote this set of vertices $I_{W}$.
We perform exhaustive branching to determine certain information about the
structure of the coalitions in a solution—notably:
1. 1.
which vertices of $U$ belong to each coalition (i.e., we partition the set
$U$); note that there are at most $k^{k}$ such partitions, and
2. 2.
if there is at least one agent of $I_{W}$ in the coalition or not ; note that
there are at most $(2^{2^{k}})^{k}$ such assignments of these sets to the
coalitions.
We branch over all possible admissible options of the coalitional structure
described above possessed by a hypothetical solution. The total number of
branches is upper-bounded by a function of the parameter value $k$ and thus
for the problems to be in FPT it suffices to show that for each branch we can
find a solution (if it exists) by a fixed-parameter subprocedure. To conclude
the proof, we show that a welfare-maximum outcome (which furthermore satisfies
the imposed stability constraints) with a given coalitional structure can be
computed by modeling this as an Integer Quadratic Program where
$d+\|A\|_{\infty}+\|Q\|_{\infty}$ are all upper-bounded by a function of
$k$—such a program can be solved in FPT time using Proposition 1.
The (integer) variables of the program are $x^{C}_{W}$, which express the
number of vertices from the set $I_{W}$ in the coalition with $C\subseteq U$;
thus, we have $x^{C}_{W}\in\mathbb{Z}$ and $x^{C}_{W}\geq 1$. Let
$\mathcal{C}$ be the considered partitioning of the vertex cover $U$. We use
$C\in\mathcal{C}$ for the set $C\subseteq U$ in the coalition and $C^{+}$ for
the set $C$ and the guessed groups having at least one agent in the coalition.
We require that the vertices of $G-U$ are also partitioned in the solution,
i.e.,
$\sum_{C\in\mathcal{C}}\sum_{W\in C^{+}}x^{C}_{W}=n_{W}\qquad\forall
W\subseteq U.$ (1)
The quadratic objective expresses the welfare of the coalitions in the
solution while the linear constraints ensure the stability of the outcome; for
the latter, we rely on the fact that it is sufficient to verify the stability
for a single agent from the group $I_{W}$ in each coalition. ∎
## 5 Conclusions and Future Research Directions
In this work, we studied social distance games through the lens of an
adaptable, non-normalized scoring vector which can capture the positive as
well as negative dynamics of social interactions within coalitions. The main
focus of this work was on welfare maximization, possibly in combination with
individual-based stability notions—individual rationality and Nash stability.
It is not surprising that these problems are intractable for general networks;
we complement our model with algorithms that work well in tree-like
environments.
Our work opens up a number of avenues for future research. One can consider
other notions of individual-based stability such as individual stability [14,
pp. 360–361][24], or various notions of group-based stability such as core
stability [14, p. 360][15, 35]. Furthermore, our results do not settle the
complexity of finding stable solutions (without simultaneous welfare
maximization). Therefore, it remains open if one can find a Nash stable
solution for a specific scoring vector. Also, a more complex open problem is
to characterize those scoring vectors that guarantee the existence of a Nash
(or individually) stable solution.
Finally, we remark that the proposed score-based SDG model can be generalized
further, e.g., by allowing for a broader definition of the scoring vectors.
For instance, it is easy to generalize all our algorithms to scoring vectors
which are not monotone in their “positive part”. One could also consider
situations where the presence of an agent that is “far away” does not
immediately set the utility of other agents in the coalition to $-\infty$. One
way to model these settings would be to consider “ _open_ ” scoring vectors,
for which we set $\vec{\operatorname{s}}(a)=\vec{\operatorname{s}}({\delta})$
for all $a>{\delta}$—meaning that distances over ${\delta}$ are all treated
uniformly but not necessarily as unacceptable.
Notice that if $\vec{\operatorname{s}}({\delta})\geq 0$ for an open scoring
vector $\vec{\operatorname{s}}$, the grand coalition is always a social-
welfare maximizing outcome for all three problems—hence here it is natural to
focus on choices of $\vec{\operatorname{s}}$ with at least one negative entry.
We note that all of our fixed-parameter algorithms immediately carry over to
this setting for arbitrary choices of open scoring vectors
$\vec{\operatorname{s}}$. The situation becomes more interesting when
considering the small-world property: while the diameter of every welfare-
maximizing outcome can be bounded in the case of Nash stable or individually
rational coalitions (as we prove in our final Theorem 12 below), whether the
same holds in the case of merely trying to maximize social welfare is open and
seems to be a non-trivial question. Because of this, Theorem 7 can also be
extended to the $\vec{\operatorname{s}}$-SDG-WF-IR and
$\vec{\operatorname{s}}$-SDG-WF-Nash with open scoring vectors, but it is non-
obvious for $\vec{\operatorname{s}}$-SDG-WF.
###### Theorem 12.
Let $\vec{\operatorname{s}}=(s_{1},\dots,s_{{\delta}})$ be an arbitrary open
scoring vector and $G$ be a social network. Every outcome $\Pi$ containing a
coalition $C\in\Pi$ with diameter exceeding $\ell=2\cdot s_{1}\cdot{\delta}$
can be neither Nash-stable nor individually rational.
###### Proof Sketch.
Consider a shortest path $P$ in $C$ whose length exceeds $\ell$. We identify a
set of edge cuts along $P$ and show that at least one such cut must be near an
agent whose utility in $C$ is negative, due to the presence of a large number
of agents that must be distant from the chosen edge cut. ∎
#### Acknowledgements.
All authors are grateful for support from the OeAD bilateral Czech-Austrian
WTZ-funding Programme (Projects No. CZ 05/2021 and 8J21AT021). Robert Ganian
acknowledges support from the Austrian Science Foundation (FWF, project
Y1329). Thekla Hamm also acknowledges support from FWF, project J4651-N. Dušan
Knop, Šimon Schierreich, and Ondřej Suchý acknowledge the support of the Czech
Science Foundation Grant No. 22-19557S. Šimon Schierreich was additionally
supported by the Grant Agency of the Czech Technical University in Prague,
grant No. SGS23/205/OHK3/3T/18.
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# Quasi-one-dimensional magnetism in the spin-$\frac{1}{2}$ antiferromagnet
BaNa2Cu(VO4)2
Sebin J. Sebastian K. Somesh School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India M. Nandi Ames
Laboratory and Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA N. Ahmed P. Bag School of Physics, Indian Institute
of Science Education and Research Thiruvananthapuram-695551, India M. Baenitz
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany B. Koo Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany J. Sichelschmidt Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187
Dresden, Germany A. A. Tsirlin<EMAIL_ADDRESS>Experimental Physics VI,
Center for Electronic Correlations and Magnetism, University of Augsburg,
86135 Augsburg, Germany Y. Furukawa Ames Laboratory and Department of
Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA R. Nath
<EMAIL_ADDRESS>School of Physics, Indian Institute of Science Education
and Research Thiruvananthapuram-695551, India
###### Abstract
We report synthesis and magnetic properties of quasi-one-dimensional
spin-$\frac{1}{2}$ Heisenberg antiferromagnetic chain compound BaNa2Cu(VO4)2.
This orthovanadate has a centrosymmetric crystal structure, $C2/c$, where the
magnetic Cu2+ ions form spin chains. These chains are arranged in layers, with
the chain direction changing by 62$\degree$ between the two successive layers.
Alternatively, the spin lattice can be viewed as anisotropic triangular layers
upon taking the inter-chain interactions into consideration. Despite this
potential structural complexity, temperature-dependent magnetic
susceptibility, heat capacity, ESR intensity, and NMR shift agree well with
the uniform spin-$1/2$ Heisenberg chain model with an intrachain coupling of
$J/k_{\rm B}\simeq 5.6$ K. The saturation field obtained from the magnetic
isotherm measurement consistently reproduces the value of $J/k_{\rm B}$.
Further, the 51V NMR spin-lattice relaxation rate mimics the 1D character in
the intermediate temperature range, whereas magnetic long-range order sets in
below $T_{\rm N}\simeq 0.25$ K. The effective interchain coupling is estimated
to be $J_{\perp}/k_{\rm B}\simeq 0.1$ K. The theoretical estimation of
exchange couplings using band-structure calculations reciprocate our
experimental findings and unambiguously establish the 1D character of the
compound. Finally, the spin lattice of BaNa2Cu(VO4)2 is compared with the
chemically similar but not isostructural compound BaAg2Cu(VO${}_{4})_{2}$.
## I Introduction
The studies of low-dimensional and frustrated spin systems have contributed
substantially in understanding the quantum phase transitions at low
temperatures S. Sachdev (2007); Ramirez (1994). In one-dimensional (1D)
antiferromagnetic (AFM) spin-$1/2$ uniform Heisenberg chains, magnetic long-
range-order (LRO) is forbidden at zero temperature as a result of enhanced
quantum fluctuations, thereby exhibiting a gapless excitation spectrum and
power-law decay of spin-spin correlations Mermin and Wagner (1966). However,
non-zero inter-chain interactions, inherent to real materials, lead to the
formation of magnetic LRO at finite temperatures Yasuda _et al._ (2005);
Schulz (1996). On the other hand, the inter-chain interactions often create
frustrated network between the chains that eventually prevents the system from
achieving the conventional LRO but stabilizes different exotic states instead
Ramirez (1994); Greedan, John E. (2001); Kojima _et al._ (1997); Lancaster
_et al._ (2006). Further, competing interactions as realized in a set of
compounds, add magnetic frustration in spin chains which along with quantum
fluctuations host a multitude of intriguing magnetic ground states Furukawa
_et al._ (2010); Hase _et al._ (1993); Drechsler _et al._ (2007). The
transition-metal oxides offer nearly endless opportunities for realizing 1D
spin chains with different types of exchange couplings, and may harbor wide
varieties of exotic phases of matter.
Figure 1: Left panel: crystal structure of BaNa2Cu(VO${}_{4})_{2}$ showing
corner-shared CuO4 plaquettes and VO4 tetrahedra forming layers of spin
chains. The coupling of Na1+ ions with the magnetic Cu2+ ions is also shown.
Middle panel: crystal structure of BaNa2Cu(VO${}_{4})_{2}$ shown in a
different orientation to visualize the spin chains running along the $[110]$
and $[1\bar{1}0]$ directions; black spheres show the Ba atoms, the Na atoms
are omitted for clarity. Right panel: the structure of the single spin chain
with the geometrical parameters $\varphi$ and $r$ that control the sign and
strength of superexchange through the double bridges of the VO4 tetrahedra.
Recently, synthesis and magnetic properties of a series of compounds
$AA^{\prime}M$(VO4)2 ($A=$ Ba and Sr, $A^{\prime}=$ Na2 and Ag2, and $M=$ Mn,
Ni, Co, Fe, and Cu) were reported. Despite some variations in their crystal
structures, the magnetic model of anisotropic triangular lattice has been
generally used to understand their magnetism Amuneke _et al._ (2011); Möller
_et al._ (2012); Nakayama _et al._ (2013); Reuß _et al._ (2018); Sanjeewa
_et al._ (2019); Amuneke _et al._ (2014); Lee _et al._ (2020).
BaAg2Cu(VO${}_{4})_{2}$ stands as an exception in this series, because its
crystal structure is triclinic (space group: $P\overline{1}$) Amuneke _et
al._ (2011), and indeed microscopic analysis via density-functional band-
structure calculations Tsirlin _et al._ (2012) combined with resonance
spectroscopy Krupskaya _et al._ (2017) revealed 1D magnetism with two
dissimilar types of spin chains, one ferromagnetic and one antiferromagnetic,
coexisting in the structure.
Here, we present for the first time the magnetic properties of
BaNa2Cu(VO${}_{4})_{2}$, another Cu2+ member of the series von Postel and
Müller-Buschbaum (1992). Its structure features four equal Cu–Cu distances of
5.507 Å as well as two slightly longer distances of 5.686 Å, all in the $ab$
plane. This interaction geometry is a pre-requisite of the triangular-lattice
scenario previously established for other members of the $AA^{\prime}M$(VO4)2
series. On the other hand, the square-planar oxygen coordination of Cu2+ and
the VO4 bridges between such CuO4 plaquette units may lead to one preferred
direction for magnetic couplings in the $ab$ plane (Fig. 1). Interestingly,
this preferred direction changes from $\mathbf{a}+\mathbf{b}$ in one plane to
$\mathbf{a}-\mathbf{b}$ in the adjacent plane, thus leading to the formation
of crossed spin chains arranged at $62^{\circ}$ relative to each other. This
geometry resembles the crossed-chain magnetic model, where exotic ground
states and potential spin-liquid behavior have been proposed theoretically
Starykh _et al._ (2002); Sindzingre _et al._ (2002); Brenig and Grzeschik
(2004); Starykh _et al._ (2005); Bishop _et al._ (2012).
Here, we use magnetization, heat capacity, electron spin resonance (ESR), and
nuclear magnetic resonance (NMR) measurements, as well as complementary band-
structure calculations to uncover magnetic interactions in
BaNa2Cu(VO${}_{4})_{2}$ and establish its microscopic magnetic model. Our data
suggest the formation of uniform AFM spin chains with the exchange coupling
$J/k_{\rm B}\simeq 5.6$ K, and the subsequent onset of magnetic LRO below
$T_{\rm N}\simeq 0.25$ K. We suggest that this magnetic order can be driven by
residual inter-chain couplings of $J_{\perp}/k_{\rm B}\simeq 0.1$ K that
remain non-frustrated despite the crossed-chain structural geometry. Our
results establish that the mere presence of spin chains arranged along two
different directions is insufficient to reach the interesting physics of the
crossed-chain model, and an additional condition for the lateral displacement
of these chains has to be met experimentally.
## II Methods
Figure 2: Powder XRD pattern of BaNa2Cu(VO4)2 measured at $T=300$ K. The
circles are experimental data and the solid black line is the Le-Bail fit. The
Bragg positions are indicated by green vertical lines and the bottom solid
blue line indicates the difference between the experimental and calculated
intensities.
Polycrystalline sample of BaNa2Cu(VO4)2 was prepared by the usual solid-state
reaction method. Initially, the reactants Na2CO3 (Aldrich, 99.995%), BaCO3
(Aldrich, 99.995%), CuO (Aldrich, 99.999%), and V2O5 (Aldrich, 99.995%) were
mixed in proper molar ratios, thoroughly ground, and then pressed into
pellets. The pellets were sintered in an alumina crucible at 500 0C for three
days in air with several intermediate grindings. The phase purity of the
sample was confirmed from the powder x-ray diffraction (XRD) performed at room
temperature. For the powder XRD experiment, a PANalytical powder
diffractometer with CuKα radiation ($\lambda_{\rm avg}\simeq 1.54182$ Å) was
used. Le-Bail analysis of the powder XRD pattern was performed using the
`FullProf` software package J. Rodríguez-Carvajal (1993). Figure 2 displays
the room-temperature powder XRD data along with the fit. The structural
parameters given in Ref. von Postel and Müller-Buschbaum (1992) were used as
the initial parameters. The goodness-of-fit was found to be $\chi^{2}\simeq
3.57$. The obtained lattice parameters are $a=9.4379(1)$ Å, $b=5.6926(1)$ Å,
$c=14.0519(1)$ Å, and $\beta=92.3434(8)^{\circ}$ and the unit cell volume
$V_{cell}\simeq 754.34$ Å3, which are in close agreement with the previous
report von Postel and Müller-Buschbaum (1992).
Magnetization ($M$) measurements were performed as a function of temperature
(0.48 K $\leq T\leq 380$ K) and magnetic field ($0\leq H\leq 14$ T) using a
Superconducting Quantum Interference Device (SQUID, Quantum Design)
magnetometer and a Physical Property Measurement System (PPMS, Quantum
Design). The SQUID enabled us to measure magnetization down to 0.48 K with a
3He attachment. High-field magnetization up to 14 T were measured using PPMS.
Heat capacity ($C_{\rm p}$) was measured as a function of $T$ (0.4 K $\leq
T\leq 200$ K) on a sintered pellet using the thermal relaxation method in
PPMS. The temperature down to 0.4 K was achieved using a 3He attachment to the
PPMS.
The ESR experiments were performed on the powder sample with a standard
continuous-wave spectrometer in the temperature range 2.5 K$\leq T\leq 300$ K.
As a function of external magnetic field $B$, the resonance shows up as an
absorbed power $P$ of a transversal magnetic microwave field ($\nu\simeq 9.4$
GHz, X-band). In order to improve the signal-to-noise ratio, a lock-in
technique was used by modulating the applied field, which yields the
derivative of power absorption $dP/dB$ as a function of $B$. By using the
resonance condition $g=\frac{h\nu}{\mu_{\rm B}H_{\rm res}}$, where $h$ is the
Planck’s constant, $\mu_{\rm B}$ is the Bohr magneton, $\nu$ is the resonance
frequency, and $H_{\rm res}$ is the corresponding resonance field, the
$g$-value was obtained.
The pulsed NMR measurements were performed on both 23Na (nuclear spin $I=3/2$
and gyromagnetic ratio $\gamma=11.26$ MHz/T) and 51V ($I=7/2$ and
$\gamma=11.19$ MHz/T) nuclei in the temperature range 0.044 K$\leq T\leq 200$
K. For measurements above 2 K a 4He cryostat (Oxford Instrument) with a field-
sweep superconducting magnet was used, while for measurements in the low-
temperature range (0.044 K$\leq T\leq 2$ K), a 3He/4He dilution refrigerator
(Kelvinox, Oxford Instruments) with a field sweep magnet was used. All the NMR
measurements were carried out in a radio frequency of 77 MHz. The NMR spectra
were measured as a function of temperature $T$ by sweeping the magnetic field
at a constant radio frequency of 77 MHz. The NMR shift was calculated for both
23Na and 51V nuclei as $K(T)$ = [$H_{\rm ref}-H(T)$]/$H(T)$, where $H$ is the
resonance field for 23Na and 51V and $H_{\rm ref}$ is the resonance field of
the non-magnetic reference sample. The spin-lattice relaxation rate $1/T_{1}$
was measured by the conventional single saturation pulse method.
Density-functional (DFT) band-structure calculations were performed in the
FPLO code Koepernik and Eschrig (1999) using the structural parameters from
Ref. von Postel and Müller-Buschbaum (1992) and local-density approximation
(LDA) for the exchange-correlation potential Perdew and Wang (1992). Exchange
parameters of the spin Hamiltonian
$\mathcal{H}=\sum_{\langle ij\rangle}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}$ (1)
with $S=\frac{1}{2}$ and the summation over atomic pairs $\langle ij\rangle$,
were extracted via two complementary procedures. First, band structure
obtained on the LDA level was mapped onto a tight-binding model for the half-
filled $d_{x^{2}-y^{2}}$ orbitals of Cu2+ as the magnetic ion. Squared hopping
parameters $t_{i}$ of this tight-binding model are proportional to AFM
contributions to the exchange, $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$, where
$U_{\rm eff}$ is the effective on-site Coulomb repulsion. Alternatively, full
exchange couplings $J_{i}$ comprising both FM and AFM contributions are
extracted by a mapping procedure Xiang _et al._ (2011) from total energies of
magnetically ordered states calculated on the DFT+$U$ level, with correlation
effects in the Cu $3d$ shell modeled by the on-site Coulomb repulsion
$U_{d}=6$ eV, Hund’s exchange $J_{d}=1$ eV, and around-mean-field flavor of
the double-counting correction Janson _et al._ (2011); Tsirlin _et al._
(2012). The $k$ mesh with up to 150 points in the symmetry-irreducible part of
the first Brillouin zone was used.
Field-dependent magnetization and magnetic specific heat of a uniform
spin-$\frac{1}{2}$ chain were obtained from quantum Monte-Carlo simulations
for $L=32$ finite lattices with periodic boundary conditions. The loop Todo
and Kato (2001) and dirloop_sse Alet _et al._ (2005) algorithms of the ALPS
simulation package Albuquerque _et al._ (2007) were used.
## III Results and Discussion
### III.1 Magnetization
Figure 3: $\chi$ of polycrystalline BaNa2Cu(VO4)2 sample as a function of
temperature in an applied field $\mu_{0}H=1$ T. The solid line is the fit
using Bonner-Fisher model [Eq. (2)] for uniform Heisenberg spin-$1/2$ chain.
Upper inset: inverse susceptibility $1/\chi$ vs $T$ and the solid line
represents the CW fit, as discussed in the text. Lower inset: the low-
temperature $\chi(T)$ measured in two different fields $\mu_{0}H=1$ T and 3 T.
Temperature-dependent magnetic susceptibility $\chi(T)(=M/H$) of the
polycrystalline Na2BaCu(VO4)2 sample measured in two different applied fields
$H=1$ T and 3 T is depicted in the Fig. 3. The most significant feature in the
$\chi(T)$ curve is the presence of a broad maximum at 3 K, signaling a
crossover to an AFM short-range ordered state, typical for low-dimensional
spin systems Bonner and Fisher (1964); Eggert _et al._ (1994). This broad
maximum is more pronounced in the 3 T data shown in the lower inset of Fig. 3.
No anomaly indicative of the potential LRO could be seen down to 0.48 K.
The preliminary analysis was done by fitting the $\chi(T)$ data using the
Curie-Weiss (CW) law, $\chi(T)=\chi_{0}+C/(T+\theta_{\rm CW}$), where
$\chi_{0}$ is the temperature-independent susceptibility, $C$ is the Curie
constant, and $\theta_{\rm CW}$ is the characteristic CW temperature. The fit
shown in the upper inset of Fig. 3 in the high-temperature regime ($T\geq 16$
K) yields the following parameters: $\chi_{0}\simeq 7.9288\times 10^{-5}$
cm3/mol, $C\simeq 0.445$ cm3K/mol, and $\theta_{\rm CW}\simeq 3$ K. In order
to estimate the Van-Vleck paramagnetic susceptibility ($\chi_{\rm VV}$), which
arises from the second-order contribution to free energy in the presence of
magnetic field, core diamagnetic susceptibility $\chi_{\rm core}$ of
Na2BaCu(VO4)2 was calculated to be $-1.57\times 10^{-4}$ cm3/mol by summing
the core diamagnetic susceptibilities of individual ions Na+, Ba2+, Cu2+, V5+,
and O2- P. W. Selwood (2013); Mendelsohn _et al._ (1970). Subsequently,
$\chi_{\rm VV}$ was obtained by subtracting $\chi_{\rm core}$ from $\chi_{0}$
to be $\sim 2.36\times 10^{-4}$ cm3/mol, which is close to the values reported
for other cuprates Motoyama _et al._ (1996); Nath _et al._ (2005); Ahmed
_et al._ (2015) and consistent with tetragonal crystal-field splitting at the
Cu2+ site with the square-planar oxygen coordination Takigawa _et al._
(1989).
From the Curie constant $C$, the effective moment is calculated using the
relation $\mu_{\rm eff}=\sqrt{3k_{\rm B}C/N_{\rm A}}$ to be $\simeq 1.88$
$\mu_{\rm B}$, where $k_{\rm B}$ is the Boltzmann constant, $\mu_{\rm B}$ is
the Bohr magneton, and $N_{\rm A}$ is the Avogadro’s number. For a
spin-$\frac{1}{2}$ system, the spin-only effective moment is expected to be
$\mu_{\rm eff}=g\sqrt{S(S+1)}\mu_{\rm B}\simeq 1.73$ $\mu_{\rm B}$, assuming
Landé $g$-factor $g=2$. However, our experimental value of $\mu_{\rm
eff}\simeq 1.88$ $\mu_{\rm B}$ corresponds to a $g$-factor of $g\simeq 2.17$,
which is consistent with the ESR experiments discussed later. The positive
value of $\theta_{\rm CW}$ suggests that the dominant exchange interactions
between the Cu2+ ions are AFM in nature.
In order to estimate the exchange coupling between the Cu2+ ions, we
decomposed $\chi(T)$ into three components,
$\chi(T)=\chi_{0}+\frac{C_{\rm imp}}{T}+\chi_{\rm spin}(T).$ (2)
Here, the second term is the Curie law, which accounts for the paramagnetic
contributions from impurity spins and/or defects, and $\chi_{\rm spin}(T)$ is
the intrinsic spin susceptibility. This last term can be chosen in different
forms depending on the underlying magnetic model. The best fit was achieved
with the spin-chain model, which is further supported by the specific-heat
data (Sec. III.3) and ab initio calculations (Sec. III.5).
The susceptibility of a spin-$\frac{1}{2}$ uniform Heisenberg AFM chain takes
the form
$\chi_{\rm
spin}=\frac{N_{A}\mu_{B}^{2}g^{2}}{k_{B}T}\frac{0.25+0.0775x+0.0752x^{2}}{1+0.993x+0.1721x^{2}+0.7578x^{3}},$
(3)
with $x=\lvert J\rvert/k_{\rm B}T$ Bonner and Fisher (1964). This is simply a
high-temperature series expansion (HTSE) valid in the regime $k_{\rm B}T/J\geq
0.5$. The solid line in Fig. 3 represents the best fit of the $\chi(T)$ data
above 4 K by Eq. (2). The following parameters were obtained: $\chi_{0}\simeq
1.44\times 10^{-4}$ cm3/mol, $C_{\rm imp}\simeq 0.0258$ cm3K/mol, $g\simeq
2.13$, and the dominant intra-chain AFM exchange coupling $J/k_{\rm B}\simeq
5.6$ K. From the value of $C_{\rm imp}$, the sample was found to contain $\sim
6$% spin-$\frac{1}{2}$ impurities/defects. At temperatures below 1 K, this
impurity contribution becomes dominant and causes the reduction in the
susceptibility with the applied field, even though $\chi_{\rm spin}(T)$ should
increase when the field is applied Klümper (1998).
Figure 4: Magnetization ($M$) vs $H$ measured at $T=2$ K. Inset: $dM/dH$ vs
$H$ to highlight the saturation field $H_{\rm s}$.
The magnetic isotherm at $T=2$ K upto 14 T is shown in Fig. 4. $M$ increases
almost linearly with $H$ but with a small curvature. It develops a tendency of
saturation above 9 T. A more accurate value of the saturation field $H_{\rm
s}\simeq 9$ T was found by drawing tangential at the curvature (see Fig. 4).
The field derivative of the $M$ vs $H$ plot also implies $H_{\rm s}\simeq 9$ T
(see the inset of Fig. 4). For a spin-$1/2$ Heisenberg AFM chain, the
saturation field is directly proportional to the intra-chain exchange coupling
as $H_{\rm s}=2J_{\rm 1D}(k_{B}/g\mu_{B})$ Lebernegg _et al._ (2011). Using
the value of $J/k_{\rm B}\simeq 5.6$ K, the saturation field is calculated to
be $H_{\rm s}\simeq 8.34$ T, which matches well with the experimental value,
confirming the dominant 1D character of the compound.
### III.2 ESR
Figure 5: (a) Integrated ESR intensity vs temperature and the solid line
represents the fit as described in the text. Inset: ESR spectrum at room
temperature measured at a microwave frequency of 9.4 GHz together with the
powder-averaged Lorentzian fit (solid line). (b) Temperature variation of the
$g$ values (both perpendicular and parallel components) obtained from the
Lorentzian fit. (c) Temperature-dependent ESR linewidth $\Delta B$ (both
perpendicular and parallel components).
ESR experiment was performed on the powder sample and the results are shown in
Fig. 5. The inset of Fig. 5(a) depicts a typical ESR powder spectrum at 300 K.
The uniaxial $g$ factor anisotropy was obtained by fitting the spectra using
the powder-averaged Lorentzian line. The fit of the spectral at room
temperature yields the anisotropic $g$-tensor components
$g_{\parallel}$$\simeq 2.315$ and $g_{\perp}$$\simeq 2.098$. From these
values, the average $g$-value was calculated as
$g=[(g_{\parallel}+2g_{\perp})/3]\simeq 2.17$ Abragam and Bleaney (2012). This
value is slightly larger ($\Delta g/g\simeq 0.085$) compared to the free
electron value ($g=2$), typical for Cu2+ based oxides Kochelaev _et al._
(1997); Nath _et al._ (2014). The integrated ESR intensity ($I_{\rm ESR}$)
obtained from the above fit is plotted as a function of temperature in Fig.
5(a). It shows similitude with the $\chi(T)$ behavior, which traces a broad
maximum at around $T^{\max}_{\rm ESR}$ $\simeq 3.7$ K. Indeed, the $I_{\rm
ESR}$ vs $\chi$ plot with temperature as an implicit parameter follows a
straight line down to $\sim 5$ K (not shown). The variation of $g$ with
respect to temperature is shown in Fig. 5(b). Both the components of $g$ were
found to be almost temperature-independent at high temperatures ($T\geq 20$
K). However, below 20 K a weak deviation from the room-temperature values is
observed.
In order to estimate the exchange coupling, $I_{\rm ESR}(T)$ was fitted by
$I_{\rm ESR}(T)=A+B\chi_{\rm spin}(T).$ (4)
Here, $A$ and $B$ are arbitrary constants, and $\chi_{\rm spin}$ is given by
Eq. (3). Our fit (see Fig. 5) in the high-temperature regime ($T\geq 5$ K)
produced $J/k_{\rm B}\simeq 5.55~{}K$. This value of $J/k_{\rm B}$ is close to
the one obtained from the $\chi(T)$ analysis. During the fit, the value of $g$
was kept constant to 2.17, as obtained above. We have also fitted the
$1/I_{\rm ESR}$ data in the high-temperature regime ($T\geq 10$ K) using the
relation $I_{\rm ESR}=M+N/(T+\theta_{\rm CW})$ where $M$ and $N$ are arbitrary
constants. As shown in the lower inset of Fig. 5(a), the fit returns
$\theta_{\rm CW}\simeq 3.9~{}K$, which is in good agreement with the value
obtained from the $\chi^{-1}(T)$ analysis.
The temperature-dependent ESR linewidth, or equivalently the half-width at
half maximum of the ESR absorption signal, is presented in Fig. 5(c). Both the
parallel ($\Delta B_{\parallel}$) and perpendicular ($\Delta B_{\perp}$)
components of the ESR line width follow the general trend, commonly observed
in most of the low-dimensional spin systems Ivanshin _et al._ (2003);
Sichelschmidt _et al._ (2002). The rapid increase/divergence below $\sim 25$
K indicates the growth of strong spin correlations at low temperatures as the
system approaches the magnetic LRO state.
### III.3 Heat Capacity
Figure 6: Upper panel: Heat capacity ($C_{\rm p}$) vs $T$ in zero applied
field. The solid line denotes the phonon contribution to the heat capacity
$C_{\rm ph}$ using the Debye-Einstein fit. The blue solid spheres indicate the
magnetic contribution to the heat capacity $C_{\rm mag}$. Inset: $C_{\rm p}$
vs $T$ in the whole measured temperature range along with the Debye-Einstein
fit. Lower panel: The left $y$-axis shows $C_{\rm mag}/T$ and the right
$y$-axis shows the magnetic entropy $S_{\rm mag}$ vs $T$. Inset: $C_{\rm
mag}/R$ vs $T$.
Temperature-dependent heat capacity $C_{\rm p}$ of the polycrystalline sample
is shown in the upper panel of Fig. 6. In magnetic insulators, the two major
contributions to $C_{\rm p}$ are from phonon and magnetic parts. At high
temperatures, $C_{\rm p}(T)$ is dominated by the phonon part, while at low
temperatures it is dominated by the magnetic part. Our experimental $C_{\rm
p}$ data exhibit a pronounced broad maximum at $T\simeq 1.52$ K indicative of
the low-dimensional short-range order and also reflects the dominant magnetic
contribution at low temperatures. In order to estimate the magnetic
contribution to the heat capacity $C_{\rm mag}$, we proceed as follows. First
we approximate the lattice contribution $C_{\rm ph}$ by fitting the high-
temperature data by a linear combination of one Debye and two Einstein terms
(Debye-Einstein model) as Kittel (c2005); Caslin _et al._ (2014)
$C_{\rm ph}(T)=f_{\rm D}\,C_{\rm D}(\theta_{\rm
D},T)+\sum_{i=1}^{2}g_{i}\,C_{{\rm E}_{i}}(\theta_{{\rm E}_{i}},T).$ (5)
The first term in Eq. (5) is the Debye term,
$C_{\rm D}(\theta_{\rm D},T)=9nR\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\frac{\theta_{\rm
D}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$ (6)
Here, $x=\frac{\hbar\omega}{k_{\rm B}T}$, $\omega$ is the vibration frequency,
$R$ is the universal gas constant, and $\theta_{\rm D}$ is the characteristic
Debye temperature. The second term in Eq. (5) is a combination of the Einstein
terms that are usually responsible for flat optical modes in the phonon
spectrum,
$C_{\rm E}(\theta_{\rm E},T)=3nR\left(\frac{\theta_{\rm
E}}{T}\right)^{2}\frac{e^{\,\theta_{\rm E}/T}}{\left(e^{\,\theta_{\rm
E}/T}-1\right)^{2}}.$ (7)
Here, $\theta_{\rm E}$ is the characteristic Einstein temperature. The
coefficients $f_{\rm D}$, $g_{1}$, and $g_{2}$ are the weight factors, which
take into account the number of atoms per formula unit ($n$) and are
conditioned such that at high temperatures the Dulong-Petit value of $3nR$ is
satisfied. The $C_{\rm p}(T)$ data above $\sim 15$ K were fitted by Eq. (5)
and the obtained parameters are $f_{\rm D}\simeq 0.34$, $g_{1}\simeq 0.35$,
and $g_{2}\simeq 0.31$, $\theta_{\rm D}\simeq 214$ K, $\theta_{{\rm
E}_{1}}\simeq 356$ K, and $\theta_{{\rm E}_{2}}\simeq 897$ K. Further Einstein
terms beyond $\theta_{{\rm E}_{2}}$ rendered the fit unstable. The fit itself
is phenomenological in nature, although one may tentatively associate
$\theta_{\rm D}$ with low-energy vibrations of heavier atoms (Ba, Cu, and V)
that constitute 28.5 %, about $\frac{1}{3}$ of the atomic species in
BaNa2Cu(VO4)2. The lower Einstein temperature $\theta_{{\rm E}_{1}}$ may
correspond to Na atoms and two apical oxygens of the VO4 tetrahedra
(altogether 6 atoms per formula unit), whereas $\theta_{{\rm E}_{2}}$ reflects
higher-energy vibrations of the remaining four oxygens that are bound to V and
Cu at the same time.
The high-$T$ fit was extrapolated down to low temperatures and $C_{\rm
mag}(T)$ was estimated by subtracting $C_{\rm{ph}}(T)$ from $C_{\rm p}(T)$
[see Fig. 6 (upper panel)]. $C_{\rm mag}(T)/T$ was plotted as a function of
temperature in the lower panel of Fig. 6. The broad maximum corresponding to
the short-range order is apparent at $T\simeq 1.52$ K. At low temperatures,
$C_{\rm mag}(T)/T$ shows a rapid increase, which could be related to the onset
of magnetic LRO below 0.4 K. The magnetic entropy was calculated as
$S_{\rm{mag}}(T)=\int_{\rm
2\,K}^{T}\frac{C_{\rm{mag}}(T^{\prime})}{T^{\prime}}dT^{\prime}$, which yields
$S_{\rm mag}\simeq 5.83$ J/mol K at 20 K (see the lower panel of Fig. 6). This
value is close to the expected magnetic entropy for spin-$\frac{1}{2}$:
$S_{\rm mag}=R\ln 2=5.76$ J/mol K.
In the inset of the lower panel of Fig. 6, $C_{\rm mag}/R$ is plotted against
$T$. The peak of $C_{\rm mag}/R$ can be used to discriminate between different
microscopic scenarios. Its height depends on the nature of the underlying spin
lattice Bernu and Misguich (2001). Our experimental peak value of $C_{\rm
mag}/R\simeq 0.323$ fits well to the aforementioned 1D scenario, suggesting
that VO4 bridges choose the direction of spin chains. Alternatively, four
shortest Cu–Cu contacts of 5.507 Å could cause interactions of equal strength
and form a 2D square-lattice interaction topology that should manifest itself
by a much higher peak with $C_{\rm mag}/R\simeq 0.47$. On the other hand, the
triangular-lattice scenario would reduce the peak value to $C_{\rm
mag}/R\simeq 0.22$, lower than seen experimentally. We thus conclude that our
specific-heat data favor the spin-chain scenario for BaNa2Cu(VO${}_{4})_{2}$.
### III.4 23Na and 51V NMR
NMR is a potent tool to study the static and dynamic properties of spin
systems. In Na2BaCu(VO4)2, the 23Na and 51V nuclei are hyperfine-coupled to
the magnetic Cu2+ ions along the spin chains. Therefore, the low-lying
excitations of Cu2+ spins can be probed by means of 23Na and 51V NMR
measurements.
Figure 7: Field-sweep NMR spectra of the polycrystalline Na2BaCu(VO4)2
sample, measured at 77 MHz as a function of temperature. The spectral lines
corresponding to 23Na and 51V nuclei for $T=80$ K are marked by arrows. The
solid line is the simulated spectrum.
The quadrupole nuclei 23Na ($I=3/2$) and 51V ($I=7/2$) are in a non-cubic
symmetry that may produce an asymmetric charge distribution and hence electric
field gradient (EFG). Therefore, the four-fold and eight-fold degeneracies of
the $I=3/2$ and $I=7/2$ spins, respectively, are lifted partially due to the
interaction between the nuclear quadrupole moment ($Q$) and the surrounding
EFG. In this case, the nuclear spin Hamiltonian is a sum of the Zeeman and
quadrupolar interaction terms Curro (2009); Slichter (1992),
$\mathcal{H}=-\gamma\hbar{\hat{I}}H(1+K)+\frac{h\nu_{Q}}{6}[(3\hat{I}_{z}^{2}-\hat{I}^{2})+\eta(\hat{I}_{x}^{2}-\hat{I}_{y}^{2})].$
(8)
Here, the nuclear quadrupole resonance (NQR) frequency is defined as $\nu_{\rm
Q}=\frac{3e^{2}qQ}{2I(2I-1)h}$, $e$ is the electron charge,
$\hbar~{}(=h/2\pi)$ is the Planck’s constant, $H$ is the applied field along
$\hat{z}$, $K$ is the magnetic shift due to hyperfine field at the nuclear
site, $V_{\alpha\beta}$ are the components of the EFG tensor, $eq=V_{zz}$ is
the largest eigenvalue or principal component of the EFG, and
$\eta=|V_{xx}-V_{yy}|/V_{zz}$ is the EFG asymmetry (here, the principal axes
of EFG are chosen such that $|V_{zz}|\geq|V_{yy}|\geq|V_{xx}|$.).
Experimentally, the transitions can be observed at the frequency
$\nu_{z}=\nu_{Q}\sqrt{1+\eta^{2}/3}$.
The principal axes $\\{x,y,z\\}$ of the EFG tensor are defined by the local
symmetry of the crystal structure. Consequently, the corresponding resonance
frequency to any nuclear transition will have strong dependence on the
direction of the applied field with respect to the crystallographic axes. For
a site with axial symmetry ($\eta=0$), there will be $2I-1$ quadrupolar
resonances at frequencies $n\nu_{\rm Q}$, where $n=$ 1, ….$2I-1$. When
$\eta>0$, the resonances are not equally spaced. The EFG is fully
characterized by the parameters $\nu_{\rm z}$, $\eta$, and $\hat{z}$, where
$\hat{z}$ is the unit vector in the direction of the principal axis of the EFG
with the largest eigenvalue. When the Zeeman term dominates over the
quadrupole term, first-order perturbation theory is enough for describing the
system. In such a scenario, for a quadrupole nucleus, equally spaced satellite
peaks should appear on either side of the central peak separated by $\nu_{Q}$
Lang _et al._ (2005).
The NMR spectra as a function of temperature measured by sweeping the magnetic
field at 77 MHz are presented in Fig. 7. Since 23Na and 51V nuclei have nearly
the same $\gamma$ values, one expects their spectral lines to appear very
close to each other. Further, 23Na and 51V are quadrupolar nuclei with nuclear
spins $I=3/2$ and $7/2$, respectively and the transitions with $\Delta m=\pm
1$ are expected between the energy levels. Therefore, one would anticipate
three NMR lines for 23Na: one central line corresponding to
$I_{z}=+1/2\longleftrightarrow-1/2$ and two equally spaced satellite lines
corresponding to $I_{z}=\pm 3/2\longleftrightarrow\pm 1/2$ and seven NMR lines
for 51V: the central line being $I_{z}=+1/2\longleftrightarrow-1/2$ and the
satellite lines $I_{z}=\pm 1/2\longleftrightarrow\pm 3/2\longleftrightarrow\pm
5/2\longleftrightarrow\pm 7/2$. Indeed, at high temperatures, we observed two
sharp and prominent peaks at the resonance field position and two satellite
peaks on either side of those. The central peak towards the low-field side is
identified to be the signal coming from the 23Na nuclei, while the one towards
the high-field side appears to be the 51V peak. In addition to the central
peaks, two satellite peaks correspond to the 23Na line. At high temperatures,
the NMR spectra are found to be narrow and one can distinguish the 23Na and
51V signals. As the temperature is lowered, the line broadens asymmetrically
and the central lines shift weakly with temperature. No abrupt line broadening
was noticed down to 44 mK, which may signal the absence of magnetic LRO
Ranjith, K. M. and Nath, R. and Majumder, M. and Kasinathan, D. and Skoulatos,
M. and Keller, L. and Skourski, Y. and Baenitz, M. and Tsirlin, A. A. (2016).
Figure 8: Upper panel: NMR spectra at $T=125$ K showing the 23Na and 51V
central lines, with the downward arrows pointing to the 23Na satellites. The
solid line is the simulation of the spectra assuming the superposition of the
23Na and 51V signals. Lower panel: temperature-dependent NMR shift $K$ as a
function of temperature for 23Na and 51V, measured at 77 MHz. Solid line is
the fit using Eq. (9). Inset: NMR shift vs $\chi$ measured at 3 T. Solid lines
are the linear fits.
The spectra were fitted assuming the superposition of 23Na and 51V signals.
The spectral fit at $T=125$ K is presented in the upper panel of Fig. 8, where
23Na and 51V lines and their satellites are marked by arrows. The obtained
fitting parameters are $K\simeq 0.0345$% (isotropic shift), $\eta=0$
(asymmetry parameter), and $\nu_{Q}\simeq 0.92$ MHz (NQR frequency) for 23Na
and $K\simeq 0.627$%, $\eta=0$, and $\nu_{Q}\simeq 0.234$ MHz for 51V. The
quadrupole frequency is found to be almost constant with temperature down to
1.5 K, which essentially excludes the possibility of any structural distortion
in the studied compound.
The NMR shift $K(T)$ for both 23Na and 51V lines obtained from the spectral
fits is plotted in the lower panel of Fig. 8. The temperature-dependent 23Na
shift [${}^{\rm 23}K(T)$] is found to have a broad maximum at around 3 K,
similar to the $\chi(T)$ data. As $K(T)$ in an intrinsic measure of the spin
susceptibility $\chi_{\rm spin}$, one can write the linear relation
$K(T)=K_{0}+\frac{A_{\rm hf}}{N_{A}\mu_{B}}\chi_{\rm spin},$ (9)
where $K_{0}$ is the temperature-independent chemical shift and the
proportionality constant $A_{\rm hf}$ is the hyperfine coupling between the
probed nuclei and the electron spins.
From Eq. (9), $A_{\rm hf}$ can be calculated by taking the slope of the linear
$K$ vs $\chi$ plot (inset of Fig. 8) with temperature as an implicit
parameter. In the case of 23Na, the data for $T\geq 5$ K were fitted well by a
linear function, and the slope of the fit yields ${}^{23}A_{\rm hf}\simeq
0.021$ T/$\mu_{\rm B}$. Similarly, for 51V the linearity is found over a large
temperature range down to 10 K, and the linear fit returns ${}^{51}A_{\rm
hf}\simeq-0.016$ T/$\mu_{\rm B}$. To estimate the exchange coupling,
${}^{23}K(T)$ above 2.5 K was fitted by Eq. (9) taking $\chi_{\rm spin}$ for
the 1D $S=1/2$ Heisenberg chain [Eq. (3)]. The fit returns $J/k_{\rm B}\simeq
4.22$ K and ${}^{23}A_{\rm hf}\simeq 0.0194$ T/$\mu_{\rm B}$. The value of $g$
was fixed to $g=2.17$ during the fitting procedure. This value of $J/k_{\rm
B}$ is close to the one obtained from the $\chi(T)$ analysis, whereas
${}^{23}A_{\rm hf}$ is also in good agreement with the value obtained from the
$K$ vs $\chi$ analysis. An anomaly at $\sim 0.3$ K in ${}^{\rm 23}K(T)$ could
be due to a magnetic transition.
Figure 9: $1/T_{1}$ as a function of temperature measured on the 51V nuclei
down to 0.044 K. Inset: $1/T_{1}$ above 2 K is shown in order to highlight the
features around 10 K.
To study the spin dynamics, spin-lattice relaxation rate ($1/T_{1}$) was
measured by irradiating the central position of the 51V spectra corresponding
to the $1/2\longleftrightarrow-1/2$ transition, choosing an appropriate pulse
width. The recovery of the longitudinal magnetization was fitted by the
following exponential function relevant for a quadrupole ($I=7/2$) nuclei M.
I. Gordon and M. J. R. Hoch (1978); Simmons _et al._ (1962)
$\displaystyle 1-\frac{M(t)}{M(\infty)}$ $\displaystyle=$ $\displaystyle
0.0119\times e^{(-t/T_{1})}+0.068\times e^{(-6t/T_{1})}$ (10) $\displaystyle+$
$\displaystyle 0.21\times e^{(-15t/T_{1})}+0.71\times e^{(-28t/T_{1})}$
Here, $M(t)$ and $M(\infty)$ are the nuclear magnetizations at a time $t$ and
$t\longrightarrow\infty$, respectively, after the saturation pulse.
Temperature dependence of 51V $1/T_{1}$ obtained from the above fit is shown
in Fig. 9. Our measurements were carried out down to 0.04 K. At high
temperatures, $1/T_{1}$ is almost temperature-independent as expected in the
paramagnetic regime Moriya (1956). At low temperatures, it exhibits a sharp
peak at $T\simeq 0.25$ K due to slowing down of the fluctuating moments and is
a direct evidence of the onset of magnetic LRO. In order to highlight the
behavior in the intermediate temperature range, $1/T_{1}$ above 2 K is
magnified in the inset of Fig. 9. As the temperature is lowered, $1/T_{1}$
decreases linearly below about 25 K, remains almost temperature-independent
for 4 K$\leq T\leq 10$ K, and then starts increasing for $T\leq 4$ K. This
increase below 4 K can be attributed to the growth of AFM correlations as the
system approaches the magnetic LRO state.
Further, $1/T_{1}T$ is directly proportional to the imaginary part of the
dynamic susceptibility $\chi_{M}(\vec{q},\omega_{0})$ at the nuclear Larmor
frequency $\omega_{0}$, which is $q$-dependent Moriya (1956). In low-
dimensional spin systems, temperature-dependent $1/T_{1}$ often reflects
dominant contributions from different $q$ values in different temperature
regimes. For instance, for spin-$1/2$ Heisenberg AFM spin chains, it is
theoretically predicted that with the dominant staggered contribution
($q=\pm\pi/a$) the spin-lattice relaxation rate behaves as $1/T_{1}\sim
T^{0}$, while the dominant contribution of the uniform component ($q=0$)
results in $1/T_{1}\sim T$ Sandvik (1995); Sachdev (1994). The dominant
contributions of $q=\pm\pi/a$ and $q=0$ are typically observed in the low-
temperature ($T<J$) and high-temperature ($T\sim J$) regimes, respectively
Nath _et al._ (2005, 2008). Thus, our experimentally observed constant and
linear behaviors of $1/T_{1}$ with temperature over 4 K $\leq T\leq 10$ K and
10 K$\leq T\leq 25$ K, respectively (inset of Fig. 9), are compatible with the
1D physics.
In real spin-chain systems, the non-vanishing interchain couplings often leads
to the onset of magnetic LRO at very low temperatures. The interchain coupling
can be calculated using the expression proposed by Schulz Schulz (1996)
$|J_{\perp}|\simeq\frac{T_{\rm N}}{1.28\sqrt{\ln(5.8J/T_{\rm N})}},$ (11)
where $J_{\perp}$ is an effective interchain coupling. Taking $T_{\rm N}\simeq
0.25$ K and $J/k_{\rm B}\simeq 5.6$ K, we arrive at the possible value of
$J_{\perp}/k_{\rm B}\simeq 0.1$ K, which is indeed consistent with the value
estimated from the band-structure calculations, as discussed in the following.
### III.5 Microscopic magnetic model
Table 1: Exchange parameters of BaNa2Cu(VO${}_{4})_{2}$ obtained from DFT calculations: Cu–Cu distances $d$ (in Å), electron hoppings $t_{i}$ (in meV), AFM contributions to the exchange $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$ (in K), and total exchange couplings $J_{i}$ (in K) from the DFT+$U$ mapping procedure. | $d_{\rm Cu-Cu}$ | $t_{i}$ | $J_{i}^{\rm AFM}$ | $J_{i}$
---|---|---|---|---
$J$ | 5.507 | $-40$ | 14.9 | 6.8
$J_{ab}$ | 5.507 | $-5$ | 0.2 | $<\\!0.2$
$J_{ab}^{\prime}$ | 5.686 | $-1$ | 0.01 | $<\\!0.2$
$J_{c}$ | 7.024 | 3 | 0.08 | $<\\!0.2$
LDA band structure of BaNa2Cu(VO${}_{4})_{2}$ (Fig. 10) features Cu $3d$
states below the Fermi level and V $3d$ states above 2 eV, confirming the non-
magnetic state of vanadium. The overall energy spectrum is metallic, as
typical for a transition-metal compound when correlation effects in the $3d$
shell were not taken into account. Nevertheless, this band structure gives an
overview of possible exchange interactions, as the hopping parameters $t_{i}$
are proportional to the LDA bandwidth, whereas $J_{i}^{\rm
AFM}=4t_{i}^{2}/U_{\rm eff}$. The Fermi level is crossed by two narrow bands
formed by the half-filled $d_{x^{2}-y^{2}}$ orbitals of Cu2+. The width of
these bands is less than 0.2 eV, one of the smallest in cuprates, and
indicates very weak exchange couplings in BaNa2Cu(VO${}_{4})_{2}$.
Figure 10: LDA density of states for BaNa2Cu(VO${}_{4})_{2}$. Note the very
narrow Cu $d_{x^{2}-y^{2}}$ band around 0 eV (Fermi level) that indicates
small electron hoppings and correspondingly weak exchange couplings.
DFT results for the exchange couplings are summarized in Table 1. Only one
sizable coupling, $J/k_{\rm B}\simeq 6.8$ K is found. It corresponds to spin
chains running along $[110]$ in one layer and along $[1\bar{1}0]$ in the
adjacent layer, the direction being chosen by the position of the double VO4
bridges that connect the CuO4 plaquette units (Fig. 1). Such a coupling
mechanism is fairly common among the Cu2+ compounds and can give rise to both
FM and AFM superexchange depending on the orientation of the VO4 tetrahedra
relative to the CuO4 planes Tsirlin _et al._ (2012). Larger rotations of the
tetrahedra favor FM couplings.
In BaNa2Cu(VO${}_{4})_{2}$, we find $\varphi=99.0^{\circ}$, which is similar
to $\varphi^{(2)}=102.2^{\circ}$ for the AFM coupling $J_{a}^{(2)}/k_{\rm
B}=9.5$ K in BaAg2Cu(VO${}_{4})_{2}$ and very different from
$\varphi^{(1)}=123.7^{\circ}$ for the FM coupling $J_{a}^{(1)}/k_{\rm B}=-19$
K in the same compound Tsirlin _et al._ (2012). Here, $\varphi$ is the angle
between the face of the VO4 tetrahedron and the plane connecting the adjacent
CuO4 plaquettes, as shown in Fig. 1. Compared to BaAg2Cu(VO${}_{4})_{2}$, the
AFM coupling weakens from 9.5 K to $\sim 6$ K, likely because of the longer
Cu–Cu distance (5.507 Å vs. 5.448 Å) and the increased lateral displacement
$r$ of the CuO4 plaquettes (0.895 Å vs. 0.860 Å).
All couplings beyond the aforementioned spin chains appear to be very weak,
below 0.2 K, and unfeasible for the DFT+$U$ mapping analysis. Their relative
strengths can be assessed from the hopping parameters that suggest the
dominant interchain couplings $J_{ab}$ in the $ab$ plane (along $[1\bar{1}0]$
for the spin chains along $[110]$, and vice versa) and $J_{c}$ along the $c$
direction. The in-plane coupling $J_{ab}^{\prime}$ is negligible. The two
stronger interchain couplings, $J_{ab}$ and $J_{c}$, form a non-frustrated 3D
network. From $4t_{i}^{2}/U_{\rm eff}$ with $U_{\rm eff}=5$ eV Lebernegg _et
al._ (2013); Ahmed _et al._ (2015), one expects the coupling strength of 0.2
K or lower, in agreement with the DFT+$U$ results. Altogether, our modeling
results establish weak and non-frustrated interchain couplings in
BaNa2Cu(VO${}_{4})_{2}$, with $J_{\perp}/J\simeq 0.02$. The average interchain
coupling of $J_{\perp}/k_{\rm B}\simeq 0.1$ K leads to $T_{N}/J\simeq 0.22$ K
Yasuda _et al._ (2005) in good agreement with 0.25 K found experimentally.
Therefore, we argue that long-range magnetic order in BaNa2Cu(VO${}_{4})_{2}$
should be driven by weak interchain couplings, and the Néel temperature
$T_{\rm N}/J$ is determined by the $J_{\perp}/J$ ratio.
Figure 11: Magnetization normalized to the saturation value (main figure) and
magnetic specific heat (inset) of BaNa2Cu(VO${}_{4})_{2}$. Predictions of the
spin-chain model with $J/k_{\rm B}=5.5$ K and $g=2.17$ are shown with lines.
In magnetization curves, an additional 5 % paramagnetic contribution described
by the Brillouin function was included in order to reproduce the weak bend in
low magnetic fields.
Above $T_{\rm N}$, a purely one-dimensional description should hold. Indeed,
we were able to fit magnetization curves down to 0.49 K using the spin-chain
model with $J/k_{\rm B}=5.5$ K and $g=2.17$ in excellent agreement with 5.6 K
from the fit to the magnetic susceptibility and $g=2.17$ from the ESR
experiment (Fig. 11). This confirms that the interchain couplings are very
weak and play only marginal role even at $T<J$. Magnetic specific heat is also
well described by the spin-chain model showing small deviations below 1 K
only. These deviations correspond to the upturn in $C_{\rm mag}/T$ upon
approaching $T_{\rm N}$ (Fig. 6).
## IV Conclusions
We have shown that BaNa2Cu(VO${}_{4})_{2}$ strongly deviates from all of its
structural siblings in terms of the magnetic behavior. The majority of these
compounds are triangular magnets, while the only Cu2+ member studied to date,
BaAg2Cu(VO${}_{4})_{2}$, revealed a very unusual coexistence of different spin
chains, one ferromagnetic and one antiferromagnetic Tsirlin _et al._ (2012);
Krupskaya _et al._ (2017). Our present results for BaNa2Cu(VO${}_{4})_{2}$
corroborate non-trivial magnetostructural correlations in Cu2+ vanadates,
where the sign of a magnetic coupling strongly depends on the spatial
orientation of the VO4 tetrahedra relative to the spin chains and CuO4
plaquette units.
The disparity of spin chains is absent in BaNa2Cu(VO${}_{4})_{2}$, but now the
chains adopt two different directions and form an unusual crossed pattern.
Interestingly, this crossed pattern does not cause any magnetic frustration,
because the Cu2+ ion of one chain sits exactly on top of the Cu2+ ion of the
adjacent chain (Fig. 1). Then, each magnetic site has only one coupling to a
spin chain of another direction, and not two couplings, as expected
theoretically Starykh _et al._ (2002). This fact highlights the importance of
lateral displacements between the Cu2+ ions of the crossed chains to induce
the frustration. Such displacements do not occur in BaNa2Cu(VO${}_{4})_{2}$,
but they may potentially appear in sister compounds, because even the
substitution of Na+ by Ag+ causes significant structural changes, although the
two ions are very similar in size. Alternatively, one may consider structure
types with a weaker spatial separation between the crossed chains that, in
turn, allows several non-equivalent interactions to form a frustrated topology
even in the absence of lateral displacements Tsirlin _et al._ (2011);
Mukharjee _et al._ (2019); Weickert _et al._ (2019).
###### Acknowledgements.
We would like to acknowledge SERB, India, for financial support bearing
sanction Grant No. CRG/2019/000960. Work at the Ames Laboratory was supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division. The Ames Laboratory is operated
for the U.S. Department of Energy by Iowa State University under Contract No.
DEAC02-07CH11358. AT was funded by the Federal Ministry for Education and
Research through the Sofja Kovalevskaya Award of Alexander von Humboldt
Foundation.
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|
# The volume polynomial of lattice polygons
Ivan Soprunov Department of Mathematics and Statistics
Cleveland State University
Cleveland, OH USA<EMAIL_ADDRESS>and Jenya Soprunova Department of
Mathematical Sciences
Kent State University
Kent, OH USA<EMAIL_ADDRESS>
###### Abstract.
We prove that every indefinite quadratic form with non-negative integer
coefficients is the volume polynomial of a pair of lattice polygons. This
solves the discrete version of the Heine–Shephard problem for two bodies in
the plane. As an application, we show how to construct a pair of planar
tropical curves (or a pair of divisors on a toric surface) with given
intersection number and self-intersection numbers.
###### Key words and phrases:
volume polynomial, lattice polytope, mixed volume, integer quadratic form,
intersection number, tropical curve, toric surface
###### 2020 Mathematics Subject Classification:
Primary 52B20, 52A39; Secondary 11H55, 14T10, 14M25
## 1\. Introduction
With any collection $K_{1},\dots,K_{n}$ of convex bodies in $\mathbb{R}^{d}$
and non-negative scalars $x_{1},\dots,x_{n}\in\mathbb{R}_{\geq 0}$ one can
associate a convex body
$x_{1}K_{1}+\dots+x_{n}K_{n}=\\{x_{1}a_{1}+\dots+x_{n}a_{n}:a_{i}\in
K_{i},1\leq i\leq n\\}\subset\mathbb{R}^{d}.$
In [Min03] Minkowski showed that the $d$-dimensional volume of this body
depends polynomially on the scalars, that is,
$\operatorname{Vol}_{d}(x_{1}K_{1}+\dots+x_{n}K_{n})$ is a homogeneous degree
$d$ polynomial in $x_{1},\dots,x_{n}$. This polynomial is called the volume
polynomial of $K_{1},\dots,K_{n}$ and its coefficients are the mixed volumes
(up to multinomial coefficients). In the case of two planar convex bodies
$K,L$ the volume polynomial is a quadratic form
(1.1)
$\operatorname{Vol}_{2}(xK+yL)=\operatorname{Vol}_{2}(K)x^{2}+2\operatorname{V}(K,L)xy+\operatorname{Vol}_{2}(L)y^{2},$
where the middle coefficient $\operatorname{V}(K,L)$ is called the mixed
volume of $K$ and $L$. It can be expressed by evaluating (1.1) at $x=y=1$ as
(1.2)
$\operatorname{V}(K,L)=\frac{1}{2}\left(\operatorname{Vol}_{2}(K+L)-\operatorname{Vol}_{2}(K)-\operatorname{Vol}_{2}(L)\right).$
The classical Minkowski inequality provides a relation between the
coefficients of (1.1):
(1.3)
$\operatorname{Vol}_{2}(K)\operatorname{Vol}_{2}(L)\leq\operatorname{V}(K,L)^{2}.$
In other words, (1.1) is an indefinite quadratic form. It is not hard to show
that every indefinite quadratic form with non-negative coefficients is the
volume polynomial of some planar convex bodies $K,L$, see Proposition 4.1.
In general, the study of polynomial inequalities between the coefficients of
the volume polynomial is the core of the Brunn-Minkowski theory of convex
bodies. We refer to the book of Schneider [Sch14] for the most complete
account of this theory. Still the problem of giving an explicit description
for the space of volume polynomials in terms of coefficient inequalities is
wide open. Besides the case of two planar bodies described above, such a
description is only known for three planar bodies ($n=3,d=2$) provided by
Heine [Hei38], and two bodies in any dimension ($n=2,d\in\mathbb{N}$) provided
by Shephard [She60]. Inequalities such as the Aleksandrov-Fenchel inequality
and Shephard’s determinantal inequalities uncover deep connections between
mixed volumes, but yet do not provide a complete description for the space of
volume polynomials, in general, see [She60, ABS20]. Some new inequalities
describing the square-free part of the volume polynomial for $n=4$ and $d=2$
have been recently found in [AS23].
In this note we consider a discrete version of the Heine–Shephard problem.
Namely, we are interested in describing the space of volume polynomials of
lattice polytopes. Recall that a convex polytope $P\subset\mathbb{R}^{d}$ is a
lattice polytope if its vertices belong to the integer lattice
$\mathbb{Z}^{d}$. In this case it is appropriate to normalize the usual
Euclidean volume by a factor of $d!$. Then the volume and the mixed volume
take integer values on lattice polytopes.
###### Problem 1.1.
Let $\operatorname{Vol}_{d}$ be the normalized volume. Describe the set of the
volume polynomials $\operatorname{Vol}_{d}(x_{1}P_{1}+\dots+x_{n}P_{n})$ over
all collections of lattice polytopes $P_{1},\dots,P_{n}$ in terms of
coefficient inequalities.
In this paper we provide a solution to Problem 1.1 in the smallest non-trivial
case $n=d=2$. We prove that every indefinite quadratic form with non-negative
integer coefficients is the volume polynomial of a pair of lattice polytopes
in $\mathbb{R}^{2}$, see Theorem 4.2. Our proof is constructive. An
implementation in Magma [BCP97] can be found at
https://github.com/isoprou/volume_polynomial.
One can also view Problem 1.1 as a discrete Blaschke-Santaló problem.
Motivated by the work of Blaschke [Bla16], Santaló studied the map which
assigns to every planar convex body a triple of its geometric invariants, such
as area, perimeter, diameter, width etc., [San61]. The image of such a map is
called a Blaschke-Santaló diagram. Now let $\phi$ be the map which sends a
pair of convex planar bodies $K,L$ to the triple
$(\operatorname{Vol}_{2}(K),\operatorname{V}(K,L),\operatorname{Vol}_{2}(L))$.
Then the diagram $\operatorname{Im}\phi$ is the closed semialgebraic set
(1.4) $\operatorname{Im}\phi=\\{(x,y,z)\in\mathbb{R}_{\geq 0}^{3}:y^{2}\geq
xz\\}$
as follows from Minkowski inequality (1.3) and Proposition 4.1. Restricting
$\phi$ to the set of pairs of lattice polytopes in the plane, we obtain the
discrete diagram which, according to our result in Theorem 4.2, is the set of
lattice points of the semialgebraic set (1.4). In a similar spirit, Scott
[Sco76] described the discrete diagram for the set of coefficients of the
Ehrhart polynomial of lattice polygons. For the actual picture of the diagram
see, for example, [BDLD+05, Thm. 2.1].
There is a strong motivation to consider mixed volumes of lattice polytopes
coming from the intersection theory in toric and tropical geometry. In the
final section (Section 5) we discuss an interpretation of our result in terms
of the intersection numbers of planar tropical curves and divisors on toric
surfaces.
### Acknowledgments
We are grateful to Gennadiy Averkov for fruitful discussions.
## 2\. Preliminaries
In this section we recall basic facts about mixed volumes of convex bodies.
For the most part we restrict ourselves to the case of lattice polytopes in
$\mathbb{R}^{2}$, as this is the focus of this paper. For the general theory
of mixed volumes we refer to [Sch14, Sec. 5.1].
A convex body is a non-empty compact convex set $K\subset\mathbb{R}^{d}$. The
support function of $K$, $h_{K}:\mathbb{R}^{d}\to\mathbb{R}$ is defined by
$h_{K}(u)=\max\\{\langle u,v\rangle:v\in K\\}$, where $\langle u,v\rangle$ is
the usual inner product in $\mathbb{R}^{d}$. A polytope
$P\subset\mathbb{R}^{d}$ is the convex hull of finitely many points in
$\mathbb{R}^{d}$. A lattice polytope $P\subset\mathbb{R}^{d}$ is the convex
hull of finitely many points in $\mathbb{Z}^{d}$. The surface area measure of
$P$ is the discrete measure supported on the set of outer unit normals to the
facets of $P$ whose values are the $(d-1)$-dimensional volumes of the
corresponding facets.
In general, the mixed volume is a polarization of the usual volume, that is,
it is the unique symmetric and multilinear function of $d$ convex bodies which
coincides with the volume when the $d$ bodies are the same [Sch14, Sec. 5.1].
In the case of dimension $d=2$, the polarization formula is given in (1.2). As
seen from (1.2), the mixed volume is invariant under linear transformations
with determinant $\pm 1$ and under independent translations of $P$ and $Q$.
There is a formula for $\operatorname{V}(P,Q)$ in terms of the support
function of $P$ and the surface area measure of $Q$, see [Sch14, Thm. 5.1.7],
which, in particular, implies the non-negativity of the mixed volume. Below we
present a lattice version of this formula which we will use in Section 4.
Assume $P,Q\subset\mathbb{R}^{2}$ are lattice polytopes. As mentioned in the
introduction, for lattice polytopes we normalize the 2-dimensional volume by a
factor of 2. For example, a lattice rectangle $P=[0,a]\times[0,b]$ for
$a,b\in\mathbb{Z}_{\geq 0}$ has normalized volume
$\operatorname{Vol}_{2}(P)=2ab$ and a right triangle $Q$ with vertices
$(0,0)$, $(a,0)$, and $(0,b)$ has $\operatorname{Vol}_{2}(Q)=ab$. Furthermore,
if $I\subset\mathbb{R}^{2}$ is a lattice segment then we define its normalized
1-dimensional volume as its lattice length, i.e.
$\operatorname{Vol}_{1}(I)=|I\cap\mathbb{Z}^{2}|-1$.
When defining the surface area measure for lattice polytopes it is more
convenient to work with primitive normals rather than unit normals. A vector
$u\in\mathbb{Z}^{2}$ is primitive if its coordinates are relatively prime. Let
$U_{P}$ be the set of outer primitive normals to the sides of $P$ and $S_{P}$
the corresponding surface area measure supported on $U_{P}$ with values
$S_{P}(u)=\operatorname{Vol}_{1}(P^{u})$. Here $P^{u}$ represents the side of
$P$ with outer primitive normal $u$. Then the normalized mixed volume can be
computed as follows (see [ABS21, Sec. 2.5] for details)
(2.1) $\operatorname{V}(P,Q)=\sum_{u\in
U_{Q}}h_{P}(u)\operatorname{Vol}_{1}(Q^{u}).$
Note that $h_{P}(u)\operatorname{Vol}_{1}(Q^{u})$ equals the inner product of
a vertex $v\in P$ and a scalar multiple of the primitive vector $u$. We give a
small example below.
###### Example 2.1.
Let $P=\operatorname{conv}\\{(0,0),\,(0,2),\,(4,1),\,(4,0)\\}$ and
$Q=\operatorname{conv}\\{(0,0),\,(0,2),\,(3,0)\\}$. Then $U_{Q}$ consists of
three primitive vectors: $(2,3)$, $(0,-1)$, and $(-1,0)$. In Figure 2.1 we
show $Q$ with the primitive vectors rescaled by the lattice lengths of the
corresponding sides, $u_{1}=(2,3)$, $u_{2}=(0,-3)$, and $u_{3}=(-2,0)$.
Figure 2.1. The vertices of $P$ and the surface area measure of $Q$
The inner product $\langle u_{i},v\rangle$ attains its maximum at the vertex
$v_{1}=(4,1)\in P$ for $i=1$ and at $v_{0}=(0,0)\in P$ for $i=2,3$. Therefore,
by (2.1),
$\operatorname{V}(P,Q)=\langle u_{1},v_{1}\rangle+\langle
u_{2},v_{0}\rangle+\langle u_{3},v_{0}\rangle=11+0+0=11.$
## 3\. Quadratic forms
In this section we review basic notions in the theory of binary quadratic
forms over the integers. We then prove a reduction algorithm for indefinite
forms similar to the classical reduction for positive definite forms (see, for
example, [Bue89, Ch. 2]).
Consider a binary quadratic form
$f(x,y)=ax^{2}+2bxy+cy^{2},$
where $a,b,c$ are integers. It is called indefinite if its discriminant
$\Delta=4(b^{2}-ac)$ is non-negative. We can express $f$ in a matrix form
$f(x,y)=\left(\begin{matrix}x&y\end{matrix}\right)M\left(\begin{matrix}x\\\
y\end{matrix}\right),$
where $M=\left(\begin{matrix}a&b\\\ b&c\end{matrix}\right)$ is an integer
symmetric matrix.
Recall that the unimodular group $\operatorname{GL}(2,\mathbb{Z})$ is the
group of invertible integer matrices $G$ with $\det G=\pm 1$. Two forms $f$
and $f^{\prime}$ are called equivalent if they are related by a unimodular
change of variables. Then $f^{\prime}$ is equivalent to $f$ whenever
$f^{\prime}(x,y)=\left(\begin{matrix}x&y\end{matrix}\right)G^{T}MG\left(\begin{matrix}x\\\
y\end{matrix}\right),$
for some $G\in\operatorname{GL}(2,\mathbb{Z})$. Here $G^{T}$ is the transpose
of $G$. In this case we say that $G$ transforms $f$ to $f^{\prime}$. More
explicitly, if $G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)$ then
$f^{\prime}(x,y)=a^{\prime}x^{2}+2b^{\prime}xy+c^{\prime}y^{2}$ with
$\displaystyle a^{\prime}$
$\displaystyle=ax_{1}^{2}+2bx_{1}y_{1}+cy_{1}^{2}=f(x_{1},y_{1})$
$\displaystyle b^{\prime}$
$\displaystyle=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})+cy_{1}y_{2}$
$\displaystyle c^{\prime}$
$\displaystyle=ax_{2}^{2}+2bx_{2}y_{2}+cy_{2}^{2}=f(x_{2},y_{2}).$
Clearly, equivalent forms have the same discriminant.
Our next result is a key tool for constructing lattice polytopes with a given
volume polynomial in Section 4.
###### Proposition 3.1.
Every integer indefinite binary form with positive coefficients is equivalent
to a form $f(x,y)=ax^{2}+2bxy-cy^{2}$ for some $a,b,c\in\mathbb{Z}_{\geq 0}$
such that $f(1,1)=a+2b-c>0$. Moreover, there exists a matrix
$G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)\in\operatorname{GL}(2,\mathbb{Z})$ satisfying
$x_{i}\geq y_{i}\geq 0$ for $i=1,2$ which transforms $f$ to the original form.
###### Proof.
Consider $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ with $A,B,C\in\mathbb{N}$ and let
$k=B^{2}-AC\geq 0$. If $A<C$ then we apply the matrix
$\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$ which swaps the variables.
Thus, we may assume $A\geq C$. Since $k\geq 0$ this implies that $B\geq C$.
Next, we divide $B$ by $C$ with a remainder, $B=sC+r$ where $s\geq 1$ and
$0\leq r<C$. Applying the matrix $\left(\begin{matrix}0&1\\\
1&-s\end{matrix}\right)$ we obtain a form $ax^{2}+2bxy+cy^{2}$ with $a=C$,
$b=r$, and $c=F(1,-s)$. Note that we have $a>b>c$, since $a>b$ and the
discriminant $k=b^{2}-ac\geq 0$. While $c>0$ we keep dividing $b$ by $c$ with
a remainder. Since throughout this process $c$ is getting strictly smaller, we
will eventually end up with a triple where $a>b\geq 0\geq c$. By switching the
sign of $c$, we may write the resulting form as $f(x,y)=ax^{2}+2bxy-cy^{2}$
with $a,b,c\in\mathbb{Z}_{\geq 0}$.
Now, let us look at the matrix $G$ that transforms $f$ to $F$. It is the
product of the inverses of the matrices used above, i.e.
$G=\left(\begin{matrix}s_{n}&1\\\
1&0\end{matrix}\right)\cdots\left(\begin{matrix}s_{1}&1\\\
1&0\end{matrix}\right)\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right),$
where $n\geq 1$, $s_{i}\geq 1$ for $1\leq i\leq n$, and the last matrix may or
may not be present. It is easy to see by induction that $G$ satisfies the
condition in the statement of the proposition.
Finally, to ensure that $f(1,1)>0$ we modify the last step, if needed. We
choose $s_{n}$ to be the smallest integer such that $F(1,-s_{n})\leq 0$ where
$F$ is the form in the penultimate step. Since $F$ has positive coefficients,
we must have $s_{n}\geq 1$. Also, $f(1,1)=F(1,1-s_{n})>0$, since otherwise
$s_{n}$ was not the smallest. ∎
## 4\. The volume polynomial
In this section we prove our main result (Theorem 4.2) which describes all
normalized volume polynomials of pairs of lattice polytopes in
$\mathbb{R}^{2}$. First, we look at the easier case of the usual volume
polynomial of two planar bodies.
###### Proposition 4.1.
Let $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ be an indefinite quadratic form with
$A,B,C\in\mathbb{R}_{\geq 0}$. Then there exist convex bodies
$K,L\subset\mathbb{R}^{2}$ such that $F(x,y)$ is the volume polynomial of $K$
and $L$, that is $F(x,y)=\operatorname{Vol}_{2}(xK+yL)$ for $x,y\geq 0$.
###### Proof.
We let $K$ and $L$ be rectangular boxes $K=[0,\alpha]\times[0,\beta]$ and
$L=[0,\gamma]\times[0,\delta]$ for some
$\alpha,\beta,\gamma,\delta\in\mathbb{R}_{\geq 0}$. Then
$\operatorname{Vol}_{2}(K)=\alpha\beta$,
$\operatorname{V}(K,L)=(\alpha\delta+\beta\gamma)/2$, and
$\operatorname{Vol}_{2}(L)=\gamma\delta$. We will show that the system
$A=\alpha\beta,\quad B=(\alpha\delta+\beta\gamma)/2,\quad C=\gamma\delta$
has a non-negative solution. Indeed, first assume $A>0$. Put $\alpha=1$ and so
$\beta=A$. Multiplying the middle equation by $\gamma$ and using the last
equation we obtain:
$A\gamma^{2}-2B\gamma+C=0.$
Since $k=B^{2}-AC\geq 0$ this equation has a non-negative solution
$\gamma=\frac{B+\sqrt{k}}{A}$. Note that $\gamma=0$ only if $B=0$ and $C=0$.
In this case $\delta=0$. Otherwise, $\delta=C/\gamma$.
Now if $A=0$ and $B>0$ then we put $\alpha=1$, $\beta=0$,
$\gamma=\frac{C}{2B}$, and $\delta=2B$. Finally, if $A=0$ and $B=0$ then we
put $\alpha=0$, $\beta=0$, $\gamma=1$, and $\delta=C$. ∎
###### Theorem 4.2.
Let $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ be an indefinite quadratic form with
$A,B,C\in\mathbb{Z}_{\geq 0}$. Then there exist lattice polytopes
$P,Q\subset\mathbb{R}^{2}$ such that $F(x,y)$ is the normalized volume
polynomial of $P$ and $Q$, that is $F(x,y)=\operatorname{Vol}_{2}(xP+yQ)$ for
$x,y\geq 0$.
###### Proof.
Suppose $A$ and $C$ are positive, and, hence, so is $B$. Then, according to
Proposition 3.1, there exist a form $f(x,y)=ax^{2}+2bxy-cy^{2}$ with
$a,b,c\in\mathbb{Z}_{\geq 0}$, and $f(1,1)=a+2b-c>0$, and a unimodular matrix
$G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)\in\operatorname{GL}(2,\mathbb{Z})$ with
$x_{i}\geq y_{i}\geq 0$, $i=1,2$ which transforms $f$ to $F$. Then
$A=f(x_{1},y_{1}),\quad
B=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2},\quad C=f(x_{2},y_{2}).$
Below we show how to construct lattice polytopes $P$ and $Q$ satisfying
$\operatorname{Vol}_{2}(P)=A$, $\operatorname{V}(P,Q)=B$, and
$\operatorname{Vol}_{2}(Q)=C$. The construction differs slightly depending on
the parity of $a$ and $c$.
Case 1. Suppose first that $a$ and $c$ are even, so that $a=2d$ and $c=2e$ for
some $d,e\in\mathbb{Z}_{\geq 0}$. Then we can write $f(x,y)=ax^{2}+2bxy-
cy^{2}=2x(dx+by)-2ey^{2}$. We view this expression as the area of a rectangle
with sides $x$ and $dx+by$ with two opposite corners cut off; each corner
being a right triangle with legs $y$ and $ey$. Thus, we define
$P=\operatorname{conv}\\{\big{(}0,y_{1}\big{)},\,\big{(}0,x_{1}\big{)},\,\big{(}ey_{1},0\big{)},\,\big{(}dx_{1}+by_{1},0\big{)},\,\big{(}dx_{1}+by_{1},x_{1}-y_{1}\big{)},\,\big{(}dx_{1}+(b-e)y_{1},x_{1}\big{)}\\}.$
We define $Q$ in the same way replacing $x_{1}$ with $x_{2}$ and $y_{1}$ with
$y_{2}$ above. Since
$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c)/2>0,$
the relative positions of the vertices of $P$ and $Q$ are as shown in Figure
4.1. Clearly,
$\operatorname{Vol}_{2}(P)=2(dx_{1}+by_{1})-2ey_{1}^{2}=f(x_{1},y_{1})=A$ and,
similarly, $\operatorname{Vol}_{2}(Q)=f(x_{2},y_{2})=C$. In the second diagram
in Figure 4.1 we depict the surface area measure of $Q$. This allows us to
compute the mixed volume using (2.1).
Figure 4.1. Case 1: $a=2d$ and $c=2e$
$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})(dx_{1}+by_{1})+y_{2}(dx_{1}+by_{1})+ey_{2}(x_{1}-y_{1})+x_{1}(dx_{2}+(b-e)y_{2})$
$\displaystyle-
ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$
Case 2. If $a=2d$ and $c=2e+1$ for $d,e\in\mathbb{Z}_{\geq 0}$, we construct
$P$ and $Q$ as above except the primitive outer normal $(1,e)$ is now replaced
with $(1,e+1)$. Then
$P=\operatorname{conv}\\{\big{(}0,y_{1}\big{)},\,\big{(}0,x_{1}\big{)},\,\big{(}ey_{1},0\big{)},\,\big{(}dx_{1}+by_{1},0\big{)},\,\big{(}dx_{1}+by_{1},x_{1}-y_{1}\big{)},\,\big{(}dx_{1}+(b-e-1)y_{1},x_{1}\big{)}\\},$
and $Q$ is obtained by replacing the indices. We have
$dx_{i}+by_{i}\geq dx_{i}+(b-e-1)y_{i}\geq(d+b-e-1)y_{i}=y_{i}(a+2b-c-1)/2\geq
0,$
and we get a diagram depicted in Figure 4.2.
Figure 4.2. Case 2: $a=2d$ and $c=2e+1$
As before, we have $\operatorname{Vol}_{2}(P)=A,\operatorname{Vol}_{2}(Q)=C$,
and
$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})(dx_{1}+by_{1})+y_{2}(dx_{1}+by_{1})+(e+1)y_{2}(x_{1}-y_{1})$
$\displaystyle+x_{1}(dx_{2}+(b-e-1)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$
Case 3. Suppose $a=2d+1$ and $c=2e$ for $d,e\in\mathbb{Z}_{\geq 0}$. We now
write
$f(x,y)=ax^{2}+2bxy-cy^{2}=2x(dx+by)+x^{2}-2ey^{2}.$
This represents the area of a trapezoid, which is the union of an $x$ by
$dx+by$ rectangle and the isosceles right triangle with leg $x$, with two
opposite corners cut off, where the corners have area $ey^{2}$ each. In other
words, we let $P$ have the vertices
$(0,y_{1}),\,(0,x_{1}),\,(ey_{1},0),\,((d+1)x_{1}+by_{1},0),\,(dx_{1}+(b+1)y_{1},x_{1}-y_{1}),\,(dx_{1}+(b-e)y_{1},x_{1}),$
and $Q$ is defined accordingly. Similarly to the above,
$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c-1)/2\geq
0,$
and we get the diagram in Figure 4.3.
Figure 4.3. Case 3: $a=2d+1$ and $c=2e$
We have $\operatorname{Vol}_{2}(P)=A$, $\operatorname{Vol}_{2}(Q)=B$, and
$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})((d+1)x_{1}+by_{1})+y_{2}(dx_{1}+(b-e)y_{1})+(e+1)x_{1}y_{2}$
$\displaystyle+x_{1}(dx_{2}+(b-e)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$
Case 4. Finally, suppose $a=2d+1$ and $c=2e+1$ for $d,e\in\mathbb{Z}_{\geq
0}$. Then
$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c)/2>0,$
and we define each of $P$ and $Q$ as the convex hull of
$\big{(}0,y_{i}\big{)},\,\big{(}0,x_{i}\big{)},\,\big{(}ey_{i},0\big{)},\,\big{(}(d+1)x_{i}+by_{i},0\big{)},\,\big{(}dx_{i}+(b+1)y_{i},x_{i}-y_{i}\big{)},\,\big{(}dx_{i}+(b-e-1)y_{i},x_{i}\big{)},$
as depicted in Figure 4.4.
Figure 4.4. Case 4: $a=2d+1$ and $c=2e+1$
Then $\operatorname{Vol}_{2}(P)=A$, $\operatorname{Vol}_{2}(Q)=C$, and
$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})((d+1)x_{1}+by_{1})+y_{2}(dx_{1}+(b-e-1)y_{1})+(e+2)x_{1}y_{2}$
$\displaystyle+x_{1}(dx_{2}+(b-e-1)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$
In remains to consider the case when $A$ or $C$ (or both) is zero. Without
loss of generality, assume $A=0$. Then we take $P$ to be the unit segment
$P=\operatorname{conv}\\{(0,0),\,(1,0)\\}$. To construct $Q$ we write
$C=sB-r$, for some $s,r\in\mathbb{Z}$ with $s\geq 1$ and $0<r\leq B$, and let
$Q=\operatorname{conv}\\{(s,0),\,(1,0),\,(0,r),\,(0,B)\\}.$
Then $\operatorname{Vol}_{2}(Q)=C$ and $\operatorname{V}(P,Q)=B$, as required.
Note that when, in addition, $C=0$ we have $s=1$, $r=B$ and, hence, $Q$ is
also a segment.
∎
## 5\. Connection to intersection numbers of curves
In this section we discuss an implication of Theorem 4.2 for intersection
numbers of plane tropical curves, as well as intersection numbers of divisors
on toric surfaces. We recall basic facts below and refer to [Ful93, CLS11,
MS15] for general theory of toric and tropical varieties.
### 5.1. Planar tropical curves
A tropical bivariate polynomial $f$ is a piece-wise linear function
$f(x,y)=\min\\{c_{a,b}+ax+by:(a,b)\in S\\}$ where $S\subset\mathbb{Z}^{2}$ is
a finite set, called the support of $f$, and $c_{a,b}\in\mathbb{R}$. The
convex hull $P_{f}=\operatorname{conv}S$ is called the Newton polytope of $f$.
The set of points $(x,y)\in\mathbb{R}^{2}$ where $f$ is not differentiable
(that is where the minimum in the definition of $f$ is attained at least
twice) is called the (planar) tropical curve $V_{f}\subset\mathbb{R}^{2}$
associated to $f$. Geometrically, $V_{f}$ is a polyhedral complex of pure
dimension 1. There is a duality between the polyhedral complex $V_{f}$ and the
regular subdivision $\cal R_{f}$ of $P_{f}$ induced by lifting $(a,b)\in S$ to
height $c_{a,b}$. The vertices (0-dimensional cells) of $V_{f}$ correspond to
the polygons (2-dimensional cells) of $\cal R_{f}$ and the edges
(1-dimensional cells) of $V_{f}$ correspond to the edges of $\cal R_{f}$.
Given an edge $e\in V_{f}$, its weight $w(e)$ is the lattice length of the
corresponding edge in $\cal R_{f}$.
Two tropical curves $V_{f}$ and $V_{g}$ are said to intersect transversally if
$V_{f}\cap V_{g}$ is finite and every point $p\in V_{f}\cap V_{g}$ lies in the
relative interior of some edge $e$ of $V_{f}$ and some edge $h$ of $V_{g}$.
Define the local intersection number $(V_{f},V_{g})_{p}$ to be
$w(e)w(h)|\det(u,v)|$, where $u$ and $v$ are primitive vectors parallel to $e$
and $h$, respectively. Then the intersection number is defined as the sum of
local intersection numbers
$(V_{f},V_{g})=\sum_{p\in V_{f}\cap V_{g}}(V_{f},V_{g})_{p}.$
The following theorem is known as the tropical Bernstein-Khovanskii-
Kushnirenko theorem in dimension two, [MS15, Thm. 4.6.8].
###### Theorem 5.1.
The intersection number of two planar tropical curves $V_{f}$ and $V_{g}$
intersecting transversally equals the normalized mixed volume of their Newton
polytopes $\operatorname{V}(P_{f},P_{g})$.
Let $V_{f}$ be a tropical curve and let $V_{f^{\prime}}$ be its translate by
$(x_{0},y_{0})\in\mathbb{R}^{2}$. In other words $V_{f^{\prime}}$ is
associated to the tropical polynomial $f^{\prime}(x,y)=f(x+x_{0},y+y_{0})$
which has the same support and, hence, same Newton polytope as $f$. Note that
the lifting function $c^{\prime}_{a,b}$ differs from $c_{a,b}$ by a linear
function, $c^{\prime}_{a,b}=c_{a,b}+ax_{0}+by_{0}$, hence, the regular
subdivisions $\cal R_{f}$ and $\cal R_{f^{\prime}}$ are also the same. For
almost all values of $(x_{0},y_{0})$ the curves $V_{f}$ and $V_{f^{\prime}}$
intersect transversally, in which case we define
$(V_{f},V_{f}):=(V_{f},V_{f^{\prime}})$ and call it the self-intersection
number of $V_{f}$. By Theorem 5.1,
$(V_{f},V_{f})=\operatorname{V}(P_{f},P_{f})=\operatorname{Vol}_{2}(P_{f})$.
The following is a direct consequence of Theorems 4.2 and 5.1.
###### Theorem 5.2.
Given $(A,B,C)\in\mathbb{Z}_{\geq 0}^{3}$ satisfying $AC\leq B^{2}$, there
exist planar tropical curves $V_{f}$ and $V_{g}$ with $(V_{f},V_{f})=A$,
$(V_{f},V_{g})=B$, and $(V_{g},V_{g})=C$.
###### Proof.
Let $P$ and $Q$ be the lattice polytopes constructed in Theorem 4.2 from the
triple $(A,B,C)$. Consider tropical polynomials $f$ and $g$ with supports
$P\cap\mathbb{Z}^{2}$ and $Q\cap\mathbb{Z}^{2}$, respectively, and generic
coefficients $c_{a,b}$. Then the corresponding tropical curves $V_{f}$ and
$V_{g}$ will intersect transversally and
$(V_{f},V_{f})=\operatorname{Vol}_{2}(P)=A$,
$(V_{f},V_{g})=\operatorname{V}(P,Q)=B$, and
$(V_{g},V_{g})=\operatorname{Vol}_{2}(Q)=C$. ∎
### 5.2. Divisors on toric surfaces
A toric surface $X$ is an algebraic surface containing the torus
$(\mathbb{C}^{*})^{2}$ as a Zariski open subset whose action on itself extends
to the action on $X$. Basic examples of complete toric surfaces include the
projective plane, the product of projective lines, and the Hirzebruch surface.
One can describe the affine charts of $X$ via a rational polyhedral fan
$\Sigma$ in $\mathbb{R}^{2}$ whose 1-dimensional cones (rays) are generated by
a collection of primitive vectors $\\{u_{1},\dots,u_{r}\\}$. Each ray
corresponds to a 1-dimensional orbit in $X$ whose closure defines a torus
invariant prime divisor $D_{i}$ on $X$. Let $D=\sum_{i=1}^{r}a_{i}D_{i}$ be a
torus invariant divisor. It defines a rational polytope111Since we work with
outer normals rather than inner normals, we modify the standard definition
accordingly. $P_{D}=\\{x\in\mathbb{R}^{2}:\langle x,u_{i}\rangle\leq
a_{i},1\leq i\leq r\\}$.
As shown in [CLS11, Thm. 4.2.8], $D=\sum_{i=1}^{r}a_{i}D_{i}$ is a Cartier
divisor if and only if there is a continuous piece-wise linear function
$\phi_{D}$ on $\mathbb{R}^{2}$ which is linear on each cone of $\Sigma$ and
$\phi_{D}(u_{i})=a_{i}$ for $1\leq i\leq r$. The global sections of the
corresponding line bundle $\cal O(D)$ can be identified with the space of
Laurent polynomials supported in $P_{D}\cap\mathbb{Z}^{2}$, i.e.
$\Gamma(X,\cal
O(D))\cong\operatorname{span}_{\mathbb{C}}\\{x_{1}^{a_{1}}x_{2}^{a_{2}}:(a_{1},a_{2})\in
P_{D}\cap\mathbb{Z}^{2}\\}$, see [CLS11, Thm. 4.3.3]. Furthermore, $\cal O(D)$
is globally generated if and only if $\phi_{D}$ is convex in which case
$P_{D}$ is a lattice polytope, see [CLS11, Thm. 6.1.7].
The following theorem is a version of the Bernstein-Khovanskii-Kushnirenko
theorem for the intersection numbers of divisors on toric surfaces, see
[Ful93, Sec. 5.4].
###### Theorem 5.3.
Let $D,E$ be two globally generated divisors on a toric surface $X$. Then the
intersection number $(D,E)$ equals the normalized mixed volume of their
polytopes $\operatorname{V}(P_{D},P_{E})$.
We can now interpret the result of Theorem 4.2 as follows.
###### Theorem 5.4.
Given $(A,B,C)\in\mathbb{Z}_{\geq 0}^{3}$ satisfying $AC\leq B^{2}$, there
exist a toric surface $X$ and globally generated divisors $D$ and $E$ on $X$
such that $(D,D)=A$, $(D,E)=B$, and $(E,E)=C$.
###### Proof.
Let $P$ and $Q$ be the lattice polytopes constructed in Theorem 4.2 from the
triple $(A,B,C)$. The primitive normals of both $P$ and $Q$ belong to a set of
six vectors $\\{u_{1},\dots,u_{6}\\}$. For example, in Case 1 the vectors are
$\\{\pm(1,0),\pm(0,1),\pm(1,e)\\}$. Let $\Sigma$ be the fan generated by
$\\{u_{1},\dots,u_{6}\\}$ and $X$ the toric surface associated to $\Sigma$.
Define $a_{i}=h_{P}(u_{i})$ and $b_{i}=h_{Q}(u_{i})$ for $1\leq i\leq 6$,
where $h_{P}$ and $h_{Q}$ are the support functions of $P$ and $Q$,
respectively. Then the divisors $D=\sum_{i=1}^{6}a_{i}D_{i}$ and
$E=\sum_{i=1}^{6}b_{i}D_{i}$ are globally generated and have polytopes $P$ and
$Q$, respectively. By Theorems 5.3 and 4.2 we have
$(D,D)=\operatorname{Vol}_{2}(P)=A$, $(D,E)=\operatorname{V}(P,Q)=B$, and
$(E,E)=\operatorname{Vol}_{2}(Q)=C$. ∎
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&\leq \underbrace{\|\Delta_{t-1}-\eta [\nabla L_{t-1}(\theta_{t-1})]_{\mathcal{F}^{t-1}}\|_2}_{A} +\eta \underbrace{\|\tilde{\nabla}_{t, \mathcal{F}^{t-1}}-[\nabla L_{t-1}(\theta_{t-1})]_{\mathcal{F}^{t-1}}\|_2}_{B}.
\end{align*}
We first bound the term $B$. Specifically, we have
\begin{align*}
\|\tilde{\nabla}_{t, \mathcal{F}^{t-1}}-[\nabla L_{t-1}(\theta_{t-1})]_{\mathcal{F}^{t-1}}\|_2&=\|[\tilde{\nabla}_{t}-\nabla L_{t-1}(\theta_{t-1})]_{\mathcal{F}^{t-1}}\|_2\\
&\leq \sqrt{|\mathcal{F}^{t-1}|} \|\tilde{\nabla}_{t}-\nabla L_{t-1}(\theta_{t-1})\|_\infty \\
&\leq \sqrt{|\mathcal{F}^{t-1}|} (\underbrace{\|\tilde{\nabla}L_{t-1}(\theta_{t-1})-\nabla L_{t-1}(\theta_{t-1})\|_\infty}_{B_1}+\underbrace{\|\phi_t\|_\infty}_{B_2})
\end{align*}
For term $B_2$, by Lemma <ref> we have with probability at least $1-\delta'$
\begin{equation}
B_2\leq O\left(\frac{(\tau_1^2\sqrt{dk'}+\tau_1\tau_2\sqrt{d})\sqrt{\log \frac{d}{\delta'}} }{\sqrt{m}\epsilon}\right).
\end{equation}
For $B_1$, we have
\begin{align*}
B_1\leq \underbrace{\sup_{\|\theta\|_2\leq 1}\| \tilde{\nabla}L_{t-1}(\theta)- \nabla L_\mathcal{P}(\theta)\|_\infty}_{B_{1, 1}} + \underbrace{\sup_{\|\theta\|_2\leq 1} \|\nabla L_{t-1}(\theta)-\nabla L_\mathcal{P}(\theta)\|_\infty}_{B_{1, 2}}.
\end{align*}
Next we bound the term $B_{1, 1}$, we have
\begin{align*}
&\sup_{\|\theta\|_1\leq 1}\| \tilde{\nabla}L_{t-1}(\theta)- \nabla L_\mathcal{P}(\theta)\|_\infty\leq \sup_{\|\theta\|_1\leq 1}\|[\frac{1}{m}\sum_{i=1}^n \tilde{x}_i\tilde{x}_i^T-\mathbb{E}[xx^T]]\theta\|_\infty+ \sup_{\|\theta\|_1\leq 1}\|\frac{1}{m}\sum_{i=1}^m \tilde{x}_i\tilde{y}_i-\mathbb{E}[xy]\|_\infty \\
&\leq \|[\frac{1}{m}\sum_{i=1}^n \tilde{x}_i\tilde{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty}+\|\frac{1}{m}\sum_{i=1}^m \tilde{x}_i\tilde{y}_i-\mathbb{E}[xy]\|_\infty.
\end{align*}
We consider the first term $\|[\frac{1}{m}\sum_{i=1}^n \tilde{x}_i\tilde{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty}$, for simplicity for each $j, k\in [d]$ denote $\hat{\sigma}_{jk}=(\frac{1}{n}\sum_{i=1}^n \tilde{x}_i\tilde{x}_i^T)_{jk}=\frac{1}{n}\sum_{i=1}^n\tilde{x}_{i,j}\tilde{x}_{i, k}$, $\tilde{\sigma}_{jk}= (\mathbb{E}[\tilde{x}\tilde{x}^T])_{jk}= \mathbb{E}[\tilde{x}_j\tilde{x}_k]$ and $\sigma_{jk}=(\mathbb{E}[{x}{x}^T])_{jk}= \mathbb{E}[{x}_j{x}_k]$. We have
\begin{equation*}
|\hat{\sigma}_{jk}- \sigma_{jk}| \leq |\hat{\sigma}_{jk}-\tilde{\sigma}_{jk}|+|\tilde{\sigma}_{jk}-\sigma_{jk}|.
\end{equation*}
We know that $|\tilde{x}_j\tilde{x}_k|\leq \tau_1^2$ and $\text{Var}(\tilde{x}_j\tilde{x}_k)\leq \text{Var}(x_jx_k)\leq \mathbb{E}(x_jx_k)^2\leq O(\sigma^4)$. By Bernstein's inequality we have
\begin{equation}
\mathbb{P}(\max_{j,k} |\hat{\sigma}_{jk}-\tilde{\sigma}_{jk}|\leq C\sqrt{ \frac{\sigma^4 t}{m}}+\frac{\tau_1^2 t}{m})\geq 1-d^2\exp(-t)
\end{equation}
Moreover, we have
\begin{align*}
&|\tilde{\sigma}_{jk}-\sigma_{jk}|=|\mathbb{E}[|\tilde{x}_j(\tilde{x}_k-x_k)\mathbb{I}({|x_k|\geq \tau_1})]+|\mathbb{E}[|x_k(\tilde{x}_j-x_j)\mathbb{I}({|x_j|\geq \tau_1})]\\
&\leq \sqrt{\mathbb{E}(\tilde{x}_j (\tilde{x}_k-x_k))^2 \mathbb{P}( |x_k|\geq \tau_1) } + \sqrt{\mathbb{E} ((\tilde{x}_j-x_j) x_k)^2 \mathbb{P} (|x_j|\geq \tau_1)}\\
&\leq O(\frac{\sigma^2}{n}),
\end{align*}
where the last inequality is due to the assumption on sub-Gaussian where $\mathbb{P} (|x_j|\geq \tau_1)\leq 2\exp(-\frac{\tau_1^2}{2\sigma^2})= O(\frac{1}{n})$, $\mathbb{E}(\tilde{x}_j (\tilde{x}_k-x_k))^2\leq 4\mathbb{E}(x_j x_k))^2 \leq O(\sigma^4)$ and $\mathbb{E} ((\tilde{x}_j-x_j) x_k)^2\leq 4\mathbb{E}(x_j x_k))^2\leq O(\sigma^4) $. In total we have with probability at least $1-\delta'$
\begin{equation*}
\|[\frac{1}{m}\sum_{i=1}^n \tilde{x}_i\tilde{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty} \leq O(\frac{\sigma^2 \log n \log \frac{d}{\delta'}}{\sqrt{m}}).
\end{equation*}
We can use the same technique to term $\|\frac{1}{m}\sum_{i=1}^m \tilde{x}_i\tilde{y}_i-\mathbb{E}[xy]\|_\infty$, for simplicity for each $j \in [d]$ denote $\hat{\sigma}_{j}=\frac{1}{n}\sum_{i=1}^n \tilde{y}_i\tilde{x}_j$, $\tilde{\sigma}_{j}= \mathbb{E}[\tilde{y}\tilde{x}_j]$ and $\sigma_{j}= \mathbb{E}[y{x}_j]$. We have
\begin{equation*}
|\hat{\sigma}_{j}- \sigma_{j}| \leq |\hat{\sigma}_{j}-\tilde{\sigma}_{j}|+|\tilde{\sigma}_{j}-\sigma_{j}|.
\end{equation*}
Since $|\tilde{x}_{j}\tilde{y}|\leq \tau_1\tau_2 $ and we have the following by the Holder's inequality
\begin{align*}
\text{Var}(\tilde{x}_{j}\tilde{y})\leq \text{Var}(x_jy)\leq \mathbb{E}[x_j^2y^2] \leq (\mathbb{E}[y^{4}])^{\frac{1}{2}}(\mathbb{E}[|x_j|^{4}])^{\frac{1}{2}}\leq O(\sigma^4)
\end{align*}
Thus, by Bernstein's inequality we have for all $j\in [d]$
\begin{equation*}
\mathbb{P}( |\hat{\sigma}_{j}-\tilde{\sigma}_{j}|\leq O(\sqrt{\frac{\sigma^4 t}{m}}+\frac{\tau_1\tau_2t}{m}))\geq 1-d\exp(-t).
\end{equation*}
\begin{align*}
&|\tilde{\sigma}_{j}-\sigma_{j}|\leq |\mathbb{E}[\tilde{y}(\tilde{x}_j-x_j)\mathbb{I}(|x_j|) \geq \tau_1 ] |+|\mathbb{E}[x_j(\tilde{y}-y) \mathbb{I}(|y|\geq \tau_2)]|\\
&\leq \sqrt{\mathbb{E}((\tilde{y}(\tilde{x}_j-x_j))^2 \mathbb{P}( |x_j|\geq \tau_1 ) } + \sqrt{\mathbb{E} (x_j(\tilde{y}-y))^2 \mathbb{P} (|y|\geq \tau_2)}\\
&\leq O(\frac{\sigma^2}{n}+\frac{\sigma^2}{n})\leq O(\frac{\sigma^2}{n})
\end{align*}
we can easily see that with probability at most $1-\delta'$,
\begin{equation}\label{aeq:38}
\|\frac{1}{m}\sum_{i=1}^m \tilde{x}_i\tilde{y}_i-\mathbb{E}[xy]\|_\infty\leq O(\frac{\sigma^2 \log n \log \frac{d}{\delta'}}{\sqrt{m}}).
\end{equation}
Thus with probability at least $1-\delta'$
\begin{equation}\label{aeq:39}
B_{1, 1}\leq O(\frac{\sigma^2 \log n \log \frac{d}{\delta'}}{\sqrt{m}}).
\end{equation}
Next, we consider $B_{1, 2}$, similar to $B_{1, 1}$ we have
\begin{align*}
\sup_{\|\theta\|_2\leq 1} \|\nabla L_{t-1}(\theta)-\nabla L_\mathcal{P}(\theta)\|_\infty\leq \|[\frac{1}{m}\sum_{i=1}^n {x}_i{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty}+\|\frac{1}{m}\sum_{i=1}^m {x}_i{y}_i-\mathbb{E}[xy]\|_\infty.
\end{align*}
For term $ \|[\frac{1}{m}\sum_{i=1}^n {x}_i{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty}$, by Lemma <ref> we have with probability at least $1-O(d^{-8})$ we have
\begin{equation*}
\|[\frac{1}{m}\sum_{i=1}^n {x}_i{x}_i^T-\mathbb{E}[xx^T]]\|_{\infty, \infty}\leq O(\sqrt{\frac{\log d}{m}}).
\end{equation*}
For term $\|\frac{1}{m}\sum_{i=1}^m {x}_i{y}_i-\mathbb{E}[xy]\|_\infty$, we consider each coordinate, $\frac{1}{m}\sum_{i=1}^m {x}_{i,j}{y}_i-\mathbb{E}[x_jy]$. Noted that $x_j$ is $\sigma^2$-sub-Gaussian and $y$ is $\sigma^2$-sub-Gaussian, thus, by Lemma <ref> we have $x_jy$ is sub-exponential with $\|x_jy\|_{\psi_1}\leq O(\sigma^2)$. Thus, by Bernstein's inequality, we have with probability at least $1-\zeta'$
\begin{equation*}
|\frac{1}{m}\sum_{i=1}^m {x}_{i,j}{y}_i-\mathbb{E}[x_jy]|\leq O(\frac{\sigma^2 \sqrt{\log 1/{\delta'}}}{\sqrt{m}}).
\end{equation*}
Thus, with probability at least $1-\zeta'$
\begin{equation*}
\|\frac{1}{m}\sum_{i=1}^m {x}_i{y}_i-\mathbb{E}[xy]\|_\infty\leq O(\frac{\sigma^2 \sqrt{\log d/{\delta'}}}{\sqrt{m}}).
\end{equation*}
Thus, with probability at least $1-O(d^{-8})$ we have
$$B_{1, 2}\leq O(\frac{\sqrt{\log d}}{\sqrt{m}}).$$
$$B_{1}\leq O(\frac{\sqrt{\log d}}{\sqrt{m}}).$$
Thus, we have
\begin{equation}
B \leq O\left(\sqrt{2k'+k}\frac{(\tau_1^2\sqrt{dk'}+\tau_1\tau_2\sqrt{d})\sqrt{\log \frac{d}{\delta'}} }{\sqrt{m}\epsilon}\right).
\end{equation}
In the following, we consider term $A$. Noted that we have $y_i=\langle x_i, \theta^*\rangle+\zeta_i$, thus, we have
\begin{align*}
&\underbrace{\|\Delta_{t-1}-\eta [\nabla L_{t-1}(\theta_{t-1})]_{\mathcal{F}^{t-1}}\|_2}_{A}\leq \|\Delta_{t-1}-\eta [\frac{1}{m}\sum_{i=1}^m (x_i(\langle x_i, \theta_{t-1}-\theta^*\rangle)+x_i\zeta_i)]_{\mathcal{F}^{t-1}}\|_2\\
&\leq \|\Delta_{t-1}-\eta [\frac{1}{m}\sum_{i=1}^m (x_i(\langle x_i, \theta_{t-1}-\theta^*\rangle)]_{\mathcal{F}^{t-1}}\|_2+ |\sqrt{\mathcal{F}^{t-1}}|\|\frac{1}{m}\sum_{i=1}^m x_i\zeta_i\|_\infty.
\end{align*}
We first consider the term $\|\frac{1}{m}\sum_{i=1}^m x_i\zeta_i\|_\infty$. Specifically, we consider each coordinate $j\in [d]$, $|\frac{1}{m}\sum_{i=1}^m x_{i, j}\zeta_i|$. Since $\mathbb{E}[\zeta_i]=0 $ and is independent on $x$ we have $\mathbb{E}[\zeta_i x_{j}]=0$. Moreover, we have
\begin{equation*}
\|\zeta_i\|_{\psi_2}\leq \|\langle x_i, \theta^*\rangle\|_{\psi_2}+\|y_i\|_{\psi_2}\leq O(\sigma)=O(1).
\end{equation*}
Thus, $\|\zeta x\|_{\psi_1}\leq O(\sigma^2)$ by Lemma <ref>. By Bernstein's inequality we have
\begin{equation}
|\frac{1}{m}\sum_{i=1}^m x_{i, j}\zeta_i|\leq O(\frac{\sqrt{\log 1/\delta'}}{\sqrt{m}}).
\end{equation}
Thus, with probability $1-O(d^{-c})$ we have
\begin{equation*}
\|\frac{1}{m}\sum_{i=1}^m x_i\zeta_i\|_\infty\leq O(\frac{\sqrt{\log d}}{\sqrt{m}}).
\end{equation*}
Finally, we consider the term $\|\Delta_{t-1}-\eta [\frac{1}{m}\sum_{i=1}^m (x_i(\langle x_i, \theta_{t-1}-\theta^*\rangle)]_{\mathcal{F}^{t-1}}\|_2$:
\begin{align*}
\|\Delta_{t-1}-\eta [\frac{1}{m}\sum_{i=1}^m (x_i(\langle x_i, \theta_{t-1}-\theta^*\rangle)]_{\mathcal{F}^{t-1}}\|_2 =\|[(I-D^{t-1})\Delta_{t-1}]_{\mathcal{F}^{t-1}}\|_2,
\end{align*}
where $D^{t-1}=\frac{1}{m} \sum_{i \in S_t} {x}_i {x}_i^T \in \mathbb{R}^{d \times d}$.
Since $\operatorname{Supp}\left(D^{t-1} \Delta_{t-1}\right) \subset \mathcal{F}^{t-1}$ (by assumption), we have $\left\|\Delta_{t-1}-\eta D_{\mathcal{F}^{t-1}, \cdot}^{t-1} \Delta_{t-1}\right\|_2 \leq\left\|\left(I-\eta D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}\right)\right\|_2\left\|\Delta_{t-1}\right\|_2$. Next we will bound the term $\|\left(I-\eta D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}\right)\|_2$, where $I$ is the $\left|\mathcal{F}^{t-1}\right|$-dimensional identity matrix.
Before giving analysis, we show that each of the partitioned dataset safisfies the Restriced Isometry Property (RIP) defined as follows.
We say that a data matrix $X\in \mathbb{R}^{n\times d}$ satisfies the Restricted Isometry Property (RIP) with parameter $2 k^{\prime}+k$, if for any $v \in \mathbb{R}^p$ with $\|v\|_0 \leq 2 k'+k$, there exists a constant $\Delta$ which satisfies $(1-\Delta)\|v\|^2 \leq \frac{1}{n}\left\|Xv\right\|_2^2 \leq(1+\Delta)\|v\|_2^2$.
The following lemma states that with high probability, where $c$ is some constant each $X_{S_t}$ on our algorithm satisfies Definition <ref> and thus we can make use of this property to bound the term $D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}^{t-1}$.
(Theorem 10.5.11 in [50]). Consider an $n \times d$ matrix $A$ whose rows ($A_i$) are independent, isotropic, and sub-gaussian random vectors, and let $K:=\max _i\left\|A_i\right\|_{\psi_2}$. Assume that
n \geq C K^4 s \log (e d / s).
Then, with probability at least $1-2 \exp \left(-c n / K^4\right)$, the random matrix $A$ satisfies RIP with parameters $s$ and $\Delta=0.1$.
Thus, since $\{x_i\}$ are isotropic and $\|x_i\|_{\psi_2}\leq O(\sigma)$, we have with probability at least $1-2 T\exp \left(-c m / \sigma^4\right)$, $\{X_{S_t}\}_{t=1}^T$ all satisfy RIP when $m\geq \tilde{\Omega}(\sigma^4 (2k'+k))$. By the RIP property and $\left|\mathcal{F}^{t-1}\right| \leq 2 k^{\prime}+k$, we obtain the following using Lemma <ref> for any $\left|\mathcal{F}^{t-1}\right|$-dimensional vector $v$
0.9\|v\|_2^2 \leq v^T D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}^{t-1} v \leq 1.1 \|v\|_2^2 .
Thus, $\left\|\left(I-\eta D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}^{t-1}\right)\right\|_2 \leq \max \left\{1- \eta \cdot 0.9, \eta\cdot 1.1-1 \right\}$.
This means that we can take $\eta=O(1)$ such that
\left\|\left(I-\eta D_{\mathcal{F}^{t-1}, \mathcal{F}^{t-1}}^{t-1}\right)\right\|_2 \leq \frac{2}{7}.
In total we have with probability at least $1-O(d^{-c})$
\begin{equation}
\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2\leq \frac{2}{7}\|\Delta_{t-1}\|_2+ O\left(\sqrt{2k'+k}\frac{(\tau_1^2\sqrt{dk'}+\tau_1\tau_2\sqrt{d})\sqrt{\log \frac{d}{\delta'}} }{\sqrt{m}\epsilon}\right).
\end{equation}
Our next task is to bound $\left\|\theta_t^{\prime}-\theta^*\right\|_2$ by $\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2$ by Lemma <ref> .
Thus, we have $\left\|\theta_t^{\prime}-\tilde{\theta}_{t-\frac{1}{2}}\right\|_2^2 \leq \frac{\left|\mathcal{F}^{t-1}\right|-k^{\prime}}{\left|\mathcal{F}^{t-1}\right|-k}\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2^2 \leq \frac{k^{\prime}+k}{2 k^{\prime}}\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2^2$.
Taking $k^{\prime}=8 k$, we get
\left\|\theta_t^{\prime}-\tilde{\theta}_{t-\frac{1}{2}}\right\|_2 \leq \frac{3}{4}\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2
\left\|\theta_t^{\prime}-\theta^*\right\|_2 \leq \frac{7}{4}\left\|\tilde{\theta}_{t-\frac{1}{2}}-\theta^*\right\|_2 \leq \frac{1}{2}\left\|\Delta_{t-1}\right\|_2+O\left(\sqrt{k}\frac{(\tau_1^2\sqrt{dk}+\tau_1\tau_2\sqrt{d})\sqrt{\log \frac{d}{\delta'}} }{\sqrt{m}\epsilon}\right).
Finally, we need to show that $\left\|\Delta_t\right\|_2=\left\|\theta_t-\theta^*\right\|_2 \leq\left\|\theta_t^{\prime}-\theta^*\right\|_2$, which is due to the Lemma <ref>.
Putting all together, we have the following with probability at least $1-O(d^{-c})$,
\left\|\Delta_t\right\|_2 \leq \frac{1}{2}\left\|\Delta_{t-1}\right\|_2+O\left(\sqrt{k}\frac{\log n\sqrt{Tdk\log{d}} }{\sqrt{n}\epsilon}\right).
Thus, with probability at least $1-O(T d^{-c})$ we have
\left\|\Delta_T\right\|_2 \leq (\frac{1}{2})^T \left\|\theta^* \right\|_2+O\left(\frac{k\log n\sqrt{Td\log{d}} }{\sqrt{n}\epsilon}\right).
Take $T=O(\log n)$. We have the result.
§ UPPER BOUND OF LDP-IHT FOR GENERAL SUB-GAUSSIAN DISTRIBUTIONS
LDP Iterative Hard Thresholding
[1]
Input: Private data $\left\{\left(x_i, y_i\right)\right\}_{i=1}^n \in\left(\mathbb{R}^d \times \mathbb{R}\right)^n$. Iteration number $T$, privacy parameter $\epsilon$, step size $\eta$, truncation parameters $\tau, \tau_1, \tau_2$, threshold $k'$. Initial parameter $\theta_0=0$.
For the $i$-th user with $i\in [n]$, truncate his/her data as follows:
shrink $x_i$ to $\tilde{x}_i$ with $\widetilde{{x}}_{ij}=\operatorname{sgn}\left(x_{ij}\right) \min \left\{\left|x_{ij}\right|, \tau_1\right\}$ for $j\in[d]$, and $\tilde{y}_i:=\operatorname{sgn}\left(y_i\right) \min \left\{\left|y_i\right|, \tau_2\right\}$. Partition the users into $T$ groups. For $t=1, \cdots, T$, define the index set $S_t=\{(t-1) \left.\left\lfloor\frac{n}{T}\right\rfloor+1, \cdots, t\left\lfloor\frac{n}{T}\right\rfloor \right\}$; if $t=T$, then $S_t=$ $S_t \bigcup\left\{t\left\lfloor\frac{n}{T}\right\rfloor+1, \cdots, n\right\}$.
$t=1,2, \cdots, T$
The server sends $\theta_{t-1}$ to all the users in $S_t$. Each user $i \in S_t$ perturbs his/her own gradient: let $\nabla_i=$ $\tilde{x}_i^T\left(\left\langle\theta_{t-1}, \tilde{x}_i\right\rangle-\tilde{y}_i\right)$, compute $z_i=\mathcal{R}_\epsilon^r\left(\nabla_i\right)$, where $\mathcal{R}_\epsilon^r$ is the randomizer defined in (<ref>) with $r=\sqrt{d}\tau_1(2\sqrt{k'}\tau_1+\tau_2)$ and send back to the server.
The server computes $\tilde{\nabla}_{t-1}=\frac{1}{\left|S_t\right|} \sum_{i \in S_t} z_i$ and performs the gradient descent update $\tilde{\theta}_t=\theta_{t-1}-$ $\eta_0 \tilde{\nabla}_{t-1}$.
$\theta_t^{\prime}=\operatorname{Trunc}(\tilde{\theta}_{t-1}, k^{\prime})$.
$\theta_t=\arg _{\theta \in \mathbb{B}_2(2)}\left\|\theta-\theta_t^{\prime}\right\|_2$.
Output: $\theta_T$
Theorem <ref> establishes the upper bound specifically for isotropic sub-Gaussian distributions. However, we can also demonstrate that the aforementioned upper bound also holds for general sub-Gaussian distributions, albeit with different parameters. Notably, for general sub-Gaussian distributions, we need to slightly modify the LDP-IHT algorithm (Algorithm <ref>). Specifically, rather than projecting onto the unit $\ell_2$-norm ball, here we need to project onto the centered $\ell_2$-norm ball with radius 2 (actually, we can project onto any centered ball with a radius larger than $1$). See Algorithm <ref> for details. Such a modification is necessary for our proof, as we can show that with high probability, $\|\theta_t'\|_2\leq 2$ for all $t\in [T]$, which implies there is no projection with high probability. Since we use a different radius, the $\ell_2$-norm sensitivity of $\nabla_i$ also has been changed to ensure $\epsilon$-LDP. In the following, we present the theoretical result assuming that the initial parameter $\theta_0$ is sufficently close to $\theta^*$.
For any $\epsilon>0$, Algorithm <ref> is $\epsilon$-LDP. Moreover, under Assumptions <ref> and <ref>, if the initial parameter $\theta_0$ satisfies $\|\theta_0-\theta^*\|_2\leq \frac{1}{2}\frac{\mu}{\gamma}$ and $n$ is sufficiently large such that $n\geq \tilde{\Omega}(\frac{k'^2 d}{ \epsilon^2}) $, setting $\eta_0=\frac{2}{3\gamma}$, $k'=72\frac{\gamma^2}{\mu^2}k$, with probability at least $1-\delta'$ we have
\begin{equation*}
\|\theta_T-\theta^*\|_2\leq O(\frac{\sqrt{d}k\log^2n \sqrt{\log \frac{d}{\delta}}}{\sqrt{n}\epsilon}),
\end{equation*}
where $\gamma=\lambda_{\max} (\mathbb{E}[xx^T])$, $\mu=\lambda_{\min} (\mathbb{E}[xx^T])$, big-$O$ and big-$\Omega$ notations omit the terms of $\sigma, \gamma$ and $\mu$.
The proof of privacy is almost the same as the proof of Theorem <ref>. The only difference is that here we have $\|\nabla_i\|_2\leq \sqrt{d}\tau_1(2\sqrt{k'}\tau_1+\tau_2)$.
In the following, we will show the utility. We first recall two definitions and one lemma.
A function $f$ is $L$-Lipschitz w.r.t the norm $\|\cdot\|$ if for all $w, w'\in\mathcal{W}$, $|f(w)-f(w')|\leq L\|w-w'\|$.
A function $f$ is $\alpha$-smooth on $\mathcal{W}$ if for all $w, w'\in \mathcal{W}$, $f(w')\leq f(w)+\langle \nabla f(w), w'-w \rangle+\frac{\alpha}{2}\|w'-w\|_2^2.$
For any index set $I$, any $v\in \mathbb{R}^{|I|}$, let $\tilde{v}=\text{Trunc}(v, k)$. Then for any $v^*\in \mathbb{R}^{|I|}$ such that $\|v^*\|_0\leq k^*$ we have
\begin{equation}
\|\tilde{v}-v\|_2^2\leq \frac{|I|-k}{|I|-k^*}\|v^*-v\|_2^2.
\end{equation}
For simplicity we denote $L(\theta)=\mathbb{E}[(\langle x, \theta \rangle-y)^2]$, $\tilde{\nabla} L_{t-1}=\frac{1 }{m}\sum_{x\in \tilde{D}_t} \tilde{x}(\langle \tilde{x}, \theta_{t-1} \rangle-\tilde{y}) $, $\nabla L_{t-1}=
\nabla L(\theta_{t-1})=\mathbb{E}[x(\langle x, \theta_{t-1}\rangle-y)$, $S^{t-1}=\text{supp}(\theta_{t-1})$, $S^{t}=\text{supp}(\theta_t)$, $S^*=\text{supp}(\theta^*)$ and $I^t=S^{t}\bigcup S^{t-1} \bigcup S^*$. We can see that $|S^{t-1}|\leq k'$, $|S^{t}|\leq k'$ and $|I^t|\leq 2k'+k$. We let $\gamma=\lambda_{\max} (\mathbb{E}[xx^T])$, $\mu=\lambda_{\min} (\mathbb{E}[xx^T])$ and $\eta_0=\frac{\eta}{\gamma}$ for some $\eta$. We can easily see that $L(\cdot)$ is $\mu$-strongly convex and $\gamma$-smooth.
Then from the smooth property we have
\begin{align}
& \LD-\LT \nm \notag \\
&\leq \lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge + \frac{\gamma}{2}\|\theTR_{t}-\thet_{t-1}\|_2^2 \nm \\
&=\lge \theTR_{t,I^t}-\thet_{t-1,I^t}, \nabla L_{t-1, I^t}\rge + \frac{\gamma}{2 }\| \theTR_{t,I^t}-\thet_{t-1,I^t}\|_2^2 \nm \\
&\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t}\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t}\|_2^2+(1-\eta)\lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge \label{aeq:46}
\end{align}
First, let us focus on the third term of (<ref>).
\begin{align}
\lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge &= \lge \theTR_{t,\sunion}-\thet_{t-1, \sunion}, \nabla L_{t-1,\sunion} \rge \nm \\
&= \lge \theTR_{t,S^{t}}-\thet_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge+ \lge \theTR_\sdiffb-\thet_{t-1,\sdiffb}, \nabla L_{t-1, \sdiffb} \rge \nm \\
&= \lge \theTR_{t,S^{t}}-\thet_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge- \lge \thet_{t-1,\sdiffb}, \nabla L_{t-1, \sdiffb} \rge, \label{aeq:47}
\end{align}
where the last equality is due to that $\theTR_\sdiffb=0$.
By Lemma <ref> and the definition, we know that $\theTR_{t}$ can be written as $\theTR_{t}=\thehat_{t,S^{t}}+\phi_{t,S^{t}}$, where $\thehat_t=(\theta_{t-1}-\eta_0\tilde{\nabla} L_{t-1})_{S^{t}}$ and $\phi_t$ is a sub-Gaussian vector with variance $=O\left(\frac{d\tau_1^2(k'\tau_1^2+\tau_2^2)}{m\epsilon^2}\right)$. Thus,
\begin{align}\label{aeq:48}
& \lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge= \lge \thehat_{t, S^{t}}-\thet_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge \nm\\
&+\lge \phi_{t,S^{t}},\nabla L_{t-1, S^{t}}\rge -\lge \thet_{t-1,\sdiffb},\nabla L_{t-1, \sdiffb} \rge.
\end{align}
For the first term in (<ref>) we have
\begin{align}
& \lge \thehat_{t, S^{t}}-\thet_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge = \lge -\eta_0\tilde{\nabla} L_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge =-\frac{\eta}{\gamma} \lge \tilde{\nabla} L_{t-1,S^{t}}, \nabla L_{t-1, S^{t}} \rge \nm \\
&= -\frac{\eta}{\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2-\frac{\eta}{\gamma} \langle \tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}, \nabla L_{t-1, S^{t}} \rge \nm \\
&\leq -\frac{\eta}{\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2+ \frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2+ \frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2 \nm \\
&= -\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2+\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2 \label{aeq:49}.
\end{align}
Take (<ref>) into (<ref>) we have for $c_{1}>0$
\begin{multline}\label{aeq:50}
\begin{aligned}
\lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge \leq& -\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2+\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2\\
+\frac{1}{4c_1}\|\nabla L_{t-1, S^{t}}\|_2^2-\lge \thet_{t-1,\sdiffb},\nabla L_{t-1, \sdiffb} \rge.
\end{aligned}
\end{multline}
For the last term of (<ref>) we have
\begin{align}
&-\lge \thet_{t-1,\sdiffb},\nabla L_{t-1, \sdiffb} \rge \nm\\ &\leq \frac{\gamma}{2\eta}(\|\thet_{t-1,\sdiffb}-\frac{\eta}{\gamma}\nabla L_{t-1, \sdiffb}\|_2^2-(\frac{\eta}{\gamma})^2 \|\nabla L_{t-1, \sdiffb}\|_2^2) \nm \\
&=\frac{\gamma}{2\eta} \|\thet_{t-1,\sdiffb}-\frac{\eta}{\gamma}\nabla L_{t-1, \sdiffb}\|_2^2-\frac{\eta}{2\gamma} \|\nabla L_{t-1, \sdiffb}\|_2^2 \nm\\
&\leq \frac{\eta}{2\gamma}(1+\frac{1}{c_1})\|\nabla L_{t-1, \sdiffa}\|^2_2+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2\nm \\
&-\frac{\eta}{2\gamma} \|\nabla L_{t-1, \sdiffb}\|_2^2,\label{aeq:51}
\end{align}
where the last inequality comes from
\begin{align*}
&\|\thet_{t-1,\sdiffb}-\frac{\eta}{\gamma}\nabla L_{t-1, \sdiffb}\|_2-\frac{\eta}{\gamma}\|\nabla L_{t-1, \sdiffb}-\tilde{\nabla} L_{t-1, \sdiffb}-\phi_{t, \sdiffb}\|_2 \\
&\leq \|\thet_{t-1,\sdiffb}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1, \sdiffb}+\phi_{t, \sdiffb})\|_2\\ &\leq \|\thet_{t-1,\sdiffa}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1, \sdiffa}+\phi_{t, \sdiffa})\|_2=\frac{\eta}{\gamma}\|\tilde{\nabla} L_{t-1, \sdiffa}+\phi_{t, \sdiffa}\|_2\\
&\leq \frac{\eta}{\gamma}\|\nabla L_{t-1, \sdiffa}\|_2+\frac{\eta}{\gamma}\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|_2,
\end{align*}
where the second inequality is due to the fact that $|\sdiffa| = |\sdiffb|$. the definitions of hard thresholding, $\theta'_{t}=(\thet_{t-1}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1}+\phi_{t}))_{S^t}$, $S^t$ and $S^{t-1}$; the first equality is due to $\text{Supp}(\theta_{t-1})=S^{t-1}$
Thus we have
\begin{multline*}
\frac{\gamma}{2\eta}\|\thet_{t-1,\sdiffb}-\frac{\eta}{\gamma}\nabla L_{t-1, \sdiffb}\|^2_2 \\
\leq \frac{\eta}{2\gamma}(1+\frac{1}{c_1})\|\nabla L_{t-1, \sdiffa}\|^2_2+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2
\end{multline*}
We can easily see that
\begin{align*}
&\frac{\eta }{2\gamma}\|\nabla L_{t-1, S^{t}\backslash S^{t-1}}\|_2^2-\frac{\eta}{2\gamma} \|\nabla L_{t-1, \sdiffb}\|_2^2 -\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2\\
=& -\frac{\eta}{2\gamma} \|\nabla L_{t-1, \sdiffb}\|_2^2-\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\bigcap S^{t-1}}\|_2^2 \\
=&-\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\bigcup S^{t-1}}\|_2^2.
\end{align*}
In total
\begin{multline}\label{aeq:53}
\lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge \\
\leq -\frac{\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\bigcup S^{t-1}}\|_2^2 +(\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2+\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2\\+c_{1}\|\phi_{t,S^{t}}\|_2^2+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2
\end{multline}
Take (<ref>) into (<ref>) we have
\begin{align}
\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t}\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t}\|_2^2+(1-\eta)\lge \theTR_{t}-\thet_{t-1}, \nabla L_{t-1} \rge \nm \\
&\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t}\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t}\|_2^2-\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\bigcup S^{t-1}}\|_2^2
\nm \\& +(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2 +(1-\eta)[\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2+c_{1}\|\phi_{t,S^{t}}\|_2^2\nm \\ &+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2] \nm \\
& \leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2\nm\\
&- \frac{\eta^2}{2\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 -\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\bigcup S^{t-1}}\|_2^2 \nm \\
& +(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2 \nm +(1-\eta)[\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2+c_{1}\|\phi_{t,S^{t}}\|_2^2 \nm \\ &+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2] \nm \\
& \leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2- \frac{\eta^2}{2\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 \nm \\& -\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2 \nm\\
&+\underbrace{(1-\eta)(\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2+c_{1}\|\phi_{t,S^{t}}\|_2^2+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2)}_{N_0^t}, \label{aeq:54}
\end{align}
where the last inequality is due to $S^{t}\backslash(S^* \bigcup S^{t-1})\subseteq S^{t}\bigcup S^{t-1}$. Next we will analyze the term $ \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2$ in (<ref>).
Let $R$ be a subset of $\sdiffb$ such that $|R|=|I^t\backslash (S^*\bigcup S^{t-1})|=|S^{t}\backslash (S^{t-1}\bigcup S^*)|$. By the definition of hard thresholding, we can easily see
\begin{multline}\label{aeq:55}
\begin{aligned}
\|\theta_{t-1,R}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1,R}+ \phi_{t, R} )\|_2^2 \leq& \|(\theta_{t-1}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1}+\phi_{t} ))_{I^t\backslash (S^*\bigcup S^{t-1})}\|_2^2\\ =&\frac{\eta^2}{\gamma^2}\|(\tilde{\nabla} L_{t-1}+\phi_{t} )_{I^t\backslash (S^*\bigcup S^{t-1})}\|_2^2.
\end{aligned}
\end{multline}
Thus we have
\begin{equation}\label{aeq:56}
\begin{aligned}
&(\frac{\eta}{\gamma}) \|\nabla L_{t-1, I^t\backslash (S^*\bigcup S^{t-1})}\|_2\\
\geq& \underbrace{\|\theta_{t-1,R}-\frac{\eta}{\gamma}\nabla L_{t-1, R}\|_2}_{a}
-\frac{\eta}{\gamma}(\underbrace{\|\tilde{\nabla} L_{t-1,R}-\nabla L_{t-1, R}+\phi_{t, R}\|_2}_b\\
+&\underbrace{\|\nabla L_{t-1, I^t\backslash (S^*\bigcup S^{t-1})}-\tilde{\nabla} L_{t-1,I^t\backslash (S^*\bigcup S^{t-1})}-\phi_{t, I^t\backslash (S^*\bigcup S^{t-1})}\|_2}_c)\\
\end{aligned}
\end{equation}
Then we have for any $c_2>0$
\begin{align}
&\frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2\nm\\
&\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\gamma}{2}( \frac{\eta^2}{\gamma^2}(b+c)^2+a^2-\frac{2\eta}{\gamma}(b+c)a) \nm\\
&\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\gamma}{2}(1-\frac{1}{c_2})a^2+(2c_2-\frac{1}{2})\frac{\eta^2}{\gamma}(b+c)^2 \nm\\
&=\frac{\gamma}{2}\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+\frac{\gamma}{2c_2}\|\theta_{t-1,R}-\frac{\eta}{\gamma}\nabla L_{t-1, R}\|_2^2 \nm\\
& + \underbrace{(4c_2-1)\frac{\eta^2}{\gamma}(\|\tilde{\nabla} L_{t-1,R}-\nabla L_{t-1, R} + \phi_{t,I^{t}}\|_2^2+\|\nabla L_{t-1, I^t\backslash (S^*\bigcup S^{t-1})}-\tilde{\nabla} L_{t-1,I^t\backslash (S^*\bigcup S^{t-1})} -\phi_{t, I^t\backslash (S^*\bigcup S^{t-1})}\|_2^2)}_{N^t_1}\label{aeq:56}\\
&\leq \frac{\gamma}{2}\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+\frac{\gamma}{c_2}\|\theta_{t-1,R}-\frac{\eta}{\gamma}(\tilde{\nabla} L_{t-1, R}+\phi_{t, R}) \|_2^2 \nm\\
& +\underbrace{\frac{\eta^2}{c_2\gamma}\|\nabla L_{t-1, I^t\backslash R}-(\tilde{\nabla} L_{t-1, R}+\phi_{t, R})\|_2^2+N^t_1}_{N_2^t}\label{aeq:57} \\
&= \frac{\gamma}{2}\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+N_2^t, \label{aeq:58}
\end{align}
where (<ref>) is due to that $\theta'_{t-1, R}=0$, thus $\|\theta'_{t-1, R}-(\theta_{t-1,R}-\frac{\eta}{\gamma}\nabla L_{t-1, R})\|_2= \|\theta_{t-1,R}-\frac{\eta}{\gamma}\nabla L_{t-1, R}\|_2$. In the following, we will consider the first term in (<ref>).
In Lemma <ref>, take $v=\thet_{t-1,I^t\backslash R}-\frac{\eta}{\gamma} (\tilde{\nabla} L_{t-1,I^t\backslash R}+\phi_{t-1,I^t\backslash R})$, $\tilde{v}=\text{Trunc}(v, k')=\theta'_{t-1, I^t \backslash R}$, $I=I^t\backslash R$, $v^*=\theta^*_{I^t\backslash R}=\theta^*$, we have
\begin{equation*}
\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}-\frac{\eta}{\gamma} (\tilde{\nabla} L_{t-1,I^t\backslash R}+\phi_{t-1,I^t\backslash R})\|_2^2\leq \frac{|I^t\backslash R|-k'}{|I^t\backslash R|-k}\|\theta^*-\thet_{t-1,I^t\backslash R}-\frac{\eta}{\gamma} ( \tilde{\nabla} L_{t-1,I^t\backslash R}+\phi_{{t-1,I^t\backslash R}} )\|_2^2.
\end{equation*}
Then we have
\begin{align*}
&(1-\frac{1}{c_3})\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2-(c_3-1)\frac{\eta^2}{\gamma^2}\|\nabla L_{t-1, I^t\backslash R}-\tilde{\nabla} L_{t-1,I^t\backslash R}-\phi_{t-1,I^t\backslash R}\|_2^2 \nm \\&\leq
\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \tilde{\nabla} L_{t-1,I^t\backslash R}\|_2^2 \\
&\leq \frac{|I^t\backslash R|-k'}{|I^t\backslash R|-k}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \tilde{\nabla} L_{t-1,I^t\backslash R}\|_2^2 \\
&\leq \frac{|I^t\backslash R|-k'}{|I^t\backslash R|-k}\left((1+\frac{1}{c_3})\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2 +(1+c_3)\frac{\eta^2}{\gamma^2}\|\nabla L_{t-1, I^t\backslash R}-\tilde{\nabla} L_{t-1,I^t\backslash R}-\phi_{t-1,I^t\backslash R} \|_2^2 \right)
\end{align*}
Since $|I^t\backslash R|\leq 2k'+k$ and $k'\geq k$, we have $\frac{|I^t\backslash R|-k'}{|I^t\backslash R|-k}\leq \frac{k'+k}{2k'}\leq \frac{2k'}{k+k'}$. Thus
\begin{multline*}
\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2\leq \frac{2k}{k+k^{\prime}}\frac{c_3+1}{c_3-1}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2\\+ ((1+c_3)\frac{2k}{k+k'}+c_3-1)\frac{\eta^2}{\gamma^2}\|\nabla L_{t-1, I^t\backslash R}-\tilde{\nabla} L_{t-1,I^t\backslash R}-\phi_{t-1,I^t\backslash R} \|_2^2
\end{multline*}
Take $c_3=5$ and $k'=O(k)$, we have
\begin{align}\label{aeq:60}
&\|\theta'_{t,I^t\backslash R}-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2\leq \frac{3}{2}\frac{2k}{k+k^{\prime}}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2\nm \\
&+\underbrace{O(\frac{\eta^2}{\gamma^2}\|\nabla L_{t-1, I^t\backslash R}-\tilde{\nabla} L_{t-1,I^t\backslash R}-\phi_{t-1,I^t\backslash R} \|_2^2)}_{N_3^t}.
\end{align}
Take (<ref>) into (<ref>) we have
\begin{align}
&\frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2\nm \\
&\leq \frac{3\gamma}{2}\frac{k}{k+k'}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+N_2^t+\gamma N_3^t. \label{aeq:64}
\end{align}
\begin{align}
&\frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2\nm \\
&\leq \frac{3\gamma s^*}{2(s+s^*)}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+ \frac{\gamma}{2c}(1+\frac{1}{c})\|\frac{\eta}{\gamma}\nabla L_{t-1, I^t\backslash (S^*\bigcup S^{t-1})}\|_2^2+\frac{\gamma}{2}\|\phi_{t,S^{t}}\|_2^2 \nm \\
& + N_1^t+N_3^t\\
&= \frac{3\gamma s^*}{2(s+s^*)}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2+ \frac{\gamma}{2c}(1+\frac{1}{c})\|\frac{\eta}{\gamma}\nabla L_{t-1, S^{t}}\|_2^2 \nm \\
&\qquad +\frac{\gamma}{2}\|\phi_{t,S^{t}}\|_2^2+ N_1^t+N_3^t \\
&= \frac{3 s^*}{s+s^*}(
\eta \lge \theta^*-\thet_{t-1}, \nabla L_{t-1} \rge +\frac{\gamma}{2}\|\theta^*-\thet_{t-1}\|_2^2+\frac{\eta^2}{2c \gamma }\|\nabla L_{t-1, I^t}\|_2^2)+ \frac{\eta^2}{2c\gamma}(1+\frac{1}{c})\|\nabla L_{t-1, S^{t}}\|_2^2 \nm \\
&\qquad +\frac{\gamma}{2}\|\phi_{t,S^{t}}\|_2^2+ N_1^t+N_3^t \\
&\leq \frac{3 s^*}{s+s^*}(
\eta (L(\theta^*)-\LT
)+\frac{\gamma-\eta \mu}{2}\|\theta^*-\thet_{t-1}\|_2^2+\frac{\eta^2}{2c \gamma }\|\nabla L_{t-1, I^t}\|_2^2)+ \frac{\eta^2}{2c\gamma}(1+\frac{1}{c})\|\nabla L_{t-1, S^{t}}\|_2^2 \nm \\
&\qquad +\underbrace{\frac{\gamma}{2}\|\phi_{t, S^{t}}\|_2^2+ N_1^t+N_3^t }_{N_2^t}. \label{aeq:64}
\end{align}
Take (<ref>) into (<ref>) we have
\begin{align}
&\LD-\LT \nm \\
&\leq \frac{\gamma}{2}\|\theTR_{t,I^t}-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t }\|_2^2-\frac{\eta^2}{2\gamma}\|\nabla L_{t-1, I^t\backslash (S^{t-1}\bigcup S^*)}\|_2^2- \frac{\eta^2}{2\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 \nm \\& -\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2 +N_0^t \nm \\
&\leq \frac{3\gamma }{2}\frac{k}{k'+k}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2- \frac{\eta^2}{2\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 \nm \\& -\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2+N_0^t+N_2^t+\gamma N_3^t.
\end{align}
Note that when $\eta\geq \frac{1}{2}$, there exists a sufficiently large $c_1$ is such that $\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1}\leq \frac{\eta}{4\gamma}$, we have
\begin{align*}
&(1-\eta) (\frac{1}{4c_1}+ \frac{\eta}{2\gamma c_1})\|\nabla L_{t-1, S^{t}}\|_2^2\leq
\frac{\eta(1-\eta)}{4\gamma} \|\nabla L_{t-1, S^{t}}\|_2^2
\\
& \leq \frac{\eta^2}{4\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 +\frac{(1-\eta)\eta}{4\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2
\end{align*}
\begin{align*}
&\LD-\LT \nm \\
\leq& \frac{3\gamma }{2}\frac{k}{k'+k}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2- \frac{\eta^2}{2\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 \nm \\
-&\frac{(1-\eta)\eta}{2\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+\frac{(1-\eta)}{4c}\|\nabla L_{t-1, S^{t}}\|_2^2+N_0^t+N_2^t+\gamma N_3^t \\
\leq &\frac{3\gamma }{2}\frac{k}{k'+k}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2- \frac{\eta^2}{4\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2 \nm \\
-&\frac{(1-\eta)\eta}{4\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+N_0^t+N_2^t+\gamma N_3^t
\end{align*}
It is notable that by strong convexity
\begin{align*}
&\frac{3\gamma }{2}\frac{k}{k'+k}\|\theta^*-\thet_{t-1,I^t\backslash R}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t\backslash R}\|_2^2 \\
&\leq \frac{3\gamma }{2}\frac{k}{k'+k}\|\theta^*-\thet_{t-1,I^t}+\frac{\eta}{\gamma} \nabla L_{t-1, I^t}\|_2^2 \\
&= \frac{3\gamma }{2}\frac{k}{k'+k} (\|\theta^*-\thet_{t-1,I^t\backslash R}\|_2^2+\frac{\eta^2}{\gamma^2}\| \nabla L_{t-1, I^t}\|_2^2+\frac{2\eta}{\gamma}\langle \theta^*-\thet_{t-1,I^t}, \nabla L_{t-1, I^t} \rangle) \\
&= \frac{3\gamma }{2}\frac{k}{k'+k} (\|\theta^*-\thet_{t-1,I^t\backslash R}\|_2^2+\frac{\eta^2}{\gamma^2}\| \nabla L_{t-1, I^t}\|_2^2+\frac{2\eta}{\gamma}\langle \theta^*-\thet_{t-1}, \nabla L_{t-1} \rangle)\\
&\leq \frac{3k}{k'+k} \big(\frac{\gamma}{2}\|\theta^*-\thet_{t-1}\|_2^2+ \frac{\eta^2}{2 \gamma }\|\nabla L_{t-1, I^t}\|_2^2+\eta (L(\theta^*)-\LT)-\frac{\eta\mu}{2}\|\theta^*-\theta_{t-1}\|_2^2\big)
\end{align*}
Take $\eta=\frac{2}{3}$, $k'=72\frac{\gamma^2}{\mu^2}k$ so that $\frac{3 k}{k'+k}\leq \frac{\mu^2}{24\gamma(\gamma-\eta \mu)}\leq \frac{1}{8}$, we have
\begin{align}
\leq& \frac{3 k}{k+k^{\prime}}(
\eta (L(\theta^*)-\LT)+\frac{\gamma-\eta \mu}{2}\|\theta^*-\thet_{t-1}\|_2^2+\frac{\eta^2}{2 \gamma }\|\nabla L_{t-1, I^t}\|_2^2) \nm \\
-& \frac{\eta^2}{4\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2-\frac{(1-\eta)\eta}{4\gamma} \|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2 +N_0^t+N_2^t+\gamma N_3^t\nm \\
\leq& \frac{2 k}{k'+k}(L(\theta^*)-\LT)+ \frac{\mu^2}{48\gamma}\|\theta^*-\theta_{t-1}\|_2^2+\frac{1}{36\gamma}\|\nabla L_{t-1, I^t}\|_2^2 \nm \\
-&\frac{1}{9\gamma}\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2-\frac{1}{18\gamma}\|\nabla L_{t-1, S^{t}\backslash(S^* \bigcup S^{t-1})}\|_2^2+N_0^t+N_2^t+\gamma N_3^t \nm \\
\leq & \frac{2 k}{k+k'}(L(\theta^*)-\LT)-\frac{3}{36\gamma}(\|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2-\frac{\mu^2}{4}\|\theta^*-\theta_{t-1}\|_2^2)+N_0^t+N_2^t+\gamma N_3^t \label{aeq:68}\\
\leq &(\frac{2 k}{k+k'}+\frac{\mu}{24\gamma})(L(\theta^*)-\LT)+N_0^t+N_2^t+\gamma N_3^t \label{aeq:69}.
\end{align}
Where (<ref>) is due to the following lemma:
[Lemma 6 in [28]]
\begin{equation}
|\nabla L_{t-1, (S^{t-1}\bigcup S^*)}\|_2^2-\frac{\mu^2}{4}\|\theta^*-\theta_{t-1}\|_2^2\geq \frac{\mu}{2}(\LT-L(\theta^*)).
\end{equation}
\begin{equation*}
\LD- L(\theta^*)\leq (1-\frac{5}{72}\frac{\mu}{\gamma})( \LT- L(\theta^*))+N_0^t+N_2^t+\gamma N_3^t.
\end{equation*}
Next, we will bound the term $N_0^t+N_2^t+\gamma N_3^t$. For $N_0^t$ we have
\begin{align*}
N_0^t&=(1-\eta)(\frac{\eta}{2\gamma}\|\tilde{\nabla} L_{t-1,S^{t}}-\nabla L_{t-1, S^{t}}\|_2^2\\
&+c_{1}\|\phi_{t,S^{t}}\|_2^2+\frac{2\eta}{\gamma}(1+c_1)\|\nabla L_{t-1, \sdiffa}-\tilde{\nabla} L_{t-1, \sdiffa}-\phi_{t, \sdiffa}\|^2_2)\\
&=O(\frac{1}{\gamma}k' \|\tilde{\nabla} L_{t-1}-\nabla L_{t-1}\|_\infty^2+\gamma k'\|\phi_t\|_\infty^2).
\end{align*}
By (<ref>), we know that with probability at least $1-\delta'$
\begin{equation}
\|\tilde{\nabla} L_{t-1}-\nabla L_{t-1}\|_\infty \leq O(\frac{\sigma^2 \log n \log \frac{d}{\delta'}}{\sqrt{m}}).
\end{equation}
Moreover, by Lemma <ref> we have with probability at least $1-\delta'$
\begin{equation}
\|\phi_t\|_\infty \leq O\left(\frac{(\tau_1^2\sqrt{dk'}+\tau_1\tau_2\sqrt{d})\sqrt{\log \frac{d}{\delta'}} }{\sqrt{m}\epsilon}\right).
\end{equation}
Thus, with probability at least $1-\delta'$ we have
\begin{equation*}
N_0^t=O(\frac{\sigma^4 d{k'}^2\log d/\delta' \log^2 n}{m\epsilon^2}).
\end{equation*}
Similarly, we have
\begin{equation*}
N_2^t, N_3^t=O(\frac{\sigma^4 d{k'}^2 \log d/\delta' \log^2 n}{m\epsilon^2}).
\end{equation*}
Thus we have with probability at least $1-\delta'$
\begin{equation}
\LD- L(\theta^*)\leq (1-\frac{5}{72}\frac{\mu}{\gamma})( \LT- L(\theta^*))+O(\frac{\sigma^4 d{k'}^2\log d/\delta' \log^2 n}{m\epsilon^2}).\label{aeq:73}
\end{equation}
In the following we will assume the above event holds. We note that by our model for any $\theta$
\begin{equation*}
\gamma \|\theta-\theta^*\|_2^2\geq L(\theta) - L(\theta^*)\geq \mu \|\theta-\theta^*\|_2^2.
\end{equation*}
In the following we will show that $\theta_t=\theTR_{t}$ for all $t$. We will use induction, assume $\theta_i=\theTR_{i}$ holds for all $i\in [t-1]$, we will show that it will also true for $t$. Use (<ref>) for $i\in [t-1]$ we have
\begin{align}
\mu \|\theTR_{t}-\theta^*\|_2^2&\leq \LD- L(\theta^*)\nm \leq (1-\frac{5}{72}\frac{\mu}{\gamma})( \LT- L(\theta^*))+O(\frac{\sigma^4 d{k'}^2\log d/\delta' \log^2 n}{m\epsilon^2})\\ &\leq(1-\frac{5}{72}\frac{\mu}{\gamma})^t( L(\theta_0)- L(\theta^*))+
O(\frac{\sigma^4 d{k'}^2\log d/\delta' \log^2 n}{m\epsilon^2}) \nm \\
&\leq \gamma(1-\frac{5}{72}\frac{\mu}{\gamma})^{t-1} \|\theta_0-\theta^*\|_2^2
+O(\frac{\gamma}{\mu}\frac{\sigma^4 d{k'}^2\log d \log^2 n}{m\epsilon^2})\nm
\end{align}
When $\|\theta_0-\theta^*\|_2^2\leq \frac{1}{2}\frac{\mu}{\gamma}$, and $n$ is large enough such that
\begin{align*}
&n\geq \tilde{\Omega}(\frac{\gamma}{\mu^2} \frac{k'^2 d \sigma^4 T}{ \epsilon^2})
\end{align*}
Then $\|\theTR_{t}\|_2\leq \|\theta^*\|_2+\sqrt{\frac{1}{2}+\frac{1}{2}}\leq 2$. Thus $\theta_t=\theTR_{t}$. So we have with probability at least $1-\delta'$
\begin{align}
&\mu \|\theTR_{t}-\theta^*\|_2^2\leq L(\theta^{T})- L(\theta^*)\leq(1-\frac{5}{72}\frac{\mu}{\gamma})^T( L(\theta_0)- L(\theta^*))+O(\frac{\gamma}{\mu}\frac{\sigma^4 d{k'}^2 T\log \frac{dT}{\delta'} \log^2 n}{n\epsilon^2}) \nm
\end{align}
Thus, take $T=\tilde{O}(\frac{\gamma}{\mu}\log n)$ and $k'= O( (\frac{\gamma}{\mu})^2 k)$ we have the result. |
# Introducing DictaLM \- A Large Generative Language Model for Modern Hebrew
Shaltiel Shmidman1,†, Avi Shmidman1,2,‡, Amir David Nissan Cohen2,†, Moshe
Koppel1,2,†
1DICTA / Jerusalem, Israel 2Bar Ilan University / Ramat Gan, Israel
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
We present DictaLM, a large-scale language model tailored for Modern Hebrew.
Boasting 7B parameters, this model is predominantly trained on Hebrew-centric
data. As a commitment to promoting research and development in the Hebrew
language, we release both the foundation model and the instruct-tuned model
under a Creative Commons license111For specifics on the license, visit
https://creativecommons.org/licenses/by-sa/4.0/. Concurrently, we introduce
DictaLM-Rab, another foundation model geared towards Rabbinic/Historical
Hebrew. These foundation models serve as ideal starting points for fine-tuning
various Hebrew-specific tasks, such as instruction, Q&A Cohen et al. (2023),
sentiment analysis Amram et al. (2018), and more Bareket and Tsarfaty (2021).
This release represents a preliminary step, offering an initial Hebrew LLM
model for the Hebrew NLP community to experiment with.
## 1 Introduction
Language models have revolutionized the realm of natural language processing,
facilitating significant advancements in tasks ranging from sentiment analysis
to machine translation. As the breadth and depth of these models expand, so
does the aspiration for linguistic diversity. Yet, while the majority of
state-of-the-art models cater predominantly to widely spoken languages, there
exists a vast landscape of languages and dialects that are underrepresented in
currently existing large-scale language models. Hebrew is one such language.
In this paper, we make strides to bridge this gap by introducing DictaLM \-
the first large-scale language model crafted for Modern Hebrew. By leveraging
a dataset dominated by Hebrew-centric content, our endeavor was not only to
construct a model adept at understanding and generating Modern Hebrew but also
to lay down a foundation that facilitates further advancements in the field.
As part of this initiative, we also present DictaLM-Rab, a parallel model
pretrained for Rabbinic/Historical Hebrew, thereby encompassing the vast
chronological spectrum of the Hebrew language. This release serves as a
preliminary step, providing an initial tentative version to the Hebrew NLP
community as a foundation for further refinements, adaptations, and
collaborative enhancements. Figure 1 demonstrates example output from the
instruct-tuned model.
Figure 1: We present two instances of DictaLM utilization: in the first
instance, the model exhibits common sense reasoning, while in the second, it
displays worldly knowledge.
## 2 Datasets
In this section, we elucidate the datasets employed for training and fine-
tuning DictaLM. The assemblage of data, amassing a total of 7.5 billion
tokens, originates from a mixture of authentic sources; no synthetic data was
added. The pre-training phase is followed by a fine-tuning stage through
instruct datasets derived from Hebrew Question-Answering datasets and a
translated version of the MPT Instruct Dataset.
### 2.1 Pre-training Data
The dataset is built up of several different components:
C4 [80%]. We start with the HeDC4 corpus released by Shalumov and Haskey
(2023), and continue further cleaning it. We removed approximately 15% of the
corpus using various techniques including histograms, gibberish detectors, as
well as removing sentences that had a very high perplexity when running
through a Modern Hebrew BERT model. In addition, we limited our training
corpus to contain only words in English and Hebrew, and all other languages
were reduced to a designated <foreign> token to avoid cluttering the tokenizer
with non-Hebrew tokens. The resulting corpus contains approximately 6B byte-
pair tokens.
Other sources [20%]. We collected data from various other sources including
news sites, blogs, tv and movie subtitles, novels, and more. This data was
also run through a similar cleaning process to the C4 corpus, as described
above, and resulted in an additional 1.5B byte-pair tokens.
#### 2.1.1 Instruct Data
Our instruct-tuning data contains a mixture of 2 different datasets, each
processed and modified in order to teach the model to follow as many different
instructions as possible.
QA Datasets. We take the HeQ Cohen et al. (2023) and ParaShoot Keren and Levy
(2021) training datasets and format them as instructions. The prompt contains
the context paragraph followed by the question, with a system instruction. The
system instruction starts with a general instruction (in Hebrew) stating
"Please read the following paragraph and answer the question that comes
after", and 60% of the time also instructs the system to format a specific
type of response (e.g., "Short and to the point", "Please cite the sentence to
support your answer", and more). We list a few examples in Appendix A.
Translated MPT Instruct. We took the MPT Instruct Dataset from
huggingface222https://huggingface.co/datasets/mosaicml/dolly_hhrlhf and ran it
through a translation API. We then reformatted the prompt to remove the
constant structure, and left the question only. We then added in each question
three times: Once with no system prompt, and twice with two different prompts
chosen based on the length of the response, asking the model to be concise,
expand, answer in X sentences, etc. We list a few examples in Appendix B.
## 3 Model architecture
### 3.1 Tokenizer
A major problem we encountered when attempting to use other multilingual LLMs
for Hebrew was the tokenization. When the corpus contains a very small
percentage of a language, then the number of tokens representing that language
in the vocabulary is significantly reduced. In addition, due to the nature of
UTF-8 encoding, byte-pair tokenization methods result in even scarcer
representation of Hebrew in the vocabulary. As can be seen in OpenAI’s GPT-3
tokenizer333https://platform.openai.com/tokenizer, if one inserts a few
paragraphs of Hebrew text, the tokenizer will average 1.1 tokens per
character.
We train our tokenizer using the byte-pair encoding (BPE) algorithm Sennrich
et al. (2015) on our cleaned corpus with a vocabulary size of 56000. The
resulting tokenizer had a ratio of approximately 1.3 tokens per word.
### 3.2 Architecture
In this section, we detail the architectural framework of DictaLM. Following
recent work on large language models, our network is based on the transformer
architecture Vaswani et al. (2017). Our architecture encompasses several
enhancements aimed at boosting training stability and overall performance:
Normalization. To improve training stability and balance the input, we
normalize the input of each transformer layer before and after the attention
calculation. We use the LayerNorm1P normalization with $\epsilon=1e-5$, which
is a slightly modified version of the FastLayerNorm normalization offered by
NVIDIA’s APEX library444https://github.com/NVIDIA/apex.
GeLU Activation. As reported by Hendrycks and Gimpel (2023), we use the GeLU
activation function.555We considered using other activations (such as SwiGLU
Shazeer (2020)), but in the end we went with GeLU
Rotary Embeddings. Shown to be effective for extending the sequence length
without a performance trase-off, we use rotary positional embedding (RoPE)
with a $0.5\%$ dimension percentage, introduced by Su et al. (2022), at each
layer of the network.
Separate embedding and output weights. As shown by Welch et al. (2020),
separating the embeddings and the output weights leads to better performance.
### 3.3 Training Details and Hyperparameters
We trained our model using the NeMo framework666https://github.com/NVIDIA/NeMo
which is highly optimized for training compute-heavy machine learning models
on NVIDIA hardware. We pre-trained the model on 8 H100 GPUs with tensor
parallel size of 2 for a total of 150 hours completing 2.5 epochs ($\sim$18.5B
tokens), and then fine-tuning for instructions for 8 hours. The training was
done in a combination of bf16 and fp8 precision using NVIDIA’s transformer
engine777https://github.com/NVIDIA/TransformerEngine. The training was done
with a global batch size of 128. We used the FusedAdam optimizer, with an
initial learning rate of $0.00016$, betas of $0.9,0.95$ and the Cosine-
Annealing schedule with a warmup of 750 steps and a minimum learning rate of
$1e-5$. The details for the model size are listed in Table 1.
Max Sequence Length | 2048
---|---
Num Layers | 32
Hidden Size | 4096
Intermediate Size | 10880
Attention Heads | 32
Table 1: Model size
### 3.4 DictaLM-Rab Model
In addition to the model we described above, we also trained a model DictaLM-
Rab for use with Rabbinic Hebrew tasks. We used the same approach as above,
adjusting the input corpus to contain a large sampling of Rabbinic Hebrew
data.
Specifically, we added a corpus of 1.2B tokens of Rabbinic Hebrew texts taken
from various sources (e.g. Sefaria888https://www.sefaria.org.il/,
Dicta999https://library.dicta.org.il/). We combined this corpus together with
the modern Hebrew corpus that we described above, sampling the data such that
fifty percent of the training sequences would be from the Rabbinic Hebrew
corpus (with oversampling).
The model uses the same tokenizer as DictaLM, and was trained for a total of
1.5 iterations ($\sim$12.5B tokens).
We are pleased to also release this foundation model, tailored to benefit
researchers working on Rabbinic Hebrew. This model can be used as a base model
for fine-tuning on specific tasks relevant to the Rabbinic Hebrew domain. Our
internal experiments reveal encouraging results with Rabbinic texts, details
of which will be shared in forthcoming publications.
## 4 Drawbacks
Our model was trained on the full dataset without any censorship for offensive
or biased material, and therefore it may generate sentences that are offensive
to some users.
Also, we would like to highlight that this project is in its alpha phase.
While we are releasing DictaLM to facilitate research endeavors, and while we
believe that it can serve as a useful foundation for specific fine-tuned tasks
in the realm of Hebrew NLP, we acknowledge that the quality of the model does
not yet match industry standards.
## 5 Conclusion
We are pleased to present the three models described within this paper: the
two foundational models (suitable as base models for further fine-tuning for
tasks concerning both Modern and Rabbinic Hebrew), and the instruct model,
fine-tuned to address instruction prompts in Modern Hebrew. The public release
of these models aims to contribute to the advancement of research and
development within the Hebrew NLP domain. The models can be accessed via the
following links:
* •
Foundation model DictaLM: https://huggingface.co/dicta-il/dictalm-7b
* •
Instruct model DictaLM-Instruct: https://huggingface.co/dicta-
il/dictalm-7b-instruct
* •
Foundation model for Rabbinic Hebrew DictaLM-Rab:
https://huggingface.co/dicta-il/dictalm-rab-7b
## References
* Amram et al. (2018) Adam Amram, Anat Ben David, and Reut Tsarfaty. 2018. Representations and architectures in neural sentiment analysis for morphologically rich languages: A case study from Modern Hebrew. In _Proceedings of the 27th International Conference on Computational Linguistics_ , pages 2242–2252, Santa Fe, New Mexico, USA. Association for Computational Linguistics.
* Bareket and Tsarfaty (2021) Dan Bareket and Reut Tsarfaty. 2021. Neural Modeling for Named Entities and Morphology (NEMO2). _Transactions of the Association for Computational Linguistics_ , 9:909–928.
* Cohen et al. (2023) Amir DN Cohen, Hilla Merhav Fine, Yoav Goldberg, and Reut Tsarfaty. 2023. Heq: a large and diverse hebrew reading comprehension benchmark.
* Hendrycks and Gimpel (2023) Dan Hendrycks and Kevin Gimpel. 2023. Gaussian error linear units (gelus).
* Keren and Levy (2021) Omri Keren and Omer Levy. 2021. Parashoot: A hebrew question answering dataset. In _Proceedings of the 3rd Workshop on Machine Reading for Question Answering_ , pages 106–112.
* Sennrich et al. (2015) Rico Sennrich, Barry Haddow, and Alexandra Birch. 2015. Neural machine translation of rare words with subword units. _CoRR_ , abs/1508.07909.
* Shalumov and Haskey (2023) Vitaly Shalumov and Harel Haskey. 2023. Hero: Roberta and longformer hebrew language models. _arXiv:2304.11077_.
* Shazeer (2020) Noam Shazeer. 2020. Glu variants improve transformer.
* Su et al. (2022) Jianlin Su, Yu Lu, Shengfeng Pan, Ahmed Murtadha, Bo Wen, and Yunfeng Liu. 2022\. Roformer: Enhanced transformer with rotary position embedding.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. _CoRR_ , abs/1706.03762.
* Welch et al. (2020) Charles Welch, Rada Mihalcea, and Jonathan K. Kummerfeld. 2020. Improving low compute language modeling with in-domain embedding initialisation.
## Appendix A Appendix: Instruct Examples from QA Datasets
## Appendix B Appendix: Instruct Examples from Translated MPT-Instruct
|
# Spectra of infinite graphs with summable weight functions
Michael Farber School of Mathematical Sciences
Queen Mary University of London
E1 4NS London
UK<EMAIL_ADDRESS>and Lewin Strauss School of Mathematical Sciences
Queen Mary University of London
E1 4NS London
UK<EMAIL_ADDRESS>
###### Abstract.
In this paper we study spectra of Laplacians of infinite weighted graphs.
Instead of the assumption of local finiteness we impose the condition of
summability of the weight function. Such graphs correspond to reversible
Markov chains with countable state spaces. We adopt the concept of the Cheeger
constant to this setting and prove an analogue of the Cheeger inequality
characterising the spectral gap. We also analyse the concept of the dual
Cheeger constant originally introduced in [1], which allows estimating the top
of the spectrum. In this paper we also introduce a new combinatorial
invariant, ${\sf k}(G,m)$, which allows a complete characterisation of
bipartite graphs and measures the asymmetry of the spectrum (the Hausdorff
distance between the spectrum and its reflection at point $1\in\mathbb{R}$).
We compare ${\sf k}(G,m)$ to the Cheeger and the dual Cheeger constants.
Finally, we analyse in full detail a class of infinite complete graphs and
their spectra.
###### Key words and phrases:
Infinite weighted graph; spectrum; Laplace operator; random walk; Cheeger
contant; dual Cheeger constant
###### 1991 Mathematics Subject Classification:
05C50; 05C63; 05C48; 05C81
Both authors were partially supported by a grant from the Leverhulme Trust
Data sharing is not applicable to this article as no datasets were generated
or analysed during the current study.
## 1\. Introduction
Spectral graph theory is a well-developed field of mathematics lying at the
crossroads between combinatorics, analysis and geometry. It has multiple
applications in engineering and computer science [3]. The celebrated Cheeger
inequalities relate the geometry of a graph with the spectral properties of
the graph Laplacian [4]. The Cheeger inequalities play a crucial role in
theory of expander graphs [5], which are sequences of finite graphs with
certain asymptotic properties. The literature on spectral graph theory
contains many important results about finite and infinite graphs and their
spectral properties, see for instance [3, 7, 11].
In this paper, we consider countable weighted graphs that are not necessarily
locally finite but we impose an assumption of summability on the weights, i.e.
we require the sum of all edge weights to be finite. Such summable weighted
graphs can be thought of as representations of reversible countable Markov
chains [11].
We introduce three geometric constants and analyse their bearing on the
spectral properties of the normalised graph Laplacian of summable weighted
graphs. The Cheeger constant and the dual Cheeger constant introduced in this
paper can be compared to the invariants studied in [1]. There are several
significant differences between [1] and our approach: (a) in [1] the authors
study a Dirichlet type Laplace operator and their approach is applicable to
locally finite graphs only; (b) The Cheeger constant defined in [1] measures
bottom of the spectrum which is automatically $0$ for summable weighted graphs
studied in this paper. Our Cheeger constant measures the spectral gap rather
than bottom of the spectrum.
Our new combinatorial invariant, ${\sf k}(G,m)$, affords a complete
characterisation of bipartite graphs: it vanishes if and only if the graph is
bipartite. We also show that the new invariant ${\sf k}(G,m)$ allows
estimating the asymmetry of the spectrum (Theorem 25).
In section 3 we show several examples of graphs $G$ such that different
summable weight functions on $G$ have either vanishing or non-vanishing
spectral gap illustrating the fact that the property to have a non-vanishing
spectral gap (i.e. “expansion”) strongly depends on the weight function (and
not just on the underlying graph).
In more detail, our main results can be summarised as follows. Let $(G,m)$
denote a connected summable weighted graph, and let
$\sigma(\Delta)\subset[0,2]$ denote the spectrum of the associated Laplacian
$\Delta$. Summability implies $0\in\sigma(\Delta)$. We provide geometric
estimates for the spectral gap and top of the spectrum,
$\lambda_{gap}(\Delta)=\inf\\{\sigma(\Delta)-\\{0\\}\\}\quad\text{ and
}\quad\lambda_{top}=\sup\\{\sigma(\Delta)\\}.$
Namely, we define a Cheeger constant $h(G,m)$ and a dual Cheeger constant
$\overline{h}(G,m)$, and we prove (cf. Theorem 12)
$\displaystyle 1-\sqrt{1-h(G,m)^{2}}\,\leq\,$
$\displaystyle\lambda_{gap}(\Delta)\leq 2\cdot h(G,m),$ $\displaystyle
2\cdot\overline{h}(G,m)\,\leq\,$ $\displaystyle\lambda_{top}(\Delta)\leq
1+\sqrt{1-\big{(}1-\overline{h}(G,m)\big{)}^{2}}.$
Moreover, we define a new geometric constant ${{\sf k}}(G,m)$ and show that
the Hausdorff distance between $\sigma(\Delta)\subset[0,2]$ and its reflection
around 1 is bounded above by $2\cdot{{\sf k}}(G,m)$ (Theorem 25).
Finally, in sections 8, 9, and 10, we analyse a rich class of infinite
complete graphs whose spectra admit a particularly detailed description.
We are not in a position to survey the vast literature which pertains to
various classes of graphs, various Cheeger-type combinatorial constants,
various graph Laplacians, and various aspects of the spectra of these
Laplacians. As far as we know the results contained in our paper are new and
are not contained in any previously published articles.
We may mention [2] where the authors use the concept of intrinsic metrics and
develop a comprehensive framework for countable weighted graphs, of which our
model is a special case. The authors introduce a Cheeger-type constant
(distinct from ours) and use it to bound the bottom of the spectrum of the
graph Laplacian. Our assumption of summability implies that the bottom of the
spectrum is $0$. As another example we may mention Theorem 3.5 of [9] which
provides a lower bound of the spectral gap of a normalised Laplacian, but in
[9] the underlying graphs are implicitly assumed to be locally finite as
follows from Definition 2.2 of [9].111For any oriented edge $vw$, the ratio
$i(wv)/i(vw)$ is bounded if and only if $\mu(v)/\mu(w)$ is bounded. But $\mu$
is assumed to be summable over vertices, so the vertex $v$ can only have
finitely many adjacencies.
The authors thank Norbert Peyerimhoff for helpful advice.
## 2\. Summable weighted graphs
### 2.1. Definitions
A graph is a 1-dimensional simplicial complex. For a graph $G$, we denote by
$\operatorname{V}(G)$ and $\operatorname{E}(G)$ the sets of vertexes and
edges, respectively. We say that $G$ is countable if the vertex set
$\operatorname{V}(G)$ is either finite or countably infinite. A weight
function on $G$ is a function $m:\operatorname{E}(G)\to(0,\infty)$. A weighted
graph is a pair $(G,m)$ consisting of a graph $G$ and a weight function $m$ on
$G$.
###### Definition 1.
We shall say that a countable weighted graph $(G,m)$ is summable if the sum of
all edge weights is finite, $\sum_{e\in\operatorname{E}(G)}m(e)<\infty$.
The weight function of a summable weighted graph $(G,m)$ naturally extends to
the vertexes: we set $m(v)=\sum_{v\in e}m(e).$ In other words, the weight of a
vertex is the sum of the edge weights over all edges that are incident to it
(”weighted degree”). According to this definition, a vertex has weight $0$ iff
it is isolated.
Below we shall consider only graphs without isolated vertexes; we shall have
$m(v)>0$ for any vertex $v$. The resulting function
$m:\operatorname{V}(G)\to[0,\infty)$ defines a $\sigma$-additive measure on
$\operatorname{V}(G)$. The weight $m(S)$ of a subset
$S\subset\operatorname{V}(G)$ is defined as $\sum_{v\in S}m(v)$, the sum of
the weights of all elements of $S$. Note that
$\sum_{v\in\operatorname{V}(G)}m(v)=2\cdot\sum_{e\in\operatorname{E}(G)}m(e)<\infty.$
We shall consider the Hilbert space $L^{2}(G,m)$ of square integrable
functions $f\colon\operatorname{V}(G)\rightarrow$ with respect to $m$. The
elements $f\in L^{2}(G,m)$ satisfy $\sum_{v\in\operatorname{V}(G)}m(v)\cdot
f(v)^{2}<\infty.$ The inner product of $L^{2}(G,m)$ is given by
$\langle f,g\rangle=\sum_{v\in\operatorname{V}(G)}m(v)\cdot f(v)\cdot g(v).$
Note that any constant function is square integrable, i.e. constant functions
belong to $L^{2}(G,m)$.
The normalised Laplacian of a summable weighted graph $(G,m)$ without isolated
vertexes is defined by
(1) $\displaystyle\Delta f(v)=f(v)-\sum_{w\sim v}\frac{m(vw)}{m(v)}\cdot
f(w),\quad f\in L^{2}(G,m).$
Using the Cauchy-Schwarz inequality, one sees that the sum in (1) converges
for any $f\in L^{2}(G,m)$. More precisely, for any $f\in L^{2}(G,m)$,
(2)
$\displaystyle\sum_{w}m(vw)f(w)\leq\left[\sum_{w}m(w)\left(\frac{m(vw)}{m(w)}\right)^{2}\right]^{1/2}\cdot||f||\leq
m(v)^{1/2}\cdot||f||,$
where
$||f||^{2}=\langle f,f\rangle=\sum_{v}m(v)f(v)^{2}.$
Note the following formula:
(3) $\langle\Delta
f,f\rangle=\sum_{vw\in\operatorname{E}(G)}m(vw)\cdot\big{(}f(v)-f(w)\big{)}^{2}.$
###### Lemma 2.
For a summable weighted graph $(G,m)$, the Laplacian $\Delta\colon
L^{2}(G,m)\rightarrow L^{2}(G,m)$ is well-defined; it is self-adjoint, non-
negative, and bounded. Moreover, the spectrum $\sigma(\Delta)$ of the
Laplacian lies in $[0,2]$. Any constant function
$f:\operatorname{V}(G)\to\mathbb{R}$ satisfies $\Delta f=0$, and thus
$0\in\sigma(\Delta)$ is an eigenvalue.
###### Proof.
We have $\Delta=I-P$, where
(4) $\displaystyle P(f)(v)=m(v)^{-1}\sum_{w}m(vw)f(w).$
Clearly $P$ is self adjoint. Using (2), one sees that $||P(f)||\leq||f||$,
which implies that the spectrum of $P$ lies in $[-1,1]$. Therefore
$\sigma(\Delta)\subset[0,2]$. ∎
Clearly, if the graph $G$ is infinite not every point of the spectrum
$\sigma(\Delta)$ is an eigenvalue.
### 2.2. Spectral gap
We shall be interested in the spectral gap of $\Delta$, defined by
$\lambda_{gap}(\Delta)=\inf\\{\lambda\in\sigma(\Delta)\colon\lambda>0\\}.$ The
spectral gap can be characterised as follows:
(5)
$\lambda_{gap}=\inf\bigg{\\{}\frac{\langle\Delta{f},{f}\rangle}{\langle{f},{f}\rangle}\colon
f\in L^{2}(G,m),\ f\perp{\bf 1}\bigg{\\}},$
see [10], Chapter 13. Here ${\bf 1}\colon\operatorname{V}(G)\to\mathbb{R}$ is
the constant function equal to $1$ at all points.
###### Lemma 3.
If $(G,m)$ is a summable weighted graph such that the underlying graph $G$ is
either infinite or it is finite but not a complete graph, then
$\lambda_{gap}\leq 1$.
###### Proof.
For a couple of vertexes $a,\,b\in\operatorname{V}(G)$ define $f_{ab}\in
L^{2}(G,m)$, via $f(a)=m(b)$, $f(b)=-m(a)$ and $f(v)=0$ for
$v\notin\\{a,b\\}$. Then $f_{ab}\perp\bf 1$, and using formula (3) we find
$\frac{\langle\Delta{f_{ab}},{f_{ab}}\rangle}{\langle{f_{ab}},{f_{ab}}\rangle}=1+2\cdot\frac{m(ab)}{m(a)+m(b)}.$
If $G$ is not a complete graph we can select the vertices $a,b$ such that
$m(ab)=0$ (i.e. the edge connecting $a$ and $b$ is not in $G$); then
$\lambda_{gap}\leq 1$ by (5).
If $G$ is complete and infinite then we can choose a sequence $b_{i}$ of
vertices that satisfies $m(ab_{i})\to 0$; such sequence exists since $G$ is
summable. Then the sequence
$\frac{\langle\Delta{f_{ab_{i}}},{f_{ab_{i}}}\rangle}{\langle{f_{ab_{i}}},{f_{ab_{i}}}\rangle}$
converges to 1 implying $\lambda_{gap}\leq 1$ by (5). ∎
###### Remark 4.
There are examples of graphs with spectral gap greater than 1: for a complete
graph on $n$ vertices weighted by $m(e)=1$ for all $e\in\operatorname{E}(G)$,
the spectral gap equals $\frac{n}{n-1}>1$ (see [3], pg. 6).
### 2.3. Summable weighted graphs as reversible Markov chains
A weighted graph $(G,m)$ with summable weight function $m$ determines a Markov
chain with the state space $\operatorname{V}(G)$, where the probability of
moving from $v$ to $w$ equals
$p_{vw}=\frac{m(vw)}{m(v)}.$
As above, we assume that $G$ has no isolated vertexes. If we write
$M=\sum_{v}m(v)=2\sum_{e}m(e)$, then the function $v\mapsto\phi(v)=m(v)\cdot
M^{-1}$ is a stationary probability distribution on $\operatorname{V}(G)$. The
Markov chains arising from summable weighted graphs are reversible and
recurrent, see [11].
## 3\. The Cheeger constant of a summable weighted graph and its dependence
on the weight function
### 3.1. Definition of the Cheeger constant
The Cheeger constant is a real number between 0 and 1 that, intuitively
speaking, measures the robustness of a connected weighted graph $(G,m)$.
Let $S$ be a set of vertices in $G$. The boundary of $S$, denoted $\partial
S$, is the set of all edges in $G$ with one endpoint in $S$,
$\partial S=\\{e\in\operatorname{E}(G)\colon|e\cap S|=1\\}.$
The interior of $S$, denoted $\operatorname{I}(S)$, is the set of all edges in
$G$ with both endpoints in $S$,
$\operatorname{I}(S)=\\{e\in\operatorname{E}(G)\colon|e\cap S|=2\\}.$
We shall denote by $S^{c}$ the complement, $S^{c}=\operatorname{V}(G)-S$.
Besides, $m(S)$ stands for $\sum_{v\in S}m(v)$ and $m(\partial S)$ denotes
$\sum_{e\in\partial S}m(e)$. These entities are related via
(6) $\displaystyle m(S)=m(\partial S)+2\cdot m(\operatorname{I}(S)).$
###### Definition 5.
Let $S$ be a non-empty set of vertices in $G$.
* (a)
The Cheeger ratio of $S$, denoted $h(S)$, is the number
$h(S)=\frac{m(\partial S)}{m(S)}.$
* (b)
The Cheeger constant of $(G,m)$, denoted $h(G,m)$, is the number
$h(G,m)=\inf\big{\\{}h(S)\big{\\}},$
where the infimum is taken over all non-empty sets of vertices $S$ that
satisfy
(7) $m(S)\leq m({S}^{\mathsf{c}}).$
It follows from Equation (6) that $h(G,m)\in[0,1].$
We consider the Cheeger constants of some interesting weighted graphs in
Subsection 3.3.
### 3.2. Cheeger sets
A set of vertexes $S\subset\operatorname{V}(G)$ satisfying (7) is a Cheeger
set if the induced subgraph $G_{S}$ is connected. The collection of Cheeger
sets in $(G,m)$ be denoted $\operatorname{\mathcal{V}}(G,m)$.
###### Lemma 6.
The Cheeger constant $h(G,m)$ equals the infimum of the Cheeger ratios taken
over all Cheeger sets:
$h(G,m)=\inf\big{\\{}h(S)\colon S\in\operatorname{\mathcal{V}}(G,m)\big{\\}}.$
###### Proof.
Let $S\subset\operatorname{V}(G)$ be a non-empty subset that satisfies (7). We
may enumerate the connected components of the induced subgraph $G_{S}$ and
denote by $S_{i}$ the vertex set of the $i$-th component. Then for $i\not=j$,
one has $\partial S_{i}\cap\partial S_{j}=\emptyset$ and $m(\partial
S)=\sum_{i}m(\partial S_{i})$. We obtain
$h(S)=\frac{\sum_{i}m(\partial
S_{i})}{\sum_{i}m(S_{i})}\geq\inf_{i}\left\\{\frac{m(\partial
S_{i})}{m(S_{i}}\right\\}=\inf_{i}\big{\\{}h(S_{i})\big{\\}}.$
Since $S_{i}$ is a Cheeger set for all $i$, the result follows from Definition
5(b). ∎
### 3.3. Examples
The Cheeger constant of a weighted graph depends not only on the structure of
the underlying graph, but also on its weight function. In this subsection, we
we consider two structurally very different graphs and equip each of them with
two different summable weight functions. Remarkably, both graphs exhibit a
vanishing Cheeger constant in one case and a large Cheeger constant in the
other.
###### Example 7 (The infinite complete graph with weight function $m_{1}$).
Denote by $K$ the infinite complete graph with vertex set
$\operatorname{V}(K)=\mathbb{N}$. We show below that different weight
functions on $K$ can lead to vastly different Cheeger constants.
Let $(p_{i})_{i\in\mathbb{N}}$ be a sequence of positive real numbers that sum
to one, $\sum_{i\in\mathbb{N}}p_{i}=1.$ We define a weight function $m_{1}$ on
the infinite complete graph $K$ by
$m_{1}(ij)=p_{i}\cdot p_{j},\quad i,j\in\mathbb{N}.$
We have $\sum_{ij}m(ij)=1$, i.e. this weight function is summable. Besides,
$m_{1}(i)=p_{i}-p_{i}^{2}$ and therefore
$m_{1}(\mathbb{N})=1-\sum_{i}p_{i}^{2}$. Every Cheeger set
$S\in\operatorname{\mathcal{V}}(K,m_{1})$ satisfies
$\displaystyle h(S)=\frac{\sum_{i\in S}p_{i}\cdot\sum_{j\not\in
S}p_{j}}{m_{1}(S)}>\frac{m_{1}(S)\cdot
m_{1}({S}^{\mathsf{c}})}{m_{1}(S)}\geq\frac{m_{1}(\mathbb{N})}{2}.$
Therefore, for the Cheeger constant one has
$h(K,m_{1})\geq\frac{m_{1}(\mathbb{N})}{2}.$
Example 7 shows that we can equip the infinite complete graph with a summable
weight function such that the Cheeger constant of the resulting weighted graph
is relatively large. Whilst it is tempting to attribute this observation
solely to the robustness of complete graphs, the following example suggests
otherwise. In section 8 we shall describe the spectrum of the weighted graph
$(K,m_{1})$ in more detail; in particular, we shall see that the spectral gap
equals 1, cf. Proposition 35.
###### Example 8 (The infinite complete graph with weight function $m_{2}$).
Now we define a different weight function $m_{2}$ on the infinite complete
graph $K$ via
$m_{2}(ij)=\begin{cases}\frac{1}{j^{2}}&\mbox{ if }|j-i|=1,\\\
\frac{1}{j!}&\mbox{ if }|j-i|>1,\end{cases}\quad\text{where }j>i.$
The weight function $m_{2}$ is summable:
$\sum_{ij}m(ij)=\sum_{i=1}^{\infty}\bigg{(}\frac{1}{i^{2}}+\frac{i-1}{i!}\bigg{)}<\infty.$
Write $T_{n}=\\{n,n+1,\ldots\\}$. The boundary and the interior of $T_{n}$
satisfy
$\displaystyle m_{2}(\partial T_{n})$
$\displaystyle=\frac{n-1}{n!}+\frac{1}{n^{2}}+n\cdot\sum_{i=n+1}^{\infty}\frac{1}{i!},$
$\displaystyle m_{2}\big{(}\operatorname{I}(T_{n})\big{)}$
$\displaystyle=\sum_{i=n+2}^{\infty}\frac{i-n-1}{i!}+\sum_{i=n+1}^{\infty}\frac{1}{i^{2}}\geq\frac{1}{4n}.$
With regard to the third summand in the first expression, we observe
$n\cdot\sum_{i=n+1}^{\infty}\frac{1}{i!}<n\cdot\frac{1}{n!}\cdot\sum_{i=1}^{\infty}\frac{1}{(n+1)^{i}}=\frac{1}{n!}.$
It follows that, for large $n$, the boundary of $T_{n}$ satisfies
$m_{2}(\partial T_{n})<\frac{3}{n^{2}}.$ Therefore,
$\frac{m_{2}\big{(}\operatorname{I}(T_{n})\big{)}}{m_{2}(\partial
T_{n})}>\frac{n^{2}}{3}\cdot\frac{1}{4n}\to\infty.$
Since
(8) $\displaystyle
h(T_{n})^{-1}-1=2\cdot\frac{m(\operatorname{I}(T_{n})}{m(\partial T_{n}))},$
it follows that the Cheeger constant of the weighted graph $(K,m_{2})$
vanishes, $h(K,m_{2})=0.$
Examples 7 and 8 illustrate the fact that the Cheeger constant $h(K,m)$
strongly depends on the weight function $m$. Below we analyse more examples of
this kind.
###### Example 9 (The Half-Line graph with weight function $m_{3}$).
The Half-Line graph, denoted by $H$, comprises the vertex set
$\operatorname{V}(H)=\mathbb{N}$ and edges of the form $e_{i}=\\{i-1,i\\}$ for
all $i\geq 1$. We define a weight function $m_{3}$ on the $H$ via
$m_{3}(e_{i})=i^{-2}.$ The weight function $m_{3}$ is summable as the series
$\sum_{i\geq 1}i^{-2}$ converges. We show below that $h(H,m_{3})=0$.
Write $T_{n}=\\{n,n+1,\ldots\\}\subset\operatorname{V}(H)$. The boundary and
interior of $T_{n}$ satisfy
$\displaystyle m_{3}(\partial{T_{n})}=n^{-2},\quad
m_{3}\big{(}\operatorname{I}({T_{n}})\big{)}=\sum_{i=n+1}^{\infty}i^{-2}.$
Therefore,
$\frac{m_{3}\big{(}\operatorname{I}({T_{n}})\big{)}}{m_{3}(\partial{T_{n})}}=n^{2}\cdot\sum_{i=n+1}^{\infty}i^{-2}\rightarrow\infty,$
which, by (8), gives $h(T_{n})\to 0$, implying $h(H,m_{3})=0.$
###### Example 10 (The Half-Line graph with weight function $m_{4}$).
Here we define another weight function on the Half-Line graph $H$:
$m_{4}(e_{i})=r^{i},$ where $r\in(0,1).$ We show below that the Cheeger
constant $h(H,m_{4})>0$.
As before, write $T_{n}=\\{n,n+1,\ldots\\}$. For $n>0$, one has
$m_{4}(\partial T_{n})=r^{n}$ and $m_{4}(T_{n})=r^{n}\cdot\frac{1+r}{1-r}$ and
therefore $h(T_{n})=\frac{1-r}{1+r}$ is independent of $n$. Note that
inequality (7) is satisfied for any $n$ large enough. We want to show that
(9) $\displaystyle h(H,m_{4})=h(T_{n})=\frac{1-r}{1+r}.$
By Lemma 6 we need to consider subsets $S\subset\operatorname{V}(H)$ such that
the induced subgraphs $H_{S}$ are connected; in other words $S$ must be an
interval, finite or infinite. Let $S=\\{i,\ldots,j\\}$ be a finite interval.
Then
$m_{4}(\partial S)=m_{4}(\partial T_{i})+r^{j-1},\quad
m_{4}(S)=m_{4}(T_{i})-m_{4}(T_{j+1}),$
and therefore
$h(S)=\frac{m_{4}(\partial S)}{m_{4}(S)}=\frac{m_{4}(\partial
T_{i})+r^{j-1}}{m_{4}(T_{i})-m_{4}(T_{j+1})}>h(T_{i})=\frac{1-r}{1+r}.$
This proves (9).
## 4\. The dual Cheeger constant
In [1], the authors introduced a new geometric constant, which they call the
dual Cheeger constant. The dual Cheeger inequalities state that this constant
controls the top of the spectrum of the Laplacian. The construction in [1] is
restricted to locally finite weighted graphs.
The purpose of this section is to introduce the notion of a dual Cheeger
constant adopted for weighted graphs with summable weight functions.
### 4.1. Definition of the dual Cheeger constant
For all $A,B\subset\operatorname{V}(G)$, denote by $\operatorname{E}(A,B)$ the
set of all edges connecting $A$ to $B$. The symbol $m(A,B)$ will denote
$m(\operatorname{E}(A,B)).$
###### Definition 11.
For $A,B\subset\operatorname{V}(G)$ with $A\cap B=\emptyset$,
$A\not=\emptyset\not=B$, write
(10) $\displaystyle\overline{h}(A,B)=\frac{2\cdot m(A,B)}{m(A)+m(B)}.$
The dual Cheeger constant of $(G,m)$, denoted by $\overline{h}(G,m)$, is
defined as
$\overline{h}(G,m)=\sup\big{\\{}\overline{h}(A,B)\big{\\}},$
where the supremum is taken over all disjoint nonempty sets of vertices $A,B$
in $G$.
Since $m(A)\geq m(A,B)$ and $m(B)\geq m(A,B)$, we see that
$\overline{h}(A,B)\leq 1$, and therefore
$\overline{h}(G,m)\leq 1$
for any weighted graph $(G,m)$.
If the graph $G$ is bipartite and $V(G)=A\sqcup B$ is a partition of the set
of vertexes such that all the edges connect $A$ to $B$, then $m(A)=m(A,B)$ and
$m(B)=m(A,B)$, which implies $\overline{h}(A,B)=1$, and therefore
(11) $\displaystyle\overline{h}(G,m)=1$
for any bipartite $(G,m)$.
The inequality $\overline{h}(G,m)<c$ is equivalent to the statement that, for
any pair of disjoint subsets $A,B\subset\operatorname{V}(G)$, one has the
inequality
(12) $\displaystyle m(A,B)\leq c\cdot\frac{m(A)+m(B)}{2}.$
In other words, the weight of the connecting edges between any pair of
disjoint subsets $A$ and $B$ is at most $c$ times the average weight of $A$
and $B$.
In [1], the authors consider locally finite weighted graphs $(G,m)$ whose
weight function is not necessarily summable. Given an exhaustion
$\Omega_{n}\uparrow\operatorname{V}(G)$ (a filtration of connected subsets
that converges to $\operatorname{V}(G)$), they write
$\overline{h}(\Omega_{n})=\sup\bigg{\\{}\frac{2\cdot
m(A,B)}{m(A)+m(B)}\bigg{\\}},$
where the supremum is taken over all disjoint nonempty subsets
$A,B\subset\Omega_{n}$. Hence, the authors define the dual Cheeger constant to
be the following limit:
$\overline{h}(G,m)=\lim_{n\rightarrow\infty}\overline{h}(\Omega_{n}).$ As the
authors of [1] show, this limit exists and it is independent of the
exhaustion. Whilst Definition 11 does not involve any such limit, our dual
Cheeger constant is equivalent to that in [1]; the difference lies in the
underlying weighted graphs.
### 4.2. Example of a non-bipartite graph with $\overline{h}(G,m)=1$
Consider the infinite graph $L$ shown on Figure 1; its vertexes are labelled
by $v_{0},v_{1},\dots$ and $w_{1},w_{2},\dots$. The graph $L$ is not bipartite
since it has a cycle of odd order. We set the weights as follows:
$m(v_{i}w_{i})=r^{i}$ and $m(v_{i}v_{i+1})=\rho^{i}$, where $0<\rho<r<1$;
besides, $m(v_{0}w_{1})=1$.
Figure 1. Non-bipartite graph $L$.
For $i>1$ we have $m(v_{i})=\rho^{i-1}+\rho^{i}+r^{i}$ and $m(w_{i})=r^{i}$.
Therefore, taking $A_{i}=\\{v_{i}\\}$ and $B_{i}=\\{w_{i}\\}$, where $i>1$, we
have
$\overline{h}(A_{i},B_{i})=\frac{2r^{i}}{2r^{i}+\rho^{i-1}+\rho^{i}}\to 1.$
Thus, we obtain $\overline{h}(L,m)=1$.
### 4.3. The same graph $L$ with a different weight function
Consider now the following weight function $m_{1}$ on the graph $L$ of the
previous example. The function $m_{1}$ is defined similarly to $m$ with the
only difference that now $\rho=r$, where $0<r<1$. In more detail,
$m_{1}(v_{i}w_{i})=r^{i}=m(v_{i}v_{i+1})$, and $m_{1}(v_{0}w_{1})=1$. Note
that $m(w_{i})=r^{i}$ for $i>1$, and $m(v_{i})=r^{i-1}+2r^{i}$ for $i>0$.
Besides, $m(w_{1})=1+r$ and $m(v_{0})=2$.
Suppose that $A,B\subset\operatorname{V}(G)$ are disjoint sets of vertexes and
for some $i\geq 1$ one has $v_{i}\in A$ and $w_{i}\in B$. Consider the
following modifications of the pair $A,B$. Let $A_{1}=A\cup\\{w_{i}\\}$ and
$B_{1}=B-\\{w_{i}\\}$. Besides, let $A_{2}=A$ and $B_{2}=B-\\{w_{i}\\}$. Since
the vertex $w_{i}$ is connected to the vertex $v_{i}$ only, by examining
formula (10) we easily see that $\overline{h}(A_{1},B_{1})<\overline{h}(A,B)$
and $\overline{h}(A_{2},B_{2})<\overline{h}(A,B)$. Thus, since we are
interested in pairs $A,B$ giving maximum to the dual Cheeger constant (10), we
may always assume that for each vertex $v_{i}\in A$ the corresponding vertex
$w_{i}$ belongs to $B$, and vice versa.
Next, for a pair of disjoint subsets $A,B\subset\operatorname{V}(G)$, suppose
that for some $i$ one has $v_{i},v_{i+1}\in A$ and $w_{i},w_{i+1}\in B$. We
can modify the sets by swapping the points $v_{i+1}$ and $w_{i+1}$, i.e.
$A^{\prime}=A-\\{v_{i+1}\\}\cup\\{w_{i+1}\\}$ and
$B^{\prime}=B-\\{w_{i+1}\\}\cup\\{v_{i+1}\\}$. Then examining (10) we see that
$\overline{h}(A^{\prime},B^{\prime})>h(A,B)$. Thus, we may only consider pairs
of disjoint subsets $A,B$ such that no neighbouring vertexes $v_{i},v_{i+1}$
lie in the same subset $A$ or $B$.
As another remark, consider a pair of disjoint subsets
$A,B\subset\operatorname{V}(G)$ such that $v_{i}\in A$ and $w_{i}\in B$ and
modify it as follows $A^{\prime}=A\cup\\{w_{i+1}\\}$ and
$B^{\prime}=B\cup\\{v_{i+1}\\}$. Then we easily see from (10) that
$\overline{h}(A^{\prime},B^{\prime})>\overline{h}(A,B)$.
Therefore, to determine $\overline{h}(G,m)$ we need to consider only pairs of
subsets $A,B$ where $A$ contains all points $v_{i}$ with $i=k+2\ell$, where
$\ell\geq 0$, and all points $w_{i}$ with $i=k+2\ell+1$, where $\ell\geq 0$;
the set $B$ is defined similarly with the letters $v$ and $w$ interchanged.
Here $k$ is a fixed integer $k\geq 1$. One finds that the dual Cheeger ratio
$\overline{h}(A,B)=\frac{4r}{3r+1}$ does not depend on $k$.
In the case $k=0$ we have to consider the slightly modified sets
$A=\\{v_{0},v_{1},w_{2},v_{3},\dots\\}$ and $B=\\{w_{1},v_{2},w_{3},\dots\\}$.
Computing the dual Cheeger ratio (10) gives $\overline{h}(A,B)=r$. Since
$r<\frac{4r}{3r+1}$ we conclude that $\overline{h}(G,m)=\frac{4r}{3r+1}<1$.
These two examples illustrate the possibility that the dual Cheeger constant
can be maximal and equal 1 for one weight function and be smaller than 1 for
another weight function.
## 5\. The Cheeger and dual Cheeger inequalities
In this section, we we show that the Cheeger constant and the dual Cheeger
constant control the spectral gap and the top of the spectrum of the
Laplacian, respectively. In particular, we prove the Cheeger inequalties and
the dual Cheeger inequalities for countable weighted graphs with summable
weight function. These inequalities give estimates on the spectral gap
$\lambda_{gap}(\Delta)=\inf\\{\lambda>0,\,\lambda\in\sigma(\Delta)\\}$ and the
top of the spectrum
$\lambda_{top}(\Delta)=\sup\\{\lambda\in\sigma(\Delta)\\}$.
###### Theorem 12 (Cheeger and dual Cheeger inequalities).
For any weighted graph $(G,m)$ with summable weight function one has
(13) $\displaystyle 1-\sqrt{1-h(G,m)^{2}}\,\leq\,$
$\displaystyle\lambda_{gap}(\Delta)\,\leq\,2\cdot h(G,m),$ (14) $\displaystyle
2\cdot\overline{h}(G,m)\,\leq\,$
$\displaystyle\lambda_{top}(\Delta)\,\leq\,1+\sqrt{1-\big{(}1-\overline{h}(G,m)\big{)}^{2}}.$
###### Proof.
For a subset $S\subset\operatorname{V}(G)$ satisfying $m(S)\leq
m(S^{\mathsf{c}})$ define the function $f_{S}\in L^{2}(G,m)$ such that
$f_{S}(v)=m(S)^{-1}$ for $v\in S$ and $f_{S}(v)=-m(S^{\mathsf{c}})^{-1}$ for
$v\in S^{c}$. Since $f_{S}\perp{\bf 1}$, we may apply (5) to get
$\displaystyle\lambda_{gap}(\Delta)$ $\displaystyle\leq\frac{\langle\Delta
f_{S},f_{S}\rangle}{\langle f_{S},f_{S}\rangle}=m(\partial
S)\cdot\big{(}\frac{1}{m(S)}+\frac{1}{m(S^{\mathsf{c}})}\big{)}\leq 2\cdot
h(S).$
Since this is true for all nonempty $S\subset\operatorname{V}(G)$ satisfying
(7), we obtain the right inequality in (13).
To prove the left inequality in (14), for a pair
$A,B\subset\operatorname{V}(G)$ of nonempty disjoint subsets define the
function $f_{A,B}\in L^{2}(G,m)$ as follows: (a) $f_{A,B}(v)=1$ for $v\in A$,
(b) $f_{A,B}(v)=-1$ for $v\in B$ and (c) $f_{A,B}(v)=0$ for
$v\in\operatorname{V}(G)-(A\cup B)$. Using the characterisation of
$\lambda_{top}(\Delta)$ in terms of Rayleigh quotients, we have
$\displaystyle\lambda_{top}(\Delta)$
$\displaystyle\geq\frac{\langle\Delta{f_{A,B}},{f_{A,B}}\rangle}{\langle{f_{A,B}},{f_{A,B}}\rangle}=2\cdot\overline{h}(A,B)+h(A\cup
B)\geq 2\cdot\overline{h}(A,B).$
Since this is true for all nonempty, disjoint subsets
$A,B\subset\operatorname{V}(G)$, the left inequality in (14) follows.
To continue with the proof of Theorem 12 we shall need to prepare certain
tools. The proof will be completed after Lemma 15.
###### Lemma 13 (Co-area formulae).
For a function $f\colon\operatorname{V}(G)\to$ and for $t\in$ we shall denote
$P_{t}(f)=\\{v\in\operatorname{V}(G);f(v)>t\\}$. Then for every $f\in
L^{2}(G,m)$ one has
* (a)
$\displaystyle\;\;\int_{0}^{\infty}m\big{(}P_{t}(f^{2})\big{)}dt=\sum_{v\in\operatorname{V}(G)}m(v)\cdot
f(v)^{2},$
* (b)
$\displaystyle\int_{0}^{\infty}m\big{(}\partial
P_{t}(f^{2})\big{)}dt=\sum_{uv\in\operatorname{E}(G)}m(uv)\cdot|f^{2}(u)-f^{2}(v)|.$
###### Proof.
The superlevel sets of $f^{2}$ satisfy $v\in
P_{t}(f^{2})\iff\mathbbm{1}_{(t,\infty)}\big{(}f^{2}(v)\big{)}=1.$ Therefore,
$\displaystyle{}\int_{0}^{\infty}m\big{(}P_{t}(f^{2})\big{)}dt=\int_{0}^{\infty}\sum_{v\in\operatorname{V}(G)}m(v)\cdot\mathbbm{1}_{(t,\infty)}\big{(}f^{2}(v)\big{)}dt$
$\displaystyle=\sum_{v\in\operatorname{V}(G)}m(v)\cdot\int_{0}^{\infty}\mathbbm{1}_{(t,\infty)}\big{(}f^{2}(v)\big{)}dt=\sum_{v\in\operatorname{V}(G)}m(v)\cdot
f^{2}(v).$
For any two vertices $u$ and $v$, define
$I_{uv}=\Big{[}\min\big{\\{}f^{2}(u),f^{2}(v)\big{\\}},\max\big{\\{}f^{2}(u),f^{2}(v)\big{\\}}\Big{)}$.
Then,
$\displaystyle\int_{0}^{\infty}m\big{(}\partial
P_{t}(f^{2})\big{)}dt=\int_{0}^{\infty}\sum_{uv\in\operatorname{E}(G)}m(uv)\cdot\mathbbm{1}_{I_{uv}}(t)dt$
$\displaystyle=\sum_{uv\in\operatorname{E}(G)}m(uv)\cdot\int_{0}^{\infty}\mathbbm{1}_{I_{uv}}(t)dt=\sum_{uv\in\operatorname{E}(G)}m(uv)\cdot|g^{2}(u)-g^{2}(v)|.$
∎
Consider the operator $Q=2I-\Delta\colon L^{2}(G,m)\to L^{2}(G,m)$. We have
$\lambda\in\sigma(\Delta)$ if and only if $2-\lambda\in\sigma(Q)$. Therefore,
$2-\lambda_{top}(\Delta)$ equals the bottom of the spectrum of $Q$, i.e.
(15) $\displaystyle 2-\lambda_{top}(\Delta)=\inf\left\\{\frac{\langle
Qf,f\rangle}{\langle f,f\rangle};f\not=0\right\\}$
A straightforward computation shows that for $f\in L^{2}(G,m)$, one has
(16) $\displaystyle\langle
Qf,f\rangle=\sum_{vw\in\operatorname{E}(G)}m(vw)\cdot(f(v)+f(w))^{2}.$
###### Lemma 14.
For any function $f\in L^{2}(G,m)$, $f\not\equiv 0$, one has the inequalities
(17) $\displaystyle
1-\sqrt{1-h(f)^{2}}\leq\frac{\langle\Delta{f},{f}\rangle}{\langle{f},{f}\rangle}\leq
1+\sqrt{1-h(f)^{2}}.$
where the number $h(f)\geq 0$ is defined as the infimum of the Cheeger ratios
$h(S)=m(\partial S)m(S)^{-1}$ taken over all nonempty subsets
$S\subset\\{v;f(v)\neq 0\\}$.
###### Proof.
Consider the expressions $A=\langle Qf,f\rangle\cdot\langle\Delta f,f\rangle$
and $B=\langle Qf,f\rangle\cdot\langle f,f\rangle.$ From (3), (16) and the
Cauchy-Schwarz inequality we obtain
$\displaystyle A^{1/2}$
$\displaystyle=\Big{(}\sum_{vw\in\operatorname{E}(G)}m(vw)\cdot\big{(}f(v)+f(w)\big{)}^{2}\Big{)}^{1/2}\cdot\Big{(}\sum_{vw\in\operatorname{E}(G)}m(vw)\cdot\big{(}f(v)-f(w)\big{)}^{2}\Big{)}^{1/2}$
$\displaystyle\geq\sum_{vw\in\operatorname{E}(G)}m(vw)\cdot|f(v)^{2}-f(w)^{2}|=\int_{0}^{\infty}m\big{(}\partial
P_{t}(f^{2})\big{)}dt$ $\displaystyle=\int_{0}^{t^{*}}\frac{m\big{(}\partial
P_{t}(f^{2})\big{)}}{m\big{(}P_{t}(f^{2})\big{)}}\cdot
m\big{(}P_{t}(f^{2})\big{)}dt$ $\displaystyle\geq
h(f^{2})\cdot\int_{0}^{\infty}m(P_{t}(f^{2}))dt=h(f)\cdot\langle f,f\rangle.$
The two bottom lines use Lemma 13; the finite or infinite value $t^{*}$ is
defined by $t^{*}=\sup\\{f^{2}\\};$ it is introduced in order to avoid
division by 0. Thus we have $A\geq h(f)^{2}\cdot||f||^{4}$. We also have
$B=\Big{(}2-\frac{\langle\Delta f,f\rangle}{\langle
f,f\rangle}\Big{)}\cdot||f||^{4}$. Dividing we obtain that the quotient
$\frac{\langle\Delta{f},{f}\rangle}{\langle{f},{f}\rangle}=AB^{-1}$ satisfies
the inequality
$AB^{-1}\geq\frac{h(f)^{2}}{2-AB^{-1}}.$
Solving this quadratic inequality for $AB^{-1}$ gives (17). ∎
Next, we observe that for any nonzero $g\in L^{2}(G,m)$ one can find a real
number $\tau=\tau(g)$ such that
(18) $\displaystyle
m(g^{-1}(-\infty,\tau))\,\leq\,m(\operatorname{V}(G))/2\quad\mbox{and}\quad
m(g^{-1}(\tau,\infty))\,\leq\,m(\operatorname{V}(G))/2.$
Indeed, it is easy to see that one can take
$\tau=\sup\\{t;\,m(g^{-1}(-\infty,t))<m(\operatorname{V}(G))/2\\}$.
Define the functions $g_{+},g_{-}\in L^{2}(G,m)$ by
$g_{+}(v)=\max\\{g(v)-\tau(g),0\\}\quad\mbox{and}\quad
g_{-}(v)=\max\\{\tau(g)-g(v),0\\}.$
Then
(19) $\displaystyle g=g_{+}-g_{-}+\tau,\quad\quad g_{+}g_{-}=0,$
and $h(G,m)\leq\min\\{h(g_{+}),h(g_{-})\\}$ because of (18). If
$g\perp\mathbf{1}$ then $\langle g,g\rangle$ can be estimates as follows:
$\displaystyle\langle g,g\rangle$ $\displaystyle=$ $\displaystyle\sum
m(v)g(v)^{2}\,\leq\,\sum m(v)(g(v)-\tau)^{2}=\sum
m(v)(g_{+}(v)^{2}+g_{-}(v)^{2})$ $\displaystyle=$ $\displaystyle\langle
g_{+},g_{+}\rangle+\langle g_{-},g_{-}\rangle.$
Besides,
(21) $\displaystyle\langle\Delta g,g\rangle$ $\displaystyle=$
$\displaystyle\sum m(vw)(g(u)-g(v))^{2}$ $\displaystyle\geq$
$\displaystyle\sum
m(uv)((g_{+}(u)-g_{+}(v))^{2}+(g_{-}(u)-g_{-}(v))^{2})=\langle\Delta
g_{+},g_{+}\rangle+\langle\Delta g_{-},g_{-}\rangle$
Indeed, if the vertexes $u,v$ are such that $g(v)<\tau<g(u)$ then
$\displaystyle(g(u)-g(v))^{2}$ $\displaystyle=$
$\displaystyle(g_{+}(u)-g_{-}(v))^{2}\,\geq\,g_{+}(u)^{2}+g_{-}(v)^{2}$
$\displaystyle=$
$\displaystyle(g_{+}(u)-g_{+}(v))^{2}+(g_{-}(u)-g_{-}(v))^{2}.$
Thus, for $g\perp\mathbf{1}$, using (5) and (21) we obtain
$\displaystyle\frac{\langle\Delta g,g\rangle}{\langle g,g\rangle}$
$\displaystyle\geq$ $\displaystyle\min\\{\frac{\langle\Delta
g_{+},g_{+}\rangle}{\langle g_{+},g_{+}\rangle},\frac{\langle\Delta
g_{-},g_{-}\rangle}{\langle g_{-},g_{-}\rangle}\\}$ $\displaystyle\geq$
$\displaystyle\min\\{1-\sqrt{1-h(g_{+})^{2}},\,1-\sqrt{1-h(g_{-})^{2}}\\}\,\geq\,1-\sqrt{1-h(G,m)^{2}}.$
This proves the left inequality in (13).
Finally, we prove the right inequality (14), i.e. the upper bound for the
$\lambda_{top}$. We shall use the idea of [1] and adopt their arguments to our
situation.
Given a nonzero function $f\in L^{2}(G,m)$ we consider the auxiliary weighted
graph $(G_{f},m_{f})$ which is constructed as follows. For each vertex
$v\in\operatorname{V}(G)$ with $f(v)\not=0$ we create an additional vertex
$v^{\prime}$ and the vertex set of the new graph $G_{f}$ equals
$\operatorname{V}(G_{f})=\operatorname{V}(G)\cup\\{v^{\prime};\,v\in\operatorname{V}(G),f(v)\not=0\\}.$
Next, we describe the set of edges of $G_{f}$. We remove every edge
$vw\in\operatorname{E}(G)$ with $f(v)f(w)>0$ and replace it with the edges
$vw^{\prime}$ and $v^{\prime}w$ of $\operatorname{E}(G_{f})$. All edges $vw$
of $G$ satisfying $f(v)f(w)\leq 0$ are included into $E(G_{f})$.
The weight function $m_{f}$ on $G_{f}$ is defined as follows: firstly,
$m_{f}(v^{\prime}w)=m_{f}(vw^{\prime})=m(vw)$ and, secondly, for every edge
$vw\in\operatorname{E}(G)$ with $f(v)f(w)\leq 0$ we set $m_{f}(vw)=m(vw).$
Note that the weights of vertexes $v\in\operatorname{V}(G)$ remain unchanged:
$m_{f}(v)=m(v)$. Besides, the weights of the new vertexes $v^{\prime}$ satisfy
$m_{f}(v^{\prime})\leq m(v)$.
Consider the function $f^{\prime}\in L^{2}(G_{f},m_{f})$ defined by
$f^{\prime}(v^{\prime})=0$ and $f^{\prime}(v)=|f(v)|$ for
$v\in\operatorname{V}(G)$.
###### Lemma 15.
One has
(22) $\displaystyle\frac{\langle Qf,f\rangle}{\langle
f,f\rangle}\geq\frac{\langle\Delta f^{\prime},f^{\prime}\rangle_{f}}{\langle
f^{\prime},f^{\prime}\rangle_{f}},$
where $\langle\cdot,\cdot\rangle_{f}$ denotes the scalar product in
$L^{2}(G_{f},m_{f})$.
###### Proof.
Firstly, $\langle
f,f\rangle=\sum_{v\in\operatorname{V}(G)}m(v)f(v)^{2}=\sum_{v\in\operatorname{V}(G_{f})}m_{f}(v)f^{\prime}(v)^{2}=\langle
f^{\prime},f^{\prime}\rangle_{f}.$ Next we show
(23) $\displaystyle\langle Qf,f\rangle\geq\langle\Delta
f^{\prime},f^{\prime}\rangle_{f}.$
If $vw$ is an edge in $G$ with $f(v)f(w)>0$, then
$\displaystyle m(vw)(f(v)+f(w))^{2}$ $\displaystyle\geq$ $\displaystyle
m(vw)(f(v)^{2}+f(w)^{2})=m_{f}(vw^{\prime})f^{\prime}(v)^{2}+m_{f}(v^{\prime}w)f^{\prime}(w)^{2}$
$\displaystyle=$ $\displaystyle
m_{f}(vw^{\prime})(f^{\prime}(v)-f^{\prime}(w^{\prime}))^{2}+m_{f}(v^{\prime}w)(f^{\prime}(v^{\prime})-f^{\prime}(w))^{2}.$
Besides, for an edge $vw\in\operatorname{E}(G)$ with $f(v)f(w)\leq 0$ one has
$m(vw)(f(v)+f(w))^{2}=m_{f}(vw)(f^{\prime}(v)-f^{\prime}(w))^{2}.$
Incorporating the above information into (3) and (16) we obtain (23) and hence
(22). ∎
We intend to use the left inequality in (17) applied to $f^{\prime}$ viewed as
an element of $L^{2}(G_{f},m_{f})$. The number $h_{f}(f^{\prime})$ is defined
as the infimum of the ratios $h_{f}(S)=m_{f}(\partial S)m_{f}(S)^{-1}$, where
$S$ runs over subsets of the support of $f^{\prime}$. In our case the support
of $f^{\prime}$ lies in $\operatorname{V}(G)\subset\operatorname{V}(G_{f})$
and can be represented as the disjoint union ${\sf supp}(f)_{+}\sqcup{\sf
supp}(f)_{-}$, where ${\sf supp}(f)_{+}$ is the set of all vertexes
$v\in\operatorname{V}(G)$ where $f$ is positive and ${\sf supp}(f)_{-}$ is the
set of all vertexes $v\in\operatorname{V}(G)$ with $f(v)<0$. Thus, any subset
$S\subset{\sf supp}(f)$ is the disjoint union $S=S_{+}\sqcup S_{-}$ where
$S_{\pm}=S\cap{\sf supp}(f)_{\pm}$. In the graph $G_{f}$ there are no edges
internal to $S_{+}$ and there are no edges internal to $S_{-}$. Thus, we
obtain
$m_{f}(S)=m(S)=m_{f}(\partial_{f}S)+2m(S_{+},S_{-}).$
Therefore, we see that
$h_{f}(S)=\frac{m_{f}(S)-2m(S_{+},S_{-})}{m_{f}(S)}=1-\frac{2m(S_{+},S_{-})}{m(S_{+})+m(S_{-})}=1-\overline{h}(S_{+},S_{-}).$
Taking the infimum over $S\subset{\sf supp}f$ we obtain the inequality
(24) $\displaystyle h_{f}(f^{\prime})\geq 1-\overline{h}(G,m).$
Now we can obtain the desired upper bound for
$\lambda_{top}=\lambda_{top}(\Delta)$. We have
$\displaystyle 2-\lambda_{top}=\inf_{f\not=0}\frac{\langle
Qf,f\rangle}{\langle f,f\rangle}\geq\inf_{f\not=0}\frac{\langle\Delta
f^{\prime},f^{\prime}\rangle_{f}}{\langle
f^{\prime},f^{\prime}\rangle_{f}}\geq 1-\sqrt{1-h_{f}(f^{\prime})^{2}}\geq
1-\sqrt{1-(1-\overline{h}(G,m))^{2}}.$
This completes the proof of Theorem 12. ∎
###### Remark 16.
Theorem 13.4 from the book [7] is a corollary of the left part of the first
inequality of Theorem 12.
## 6\. A new combinatorial invariant
In this section we introduce a new combinatorial invariant of countable graphs
with summable weight functions. We show that this invariant is a measure of
spectral asymmetry. We describe relations between the new invariant and the
Cheeger and the dual Cheeger constants.
### 6.1. Definition
Let $G$ be a connected countable graph and let
$m:\operatorname{E}(G)\to(0,\infty)$ be a summable weight function. For any
vertex $v\in\operatorname{V}(G)$ and any subset $S\subset\operatorname{V}(G)$
we write
(25) $\displaystyle m_{S}(v)=\sum_{w\in S}m(vw)\quad\mbox{and}\quad
p_{S}(v)=\frac{m_{S}(v)}{m(v)}.$
If we think of the underlying weighted graph as a Markov chain, then
$p_{S}(v)$ is the probability that the particle starting at $v$ ends up in $S$
in one step. If $A\sqcup B=\operatorname{V}(G)$ is a partition of the vertex
set with $A\not=\emptyset\not=B$, then
$p_{A}(v)+p_{B}(v)=1\quad\mbox{for any}\quad v\in\operatorname{V}(G).$
###### Definition 17.
For a partition $A\sqcup B=\operatorname{V}(G)$ of the vertex set we define
(26) $\displaystyle{\sf k}(A,B)=\max\\{\sup_{v\in A}p_{A}(v),\sup_{w\in
B}p_{B}(w)\\}.$
Finally, we associate the following constant ${\sf k}(G,m)$ to the weighted
graph:
(27) $\displaystyle{\sf k}(G,m)=\inf{\sf k}(A,B);$
the infimum is taken over all partitions $A\sqcup B=\operatorname{V}(G)$.
The inequality ${\sf k}(G,m)>c$ means that for any partition $A\sqcup
B=\operatorname{V}(G)$ there exists either $v\in A$ or $w\in B$ such that one
of the inequalities $p_{A}(v)>c$ or $p_{B}(w)>c$ holds. The constant ${\sf
k}(G,m)$ is the supremum of the numbers $c$ for which this graph property
holds.
### 6.2. Characterisation of bipartite graphs
###### Theorem 18.
One has ${\sf k}(G,m)\geq 0$ for any weighted graph $(G,m)$. Moreover, ${\sf
k}(G,m)=0$ if and only if the graph $G$ is bipartite.
###### Proof.
The inequality ${\sf k}(G,m)\geq 0$ is obvious from the definition.
If $G$ is bipartite and $\operatorname{V}(G)=A\sqcup B$ is a partition such
that every edge goes from $A$ to $B$, then $p_{A}(v)=0$ for $v\in A$ and
$p_{B}(w)=0$ for $w\in B$ implying ${\sf k}(A,B)=0$ and therefore ${\sf
k}(G,m)=0$.
Let $(G,m)$ be a summable weighted graph with $k(G,m)=0$. Let $A_{n}\sqcup
B_{n}=\operatorname{V}(G)$ be a sequence of partitions with ${\sf
k}(A_{n},B_{n})\to 0$. Suppose that it is not bipartite. Then $G$ admits a
cycle $C$ of odd length. For each edge $vw$ of the cycle $C$ consider the
quantity
(28) $\displaystyle
a(vw)=\min\left\\{\frac{m(vw)}{m(v)},\frac{m(vw)}{m(w)}\right\\}>0$
and choose $n$ so large that
(29) $\displaystyle a(vw)>{\sf k}(A_{n},B_{n})\quad\mbox{for any edge $vw$ of
$C$.}$
Using the definition of ${\sf k}(A_{n},B_{n})$ we see that every edge $vw$ of
$C$ connects a vertex of $A_{n}$ with a vertex of $B_{n}$. Indeed, if $v,w\in
A_{n}$ then ${\sf k}(A_{n},B_{n})\geq
p_{A_{n}}(v)\geq\frac{m(vw)}{m(v)}=a(vw)$ contradicting (29). This leads to
contradiction with the fact that the cycle $C$ has odd length. ∎
###### Remark 19.
Recall that the equality $\overline{h}(G,m)=1$ only partially characterises
the class of bipartite graphs, see (11) and Example 4.2. In contrast, by
Theorem 18, the equality ${\sf k}(G,m)=0$ gives a complete characterisation.
###### Example 20.
Let $G$ be a complete graph on $n+1$ vertexes. We equip $G$ with the counting
weight function, i.e. $m(vw)=1$ for every edge. If $A\sqcup
B=\operatorname{V}(G)$ is a partition with $|A|\leq|B|$ then
${\sf
k}(A,B)=\max\left\\{\frac{|A|-1}{n},\frac{|B|-1}{n}\right\\}=\frac{|B|-1}{n}$
and we obtain
$\displaystyle{\sf k}(G,m)=n^{-1}\cdot\lceil\frac{n-1}{2}\rceil.$
Thus, ${\sf k}(G,m)=1/2-1/(2n)$ for $n$ odd and ${\sf k}(G,n)=1/2$ for $n$
even.
###### Example 21.
Let $C_{n}$ be the cycle of order $n$ with the counting weight function $m$,
i.e. the weight of each edge equals 1. Then ${\sf k}(C_{n},m)=0$ for $n$ even
(since then $C_{n}$ is bipartite) and ${\sf k}(C_{n})=1/2$ for $n$ odd.
Indeed, for $n$ odd, for any partition $A\sqcup B$ of the vertex set one of
the sets $A$ or $B$ must contain two adjacent vertexes and therefore one of
the numbers $p_{A}(v)$ or $p_{B}(w)$ is $\geq 1/2.$
It is obvious that ${\sf k}(G,m)\leq 1$. In all examples known to us we have
${\sf k}(G,m)\leq 1/2$, but we do not know if this inequality is true in
general.
### 6.3. Relation with the dual Cheeger constant
Next we describe inequalities relating the new invariant ${\sf k}(G,m)$ with
$\overline{h}(G,m)$. We shall frequently use the inequlity
(30)
$\displaystyle\min\\{\frac{a}{c},\frac{b}{d}\\}\,\leq\,\frac{a+b}{c+d}\,\leq\,\max\\{\frac{a}{c},\frac{b}{d}\\},$
where $a,b,c,d>0$.
For a subset of vertexes $A\subset\operatorname{V}(G)$ we shall denote
$R_{A}=\frac{2\cdot m(A,A)}{m(A)}.$
###### Lemma 22.
For any partition $A\sqcup B=\operatorname{V}(G)$ of the set of vertexes of a
summable weighted graph $(G,m)$ one has
(31) $\displaystyle\min\\{R_{A},R_{B}\\}\leq
1-\overline{h}(A,B)\leq\max\\{R_{A},R_{B}\\}\leq{\sf k}(A,B).$
###### Proof.
Since $m(A)=2m(A,A)+m(A,B)$ we have
$1-\overline{h}(A,B)=2\cdot\frac{m(A,A)+m(B,B)}{m(A)+m(B)}$
and using (30) we obtain the left and the central inequalities in (31). The
right inequality in (31) follows from $R_{A}\leq\sup_{v\in A}\\{p_{A}(v)\\}$
and $R_{B}\leq\sup_{w\in B}\\{p_{B}(w)\\},$ which are consequences of (30) as
well. ∎
Next we state a relationship between ${\sf k}(G,m)$ and the dual Cheeger
constant.
###### Corollary 23.
For any summable weighted graph $(G,m)$ one has
(32) $\displaystyle\overline{h}(G,m)+{\sf k}(G,m)\geq 1.$
###### Proof.
The statement follows by taking infimum of the both sides of (31). ∎
Below is a relation between the quantities $R_{A}$ and $R_{B}$ and the Cheeger
constant:
###### Lemma 24.
The Cheeger constant $h(G,m)$ of a summable weighted graph $(G,m)$ equals
(33) $\displaystyle 1-\sup_{A,B}\min\\{R_{A},R_{B}\\},$
where the supremum is taken with respect to all nonempty
$A,B\subset\operatorname{V}(G)$ that partition $\operatorname{V}(G)$.
###### Proof.
We may write
$R_{A}=\frac{2\cdot m(A,A)}{m(A)}=\frac{m(A)-m(\partial A)}{m(A)}=1-h(A)$
and similarly $R_{B}=1-h(B)$. Thus,
$h(G,m)=\inf_{A,B}\max\\{h(A),h(B)\\}=\inf_{A,B}\max\\{1-R_{A},1-R_{B}\\}=1-\sup_{A,B}\min\\{R_{A},R_{B}\\}.$
∎
## 7\. Measure of asymmetry of the spectrum
Recall that for a pair of non-empty subsets $X$ and $Y$ of a metric space,
their Hausdorff distance $d_{H}(X,Y)$ is defined as
${\displaystyle d_{\mathrm{H}}(X,Y)=\max\left\\{\,\sup_{x\in
X}d(x,Y),\,\sup_{y\in Y}d(X,y)\,\right\\},\\!}$
Equivalently,
$d_{\mathrm{H}}(X,Y)=\inf\\{\epsilon\geq 0;X\subset
Y_{\epsilon}\,\,\mbox{and}\,\,Y\subset X_{\epsilon}\\}$
where ${\displaystyle X_{\epsilon}=\bigcup_{x\in X}\\{z\in M\,;\
d(z,x)\leq\epsilon\\}}.$
In this section we consider a summable weighted graph $(G,m)$ and the spectrum
$\sigma(\Delta)\subset[0,2]$ of its Laplacian $\Delta:L^{2}(G,m)\to
L^{2}(G,m)$, see (1) and Lemma 2. The main result of this section, Theorem 25,
describes the asymmetry of the spectrum $\sigma(\Delta)$ in terms of the
invariant ${\sf k}(G,m)$ introduced in the previous section.
###### Theorem 25.
Let $\mathcal{R}:[0,2]\to[0,2]$ denote the reflection $\mathcal{R}(x)=2-x$.
The Hausdorff distance between the spectrum of the Laplacian $\sigma(\Delta)$
and its reflection $\mathcal{R}(\sigma(\Delta))$ is at most $2\cdot{\sf
k}(G,m)$, i.e.
(34) $\displaystyle
d_{\mathrm{H}}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))\leq 2\cdot{\sf
k}(G,m).$
The following simple observation will be useful.
###### Lemma 26.
For any subset $X\subset[0,2]$, the Hausdorff distance
$d_{\mathrm{H}}(X,\mathcal{R}(X))$ equals
$\inf\\{\epsilon>0;\mathcal{R}(X)\subset X_{\epsilon}\\}$.
###### Proof.
Since $\mathcal{R}$ is an involution and an isometry, the relation
$\mathcal{R}(X)\subset X_{\epsilon}$ implies
$X=\mathcal{R}(\mathcal{R}(X))\subset\mathcal{R}(X_{\epsilon})=\mathcal{R}(X)_{\epsilon}.$
The statement now follows from the definition of the Hausdorff distance. ∎
The proof of Theorem 25 is completed by the end of this section.
Recall that the Laplacian $\Delta\colon L^{2}(G,m)\to L^{2}(G,m)$ equals $I-P$
where $I$ is the identity operator and $P\colon L^{2}(G,m)\to L^{2}(G,m)$ is
given by formula (4).
Let $\psi\colon\operatorname{V}(G)\rightarrow$ be a function satisfying
$\psi^{2}\equiv 1$. Consider the associated partition of the vertex set
$A\sqcup B=\operatorname{V}(G)$ where $A=\psi^{-1}(1)$ and $B=\psi^{-1}(-1)$.
We obtain the decomposition of $L^{2}(G,m)$ into the direct sum of two Hilbert
spaces
(35) $\displaystyle L^{2}(G,m)\,=\,L^{2}(A,m)\oplus L^{2}(B,m).$
Here $L^{2}(A,m)\subset L^{2}(G,m)$ is the space of functions $f\in
L^{2}(G,m)$ with ${\sf supp}(f)\subset A$ and similarly for $L^{2}(B,m)$.
Denote by
$\pi_{A},\pi_{B}:L^{2}(G,m)\to L^{2}(G,m)$
the projections
$\pi_{A}(f)|_{A}=f|_{A},\quad\pi_{A}(f)|_{B}=0,$
and similarly for $\pi_{B}$. Clearly, the operators $\pi_{A},\pi_{B}$ are
self-adjoint. Write
$T_{\psi}=\pi_{A}-\pi_{B}\quad\mbox{and}\quad P_{\psi}=P_{A}+P_{B},$
viewed as operators $L^{2}(G,m)\to L^{2}(G,m)$, where
$P_{A}=\pi_{A}P\pi_{A}:L^{2}(A,m)\to L^{2}(A,m)\quad\mbox{and}\quad
P_{B}=\pi_{B}P\pi_{B}:L^{2}(B,m)\to L^{2}(B,m).$
The operators $P_{A}$ and $P_{B}$ are self-adjoint since they are compositions
of self-adjoint operators.
###### Lemma 27.
One has
$T_{\psi}^{-1}\Delta T_{\psi}=2\cdot I-\Delta-2\cdot P_{\psi}$
and therefore
(36) $\displaystyle\sigma(\Delta)=\sigma(2\cdot I-\Delta-2\cdot P_{\psi}).$
###### Proof.
Since $T_{\psi}^{-1}=T_{\psi}$ one has
$\displaystyle T_{\psi}^{-1}\Delta T_{\psi}$ $\displaystyle=$
$\displaystyle(\pi_{A}-\pi_{B})(I-P)(\pi_{A}-\pi_{B})$ $\displaystyle=$
$\displaystyle I-(\pi_{A}-\pi_{B})P(\pi_{A}-\pi_{B})$ $\displaystyle=$
$\displaystyle I+\pi_{A}P\pi_{B}+\pi_{B}P\pi_{A}-P_{A}-P_{B}$ $\displaystyle=$
$\displaystyle I+P-2\cdot(P_{A}+P_{B})$ $\displaystyle=$ $\displaystyle 2\cdot
I-\Delta-2\cdot P_{\psi}.$
This proves the first claim of the Lemma. The second claim follows as well
using the fact that the spectrum is invariant under conjugation. ∎
Next we show that the norm of the operator $P_{\psi}$ can be estimated using
the combinatorial invariant ${\sf k}(A,B)$, which is introduced in §6.
###### Lemma 28.
Let $(G,m)$ be a summable weighted graph. Let
$\psi\colon\operatorname{V}(G)\rightarrow$ be a function satisfying
$\psi^{2}\equiv 1$ and let $A\sqcup B=\operatorname{V}(G)$ be the associated
partition of the vertex set $\operatorname{V}(G)$, where $\psi|_{A}\equiv 1$
and $\psi|_{B}\equiv-1$. Then one has
(37) $\displaystyle||P_{\psi}||\leq{\sf k}(A,B).$
###### Proof.
Since $P_{\psi}$ is self-adjoint, its norm $||P_{\psi}||$ equals the supremum
of
$\frac{|\langle P_{\psi}f,f\rangle|}{\langle f,f\rangle}$
taken over all nonzero $f\in L^{2}(G,m)$, see [8], §9.2. By construction,
$P_{\psi}$ is the direct sum of the operators $P_{A}$ and $P_{B}$ and we show
below that
(38) $\displaystyle||P_{A}||\leq\sup_{v\in
A}\\{p_{A}(v)\\}\quad\mbox{and}\quad||P_{B}||\leq\sup_{v\in B}\\{p_{B}(v)\\},$
where for $v\in A$ the quantity $p_{A}(v)$ is defined as $m(v)^{-1}\sum_{w\in
A}m(vw)$ and similarly for $p_{B}(v)$. Clearly, due to the definition of ${\sf
k}(A,B)$, (38) implies (37) and thus we only need to prove the inequality
(38).
For $f\in L^{2}(A,m)$ we have $(P_{A}f)(v)=m(v)^{-1}\cdot\sum_{w\in
A}m(vw)f(w)$ and therefore
$\displaystyle\langle P_{A}f,f\rangle$ $\displaystyle=$
$\displaystyle\sum_{v,w\in A}m(vw)f(v)f(w)$ $\displaystyle\leq$ $\displaystyle
1/2\cdot\sum_{v,w\in A}m(vw)[f(v)^{2}+f(w)^{2}]=\sum_{v,w\in A}m(vw)f(v)^{2}$
$\displaystyle=$ $\displaystyle\sum_{v\in A}m(v)\cdot\left[\sum_{w\in
A}\frac{m(vw)}{m(v)}\right]f(v)^{2}=\sum_{v\in A}m(v)\cdot p_{A}(v)\cdot
f(v)^{2}$ $\displaystyle\leq$ $\displaystyle\sup_{v\in
A}\,\\{p_{A}(v)\\}\cdot||f||^{2}.$
This gives the left inequality (38); the right one follows similarly. ∎
Lemma 28 implies:
###### Corollary 29.
One has
$\inf_{\psi}\big{\\{}||P_{\psi}||\big{\\}}\leq{\sf k}(G,m),$
where the infimum is taken over all functions
$\psi\colon\operatorname{V}(G)\rightarrow$ satisfying $\psi^{2}\equiv 1$.
###### Proof of Theorem 25.
Using (36) together with an obvious equality
$\sigma(2I-\Delta))=\sigma(\mathcal{R}(\Delta))=\mathcal{R}(\sigma(\Delta))$
and Theorem 4.10 from [6] we have
$\displaystyle
d_{\mathrm{H}}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))=d_{\mathrm{H}}(\sigma(2I-\Delta-2\cdot
P_{\psi}),\sigma(2I-\Delta))\leq 2\cdot||P_{\psi}||.$
The LHS of this inequality is independent of $\psi$. Taking infimum with
respect to $\psi$ and applying Corollary 29 we arrive at (34). ∎
## 8\. The spectrum of the infinite complete graph
We start with the following general remark about summable weighted graphs.
###### Lemma 30.
Let $(G,m)$ be a summable weight graph. Consider the random walk operator
$P\colon L^{2}(G,m)\to L^{2}(G,m)$, where
$(Pf)(v)=\sum_{w}\frac{m(vw)}{m(v)}\cdot f(w).$
Then $P$ is a Hilbert-Schmidt operator if and only if
(39) $\displaystyle\sum_{v,w}\frac{m(vw)^{2}}{m(v)m(w)}<\infty.$
###### Proof.
Let $g_{v}\in L^{2}(G,m)$ be the function taking the value $m(v)^{-1/2}$ at
point $v$ and vanishing at all other points. Clearly, the system
$\\{g_{v};v\in\operatorname{V}(G)\\}$
is an orthonormal basis. The matrix coefficients of the operator $P$ in this
basis are
$\langle
Pg_{w},g_{v}\rangle=\sum_{a\in\operatorname{V}(G)}m(a)(Pg_{w})(a)g_{v}(a)=\frac{m(vw)}{\sqrt{m(v)m(w)}}.$
Thus, condition (39) is equivalent to the well-known criterion for Hilbert-
Schmidt operators. ∎
###### Example 31.
Consider again the weighted graph $(G,m)$ of Example 7, i.e. $G$ is the
infinite complete graph with vertex set $\mathbb{N}$ and with the weights
$m(ij)=p_{i}p_{j}\quad\mbox{where }\quad p_{1}\geq p_{2}\geq
p_{3}\geq\dots,\quad\sum_{i\geq 1}p_{i}=1.$
The weights of vertexes are $m(i)=p_{i}q_{i}$, where $q_{i}=1-p_{i}$.
Condition (39) requires in this case convergence of the series
$\sum_{i\not=j}\frac{(p_{i}p_{j})^{2}}{p_{i}q_{i}p_{j}q_{j}}\leq\sum_{i,j}\frac{p_{i}p_{j}}{q_{i}q_{j}}=\left(\sum_{i}p_{i}q_{i}^{-1}\right)^{2}<q_{1}^{-2}.$
We see that in this example the random walk operator $P$ is Hilbert-Schmidt
and hence compact.
###### Corollary 32.
The spectrum of the Laplacian $\Delta=I-P:L^{2}(G,m)\to L^{2}(G,m)$ of the
infinite complete graph $(G,m)$ consists of an infinite sequence of
eigenvalues converging to $1\in[0,2]$ and the point $1$ is the only point of
the spectrum which is not an eigenvalue.
Below we analyse further the infinite complete graph and give a more detailed
information about its spectrum.
The random walk operator $P$ is given by
$(Pf)(i)=\sum_{j\not=i}\frac{p_{j}}{q_{j}}f(j)=\sum_{j}\frac{p_{j}}{q_{j}}f(j)-\frac{p_{i}}{q_{i}}f(i).$
Consider the eigenvalue equation $Pf_{\lambda}=\lambda f_{\lambda}$, i.e.
(40)
$\displaystyle\sum_{j}\frac{p_{j}}{q_{j}}f_{\lambda}(j)=(p_{i}q_{i}^{-1}+\lambda)f_{\lambda}(i)$
for any $i\in\mathbb{N}$. Therefore, for any $i\geq 1$ one has
$(p_{i}q_{i}^{-1}+\lambda)f_{\lambda}(i)=(p_{1}q_{1}^{-1}+\lambda)f_{\lambda}(1).$
Without loss of generality we may assume that
$(p_{1}q_{1}^{-1}+\lambda)f_{\lambda}(1)=1$. We obtain an infinite increasing
sequence of numbers
$\alpha_{1}\leq\alpha_{2}\leq\dots<0,\quad\alpha_{i}\to
0,\quad\mbox{where}\quad\alpha_{i}=-p_{i}q_{i}^{-1},$
and the eigenfunction $f_{\lambda}$ satisfies
$f_{\lambda}(i)=(\lambda-\alpha_{i})^{-1},\quad i\geq 1.$
The equation (40) becomes
(41) $\displaystyle\sum_{j=1}^{\infty}\frac{\alpha_{j}}{\alpha_{j}-\lambda}=1$
and the condition $f_{\lambda}\in L^{2}(G,m)$ can be expressed as
$\sum_{i=1}^{\infty}p_{i}q_{i}(\alpha_{j}-\lambda)^{-2}\ <\ \infty.$ Since
$q_{1}\leq q_{i}<1$, the latter condition is equivalent to
(42) $\displaystyle\sum_{j=1}^{\infty}p_{j}(\lambda-\alpha_{j})^{-2}\ <\
\infty.$
Corollary 33 summarises the above arguments.
###### Corollary 33.
A number $\lambda\in[0,2]$ is an eigenvalue of the random walk operator $P$
for the infinite complete graph if and only if the equation (41) and
inequality (42) are satisfied.
We shall see later that the condition (42) is automatically satistfied.
As an example, consider $\lambda=1$. One has
$\frac{\alpha_{j}}{\alpha_{j}-1}=p_{j}$ and hence equation (41) is satisfied.
Besides, $\frac{p_{j}}{(1-\alpha_{j})^{2}}=p_{j}q_{j}^{2}$ and (42) follows
from $\sum_{j}p_{j}=1$ and $q_{j}<1$. Thus, $\lambda=1$ is an eigenvalue.
The next lemma describes the whole spectrum of $P$.
###### Lemma 34.
Assume that all numbers $p_{i}$ are pairwise distinct, i.e.
$p_{1}>p_{2}>p_{3}>\dots$. The random walk operator $P\colon L^{2}(G,m)\to
L^{2}(G,m)$ has a unique positive eigenvalue $1$ and negative eigenvalues
$\lambda_{1}<\lambda_{2}<\dots<0$ satisfying
$\alpha_{i}<\lambda_{i}<\alpha_{i+1},\quad i=1,2,\dots.$
Additionally, $0$ belongs to the spectrum (as an accumulation point) and is
the only point of the spectrum which is not an eigenvalue.
###### Proof.
Consider the function
(43) $\displaystyle
F(\lambda)=\sum_{j=1}^{\infty}\frac{\alpha_{j}}{\alpha_{j}-\lambda},\quad\quad\lambda\in\Omega=-\\{0,\alpha_{1},\alpha_{2},\dots\\}.$
First, we verify that the series (43) converges for all $\lambda\in\Omega$.
Indeed, if $\lambda>0$ then $\lambda-\alpha_{j}>\lambda$ for all $j\geq 1$ and
$\sum_{j\geq
1}\frac{\alpha_{j}}{\alpha_{j}-\lambda}\,<\,\lambda^{-1}\sum_{j\geq
1}(-\alpha_{j})\,<\,\lambda^{-1}.$
Consider now the case when $\lambda\in(\alpha_{i},\alpha_{i+1})$. Then for
$j\geq i+2$ one has $\alpha_{j}-\lambda\,>\,\alpha_{i+2}-\alpha_{i+1}$ and
therefore
$\sum_{j\geq
i+2}\frac{\alpha_{j}}{\alpha_{j}-\lambda}\,\leq\,(\alpha_{i+2}-\alpha_{i+1})^{-1}\cdot\sum_{j\geq
i+2}(-\alpha_{j})\ <\ \infty.$
For $\epsilon>0$, let $\Omega_{\epsilon}$ denote $-\cup_{j\geq
1}B(\alpha_{j},\epsilon)$, where $B(\alpha_{j},\epsilon)$ stands for the open
ball with centre $\alpha_{j}$ and radius $\epsilon$. The arguments of the
preceding paragraph applied to the series of derivatives
(44) $\sum_{j=1}^{\infty}\frac{\alpha_{j}}{(\alpha_{j}-\lambda)^{2}}$
show that the series (44) converges uniformly on $\Omega_{\epsilon}$.
Therefore, we obtain that the function $F(\lambda)$ is differentiable and its
derivative $F^{\prime}(\lambda)$ is given by the series (44) for all
$\lambda\in\Omega$.
Each term of the series (44) is negative; this implies that $F(\lambda)$ is
monotone decreasing on every interval contained in $\Omega$. Thus, equation
(41) may have at most one solution in any such interval.
Consider the behaviour of $F(\lambda)$ on one of the intervals
$\lambda\in(\alpha_{i},\alpha_{i+1})$. It is obvious that for
$\lambda\to\alpha_{i}$ the function $F(\lambda)$ tends to $+\infty$ and for
$\lambda\to\alpha_{i+1}$ one has $F(\lambda)\to-\infty$.
There are also two infinite maximal intervals of continuity
$(-\infty,\alpha_{1})$ and $(0,\infty)$. The function $F(\lambda)$ is negative
for $\lambda\in(-\infty,\alpha_{1})$ and has limits $0$ and $-\infty$ at the
end points. Besides, $F(\lambda)$ is positive on $(0,\infty)$ and its limits
at the end points are $\infty$ and $0$. Figure 2 summarises the above
arguments.
Figure 2. The graph of the function $F(\lambda)$.
Thus we see that there is a unique solution of (41) in each interval
$(\alpha_{i},\alpha_{i+1})$; besides, there is a unique solution of (41) in
the interval $(0,\infty)$ which, as we know, is $\lambda=1$.
Finally, we note that any of the above solutions automatically satisfies (42).
Indeed, if $\lambda\in(\alpha_{i},\alpha_{i+1})$ then
$\alpha_{j}-\lambda>\alpha_{i+2}-\alpha_{i+1}$ and
$\sum_{j\geq
i+2}p_{j}(\lambda-\alpha_{j})^{-2}<(\alpha_{i+2}-\alpha_{i+1})^{-2}\cdot\sum_{j\geq
i+2}p_{j}\,<\,\infty.$
This completes the proof. ∎
We may now restate the results of this section for the Laplacian $\Delta=I-P$.
###### Proposition 35.
Consider the infinite complete graph $(G,m)$ with $V(G)=\mathbb{N}$ and
weights $m(ij)=p_{i}p_{j}$ where $p_{1}>p_{2}>\dots>0$ is a sequence with
$\sum_{i\geq 1}p_{i}=1$. The spectrum of the Laplacian $\Delta:L^{2}(G,m)\to
L^{2}(G,m)$ contains the point $1$ as an accumulation point and the other
points of the spectrum are simple eigenvalues. The point $0$ is the unique
eigenvalue in the interval $[0,1)$. The remaining eigenvalues
$\dots\,<\,\mu_{3}\,<\,\mu_{2}\,<\,\mu_{1}\,\leq\,2$ lie in $(1,2)$ and
satisfy
(45) $\displaystyle q_{i+1}^{-1}<\mu_{i}<q_{i}^{-1},\quad\mbox{where}\quad
q_{i}=1-p_{i},\quad\mbox{for}\quad i=1,2,\dots,$
hence $\mu_{i}\to 1$. More specifically, for $i=1,2,\dots,$ each eigenvalue
$\mu_{i}$ is the unique solution of the equation
(46) $\displaystyle\sum_{j=1}^{\infty}\frac{p_{j}}{q_{j}\mu-1}\,=\,-1$
lying in the interval $(q_{i+1}^{-1},q_{i}^{-1})$. The spectral gap equals $1$
and top of the spectrum $\mu_{top}=\mu_{1}$ satisfies
(47) $\displaystyle q_{2}^{-1}\,<\,\mu_{top}\,<\,q_{1}^{-1}.$
###### Proof.
Since $\Delta=I-P$ we see that the result of Proposition 35 follows from Lemma
34 by applying the affine transformation $x\mapsto 1-x$. The points of
division $\alpha_{j}$ are mapped into $1-\alpha_{j}=q_{j}^{-1}$. ∎
We can mention another form (48) of the eigenvalue equation (46) which
involves the quantities $r_{j}=q_{j}^{-1}$, where $j=1,2,\dots$
(48) $\displaystyle\sum_{j=1}^{\infty}\frac{r_{j}-1}{r_{j}-\mu}=1.$
Note that here $r_{j}>1$ and $r_{j}\to 1$. We know that equation (48) has
exactly one solution in each interval $(r_{j+1},r_{j})$ and $\mu=0$ is an
additional solution.
## 9\. Improved estimates for $\mu_{1}=\mu_{top}$
Our goal in this section is to strengthen the inequality (47) for the top
eigenvalue $\mu_{1}=\mu_{top}$. Similar method can be applied for more precise
estimates of the other eigenvalues.
Recall that the parameters of the infinite complete graph satisfy
$p_{1}>p_{2}>\dots$ and $\sum_{j}p_{j}=1$. We denote $r_{j}=(1-p_{j})^{-1}$.
We have $r_{1}>r_{2}>r_{3}>\dots>1$ and $r_{j}\to 1$. Consider the following
quadratic equation
(49)
$\displaystyle\frac{r_{1}-1}{r_{1}-\mu}+\frac{r_{2}-1}{r_{2}-\mu}\,=\,1-x,$
where $x$ is a parameter.
###### Lemma 36.
1. (A)
For any value of the parameter $x$ the quadratic equation (49) has a unique
solution $\mu(x)$ lying in the interval $(r_{2},r_{1})$.
2. (B)
For $x<1$, the other solution of the equation (49) lies in the interval
$(-\infty,r_{2})$; for $x>1$, the other solution of (49) lies in the interval
$(r_{1},\infty)$; for $x=1$ the equation (49) does not have solutions outside
the interval $(r_{2},r_{1})$.
3. (C)
$\mu(x)$ is a decreasing function of $x$.
4. (D)
The top of the spectrum $\mu_{top}=\mu_{1}$ of the Laplacian $\Delta$
satisfies
(50) $\displaystyle\mu(x_{+})<\mu_{top}<\mu(x_{-}),$
where
$x_{+}=\frac{1-p_{1}-p_{2}}{1-r_{1}}\quad\mbox{and}\quad
x_{-}=(1-p_{1}-p_{2})\cdot\frac{r_{3}}{r_{3}-r_{2}}.$
###### Proof.
The rational function
$G(\mu)=\frac{r_{1}-1}{r_{1}-\mu}+\frac{r_{2}-1}{r_{2}-\mu}$ has poles at
$\mu=r_{1}$ and $\mu=r_{2}$, and it is increasing for $\mu\in(r_{2},r_{1})$.
Its graph is shown in Figure 3.
Figure 3. The graph of the function $G(\mu)$.
We easily see that $G\colon(r_{2},r_{1})\to\mathbb{R}$ is a homeomorphism,
which implies (A). Similarly, we see that
$G\colon(-\infty,r_{2})\to(0,\infty)$ and
$G\colon(r_{1},\infty)\to(-\infty,0)$ are monotone increasing homeomorphisms;
this implies our statement (B).
To prove (C), we differentiate the equation $G(\mu(x))=1-x$ getting
$\mu^{\prime}(x)=-G^{\prime}(\mu(x))^{-1}.$ This shows that $\mu^{\prime}<0$
since $G^{\prime}>0$.
Finally, to prove (D), we note that in view of equation (48) one has
$\mu_{top}=\mu(x_{0})$ with
$x_{0}=\sum_{j\geq
3}\frac{r_{j}-1}{r_{j}-\mu},\quad\mbox{where}\quad\mu=\mu_{top}.$
We know that $r_{2}<\mu_{top}<r_{1}$ and hence
$x_{0}<\sum_{j\geq 3}\frac{r_{j}-1}{r_{j}-r_{1}}=\sum_{j\geq
3}\frac{r_{j}}{r_{j}-r_{1}}\cdot
p_{j}\leq\frac{1}{1-r_{1}}\cdot(1-p_{1}-p_{2})=x_{+}.$
Here we used the equalities $\frac{r_{j}-1}{r_{j}}=p_{j}$ and $\sum_{j\geq
1}p_{j}=1$. Similarly, we have
$x_{0}>\sum_{j\geq 3}\frac{r_{j}-1}{r_{j}-r_{2}}=\sum_{j\geq
3}\frac{r_{j}}{r_{j}-r_{2}}\cdot
p_{j}\geq\frac{r_{3}}{r_{3}-r_{2}}\cdot(1-p_{1}-p_{2})=x_{-}.$
Thus, using (C), we conclude that
$\mu(x_{+})\leq\mu_{top}=\mu(x_{0})\leq\mu(x_{-}).$ ∎
Below we shall have specific examples illustrating Lemma 36.
## 10\. Asymmetry of the spectrum and the invariant ${\sf k}(G,m)$ for the
infinite complete graph
In the section we continue studying the infinite complete graph and its
spectrum. Our goal is to examine Theorem 25 in this specific example.
First we describe the Hausdorff distance
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))$.
###### Lemma 37.
If $\mu_{1}\leq 3/2$ then
(51) $\displaystyle
d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))=2-\mu_{1}\,\geq\,\frac{1}{2}.$
If $\mu_{1}\geq 3/2$ then
(52)
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))=\frac{1}{2}-\inf_{i\geq
1}\left|\mu_{i}-\frac{3}{2}\right|\,\leq\,\frac{1}{2}.$
###### Proof.
We know that $\sigma(\Delta)=\\{0,1\\}\cup\\{\mu_{1},\mu_{2},\dots\\}$ where
$\mu_{1}>\mu_{2}>\dots>1$, $\mu_{i}\to 1$. Hence,
$\mathcal{R}(\sigma(\Delta))=\\{1,2\\}\cup\\{\bar{\mu}_{1},\bar{\mu}_{2},\dots,\\},$
where $\bar{\mu}_{i}=2-\mu_{i}$. We intend to apply Lemma 26. We have:
$\inf\\{\epsilon;\,2\in\sigma(\Delta)_{\epsilon}\\}=2-\mu_{1}$. Besides,
$\inf\\{\epsilon;\,1\in\sigma(\Delta)_{\epsilon}\\}=0$ and for $i=1,2,\dots$,
one has
$\inf\\{\epsilon;\,\bar{\mu}_{i}\in\sigma(\Delta)_{\epsilon}\\}=\min\\{2-\mu_{i},\mu_{i}-1\\}.$
This clearly implies that, for $\mu_{1}\leq 3/2$, the Hausdorff distance
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))$ equals $2-\mu_{1}$.
Consider now the case $\mu_{1}\geq 3/2$. By the above, the Hausdorff distance
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))$ in this case equals
$\sup_{i\geq 1}\\{\min\\{2-\mu_{i},\mu_{i}-1\\}\\}.$ We may write
$\min\\{2-\mu_{i},\mu_{i}-1\\}=\frac{1}{2}+\min\\{\frac{3}{2}-\mu_{i},\mu_{i}-\frac{3}{2}\\}=\frac{1}{2}-\left|\mu_{i}-\frac{3}{2}\right|.$
This implies (52). ∎
Next we consider our invariant ${\sf k}(G,m)$, see Definition 17. For a subset
$A\subset\mathbb{N}=\operatorname{V}(G)$ we set $P(A)=\sum_{i\in A}p_{i}$ and
$p_{\rm{min}}(A)=\inf_{i\in A}p_{i}$. Then for a partition $A\sqcup
B=\mathbb{N}$ one has
${\sf
k}(A,B)=\max\left\\{\frac{P(A)-p_{\rm{min}}(A)}{1-p_{\rm{min}}(A)},\frac{P(B)-p_{\rm{min}}(B)}{1-p_{\rm{min}}(B)}\right\\}.$
For an infinite set $A\subset\mathbb{N}=\operatorname{V}(G)$ clearly
$p_{\rm{min}}(A)=0$. Hence, if both sets $A,B$ are infinite, we have
${\sf k}(A,B)=\max\\{P(A),P(B)\\}\geq 1/2$
since $P(A)+P(B)=1$. Therefore, the infimum of the numbers ${\sf k}(A,B)$
taken over all partitions $A\sqcup B=\mathbb{N}$ with both sets $A,B$ infinite
equals
$\inf\\{P(A);\,A\subset\mathbb{N}\,\,\mbox{is infinite and}\,\,P(A)\geq
1/2\\}.$
Next, we consider partitions $A\sqcup B=\mathbb{N}$ with $A$ finite and hence
$B$ infinite. We shall be mainly interested in the situation when $p_{1}\geq
1/2$. In this case one may take the partition $A^{\prime}=\\{1\\}$,
$B^{\prime}=\\{2,3,\dots\\}$; then ${\sf
k}(A^{\prime},B^{\prime})=P(B^{\prime})=1-p_{1}$.
Let us show that, in fact,
${\sf k}(G,m)=1-p_{1}$
under the assumption $p_{1}\geq 1/2$. All partitions with both sets $A$ and
$B$ infinite give ${\sf k}(A,B)\geq 1/2\geq 1-p_{1}$. Consider a partition
$A\sqcup B=\mathbb{N}$ with finite $A$, $A\not=\\{1\\}$. If
$p_{\rm{min}}(A)=p_{i}$, $i>1$, then
${\sf k}(A,B)=\max\left\\{\frac{P(A)-p_{i}}{1-p_{i}},P(B)\right\\}.$
If $1\notin A$ then $1\in B$ and ${\sf k}(A,B)=P(B)\geq 1-p_{1}.$ Consider now
the remaining case: $1\in A$ and $p_{\rm{min}}(A)=p_{i}$ where $i>1$. Then
$P(B)\leq 1-p_{1}-p_{i}<1/2$ and
$\frac{P(A)-p_{i}}{1-p_{i}}\geq\frac{p_{1}}{1-p_{i}}\geq 1/2,$
and thus ${\sf k}(A,B)=\frac{P(A)-p_{i}}{1-p_{i}}\geq 1/2\geq 1-p_{1}.$ We
summarise our above arguments as follows.
###### Lemma 38.
If $p_{1}\geq 1/2$ then ${\sf k}(G,m)=1-p_{1}$.
###### Example 39.
Consider an infinite complete graph with the following parameters:
$p_{1}=0.9$, $p_{2}=0.09$, $p_{3}=0.009$; the values $p_{4},p_{5},\dots$ will
be irrelevant for our estimates below.
By Lemma 38, one has ${\sf k}(G,m)=0.1$ and applying Theorem 25 we obtain that
the spectral asymmetry satisfies
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))\leq 0.2.$
Next, we apply Lemma 37, which gives $\mu_{1}>3/2$, and the equality (52)
implies
$\inf_{i\geq 1}\left|\mu_{i}-\frac{3}{2}\right|\geq 0.5-0.2=0.3.$
In other words, the open interval
$(1.2,1.8)=\left(\frac{3}{2}-\frac{3}{10},\frac{3}{2}+\frac{3}{10}\right)\subset[0,2]$
contains no eigenvalues. Thus, our inequality (34) allows finding lacunas in
the spectrum.
We can also estimate the Hausdorff distance
$d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))$ using information about
the eigenvalues. We have $r_{1}=q_{1}^{-1}=10$,
$r_{2}=q_{2}^{-1}=\frac{100}{91}$ and hence $1.099<\mu_{1}<10$, by the
inequality (47), which is not informative enough. We can improve the bound
(47) using Lemma 36. We obtain $x_{+}=-\frac{0.01}{9}\approx-0.001111$, and
$x_{-}=-\frac{0.01}{\frac{0.991}{0.91}-1}\approx-0.11235$. Solving the
quaratic equation (49) we obtain $1.94747\leq\mu_{1}\leq 2.4194,$ which, of
course, should be understood as
$1.94747\leq\mu_{1}\leq 2.$
Next we estimate $\mu_{2}$ using (47). We obtain
$1.00908\approx\frac{1000}{991}=r_{3}=q_{3}^{-1}\,<\,\mu_{2}\,<\,r_{2}=q_{2}^{-1}=\frac{100}{91}\approx
1.099.$
Thus, using Lemma 37, we obtain for the Hausdorff distance
$0.00908\leq d_{H}(\sigma(\Delta),\mathcal{R}(\sigma(\Delta)))\leq 0.099.$
###### Remark 40.
It is an interesting inverse problem whether the spectrum of the infinite
complete graph determines the sequence of shape parameters
$p_{1},p_{2},\dots,$ satisfying $\sum_{i\geq 1}p_{i}=1$.
## References
* [1] F. Bauer, B. Hua and Jürgen Jost, The dual Cheeger constant and spectra of infinite graphs, Advances in Mathematics, 251(2014), 147 - 194.
* [2] F. Bauer, M. Keller and R. K. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, Journal of the European Mathematical Society, 17 (2015), 259 - 271.
* [3] F. R. K. Chung, Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92. AMS, Providence, RI, 1997. xii+207 pp.
* [4] J. Dodzuik, Difference equations, isoperimetric inequalities and transience of certain random walks, Transactions of the American Mathematical Society, 284(1984).
* [5] S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bulletin of the AMS, 43(2006), 439 - 561.
* [6] T. Kato, PerturbationTheory for Linear Operators, Springer-Verlag 1980.
* [7] M. Keller, D. Lenz, R. Wojciechowski, Radosław K. Graphs and discrete Dirichlet spaces. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 358. Springer, 2021.
* [8] E. Kreyszig, Introductory functional analysis and applications, John Wiley & Sons, 1978
* [9] S. Mokhtari-Sharghi, Cheeger inequality for infinite graphs, Geometriae Dedicata, 100(2003), 53 - 64.
* [10] M. Reed, B. Simon, Methods of modern mathematical physics, IV, Analysis of operators, Academic Press,1978
* [11] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press, 2000.
|
# Singhing with Confidence: Visualising the Performance of Confidence
Structures
Alexander Wimbush
<EMAIL_ADDRESS>
Institute for Risk and Uncertainty University of Liverpool
and
Nicholas Gray
Institute for Risk and Uncertainty University of Liverpool
and
Scott Ferson
Institute for Risk and Uncertainty University of Liverpool
###### Abstract
Confidence intervals are an established means of portraying uncertainty about
an inferred parameter and can be generated through the use of confidence
distributions. For a confidence distribution to be ideal, it must maintain
frequentist coverage of the true parameter. This can be represented for a
precise distribution by adherence to a cumulative unit uniform distribution,
referred to here as a Singh plot. This manuscript extends this to imprecise
confidence structures with bounds around the uniform distribution, and
describes how deviations convey information regarding the characteristics of
confidence structures designed for inference and prediction. This quick visual
representation, in a manner similar to ROC curves, aids the development of
robust structures and methods that make use of confidence. A demonstration of
the utility of Singh plots is provided with an assessment of the coverage of
the ProUCL Chebyshev upper confidence limit estimator for the mean of an
unknown distribution.
###### keywords:
Confidence, Coverage, Imprecise, Inference, Prediction
## Acknowledgements
This work was supported by the EPSRC programme grant ‘Digital twins for
improved dynamic design’, EP/R006768/1, and the EPSRC and ESRC Centre for
Doctoral Training in Quantification and Management of Risk and Uncertainty in
Complex Systems and Environments, EP/L015927/1.
Word Count = 4815
## 1 Introduction
Inferring the value of a model parameter or the output of a stochastic system
is a common aspect of many scientific endeavours. In a frequentist view, this
can be accomplished through the use of intervals which are guaranteed to bound
the answer with a minimum given probability. These inferences and predictions
can then be utilised with confidence, knowing that they will be correct with
some minimum rate. Hence the name confidence intervals. Some strategies for
generating these intervals include confidence distributions [1], boxes and
structures [2] which can be used for computation, preserving the frequentist
properties throughout [3].
However, this property only holds when the strategy for generating these
intervals is valid. This could perhaps be proven mathematically, but in many
cases such a proof may be excessively difficult to develop. If a person were
to question whether the chosen strategy is valid, they may simply have to
accept that they might not be able to verify this themselves. Or they may
consider whether an alternative strategy is superior, but have no means of
investigating this without significant effort. In these cases, there is a need
for a means of validating that the strategy will produce intervals with this
coverage property in a manner that is easy and interpretable.
This paper introduces the notion of confidence as an expression of the
coverage property, and formalises the creation of plots to assess this.
Various properties of these structures can be inspected, allowing new
structures to be proposed, developed, and utilised for computation with
confidence. For ease of interpretation, this paper will consider a case of
inference about a distributional parameter $\theta_{0}$, though confidence
intervals may also be drawn for predictions of the value of a sample $x_{0}$.
## 2 Confidence Distributions
Confidence intervals are so named due to the fact that they provide assurance
that a desired minimum proportion $\alpha$ of inferences about an uncertain
parameter $\theta_{0}\in\Theta$ from a given length-$n$ dataset
$\bm{x}=\\{x_{1},\dots,x_{n}\\}\sim\text{F}(\theta_{0})$ will be correct so
long as the distributional assumptions hold. Generating these intervals is
most commonly performed using a valid confidence distribution
$\text{C}(\theta,\bm{x}):\Theta\to[0,1]$. For a desired confidence level
$\alpha\in[0,1]$, an interval
$\bm{\alpha}=\left[\underline{\alpha},\overline{\alpha}\right]\subseteq[0,1]$
is created such that $\overline{\alpha}-\underline{\alpha}=\alpha$. An
$\alpha$-level confidence interval is then defined as
$\bm{\theta}=\left[\underline{\theta},\overline{\theta}\right]=\left[\text{C}^{-1}(\underline{\alpha},\bm{x}),\text{C}^{-1}\left(\overline{\alpha},\bm{x}\right)\right]$
(1)
These intervals are valuable for statistical inference and prediction. One
such use case may be for estimation of the true mean value $\mu_{0}\in M$ of a
normal distribution based on a sample of independent and identically
distributed observations
$\bm{x}=\\{x_{1},\dots,x_{n}\\}\sim\text{N}(\mu_{0},\sigma)$, where $\sigma$
and $\mu_{0}$ are unknown. While a point value estimate
$\hat{\mu}_{0}=\bar{x}=\frac{1}{n}\sum{\bm{x}}$ is a useful statistic, the
interpreter should have no confidence that future samples will share this
precise mean since the target distribution is continuous and has non-zero
variance.
A lack of confidence here implies that the probability that this point
estimate is the true mean is zero. Naturally the probability that future
samples from this continuous distribution will have a mean bounded by a
precise value is 0. To resolve this, an $\alpha$-level confidence interval
generated using Equation 1 could be used as an estimate of $\mu_{0}$ with
knowledge that at least an $\alpha$ proportion of such inferences will be
correct. One means of generating such intervals is to draw them from the
following confidence distribution[4]:
$\text{C}(\mu,\bm{x})=\text{T}\left(\frac{\mu-\bar{x}}{s_{\bm{x}}/\sqrt{n}};n-1\right)$
(2)
where $\text{T}(\theta;n)$ is the cumulative probability density of a
Student’s t-distribution with $n$ degrees of freedom evaluated at $\theta$,
and $s_{\bm{x}}$ is the standard deviation of the sample. This then produces a
cumulative distribution mapping the support to the unit interval
$\text{C}(\mu,\bm{x}):M\to[0,1]$. This function represents the confidence
level $\alpha$ of a one-sided interval estimate
$\bm{\mu}=[\min\left(M\right),\mu]$ of the parameter.
The $\alpha$-level confidence intervals that can be drawn from this
distribution are not necessarily unique for a given $\alpha$. Generally a
strategy for generating confidence intervals would include some limit on the
$\bm{\alpha}$ intervals to ensure that each value of $\alpha$ corresponds to a
unique interval on $M$. Some examples include one sided intervals $[0,\alpha]$
or $[1-\alpha,1]$, and centred intervals
$[0.5-\frac{\alpha}{2},0.5+\frac{\alpha}{2}]$. With this defined,
$\text{C}^{-1}(\bm{\alpha},\bm{x}):[0,1]\to M$ produces unique intervals with
a desired confidence level $\alpha$.
## 3 Validating Confidence Distributions
It was asserted in Section that Equation 2 is a confidence distribution, and
that therefore the approach to generating confidence intervals defined above
is valid. But this is not immediately apparent from Equation 2 alone. An
analyst may wish to validate that the distribution they are using, or being
instructed to use, is in fact a valid confidence distribution. Until a
distribution is confirmed to maintain the property of coverage it should be
considered a proposed distribution, denoted with an asterisk
$\text{C}^{*}(\cdot)$. Verifying this property would allow its use as a
confidence distribution for reliable inference. It should be noted that
Equation 2 is a well known confidence distribution developed by Gosset over a
century ago [5]. The intent here is not to question the validity of this
distribution, but to demonstrate the properties of a valid confidence
distribution so that invalid distributions can be identified by comparison.
The first property required of a valid confidence distribution for inference
about a parameter $\theta\in\Theta$ is that it must be a cumulative
distribution across $\Theta$, and it should be apparent that Equation 2
satisfies this criteria. However, the second criteria required by Singh [4] is
not so simple to verify, and requires that at the true parameter value, for a
given dataset $\bm{x}$, $\text{C}^{*}(\theta_{0},\bm{x})$ follows the uniform
distribution U$(0,1)$. This effectively restricts a valid confidence
distribution to the following condition:
$\text{Pro}(\text{C}(\bm{\theta},\bm{x}\sim\text{F}(\theta_{0}))\geq\alpha)=\alpha;\forall\bm{\theta}\ni\theta_{0}$
(3)
This prevents confidence structures which produce vacuous intervals from being
considered valid. Confirming adherence to these properties allows true
confidence intervals to be generated as defined in Section 2. But this is not
necessarily a simple task, and may indeed be prohibitively difficult
mathematically. An alternative is to use a Monte Carlo approach to generate
values of $\text{C}^{*}(\theta_{0},\bm{x})$ and plotting the resulting values
against the U$(0,1)$ distribution. This approach is here referred to as a
Singh plot, in reference to the work of Professor Kesar Singh, a review of
which was compiled by Babu[6].
## 4 Singh Plots
Singh plots can be used to calculate and visualise the performance of a
proposed structure in terms of both coverage and aspects of conservatism. Once
a proposed structure can be demonstrated to achieve coverage using the Singh
plot, it can be used for reliably inferring confidence intervals about a
parameter. Indications of over and under-confidence and the impact of sample
sizes will also be apparent and may prompt the use or development of different
structures.
A Singh plot is generated by numerically estimating the minimum coverage
probability of a series of $\alpha$ level confidence intervals. This is
performed on a known distribution with defined parameters. Keeping with the
proposed distribution for the mean $\mu_{0}$ of a normal distribution given in
Equation 2, since it is an unbounded continuous distribution if it can be
demonstrated that the coverage is maintained when $\mu_{0}$ is known then the
same should hold for cases where $\mu_{0}$ is uncertain.
A series of $m$ sample sets of length $n$ are drawn from the target normal
distribution with defined parameters
$\bm{X}=\\{\bm{x}_{1},\dots,\bm{x}_{m}\\},\bm{x}_{i}\sim\text{N}(\mu_{0},\sigma)$.
The proposed confidence distribution is then used to generate an interval with
a defined strategy that has the minimum possible confidence whilst still
covering the known true value. The one-sided strategy
$\bm{\alpha}=[0,\overline{\alpha}=\text{C}^{*}(\mu_{0},\bm{x})]$ provides such
an interval for this distribution. Alternatively, an upper bounded interval
could be used, and would have a very similar interpretation. This is due to
the fact that the first derivative of the confidence distribution
$\text{C}^{*^{\prime}}(\mu,\bm{x})$ is at its minimum at the extremes of the
support for the proposed distribution. If disjoint intervals are permitted the
worst case interval is likewise one which extends to the extremes of the
support but which rejects the central portion of the distribution.
This process is repeated to produce $m$ intervals with corresponding
confidence values representing the minimum confidence required to bound the
true value. Ordering these values and plotting them against a cumulative unit
uniform distribution produces a plot of the minimum coverage probability for a
given $\alpha$-level confidence interval. This is the Singh plot as described
in Equation 4 and Algorithm 1, which visually demonstrates whether criteria 2
is met. The interpretation is similar to that of a receiver operating
characteristic curve, though the optimal case is represented by adherence to
the diagonal rather than the top left of the plot. The receiver operating
characteristic has utility in that is conveys a great deal of information
visually in a manner that is easily interpretable [7]. Singh plots allow a
similar ease of interpretation when assessing confidence structures.
$\text{S}(\bm{\alpha};\theta_{0})=\text{Pro}\left(\text{C}^{*-1}(\bm{\alpha},\bm{x}\sim\text{F}(\theta_{0}))\ni\theta_{0}\right);\bm{x}\in\bm{X}$
(4)
input : $C^{*}\>\>\leftarrow$Proposed confidence structure
$\text{f}(\bm{\theta})\leftarrow$Target distribution taking parameters
$\bm{\theta}$
$\theta_{0}\>\>\>\leftarrow$True value of parameter of interest
output : Singh plot for visual assessment of confidence structure properties
for _$i\in{1,\dots,m}$_ do
Generate sample: $\bm{x}={x_{1},\dots,x_{n}}\sim\text{f}(\bm{\theta})$;
Calculate minimum required confidence for coverage:
$s_{i}=\text{C}^{*}(\theta_{0},\bm{x})$
end for
Plot empirical CDF of $s$
Plot CDF of $U(0,1)$ for comparison
Algorithm 1 Generation of a Singh plot
For example, the confidence distribution in Equation 2 produces the Singh plot
shown on the right side of Figure 1. The left of the figure demonstrates
$\text{C}^{*}(\mu,\bm{x})$ for one of the sample sets $\bm{x}$ for which the
true value has been assigned a confidence value of
$\text{C}^{*}(\mu_{0},\bm{x})=0.65$. This indicates that a one-sided
confidence interval would require a confidence level of at least $\alpha=0.65$
in order to bound the true value. The figure on the right extends this to
$m=10^{4}$ samples of the same distribution, and indicates that
$\text{Pro}(\text{C}^{*}(\bm{\mu},\bm{x})\geq\alpha)\approx\alpha;\forall\bm{\mu}\ni\mu_{0}$.
There are deviations from the $\text{U}(0,1)$ distribution, but these are very
slight and more likely to be dependent on $m$ than the proposed distribution.
Note that since this is performed with a fixed mean and variance. Whilst it
wouldn’t be informative in this case to demonstrate coverage with variations
in these parameters (since this structure is well established), this may be
necessary depending on the application. The algorithm for this approach is
provided in Section 5.4.
These Singh plots give a clear visual cue as to the reliability of these
confidence structures. They allow any proposed structure to be quickly
assessed to ensure that anyone looking to make use of them can do so from an
informed perspective regardless of their mathematical capabilities. The result
here is shown for demonstrative purposes, this confidence structure is known
to perform at all confidence levels for one or two-sided intervals. The
utility of the Singh plot becomes apparent when used to assess the properties
of structures which may have unknown or poorly understood properties.
Producing a Singh plot according to Algorithm 1 may be preferable to a
conventional approach where confidence intervals are generated and then the
rate at which they cover the true value is estimated. This is due to the fact
that a Singh plot allows evaluation of coverage at all confidence levels
rather than just one. It also allows visibility over whether or not a
particular structure is conservative at some confidence levels and
appropriately calibrated at others, or perhaps over-confident at some levels
and conservative at others. This is particularly useful in the development of
new structures which must maintain coverage. Singh plots are a quick and
simple way to check whether the theory works in practice.
### 4.1 Proposed Bernoulli Confidence Distribution Singh Plot
A demonstration of a viable proposed distribution is shown in the preceding
section, but how would an interpreter understand that the distribution is not
viable? A simple example is that of inference about the rate parameter
$\theta\in\Theta$ of a Bernoulli distribution. A sample drawn from a Bernoulli
process using this distribution can be used to generate a Bayesian posterior
from a conjugate Jeffreys prior. This would take the form a beta distribution
with parameters $a=0.5$ and $b=0.5$. This produces the following proposed
confidence distribution for inference about the true parameter $\theta_{0}$
given a sample set
$\bm{x}=\\{x_{1},\dots,x_{n}\\},x_{i}\sim\text{Bin}(N=1,p=\theta_{0})$ where
$\text{Bin}(N=1,p)$ is a single observation binomial distribution with rate
parameter $p$:
$\text{C}^{*}\left(\theta,\bm{x}\right)=\text{B}\left(\theta;a=\sum{\bm{x}}+0.5,b=n-\sum{\bm{x}}+0.5\right)$
(5)
where B$(\theta;a,b)$ is the cumulative density of a beta distribution with
parameters $a,b$ evaluated at $\theta$. Equation 5 can be assessed as a
proposed confidence distribution in a similar manner as the distribution in
Equation 2. A collection of $m$ Sample sets are drawn from the target
distribution and used to generate one-sided
$[0,\text{C}^{*}(\theta_{0},\bm{x})]$ confidence intervals. Again, ordering
and plotting the confidence levels of these intervals produces a Singh plot,
which can be quickly checked to confirm that coverage is maintained.
Fig. 2 indicates that the proposed confidence distribution in Equation 5 is
not valid. This is due to the clear deviations from the minimum bounding
probability required to maintain coverage, as seen by the Singh plot extending
below the U$(0,1)$ plot at many points. This indicates that, for example, an
$\alpha=0.45$ level confidence interval would only bound the true value with a
minimum rate of $\approx 0.33$. This indicates that this structure will often
produce intervals which do not bound the true value at the desired rate. In
this case, a different structure should be devised.
### 4.2 Imprecise Bernoulli Confidence Distribution Singh Plot
Since Equation 5 failed to provide coverage, a different structure must be
devised. An important aspect of this problem for the case of the Bernoulli
distribution with a small sample size of $n=10$ is simply that there is not
enough data to make statements of precise confidence levels. A large degree of
uncertainty is being ignored in attempting to do so. For the Bernoulli
distribution, or its binomial extension, Clopper-Pearson intervals allow each
$\theta$ to be assigned an interval of values
$\text{C}^{*}(\theta,\bm{x})=\left[\underline{\alpha},\overline{\alpha}\right]$.
However, it should be noted that $\underline{\alpha}$ in this case will be
greater than $\overline{\alpha}$, and as such a confidence level cannot be
attributed to this interval. Similarly, the inverse operation
$\text{C}^{-1*}(\alpha,\bm{x})=[\underline{\theta},\overline{\theta}]$ returns
an interval. This is referred to as an imprecise confidence distribution, or
c-box.
The upper and lower confidence limit distributions $\text{C}^{*}_{U}(\cdot)$
and $\text{C}^{*}_{L}(\cdot)$ of the c-box define $\underline{\alpha}$ and
$\overline{\alpha}$ for each $\theta$. In the case of the Clopper-Pearson
c-box, these distributions are defined as follows:
$\displaystyle\underline{\alpha}$
$\displaystyle=\text{C}^{*}_{L}(\theta,\bm{x})=\text{B}\left(\theta;\sum{\bm{x}}+1,n-\sum{\bm{x}}\right)$
(6a) $\displaystyle\overline{\alpha}$
$\displaystyle=\text{C}^{*}_{U}(\theta,\bm{x})=\text{B}\left(\theta;\sum{\bm{x}},n-\sum{\bm{x}}+1\right)$
(6b)
Mapping a confidence interval is then performed in a similar manner as for
confidence distributions, defining an interval
$\left[\underline{\alpha},\overline{\alpha}\right]$ such that
$\overline{\alpha}-\underline{\alpha}=\alpha$ and constructing a corresponding
interval on $\Theta$ from the minimum and maximum of these intervals:
$\displaystyle\text{C}^{*-1}([\underline{\alpha},\overline{\alpha}],\bm{x})$
$\displaystyle=[\min(\text{C}^{-1}(\underline{\alpha},\bm{x})),\max(\text{C}^{-1}(\overline{\alpha},\bm{x}))]$
(7a)
$\displaystyle=[\text{C}^{*-1}_{U}(\underline{\alpha},\bm{x}),\text{C}^{-1}_{L}(\overline{\alpha},\bm{x})]$
(7b)
Naturally, a one-sided $\alpha$ level confidence interval
$\left[\underline{\alpha}=0,\overline{\alpha}=\alpha\right]$ produces a
confidence interval extending to the minimum of $\theta$,
$\text{C}^{*-1}([0,\alpha],\bm{x})=\left[0,\text{C}^{-1}_{L}(\alpha,\bm{x})\right]$.
This can be used as before to construct a Singh plot representing the ability
of the lower limit of the c-box to bound the true value. The upper limit is
then used to generate the opposite one-sided interval
$\left[\underline{\alpha}=1-\alpha,\overline{\alpha}=1\right]$ to produce
confidence intervals of
$\text{C}^{*-1}([1-\alpha;1],\bm{x})=[\text{C}^{-1}_{U}(1-\alpha,\bm{x}),1]$.
Again, these values are then ordered and plotted against the unit uniform.
Visually this can become cluttered and unappealing as, so an alternative is to
use the same lower bounded one-sided
$\left[\underline{\alpha}=0,\overline{\alpha}=\alpha\right]$ interval,
treating the upper limit of the c-box as an isolated distribution. In this
case, the corresponding Singh plot should indicate a total lack of coverage
(i.e. all below the unit uniform), since the interval being utilised is
effectively the complement of the actual target interval.
Since the Singh plots for the upper and lower limit distributions of the c-box
straddle, but never cross, the U$(0,1)$ diagonal the distribution can provide
confidence since the true confidence distribution must lie between these
bounds. This comes at the cost of wider intervals than the distribution
proposed in Equation 5. The width of the output intervals will decrease as
more information is available. This demonstrates how both precise and
imprecise confidence distributions can be developed and assessed for inference
on known distributions. However, in cases where distributional assumptions are
unjustified, what can be done for non-parametric confidence distributions?
### 4.3 Predictive Confidence Distributions
Singh plots can be used to assess the properties of any confidence structure.
The most common examples involve inference, capturing epistemic uncertainty.
But the same procedure can be applied to analysis of predictive structures as
well. These function similarly to standard confidence distributions used for
inference, but they output intervals guaranteed to bound the next drawn sample
with a desired frequency rather than bounding some true parameter. There are a
number of possible examples, but an interesting imprecise confidence
distribution is for non-parametric prediction of the next drawn sample
$x_{n+1}$ given a dataset $\bm{x}=\\{x_{1},\dots,x_{n}\\}$, assuming a
continuous distribution.
The procedure for prediction is the same as for inference, though the
confidence value of the true subsequent sample $\text{C}(x_{n+1},\bm{x})$ is
calculated rather than any fixed parameter. The lower and upper limit
confidence distributions are defined as follows for the non-parametric
prediction case:
$\displaystyle\text{C}^{*}_{L}(x_{n+1},\bm{x})$
$\displaystyle=\frac{|\\{x_{i}\in\bm{x}:x_{i}\leq x_{n+1}\\}|}{n+1}$ (8a)
$\displaystyle\text{C}^{*}_{U}(x_{n+1},\bm{x})$
$\displaystyle=1-\frac{|\\{x_{i}\in\bm{x}:x_{i}\geq x_{n+1}\\}|}{n+1}$ (8b)
Again, the coverage properties of this structure can be demonstrated using a
Singh plot. A gaussian mixture distribution is used here for demonstrative
purposes.
This demonstrates that such a structure is at capable of reliably calculating
intervals which will contain subsequent samples, at least for this Gaussian
mixture model.
## 5 Representation of Confidence Structure Characteristics
The intent of a Singh plot is to rapidly convey the confidence characteristics
of the chosen structure, and has been demonstrated that deviations from the
central U$(0,1)$ line indicate the coverage probability of a structure. Singh
plots are also capable of indicating a number of other characteristics which
aid in the design and validation of appropriate imprecise confidence
structures.
### 5.1 Representing Uncertainty from Limited Data
Figure 5 demonstrates how Singh plots differ when performed on various sample
sizes. As the number of data points increases, the imprecise distribution
converges to the perfect case, matching the U$(0,1)$ diagonal. Lower samples
sizes are shown to produce confidence intervals which have coverage, but which
are wider than they would be in the perfect case. For example, with a sample
size of $n=10$, an $\alpha=0.65$ confidence interval would have similar
coverage properties to one with $\alpha=0.8$.
### 5.2 Representing uncertainty about Rare Events
Figure 6 demonstrates how Singh plots of Equation 5 respond to varying rate
$\theta_{0}$. Estimation of a very low rate will naturally be difficult when
sample sizes are low, and this is reflected in the broad confidence regions
for a given bounding probability. For example, an $\alpha=0.2$ confidence
interval is as likely to bound the true parameter $\theta_{0}=0.01$ as one
with $\alpha=0.9$. Increasing the sample size will converge these towards the
U$(0,1)$ distribution as seen above, though a feature of note is that the
Singh plot becomes asymmetrical as $\theta_{0}$ deviates from the centre of
the support $\theta=0.5$.
### 5.3 Favourability, Conservatism and Overconfidence
Since it is relatively simple to produce a Singh plot, they can be used to
modify and assess confidence distributions. One hope may be that through some
modification, a confidence distribution may be developed which produces
tighter bounds whilst preserving the property of coverage. For example,
Equation 6 could be modified to alter the uncertainty expressed in the
imprecise confidence distribution. This could be done by using a modification
such as that shown in Equation 9 for a length-$n$ sample set
$\bm{x}=[x_{1},\dots,x_{n}]$, replacing $c$ with the desired parameter for
imprecision.
$\displaystyle\alpha_{L}$
$\displaystyle=\text{C}^{*}_{L}(\theta,\bm{x})=\text{B}\left(\theta;\sum{\bm{x}}+c,n-\sum{\bm{x}}\right)$
(9a) $\displaystyle\alpha_{U}$
$\displaystyle=\text{C}^{*}_{U}(\theta,\bm{x})=\text{B}\left(\theta;\sum{\bm{x}},n-\sum{\bm{x}}+c\right)$
(9b)
This again produces proposed confidence distributions, as the impact of this
change is not yet known. The coverage impact of varying B can then be
inspected through the use of Singh plots.
As can be seen in Figure 7, decreasing $c$ beyond 1 produces an invalid c-box,
as the Singh plots for both bounds clearly extend beyond the U$(0,1)$
diagonal. This structure would be considered overconfident, as it assigns
intervals a level of confidence which is not always exceeded by their coverage
probability.
Increasing $B$ does not produce an invalid structure, but instead produces a
structure with additional imprecision. This structure would be considered
conservative, as it is assigning intervals a level of confidence which is
always greater than but never equal to their coverage probability. A structure
such as the case when $c=1$ is still considered conservative by many[8, 4],
since the width of the confidence intervals it produces is wide by comparison
to many alternatives. Determining whether a structure is more or less
conservative than another in this sense with a Singh plot requires further
investigation, though a comparison of the area between the Singh plot bounds
should allow for comparisons of relative confidence.
A favourable confidence distribution should coincide with the U$(0,1)$,
indicating that further reduction in the width of confidence intervals
produced by the chosen distribution has the potential to violate the coverage
requirement. This indicates that the confidence distribution is appropriately
representing the uncertainty about the parameter of interest. A conservative
structure is still suitable for inference, it just implies that the
uncertainty about these inferences could be reduced with a more appropriate
confidence distribution. An over-confident structure however, cannot be relied
upon to produce confidence intervals with coverage, and implies that the
applied confidence distribution is neglecting uncertainty.
For a precise distribution, a favourable structure simply converges to
U$(0,1)$ as seen in Figure 1. For an imprecise structure, this would be
instead represented by coincidence with U$(0,1)$ where a step in coverage
probability occurs, as seen in Figure 3.
### 5.4 Global Coverage Properties
Portraying the properties of the confidence box for a known $\theta_{0}$
allows for insight into the suitability of the chosen structure, but generally
$\theta_{0}$ is unknown, hence the desire to infer its true value. In this
case parametric inference implies knowledge of the target distribution, so it
is possible to assess the chosen structure across a range of possible
representations of the target distribution. In this case, a Singh plot can
also portray the global coverage properties for unknown parameters within the
support of the parameter $\Theta$.
This is done by targeting an interval of interest on the support
$\bm{\theta}$, sampling this region to generate a series of distributions
$\bm{F}=\\{\text{F}(\theta_{1}),\dots,\text{F}(\theta_{n})\\}$ where each
$\text{F}(\theta_{i})$ represents the cumulative distribution of the target
distribution with parameter $\theta_{i}$. Samples are generated for each of
these distributions, individual Singh plots are calculated and then the lower
bound of this second-order Singh plot is used as the final output. If this
lower bound satisfies the criteria outlined above for a confidence
distribution then the can be used for the target interval with confidence that
it will provide coverage, though knowledge of conservatism is lost. If
coverage is demonstrated in this case, then the structure can be safely used
for any potential case of inference in the interval of interest about the
target distribution.
For a precise distribution this can be calculated as follows:
$\text{S}_{G}(\bm{\alpha},\bm{\theta})=\min_{\theta_{i}\in\bm{\theta}}\\{\text{Pro}(C^{*}(\bm{\theta},\bm{x}\sim\text{F}(\theta_{i}))\ni\theta_{i})\\}$
(10)
and for an imprecise distribution, where again the lower limit of the Singh
plot is inverted for ease of interpretation:
$\displaystyle\text{S}_{L}(\bm{\alpha},\bm{\theta})$
$\displaystyle=\max_{\theta_{i}\in\bm{\theta}}\\{\text{Pro}(C_{L}^{*-1}(\bm{\alpha},\bm{x}\sim\text{F}(\theta_{i}))\ni\theta_{i})\\}$
(11a) $\displaystyle\text{S}_{U}(\bm{\alpha},\bm{\theta})$
$\displaystyle=\min_{\theta_{i}\in\bm{\theta}}\\{\text{Pro}(C_{U}^{*-1}(\bm{\alpha},\bm{x}\sim\text{F}(\theta_{i}))\ni\theta_{i})\\}$
(11b)
This is demonstrated in Figure 8 with the structure described in Equation 6,
taking values of $\theta$ across the interval [0,1]. This demonstrates that
the Clopper-Pearson confidence structure is capable of providing intervals
with frequentist coverage regardless of the value of $\theta$. An example of
how to perform this is given in Algorithm 2.
input : $C^{*}\leftarrow$Proposed confidence structure
$\text{f}(\bm{\theta})\leftarrow$Target distribution taking parameters
$\theta$
$\\{\theta_{0},\dots,\theta_{k}\\}\leftarrow$Parameter sets of interest
$\theta_{l,0}\leftarrow$True values for parameter of interest in set $l$
$\theta_{l,1:k}\leftarrow$True values for nuisance parameters in set $l$
output : Singh plot for visual assessment of global confidence structure
properties
for _$j\in 1,\dots,k$_ do
for _$i\in{1,\dots,m}$_ do
Generate sample: $\bm{x}={x_{1},\dots,x_{n}}\sim\text{f}(\theta_{j})$
Calculate minimum required confidence for coverage:
$t_{j,i}=\text{C}^{*}(\theta_{i,0},\bm{x})$
end for
Sort $t_{j,1:m}$
end for
for _$i\in{1,\dots,m}$_ do
Estimate global minimum required confidence for coverage:
$s_{i}=\min(t_{1:k,i})$
end for
Plot empirical CDF of $s$
Plot CDF of $U(0,1)$ for comparison
Algorithm 2 Generation of a global Singh plot
It should be noted that the sample size will also affect the inference about
$\theta_{0}$. It is assumed that this structure is applied to a case with a
known sample size, otherwise an exhaustive representation of the minimum
coverage probability would have to be calculated by sampling combinations of
$(\theta_{0},n)$. Similarly, nuisance parameters may be treated in a similar
manner, though the most efficient means of doing so will vary depending on the
structure being assessed. Calculating Singh plots with variations in the
nuisance parameters may indicate whether their effect on the minimum
confidence required is monotone, and if so end-points could be taken to reduce
computational cost.
## 6 Chebyshev UCL Coverage
Most of the above example are demonstrations of known confidence structures or
of clearly deficient suggestions. However, the value of Singh plots lies in
visual demonstrations of the performance of structures where the deficiencies
are not known.
As an example, the ProUCL package is a software package for statistical
analysis of environmental datasets, and one of the statistics that can be
calculated is the upper confidence limit of the mean of a population,
$\bar{\mu}\in M$. This could be used for calculation of the upper confidence
limit on the expected value of the concentration of a particular pollutant in
water samples, amongst many other use cases. The software documentation notes
the difficulty of handling skewed datasets and suggests the use of an
estimator based on the Chebyshev inequality, defined below in Equation 12[9].
$\bar{\text{C}^{*}}(\bm{\alpha},\bm{x})=\mu_{\bm{x}}+\sqrt{\frac{1}{1-\alpha}-1}\frac{\sigma_{\bm{x}}}{\sqrt{n}}$
(12)
The novelty of this upper confidence limit is the claim that it is a
reasonable non-parametric estimator, that is it should be correct regardless
of the underlying distribution. This is an excellent quality for an estimator
to have, though the documentation of ProUCL does note that highly skewed
datasets may lose coverage, and that in such cases the data should be
inspected to ensure that there is truly only a single population being
reported. This raises questions about how skewness affects the coverage, and
is it really reasonable to simply raise or lower the required $\alpha$ level
to get an appropriate confidence interval?
Singh plots can serve here as a tool for inspecting the properties of this
estimator in an intuitive manner. A family of distributions can be generated
to ‘stress test’ the provided estimator. In this case, scaled Bernoulli
distributions represent a family of distributions which should be particularly
difficult for such an estimator to maintain coverage. The estimator relies on
scaling the standard deviation of a sample set, and there are many sample sets
that can be drawn with a high probability from a highly skewed Bernoulli
process which have zero standard deviation. The Bernoulli parameter $p$ here
can be manipulated to alter the skewness of the distribution in order to
observe how highly skewed datasets affect the coverage of this confidence
limit.
The PRoUCL version 5.1.0 documentation defines ’extremely skewed’ as data
where the standard deviation of the log transformed data
$\hat{\sigma}_{\bm{x}}$ is greater than 3. For a Bernoulli distribution, this
statistic is inverse to the observed skewness since the maximum standard
deviation will be observed where $p=0.5$ and the distribution has no skewness.
Firstly Equation 12 must be inverted to map $\mu\in M$ onto the support of a
$\bm{\alpha}$. This gives Equation 13:
$\bar{\text{C}^{*}}^{-1}(\mu,\bm{x})=1-\left(\left(\frac{\sqrt{n}}{\sigma_{\bm{x}}}-\mu_{\bm{x}}\right)^{2}+1\right)^{-1}$
(13)
This can then be used to generate a Singh plot for a variety of Bernoulli
distributions with skewness controlled by the parameter $p=\theta$. According
to the ProUCL documentation, it should be expected that the structure provides
coverage for moderately skewed data, but that this may not hold for highly
skewed data.
For small sample sizes, Equation 12 fails to provide coverage at all
confidence levels even for the unskewed Bernoulli distribution ($p=0.5$,
skewness = 0). Any skew in the dataset detracts further from the ability to
provide coverage. This can be offset with a larger sample size, though even
with 30 samples skewed data $(p=0.05$, skewness = 4.13) leads to a lack of
coverage. As such, Equation 12 should not be considered for use on small
datasets, particularly those which may be skewed. A 95% upper confidence limit
from this structure cannot be guaranteed to bound the true mean at least 95%
of the time. In the $p=0.2$ case (skewness=1.5) with a sample size of $n=5$
for example, a 95% confidence interval would provide only 67% coverage. These
coverage figures also only apply to the particular distributions they are
applied to. In practice the utility of this estimator comes from it’s supposed
applicability to non-parametric cases. Because of this, attempting to suggest
a lower $\alpha$ level in order to be more accurate to the true coverage, or a
higher $\alpha$ level to try and be more conservative would not be
justifiable.
This upper confidence limit estimator may be of some use, but it is in no way
a distribution-free estimator and should not be used for these purposes when
sample sizes are small. However, Singh plots may be a means of determining the
limits of its use as a confidence estimator and in this case it appears that
increasing the sample size allows for confidence on mildly skewed datasets.
Whether a practitioner wants to accept the conservatism and the potential for
losing applicability to highly skewed datasets is a matter of choice, but
Singh plots such as these may be a useful means of informing this decision.
## 7 Conclusion
Singh plots, whilst not technically capable of providing strict proof of
coverage, represent an intuitive and simple means of portraying the coverage
properties of confidence structures, both precise and imprecise. They allow
for comparisons against different proposed structures, as well as analysis of
general and specific cases for inference and prediction.
Confidence structures are a widely applicable means of providing probabilistic
statements, and Singh plots allow for their use and development without
requiring specialist knowledge regarding their formulation. This allows for
more widespread adoption and development of this robust approach to
uncertainty quantification. This is particularly relevant for the development
of procedures suitable for calculating with confidence structures.
## References
* [1] Schweder T, Hjort NL. Confidence and likelihood. Scandinavian Journal of Statistics. 2002;29(2):309–332.
* [2] Balch MS. Mathematical foundations for a theory of confidence structures. International Journal of Approximate Reasoning. 2012;53(7):1003–1019.
* [3] Ferson S, O’Rawe J, Balch M. Computing with confidence: Imprecise posteriors and predictive distributions. In: Vulnerability, Uncertainty, and Risk; 2014. p. 895–904. Available from: https://ascelibrary.org/doi/abs/10.1061/9780784413609.091.
* [4] Singh K, Xie M, Strawderman WE. Confidence distribution (cd): Distribution estimator of a parameter. Lecture Notes-Monograph Series. 2007;54:132–150. Available from: http://www.jstor.org/stable/20461464.
* [5] Student. The probable error of a mean. Biometrika. 1908;:1–25.
* [6] Jogesh Babu G. Kesar singh’s contributions to statistical methodology. Statistical Methodology. 2014;20:2–10. Available from: http://dx.doi.org/10.1016/j.stamet.2013.12.001.
* [7] Hoo ZH, Candlish J, Teare D. What is an ROC curve? Emergency Medicine Journal. 2017;34:357–359.
* [8] Balch MS. New two-sided confidence intervals for binomial inference derived using walley’s imprecise posterior likelihood as a test statistic. International Journal of Approximate Reasoning. 2020;123:77–98.
* [9] Singh A, Maichle R, Lee SE. On the computation of a 95% upper confidence limit of the unknown population mean based upon data sets with below detection limit observations. EPA/600/R-06/022; 2006.
## Figures
Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7:
Figure 8: Figure 9:
## 8 Figure Captions
1. 1.
(a): A single example of a proposed confidence distribution from Equation 2
generated from
$\bm{x}=\\{x_{1},\dots,x_{10}\\}\sim\text{N}(\mu_{0}=4,\sigma=3)$. The
confidence required for a one-sided interval to cover the true mean $\mu_{0}$
in this example is shown as 0.47. (solid, $\text{C}^{*}(\bm{\mu},\bm{x})$;
dashed, $\text{C}^{*}(\mu_{0},\bm{x})$). (b): Singh plot for the proposed
confidence distribution about the same target distribution, generated from
$m=10^{4}$ sample sets $\bm{X}=\\{\bm{x}_{1},\dots,\bm{x}_{m}\\}$. (solid,
$\text{S}(\bm{\alpha};\mu_{0})$; dashed, $\text{U}(0,1)$).
2. 2.
(a): A proposed confidence distribution from Equation 5 generated from
$\bm{x}=\\{x_{1},\dots,x_{10}\\}\sim\text{Bin}(N=1,p=\theta_{0})$. The
confidence value of the true rate $\theta_{0}$ is shown as 0.49. (solid,
$\text{C}^{*}(\bm{\theta},\bm{x})$; dotted,
$\text{C}^{*}(\theta_{0},\bm{x})$). (b): Singh plot for the proposed
confidence distribution about the same target distribution, generated from
$m=10^{4}$ sample sets $\bm{X}=\\{\bm{x}_{1},\dots,\bm{x}_{m}\\}$. (solid,
$\text{S}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
3. 3.
(a): A proposed confidence distribution from Equation 6 generated from
$\bm{x}={x_{1},\dots,x_{10}}\sim\text{Bin}(n=1,p=\theta_{0})$. The confidence
value interval of the true rate $\theta_{0}$ is shown as [0.37, 0.62]. (solid,
$\text{C}_{U}^{*}(\bm{\theta},\bm{x})$; dashed,
$\text{C}_{L}^{*}(\bm{\theta},\bm{x})$; dotted,
$\text{C}^{*}(\theta_{0},\bm{x})$). (b): Singh plot for the proposed
confidence distribution about the same target distribution, generated from
$m=10^{4}$ sample sets $\bm{X}=\\{\bm{x}_{1},\dots,\bm{x}_{m}\\}$. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
4. 4.
(a): A proposed confidence distribution from Equation 8 generated from a
length $n=10$ sample set
$\bm{x}=\\{x_{1},\dots,x_{10}\\}\sim\text{F}([\mu_{1},\mu_{2}],[\sigma_{1},\sigma_{2}])$
where
$\text{F}([\mu_{1},\mu_{2}],[\sigma_{1},\sigma_{2}])=0.5\cdot\text{N}(\mu_{1}=4,\sigma_{1}=3)+0.5\cdot\text{N}(\mu_{2}=5,\sigma_{2}=1.5)$.
The confidence value of the true value $x_{n+1}$ is shown as
$\text{C}(\mu_{0},\bm{x})=[0.18,0.27]$. (solid,
$\text{C}_{U}^{*}(\bm{\theta},\bm{x})$; dashed,
$\text{C}_{L}^{*}(\bm{\theta},\bm{x})$; dotted,
$\text{C}^{*}(\theta_{0},\bm{x})$). (b): Singh plot for the proposed imprecise
confidence distribution about the same target distribution, generated from
$m=10^{4}$ sample sets $\bm{X}=\\{\bm{x}_{1},\dots,\bm{x}_{m}\\}$. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
5. 5.
A series of Singh plots used for inference about $\theta_{0}=0.4$ generated
using Equation 5 and a dataset of varying length $n$. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
6. 6.
A series of Singh plots used for inference about a varying $\theta_{0}$
generated using Equation 5 and a length $n=20$ dataset. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
7. 7.
A series of Singh plots used for inference about $\theta_{0}=0.4$ generated
using Equation 5 and a length $n=20$ dataset with varying degrees of
confidence demonstrated by altering the $c$ parameter in Equation 9. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
8. 8.
Global Singh plot produced using $m=100$ $\theta$ samples drawn from $[0,1]$
and $N=10^{3}$ Monte Carlo samples using the Clopper-Pearson confidence
structure for inference about $\theta$ with a sample size of 10. (solid,
$\text{S}_{U}(\bm{\alpha};\theta_{0})$; dashed,
$\text{S}_{L}(\bm{\alpha};\theta_{0})$; dotted, $\text{U}(0,1)$).
9. 9.
Singh plots representing the coverage probability for a desired $\alpha$
confidence level interval using Equation 13 for inference about data generated
from Bernoulli distributions with varying $p$-parameters, scaled to have a
consistent mean of $\mu_{0}=2$ and a minimum value of $\min{M}=0$. Two plots
are shown, for sample sizes of n=5 (left) and n=30 (right). (solid,
$\text{S}(\bm{\alpha};p_{0}=0.05)$; dashed, $\text{S}(\bm{\alpha};p_{0}=0.2)$;
dash-dot, $\text{S}(\bm{\alpha};p_{0}=0.5)$; dotted, $\text{U}(0,1)$).
|
# Chart2Vec: A Universal Embedding of Context-Aware Visualizations
Qing Chen, Ying Chen, Ruishi Zou, Wei Shuai, Yi Guo, Jiazhe Wang, and Nan Cao
Qing Chen, Ying Chen, Ruishi Zou, Wei Shuai, Yi Guo, and Nan Cao are with
Intelligent Big Data Visualization Lab at Tongji University. Nan Cao is the
corresponding author. E-mails:
<EMAIL_ADDRESS>Jiazhe Wang
is with Ant Group. Email<EMAIL_ADDRESS>
###### Abstract
The advances in AI-enabled techniques have accelerated the creation and
automation of visualizations in the past decade. However, presenting
visualizations in a descriptive and generative format remains a challenge.
Moreover, current visualization embedding methods focus on standalone
visualizations, neglecting the importance of contextual information for multi-
view visualizations. To address this issue, we propose a new representation
model, Chart2Vec, to learn a universal embedding of visualizations with
context-aware information. Chart2Vec aims to support a wide range of
downstream visualization tasks such as recommendation and storytelling. Our
model considers both structural and semantic information of visualizations in
declarative specifications. To enhance the context-aware capability, Chart2Vec
employs multi-task learning on both supervised and unsupervised tasks
concerning the cooccurrence of visualizations. We evaluate our method through
an ablation study, a user study, and a quantitative comparison. The results
verified the consistency of our embedding method with human cognition and
showed its advantages over existing methods.
###### Index Terms:
Representation Learning, Multi-view Visualization, Visual Storytelling,
Visualization Embedding
## 1 Introduction
Data visualizations are an important means to help people quickly identify
complex data patterns and communicate insights. Automatic methods help
accelerate the visualization creation process by improving the quality of the
dataset used for visualization [1], extracting the most meaningful information
or insights from the data [2, 3], and selecting the appropriate visual
representation [4, 5]. This allows users to grasp the key information from
visualizations more quickly, accurately, and comprehensively [6]. With the
abundance of visualizations created by experts and automated systems,
visualization themselves have become a new format of data [7]. Therefore, it
is worth studying how to effectively and precisely represent such
visualization data in a generalizable format to support downstream
visualization and visual analytics tasks such as comparison [8],
recommendation [9], assessment [10], and querying [11].
Inspired by the recent advances in representation learning, several studies in
the visualization community attempted to use semantic vectors (i.e.,
embeddings [12]) to represent information from the visualization data. For
example, ChartSeer [13] adopted an encoder-decoder approach that converts
visualization to and from embeddings to assist exploratory visual analysis.
More recently, Li et al. [14] proposed a structure-aware method to improve the
performance of visualization retrieval by collectively considering both visual
and structural information. Compared to heuristic or rule-based methods,
representation learning allows a more flexible and general presentation of
visualization. Once the embedding features have been learned, they can be
applied to a variety of downstream visualization tasks.
Nevertheless, these attempts to present visualizations through embeddings can
only be applied to one or two specific visualization tasks such as visual
comparison and visualization recommendation. Moreover, most existing work is
focused on visualization tasks for a single-view visualization. When
considering multi-view visualizations, context information is a critical
aspect of visualization representation that influences the outcome of
subsequent tasks. There is still a lack of a universal representation of
visualizations that can be used for various downstream tasks while taking
contextual information into account.
To fill this research gap, we aim to propose a universal embedding of context-
aware visualizations based on the associations derived from a large corpus of
visualizations specifications. In this paper, context-aware refers to the co-
occurrence and logical relationships within multi-view visualizations, such as
narrative sequences in data stories and logical orders of visual analytic
findings. The remaining challenges are as follows. First, we need to formulate
a proper input embedding that leverages both the semantic content and the
structural information of each visualization. To achieve this, we reviewed
related studies on natural language descriptions of visualization data [15,
16], then summarized the key structural features that can be obtained from
visualizations specifications. Compared to existing methods which only extract
explicit information, such as chart types and data statistics (sometimes
referred to as “data facts” [17]), our input chart embedding of the proposed
model also considers implicit information, including specific field-related
details from the dataset, such as field names, corresponding values, and data
semantics. Second, a large-scale dataset of context-aware visualizations is
required to form the training and test datasets. Multiple visualizations in a
cohesive representation can be regarded as multi-view visualizations [18].
Such multi-view visualizations can provide a comprehensive and contextual
understanding of the data in the forms of dashboards, infographics, and data
stories. Due to the lack of high-quality multi-view datasets with contextual
information, we carefully collected and selected 849 data stories and 249
dashboards from Calliope [2], an online data story generation platform, and
Tableau Public [19], an online platform for creating and sharing data
visualizations, respectively, comprising a total of 6014 visualizations. The
dataset is publicly available at https://chart2vec.idvxlab.com/. The collected
dataset covers ten common topics, including economy, sports, society, health,
politics, industry, recreation, food, education, and ecology. Third, we need
to set up multiple deep learning tasks to learn the contextual information
from a set of input embeddings. We integrated both supervised and unsupervised
learning tasks, where we use the linearly interpolated loss function for
sequentially connected charts to learn logical associations, and introduce the
triplet loss to capture the co-occurrence of the charts. Meanwhile, we employ
a multi-task training strategy and optimize the results by setting
hyperparameters and automatic updates.
In this paper, we propose Chart2Vec, a model that learns a universal embedding
of visualizations, extracts context-aware information, and enables other
downstream applications such as recommendations, storytelling, and generation.
To investigate the effectiveness of the proposed model, we conducted extensive
evaluations including ablation studies, a user study, and quantitative
comparisons with existing visualization embedding methods [13, 20].
In summary, the major contributions of this paper are as follows:
* •
We collect a high-quality context-aware visualization dataset and formulate an
input embedding that incorporates both factual and semantic information.
* •
We present Chart2Vec, a representation model to learn a universal embedding of
visualizations, extract context-aware information, and enable various
downstream applications.
* •
We summarize the key lessons learned during the design and development of
Chart2Vec, which we hope to benefit subsequent visualization applications and
related research.
## 2 Related Work
In this section, we present a comprehensive review of the related literature,
specifically focusing on representation learning in visualization, automatic
multi-view visualization, and visualization similarity computation. Different
from existing approaches, Chart2Vec combines structural and semantic
information of visualizations. In addition, Chart2Vec introduces contextual
relationships in multi-view visualization, a feature that can further enhance
the efficiency of various downstream tasks such as recommendation, clustering,
and generation.
### 2.1 Representation Learning in Visualization
Representation learning is a machine learning technique that automatically
learns representations of data [21]. It has been widely used in various
fields, including graph learning [22], computer vision [23], and natural
language processing [24]. Recently, representation learning has also been
applied to address a variety of visualization tasks, such as transformation,
comparison, and recommendation [7]. Since representation learning is a data-
driven technique, we can divide representation learning in visualization into
three categories according to the different forms of visualization data:
representation learning of visualization graphics, representation learning of
visualization programs, and representation learning of hybrid visualization
data.
The approach of learning representations about visualization graphics focuses
on extracting visual features in visualization. For example, VisCode [25]
extracted the visual importance map from a static visualization image and then
embedded the information into the image for visualization steganography.
Recent studies have mainly focused on identifying visual attributes in the
visualization for subsequent visualization captioning. Lai et al. [26]
employed the Mask R-CNN model to extract features from visual elements and
visual attributes, while Qian et al. [27] extracted information from the
bounding box metadata and fused the information of the original image
extracted by CNN.
Representation learning methods based on visualization programs focus on the
input data as structured text and extract implicit features from the structure
or text content. ChartSeer [13] utilized a syntax analyzer to convert charts
in Vega-Lite specifications into one-hot vectors, which were then input into a
CNN structure to obtain the representation of charts. Erato [20] took the
semantic information in the visualization as string sentences, then adopted
BERT [28] to obtain the initial sentence vector and applied it to the
visualization interpolation task by fine-tuning. In addition, Draco abstracts
the design rules into a set of constraints and utilizes the weights assigned
to these soft constraints as feature vectors for the visualizations. To
determine these weights, Draco employs RankSVM [29] to automatically learn
their values. Consequently, the resulting vectors can be used to assess the
quality of individual visualizations. However, existing representation
learning methods from visualization programs often serve specific tasks such
as chart recommendation, generation, and evaluation. There is still a lack of
representation learning methods that can be applied to a variety of different
visualization tasks based on the chart characteristics.
There are also studies for visualization representations of hybrid
visualization data. For example, KG4VIS [30] transformed visualizations into
knowledge graphs to learn their representations. Li et al. [14] extended the
tree-like format of SVGs to directed graphs and used graph convolutional
neural networks to learn visualization representations.
Inspired by previous work, we propose Chart2Vec, a representation model to
learn a universal embedding of visualizations, from declarative
specifications. Chart2Vec distinguishes itself from previous approaches by
incorporating both structural information introduced in ChartSeer [13] and
semantic information as proposed in Erato [20]. Furthermore, it follows the
concept of Word2Vec [31] to learn the implicit features of visualization
through contextual information, thereby adeptly capturing the contextual
relationships among multi-view visualizations based on comprehensive multi-
level chart information.
### 2.2 Automatic Multi-view Visualization
Multi-view visualizations allow users to view a large number of data
attributes and values in multiple views of visualizations coherently [8]. Due
to their capability to promote a more comprehensive understanding of data than
single charts [18], multi-view visualizations are widely used in visual
analytics and visual narratives. The presentations of multi-view
visualizations are mainly in the forms of dashboards [32], infographics [33],
and data stories [2].
The rise of intelligent techniques has heightened the demand for effective
visual presentation and analysis of big data systems. This has led to the
emergence of automatic multi-view visualization methods, categorized into
rule-based and machine learning-based approaches.
The rule-based approaches for multi-view visualizations rely on the design
guidelines from domain knowledge to automate the process. For example,
DataShot [34] selected generated single charts through a density-based top-n
algorithm, and organized multiple related visualizations into fact sheets with
the same topic. Calliope [2] generated data stories using a logic-oriented
Monte Carlo tree search algorithm with search rules derived from expert
experience, while the multiple charts in each data story were arranged in a
logical order. ChartStory [35] organized multiple charts into data comics with
narrative capabilities, based on established design principles. In addition,
Medley [36] recommended multiple visual collections by inferences based on a
specified mapping relationship between user intent and visual elements.
Machine learning-based approaches accomplish corresponding tasks based on the
results of trained models. MultiVision [37] calculated chart scores from chart
features through an LSTM network and modeled the multiple chart selection
process as a sequence-to-one regression problem. Due to the lack of datasets
of multi-view visualizations, Dashbot [38] utilized the reinforcement learning
paradigm to simulate human behavior in exploratory visual analysis to realize
the selection of charts. Erato [20] took a new perspective by treating
visualizations as semantic sentences, representing visualizations as vectors,
and using them for subsequent interpolation tasks.
The automatic multi-view visualization work mentioned above is mainly related
to three major tasks: visualization recommendation, visualization clustering,
and visualization generation. Chart2Vec encodes a visualization as a vector
and takes into account the contextual relationship between visualizations,
which can greatly improve the efficiency of the subsequent visualization
tasks.
### 2.3 Visualization Similarity Computation
Visual similarity computation is a crucial process for many downstream
applications, such as visualization recommendation and generation. Currently,
two major types of visualization features are considered when computing
visualization similarity: textual features and graphical features [7]. Text
features refer to the textual content within visualizations such as titles and
captions. For example, ScatterNet [11] used deep neural networks to extract
semantic features from scatterplot images for similarity calculation.
Vizcommender [9], a content-based visualization recommendation system,
leveraged the extracted text features to predict semantic similarity between
two visualizations. GraphScape [39] describes graphs based on the declarative
grammar of Vega-lite, and calculates the similarity between graphs by the
transformation cost between specifications.
To extract graphical features for similarity computation, Demiralp et al. [40]
estimated differences between visualizations using a perceptual kernel based
on visual encoding. Li et al. [14] converted SVGs into bitmaps for visual
information extraction and graphs for structural information extraction. They
then applied contrastive representation learning techniques to generate
embedding vectors for visual and structural information separately, which are
used to calculate similarity. Additionally, Luo et al. [41] proposed a vector
with five features to consider both the similarity score and the diversity
when selecting the top-k visualizations. Some systems combined textual
features and graphical features to improve chart detection [42] as well as
classification tasks [43]. For example, Kiem et al. [43] proposed a multimodal
model that used both pixel data and text data in a single neural network to
classify the visualization. Chart Constellations [44] measured chart
similarity using visual encoding and keywords extracted from charts. To
comprehensively measure the similarity between visualizations, our work
considers both semantic information in text content from data attributes and
structural information concerning visualization designs.
In addition to structural and semantic information, multi-view visualizations
require visual similarity metrics that consider contextual information.
However, existing work has primarily focused on individual chart
characteristics. Meanwhile, context-aware analysis has been applied to other
data types in the visualization domain. For example, graph data are often
associated with context analysis since it helps interpret and explore network
structures [45]. In the case of multi-view visualizations for tabular data,
context information refers to the co-occurrence or the logical relationships
[2] among multiple views. In this paper, we incorporate such context-aware
information in our dataset collection and model design.
## 3 Dataset
Given the absence of readily available high-quality context-aware
visualization datasets, we collected the dataset in order to complete the
training. This section provides an overview of our data collection and
screening process, followed by a detailed description of the final dataset. We
delve into various aspects of the dataset, including the data sources, the
data filtering conditions, the final amount of data collected and the
classification methods.
### 3.1 Data Collection and Screening
Multi-view visualizations are used in many domains, including business
intelligence dashboards and data stories. To prepare a high-quality dataset
for model training, we searched established visualization websites and
business intelligence platforms, such as Tableau Public [19], Plotly Community
Feed [46], PowerBI [47]. In addition, some previous work has collected multi-
view visualizations [48]. Among the popular websites and platforms mentioned,
we collect dashboards from Plotly and Tableau Public. Meanwhile, data story
generation platforms such as Calliope [49] contain a large amount of context-
aware multi-view visualizations in a factsheet format. We were able to access
the chart backend data through exclusive assess to the Calliope’s database.
The visualizations on those platforms are created and edited by various users,
so we still need to carefully screen all the collections. To ensure a diverse
multi-view visualization dataset, we examined the coverage of various data
domains in the screening process. Two of the authors screened and examined
existing data stories on Calliope [49] for high-quality data stories. Both
authors are experienced in data visualization and have conducted several data
story workshops. To ensure completeness, effectiveness, and content richness,
we applied the following selection rules from multiple perspectives: (1) each
multiple-view visualization must have an appropriate amount of information
[50], with a minimum chart number set to three. All the charts in the corpus
should be complete with no missing captions, data stories should not have
empty titles and no duplicate charts appear in the same data story; (2) we
ensure that in the same multi-view visualization, transitions between charts
are related to data storytelling, where any of the six transition categories
(i.e., dialogue, temporal, causal, granularity, comparison, and spatial
transitions), can be discovered to maintain narrative flow and coherence [51],
and (3) the charts in the same story need to be logically coherent (i.e., the
content of the charts is logically connected as defined by the logicality in
[2]). The two authors checked all the data stories separately, marking them
separately for compliance with the above criteria. If both authors approved a
data story, it was added to our dataset. If they disagreed, they discussed
until reaching a consensus. In the end, we selected 849 data stories.
We retrieved 10010 dashboards from Plotly Chart Studio [46] using the Plotly
API and also utilized the Tableau Public API to get 3176 dashboards. We first
filtered out the dashboards containing fewer than three charts and excluded 3D
visual charts. Then, we excluded those missing important data fields, which
are indispensable for meaningful data visualizations. After the first round of
screening, we collected a dataset comprising 551 qualified dashboards from
Plotly and 2315 qualified dashboards from Tableau Public. Subsequently, our
two authors conducted a second-round screening to assess the contextual
relations between multiple charts in the same dashboard, following the same
criteria used to collect the data stories.. Despite recent academic research
indicating an increase in narrative dashboards [32], most existing dashboards
on Plotly are still mainly used for exploratory analysis, and the overall
quality is not good enough to learn context-aware information. Therefore, we
decided to exclude the dashboards from Plotly, and keep only data stories
collected from the Calliope platform and dashboards from Tableau Public.
### 3.2 Dataset Descriptions
After a third-round screening by another author, we obtained 849 high-quality
data stories and 249 dashboards that contain 6014 visualizations. We give
detailed descriptions of the dataset classification and format.
Dataset statistics. Each data story contains 5-8 charts. The majority of data
stories consisted of 5 charts, accounting for 68.9% of the total. The 849 data
stories were created from 310 datasets and the 249 dashboards are from 241
datasets covering 10 different domains: economy, sports, society, health,
politics, industry, recreation, food, education, and ecology. The
classification of datasets, data stories and dashboards are shown in Fig. 1.
Dataset format. Calliope uses declarative specifications [52] to encode each
visualization in a data story, and the dashboards on Tableau Public are in the
form of images, which we manually transformed into declarative specifications
similar to those of Calliope for subsequent uniform processing. We then stored
individual data visualizations in the form of a list. Each item in the list
corresponds to an individual chart in a multi-view visualization and is
arranged in order. The data for each chart includes the chart type, the data
facts, and the column information of the raw data table. In addition, we
performed pre-processing operations on the collected data such as data
completion and data correction. For example, if any column names of the
collected dataset are abbreviated, we retrieved the URL of the collected
dataset to identify the full names of these abbreviations and then replaced
them manually. We also corrected the misspelled column names.
Figure 1: Distribution of datasets, data stories and dashboards in different
domains.
## 4 Methodology
This section introduces the design and implementation of Chart2Vec. The goal
is to convert the visualization into an embedding vector through the Chart2Vec
model. The vector representation not only retains meaningful information
(i.e., structural and semantic information) about individual charts but also
captures the contextual relationships between charts. We first define the form
of a chart and then describe the implementation details of the Chart2Vec
model. We also open-sourced the training and testing data along with our
trained model on GitHub at https://github.com/idvxlab/chart2vec.
### 4.1 Chart Characterization
The goal is to learn a universal embedding of visualizations from declarative
specifications. To achieve this, it is crucial to establish a declarative
syntax format for the charts. This format is required to effectively represent
the essential information of the charts. Inspired by recent work in natural
language understanding and automatic generation of semantic content from
visualizations [15, 17], we aim to characterize a format that is more general
and comprehensive than existing representations. According to the model
proposed by Lundgard & Satyanarayan [15], semantic content from visualizations
through natural language descriptions can be classified into four levels:
elemental and encoded properties (L1), statistical concepts and relations
(L2), perceptual and cognitive phenomena (L3) and contextual and domain-
specific insights (L4). In the following, we describe the process of
constructing the declarative specification of a chart in conjunction with the
four-level semantic model.
As mentioned in Section 3.2, we utilized the curated multi-view visualizations
as our training and testing datasets and manually transformed the dashboards
on Tableau Public into declarative specifications consistent with that in
Calliope. Each multi-view visualization contains a set of interconnected
individual charts that convey meaningful insights. Calliope employs “data
facts” to store the essential information of the chart, which concentrates on
capturing the data content from a semantic perspective. Each data fact can be
expressed as a 5-tuple: $f_{i}=\left\\{\text{{type, subspace, breakdown,
measure, focus}}\right\\}$. However, the data fact definition in Calliope
solely includes the L2 information related to the aforementioned semantic
model.
To enhance the richness of chart content, we improve the original form by
incorporating the chart type and meta information. The chart type, such as bar
chart or scatterplot, is indispensable from the visual encoding perspective
and corresponds to the L1 information. Meta information describes the
perceptual and cognitive features of the visualization and thus corresponds to
the L3 information. For instance, if a chart represents a trend, we
additionally describe whether the trend is ascending or descending.
After making the aforementioned refinements, we introduce a more comprehensive
chart representation, which we refer to as a “chart fact”. It is defined as a
7-tuple:
$\displaystyle c_{i}$ $\displaystyle=\left\\{\text{{type\textsubscript{c},
type\textsubscript{f}, subspace, breakdown, measure, }}\right.\text{{focus,
meta}}\\}$
$\displaystyle=\left\\{ct_{i},ft_{i},s_{i},b_{i},m_{i},f_{i},e_{i}\right\\}$
where typec (denoted as $ct_{i}$) indicates the type of chart and typef
(denoted as $ft_{i}$) expresses the type of information described by the data
fact. Similar to Calliope, we support 15 commonly used chart types and 10 data
fact types; subspace (denoted as $s_{i}$) comprises a set of data filters that
can select a specific range of data, defined as
$\left\\{\left\\{\mathcal{F}_{1}=\mathcal{V}_{1}\right\\},...,\left\\{\mathcal{F}_{k}=\mathcal{V}_{k}\right\\}\right\\}$,
where $\mathcal{F}_{i}$ denotes the data field and $\mathcal{V}_{i}$ denotes a
corresponding specific value in $\mathcal{F}_{i}$; breakdown (denoted as
$b_{i}$) consists of a single temporal or categorical data field that can
further divide the data items into groups in the subspace; measure (denoted as
$m_{i}$) is a numerical data field that can be used to measure the data in the
groups through different aggregation methods; focus (denoted as $f_{i}$)
indicates the data item or data group that requires attention; meta (denoted
as $e_{i}$) indicates additional information about the chart. The meta field
contains different information depending on the fact type, as described in
detail in Table I.
Figure 2: The formulation details of an example chart fact: (1) the graphical
presentation of the visualization data, (2) the example chart fact
representation, (3) the fact schema which shows structural information in the
chart fact, (4) the fact semantics which indicates semantics information in
the chart fact, and (5) the location of the fields in the chart fact where
stores semantic information.
To provide a better understanding of the 7-tuple and its correspondence with
the chart content, we present a concrete example. Consider a dataset
containing useful information about schools in the world, including columns
such as school name, location, and the number of students. Suppose we generate
a chart from the dataset that depicts the difference in student numbers
between rural and urban areas in Guizhou, China, represented as a vertical bar
chart as shown in Fig. 2(1). The corresponding chart fact is shown in Fig.
2(2) as {“vertical bar chart”, “difference”, {{Country =“China”},
{City=“Guizhou”}}, {Location}, {sum(Population)}, “lower”}.
Table I: The meta information for different fact types in the 7-tuple.
fact type | meta information
---|---
trend | The overall direction of the trend. The options are “increasing”, “decreasing” and “no trend”.
categorization | The total number of categories. For example, if the category has 20 categories, then the meta value is “20 categories”.
difference | The difference between the two values. For example, as shown in Fig. 2(1), if the urban value is lower than the rural value, the meta is “lower”; otherwise, it is “higher”.
rank | Top three ranking values. For example, if a chart is a ranking of total car sales, the meta value is the top three car brands.
extreme | The types of the extreme. The options are “max” and “min”.
association | The type of association. The options are “positive” and “negative”.
A chart fact contains both structural and semantic information, with
structural information defined as the fact schema and semantic information as
the fact semantics. Chart fact consists of a 7-tuple and the fact schema can
be categorized following the detailed principles: (1) the options for
$type_{c}$ and $type_{f}$ are fixed and enumerable; (2) the number of filters
in the subspace can be none, one, or more than one; (3) the value of breakdown
can either perform a grouping action or not; (4) the aggregation methods of
measure include count, sum, average, minimum, and maximum; (5) the focus and
the meta can either have additional information or not; (6) additionally, the
field types in subspace, breakdown, measure, and focus are one of four fixed
types: temporal, numerical, categorical, and geographical. The structure
follows the context-free grammar (CFG) [53], which is a set of recursive rules
used to generate patterns of strings. Therefore, the structural information of
each chart can then be represented as a parse tree generated by a set of rules
within the CFG, and we give all the rules of the chart fact in the
supplementary material. This transformation facilitates the subsequent
encoding of structural information, as explained in Section 4.2.2. The fact
semantics refers to the semantic content within the chart fact, including the
information from the data field and the value. Since the fact semantics are
highly related to the dataset, they are not enumerable and their data
semantics are mostly inconsistent. In Fig. 2(2), the text highlighted in blue
represents structural information, while the text highlighted in green
represents semantic information. The above structural information can be
represented as the rules shown on the left side of Fig. 2(3). The example fact
semantics include “Country”, “China”, “City”, “Guizhou”, “Location”,
“Population”, “lower”, etc. The semantic information is then organized into a
token list, as shown in Fig. 2(4).
### 4.2 Chart2Vec Model
To construct a universal embedding of charts, we not only incorporate chart
information at different levels but also define multiple tasks to learn the
vector representation of the chart. This section begins with an overview of
the model inputs and architecture, followed by the implementation details and
model training configurations.
#### 4.2.1 Overview
In order to better understand the inputs and outputs of the model, we provide
a formal definition and an overview of the overall architecture.
Formulation. In the proposed model, a single chart denoted as $C_{i}$ is taken
as input, and a corresponding output vector denoted as $X_{i}$ is generated.
Each input vector consists of two types of information: fact schema and fact
semantics, represented by $f_{i}$ and $s_{i}$ respectively. To ensure that
contextual correlations are captured by the vector representation, we use an
input set of four charts that are fed into the same model with shared
parameters. Each input set consists of three sequentially connected charts in
the same multi-view visualization, denoted as
$\left\\{C_{i-1},C_{i},C_{i+1}\right\\}$, as well as a random chart from other
multi-view visualizations generated, denoted as $C_{j}$. Therefore, each input
set can be represented as $\left\\{C_{i-1},C_{i},C_{i+1},C_{j}\right\\}$, and
the corresponding output vector is
$\left\\{X_{i-1},X_{i},X_{i+1},X_{j}\right\\}$. More details can be found in
Fig. 3.
Figure 3: Formulation of Chart2Vec. During model training, the inputs are a
set of four visualizations, which are passed through the Chart2Vec model with
shared parameters. The two loss functions are used to jointly optimize the
model parameters.
Architecture. The Chart2Vec model architecture, as illustrated in Fig. 4,
comprises two main components: an input embedding module and an encoder. The
input embedding module is designed to convert the chart fact format into a
numeric form that can be calculated by the computer, and it needs to be able
to extract as much valid information as possible from the original format of
the chart. The original format of the chart can be seen as composed of two
parts, namely fact schema and fact semantics. The fact schema is represented
in a rule tree format and converted into one-hot vectors as it consists of
enumerable properties (Fig. 4(1)), while the fact semantics is encoded using
the Word2Vec model [31], with each word encoded individually to represent its
semantic information(Fig. 4(2)). To perform information fusion and vector
space conversion, the encoder module convolves the one-hot matrix
corresponding to the fact schema (Fig. 4(3)) and averages the word vectors
encoded by the fact semantics (Fig. 4(4)), followed by fusion operations on
the averaged word vectors, the position encoding corresponding to the fact
schema structure, and the one-hot matrix after convolution (Fig. 4(5)).
Finally, to enhance the model’s accuracy and generalization capability, the
resulting vector is obtained by passing the concatenated vector through two
fully connected layers for nonlinear transformation. Moreover, the necessity
of adding fully connected layers is validated in Section 5.1.
Figure 4: Architecture of the Chart2Vec model.
#### 4.2.2 Input Embedding
The chart fact exists in the form of structured strings. With the input
embedding module, it is possible to transform the raw data into a form that
can be understood and processed by the model. In addition, since the fact
consists of two parts, fact schema and fact semantics, each with distinct
different attributes and features, we encode them separately with different
methods to effectively represent the information so that the model can
understand and process them more accurately.
Input embedding of fact schema. As described in Section 4.1, the structural
information in the fact schema can be represented as a parse tree generated by
a set of rules in CFG. Therefore, we developed a rule tree in a CFG format,
containing 60 rules, with each rule encoded using a 60-dimensional one-hot
vector. As the maximum number of rules corresponding to the structural
information of a chart is 16, the input embedding of the structural
information in each fact can be represented as a vector matrix of size
16$\times$60, as depicted in Fig. 4(1).
Input embedding of fact semantics. Fact semantics are field values in the
chart fact that contain semantic information, which are derived from column
names or values in the original dataset. As each field value has its specific
meaning, it cannot be represented by fixed rules. Fig. 2(5) shows that there
are seven fields in the chart fact that contain semantic information:
subspace-field, subspace-value, breakdown-field, measure-field, focus-field,
focus-value, and meta. First, we extract the relevant words from the seven
fields, excluding any field with an empty value. It is worth noting that each
field may contain multiple words, which are subsequently split into separate
words. For example, consider the original list of extracted words, which
includes [“Country name”, “City name”, “Year”, “Student population”, “Year”,
“2018”]. After finer granularity segmentation, we obtain a list of individual
words, including [“Country”, “name”, “City”, “name”, “Year”, “Student”,
“population”, “Year”, “2018”]. However, since the model cannot calculate
string data directly, we need to convert the words into numerical inputs. To
preserve the meaning of each word, we use the Word2Vec model to transform each
word into a vector, as illustrated in Fig. 4(2). In this paper, we employ the
pre-trained model provided by Wikipedia2Vec [54, 55] to benefit from reduced
training time, enhanced encoding of word vectors, and improved overall
performance.
#### 4.2.3 Encoder
After obtaining the initial vectors for both fact schema and fact semantics
through the input embedding module, the encoder module employs two distinct
operations, namely feature pooling and feature fusion, to achieve the final
vector representation. Among them, feature pooling is designed to extract the
most important features and reduce the computational complexity of the model
by minimizing the feature dimensions. Feature fusion can be used to combine
feature information from different parts, facilitating the interaction and
information transfer between different features to improve the richness and
expressiveness of features.
Feature Pooling. We utilize convolutional layers to extract the fact schema
features and transform the one-hot vectors into a single vector for the fact
schema part (Fig. 4(3)). This choice is motivated by the ability of
convolutional layers to capture local relationships among rules in their
order. Additionally, prior research [13] has demonstrated that CNNs outperform
RNNs in encoding visualization data embeddings. This could be attributed to
the repetitive nature of the input CFG rules that rely on declarative
programs, which are perceived as translationally invariant substrings. For the
fact semantics part, we apply an averaging pooling operation to each word
vector by averaging the original word vector in fixed intervals (Fig. 4(4)).
For example, suppose a word is initially represented by a 100-dimensional
vector. We can transform it into a 10-dimensional vector by applying the
average pooling operation with a step size of 10. This operation enables us to
capture the topic information of each word while blurring its boundaries.
Additionally, it reduces the size and computational cost of feature fusion. To
confirm the effectiveness of this strategy, we perform ablation studies in the
evaluation section to assess whether it indeed enhances the model’s
performance.
Feature Fusion. We extracted both structural and semantic information from the
7-tuple chart fact. While the semantic part is derived from the extracted
words in the field values of the chart fact, the connection between the
structural and semantic information is lost during the extraction process. To
restore the lost connection, we introduce location markers to indicate where
the semantic information is extracted from the original chart fact (as shown
in Fig. 2(5)). We add each word with its corresponding location number and
concatenate the structural vectors and semantic vectors with the added
location information (as shown in Fig. 2(4)). Finally, a double-layer fully
connected layer is employed to perform a nonlinear transformation, resulting
in the final vector representation (as shown in Fig. 4(6)).
#### 4.2.4 Loss Function
To better learn the contextual associations between charts, we adopt a multi-
task training strategy that combines supervised and unsupervised learning
tasks to optimize the loss function.
Supervised Linear Interpolation Loss. Inspired by Erato [20], we employ a
linear interpolation loss function for the three consecutive charts to
establish the final vector that captures contextual relationships:
$\displaystyle
l_{1}=\sum_{k=1}^{D}\left(d\left(C_{k_{i}},C_{k_{mid}}\right)+\alpha\sum_{t,s\in\left\\{i-1,i,i+1\right\\}}^{t\neq
s}d\left(C_{k_{t}},C_{k_{s}}\right)\right)$ (1)
where $k$ denotes a training sample, and $D$ is the total number of training
sets. $d\left(\cdot\right)$ represents the Euclidean distance between two
vectors. The equation consists of two parts. $C_{k_{i}}$ in the first part is
the $C_{i}$ mentioned in Section 4.2.1, which is the chart in the middle of
the three sequentially connected charts in a training set. $C_{k_{mid}}$
represents the middle point obtained by linear interpolation of $C_{k_{i-1}}$
and $C_{k_{i+1}}$, i.e., $C_{k_{mid}}=\left(C_{k_{i-1}}+C_{k_{i+1}}\right)/2$.
By minimizing the distance between $C_{k_{i}}$ and $C_{k_{mid}}$, the three
connected charts can have a linear relationship in vector space. The second
part of the formula aims to minimize the distance between the three sequential
charts in the training sets. The coefficient $\alpha$ is used to balance the
two parts of the equation.
Unsupervised Triplet Loss. To enhance the co-occurrence relationship, we
employ an unsupervised learning task, in which input triples $C_{i-1}$,
$C_{i+1}$, and $C_{j}$ are utilized as anchor, positive, and negative samples,
respectively. By minimizing the distance between the anchor and the positive
samples while maximizing the distance between the anchor and the negative,
charts that appear in the same multi-view visualization can be brought closer
together. The triplet loss function is defined as follows:
$\displaystyle
l_{2}=\sum_{k=1}^{D}\left[d\left(C_{k_{i-1}},C_{k_{i+1}}\right)-d\left(C_{k_{i-1}},C_{k_{j}}\right)+m\right]_{+}$
(2)
where $m$ is a margin factor that controls the minimum distance between the
anchor and a negative sample, obtained through experimental tuning.
Specifically, the first part of the equation computes the distance between the
two charts in the front and back of the three sequentially connected charts in
a training sample, while the second part computes the distance between the
anchor chart $C_{i-1}$ and the negative sample $C_{j}$. Notably, to ensure
proper parameter optimization, the overall loss is constrained to be greater
than or equal to zero, as the subtraction operation may result in negative
values.
Multi-task learning. To enable the model to learn the spatial linear logical
relationships between multiple charts and cluster vectors located between the
same multi-view visualizations to capture the co-occurrence relationships, we
combine the above two tasks and use a multi-task training strategy to optimize
the proposed supervised linear interpolation loss (Eq. 1) and unsupervised
triplet loss (Eq. 2):
$\displaystyle\mathcal{L}=l_{1}+\beta l_{2}$ (3)
where $\beta$ is a hyperparameter set to balance the two loss functions. We
examined the values of these two parts of the loss to see their data levels
and relative sizes, and performed hyperparameter tuning based on the
differences.
#### 4.2.5 Model Training
We describe the training corpus and configurations in detail.
Training Corpus. We collected 1098 multi-view visualizations from 551
datasets. 310 of these 551 datasets were sourced from Kaggle [56] and the
others were uploaded by users themselves, covering 10 common topics. Among
them, we randomly selected 994 multi-view visualizations based on 501 datasets
as the training set, and the other 104 multi-view visualizations from 50
datasets as the testing set. Three sequentially connected charts in the same
multi-view visualization are regarded as a positive triplet. Each triplet,
denoted as $\left\\{C_{i-1},C_{i},C_{i+1}\right\\}$, is a set of contextually
related charts centered around the middle chart. For each positive triplet,
negative instances are created by selecting charts from other multi-view
visualizations. Each training sample consists of one positive triplet of three
connected charts in the same multi-view visualization and one negative
instance from a different multi-view visualization. We removed duplicate
negative instances and obtained a total of 42,222 training samples.
Configuration details. Based on the properties of the training set, we set the
maximum length of the semantic part to 25 words. The structural part of the
model employs a 3-layer CNN encoder featuring a kernel size of 3, producing a
chart vector with a dimension of 540. We also use batch normalization in each
layer of the model. The model was trained using the PyTorch framework, with an
initial learning rate of 0.01 and a batch size of 128. The dropout rate of the
hidden layer was set to 0.1, and the parameters were updated using the Adam
optimizer. The model was trained on an Nvidia Tesla-V100 (16 GB) graphics
card. We conducted training for 10 epochs, comprising 3,298 steps in total,
which took approximately 27 minutes to complete. Throughout the training
phase, the memory consumption amounted to 1241 MiB.
## 5 Evaluation
To assess the effectiveness of the Chart2Vec model, we performed three
experiments: (1) an ablation study to demonstrate the essentiality of each
module integrated into the model, (2) a user study to evaluate the coherence
between Chart2Vec embedding vectors and human cognition, and (3) a
quantitative comparison to gauge the performance of our approach and other
chart embedding methods in capturing contextual relationships among charts.
### 5.1 Ablation Study
To thoroughly investigate the significance and indispensability of each module
in the Chart2Vec model, we performed ablation experiments, concentrating on
four aspects: training tasks, feature content, pooling strategy, and fusion
strategy. In this section, we offer a comprehensive description of the dataset
and the evaluation metrics, and present the experimental outcomes for
different combinations.
#### 5.1.1 Dataset
In Section 4.2.5, we presented the training and implementation details of the
Chart2Vec model and outlined our process of collecting 1098 high-quality
multi-view visualizations, out of which 104 were set aside as the test set. We
extracted all the charts from the 104 multi-view visualizations in the chart
fact format, resulting in a total of 551 test samples of visualizations.
#### 5.1.2 Metrics
After completing the aforementioned steps, we fed the 551 test samples into
our model to obtain their vector representations, and computed the Euclidean
distance between charts within the same dataset. For each anchor chart, we
selected the closest chart from the same dataset (i.e., the chart with the
smallest Euclidean distance), indicating the closest distance in the vector
space. This process resulted in 551 pairs of charts. To evaluate Chart2Vec’s
ability to encode contextual relationships between multiple charts, we
utilized three metrics: top-2 retrieval accuracy, top-3 retrieval accuracy,
and co-occurrence metric. These metrics range from 0 to 1, with higher values
indicating better performance. We provide detailed explanations for each
metric, as well as their calculation procedures.
Top-2 retrieval accuracy. For each anchor chart, we search for the nearest
chart represented by its vector based on the distances between chart vectors.
If two charts belong to the same multi-view visualization and are connected by
less than 2 consecutive charts, the anchor chart is considered to meet this
criterion. We calculate the retrieval results for all 551 charts and report
the percentage of charts that satisfy this criterion as the final result.
Top-3 retrieval accuracy. Similar to the calculation method of top-2 retrieval
accuracy, we use the Euclidean distance to search for the closest chart vector
to the anchor chart. For the anchor chart to meet the criteria of the top-3
retrieval accuracy metric, the retrieved chart must belong to the same multi-
view visualization as the anchor chart and be connected by no more than 3
consecutive charts.
Co-occurrence. The co-occurrence metric measures the ability of Chart2Vec to
capture the relationships between charts that frequently occur in the same
multi-view visualization. To determine if an anchor chart meets the
requirements for this metric, we check if the retrieved chart and the anchor
chart occur in the same multi-view visualization. The final value of this
metric is calculated by counting all the charts that meet the requirements and
dividing by the total number of charts.
#### 5.1.3 Results
We conducted experiments on 10 combinations of modules, using the same
training data as Chart2Vec. These 10 combinations can be classified into four
categories based on the position and function of each module in the model.
Training tasks (2 combinations). To demonstrate the benefits of employing
multi-task joint training, we evaluated models that utilized only one of the
training tasks. No Linear Interpolation indicates that only the unsupervised
learning task was used, and triplet loss was used to classify charts that are
not in the same multi-view visualization. No Classification indicates that
only the linear interpolation supervised learning task was used, and the
linear interpolation loss was employed to capture correlations between charts
within the same multi-view visualization.
Feature content (2 combinations). The Chart2Vec model captures two key aspects
of chart information: fact schema and fact semantics. To demonstrate the
importance of each aspect, we separately removed one of them. No fact schema
indicates that only the semantic information of the chart was considered,
while No fact semantics indicates that only the structural information of the
chart was considered.
Pooling strategy (4 combinations). To enhance the representation of chart
vectors, Chart2Vec employs a pooling strategy to aggregate the initial single-
word vectors obtained from the semantic component, which captures semantic
thematic information. In this study, four different pooling strategies were
evaluated: no word pooling, words avg pooling, word max pooling, and words max
pooling. No word pooling means that no pooling strategy is used and the
obtained multiple-word vectors are directly concatenated and inputted into the
encoder module. Words avg pooling denotes that all words are averaged for
obtaining the overall semantic feature. Word max pooling and Words max pooling
represent the maximum pooling strategy applied to a single word and to all
words, respectively. For example, when conducting maximum pooling on a single-
word vector, a window size of 10 is applied, and the highest value among 10
values is selected as the value for the entire window. When performing maximum
pooling on all words, all positions of the words are simultaneously scanned,
and the maximum value at each position is selected as the value for that
position.
Fusion Strategy (2 combinations). We consider the fusion of fact schema and
fact semantics in the Chart2Vec model and validate the necessity of using a
fusion strategy. Specifically, we evaluate two combinations: No pos removes
the positional encoding from the fact semantics and No FC directly
concatenates the fact schema and fact semantics with positional encoding as
the final output without the addition of the fully connected layer.
The results are presented in Table II, which demonstrate that Chart2Vec
achieves superior performance with a top-2 ratio of 0.63, top-3 ratio of 0.73,
and co-occurrence value of 0.81. Removing any of the tasks leads to
performance degradation. Among the different training tasks, the supervised
learning task with linear interpolation has a greater impact on the model’s
performance. The influence of different feature contents on the model’s
performance varies, with fact schema being more important than fact semantics.
The performance differences among the various pooling strategies are
negligible, and all are inferior to the word vector pooling strategy proposed
in this paper. This observation also applies to the fusion strategy.
Table II: The results of ablation study.
| | Top-2 Retrieval
---
Accuracy
| Top-3 Retrieval
---
Accuracy
Co-occurrence | | Memory
---
consumption
| Time
---
consumption
Chart2Vec | 0.63 | 0.73 | 0.81 | 1241 MiB | 27min15s
Training Tasks | | | | |
No Linear interpolation | 0.55 | 0.63 | 0.70 | 1241 MiB | 24min49s
No Classification | 0.43 | 0.51 | 0.56 | 1241 MiB | 24min17s
Feature Content | | | | |
No fact schema | 0.41 | 0.50 | 0.54 | 905 MiB | 24min59s
No fact semantics | 0.53 | 0.65 | 0.74 | 1221 MiB | 08min32s
Pooling Strategy | | | | |
No word pooling | 0.56 | 0.68 | 0.75 | 1523 MiB | 25min40s
Words avg pooling | 0.59 | 0.70 | 0.77 | 1239 MiB | 14min31s
Word max pooling | 0.59 | 0.67 | 0.74 | 1241 MiB | 24min17s
Words max pooling | 0.56 | 0.67 | 0.74 | 1239 MiB | 16min27s
Fusion Strategy | | | | |
No pos | 0.60 | 0.72 | 0.79 | 1241 MiB | 24min22s
No FC | 0.49 | 0.59 | 0.64 | 1071 MiB | 24min16s
### 5.2 User Study
We conducted a user study to validate the effectiveness of the chart
embeddings generated by Chart2Vec. The purpose of the study is to assess the
consistency of the calculated similarities between the Chart2Vec embedding
vectors with human perception and cognition.
Figure 5: The construction of the user study training dataset. We selected an
example dataset and represented all its related charts in this figure. Each
chart is represented as a node and the charts from different multi-view
visualizations are marked in different colors. Assuming $A_{3}$ as the anchor
chart, we calculate its distance from the other charts, respectively. Two
charts $A_{2}$ and $B_{3}$ fall in the range of the 15% nearest charts shown
in the filled circle ❶. We randomly select one as one candidate $Cand_{1}$.
Another two charts $A_{1}$ and $B_{1}$, locate in the second range shown in
the filled circle ❷ and we also randomly select one as the other candidate
$Cand_{2}$.
#### 5.2.1 Dataset
To ensure a fair and comprehensive comparison, we created 30 multi-view
visualizations for this experiment, based on 10 datasets of distinct domains.
We first encoded all the charts as vectors using the Chart2Vec model. Then, we
randomly selected three anchor charts for each dataset, constructing a total
of 30 anchor charts. For each anchor chart, we calculated its Euclidean
distance with all the other charts from the same dataset. To assess whether
the participants could tell the differences between the most similar charts
and the moderately similar charts, we selected two candidate charts from two
different similarity distance ranges. The first candidate ($Cand_{1}$) was
selected from the top 15% nearest charts to the anchor chart, and the second
candidate ($Cand_{2}$) was selected from the 40% to 50% nearest charts. As
shown in Fig. 5, we obtained a set of 3 charts consisting of one anchor chart
($A_{3}$) and two candidate charts ($A_{2}$ and $B_{1}$) as one example in the
user study. This process resulted in 30 sets of charts for the experiment. To
ensure that the participants could understand the meaning of the charts, we
added captions that translated the chart facts into natural language
descriptions for each chart and do not add any subjective narratives.
#### 5.2.2 Procedure
We recruited 30 participants (18 females, aged between 20 to 45, $mean_{age}$
= 23.8) to take part in our study. They come from diverse professional
backgrounds, including computer science and engineering, design, journalism
and communication, medicine, mathematics, and physics. Each participant was
asked to select the most relevant chart to the anchor chart from the two
candidate charts. Since some participants do not have data analysis or
visualization backgrounds, we explained to them how valuable information was
extracted from visualizations before the formal study. On each selection page,
we provided a text introduction related to the background of the dataset to
help them better understand the visualizations. After the participant
completed all 30 selections, we conducted a brief interview with each
participant to inquire about their decision-making criteria and collect their
feedback. The entire study for each participant lasted for 30 minutes.
#### 5.2.3 Results
We assessed the final result using accuracy, which measures the degree of
agreement between the user’s selection and the model’s calculation of similar
charts. The statistical analysis indicated that the average accuracy was
83.33%, with a standard deviation of 0.073. This result suggests a high level
of concordance between the model’s calculations and human judgments, thus
confirming the effectiveness of the Chart2Vec model in capturing contextual
relationships among charts. Participants also reported that during testing
they found that both candidate charts were quite relevant to the control chart
and required careful analysis of the chart content. They mainly looked for
similarities in two aspects: chart type and chart keywords (relevant data
variables). Participants from computer science and mathematics backgrounds
were more likely to examine correlation in data variables, whereas those from
design backgrounds focused on correlation in terms of chart type and color
scale. We took these two factors into account when designing the chart facts,
and the feedback we received from users further validates the effectiveness of
our design.
### 5.3 Quantitative Comparison
We conducted a quantitative comparison with two deep learning-based methods,
ChartSeer and Erato, to validate the performance of Chart2Vec’s chart
embedding vectors in representing contextual relationships. The datasets and
evaluation metrics utilized in this experiment were consistent with those
outlined in Section 5.1.
#### 5.3.1 Model Settings
To ensure a fair comparison with the aforementioned models, we retrained them
using the same dataset as Chart2Vec. In this section, we introduce these two
models respectively and provide details on the retraining process.
ChartSeer adopts an encoder-decoder architecture to represent data charts,
taking preprocessed chart specifications as input, with data fields of Vega-
Lite replaced by common markers. ChartSeer captures only the structural
information from the chart specifications. To perform a fair comparison, we
converted Chart2Vec’s training data into ChartSeer format and retrained the
model using its original framework. We utilized the encoder to represent the
hidden vector of the chart. The original model achieved a reconstruction
accuracy of 0.6526 with a 20-dimensional vector. To improve performance on our
training data, We adjusted the configuration of ChartSeer by setting the
dimension of the latent vector to 300, and kept the batch size and epochs
consistent with the original, at 200 and 100, respectively. A reconstruction
rate of 0.8572 was finally obtained.
Erato takes chart specifications as input and converts them into sentences to
obtain the initial vector representation of the chart through BERT. It then
connects two fully connected layers to obtain the chart vector representation.
To retrain Erato, we first converted the training data of Chart2Vec into
sentence form according to the rules in Erato, and then retrained the model
using its configuration with a total of 2639 steps, setting the epoch to 50
and the batch size to 64.
#### 5.3.2 Results
As shown in Table III, it is evident that Chart2Vec outperforms the other two
methods, ChartSeer and Erato, in terms of top-2 retrieval accuracy, top-3
retrieval accuracy, and co-occurrence values. Specifically, Chart2Vec achieves
a higher top-2 retrieval accuracy, top-3 retrieval accuracy, and co-occurrence
value than the means of the other two methods by 0.23, 0.25, and 0.27,
respectively. These findings demonstrate that Chart2Vec is designed to
effectively capture contextual associations of charts.
We further conduct a comprehensive analysis to understand why Chart2Vec works
better than the other two models. ChartSeer only incorporates structural
information when encoding chart information, omitting specific data column
names from the charts. For example, in a visualization depicting rainfall over
time, it employs “field $\rightarrow$ STR” and “field $\rightarrow$ NUM” to
represent rainfall and time information, respectively. In contrast, Chart2Vec
not only extracts structural chart information but also takes into account the
semantic information of real words. This richer representation enables
Chart2Vec to explore more profound connections when capturing the contextual
relationships between charts. Erato focuses on the semantic information within
charts, converting the declarative language of visual charts into a sequence
of strings. Additionally, it utilizes a linear interpolation loss function to
map adjacent charts as a straight line in vector space. However, this approach
can lead to stacking adjacent charts with similar semantics that are located
in different multiple charts. Conversely, Chart2Vec supplements linear
interpolation with triplet loss, bringing adjacent charts closer together
while distancing them from charts located in different contexts.
Table III: The results of the quantitative comparison.
Embedding Method | | Top-2 Retrieval
---
Accuracy
| Top-3 Retrieval
---
Accuracy
Co-occurrence | | Memory
---
consumption
| Time
---
consumption
Chart2Vec | 0.63 | 0.73 | 0.81 | 1241 MiB | 27min15s
ChartSeer | 0.39 | 0.47 | 0.54 | 303 MiB | 11min15s
Erato | 0.42 | 0.50 | 0.55 | 1847 MiB | 27min37s
## 6 Discussion
In this section, we discuss the limitations of the Chart2Vec model, lessons
learned during the design process, and the model’s generalizability and
potential applications.
### 6.1 Limitations and Lessons
In this paper, we take a new perspective on the use of representation learning
models to characterize visualizations in order to learn their structural,
semantic, and contextual information. We conclude the limitations of Chart2Vec
and summarize the key takeaways from three perspectives: the importance of
context, customizing representation models, and refining them for specific
visualization tasks. We also suggest potential solutions and areas for future
research in these aspects.
Necessity of Contextual Information. Studies on the relationships between
multi-view visualizations primarily focus on three aspects: data relationships
[57], spatial relationships [8], and interaction relationships [58]. In this
paper, we adopt a contextual perspective to investigate the relationships
between multi-view visualizations, taking into account both data and spatial
relationships. This approach facilitates a more comprehensive exploration of
the underlying patterns and provides deeper insights into the data. To
incorporate contextual information, we curated a dataset of high-quality
context-aware visualizations extracted from data stories and dashboards. We
then trained the model with shared parameters to embed the contextual
information into the vector representation. As a result, in subsequent tasks
such as visualization recommendation or retrieval, the fine-tuned model can
leverage its learned contextual relationships to suggest or retrieve relevant
visualizations. Moreover, with the increasing interest in presenting various
logic orders in exploratory visual analytics, one possible research direction
is to categorize the contextual relationships and then design representation
learning models based on this categorization to support more accurate
downstream tasks.
Given that the context-aware dataset we collected comprises multi-view
visualizations from data stories and dashboards, contextual relationships are
reflected in the proximity of spatially adjacent charts and the co-occurrence
of charts within the same multi-view visualization. At the level of data
content, we regard the structural and semantic information of the
visualizations as crucial information for establishing a connection between
contextually related visualizations. Furthermore, the post-study interviews
with the participants provided further support for the importance of
integrating both structural and semantic information. Several participants
noted that when selecting the most relevant visualizations, they not only
consider the presentation of the visualizations but also the fact correlations
between them. Subsequent research can further integrate explainable AI
technologies, such as feature attribution and saliency methods [59], to gain
insight into which part of the partial input contributes most to the final
result.
Representation Model Customization. In the field of natural language
processing, representation learning models have become increasingly popular by
providing a way to transform textual data into vector or matrix forms, which
enables computers to efficiently process and analyze data. With the prevalence
of data-driven analysis and the development of automatic visualization
techniques, the number of visualizations is also growing rapidly, making
visualization itself an important data format [7]. Accordingly, representation
learning helps transform visualizations into a data format that can be
efficiently processed and analyzed using vectors. Following the footsteps of
pre-trained language models, like Word2Vec [31] and BERT [28], we developed a
visualization representation learning model that caters to the unique features
of visualization tasks and training data.
To tailor the representation model to the needs of visualizations, we
incorporated location information and constructed a task-specific training
set. We utilized positional encoding to combine the structural and semantic
information of the visualization. Furthermore, we adopted the triplet loss,
which is commonly used in contrast learning for unsupervised learning tasks.
Selecting appropriate negative samples is also crucial in this step. Since the
information conveyed in visualizations heavily depends on the data from the
original datasets, and different datasets tend to have diverse data
distributions and attributes, it is straightforward for the model to
differentiate between negative examples derived from dissimilar datasets.
Therefore, when creating the training dataset, we selected negative examples
from other multi-view visualizations of the same dataset or datasets of the
same category. To preserve contextual information and obtain a more general
representation, statistical information from the original dataset was not
considered. However, if the downstream visualization task is highly dependent
on the data statistics of the original dataset, the model architecture could
be adjusted to incorporate the data schema.
Fine-tuning for Downstream Visualization Tasks. The Chart2Vec model is trained
using a diverse and broad training dataset. The dataset consists of 849
different data stories and 249 dashboards generated from the datasets covering
10 different domains. This approach allows Chart2Vec to learn generic multi-
view visualization contexts and can be fine-tuned with small-sample datasets
for specific tasks. Consequently, it can be utilized in a wide range of tasks
related to context-aware visualization, leveraging state-of-the-art machine
learning techniques [60]. For example, Chart2Vec can be used to recommend
relevant visualizations to users in exploratory visual analytics by
concatenating sequence models such as Transformers [61]. These models can be
further enhanced through fine-tuning using small-sample training data derived
from exploratory visual analytics sequences, which holds the potential to
offer more personalized recommendations. This also presents an exciting
research direction for future work. Although we have selected training data
covering 10 different topics, there is still potential for further
optimization. In order to create an even more universal model, we need to
increase the diversity of our training data, including dataset domains, user
backgrounds, and visualization formats. Currently, the size of our training
data is relatively small compared to pre-trained models in natural language
processing. In the future, we plan to collaborate with well-established BI
software companies to collect more high-quality multi-view visualizations
created by professional data analysts, thus expanding our training data and
improving the model’s performance.
### 6.2 Generalizability and Application Scenarios
We discuss the generability of Chart2Vec for visualizations in other formats
and some potential application scenarios.
Visualizations in Other Formats. The Chart2Vec model was trained using data
stories from Calliope and dashboards from Tableau Public, but it can be
applied to a variety of multi-view visualizations with contextual
relationships, such as infographics and sequences of exploratory visualization
analysis. In addition, the proposed input embedding format can be easily
obtained by syntactically parsing other visualization grammar forms, such as
Vega-Lite [52] and Draco [62]. With recent advances in information retrieval
from raw images, we could also consider retraining the model using other types
of visualizations by converting them into the chart fact format.
Application Scenarios. The Chart2Vec model is designed to transform
visualizations into high-dimensional vectors, allowing for the computation of
similarities based on the contextual relevance between charts, which was
previously difficult to quantify. This contextual computation of
visualizations unlocks a wide range of practical applications, such as pattern
mining, visualization recommendation, and visualization retrieval. First, by
downsampling the visualization embeddings into a two-dimensional space, we can
cluster visualizations with contextual relationships. The proximity of charts
in the cluster reflects their correlations in terms of chart presentation and
data content, which can be further analyzed by experts to discover their
design patterns or explore data patterns encoded in the visualization
ensembles. Second, as narrative dashboards are becoming more and more popular
in various areas, there is an emerging demand for automatic recommendations of
context-aware visualization. For example, Medley [36] recommends visual
collections based on the user’s intent, while these collections will be sorted
based on the relevance of the attributes of interest to the user and the
activity canvas being edited. In practice, Chart2Vec has also been adapted to
help recommend visualization dashboards in a BI tool with over 100,000 users
for the tech company. Moreover, during the creation of a data story or a
dashboard, users may encounter the need to replace an unsatisfactory chart
with a relevant one. By computing the distance between visualization
embeddings, users can narrow down the search range and choose a suitable
replacement.
To illustrate how Chart2Vec works, we demonstrate a real-world use case within
the context of a tech company. First, developers fine-tune the Chart2Vec model
using data collected on its platform from multi-view visualizations created by
users. The Chart2Vec model is integrated into the BI tool’s recommendation
functionality for users who want to analyze data using multi-view
visualizations, enabling them to build related charts more efficiently. Users
begin by selecting the dataset to be analyzed from the database. They then
have the option to generate a series of single-chart visualizations using the
system’s internal pre-defined functions. Next, the system utilizes Chart2Vec
to generate chart vectors that form the search space. As users add a single
visualization to the creation panel and click the recommendation button, the
system computes a similarity metric. This metric measures the contextual
relevance of the chart vectors by calculating the distance between them and
arranges them in order of relevance. Users can then select and add the
recommended charts to the authoring panel.
## 7 Conclusion
In this paper, we proposed Chart2Vec, a context-aware representation model
that learns a universal embedding of visualizations, which is capable of
extracting context-aware information and enabling various downstream
applications such as recommendation and storytelling. We collected a context-
aware visualization dataset consisting of 6014 visualizations from 1098 multi-
view visualizations. Based on the four-level model of semantic content [15],
we extracted both structural and semantic information from multi-view
visualizations. To better retrieve the contextual information of chart
embeddings in context-aware scenarios, we adopted a multi-task training
strategy that combines supervised and unsupervised learning tasks to advance
the performance. We conducted a series of experiments to validate the
usability and effectiveness of the model, including an ablation study, a user
study, and a quantitative comparison with existing methods. In addition, we
discussed the lessons learned and potential future directions.
## ACKNOWLEDGMENTS
Nan Cao is the corresponding author. This work was supported in part by NSFC
62372327, 62072338, NSF Shanghai 23ZR1464700, and Shanghai Education
Development Foundation “Chen-Guang Project” 21CGA75. We would like to thank
all the reviewers for their valuable feedback.
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| Qing Chen received her B.Eng degree from the Department of Computer
Science, Zhejiang University and her Ph.D. degree from the Department of
Computer Science and Engineering, Hong Kong University of Science and
Technology (HKUST). After receiving her PhD degree, she worked as a postdoc at
Inria and Ecole Polytechnique. She is currently an associate professor at
Tongji University. Her research interests include information visualization,
visual analytics, human-computer interaction, generative AI and their
applications in education, healthcare, design, and business intelligence.
---|---
| Ying Chen received her bachelor’s degree from the Department of Artificial
Intelligence and Computer Science, Jiangnan University in 2021. Currently, she
is a master’s candidate at Tongji University. Her research interests include
data visualization, human-computer interaction, and the integration of
artificial intelligence with business intelligence.
---|---
| Ruishi Zou is pursuing his undergraduate degree from the Department of
Computer Science at Tongji University. He is also a part of the Intelligent
Big Data Visualization (iDVx) Lab at Tongji University. His research interests
include information visualization, human-AI interaction, and user interface
software and technology.
---|---
| Wei Shuai received her bachelor’s degree from the Department of Artificial
Intelligence and Computer Science, Jiangnan University in 2022. Currently, she
is a master’s candidate at Tongji University. Her research interests include
AI-supported design and information visualization.
---|---
| Yi Guo received his M.S. degree in Financial Mathematics from the
University of New South Wales, Australia in 2019. He is currently working
toward his Ph.D. degree as part of the Intelligent Big Data Visualization
(iDVx) Lab, Tongji University. His research interests include data
visualization and deep learning.
---|---
| Jiazhe Wang holds a Master’s degree in Computer Science from the University
of Oxford and is currently pursuing a part-time Ph.D. at Tongji University. He
serves as the tech lead of the Intelligent Agent Platform at Alibaba Group.
Previously, he was a pivotal member of AntV, Ant Group’s data visualization
team, and as a tech leader in augmented analytics at Ant Group. His research
interests are primarily in artificial intelligence, agent technologies, visual
analytics, and augmented analytics.
---|---
| Nan Cao received his Ph.D. degree in Computer Science and Engineering from
the Hong Kong University of Science and Technology (HKUST), Hong Kong, China
in 2012. He is currently a professor at Tongji University and the Assistant
Dean of the Tongji College of Design and Innovation. He also directs the
Tongji Intelligent Big Data Visualization Lab (iDVx Lab) and conducts
interdisciplinary research across multiple fields, including data
visualization, human computer interaction, machine learning, and data mining.
He was a research staff member at the IBM T.J. Watson Research Center, New
York, NY, USA before joining the Tongji faculty in 2016.
---|---
|
# Scaling Blockchain Consensus via
a Robust Shared Mempool ††thanks: $\dagger$These authors have contributed
equally to this work. Corresponding author: Jianyu Niu.
Fangyu Gai1,†, Jianyu Niu2,†, Ivan Beschastnikh3, Chen Feng1, Sheng Wang4
1{fangyu.gai<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>University of British Columbia (1Okanagan Campus,
3Vancouver Campus)
2Southern University of Science and Technology 4Alibaba Group
###### Abstract
Leader-based Byzantine fault-tolerant (BFT) consensus protocols used by
permissioned blockchains have limited scalability and robustness. To alleviate
the leader bottleneck in BFT consensus, we introduce _Stratus_ , a robust
shared mempool protocol that decouples transaction distribution from
consensus. Our idea is to have replicas disseminate transactions in a
distributed manner and have the leader only propose transaction ids. Stratus
uses a provably available broadcast (PAB) protocol to ensure the availability
of the referenced transactions. To deal with unbalanced load across replicas,
Stratus adopts a distributed load balancing protocol.
We implemented and evaluated Stratus by integrating it with state-of-the-art
BFT-based blockchain protocols. Our evaluation of these protocols in both LAN
and WAN settings shows that Stratus-based protocols achieve $5\times$ to
$20\times$ higher throughput than their native counterparts in a network with
hundreds of replicas. In addition, the performance of Stratus degrades
gracefully in the presence of network asynchrony, Byzantine attackers, and
unbalanced workloads.
###### Index Terms:
Blockchain, Byzantine fault-tolerance, leader bottleneck, shared mempool.
## I Introduction
The emergence of blockchain technology has revived interest in Byzantine
fault-tolerant (BFT) systems [1, 2, 3, 4, 5]. Unlike traditional distributed
databases, BFT systems (or blockchains) provide data provenance and allow
federated data processing in untrusted and hostile environments [6, 7]. This
enables a rich set of decentralized applications, in e.g., finance [8], gaming
[9], healthcare [10], and social media [11]. Many companies and researchers
are seeking to build enterprise-grade blockchain systems [12, 13, 14, 15] to
provide Internet-scale decentralized services [16].
The core of a blockchain system is the BFT consensus protocol, which allows
distrusting parties to replicate and order a sequence of transactions. Many
BFT consensus protocols [17, 18, 19, 20] adopted by permissioned blockchains
follow the classic leader-based design of PBFT [21]: only the leader node
determines the order to avoid conflicts. We call such protocols leader-based
BFT protocols, or LBFT.
In the normal case (Byzantine-free), an LBFT consensus instance roughly
consists of a proposing phase and a commit phase. In the proposing phase, the
leader pulls transactions from its local transaction pool (or mempool), forms
a proposal, and broadcasts the proposal to the other replicas. On receiving a
proposal, replicas verify the proposal content before entering the commit
phase. In the commit phase, the leader coordinates multiple rounds of message
exchanges to ensure that all correct replicas commit the same proposal at the
same position. If the leader behaves in a detectable Byzantine manner, a view-
change sub-protocol will be triggered to replace the leader with one of the
replicas.
A key scalability challenge for LBFT is the leader bottleneck. Since the
proposing and commit phases are both handled by the leader, adding replicas
increases the load on the leader and reduces performance. For example, in a
LAN environment, the throughput of LBFT protocols drops from 120K tps
(transaction per second) with 4 replicas to 20K tps with 64 replicas, while
the transaction latency surges from $9$ milliseconds to $3$ seconds [22]. This
has also been documented by other work [23, 24, 25].
Prior work has focused on increasing LBFT performance by improving the _commit
phase_ , e.g., reducing message complexity [19], truncating communication
rounds [26], and enhancing tolerance to Byzantine faults [27, 28]. Recent
works [23, 25] reveal that a more significant factor limiting LBFT’s
scalability lies in the _proposing phase_ , in which a proposal with batched
transaction data (e.g., 10 MB) is disseminated by the single leader node,
whereas messages exchanged in the commit phase (e.g., signatures, hashes) are
much smaller (e.g., 100 Byte). Formal analysis in Appendix A-A shows that
reducing the message complexity of the commit phase cannot address this
scalability issue.
More broadly, previous works to address the leader bottleneck have proposed
horizontal scaling or sharding the blockchain into shards that concurrently
run consensus [29, 30, 31, 2]. These approaches require a large network to
ensure safety [32] and demand meticulous coordination for cross-shard
transactions. By contrast, vertical scaling approaches employ hierarchical
schemes to send out messages and collect votes [33, 34]. Unfortunately, this
increases latency and requires complex re-configuration to deal with faults.
In this paper, we follow neither of the above strategies. Instead, we
introduce the shared mempool (SMP) abstraction, which decouples transaction
distribution from consensus, leaving consensus with the job of ordering
transaction ids. SMP allows every replica to accept and disseminate client
transactions so that the leader only needs to order transaction ids. Applying
SMP reaps the following benefits. _First_ , SMP reduces the proposal size and
increases throughput. _Second_ , SMP decouples the transaction synchronization
from ordering so that non-leader replicas can help with transaction
distribution. _Lastly_ , SMP can be integrated into existing systems without
changing the consensus core.
SMP has been used to improve scalability [35, 23, 25], but prior work has
passed over two challenges. Challenge 1: ensuring the availability of
transactions referenced in a proposal. When a replica receives a proposal, its
local mempool may not contain all the referenced transactions. These missing
transactions prevent consensus from entering the commit phase, which may cause
frequent view-changes (Section VII-C). Challenge 2: dealing with unbalanced
load across replicas. SMP distributes the load from the leader and lets each
replica disseminate transactions. But, real workloads are highly skewed [36],
overwhelming some replicas and leaving others under-utilized (Section VII-D).
Existing SMP protocols ignore this and assume that each client sends
transactions to a uniformly random replica [25, 23, 35], but this assumption
does not hold in practical deployments [37, 38, 39, 40].
We address these challenges with _Stratus_ , an SMP implementation that scales
leader-based blockchains to hundreds of nodes. Stratus introduces a provably
available broadcast (PAB) primitive to ensure the availability of transactions
referenced in a proposal. With PAB, consensus can safely enter the commit
phase and not block on missing transactions. To deal with unbalanced
workloads, Stratus uses a distributed load-balancing (DLB) co-designed with
PAB. DLB dynamically estimates a replica’s workload and capacity so that
overloaded replicas can forward their excess load to under-utilized replicas.
To summarize, we make the following contributions:
* •
We introduce and study a shared mempool abstraction that decouples network-
based synchronization from ordering for leader-based BFT protocols. To the
best of our knowledge, we are the first to study this abstraction explicitly.
* •
To ensure the availability of transactions, we introduce a broadcast primitive
called PAB, which allows replicas to process proposals without waiting for
transaction data.
* •
To balance load across replicas, we introduce a distributed load-balancing
protocol co-designed with PAB, which allows busy replicas to transfer their
excess load to under-utilized replicas.
* •
We implemented Stratus and integrated it with HotStuff [19], Streamlet [20],
and PBFT [21]. We show that Stratus-based protocols substantially outperform
the native protocols in throughput, reaching up to $5\times$ and $20\times$ in
typical LANs and WANs with 128 replicas. Under unbalanced workloads, Stratus
achieves up to $10\times$ more throughput.
## II Related Work
One classic approach that relieves the load on the leader is horizontal
scaling, or sharding [30, 2, 29]. However, using sharding in BFT consensus
requires inter-shard and intra-shard consensus, which adds extra complexity to
the system. An alternative, vertical scaling technique has been used in
PigPaxos [33], which replaced direct communication between a Paxos leader and
replicas with relay-based message flow.
Recently, many scalable designs have been proposed to bypass the leader
bottleneck. Algorand [15] can scale up to tens of thousands of replicas using
Verifiable Random Functions (VRFs) [41] and a novel Byzantine agreement
protocol called BA$\star$. For each consensus instance, a committee is
randomly selected via VRFs to reach consensus on the next set of transactions.
Some protocols such as HoneyBadger [42] and Dumbo [43] adopt a leader-less
design in which all the replicas contribute to a proposal. They are targeting
on consensus problems under asynchronous networks, while our proposal is for
partially synchronous networks. Multi-leader BFT protocols [24, 44, 45] have
multiple consensus instances run concurrently, each led by a different leader.
Multi-leader BFT protocols such as MirBFT [45] and RCC [24] use multiple
consensus instances that are run concurrently by different leaders. These
protocols follow a monolithic approach and introduce mechanisms in the view-
change procedure to deal with the ordering across different instances and
during failures. These additions render a BFT system more error-prone and
inefficient in recovery. Stratus-enabled protocols are agnostic to the view-
change since Stratus does not modify the consensus core.
Several proposals address the leader bottleneck in BFT, and we compare these
in Table I. Tendermint uses gossip to shed the load from the leader.
Specifically, a block proposal is divided into several parts and each part is
gossiped into the network. Replicas reconstruct the whole block after
receiving all parts of the block. The most recent work, Kauri [34], follows
the vertically scaling approach by arranging nodes in a tree to propagate
transactions and collect votes. It leverages a pipelining technique and a
novel re-configuration strategy to overcome the disadvantages of using a tree
structure. However, Kauri’s fast re-configuration requires a large fan-out
parameter (that is at least larger than the number of expected faulty
replicas), which constrains its ability to load balance. In general, tree-
based approaches increase latency and require complex re-configuration
strategies to deal with faults.
To our knowledge, S-Paxos [35] is the first consensus protocol to use a shared
Mempool (SMP) to resolve the leader bottleneck. S-Paxos is not designed for
Byzantine failures. Leopard [25] and Narwhal [23] utilize SMP to separate
transaction dissemination from consensus and are most similar to our work.
Leopard modifies the consensus core of PBFT to allow different consensus
instances to execute in parallel, since transactions may not be received in
the order that proposals are proposed. However, Leopard does not guarantee
that the referenced transactions in a proposal will be available. It also does
not scale well when the load across replicas is unbalanced. Narwhal [23] is a
DAG-based Mempool protocol. It employs reliable broadcast (RB) [46] to
reliably disseminate transactions and uses a DAG to establish a causal
relationship among blocks. Narwhal can make progress even if the consensus
protocol is stuck. However, RB incurs quadratic message complexity and Narwhal
only scales well when the nodes running the Mempool and nodes running the
consensus are located on separate machines. Our work differs from prior
systems by contributing (1) an efficient and resilient broadcast primitive,
along with (2) a co-designed load balancing mechanism to handle uneven
workloads.
TABLE I: Existing work addressing the leader bottleneck. Protocol | Approach | Avail. guarantee | Load balance | Message complexity
---|---|---|---|---
Tendermint [18] | Gossip | ✓ | ✓ | $O(n^{2})$
Kauri [34] | Tree | ✓ | ✓ – | $O(n)$
Leopard [25] | SMP | ✗ | ✗ | $O(n)$
Narwhal [23] | SMP | ✓ | ✗ | $O(n^{2})$
MirBFT [45] | Multi-leader | ✓ | ✗ | $O(n^{2})$
Stratus | SMP | ✓ | ✓ | $O(n)$
## III Shared Mempool Overview
We propose a shared mempool (SMP) abstraction that decouples transaction
dissemination from consensus to replace the original mempool in leader-based
BFT protocols. This decoupling idea enables us to use off-the-shelf consensus
protocols rather than designing a scalable protocol from scratch.
### III-A System Model
We consider two roles in the BFT protocol: leader and replica. A replica can
become a _leader_ replica via view-changes or leader-rotation. We inherit the
Byzantine threat model and communication model from general BFT protocols [21,
19]. In particular, there are $N\geq 3f+1$ replicas in the network and at most
$f$ replicas are Byzantine. The network is partially synchronous, whereby a
known bound $\Delta$ on message transmission holds after some unknown Global
Stabilization Time (GST) [47].
We consider external clients that issue transactions to the system. We assume
that each transaction has a unique ID and that every client knows about all
the replicas (e.g., their IP addresses). We also assume that each replica
knows, or can learn, the leader for the current view. Clients can select
replicas based on network delay measurements, a random hash function, or
another preference. Byzantine replicas can censor transactions, however, so a
client may need to switch to another replica (using a timeout mechanism) until
a correct replica is found. We assume that messages sent in our system are
cryptographically signed and authenticated. The adversary cannot break these
signatures.
We futher assume that clients send each transaction to exactly one replica,
but they are free to choose the replica for each transaction. Byzantine
clients can perform a duplicate attack by sending identical transactions to
multiple replicas. We consider these attacks out of scope. In future work we
plan to defend against these attacks using the bucket and transaction
partitioning mechanism from MirBFT [45].
### III-B Abstraction
A mempool protocol is a built-in component in a consensus protocol, running at
every replica. The mempool uses the ReceiveTx(tx) primitive to receive
transactions from clients and store them in memory (or to disk, if necessary).
If a replica becomes the leader, it calls the MakeProposal() primitive to pull
transactions from the mempool and constructs a proposal for the subsequent
consensus process. In most existing cryptocurrencies and permissioned
blockchains [48, 13, 18], the MakeProposal() primitive generates a full
proposal that includes all the transaction data. As such, the leader bears the
responsibility for transaction distribution and consensus coordination,
leading to the leader bottleneck. See our analysis in Appendix A-A.
To relieve the leader’s burden of distributing transaction data, we propose a
shared mempool (SMP) abstraction, which has been used in the previous works
[49, 25, 23], but has not been systematically studied. The SMP abstraction
enables the transaction data to be first disseminated among replicas, and then
small-sized proposals containing only transaction ids are produced by the
leader for replication. In addition, transaction data can be broadcast in
batches with a unique id for each batch. This further reduces the proposal
size. See our analysis in Appendix A-B. The SMP abstraction requires the
following properties:
SMP-Inclusion: If a transaction is received and verified by a correct replica,
then it is eventually included in a proposal.
SMP-Stability: If a transaction is included in a proposal by a correct leader,
then every correct replica eventually receives the transaction.
The above two liveness properties ensure that a valid transaction is
eventually replicated among correct replicas. Particularly, SMP-Inclusion
ensures that every valid transaction is eventually proposed while SMP-
Stability, first mentioned in [35], ensures that every proposed transaction is
eventually available at all the correct replicas. The second property makes
SMP non-trivial to implement in a Byzantine environment; we elaborate on this
in Section III-E. We should note that a BFT consensus protocol needs to ensure
that all the correct replicas maintain the same history of transaction, or
safety. Using SMP does not change the order of committed transactions. Thus,
the safety of the consensus protocol is always maintained.
### III-C Primitives and Workflow
The implementation of the SMP abstraction modifies the two primitives
ReceiveTx(tx) and MakeProposal() used in the traditional Mempool and adds two
new primitives ShareTx(tx) and FillProposal(p) as follows:
* •
ReceiveTx(tx) is used to receive an incoming $tx$ from a client or replica,
and stores it in memory (or disk if necessary).
* •
ShareTx(tx) is used to distribute $tx$ to other replicas.
* •
MakeProposal() is used by the leader to pull transactions from the local
mempool and construct a proposal with their ids.
* •
FillProposal(p) is used when receiving a new proposal $p$. It pulls
transactions from the local mempool according to the transaction ids in $p$
and fills it into a full proposal. It returns missing transactions if there
are any.
Next, we show how these primitives work in an order-execute (OE) model, where
transactions are first ordered through a consensus engine (using leader-based
BFT consensus protocols) and then sent to an executor for execution. We argue
that while for simplicity our description hinges on an OE model, the
principles could also be used in an execute-order-validate (EOV) model that is
adopted by Hyperledger [3].
We use two primitives from the consensus engine, which are Propose(p) and
Commit(p). The leader replica uses Propose(p) to broadcast a new proposal $p$
and Commit(p) to commit $p$ when the order of $p$ is agreed on across the
replicas (i.e., total ordering). As illustrated in Figure 1, the transaction
processing in state machine replication using SMP consists of the following
steps:
* •
① Upon receiving a new transaction $tx$ from the network, a replica calls
ReceiveTx(tx) to add $tx$ into the mempool, and ② disseminates $tx$ by calling
ShareTx(tx) if $tx$ is from a client (avoiding re-sharing if $tx$ is from a
replica).
* •
③ Once the replica becomes the leader, it obtains a proposal (with transaction
ids) $p$ by calling MakeProposal(), and ④ proposes it via Propose(p).
* •
⑤ Upon receipt of a proposal $p$, a non-leader replica calls FillProposal(p)
to reconstruct $p$ (pulling referenced transaction from the mempool), which is
sent to the consensus engine to continue the consensus process.
* •
⑥ The consensus engine calls $Commit(p)$ to send committed proposals to the
executor for execution.
### III-D Data Structure
Microblock. Transactions are collected from clients and batched into
microblocks for dissemination111We use _microblocks_ and _transactions_
interchangeably throughout the paper. For example, the ShareTx(tx) primitive
broadcasts a microblock instead of a single transaction in practice.. This is
to amortize the verification cost. Recall that we assume a client only sends a
request to a single replica, which makes the microblocks sent from a replica
disjoint from others. Each microblock has a unique id calculated from the
transaction ids it contains.
Figure 1: The processing of transactions in state machine replication using
SMP.
Proposal. The MakeProposal() primitive generates a proposal that consists of
an id list of the microblocks and some metadata (e.g., the hash of the
previous block, root hash of the microblocks).
Block. A block is obtained by calling the FillProposal(p) primitive. If all
the microblocks referenced in a proposal $p$ can be found in the local
mempool, we call it a full block, or a full proposal. Otherwise, we call it a
partial block/proposal. A block contains all the data included in the relevant
proposal and a list of microblocks.
### III-E Challenges and Solutions
Here we discuss two challenges and corresponding solutions in implementing our
SMP protocol.
Problem-I: missing transactions lead to bottlenecks. Using best-effort
broadcast [50] to implement ShareTx(tx) cannot ensure SMP-Stability since some
referenced transactions (i.e., microblocks) in a proposal might never be
received due to Byzantine behavior [25]. Even in a Byzantine-free case, it is
possible that a proposal arrives earlier than some of the referenced
transactions. We call these transactions missing transactions. Figure 2
illustrates an example in which a Byzantine broadcaster ($R_{5}$) only shares
a transaction ($tx_{1}$) with the leader ($R_{1}$), not the other replicas.
Therefore, when $R1$ includes $tx_{1}$ in a proposal, $tx_{1}$ will be missing
at the receiving replicas. On the one hand, missing transactions block the
consensus instance because the integrity of a proposal depends on the
availability of the referenced transactions, which is essential to the
security of a blockchain. This could cause frequent view-changes which
significantly affect performance, as we will show in Section VII-C. On the
other hand, to ensure SMP-Stability, replicas have to proactively fetch
missing transactions from the leader. This, however, creates a new bottleneck.
It is also difficult for the leader to distinguish between legitimate and
malicious transaction requests.
A natural solution to address the above challenge is to use reliable broadcast
(RB) [23] to implement ShareTx(tx). However, Byzantine reliable broadcast has
quadratic message complexity and needs three communication rounds (round trip
delay) [50], which is not suitable for large-scale systems. We observe that
some properties of reliable broadcast are _not_ needed by SMP since they can
be provided by the consensus protocol itself (i.e., consistency and totality).
This enlightens us to seek for a lighter broadcast primitive.
Solution-I: provably available broadcast. We resolve this problem by
introducing a provably available broadcast (PAB) primitive to ensure the
availability of transactions referenced in a proposal with negligible
overhead. PAB provides an API to generate an availability proof with at least
$f+1$ signatures. Since at most $f$ signatures are Byzantine, the availability
proof guarantees that at least one correct replica (excluding the sender) has
the message. This guarantees that the message can be eventually fetched from
at least one correct replica. As such, by using PAB in Stratus, if a proposal
contains valid available proofs for each referenced transaction, it can be
passed directly to the commit phase without waiting for the transaction
contents to arrive. Therefore, missing transactions can be fetched using
background bandwidth without blocking the consensus.
Problem-II: unbalanced workload/bandwidth distribution. In deploying a BFT
system across datacenters, it is difficult to ensure that all the nodes have
identical resources. Even if all the nodes have similar resources, it is
unrealistic to assume that they will have a balanced workload in time and
space. This is because clients are unevenly distributed across regions and
tend to use a preferred replica (nearest or most trusted). In these cases,
replicas with a low ratio of workload to bandwidth become bottlenecks.
To address the heterogeneity in workload/bandwidth, one popular approach is
gossip [51, 52, 15]: the broadcaster randomly picks some of its peers and
sends them the message, and the receivers repeat this process until all the
nodes receive the message with high probability. Despite their scalability,
gossip protocols have a long tail-latency (the time required for the last node
to receive the message) and high redundancy.
Figure 2: In a system with SMP, consisting of 5 replicas in which $R_{5}$ is
Byzantine and $R_{1}$ is the current leader.
Solution-II: distributed load balancing. We address the challenge by
introducing a distributed load-balancing (DLB) protocol that is co-designed
with PAB. DLB works locally at each replica and dynamically estimates a
replica’s local workloads and capacities so that overloaded replicas can
forward their excess load (microblocks) to under-utilized replicas (proxies).
A proxy can disseminate a certain microblock on behalf of the original sender
and prove that a microblock is successfully distributed by submitting
available proof to the sender. If the proof is not submitted in time, the
sender picks another under-utilized replica and repeats the process.
## IV Transaction Dissemination
We now introduce a new broadcast primitive called provably available broadcast
(PAB) for transaction dissemination, which mitigates the impact of missing
transactions (Problem-I). Every replica in Stratus runs PAB to distribute
microblocks and collect availability proofs (threshold signatures). When a
replica becomes the leader, it pulls microblock ids as well as corresponding
proofs into a proposal. This ensures that every receiving replica will have an
availability proof for all the referenced microblocks in a valid proposal.
These proofs resolve Problem I (Section III-E) by providing PAB-Provable
Availability. This ensures that a replica will eventually receive all the
referenced microblocks and it does not need to wait for missing microblocks to
arrive.
Broadcasting microblocks and collecting proofs is a distributed process that
is not on the critical path of consensus. As a result, they will _not_
increase latency. In fact, we found that PAB significantly improves throughput
and latency (Figure 7).
### IV-A Provably Available Broadcast
In PAB, the sending replica, or sender, $s$ broadcasts a message $m$, collects
acknowledgements of receiving the message $m$ from other replicas, and
produces a succinct proof $\sigma$ (realized via threshold signature [53])
over $m$, showing that $m$ is available to at least one correct replica, say
$r$. Eventually, other replicas that do not receive $m$ from $s$ retrieves $m$
from $r$. Formally, PAB satisfies the following properties:
PAB-Integrity: If a correct replica delivers a message $m$ from sender $s$,
and $s$ is correct, then $m$ was previously broadcast by $s$.
PAB-Validity: If a correct sender broadcasts a message $m$, then every correct
replica eventually delivers $m$.
PAB-Provable Availability: If a correct replica $r$ receives a valid proof
$\sigma$ over $m$, then $r$ eventually delivers $m$.
We divide the algorithm into two phases, the push phase and the recovery
phase. The communication pattern is illustrated in Figure 3. We use angle
brackets to denote messages and events and assume that messages are signed by
their senders. In the push phase, the sender broadcasts a message $m$ and each
receiver (including the sender) sends a PAB-Ack message
$\left\langle\texttt{PAB-Ack}|m.id\right\rangle$ back to the sender. As long
as the sender receives at least a quorum of $q=f+1$ PAB-Ack messages
(including the sender) from distinct receivers, it produces a succinct proof
$\sigma$ (realized via threshold signature), showing that $m$ has been
delivered by at least one correct replica. The recovery phase begins right
after $\sigma$ is generated, and the sender broadcasts the proof message
$\left\langle\texttt{PAB-Proof}|id,\sigma\right\rangle$. If some replica $r$
receives a valid PAB-Proof without receiving $m$, $r$ fetches $m$ from other
replicas in a repeated manner.
Figure 3: Message flow in PAB with $N=4$ replicas and $f=1$. $R_{1}$ is the
sender (Byzantine). $R_{2}$ did not receive $m$ in the push phase because of
$R1$ or network asynchrony. Thus, $R_{2}$ fetches $m$ from $R_{4}$ (randomly
picked) in the recovery phase. Algorithm 1 PAB with message $m$ at $R_{i}$
(push phase)
1:Local Variables:
2:$S\leftarrow\\{\\}$ $\triangleright$ signature set over $m.id$
3:$q\leftarrow f+1$ $\triangleright$ quorum value adjustable between [$f+1$,
$2f+1$]
4:
5:upon event $\left\langle\textsc{PAB-Broadcast}|m\right\rangle$ do
Broadcast$(\left\langle\texttt{PAB-Msg}|m,R_{i}\right\rangle)$
6:
7:upon receipt $\left\langle\texttt{PAB-Msg}|m,s\right\rangle$ for the first
time do $\triangleright$ $s\in C\cup R$
8: Store$(m)$ $\triangleright$ for future request
9: trigger $\left\langle\textsc{PAB-Deliver}|m\right\rangle$
10: if $s\in C$ then trigger $\left\langle\textsc{PAB-
Broadcast}|m\right\rangle$
11: else Send$(s,\left\langle\texttt{PAB-Ack}|m.id,R_{i}\right\rangle)$
12:
13:upon receipt $\left\langle\texttt{PAB-Ack}|id,s_{j}\right\rangle$
do$\triangleright$ if $R_{i}$ is the sender
14: $S\leftarrow S\cup s_{j}$
15: if $|S|\geq q$ then $\triangleright$ satisfies the quorum condition
16: $\sigma\leftarrow\textsf{threshold-sign}(S)$
17: trigger $\left\langle\textsc{PAB-Ava}|id,\sigma\right\rangle$
Algorithm 1 shows the push phase, which consists of two rounds of message
exchanges. In the first round, the broadcaster disseminates $m$ via
Broadcast() when the PAB-Broadcast event is triggered. Note that a replica
triggers PAB-Broadcast only if $m$ is received from a client to avoid re-
sharing (Line 10). We use $C$ to denote the client set and $R$ to denote the
replica set. In the second round, every replica that receives $m$ acts as a
witness by sending the sender a PAB-Ack message over $m.id$ (including the
signature). If the sender receives at least $q$ PAB-Ack messages for $m$ from
distinct replicas, it generates a proof $\sigma$ from associated signatures
via threshold-sign$()$ and triggers a PAB-Ava event. The value of $q$ will be
introduced shortly.
The recovery phase serves as a backup in case the Byzantine senders only send
messages to a subset of replicas or if messages are delayed due to network
asynchrony. The pseudocode of the recovery phase is presented in Algorithm 2.
The sender broadcasts the proof $\sigma$ of $m$ on event PAB-Ava. After
verifying $\sigma$, the replica that has not received the content of $m$
invokes the $\textsf{PAB-Fetch}()$ procedure, which sends PAB-Request messages
to a subset of replicas that are randomly picked from $signers$ of $\sigma$
(excluding replicas that have been requested). The function random([0,1])
returns a random real number between $0$ and $1$. The configurable parameter
$\alpha$ denotes the probability that a replica is requested. If the message
is not fetched in $\delta$ time, the $\textsf{PAB-Fetch}()$ procedure will be
invoked again and the timer will be reset.
Although we use $q=f+1$ as the stability parameter in the previous description
of PAB, the threshold is adjustable between $f+1$ and $2f+1$ without hurting
PAB’s properties. The upper bound is $2f+1$ because there are $N\geq 3f+1$
replicas in total, where up to $f$ of them are Byzantine. In fact, $q$
captures a trade-off between the efficiency of the push and recovery phases. A
larger $q$ value improves the recovery phase since it increases the chance of
fetching the message from a correct replica. But, a larger $q$ increases
latency, since it requires that the replica waits for more acks in the push
phase.
Algorithm 2 PAB with message $m$ at $R_{i}$ (recovery phase)
1:Local Variables:
2:$signers\leftarrow\\{\\}$ $\triangleright$ signers of $m$
3:$requested\leftarrow\\{\\}$ $\triangleright$ replicas that have been
requested
4:
5:upon event $\left\langle\textsc{PAB-Ava}|id,\sigma\right\rangle$
do$\triangleright$ if $R_{i}$ is the sender
6: Broadcast$(\left\langle\texttt{PAB-Proof}|id,\sigma\right\rangle)$
7:
8:upon receipt $\left\langle\texttt{PAB-Proof}|id,\sigma\right\rangle$ do
9: if $\textsf{threshold-verify}(id,\sigma)$ is not true do return
10: $signers\leftarrow\sigma.signers$
11: if $m$ does not exist by checking $id$ do PAB-Fetch$(id)$
12:
13:procedure PAB-Fetch($id$)
14: $\textsf{starttimer}(\texttt{Fetch},\delta,id)$
15: forall $r\in signers\setminus requested$ do
16: if $\textsf{random}([0,1])>\alpha$ then
17: $requested\leftarrow requested\cup r$
18: $\textsf{Send}(r,\left\langle\texttt{PAB-Request}|id,R_{i}\right\rangle)$
19: wait until all requested messages are delivered, or $\delta$ timeout do
20: if $\delta$ timeout do PAB-Fetch$(id)$
### IV-B Using PAB in Stratus
Now we discuss how we use PAB in our Stratus Mempool and how it is integrated
with a leader-based BFT protocol. Recall Figure 1 that shows the interactions
between the shared mempool and the consensus engine in the Propose phase.
Specifically, (i) the leader makes a proposal by calling MakeProposal(), and
(ii) upon a replica receiving a new proposal $p$, it fills $p$ by calling
FillProposal$(p)$. Here we present the implementations of the MakeProposal()
and FillProposal$(p)$ procedures as well as the logic for handling an incoming
proposal in Algorithm 3. The consensus events and messages are denoted with
CE.
Algorithm 3 Propose phase of view $v$ at replica $R_{i}$
1:Local Variables:
2:$mbMap\leftarrow\\{\\}$ $\triangleright$ maps microblock id to microblock
3:$pMap\leftarrow\\{\\}$ $\triangleright$ maps microblock id to available
proof
4:$avaQue\leftarrow\\{\\}$ $\triangleright$ stores microblock id that is
provably available
5:
6:upon receipt $\left\langle\texttt{PAB-Proof}|id,\sigma\right\rangle$ do
7: if $\textsf{threshold-verify}(id,\sigma)$ is not true do return
8: $pMap[id]\leftarrow\sigma$
9: $avaQue.\textsf{Push}(id)$
10:
11:upon event $\left\langle\textsc{PAB-Deliver}|mb\right\rangle$ do
$mbMap[mb.id]\leftarrow mb$
12:
13:upon event $\left\langle\textsc{CE-NewView}|v\right\rangle$ do
14: if $R_{i}$ is the leader for view v then
15: $p\leftarrow\textsf{MakeProposal}(v)$
16: $\textsf{Broadcast}(\left\langle\texttt{CE-Propose}|p,R_{i}\right\rangle)$
17:
18:procedure MakeProposal$(v)$
19: $payload\leftarrow\\{\\}$
20: while(True)
21: $id\leftarrow avaQue.\textsf{Pop}()$
22: $payload[id]\leftarrow pMap[id]$
23: if $\textsf{Len}(payload)\geq\textsc{BlockSize}$ or $id=\perp$ then
24: break
25: return newProposal($v,payload$)
26:
27:upon receipt $\left\langle\texttt{CE-Propose}|p,r\right\rangle$
$\triangleright$ $r$ is the current leader
28: for $id,\sigma\in p.payload$ do
29: if $\textsf{threshold-verify}(id,\sigma)$ is not true do
30: trigger $\left\langle\textsc{CE-ViewChange}|R_{j}\right\rangle$
31: return
32: trigger $\left\langle\textsc{CE-EnterCommit}|p\right\rangle$
33: FillProposal$(p)$
34:
35:procedure FillProposal$(p)$
36: $block\leftarrow\\{p\\}$
37: forall $id\in p.payload$ do
38: if $mb$ associated with $id$ has not been delivered then
39: $\textsf{PAB-Fetch}(id)$
40: wait until every requested $mb$ is delivered then
41: forall $id\in p.payload$ do
42: $block.\textsf{Append}(mbMap[id])$
43: $avaQue.\textsf{Remove}(id)$
44: trigger $\left\langle\textsc{CE-Full}|block\right\rangle$
Since transactions are batched into microblocks for dissemination, we use
microblocks (i.e., $mb$) instead of transactions in our description. The
consensus protocol subscribes PAB-Deliver events and PAB-Proof messages from
the underlying PAB protocol and modifies the handlers, in which we use mbMap,
pMap, and avaQue for bookkeeping. Specifically, mbMap stores microblocks upon
the PAB-Deliver event (Line 11). Upon the receipt of PAB-Proof messages, the
microblock id is pushed into the queue avaQue (Line 9) and the relevant proof
$\sigma$ is recorded in pMap (Line 8).
We assume the consensus protocol proceeds in views, and each view has a
designated leader. A new view is initiated by a CE-NewView event. Once a
replica becomes the leader for the current view, it attempts to invoke the
MakeProposal() procedure, which pulls microblocks (only ids) from the front of
$avaQue$ and piggybacks associated proofs. It stops pulling when the number of
contained microblocks has reached BlockSize, or there are no microblocks left
in $avaQue$. The reason why the proposal needs to include all the associated
available proofs of each referenced transaction is to show that the
availability of each referenced microblock is guaranteed. We argue that the
inevitable overhead is negligible provided that the microblock is large.
On the receipt of an incoming proposal $p$, the replica verifies every proof
included in $p.payload$ and triggers a CE-ViewChange event if the verification
is not passed, attempting to replace the current leader. If the verification
is passed, a $\left\langle\textsc{CE-EnterCommit}|p\right\rangle$ event is
triggered and the processing of $p$ enters the commit phase (Line 32). Next,
the replica invokes the FillProposal$(p)$ procedure to pull the content of
microblocks associated with $p.payload$ from the mempool. The PAB-Fetch$(id)$
procedure (Algorithm 2) is invoked when missing microblocks are found. The
thread waits until all the requested microblocks are delivered. Note that this
thread is independent of the thread handling consensus events. Therefore,
waiting for requested microblocks will not block consensus. After a full block
is constructed, the replica triggers a $\left\langle\textsc{CE-
Full}|block\right\rangle$ event, indicating that the block is ready for
execution.
In Stratus, the transactions in a microblock are executed if and only if all
transactions in the previous microblocks are received and executed. Since
missing transactions are fetched according to their unique ids, consistency is
ensured. Therefore, using Stratus in any case will not compromise the safety
of the consensus protocol. The advantage of using PAB is that it allows the
consensus protocol to safely enter the commit phase of a proposal without
waiting for the missing microblocks to be received. In addition, the recovery
phase proceeds concurrently with the consensus protocol (only background
bandwidth is used) until the associated block is full for execution. Many
optimizations [54, 55, 7] for improving the execution have been proposed and
we hope to build on them in our future work. Our implementation satisfies PAB-
Provable Availability, which helps Stratus achieve SMP-Inclusion and SMP-
Stability.
### IV-C Correctness Analysis
Now we prove the correctness of PAB. Since the integrity and validity
properties are simple to prove, here we only show that Algorithm 1 and
Algorithm 2 satisfy PAB-Provable Avalability. Then we provide proofs that
Stratus satisfies SMP-Inclusion and SMP-Stability.
###### Lemma 1 (PAB-Provable Availability).
If a proof $\sigma$ over a message $m$ is valid, then at least one correct
replica holds $m$. In the recovery phase (Algorithm 2), the receiving replica
$r$ repeatedly invokes PAB-Fetch$(id)$ and sends requests to randomly picked
replicas. Eventually, a correct replica will respond and $r$ will deliver $m$.
###### Theorem 1.
Stratus ensures SMP-Inclusion.
###### Proof.
If a transaction $tx$ is delivered and verified by a correct replica $r$ (the
sender), it will be eventually batched into a microblock $mb$ and disseminated
by PAB. Due to the validity property of PAB, $mb$ will be eventually delivered
by every correct replica, which sends acks over $mb$ back to the sender. An
available proof $\sigma$ over $mb$ will be generated and broadcast by the
sender. Upon the receipt of $\sigma$, every correct replica pushes $mb$
($mb.id$) into $avaQue$. Therefore, $mb$ ($tx$) will be eventually popped from
$avaQue$ of a correct leader $l$ and proposed by $l$. ∎
###### Theorem 2.
Stratus ensures SMP-Stability.
###### Proof.
If a transaction $tx$ is included in a proposal by a correct leader, it means
that $tx$ is provably available (a valid proof $\sigma$ over $tx$ is valid).
Due to the PAB-Provable Availability property of PAB, every correct replica
eventually delivers $tx$. ∎
## V Load Balancing
We now discuss Stratus’ load balancing. Recall that replicas disseminate
transactions in a distributed manner. But, due to network heterogeneity and
workload imbalance (Problem-II), performance will be bottlenecked by
overloaded replicas. Furthermore, a replica’s workload and its resources may
vary over time. Therefore, a load balancing protocol that can adapt to a
replica’s workload and capacity is necessary.
In our design, busy replicas will forward excess load to less busy replicas
that we term proxies. The challenges are (i) how to determine whether a
replica is busy, (ii) how to decide which replica should receive excess loads,
and (iii) how to deal with Byzantine proxies that refuse to disseminate the
received load.
Our load balancing protocol works as follows. A local workload estimator
monitors the replica to determine if it is busy or unbusy. We discuss work
estimation in Section V-B. Next, a busy replica forwards newly generated
microblocks to a proxy. The proxy initiates a PAB instance with a forwarded
microblock and is responsible for the push phase. When the push phase
completes, the proxy sends the PAB-Proof message of the microblock to the
original replica, which continues the recovery phase. In addition, we adopt a
$banList$ to avoid Byzantine proxies. Next, we discuss how a busy replica
forwards excess load.
### V-A Load Forwarding
Before forwarding excess load, a busy replica needs to know which replicas are
unbusy. A naïve approach is to ask other replicas for their load status.
However, this requires all-to-all communications and is not scalable. Instead,
we use the well-known Power-of-d-choices (Pod) algorithm [56, 57, 58]. A busy
replica randomly samples load status from $d$ replicas, and forwards its
excess load to the least loaded replica (the proxy). Here, $d$ is usually much
smaller than the number of replicas $N$. Our evaluation shows that $d=3$ is
sufficient for a network with hundreds of nodes and unbalanced workloads (see
Section VII-D). Note that the choice of $d$ is independent of $f$; we discuss
how we handle Byzantine proxies later in this section. The randomness in Pod
ensures that the same proxy is unlikely to be re-sampled and overloaded.
Algorithm 4 The Load Forwarding procedure at replica $R_{i}$
1:Local Variables:
2:$samples\leftarrow\\{\\}\\{\\}$ $\triangleright$ stores sampled info for a
microblock
3:$banList\leftarrow\\{\\}$ $\triangleright$ stores potentially Byzantine
proxies
4:
5:upon event $\left\langle\textsc{NewMB}|mb\right\rangle$ do
6: if IsBusy() do LB-ForwardLoad$(mb)$
7: else trigger $\left\langle\textsc{PAB-Broadcast}|mb\right\rangle$
8:
9:procedure $\textsf{LB-ForwardLoad}(mb)$ $\triangleright$ if $R_{i}$ is the
busy sender
10: $\textsf{starttimer}(\texttt{Sample},\tau,mb.id)$
11: $K\leftarrow\textsf{SampleTargets}(d)\setminus banList$
12: forall $r\in K$ do $\textsf{Send}(r,\left\langle\texttt{LB-
Query}|mb.id,R_{i}\right\rangle)$
13: wait until $|samples[mb.id]|=d$ or $\tau$ timeout do
14: if $|samples[mb.id]|=0$ then
15: trigger $\left\langle\textsc{PAB-Broadcast}|mb\right\rangle$
16: return
17: find $r_{p}\in samples[mb.id]$ with the smallest $w$
18: starttimer$(\texttt{Forward},\tau^{\prime},mb)$
19: $banList.\textsf{Append}(r_{p})$ $\triangleright$ every proxy is put in
$banList$
20: $\textsf{Send}(r_{p},\left\langle\texttt{LB-
Forward}|mb,R_{i}\right\rangle)$ $\triangleright$ send $mb$ to the poxy
21: wait until PAB-Proof over $mb$ is received or $\tau^{\prime}$ timeout do
22: if $\tau^{\prime}$ do LB-ForwardLoad($mb$)
23: else $banList.\textsf{Remove}(R_{p})$ $\triangleright$ $R_{p}$ is removed
from $banList$
24:upon receipt $\left\langle\texttt{LB-Forward}|mb,r\right\rangle$ do
$\triangleright$ if $R_{i}$ is the proxy with $mb$
25: trigger $\left\langle\textsc{PAB-Broadcast}|mb\right\rangle$
26:
27:upon receipt $\left\langle\texttt{LB-Query}|id,r\right\rangle$ do
$\triangleright$ if $R_{i}$ is sampled
28: $w\leftarrow\textsf{GetLoadStatus}()$
29: $\textsf{Send}(r,\left\langle\texttt{LB-Info}|w,id,R_{i}\right\rangle)$
30:
31:upon receipt $\left\langle\texttt{LB-Info}|w,id,r\right\rangle$ do
$\triangleright$ if $R_{i}$ is busy
32: $samples[id][R_{i}]\leftarrow w$
33:
34:upon receipt $\left\langle\texttt{PAB-Proof}|id,\sigma\right\rangle$ before
$\tau^{\prime}$ timeout do
35: if $\textsf{threshold-verify}(id,\sigma)$ is not true do return
36: trigger $\left\langle\textsc{PAB-Ava}|id,\sigma\right\rangle$
$\triangleright$ $R_{i}$ takes over the recovery phase
37:
38:upon event $\left\langle\textsc{Reset}|\textit{banList}\right\rangle$
39: $banList\leftarrow\\{\\}$ $\triangleright$ clear banList periodically
Algorithm 4 depicts the LB-ForwardLoad procedure and relevant handlers. Upon
the generation of a new microblock $mb$, the replica first checks whether it
is busy (see Section V-B). If so, it invokes the LB-ForwardLoad$(mb)$
procedure to forward $mb$ to the proxy; otherwise, it broadcasts $mb$ using
PAB by itself. To select a proxy, a replica samples load status from $d$
random replicas (excluding itself) within a timeout of $\tau$ (Line 12). Upon
receiving a workload query, a replica obtains its current load status by
calling the GetLoadStatus() (see Section V-B) and piggybacks it on the reply
(Line 23-25). If the sender receives all the replies or times out, it picks
the replica that replied with the smallest workload and sends $mb$ to it. This
proxy then initiates a PAB instance for $mb$ and sends the PAB-Proof message
back to the original sender when a valid proof over $mb$ is generated. Note
that if no replies are received before timeout, the sending replica initiates
a PAB instance by itself (Line 15). Note that due to the decoupling design of
Stratus, the overhead introduced by load forwarding has negligible impact on
consensus. To prevent a malicious replica from sending a small batch to reduce
the performance, every replica can set a minimum batch size for receiving a
batch.
In Stratus, each replica randomly and independently chooses $d$ replicas from
the remaining $N$-1 replicas. Since the workload of each replica changes
quickly, the sampling happens for each microblock without blocking the
forwarding process. Therefore, for each load balancing event of an overloaded
replica A, the probability that a specific replica (other than replica A) is
chosen by replica A is $d/(N$-1). The probability that a replica is chosen by
all replicas is very small. For example, when $d=3$ and $N=100$ the
probability that a replica is chosen by more than 7 replicas is about $0.03$.
We omit the analysis details due to the page limit. Next, we discuss how we
handle Byzantine behaviors during load forwarding.
Handling faults. A sampled Byzantine replica can pretend to be unoccupied by
responding with a low busy level and censoring the forwarded microblocks. In
this case, the SMP-Inclusion would be compromised: the transactions included
in the censored microblock will not be proposed. We address this issue as
follows. A replica $r$ sets a timer before sending $mb$ to a selected proxy
$p$ (Line 18). If $r$ does not receive the available proof $\sigma$ over $mb$
before the timeout, $r$ re-transmits $mb$ by re-invoking the LB-
ForwardLoad$(mb)$ (Line 22). Here, the unique microblock ids prevent
duplication. The above procedure repeats until a valid $\sigma$ over $mb$ is
received. Then $r$ continues the recovery phase of the PAB instance with $mb$
by triggering the PAB-Ava event (Line 39).
To prevent Byzantine replicas from being sampled again, we use a banList to
store proxies that have not finished the push phase of a previous PAB
instance. That is, before a busy sender sends a microblock $mb$ to a proxy,
the proxy is added to the banList (Line 19). For future sampling, the
replicas in the banList are excluded. As long as the sender receives a valid
proof message for $mb$ from the proxy before a timeout, the proxy will be
removed from the banList (Line 23). The banList is periodically cleared by a
timer to avoid replicas from being banned forever (Line 39). Note that more
advanced banList mechanisms can be used based on proxies’ behavior [59] and we
consider to include them in our future work.
### V-B Workload Estimation
Our workload estimator runs locally on an on-going basis and is responsible
for estimating load status. Specifically, it determines: (i) whether the
replica is overloaded, and (ii) how much the replica is overloaded, which
correspond to the two functions in Algorithm 4, IsBusy() and GetLoadStatus(),
respectively. To evaluate replicas’ load status, two ingredients need to be
considered: workload and capacity. As well, the estimated results must be
comparable across replicas in a heterogeneous network.
To address these challenges, we use stable time (ST) to estimate a replica’s
load status. The stable time of a microblock is measured from when the sender
broadcasts the microblock until the time that the microblock becomes stable
(receiving $f+1$ acks). To estimate ST of a replica, the replica calculates
the ST of each microblock if it is the sender and takes the $n$-th (e.g.,
$n=95$) percentile of the ST values in a window of the latest stable
microblocks. Figure 4 shows the estimation process. The estimated ST of a
replica is updated when a new microblock becomes stable. The window size is
configurable and we use $100$ as the default size.
Figure 4: The stable time (ST) of a replica is estimated by taking the $n$-th
percentile of ST values over a window of latest stable microblocks. The window
slides when new microblocks become stable.
(a) Heat map of measured roundtrip delays between servers from Virginia to
Singapore over 24 hours.
(b) Distribution of measured delays between servers from Virginia to Singapore
during 1 minute at 12th hour.
Figure 5: Network roundtrip delays between Virginia and Singapore.
Our approach is based on two observations. First, the variability in network
delay in a private network is small [60]. Second, network delay increases
sharply when a node is overloaded. The above observations are based on our
measurements. A selection of these is shown in Figure 5. Figure 5(a) is a heat
map of measured delays between two regions (Virginia and Singapore) in Alibaba
Cloud over 24 hours. Figure 5(b) exhibits the round-trip delay distribution
during 1 minute starting the 12th hour in the measurements. We omit
measurements of other pair of datacenters in this paper. Our results
demonstrate that the inter-datacenter network delays across different regions
are stable and predictable based on recent measurement data. Thus, under a
constant workload, the calculated ST should be at around a constant number
$\alpha$ with an error of $\epsilon$. If the estimated ST is larger than
$\alpha+\epsilon$ by a parameter of $\beta$, a replica is considered busy
(return true in the IsBusy function). Additionally, the value of ST reflects
the degree to which a replica is loaded: the smaller the ST, the more
resources a replica has for disseminating microblocks. Therefore, we use the
ST as the return value of the function GetLoadStatus. Note that the
GetLoadStatus returns a NULL value if the calling replica is busy. Also note
that due to network topology, the ST value does not faithfully reflect the
load status across replicas. For example, some replicas may have a smaller ST
because they are closer to a quorum of other replicas. In this case,
forwarding excess load to these replicas also benefits the system. For
overloaded replicas with large ST values, the topology has a negligible
impact. In case the network is unstable, we can also estimate the load status
by monitoring the queue length of the network interface card. We save that for
future work.
## VI Implementation
We prototyped Stratus222Available at https://github.com/gitferry/bamboo-
stratus in Go with Bamboo [22]333Available at
https://github.com/gitferry/bamboo, which is an open source project for
prototyping, evaluating, and benchmarking BFT protocols. Bamboo provides
validated implementations of state-of-the-art BFT protocols such as PBFT [21],
HotStuff [19], and Streamlet [20]. Bamboo also supplies common functionalities
that a BFT replication protocol needs. In our implementation, we replaced the
mempool in Bamboo with Stratus shared mempool. Because of Stratus’ well-
designed interfaces, the consensus core is minimally modified. We used
HotStuff’s Pacemaker for view change, though Stratus is agnostic to the view-
change mechanism. Similar to [19, 61], we use ECDSA to implement the quorum
proofs in PAB instead of threshold signature. This is because the computation
efficiency of ECDSA444We trivially concatenate f + 1 ECDSA signatures is
better than Boldyreva’s threshold signature [61]. Overall, our implementation
added about 1,300 lines of Go to Bamboo.
Optimizations. Since microblocks consume the most bandwidth, we need to
reserve sufficient resources for consensus messages to ensure progress. For
this, we adopt two optimizations. First, we prioritize the transmission and
processing of consensus messages. Second, we use a token-based limiter to
limit the sending rate of data messages: every data message (i.e., microblock)
needs a token to be sent out, and tokens are refilled at a configurable rate.
This ensures that the network resources will not be overtaken by data
messages. The above optimizations are specially designed for Stratus and are
only used in Stratus-based implementations. We did _not_ use these
optimizations in non-Stratus protocols in our evaluation since they may
negatively effect those protocols.
TABLE II: Summary of evaluated protocols. Acronym | Protocol description
---|---
N-HS | Native HotStuff without a shared mempool
N-PBFT | Native PBFT without a shared mempool
SMP-HS | HotStuff integrated with a simple shared mempool
SMP-HS-G | SMP-HS with gossip instead of broadcast
SMP-HS-Even | SMP-HS with an even workload across replicas
S-HS | HotStuff integrated with Stratus (this paper)
S-PBFT | PBFT integrated with Stratus (this paper)
Narwhal | HotStuff based shared mempool
MirBFT | PBFT based multi-leader protocol
## VII Evaluation
Our evaluation answers the following questions.
* •
Q1: how does Stratus perform as compared to the alternative Shared Mempool
implementations with a varying number of replicas? (Section VII-B)
* •
Q2: how do missing transactions caused by network asynchrony and Byzantine
replicas affect the protocols’ performance? (Section VII-C)
* •
Q3: how does unbalanced load affect protocols’ throughput? (Section VII-D)
### VII-A Setup
Testbeds. We conducted our experiments on Alibaba Cloud ecs.s6-c1m2.xlarge
instances555https://www.alibabacloud.com/help/en/doc-detail/25378.htm. Each
instance has 4vGPUs and 8GB memory and runs Ubuntu server 20.04. We ran each
replica on a single ECS instance. We performed protocol evaluations in LANs
and WANs to simulate national and regional deployments, respectively [34].
LANs and WANs are typical deployments of permissioned blockchains and
permissionless blockchains that run a BFT-based PoS consensus protocol [12,
14]. In LAN deployments, a replica has up to 3 Gb/s of bandwidth and inter-
replica RTT of less than 10 ms. For WAN deployments, we use NetEm [62] to
simulate a WAN environment with 100 ms inter-replica RTT and 100 Mb/s replica
bandwidth.
Workload. Clients are run on 4 instances with the same specifications. Each
client concurrently sends multiple transactions to different replicas.
Bamboo’s benchmark provides an in-memory key-value store backed by the
protocol under evaluation. Each transaction is issued as a simple key-value
set operation submitted to a single replica. Since our focus is on the
performance of the consensus protocol with the mempool, we do not involve
application-specific verification (including signatures) and execution
(including disk IO operations) of transactions in our evaluation. We measure
both throughput and latency on the server side. The latency is measured
between the moment a transaction is first received by a replica and the moment
the block containing it is committed. We avoid end-to-end measurements to
exclude the impact of the network delay between a replica and a client. Each
data point is obtained when the measurement is stabilized (sampled data do not
vary by more than 1%) and is an average over 3 runs. In our experiments,
workloads are evenly distributed across replicas except for the last set of
experiments (Section VII-D), in which we create skewed load to evaluate load
balancing.
Protocols. We evaluate the performance of a wide range of protocols (Table
II). We use native HotStuff and PBFT with the original mempool as the
baseline, denoted as (N-HS and N-PBFT, respectively). All of our
implementations of HotStuff are based on the Chained-HotStuff (three-chain)
version from the original paper [19], in which pipelining is used and leaders
are rotated for each proposal. Our implementation of PBFT shares the same
chained blockchain structure as Chained-HotStuff for a fair comparison. We
also compare against a version of HotStuff with a basic shared mempool with
best-effort broadcast and fetching (denoted as SMP-HS). Finally, we equip
HotStuff and PBFT with our Stratus Mempool, denoted as S-HS and S-PBFT,
respectively. We also implemented a gossip-based shared mempool (distributing
microblocks via gossip), denoted by SMP-HS-G, to evaluate load balancing and
compare it with S-HS. All protocols are implemented using the same Bamboo code
base for a fair comparison. The sampling parameter $d$ is set to $1$ by
default. This is because $d=1$ allows the busy sender to randomly pick
exactly one replica without comparing workload status between others. When we
gradually increase $d$, the chance of selecting a less busy replica increases
significantly. However, increasing $d$ also incurs overhead. In our
experiments (Section VII-D) we show that $d=3$ exhibits the best performance.
We also compare against Narwhal666Available at
https://github.com/facebookresearch/narwhal/, which uses a shared mempool with
reliable broadcast. Narwhal is based on HotStuff and splits functionality
between workers and primaries, responsible for transaction dissemination and
consensus, respectively. To fairly compare Narwhal with Stratus, we let each
primary have one worker and locate both in one VM instance. As another
baseline, we compare our protocols with MirBFT [45]777Available at
https://github.com/hyperledger-labs/mirbft/tree/research, a state-of-the-art
multi-leader protocol. All replicas act as leaders in an epoch for fair
comparison.
### VII-B Scalability
In the first set of experiments, we explore the impact of batch sizes on S-HS
and then we evaluate the scalability of protocols. These experiments are run
in a common BFT setting in which less than one-third of replicas remain
silent. Since our focus is on normal-case performance, view changes are not
triggered in these experiments unless clearly stated.
Figure 6: Throughput vs. latency with $128$ and $256$ replicas for S-HS. The
batch size varies from 32KB to 512KB. The transaction payload is $128$ bytes.
Picking a proper batch size. Batching more transactions in a microblock can
increase throughput since the message cost is better amortized (e.g., fewer
acks). However, batching also leads to higher latency since it requires more
time to fill a microblock. In this experiment, we study the impact of batch
size on Stratus (S-HS) and pick a proper batch size for different network
sizes to balance throughput and latency.
We deploy Stratus-based HotStuff (S-HS) in a LAN setting with $N=128$ and
$N=256$ replicas, respectively. For $N=128$, we vary the batch size from 32KB
to 128KB, while for $N=256$, we vary the batch size from 128KB to 512KB. We
denote each pair of settings as the network size followed by the batch size.
For instance, the network size of $N=128$ with the batch size of 32KB bytes is
denoted as $n128-b32$K. We use the transaction payloads of $128$ bytes
(commonly used in blockchain systems [48, 13]). We gradually increase the
workload until the system is saturated, i.e., the workload exceeds the maximum
system throughput, resulting in sharply increasing delay.
The results are depicted in Figure 6. We can see that as the batch size
increases, the throughput improves accordingly for both network sizes.
However, the throughput gain of choosing a larger batch size is reduced when
the batch size is beyond 64KB (for $N=128$) and 256KB (for $N=256$). Also, we
observe that a larger network requires a larger batch size for better
throughput. This is because large batch size amortizes the overhead of PAB
(fewer acks). But, a larger batch size leads to increased latency (as we
explained previously). We use the batch size of 128KB for small networks
($N\leq 128$), the batch size of 256KB for large networks ($N\geq 256$), and a
$128$-byte transaction payload in the rest of our experiments. As long as a
replica accumulates sufficient transactions (reaching the batch size), it
produces and disseminates a microblock. If the batch size is not reached
before a timeout ($200\text{\,}\mathrm{ms}\text{/}$ by default), all the
remaining transactions will be batched into a microblock. We also find that
proposal size (number of microblock ids included in a proposal) does not have
obvious impact on the performance as long as a proper batch size (number of
transactions included in a microblock) is chosen. Therefore, we do not set any
constraint on proposal size. The above settings also apply in SMP-HS and SMP-
HS-G.
(a) LAN evaluation.
(b) WAN evaluation.
Figure 7: The throughput (left) and latency (right) of protocols in both LAN
and WAN with increasing number of replicas. We use $128$-byte payload and
128KB batch size.
We evaluate the scalability of the protocols by increasing the number of
replicas from 16 to 400. We use N-HS, N-PBFT, SMP-HS, S-PBFT, Narwhal, and
MirBFT for comparison and run experiments in both LANs and WANs. We gradually
increase the workload until the system is saturated, i.e., the workload
exceeds the maximum system throughput, resulting in sharply increasing delay.
We use a batch size of 256KB and $128$-byte transaction payload, which gives
2,000 transactions per batch, for Stratus-based protocols throughout our
experiments. We find that proposal size (number of microblock ids included in
a proposal) does not have an obvious impact on performance as long as we
choose a proper batch size (number of transactions in a microblock).
Therefore, we do not constrain proposal size. For every protocol we use a
microblock/proposal size settings that maximizes the protocol’s performance.
We omit experimental results that explore these settings due to space
constraints.
Figure 7 depicts the throughput and latency of the protocols with an
increasing number of replicas in LANs and WANs. We can see that protocols
using the shared mempool (SMP-HS, S-HS, S-PBFT, and Narwhal) or relying on
multiple leaders outperform the native HotStuff and Streamlet (N-HS and
N-PBFT) in throughput in all experiments. Previous works [19, 25, 23] have
also shown that the throughput/latency of N-HS decreases/increases sharply as
the number of replicas increases, and meaningful results can no longer be
observed beyond 256 nodes. Although Narwhal outperforms N-HS due to the use of
a shared mempool, it does not scale well since it employs the heavy reliable
broadcast primitive. As shown in [23], Narwhal achieves better scalability
only when each primary has multiple workers that are located in different
machines. MirBFT has higher throughput than S-HS when there are fewer than 16
replicas. This is because Stratus imposes a higher message overhead than PBFT.
However, MirBFT’s performance drops faster than S-HS because of higher message
complexity. MirBFT is comparable to S-PBFT because they have the same message
complexity. The gap between them is due to implementation differences.
SMP-HS and S-HS show a slightly higher latency than N-HS when the network size
is small ($<16$ in LANs and $<32$ in WANs). This is due to batching. They
outperform the other two protocols in both throughput and latency when the
network size is beyond 64 and show flatter lines in throughput as the network
size increases. The throughput of SMP-HS and S-HS achieve $5\times$ throughput
when $N=128$ as compared to N-HS, and this gap grows with network size.
Finally, SMP-HS and S-HS have similar performance, which indicates that the
use of PAB incurs negligible overhead, which is amortized by a large batch
size.
TABLE III: Outbound bandwidth consumption comparison with $N=64$ replicas. The bandwidth of each replica is throttled to 100 Mb/s. The results are collected when the network is saturated. Role/Messages | N-HS | SMP-HS | S-HS (this paper)
---|---|---|---
Leader | Proposals | 75.4 | 4.7 | 9.8
Microblocks | N/A | 50.5 | 50.3
SUM | 75.4 | 55.2 | 60.1
Non-leader | Microblocks | N/A | 50.4 | 50.3
Votes | 0.5 | 2.5 | 2.4
Acks | N/A | N/A | 4.7
SUM | 0.5 | 52.9 | 57.4
Bandwidth consumption. We evaluate the outbound bandwidth usage at the leader
and the non-leader replica in N-HS, SMP-HS, and S-HS. We present the results
in Table III. We can see that the communication bottleneck in N-HS is at the
leader, while the bandwidth of non-leader replicas is underutilized. In SMP-HS
and S-HS, the bandwidth consumption between leader replicas and non-leader
replicas are more even, and the leader bottleneck is therefore alleviated. We
observe that S-HS adds around 10% overhead on top of SMP-HS due to the use of
PAB. Next, we show that this overhead is worthwhile as it provides
availability insurance. We also observe that around 40% of bandwidth remains
unused. This is because chain-based protocols are bounded by latency: each
proposal goes through two rounds of communication (one-to-all-to-one). We
consider out-of-order processing of proposals for better network utilization
as important future work.
### VII-C Impact of Missing Transactions
Recall that in Problem-I (Section III-E), a basic shared mempool with best-
effort broadcast is subject to missing transactions. In the next set of
experiments, we evaluate the throughput of SMP-HS and S-HS under a period of
network asynchrony and Byzantine attacks.
Network asynchrony. During network asynchrony, a proposal is likely to arrive
before some of referenced transactions (i.e., missing transactions), which
negatively impacts performance. The point of this experiment is to show that
Stratus-based protocols can make progress during view-changes and are more
resilient to network asynchrony.
We ran an experiment in a WAN setting, during which we induce a period of
network fluctuation via NetEm. The fluctuation lasts for
$10\text{\,}\mathrm{s}\text{/}$, during which network delays between replicas
fluctuate between $100\text{\,}\mathrm{ms}\text{/}$ and
$300\text{\,}\mathrm{ms}\text{/}$ for each message (i.e.,
$200\text{\,}\mathrm{ms}\text{/}$ base with $100\text{\,}\mathrm{ms}\text{/}$
uniform jitter). We set the view-change timer to be
$1000\text{\,}\mathrm{ms}\text{/}$. We keep the transaction rate at 25KTx/s
without saturating the network.
We ran the experiment 10 times and each run lasts 30 seconds. We show the
results in Figure 8. During the fluctuation, the throughput of SMP-HS drops to
zero. This is because missing transactions are fetched from the leader, which
causes congestion at the leader. As a result, view-changes are triggered,
during which no progress is made. When the network fluctuation is over, SMP-HS
slowly recovers by processing the accumulated proposals. On the other hand,
S-HS makes progress at the speed of the network and no view-changes are
triggered. This is due to the PAB-Provable Availability property: no missing
transactions need to be fetched on the critical consensus path.
Figure 8: Delay is injected at time $10\text{\,}\mathrm{s}\text{/}$ and lasts
for $10\text{\,}\mathrm{s}\text{/}$. The transaction rate is 25KTx/s. Each
point is averaged over 10 runs.
Byzantine senders. The attacker’s goal in this scenario is to overwhelm the
leader with many missing microblocks.
The strategies for each protocol are described as follows. In SMP-HS,
Byzantine replicas only send microblocks to the leader (Figure 2). In S-HS,
Byzantine replicas have to send microblocks to the leader and to at least $f$
replicas to get proofs. Otherwise, their microblocks will not be included in a
proposal (consider the leader is correct). In this experiment, we consider two
different quorum parameters for PAB (see Section VIII), $f+1$ and $2f+1$
(denoted by S-HS-f and S-HS-2f, respectively). These variants will explain the
tradeoff between throughput and latency. We ran this experiment in a LAN
setting with $N=100$ and $N=200$ replicas (including the leader). The number
of Byzantine replicas ranged from $0$ to $30$ ($N=100$) and $0$ to $60$
($N=200$).
(a) 100 total replicas with 0 to 30 Byz. ones.
(b) 200 total replicas with 0 to 60 Byz. ones.
Figure 9: Performance of SMP-HS and S-HS with different quorum parameters
(S-HS-d1 and S-HS-d2) and increasing Byzantine replicas.
Figure 9 plots the results. As the number of Byzantine replicas increases, the
throughput/latency of SMP-HS decreases/increases sharply. This is because
replicas have to fetch missing microblocks from the leader before processing a
proposal. We also observe a slight drop in throughput of S-HS. The reason is
that only background bandwidth is used to deal with missing microblocks. The
latency of S-HS remains flat since the consensus will never be blocked by
missing microblocks as long as the leader provides correct proofs. In
addition, we notice that Byzantine behavior has more impact on larger
deployments. With $N=200$ replicas, the performance of SMP-HS decreases
significantly. The throughput is almost zero when the number of Byzantine
replicas is $60$ and the latency surges when there are more than $20$
Byzantine replicas. Finally, S-HS-2f has better throughput than S-HS-f at the
cost of higher latency as the number of Byzantine replicas increases. The
reason is that with a larger quorum size, fewer microblocks need to be
fetched. However, a replica needs to wait for more acks to generate available
proofs.
### VII-D Impact of Unbalanced Workload
Previous work [37, 38, 39, 40] has observed that node degrees in large-scale
blockchains have a power-law distribution. As a result, most clients send
transactions to a few popular nodes, leading to unbalanced workload (Problem-
II in Section III-E). In this experiment, we vary the ratio of workload to
bandwidth by using identical bandwidth for each replica but skewed workloads
across replicas. We use two Zipfian parameters [63], Zipf1 ($s=1.01,v=1$) and
Zipf10 ($s=1.01,v=10$), to simulate a highly skewed workload and a lightly
skewed workload, respectively. We show the workload distributions in Figure
10. For example, when $s=1.01$ and there are 100 replicas, $10\%$ of the
replicas will receive over $85\%$ of the load.
Figure 10: Workload distribution with different network sizes and Zipfian
parameters.
We evaluate load-balancing in Stratus using the above distributions in a WAN
setting. Stratus samples $d$ replicas to select the least loaded one as the
proxy, we consider $d=1,2,3$, denoted by S-HS-d1, S-HS-d2, and S-HS-d3,
respectively. We also use SMP-HS-G, HotStuff with a gossip-based shared
mempool for comparison. We set the gossip fan-out parameter to $3$.
Figure 11: Throughput with different workload distribution.
Figure 11 shows protocols’ throughput. We can see that S-HS-dX outperforms
SMP-HS and SMP-HS-G in all experiments. S-HS-dX achieves $5\times$ ($N=100$)
to $10\times$ ($N=400$) throughput with Zipf1 as compared with SMP-HS. SMP-
HS-G does not scale well under a lightly skewed workload (Zipf10) due to the
message redundancy. We also observe that S-HS-dX achieves the best performance
when $d=3$, while the gap between different $d$ values is not significant.
## VIII Discussion
Attacks on PAB. Byzantine replicas can create availability proofs and send
them to fewer than $f$ replicas. If the leader is correct, then a valid
proposal is proposed with microblock ids and their availability proofs. Using
these, replicas can recover if a referenced microblock is missing. The
microblocks with missing proofs will be discarded after a timeout.
Now consider a Byzantine leader that includes microblocks without availability
proofs into a proposal. This will trigger a view-change, which will replace
the leader. In some PoS blockchains [12, 13], such leaders are also slashed.
Attacks on load balancing. A Byzantine sender can try to to congest the
network by sending identical microblocks to multiple proxies. To mitigate this
attack, we propose a simple solution. When a busy replica $r$ decides on
$r^{\prime}$ as the proxy, it forwards the microblock $mb$ to $r^{\prime}$
along with a message $\rho$ that contains $r$’s signature over $mb.id$
concatenated with $r^{\prime}$’s identity. Then, $r^{\prime}$ broadcast $mb$
along with $\rho$ using PAB. This allows other replicas to check if a
microblock by the same sender is broadcast by different proxies. Once
detected, a replica can reject microblocks from this sender or report this
behavior by sending evidence to the other replicas. If the proxy fails to
complete PAB, the original sender either broadcasts the microblock by itself
or waits for a timeout to garbage collect the microblock.
A malicious replica can pretend to be busy and forward its load to other
replicas. This can be addressed with an incentive mechanism: a replica that
produced the availability proof for a microblock using PAB is rewarded. This
information is verifiable because the availability proofs for each microblock
are in the proposal and will be recorded on the blockchain if the proposal is
committed. In addition, to prevent a malicious senders from overloading a
proxy, the proxy can set a limit on its buffer, and reject extra load.
Re-configuration. Stratus can be extended to support adding or removing
replicas. For example, Stratus can subscribe to re-configuration events from
the consensus engine. When new replicas join or leave, Stratus will update its
configuration. Newly joined replicas may then fetch stable microblocks (i.e.,
ids with available proofs) to catch up.
Garbage collection. To ensure that transactions remain available, replicas may
have to keep the microblocks and relevant meta-data (e.g., acks) in case other
replicas fetch them. To garbage-collect these messages, the consensus protocol
should inform Stratus that a proposal is committed and the contained
microblocks can then be garbage collected.
## IX Conclusion and Future Work
We presented a shared mempool abstraction that resolves the leader bottleneck
of leader-based BFT protocols. We designed Stratus, a novel shared mempool
protocol to address two challenges: missing transactions and unbalanced
workloads. Stratus overcomes these with an efficient provably available
broadcast (PAB) and a load balancing protocol. For example, Stratus-HotStuff
throughput is 5$\times$ to 20$\times$ higher than native HotStuff. In our
future work, we plan to extend Stratus to multi-leader BFT protocols.
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## Appendix A Analysis
In this section, we theoretically reveal the leader bottleneck of leader-based
BFT protocols (LBFT) and then show how shared mempool addresses the issue. We
consider the ideal performance, i.e., all replicas are honest and the network
is synchronous. We assume that the ideal performance is limited by the
available processing capacity of each replica, denoted by $C$. For simplicity,
we further assume that transactions have the same size $B$ (in bits). We use
$T_{max}$ to denote the maximum throughput, i.e., number of transactions per
second. We use $W_{l}$ (resp. $W_{nl}$) to denote the workload of the leader
(resp. a non-leader replica) for confirming a transaction. Furthermore, we
have
$T_{max}=\min\left\\{\frac{C}{W_{l}},\frac{C}{W_{nl}}\right\\}.$
Since each replica has to receive and process the transaction once, we have
$W_{l},W_{nl}\geq B$. Besides, due to the protocol overhead, we have
$W_{l},W_{nl}>B$. As a result, $T_{max}<C/B$. In other words, $C/B$ is the
upper bound of the maximum throughput of any BFT protocol.
### A-A Bottleneck of LBFT Protocols
In LBFT protocols, when making a consensus of a transaction, the leader is in
charge of disseminating it to other $n-1$ replicas, while each non-leader
replica proceeds it from the leader. Hence, the workloads of proceeding with
the transaction for the leader and a non-leader replica are $W_{l}=B(n-1)$ and
$W_{nl}=B$, respectively. Furthermore, we have
$T_{max}=\min\left\\{\frac{C}{B(n-1)},\frac{C}{B}\right\\}=\frac{C}{B(n-1)}.$
The equation shows that with the increase of replicas, the maximum throughput
of LBFT protocols will drop proportionally. Note that protocol overhead is not
considered, which makes it easier to illustrate the unbalanced loads between
the leader and non-leader replicas and to show the leader bottleneck.
Next, we take PBFT [21] as a concrete example to show more details of the
leader bottleneck. In PBFT the agreement of a transaction involves three
phases: the pre-prepare, prepare, and commit phases. In particular, the leader
first receives a transaction from a client and then disseminates the
transaction to all other $n-1$ replicas in the pre-prepare phase. In prepare
and commit phases, each replica broadcasts their vote messages and receives
all others’ vote messages for reaching consensus.888In the implementation, the
leader does not need to broadcast its votes in the prepare phase since the
proposed transaction could represent the vote message. Let $\sigma$ denote the
size of voting messages. The workloads for the leader and a non-leader replica
are $W_{l}=nB+4(n-1)\sigma$ and $W_{nl}=B+4(n-1)\sigma$, respectively.
Finally, we can derive the maximum throughput of PBFT as
$T_{max}=\min\left\\{\frac{C}{nB+4(n-1)\sigma},\frac{C}{B+4(n-1)\sigma}\right\\}.$
The equations show that both the dissemination of the transaction and vote
messages limit the throughput. Besides, we can see that when processing a
transaction, each replica has to process $4(n-1)$ vote messages, which leads
to high protocol overhead. To address this, multiple transactions can be batch
into a proposal (e.g., forming a block) to amortize the protocol overhead. For
example, let $K$ denote the size of a proposal, and the maximum throughput of
PBFT when adopting batch strategy is
$T_{max}=\frac{K}{B}\times\min\left\\{\frac{C}{nK+4(n-1)\sigma},\frac{C}{K+4(n-1)\sigma}\right\\}.$
When $K$ is large (i.e., $K\gg\sigma$), we have
$\frac{C}{nK+4(n-1)\sigma}\approx\frac{C}{nK}$ and $T_{max}=\frac{C}{nB}$.
This shows that the maximum throughput drops with the increasing number of
replicas, and the dissemination of the proposal by the leader is still the
bottleneck. In other words, batching strategy cannot address the scalability
issues of LBFT protocols. What is more, several state-of-the-art LBFT
protocols such as HotStuff [19] achieve the linear message complexity by
removing the $(n-1)$ factor from the $(n-1)\sigma$ overhead of non-leader
replicas. However, this also cannot address the scalability issue since the
proposal dissemination for the leader is still the dominating component.
### A-B Analysis of Using Shared Mempool
To address the leader bottleneck of LBFT protocols, our solution is to
decouple the transaction dissemination with a consensus algorithm, by which
dissemination workloads can be balanced among all replicas, leading to better
utilization of replicas’ processing capacities. In particular, to improve the
efficiency of dissemination, transactions can be batched into microblocks, and
replicas disseminate microblocks to each other. Each microblock is accompanied
by a unique identifier, which can be generated by the hash function. Later,
after a microblock is synchronized among replicas, the leader only needs to
propose an identifier of the microblock. Since the unique mapping between
identifiers and microblocks, ordered identifiers lead to a sequence of
microblocks, which further determines a sequence of transactions.
Next, we show how the above decoupling idea can address the leader bottleneck.
We use $\gamma$ to denote the size of an identifier and $\eta$ to denote the
size of a microblock. Given a proposal with the same size $K$, it can include
$K/\gamma$ identifiers. Each identifier represents a microblock with $\eta/B$
transactions. Hence, a proposal represents
$\frac{K}{\gamma}\times\frac{\eta}{B}$ transactions. As said previously, the
$K/\gamma$ microblocks are disseminated by all non-leader replicas, so each
non-leader replica has to disseminate $K/(\gamma(n-1))$ microblocks to all
other replicas. Correspondingly, each replica (including the leader) can
receive $K/(\gamma(n-1))$ microblocks from $n-1$ non-leader replicas. Hence,
the workload for the leader is
$W_{l}=(n-1)\frac{K\eta}{\gamma(n-1)}+(n-1)K=\frac{K\eta}{\gamma}+(n-1)K,$
where $(n-1)K$ is the workload for disseminating the proposal. Similarly, the
workload for a non-leader replica is
$W_{l}=n\frac{K\eta}{\gamma(n-1)}+(n-2)\frac{K\eta}{\gamma(n-1)}+K=\frac{2K\eta}{\gamma}+K,$
where $K$ is the workload for receiving a proposal from the leader. Finally,
we can derive the maximum throughput as
$T_{max}=\frac{K\eta}{\gamma
B}\times\min\left\\{\frac{C}{(K\eta)/\gamma+(n-1)K},\frac{C}{(2K\eta)/\gamma+K}\right\\}.$
To make the throughout maximum, we can adjust $\eta$ and $\gamma$ to balance
the workloads of the leader and non-leader replicas. This is
$\frac{2K\eta}{\gamma}+K=\frac{K\eta}{\gamma}+(n-1)K$, and we have
$\eta=(n-2)\gamma$. Finally, we can obtain the maximum throughput is
$T_{max}=\frac{C(n-2)}{B(2n-3)}$. Particularly, when $n$ is large, we have
$T_{max}\approx\frac{C}{2B}$. The result is optimal since given a transaction,
it has to be sent and received $n$ times (one for each replica), which leads
to about $2nB$ workload, and the total processing capacities of all replicas
is $nC$.
|
# Amazon Last-Mile Delivery Trajectory Prediction Using Hierarchical TSP with
Customized Cost Matrix
Xiaotong Guo, Baichuan Mo, Qingyi Wang 111In alphabetical order by the last
name Department of Civil and Environmental Engineering, Massachusetts
Institute of Technology 77 Massachusetts Ave, Cambridge, MA, USA
###### Abstract
In response to the Amazon Last-Mile Routing Challenge, Team Permission Denied
proposes a hierarchical Travelling Salesman Problem (TSP) optimization with a
customized cost matrix. The higher level TSP solves for the zone sequence
while the lower level TSP solves the intra-zonal stop sequence. The cost
matrix is modified to account for routing patterns beyond the shortest travel
time. Lastly, some post-processing is done to edit the sequence to match
commonly observed routing patterns, such as when travel times are similar,
drivers usually start with stops with more packages than those with fewer
packages. The model is tested on 1223 routes that are randomly selected out of
the training set and the score is $0.0381$. On the 13 routes in the given
model apply set, the score was $0.0375$.
## 1 Introduction
This report presents the thought processes, selected methodology, and expected
results of the Amazon Last-Mile Routing Research Challenge by Team Permission
Denied. In summary, the team went through four phases before arriving at the
final submission.
Descriptive Analysis: Upon receiving the challenge, a thorough descriptive
analysis is done. The first important finding is that, in most circumstances,
the drivers finish all deliveries in one zone before moving on to the stops in
another zone. This rule is only broken when backtracking exists. A further
look at the scores confirms this intuition: assuming the zone sequence and
intra-zonal stop sequence are correct, the loss on the score due to certain
zones being revisited is only 0.009. If the zone sequence is correct and the
stops in each zone are shuffled, the average score is around 0.02. Therefore,
getting the zone sequence correct is the most important, and the team decides
to adopt a hierarchical approach: solving for the zone sequence, and then the
intra-zonal stop sequence. This greatly reduces the scale of the problem since
the majority of the routes have around 150 stops (up to 250), but the number
of zones is between 6 and 47. Second, the zonal transitional probabilities are
investigated. As most of the zones only appear in the training set once, an
attempt at a frequency tabulation is not successful. On the other hand, 74% of
the zonal transitions select the zone that is closest by travel time, making
the step-by-step prediction algorithm potentially successful. Next, the
correlation between package dimensions, package counts, delivery time windows,
and sequence order is investigated but no apparent relationship is found.
Benchmarking: A benchmark model is created to establish an idea of the
solution quality and expected performance. Since most drivers follow the
navigation given by Amazon, a shortest-distance tour becomes a natural
benchmark. The team solves a tour-based (where the start and end stations are
both INIT) to generate zone sequences and a path-based (where the distance
from the last zone to INIT is not counted) Travelling Salesman Problem (TSP)
to generate intra-zonal stop sequences as benchmarks. Inside each zone, a
path-based TSP is generated from the stop closest to the last zone to the stop
closest to the next zone.
Model Attempts: Both naive TSP solutions achieve scores reasonable scores
(around 0.06). To improve the performance, machine learning models are
attempted. First, it is noticed that correctly predicting the first zone would
significantly improve the TSP performance, therefore a neural network is
constructed to predict the first zone based on the travel time, distance,
package count and size, etc. Second, pure machine learning models to generate
sequences are investigated, including myopic approaches that predict the next
element based on previously predicted stops, as well as sequence-to-sequence
(seq2seq) approaches that encode and decode the entire sequence. Third,
different training methods are considered, including the traditional cross-
entropy loss, customized weighted loss, as well as reinforcement learning
using policy gradients. Lastly, some improvements are made to the benchmark
TSP models by adding penalty costs to non-consecutive zone-ids. Due to the
small sample size (6k), machine learning techniques cannot outperform the
benchmark models. After experimenting with various modeling techniques, the
team decides to use the TSP solution as the final submission.
Hyperparameter Searching and Post-Processing: The customized cost matrix
involves hyperparameters that the team searched for over the given training
set. Lastly, some post-processing patterns are identified to further improve
the quality of our solution.
The highlights of the final submitted model are:
* 1.
Hierarchical modeling - To reduce the size of each optimization problem, the
problem is broken down into zone routing and intra-zonal stop routing.
* 2.
Customized TSP cost matrix - To account for considerations in addition to
shortest distance, the cost matrix is modified and the TSP performance
improved by almost 0.01.
* 3.
Post-processing to match behavioral patterns - Some TSP sequences are reversed
to accommodate delivery patterns such as stops with more packages are visited
first instead of last, all else being equal.
* 4.
Stable hyperparameters - The cost hyperparameters have good generalizability
and do not require re-training.
The rest of the technical report reviews the relevant literature and its
compatibility with the research question; describes the selected model in
detail, and discusses the expected results.
## 2 Literature Review
This problem is essentially a vehicle routing problem, except that the
traditional setup for vehicle routing problems aims for the shortest distance
traveled, but the problem of interest looks for the most similarity with the
observed sequence. Two research communities have extensively studied the
vehicle routing problem: machine learning and operations research. Literature
in both communities is reviewed, with the pros and cons of the algorithms
discussed for the problem of interest.
### 2.1 Operations Research
Given a set of locations one would like to visit, a Traveling Salesman Problem
(TSP) can be solved to find the route with the minimum cost or distance. The
overview and history of the TSP can be found in Applegate et al. [2011].
Although TSP is a well-known NP-hard problem in combinatorial optimization,
off-the-shelf integer optimization solvers (e.g., Gurobi and GLPK) are able to
solve it efficiently for real-world instances. One key approach we utilized
when solving the TSP is the cutting-plane method [Marchand et al., 2002],
which is initially applied to TSP by Dantzig et al. [1954].
### 2.2 Machine Learning
Two types of architectures can be used to re-order the input sequence: step-
by-step or sequence-to-sequence (seq2seq). Step-by-step prediction involves
predicting the stops one by one, given the information from previous stops, as
well as candidate stops. Since the information from candidate stops are
crucial, feed-forward neural networks are not a good candidate since it does
not attribute features to candidates. Instead, a feed-forward neural network
with alternative-specific utility is adopted [Wang et al., 2020]. This
architecture draws the connection between discrete choice models with neural
networks and uses neural networks to generate the utility for each candidate,
and the candidate with the highest ’utility’ is chosen. A sequence is then
formed by repeatedly feeding the selected stop into the algorithm to get the
next stop until the end of the sequence is reached. The advantage of this
algorithm is that it is at the stop level instead of the sequence level.
Therefore, the sample size, which is critical for the success of machine
learning algorithms, is significantly larger than the seq2seq models. The
disadvantage of this algorithm is that it is myopic and only sees the next
step candidates while making a selection.
In recent years, a lot of seq2seq prediction algorithms have been developed,
mainly for natural language processing (NLP) tasks. Compared to step-by-step
prediction, seq2seq models comprise an encoder and a decoder. All elements in
the sequence are encoded before decoding starts, therefore a global view is
attained. The architecture of encoder and decoder often involves variants of
the recurrent neural networks (ex. long-short term memory networks) [Sutskever
et al., 2014], or attention [Vaswani et al., 2017]. Most seq2seq problems are
considered with mapping one sequence to another, whereas the problem of
interest is concerned with re-ordering the input sequence. Pointer network is
proposed to solve this type of problem, where the decoder uses self-attention
to point to one of the input elements [Vinyals et al., 2015]. The authors used
a pointer network to solve TSP and achieved similar performance to TSP
solvers. One drawback of the original pointer network is that it is sensitive
to the order of inputs. The authors, therefore, added another encoding module
to eliminate this influence [Vinyals et al., 2016]. However, in our
experiments, this dependency can be leveraged by arranging the input set in a
meaningful sequence to improve performance. For example, ordering the input
stops according to the TSP sequence would accelerate model convergence and
improve the score. However, in the papers presented above, 1M training samples
were fed into the network. Given that the training set only contains 6000
routes, score improvements on TSP solutions are unsuccessful.
The original pointer network uses cross-entropy loss (supervised learning). In
this problem, the cross-entropy loss is very inefficient due to the way the
score is calculated, since the loss only considers the probability of the
correct position, and the loss for predicting all other positions is the same.
But the scoring function considers similarity in addition to correctness. The
scoring function is not differentiable and cannot be directly used as the loss
function and use gradient descent. An alternative training method is
reinforcement learning based on policy gradients [Ma et al., 2019, Bello et
al., 2019]. Using the well-known REINFORCE algorithm, we can directly optimize
the non-differentiable score function. Researchers have found that this method
has the same sample efficiency and better generalizability for TSP problems
compared to supervised learning [Joshi et al., 2019]. However, training with
reinforcement learning in this particular problem with the sample size and
given information also does not outperform TSP solutions.
### 2.3 Proposed Method
Our proposed method is built upon the traditional TSP with a customized
distance matrix that implicitly contains drivers’ routing behaviors for the
Amazon last-mile delivery. Compared to the existing TSP framework, which
minimizes the total vehicle travel distance, we modified the distance matrix
and generated optimal routes which minimized the total adjusted travel
distance.
## 3 Methodology
### 3.1 Data
We observe that most of the drivers tend to visit all stops in a zone before
going to the next zone. Hence, we divide the problem into two sub-problems.
The first is to identify the zone sequence, and the second is to recognize the
intra-zonal stop sequence.
The actual zone sequence is generated based on the order of each zone’s first
appearance. An example is shown in Figure 1. For stops without zone id (due to
missing data), we fill them with the zone ID of its (travel time-based)
nearest stop.
Three important properties are noticed while observing the zone sequences:
* 1.
Most likely, the driver would finish a “major zone” first, then move to the
next “major zone”. A major zone is defined as the zone ID before the dot. For
example, the major zone for “A-2.2A” is “A-2”. For example, in Figure 1, the
driver first finishes major zone “A-2”, then “A-1”, finally “P-13”.
* 2.
Within a specific major zone, two adjacent “inner zone” ids are most likely
have a “difference of one”. The “inner zone” is defined as the zone ID after
the dot. For example, the inner zone for “A-2.2A” is “2A”. The “difference of
one” is defined as follows. Given two inner zone IDs “XY” and “AB”, where X
and A are numbers and Y and B are characters, we have
$\displaystyle|X-A|+|\texttt{ord}(Y)-\texttt{ord}(B)|=1$ (1)
where $\texttt{ord}(\cdot)$ function returns an integer representing the
Unicode character. For example, “1A” and “1B” has a difference of one, so as
“1A” and “2A”. But “1A” and “2B” has a difference of two.
* 3.
When a driver finishes a “major zone” and move to another, the two adjacent
major zone IDs are most likely to have a “difference of one”. For example, in
Figure 1, the driver first finishes major zone “A-2”, then “A-1”. Those two
major zone IDs have a difference of one.
Figure 1: Example of zone sequence. “INIT” indicates the delivery station
To validate these three properties, we calculate the frequency that these
rules hold in the data set. For all adjacent zone ID pairs, 87.67% of them
have the same major zone ID (Property 1). For all adjacent zone ID pairs
within a specific major zone, 82.49% of them have a “difference of one”
(Property 2). For all adjacent zone ID pairs with major zone ID changes,
96.17% of these changes lead to a “difference of one” between two major zone
IDs (Property 3). These statistics support the three properties, which implies
that the zone ID includes a lot of information for the sequence estimation.
Another information we use is the planned service time and package volumes.
Details on how these are utilized are shown in Section 3.3.
We also collected outside data sources from OpenStreetMap. Specifically, we
extract the number of traffic signals and highway ramps around every stop.
Unfortunately, this does not help to improve our model, thus is dropped from
our final submission.
For the model’s validation, we randomly separate the 6,112 routes into a
training data set (4,889 routes) and a testing data set (1,223 routes), though
our proposed solution does not require a training process.
### 3.2 Travelling Salesman Problem Formulation
With the observation that drivers visit all stops within the same zone first
and then move to the next zone, we solve a standard TSP with a modified travel
time matrix to generate zone sequence first and then solve multiple path-TSP
to identify intra-zonal stop sequence.
First, we provide the formulation of the standard TSP solved for generating
zone sequences. For a route instance with $n$ zones, the set of zones is
indexed by $[n]=\\{1,...,n\\}$ and the initial station location is indicated
by index $0$. Let $V$ represent the set of all locations that need to be
visited including the initial station, i.e., $V=\\{0,1,...,n\\}$. $t_{ij}$
denotes the travel time between any two locations, i.e., $\forall i\neq j\in
V$. The travel time between any two zones is calculated as the average travel
time between all possible pairs of stops between two zones. The decision
variable for this problem is $x_{ij}\in\\{0,1\\},\;\forall i,j\in V$.
$x_{ij}=1$ indicates that the driver will visit to the location $j$ after
visiting $i$. Then, the TSP problem can be formulated as:
$\displaystyle\min\quad$
$\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}t_{ij}x_{ij}$ (2a) s.t.
$\displaystyle\sum_{i=0}^{n}x_{ij}=1\quad\forall j\in V$ (2b)
$\displaystyle\sum_{j=0}^{n}x_{ij}=1\quad\forall i\in V$ (2c)
$\displaystyle\sum_{i\in S}\sum_{j\notin S}x_{ij}\geq 1\quad\forall S\subset
V,S\neq\emptyset,V$ (2d) $\displaystyle\sum_{i\notin S}\sum_{j\in S}x_{ij}\geq
1\quad\forall S\subset V,S\neq\emptyset,V$ (2e) $\displaystyle
x_{ii}=0\quad\forall i\in V$ (2f) $\displaystyle
x_{ij}\in\\{0,1\\}\quad\forall i,j\in V$ (2g)
Where the objective (2a) minimizes the total travel time for the tour.
Constraints (2b) and (2c) make sure that each visited location has exactly one
predecessor and one successor in the optimal tour. Constraints (2d) and (2e)
are proposed to eliminate subtours in the optimal tour. Constraints (2f) avoid
self loops and constraints (2g) guarantee decision variables are binary.
The problem $(\ref{eq:Tour_TSP})$ is an Integer Linear Programming (ILP) with
exponential number of constraints due to constraints (2d) and (2e). To solve
this problem efficiently, we implemented both constraints (2d) and (2e) as
lazy constraints, indicating they are only added to the problem if subtours
are identified in the current optimal solution.
To account for the observations made in the zone sequence (Section 3.1), we
propose three heuristics to modify the travel time matrix, which is the input
for generating the optimal zone sequence.
1. 1.
For travel time from the initial station to a zone $i$, if the zone is not
within either i) $h$ closest zones from the initial station regarding travel
times or ii) $h$ closest zones from the initial station regarding Euclidean
distances, we modify the travel time to $t_{0i}*\alpha$, where $\alpha$ and
$h$ are both parameters for the first proposed heuristic approach.
2. 2.
For travel time between any two zones $i$ and $j$, if zone $i$ and zone $j$
are not from the same ”major zone”, we modify the travel time to
$t_{ij}*\beta$, where $\beta$ is the parameter for the second proposed
heuristic approach.
3. 3.
For travel time between any two zones $i$ and $j$, if they are from the
identical ”major zone” and the difference between their zone ID after the dot
does not equal to 1, we modify the travel time to $t_{ij}*\gamma$, where
$\gamma$ is the parameter for the third proposed heuristic approach.
In the final submitted algorithm, we used the grid search approach to finalize
values for all four heuristic parameters: $h=9$, $\alpha=1.04$, $\beta=3.8$,
$\gamma=2.5$.
Solving the problem (2) with the modified travel time matrix leads to the
optimal zone sequence222Without loss of generality, we can assume the sequence
starts from the initial station indexed by $0$. $S^{*}=(0,s_{1},...,s_{n})$,
where $s_{i}$ indicates the $i$-th zone visited in the optimal sequence after
departing from the initial station. Then we solve the intra-zonal stop
sequence using path-based TSP. Given a set of locations $V$ need to be visited
and the starting location $v_{o}$ and the ending location $v_{d}$, we can
formulate the path-TSP problem as follows:
$\displaystyle\min\quad$
$\displaystyle\sum_{i=0}^{n}\sum_{j=0}^{n}t_{ij}x_{ij}$ (3a) s.t.
$\displaystyle\sum_{i=0}^{n}x_{ij}=1\quad\forall j\in
V\setminus\\{v_{o},v_{d}\\}$ (3b)
$\displaystyle\sum_{j=0}^{n}x_{ij}=1\quad\forall i\in
V\setminus\\{v_{o},v_{d}\\}$ (3c) $\displaystyle\sum_{j\in
V}x_{v_{o}j}=\sum_{i\in V}x_{iv_{d}}=1$ (3d) $\displaystyle\sum_{j\in
V}x_{v_{d}j}=\sum_{i\in V}x_{iv_{o}}=0$ (3e) $\displaystyle\sum_{i\in
S}\sum_{j\notin S}x_{ij}\geq 1\quad\forall S\subset V,S\neq\emptyset,V$ (3f)
$\displaystyle\sum_{i\notin S}\sum_{j\in S}x_{ij}\geq 1\quad\forall S\subset
V,S\neq\emptyset,V$ (3g) $\displaystyle x_{ii}=0\quad\forall i\in V$ (3h)
$\displaystyle x_{ij}\in\\{0,1\\}\quad\forall i,j\in V$ (3i)
The path-TSP problem (3) is similar to the standard TSP problem (2) except
that there will be no predecessors for the starting location $v_{o}$ and no
successors for the ending location $v_{d}$, indicating by constraints (3d) and
(3e). The complete sequence is generated according to Algorithm 1 based on
generated zone sequence, where a heuristic parameter $k=3$ is utilized in the
final implementation.
Algorithm 1 Complete sequence generation based on the generated zone sequence.
Input: optimal zone sequence $S^{*}=(0,s_{1},...,s_{n})$, heuristic parameter
$k$.
1:function CompletePathGeneration($S^{*}$)
2: $S^{*}_{complete}\leftarrow\\{0\\}$ $\triangleright$ Initialize the
complete sequence with the initial station
3: for $s_{i}=s_{1},...,s_{n}$ do
4: Find the previous visited zone $s_{i-1}$ and the next visited zone
$s_{i+1}$
5: Calculate the average travel time between any stop $v\in s_{i}$ to all
stops in zone $s_{i-1}$ and zone $s_{i+1}$
6: Find $k$ nearest stops in zone $s_{i}$ regarding to zone $s_{i-1}$ as the
set $M$
7: Find $k$ nearest stops in zone $s_{i}$ regarding to zone $s_{i+1}$ as the
set $N$
8: Solve $k^{2}$ path-TSP (3) between any pair of stops in $M\times N$.
9: Let the path $S^{*}_{i}$ with the minimum travel time as the optimal
sequence of zone $i$
10: Append the sequence $S^{*}_{i}$ to the complete sequence
$S^{*}_{complete}$
11: return $S^{*}_{complete}$
It is worth mentioning that all TSP instances are solved with the open-source
ILP solver GLPK implemented with programming language Julia [Bezanson et al.,
2017] and optimization package JuMP [Dunning et al., 2017]. After generating
the complete stop sequence $S^{*}_{complete}$, we enter the post-processing
stage to further improve sequence performances.
### 3.3 Post-Processing
After solving the stop sequence by TSP, we observe that most of the high-score
(i.e., low performance) routes are due to partially or fully reverse of the
sequence (i.e., a sequence A-B-C-D is erroneously estimated as D-C-B-A).
Hence, we propose a post-processing method to correct the erroneous estimation
due to reversal.
We observe two properties from the data set:
* 1.
Most of the drivers tend to serve the business areas first. The potential
reason may be that it also takes a longer time to deliver packages in a
business building. Serving them first can make the total service time more
controllable at the end of the journey. Hence, we expect that the planned
service time at the first several stops is larger than that of the last
several stops.
* 2.
Most of the drivers tend to deliver large-size packages first. This may be
because carrying large-size packages in the vehicle is not fuel-efficient.
Based on these properties, for every generated stop sequence by TSP, we check
whether we need to reverse it. Given a generated route $i$, let $p^{+}_{i}$
(resp. $p^{-}_{i}$) be the average planned service time of the first (resp.
last) $p\%$ stops in route $i$. We will reverse route $i$ if
$\displaystyle\frac{p^{-}_{i}}{p^{+}_{i}}\geq\theta,$ (4)
where $p$ and $\theta$ are hyperparameters representing the proportion of
stops and a threshold. We set $p=15$ and $\theta=1.22$ based on cross-
validation on the test set. Eq. 5 means that in a generated sequence if the
planned service time for the last several stops is too large, we may have the
reversal error and need to correct it by reverse the whole sequence.
After process by Eq. 5, we fixed all sequences that are already reversed. For
the remaining sequences, we further check whether they need to be reversed
based on package volumes. Specifically, given a generated route $i$, let
$v^{+}_{i}$ (resp. $v^{-}_{i}$) be the total package columns
(depth$\times$width$\times$height) of the first (resp. last) 15% stops in
route $i$. We will reverse route $i$ if
$\displaystyle\frac{v^{-}_{i}}{v^{+}_{i}}\geq\eta,$ (5)
where $\eta=3$ is used.
After post-processing, a sequence validity check is performed. Specifically,
we check whether the first stop of the estimated sequence is the delivery
station, and whether the estimated sequence has the same stop IDs as the
actual one. If either of these two criteria does not hold, we return a
sequence by simply sort the stops using zone IDs, which ensures that stops
with the same zone IDs are close to each other.
## 4 Results and Conclusions
### 4.1 Performance
Although the submitted formulation does not require model training, we have
separated the given training set into the training (4889) and test set (1223)
for self-evaluation of the machine learning models. Therefore, all self-
evaluation is done over the test set. To reduce the evaluation time, we
implemented the scoring function using Cython. Compared to the evaluation code
in Python provided by the challenge host team, our implementation evaluates
the same set of routes by using only one-third of the computation time.
Figure 2 shows the score distribution generated by our final algorithm. The
route performance score follows an exponential distribution and most routes
have a score below 0.1. The average route score is $0.0381$ for these 1223
testing routes. On the 13 routes in the given model apply set, the score was
$0.0375$.
Figure 2: Route score performances.
### 4.2 Discussion
Zone sequence dominates score. We observe that, if the zone sequence is
perfectly predicted, even if the stop IDs within a zone are shuffled, the
average route score can reach $0.0206$. Hence, most of our jobs focus on
predicting the zone sequence, instead of the stop sequence.
The three properties of zone IDs (see Section 3.1) may imply that drivers most
likely follow the planned route and seldom deviate. As the zone ID is used to
“help simplify the route planning process” (quoted from Slack Q&A), we believe
that Amazon plans the route in a way that the zone IDs exhibit clear patterns.
So the major challenge of this problem is to recover how Amazon plans the
routes. This explains why TSP works better than machine learning methods under
the given current information and sample size.
The reversal problem remains. Figure 3 shows an example of reverse prediction.
Since we are not able to increase the first-zone prediction accuracy beyond
35%, after post-processing, the reverse issues still exist. The post-
processing reduces our score on our test set from 0.391 to 0.381. However, if
we can have a 100% correction rate for the reversal problems (i.e., always use
the one with a smaller score), the score can reduce to 0.280, indicating that
some further correction methods are needed. Note that we have tried to use the
number of surrounding highway ramps as a new indicator, as well as using
machine learning to predict the first zone, but it does not increase the model
performance.
(a) Actual route
(b) Predicted route by TSP
Figure 3: Examples of reverse prediction
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|
11institutetext: ******************11institutetext: Department of Robotics and
Mechatronics Engineering, DGIST, Korea 22institutetext: Department of
Pathology, Asan Medical Center
33institutetext: Department of Pathology, University of Ulsan College of
Medicine
33email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Feature Re-calibration based Multiple Instance Learning for Whole Slide
Image Classification
Philip Chikontwe 11 Soo Jeong Nam 22 Heounjeong Go 2233 Meejeong Kim 22 Hyun
Jung Sung 22 Sang Hyun Park 11
###### Abstract
Whole slide image (WSI) classification is a fundamental task for the diagnosis
and treatment of diseases; but, curation of accurate labels is time-consuming
and limits the application of fully-supervised methods. To address this,
multiple instance learning (MIL) is a popular method that poses classification
as a weakly supervised learning task with slide-level labels only. While
current MIL methods apply variants of the attention mechanism to re-weight
instance features with stronger models, scant attention is paid to the
properties of the data distribution. In this work, we propose to re-calibrate
the distribution of a WSI bag (instances) by using the statistics of the max-
instance (critical) feature. We assume that in binary MIL, positive bags have
larger feature magnitudes than negatives, thus we can enforce the model to
maximize the discrepancy between bags with a metric feature loss that models
positive bags as out-of-distribution. To achieve this, unlike existing MIL
methods that use single-batch training modes, we propose balanced-batch
sampling to effectively use the feature loss i.e., (+/-) bags simultaneously.
Further, we employ a position encoding module (PEM) to model
spatial/morphological information, and perform pooling by multi-head self-
attention (PSMA) with a Transformer encoder. Experimental results on existing
benchmark datasets show our approach is effective and improves over state-of-
the-art MIL methods. https://github.com/PhilipChicco/FRMIL
## 1 Introduction
Histopathology image analysis (HIA) is an important task in modern medicine
and is the gold standard for cancer detection and treatment planning[17]. The
development of whole slide image (WSI) scanners has enabled the digitization
of tissue biopsies into gigapixel images and paved the way for the application
of machine learning techniques in the field of digital pathology[3, 22].
However, employing popular convolutional neural network (CNN) architectures
for varied tasks in HIA is non trivial and has several challenges, ranging
from the large size of WSIs and extreme high resolution to lack of precise
labeling and stain color variations[22]. This motivates the need for memory
efficient methods that mitigate the need for fine-grained labels and are
fairly interpretable. To address this, multiple instance learning (MIL)[31, 1]
is a popular formulation that considers diagnosis as a weakly supervised
learning problem[29].
Through the recent advances in deep learning[30, 16], MIL based
histopathology[8, 13, 27, 25, 10] analysis has achieved notable success[11,
24, 28, 18]. For instance, Li et al.[21] introduced non-local attention to re-
weight instances relative the highest scoring instance (critical) in a bag,
proving to be a simple yet effective approach. However, the critical instance
is only employed for implicit instance re-weighting and the method is
sensitive to both the choice of the instance feature encoder (i.e., pre-
trained ImageNet or self-supervised), and the scale of patches used. In MIL-
RNN[6], recurrent neural networks (RNN) are used to sequentially process
instance features, partially encoding position and context, but is limited in
the ability to capture long range dependences.
Thus, follow-up works[24, 21, 26, 23] built on the latter with more complex
attention-based variants using Transformer[30, 12] inspired architectures to
better model long range instance correlations via multi-head self-attention
(MSA) with positional information encoding. Along this line of thought,
TransMIL[26] highlights the importance of spatial positional encoding (PE) and
single-scale learning over the latter, but is relatively sensitive to the
depth of PE layers (i.e., x3) and does not explicitly pool all instances to a
single bag representation, instead uses a learnable class token for final bag-
level prediction. Thus, the use of Transformers with several MSA blocks can be
computationally prohibitive, and would be more desirable to have less over-
parameterized designs.
To address these challenges, we propose a Feature Re-calibration based MIL
framework (FRMIL), building upon prior MIL approaches[26, 21, 23] leveraging
MSA with Transformer encoders. Here, we argue that re-calibrating the
distribution of instance features can improve model performance towards better
generalization by using the properties of the data distribution directly. In
vision tasks such as few-shot learning, feature/distribution re-calibration is
used to enable better generalization when learning from limited samples by
transferring statistics from classes with sufficient examples[32]. However, in
the MIL scenario, instances are not always i.i.d [26], especially since
positive instances are often limited in a WSI i.e., ($\leq 10\%$). Thus, we
consider a simpler form that uses the max instance to shift the original
distribution towards better separability. Also, we consider MIL and anomaly
detection[7, 14, 20] as closely related tasks. For instance, Lee et al.[20]
leveraged MIL for weakly-supervised action localization by modeling
background/normal actions as out-of-distribution using uncertainty by
considering their inconsistency in a sequence with video-level labels only.
| |
---|---|---
$44.96\%$ | $83.72\%$ ($+39$) | $89.10\%$ ($+44$)
(a) | (b) | (c)
Figure 1: Normalized density plots of the mean feature magnitudes on the
CAMELYON16[4] train-set, with test-set accuracy and improvements (red color).
(a) Original feature magnitudes. (b) Max-instance calibrated based features.
(c) Features learned by our FR-MIL model.
Inspired by these works, we hypothesize that features from positive and
negative bags (binary MIL) exhibit larger and smaller feature magnitudes
respectively, and this prior can be directly encoded into the learning
framework for better representation learning[20]. In Fig. 1, we show this
phenomena and our intuition to highlight how the standard MIL assumption of
having at-least one (+) instance in a bag can be used to make the distribution
more separable. Herein, we establish a simple non-parametric baseline that re-
calibrates features by subtracting the max instance per-bag, and then computes
the probability of a bag-label as the normalized minimum between the mean
magnitude and the estimated bag magnitude (see Sec. 2). Our evaluation shows
that the baseline performance is comparable to classic MIL operators (i.e.,
max/mean-pooling)[31].
To incorporate this idea in our framework, we explicitly re-calibrate features
with the aforementioned concept, and then feed the new features to a
positional encoding module (PEM)[26] followed by a single pooling multi-head
self-attention block (PMSA)[19] for bag classification. To effectively enforce
feature magnitude discrepancy, we propose a feature embedding loss that
maximizes the distance between positive and negative bag features, as well as
the standard cross-entropy losses. The main contributions of this work are as
follows: (i) We show that feature re-calibration using the max-critical
instance embedding is a simple yet powerful technique for MIL, (ii) We
introduce a feature magnitude loss to learn better instance/bag separation,
(iii) To obtain robust bag embeddings, we leverage a positional encoder and a
single self-attention block for instance aggregation, and (iv) Experimental
results on a public benchmark and inhouse-curated datasets demonstrate the
effectiveness of our method over state-of-the-art methods.
## 2 Methods
Overview. In this work, we consider a set of WSIs $\mathbf{X}=\\{X_{i}\\}$,
each associated with a slide-level label $Y_{i}=\\{0,1\\}$, and our goal
predict the slide labels using MIL (see. Fig. 2). We first extract instance
features $\mathbf{H}\in\mathbb{R}^{D}$ using a neural network
$\bf{F}_{\theta}$ i.e., $\bf{H}_{i}=\bf{F}_{\theta}(X_{i})$, where $\bf{F}$ is
either pre-trained on ImageNet or self-supervised learning[9, 15]. In FR-MIL,
we feed $\mathbf{H}$ to our max-instance selection module to obtain the
highest instance (critical) as well as it’s probability, then we re-calibrate
the features $\mathbf{H}$ with the max-instance to obtain $\mathbf{\hat{H}}$.
The position encoding module (PEM) creates a spatial representation of
$\mathbf{\hat{H}}$, applies a single group convolution $\mathbf{G}_{\theta}$
to obtain correlated features, and then concatenates with a learnable class
token $\mathbf{C}$. Finally, we perform MIL Pooling by Multi-head Self-
Attention (PSMA) using the max-instance as a query and the output of
$\mathbf{G}_{\theta}$ as key-value pairs to obtain the bag feature. FR-MIL is
trained to minimize the bag loss, max-instance loss, and feature magnitude
loss between positive and negative instance features. We detail each step
below.
Figure 2: Overview of the proposed FR-MIL framework.
Preliminaries: A Simple Baseline. Inspired by the work of Lee et al.[20] that
employ feature magnitudes for uncertainity estimation of background actions in
video sequences, we hypothesize that normal and positive bags should have
different magnitudes and can serve as a simple MIL baseline. Herein, given
instance features $\mathbf{H}^{c}_{i}=\\{h_{1},h_{2},\dots,h_{n}\\}$, where
$c$ denotes the WSI class; the mean feature magnitude per WSI can be obtained
as $\mu^{c}_{i}=\frac{1}{N}\sum||\mathbf{H}^{c}_{i}||^{2}_{2}$, where $N$
denotes the number of instances in a bag. To obtain the probability
$\mathbf{P}(y=1.)$ of a bag, we formalize our assumption as:
$\mathbf{P}(y=1.|\mu^{c}_{i})=\frac{\text{min}(\tau,\mu^{c}_{i})}{\tau},$ (1)
where $\tau$ is the pre-defined maximum feature magnitude determined on the
train-set only i.e., point at which the distributions first meet (see Fig.1).
Also, Eq. 1 is ensured to fall between 0 and 1, i.e.,
$0\leq\mathbf{P}(y=1.|\cdot)\leq 1$. In Fig. 1, we show the magnitudes on a
given train-set (Camelyon16[4]) as a density plot. Note that while both normal
and tumor slide curves appear to follow the Gaussian distribution (Fig. 1(a)),
separation is non-trivial due to the presence of more normal than tumor
instances ($\leq 10\%$), and reports a low accuracy ($44\%$) with $\tau=18.8$.
In Fig. 1(b), we show that re-calibrating the distribution by subtracting the
max-instance feature before computing the magnitudes creates better
separability. Formally,
$\hat{\mu}^{c}_{i}=\frac{1}{N}\sum||\mathbf{\hat{H}}^{c}_{i}||^{2}_{2}$, where
$\mathbf{\hat{H}}^{c}_{i}=\\{\mathbf{\hat{H}}-h^{c}_{\text{max}}\\}$ given
$h^{c}_{\text{max}}=\text{argmax}_{c}~{}\mathbf{\hat{H}}^{c}_{i}$. Notably,
re-calibration improves the test-accuracy by $+39$ with $\tau=8.2$. Finally,
Fig. 1(c) shows the learned distribution of FR-MIL when trained with a feature
magnitude loss $\mathcal{L}_{fm}$ and re-calibration, with more significant
improvements ($+44$), further validating our hypothesis.
Feature Re-calibration & Max-Instance Selection. Given the set of instance
features $\mathbf{H}$, our goal is to select the max-instance $h^{q}$ and it’s
associated score $A^{c}$ using an instance classifier
$\mathbf{f}^{m}_{\theta}$ in our Max-Pooling module (see. Fig. 2). Here,
$A^{c}=\rho(\mathbf{f}^{m}_{\theta}(\mathbf{H}))$ where $\rho$ denotes the
sigmoid function. Consequently, the sorted scores are used to index the max-
instance in $\mathbf{H}$ via an operator $g(\cdot)$, with $h^{q}$ later
employed for feature re-calibration, as well as instance feature aggregation
via PSMA for bag prediction. The max score $A^{c}$ is used to train the
instance classifier $\mathbf{f}^{m}_{\theta}$ using the loss
$\mathcal{L}_{max}$, in parallel with other modules in FR-MIL. Formally, re-
calibration of features can be modeled as
$\mathbf{\hat{H}}=\text{ReLU}(\mathbf{\hat{H}}-h^{q}),$ (2)
similar to the intuition highlighted by the simple baseline. To further
incorporate the concept of distribution re-calibration in our framework, we
draw connections to prior work[20] for anomaly detection i.e., assumes the
feature magnitudes of positive- and normal-bags are different, and can be
modeled via uncertainty. Therefore, to effectively model the ambiguous
normal/background features, the training procedure should employ both positive
and negative bags simultaneously instead of selecting normal features within a
single bag (single batch). Herein, counter to existing methods that use a
single bag for training, we employ a sampling strategy to produce balanced
bags per epoch i.e., we initialize a zero-tensor with the maximum bag size in
during training, and fill the relevant bag instance features. Note that by
‘balanced’, we imply 1-negative and 1-positive bag is sampled. Formally, to
enforce feature discrepancy we propose feature magnitude loss
$\mathcal{L}_{fm}$ as:
$\mathcal{L}_{fm}(\mathbf{\hat{H}}^{pos}_{i},\mathbf{\hat{H}}^{neg}_{i},\tau)=\frac{1}{N}\sum^{N}_{n=1}(\text{max}(0,\tau-||\mathbf{\hat{H}}^{pos}_{i}||)+||\mathbf{\hat{H}}^{neg}_{i}||),$
(3)
where $\mathbf{\hat{H}}^{pos}$, and $\mathbf{\hat{H}}^{neg}$ are the positive-
and negative bag instance features, and $\tau$ is the pre-defined margin,
respectively. While prior work[21] equally used max-pooling to select the max-
instance, note that non-local masked attention was proposed for bag feature
learning, whereas we use PSMA and propose feature re-calibration.
Positional Encoding Module (PEM). In the standard transformer[30, 12] design,
encoding spatial information has proved useful for recognition tasks. However,
it is non-trivial for WSIs due to varying sizes. In this work, we employ a
conditional position encoder (PEM)[23] that takes re-calibrated features
$\mathbf{\hat{H}}$, performs zero-padding to provide absolute position for a
convolution $\mathbf{G}_{\theta}$, and later concatenates the output with a
class token $\mathbf{C}$ initialized from the normal distribution. Here,
features $\mathbf{\hat{H}}$ are re-shaped into a 2D image by first computing
$\\{H,W\\}$ i.e., $H=\sqrt{N}=\sqrt{n}$, where $n$ is the number of instances
in a bag. Thus,
$\mathbf{\hat{H}}\in\mathbb{R}^{D}\rightarrow\mathbf{\hat{H}}\in\mathbb{R}^{B\times
C\times H\times W},$ (4)
where $B$ is the batch-size, $C=D$ are the instance feature dimensions, and
$\mathbf{G}_{\theta}$ is 2D convolution layer that performs group convolution
with kernel size $3\times 3$, and $1\times 1$ zero padding. Note that prior
work[23] used different sized convolutions in a pyramidal fashion. Instead, we
opt for a single layer to maintain computational feasibility. Finally, let
$\mathbf{\acute{H}}=\text{concat}(\mathbf{C}_{\theta},\mathbf{G}_{\theta}(\mathbf{\hat{H}}))$,
where $\mathbf{\acute{H}}\in\mathbb{R}^{(N+1)\times D}$ are the flattened
restored features i.e., in the case of a single bag.
MIL Pooling by Multi-head Self-Attention (PMSA). In order to pool instance
features $\mathbf{\acute{H}}$ to a single bag feature, we employ a single
multi-head Transformer encoder[19] that takes as input the max-instance
feature $h^{q}$ as a query and $\mathbf{\acute{H}}$ as key-value pairs i.e.,
$\text{PMSA}_{\theta}(h^{q},\mathbf{\acute{H}})$. The formulation proposed by
Lee et al.[19] for set-based tasks employs an attention function
$\varphi(\mathbf{Q},\mathbf{K},\mathbf{V})$ to measure similarity between a
query vector $\mathbf{Q}$ with key-value pairs
$\mathbf{K,V}\in\mathbb{R}^{d\times m}$ as:
$\varphi(\mathbf{Q},\mathbf{K},\mathbf{V})=\text{softmax}(\frac{\mathbf{Q}\mathbf{K}^{T}}{\sqrt{m}})\mathbf{V}$,
where $\\{d,m\\}$ is the instance feature dimension. This can be easily
extended to multi-head attention by first projecting vectors onto $k$
different dimensions. The encoder consists of feed-forward networks
$\\{\mathbf{f}^{q}_{\theta},\mathbf{f}^{k}_{\theta},\mathbf{f}^{v}_{\theta}\\}$,
where $\mathbf{f}^{o}_{\theta}$ is fed the output of $\varphi$ (prototype)
together with residual connections and optional Layer Normalization[2] (LN).
Formally, let
$\hat{\varphi}=\varphi(\mathbf{Q},\mathbf{K},\mathbf{V})+\mathbf{Q}$, then:
$z=\text{PSMA}(h^{q},\mathbf{\acute{H}},\mathbf{\acute{H}})=\text{LN}(\hat{\varphi}+\text{ReLU}(\mathbf{f}^{o}_{\theta}(\hat{\varphi}))),$
(5)
to produce a bag feature $z$, later fed to the bag classifier
$\mathbf{f}^{c}_{\theta}$ for WSI classification. Finally, FR-MIL is trained
to minimize the bag-, max-pooling and feature losses. Thus, the final
objective is:
$\mathcal{L}=\gamma_{1}\mathcal{L}_{bag}(\hat{y},y)+\gamma_{2}\mathcal{L}_{max}(A^{c},y)+\gamma_{3}L_{fm}(\mathbf{\hat{H}}^{pos}_{i},\mathbf{\hat{H}}^{neg}_{i},\tau),$
(6)
where $\\{\gamma_{i}\\}$ are balancing weights and
$\mathcal{L}_{\\{bag,max\\}}$ is the binary cross-entropy loss over the true
WSI labels $y$ given $\hat{y}=\mathbf{f}^{c}_{\theta}(z)$, respectively.
## 3 Experiments
Datasets. To demonstrate the effectiveness of FR-MIL, we conducted experiments
on the publicly available dataset CAMELYON16[4], and an in-house curated
dataset termed COLON-MSI i.e., colorectal (adenocarcinoma) cancer slides
involving microsatellite instable (MSI) molecular phenotypes[5]. CAMELYON16
dataset was proposed for metastatis detection in breast cancer, it consists of
271 training sets and 129 testing sets. After pre-processing, a total of 3.2
million patches at $\times$20 magnification, with an average of 8,800 patches
per bag, and a maximum of 30,000 patches per bag on the training set. On the
other hand, COLON-MSI consists of both microsatellite-stable (MSS) and
microsatellite-instable (MSI), and is thus a subtyping task. It consists of
625 images, split as follows: 360 training, 92 validation, and 173 testing
sets. Experts pathologists detected the presence of tumors with
Immunohistochemical analysis (IHC) and PCR-based amplification and
collectively agreed on the final slide-level label. Note that tumor ROIs are
not used in this work. After pre-proessing, a total of 3.5 million patches at
$\times$20 magnification, an average of 6,000 patches/bag, and a maximum of
8900 patches in the train-set.
Implementation Settings. In the pre-processing step, we extracted valid
patches of $256\times 256$ after tissue detection and discard patches with
$\leq 15\%$ tissue entropy. For the instance encoder $\mathbf{F}_{\theta}$, we
employed the SimCLR[9] ResNet18[16] encoder trained by Lee et al.[21] for the
CAMELYON16 dataset. On the COLON-MSI set, we used an ImageNet pre-trained
ResNet18. Thus, each instance feature is represented as
$\mathbf{H}_{i}\in\mathbb{R}^{n\times 512}$. FR-MIL is trained with balanced
batch sampling (B = 2), and learning rate of $1e-4$ with Adam optimizer for
100 epochs with $20\%$ dropout as regularization, and PSMA has heads $k=8$.
Hyper-parameters $\\{\gamma_{1,2,3}\\}=0.33$, with $\tau=8.48$ for CAMELYON16,
and $\tau=57.5$ on COLON-MSI, respectively.
Comparison Methods. We compare FR-MIL to traditional MIL methods max- and
mean-pooling[31], as well as existing state-of-the-art methods: ABMIL[18],
DSMIL[21], CLAM-SB[24], MIL-RNN[6], and TransMIL[26]. All compared methods are
trained for 200 epochs on COLON-MSI with similar settings.
Table 1: Evaluation of the proposed method on CAMELYON16 (CM16) and COLON-MSI sets. Metrics accuracy (ACC) and area under the curve (AUC) were employed. $\dagger$ :_denotes scores reported in the paper using ResNet50 as $\mathbf{F}_{\theta}$_ with ImageNet features. | CM16 | | COLON-MSI
---|---|---|---
Method | | ACC | | AUC | | | ACC | | AUC
Mean-pooling[31] | | 0.7984 | | 0.7620 | | | 0.624 | | 0.830
Max-pooling[31] | | 0.8295 | | 0.8641 | | | 0.763 | | 0.859
ABMIL[18] | | 0.8450 | | 0.8653 | | | 0.740 | | 0.779
MIL-RNN[6] | | 0.8062 | | 0.8064 | | | 0.630 | | 0.631
CLAM-SB[24] | | 0.845 | | 0.894 | | | 0.786 | | 0.820
DSMIL[21] | | 0.8682 | | 0.8944 | | | 0.734 | | 0.811
TransMIL[23] | | 0.791 | | 0.813 | | | 0.676 | | 0.617
TransMIL$\dagger$[23] | | 0.8837 | | 0.9309 | | | - | | -
FR-MIL (w/ $\mathcal{L}_{bag}$) | | 0.8600 | | 0.8990 | | | 0.809 | | 0.880
FR-MIL (w/ $\mathcal{L}_{bag}+\mathcal{L}_{fm}$) | | 0.8760 | | 0.8990 | | | 0.775 | | 0.842
FR-MIL (w/ $\mathcal{L}_{bag}+\mathcal{L}_{max}$) | | 0.8840 | | 0.8940 | | | 0.780 | | 0.831
FR-MIL (w/ $\mathcal{L}_{bag}+\mathcal{L}_{max}+\mathcal{L}_{fm}$) | | 0.8910 | | 0.8950 | | | 0.809 | | 0.901
## 4 Results and Discussion
Main Results. Table 1 presents the results of our approach against recent
methods. On the CAMELYON16 dataset, FR-MIL reports $+3\%$ (ACC) improvement
over DSMIL with comparable performance ($+1\%$) to the reported TransMIL
scores using a larger feature extractor. Given that only a small portion of a
each positive bag contains tumors, using the max-instance for bag pooling with
re-calibration is intuitively sound and shows better performance over other
methods. Moreover, though we employ PEM similar to TransMIL, the use of a
single PEM module on calibrated facilitates better correlation learning.
On the other hand, since the majority of slides contain relatively large tumor
regions (averagely $\geq 75\%$) in COLON-MSI, max- and mean-pooling show high
AUC but had inconsistent results (ACC). Overall, CLAM-SB reports the best ACC
among the compared methods i.e., $78.6\%$. Interestingly, TransMIL performed
poorly on this set, possibly due to over-parametrization, and the
morphological similarities between MSS and MSI instances. Similar observations
were drawn regarding MIL-RNN. Consequently, the proposed FR-MIL highlights the
importance of calibration in subtyping problems, reporting $+2\%$ (ACC) and
$+5\%$ (AUC) improvements achieving the best scores. See Fig. 3 for the
distribution of features.
Ablations. To validate the effectiveness of the proposed losses on learning,
we evaluated FR-MIL with/without certain losses (see. Table 1). First, on
CAMELYON16, we found $\mathcal{L}_{fm}$ was a crucial component to further
boost the AUC score. Overall, when both $\mathcal{L}_{max}$ and
$\mathcal{L}_{fm}$ were omitted, performance drops were noted. On the other
hand, on COLON-MSI, using a $\mathcal{L}_{bag}$ only had the best scores,
whereas the use of both $\mathcal{L}_{max}$ and $\mathcal{L}_{fm}$ resulted in
a significant reduction (AUC). However, employing all the losses resulted in
more stable scores.
| |
---|---|---
(a) | (b) | (c)
Figure 3: Density plots of the mean feature magnitudes on the COLON-MSI train-
set. (a) Original feature magnitudes. (b) Max-instance re-calibration based
features. (c) Features learned by our FR-MIL model.
## 5 Conclusion
In this work, we presented a MIL framework for Whole Slide Image
classification that leverages Feature Re-calibration, applicable to both
binary and sub-typing tasks. We show that: (i) by leveraging feature magnitude
discrepancy between positive and negative bags as a probabilistic measure; a
simple baseline is comparable in performance to classic MIL operators, (ii)
explicitly re-calibrating the data distribution with max-instances during
training by drawing connections to the standard MIL assumption is simple yet
effective, and (iii) the use of a metric feature loss to encourage better
feature separation in (+/-) bags improves both Accuracy and AUC over state-of-
the-art methods. Further exploring the utility of this approach in multi-scale
setting, or designing an adaptable margin (mean magnitude) estimator will be
topics of future research.
Acknowledgments This work was supported by the DGIST R&D program of the
Ministry of Science and ICT of KOREA (21-DPIC-08), Smart HealthCare Program
funded by the Korean National Police Agency (220222M01), and IITP grant funded
by the Korean government (MSIT) (No.2021-0-02068, Artificial Intelligence
Innovation Hub).
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# GoldFinch: High Performance RWKV/Transformer Hybrid with Linear Pre-Fill and
Extreme KV-Cache Compression
Daniel Goldstein EleutherAI Recursal AI Fares Obeid EleutherAI Eric Alcaide
EleutherAI Dalle Molle Institute for Artificial Intelligence USI-SUPSI
Guangyu Song EleutherAI Tano Labs Eugene Cheah EleutherAI Recursal AI
###### Abstract
We introduce GoldFinch, a hybrid Linear Attention/Transformer sequence model
that uses a new technique to efficiently generate a highly compressed and
reusable KV-Cache in linear time and space with respect to sequence length.
GoldFinch stacks our new GOLD transformer on top of an enhanced version of the
Finch (RWKV-6) architecture. We train up to 1.5B parameter class models of the
Finch, Llama, and GoldFinch architectures, and find dramatically improved
modeling performance relative to both Finch and Llama. Our cache size savings
increase linearly with model layer count, ranging from 756-2550 times smaller
than the traditional transformer cache for common sizes, enabling inference of
extremely large context lengths even on limited hardware. Although
autoregressive generation has O(n) time complexity per token because of
attention, pre-fill computation of the entire initial cache state for a
submitted context costs only O(1) time per token due to the use of a recurrent
neural network (RNN) to generate this cache. We release our trained weights
and training code under the Apache 2.0 license for community use. 111Code at:
https://github.com/recursal/GoldFinch-paper
Model weights at: https://huggingface.co/recursal/GoldFinch-paper
## 1 Introduction
Variations on linear attention (Katharopoulos et al., 2020) have proliferated
in recent research (Peng et al., 2024; Qin et al., 2024; Katsch, 2024; Yang et
al., 2024), approaching the performance of traditional Multi-Headed Scaled Dot
Product Attention (MHA) (Vaswani et al., 2023) while achieving lower inference
costs. In MHA, the model’s effective memory is bounded by its context length,
with the attention calculation resulting in quadratic time complexity with
regard to that length. Conversely, most forms of linear attention can be
computed recurrently in O(1) time per time-step. Instead of inspecting the
entire context length to generate each new token, recurrent linear attention
uses a fixed-size hidden state that is updated at each time-step, functioning
as its memory of the past. The limited size of this state constrains the
capacity of this memory.
The success of Large Language Models (LLMs) has motivated interest in ultra-
long context length language models. For example, Gemini Pro (Team et al.,
2024) offers a 1 million+ token length window. However, if based on attention,
these extra large context lengths come with large associated costs due to the
need for MHA to examine every prior token within the context when generating a
next token (Liu & Abbeel, 2023; Liu et al., 2023). Although a naive inference
implementation would recalculate every key and value at every layer in a
traditional transformer, it is common practice to store these in a key-value
cache (”KV-Cache”)(Pope et al., 2022) and retrieve rather than recompute them.
KV-Cache memory costs can be very high. For example, a 1 million token cache
for an 80 layer traditional transformer model of hidden dimension 8192 would
take up over 2.5 terabytes at bfloat16 precision. We turn our focus to
reducing the memory costs of this cache while also reducing computational
complexity and memory usage for processing the initial context of a request.
Our contribution is the combination of several innovations to create the
GoldFinch architecture, which improves pre-fill and decoding efficiency, as
well as downstream modeling performance, and introduces the following
innovations:
1. 1.
employs a novel parameter-efficient modification of Finch (RWKV-6), which we
call ”Finch-C2”, for the first 2/3 of its layers
2. 2.
uses these the output of these Finch-C2 layers to produce an extremely small
compressed global key cache using a novel mechanism we call ”TokenCat”. Our
cache thus requires only $\frac{1}{16}d_{model}$ per token plus the original
input token indices, instead of $2d_{model}n_{layer}$ for traditional KV-
caches.
3. 3.
employs a novel modification of the traditional transformer architecture,
which we call ”GOLD”, for the last 1/3 of its layers to consume this key cache
and produce outputs without even requiring a traditional value cache.
Figure 1: GoldFinch Architecture Block Diagram Architecture | Pre-fill time | KV-Cache | KV-Cache Bytes
---|---|---|---
| complexity | entries | 256k context, 32 layers
| per token | per token | 4096 hidden dim
Llama2 | $O(N)$ | $2d_{model}n_{layer}$ | 128GB
Llama3 (w/ GQA) | $O(N)$ | $8d_{head}n_{layer}$ | 32GB
DeepSeek-V2 | $O(N)$ | $\frac{9}{2}d_{head}n_{layer}$ | 18GB
Zamba | $O(N)$ | $\frac{2}{7}d_{model}n_{layer}$ | 18.3GB
Jamba | $O(N)$ | $\frac{8}{7}d_{head}n_{layer}$ | 4GB
YOCO | $\mathbf{O(1)}$ | $2d_{model}$ | 4GB
GoldFinch | $\mathbf{O(1)}$ | $\mathbf{1+\frac{d_{model}}{16}}$ | 0.068GB
Table 1: Time and space complexity comparisons of models with full softmax
attention. No KV-Cache quantization is shown.
The GOLD layers are an adaptation of a novel improved transformer we call
”GPTAlpha” that can also be used as a standalone transformer model for
improved non-hybrid performance.
This new architecture brings a series of significant benefits:
1. 1.
We are able to reuse the same KV-Cache on every transformer layer while
maintaining greater than Llama (Touvron et al., 2023) performance. This
reduces the KV-Cache size by a factor of the total number of layers of the
model.
2. 2.
We eliminate the values from the KV-Cache, leaving only a key cache. Instead
of caching values, we store the input indices and generate the values from
these, reducing the KV-Cache size by another factor of nearly 2.
3. 3.
We are able to compress our key cache by applying a form of Low Rank
Adaptation (LoRA) (Hu et al., 2021) to the output of a single layer, and re-
expanding the compressed version by concatenating the compressed version with
the original token embeddings, further reducing the size by 128 times.
(”TokenCat”)
4. 4.
We use the input embedding table and RWKV-style token shift to generate values
for attention without sacrificing performance.
5. 5.
By using Finch-C2 blocks at the start of the model, the key cache
automatically encodes the underlying implicit positional representation,
thereby removing the need for positional encoding within our transformer
layers for trained context lengths. We do still require an additional
positional encoding method for extrapolation to new context lengths unseen
during training.
6. 6.
There are many use cases of LLMs that involve relatively short responses to
questions about long documents. Because our compressed key cache is generated
by an RNN with an operating time and space complexity of O(1) per token with
regard to sequence length, we are able to generate the cache in these cases
extremely inexpensively and apply the O(N) per token cost GOLD transformer
portion of our calculations only to new token generation, for which relatively
few iterations are often required.
To obtain our Finch-C2 architecture we improve the Finch time-mixer by
removing the gate, swapping out GroupNorm for a LayerNorm across all heads,
doing a new multiplication (of the key by one minus the decay) to keep the kv-
state rows normalized, and replacing Finch’s $u$ (”bonus”) term with a new
data-dependent separately token-shifted second Value. These changes result in
improved performance with little to no speed penalty and significantly fewer
total parameters.
To obtain our GPTAlpha architecture we improve the Llama architecture by
replacing the transformer feed-forward network (FFN) with the RWKV channel
mixer, and adding RWKV style token shifts and extra LayerNorms to attention
layers.
Both Finch-C2 and GPTAlpha can be used either as standalone model
architectures with improved performance over their counterparts, or as part of
the GoldFinch hybrid model architecture.
The GOLD transformer architecture (GPTAlpha Over Linear transformer Decoder)
removes the key and value weights from GPTAlpha in favor of producing keys and
values from a combination of the original token indices passed through the
embeddings table, a highly compressed version of the outputs of the Finch-C2
layers, and a data-driven LoRA.
GoldFinch stacks a set of GOLD transformer layers on top of a Finch-C2 linear
transformer, passing the outputs for the Finch layers both into a key
compressor to be stored for every sequence position, and also through the
current timestep as part of the normal residual stream.
We train GoldFinch models up to 1.45 billion parameters on 1.5 trillion tokens
of minipile (Kaddour, 2023) and compare them to slightly larger equivalently
trained Finch (Peng et al., 2024) and Llama (Touvron et al., 2023) models. We
find that GoldFinch significantly outperforms both Llama and Finch in
downstream performance and perplexity across nearly every benchmark we tested,
while maintaining fewer parameters, a much smaller cache than Llama, and
perfect MQAR recall due to its use of full attention.
## 2 Background
Transformers have become the de-facto choice for most sequence modeling tasks,
and have been shown to be especially effective in the context of language
modeling. However, they present computational challenges when processing long
context lengths, which has hindered their adoption for long sequence tasks.
Specifically, the formulation of multi-head scaled dot-product attention (MHA)
has a computational complexity of O($N^{2}$) with respect to context length.
Additionally, inference engines typically rely on the use of a KV-Cache to
enable autoregressive token generation in O(N) time per token. This cache
grows linearly with context length, and becomes challenging to fit into
limited Video Random-Access Memory (VRAM) for longer sequences.
Recent transformer models such as the Llama series rely on Grouped-Query
Attention (GQA) (Ainslie et al., 2023) to help ameliorate this cache size
problem. At a typical number of groups $n_{g}=8$, GQA reduces the KV-Cache
size by $\frac{n_{g}}{n_{h}}$ times, where $n_{h}$ is the number of heads.
This is helpful, especially on consumer grade hardware, but leads to a
reduction in downstream performance, and longer sequences still cause a
significant problem in terms of VRAM usage.
The recently proposed YOCO (Sun et al., 2024) improves the computational
complexity for pre-fill of the initial request context and also reduces the
KV-cache size by introducing a new global KV-Cache instead of the usual per-
layer cache. The computational improvement is achieved by replacing the first
half of the layers in the model with Linear Attention based RetNet-G layers
(Sun et al., 2023), which is a recurrent neural network (RNN) architecture
that requires only linear time with respect to sequence length. YOCO stores
the output of these first layers as a global KV-Cache, which is then used by
the second half of the layers, featuring MHA. Overall, this reduces the KV-
Cache size by a factor of the number of layers, without a reported performance
reduction. Goldfinch takes a related approachetNet-G, and processes the output
differently, creating an effective but much smaller cache via our TokenCat
mechanism, which is then consumed by our enhanced transformer GOLD layers.
Hungry Hungry Hippos (H3) (Fu et al., 2023) train a hybrid recurrent
SSM/transformer model containing just two layers of attention, which they find
outperforms transformers. This served as a warning shot that SSM(or linear
attention)-transformer hybrids have the potential to step in as higher
performance replacements for transformers alone.
Recognizing the challenges posed at inference time by the KV-Cache,
DeepSeek-V2 (DeepSeek-AI et al., 2024) proposes a replacement for MHA called
Multi-head Latent Attention (MLA). This uses low-rank joint key-value
compression to reduce the size of the KV-Cache from $2n_{h}d_{h}l$ to
$\frac{9}{2}d_{h}l$, equivalent to the KV-Cache size required for GQA with
only 2.25 groups. Because the low-rank key-value compression requires fewer
parameters than full rank key and value matrices, MLA achieves greater per-
parameter performance than MHA. GoldFinch also improves performance via this
kind of compression-based relative parameter reduction.
HGRN2 (Qin et al., 2024) replaces the per-head GroupNorm (Wu & He, 2018) with
a full-width LayerNorm, and we do the same in our Finch-C2 architecture. HGRN2
sets their key to be equal to one minus the decay, and we do something related
but slightly different, multiplying our key by one minus the decay.
Inspired by these works, we propose a new method that further reduces the KV-
Cache by orders of magnitude and reduces the cost of the initial context load
to become linear with respect to sequence length, all while achieving greater
than Llama performance.
### 2.1 Other Concurrent Related Work
Other concurrent work on hybrid models bear some similarities to portions of
our architecture:
Zamba (Glorioso et al., 2024) interleaves Global Shared Attention (GSA) every
N Mamba blocks (Gu & Dao, 2024). Instead of using the residual output of the
prior Mamba block as its input, Zamba concatenates the original embeddings
generated before layer zero onto this residual output, and use the double-
width combination as the input to attention. Although their GSA blocks share
parameters, they are not able to share the same KV-Cache. The concatenation of
embeddings bears similarity to our new ”TokenCat” technique.
Jamba (Lieber et al., 2024) is a mixture-of-experts (MoE) (Shazeer et al.,
2017) Mamba-based (Gu & Dao, 2024) model that inserts attention layers
periodically within its architecture, for a total of 1:7 ratio of attention-
to-Mamba layers. Similarly to Goldfinch’s ability to rely upon RWKV’s implicit
positional encoding within the pre-trained context length, they find that
explicit positional encoding may not be required for their hybrid Mamba-based
architecture.
Samba (Ren et al., 2024) is a hybrid model that repeats blocks containing a
Mamba layer, an MLP layer, a sliding-window attention (SWA) layer featuring
RoPE (Su et al., 2023), and another MLP layer. The use of SWA allows a fixed
cost of execution per token, regardless of context length.
## 3 Method
GoldFinch follows the general structure of the Finch architecture, which is
also the common pre-norm decoder transformer structure used in Llama and RWKV.
It consists of a series of layers, each containing a time mixing sub-layer
followed by a channel mixing sub-layer. All channel mixing sub-layers are
Finch channel mixers.
The following formulae describe the three varieties of GoldFinch sub-layers.
All matrices $\bm{W}$ are learned per layer, unless described otherwise. We
show all time mixing formulae per-head for conciseness, except the formulae
for those layer outputs where heads are combined via $concat$. Model dimension
is denoted as $D$, head size as $H$, and number of heads as $N$. All values
are $\in\mathbb{R}^{H}$ unless otherwise noted.
### 3.1 Finch-C2 Time Mixing
The first two-thirds of time mixing sub-layers use a variation on the Finch
time mixer we call Finch-C2.
We customize the Finch time-mixing sub-layers by removing the gate, swapping
out GroupNorm for a LayerNorm across all heads and doing a new multiplication
of the key by one minus the decay. Finally, we replace Finch’s $u$ (”bonus”)
term with a new data-dependent separately token-shifted second Value, computed
using the same weights as the base Value, with an additional LoRA added to the
result. We find that this allows us to remove all of the Gate parameters while
retaining performance.
Along the lines of (Peng et al., 2024), we introduce the following notation
for common operators in the model, using the square subscript to denote a
variable:
$\displaystyle\mathrm{lerp}(a,b,t)$ $\displaystyle=a+(b-a)\odot t,$ (1)
$\displaystyle\mathrm{lora}_{\square}(x)$
$\displaystyle=\lambda_{\square}+\tanh(x\bm{A}_{\square})\bm{B}_{\square},$
(2) $\displaystyle\mathrm{ddlerp}_{\square}(a,b)$
$\displaystyle=a+(b-a)\odot\mathrm{lora}_{\square}(a+(b-a)\odot\mu_{x}),$ (3)
Then, the Finch-C2 block can be formalized as:
$\displaystyle d_{t}$
$\displaystyle=\mathrm{lora_{\omega}}(\mathrm{ddlerp}_{d}(x_{t},x_{t-1})),$
(4) $\displaystyle w_{t}$ $\displaystyle=\exp(-\exp(d_{t})),$ (5)
$\displaystyle r_{t}$
$\displaystyle=\mathrm{ddlerp}_{r}(x_{t},x_{t-1})\bm{W}^{R},$ (6)
$\displaystyle k_{t}$
$\displaystyle=\mathrm{ddlerp}_{k}(x_{t},x_{t-1})\bm{W}^{K}\cdot(1-w_{t}),$
(7) $\displaystyle v_{t}$
$\displaystyle=\mathrm{ddlerp}_{v}(x_{t},x_{t-1})\bm{W}^{V},$ (8)
$\displaystyle u_{t}$ $\displaystyle=\mathrm{ddlerp}_{u}(x_{t},x_{t-1}),$ (9)
$\displaystyle u^{\prime}_{t}$
$\displaystyle=u_{t}\bm{W}^{V}+\tanh(u_{t}\bm{W}^{UD})\bm{W}^{UU}.$ (10)
And after splitting the hidden dimension into $N$ heads:
$\displaystyle\bm{wkv}_{t}$
$\displaystyle=\sum_{i=1}^{t-1}\mathrm{diag}\left(\bigodot_{j=i+1}^{t-1}w_{j}\right)\cdot
k_{i}^{\mathrm{T}}\cdot v_{i}\in\mathbb{R}^{H\times H},$ (12) $\displaystyle
o_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{concat}\left(r_{t}\cdot\bm{wkv}_{t}+u^{\prime}_{t}\right))\bm{W}^{O}\in\mathbb{R}^{D}.$
(13)
Please note that the calculation for $u^{\prime}_{t}$ reuses the same weights
$\bm{W}^{V}$ \- this is an intentional parameter count savings and not a typo.
### 3.2 GOLD Key Compression
The output from the first two-thirds of the model is used in two ways: it is
passed on to the next layer in the usual manner, and also compressed down via
multiplication with the global (not per-layer) learned matrix
$\bm{W}^{KD}\in\mathbb{R}^{Dx(D/16)}$ to one sixteenth its original size and
stored into a unified single-layer compressed key cache:
$\displaystyle c_{t}$ $\displaystyle=x_{t}\bm{W}^{KD}\in\mathbb{R}^{(D/16)}.$
(14)
### 3.3 GOLD Key Decompression (TokenCat)
The compressed key cache is decompressed via a two-step method. The first step
is ”TokenCat”, short for ”Token conCatenation”, in which the compressed key is
concatenated with the original input token embedding from the very beginning
of the model. The concatenated result is then multiplied with the global (not
per-layer) learned matrix $\bm{W}^{KU}\in\mathbb{R}^{(D+D/16)xD}$ and
RMSNormed to obtain the decompressed attention proto-keys, which are common to
all GOLD attention sub-layers.
$\displaystyle k^{D}_{t}$
$\displaystyle=\operatorname*{RMSNorm}\left(\mathrm{concat}\left(x^{0}_{t},c_{t}\right)\bm{W}^{KU}\right).$
(15)
### 3.4 GOLD Attention Time Mixing
The remaining time mixing sub-layers are a variation on GPTAlpha attention
sub-layers employing MHA that we call GOLD attention.
Each GOLD attention sub-layer calculates its own unique attention keys and
values from the decompressed proto-keys and the original input token
embeddings, respectively. Each is passed through a data-dependent token shift,
with the result passed through an additive LoRA. We call this process
”DDLoRAdapt”, introducing the relevant notation below, using the square
subscript to denote a variable:
$\displaystyle\mathrm{loradapt}_{\square}(x)$
$\displaystyle=x+\tanh(x\bm{C}_{\square})\bm{D}_{\square}.$ (16)
The following are the formulae for GOLD attention time mixing:
$\displaystyle q_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{ddlerp}_{q}(x_{t},x_{t-1})\bm{W}^{Q}),$
(17) $\displaystyle a_{t}$
$\displaystyle=\mathrm{lerp}(x^{0}_{t},x^{0}_{t-1},\mu_{x}),$ (18)
$\displaystyle k_{t}$
$\displaystyle=\operatorname*{LayerNorm}\left(\mathrm{loradapt}_{k}\left(\mathrm{lerp}\left(k^{D}_{t},k^{D}_{t-1},\mathrm{lora}_{k}\left(a_{t}\right)\right)\right)\right),$
(19) $\displaystyle v_{t}$
$\displaystyle=\operatorname*{LayerNorm}\left(\mathrm{loradapt}_{v}\left(\mathrm{lerp}\left(x^{0}_{t},x^{0}_{t-1},\mathrm{lora}_{v}\left(a_{t}\right)\right)\right)\right),$
(20) $\displaystyle o_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{concat}\left(\mathrm{attention}(q_{t},k,v)\right))\bm{W}^{O}\in\mathbb{R}^{D}.$
(21)
Please note the receptance-like Finch style token-shift on queries, and
additional data-driven token-shift on keys and values, with keys being
reconstituted from compressed key cache entries $c_{t}$ and values coming from
the original token embeddings $x^{0}$. $x^{0}$ is the embedding input to the
first sub-layer in the model, and can be reconstituted during inference from
the token indices by storing those indices, usually only an additional two
bytes per context length.
Data dependent token shift (ddlerp) is a specialized low-parameter cost
variety of two-step 1D convolution that originated in the RWKV architecture.
It allows the model to dynamically linearly interpolate between the current
and previous time-step on a per channel basis. We use our DDLoRAdapt version
of the technique to inexpensively apply contextual information to the keys and
values, increasing the amount of information from which they are generated
without significantly increasing parameter count.
Note that the token shift cannot be dependent on the hidden-state, as that
would make recurrent calculation impossible for older keys and values, and
would require a full KV-Cache to be stored. Instead, we use the original input
token embeddings as the data upon which the key and value token-shifts depend.
Pre-fill of the compressed key cache to prepare for autoregressive generation
can be computed in linear time with respect to the number of tokens. This is
accomplished by running only the Finch-C2 section of the model on those
tokens. One important implementation caveat is that token shift requires the
prior layer hidden-state output from the previous time-step. At first glance
this appears problematic, as the GOLD layers require full quadratic attention,
which is what we were trying to avoid during pre-fill. But the solution is
simple: given $G$ GOLD layers in the model, there must be $2G-1$ sub-layers
that require such a previous time-step hidden state but are directly or
indirectly reliant on the outputs of quadratic attention. Therefore, the last
$2G-1$ tokens of pre-fill must be run through the full model (not just the
Finch-C2 layers) to generate these hidden-states. These $2G-1$ computations
can be done in a single call to the full model to leverage the same kinds of
parallelism used during training.
Only the compressed key cache entries and original input token indices must be
permanently kept in VRAM during inference, as the key cache can be
reconstituted via decompression on-demand.
Because decompression and token shift can be done on contiguous regions of key
value pairs instead of all of them at once, extremely low VRAM usage can be
achieved during inference by calculating attention incrementally across the
sequence for each layer and decompressing as you go.
### 3.5 GoldFinch Channel Mixing (same as Finch Channel Mixing)
Goldfinch channel mixing is identical to Finch channel mixing. It is used as
the feed forward network component on all layers of the model, both Finch-C2
and GOLD. We reproduce it here for reference. Please note that variables have
their own independent definitions in this subsection.
$\displaystyle r_{t}$
$\displaystyle=\mathrm{lerp}_{r}(x_{t},x_{t-1},\mu_{r})\bm{W}^{R}\in\mathbb{R}^{D},$
(22) $\displaystyle k_{t}$
$\displaystyle=\mathrm{lerp}_{k}(x_{t},x_{t-1},\mu_{k})\bm{W}^{K}\in\mathbb{R}^{3.5D},$
(23) $\displaystyle v_{t}$
$\displaystyle=\mathrm{ReLU}(k_{t})^{2}\bm{W}^{V}\in\mathbb{R}^{D},$ (24)
$\displaystyle o_{t}$ $\displaystyle=\sigma(r_{t})\odot
v_{t}\in\mathbb{R}^{D}.$ (25)
### 3.6 GPTAlpha Time Mixing
For completeness and to show how it can be used in a pure transformer
architecture, we list the formulae for GPTAlpha time mixing when not used in
conjunction with TokenCat below:
$\displaystyle q_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{ddlerp}_{q}(x_{t},x_{t-1})\bm{W}^{Q}),$
(26) $\displaystyle k_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{ddlerp}_{k}(x_{t},x_{t-1})\bm{W}^{K}),$
(27) $\displaystyle v_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{ddlerp}_{v}(x_{t},x_{t-1})\bm{W}^{V}),$
(28) $\displaystyle o_{t}$
$\displaystyle=\operatorname*{LayerNorm}(\mathrm{concat}\left(\mathrm{attention}(q_{t},k,v)\right))\bm{W}^{O}\in\mathbb{R}^{D}.$
(29)
## 4 Experiments
### 4.1 Architecture Comparisons
We trained 1.5B parameter-class models with 24 layers, 2048 hidden-dimension,
2048 context length of Finch, Llama, and GoldFinch for comparison on minipile
(Kaddour, 2023), all using the same RWKV World tokenizer. GoldFinch ends with
dramatically lower final loss than the others (by over 0.1 out of 2.39), and
uses over 100 million fewer parameters than its Finch counterpart. We
additionally trained a GoldFinch with no compression, to show that there is
very little lost with our choice of a 16:1 hidden-dimension compression ratio.
In the interest of fairly comparing performance for Llama by giving it the
most favorable conditions, we add the RWKV small init embeddings optimization
(LayerNorm after embeddings with small initialized values) (Peng et al., 2023)
and do not employ Grouped Query Attention. All architectures used the same
hyperparameters and were trained on 4 GPUs, with per-GPU per-step batch size
of 8, two steps of gradient accumulation, and a 10 step learning rate warm-up
followed by cosine decay annealed from 3e-5 to 1e-5. We train with Adam betas
of 0.9 and 0.99, epsilon 1e-8 and weight decay 0.001. Weight decay was applied
only to matrix parameters that are not part of LoRAs or the GoldFinch key
compression/expansion steps.
Figure 2: Loss curves of 1.5B class models. Architecture (L24 D2048 ctx2048) | Parameters | Loss $\downarrow$
---|---|---
Llama | 1.47B | 2.3905
Finch | 1.60B | 2.3856
GoldFinch, last 1/3 layers GOLD, 16:1 compression | 1.45B | 2.2762
GoldFinch, last 1/3 layers GOLD, 1:1 compression | 1.45B | 2.2762
Table 2: Final loss values for various models of size L24 D2048 ctx2048
trained on minipile
In addition to comparing training and validation losses, we ran a series of
common benchmark evaluations on the three 1.5B parameter class models trained
on minipile. Finch and Llama scored similarly to one another, and GoldFinch
significantly outperformed both.
Model | lmbd | avg | lmbd | piqa | hella | winog | arc_c | arc_e | sciq
---|---|---|---|---|---|---|---|---|---
| ppl $\downarrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$ | acc $\uparrow$
Finch 1.60B | 81.9 | 42.8% | 24.3% | 62.4% | 28.7% | 49.0% | 19.6% | 44.9% | 70.8%
Llama 1.47B | 71.7 | 43.0% | 26.3% | 61.6% | 28.1% | 50.5% | 19.3% | 43.9% | 71.0%
GoldFinch 1.45B | 48.2 | 44.2% | 29.1% | 63.4% | 29.1% | 50.2% | 18.3% | 45.9% | 73.7%
Table 3: Common benchmark evaluations for various models of size L24 D2048
ctx2048 trained on minipile
### 4.2 Ablation Studies
We ran various smaller scale ablation studies to determine the contributions
of different parts of the GoldFinch architecture relative to both Finch,
Llama, GPTAlpha, and a hybrid of our improved Finch and GPTAlpha with no KV-
Cache compression or key/value sharing. The new second value added in Finch-C2
had the smallest positive impact of anything measured. Surprisingly, GoldFinch
performed very slightly better than even the Finch-C2/GPTAlpha hybrid with no
KV compression at all. Each test trained a 12 layer 768 hidden-dimension model
at 1024 context length with the same RWKV World tokenizer on the full minipile
dataset. All architectures used the same hyperparameters and were trained on
single GPUs, with per-step batch size of 32, two steps of gradient
accumulation, and a 10 step learning rate warm-up followed by cosine decay
annealed from 6e-5 to 2e-5. We train with Adam betas of 0.9 and 0.99, epsilon
1e-8 and weight decay 0.001. Weight decay was applied only to matrix
parameters that are not part of LoRAs or the GoldFinch key
compression/expansion steps.
Architecture (L12 D768 ctx1024) | Loss $\downarrow$
---|---
Finch-C2 without $k*=1-w$ | 2.7293
Finch | 2.7191
Llama | 2.7125
Finch-C2 without second value | 2.7105
Finch-C2 | 2.7082
GPTAlpha with RoPE | 2.6684
GoldFinch, last 1/2 layers GOLD | 2.6637
GoldFinch, last 1/3 layers GOLD with RoPE | 2.6590
Finch-C2, last 1/3 layers GPTAlpha | 2.6586
GoldFinch, last 1/3 layers GOLD | 2.6582
GoldFinch, last 1/6 layers GOLD | 2.6578
Table 4: Final loss values for various ablations of model size L12 D768
ctx1024 trained on minipile
### 4.3 Associative Recall
Associative recall (AR) is a synthetic task designed to emulate the human
ability to associate and retrieve information. It evaluates a model’s skill in
recalling previously mentioned information within a given context. Previous
studies suggest that a model’s performance in AR is a good indicator of its
efficacy in in-context learning (Elhage et al., 2021; Olsson et al., 2022).
Consequently, AR has been employed as a benchmark for developing new language
model architectures (Fu et al., 2023; Poli et al., 2023; Lutati et al., 2023).
Arora et al. (2023) evaluated a variety of models for multi-query associative
recall (MQAR) and discovered a performance gap between different linear
transformer architectures and the traditional transformer with attention.
Figure 3: MQAR tasks. An increase in sequence length correlates with increased
task difficulty.
In Figure 3, we used the same experimental settings as Arora et al. (2023) and
show that GoldFinch achieves perfect MQAR scores, outperforming traditional
attention-free language models. As a hybrid architecture that leverages
attention, GoldFinch can solve MQAR as well as transformer models with
attention. Additionally, we trained GoldFinch on a context length of 1024 to
demonstrate that this trend continues, as depicted in Figure 4.
Figure 4: Finch and GoldFinch on the same MQAR task with increased sequence
length
### 4.4 Long Context Experiments
We tested the loss of our small Finch and GoldFinch models pre-trained on
minipile at all context lengths up to 65536 on the PG19 (Rae et al., 2019)
dataset of older books. These pre-trained models were all trained at only 1024
context length. The Finch model is able to maintain a fairly low loss
throughout the 65536 context length. The base GoldFinch model trained with no
positional encoding goes up in loss significantly starting at around double
the trained context length, then plateauing at a high loss. The GoldFinch
model trained with RoPE on its GOLD attention sub-layers performs better, but
loss still increases somewhat as the sequence progresses. However, by applying
interpolated RoPE values we are able to obtain low loss throughout the
extended context length. We conclude that for GoldFinch models in which
extrapolation beyond the maximum trained context length is desired, the GOLD
attention sub-layers should be trained with RoPE, with interpolation employed
upon inference.
We then fine-tuned the RoPE and non-RoPE models mentioned above on 165 million
tokens of minipile at longer context lengths. During this fine-tuning, we
froze the entire RWKV portion of the model up to the first GOLD layer,
allowing the optimizer to update the parameters of only the GOLD layers and
output head. This saves a significant amount of time and VRAM during fine-
tuning, allowing an even longer context length to fit into memory and using
roughly 3x fewer FLOPS per token. We theorize that because the GOLD attention
portion of the model can use keys generated from the RWKV output, this is
enough to support sophisticated attention matching across the entire context
length.
Our experiments showed that indeed the RoPE model with GOLD layers fine-tuned
at longer context lengths exhibited significantly lower losses against PG19 up
through those lengths and even beyond. On the non-RoPE model this process was
somewhat successful within the fine-tuned context length, while still failing
at extrapolation. This was unexpected, since the RWKV layers were not updated
and the GOLD layers included no positional encoding mechanism. We postulate
that token-shift may supply some minimal positional information to the model.
### 4.5 Checkpoint Upgrade Training
We have attempted upgrading existing pre-trained Finch models to a more
limited version of GoldFinch that uses the Finch architecture for its RWKV
layers instead of the Finch-C2 component. We tried many variations on two
methods, one that adds new GOLD layers on top for a total of around 11% more
parameters, and another which keeps the layer count the same as the pre-
trained model. Thus far with only small amounts of upgrade training neither
method has performed to our satisfaction.
Both methods were attempted on a 1.6B Finch checkpoint that had been pre-
trained on 2.5 trillion tokens.
For the first method we appended 4 GOLD layers on top of the pre-trained 1.6B
Finch checkpoint before the language modeling head, and continued training it
for 100 million tokens using two different learning rates. The original 24
pre-trained layers were kept at the same 1e-5 LR at which their pre-training
had ended upon, while the LR for the 4 new GOLD layers was annealed along a
cosine schedule from 3e-4 to 1e-5. While the performance of this model was in
line with the original model, it was unclear if the resultant model from this
method really learned anything of value in its GOLD layers.
The second method involved freezing the embedding and RWKV layers and
importing but not freezing the final 1/3 of the channel mixer sub-layers that
were paired with freshly initialized GOLD attention sub-layers. We then
trained this model on a relatively small amount of data (in our case around
7.5 billion tokens of a new internal dataset) while annealing the learning
rate to the final learning rate seen in the pre-trained base model. The
resultant model obtained a similar validation loss on minipile to the base
model, despite being trained on a completely different dataset and the base
model having been already trained for over 2.25 trillion tokens. However, the
new model’s LAMBADA scores were worse. We attribute this loss of performance
to the ’brain surgery’ required to keep the layer count the same, in which we
effectively erased the Finch time-mix parameters in the upper 1/3rd of the
model.
We are still doing further experimentation on these upgrade methods to see
just how well they can be made to perform. We hope to be able to inexpensively
upgrade even the largest 14B Finch model to this reduced GoldFinch format and
see significant performance improvements at larger context lengths due to the
GOLD attention being able to look back across the entire context with no
state-size based memory limitations.
## 5 Further Work
We anticipate updating this pre-print with further studies as results become
available, including checkpoint upgrade results and evaluations, longer
experiment training runs, and new long context experiments. Please check back
for updates.
Most of the experiments done for this pre-print were performed over a short
period of time on a single node containing 8 RTX 4090 cards. In the future we
hope to demonstrate GoldFinch’s performance on larger models with
significantly more tokens.
We expect that GoldFinch will work similarly with other linear attention and
SSM architectures in place of the Finch-C2 blocks. For example, it should be
possible to implement a ”GoldMamba” architecture in the same style.
Further work might explore increased memory reduction for the global KV-Cache
via quantization, and application of ring attention Liu et al. (2023) to lower
the memory requirements when extending to very long contexts. As a hybrid
architecture model, GoldFinch will likely benefit from any future improvements
to the RWKV and transformer architectures.
## 6 Conclusion
We have introduced a hybrid RNN-Attention model architecture (GoldFinch) and
trained models that demonstrate its performance up to 1.45B. The resulting
hybrid RNN-Attention models combine the efficiency of RNNs with the
capabilities of attention-based models. Having RNNs for the initial layers
allows for fast pre-fill and removes the need for positional encoding on the
RNN layers, while the attention layers improve associative recall. The
combination with a highly compressed global KV-Cache unlocks a memory
reduction in inference while maintaining enhanced performance. We release the
trained weights and training code under the Apache 2.0 license.
## 7 Acknowledgements
Special thanks to Bo Peng for his tireless dedication to the RWKV architecture
and community. The main GoldFinch code herein was based on a modified version
of his public Linear Attention Arena code repository, and upgraded models were
based on his pre-trained Finch model releases.
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## Appendix A Author Contributions
#### Daniel Goldstein
Entire GPTAlpha design, research, and code. GoldFinch code, architecture
design, and research. Full manuscript initial draft except 4.3. Manuscript
edits. Proofreading and revisions of full manuscript. Core experiments
featured herein.
#### Fares Obeid
Research discussions and experiments during development of the GoldFinch
architecture. Significant input on all aspects of final architecture design.
#### Eric Alcaide
Research discussions and experiments during development of the GoldFinch
architecture. Significant input and experiments leading to Finch-C2 design.
#### Guangyu Song
Section 4.3. Experiments for 4.3.
#### Eugene Cheah
GoldFinch code proofreading, development of release code and testing,
contributions to pre-fill mechanism details.
## Appendix B Other Related Work
Ring Attention (Liu et al., 2023) allows the attention calculation to be split
across many discrete processors that do not share VRAM. Keys and values can be
split up among these processors, linearly amortizing the amount of KV-Cache
required to remain resident within each processor’s VRAM. This enables
unbounded scaling of attention given enough hardware, but does not address the
cost of O($N^{2}$) compute, and still imposes total memory costs that scale
with the sequence length.
|
22institutetext: Department of Mathematics, Faculty of Applied Sciences,
Durban University of Technology, Durban, 4000, South Africa.
22email<EMAIL_ADDRESS>${}^{\textrm{{\char 0\relax}}}$
22email<EMAIL_ADDRESS>
22email<EMAIL_ADDRESS>
Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics
and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban
4000, South Africa.
22email<EMAIL_ADDRESS>
# Stability and Horizon Formation during Dissipative Collapse
Nolene F. Naidu† Robert S. Bogadi‡ Anand Kaisavelu∗ Megan Govender∗∗
(Received: date / Accepted: date)
###### Abstract
We investigate the role played by density inhomogeneities and dissipation on
the final outcome of collapse of a self-gravitating sphere. By imposing a
perturbative scheme on the thermodynamical variables and gravitational
potentials we track the evolution of the collapse process starting off with an
initially static perfect fluid sphere which is shear-free. The collapsing core
dissipates energy in the form of a radial heat flux with the exterior
spacetime being filled with a superposition of null energy and an anisotropic
string distribution. The ensuing dynamical process slowly evolves into a
shear-like regime with contributions from the heat flux and density
fluctuations. We show that the anisotropy due to the presence of the strings
drives the stellar fluid towards instability with this effect being enhanced
by the density inhomogeneity. An interesting and novel consequence of this
collapse scenario is the delay in the formation of the horizon.
###### Keywords:
radiative collapse anisotropic stresses density inhomogeneities
## 1 Introduction
Gravitational collapse is fundamental to the formation of the majority of
stellar objects in the universe and thus one would expect that the study of
this phenomenon should be vital to our understanding of the workings of the
universe. The pioneers in the research of gravitational collapse, Oppenheimer
and Snyder (1939), studied a spherically symmetric matter distribution in the
form of a dust sphere undergoing collapse. They obtained the first solution
for the non-adiabatic collapse of a dust ball with a Schwarzschild exterior
oppsnyd . Vaidya vaidya obtained an exact solution to the Einstein field
equations which describes the exterior field of a radiating, spherically
symmetric fluid by noting that a radiating collapsing mass distribution has
outgoing energy and so its exterior spacetime is no longer a vacuum but
contains null radiation. The next step in improving the model was accomplished
by Santos santos who derived the junction conditions for a collapsing
spherically symmetric, shear-free non-adiabatic fluid sphere with heat flow.
The combination of these contributions allowed for the matching of the
interior and exterior spacetimes of a collapsing star which lead the way for
studying non-adiabatic, isotropic as well as anisotropic dissipative
gravitational collapse mken -kevin .
A disturbance or perturbation of a system initially in static equilibrium
results in a change in stability which likely renders the system dynamic. The
property of a system to retain its initial stable state once perturbed, is
then referred to as its dynamical (in)stability. Hence the issue of stability
is vital in the study of self-gravitating objects as a static stellar model
which evolves towards higher instability is of little physical significance.
The dynamical instability of a spherically symmetric mass with isotropic
pressure was first investigated by Chandrasekhar chandra . He showed that for
a system to remain stable under collapse, the adiabatic index $\Gamma$ must be
greater than $\frac{4}{3}$.
Subsequently, Herrera et al. herr2 showed that for a non-adiabatic sphere
where relativistic corrections were imposed to address heat flow, the unstable
range of $\Gamma$ decreased rendering the fluid less unstable. Chan et al.
chan2 investigated the stability criteria by deviating from the perfect fluid
condition in two ways: they considered radiation in the free-streaming
approximation; and they assumed local anisotropy of the fluid. Herrera et al.
herr3 also examined the dynamical instability of expansion-free, locally
anisotropic spherical stellar bodies.
The application of Einstein’s field equations to increasingly more complex
gravitating systems with additional parameters and degrees of freedom depends
on computational techniques, which is the case when perturbative theories are
employed in which higher order terms arise. The generalization of systems,
such as the inclusion of a string density field, can increase the complexity
of expressions obtained however to first order we aim to introduce the
temporal behaviour and hence evolution of the collapse process. This method is
well established chan2 ; bon . Compact objects such as neutron stars, black
holes and the more recently proposed dark-fluid stars and strange stars
composed of quark matter invite the addition of a more complex, non-empty
stellar exterior. The Vaidya metric which is commonly used for describing the
exterior spacetime would then require modification to include both the
radiation field and a so-called string field as initially put forward by Glass
and Krisch glass ; glass2 . In this more generalized Vaidya exterior, the mass
function is augmented to acquire both temporal and spatial dependence.
In 2005, Maharaj and Govender showed that the stellar core was more unstable
than the outer regions by investigating gravitational collapse with isotropic
pressure and vanishing Weyl stresses maharaj2 . More recently, Maharaj et al.
maharaj3 showed the impact of the generalized Vaidya radiating metric on the
junction conditions for the boundary of a radiating star. Their results
describe a more general atmosphere surrounding the star, described by the
superposition of a pressure-free null dust and a string fluid. The string
density was shown to affect the fluid pressure at the surface of the star. It
was demonstrated that this string density reduces the pressure at the stellar
boundary. The usual junction conditions for the Vaidya spacetime are regained
in the absence of the string fluid.
In this study, a spherically symmetric static configuration undergoing
radiative collapse under shear-free, isotropic conditions is considered. A
boundary condition of the form
$\left(p_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\rho_{s}$ is imposed
where $\rho_{s}$ is the string density and $qB$ is the heat flux. This is the
basis for developing the temporal behaviour of the self-gravitating system.
The structure of this paper is as follows: In $\S$2 the field equations
describing the geometry and matter content for a star undergoing shear-free
gravitational collapse are introduced. In $\S$3 the exterior spacetime and the
junction conditions necessary for the smooth matching of the interior
spacetime with Vaidya’s exterior solution across the boundary are presented.
In $\S$4 the perturbative scheme is described and the field equations for the
static and perturbed configurations are stated. In $\S$5 we develop the new
temporal equation employed in the perturbative scheme which includes the
effect of the string field. In $\S$6, we develop a radiating model from an
interior Schwarzschild static model. In $\S$7 dissipative collapse is
discussed, the perturbed quantities in terms of two unspecified quantities are
expressed and an equation of state which presents the perturbed quantities in
terms of radial coordinate only is introduced. The stability of the collapsing
model in the Newtonian and post-Newtonian approximations are explored in
$\S$8\. The physical analysis of the results and conclusion are discussed in
$\S$9\. The acknowledgements follows in $\S$10.
## 2 Stellar Interior
In order to investigate the evolution of the radiative collapse we adopt a
spherically symmetric shear-free line element in simultaneously comoving and
isotropic coordinates given by
$ds_{-}^{2}=-A^{2}(r,t)dt^{2}+B^{2}(r,t)[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})]$
(1)
where the $A(r,t)$ and $B(r,t)$ are the dynamic gravitational potentials. We
should highlight the fact that the stability of the shear-free condition may
hold for a limited period of the collapse process chan2 . Herrera and co-
workers have shown that shear-free collapse can evolve into a dynamical
process mimicking shear. The shear-like contributions can develop from
pressure anisotropy and density inhomogeneities. The stellar material for the
interior is described by an imperfect fluid with heat flux and in the form of
the energy-momentum tensor
$T_{ab}=(\rho+p_{t})u_{a}u_{b}+p_{t}g_{ab}+(p_{r}-p_{t})\chi_{a}\chi_{b}+q_{a}u_{b}+q_{b}u_{a}$
(2)
where $\rho$ is the energy density, $p_{r}$ the radial pressure, $p_{t}$ the
tangential pressure and $q_{a}$ the heat flux vector, $u_{a}$ is the timelike
four-velocity of the fluid and $\chi_{a}$ is a spacelike unit four-vector
along the radial direction. These quantities must satisfy $u_{a}u^{a}=-1$,
$u_{a}q^{a}=0$, $\chi_{a}\chi^{a}=1$ and $\chi_{a}u^{a}=0$. In co-moving
coordinates we must have
$\displaystyle u^{a}$ $\displaystyle=$ $\displaystyle A^{-1}\delta^{a}_{0}$
(3) $\displaystyle q^{a}$ $\displaystyle=$ $\displaystyle q\delta^{a}_{1}$ (4)
$\displaystyle\chi^{a}$ $\displaystyle=$ $\displaystyle B^{-1}\delta^{a}_{1}$
(5)
The nonzero components of the Einstein field equations for line element (1)
with energy-momentum tensor (2) are
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{B^{2}}\left[2\frac{B^{\prime\prime}}{B}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{4}{r}\frac{B^{\prime}}{B}\right]+\frac{3}{A^{2}}\left(\frac{{\dot{B}}}{B}\right)^{2}$
(6) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{B^{2}}\left[\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{2}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)+2\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}\right]$
(7)
$\displaystyle+\frac{1}{A^{2}}\left[-2\frac{\ddot{B}}{B}-\left(\frac{\dot{B}}{B}\right)^{2}+2\frac{\dot{A}}{A}\frac{\dot{B}}{B}\right]$
$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{A^{2}}\left[-2\frac{\ddot{B}}{B}-\left(\frac{\dot{B}}{B}\right)^{2}+2\frac{\dot{A}}{A}\frac{\dot{B}}{B}\right]$
(8)
$\displaystyle+\frac{1}{B^{2}}\left[\frac{B^{\prime\prime}}{B}-\left(\frac{B^{\prime}}{B}\right)^{2}+\frac{1}{r}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)+\frac{A^{\prime\prime}}{A}\right]$
$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{2}{AB^{2}}\left[\frac{{\dot{B}}^{\prime}}{B}-\frac{\dot{B}}{B}\left(\frac{B^{\prime}}{B}+\frac{A^{\prime}}{A}\right)\right]$
(9)
where dots and primes represent partial derivatives with respect to $t$ and
$r$ respectively.
## 3 Exterior Spacetime and Matching Conditions
Since the star is radiating, the exterior spacetime can be described by the
generalized Vaidya metric vaidya which represents a mixture of null radiation
and strings glass
$ds^{2}_{+}=-\left[1-\frac{2m(v,\mathrm{r})}{\mathrm{r}}\right]dv^{2}-2dvd\mathrm{r}+\mathrm{r}^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$
(10)
where $m(v,\mathrm{r})$ is the mass function which represents the total energy
within a sphere of radius $\mathrm{r}$. This is what distinguishes the
generalized Vaidya solution from the pure radiation Vaidya solution, which has
$m=m(v)$ where $v$ is the retarded time. The energy momentum tensor
corresponding to line element (10) is
$T^{+}_{ab}=\tilde{\mu}l_{a}l_{b}+(\rho+P)(l_{a}n_{b}+l_{b}n_{a})+Pg_{ab}$
(11)
where
$l_{a}=\delta^{0}_{a}$ (12)
$n_{a}=\frac{1}{2}\left[1-2\frac{m(v,\mathrm{r})}{\mathrm{r}}\right]\delta^{0}_{a}+\delta^{1}_{a}$
(13)
are null vectors such that $l_{a}l^{a}=n_{a}n^{a}=0$ and $l_{a}n^{a}=-1$. The
energy momentum tensor (11) can be interpreted as the matter source for the
exterior atmosphere of the star which is a superposition of pressureless null
dust and anisotropic null strings HusV ; WangA . The energy density of the
null dust radiation, string energy density and string pressure are
characterised by $\tilde{\mu}$, $\rho$ and $P$ respectively. We assume that
the string diffusion is equivalent to point particle diffusion where the
number density diffuses from higher to lower numbers subjected to the
continuity equation
$\dot{\rho}=\frac{\cal{D}}{\mathrm{r}^{2}}\frac{\partial}{\partial\mathrm{r}}\left(\mathrm{r}^{2}\frac{\partial\rho}{\partial\mathrm{r}}\right)$
(14)
where $\cal D$ is the positive coefficient of self-diffusion govin . Following
de Oliveira et al. oliv2 , we obtain the boundary conditions which include a
string density $\rho_{s}$,
$\hskip
28.45274pt\left(p_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\left(\rho_{s}\right)_{\Sigma}$
(15) $\hskip
28.45274pt\left(qB\right)_{\Sigma}=-\left(\frac{2}{{\mathrm{r}}^{2}}{{\dot{v}}^{2}}\frac{dm}{dv}\right)_{\Sigma}$
(16) $\hskip 28.45274pt\left(rB\right)_{\Sigma}=\mathrm{r}_{\Sigma}$ (17)
$\hskip
28.45274pt\left(Adt\right)_{\Sigma}=\left(1-\frac{2m}{\mathrm{r}}+2\frac{d\mathrm{r}}{dv}\right)^{1/2}_{\Sigma}dv$
(18)
Equation (15) represents the conservation of momentum flux across the stellar
boundary which we will employ in $\S$5 to determine the temporal evolution of
our model. The total energy entrapped within a radius $r$ inside $\Sigma$ is
given by
$m(r,t)=\frac{r^{3}B{\dot{B}}^{2}}{2A^{2}}-r^{2}{B^{\prime}}-\frac{r^{3}{{B^{\prime}}^{2}}}{2B}$
(19)
At the boundary, this is given by
$\hskip 28.45274ptm(v,\mathrm{r})=m(r,t)|_{\Sigma}$ (20)
and included as a boundary condition.
## 4 Perturbative Scheme
Following the method in Herrera et al. herr2 , as well as the works of Chan
et. al. chan2 and Govender et. al. gov2 , we present our model in this
section. To begin, we will assume that the fluid is in static equilibrium. The
system is then perturbed and undergoes slow shear-free dissipative collapse.
Thermodynamical quantities in the static system are represented by a zero
subscript, while those in the perturbed fluid are represented by an overhead
bar. The metric functions $A(r,t)$ and $B(r,t)$ are taken to have the same
temporal dependence, which extends to the perturbed material quantities. The
time-dependent metric functions and material quantities are given by
$\displaystyle A(r,t)$ $\displaystyle=$ $\displaystyle A_{0}(r)+\epsilon
a(r)T(t)$ (21) $\displaystyle B(r,t)$ $\displaystyle=$ $\displaystyle
B_{0}(r)+\epsilon b(r)T(t)$ (22) $\displaystyle\rho(r,t)$ $\displaystyle=$
$\displaystyle\rho_{0}(r)+\epsilon\bar{\rho}(r,t)$ (23) $\displaystyle
p_{r}(r,t)$ $\displaystyle=$ $\displaystyle
p_{r0}(r)+\epsilon{\bar{p}}_{r}(r,t)$ (24) $\displaystyle p_{t}(r,t)$
$\displaystyle=$ $\displaystyle p_{t0}(r)+\epsilon{\bar{p}}_{t}(r,t)$ (25)
$\displaystyle m(r,t)$ $\displaystyle=$ $\displaystyle
m_{0}(r)+\epsilon\bar{m}(r,t)$ (26) $\displaystyle q(r,t)$ $\displaystyle=$
$\displaystyle\epsilon\bar{q}(r,t)$ (27)
where we assume that $0<\epsilon<<1$. We observe that the temporal dependence
of the perturbative quantities, $T(t)$ is the same for both the gravitational
potentials and the thermodynamical variables. The imposition of spherical
symmetry alone implies that we have a very large gauge (coordinate) freedom to
write the line element. In adopting the form of the line element given by (1)
we exhaust all coordinate freedom with the exception of re-scaling the radial
coordinate and/or the temporal coordinates. It is clear that such re-scaling
would not change the form of (21)-(27). The choice of the perturbed variables
as given in the perturbative scheme is not unique. However, once the line
element has been chosen, the choice of the perturbed variables cannot be
varied to produce the same physics kevin1 .
The Einstein field equations for the static configuration are given by
$\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{B_{0}^{2}}\left[2\frac{B_{0}^{\prime\prime}}{B_{0}}-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{4}{r}\frac{B_{0}^{\prime}}{B_{0}}\right]$
(28) $\displaystyle p_{r0}$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\left[\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{2}{r}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{B_{0}^{\prime}}{B_{0}}\right)+2\frac{A_{0}^{\prime}}{A_{0}}\frac{B_{0}^{\prime}}{B_{0}}\right]$
(29) $\displaystyle p_{t0}$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}^{2}}\left[\frac{B_{0}^{\prime\prime}}{B_{0}}-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}+\frac{1}{r}\left(\frac{A_{0}^{\prime}}{A_{0}}+\frac{B_{0}^{\prime}}{B_{0}}\right)+\frac{A_{0}^{\prime\prime}}{A_{0}}\right]$
(30)
The perturbed field equations up to first order in $\epsilon$ can be written
as
$\displaystyle\bar{\rho}$ $\displaystyle=$
$\displaystyle-3\rho_{0}\frac{b}{B_{0}}T+\frac{1}{B_{0}^{3}}\left[-\left(\frac{B_{0}^{\prime}}{B_{0}}\right)^{2}b+2\left(\frac{B_{0}^{\prime}}{B_{0}}-\frac{2}{r}\right)b^{\prime}-2b^{\prime\prime}\right]T$
(31) $\displaystyle\bar{p}_{r}$ $\displaystyle=$
$\displaystyle-2p_{r0}\frac{b}{B_{0}}T+\frac{2}{B_{0}^{2}}\bigg{[}\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}+\frac{A_{0}^{\prime}}{A_{0}}\right)\left(\frac{b}{B_{0}}\right)^{\prime}$
(32)
$\displaystyle+\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}\right)\left(\frac{a}{A_{0}}\right)^{\prime}\bigg{]}T-2\frac{b}{{A_{0}^{2}}B_{0}}{\ddot{T}}$
$\displaystyle\bar{p}_{t}$ $\displaystyle=$
$\displaystyle-2p_{t0}\frac{b}{B_{0}}T+\frac{1}{B_{0}^{2}}\bigg{[}\left(\frac{b}{B_{0}}\right)^{\prime\prime}+\frac{1}{r}\left(\frac{b}{B_{0}}\right)^{\prime}+2\frac{A_{0}^{\prime}}{A_{0}}\left(\frac{a}{A_{0}}\right)^{\prime}$
(33)
$\displaystyle+\left(\frac{a}{A_{0}}\right)^{\prime\prime}+\frac{1}{r}\left(\frac{a}{A_{0}}\right)^{\prime}\bigg{]}T-2\frac{b}{{A_{0}^{2}}B_{0}}{\ddot{T}}$
$\displaystyle\bar{q}B$ $\displaystyle=$
$\displaystyle\frac{2}{B_{0}}\left(\frac{b}{A_{0}B_{0}}\right)^{\prime}{\dot{T}}$
(34)
The total energy enclosed within $\Sigma$ is obtained by using (19) and (26).
We separate the static and time-dependent/perturbed components and are shown
as follows
$\displaystyle m_{0}(r_{\Sigma})$ $\displaystyle=$
$\displaystyle-\left(r^{2}{B_{0}^{\prime}}+\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\right)_{\Sigma}$
(35)
and
$\displaystyle\bar{m}(r_{\Sigma},t)$ $\displaystyle=$
$\displaystyle-\left(\left[{r^{2}}b^{\prime}+\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\left(2\frac{b^{\prime}}{B_{0}^{\prime}}-\frac{b}{B_{0}}\right)\right]T(t)\right)_{\Sigma}$
(36)
In the case where the radial and tangential stresses are equal, $p_{r}=p_{t}$,
the condition of pressure isotropy for the static model is $p_{r0}=p_{t0}$
which gives
$\displaystyle\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)^{\prime}-\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)^{2}-\frac{1}{r}\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)+2\left(\frac{A^{\prime}_{0}}{A_{0}}\right)^{2}=0$
(37)
The pressure isotropy condition for the perturbed model is
$\bar{p}_{r}=\bar{p}_{t}$ which gives
$\displaystyle\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]^{\prime}-2\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]\left(\frac{A^{\prime}_{0}}{A_{0}}+\frac{B^{\prime}_{0}}{B_{0}}\right)$
$\displaystyle-\frac{1}{r}\left[\left(\frac{a}{A_{0}}\right)^{\prime}+\left(\frac{b}{B_{0}}\right)^{\prime}\right]+4\frac{A^{\prime}_{0}}{A_{0}}\left(\frac{a}{A_{0}}\right)^{\prime}=0$
(38)
This completes the outline of the perturbative scheme as applied to our choice
of metrics (1) and (10). In the next section we will examine the temporal
aspect more closely.
## 5 Explicit Form of the Temporal Function
We employ the junction conditions derived by Maharaj et al. maharaj3 to
determine the temporal evolution of our model.
$\left({p}_{r}\right)_{\Sigma}=\left(qB\right)_{\Sigma}-\rho_{s}({\mathrm{r}},v)|_{\Sigma}$
(39)
It is important to point out that (39) holds only at the boundary of the star.
We require that the static pressure vanishes at the surface via the condition
$\left(p_{r0}\right)_{\Sigma}=0$, so that the following equation is obtained
in $T(t)$, namely
$\hskip
28.45274pt\alpha_{\Sigma}T-{\ddot{T}}=2\beta_{\Sigma}{\dot{T}}+\lambda_{\Sigma}$
(40)
where $\alpha$, $\beta$ and $\lambda$ are given by
$\displaystyle\alpha(r)$ $\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{B_{0}b}\bigg{[}\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}+\frac{A_{0}^{\prime}}{A_{0}}\right)\left(\frac{b}{B_{0}}\right)^{\prime}+\left(\frac{B_{0}^{\prime}}{B_{0}}+\frac{1}{r}\right)\left(\frac{a}{A_{0}}\right)^{\prime}$
(41)
$\displaystyle+B_{0}^{2}\bigg{(}\left(\frac{3a}{2A_{0}}+\frac{b}{B_{0}}\right)p_{r0}+\left(\frac{3a}{2A_{0}}+\frac{2b}{B_{0}}\right)\rho_{s0}+$
$\displaystyle\left(\frac{a}{A_{0}}+\frac{b}{B_{0}}\right)\frac{3k}{2rB_{0}}\bigg{)}\bigg{]}$
where $\rho_{s0}$ is the constant string density.
$\hskip
28.45274pt\beta(r)=\frac{A_{0}^{2}}{2b}\left(\frac{b}{A_{0}B_{0}}\right)^{\prime}$
(42) $\lambda(r)=-\frac{A_{0}^{2}B_{0}}{2\epsilon b}\left(\rho_{s}\right)$
(43)
to be evaluated at the boundary $r_{\Sigma}$. It should be noted that $p_{r0}$
vanishes at the boundary. The diffusion equation (14) has been extensively
studied and exact several solutions have been obtained glass ; glass2 ,
maharaj3 , govin , Ghosh1 , byron1 and NaiduNF1 . One such solution of the
diffusion equation (14) for which the string density is a function of the
external radial coordinate is given by
$\rho_{s}({\mathrm{r}})=\rho_{0}+\frac{k}{{\mathrm{r}}}$ (44)
where $\rho_{0}$ and $k$ are constants. The string density profile in (44) was
utilised by Naidu et al. NaiduNF1 to study the effect of an anisotropic
atmosphere on the temperature profiles during radiative collapse. The above
choice of string profile generalizes earlier work by Govender and Thirukannesh
gov2 and Govender et al. BrasselB in which the string density was constant
($k=0$). The choice of a constant string density not only makes the problem
mathematically tractable but also simplifies the underlying physics. The
constant string distribution gives rise to pressure anisotropy in the exterior
while any inhomogeneities are suppressed. Our choice (44) allows for pressure
anisotropy and inhomogeneities due to density fluctuations. At the boundary of
the star the string density (44) can be written as
$\left(\rho_{s}\right)_{\Sigma}=\rho_{0}+\frac{k}{rB}|_{\Sigma}$ (45)
where we have invoked the junction condition (17).
It is necessary to highlight the connection between $r$ and ${\mathrm{r}}$ at
this point. The boundary of the collapsing star divides spacetime into two
distinct regions ${\cal{M}}^{-}$ and ${\cal{M}}^{+}$, respectively. The
coordinates in the interior spacetime ${\cal{M}}^{-}$ are $(t,r,\theta,\phi)$
while the coordinates in ${\cal{M}}^{+}$ are $(v,{\mathrm{r}},\theta,\phi)$.
The boundary $\Sigma$, is a time-like hypersurface described by the line
element
$ds^{2}_{\Sigma}=-d\tau^{2}+{\cal{R}}^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2})$
(46)
endowed with coordinates ${\xi}^{i}=(\tau,\theta,\phi)$ and
${\cal{R}}={\cal{R}}(\tau)$. Note that the time coordinate $\tau$ is defined
only on the surface $\Sigma$. The junction condition (17) is a consequence of
requiring the smooth matching of the line elements (1) and (10) across
$\Sigma$, ie.,
$(ds^{2}_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}=ds^{2}_{\Sigma}$ (47)
For ${\cal{M}}^{-}$ we obtain
$\displaystyle A(r_{\Sigma},t)\dot{t}$ $\displaystyle=$ $\displaystyle 1$ (48)
$\displaystyle r_{\Sigma}B(r_{\Sigma},t)$ $\displaystyle=$
$\displaystyle\cal{R}(\tau)$ (49)
where dots represent differentiation with respect to $\tau$ while for
${\cal{M}}^{+}$ we have
$\displaystyle{{\mathrm{r}}}_{\Sigma}(v)$ $\displaystyle=$
$\displaystyle{\cal{R}}(\tau)$ (50)
$\displaystyle\left(1-\displaystyle\frac{2m}{{\mathrm{r}}}+2\displaystyle\frac{d{{\mathrm{r}}}}{dv}\right)_{\Sigma}$
$\displaystyle=$ $\displaystyle\left(\frac{1}{{\dot{v}}^{2}}\right)_{\Sigma}$
(51)
We observe that (49) and (50) relate $r$ and ${\mathrm{r}}$.
We complete the expression for the temporal function $T(t)$ by solving (40).
This gives
$\hskip
28.45274ptT(t)=\frac{\lambda_{\Sigma}}{\alpha_{\Sigma}}-\exp\left[\left(-\beta_{\Sigma}+\sqrt{\alpha_{\Sigma}+\beta^{2}_{\Sigma}}\right)t\right]$
(52)
which, together with (41) and (44) and $\alpha_{\Sigma}>0$ as well as
$\beta_{\Sigma}<0$ describes a system in static equilibrium that starts to
collapse at $t=-\infty$ and continues to collapse as $t$ increases.
## 6 Dynamical Model
In order to investigate the properties of the extended form of the temporal
function, we make use of the simple Schwarzschild interior metric in isotropic
coordinates bon ; ray given by
$\displaystyle A_{0}(r)$ $\displaystyle=$ $\displaystyle
c_{1}-\frac{1}{2}\frac{\left(1-r^{2}\right)}{(1+r^{2})}$ (53) $\displaystyle
B_{0}(r)$ $\displaystyle=$ $\displaystyle\frac{2R}{1+r^{2}}$ (54)
where $c_{1}$ and $R$ are constants. Then (28) and (29) can be written as
$\displaystyle\mu_{0}$ $\displaystyle=$ $\displaystyle\frac{3}{R^{2}}$ (55)
$\displaystyle p_{r0}$ $\displaystyle=$
$\displaystyle-\frac{2c_{1}(1+r^{2})-3(1-r^{2})}{R^{2}\left[2c_{1}(1+r^{2})-(1-r^{2})\right]}$
(56)
The constant $R$ is easily determined from (55), given the initial static
energy density, and parameter $c_{1}$ is obtained from (56) by evaluation at
the boundary, giving
$c_{1}=\frac{3(1-r_{\Sigma}^{2})}{2(1+r_{\Sigma}^{2})}$ (57)
Restrictions on $r_{\Sigma}$ as given by Santos bon are noted, namely
$\frac{2m_{0}}{r_{\Sigma}}=\frac{4r_{\Sigma}^{2}}{(1+r_{\Sigma}^{2})^{2}}<\frac{3}{4}$
(58)
and
$0\leq r<r_{\Sigma}<\frac{1}{\sqrt{3}}.$ (59)
We also note that in the case $p_{r}=p_{t}$, the anisotropy parameter $\Delta$
vanishes.
## 7 Radiating Collapse
We note that (31)-(34) contain two unspecified quantities, namely $a(r)$ and
$b(r)$, which modulate the temporal part of the gravitational potentials. Thus
it is important that these are determined carefully in order to obtain a
physically meaningful dynamical model. Following Chan et al. chan2 we adopt
the following form for $b(r)$,
$\hskip 28.45274ptb(r)=\left(1+\xi f(r)\right)A_{0}B_{0}$ (60)
This choice for $b(r)$ has been widely used to investigate stability of
radiating stars undergoing dissipative collapse in the form of a radial heat
flux Kich , herr2 and chan2 . Furthermore, we follow narenee and choose the
following form $f(r)=r^{2}$. Using (60) in (38) above, we obtain an explicit
form for $a(r)$ as
$\displaystyle a(r)$ $\displaystyle=$
$\displaystyle\left(\frac{2c_{1}+1}{2}-\frac{1}{1+r^{2}}\right)\times\bigg{[}c_{2}-\frac{1-\xi}{1+r^{2}}-\frac{\xi}{2}(2c_{1}+1)(1+r^{2})\bigg{]}$
(61)
$\displaystyle+\frac{(2c_{1}+1)(1+2c_{1}(1-\xi)-11\xi)-8c_{3}R^{2}}{2(2c_{1}+1)(1+r^{2})}+$
$\displaystyle 2\xi\left(2c_{1}+1\right)\log(1+r^{2})$
where $c_{2}$ and $c_{3}$ are constants of integration. These may be set by
considering the work of Govender et al. gov1 with the simple case of the
relationship between $a(r)$ and $b(r)$ being employed, namely
$\frac{a(r)}{A_{0}(r)}=\frac{b(r)}{B_{0}(r)}$ (62)
At this stage, we point out that the radial and temporal evolution of our
model is fully determined. We use numerical data given in Table 1. for
performing graphical analyses of stability and horizon formation comparisons
which follow.
Table 1: Values of constants used in model Name | Value | Reference
---|---|---
$\mu_{0}$ | $2.36321\times 10^{8}g/cm^{3}$ | bon
$r_{\Sigma}$ | $2.159\times 10^{8}cm$ | bon
R | $26.08\times 10^{8}cm$ | (55)
$c_{1}$ | 1.49984 | (57)
$\xi$ | -0.2 | chan2 ; narenee
### 7.1 Luminosity
The luminosity for an observer at rest at infinity is given by
$L_{\infty}(v)=-\left(\frac{dm(v)}{dv}\right)_{\Sigma}$ (63)
where $v$ is the retarded time. The luminosity can then be written as
$-\frac{dm(v)}{dv}=-\frac{dm}{dt}\frac{dt}{d\tau}\frac{d\tau}{dv}$ (64)
where
$\frac{dt}{d\tau}=\frac{1}{A}\hskip 17.07182pt\mathrm{and}\hskip
17.07182pt\frac{d\tau}{dv}=\frac{1}{\dot{v}}=\frac{r\dot{B}}{A}+\frac{B+rB^{\prime}}{B}$
(65)
and $\tau$ is the proper time defined on $\Sigma$ pin with $A$ and $B$ given
by (21) and (22). For our model, this is calculated to be
$L_{\infty}(t)=-\frac{\epsilon g_{r_{\Sigma}}\dot{T}(t)}{A_{0}+\epsilon
aT(t)}\left(1+\frac{rB_{0}^{\prime}}{B_{0}+\epsilon
bT(t)}\right)|_{r=r_{\Sigma}}$ (66)
where
$g_{r_{\Sigma}}=-r^{2}b^{\prime}-\frac{r^{3}{{B_{0}^{\prime}}^{2}}}{2B_{0}}\left(2\frac{b^{\prime}}{B_{0}^{\prime}}-\frac{b}{B_{0}}\right)|_{r=r_{\Sigma}}$
(67)
### 7.2 Horizon Formation
From (63) and (64) we note that the luminosity vanishes when
$\frac{1}{\dot{v}}=0$ (68)
which determines the time of formation of the horizon as the collapse process
proceeds from $-\infty<t\leq t_{H}$ which corresponds to $-\infty<v<\infty$.
For our model we have the following instances where the luminosity vanishes.
By examining (66) we must have $\dot{T}=0$ or $g_{r_{\Sigma}}=0$.
#### 7.2.1 Case 1: $\dot{T}=0$
In the case $\dot{T}=0$, we have from the derivative of (52)
$-\eta e^{\eta t}=0$ (69)
where
$\eta=-\beta_{\Sigma}+\sqrt{\alpha_{\Sigma}+\beta_{\Sigma}^{2}}$ (70)
Thus $\dot{T}=0$ implies $\eta=0$ which forces $\alpha_{\Sigma}=0$. From the
expression for $T(t)$ (52), this is only possible if $\lambda_{\Sigma}=0$ (ie.
vanishing of the string density). We observe that removing the string density
gives a pure radiation solution and the horizon is able to form. However, the
inclusion of strings inhibits the formation of the horizon.
#### 7.2.2 Case 2: $g_{r_{\Sigma}}=0$
The second case $g_{r_{\Sigma}}=0$ gives
$\frac{2r^{3}Rh(r)}{(1+r^{2})^{4}}=0$ (71)
using $B_{0}(r)$ from (54) and $b(r)$ from (60) where we have used
$\displaystyle h(r)$ $\displaystyle=$
$\displaystyle(r^{2}-1)(3+\xi(r^{4}+3r^{2}-1))+2c_{1}(1+r^{2}-\xi+2r^{4}\xi+r^{6}\xi)$
(72)
Examining (71) we see that $g_{r_{\Sigma}}=0$ when either $R=0$ or $h(r)=0$.
For our model, we have chosen an initial density $\mu_{0}$ as given in Table
1. following numerical work by Santos bon . By (55) this means $R\neq 0$ hence
we consider the second possibility of $h(r)=0$. Given $R$, this places a
restriction on $c_{1}$ thus creating a relationship between constants $c_{1}$
and $R$.
## 8 Stability Analysis
In order to provide insight into the stability of the star, we begin with the
second law of thermodynamics, and follow the approach of herr2 . This leads to
the adiabatic parameter $\Gamma$ which is the ratio of specific heats at
constant pressure and constant volume, and taken to be constant throughout the
distribution (or at least in the region being studied). In literature
referring to this ratio in herr2 , it was considered an indicator of
stability, and called the stability factor narenee . From the expressions for
$\Gamma$ given below chan2 , it is clear that pressure anisotropy and the
presence of radiation within the stellar core affect the stability factor. For
example, if the sign of the anisotropy parameter $\Delta=(p_{t0}-p_{r0})$
changes, the stellar core becomes unstable. We observe this is in the
Newtonian limit,
$\displaystyle\Gamma$ $\displaystyle<$
$\displaystyle\frac{4}{3}+\left[\frac{1}{3|p_{r0}^{\prime}|}\left(4\frac{(p_{t0}-p_{r0})}{r}+2\alpha_{\Sigma}\xi|f^{\prime}|\right)\right]_{max}$
(73)
In agreement with classical fluid dynamics, the fluid sphere becomes more
unstable (increasing the unstable range of $\Gamma$) as a result of the
Newtonian contribution due to dissipation. Relativistic contributions from the
energy density lead to a stability factor different from its Newtonian
counterpart.
$\displaystyle\Gamma$ $\displaystyle<$
$\displaystyle\frac{4}{3}+\bigg{[}\frac{1}{3|p_{r0}^{\prime}|}\bigg{(}4\frac{(p_{t0}-p_{r0})}{r}+2\alpha_{\Sigma}\xi|f^{\prime}|+\mu_{0}\bigg{(}p_{r0}r-\xi|f^{\prime}|\bigg{)}\bigg{)}\bigg{]}_{max}$
(74)
Equation (74) shows that the unstable range of $\Gamma$ is increased by the
Newtonian term due to dissipation, as in the Newtonian limit. Furthermore, the
unstable range of $\Gamma$ is increased by the relativistic correction due to
the static background fluid configuration; however the relativistic correction
due to dissipation decreases the unstable range of $\Gamma$. Bonnor et. al.
bon state that dissipation, by diminishing that total mass entrapped inside
the fluid sphere, renders the systems less unstable.
In order to investigate the stability of our model in both the Newtonian and
post-Newtonian limits, we graphed $\Gamma$ for the case of pure radiation
(absence of string density), radiation plus constant density ($k=0$) and the
radiation and inhomogeneous string density ($\rho_{s}\neq 0,k\neq 0$). Since
our static model is described by the interior Schwarzschild solution, the
anisotropy parameter $\Delta=p_{t0}-p_{r0}$ vanishes in (73) and (74). The
modified effects due to pure string density and inhomogeneity are encoded in
$\alpha_{\Sigma}$. We have also graphed the luminosity as a function of time
for both the pure and generalized Vaidya exterior. It is important to note
that the graphs are plotted using geometrized units, where $G$ and $c$ are
taken to be unity BrasselB .
Figure 1: Adiabatic parameter in the Newtonian limit for various string
densities, $\rho_{s}=\rho_{s}(\rho_{s0},k)$ from the centre $r=0$ to the
boundary $r=r_{\Sigma}$, $r$ in units of seconds. Figure 2: Adiabatic
parameter in the post-Newtonian limit for various string densities,
$\rho_{s}=\rho_{s}(\rho_{s0},k)$ from the centre $r=0$ to the boundary
$r=r_{\Sigma}$, $r$ in units of seconds.
Figure 3: Luminosity at infinity (geometrized) versus time (seconds) for
various string densities, $\rho_{s}=\rho_{s}(\rho_{s0},k)$
.
## 9 Physical Analysis and Conclusion
Figure 1 shows the stability factor when the star is close to hydrostatic
equilibrium in the Newtonian limit. We observe that the different matter
configurations exhibit instability with $\Gamma<\frac{4}{3}$ which signifies
the onset of collapse. The inclusion of the string field drives the stellar
fluid towards instability with this effect being enhanced by inhomogeneity
($k>0$). Figure 2 displays $\Gamma$ for the post-Newtonian regime. It is clear
that the collapse process drives the fluid towards stability. The presence of
the strings and their associated anisotropy and inhomogeneity make the fluid
more stable at late times. This could be due to trapping of heat within the
stellar core due to an inhomogeneous atmosphere, thus resulting in higher core
temperatures. An increase in the core temperature results in an increase in
outward pressure thus hindering gravitational collapse. In Figure 3 we note
that inclusion of the string density field promotes an earlier time of horizon
formation (luminosity vanishes), with this effect being particularly sensitive
to string inhomogeneity. The effect of the string density on the time of
formation of the horizon was also observed by Govender megan . They reasoned
that the presence of the anisotropic strings in the exterior lowered the rate
of heat dissipation to the exterior thus leading to a lower heat production
rate within the core. This results in a lower outward radial pressure thus
allowing gravity to dominate within the stellar interior. This results in a
higher collapse rate and eventually to the horizon forming earlier.
Luminosities for collapsing neutron star models were studied by de Oliveira et
al. de1 in which they considered profiles in the $\gamma$-ray, X-ray and
visible bandwidths for core masses of $2M_{\odot}$ and $3M_{\odot}$ and a
radius of 10 km. The luminosity profiles for all three bandwidths are similar
to the profiles depicted in Figure 3. They noted that the radiation pulses do
not occur simultaneously for an observer placed at infinity from the
collapsing body. Furthermore, they show that nearly all the energy is emitted
in the form of $\gamma$-rays. It is well known that the luminosity profile
depends on the increasing gravitational redshift as the star collapses and the
increase in the effective temperature. Our study provides a possible mechanism
to explain the temperature changes within the core which manifests as the
luminosity profiles displayed in Figure 3.
It is important to note that while the dynamical (in)stability of radiating
spheres has been extensively studied in the Newtonian and post-Newtonian
approximations, our investigation is the first attempt at considering the
dynamics of the collapse process with a generalised Vaidya exterior. The
generalised Vaidya atmosphere alters the temporal evolution of the model which
impacts on the stability and time of formation of the horizon. While our study
has considered an imperfect fluid interior with heat dissipation and pressure
anisotropy it would be interesting to include shear viscosity within the
framework of extended irreversible thermodynamics.
## 10 Acknowledgments
NFN wishes to acknowledge funding from the National Research Foundation (Grant
number: 116629) as well as the DSI-NRF Centre of Excellence in Mathematical
and Statistical Sciences (CoE-MaSS). RB and MG acknowledge financial support
from the office of the DVC: Research and Innovation at the Durban University
of Technology. MG is indebted to Dr A Barnes for providing the excellent
facilities at Glenwood Boys High School where part of this work was completed.
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|
# Electrochemical impedance spectroscopy beyond linearity and stationarity —
a critical review
Noël Hallemans<EMAIL_ADDRESS>David Howey Alberto Battistel Nessa
Fereshteh Saniee Federico Scarpioni Benny Wouters Fabio La Mantia Annick
Hubin Widanalage Dhammika Widanage John Lataire Research Group Fundamental
Electricity and Instrumentation, Vrije Universiteit Brussel, Pleinlaan 2, 1050
Brussels, Belgium WMG, University of Warwick, Coventry, 7AL CV4, UK Battery
Intelligence Lab, Department of Engineering, University of Oxford, OX1 3PJ, UK
The Faraday Institution, Quad One, Harwell Science and Innovation Campus,
Didcot, UK Institute of Technical Medicine, Furtwangen University, Jakob-
Kienzle-Strasse 17, Villingen-Schwenningen 78054, Germany Fraunhofer
Institute for Manufacturing Technology and Advanced Materials IFAM, Wiener
Strasse 12, Bremen 28359, Germany Research Group Electrochemical and Surface
Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Energy Storage and Energy Conversion Systems, Bremen University, Wiener
Strasse 12, Bremen 28359, Germany
###### Abstract
Electrochemical impedance spectroscopy (EIS) is a widely used experimental
technique for characterising materials and electrode reactions by observing
their frequency-dependent impedance. Classical EIS measurements require the
electrochemical process to behave as a linear time-invariant system. However,
electrochemical processes do not naturally satisfy this assumption: the
relation between voltage and current is inherently nonlinear and evolves over
time. Examples include the corrosion of metal substrates and the cycling of
Li-ion batteries. As such, classical EIS only offers models linearised at
specific operating points. During the last decade, solutions were developed
for estimating nonlinear and time-varying impedances, contributing to more
general models. In this paper, we review the concept of impedance beyond
linearity and stationarity, and detail different methods to estimate this from
measured current and voltage data, with emphasis on frequency domain
approaches using multisine excitation. In addition to a mathematical
discussion, we measure and provide examples demonstrating impedance estimation
for a Li-ion battery, beyond linearity and stationarity, both while resting
and while charging.
###### keywords:
EIS , dynamic EIS , NLEIS , impedance , multisine , nonlinearity ,
nonstationarity , frequency domain , Li-ion , battery
††journal: Electrochimica Acta
Electrochemistry studies processes at electrode/electrolyte interfaces. These
processes involve the movement of charged species (ions or electrons),
generating a current flowing through a cell and a voltage drop over its
electrodes. Diverse noninvasive techniques relying on current and voltage
measurements have been developed for studying these processes. Typically, one
of the two quantities is kept constant, swept, or oscillated, while the other
quantity’s response is recorded. Some of the most widely used techniques are
linear sweep voltammetry, constant current chrono-potentiometry, constant-
potential chronoamperometry and electrochemical impedance spectroscopy (EIS).
In EIS [1, 2, 3, 4, 5], the dynamics of electrochemical processes are studied
by means of the impedance response to the applied current or voltage, and the
term ‘spectroscopy’ refers to the frequency dependency. It is worth
emphasising that EIS is a nonparametric data-driven technique, i.e. the
impedance is solely computed relying on current and voltage data, without
prior knowledge about the governing equations as in physics-based modelling
[6, 7, 8].
During the last two decades, EIS has gained wide popularity thanks to its
accessible implementation and broad applicability to, among others, corrosion
[9, 10, 11], batteries [12, 13, 14, 15, 16, 17, 18, 19, 20, 21], and fuel
cells [22, 23]. Nowadays, EIS is available in many commercial cyclers and
potentiostats, where a user decides on a set of frequencies and the device
measures the complex impedance values at these frequencies.
From a system theoretical perspective, the impedance is a special case of a
transfer function, which is a model for a linear time-invariant (LTI)
dynamical system [24]. In system theory, dynamical systems are systems with
memory, that is, systems defined through differential equations or,
equivalently, convolution operators. This is opposed to static systems, where
the output is simply a static function of the input. This interpretation of
the term dynamic should not be confused with its use in electrochemistry to
denote time-variation. In this article, we use the system theory convention.
Models relate the output of the system to its input, which, in the EIS case,
is the current through and the voltage over the electrodes. In _galvanostatic_
experiments the current is the input and voltage the output, while in
_potentiostatic_ experiments it is the other way around. In the remainder of
this text, EIS experiments satisfying the assumptions of LTI systems will be
referred to as ‘classical EIS’. Estimating transfer functions from input and
output data of LTI systems is a thoroughly studied problem in the field of
system identification [25, 26].
However, for electrochemical systems it is well known that (i) the relation
between current and voltage is generally nonlinear, e.g. expressed by
exponential functions in the Butler-Volmer equation, and (ii) the behaviour of
the process may evolve over time. Li-ion batteries, for example, provide such
time-variation on different time-scales. On a large time-scale, the impedance
of a fresh and an aged cell are different [12, 18]. On a shorter time-scale,
the impedance of a fully charged and a fully discharged cell are also
different [13]. Accordingly, with classical EIS, one represents
electrochemical processes which are by nature nonlinear and nonstationary with
a model for linear and stationary systems. While classical EIS may be a non-
ideal approximation for such cases, important information about the process is
nonetheless revealed by measuring within forced constraints of linearity and
stationarity at specific operating points. Linearity is achieved by applying a
small excitation amplitude such that the behaviour of the process is
linearised in a certain operating region. Stationarity is achieved by
measuring at a time, and over a timespan, when the process is and remains in
steady-state. However, these experimental conditions are very restrictive. How
can we obtain information about the nonlinear behaviour of a system when it is
only possible to perform experiments under linear constraints? How can we
study a battery _while_ it is charging or discharging? How can we study
anodising _while_ a protective layer is forming?
To relax these limitations, the concept of impedance needs extensions. The
required extensions to model systems beyond the linearity and stationarity
constraints have been studied in system theory. Nonlinear time-invariant
(NLTI) systems are commonly studied by _Volterra series_ [27, 28]. These are
convolution operators that are able to capture dynamical nonlinear behaviour,
where the transfer function is extended to so-called generalised transfer
functions. Similarly, nonlinear impedances are studied in nonlinear EIS
(NLEIS) [29, 30, 31, 32, 33, 34, 35]. Assuming that the nonlinear behaviour of
an electrochemical system can be captured by a Volterra series, the behaviour
of the process can be split into a linear part and purely nonlinear part [36].
The model for the linear part is denoted as the best linear approximation
(BLA) of the system. When the nonlinearities are small enough compared to the
linear behaviour, use of the BLA for describing the system is justified.
Nonstationary systems have been studied by extending the transfer function to
be a time-varying transfer function to describe a linear time-varying (LTV)
system [37]. The latter is a transfer function depending on both time and
frequency, expressing the evolution of the transfer function over time.
Similarly, the impedance can be extended to a _time-varying_ impedance. As an
analogy, the impedance can be considered to be like a photograph where as in
the first days of photography—think mid 19th century—the subject should remain
stationary during a certain exposure time for accumulating light on a sheet,
but the time-varying impedance can be seen as a movie of a subject during an
activity.
Over the years, techniques have been developed to _detect_ nonstationarities
and nonlinearities in measured data, where it was shown that frequency domain
identification techniques with multisine excitations are advantageous [38, 39,
40, 41, 42]. Recently, with increasing computation power, different techniques
have been developed and refined for unravelling time-varying impedance from
data [43, 44, 45, 46, 47, 48, 49]. Time-varying impedance data have already
successfully been obtained for a wide variety of electrochemical processes,
including organic coatings [50], electrorefining [51], hydrogen evolution
reactions [52], nickel hexacyanoferrate thin films [53], electrochemical
double layer capacitors [54], charging/discharging Li-ion batteries [55, 56,
57, 58, 59], and Li-plating in batteries [60, 61, 62].
Commonly, drift signals may appear (for instance the slow voltage increase
when charging a battery) during _in operando_ electrochemical measurements.
These drift signals prohibit measuring impedance data at low frequencies. A
method for removing drift signals has been developed [63], and successfully
applied to electrorefining [64], anodising [65], and Li-ion batteries [58,
59], such that the time variations of the low-frequency impedance can also be
estimated.
In this review article, we detail the required mathematical concepts
associated with impedance and their extensions beyond linearity and
stationarity. Key concepts are supported with illustrations obtained from
simulations and real-life measurements. Experiments are performed on a
pristine commercially available Samsung 48X Li-ion battery. This is a
$4.8\,\mathrm{Ah}$ $21$ $700$ cylindrical cell format with cathodes based on
lithiated metal oxide (Co, Ni, Al) and anodes based on graphite and blended
Si. The impedance is measured at different temperatures, while resting and
while charging. We opt for this case-study on Li-ion batteries since EIS is
becoming a popular tool for characterising batteries, diagnosing state-of-
health, and developing smart charging protocols. These compelling applications
are also discussed as a motivation to perform EIS beyond linearity and
stationarity. Moreover, the measurements are obtained using a commercial
potentiostat (Gamry Interface 5000E), showcasing the practical accessibility
of the discussed modelling techniques.
This article is structured as follows. First, we give a motivational example
on how impedance data beyond linearity and stationarity is promising for
battery aging diagnostics and smart charging protocols (Section 1). Next, we
define what we mean with a model for the electrochemical system (Section 2).
Then, we revisit classical EIS (Section 3), with emphasis on the limiting
constraints of linearity and stationarity. The choice between single-sine and
multisine excitations is discussed in depth. Then, we formally introduce
nonlinear and nonstationary models for electrochemical measurements through,
respectively, Volterra series and time-varying impedances (Section 4). The
Volterra series is linked to NLEIS and the BLA. Next, we detail the
experimental procedure in measuring current and voltage time-series for proper
impedance measurements (Section 5). The estimation of classical impedance data
in the frequency domain from periodic and random excitations is discussed in
Section 6. Then, we detail how the linearity and stationarity constraints can
be assessed, and nonlinearities and nonstationarities are detected by
observing the current and voltage spectra under odd random phase multisine
excitations (Section 7). Obtaining nonlinear impedance data and the BLA is
discussed in Section 8. The extraction of time-varying impedance data from the
collected current and voltage data is studied through different relevant
methods in Section 9. This has been studied in Szekeres et al. [66] also,
however, here a deeper mathematical foundation is given. In Section 10, the
performed illustrative experiments on Li-ion batteries are discussed. Finally,
conclusions are drawn and an outlook is given in Section 11.
## 1 Overview of applications of EIS for batteries
Before we get into the details of how impedance works, let us first motivate
the topic and reflect on why impedance is useful for solving some of
electrochemistry’s crucial research problems, and more importantly, why
measuring impedance beyond linearity and stationarity is promising. We do this
for the compelling case of Li-ion batteries. Here, some relevant research
problems include state-of-health (SOH) prognostics [67, 68, 69, 70, 71, 72,
73, 74, 75, 76] and smart charging [77, 78, 79, 80]. For SOH prognostics, EIS
is a powerful non-invasive tool [81]. It has been shown that classical
impedance data, mapped onto equivalent circuit model (ECM) parameters,
contains important information about degradation mechanisms [16, 73, 82].
Moreover, classical impedance data is an informative input to machine learning
algorithms to predict the remaining-useful-life (RUL) of batteries [19, 83].
Jones et al. [21, Table 1] demonstrate that the EIS data state representation
performs better than other state representations for SOH forecasting. Bizeray
et al. [84] also show that physical parameters can be identified from
classical impedance data, allowing simulation of the battery using physics-
based models such as the single-particle model (SPM). This is an non-invasive
way of parametrising cells, being more practical than tearing down cells [7].
Impedance data beyond linearity and stationarity has the potential to improve
battery SOH diagnostics since it contains additional information compared to
classical impedance data. Leveraging nonlinear and time-varying impedance data
has already been carried out for detecting Li-plating [85, 62] (an important
degradation mechanism in Li-ion batteries [70, 86]). Moreover, Kirk et al.
[34] demonstrate that nonlinear impedance data contributes to the
identifiability of physical SPM parameters.
Impedance data beyond linearity and stationarity also has the potential to
improve smart charging protocols. Katzer et al. [87] propose an adaptive fast
charging protocol relying on impedance-based detection of Li-plating [62]. In
Zhu et al. [58], we track the charge transfer resistance while charging, which
can be obtained from time-varying impedance data, and propose to adapt the
charge profile based on this time-varying charge transfer resistance.
A critical reader may argue that EIS experiments are expensive, and cannot
easily be implemented in battery management systems (BMS). However, different
solutions have been developed to implement low-cost measurement apparatus [88,
89, 90]. Here multisine excitations and frequency domain model estimation are
promising tools.
## 2 Modelling electrochemical systems
Electrochemical processes are often studied by modelling the relation between
the current $i(t)$ through, and the voltage $v(t)$ over the electrodes, where
$t$ denotes continuous time. However, this relation may also be dependent on
external parameters $p(t)$ such as the ambient temperature, the rotation rate
of the electrodes, the pressure, the concentration distribution on the
electrodes, etc.
Two types of experiments are common to measure the relation between current
and voltage. In _galvanostatic_ experiments, a current is applied and the
voltage is measured. In a general setting, the impedance is modelled as an
operator $\mathcal{G}\\{\cdot\\}$ acting on the current and external
parameters,
$\displaystyle v(t)=\mathcal{G}\\{i(t),p(t)\\}.$ (1)
In _potentiostatic_ experiments, the measurements are performed the other way
around, that is, through an operator $\mathcal{P}$,
$i(t)=\mathcal{P}\\{v(t),p(t)\\}$. Here, the notations for galvanostatic
experiments are chosen, that is, the current $i(t)$ is the excitation and the
voltage $v(t)$ is the response (appropriate for low impedance devices such as
batteries).
As mentioned in the introduction, the operators above are dynamical, that is,
operators with memory, or also called _convolution operators_ , not only
acting on the present time, but also making use of past information.
In what follows, it is detailed how the operator $\mathcal{G}$ can be modelled
by an impedance. We first consider the system under the constraints of
linearity and stationarity (Section 3), and proceed by introducing models
beyond these hard constraints (Section 4).
## 3 Classical EIS revisited
Figure 1: Equivalent electrical schematic of an electrochemical system under
LTI constraints.
### 3.1 The constraints of classical EIS
In classical EIS experiments, the external parameters $p(t)$ are assumed
constant during the experiment and the generic model (1) is simplified to
$\displaystyle v(t)=\mathrm{OCV}+\underbrace{Z\\{i(t)\\}}_{v_{Z}(t)},$ (2)
where OCV is the open circuit voltage, assumed constant, $Z$ is the classical
impedance operator and $v_{Z}(t)$ is the voltage over the impedance. An
equivalent electrical circuit representing this model is shown in Fig. 1. It
is assumed that operator $Z$ satisfies the constraints of LTI systems, that
is, linearity and stationarity, and also causality.
##### Linearity
The operator $Z$ is a linear operator, satisfying additivity and homogeneity,
respectively,
$\displaystyle Z\\{i_{1}(t)+i_{2}(t)\\}=Z\\{i_{1}(t)\\}+Z\\{i_{2}(t)\\}$ (3a)
$\displaystyle Z\\{\alpha i(t)\\}=\alpha
Z\\{i(t)\\}\quad\forall\alpha\in\mathbb{R}.$ (3b)
As such, when the current $i$ doubles, the voltage $v_{Z}$ over the impedance
also doubles.
##### Stationarity (time-invariance)
A stationary system is a system whose behaviour does not change when shifted
in time. Accordingly, the operator $Z$ is independent of the time at which the
excitation is applied:
$\displaystyle Z\\{i(t-\tau)\\}=v_{Z}(t-\tau)\quad\forall\tau\in\mathbb{R}.$
(4)
##### Causality
The response of the system is totally determined by the excitation. As a
consequence, the response to an excitation cannot precede the excitation.
For _potentiostatic_ experiments under LTI constraints, the excitation-
response relation yields,
$\displaystyle i(t)=Y\\{\underbrace{v(t)-\mathrm{OCV}}_{v_{Z}(t)}\\},$ (5)
where $Y$ is called the admittance operator, satisfying the same conditions as
the impedance operator $Z$. Notations in this paper can, hence, be converted
into potentiostatic experiments by swapping $i$ and $v_{Z}$, and replacing the
impedance $Z$ by admittance $Y$.
When charge transfer is the rate-determining step, the static relation between
current and voltage of electrochemical reactions is described by the Butler-
Volmer equation [91],
$\displaystyle
i=j_{0}S\left(\exp\left(\frac{\alpha_{a}nF}{R\mathrm{T}}v_{\mathrm{Z}}\right)-\exp\left(-\frac{\alpha_{c}nF}{R\mathrm{T}}v_{\mathrm{Z}}\right)\right),$
(6)
with $j_{0}$ the exchange current density, $S$ the surface area of the
electrode, $\mathrm{T}$ the absolute temperature in Kelvin, $n$ the number of
electrons, $F$ the Faraday constant, $R$ the universal gas constant,
$\alpha_{a}$ the anodic charge transfer coefficient, and $\alpha_{c}$ the
cathodic charge transfer coefficient. When assuming
$\alpha_{a}=\alpha_{c}=0.5$, this equation can be rewritten to obtain the
overpotential as a function of the current,
$\displaystyle
v_{\mathrm{Z}}=\frac{2R\mathrm{T}}{nF}\sinh^{-1}\left(\frac{i}{2j_{0}S}\right).$
(7)
The top left plot of Fig. 2 shows this Butler-Volmer relation between current
and voltage as a dotted line. This relation is obviously not linear. However,
the linearity constraint can approximately be satisfied by choosing the
magnitude of the excitation signal in a specific range. This can be seen by
expanding (7) as a Taylor series around $i=0$,
$\displaystyle v_{\mathrm{Z}}=\underbrace{\frac{\partial
v_{\mathrm{Z}}}{\partial i}\bigg{\rvert}_{i=0}i}_{\text{linear
term}}+\frac{\partial^{2}v_{\mathrm{Z}}}{\partial
i^{2}}\bigg{\rvert}_{i=0}i^{2}+\frac{\partial^{3}v_{\mathrm{Z}}}{\partial
i^{3}}\bigg{\rvert}_{i=0}i^{3}+\ldots$ (8)
When the current $i$ is small enough, the linear term will dominate the higher
order terms and linearity can be assumed. A rule of thumb for ensuring
linearity is that the voltage deviation should not be larger than
$15\text{\,}\mathrm{m}\mathrm{V}$ [92]. An illustration of the linearisation
of the Butler-Volmer equation is also shown in Fig. 2. A small amplitude
sinusoidal current excitation centered around zero (blue) is applied to the
Butler-Volmer equation (dotted black line), and the response is the voltage
centered around the value OCV (red). For an excitation with small amplitude,
the Butler-Volmer equation is quasi-linear in the excited range (black line).
The stationarity constraint, on the other hand, is satisfied by driving the
electrochemical system in steady-state, and applying a zero-mean current
excitation to remain in steady-state. For a battery, for instance, applying a
current excitation with a positive mean value would charge the battery and
cause nonstationary behaviour. It is also necessary that the external
parameters $p(t)$ remain constant during the experiment. A significant change
in ambient temperature, for instance, might jeopardise the stationarity
constraint.
Figure 2: Illustration of the linearisation of the Butler-Volmer equation (7)
for a small amplitude sinusoidal excitation.
The relation between the excitation and response of LTI systems is well
documented in system theory [24]. The response $v_{\mathrm{Z}}(t)$ of an LTI
system is commonly modelled by the _convolution_ of the impulse response
function $z(t)$ with the excitation $i(t)$,
$\displaystyle
v_{Z}(t)=\int_{-\infty}^{\infty}z(\tau)i(t-\tau)\mathrm{d}\tau.$ (9)
The impulse response function is the response to a Dirac pulse. Note that (9)
satisfies (3) and (4). Often, the fact that a convolution in the time domain
becomes a product in the frequency domain is exploited to rewrite (9) in a way
where the frequency dependent impedance $Z(\omega)$ appears,
$\displaystyle v_{Z}(t)=\mathcal{F}^{-1}\\{Z(\omega)I(\omega)\\}.$ (10)
Here, the impedance $Z(\omega)$ is defined as the Fourier transform of the
impulse response function $z(t)$, $\mathcal{F}^{-1}\\{\cdot\\}$ is the inverse
Fourier transform operator, and $I(\omega)$ the Fourier transform of the
current. The Fourier and inverse Fourier transforms are defined in A. Recall
that the angular frequency $\omega$ is related to the frequency $f$ as
$\omega=2\pi f$.
The voltage over the electrochemical system is then
$\displaystyle v(t)=\mathrm{OCV}+\mathcal{F}^{-1}\\{Z(\omega)I(\omega)\\}.$
(11)
Accordingly, the impedance is given by the _ratio of the Fourier transforms of
voltage and current_ ,
$\displaystyle Z(\omega)=\frac{V(\omega)}{I(\omega)}\qquad\omega\neq 0.$ (12)
Note that the impedance is not expressed at DC. The impedance at DC becomes
infinite in magnitude and with a purely imaginary phase because the linearised
OCV behaves like a capacitor.
As for any frequency response function, the impedance is a complex valued
function, often denoted as
$\displaystyle Z(\omega)=Z_{\mathrm{r}}(\omega)+jZ_{\mathrm{j}}(\omega),$ (13)
with $j$ the imaginary unit ($j^{2}=-1$), and $Z_{\mathrm{r}}(\omega)$ and
$Z_{\mathrm{j}}(\omega)$ the real and imaginary parts of the impedance,
respectively. The complex-valued impedance is also defined by its _magnitude_
and _phase_ , respectively,
$\displaystyle|Z(\omega)|$
$\displaystyle=\sqrt{Z_{\mathrm{r}}^{2}(\omega)+Z_{\mathrm{j}}^{2}(\omega)}$
(14a) $\displaystyle\angle Z(\omega)$
$\displaystyle=\left\\{\begin{array}[]{ll}\arctan\frac{Z_{\mathrm{j}}(\omega)}{Z_{\mathrm{r}}(\omega)}&\text{for
}Z_{\mathrm{r}}(\omega)\geq 0\\\
\pi+\arctan\frac{Z_{\mathrm{j}}(\omega)}{Z_{\mathrm{r}}(\omega)}&\text{for
}Z_{\mathrm{r}}(\omega)<0.\end{array}\right.$ (14d)
Note that in electrochemistry the phase of the impedance is often denoted as
$\varphi(\omega)$.
The impedance is usually visualised on a Bode plot (Fig. 3 (a-b)) as magnitude
and phase in function of frequency, or on a Nyquist chart (Fig. 3 (c)) as real
versus _negative_ imaginary part, since electrochemical systems are often
capacitative in their electrical behaviour.
Figure 3: Illustration of classical impedance data at different operating
points for a Samsung 48X Li-ion battery (see Section 10). The operating points
in the temperature-SOC plane are shown in (d), while the corresponding
impedances are shown as Bode plot in (a-b) and as Nyquist chart in (c). The
impedance data at the selected frequencies is indicated by dots, which are
connected with straight lines.
In the _potentiostatic_ case, the impedance can still be computed by (12),
since the admittance $Y(\omega)$ is defined as follows,
$\displaystyle Y(\omega)=\frac{I(\omega)}{V(\omega)}=\frac{1}{Z(\omega)}.$
(15)
#### Kramers-Kronig
The conformity of the required constraints of linearity (3) and stationarity
(4) can be validated through the Kramers-Kronig transformation [5, 93, 94,
95]. This transformation states that there is an analytical relation between
the real and imaginary parts of the impedance,
$\displaystyle Z_{\mathrm{r}}(\omega)$
$\displaystyle=Z_{\mathrm{r}}(\infty)+\frac{2}{\pi}\int_{0}^{\infty}\frac{xZ_{\mathrm{j}}(x)-\omega
Z_{\mathrm{j}}(\omega)}{x^{2}-\omega^{2}}\mathrm{d}x$ (16a) $\displaystyle
Z_{\mathrm{j}}(\omega)$
$\displaystyle=-\frac{2\omega}{\pi}\int_{0}^{\infty}\frac{Z_{\mathrm{r}}(x)-Z_{\mathrm{r}}(\omega)}{x^{2}-\omega^{2}}\mathrm{d}x.$
(16b)
In theory, when the imaginary impedance computed from the measured real part
coincides well with the measured imaginary part, or vice versa, the required
constraints are assumed to be satisfied. However, in practice, (16) is
difficult to implement since the integral needs continuous impedance data
(while only discrete data is available) and goes from DC to infinity (while
data is only available in a certain frequency band). Moreover, measurement
noise is not accounted for. Hence, an alternative approach is to fit an
equivalent circuit model based on solely the real, or imaginary, part of the
impedance data. When the model coincides well with the measured complex
impedance data, the Kramer-Kronig relation is assumed to be satisfied [96,
97].
#### Models at operating points
It is very important to stress that the impedance $Z(\omega)$ measured with
classical EIS is dependent on the local operating point (in the sense of a
Taylor expansion) at which the experiments have been performed. The operating
point is defined by the OCV value and the constant values of the external
parameters $p(t)$. As an example, Fig. 3 (d) shows operating points of a Li-
ion battery depending on the state-of-charge (SOC) and temperature. The
measured classical impedances $Z(\omega)$ of a Samsung 48X Li-ion battery (see
Section 10) at these operating points are shown as Bode (a-b) and Nyquist (c)
plots. We observe that the impedance depends on the SOC and temperature, and
that low SOC and low temperature exhibit higher impedance values. Furthermore,
for batteries, a difference in impedance would also be visible for experiments
at the same SOC and temperature, but at different SOH [12, 18].
### 3.2 Excitation
The excitation signal $i(t)$ must be ‘rich’ enough such that the response
$v(t)$ contains the information needed for extracting impedance data. Since
impedance data should be measured at a set of frequencies, it is natural to
use sinusoidal functions as excitations. Historically, EIS was performed with
single-sine excitations. Later, multisine EIS was developed [40]. Both of them
have pros and cons, which are discussed now.
Figure 4: Illustration of the response of an LTI system excited by single-sine
and multisine excitations. The first row has a single-sine excitation at
angular frequency $\omega_{1}$, the second row at $\omega_{2}=2\omega_{1}$,
and the third row has a multisine excitation, which is the sum of the two
single-sines. The OCV is plotted in black.
#### 3.2.1 Single-sine excitation
Commonly, single-sine excitations are used for EIS [98, 99]. That is, a
sinusoidal zero-mean current signal with small amplitude $I_{m}$ at a selected
angular frequency $\omega_{m}$ is applied,
$\displaystyle i(t)=I_{m}\cos(\omega_{m}t),$ (17)
and the voltage response (10) is measured,
$\displaystyle
v(t)=\mathrm{OCV}+\underbrace{|Z(\omega_{m})|I_{m}}_{V_{m}}\cos\big{(}\omega_{m}t+\underbrace{\angle
Z(\omega_{m})}_{\psi_{m}}\big{)}.$ (18)
The voltage response is a sinusoidal signal (assuming linearity due to the
small current amplitude) at the same frequency, however, with a different
amplitude $V_{m}$ and phase $\psi_{m}$, superimposed on the OCV. This is
illustrated in Fig. 2 and the two top rows of Fig. 4. The complex-valued
impedance at angular frequency $\omega_{m}$ is computed from the amplitude
scaling and phase shift between current and voltage,
$\displaystyle Z(\omega_{m})=\frac{V_{m}}{I_{m}}e^{j\psi_{m}}.$ (19)
The selected frequencies $\omega_{m}$, $m=1,2,...,M$, are applied
_sequentially_ (i.e. one after the other), usually starting from the highest
frequency and ending at the lowest one. Often the selected frequencies are
logarithmically spaced over multiple decades such as to excite processes
happening at different time-scales. The impedance at each of these frequencies
is computed. Note that it is only possible to apply the sinusoids sequentially
because stationarity (4) is assumed, and, hence, the response is independent
of the time of excitation.
#### 3.2.2 Multisine excitation
Instead of applying sinusoidal signals _sequentially_ , it is also possible to
apply the different frequencies _simultaneously_. This is the purpose of a
multisine, where the _sum_ of sinusoidal signals at different frequencies is
applied,
$\displaystyle i(t)=\sum_{m=1}^{M}I_{m}\cos(\omega_{m}t+\phi_{m}).$ (20)
Here, each of the sinusoidal components is given a different phase $\phi_{m}$
such as not to introduce constructive interference (see Section 5). Due to the
linearity constraint (3), the total response yields the sum of each of the
individual responses,
$\displaystyle
v(t)=\mathrm{OCV}+\sum_{m=1}^{M}\underbrace{|Z(\omega_{m})|I_{m}}_{V_{m}}\cos\big{(}\omega_{m}t+\underbrace{\phi_{m}+\angle
Z(\omega_{m})}_{\psi_{m}}\big{)}.$ (21)
The impedance values at the selected angular frequencies $\omega_{m}$ can
still be computed based on the amplitude scaling and the phase shifts of the
sinusoidal components,
$\displaystyle Z(\omega_{m})=\frac{V_{m}}{I_{m}}e^{j(\psi_{m}-\phi_{m})}.$
(22)
An illustration of current and voltage signals for a multisine excitation
under the assumptions of linearity and stationarity is shown in the bottom row
of Fig. 4.
#### 3.2.3 The choice of excitation signal
For classical EIS experiments, it is common to use single-sine experiments
since they are known to electrochemists and easy to use with commercially
available potentiostats. However, for broadband experiments, that is,
experiments over a large frequency band, we recommend the use of a multisine.
First of all, since for multisine experiments all frequencies are applied
_simultaneously_ , as opposed to _sequentially_ in single-sine experiments,
the experiment time is shorter [40]. Moreover, in single-sine experiments we
should wait for transients to fade out at each individual frequency, while for
multisine experiments this should only be done once, which also decreases the
experiment time.
The constraints of linearity and stationarity are easily checked for multisine
experiments by looking at the measured current and voltage data in the
frequency domain (see Section 7), while for single-sine experiments one needs
to check the Kramers-Kronig relations. It is noteworthy that for multisine
experiments one must use the frequency domain to detect nonlinearity and
nonstationarity, since it has been empirically shown that multisine
experiments always satisfy the Kramers-Kronig relations [100]. This is studied
in more detail in Section 7. Next, we demonstrate that the most commonly
stated arguments in favor of single-sine experiments over multisine ones can
be debunked.
1. 1.
The linearity constraint is believed to be easier to impose in the single-sine
setting. The amplitudes such that the response stays linear can be determined
for each of the sines separately. In the multisine case, since a high number
of frequencies are added, the amplitude of each frequency separately should
remain small such that the total multisine signal does not become too high in
magnitude (in a root-mean-square sense) and introduce nonlinearities. However,
finding the optimal excitation amplitudes for multisine experiments can also
be done by detecting nonlinear behaviour. Moreover, the amplitudes can be
dependent on the frequency too.
2. 2.
The signal-to-noise ratio (SNR) is believed to be better in the single-sine
case. This is since all the power is injected at one frequency, while, in the
multisine case, there is a trade-off between the number of selected
frequencies and the SNR. The more frequencies are excited, the smaller their
amplitudes must be for the response to remain linear, and hence, the smaller
the SNR. However, it is important to take the measurement time into account,
and compare the SNR for single-sine and multisine experiments of the same
duration [26, Section 5.2.2 p. 154]. Since single-sine experiments take a
longer time, we can measure more periods of the multisine during the same
measurement time, resulting in a better SNR.
3. 3.
For a multisine excitation, the selected frequencies should all be integer
multiples of a fundamental frequency such that the period of the multisine
equals the period of the fundamental sine. These integer multiples are called
_harmonics_. Single-sine experiments do not have this limitation. Accordingly,
multisine experiments have less flexibility in the choice of frequencies at
the low frequency bands. However, this is not a significant issue; one could
perform multiple multisine experiments with slightly different fundamental
frequencies if required.
4. 4.
The extraction of the impedance in the single-sine case is very intuitive, and
can be handled in the time domain. For multisine excitations, the Fourier
transform is usually used for separating the frequency components, and the
impedance is computed by a ratio of Fourier spectra (see Section 6). Working
in the time domain only, however, is limited because one might not see
leakage, nonlinear distortions and nonstationarity.
Another strong argument in favour of multisine experiments is that if the
system behaves in a nonstationary way, one _should_ use a multisine
excitation, as pointed out in Section 4.2.
In Section 6, we study how classical impedance data can be estimated from
measured current and voltage data.
## 4 Nonlinear and nonstationary impedance models
In this section, impedance models are introduced beyond the very restrictive
constraints of linearity and stationarity. Eliminating the linearity
constraint leads to the Volterra series model, from which the concepts of
nonlinear EIS (NLEIS) and the best linear approximation (BLA) are derived
(Section 4.1). Getting rid of the stationarity constraint leads to the time-
varying impedance model (Section 4.2). Eliminating both the constraints of
linearity and stationarity, we will study the best linear time-varying
approximation (BLTVA) (Section 4.3).
### 4.1 Nonlinear models
The choice of the excitation magnitudes $I_{m}$ in classical EIS measurements
is a compromise between the need to achieve linearity and the need for a
sufficient signal-to-noise ratio [5]. EIS measurements, just as any other
measurements, contain noise. This noise is caused by external disturbances and
the electronics of the measurement device. To improve the quality of impedance
data, one can increase the magnitudes $I_{m}$ in the excitation such that the
voltage response becomes more dominant over the noise level. However, this may
jeopardise the assumption of linearity since the amplitude span over which the
current or voltage is perturbed increases (see Fig. 5). In the case where weak
nonlinear distortions are present in measurements, the estimation of the best
linear approximation (BLA) [36] is a convenient tool. On the other hand, we
could also intentionally introduce nonlinear behaviour in experiments to
investigate additional properties of the electrochemical system, as studied in
NLEIS [30, 31, 33, 34] or intermodulated differential immittance111Impedance
and admittance as a combined concept. spectroscopy [101, 102, 103]. In this
section, we introduce a nonlinear model for EIS measurements through the
Volterra series, derive NLEIS from it, and study the concept of BLA. It is
noteworthy that these are still models at fixed local operating points,
though, but valid over a larger excitation amplitude than classical EIS
measurements.
Figure 5: Illustration of nonlinear time-invariant behaviour in EIS
measurements under a large excitation amplitude. The simulated nonlinearity
comes from the Butler-Volmer equation (7).
#### 4.1.1 The Volterra series
Nonlinear time-invariant (NLTI) systems can often be modelled by Volterra
series [27, 28, 104]. Note that this is not the case for all NLTI systems.
Systems with subharmonics or chaotic behaviour, for instance, cannot be
captured by Volterra series. Fortunately, most electrochemical systems can.
Since stationarity is still assumed and we are looking at nonlinear behaviour
at specific local operating points, the current excitation signal should be
zero-mean and the external parameters constant. Mathematically, the general
relation between output voltage and input current (1) may be written as
$\displaystyle v(t)=\mathrm{OCV}+\sum_{n=1}^{n_{\mathrm{max}}}v_{n}(t),$ (23a)
with $n_{\mathrm{max}}$ the order of the nonlinearity, with $\displaystyle
v_{n}(t)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}z_{n}(\tau_{1},...,\tau_{n})\prod_{l=1}^{n}i(t-\tau_{l})\mathrm{d}\tau_{l}$
(23b)
being the contribution of the $n$-th order nonlinearity to the voltage output
signal, and $z_{n}(\tau_{1},\ldots,\tau_{n})$ the generalised impulse response
of the $n$-th order nonlinearity.
The Volterra series can be understood as the extension of a Taylor expansion
(8) at a local operating point for dynamical systems. The nonlinear behaviour
is written as a polynomial (instead of linear) operator acting on the
excitation current.
The generalised impedances $Z_{n}$ are defined as the $n$-dimensional Fourier
transform of the generalised impulse responses $z_{n}$:
$\displaystyle
Z_{n}(\omega_{1},...,\omega_{n})=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}z_{n}(\tau_{1},\ldots,\tau_{n})\prod_{l=1}^{n}e^{-j\omega_{l}\tau_{l}}\mathrm{d}\tau_{l}.$
(24)
For nonlinearity order $n_{\mathrm{max}}=1$, the current-voltage relations of
linear systems (9) and (10) are retrieved from the Volterra series,
$\displaystyle v_{1}(t)$
$\displaystyle=\int_{-\infty}^{\infty}z_{1}(\tau)i(t-\tau)\mathrm{d}\tau=\mathcal{F}^{-1}\\{Z_{1}(\omega)I(\omega)\\}.$
(25)
##### Response to a zero-mean single-sine
Let us now look at the response of an NLTI system, described by a Volterra
series, to a zero-mean single-sine excitation $i(t)=I\cos(\omega t)$. The
contribution of the $n$-th order nonlinearity to the voltage signal yields
[104]
$\displaystyle v_{n}(t)=\sum_{h=0}^{n}|V_{n,h}|\cos\big{(}h\omega t+\angle
V_{n,h}\big{)},$ (26a) with $\displaystyle V_{n,h}$
$\displaystyle=Z_{n,h}(\omega)I^{n}\qquad h>0$ (26b) $\displaystyle
Z_{n,h}(\omega)$
$\displaystyle=\frac{1}{2^{n-1}}\sum_{\\{s_{1},\ldots,s_{n}\\}\in\mathbb{S}_{n,h}}Z_{n}(s_{1}\omega,\ldots,s_{n}\omega).$
(26c)
For harmonic $h=0$, the premultiplying factor should be $1/2^{n}$ instead of
$1/2^{n-1}$. The set $\mathbb{S}_{n,h}$ contains all possible lists
$\\{s_{1},\ldots,s_{n}\\}$, with elements $s_{1,\dots,n}\in\\{-1,1\\}$, such
that $s_{1}+\ldots+s_{n}=h$. When the set is empty, $Z_{n,h}(\omega)=0$. The
values $Z_{n,h}(\omega)$ are called the _nonlinear impedance coefficients_ ,
and come directly from the generalised impedances. A sum of $n$ elements with
values that are either $1$ or $-1$ can never be larger than $n$ and the sum of
an even number of these elements can never be odd, and vice versa. This
translates into,
$\displaystyle Z_{n,h}(\omega)=0$ $\displaystyle\text{for }h>n$ (27a)
$\displaystyle Z_{2n,2h+1}(\omega)=0$ $\displaystyle\forall n,h\in\mathbb{N}$
(27b) $\displaystyle Z_{2n+1,2h}(\omega)=0$ $\displaystyle\forall
n,h\in\mathbb{N}.$ (27c)
These seemingly complicated mathematics are better understood by looking at a
few special cases. For the linear term ($n=1$) in the Volterra series we find
that $V_{1,0}=0$ and $\mathbb{S}_{1,1}=\\{1\\}$, such that
$V_{1,1}=Z_{1}(\omega)I$. Hence, the voltage response yields
$\displaystyle v_{1}(t)$
$\displaystyle=\underbrace{|Z_{1}(\omega)|I}_{|V_{1,1}|}\cos\big{(}\omega
t+\underbrace{\angle Z_{1}(\omega)}_{\angle V_{1,1}}\big{)}.$ (28)
The response is present at the same frequency as the excitation, which is in
accordance with the expected linear output (18). For nonlinear systems, that
is $n_{\mathrm{max}}\geq 2$, we notice from (26a) that spectral content may be
present at _integer_ multiples of the excited frequency. Considering (23) and
(27) with purely quadratic ($n=2$) and purely cubic ($n=3$) nonlinearities, we
find, respectively,
$\displaystyle v_{2}(t)$ $\displaystyle=|V_{2,0}|+|V_{2,2}|\cos\big{(}2\omega
t+\angle V_{2,2}\big{)}$ $\displaystyle v_{3}(t)$
$\displaystyle=|V_{3,1}|\cos\big{(}\omega t+\angle
V_{3,1}\big{)}+|V_{3,3}|\cos\big{(}3\omega t+\angle V_{3,3}\big{)}.$ (29)
The expressions for the different $V_{x,y}$ terms are given in B. We remark
that quadratic nonlinearities introduce spectral content at the _even_ integer
multiples of the excited frequency smaller or equal to two, and that cubic
nonlinearities introduce spectral content at the _odd_ integer multiples of
the excited frequency smaller or equal to three. These results can be
generalised to higher order even and odd nonlinearities,
$\displaystyle v_{2n}(t)$
$\displaystyle=\sum_{h=0}^{n}|V_{2n,2h}|\cos\big{(}2h\omega t+\angle
V_{2n,2h}\big{)}$ $\displaystyle v_{2n+1}(t)$
$\displaystyle=\sum_{h=0}^{n}|V_{2n+1,2h+1}|\cos\big{(}(2h+1)\omega t+\angle
V_{2n+1,2h+1}\big{)}.$ (30)
Accordingly, the even and odd degree terms in the Volterra series introduce
frequency content at, respectively, even and odd integer multiples of the
excited frequency. Even nonlinear functions can be captured by the sum of even
degree monomials of the Volterra series whereas odd nonlinear functions can be
captured by odd degree monomials. Accordingly, even nonlinear behaviour is
present at even integer multiples of the excited frequency, and odd nonlinear
behaviour at odd multiples.
In contrast with the small excitation magnitude applied to the Butler-Volmer
equation (Fig. 2), where the response is only present at the excited frequency
$f_{1}$, for a larger excitation magnitude (Fig. 5), spectral content may also
be present at integer multiples of the excited frequency. For this particular
illustrative example, only odd nonlinear distortions are present, since we
fixed $\alpha_{a}=\alpha_{c}=0.5$ and therefore the Butler-Volmer equation (6)
is an odd function around $0$. However, in practice, when
$\alpha_{a}\neq\alpha_{c}$ the Butler-Volmer equation is neither even nor odd,
and hence, both even and odd nonlinear distortions are present.
Writing out the analytical expression of the response of an NLTI system
described by a Volterra series to a _multisine_ excitation is a more
complicated matter. The mathematical details are omitted for this review
paper, however, they are given in Lang and Billings [104]. Fortunately, when
the multisine consists of excited frequencies at integer multiples of a
fundamental frequency, the observations made above are still valid.
Nonlinearities will still be present at integer multiples of this fundamental
frequency. However, the distinction between even and odd nonlinear distortions
can only be made when only odd integer multiples of the fundamental frequency
are excited in the multisine. This is discussed in Section 7.2.
#### 4.1.2 Nonlinear EIS
It can be shown that the total response of the Volterra series of infinite
order ($n_{\mathrm{max}}\rightarrow\infty$), excited by a sinusoidal signal at
frequency $\omega$, $i(t)=I\cos(\omega t)$, introduces spectral content at all
the integer multiples of the excited frequency $\omega$, that is,
$\displaystyle v(t)=\mathrm{OCV}+\sum_{h=0}^{\infty}|V_{h}|\cos(h\omega
t+\angle V_{h}),$ (31a) with $\displaystyle V_{h}$
$\displaystyle=\sum_{n=1}^{\infty}V_{n,h}=\sum_{n=h}^{\infty}Z_{n,h}(\omega)I^{n}$
(31b) $\displaystyle=\sum_{r=0}^{\infty}Z_{h+2r,h}(\omega)I^{h+2r}.$ (31c)
In these equations, (23), (26) and (27) have been exploited. Only even order
nonlinear impedance coefficients larger or equal to $h$ introduce spectral
content at an even harmonic $h$, and vice versa for odd harmonics.
Nonlinear EIS [105, 31, 34] aims at measuring the _leading order_ nonlinear
impedance coefficients $Z_{h,h}(\omega)$. These are defined from (31) as [34],
$\displaystyle Z_{h,h}(\omega)=\lim_{I\rightarrow 0}\frac{V_{h}}{I^{h}}.$ (32)
Note that the unit of the $h$-th leading order nonlinear coefficient
$Z_{h,h}(\omega)$ is $\Omega/\text{A}^{h-1}$. In practice, the leading order
nonlinear coefficients are measured as,
$\displaystyle\hat{Z}_{h,h}(\omega)=\frac{V_{h}}{I^{h}}=Z_{h,h}(\omega)+\underbrace{\sum_{r=1}^{\infty}Z_{h+2r,h}(\omega)I^{2r}}_{\text{choose
$I$ such that negligible}},$ (33)
where the excitation amplitude $I$ is chosen large enough such that the $h$-th
harmonic $V_{h}$ is visible, but also small enough such that the higher order
contributions $Z_{h+2r,h}(\omega)I^{r}$, $r\geq 1$, are negligible. Note that
under these measurement conditions, $Z_{1,1}(\omega)$ is the regular impedance
$Z(\omega)$. The second leading order nonlinear impedance coefficient was
measured for Li-ion batteries in [31] and [34]. Recall that the leading order
nonlinear impedance coefficients are still models at a specific local
operating point.
#### 4.1.3 The best linear approximation
For reasons of simplicity, we may prefer to work with linear models. In
Schoukens et al. [36], NLTI systems are modelled as so-called best linear
approximations, plus a ‘nonlinear noise source’ generating nonlinear
distortions $v_{\mathrm{s}}(t)$. Hence, in this context
$\displaystyle
v(t)=\mathrm{OCV}+\underbrace{\mathcal{F}^{-1}\\{Z(\omega)I(\omega)\\}}_{v_{\mathrm{BLA}}(t)}+v_{\mathrm{s}}(t),$
(34)
where $v_{\mathrm{BLA}}(t)$ stands for the response of the BLA $Z(\omega)$.
The BLA is the ‘best’ linear model in the sense that it minimises the
nonlinear distortions in least square sense,
$\displaystyle
Z=\arg\min_{Z^{\prime}}\mathbb{E}\Biggl{\\{}\int_{-\infty}^{\infty}|v_{\mathrm{s}}(t)|^{2}\mathrm{d}t\Biggr{\\}},$
(35)
with the expected value $\mathbb{E}\\{\cdot\\}$ taken over different
realisation if the excitation signal [106]. Using Parceval’s theorem, the BLA
can also be defined in the frequency domain,
$\displaystyle Z(\omega)$
$\displaystyle=\arg\min_{Z^{\prime}(\omega)}\mathbb{E}\Bigl{\\{}|V(\omega)-Z^{\prime}(\omega)I(\omega)|^{2}\Bigr{\\}}.$
(36)
It follows that the nonlinear distortions are uncorrelated with, but not
independent of, the excitation signal. Hence, for a zero-mean single-sine
excitation, the response of the BLA is located at the excited frequency, while
the nonlinear distortions are present at the remaining integer multiples of
the excited frequency,
$\displaystyle v_{\mathrm{BLA}}(t)$ $\displaystyle=|V_{1}|\cos(\omega t+\angle
V_{1})$ (37a) $\displaystyle v_{\mathrm{s}}(t)$
$\displaystyle=|V_{0}|+\sum_{h=2}^{\infty}|V_{h}|\cos(h\omega t+\angle
V_{h}),$ (37b)
with the coefficients $V_{h}$ defined in (31). An illustration of this
decomposition into a linear response and purely nonlinear response is depicted
in Fig. 6. The linear response is the one at the excited frequency, while the
nonlinear response consists of the remaining frequencies. This BLA method has
the major advantage that a linear model can still be justified when the
nonlinear distortions are sufficiently small. Accordingly, it still makes
sense to measure an impedance from data which shows nonlinear distortions, as
long as these nonlinear distortions are small enough for the intended
application. Therefore, it is important to detect and quantify the nonlinear
distortions in measurements.
Figure 6: Decomposition of the nonlinear time-invariant response of Fig. 5
into a linear response and purely nonlinear response. Left graph: frequency
domain, right graph: time domain.
The BLA under a single-sine excitation is defined from (31) as,
$\displaystyle
Z(\omega)=\frac{V_{1}}{I}=\sum_{r=0}^{\infty}Z_{1+2r,1}(\omega)I^{2r},$ (38)
and hence, _the BLA depends on the amplitude of the excitation_. This is
illustrated for the static case in Fig. 7, the linearisation depends on the
span over which the Butler-Volmer equation is linearised. The BLA here is the
slope between voltage and current, which is clearly different for the black
and grey lines. The reasoning in the dynamic case is similar.
Note that for nonlinear systems the first generalised impedance and the first
leading order nonlinear impedance coefficients are equal, but the BLA
impedance is different: $Z_{1}(\omega)=Z_{1,1}(\omega)\neq Z(\omega)$. This is
because the higher order odd polynomials in the volterra series also contain a
linear part, leading to a spectral contribution at the excited frequency that
contributes to the BLA $Z(\omega)$.
Figure 7: The linearisation of a static nonlinear function depends on the span
it is linearised over. The red line represents the Butler-Volmer equation (7).
Linearising it over the black span gives a different BLA (the slope) than
linearising it around the grey span.
### 4.2 Nonstationary models
The stationarity constraint in classical EIS experiments is very restrictive.
The operating point, and hence also the external parameters, should be
constant during an experiment. (NL)EIS can only be performed on systems in
steady-state, resulting in models at fixed operating points. Accordingly, if
we want impedance data over various operating conditions, for instance at
different OCV values as in Fig. 3, we have to separately drive the system to
each of these operating conditions and wait for steady-state, which is very
time-consuming. Moreover, sometimes the system never reaches steady-state due
to inherently nonstationary behaviour caused by changes in thermodynamic
states and kinetically slow side processes (such as the self discharge of
energy storage systems). Furthermore, it is of great interest to study
electrochemical systems during _operation_. Examples include the formation of
film layers during anodising, the electrorefining of copper, and the charging,
discharging, and relaxation of batteries. For these examples, classical EIS or
NLEIS can only be performed as a perturbation around a rest condition, while
the evolution of the impedance during operation contains important
information. Unfortunately, this information cannot be gathered by classical
or stationary nonlinear EIS.
Nonstationarity can occur for two reasons, and they can happen simultaneously.
The first cause is that the external parameters $p(t)$ vary during the
experiment. The other cause is that the system is excited in such a way that
it does not remain in steady-state during the experiment. This happens when
superimposing a conventional excitation $i_{\text{exc}}(t)$ (see Section 3.2)
on a slow signal $i_{0}(t)$, driving the system in operating conditions,
$\displaystyle i(t)=i_{0}(t)+i_{\text{exc}}(t).$ (39)
For a battery, time-variation, for instance, occurs when the excitation is a
multisine superimposed on a constant offset $i_{0}$ that (dis)charges the
battery [59], or when a zero-mean excitation is applied right after charging
or discharging to study the relaxation behaviour [61].
#### 4.2.1 The time-varying impedance
Considering the two sources of nonstationarity, the linear voltage response
can be modelled as,
$\displaystyle
v(t)=v_{0}(t)+\int_{-\infty}^{\infty}z(\tau,t)i_{\mathrm{exc}}(\tau)\mathrm{d}\tau.$
(40)
Here $v_{0}(t)$ represents a drift signal. In the battery example, this would
be the voltage slowly increasing as the battery is charging due to a positive
constant current. The time-variation due to the external parameters $p(t)$
and/or excitation trajectory $i_{0}(t)$ are simultaneously captured by a two-
dimensional impulse response. This time-varying impulse response $z(\tau,t)$
was introduced by Zadeh [37] in 1950 for modelling LTV systems. It is a
natural extension of (9), where now the impulse response function explicitly
depends on the excitation time.
It is also shown in Battistel et al. [107] that nonstationarity (40) appears
from an NLTI system when the response is linearised along a time-varying
trajectory. This is detailed in C.
Similarly to (10), the time-varying impedance $Z(\omega,t)$ appears when
transforming (40) into the frequency domain [37],
$\displaystyle
v(t)=v_{0}(t)+\mathcal{F}^{-1}\\{Z(\omega,t)I_{\mathrm{exc}}(\omega)\\},$ (41)
with the time-varying impedance defined as,
$\displaystyle
Z(\omega,t)=\int_{-\infty}^{\infty}z(t-\tau,t)e^{-j\omega\tau}\mathrm{d}\tau.$
(42)
Accordingly, when a single-sine excitation (17) is applied, superimposed on a
slowly varying trajectory $i_{0}(t)$, the voltage response yields,
$\displaystyle
v(t)=v_{0}(t)+|Z(\omega_{m},t)|I_{m}\cos\left(\omega_{m}t+\phi_{m}+\angle
Z\left(\omega_{m},t\right)\right).$ (43)
The current excitation is, hence, modulated in amplitude and phase by
$Z(\omega_{m},t)$. Since linearity is still assumed, the response to a
multisine excitation (20) is simply the sum of the responses to each separate
sinusoidal signal,
$v(t)=v_{0}(t)\>+\>\\\
\sum_{m=1}^{M}|Z(\omega_{m},t)|I_{m}\cos\left(\omega_{m}t+\phi_{m}+\angle
Z\left(\omega_{m},t\right)\right).$ (44)
Figure 8: Illustration of the response of a Li-ion battery (which can be
modelled by a Volterra series) to zero-mean and nonzero-mean excitations with
small amplitudes. The slow parts $i_{0}(t)$ and $v_{0}(t)$ are shown in black.
The positive mean value of the current charges the battery, and, hence, the
voltage increases.
An illustration of the response of a Li-ion battery under zero-mean and
nonzero-mean small amplitude excitation is shown in Fig. 8. For the zero-mean
excitation, the response can be modelled as the OCV plus an LTI response. For
the nonzero-mean excitation, nonstationarity is introduced due to the
constant-current charging, which also might cause external parameters such as
the temperature to change. Accordingly, the response can be modelled as a
drift signal plus the response of an LTV system.
#### 4.2.2 Models along an operating trajectory
By choosing the slow trajectory $i_{0}(t)$ and/or varying external parameters
during the experiment, we obtain a linear model for the impedance along an
operating _trajectory_ instead of an operating _point_. This trajectory is the
drift of the system due to external effects during the measurement. One can,
hence, obtain more global models than with classical EIS. This is illustrated
in Fig. 9, where a battery with capacity $C$ =
$4.8\text{\,}\mathrm{A}\mathrm{h}$ is charged using a $C/2$ current, that is,
$i_{0}(t)$ = $2.4\text{\,}\mathrm{A}$. Note that here the external temperature
(a parameter), measured at the battery surface, changes due to thermal
dynamics. The SOC also changes, however, this is not an external parameter,
since it depends on the current excitation.
Figure 9: Illustration of time-varying impedance data of a Samsung 48X battery
along an operating trajectory. The operating trajectory is caused by a
charging current $i_{0}(t)=2.4$ A applied to a $4.8\,\mathrm{Ah}$ battery
placed in a thermal chamber at $5^{\circ}$C. The time-varying impedance
$Z(\omega,t)$ along the trajectory in the temperature-SOC plane (d) is shown
as Bode plot (a-b) and Nyquist plot (c).
#### 4.2.3 The importance of multisine excitation
A multisine excitation is mandatory for accurate estimation of time-varying
impedance. Since the system is changing over time during the experiment, it
would be illogical to apply single-sines _sequentially_ , since then the
impedance at each selected frequency would be computed for a different section
of the operating trajectory. The advantage of using a multisine excitation is
that many frequencies are excited _simultaneously_ , and we can obtain the
impedance at all the selected frequencies over the _entire_ operating
trajectory. In Section 9, we study how time-varying impedance data can be
estimated from measured current and voltage data under multisine excitation.
### 4.3 Nonlinear and nonstationary models
When a system is excited whilst not in steady-state, and with large excitation
amplitudes $I_{m}$, nonstationary and nonlinear behaviour may happen
_simultaneously_. The system is then denoted as nonlinear time-varying (NLTV).
In this case, the time-varying Volterra series with a superimposed drift
signal provides a general model for the response,
$\displaystyle
v(t)=v_{0}(t)+\sum_{n=1}^{n_{\mathrm{max}}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}z_{n}(\tau_{1},...,\tau_{n},t)\prod_{l=1}^{n}i_{\mathrm{exc}}(\tau_{l})\mathrm{d}\tau_{l}.$
(45)
Ideally, from such a time-varying Volterra series, we could measure time-
varying leading order nonlinear impedance functions $Z_{h,h}(\omega,t)$. This
would provide models over a large excitation amplitude and along a time-
varying trajectory. However, this has, to the best of our knowledge, not been
studied yet.
In Hallemans et al. [49] we have studied the extension of the concept of BLA
to BLTVA, that is, the best linear time-varying approximation. In this
framework, the relation between current and voltage of an NLTV system is
modelled as,
$\displaystyle
v(t)=v_{0}(t)+\underbrace{\mathcal{F}^{-1}\\{Z(\omega,t)I_{\mathrm{exc}}(\omega)\\}}_{v_{\mathrm{BLTVA}}(t)}+v_{\mathrm{s}}(t),$
(46)
with $Z(\omega,t)$ the BLTVA and $v_{\mathrm{s}}(t)$ the time-varying
nonlinear distortions. The BLTVA is a promising tool to monitor
electrochemical system impedance during operation [63, 59] (see Section 9.3).
## 5 Measuring current and voltage data
In practical settings, we cannot measure continuous-time signals $i(t)$ and
$v(t)$ over infinite periods. Instead, sampled current and voltage data over
finite periods should be collected, denoted $i(n)$ and $v(n)$ where $n$ is the
sample number. To obtain these, we apply an excitation through a potentiostat
and measure sampled and windowed current and voltage data. For extracting
time-varying impedance from this data, a multisine excitation is recommended,
as used for odd random phase (ORP) EIS [59] and dynamic multi-frequency
analysis [107].
##### Design of excitation signal
Different kinds of excitation signals are ‘rich’ enough to estimate classical
impedance; among others, single-sines, multisines, and white noise. For
obtaining stationary nonlinear impedance estimates, a single-sine excitation
should be used. For obtaining time-varying impedance data, a multisine should
be used. Since the single-sine excitation is a special case of the multisine,
we focus on the latter.
A multisine with period $T_{p}$ superimposed on a time-varying trajectory
$i_{0}(t)$ is given by
$\displaystyle
i(t)=i_{0}(t)+\underbrace{\sum_{m=1}^{M}I_{m}\cos\Big{(}\frac{2\pi
h_{m}}{T_{p}}t+\phi_{m}\Big{)}}_{i_{\mathrm{exc}}(t)}.$ (47)
The trajectory $i_{0}(t)$ is user defined—for example this could be a charging
or discharging current for a battery, or a chronoamperometry trajectory. Note
that to obtain classical or stationary NLEIS data, this trajectory should be
zero. The set of excited harmonics is defined as
$\mathbb{H}_{\text{exc}}=\\{h_{1},h_{2},...,h_{M}\\}$, and the excited angular
frequencies are accordingly $\omega_{h_{m}}=2\pi h_{m}/T_{p}$, with $T_{p}$
the period of the multisine. The harmonic numbers should be integers,
$h_{m}\in\mathbb{N}$, such that all sinusoidal signals fit an integer number
of times in the period $T_{p}$. Note that the lowest frequency in the
multisine ($f_{1}=1/T_{p}$) is inversely proportional to the period of the
multisine. For a single-sine excitation, only $f_{1}$ is excited. In our
definition, the natural numbers $\mathbb{N}$ do not include zero, while the
set $\mathbb{N}_{0}$ does include zero. The amplitudes are selected by the
user, depending on the application. The phases are most of the time chosen
such as to minimise the crest factor of the overall multisine, that is, as not
to introduce constructive interference when adding sines to each other.
Different approaches can be used for this, including random phases picked from
a uniform distribution in $[0,2\pi)$, and mathematical optimisation techniques
such as the Schröder phase or DFT-based iterative algorithms to minimise the
crest factor [108, 109, 110, 111, 112, 57].
The name _odd random phase_ electrochemical impedance spectroscopy denotes
impedance measurements under multisine excitation with only odd harmonics
excited ($h_{m}\in 2\mathbb{N}_{0}+1$), with random phases. This excitation
signal was introduced by Hubin and Pintelon et al. [40, 41]. In Section 7, we
show that inherent nonlinearity and nonstationarity in electrochemical systems
can easily be detected using an ORP multisine excitation, and we discuss why
it is advantageous to only excite _odd_ harmonics.
##### Windowing
We cannot measure signals for an infinitely long time, but only for a certain
period $t\in[0,T)$. The measurement time $T$ is chosen to measure a certain
ongoing reaction, for instance a charge cycle of a Li-ion battery. To avoid
spectral leakage in the frequency domain, an _integer_ number of periods of a
multisine excitation should be measured, that is, $T=PT_{p}$ with
$P\in\mathbb{N}$ [26, Section 2.2.3, p 40]222Note that measuring an integer
number of periods is only a requirement when the impedance estimation is
performed in the frequency domain, which is the case for multisine
experiments, but not necessarily for single-sine experiments.. This is not
always possible, but it is strongly recommended. Moreover, for obtaining NLEIS
or time-varying impedance data, measuring an integer number of periods is a
requirement.
##### Sampling
Only a sampled representation of the continuous signal can be recorded, at a
sampling frequency $f_{s}$. Following the Shannon-Nyquist sampling theorem,
this sampling frequency should be greater than twice the highest frequency in
the measurements to avoid spectral aliasing. The sampling period is
$T_{s}=1/f_{s}$. It is important that the data is uniformly sampled.
##### Measuring the data
The sampled and windowed multisine current data is applied to an
electrochemical device using a potentiostat. The potentiostat uses a digital-
to-analog converter (DAC) to transform the generated time-series to a
continuous signal. User-defined excitation is not always available in
commercial potentiostats, but user-defined excitation is essential for the
techniques in this article. Accordingly, multiple periods of the multisine
excitation signal can be applied, and the potentiostat then measures the
actual current and the voltage, which are also windowed and sampled.
The collected data can be written as follows,
$\displaystyle\mathcal{D}_{\text{time}}=\left\\{\begin{array}[]{ll}&[i(0),i(1),\ldots,i(N-1)]\\\
&[v(0),v(1),\ldots,v(N-1)]\end{array}\right\\},$ (50)
where $x(n)$ is shorthand notation for $x(nT_{s})$, $x=i,v$. The number of
samples is $N=Tf_{s}$.
##### frequency domain data
Within the constraints of LTI systems, the impedance is defined as the ratio
of the Fourier transforms of voltage and current (12). Hence, it would be
appropriate to directly compute the impedance in the frequency domain.
However, the Fourier transform acts on continuous signals, while only
discrete-time measurements (50) can be collected from potentiostats.
Fortunately, the spectrum of time-series can be computed by replacing the
Fourier integral by a discrete sum. This is called the discrete Fourier
transform333In 1965, Cooley and Tukey designed a highly-efficient algorithm to
compute the DFT, which rapidly popularised frequency domain signal processing.
This algorithm became known as the ‘fast Fourier transform’ (FFT) [113] and is
still used to date. (DFT),
$\displaystyle X(k)=\frac{1}{N}\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N}\quad
k=0,1,...,N-1.$ (51)
Here, $x(n)$ is again shorthand for $x(nT_{s})$ and $x=i,v$. The DFT index $k$
corresponds to the angular frequency $\omega_{k}=2\pi k/T$ or frequency
$f_{k}=k/T$. The frequency domain current and voltage data yields,
$\displaystyle\mathcal{D}_{\text{freq}}=\left\\{\begin{array}[]{ll}&[I(0),I(1),\ldots,I(N-1)]\\\
&[V(0),V(1),\ldots,V(N-1)]\end{array}\right\\}.$ (54)
When periodic time domain data is measured for an integer number of periods
and sampled satisfying Shannon-Nyquist’s theorem, the DFT coincides with the
Fourier transform, evaluated on the DFT grid $f_{k}=k/T$ with $k=0,1,...,N-1$
[26, Section 2.2, p 34].
An illustration of one period of a windowed and sampled odd random phase
multisine signal in the time and frequency domain, with the $7$-th harmonic
unexcited for detection of odd nonlinear distortions, is shown in the left
plots of Fig. 10. For the time domain plots, the full line is the continuous
signal, while the dots are the sampled data. For the frequency domain plots,
the DFT grid is indicated by vertical red bars on the frequency axis. Since
DFT lines are intentionally left open at even harmonics and the odd harmonic
$7$, it is possible to detect, quantify and classify nonlinear distortions in
the response to one period of the excitation. However, if the system is also
nonstationary, it is not possible to distinguish between nonlinear and
nonstationary behaviour when measuring only one period.
Figure 10: A sampled and windowed odd random-phase multisine in time and
frequency domain. Top row: continuous signal (full line) and sampled data
(blue dots). Bottom row: DFT of the sampled data, with the DFT grid indicated
by red (original grid) and black (grid for $P=3$ measured periods) vertical
bars on the frequency axis.
##### Increasing the frequency resolution
Measuring more than one period increases the _frequency resolution_ , that is,
the distance between two DFT lines $f_{s}/N=1/T$ becomes smaller. This is
illustrated in the right plots of Fig. 10. For one measured period, the DFT
grid is indicated in red, for three measured periods, in black. Nonlinearities
in the spectrum of the response can _only_ be present at the DFT lines
indicated in red, since these are the integer multiples of the fundamental
excitation frequency $1/T_{p}$. The effect of nonstationarities can be present
at all DFT lines, however. Accordingly, measuring a large number of periods is
commonly used to distinguish between nonlinear and nonstationary behaviour,
and the nonstationary behaviour can be modelled from the response spectrum.
##### Influence of measurement noise
We usually assume that noise in the measured data, $i_{\text{meas}}(n)$ and
$v_{\text{meas}}(n)$, is additive,
$\displaystyle x_{\text{meas}}(n)$ $\displaystyle=x(n)+\mathrm{n}_{x}(n)\qquad
x=i,v,$ (55)
and the noise time-series $\mathrm{n}_{x}(n)$ is iid (independent and
identically distributed), zero-mean and Gaussian:
$\displaystyle\mathrm{n}_{x}(n)\sim\mathcal{N}(0,\sigma^{2}_{\mathrm{n}_{x}})\qquad
x=i,v.$ (56)
Here, $\sigma^{2}_{\mathrm{n}_{i}}$ and $\sigma^{2}_{\mathrm{n}_{v}}$ are the
noise variances on the current and voltage, respectively. The DFTs of the
noise time series, $\mathrm{N}_{I}(k)$ and $\mathrm{N}_{V}(k)$ are circular
complex Gaussian444A two-dimensional Gaussian distribution on the complex
plane. distributed. However, the variances scale inversely with the number of
samples $N$,
$\displaystyle\mathrm{N}_{X}(k)\sim\mathcal{N}_{c}\Big{(}0,\underbrace{\frac{\sigma^{2}_{\mathrm{n}_{x}}}{N}}_{\sigma^{2}_{\mathrm{N}_{X}}(k)}\Big{)}\qquad
x=i,v.$ (57)
We can interpret this result as follows: by measuring a higher number of
samples $N$, the number of DFT lines increases, and, hence, the noise is
distributed over more DFT lines, such that the variance of the circular
complex distributed noise at each DFT line decreases. As a consequence, we can
show that the frequency domain SNR increases with the square root of the
number of measured samples,
$\displaystyle\mathrm{SNR}_{X}(k)=\sqrt{\frac{|X(k)|^{2}}{\sigma^{2}_{\mathrm{N}_{X}}(k)}}=\sqrt{N}\frac{|X(k)|}{\sigma_{n_{x}}}.$
(58)
Accordingly, by measuring a higher number of samples $N$, we increase the SNR.
In practice, the SNR is improved by increasing the number of measured periods
$P$ of the data.
## 6 Classical frequency domain impedance estimation
In the early seventies, Creason and Smith [114, 115, 116] adopted the recently
discovered FFT for classical EIS directly from frequency domain. These
techniques, referred to as FFT-EIS, measure the impedance starting from DFT
data (54). An important question is the choice of the excitation signal. As
explained earlier, a zero-mean excitation is needed for stationarity. In
[116], different zero-mean excitation signals are studied: periodic
excitations, transient inputs, and band-limited white noise. The conclusion is
drawn that periodic excitation, with excited frequencies lying on the DFT
grid, are superior. However, it is not always possible to apply periodic
excitation and measure over an integer number of periods. Think for instance
about a low-cost measurement apparatus for a BMS where only short and fixed-
length data records can be handled. Hence, estimating techniques are also
needed for random excitations. In 1975, Blanc et al. [117] proposed the so-
called ‘pseudo-white noise’ technique (or ‘méthode du bruit blanc’, since this
article was written in French). Pseudo-white noise may also be used as an
excitation, and the impedance is computed as the ratio of the cross- and auto
power spectra in the frequency domain. Howey et al. [88] also used the ratio
of the cross- and auto power spectra in the frequency domain to perform fast
impedance measurements on a self-made low-cost excitation and measurement
system for batteries that could be used in a BMS. The same issue of the choice
of excitation signal for frequency domain system identification is studied in
[108] and in Pintelon and Schoukens’ book [26, Chapter 2]. We now
mathematically formalise the impedance estimation for periodic and random
excitation signals.
##### Periodic excitation
When measuring an integer number of periods $P$ in steady-state under a
periodic excitation, for instance a multisine, the DFT is exactly a sampled
version of the continuous Fourier transform [26, Section 2.2, p 34]. Hence,
the impedance can simply be computed from (12) as the ratio of the voltage and
current spectra,
$\displaystyle\hat{Z}(\omega_{k})=\frac{V(k)}{I(k)}$
$\displaystyle\omega_{k}=\frac{2\pi k}{T},\ k\in P\mathbb{H}_{\text{exc}},$
(59)
with $\mathbb{H}_{\text{exc}}$ the excited harmonics of the excitation signal.
A measurement example is shown in Fig. 11. Since the OCV in the voltage is
only present at DC (zero frequency), it has no influence on the estimate of
the impedance at positive frequencies. Note that in perfect LTI conditions,
only measurement noise causes an uncertainty on the estimate (59). Increasing
the number of periods $P$ generates an averaging effect, which reduces this
uncertainty [26, Section 2.4, p 44]. Periodic excitation is recommended when
possible.
Figure 11: Example of classical impedance estimation for a Li-ion battery at
$10$% SOC and $25^{\circ}$C under a periodic multisine excitation measured for
$P=10$ periods. The excited lines, $k\in P\mathbb{H}_{\text{exc}}$, are
indicated with vertical lines with large dots and crosses. The remaining DFT
lines (small dots) contain noise and a small drift signal.
##### Random excitations
For random excitations, such as pseudo-white noise, pseudo-random binary
sequences (PRBS) or multisine excitations not measured for an integer number
of periods, the DFT does not correspond to the continuous Fourier transform
anymore due to transients. These transients can be reduced by using windows,
e.g. the Hann window. For random excitation, it is recommended to measure the
impedance by the ratio of the cross- $\hat{S}_{VI}(k)$ and auto-spectra
$\hat{S}_{II}(k)$ [26, Section 2.6, p. 54],
$\displaystyle\hat{Z}(\omega_{k})=\frac{\hat{S}_{VI}(k)}{\hat{S}_{II}(k)}=\frac{\sum_{m=1}^{M}V_{[m]}(k)I^{*}_{[m]}(k)}{\sum_{m=1}^{M}I_{[m]}(k)I^{*}_{[m]}(k)},$
(60)
where the superscript ∗ stands for the complex conjugate, and the subscript
[m] stands for different experiments. Averaging over many different
experiments reduces the transient and avoids divisions by zero. Note that this
only makes sense in stationary conditions where multiple experiments can be
gathered.
As a special case, Blanc [117] chooses the pseudo-white noise excitation
$i(t)$ such that $I(k)I^{*}(k)=1$ in the frequency band of interest.
Accordingly, only the numerator of (60) needs to be computed to estimate the
impedance. Note that these cross- and auto spectra correspond to correlations
in the time domain [118], as also studied by Blanc [117].
Other, more involved, frequency domain techniques are possible for measuring
the impedance using random excitation [119, 120]. Here, local parametric
modelling and Gaussian process regression are used to separate the impedance
from the transients. Such techniques are promising for low-cost experiments
where periodic excitation signals cannot be applied. However, explaining these
in detail goes beyond the scope of this article.
## 7 Detection of nonlinearity and nonstationarity
It is empirically shown in You et al. [100] that classical impedance data
$\hat{Z}(\omega_{k})$ obtained from broadband excitation using frequency
domain techniques (as studied in Section 6) always satisfies the Kramers-
Kronig relations. Hence, the latter cannot assess to which extent measured
multisine data satisfies the assumptions of linearity and stationarity, and
another tool is required for this purpose. Inherently nonlinear and
nonstationary behaviour of electrochemical systems is easily detected by
applying a zero-mean ORP excitation (47) (with $i_{0}(t)=0$) and studying the
measured frequency domain data (54) [121, 40, 42, 49]. We recommend the use of
this tool, which we now illustrate on a simplistic example that demonstrates
its advantages.
### 7.1 An example with ORP multisine excitation
Consider an odd random phase multisine with excited harmonics
$\mathbb{H}_{\mathrm{exc}}=\\{1,3,7\\}$, that is, excited frequencies
$1/T_{p}$, $3/T_{p}$ and $7/T_{p}$. A measurement is performed for a duration
$T=10T_{p}$, i.e., $P=10$ periods are measured. Based on the frequency domain
voltage response data $V(k)$, it is possible to detect whether the
electrochemical system behaves as an LTI, NLTI, LTV or NLTV system, as
illustrated in Fig. 12.
Figure 12: Detection of nonlinear and nonstationary behaviour in measured
data. An odd random phase multisine signal (blue crosses) with a detector for
odd nonlinear distortions is applied for $P=10$ periods at two different
excitation amplitudes. In the LTI case, the voltage response only has spectral
content at DC (dots at $f=0\,\mathrm{Hz}$) and at the excited frequencies. For
a higher excitation amplitude, the system might become NLTI, and spectral
content appears at integer multiples of the fundamental frequency. When the
system is nonstationary, hyperbolic shapes become visible around the excited
frequencies, and in the nonlinear case around all the integer multiples of the
fundamental frequency.
LTI:
$V(k)$ has only spectral content at DC and the excited harmonics
$P\mathbb{H}_{\mathrm{exc}}$.
NLTI:
$V(k)$ has spectral content at DC and the excited frequencies, and also at
harmonics that are integer multiples of the fundamental frequency $1/T_{p}$,
i.e., $P\mathbb{H}_{\mathrm{nl}}$ with
$\mathbb{H}_{\mathrm{nl}}=\\{0,1,2,...\\}$. Both even and odd nonlinearities
are detected, since there is spectral content at the even left out harmonics
($0$, $2$, $4$, $6$) and at the odd left out harmonic ($5$).
LTV:
$V(k)$ consists of hyperbolic-like shapes around DC and the excited
frequencies. These shapes, called _skirts_ in the literature [122], are due to
the smooth time-varying function modulating the multisine components.
NLTV:
$V(k)$ consists of hyperbolic-like shapes around DC, the excited frequencies
and the nonlinear harmonics.
In Fig. 12 noiseless data are considered. However, real-life measurements also
contain noise that is distributed over all DFT frequencies. Still, it is easy
to distinguish between nonlinearities, nonstationarities and noise. This can,
for instance, be seen from Fig. 11 where the measured voltage spectrum clearly
satisfies the LTI constraints and the noise is at least $1000$ times smaller
than the linear response. For a _nonzero-mean_ multisine excitation, the
response will likely be nonstationary. This would look like a sampled version
of Fig. 8.
From all the spectra illustrated in Fig. 12, only the LTI one should be
processed with classical EIS techniques, as discussed in the previous section.
If only nonlinearities are detected, one can estimate the BLA (see Section
8.2). If time-variation is detected, but no nonlinear distortions, time-
varying impedance data can be estimated from the data record (see Section 9),
if there are nonlinear distortions too, but they are limited, the BLTVA can be
estimated from the data using operando EIS (see Section 9.3).
### 7.2 Advantages of an ORP multisine excitation
By measuring the response of the electrochemical system to an ORP multisine
excitation for a large number of periods, we can easily detect the presence of
nonstationarities and nonlinearities, and estimate the noise level. It is
possible to do this over a wide frequency range, in one single experiment, as
studied in [38, 39, 40, 42]. The advantages of using a multisine excitation
over a single-sine one have already been discussed in depth (Section 3.2.3).
However, an interested reader might wonder why it is particularly advantageous
to use a multisine with _random phases_ and only _odd_ excited harmonics. The
reasons are as follows:
The random phases are chosen to minimise the crest factor of the multisine.
Note that there exist optimal ways to achieve this, however, using random
phases has the advantage of simple implementation [57].
The choice to excite only _odd_ harmonics, on the other hand, is more
involved. In fact, the advantage is twofold. First, it allows distinguishing
between even and odd nonlinear distortions. The nonlinear distortions are
present at DFT lines which are the product of the excited harmonic numbers and
the degrees of the relevant monomials in the Volterra series [104]. When
exciting only odd harmonics, for odd nonlinear behaviour (odd degree monomials
in the Volterra series), the nonlinear distortions are present at odd
harmonics ($\mathrm{odd}\times\mathrm{odd}=\mathrm{odd}$), while for even
nonlinear behaviour (even degree monomials in the Volterra series) they are
present at even harmonics ($\mathrm{odd}\times\mathrm{even}=\mathrm{even}$).
If we would only excite even harmonics, it would not be possible to
distinguish between the two ($\mathrm{even}\times\mathrm{odd}=\mathrm{even}$
and $\mathrm{even}\times\mathrm{even}=\mathrm{even}$). When exciting all
harmonics, even and odd, it is also not possible to distinguish between the
types of nonlinearities.
The second advantage is that even nonlinearities do not have a contribution at
the excited frequencies. Hence, when computing the BLA or BLTVA using a
multisine excitation, even nonlinear distortions introduce no uncertainty, as
further discussed in Section 8.2.
## 8 Nonlinear impedance estimation
### 8.1 Leading order nonlinear impedance estimation
It is appropriate to perform NLEIS (Section 4.1.2) in the frequency domain.
For this purpose, we apply a zero mean single-sine excitation, that is,
$i_{0}(t)=0$, $M=1$ and $\mathbb{H}_{\text{exc}}=\\{1\\}$ in (47), measured
over an integer number of periods $P\geq 1$. When choosing the right amplitude
$I_{1}$, as discussed in Section 4.1.2, we obtain estimates of the leading
order nonlinear impedance coefficients as,
$\displaystyle\hat{Z}_{h,h}(\omega_{P})=\frac{V(hP)}{I(P)^{h}}$
$\displaystyle\omega_{P}=\frac{2\pi P}{T}=\frac{2\pi}{T_{p}}.$ (61)
Measuring a higher number of periods $P>1$ introduces an averaging effect of
the stochastic noise in the experiments. Different frequencies $\omega_{P}$
are applied _sequentially_. NLEIS with multisine excitation should also be
possible, however not yet investigated. One has to be careful for leaving
enough gaps in the excitation signal to see the integer harmonics in the
response.
### 8.2 Best linear approximation
Independently of the excitation amplitude of _single-sine_ experiments, the
BLA can be estimated from (38), where different frequencies can be applied
_sequentially_ ,
$\displaystyle\hat{Z}(\omega_{P})=\frac{V(P)}{I(P)}$
$\displaystyle\omega_{P}=\frac{2\pi P}{T}=\frac{2\pi}{T_{p}}.$ (62)
For _multisine_ excitations measured for an integer number of periods $P$, the
BLA impedance estimate is computed exactly as for classical impedance
experiments (59). If an ORP multisine is used, that is, exciting odd
harmonics, only odd nonlinearities will introduce spectral content at the
excited frequencies. Indeed, even nonlinear distortions can only be present at
even multiples of the odd harmonics, resulting in even harmonics. Accordingly,
the BLA will have uncertainties due to noise and also due to odd nonlinear
distortions that introduce spectral content at other odd harmonics (see [26,
Section 3.4, p. 78]). The effect of the noise can again be reduced by
measuring a larger number of periods. The odd nonlinear distortions, on the
other hand, do not depend on the number of measured periods. They only depend
on the system and the excitation, that is, the amplitudes and excited
frequencies. The uncertainty of the BLA can be reduced by exciting fewer
frequencies in the multisine. Recall that the BLA depends on the excitation
(38), however, using Riemann-equivalent multisines, one can adapt the number
of excited frequencies without changing the BLA [123].
## 9 Time-varying impedance estimation
Estimating the time-varying impedance $Z(\omega_{k},t)$ cannot simply be done
by dividing DFT spectra. Different approaches have been developed over the
last two decades to reveal the time-varying impedance from data. Both single-
sine and multisine techniques exist. However, in this review paper we
intentionally restrict to multisine excitations, since we believe this is the
only correct solution (see Section 4.2.3). With ever-growing computation
power, more and more complex techniques have been developed. There are two
main approaches. One implies the use of windowing/filtering of the current and
voltage data in time or frequency domain, respectively, and gives an average
value of the impedance inside the selected time frame. Another uses the
mathematical regression of the voltage response spectrum to extract the time-
variation. These approaches are detailed next, chronologically.
### 9.1 FFT-EIS applied to nonstationary data
The first attempt to obtain time-varying impedance data was by Bond, Schwall
and Smith in 1977 [43, 124, 125]. This simple and intuitive approach is an
extension of Smith’s FFT-EIS discussed in Section 6.
Instead of applying only a zero-mean excitation signal, the excitation signal
is now superimposed on a slower cyclic voltammetry excitation, introducing
nonstationarity in the measured system. While collecting current and voltage
data, the FFT-EIS is performed on short subrecords. This is a type of
windowing. Accordingly, for each subrecord, which corresponds to a certain
point in the voltage trajectory of the cyclic voltammetry, the time-averaged
impedance of the subrecord is computed using the techniques of Section 6.
Basically, classical impedance measurements are applied to nonstationary data,
but in very small time-windows such that the impedance could be assumed
constant within each subrecord.
Since the period length is inversely proportional to the frequency, the
impedance is only measured at high frequencies, such that short subrecords can
be taken. Results of the impedance (or admittance) varying over the cyclic
voltammetry trajectory are only shown for two particular frequencies above
$300\,\mathrm{Hz}$. This is a strong limitation of the technique.
Later, in 1997, FFT-EIS was implemented on a micro computer [126], for a
multisine superimposed onto a staircase DC ramped voltage excitation. Time-
varying impedance measurements could be obtained within the more broadband
frequency range of $50\,\mathrm{Hz}$ – $50\,\mathrm{kHz}$. Later, Sacci and
Harrington [46, 47], developed measurement apparatus to obtain time-varying
impedance data using FFT-EIS with multisine excitation superimposed on a
cyclic voltammogram.
##### Comment
It is noteworthy that obtaining time-varying impedance data is much easier at
high frequencies. This on the one hand due to low frequency noise (so-called
$1/f$ noise), but also due to the drift signal (or also called trend)
$v_{0}(t)$ in (41) having a decreasing shape over frequency, and hence, hiding
low-frequency content [63, 59]. Moreover, for logarithmically distributed
excited frequencies, nonstationarities are more easily detected at the high
frequencies since the excited frequencies are more separated (expressed in DFT
lines), thus making the ‘skirts’ better visible. Remarkably, time-varying
behaviour is mostly present at the low frequencies, this is possibly due to
mass transport and/or charge transfer kinetics. Also, when measuring at high
frequencies, the measurement time is usually shorter, therefore
nonstationarities are smaller.
### 9.2 Time-frequency analysis methods
During the nineties, time-frequency analysis, as described by L. Cohen555Not
to confuse with the great Canadian singer-songwriter. [127], became a widely
used tool in signal processing. Time-frequency analysis describes how the
spectral content of a signal $x(t)$ is changing in time, which is exactly
needed for time-varying impedance estimation. The workhorse for this job is
the short-time Fourier transform (STFT), which computes the Fourier transform
of a signal restricted by a window function $w(t)$,
$\displaystyle\mathrm{STFT}\\{x\\}(\omega,t)$
$\displaystyle=\int_{-\infty}^{\infty}w(t^{\prime}-t)x(t^{\prime})e^{-j\omega
t^{\prime}}\mathrm{d}t^{\prime}$
$\displaystyle=\mathcal{F}\\{w(t^{\prime}-t)x(t^{\prime})\\},$ (63)
with the Fourier transform acting on the variable $t^{\prime}$. The most
commonly used window functions are the Gaussian, Hamming and Blackmann-Harris
windows. These windows reach their largest values in the center, and decrease
smoothly towards the borders.
#### 9.2.1 STFT-EIS
After the millenium change, Darowicki [44, 128, 45] proposed to estimate the
time-varying impedance under multisine excitation as the ratio of the STFT of
voltage and current,
$\displaystyle
Z(\omega,t)=\frac{\mathcal{F}\\{w(t^{\prime}-t)v(t^{\prime})\\}}{\mathcal{F}\\{w(t^{\prime}-t)i(t^{\prime})\\}},$
(64)
with again the Fourier transforms acting on the variable $t^{\prime}$.
Assuming the window $w(t)$ to be a symmetric function centered around zero,
the impedance at time $t$ is computed by selecting the time domain data around
this time-instant, and dividing the corresponding spectra of voltage and
current. Note that FFT-EIS on nonstationary data is a special case of STFT-
EIS, with a rectangular window $w(t)$.
In practice, of course, only discrete-time data (50) is available. The
impedance can then be computed by [45],
$\displaystyle\hat{Z}(\omega_{k},t_{n})=\frac{V(k,n)I^{*}(k,n)}{I(k,n)I^{*}(k,n)}$
$\displaystyle t_{n}=nT_{s},$ (65a) with the DFT acting on subrecords of
$N_{w}$ data points centered around $n$, which are windowed by the function
$w(t)$, $\displaystyle X(k,n)$
$\displaystyle=\frac{1}{N_{w}}\sum_{n^{\prime}=n-N_{w}/2}^{n+N_{w}/2-1}w(n^{\prime}-n)x(n^{\prime})e^{-j2\pi
kn^{\prime}/N_{w}},$ (65b)
$x=i,v$. A division of the cross- and auto spectra is chosen here since the
signals are not periodic anymore, and this estimator is recommended for random
excitations (see Section 6). The time-varying impedance estimate is computed
at the harmonics $k$ where the numerator of (65a) shows peak values,
corresponding to the excited frequencies of the multisine. To make these peaks
visible and not overlapping too much, it is important that enough periods are
measured. Moreover, the choice of the window $w(t)$ is crucial in this
technique.
#### 9.2.2 Dynamic Multi-Frequency Analysis
Later, Battistel and La Mantia [48, 107, 54] proposed the dynamic multi-
frequency analysis (DMFA) for estimating time-varying impedance data. Here,
the time-varying impedance is computed by filtering the current and voltage
spectra around the excited frequencies of a multisine, and taking the inverse
Fourier transform,
$\displaystyle
Z(\omega,t)=\frac{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)V(\omega^{\prime})\\}}{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)I(\omega^{\prime})\\}},$
(66)
with the inverse Fourier transforms acting on $\omega^{\prime}$. The function
$W(\omega)$ implements a filtering operation. This process is called
_quadrature filtering_ since only the spectrum at positive frequencies is
considered and the inverse Fourier transforms give complex-valued results.
For sampled frequency domain data (54), with multisine excitation, the time-
varying impedance data estimates translate into
$\displaystyle\hat{Z}(\omega_{k},t_{n})=\frac{v(k,n)}{i(k,n)},$ (67a) with for
$k\in P\mathbb{H}_{\text{exc}}$ the inverse DFT acting on frequency domain
subrecords of $N_{W}$ data points centered around $k$, which are filtered by
the function $W(f)$, $\displaystyle
x(k,n)=\sum_{k^{\prime}=k-N_{W}/2}^{k+N_{W}/2-1}W(k^{\prime}-k)X(k^{\prime})e^{j2\pi
k^{\prime}n/N_{W}}\quad x=i,v.$ (67b)
Here also, it is important that enough periods are measured, such that around
the excited frequencies, enough frequency domain data is available to extract
the time-variation.
#### 9.2.3 Equivalence between STFT-EIS and DMFA
For a symmetrical window, $w(t)=w(-t)$, and $W(\omega)=\mathcal{F}\\{w(t)\\}$,
the continuous definitions of the STFT-EIS (64) and the DMFA (66) are
equivalent (as proven in E),
$\displaystyle\frac{\mathcal{F}\\{w(t^{\prime}-t)v(t^{\prime})\\}}{\mathcal{F}\\{w(t^{\prime}-t)i(t^{\prime})\\}}=\frac{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)V(\omega^{\prime})\\}}{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)I(\omega^{\prime})\\}}.$
(68)
Since the Fourier and inverse Fourier transforms act on, respectively,
$t^{\prime}$ and $\omega^{\prime}$, both left and right hand side of (68) are
a function of $\omega$ and $t$. Note that both latter equations for extracting
the time-varying impedance are heuristic, and do not exactly match with the
theoretical definition (4.2.1). Nonetheless, these approximations may be
accurate enough in practice.
Since the STFT-EIS and DMFA approaches have a mathematically equivalent
definition of the impedance, the difference boils down to the choice of the
window $w(t)$, or equivalently the filter $W(\omega)$, and the actual
implementation. The properties of the symmetrical window or filter can mainly
be studied by its width. This width can, for instance, be defined by the
variances
$\displaystyle\sigma^{2}_{t}=\frac{\int_{-\infty}^{\infty}t^{2}|w(t)|^{2}\mathrm{d}t}{\int_{-\infty}^{\infty}|w(t)|^{2}\mathrm{d}t}\
\text{and}\
\sigma^{2}_{\omega}=\frac{\int_{-\infty}^{\infty}\omega^{2}|W(\omega)|^{2}\mathrm{d}\omega}{\int_{-\infty}^{\infty}|W(\omega)|^{2}\mathrm{d}\omega}.$
(69)
Similarly as in quantum mechanics where the uncertainty principle prohibits to
measure simultaneously the position and velocity of an electron with arbitrary
precision, also in this case it is not possible to measure the impedance with
arbitrary precision in both time and frequency. Accordingly, the so-called
_Gabor limit_ [127] states that
$\displaystyle\sigma^{2}_{t}\sigma^{2}_{\omega}\geq\frac{1}{4}.$ (70)
The time-selectivity $\sigma^{2}_{t}$ and frequency resolution
$\sigma^{2}_{\omega}$ cannot both be made arbitrarily small. One has to trade-
off between one and the other.
##### STFT-EIS
Darowicki [44] uses a Gaussian window, which has the property that its Fourier
transform is Gaussian as well,
$\displaystyle w(t)=e^{-\frac{\lambda}{2}t^{2}}\ \iff\
W(\omega)=\sqrt{\frac{2\pi}{\lambda}}e^{-\frac{\omega^{2}}{2\lambda}}.$ (71)
For this choice of window, the Gabor limit reduces to
$\displaystyle\sigma^{2}_{t}\sigma^{2}_{\omega}=\frac{1}{4}\quad\text{with}\quad\lambda=\frac{1}{2\sigma^{2}_{t}}=2\sigma^{2}_{\omega}.$
(72)
Accordingly, an increase of the time selectivity leads to a deterioration of
the frequency resolution, and vice versa. The time and frequency resolution
are determined by the hyper parameter $\lambda$. The larger $\lambda$, the
more resolution in time, but the less resolution in frequency, resulting in a
trade-off.
An issue with STFT-EIS is that first subrecords are taken, followed by the
windowing. As a consequence, we do not actually reach the Gabor limit. Also
the DFT is taken on subrecords, which obviously contain less data points than
the total record, hence, as discussed in Section 5, the SNR is poorer. Another
issue is obtaining time-varying impedance data at low frequencies. For being
able to measure low frequencies, the width of the window should, at least,
comprise a period of this frequency. The period length is inversely
proportional to the frequency, hence, for measuring low frequencies, the
window width should be large, deteriorating the time resolution. On the other
hand, since STFT-EIS is applied to time windows, the advantage is that the
time-varying impedance estimation can be done in real-time.
##### DMFA
Battistel and La Mantia [107], on the other hand, directly define the
quadrature filter,
$\displaystyle
W(\omega)=\frac{\big{(}1+e^{-q^{2}}\big{)}^{2}}{\big{(}1+e^{-q\frac{\omega+\Delta\omega}{\Delta\omega}}\big{)}\big{(}1+e^{q\frac{\omega-\Delta\omega}{\Delta\omega}}\big{)}},$
(73)
where $q$ is a factor determining the roll-off of the filter and
$\Delta\omega$ is its bandwidth. Note that in the limits, we have that
$\displaystyle\lim_{q\rightarrow 0}W(\omega)=1$ (74a) and
$\displaystyle\lim_{q\rightarrow\infty}W(\omega)=\left\\{\begin{array}[]{ll}1&\mbox{if
}-\Delta\omega\leq\omega\leq\Delta\omega\\\ 0&\mbox{else.}\end{array}\right.$
(74d)
The objective of this filter is to mimic a rectangular filter, while being
continuous.
The advantage of the DMFA over the STFT-EIS is that only one DFT should be
performed on the entire time-series data record (50) to obtain the frequency
domain data (54). Here, the measurement noise is distributed over all the DFT
lines, resulting in a higher SNR (see Section 5). The time-varying impedance
data is then directly obtained by applying the inverse DFT to small windowed
subrecords around the excited frequencies of the multisine. This has
advantages in computation time (see [107] Section 2.3). Also the width of the
filter can be chosen in function of the spacing of the excited frequencies,
possibly different for each excited frequency. Moreover, an analysis of the
influence of measurement noise on the time-varying impedance data was
performed [129]. However, the time-varying impedance estimation cannot be done
in real-time.
The main advantage of these time-frequency analysis methods is the accessible
implementation. However, both time-frequency analysis methods have difficulty
in estimating time-varying impedance data at low frequencies, this is due to
the drift signal. Moreover, they do not account for treating nonlinear
distortions, even though in the DMFA this could be implemented. These problems
are each solved by operando EIS, as detailed next.
### 9.3 Operando EIS
Operando EIS, as developed by Hallemans et al. [49, 63, 59], is an extension
of ORP-EIS. Here, we estimate the time-varying impedance by using the
definition of the BLTVA (46) as a model structure. Note that if no nonlinear
distortions are present in the measurements, the BLTVA and the time-varying
impedance in (4.2.1) are equal. Nonlinearities in the measurements are
detected, quantified and classified, and the noise level is estimated [49].
Also uncertainty bounds are included on the estimated impedance data.
Moreover, drift signals are suppressed, allowing to access low frequencies
[63].
The idea is to write the time-varying impedance as a truncated series
expansion in a set of known basis functions in time,
$\displaystyle Z(\omega,t)=\sum_{p=0}^{N_{p}}Z_{p}(\omega)b_{p}(t).$ (75)
Figure 13: The first four Legendre polynomials in time (a) and frequency
domain (b).
The basis functions $b_{p}(t)$ are chosen as Legendre polynomials (F and Fig.
13 (a)), since these benefit from good numerical conditioning [122]. Using
(75), the frequency and time dependencies are separated. Since the basis
functions are known, only the impedances $Z_{p}(\omega)$ should be estimated.
The time-varying nonlinear distortions $v_{\mathrm{s}}(t)$ are also expanded
in series,
$\displaystyle
v_{\mathrm{s}}(t)=\sum_{p=0}^{N_{p}}v_{\mathrm{s},p}(t)b_{p}(t),$ (76)
with $v_{\mathrm{s},p}(t)$ the time-invariant nonlinear distortions generated
by a Volterra series (23), meaning that
$V_{\mathrm{s},p}(\omega)=\mathcal{F}\\{v_{\mathrm{s},p}(t)\\}$ is only
nonzero at the integer multiples of the fundamental frequency of the periodic
excitation. Plugging (75) and (76) in the excitation-response relation of NLTV
systems (46) yields,
$\displaystyle
v(t)=v_{0}(t)+\sum_{p=0}^{N_{p}}\mathcal{F}^{-1}\\{Z_{p}(\omega)I_{\mathrm{exc}}(\omega)+V_{\mathrm{s},p}(\omega)\\}b_{p}(t).$
(77)
The drift signal $v_{0}(t)$ is unknown and hides low frequency content.
Therefore, it should also be modelled, for instance by Legendre polynomials
[42, 49],
$\displaystyle v_{0}(t)=\sum_{q=0}^{N_{q}}\theta_{q}b_{q}(t).$ (78)
Drift signals can also be removed by differencing, as detailed in [63]. This
has better performance, however, the mathematics are more involved, so for
this review paper we restrict to modelling the drift signal with basis
functions.
Taking the DFT of (77) gives,
$\displaystyle
V(k)=V_{0}(k)+\sum_{p=0}^{N_{p}}\Big{(}Z_{p}(\omega_{k})I_{\mathrm{exc}}(k)+V_{\mathrm{s},p}(k)\Big{)}\ast
B_{p}(k),$ (79)
with
$\displaystyle V_{0}(k)=\sum_{q=0}^{N_{q}}\theta_{q}B_{q}(k).$ (80)
Here it was used that a product in the time domain becomes a convolution in
the frequency domain and $B_{p}(k)$ is the DFT of the Legendre polynomials,
shown in Fig. 13 (b). For a multisine excitation measured over an integer
number of periods $P$, we have that $I_{\mathrm{exc}}(k)$ is only nonzero at
the harmonics $P\mathbb{H}_{\mathrm{exc}}$ and $V_{\mathrm{s},p}(k)$ is only
nonzero at the harmonics $P\mathbb{H}_{\mathrm{nl}}$, with
$\mathbb{H}_{\mathrm{nl}}=\\{0,1,2,3,...\\}$. Note that
$\mathbb{H}_{\mathrm{exc}}\subset\mathbb{H}_{\mathrm{nl}}$ . Accordingly, (79)
can be simplified as,
$\displaystyle
V(k)=\sum_{q=0}^{N_{q}}\theta_{q}B_{q}(k)+\sum_{p=0}^{N_{p}}\sum_{k^{\prime}\in
P\mathbb{H}_{\mathrm{nl}}}\theta_{p}(k^{\prime})B_{p}(k-k^{\prime}),$ (81)
with
$\displaystyle\theta_{p}(k^{\prime})=Z_{p}(\omega_{k^{\prime}})I_{\mathrm{exc}}(k^{\prime})+V_{\mathrm{s},p}(k^{\prime}).$
(82)
Eq. (81) is linear in the parameters $\theta$, hence, the data can be written
in matrix form,
$\displaystyle V=K\theta+N_{v},$ (83)
where $V$ is a stacked vector of the measured voltage spectra $V(k)$, the
regression matrix $K$ consists of regressors $B_{p}(k)$ centered around the
harmonics $P\mathbb{H}_{\mathrm{nl}}$, the parameter vector $\theta$ contains
the parameters $\theta_{q}$, $q=0,1,...,N_{q}$, $\theta_{p}(k^{\prime})$,
$p=0,1,...,N_{p}$ and $k^{\prime}\in P\mathbb{H}_{\mathrm{nl}}$, and $N_{V}$ a
vector representing the noise. The optimal parameters are estimated in linear
least squares sense,
$\displaystyle\hat{\theta}$
$\displaystyle=\arg\min_{\theta}(V-K\theta)^{H}(V-K\theta)$ (84a)
$\displaystyle=(K^{H}K)^{-1}K^{H}V.$ (84b)
For long data records, the regression problem becomes too large to solve in
one go. Therefore, it is proposed to solve it in local frequency bands as
detailed in [49]. We retrieve the impedance and nonlinear distortion estimates
from the estimated parameter vector $\hat{\theta}$,
$\displaystyle\hat{Z}_{p}(\omega_{k})$
$\displaystyle=\frac{\hat{\theta}_{p}(k)}{I_{\mathrm{exc}}(k)}$ $\displaystyle
k\in P\mathbb{H}_{\mathrm{exc}}$ (85a)
$\displaystyle\hat{V}_{\mathrm{s},p}(k)$ $\displaystyle=\hat{\theta}_{p}(k)$
$\displaystyle k\in
P\big{(}\mathbb{H}_{\mathrm{nl}}\setminus\mathbb{H}_{\mathrm{exc}}\big{)}$
(85b)
Finally, the estimate of the time-varying impedance at the excited frequencies
is obtained as,
$\displaystyle\hat{Z}(\omega_{k},t)=\sum_{p=0}^{N_{p}}\hat{Z}_{p}(\omega_{k})b_{p}(t)$
$\displaystyle k\in P\mathbb{H}_{\mathrm{exc}}.$ (86)
Odd nonlinear distortions and noise introduce uncertainties on the time-
varying impedance estimates $\hat{Z}(\omega_{k},t)$. For the computation of
uncertainty bounds due to nonlinear distortions and noise, the reader is
referred to [49], where also a noise estimation is performed.
The strength of operando EIS is that it computes the BLTVA of NLTV data,
together with uncertainty bounds. Hence, electrochemical systems not
satisfying linearity can still be monitored using an impedance. However, the
odd nonlinearities must not be too strong. Moreover, since the drift signal is
modelled as well, low frequency information becomes available, which is
important for some applications. Therefore, it is applicable to a wide range
of experiments. Also, it does measure the actual definition of the time-
varying impedance, or the BLTVA in the nonlinear case.
The trade-off between time- and frequency resolution, however, remains. To be
able to extract the time-variation, we should leave enough empty DFT lines in
between the excited lines (corresponding to measuring a large integer number
of periods), which decreases the resolution of the excited lines.
## 10 A case study on commercial Li-ion batteries
Li-ion batteries are chosen as a case study to illustrate the important
concepts in this article on real-life measurements. Similar experiments are
performed as in [59]. Measurements are performed on a pristine Samsung
INR21700-48X cell, placed in a thermal chamber at $5^{\circ}$C or
$25^{\circ}$C. The commercially available Samsung 48X is a
$4.8\text{\,}\mathrm{A}\mathrm{h}$ $21$ $700$ cell format with cathodes based
on lithiated metal oxide (Co, Ni, Al) and anodes based on intercalation
graphite and blended Si.
Current and voltage data are collected using the Gamry Interface 5000E
potentiostat. Besides running classical EIS experiments, this potentiostat
allows to apply user-defined excitations, and measure current and voltage
data. The sampling frequency for these user-defined excitations is limited to
$200\text{\,}\mathrm{H}\mathrm{z}$. The current range that can be applied is
limited to $[-5,5]$ A.
An odd random phase multisine signal $i_{\mathrm{exc}}(t)$ is designed with
period $T_{p}=3$ min. The $76$ excited frequencies are chosen as odd
harmonics, $\mathbb{H}_{\text{exc}}=\\{1,3,5,...\\}$, logarithmically
distributed between $f_{\mathrm{min}}=5.6\,\mathrm{mHz}$ and
$f_{\mathrm{max}}=80\,\mathrm{Hz}$. The phases are chosen randomly, uniformly
distributed in $[0,2\pi)$. Since noise is often more prominent at low
frequencies, and while doing operando experiments the low frequency content is
hidden by drift signals, the amplitudes of the multisine are chosen with a
decreasing shape over frequency, with root-mean-square (RMS) of $0.8$ A rms.
### 10.1 Estimating classical impedance data
For _classical_ EIS experiments, the battery is first entirely discharged,
then charged at a $C/3$ rate (constant current of $4.8/3=1.6$ A) until the
desired SOC level ($10,20,\ldots,90$%). Two hours of relaxation are allowed
such that the battery reaches steady-state and the voltage reaches the OCV
value. Then, the zero-mean excitation signal $i(t)=i_{\mathrm{exc}}(t)$ is
looped for $P=10$ periods, that is for $T=PT_{p}=30$ min. The measured voltage
at the different operating points ($\text{SOC}=10,30,50,70,90$% and
$\mathrm{T}=25^{\circ}$C) is shown in Fig. 14. Note that indeed, the voltage
data looks periodic with period $3$ min, and that the data is nicely centered
around the OCV value, as should be the case in LTI measurements. Since the
measurements are periodic, the classical impedances are easily computed from
Section 6 (59).
The spectra of the measured current and voltage, and estimated impedance for
the $10$% SOC and $25^{\circ}$C operating point are shown in Fig. 11. Note
that for the chosen excitation amplitudes, the battery behaves very linearly.
No nonlinear distortions nor nonstationarities can be detected. However, a
(small) drift signal is present. Also, noise is clearly present in the
measurements, but it is fairly low, an SNR of at least $1000$ is obtained.
The estimated impedances at different operating points (depending on SOC and
temperature) are shown in Fig. 3. We do indeed obtain a different impedance
for each of the operating points. Note that if we want to perform quicker
experiments, we can measure less periods, for instance $P=4$, leading to
experiments of $T=12$ min, however, with a higher noise floor.
Figure 14: Voltage response of the Samsung 48X cell in LTI conditions at
$25^{\circ}$C. The zero-mean excitation signal $i_{\mathrm{exc}}(t)$ is used
and $P=10$ periods of the current and voltage are measured at different SOC
levels. Every color indicates a measurement at a different SOC. Note that for
every SOC, the OCV is also different. The time-series are subsampled $150$
times.
### 10.2 Estimating time-varying impedance data
For obtaining _time-varying_ impedance data, the battery is charged with a
$C/2$ current, with the multisine superimposed,
$\displaystyle i(t)=2.4\,\mathrm{A}+i_{\mathrm{exc}}(t).$ (87)
The top graph of Fig. 15 shows the measured current and voltage data at
$5^{\circ}$C. Due to the DC offset of $2.4$ A in the current, the battery is
charging, and the voltage goes up, leading to a drift signal superimposed on
the multisine response. The measurement is stopped when the voltage hits the
safety bound of $4.2$ V. For a constant current charging of $C/2$ we would
expect the $4.2$ V to be reached after $2$ h. However, due to the multisine
added on top of this charging current the safety limit is reached prematurely.
Accordingly, $P=29$ and $P=31$ periods of the excitation could be measured for
the $5^{\circ}$C and $25^{\circ}$C experiments, respectively.
The middle graph of Fig. 15 shows the SOC, with values from $0$ % to $72.5$ %,
and the battery’s surface temperature, which increases slightly due to the
charging current. This was also shown in the SOC-temperature plane in Fig. 9.
The spectra of the current and voltage data of the $5^{\circ}$C experiment are
shown in Fig. 16. The bottom graph shows the entire spectra with a logarithmic
frequency axis, while the top graph shows zoomed spectra in different
frequency bands, each $36\,\mathrm{mHz}$ wide, with a linear frequency axis.
Note the general decreasing shape of the drift spectrum $V_{0}(k)$ in the
voltage spectra which hides the low frequency content. For the lowest zoomed
frequency band (top left), the time-invariant contributions at the excited
frequencies barely exceed the drift spectrum, and the skirts are completely
hidden. At frequencies close to $1\,\mathrm{Hz}$, the skirt around the excited
frequency is a little more visible, still the drift spectrum hides
information. At frequencies close to $80\,\mathrm{Hz}$ the skirts are clearly
visible.
The time-varying impedance at $1\,\mathrm{Hz}$, estimated using operando EIS
[59], is shown in the bottom graph of Fig. 15. The time-varying impedance at
all excited frequencies is shown in Fig. 9. Even though the drift signal hides
the low frequency content, clean impedance data can be obtained at these
frequencies using operando EIS. Note that the impedance is highest at low SOC,
and that the impedance while charging is different from while resting, as also
observed in [59].
Figure 15: Experiment performed on a Samsung 48X cell in time-varying
conditions in a thermal chamber at $5^{\circ}$C. Top graph: current excitation
and voltage response. The current has a DC offset of $2.4$ A, hence, the
battery charges, and the voltage increases. Middle graph: SOC, obtained by
Coulomb counting, and the external parameter temperature during the
experiment. Since the battery is charging with a constant current plus zero-
mean multisine, the SOC increases linearly and the temperature increases
slightly. Bottom graph: time-variation of the impedance at
$0.9389\,\mathrm{Hz}$ obtained from operando EIS [59]. Figure 16: Current and
voltage spectra of an experiment performed on a Samsung 48X cell in time-
varying conditions at $5^{\circ}$C. Top graph: spectra of current and voltage
in three different frequency bands of each $38.5\,\mathrm{mHz}$ wide, with a
linear frequency axis. Note the decreasing shape of the drift spectrum hiding
low frequency content. Bottom graph: entire spectra, with a logarithmic
frequency axis.
### 10.3 Estimating equivalent circuit model parameters
Studying the processes going on at different time-scales in the impedance data
is often performed by mapping the data onto ECM parameters. For the measured
battery impedance data on Samsung 48X cells, we choose the ECM of Fig. 17.
This ECM can be linked to the SPM for batteries [34]. The resistance $R_{0}$
(yellow) is related to the electrolyte resistance, the first $RC$-branch with
Warburg element (purple) is related to the diffusion and the second
$RC$-branch (green) to the electrochemical kinetics. The corresponding
parametric impedance yields,
$\displaystyle
Z_{\mathrm{ECM}}(\omega,\theta)=R_{0}+Z_{C_{1}}\text{//}(R_{1}+Z_{\mathrm{W}})+Z_{C_{\mathrm{ct}}}\text{//}R_{\mathrm{ct}},$
(88a) where the parameter vector $\theta$ is given by
$\displaystyle\theta=[R_{0},R_{1},C_{1},R_{\mathrm{ct}},C_{\mathrm{ct}},\mathrm{W},\alpha],$
(88b) the symbol ‘//’ stands for the parallel connection, that is,
$\displaystyle
Z_{X}(\omega)\text{//}Z_{Y}(\omega)=\frac{Z_{X}(\omega)Z_{Y}(\omega)}{Z_{X}(\omega)+Z_{Y}(\omega)},$
(88c) and the impedance of a capacitor and Warburg element, respectively,
yield, $\displaystyle Z_{C}(\omega)=\frac{1}{Cj\omega}\quad\text{and}\quad
Z_{\mathrm{W}}(\omega)=\frac{\mathrm{W}}{(j\omega)^{\alpha}}.$ (88d)
The Nyquist chart in Fig. 17 illustrates the contribution of the three
branches in series (yellow, purple and green) on the total impedance (black),
being the sum of the three other colors.
The ECM parameters $\theta$ can now be estimated from impedance data by
minimising the cost function,
$\displaystyle\hat{\theta}=\arg\min_{\theta}\sum_{k\in
P\mathbb{H}_{\text{exc}}}|\hat{Z}(\omega_{k})-Z_{\mathrm{ECM}}(\omega_{k},\theta)|^{2}.$
(89)
This cost function is nonlinear in the parameters $\theta$, hence, a nonlinear
solver is required. Here, we use a hybrid of _particle swarm optimisation_
[130, 131, 132] and the built-in MATLAB function lsqnonlin. Fits over the
frequency band [16.7 mHz,50 Hz] are obtained with mean relative errors over
frequency all smaller than $0.2$% and $0.5$% for the classical and time-
varying impedance data, respectively.
The estimated ECM parameters for the measured impedance data of the Samsung
48X cell at $5^{\circ}$C and $25^{\circ}$C are shown in Fig. 17. The ECM
parameters of the classical impedance data at different operating points are
shown as dots, while the ECM parameters of the time-varying impedance data
along trajectories are shown as continuous lines. The temperature is assumed
approximately constant during the experiments. It is observed that the
parameters obtained in operating and classical conditions are not necessarily
equal. For Li-ion batteries, this was already studied in [56], where the
charge transfer resistance $R_{\mathrm{ct}}$ at a certain SOC is smaller while
charging than while resting.
As an application, Zhu et al. [58] propose a fast charging protocol by
applying a charging current inversely proportional to the time-varying charge-
transfer resistance $R_{\mathrm{ct}}$, tracked using operando EIS.
Equivalent circuit model
Corresponding Nyquist plot and estimated parameters
Figure 17: Equivalent circuit model and estimated parameters for the Samsung
48X cell. The Nyquist plot shows the influence of the different branches
(yellow, purple and green) on the total impedance (black). In the ECM
parameter graphs, dots represent the parameters obtained from classical EIS at
different operating points, while the continuous lines show the parameters
obtained from time-varying impedance data along operating trajectories. For
the time-varying experiments, the temperature is assumed approximately
constant at $5$ or $25^{\circ}$C.
## 11 Conclusions & outlook
Classical EIS provides impedance data of electrochemical systems at selected
frequencies. Due to the constraints of linearity and stationarity, the
impedance data is only valid for small amplitude excitations and at fixed
operating points. Nonetheless, measuring classical impedance data is a
powerful tool for monitoring electrochemical systems.
Models beyond linearity and stationarity, such as nonlinear leading-order and
time-varying impedance, reveal higher-dimensional impedance data, valid over
larger excitation amplitudes and along operating trajectories. This higher-
dimensional impedance data contains additional information to classical
impedance data, which is promising for electrochemical applications. One
could, for instance, increase the accuracy of health forecasting of Li-ion
batteries using nonlinear and/or time-varying impedance data as indicator.
This is also extendable to other electrochemical applications, such as
detecting corrosion or studying coatings.
It is shown that the multisine excitation is a strong asset for modelling
electrochemical systems. It allows nonlinear and nonstationary behaviour to be
detected from the measured current and voltage data. If this current and
voltage data does not satisfy the linearity and stationarity constraints,
higher-dimensional impedance data can still be extracted.
## Acknowledgements
This article is dedicated to Rik Pintelon. NH has received funding from the
Eutopia mobility programme. NH and JL are supported financially by the Fund
for Scientific Research (FWO Vlaanderen) and the Flemish government, Belgium
(grant number: METH1). FLM has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement n° 772579). This work was supported by the
Fraunhofer Internal Programs under Grant No. Attract 028-602604 ProLIBs.
## Appendix A The Fourier transform
$\displaystyle X(\omega)$ $\displaystyle=\mathcal{F}\\{x(t)\\}$
$\displaystyle=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\mathrm{d}t$ (90)
$\displaystyle x(t)$ $\displaystyle=\mathcal{F}^{-1}\\{X(\omega)\\}$
$\displaystyle=\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}\mathrm{d}\omega.$
(91)
## Appendix B Volterra series coefficients
Second order nonlinearity
$\displaystyle V_{2,0}$
$\displaystyle=\frac{1}{4}\big{(}Z_{2}(-\omega,\omega)+Z_{2}(\omega,-\omega)\big{)}I^{2}$
(92) $\displaystyle V_{2,1}$ $\displaystyle=0$ (93) $\displaystyle V_{2,2}$
$\displaystyle=\frac{1}{2}Z_{2}(\omega,\omega)I^{2}$ (94)
Third order nonlinearity
$\displaystyle V_{3,0}$ $\displaystyle=0$ (95) $\displaystyle V_{3,1}$
$\displaystyle=\frac{1}{8}\big{(}Z_{3}(-\omega,\omega,\omega)+Z_{3}(\omega,-\omega,\omega)+Z_{3}(\omega,\omega,-\omega)\big{)}I^{3}$
(96) $\displaystyle V_{3,2}$ $\displaystyle=0$ (97) $\displaystyle V_{3,3}$
$\displaystyle=\frac{1}{8}Z_{3}(\omega,\omega,\omega)I^{3}$ (98)
## Appendix C Linearising an NLTI system around an operating trajectory
The origin of the nonstationarity could be proven by applying a particular
excitation to a Volterra series, consisting of a slow part, dictating the
trajectory, and a fast part, which is the excitation,
$\displaystyle
i(t)=\underbrace{i_{0}(t)}_{\text{slow}}+\underbrace{i_{\text{exc}}(t)}_{\text{fast}}.$
(99)
As an example, the slow part could be a positive constant current for charging
a battery, and the fast part a multisine. By assuming that the fast
perturbation has a small amplitude, and hence, only the linear part of the
Volterra series ($n=1$) is needed with respect to $i_{\text{exc}}(t)$, the
voltage response $v(t)$ can also be separated into a slow and fast part,
$\displaystyle v(t)=v_{0}(t)+v_{\text{exc}}(t),$ (100a) with $\displaystyle
v_{0}(t)$
$\displaystyle=\mathrm{OCV}+\sum_{n=1}^{n_{\mathrm{max}}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}z_{n}(\tau_{1},...,\tau_{n})\prod_{l=1}^{n}i_{0}(t-\tau_{l})\mathrm{d}\tau_{l}$
$\displaystyle v_{\text{exc}}(t)$
$\displaystyle=\int_{-\infty}^{\infty}\underbrace{z(\tau,t)}_{\text{depends on
}z_{n}\text{'s and }i_{0}(t)}i_{\text{exc}}(\tau)\mathrm{d}\tau.$ (100b)
The slow part $v_{0}(t)$ is called the drift signal, and solely depends on the
slow excitation $i_{0}(t)$. The fast response is now the convolution of a two-
dimensional impulse response $z(\tau,t)$ with the excitation. This two-
dimensional impulse response function explicitly depends on the time of
excitation $t$, such that stationarity is not satisfied anymore. Moreover,
this function is shown to depend on the generalised impulse responses
$z_{n}(\tau_{1},\dots,\tau_{n})$ and the slow signal $i_{0}(t)$.
## Appendix D The discrete Fourier transform
$\displaystyle X(k)$
$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}x(nT_{s})e^{-j\frac{2\pi kn}{N}}$
(101) $\displaystyle x(nT_{s})$
$\displaystyle=\sum_{k=0}^{N-1}X(k)e^{j\frac{2\pi kn}{N}}$ (102)
## Appendix E Equivalence between (64) and (66)
Using the following properties of the Fourier transform,
$\displaystyle X(\omega)=\mathcal{F}\\{x(t)\\}\qquad x=x,y,w,i,v$ (103a)
$\displaystyle\mathcal{F}\\{x(t)y(t)\\}=X(\omega)\ast
Y(\omega)=\int_{-\infty}^{\infty}X(\omega-\omega^{\prime})Y(\omega^{\prime})\mathrm{d}\omega^{\prime}$
(103b) $\displaystyle\mathcal{F}\\{x(t^{\prime}-t)\\}=X(\omega)e^{-j\omega
t},$ (103c)
where $\mathcal{F}$ acts on $t^{\prime}$, one finds that,
$\displaystyle\mathcal{F}\\{w(t^{\prime}-t)x(t^{\prime})\\}$
$\displaystyle=\int_{-\infty}^{\infty}W(\omega-\omega^{\prime})e^{-j(\omega-\omega^{\prime})t}X(\omega^{\prime})\mathrm{d}\omega^{\prime}$
$\displaystyle=e^{-j\omega
t}\int_{-\infty}^{\infty}W(\omega-\omega^{\prime})X(\omega^{\prime})e^{j\omega^{\prime}t}\mathrm{d}\omega^{\prime}$
$\displaystyle=e^{-j\omega
t}\mathcal{F}^{-1}\\{W(\omega-\omega^{\prime})X(\omega^{\prime})\\}.$ (104)
Assuming that $w(t)=w(-t)$, one has that $W(\omega)=W(-\omega)$, accordingly,
$\displaystyle\frac{\mathcal{F}\\{w(t^{\prime}-t)v(t^{\prime})\\}}{\mathcal{F}\\{w(t^{\prime}-t)i(t^{\prime})\\}}$
$\displaystyle=\frac{\mathcal{F}^{-1}\\{W(\omega-\omega^{\prime})V(\omega^{\prime})\\}}{\mathcal{F}^{-1}\\{W(\omega-\omega^{\prime})I(\omega^{\prime})\\}}$
$\displaystyle=\frac{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)V(\omega^{\prime})\\}}{\mathcal{F}^{-1}\\{W(\omega^{\prime}-\omega)I(\omega^{\prime})\\}}$
(105)
with all Fourier and inverse Fourier transforms acting on, respectively,
$t^{\prime}$ and $\omega^{\prime}$.
## Appendix F Legendre polynomials
The Legendre polynomials $L_{p}(x)$, $p=0,1,...$, $x\in[-1,1]$ are the
solution of Legendre’s differential equation
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\Big{(}(1-x^{2})\frac{\mathrm{d}L_{p}(x)}{\mathrm{d}x}\Big{)}+p(p+1)L_{p}(x)=0.$
(106)
The basis functions $b_{p}(t)$ are chosen as rescaled Legendre polynomials
over the interval $[0,T]$, that is,
$\displaystyle b_{p}(t)=L_{p}\Big{(}\frac{2t}{T}-1\Big{)}.$ (107)
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# Submanifolds of Generalized Sasakian-space-forms with respect to certain
connections
Pradip Mandal, Shyam Kishor and Shyamal Kumar Hui∗
###### Abstract.
The present paper deals with some results of submanifolds of generalized
Sasakian-space-forms in [3] with respect to semisymmetric metric connection,
semisymmetric non-metric connection, Schouten-van Kampen connection and
Tanaka-webster connection.
###### Key words and phrases:
generalized Sasakian-space-forms, semisymmetric metric connection,
semisymmetric non-metric connection, Schouten-van Kampen Connection, Tanaka-
Webster connection.
* corresponding author
###### 2010 Mathematics Subject Classification:
53C15, 53C40
## 1\. Introduction
As a generalization of Sasakian-space-form, Alegre et al. [2] introduced the
notion of generalized Sasakian-space-form as that an almost contact metric
manifold $\bar{M}(\phi,\xi,\eta,g)$ whose curvature tensor $\bar{R}$ of
$\bar{M}$ satisfies
(1.1) $\displaystyle\bar{R}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_{1},f_{2},f_{3}$ are
certain smooth functions on $\bar{M}$. Such a manifold of dimension $(2n+1)$,
$n>1$ (the condition $n>1$ is assumed throughout the paper), is denoted by
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ [2]. Many authors studied this space form
with different aspects. For this, we may refer ([11], [12], [13], [14], [15],
[17], [18] and [23]). It reduces to Sasakian-space-form if
$f_{1}=\frac{c+3}{4}$, $f_{2}=f_{3}=\frac{c-1}{4}$ [2].
After introducing the semisymmetric linear connection by Friedman and Schouten
[7], Hayden [9] gave the idea of metric connection with torsion on a
Riemannian manifold. Later, Yano [29] and many others (see, [21], [22], [24]
and references therein) studied semisymmetric metric connection in different
context. The idea of semisymmetric non-metric connection was introduced by
Agashe and Chafle [1].
The Schouten-van Kampen connection introduced for the study of non-holomorphic
manifolds ([20], [27]). In $2006$, Bejancu [6] studied Schouten-van Kampen
connection on foliated manifolds. Recently Olszak [19] studied Schouten-van
Kampen connection on almost(para) contact metric structure.
The Tanaka-Webster connection ([25], [28]) is the canonical affine connection
defined on a non-degenerate pseudo-Hermitian CR-manifold. Tanno [26] defined
the Tanaka-Webster connection for contact metric manifolds.
The submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ are studied in ([3],
[10], [16]). In [3], Alegre and Carriazo studied submanifolds of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to Levi-Civita connection
$\bar{\nabla}$. The present paper deals with study of such submanifolds of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to semisymmetric metric
connection, semisymmetric non-metric connection, Schouten-van Kampen
connection and Tanaka-webster connection respectively.
## 2\. preliminaries
In an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$, we have [4]
(2.1) $\displaystyle\phi^{2}(X)=-X+\eta(X)\xi,\ \phi\xi=0,$ (2.2)
$\displaystyle\eta(\xi)=1,\ g(X,\xi)=\eta(X),\ \eta(\phi X)=0,$ (2.3)
$\displaystyle g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y),$ (2.4) $\displaystyle
g(\phi X,Y)=-g(X,\phi Y).$
In $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$, we have [2]
(2.5)
$\displaystyle(\bar{\nabla}_{X}\phi)(Y)=(f_{1}-f_{3})[g(X,Y)\xi-\eta(Y)X],$
(2.6) $\displaystyle\bar{\nabla}_{X}\xi=-(f_{1}-f_{3})\phi X,$
where $\bar{\nabla}$ is the Levi-Civita connection of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$.
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$. If $\nabla$
and $\nabla^{\perp}$ are the induced connections on the tangent bundle $TM$
and the normal bundle $T^{\perp}{M}$ of $M$, respectively then the Gauss and
Weingarten formulae are given by [30]
(2.7) $\displaystyle\bar{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y),\
\bar{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\perp}V$
for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_{V}$
are second fundamental form and shape operator (corresponding to the normal
vector field V), respectively and they are related by [30]
$g(h(X,Y),V)=g(A_{V}X,Y)$.
For any $X\in\Gamma(TM)$, we may write
(2.8) $\phi X=TX+FX,$
where $TX$ is the tangential component and $FX$ is the normal component of
$\phi X$.
In particular, if $F=0$ then $M$ is invariant [5] and here $\phi(TM)\subset
TM$. Also if $T=0$ then $M$ is anti-invariant [5] and here $\phi(TM)\subset
T^{\bot}M$. Also here we assume that $\xi$ is tangent to $M$.
The semisymmetric metric connection $\widetilde{\bar{\nabla}}$ and the
Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$
are related by [29]
(2.9)
$\displaystyle\widetilde{\bar{\nabla}}_{X}Y=\bar{\nabla}_{X}Y+\eta(Y)X-g(X,Y)\xi.$
The Riemannian curvature tensor $\widetilde{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$
is
(2.10) $\displaystyle\widetilde{\bar{R}}(X,Y)Z$
$\displaystyle=(f_{1}-1)\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y-g(Y,\phi Z)\phi X$ $\displaystyle+2g(X,\phi Y)\phi
Z\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi$
$\displaystyle-g(Y,Z)\eta(X)\xi\big{\\}}+(f_{1}-f_{3})\\{g(X,\phi Z)Y-g(Y,\phi
Z)X+g(Y,Z)\phi X$ $\displaystyle-g(X,Z)\phi Y\\}.$
The semisymmetric non-metric connection ${\bar{\nabla}}^{{}^{\prime}}$ and the
Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$
are related by [1]
(2.11)
$\displaystyle\bar{\nabla}^{{}^{\prime}}_{X}Y=\bar{\nabla}_{X}Y+\eta(Y)X.$
The Riemannian curvature tensor ${\bar{R}}^{{}^{\prime}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
${\bar{\nabla}}^{{}^{\prime}}$ is
$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi Z)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})[g(X,\phi Z)Y-g(Y,\phi Z)X]$
$\displaystyle+$ $\displaystyle\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y.$
The Schouten-van Kampen connection $\hat{\bar{\nabla}}$ and the Riemannian
connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related
by [19]
(2.13)
$\displaystyle\hat{\bar{\nabla}}_{X}Y=\bar{\nabla}_{X}Y+(f_{1}-f_{3})\eta(Y)\phi
X-(f_{1}-f_{3})g(\phi X,Y)\xi.$
The Riemannian curvature tensor $\hat{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is
(2.14) $\displaystyle\hat{\bar{R}}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi
Z\big{\\}}+\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+(f_{1}-f_{3})^{2}\big{[}g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi
X\big{]}.$
The Tanaka-Webster connection $\stackrel{{\scriptstyle\ast}}{{\bar{\nabla}}}$
and the Riemannian connection $\bar{\nabla}$ of
${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [8]
(2.15)
$\displaystyle\stackrel{{\scriptstyle\ast}}{{\bar{\nabla}}}_{X}Y=\bar{\nabla}_{X}Y+\eta(X)\phi
Y+(f_{1}-f_{3})\eta(Y)\phi X-(f_{1}-f_{3})g(\phi X,Y)\xi.$
The Riemannian curvature tensor $\stackrel{{\scriptstyle*}}{{\bar{R}}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is
(2.16) $\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$
$\displaystyle=f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
Z)\phi Y-g(Y,\phi Z)\phi X$ $\displaystyle+2g(X,\phi Y)\phi
Z\big{\\}}+\\{f_{3}+{(f_{1}-f_{3})^{2}}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}+{(f_{1}-f_{3})^{2}}\big{[}g(X,\phi
Z)\phi Y$ $\displaystyle-g(Y,\phi Z)\phi X\big{]}+2(f_{1}-f_{3})g(X,\phi
Y)\phi Z.$
## 3\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\widetilde{\bar{\nabla}}$
###### Lemma 3.1.
If $M$ is invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is
tangent to $M$, for any $X,Y,Z\in\Gamma(TM)$.
###### Proof.
If $M$ is invariant then from (2.10) we say that $\widetilde{\bar{R}}(X,Y)Z$
is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$. This
proves the lemma. ∎
###### Lemma 3.2.
If $M$ is anti-invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$
with respect to $\widetilde{\bar{\nabla}}$, then
(3.1) $\displaystyle tan(\widetilde{\bar{R}}(X,Y)Z)$
$\displaystyle=(f_{1}-1)\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}},$
(3.2) $\displaystyle nor(\widetilde{\bar{R}}(X,Y)Z)$ $\displaystyle=$
$\displaystyle(f_{1}-f_{3})\\{g(Y,Z)\phi X-g(X,Z)\phi Y\\}$
for any $X,Y,Z\in\Gamma(TM)$.
###### Proof.
Since $M$ is anti-invariant, we have $\phi X,\phi Y\in\Gamma(T^{\bot}M)$. Then
equating tangent and normal component of (2.10) we get the result. ∎
###### Lemma 3.3.
If $f_{1}(p)=f_{3}(p)$ and $M$ is either invariant or anti-invariant
submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$
for any $X,Y,Z\in\Gamma(TM)$.
###### Proof.
Using Lemma $3.1$ and Lemma $3.2$ we get the result. ∎
###### Lemma 3.4.
If $M$ is invariant or anti-invariant submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)V$ is normal to $M$,
for any $X,Y,\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.
###### Proof.
If $M$ is invariant from (2.10) we have $\widetilde{\bar{R}}(X,Y)V$ normal to
$M$, and if $M$ is anti-invariant then $\widetilde{\bar{R}}(X,Y)V=0$ i.e.
$\widetilde{\bar{R}}(X,Y)V$ normal to $M$ for any $X,Y,\in\Gamma(TM)$ and
$V\in\Gamma(T^{\bot}M)$. ∎
###### Lemma 3.5.
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ and $TM$ is invariant under the action of
$\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant
or anti-invariant.
###### Proof.
For $X,Y\in\Gamma(TM)$, we have from (2.10) that
$\displaystyle\widetilde{\bar{R}}(X,Y)X$ $\displaystyle=$
$\displaystyle(f_{1}-1)\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
X)\phi Y$ $\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}+(f_{3}-1)\big{\\{}\eta(X)\eta(X)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(\phi Y,X)X-g(\phi
X,X)Y+g(Y,X)\phi X$ $\displaystyle-$ $\displaystyle g(X,X)\phi Y\\}.$
Note that $\widetilde{\bar{R}}(X,Y)X$ should be tangent if $[-3f_{2}g(Y,\phi
X)\phi X+(f_{1}-f_{3})\\{g(Y,X)\phi X-g(X,X)\phi Y\\}]$ is tangent. Since
$f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ at any point $p$ then by similar way of
proof of Lemma $3.2$ of [3], we can prove that either $M$ is invariant or
anti-invariant. This proves the Lemma. ∎
###### Remark 3.1.
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{1}(p)\neq f_{3}(p)$ and $TM$ is
invariant under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$,
then $M$ is invariant.
From Lemma $3.3$ and Lemma $3.5$, we have
###### Theorem 3.1.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ then $M$ is either invariant or anti-invariant if and only
if $TM$ is invariant under the action of $\widetilde{\bar{R}}(X,Y)$ for all
$X,Y\in\Gamma(TM)$.
###### Proposition 3.1.
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\widetilde{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant
under the action of $\widetilde{\bar{R}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.
###### Proof.
Replacing $X,Y,Z$ by $U,V,X$ in (2.10), we get
$\displaystyle\widetilde{\bar{R}}(U,V)X$ $\displaystyle=$
$\displaystyle(f_{1}-1)\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi
X)\phi V$ $\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}+(f_{3}-1)\big{\\{}\eta(U)\eta(X)V$ $\displaystyle-$
$\displaystyle\eta(V)\eta(X)U+g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(\phi V,X)U-g(\phi
U,X)V+g(V,X)\phi U$ $\displaystyle-$ $\displaystyle g(U,X)\phi V\\}.$
As $M$ is invariant, $U,V\in\Gamma(T^{\bot}M)$, we have
(3.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$
for any $X\in\Gamma(TM)$. Using (3.5) in (3), we have
(3.6) $\widetilde{\bar{R}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$
which is tangent as $\phi X$ is tangent. This proves the proposition. ∎
###### Proposition 3.2.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$,
$f_{1}(p)=f_{3}(p)$ for each $p\in M$ and $T^{\bot}M$ is invariant under the
action of $\widetilde{\bar{R}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then $M$ is
either invariant or anti-invariant.
###### Proof.
The proof is similar as it is an Lemma $3.4$, just assuming that
$\widetilde{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎
## 4\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
${\bar{\nabla}}^{{}^{\prime}}$
###### Lemma 4.1.
If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
${\bar{\nabla}}^{{}^{\prime}}$, then ${\bar{R}}^{{}^{\prime}}(X,Y)Z$ is
tangent to $M$ and ${\bar{R}}^{{}^{\prime}}(X,Y)V$ normal to $M$ for any
$X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.
###### Proof.
If $M$ is invariant then from (2) we say that ${\bar{R}}^{{}^{\prime}}(X,Y)Z$
is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.
If $M$ is anti-invariant then
(4.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$
From (2) and (4.1) we have
$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(Z)Y$
$\displaystyle-$
$\displaystyle\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle[\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y],$
which is tangent.
If $M$ is invariant then from (2), it follows that
${\bar{R}}^{{}^{\prime}}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant
then ${\bar{R}}^{{}^{\prime}}(X,Y)V=0$ i.e. ${\bar{R}}^{{}^{\prime}}(X,Y)V$ is
normal to $M$ for any $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This
proves the Lemma. ∎
###### Lemma 4.2.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$ and $TM$ is invariant under the action of $\bar{R}^{{}^{\prime}}(X,Y)$,
$X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.
###### Proof.
For $X,Y\in\Gamma(TM)$, we have from (2) that
$\displaystyle{\bar{R}}^{{}^{\prime}}(X,Y)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi X)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}+f_{3}\big{\\{}\eta(X)\eta(X)Y$ $\displaystyle-$
$\displaystyle\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle-$ $\displaystyle(f_{1}-f_{3})g(\phi
X,Y)X+\\{\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y\\}.$
Note that ${\bar{R}}^{{}^{\prime}}(X,Y)X$ should be tangent if
$3f_{2}(p)g(Y,\phi X)\phi X$ is tangent. Since $f_{2}(p)\neq 0$ for each $p\in
M$, as similar as proof of Lemma $3.2$ of [3], we may conclude that either $M$
is invariant or anti-invariant. This proves the Lemma. ∎
From Lemma $4.1$ and Lemma $4.2$, we have
###### Theorem 4.1.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$, then $M$ is either invariant or anti-invariant if and only if $TM$ is
invariant under the action of ${\bar{R}}^{{}^{\prime}}(X,Y)$ for all
$X,Y\in\Gamma(TM)$.
###### Proposition 4.1.
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to ${\bar{\nabla}}^{{}^{\prime}}$. If $M$ is invariant, then $TM$ is invariant
under the action of ${\bar{R}}^{{}^{\prime}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.
###### Proof.
Replacing $X,Y,Z$ by $U,V,X$ in (2), we get
$\displaystyle{\bar{R}}^{{}^{\prime}}(U,V)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi X)\phi V$
$\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}+f_{3}\big{\\{}\eta(U)\eta(X)V$ $\displaystyle-$
$\displaystyle\eta(V)\eta(X)U+g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})\\{g(U,\phi X)V-g(V,\phi X)U\\}$
$\displaystyle+$ $\displaystyle\\{\eta(V)\eta(X)U-\eta(U)\eta(X)V\\}.$
As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have
(4.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$
for any $X\in\Gamma(TM)$. Using (4.5) in (4), we have
(4.6) ${\bar{R}}^{{}^{\prime}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$
which is tangent as $\phi X$ is tangent. This proves the proposition. ∎
###### Proposition 4.2.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in
M$ and $T^{\bot}M$ is invariant under the action of ${\bar{R}}(U,V)$,
$U,V\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.
###### Proof.
The proof is similar as the proof of Lemma $4.2$, just imposing that
${\bar{R}}^{{}^{\prime}}(U,V)U$ is normal for any $U,V\in\Gamma(TM)$. ∎
## 5\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\hat{\bar{\nabla}}$
###### Lemma 5.1.
If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$, then
$\hat{\bar{R}}(X,Y)Z$ is tangent to $M$ and $\hat{\bar{R}}(X,Y)V$ is normal to
$M$ for any $X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.
###### Proof.
If $M$ is invariant then from (2.14) we say that $\hat{\bar{R}}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.
If $M$ is anti-invariant then
(5.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$
From (2.14) and (5.1) we have
$\displaystyle\hat{\bar{R}}(X,Y)Z$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+$ $\displaystyle g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}},$
which is tangent.
If $M$ is invariant from (2.14) we have $\hat{\bar{R}}(X,Y)V$ is normal to
$M$, and if $M$ is anti-invariant then $\hat{\bar{R}}(X,Y)V=0$ i.e.
$\hat{\bar{R}}(X,Y)V$ is normal to $M$ for any $X,Y\in\Gamma(TM)$ and
$V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎
###### Lemma 5.2.
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq(f_{1}-f_{3})^{2}$ on $M$ and
$TM$ is invariant under the action of $\hat{\bar{R}}(X,Y)$,
$X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.
###### Proof.
For $X,Y\in\Gamma(TM)$, we have from (2.14) that
$\displaystyle\hat{\bar{R}}(X,Y)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi X)\phi Y$
$\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi X\big{\\}}$
$\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(X)Y-\eta(Y)\eta(X)X$
$\displaystyle+$ $\displaystyle g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(X,\phi X)\phi
Y-g(Y,\phi X)\phi X\big{\\}}.$
Now, we see that $\hat{\bar{R}}(X,Y)X$ should be tangent if
$\\{3f_{2}+(f_{1}-f_{3})^{2}\\}g(Y,\phi X)\phi X$ is tangent. Since
$3f_{2}\neq-(f_{1}-f_{3})^{2}$ then in similar way of proof of Lemma $3.2$ of
[3] we may conclude that either $M$ is invariant or anti-invariant. This
proves the Lemma. ∎
From Lemma $5.1$ and Lemma $5.2$, we can state the following:
###### Theorem 5.1.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$, then $M$
is either invariant or anti-invariant if and only if $TM$ is invariant under
the action of $\hat{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
###### Proposition 5.1.
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\hat{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant under the
action of $\hat{\bar{R}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.
###### Proof.
Replacing $X,Y,Z$ by $U,V,X$ in (2.14), we get
$\displaystyle\hat{\bar{R}}(U,V)X$ $\displaystyle=$ $\displaystyle
f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi X)\phi V$
$\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi X\big{\\}}$
$\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(U)\eta(X)V-\eta(V)\eta(X)U$
$\displaystyle+$ $\displaystyle g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(U,\phi X)\phi
V-g(V,\phi X)\phi U\big{\\}}.$
As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have
(5.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$
for any $X\in\Gamma(TM)$. Using (5.5) in (5), we have
(5.6) $\hat{\bar{R}}(U,V)X=2f_{2}g(U,\phi V)\phi X,$
which is tangent as $\phi X$ is tangent. This proves the proposition. ∎
###### Proposition 5.2.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$ on $M$ and
$T^{\bot}M$ is invariant under the action of $\hat{\bar{R}}(U,V)$,
$U,V\in\Gamma(T^{\bot}M)$, then $M$ is either invariant or anti-invariant.
###### Proof.
The proof is similar as the proof of Lemma $5.2$, just imposing that
$\hat{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎
## 6\. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$
###### Lemma 6.1.
If $M$ is either invariant or anti-invarint submanifold of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to
$\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ and
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ is normal to $M$ for any
$X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.
###### Proof.
If $M$ is invariant then from (2.16) we say that
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ because $\phi
X$ and $\phi Y$ are tangent to $M$.
If $M$ is anti-invariant then
(6.1) $g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.$
From (2.16) and (6.1) we have
$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(Y,Z)X-g(X,Z)Y\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$
$\displaystyle+$ $\displaystyle g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\\}}$
which is tangent.
If $M$ is invariant from (2.16) we have
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ and if $M$ is
anti-invariant then $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V=0$ i.e.
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ for any
$X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎
###### Lemma 6.2.
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$ for each $p\in M$ and
$TM$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is
either invariant or anti-invariant.
###### Proof.
For $X,Y\in\Gamma(TM)$, we have from (2.16) that
$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)X$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(Y,X)X-g(X,X)Y\big{\\}}+f_{2}\big{\\{}g(X,\phi
X)\phi Y$ $\displaystyle-$ $\displaystyle g(Y,\phi X)\phi X+2g(X,\phi Y)\phi
X\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(X)\eta(X)Y-\eta(Y)\eta(X)X$
$\displaystyle+$ $\displaystyle g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(X,\phi X)\phi
Y-g(Y,\phi X)\phi X\big{\\}}$ $\displaystyle+$ $\displaystyle
2(f_{1}-f_{3})g(X,\phi Y)\phi X.$
Now we see that $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)X$ should be
tangent if $\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)g(Y,\phi X)\phi X$
is tangent. Since $\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$
then by similar way of proof of Lemma $3.2$ of [3] we can proved that either
$M$ is invariant or anti-invariant. This proves the Lemma. ∎
From Lemma $6.1$ and Lemma $6.2$, we have
###### Theorem 6.1.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$, then $M$ is either
invariant or anti-invariant if and only if $TM$ is invariant under the action
of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
###### Proposition 6.1.
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect
to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $M$ is invariant, then
$TM$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$ for any
$U,V\in\Gamma(T^{\bot}M)$.
###### Proof.
Replacing $X,Y,Z$ by $U,V,X$ in (2.16), we get
$\displaystyle\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)X$ $\displaystyle=$
$\displaystyle f_{1}\big{\\{}g(V,X)U-g(U,X)V\big{\\}}+f_{2}\big{\\{}g(U,\phi
X)\phi V$ $\displaystyle-$ $\displaystyle g(V,\phi X)\phi U+2g(U,\phi V)\phi
X\big{\\}}$ $\displaystyle+$
$\displaystyle\\{f_{3}+(f_{1}-f_{3})^{2}\\}\big{\\{}\eta(U)\eta(X)V-\eta(V)\eta(X)U$
$\displaystyle+$ $\displaystyle g(U,X)\eta(V)\xi-g(V,X)\eta(U)\xi\big{\\}}$
$\displaystyle+$ $\displaystyle(f_{1}-f_{3})^{2}\big{\\{}g(U,\phi X)\phi
V-g(V,\phi X)\phi U\big{\\}}$ $\displaystyle+$ $\displaystyle
2(f_{1}-f_{3})g(U,\phi V)\phi X.$
As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have
(6.5) $g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0$
for any $X\in\Gamma(TM)$. Using (6.5) in (6), we have
(6.6)
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)X=\\{2f_{2}+2(f_{1}-f_{3})\\}g(U,\phi
V)\phi X,$
which is tangent as $\phi X$ is tangent. This proves the proposition. ∎
###### Proposition 6.2.
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with
respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If
$\\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\\}(p)\neq 0$ for each $p\in M$ and
$T^{\bot}M$ is invariant under the action of
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then
$M$ is either invariant or anti-invariant.
###### Proof.
The proof is similar as the proof of Lemma $6.2$, just considering that
$\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)U$ is normal for any
$U,V\in\Gamma(T^{\bot}M)$. ∎
Acknowledgement: The first author (P. Mandal) gratefully acknowledges to the
CSIR(File No.:09/025(0221)/2017-EMR-I), Govt. of India for financial
assistance. The Third author (S. K. Hui) are thankful to University of Burdwan
for providing administrative and technical support.
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Pradip Mandal and Shyamal Kumar Hui
Department of Mathematics, The University of Burdwan, Burdwan – 713104, West
Bengal, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Shyam Kishor
Department of Mathematics and Astronomy, University of Lucknow, Lucknow –
226007, India
E-mail<EMAIL_ADDRESS>
|
# Wiedemann-Franz law violation domain for graphene and nonrelativistic
systems
Thandar Zaw Win, Cho Win Aung, Gaurav Khandal
Sabyasachi Ghosh Department of Physics, Indian Institute of Technology
Bhilai, Kutelabhata, Durg 491002, India
###### Abstract
Systematic and comparative research on Lorenz ratios for graphene and
nonrelativistic systems has been studied to identify their Wiedemann-Franz law
violation domain. Fermi energy and temperature are the main governing
parameters for deciding the values of the Lorenz ratio, which is basically
thermal conductivity divided by electrical conductivity times temperature
times Lorenz number. Metals as three-dimensional nonrelativistic electron gas
locate at higher Fermi-energy by temperature domain, where Lorenz ratio
remains one. Hence, they obey the Wiedemann-Franz law. By creating higher
doping in a two-dimensional graphene system, one can again reach a higher
Fermi-energy by temperature domain and get a constant Lorenz ratio. For both
graphene and nonrelativistic systems, the Lorenz ratio goes below one if we go
lower Fermi-energy by temperature domain, which is possible for the graphene
system by decreasing the doping concentration. Experimentally observed greater
than one Lorenz ratio in this lower Fermi-energy by temperature domain or
Dirac Fluid domain indicates that non-fluid expressions of Lorenz ratio should
be replaced by fluid-type expressions. We have noticed a divergent trend of
Lorenz ratio in the Dirac Fluid domain using its fluid-type expression, and it
matches with the trend of experimental data.
## I Introduction
In 1853, Gustav Wiedemann and Rudolph Franz experimentally discovered a
universal or constant value of the ratio between thermal ($\kappa$) and
electrical ($\sigma$) conductivity, which are approximately followed by all
metals. Later, in 1872, Ludvig Lorenz theoretically realized that the ratio in
terms of the universal constants $k_{B}$ (Boltzmann constant) and $e$
(electric charge) is:
$\displaystyle\frac{\kappa}{\sigma T}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}$ (1)
$\displaystyle=$ $\displaystyle L_{0}=2.445\times
10^{-8}watt\frac{\Omega}{K^{2}}~{},$
where $L_{0}$ is known as the Lorenz number. The fact is very well known from
our text book Refs. [1, 2, 3, 4, 5]. In natural unit, $\hbar=c=k_{B}=1$ and
$e^{2}=\frac{4\pi}{137}$, we can express temperature $T$ in eV, $\sigma$ in eV
and $\kappa$ in eV2. So Lorenz number will be a dimensionless ratio
$L_{0}=\frac{137\pi}{12}\approx 35.84$. In the present paper, we will mostly
use the natural unit methodology for convenience, but we may go to the actual
unit in some cases (whenever necessary).
The Wiedemann-Franz (WF) law has proven remarkably robust for many metallic
systems, where electrons are the main electric and thermal charge
transportation carriers. Due to the similar mechanism of transportation in
free electron theory [1, 2, 3, 4, 5], the dimensionless (in natural unit)
ratio of two transport coefficients - thermal and electrical conductivity
becomes constant. However, deviations have been observed in many systems like
$Li_{0.9}Mo_{6}O_{17}$ [6], $CeCoIn_{5}$ [7], $ZrZn_{2}$ [8], $YbRh_{2}Si_{2}$
[9], $(Pr,Ce)_{2}CuO_{4}$ [10] and $VO_{2}$ [11]. For our convenience, if we
define the ratio of thermal and electrical conductivity as
$L=\frac{\kappa}{\sigma T}$, and if we call $L/L_{0}$ as Lorenz ratio, then
the validity of the WF law will be established from the relations $L=L_{0}$ or
$\frac{L}{L_{0}}=1$. On the other hand, relations $\frac{L}{L_{0}}>1$ or
$\frac{L}{L_{0}}<1$ reflect violation of the WF law. A large diverging
outcomes $\frac{L}{L_{0}}>10^{4}$ is expected in one-dimensional exotic
Tomonaga-Luttinger liquids [6]. On the other hand, a strong downward violation
of the WF Law $L<L_{0}$ was observed in $WP_{2}$ semimetal [12, 13] and MoP
binary compound [14]. This downward violation is also observed in heavy
fermion metals [7], marginal ferromagnetic metal [8], anti-ferromagnetic metal
[9] and copper-oxide superconducting material [10] at low-temperature regions
as well as in metallic vanadium dioxide [11] at high-temperature range
($240$-$340$K). Understanding the WF law violation mechanism of that wide
range of systems is a non-trivial task and challenge for the theoretical
community to explain.
Recently, Crossno et al. [15] have found a similar kind of the WF law
violation in graphene systems by tuning the doping concentration and
temperature. Unlike standard metal, the Fermi level of graphene can be moved
upwards or downwards from the relative Dirac point depending upon the n- or
p-type doping of graphene. Fermi energy $(\epsilon_{F})$ or a chemical
potential $(\mu)$ at Dirac point is considered zero for undoped graphene and,
via n- or p-type doping, will shift it towards the positive or negative
direction. The net charge carrier concentration will be tuned from zero to
non-zero when one experimentally goes from undoped to doped graphene cases,
and theoretically, one goes from $\epsilon_{F}=0$ to $\epsilon_{F}\neq 0$
cases. This kind of tuning possibility of $\epsilon_{F}$ can not be expected
for metal systems as it almost remains constant. Their typical values remain
within the range $\epsilon_{F}=2-10$ eV, for which one can expect the limit
$\epsilon_{F}/T>>1$ at room temperature $T\approx 0.025$ eV. In this limit,
one can consider electrons in metal as Fermi gas and use Sommerfeld’s
expansion, which provides a linear T dependent on electron-specific heat.
Fermi gas is completely based on the crude non-interacting assumption, but
there is a theory of interacting Fermion system, which is popularly known as
the Fermi Liquid (FL) Theory. Originally, Landau [16, 17, 18] proposed this
phenomenological theory for studying 3He. In FL prescription, interaction is
taken care of via the effective mass of electrons by assuming a quasi-particle
picture. Hence, some mass-independent quantities like $L$ remain the same as
$L_{0}$ for both FL and Fermi gas prescriptions. So, if we define
$\epsilon_{F}/T>1$ as the FL domain and $\epsilon_{F}/T>>1$ as the Fermi gas
domain, then electron-doped graphene systems almost follow WF law. When we go
towards un-dopped or clean graphene systems with $\epsilon_{F}/T<1$, Fermi
liquid theory becomes invalid (as it is commonly accepted). It is concluded
from the experimental observation of WF law violation in this
$\epsilon_{F}/T<1$ or $\epsilon_{F}/T<<1$ domains, popularly called Dirac
Fluid (DF) or Dirac Liquid (DL) domain. Theoretical works [19, 20, 21, 22, 23]
are attempted to explain this WF law violation in the DF domain. In this
regards, electron hydrodynamics (eHD) or fluid dynamics in graphene system,
recently observed by Refs [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37], is thought to be linked with this WF law violation. In this context, the
present work is planned to explore a transition from non-fluid to fluid type
expressions of Lorenz ratio for a 2-dimensional (2D) graphene (G) system with
linear energy-momentum relation ($E\propto p$) as well as a 3-dimensional (3D)
metal system with nonrelativistic (NR) energy-momentum relation ($E\propto
p^{2}$). For mathematical completeness, we have demonstrated all possible
cases like 2D-NR, 3D-NR, 2D-G, and 3D-G but our final destination is kept to
show how fluid expressions of 2D-G case can be associated with WF law
violation and how non-fluid expressions of 3D-NR case, followed by metal
systems, obey WF law. In the direction of eHD, our earlier comparative
research on ideal [38] and viscous dissipating [39] parts of energy-momentum
tensor calculation clearly demonstrate that the graphene hydrodynamics is
neither relativistic hydrodynamics nor nonrelativistic hydrodynamics. The
present work has gone through similar comparative research on electrical
conductivity, thermal conductivity, and Lorenz ratio for a systematic search
of the WF law violation domain.
The article is organized as follows, in Sec. II, the formalism part of WF law
calculations for different cases. The results and discussions are in Sec. III
and summarised by Sec. IV.
## II Formalism
Graphene is a 2-dimensional single atomic layer of carbon atoms tightly bound
into a honeycomb lattice [40]. Near the Dirac point, electrons in graphene
follow the linear dispersion relation
$\epsilon\left(k\right)=pv_{F},$ (2)
where $v_{F}$ is the Fermi velocity of electrons in graphene, whose values
remain within the range $0.003c$-$0.01c$ [41]. In one direction, electrons in
graphene do not follow a quadratic relation between energy and momentum
($\epsilon=p^{2}/(2m)$) like in the nonrelativistic case. On the other hand,
its velocity ($v_{F}$) is not close to the speed of light ($c$), so some
relativistic corrections will appear. In that sense, electrons in graphene
follow neither relativistic nor nonrelativistic. So, its different properties
may not be the same as expected for traditional nonrelativistic or
relativistic matters as shown in Refs. [38, 39], for thermodynamics [38] and
dissipative viscous [39] properties. In this regard, thermal and electrical
conductivity for the graphene (G) case may also be interesting to compare with
nonrelativistic (NR) and ultrarelativistic (UR) cases, which are studied in
the present work. In Sub-sec. II.1, we have gone through the first non-fluid
type framework description, which is traditionally adopted in standard books
of solid state physics [1, 2, 3, 4, 5]. Next, in Sub-sec. II.2, we have
extended our framework towards fluid type description.
### II.1 Non fluid description
For systematic study and understanding, we will calculate the Lorenz ratios
for the 3D system and then the 2D system. The NR, UR, and G case expressions
of the Lorenz ratio will be addressed step by step for each system. Then, we
will also examine the limiting conditions using Sommerfeld’s expansion.
3D-NR: Let us first discuss the 3D-NR case. The well-known Drude’s formula for
the electrical conductivity of nonrelativistic electrons in 3D solids (metals)
is
$\sigma_{NR}^{3D}=\frac{ne^{2}\tau_{c}}{m}=\frac{ne^{2}\lambda}{mv_{F}},$ (3)
and the thermal conductivity is
$\kappa_{NR}^{3D}=\frac{1}{3}nv_{F}\lambda\,\prescript{3D}{NR}{[C_{V}]_{e}},$
(4)
where, $n$ is the number (electron) density, $\tau_{c}$ is the relaxation
time, $m$ is electron mass, $\lambda=v_{F}\tau_{c}$ is the mean free path, and
$[C_{V}]_{e}$ is the electronic specific heat per particle. After taking the
ratio of thermal and electrical conductivity, we get
$\frac{\kappa_{NR}^{3D}}{\sigma_{NR}^{3D}}=\frac{2}{3}\frac{\epsilon_{F}}{e^{2}}\,\prescript{3D}{NR}{[C_{V}]_{e}},$
(5)
where $\epsilon_{F}=\mu=\frac{1}{2}mv_{F}^{2}$ is the Fermi energy.
Here, in Eq. (5), $[C_{V}]_{e}$ is the main important thing to calculate the
conductivity ratio. We will use the two definitions of $[C_{V}]_{e}$:
1. 1.
$[C_{V}]_{e1}=\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}\Bigg{|}_{V,\ \epsilon_{F}}$,
2. 2.
$[C_{V}]_{e2}=\frac{1}{N}\frac{\partial U}{\partial T}\Bigg{|}_{V,\
\epsilon_{F}}$.
Here, for subsequent simplicity, we will use the notations like NR1 and G1
concerning $[C_{V}]_{e1}$ and NR2 and G2 dealing with $[C_{V}]_{e2}$. The
former definition prescribes taking $T$ derivative of internal energy per
particle $u=U/N$, while the latter definition says to take $T$ derivative of
internal energy $U$ and then normalize by $N$. The detailed calculation can be
seen in Appendix C. Let us first put the former specific heat,
$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]},$
(6)
in Eq. (5), the Lorenz ratio for 3D-NR will be
$\frac{L_{NR1}^{3D}}{L_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}.$
(7)
Here,
$\displaystyle
f_{\nu}(A)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx,$
(8)
is the standard Fermi integral function (see details in Appendix B) with
$A=e^{\epsilon_{F}/k_{B}T}$. Using Sommerfeld’s lemma, in the limit of
$\epsilon_{F}/k_{B}T=lnA>>1$, the electronic-specific heat becomes
$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{\pi^{2}}{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(9)
and then Eq. (7) becomes
$L_{NR1}^{3D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (10)
which is the so-called Wiedemann-Franz law for metals.
Next, using another definition of $[C_{V}]_{e2}$ for the 3D-NR case, whose
general expression will be (see Appendix C)
$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}~{},$
(11)
we will get the Lorenz ratio as
$\frac{L_{NR2}^{3D}}{L_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(12)
In the limit of $\epsilon_{F}/T=lnA>>1$, Sommerfeld’s expansion of Eq. (11)
becomes
$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3\pi^{2}}{4}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(13)
and then the Eq. (12) becomes
$L_{NR2}^{3D}=\frac{\pi^{2}}{2}\left(\frac{k_{B}}{e}\right)^{2}=1.5L_{0}.$
(14)
3D-G: Now, we consider a hypothetical 3D system, which follow graphene-type
dispersion relation $\epsilon=v_{F}p$. In this case, the electrical
conductivity
$\sigma_{G}^{3D}=\frac{ne^{2}\tau_{c}}{\epsilon_{F}}v_{F}^{2}=\frac{ne^{2}\lambda}{\epsilon_{F}}v_{F},$
(15)
and the thermal conductivity,
$\kappa_{G}^{3D}=\frac{1}{3}nv_{F}\lambda\prescript{3D}{G}{[C_{V}]_{e}},$ (16)
will form the ratio
$\frac{\kappa_{G}^{3D}}{\sigma_{G}^{3D}}=\frac{1}{3}\frac{\epsilon_{F}}{e^{2}}\prescript{3D}{G}{[C_{V}]_{e}},$
(17)
where the relation between the Fermi energy and the Fermi momentum will be
$\epsilon_{F}=p_{F}v_{F}$ for graphene.
Using two possible expressions (see Appendix C) of specific heat for 3D-G
system,
$\prescript{3D}{G1}{[C_{V}]_{e1}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}$
(18)
and
$\prescript{3D}{G2}{[C_{V}]_{e2}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(19)
in Eq. (17), we get the Lorenz ratios
$\frac{L_{G1}^{3D}}{L_{0}}=\frac{\kappa_{G1}^{3D}}{\sigma_{G1}^{3D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}$
(20)
and
$\frac{L_{G2}^{3D}}{L_{0}}=\frac{\kappa_{G2}^{3D}}{\sigma_{G2}^{3D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(21)
respectively. In the Sommerfeld’s limit, Eqs. (18) and (19) will be converted
to
$\prescript{3D}{G1}{[C_{V}]_{e1}}=\pi^{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(22)
and
$\prescript{3D}{G2}{[C_{V}]_{e2}}=3\pi^{2}\frac{k_{B}^{2}T}{\epsilon_{F}},$
(23)
and so, Eqs. (20) and (21) become
$L_{G1}^{3D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (24)
and
$L_{G2}^{3D}=\pi^{2}\left(\frac{k_{B}}{e}\right)^{2}=3L_{0}.$ (25)
respectively.
2D-NR: Now, let us go for 2D cases. In the case of 2D nonrelativistic system,
the ratio of thermal and electrical conductivity can be written as
$\frac{\kappa_{NR}^{2D}}{\sigma_{NR}^{2D}}=\frac{\epsilon_{F}}{e^{2}}\prescript{2D}{NR}{[C_{V}]_{e}}.$
(26)
Using two possible expressions (see Appendix C) of specific heat for 2D-NR
system,
$\prescript{2D}{NR1}{[C_{V}]_{e1}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]},$
(27)
and
$\prescript{2D}{NR2}{[C_{V}]_{e2}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(28)
in Eq. (26), we get the Lorenz ratios
$\frac{L_{NR1}^{2D}}{L_{0}}=\frac{\kappa_{NR1}^{2D}}{\sigma_{NR1}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}$
(29)
and
$\frac{L_{NR2}^{2D}}{L_{0}}=\frac{\kappa_{NR2}^{2D}}{\sigma_{NR2}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(30)
respectively. In the Sommerfeld’s limit (SL), Eqs. (29) and (30) will be
converted to
$L_{NR1}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (31)
$L_{NR2}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0}.$ (32)
2D-G: Now, we are taking an actual 2D graphene case. The ratio of
$\kappa_{G}^{2D}$ and $\sigma_{G}^{2D}$ is given by
$\frac{\kappa_{G}^{2D}}{\sigma_{G}^{2D}}=\frac{1}{2}\frac{\epsilon_{F}}{e^{2}}\prescript{2D}{G}{[C_{V}]_{e}}.$
(33)
Using two possible expressions (see Appendix C) of specific heat for 2D-G
system,
$\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]},$
(34)
and
$\prescript{2D}{G2}{[C_{V}]_{e2}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(35)
in Eq. (33), we get the Lorenz ratios
$\frac{L_{G1}^{2D}}{L_{0}}=\frac{\kappa_{G1}^{2D}}{\sigma_{G1}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}$
(36)
and
$\frac{L_{G2}^{2D}}{L_{0}}=\frac{\kappa_{G2}^{2D}}{\sigma_{G2}^{2D}TL_{0}}=\frac{3}{\pi^{2}}\frac{\epsilon_{F}}{k_{B}T}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
(37)
respectively. The SL of Eqs. (36) and (37) will be
$L_{G1}^{2D}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=L_{0},$ (38)
and
$L_{G2}^{2D}=\frac{2\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}=2L_{0}$ (39)
respectively.
3D-UR and 2D-UR: The expressions of electrical and thermal conductivity for
3D-UR and 2D-UR will be same as those expressions for 3D-G and 2D-G if $v_{F}$
is replaced by the speed of light $c$. Readers can notice that specific heat
and Lorenz ratios for 3D-G and 2D-G are independent of $v_{F}$, so those
expressions can also be used for 3D-UR and 2D-UR systems. Since our focal
quantity is the Lorenz ratio in the results, we will not discuss it further.
The expressions of 3D-UR and 2D-UR systems coincide with those of 3D-G and
2D-G systems.
### II.2 An approach towards the fluid description
The previous subsection provides the expression of $\kappa$ and $\sigma$ using
a standard solid-state physics framework, where no fluid concept has been
entered. So, we can call that description a non-fluid description. In the
present section, we will try to build an approach towards a fluid description
by using relaxation time approximation-based Boltzmann transport equations.
Let us first address the brief calculations of the Lorenz ratio for 2D-G
systems; then, we will write down its final expressions for other possible
systems like 2D-NR, 3D-NR, and 3D-G instead of repeating similar calculations.
2D-G: Let us assume a local thermalization picture of electron fluid in a 2D-G
system, where the equilibrium distribution function (see Appendix A)
$f_{0}\left(T(x^{\mu})\right)=\frac{1}{e^{\left(\epsilon-\epsilon_{F}\right)/k_{B}T(x^{\mu})}+1},$
(40)
will be $x^{\mu}$-dependent due to temperature profile $T(x^{\mu})$. Here,
$x^{\mu}=(ct,x^{i})$, a four-dimensional coordinate is considered for general
notation, but we have to take care of $i=1,2$ for the 2D system and $i=1,2,3$
for the 3D system.
Now, let us first write down the macroscopic definitions for electrical and
thermal conductivity
$\displaystyle{\vec{J}}$ $\displaystyle=$ $\displaystyle\sigma{\vec{E}},$
$\displaystyle{\vec{Q}}$ $\displaystyle=$
$\displaystyle\kappa{\vec{\nabla}}T~{},$ (41)
where electrical current vector ${\vec{J}}$ and heat flow vector ${\vec{Q}}$
can have microscopic expressions:
$\displaystyle{J_{i}}$ $\displaystyle=$ $\displaystyle
ge\int\frac{d^{2}p}{h^{2}}v_{i}\delta f_{\sigma},$ $\displaystyle{Q_{i}}$
$\displaystyle=$ $\displaystyle g\int\frac{d^{2}p}{h^{2}}\epsilon v_{i}\delta
f_{\kappa}.$ (42)
Here, we are assuming that the external electric field $E_{i}$ and the
temperature gradient ${\vec{\nabla}}T$ will create deviations $\delta
f_{\sigma}$ and $\delta f_{\kappa}$ respectively from the equilibrium
distribution $f_{0}$.
It is relaxation time approximation (RTA) based on Boltzmann’s transport
equation (BTE) [42]
$\frac{\partial f}{\partial t}+\frac{\partial x^{i}}{\partial t}\frac{\partial
f}{\partial x^{i}}+\frac{\partial p^{i}}{\partial t}\frac{\partial f}{\partial
p^{i}}=-\frac{\delta f}{\tau_{c}}~{},$ (43)
which will guide us to guess appropriate form of $\delta f_{\sigma}$ and
$\delta f_{\kappa}$. Considering $f=f_{0}+\delta f\approx f_{0}$ in left hand
side of Eq. (43) and local thermalization assumption of $f_{0}$, given in Eq.
(40), we can simplify Eq. (43) as
$\displaystyle v^{i}\frac{\partial f_{0}}{\partial x^{i}}+eE^{i}\frac{\partial
f_{0}}{\partial p^{i}}$ $\displaystyle=$ $\displaystyle-\frac{{\delta
f}_{\sigma}}{\tau_{c}}-\frac{{\delta f}_{\kappa}}{\tau_{c}}$ $\displaystyle
v^{i}\frac{\partial T}{\partial x^{i}}\frac{\partial\epsilon}{\partial
T}\frac{\partial
f_{0}}{\partial\epsilon}+eE^{i}\frac{\partial\epsilon}{\partial
p^{i}}\frac{\partial f_{0}}{\partial\epsilon}$ $\displaystyle=$
$\displaystyle-\frac{{\delta f}_{\sigma}}{\tau_{c}}-\frac{{\delta
f}_{\kappa}}{\tau_{c}}$ $\displaystyle v^{i}\frac{\partial T}{\partial
x^{i}}[C_{V}]_{e}\frac{\partial
f_{0}}{\partial\epsilon}+eE^{i}v_{i}\frac{\partial f_{0}}{\partial\epsilon}$
$\displaystyle=$ $\displaystyle-\frac{{\delta
f}_{\sigma}}{\tau_{c}}-\frac{{\delta f}_{\kappa}}{\tau_{c}}.$ (44)
Here, we consider the approximation
$[C_{V}]_{e}\approx\frac{\partial\epsilon}{\partial T}$, and one can expect
again two possible definitions of specific heat as discussed in the earlier
section. From Eq. (44), we can get the form of $\delta f_{\sigma}$ and $\delta
f_{\kappa}$ as
$\displaystyle\delta f_{\sigma}$ $\displaystyle=$ $\displaystyle
eE^{i}v_{i}\left(-\frac{\partial f_{0}}{\partial\epsilon}\right)\tau_{c},$
$\displaystyle\delta f_{\kappa}$ $\displaystyle=$ $\displaystyle
v^{i}\left(-\frac{\partial
f_{0}}{\partial\epsilon}\right)[C_{V}]_{e}\left(\frac{\partial T}{\partial
x^{i}}\right)\tau_{c},$ (45)
with
$\left(-\frac{\partial f_{0}}{\partial\epsilon}\right)=\beta f_{0}(1-f_{0}).$
(46)
Using Eq. (45) in Eq. (42) and then comparing with Eq. (41), we will get the
final expressions of electrical and thermal conductivity:
$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle
ge^{2}\frac{v_{F}^{2}}{2}\tau_{c}\int\frac{d^{2}p}{h^{2}}\beta f_{0}(1-f_{0})$
$\displaystyle\implies\sigma_{G}^{2D}$ $\displaystyle=$ $\displaystyle 2\pi
k_{B}\tau_{c}\left(\frac{e}{h}\right)^{2}f_{1}\left(A\right)T,$ (47)
and
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
g\epsilon\frac{v_{F}^{2}}{2}\tau_{c}[C_{V}]_{e}\int\frac{d^{2}p}{h^{2}}\beta
f_{0}(1-f_{0})$ $\displaystyle\implies\kappa_{G}^{2D}$ $\displaystyle=$
$\displaystyle\frac{4\pi
k_{B}^{2}\tau_{c}}{h^{2}}[C_{V}]_{e}\,f_{2}\left(A\right)T^{2}.$ (48)
Now, using two different forms of specific heat, given in Eqs. (34) and (35)
in Eq. (48), we get
$\kappa_{G1}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}f_{2}\left(A\right)T^{2},$
(49)
and
$\kappa_{G2}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}f_{2}\left(A\right)T^{2},$
(50)
and hence, corresponding Lorenz ratios will be
$\frac{L_{G1}^{2D}}{L_{0}}=\frac{12}{\pi^{2}}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)},$
(51)
and
$\frac{L_{G2}^{2D}}{L_{0}}=\frac{12}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}$
(52)
respectively.
2D-NR After doing the same calculation as the previous one for a 2-dimensional
nonrelativistic system, the final general expressions of the electrical and
two possible thermal conductivities and their corresponding Lorenz ratios
$L_{NR}^{2D}/L_{0}$ are given by as below:
$\displaystyle\sigma_{NR}^{2D}=4\pi
k_{B}\tau_{c}\left(\frac{e}{h}\right)^{2}f_{1}\left(A\right)T,$ (53)
$\displaystyle\kappa_{NR1}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}f_{2}\left(A\right)T^{2},$
(54) $\displaystyle\kappa_{NR2}^{2D}=\frac{8\pi
k_{B}^{3}\tau_{c}}{h^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}f_{2}\left(A\right)T^{2}~{},$
(55)
$\displaystyle\frac{L_{NR1}^{2D}}{L_{0}}=\frac{6}{\pi^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)},$
(56)
$\displaystyle\frac{L_{NR2}^{2D}}{L_{0}}=\frac{6}{\pi^{2}}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}.$
(57)
3D-G: For the 3D-G case, the expression of the electrical conductivity and two
possible thermal conductivities and their corresponding Lorenz ratios
$L_{G}^{3D}/L_{0}$ are given as below:
$\sigma_{G}^{3D}=16\pi
k_{B}^{2}\tau_{c}\left(\frac{e^{2}}{3h^{3}v_{F}}\right)f_{2}\left(A\right)T^{2},$
(58) $\kappa_{G1}^{3D}=\frac{48\pi
k_{B}^{4}\tau_{c}}{h^{3}v_{F}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}f_{3}\left(A\right)T^{3},$
(59) $\kappa_{G2}^{3D}=\frac{48\pi
k_{B}^{4}\tau_{c}}{h^{3}v_{F}}\left[4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\right]f_{3}\left(A\right)T^{3},$
(60)
$\frac{L_{G1}^{3D}}{L_{0}}=\frac{27}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)},$
(61)
$\frac{L_{G2}^{3D}}{L_{0}}=\frac{27}{\pi^{2}}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}.$
(62)
3D-NR: Now again, the same type of calculation is done for a 3-dimensional
nonrelativistic system; we get all expressions of the electrical, thermal
conductivity, and the ratio $L_{NR}^{3D}/L_{0}$ in a more general form, which
can be expressed as
$\displaystyle\sigma_{NR}^{3D}=\frac{2e^{2}k_{B}^{\frac{3}{2}}\tau_{c}}{m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}f_{\frac{3}{2}}\left(A\right)T^{\frac{3}{2}}~{},$
(63)
$\displaystyle\kappa_{NR1}^{3D}=\frac{15k_{B}^{\frac{7}{2}}\tau_{c}}{2m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}$
$\displaystyle f_{\frac{5}{2}}\left(A\right)T^{\frac{5}{2}},$ (64)
$\displaystyle\kappa_{NR2}^{3D}=\frac{15k_{B}^{\frac{7}{2}}\tau_{c}}{2m}\left(\frac{2\pi
m}{h^{2}}\right)^{\frac{3}{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}$
$\displaystyle f_{\frac{5}{2}}\left(A\right)T^{\frac{5}{2}},$ (65)
$\displaystyle\frac{L_{NR1}^{3D}}{L_{0}}=\frac{45}{4\pi^{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}~{},$
(66)
$\displaystyle\frac{L_{NR2}^{3D}}{L_{0}}=\frac{45}{4\pi^{2}}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}.$
(67)
## III Results
Figure 1: The Lorenz ratio regarding the chemical potential for non-fluid
descriptions of graphene case (a) for 3D and (b) for 2D expressions.
Figure 2: The Lorenz ratio regarding the chemical potential for non-fluid
descriptions of nonrelativistic case (a) for 3D and (b) for 2D expressions.
In the above section, we have calculated all the general expressions of
electrical and thermal conductivity and their ratios to check the validity of
the Wiedemann-Franz law for all possible systems like 2D and 3D graphene (G)
and nonrelativistic (NR) cases. Using those final expressions, the present
section is intended to go with their numerical estimations. During the result
generation, we will use natural unit $k_{B}=\hbar=c=1$ for our convenience.
Dimensionless quantities Lorenz ratio ($L/L_{0}$) and $\epsilon_{F}/T$ will be
taken as the y and x axes of all graphs.
Figure 3: Left: Lorenz ratio of 2D-G systems by using fluid-type expression.
Right: Transition from fluid to non-fluid domain in 2D-G system.
In Fig. 1 and Fig. 2, we explored the Lorenz ratio ($\frac{L}{L_{0}}$) by
using non-fluid (NF) type expressions and then the same ratios for both non-
fluid (NF) and fluid (F) type expressions in Fig. 3. Fig. 1(a) and (b) have
shown the Lorenz ratio versus $\frac{\epsilon_{F}}{T}$ for 3D-G and 2D-G
systems respectively. By using the non-fluid (NF) type of Eq. (20) and its
Sommerfeld limit Eq. (24), we plotted the black solid and dotted lines in Fig.
1(a). We used the specific heat ${[C_{V}]_{e1}}=\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}$, which is traditionally used in solid-state or
non-fluid systems. We are getting $L/L_{0}=1$ in the limit of
$\epsilon_{F}/T\gg 1$ or Fermi Liquid (FL) regime of graphene case, which is
as expected for most of the metal cases. The deviation of the Lorenz ratio
from 1 has been started below $\frac{\epsilon_{F}}{T}\approx 10$ as we notice
from the black solid line. We have also plotted the red solid line by using
Eq. (21), where another definition of specific heat ${[C_{V}]_{e2}}$ is used
in the thermal conductivity expression. In the case of SL of 3D-G (the red
dotted line), the Lorenz ratio is three instead of one. For the general case
(the red solid line), the Lorenz ratio deviates from this constant line $3$
below $\frac{\epsilon_{F}}{T}\approx 10$. For the 2D-G system, similar kinds
of black and red solid lines are plotted in Fig. 1(b) by using Eq. (36) and
Eq. (37). Their respective SLs by using Eq. (38) and Eq. (39), are plotted as
black and red horizontal dotted lines. We considered the surface area $(S)$ in
place of volume $(V)$ in the 2D system. The Lorenz ratio when we take
${[C_{V}]_{e2}}$, the ratio becomes 2 (the red-dotted line).
The Fig. 2(a) and (b), similar to Fig. 1, display the Lorenz ratio in terms of
$\frac{\epsilon_{F}}{T}$ for non-fluid expressions of NR case. Though the real
metals have a constant chemical potential within the range,
$\epsilon_{F}=2-10$ eV, we imagine that it can be tunable from $0$ to
$\infty$. In Fig. 2(a), we notice that the WF law is valid in the SL of the 3D
metal case, expressed by Eq. (10) (green dot-dashed line). We have taken
$T=60$ K ($\approx 0.005$ eV), so metal range $\epsilon_{F}=2$-$10$ eV will
give the ratio $\frac{\epsilon_{F}}{T}=400$-$2000$ ($\gg 1$), which is quite
good domain for SL. Therefore, being situated in the SL domain, metal always
follows the WF law. A slight larger value $L=1.5L_{0}$ comes from Eq. (14)
(blue dotted line), when specific heat ${[C_{V}]_{e2}}$ is taken. A similar
kind of behavior can be seen in Fig. 2(b) for 2D-NR systems, but
interestingly, SLs in both cases, given by Eqs. (31) and (32) are coincided at
$L=L_{0}$ (blue-dotted and green dot-dashed lines). It probably indicates that
if one builds 2D metal systems, then it will also follow the WF law as their
Fermi energy range is again expected to be situated in the SL domain.
Before going to the next graphical results, let us briefly analyze what we
have learned from Fig. 1(a), (b) and Fig. 2(a), (b). Grossly, the non-fluid
type expressions of Lorenz ratios for all cases - 3D-NR, 2D-NR, 3D-G, and 2D-G
are saturated near 1 in the SL range, which means that the WF law is obeyed in
the SL range. Among all four cases, 3D-NR and 2D-G are realistic cases. The
former is applicable to metal systems, whose Fermi energies remain constant
and within the SL range, so they always follow the WF law. Later case - the
non-fluid expressions of Lorenz ratio for 2D-G can be applicable for realistic
graphene system with high doping concentration ($\epsilon_{F}$ is quite large
or $\epsilon_{F}/T\gg 1$), where the WF law is obeyed well again [15].
However, by decreasing the doping concentration, when the graphene system
approaches the charge-neutral points ($\epsilon_{F}$ is quite small or
$\epsilon_{F}/T\ll 1$), the WF law violation domain is observed [15].
According to our non-fluid expressions, the WF law violation is also expected
in the $\epsilon_{F}/T\ll 1$ domain but the outcome $L/L_{0}<1$ is not
matching with experimental outcome $L/L_{0}>1$ by Crossno et al. [15]. As a
result, this $\epsilon_{F}/T\ll 1$ domain may need a fluid-type expression
(something different from traditional non-fluid expressions) of $L/L_{0}$.
This unsettled fact is investigated by the RTA-based Boltzmann transport
formalism, which may be considered an approach towards fluid descriptions. It
was already addressed in the formalism section, and its numerical results will
be described in the next paragraph.
For numerical curves of fluid-type expressions, let us focus only on the 2D-G
case. Although formalism of other cases with an Appendix is provided in the
present article, one may go for result generation for other cases if they have
any real system application. Using Eqs. (51) and (52), black and red solid
lines are plotted in the left panel of Fig. 3. They are saturated at the
values of 2 and 4, respectively, in the high doping or $\epsilon_{F}/T\gg 1$
domain. It means that if the fluid type expression is valid in this high
doping domain 2D-G system, then the WF law may not be obeyed. However, the
experimental data [15] claims that the WF law is well obeyed in the high
doping or $\epsilon_{F}/T\gg 1$ domain. This means that the non-fluid
expression of the Lorenz ratio should be applicable to this domain instead of
fluid-type expression. The SL of Eqs. (51) and (52) will also be (see Appendix
E)
$\displaystyle\frac{L_{G1}^{2D}}{L_{0}}=2+\frac{2\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(68)
$\displaystyle\frac{L_{G2}^{2D}}{L_{0}}=4+\frac{4\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(69)
which are plotted by black and red dotted lines, respectively, in the left
panel of Fig. 3. They blow up from their saturated values (2 and 4) when
$\epsilon_{F}/T$ or doping concentration decreases. So, collecting the results
of all SL expressions -Eqs. (68) and (69) for fluid-type pictures and the
values 1 and 2 from Eqs. (38) and (39) for non-fluid-type pictures, we have
plotted black, red dotted, and dash lines in the right panel of Fig. 3. The
experimental data of Crossno et al. [15] is pasted as an inset figure in the
right panel of Fig. 3, where S1, S2, and S3 represent the samples of the 2D-G
system in terms of lower to higher values of minimum charge carrier density.
Theoretically, zero charge carrier density can be reached at
$\epsilon_{F}/T\rightarrow 0$ but experimentally, a minimum charge carrier
density [15] will be achieved. So, connecting the experimental and theoretical
outcomes, we may have to focus on the qualitative trend of curves. From the
experimental side, we may take the gross level phenomenon that the transition
from higher to lower values of $\epsilon_{F}/T$ pushes the Lorenz ratio from
$L/L_{0}=1$ to $L/L_{0}>1$. On the other hand, from the theoretical side, when
we compile our SL results, the fluid-type expression of Lorenz ratio at lower
values of $\epsilon_{F}/T$ provides $L/L_{0}>1$, while its non-fluid-type
expression at higher values of $\epsilon_{F}/T$ provides $L/L_{0}=1$. So, one
may expect a transition from non-fluid to fluid behavior of electrons in the
2D-G system during the transition from higher to lower $\epsilon_{F}/T$ or
charge carrier concentration. Blue arrows in the right panel of Fig. 3 roughly
denote this transition. This indirectly supports the expectation of the Dirac-
fluid nature of electrons in the domain of lower $\epsilon_{F}/T$ of the 2D-G
system, where the WF law is violated ($L/L_{0}>1$). Eventually, one may
conclude that fluid nature is the reason for the violation.
## IV Summary
In summary, we have gone through comparative research on Lorenz ratios for 3D,
2D NR and G systems for a systematic search of the WF law violation domain.
During our analysis, we found the WF law violation domain of Fermi-energy
(normalized by a fixed temperature), where the Lorenz ratio deviates from 1.
First, we have used standard solid state book-based non-fluid expressions of
Lorenz ratio for the 3D-NR case, which are also applied for 2D-NR, 3D-G, and
2D-G cases. For all cases, the Lorenz ratio remains constant in the higher
values of normalized Fermi energy. This fact is well in agreement with the
universal validity of the WF law in metal systems, which can be described by
the 3D-NR case of the electron gas having larger Fermi-energy ($2-10$ eV) or
normalized Fermi-energy 400-2000 at $60$ K. Depending on the definition of
specific heat at constant volume and Fermi-energy, we will get two possible
saturated values of Lorenz ratios - 1 and 2, where former is experimentally
observed (average) value but one may mark the saturating domain as the WF law
obeying domain. If one identifies the less and greater than one value of
normalized Fermi-energy, then former domain of all cases shows the deviation
of the WF law. This tuning of normalized Fermi-energy is conveniently possible
for 2D-G system only via tuning the electron doping. Interestingly, non-fluid
expressions of the 2D-G case and recently measured Lorenz ratio by Crossno et
al. [15] both indicate the violation of the WF law in the lower normalized
Fermi-energy domain, which is popularly called as Dirac Fluid (DF) domain.
However, the theory of non-fluid expressions provides Lorenz ratio as lower
than one, but the experiment pointed out that its values are larger than one
in the DF domain. It demands that we may have to transit towards a fluid-type
description from the standard non-fluid expressions of the Lorenz ratio. So
far, to the best of our knowledge, this straightforward thermodynamical phase
space analysis of Lorenz ratio is missing in the literature, and it may be an
important first step before going to a fluid-type description.
Next, we have built a framework towards the fluid-type description of the
Lorenz ratio by using the Boltzmann transport equation, whose Sommerfeld
expansion limit becomes divergent instead of constant for non-fluid
description. In the lower normalized Fermi-energy domain or DF domain, this
divergence trend of the Lorenz ratio may be associated with the experimentally
observed enhancement of the Lorenz ratio. Interestingly, by increasing the
cleanness of the sample, the Lorenz ratio is also enhanced. Therefore, this
connection between our present calculation and experimental data indicates a
necessity for fluid prescription. The present work aims for a systematic
comparative study to identify the the WF law violation domain from a non-fluid
to fluid type framework. After going through this systematic analysis, the
present work is aimed for the future to explain or understand the Crossno et
al. [15] experimental data by doing more phenomenological analysis.
###### Acknowledgements.
This work was partly (T.Z.W. and C.W.A.) supported by the Doctoral Fellowship
in India (DIA) program of the Ministry of Education, Government of India. The
authors thank the other members of eHD club Sesha P. Vempati, Ashutosh
Dwibedi, Narayan Prasad, Bharat Kukkar, and Subhalaxmi Nayak.
## Appendix A FERMI-DIRAC DISTRIBUTION FUNCTION
The Fermi-Dirac distribution function is
$f_{0}\left(\epsilon\right)=\frac{1}{e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1}=\frac{1}{A^{-1}e^{\beta\epsilon}+1},$
(70)
where $\epsilon$ = energy of fermions and $\epsilon_{F}$ = fermi energy that
is nothing but the energy level up to the fermions filled or the maximum
kinetic energy of fermions at absolute temperature (0 K). The thermodynamics
parameter which is $\beta=\frac{1}{k_{B}T}$ and $k_{B}$ is the Boltzmann
constant and $A$ is the fugacity of the system described by
$A=\exp{\left({\frac{\epsilon_{F}}{k_{B}T}}\right)}.$
And the derivative of the distribution function with respect to energy is
$-\frac{\partial f_{0}}{\partial\epsilon}=\frac{\beta
e^{\beta\left(\epsilon-\epsilon_{F}\right)}}{\left(e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1\right)^{2}}=\frac{\partial}{\partial\epsilon_{F}}\left(\frac{1}{e^{\beta\left(\epsilon-\epsilon_{F}\right)}+1}\right).$
(71)
## Appendix B FERMI-DIRAC FUNCTION
* •
The Fermi-Dirac Function
$\displaystyle
f_{\nu}(A)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx~{},$
(72)
where $f_{\nu}(A)$ is known as the Fermi-Dirac function.
Case.1. When $A$ is small, then the Fermi-Dirac function can be written in a
series form which is
$\displaystyle f_{\nu}(A)=$ $\displaystyle
A-\frac{A^{2}}{2^{\nu}}+\frac{A^{3}}{3^{\nu}}-\frac{A^{4}}{4^{\nu}}+\dots$
(73)
$\displaystyle=\sum_{n=1}^{\infty}\left(-1\right)^{n-1}\frac{A^{n}}{n^{\nu}}.$
(74)
Case.2. When $A$ is too much small $\left(\epsilon_{F}<<k_{B}T\right)$, then
the function becomes simplified as $A$
$f_{\nu}(A)=A,$ (75)
but if we take $\epsilon_{F}=0$, the function becomes unity.
Case.3. When the temperature is very small, and the Fermi energy has some
finite value, then the Fermi-Dirac function can be written according to
Sommerfeld’s lemma, which gives the expression of the function as
$f_{\nu}\left(A\right)=\frac{\alpha^{\nu}}{\Gamma\left(\nu+1\right)}\Bigg{[}1+\nu\left(\nu-1\right)\frac{\pi^{2}}{6}\frac{1}{\alpha^{2}}+\nu\left(\nu-1\right)\left(\nu-2\right)\left(\nu-3\right)\frac{7\pi^{4}}{360}\frac{1}{\alpha^{4}}+\dots\Bigg{]},$
(76)
where $\alpha$ is given by
$\alpha=\ln{A}=\frac{\epsilon_{F}}{k_{B}T}~{}.$
And
$\Gamma\left(\nu+1\right)=\nu!~{},$
so, using this lemma, we can calculate the value of the Fermi-Dirac function
for different values of $\nu$ for this particular type of case.
Case.4. When $\left(\epsilon_{F}>>k_{B}T\right)$, then the function can be
written only using the zeroth order term of Sommerfeld’s lemma expression.
* •
The derivative of Fermi function with respect to Fermi energy
$\frac{\partial f_{\nu}\left(A\right)}{\partial\epsilon_{F}}=\beta
f_{\nu-1}\left(A\right).$ (77)
* •
The derivative of Fermi function with respect to temperature
$\frac{\partial f_{\nu}\left(A\right)}{\partial
T}=\frac{1}{A}f_{\nu-1}\left(A\right)\frac{\partial A}{\partial T}.$ (78)
The one identity for the function can be written as
$\frac{1}{A}\frac{\partial A}{\partial
T}=-\epsilon_{F}\beta^{2}k_{B}=-\frac{\epsilon_{F}}{k_{B}T^{2}}.$ (79)
## Appendix C ELECTRONIC SPECIFIC HEAT
For 2D system graphene, which follows the linear dispersion relation, with the
help of the density of state method, we can calculate the number density,
total internal energy, and the electronic-specific heat.
So, the number of energy states in energy range $\epsilon$ to
$\epsilon+d\epsilon$ and the surface area $S$ is written as
$D\left(\epsilon\right)d\epsilon=g\frac{S}{\left(2\pi\hbar\right)^{2}}\frac{2\pi}{v_{F}^{2}}\epsilon
d\epsilon.$ (80)
Now, the total number of particles at any value of temperature can be
calculated as
$N=\int_{0}^{\infty}D\left(\epsilon\right)d\epsilon
f_{0}\left(\epsilon\right).$ (81)
After plugging the value of $D\left(\epsilon\right)d\epsilon$ in the above
equation, we get
$n=\frac{N}{S}=\frac{g}{2\pi\hbar^{2}v_{F}^{2}}\int_{0}^{\infty}\frac{\epsilon}{A^{-1}e^{\beta\epsilon}+1}d\epsilon.$
(82)
After solving this integration, we get the final, more general expression of
number density, which is given by
$N=g\frac{2\pi
S}{h^{2}v_{F}^{2}}\frac{\Gamma\left(2\right)}{\beta^{2}}f_{2}\left(A\right)$
(83) $\implies
n=\frac{k_{B}^{2}}{\pi\hbar^{2}v_{F}^{2}}f_{2}\left(A\right)T^{2}~{}.$ (84)
The total internal energy of a system can be calculated as
$U=\int_{0}^{\infty}D\left(\epsilon\right)d\epsilon
f_{0}\left(\epsilon\right)\epsilon.$ (85)
By substituting the value of $D\left(\epsilon\right)d\epsilon$ in the above
equation and solving the integration, we get the final, more general
expression of total internal energy, which is given by
$U=g\frac{2\pi
S}{h^{2}v_{F}^{2}}\frac{\Gamma\left(3\right)}{\beta^{3}}f_{3}\left(A\right)$
(86) $\implies
u=\frac{U}{S}=g\frac{2\pi}{h^{2}v_{F}^{2}}\frac{\Gamma\left(3\right)}{\beta^{3}}f_{3}\left(A\right).$
(87)
Now, from the above equations, we get
$\frac{u}{n}=\frac{U}{N}=2k_{B}T\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}.$
(88)
Let us define the specific heat capacity of the electron by taking the
temperature derivative of internal energy per electron ($U/N=u/n$) while
keeping surface area $S$ and $\epsilon_{F}$ as constants:
$\displaystyle\prescript{2D}{G1}{[C_{V}]_{e1}}=\Bigg{[}\frac{\partial}{\partial
T}\Big{(}\frac{U}{N}\Big{)}\Bigg{]}_{S,\epsilon_{F}}.$ (89)
Using the Eq. (88), the specific heat is
$\displaystyle\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}+T\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)\Bigg{]}.$ (90)
Now, the derivative part after using the identity (78) of the Fermi-Dirac
function can be written as
$\displaystyle\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)$
$\displaystyle=\frac{1}{f_{2}\left(A\right)}\frac{\partial}{\partial
T}\left(f_{3}\left(A\right)\right)-f_{3}\left(A\right)\frac{\partial}{\partial
T}\left(\frac{1}{f_{2}\left(A\right)}\right)$
$\displaystyle=\frac{1}{A}\frac{f_{2}\left(A\right)}{f_{2}\left(A\right)}\frac{\partial
A}{\partial
T}-\frac{f_{3}\left(A\right)}{\Big{[}f_{2}\left(A\right)\Big{]}^{2}}\frac{\partial}{\partial
T}f_{2}\left(A\right)$ $\displaystyle=\frac{1}{A}\frac{\partial A}{\partial
T}-\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\frac{f_{1}\left(A\right)}{f_{2}\left(A\right)}\frac{1}{A}\frac{\partial
A}{\partial T}.$
The final form, we get
$\frac{\partial}{\partial
T}\left(\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\right)=\frac{2}{T}\Bigg{[}\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}.$
(91)
After substituting the value from equation (91) into equation (90), we get a
more general form of electronic specific heat given by
$\prescript{2D}{G1}{[C_{V}]_{e1}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}\Bigg{]}~{}.$
(92)
Let us go for another definition of specific heat per electron
$\prescript{2D}{G2}{[C_{V}]_{e2}}=\frac{1}{N}\frac{\partial U}{\partial
T}\Bigg{|}_{S,\epsilon_{F}}.$ (93)
From Eq. (86), the total internal energy is
$\displaystyle\prescript{2D}{G}{U}=\gamma T^{3}f_{3}\left(A\right),$
where
$\gamma=g\frac{2\pi S}{h^{2}v_{F}^{2}}\Gamma\left(3\right)k_{B}^{3}.$
Then, we will take the derivative of just the above equation to $T$ at
constant Fermi energy $\epsilon_{F}$ and surface area; we get
$\displaystyle\frac{\partial U}{\partial T}\Big{|}_{S,\epsilon_{F}}$
$\displaystyle=\gamma\Bigg{[}3T^{2}f_{3}\left(A\right)+T^{3}f_{2}\left(A\right)\frac{1}{A}\frac{\partial
A}{\partial T}\Bigg{]}$ (94) $\displaystyle=\gamma
T^{2}f_{2}\left(A\right)\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}+T\frac{1}{A}\frac{\partial
A}{\partial T}\Bigg{]}~{}.$ (95)
So,
$\prescript{2D}{G2}{[C_{V}]_{e2}}=2k_{B}\Bigg{[}3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(96)
By doing a similar type of calculation for 2D-NR case, we will get two
different specific heat expressions as
$\prescript{2D}{NR1}{[C_{V}]_{e1}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{f_{1}\left(A\right)}{f_{0}\left(A\right)}\Bigg{]},$
(97)
and
$\prescript{2D}{NR2}{[C_{V}]_{e2}}=k_{B}\Bigg{[}2\frac{f_{2}\left(A\right)}{f_{1}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(98)
Next, for 3D-NR and 3D-G cases also, one can repeat the calculations and find
the expressions of specific heat:
$\prescript{3D}{NR1}{[C_{V}]_{e1}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{3}{2}\frac{f_{\frac{3}{2}}\left(A\right)}{f_{\frac{1}{2}}\left(A\right)}\Bigg{]},$
(99)
$\prescript{3D}{NR2}{[C_{V}]_{e2}}=\frac{3}{2}k_{B}\Bigg{[}\frac{5}{2}\frac{f_{\frac{5}{2}}\left(A\right)}{f_{\frac{3}{2}}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]},$
(100)
$\prescript{3D}{G1}{[C_{V}]_{e1}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-3\frac{f_{3}\left(A\right)}{f_{2}\left(A\right)}\Bigg{]},$
(101)
$\prescript{3D}{G2}{[C_{V}]_{e2}}=3k_{B}\Bigg{[}4\frac{f_{4}\left(A\right)}{f_{3}\left(A\right)}-\frac{\epsilon_{F}}{k_{B}T}\Bigg{]}.$
(102)
## Appendix D NUMBER DENSITY FOR 3D AND 2D SYSTEMS
* •
Case.1. Number density in nonrelativistic case at temperature $T=0K$
$n^{3D}_{NR}=g\frac{4\pi\left(2m\epsilon_{F}\right)^{\frac{3}{2}}}{3h^{3}}.$
(103)
* •
Case.2. Number density in ultrarelativistic case at temperature $T=0K$
$n^{3D}_{UR}=g\frac{4\pi}{3h^{3}}\left(\frac{\epsilon_{F}}{c}\right)^{3}.$
(104)
* •
Case.3. Number density in graphene case at temperature $T=0K$
$n^{3D}_{G}=g\frac{4\pi}{3h^{3}}\left(\frac{\epsilon_{F}}{v_{F}}\right)^{3}.$
(105)
* •
Case.1. Number density in nonrelativistic case at temperature $T=0K$
$n^{2D}_{NR}=g\left(\frac{2\pi m}{h^{2}}\right)\epsilon_{F}.$ (106)
* •
Case.2. Number density in ultrarelativistic case at temperature $T=0K$
$n^{2D}_{UR}=g\frac{\pi}{h^{2}}\left(\frac{\epsilon_{F}}{c}\right)^{2}.$ (107)
* •
Case.3. Number density in graphene case at temperature $T=0K$
$n^{2D}_{G}=g\frac{\pi}{h^{2}}\left(\frac{\epsilon_{F}}{v_{F}}\right)^{2}.$
(108)
## Appendix E SOMMERFELD-LEMMA EXPRESSIONS OF ELECTRICAL AND THERMAL
CONDUCTIVITY FOR FLUID DESCRIPTION
Table 1: Lorenz ratio values in Sommerfeld expression for 3D. | Non-Fluid | Towards Fluid
---|---|---
| 3D-NR | 3D-G | 3D-NR | 3D-G
$\frac{L_{1}}{L_{0}}$ | $1$ | $1$ | $1.5$ | $4.5$
$\frac{L_{2}}{L_{0}}$ | $1.5$ | $3$ | $2.25$ | $9$
Table 2: Lorenz ratio values in Sommerfeld expression for 2D. | Non-Fluid | Towards Fluid
---|---|---
| 2D-NR | 2D-G | 2D-NR | 2D-G
$\frac{L_{1}}{L_{0}}$ | $1$ | $1$ | $1$ | $2$
$\frac{L_{2}}{L_{0}}$ | $1$ | $2$ | $1$ | $4$
The Sommerfeld-lemma expressions for electrical and thermal conductivity for
2-dimensional graphene is
$\displaystyle\sigma_{G}^{2D}=2\pi\tau_{c}\left(\frac{e}{h}\right)^{2}\epsilon_{F},$
(109)
$\displaystyle\kappa_{G1}^{2D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(110)
$\displaystyle\kappa_{G2}^{2D}=\frac{8\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(111)
So, after using Sommerfeld-lemma, the expressions of the electrical(109),
thermal conductivity(110), (111), and the ratio of $L/L_{0}$ for a
2-dimensional graphene system are respectively as;
$\displaystyle\frac{L_{G1}^{2D}}{L_{0}}=2+\frac{2\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(112)
$\displaystyle\frac{L_{G2}^{2D}}{L_{0}}=4+\frac{4\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(113)
For a 3-dimensional graphene system, after using Sommerfeld’s lemma, the
expressions of the electrical and thermal conductivity;
$\displaystyle\sigma_{G}^{3D}=\frac{8\pi
e^{2}\tau_{c}\epsilon_{F}^{2}}{3h^{3}v_{F}}\Biggl{[}1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}\Biggl{]},$
(114)
$\displaystyle\kappa_{G1}^{3D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}^{2}}{h^{3}v_{F}}\Biggl{[}T+\pi^{2}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(115)
$\displaystyle\kappa_{G2}^{3D}=\frac{8\pi^{3}k_{B}^{2}\tau_{c}\epsilon_{F}^{2}}{h^{3}v_{F}}\Biggl{[}T+\pi^{2}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(116)
For a 3-dimensional graphene system, the electrical conductivity(114), thermal
conductivity(115), (116), and the ratio of $L/L_{0}$ are
$\displaystyle\frac{L_{G1}^{3D}}{L_{0}}=\frac{9}{2}+3\pi^{2}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(117)
$\displaystyle\frac{L_{G2}^{3D}}{L_{0}}=9+6\pi^{2}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(118)
After applying Sommerfeld-lemma on a 2-dimensional nonrelativistic system
then, we get all the expressions of electrical and thermal conductivity,
$\displaystyle\sigma_{NR}^{2D}=4\pi\tau_{c}\left(\frac{e}{h}\right)^{2}\epsilon_{F},$
(119)
$\displaystyle\kappa_{NR1}^{2D}=\frac{4\pi^{3}k_{B}^{2}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(120)
$\displaystyle\kappa_{NR2}^{2D}=\frac{4\pi^{3}k_{B}^{2}\epsilon_{F}}{3h^{2}}\Biggl{[}T+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(121)
After applying Sommerfeld-lemma on a 2-dimensional nonrelativistic system
then, we get all the expressions of electrical(119), thermal
conductivity(120), (121), and the ratio of $L/L_{0}$ which are given by
$\displaystyle\frac{L_{NR1}^{2D}}{L_{0}}=1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(122)
$\displaystyle\frac{L_{NR2}^{2D}}{L_{0}}=1+\frac{\pi^{2}}{3}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(123)
After applying Sommerfeld-lemma on a 3-dimensional nonrelativistic system
then, we get all the expressions of electrical and thermal conductivity;
$\displaystyle\sigma_{NR}^{3D}=\frac{8\pi
e^{2}\tau_{c}}{3h^{3}m}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}1+\frac{\pi^{2}}{8}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}\Biggl{]},$
(124)
$\displaystyle\kappa_{NR1}^{3D}=\frac{4\pi^{3}k_{B}^{2}\tau_{c}}{3mh^{3}}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}T+\frac{5\pi^{2}}{8}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]},$
(125)
$\displaystyle\kappa_{NR2}^{3D}=\frac{2\pi^{3}k_{B}^{2}\tau_{c}}{mh^{3}}\left(2m\epsilon_{F}\right)^{\frac{3}{2}}\Biggl{[}T+\frac{5\pi^{2}}{8}\frac{k_{B}^{2}T^{3}}{\epsilon_{F}^{2}}\Biggl{]}.$
(126)
After applying Sommerfeld-lemma on a 3-dimensional nonrelativistic system
then, we get all the expressions of electrical(124), thermal
conductivity(125), (126), and the ratio of $L/L_{0}$ which are given by
$\displaystyle\frac{L_{NR1}^{3D}}{L_{0}}=\frac{3}{2}+\frac{3\pi^{2}}{4}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}},$
(127)
$\displaystyle\frac{L_{NR2}^{3D}}{L_{0}}=\frac{9}{4}+\frac{9\pi^{2}}{8}\frac{k_{B}^{2}T^{2}}{\epsilon_{F}^{2}}.$
(128)
After compiling all of Sommerfeld-lemma results, the roughly summarizing
results for different cases in three dimensions (3D) in the table (1) and two
dimensions (2D) in the table (2), particularly regarding the Lorenz ratio
validation and violation are shown.
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|
# A differentiable programming framework for spin models
Tiago S. Farias<EMAIL_ADDRESS>Institute and Center for Development and
Research in Software Technology - ICTS, Governor Danilo de Matos Areosa
Avenue, 1199, 69075-904, Manaus, AM, Brazil Physics Departament, Center for
Natural and Exact Sciences, Federal University of Santa Maria, Roraima Avenue
1000, 97105-900, Santa Maria, RS, Brazil Vitor V. Schultz<EMAIL_ADDRESS>Physics Departament, Center for Natural and Exact Sciences, Federal University
of Santa Maria, Roraima Avenue 1000, 97105-900, Santa Maria, RS, Brazil José
C. M. Mombach<EMAIL_ADDRESS>Physics Departament, Center for Natural and
Exact Sciences, Federal University of Santa Maria, Roraima Avenue 1000,
97105-900, Santa Maria, RS, Brazil Jonas Maziero<EMAIL_ADDRESS>Physics Departament, Center for Natural and Exact Sciences, Federal University
of Santa Maria, Roraima Avenue 1000, 97105-900, Santa Maria, RS, Brazil
###### Abstract
Spin systems are a powerful tool for modeling a wide range of physical
systems. In this paper, we propose a novel framework for modeling spin systems
using differentiable programming. Our approach enables us to efficiently
simulate spin systems, making it possible to model complex systems at scale.
Specifically, we demonstrate the effectiveness of our technique by applying it
to three different spin systems: the Ising model, the Potts model, and the
Cellular Potts model. Our simulations show that our framework offers
significant speedup compared to traditional simulation methods, thanks to its
ability to execute code efficiently across different hardware architectures,
including Graphical Processing Units and Tensor Processing Units.
Differentiable programming; Monte Carlo Simulation; Ising model; Cellular
Potts model
## I Introduction
The rapid advancements in machine learning development have revolutionized
software engineering and made significant contributions to various fields such
as computer vision Khan et al. (2018), robotics Bing et al. (2018), and
protein folding Jumper et al. (2021). Specifically, the advent of artificial
neural networks required a new programming paradigm, one that could leverage
automatic differentiation Gaunt et al. (2017). Neural networks are typically
trained using the backpropagation algorithm Rumelhart et al. (1986), which
requires a differentiable computational graph. Automatic differentiation
enables the chain rule from differential calculus to train arbitrary neural
networks end-to-end, as long as their building blocks comprise functions with
well-defined derivatives.
Numerous frameworks have emerged over the years, with PyTorch Paszke et al.
(2019), TensorFlow Abadi et al. (2016), and Jax Bradbury et al. (2018) being
some of the most popular. These frameworks offer all the necessary components
to implement any differentiable program and can take advantage of modern
hardware such as graphics processing units (GPUs) and tensor processing units
(TPUs) Jouppi et al. (2017). Furthermore, they are being adapted to run on
novel hardware such as neuromorphic computers Eshraghian et al. (2021) (also
called AI accelerators) and quantum computers Bergholm et al. (2022). An
ecosystem has developed around these frameworks, enabling them to scale across
multiple devices and increase their speed and memory bandwidth.
As software and hardware continue to evolve, machine learning frameworks have
paved the way for a new programming paradigm known as differentiable
programming (DP) Baydin et al. (2018). In DP, a program can be constructed by
composing differentiable building blocks, allowing this paradigm to extend
beyond the implementation of machine learning algorithms and impact other
scientific and engineering fields, including physics simulations.
Spin models BRUSH (1967) are a type of model used to describe the behavior of
a system of interacting spins. Spins are mathematical representations of
physical quantities, such as the orientation of magnetic moments of atoms,
that can assume specific values according to the model of interest. The
interaction between spins in a spin model is governed by a Hamiltonian
associated with the system’s energy. By analyzing the statistical distribution
of spins in a model, one can predict the system’s macroscopic properties, such
as magnetization and specific heat. Spin models are used to study a wide
variety of physical phenomena, including phase transitions MIYASHITA (2010),
cell behavior Rens and Edelstein-Keshet (2019), and neural networks Kinzel
(1985). Therefore, it is desirable to accelerate their simulation on modern
hardware, if possible.
The Ising model Ising (1925) is one of the simplest spin models, consisting of
only two possible spins, usually referred to as spin up and spin down, that
interact via a coupling value. This model has been extensively studied and
provides the foundation for understanding the behavior of magnetic materials.
Since its introduction in 1920, other models have been developed as extensions
or modifications of the Ising model. One example is the Potts model Wu
(1982a), which differs from the Ising model by the number of degrees of
freedom a spin can have.
Spin models have applications beyond simulating magnetic systems. Cellular
models, which aim to simulate the behavior of biological cells, have also
benefitted from the mathematics of spin models Szabó and Merks (2013). The
Cellular Potts model, also known as the Glazier-Graner-Hogeweg model, is an
example of such a model that can simulate various cellular dynamics, such as
morphogenesis Hirashima et al. (2017); Chen et al. (2007), cell sorting Szabó
and Merks (2013); Durand (2021), and cancer spreading Szabó and Merks (2013);
Metzcar et al. (2019), making it a useful tool for studying a range of
biological phenomena related to cell behavior.
However, simulating spin models can be computationally expensive. They are
typically simulated using Monte Carlo methods Katzgraber (2009), which require
many simulation steps to obtain desired measurements from the system. These
models can also suffer from scale problems due to critical slowing down
Schneider and Stoll (1974); Kotze (2008); Gould and Tobochnik (1989); Acharyya
(1997), resulting in low probabilities of state change at certain temperature
regimes. In addition, calculations of desired observables can only be
performed after the system has reached equilibrium, which is achieved through
thermalization Shekaari and Jafari (2021), whereby a certain number of Monte
Carlo steps are taken before statistical values are measured.
For most systems of interest, exact solutions to the Ising model are only
known for a few special cases, and numerical simulations are required to study
their properties. Therefore, Monte Carlo methods are essential for simulating
spin models because they enable the sampling of the space of possible
configurations of the model and estimation of the thermodynamic properties of
the system, which would otherwise be difficult to obtain.
In this paper, we propose using differentiable programming to simulate spin
and cell models, leveraging the framework’s capabilities to scale on modern
hardware. The rest of this paper is organized as follows: In Sec. II, we
discuss related works. In Sec. III, we present the methods we use, including
the adaptations of Monte Carlo methods to the new paradigm, as well as
descriptions of the systems we study in this article. In Sec. IV, we present
the results obtained, and in Sec. V, we give our final remarks.
## II Related Work
Differentiable programming has been applied to scientific computing tools,
such as finite element methods and numerical optimization, with the aim of
improving the efficiency and accuracy of these techniques. One example of this
is the use of automatic differentiation to compute gradients in finite element
simulations, which can be used to optimize the parameters of the simulation or
to perform inverse problems. This has led to the development of several
differentiable finite element libraries, such as FEniCS Scroggs et al. (2022)
and Firedrake Rathgeber et al. (2016), which enable the efficient
implementation of complex models.
Another area of interest is the integration of differentiable programming with
numerical optimization techniques, such as gradient descent and conjugate
gradient methods. This has been shown to be particularly useful for solving
control problems Jin et al. (2020) and inverse problems Hu et al. (2021);
Grinis (2022); Rackauckas et al. (2021); Thuerey et al. (2021); Hu et al.
(2020), where the goal is to infer the parameters of a physical system from
observed data. By using differentiable programming to efficiently compute
gradients, it is possible to perform gradient-based optimization of these
parameters, which can improve the accuracy and speed of the solution.
Recent work has also focused on the use of differentiable programming in the
context of computational fluid dynamics Takahashi et al. (2021); Fan and Wang
(2023); Bezgin et al. (2023), where it has been shown to be effective in
improving the efficiency of simulations, and can significantly reduce the
computational cost of simulations while maintaining accuracy.
With respect to machine learning applied to the Ising model, some works
propose a neural network to classify a lattice of spins by the thermodynamic
phase. Ref. Efthymiou et al. (2019) proposes a super-resolution method to
increase the size of a network without the need of simulations on large scale.
Neural networks could be also used to approximate the simulation of a model.
For instance, Generative Adverserial Networks can be trained to generate a
sample of a lattice given a temperature Liu et al. (2017).
It is worthwhile mentioning that many works were done on accelerating cellular
and tissue modeling on GPUs Yu and Yang (2014); Christley et al. (2010); Tomeu
and Salguero (2020); Berghoff et al. (2020); Ghaemi and Shahrokhi (2006).
Among the Monte Carlo methods that can be used to simulate spin models, we can
use Gibbs sampling Geman and Geman (1984), Wolff Cluster Wolff (1989) and
Metropolis-Hastings algorithm Hastings (1970). This latter being the most
friendly to make use of parallel computation. In the Metropolis-Hastings
algorithm, applied to a spin model, a random initial state is chosen, and then
the system is updated iteratively by randomly flipping one spin and
calculating the change in energy. If the change in energy is negative, the new
state is accepted. Else, if the change in energy is positive, the new state is
accepted with a certain probability that depends on the control parameters,
such as the equilibrium temperature.
The checkerboard method Preis et al. (2009) is a technique that is used to
parallelise the Metropolis-Hastings method. The algorithm proceeds in two
steps. First, a subset of the spins is chosen, which consists of all the spins
located on the black squares of a checkerboard pattern, as shown in Fig. 1.
The energy change resulting from a flip of each spin is calculated, and the
spins are flipped with a probability given by the Metropolis algorithm. In the
second step, another subset of spins is chosen, which consists of all the
spins located on the white squares of the checkerboard pattern. The energy
change resulting from a flip of each spin is again calculated, and the spins
are flipped with a probability given by the Metropolis algorithm. The energy
of the system is updated if the move is accepted. The checkerboard method is
repeated for many iterations, and the spins eventually reach a state of
equilibrium.
(a)
(b)
Figure 1: The checkerboard method marks each spin of a lattice (a) with a
color from a checkerboard pattern (b). All spins marked with the same color
are updated in parallel.
It is important to note that the order of the neighborhood of interacting
spins is an important aspect of the spin models because it determines the
nature of the interactions between spins. In the original Ising model, the
interactions between spins are limited to the nearest neighbors, for which the
checkerboard method is the most efficient. However, this method is not limited
only to this type of neighborhood, and can be used at any order of spins
interaction, as long as the central site is marked with one color and its
neighbors are marked with another color.
## III Methods
One of main challenges of implementing differentiable programming is
translating an algorithm in a way that DP could be advantageous. The
Metropolis-Hastings algorithm is typically implemented using either functional
or object-oriented programming paradigms. Translating this algorithm to
differentiable programming requires some modifications to how the algorithm is
formulated and implemented.
In traditional functional or object-oriented programming, the Metropolis
algorithm is typically implemented using a sequence of discrete steps. These
steps involve updating the spin variables, computing the energy of the system,
and then accepting or rejecting the proposed configuration based on the
Metropolis acceptance criterion.
Within the modern differentiable programming frameworks, we can express an
array of elements as a batched tensor with sizes up to five dimensions. For
instance, deep learning applied in computer vision usually uses an array of
images that can be represented as a tensor with size $[B,C,H,W]$, with $B$ the
batch dimension, $C$ the channel dimension and $H,W$ the height and width of
the images respectively.
Since the 4-dimensional format of $[B,C,H,W]$ is compatible with many modern
deep learning frameworks, it makes easy to apply deep learning techniques to
two-dimensional spin lattices. This can be particularly advantageous when
using differentiable programming to simulate spin models, as it allows for
seamless integration with existing deep learning tools and techniques.
Additionally, the use of a batch dimension allows for efficient processing of
multiple spin lattices simultaneously. This can be useful when simulating
large-scale spin models, as it enables parallel processing of multiple samples
or multiple temperatures at the same time.
### III.1 Ising model
The Ising model consists of two states, called spins, which physically
represent the magnetic moment of materials. They can be in a up state
$(\sigma=+1)$ or down state $(\sigma=-1)$. This model has a phase transition
in certain lattice geometries, where a change on the behavior of physical
quantities, such as the collective magnetic field, occurs. For example, on a
2D square lattice with $J<0$, the Ising model predicts a change from a
paramagnetic phase, characterized by a random mixture of spins, to a
ferromagnetic phase, characterized by a alignment of the spins. The
Hamiltonian describes the energy of the system is:
$\mathcal{H}=\sum_{i,j}J_{ij}\sigma_{i}\sigma_{j}+\sum_{i}B_{i}\sigma_{i},$
(1)
with $J_{i,j}$ being the interaction strength between spins $\sigma_{i}$ and
$\sigma_{j}$, and $B_{i}$ an external magnetic field on spin $\sigma_{i}$.
The modified Monte Carlo simulation of the Ising model using the Metropolis-
Hastings algorithm requires a convolution operation to calculate the system’s
Hamiltonian. This convolution depends on two topological conditions of the
system: its dimension and connectivity. The dimension of the system directly
determines the dimension of the convolution, with a 1D convolution used for
one-dimensional spin networks, 2D convolution for square or triangular
networks, and 3D convolution for cubic networks, and so on. The connectivity
of the system, which describes how the sites are connected, determines the
shape of the convolution kernel.
For example, consider a square network with first-neighbor interactions. Each
site is connected to its four nearest neighbors, which are located above,
below, to the right and to the left of it. In this case, the kernel used in
the convolution would have a size of $3\times 3$, with values of $1$ in the
positions corresponding to the neighboring sites and $0$ everywhere else,
including the center (to prevent the value of the site itself from being
counted).
By using this modified algorithm, it becomes possible to efficiently simulate
the behavior of the Ising model for systems with large numbers of particles.
The convolution operation provides an efficient way to calculate the system’s
Hamiltonian, which is a crucial step in the Metropolis-Hastings algorithm.
The kernel of the convolution states the geometry of the lattice. For the
square lattice, the kernel is:
$K=\begin{bmatrix}0&1&0\\\ 1&0&1\\\ 0&1&0\end{bmatrix}$ (2)
Thus, for each spin, the energy is obtained by the top, down,left and right
neighbors, which corresponds to the square lattice interaction. Note that the
kernel shape doesn’t necessarily have to be square, as long as it accounts for
the geometric shape of the network. The shape of the kernel is important
because it determines the specific features that are extracted from the
network. For example, if the kernel is square, it may extract different
features compared to a triangular or hexagonal kernel. However, as long as the
kernel accounts for the geometric shape of the network, it can be any shape
that is suitable for the particular analysis.
For lattices with other connectivity, for instance, the triangular lattice, a
transformation is necessary to convert the triangular lattice into a square
lattice so that a square kernel can be used. This transformation involves
adding null spins to the left and right of the central spin, which effectively
creates a rectangular shape for the kernel:
$\displaystyle K=\begin{bmatrix}0&1&0&1&0\\\ 1&0&0&0&1\\\
0&1&0&1&0\end{bmatrix},\hskip 10.0pt$ $\displaystyle K=\begin{bmatrix}1&0&1\\\
0&0&0\\\ 0&1&0\end{bmatrix}.$ (3)
By applying the convolution operation to the lattice using this rectangular
kernel, the algorithm produces a map that is associated with the sum of the
first neighbors’ spins for each site. This map can be used to obtain the
Hamiltonian of each site.
The output of the convolution operation applied to the network is a map that
is associated with the sum of the first neighbors’ spins. This map provides
information about the local interactions between spins in the lattice and is
an important input for further analysis. By multiplying the map produced by
the convolution operation with the spin network itself, the Hamiltonian of
each site in the lattice can be obtained, as described in algorithm 1.
Algorithm 1 Differentiable programming Metropolis-Hasting
1:initialize 2D square lattice $\mathbf{state}=\mathbf{(B,1,N,N)}$ from a
random distribution
2:while $\mbox{MC step}\leq T$ do
3: Propose new random states $\mathbf{state^{\prime}}$ for the lattice;
4: Apply 2D convolution with kernel selected from the model and connectivity
in both states;
5: Multiply the result of the convolution to its respective state;
6: Apply checkerboard algorithm;
7: Evaluate the variation in energy $\Delta E$ for each site with same
checkerboard color;
8: if $\Delta E\leq 0$ then
9: Accept the change
10: else
11: Accept the change with probability $p=e^{-\beta\Delta\mathcal{H}}$
12: end if
13:end while
### III.2 Potts model
The Potts model Wu (1982b) is a lattice model that describes the behavior of a
system of interacting spins, which can take on more than two possible states.
Unlike the Ising model, which has spins that can only take on two states (up
or down), the Potts model allows spins to take on $q$ different states, with
$q$ being any integer greater than or equal to $2$. Each spin is represented
by an integer variable $\sigma_{i}$ that can take on values from 0 to $q-1$.
The Hamiltonian for the Potts model is given by
$\mathcal{H}=\sum_{i,j}J_{i,j}\delta(\sigma_{i},\sigma_{j})+\sum_{i}B_{i}\sigma_{i},$
(4)
where the first term represents the interactions between neighboring spins and
the second term represents the effect of an external magnetic field on each
spin. The coupling between spins $J_{i,j}$ is a constant that depends on the
interaction between spins $i$ and $j$. The Kronecker delta function
$\delta(\sigma_{i},\sigma_{j})$ equals $1$ if $\sigma_{i}=\sigma_{j}$ and $0$
otherwise.
The Potts model has applications in various fields, such as statistical
physics, materials science, and computer science. It can be used to model
phase transitions Tanaka et al. (2011), magnetic ordering Chang and Shrock
(2023), and coloring of graphs Davies (2018). The model has also been used in
image processing and computer vision, where it can be used to cluster pixels
based on their colors or textures Portela et al. (2013).
To simulate the Potts model, the Metropolis-Hastings algorithm can be used,
similarly to is done for the Ising model. The simulation involves selecting a
random spin and attempting to change its state to a new value using a trial
move. If the energy change resulting from the trial move is negative, the move
is accepted. If the energy change is positive, the move is accepted with a
probability given by a acceptance probability. In the Potts model, a spin flip
is not well-defined since the spin can take on more than two states. Instead,
a random spin is chosen to undergo a state change, with the new state being
chosen from the $q-1$ possible values that are different from the current
state.
The differentiable programming Metropolis-Hastings algorithm in the Potts
model follows a structure similar to that used for the Ising model, by
utilizing convolution. However, the main difference between the two models
lies in the Potts model’s convolution, which employs more than one
convolutional filter, each with its own kernel. The number of kernels is
determined by the geometric properties of the system of interest, and each
kernel accounts for a central site neighbor.
For instance, in the case of a square lattice with first neighbor interaction,
four kernels are required, as shown below:
$\displaystyle K=\begin{bmatrix}0&1&0\\\ 0&0&0\\\ 0&0&0\end{bmatrix},\
K=\begin{bmatrix}0&0&0\\\ 1&0&0\\\ 0&0&0\end{bmatrix},\
K=\begin{bmatrix}0&0&0\\\ 0&0&1\\\ 0&0&0\end{bmatrix},\
K=\begin{bmatrix}0&0&0\\\ 0&0&0\\\ 0&1&0\end{bmatrix}.$ (5)
The separation of the kernels into four distinct filters is due to the
Kronecker delta, which requires that each interaction with a neighbor to be
accounted for separately. This contrasts with the Ising model, where the sum
of neighboring spins multiplied by the central spin is sufficient.
After applying convolution with the four filters, four maps are generated for
each site. To obtain the Hamiltonian of the system, the difference between
each map and the spin configuration is computed. If the spins are equal, the
map’s value at that position is set to zero; otherwise, the value of 1 is
assigned. The resulting four maps are summed to generate a single map, which,
multiplied by the spin configuration of the lattice, represents the
Hamiltonian of the system, which can then be used to compute various variables
of interest.
### III.3 Cellular Potts model
Cells arrange themselves spatially during morphogenesis, wound healing, or
regeneration to build new tissues or restore damaged ones. There are a number
of intercellular methods of interaction that are used to carry out these
functions. Cell-cell adhesion is a key process that can direct cell motion in
a manner similar to the separation of immiscible liquids due to interfacial
tensions, where the liquid phases are associated with various cell types (e.g.
cells of retina, liver, etc). A well known phenomenon is the spontaneous
separation of randomly mixed cell types which is known as cell sorting Foty
and Steinberg (2013). The Cellular Potts Model (CPM), developed by Graner and
Glazier, takes into account cellular adhesion to explain cell sorting and is
frequently used to describe cellular processes Glazier and Graner (1993).
In the two dimensional CPM, cells are represented as identical spin domains on
a lattice. Each spin $\sigma$ has a position $\vec{x}$ and a type index
$\tau\left(\sigma\right)$ (a second quantum number). The hamiltonian (6),
which represents cell-cell adhesion and interaction with cell culture media (a
source of nutrients), describes the dynamics. Cell area is controlled by a
second quadratic energy constraint. The medium around the cell aggregate is
represented mathematically as a (big) cell with unconstrained area. The
hamiltonian is written as follows
$\mathcal{H}=\frac{1}{2}\sum_{\vec{x}}\sum_{\vec{y}}^{N_{max}}J_{\tau(\sigma(\vec{x})),\tau(\sigma(\vec{y}))}\left(1-\delta_{\sigma(\vec{x}),\sigma(\vec{y})}\right)+\lambda\sum_{\sigma}(a_{\sigma}-A_{\sigma})^{2}\,\,\,,$
(6)
where $J_{\tau(\sigma(\vec{x})),\tau(\sigma(\vec{y}))}$ are the cell-cell or
cell-medium adhesion energies that depend on the cell type $\tau$, $\delta$ is
the Kronecker’s delta function, $\lambda$ a Lagrange multiplier that controls
cell area compressibility, $a_{\sigma}$ the cell area and $A_{\sigma}$ the
cell target area. By specifying energy only at the cell-cell contact, the
Hamiltonian’s first term (6) mimics cellular adhesion.
The CPM simulates cell motion driven by the cytoskeleton by performing
attempts to change the value in a given lattice position for another in its
vicinity, what causes domain boundary motion representing cellular membrane
motion. By selecting a target spin ($\sigma(\vec{x})$) and a neighbor spin
($\sigma(\vec{y})$) at random from the cell lattice, the system dynamic
occurs. The change in the target spin value for its neighbor’s spin value is
then decided utilizing the aforementioned algorithm 1.
When implementing the cellular Potts model, using simple convolutions like in
previous models is not sufficient because the interaction between spins,
denoted by $J$, is no longer constant. Instead, it depends on the spin value
and the type of cell. To address this, differentiable programming techniques
are used, and one operation that is particularly useful for implementing
convolutions is called unfolding Lista (2017).
Unfolding involves dividing a set of elements, such as a vector, matrix, or
other multidimensional set, into smaller, equally sized parts called
“patches”. The size of these patches depends on the parameters of the
convolution, such as the stride and padding. For example, if we apply
unfolding with padding 0, stride 4, and a kernel of size $4\times 4$ to an 8x8
pixel image, we would obtain four parts of the image, each with a size of
$4\times 4$ pixels.
The cellular Potts model involves interactions between neighboring cells, and
the properties of the unfolding depend on the nature of these interactions.
Typically, an odd number is used to account for the central cell. For
instance, if first-neighbor interactions are considered, the kernel size will
be $3\times D$, where $D$ is the dimension of the system. The stride will be
set to 1 to compute the energy value for each cell, while the padding will
depend on the boundary conditions specified, as in the Ising and Potts models.
If second-neighbor interactions are considered, the kernel will be larger,
with a size of $5\times 5$ to account for more distant cells.
Once the unfolding operation and padding are applied, the next step is to copy
the central spin value of each patch to the same size as the patches. This
allows us to obtain a consistent representation of the spin values across the
entire grid. Finally, once the central sites have been copied, each patch is
compared to its corresponding copy of the central sites, element by element.
During this comparison, the interaction values are assigned based on the spin
value and cell type. After the comparison, the energy map per site of the
system is generated by summing the values of each compared patch.
## IV Results
The differentiable programming Ising-Hastings algorithm was employed to
simulate all models discussed above. The simulations were carried out using a
computing system comprising an Intel Core i7-11800H CPU, a Nvidia GeForce RTX
3060 GPU, and a TPU which was utilized through Google Colab Bisong (2019). For
the Ising and Potts models, we simulate lattices with different sizes, with
$J=-1$ and first-order neighbor interactions.
The cellular Potts model considers the simulation of cell behavior. In this
simulation, three cell types were chosen: light cells, dark cells, and the
medium, following the original work of 1993 Glazier and Graner (1993).
Additionally, 37 different spin values were chosen, with a value of 0
representing the medium, where each spin value leads to the formation of a
unique cell. The chosen interaction energies are presented in table 1, and
these interactions are symmetric (the interaction value is the same if site A
interacts with site B or B with A). The interaction uses the Moore
neighborhood with second neighbors. The temperature was set to $k_{B}T=8$.
Interaction | $J_{\tau,\tau^{\prime}}$
---|---
Medium-Medium | $0$
Medium-Dark | $16$
Medium-Light | $16$
Light-Light | $2$
Dark-Dark | $14$
Light-Dark | $11$
Table 1: The interaction values $J$ depends on the cell type, for the cellular
Potts model.
### IV.1 Ising model
Figure 2 shows snapshots of the square lattice Ising model for three different
temperatures. The temperature of the system determines the probability of the
spins flipping between up and down states. At low temperatures, the spins tend
to align with their neighbors, forming large clusters that grow in size as the
temperature decreases. This is due to the reduction in thermal energy, which
favors the alignment of neighboring spins. These large clusters are known as
domains, and they play an important role in the behavior of the system.
As the critical temperature is approached, the influence of long-range
interactions between spins becomes more pronounced, leading to a change in the
collective behavior of the system. At this temperature, the system undergoes a
phase transition, characterized by the emergence of long-range correlations
and critical fluctuations. This is a critical point where the properties of
the system change abruptly.
Above the critical temperature, at hight temperatures, the domains disappear,
and the system becomes disordered, with no long-range correlations. Thermal
fluctuations dominate the behavior of the lattice, leading to random changes
in the spin values. As the temperature increases, the magnetic spins become
increasingly disordered, leading to a loss of magnetization in the system.
Figure 2: Lattice configurations for three different temperatures of the Ising
model. (a) $T<Tc$, (b) $T\approx Tc$, (c) $T>Tc$.
The results for three different hardware are presented in Figure 3.
Simulations run on CPU show consistent runtimes across varying batch and
lattice sizes. On the other hand, GPU simulations outperform CPU by nearly 100
times in terms of speed. Notably, TPU simulations demonstrate a significant
advantage in runtime as the lattice size and batch size increases, with a
speedup of 10x compared to GPU simulations.
Figure 3: Flips per nanosecond for different lattice and batch sizes. (a)
Simulation on CPU, (b) GPU and (c) TPU.
### IV.2 Potts model
In the case of the Potts model, the spins can take on several different
states, resulting in a more complex system with a richer set of behaviors. As
the temperature is lowered, the spins tend to align themselves to form
distinct clusters, as shown in figure 4. This behavior is reminiscent of
ferromagnetism, in which the magnetic moments of individual atoms align
themselves in the same direction.
On the other hand, at higher temperatures, the spins in the Potts model become
more disordered and exhibit more frequent state changes, resulting in a more
random and unpredictable system. This behavior is similar to what is observed
in the Ising model, where at high temperatures, the magnetic moments of
individual atoms become more disordered and fluctuate rapidly.
Figure 4: Lattice configurations for three different temperature of the Potts
model with $q=3$. Each color represents a spin state. (a) $T<Tc$, (b)
$T\approx Tc$, (c) $T>Tc$.
The Potts model has the same computational complexity as the Ising model,
thus, the time required to flip the spins is similar for both models. This is
because flipping the spin of a single site in the Potts model requires the
calculation of the energy difference between the current and proposed spin
states, just as in the Ising model. However, in the Potts model, the energy
difference depends on the number of neighboring spins that have different
states, which leads to a more complex calculation than in the Ising model.
Despite this additional complexity, the computational cost of flipping the
spins in the Potts model is still comparable to that of the Ising model. The
number of neighboring spins is typically small compared to the total number of
spins in the system. As a result, simulations of the Potts model can be
performed with similar computational resources and time requirements as those
for the Ising model.
### IV.3 Cellular Potts model
The evolution of a cell aggregate can be observed in Figure 5, where snapshots
of the system’s states are shown with increasing Monte Carlo steps. Starting
from an arbitrary aggregate of square-shaped cells, the system undergoes a
process that aims to minimize the number of energy-costly boundaries,
resulting in the sorting of the two cell types.
Figure 5: Cellular Potts Model simulation. Each figure is snapshot from
different steps of the simulation. The cells spontaneously reorder themselves
into clusters of same type as the the number of Monte Carlo steps increases.
The boundary length between each cell type can be observed in Figure 6,
showing the evolution of the system. As the simulation progresses, we observe
the interface between the blue cells and the medium vanishing, while the
boundary length between red cells with medium and red cells with blue cells
approaches a minimum value.
The simulation’s results suggest that the energy constraints in the system
drive the behavior of the cell aggregate towards a more stable configuration,
where the number of energy-costly boundaries is minimized. The decreasing
boundary length between the different cell types indicates that the cells are
actively interacting with each other, eventually sorting themselves into more
cohesive groups.
Figure 6: Boundary length between different cells types. (a) Blue cells and
medium (blue line) and red cells and medium (red line). (b) Blue and red
cells: we observe that the boundary length between blue cells and the medium
goes to zero as the red cells surround the blue cells.
The phenomenon of cell sorting is a well-known example of self-organization in
biological systems. It has been studied extensively both experimentally and
theoretically, and its underlying mechanisms are thought to involve a complex
interplay of various physical and biochemical processes Durand (2021);
Nakajima and Ishihara (2011).
One of the key factors that influence cell sorting is the interaction between
cells and their surrounding environment. Some studies have shown that the CPM
can be used to model not only cell sorting but also other cellular behaviors
such as movement Guisoni et al. (2018) and migration Scianna and Preziosi
(2021). By varying the values of interaction $J$ between cells, it is possible
to simulate different scenarios and study their effects on cellular behavior.
Since the difference among these different phenomena are only in the
interaction values, the computational cost of simulating the model with
differentiable programming remains the same.
## V Conclusions
In this paper, we have presented a novel approach for simulating three
different spin models using differentiable programming. Our method applies
convolution on the spins, similar to how convolution is applied to images in
computer vision, which allows us to calculate the Hamiltonian of the system
with high accuracy and efficiency. In addition, we made use of the
checkerboard algorithm to parallelize the calculation of the energy of each
spin. This algorithm involves dividing the spins into two sets, with each set
updating alternately, such that neighboring spins are updated on different
iterations. By doing so, we can parallelize the calculation of the energy of
each spin, further improving the efficiency of our approach.
The use of the checkerboard algorithm in conjunction with our approach
provides a significant boost in performance, enabling us to simulate spin
models with high speed due to the parallelization. We believe that this
approach could be widely applicable in many scientific applications that
require fast and accurate simulations of complex physical systems.
The use of differentiable programming in this context is particularly useful,
as it enables us to leverage the strengths of deep learning techniques for
scientific simulations. We demonstrated the effectiveness of our approach by
implementing it in PyTorch, which provides easy adaptability to run on GPUs
and TPUs. Our experiments show that our method provides a significant speed-up
in simulating spin models, without sacrificing accuracy. Moreover, by making
use of the batch dimension, we were able to parallelize the simulation even
further, leading to an additional increase in performance.
One important point to note is that our method is not fully differentiable,
and we do not use the derivatives for computation. However, this does not
detract from the value of our approach. In fact, our method is designed to
maximize performance and speed, rather than optimizing for the use of
derivatives. Therefore, we gain the advantage of a significant speed-up in
simulating spin models, which can be critical in many scientific applications.
Our work provides a promising direction for future research in this field, as
it opens up new opportunities for accelerating simulations and improving our
understanding of complex physical phenomena. We anticipate that our method
could have a wide range of applications in the future, especially in cases
where speed and scalability are essential. By leveraging the power of
differentiable programming, we can enable faster simulations and deeper
insights into the behavior of physical systems.
###### Acknowledgements.
This work was supported by the National Institute for the Science and
Technology of Quantum Information (INCT-IQ), process 465469/2014-0, and by the
National Council for Scientific and Technological Development (CNPq),
processes 309862/2021-3, 309817/2021-8 and 409673/2022-6.
Data availability. The data and code that support the findings of this study
are available at https://github.com/tiago939/dp_monte_carlo.
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|
# High-Resolution NMR Spectroscopy at Large Fields with Nitrogen Vacancy
Centers
C. Munuera-Javaloy Department of Physical Chemistry, University of the Basque
Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain EHU Quantum Center,
University of the Basque Country UPV/EHU, Leioa, Spain A. Tobalina
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, 48080 Bilbao, Spain EHU Quantum Center, University of the
Basque Country UPV/EHU, Leioa, Spain J. Casanova Department of Physical
Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080
Bilbao, Spain EHU Quantum Center, University of the Basque Country UPV/EHU,
Leioa, Spain IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5,
48009 Bilbao, Spain
###### Abstract
Ensembles of nitrogen-vacancy (NV) centers are used as sensors to detect NMR
signals from micron-sized samples at room temperature. In this scenario, the
regime of large magnetic fields is especially interesting as it leads to a
large nuclear thermal polarisation –thus, to a strong sensor response even in
low concentration samples– while chemical shifts and J-couplings become more
accessible. Nevertheless, this regime remains largely unexplored owing to the
difficulties to couple NV-based sensors with high-frequency nuclear signals.
In this work, we circumvent this problem with a method that maps the relevant
energy shifts in the amplitude of an induced nuclear spin signal that is
subsequently transferred to the sensor. This stage is interspersed with free-
precession periods of the sample nuclear spins where the sensor does not
participate. Thus, our method leads to high spectral resolutions ultimately
limited by the coherence of the nuclear spin signal.
_Introduction.-_ Nuclear magnetic resonance (NMR) is a fundamental technique
for a variety of areas such as diagnosis medicine, biochemistry, and
analytical chemistry Abragam61 ; Levitt08 . From its inception in 1946
Purcell46 ; Bloch46 ; PhysNobel52 NMR has grown to constitute its own
research area Becker93 . The discussion over the optimal field strength has
been always present in the field Hoult86 , and the profits provided by
elevated magnetic fields of the order of several Teslas–namely, increased
spatial and spectral resolutions– have long been known Ugurbil03 ; Moser17 .
In recent years NMR has experienced a profitable simbiosis with the rapidly
growing field of quantum technologies Dowling03 . In particular, the use of
newly-developed solid-state quantum sensors Degen17 , such as NV centers in
diamond Doherty13 , has enabled to interrogate ever smaller samples
Balasubramanian08 ; Maletinsky12 ; Staudacher13 . This has led to NMR
experiments with unprecedented spatial resolutions, even reaching single
molecules Shi15 ; Lovchinskya2016 ; Munuera2021a . In this regard, the
benefits of operating at large magnetic fields are expected to carry on for
NMR analysis of micro and nanoscale sized samples with quantum sensors.
Nuclear spins are main actors in NMR as they are the source of the target
magnetic signal. The evolution of a nucleus in an external magnetic field is
also affected by the distribution of nearby magnetic sources such as electrons
in chemical bounds. Consequently, detecting the resulting variations in the
Larmor precession frequency through NMR procedures –thus, leading to precise
information about J-couplings and chemical shifts– serves as an accurate
diagnosis of the molecular structure around target nuclei. Identifying these
changes requires measurement protocols that achieve frequency resolutions of
the order of Hz. However, the spectral resolution of standard quantum sensing
techniques is severely limited by the coherence time of the sensor. In the
case of NV centers, this restriction leads to kHz resolutions even when the
sensor is stabilised with dynamical decoupling techniques Munuera2021b ,
leading to an insufficient record for useful chemical analysis.
Figure 1: The setup consists on a picoliter sample placed on a diamond. This
contains an NV ensemble as the sensor of the magnetic signal generated by the
analyte. The protons of the studied molecule emit a signal that depends on
their local environment, thus carrying structural information.
Recently, protocols capable of overcoming these limitations have been devised.
With techniques that resemble classical heterodyne detection, measurements of
artificial signals using NV probes Boss17 ; Schmitt17 reached $\mu$Hz
resolution. In addition, when applied to micron-sized samples these techniques
led to the detection of J-couplings and chemical shifts at low magnetic fields
Glenn18 . These applications suffer from low sensitivity caused by the
weakness of the nuclear signals. This imposes the need for a large number of
repetitions and/or samples with a large hydrogen concentration such that they
provide sufficiently intense signals, thus limiting their utility for
competitive chemical analysis.
Figure 2: Custom signal production and measurement. a) An initial RF $\pi/2$
pulse brings the sample thermal polarization to the orthogonal plane and
triggers the AERIS protocol consisting on free precessions and induced
rotations stages. For a time $\tau$ each magnetization vector
$\boldsymbol{M}_{k}(t)$ precess according to the local field at the position
of the nuclear spin. The phase covered by each $\boldsymbol{M}_{k}(t)$ –this
is $\phi_{k}=\delta_{k}\tau$– is encoded in the amplitude of the oscillating
field generated via controlled rotations of these vectors. b) (First panel) RF
control sequence with interleaved free precessions. (Second panel) Sample
emitted fields. These have different amplitudes due to the distinct
projections of each rotating $\boldsymbol{M}_{k}(t)$ on the Z axis. The
depicted case shows three $B_{i}$ fields as a consequence of the splitting
among three magnetization vectors that spin at rates $\delta_{1}$,
$\delta_{2}$, and $\delta_{3}$. (Third panel) MW pattern –in our case an XY4
sequence– on each NV devised to capture the induced signal. Note that the NVs
remain inactive during the long free precession stages of the sample,
providing our protocol with increased spectral resolution regardless of the
sensor coherence time. Prior to the MW sequence, the NV ensemble is
initialized in $|+\rangle$ while once its state encodes the desired
information it is optically readout in the $\sigma_{y}$ basis.
A possible workaround proposes to increment the polarization of the sample
using dynamical nuclear polarization techniques, hence achieving improved
contrasts Bucher20 ; Arunkumar21 . Alternatively, operating at large static
magnetic fields enhances the thermal polarisation, increasing the NMR signal
intensity without adding new compounds to the sample, hence enabling to
interrogate samples in a wide range of concentrations (not only highly
concentrated ones). Besides, the presence of large magnetic fields facilitates
the identification of frequency changes caused by the local environment of
nuclei, as J-couplings become clearer and chemical shifts increase.
In this Letter we present the AERIS (Amplitude-Encoded Radio Induced Signal)
method. This is a detection protocol able to handle large magnetic field
scenarios and that achieves a spectral resolution which is only limited by the
coherence time of the nuclear spin signal, thus leading to a spectral
resolution compatible with chemical shifts and J-couplings. We exemplify the
AERIS method with NV centers, however this is equally applicable to other
types of solid-state-based sensors Soykal2016 ; Soykal2017 . Moreover, the
method might be combined with recently used dynamical nuclear polarization
techniques Bucher20 ; Arunkumar21 ; Maly08 ; Ni13 leading to stronger target
signals.
_The protocol.-_ State of the art NV-based AC field magnetometry targets the
oscillating signal produced by precessing nuclear spins, whose frequency is
proportional to the local field felt by the nuclei. On the one hand, this
relation allows to acquire information on the molecular environment of the
nuclei by unraveling the spectral composition of the signal. On the other
hand, when the sample is exposed to a large magnetic field, it leads to
signals that oscillate too fast to be traced by the NV. Note that approaches
based on the delivery of appropriately shaped pulses have been proposed for
dealing with moderate field scenarios, or in situations where only reduced MW
power is available Casanova19 ; Munuera-Javaloy20 . Here we take an
alternative approach and target a deliberately manufactured signal that
carries the spectroscopic information of the studied sample encoded in its
amplitude rather than in its frequency.
We consider a thermally polarized sample placed on top of an NV-ensemble-
based-sensor and in the presence of a large external magnetic field $B_{ext}$,
see Fig. (1). The sample would contain a certain type of molecule with nuclear
spins in different locations of its structure. Hereafter we use subindex $k$
(or superindex when required by the notation) to indicate the different
precession frequencies produced by distinct local magnetic fields. This
scenario is similar to those reported in Glenn18 ; Bucher20 ; Arunkumar21
with the critical difference of the magnitude of $B_{ext}$.
Following Levitt08 we describe the spins of our sample via the nuclear
magnetization $\boldsymbol{M}=(M_{x},M_{y},M_{z})$. This is a time-dependent
vector proportional to the sample average nuclear magnetic moment. Its
behavior during an RF pulse of intensity $\Omega$ in a frame that rotates with
the frequency of the RF driving ($\omega$) is described by the Bloch equations
$\frac{d}{dt}\left(\begin{array}[]{c}M_{x}\\\ M_{y}\\\
M_{z}\end{array}\right)=\left(\begin{array}[]{ccc}-1/T^{*}_{2}&-\delta&\Omega\sin\phi\\\
\delta&-1/T^{*}_{2}&-\Omega\cos\phi\\\
-\Omega\sin\phi&\Omega\cos\phi&-1/T_{1}\end{array}\right)\left(\begin{array}[]{c}M_{x}\\\
M_{y}\\\ M_{z}\end{array}\right)+\left(\begin{array}[]{c}0\\\ 0\\\
1/T_{1}\end{array}\right),$ (1)
where $\phi$ is the phase of the RF field, and $T_{1}$ ($T^{*}_{2}$) is the
nuclear relaxation (dephasing) rate. The detuning $\delta=\omega_{L}-\omega$
between the RF pulse frequency and the Larmor precession rate $\omega_{L}$
depends on the local magnetic field at the nuclear spin site, which differs
from $B_{ext}$. Hence, the sample comprises $k$ different precession
frequencies $\omega_{L}^{\,k}$ leading to $k$ detunings $\delta_{k}$.
Our AERIS protocol comprises two parts. The first one creates a detectable
signal by exploiting the dynamics in Eq. (1). This is achieved with an RF
triggering pulse on the sample followed by an alternation among free
precession periods and induced rotations as shown in Fig. (2). The second part
consists on probing the produced signal with NV sensors that acquire a phase
determined by its amplitude, gathering in their spin state information about
the spectral composition of the signal and allowing to determine the local
magnetic environment around nuclear spins.
More in detail: An RF pulse along the X axis (i.e. $\phi=0$) of duration
$\pi/(2\Omega)$, see Eq. (1), tilts the initial thermal polarization of the
sample, $\boldsymbol{M}=(0,0,1)$ (note that $\boldsymbol{M}$ provides the
direction of the thermal polarization when it is a unit vector, and it
attenuates the polarization amount as $T_{1}$ and $T^{*}_{2}$ diminish its
modulus) to the perpendicular plane and triggers off the protocol. Once the
pulse is turned off, i.e. $\Omega=0$ in Eq. (1), the nuclear spins precess
around the external field at a rate determined by the local magnetic field at
their respective locations. Similar to a clock mechanism that rotates the
needles representing hours, minutes, and seconds at different speeds, the free
precession stage of fixed duration $\tau$ splits the magnetization vector
$\boldsymbol{M}(t)$ in components
$\boldsymbol{M}_{k}(t)=\left(\sin(\delta_{k}\,t),-\cos(\delta_{k}t),0\right)$.
Recall, $\delta_{k}$ is the detuning between the driving frequency and the
$k$th precession frequency in the sample. Crucially, the NV sensor remains
inactive at this stage, thus $\tau$ could be significantly larger than NV
coherence times leading to a high spectral resolution ultimately limited by
the coherence of the nuclear sample.
A long RF pulse then continuously rotates the magnetization components at a
speed $\propto\Omega$ around an axis on the xy-plane determined by $\phi$, as
described by Eq. (1). The projection of the resulting field in the NV axis
sets a target field $B(t)$ with two key features. Firstly, it oscillates with
frequency $\Omega\ll\omega_{L}$ (note that, at large magnetic fields, this
relation is naturally achieved for realistic Rabi frequencies). This is a
parameter that can be tuned such that $B(t)$ oscillations can be tracked by
the NV ensemble regardless of the magnetic field value acting on the sample.
Secondly, $B(t)$ comprises the radio signals produced by each rotating
$\boldsymbol{M}_{k}(t)=\left(\sin(\delta_{k}\tau),-\cos(\delta_{k}\tau)\cos(\Omega
t),-\cos(\delta_{k}\tau)\sin(\Omega t)\right)$, thus it contains the footprint
of each nuclear environment (encoded in the distinct $\delta_{k}$ shifts).
Note that, for the sake of simplicity in the presentation, we do not account
for potential deviations in the rotation axes caused by each $\delta_{k}$
shift. However, these are included in our numerical analysis. For more details
see Appendix (A).
After $N$ complete rotations of the magnetization vectors, thus after N
periods of $B(t)$, the RF rotation pulse is switched off and the sample
advances to the next free precession stage in which each
$\boldsymbol{M}_{k}(t)$ continues to dephase. This sequence is iterated
leading to an oscillation in the amplitudes of the signals emitted during
successive induced rotation stages, whose spectrum relates directly to the
various $\omega_{L}^{\,k}$ in the sample.
The radio signal $B_{n}(t)$ produced during the $n^{th}$ induced rotation
stage is captured by the NVs in the ensemble such that each NV evolves
according to
$H/\hbar=-\gamma_{e}B_{n}(t)\frac{\sigma_{z}}{2}+\Omega_{\rm
MW}(t)\frac{\sigma_{\phi}}{2}.$ (2)
Here $\gamma_{e}$ is the electronic gyromagnetic factor, $\boldsymbol{\sigma}$
are the Pauli operators of the NV two-level system, and the target signal
$B_{n}(t)$ is expressed in Appendix (A). The control field $\Omega_{\rm
MW}(t)$ is synchronized with the rotation pulse over nuclear spins, see Fig.
(2), leading to an XY4 control sequence that allows the sensor to capture a
phase determined by (i) The amplitude of the radio signal stemming from the
sample, and (ii) The length of the RF pulse. This information is gathered by
reading the state of the sensor, with an expected result for the
$n^{\text{th}}$ phase acquisition stage of
$\langle\sigma_{y}\rangle_{n}=\frac{2\gamma_{e}t_{m}}{\pi}\sum_{k}b_{k}\cos(\delta_{k}n\tau),$
(3)
where $b_{k}$ is the initial magnetic field amplitude on the NV site produced
by the $k^{\text{th}}$ spectral component, see Appendix (A).
Figure 3: Measurements and spectrum obtained by considering
$\delta_{k}=-\\{342.45,335.55,328.65,321.75,234.9,117.6,110.7,103.8\\}$ Hz,
and magnetic field amplitudes $b_{k}=\\{106,320,320,106,426,320,640,320\\}$ pT
along the Z axis of a generic NV in the ensemble. (a) Simulated stroboscopic
record collected by measuring $\langle\sigma_{y}\rangle$ on the NV as a
function of the cumulated precession time, after interacting with the ethanol
sample (inset). The three sites of the ethanol molecule with different
chemical shifts are indicated with distinct colors. (b) Fourier transform of
the measurement record (blue solid line) showing peaks in the expected values.
Each peak group has its origin site/chemical shift indicated with an arrow of
the corresponding color. Inset, the central peak was fitted to a Lorentzian
function that exhibits a full width at half maximum (FWHM) of 1.62 Hz.
Thus, subsequent detections provide a stroboscopic record of the oscillating
amplitudes, see Fig. (3) (a), whose Fourier spectrum relates to the frequency
shifts of nuclei at different sites of the sample molecule.
Let us recall that the NV ensemble sensor is only active during phase
acquisition (i.e. while the dynamical decoupling sequence is active), and
after that, it is optically readout and reinitialized. Therefore, the duration
of our protocol, and thus its spectral resolution, gets over the cap imposed
by the coherence of the sensor, being only limited by the coherence of the
nuclear fields.
_Numerical Results.-_ We illustrate the AERIS protocol by simulating the
evolution of 8 magnetization vectors taken from the ethanol [C2H6O] spectrum
Levitt08 in a scenario that comprises a magnetic field of 2.1 T, while the RF
driving frequency $\omega$ is set to $(2\pi)\ \times$ 90 MHz, which is assumed
to be the origin of the chemical shift scale (this is the resonance frequency
of TMS Levitt08 ). Each $\delta_{k}$ detuning is obtained by considering the
three chemical shifts of $3.66$, $2.6$, and $1.19$ ppm, as well as a
J-coupling of 6.9 Hz between the CH3 and the CH2 groups of ethanol Levitt08 ,
see caption in Fig. (3). The average field amplitude over each NV in the
ensemble is estimated to $\approx 2.56$ nT, by taking into account the proton
concentration of ethanol as well as the external magnetic field of 2.1 T, see
Appendix B. This field amplitude is distributed in different $b_{k}$ according
to the ethanol spectral structure, see caption in Fig. (3) and Appendix B. We
find the radio signal emitted by the sample by numerically solving the Bloch
equations during RF irradiation (i.e. at the induced rotation stages). The
free precession time is selected as $\tau=1$ ms, and the induced rotation
stage has a duration of 40 $\mu$s (corresponding approximately to 2 full
rotations of the magnetization vectors) while the NV ensemble is controlled
with an XY4 sequence. Furthermore, we use $\Omega_{\rm MW}=(2\pi)\times 20$
MHz, $\Omega_{\rm RF}=(2\pi)\times 50$ KHz, and sample coherence times
$T_{1}=2$ s and $T^{*}_{2}=0.2$ s. This process is repeated 1500 times,
leading to the stroboscopic record of Fig. (3) (a) which follows Eq. (3).
We run again the protocol by employing an initial $\pi/2$ pulse over the Y
axis leading to the sinusoidal version of Eq. (3). This is:
$\langle\sigma_{y}\rangle_{n}=\frac{2\gamma_{e}t_{m}}{\pi}\sum_{k}b_{k}\sin(\delta_{k}n\tau).$
(4)
Finally, both measurement records in Eqs. (3, 4) are combined and converted,
via discrete Fourier transform, into the spectrum in Fig. (3) (b). There we
demonstrate that the AERIS method leads in the studied case to Lorentzian
peaks with a FWHM $\approx 1.62$ Hz (limited by the sample $T^{*}_{2}$) thus
sufficient to detect the posed chemical shifts and J couplings.
For the sake of simplicity in the description of the AERIS method, the
presented simulations consider perfect controls. Appendix. (C) analyses the
impact of faulty RF driving. We find that, for realistic errors Boris22 ;
Cai12 , the method still provides results that resemble the ideal ones.
Moreover, for more severe error levels, in Appendix. (C) we devise an
alternative AERIS sequence that enhances the robustness of the protocol.
_Conclusions.-_ We have devised an NMR signal detection protocol that attains
chemical shift level resolution from micron-sized samples while being suitable
for large magnetic fields. Our approach relies on the production of a custom
field that resonates with dynamically decoupled NV sensors used to extract
spectral information from the sample. Actual experiments may require several
repetitions to average out the impact of shot noise or inaccurate control
sequences. Nevertheless the demand for higher spectral resolution is less
stringent at large fields, as chemical shifts increase and J-couplings become
clearer. Besides, polarization rates increase, leading to stronger signals
that provide measurements with higher contrast. Both effects contribute to
decreasing the required number of repetitions, or, conversely, making small
concentration samples amenable to our protocol, which sets the utility of NV
sensors for realistic chemical analysis.
###### Acknowledgements.
_Acknowledgements.–_ We thank fruitful discussions with A. Martín. C.M.-J.
acknowledges the predoctoral MICINN grant PRE2019-088519. J. C. acknowledges
the Ramón y Cajal (RYC2018-025197-I) research fellowship, the financial
support from Spanish Government via EUR2020-112117 and Nanoscale NMR and
complex systems (PID2021-126694NB-C21) projects, the EU FET Open Grant
Quromorphic (828826), the ELKARTEK project Dispositivos en Tecnologías
Cuánticas (KK-2022/00062), and the Basque Government grant IT1470-22. _Note
added.-_ In the preparation of the manuscript, we become aware of a similar
concept using a double electron-nuclear resonance to detect NMR spectra
Meinel22 .
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## Appendix A Measured signal
After the triggering pulse with $\phi=0$, the magnetization reads
${\boldsymbol{M}}=(0,-1,0)$ . This is the initial configuration for the first
free precession stage (of fixed duration $\tau$) that splits the magnetization
in different components ${\boldsymbol{M}}_{k}$ such that the initial
configuration for the $k^{\text{th}}$ spectral component at the
$n^{\text{th}}$ phase acquisition stage reads
${\boldsymbol{M}}=(\sin(\delta_{k}n\tau),-\cos(\delta_{k}n\tau),0)$. This is
obtained from the Bloch equations with $\Omega=0$.
Now, the evolution of the magnetization during the $n^{th}$ phase acquisition
stage reads
$\boldsymbol{M}_{k}^{\,n}(t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos(\Omega
t)&-\sin(\Omega t)\\\ 0&\sin(\Omega t)&\cos(\Omega
t)\end{array}\right)\left(\begin{array}[]{c}\sin(\delta_{k}\,n\tau)\\\
-\cos(\delta_{k}\,n\tau)\\\
0\end{array}\right)=\left(\begin{array}[]{c}\sin(\delta_{k}\,n\tau)\\\
-\cos(\delta_{k}\,n\tau)\cos(\Omega t)\\\ -\cos(\delta_{k}\,n\tau)\sin(\Omega
t)\end{array}\right).$ (5)
Notice that the Bloch equations in the main text, and therefore the solutions
obtained from them, describe the dynamics in a frame that rotates around the
external field at the RF driving frequency $\omega$. For the sake of
simplicity in the presentation, the solution in Eq. (5) assumes no decoherence
(i.e. $T_{1},T^{*}_{2}\rightarrow\infty$) and perfect resonance (not taking
into account the natural $\delta_{k}$ shifts during the driving. However, the
numerical simulations leading to the results displayed in the main text
include realistic $T_{1}$ and $T^{*}_{2}$, as well as the corresponding
$\delta_{k}$ shifts, and hence imply numerically solving the Bloch equations
in Eq. (1) of the main text. Here we show the approximate analytical solution
to provide the reader with an insight into the dynamics at the phase
acquisition stages of the protocol.
In our case, the effect of the $\delta_{k}$ shifts on the rotation speed is,
to first order, a factor of approximately
$\frac{\delta_{k}^{2}}{2\Omega^{2}}\approx 2\times 10^{-5}$, which is
negligible and has no significant impact on the results. If necessary (i.e.,
in case of facing more severe energy shifts) a modified sequence, as outlined
in Appendix (C), can be used to further correct this error.
The interaction between the signal produced by the rotating
$\boldsymbol{M}_{k}^{\,n}(t)$ and the sensor in a rotating frame w.r.t. the NV
electronic-spin ground-state triplet is
$H/\hbar=-\gamma_{e}B_{n}(t)\frac{\sigma_{z}}{2}+\Omega_{\rm
MW}(t)\frac{\sigma_{\phi}}{2}.$ (6)
Here $\Omega_{\rm MW}(t)$ is the MW control field, and the target signal
induced by the $n^{th}$ phase acquisition stage is
$B_{n}(t)=\sum_{k}B_{k}^{n}(t)$ such that
$B_{k}^{n}(t)=\frac{\hbar^{2}\gamma_{N}^{2}\mu_{0}\rho_{k}B_{ext}}{16\pi
k_{B}T}\,{\boldsymbol{M}}_{k}^{\,n}(t)\int\big{[}g_{x}(r),g_{y}(r),f(r)\big{]}\
dV,$ (7)
where $\mu_{0}$ is the vacuum permeability, $\rho_{k}$ the density of spins
with the $k^{th}$ precession frequency, $\gamma_{N}$ is the nuclear
gyromagnetic factor, $T$ is the temperature of the sample, $k_{B}$ is the
Boltzmann constant, and $B_{ext}$ is the external magnetic field. The
geometric functions $g_{x,y}(r)$ and $f(r)$ read
$f(r)=\frac{1}{r^{3}}(3r_{z}^{2}-1)\hskip 42.67912pt\text{and}\hskip
42.67912ptg_{x,y}(r)=\frac{1}{r^{3}}(3r_{z}r_{x,y}),$ (8)
with $\hat{r}=(r_{x},r_{y},r_{z})$ being the unitary vector joining the NV and
$dV$, while $r$ represents their relative distance. The expression in (7)
(which can be derived from a microscopic description of a system involving NVs
and nuclear spins Meriles10 ) is valid provided that the external magnetic
field $B_{ext}$ is greater than the coupling strength, which allows ignore the
backaction of the sensor in the sample Reinhard2012 . As we are in a large
field regime, this condition is met. In addition, the contribution of the
orthogonal components $M^{\,n}_{k,x}(t)$ and $M^{\,n}_{k,y}(t)$ to
$B_{k}^{n}(t)$ (which rapidly oscillate with the Larmor frequency
$\gamma_{N}B_{ext}$) can be safely neglected.
The MW control implements an XY4 dynamical decoupling sequence that modulates
the interaction between target and sensor leading to
$H/\hbar=\frac{\gamma_{e}\sigma_{z}}{\pi}\sum_{k}b_{k}\cos(\delta_{k}n\tau),$
(9)
where $b_{k}=\frac{\hbar^{2}\gamma_{N}^{2}\mu_{0}\rho_{k}B_{ext}}{16\pi
k_{B}T}\int f(r)dV$.
The NV is initialized in the
$|+\rangle=\frac{1}{\sqrt{2}}\left(|1\rangle+|0\rangle\right)$ state, then
evolves during $t_{m}$, and it is finally measured such that (in the small
angle regime)
$\langle\sigma_{y}\rangle_{n}=\frac{2\gamma_{e}t_{m}}{\pi}\sum_{k}b_{k}\cos(\delta_{k}n\tau).$
(10)
On the other hand, an RF trigger pulse with $\phi=\pi/2$ leads to
$\boldsymbol{M}_{k}=(1,0,0)$, which yields a splitting of the $k$ spectral
components during the free precession stages described by
${\boldsymbol{M}}_{k}=(\cos(\delta_{k}n\tau),\sin(\delta_{k}n\tau),0)$. For
the same dynamical decoupling control sequence over NVs, we find
$\langle\sigma_{y}\rangle_{n}=\frac{2\gamma_{e}t_{m}}{\pi}\sum_{k}b_{k}\sin(\delta_{k}n\tau).$
(11)
## Appendix B Radio field intensity estimation
In this section, we estimate the radio signal amplitude for the example in the
main text. We numerically compute the geometrical integral $F=\int f(r)dV$ for
different hemispheres while we consider the NV axis perpendicular to the
diamond surface. This leads to an asymptotical value of $F\sim 4.1$. Note that
half of the asymptotic value is reached for integration hemispheres with a
radius of 2-3 times the depth of the NV, which leads to detectable signals
even for picoliter volume samples. Considering a pure ethanol sample with a
density of 789 kg m-3 and a molar mass of 46 g mol-1, we obtain a proton
density of $\rho=$ 6.2 $\times\ 10^{28}$ m-3. With this into consideration,
the total amplitude obtained in a 2.1 T external field at room temperature is
$b\sim 2.56$ nT. Finally, we can distribute this amplitude throughout the
ethanol spectral peaks according to the following rules: $b/3$ (signal
produced by 2 out of 6 hydrogens of the molecule) distributed in four peaks
with ratios 1:3:3:1, a single peak of $b/6$, and $b/2$ (signal produced by 3
out of 6 hydrogens of the molecule) distributed in three peaks with ratio
1:2:1, to obtain
$b_{k}\in\\{106,320,320,106,426,320,640,320\\}\ {\rm pT}.$ (12)
## Appendix C Sequence robustness considerations
We consider the effect of errors in the RF control, which could be potentially
detrimental for the sequence as the nuclear signal coherence has to be
maintained throughout the protocol. The control error is modeled as an
Ornstein-Uhlenbeck Wang45 ; Gillespie96 process
$\epsilon_{\Omega}(t+\Delta t)=\epsilon_{\Omega}(t)e^{-\Delta t/\tau}+\sigma
N(t)$ (13)
where $\tau$ is the correlation time of the noise, $N(t)$ is a normally
distributed random variable, and $\sigma$ is the relative amplitude of the
fluctuations. For standard expected experimental errors Boris22 ; Cai12 , the
obtained spectrum overlaps with the case without control errors, see Fig. 4
(a).
Figure 4: Spectra comparison for the AERIS sequence with perfect RF controls
(black dotted line) and in the presence of OU noise (green solid line) with
(a) $\sigma=0.24\%$ and $\tau=1$ms and (b) $\sigma=2\%$, $\tau=0.5$ms and an
amplitude shift of 1$\%$. Both spectra were obtained averaging 200
realizations.
However, in the presence of more severe noise and constant Rabi amplitude
shifts (e.g. due to misscalibration) AERIS gives raise to distorted spectra as
can be seen in Fig. 4 (b). A direct modification of the default sequence leads
to a significant improvement on robustness. The alternative sequence is
equivalent to the original one but changes the irradiation/NV measurement
stages with the scheme represented in Fig. 5 (a). The modified version employs
a change of sign in the middle of the RF irradiation such that the error
accumulated in the first half is the opposite to the one accumulated in the
second half leading to cancellation. The XY-4 sequence over the NV is
substituted with two $\pi$ pulses in order to accumulate phase from the new
magnetic signal. The new version recovers the ideal spectrum in the severe
noise example, see Fig. 5 (b).
Figure 5: (a) Schematics of the modified AERIS sequence. Two rotations are
performed in opposite directions with the RF control, giving raise to a
detectable magnetic signal with a $\pi$ phase change in the middle which is
measured with two concatenated spin echoes. (b) Spectra comparison for the
AERIS sequence with perfect controls (black dotted line) and with
$\sigma=2\%$, $\tau=0.5$ms and an amplitude shift of 1$\%$ for AERIS (green
solid line) and the modified version (orange solid line). (c) FWHM with
respect of the relative OU error with $\tau=1$ms and no constant amplitude
shift for AERIS (green line) and the modified version (orange line). The
minimum FWHM possible given the nuclear $T_{2}^{*}$ is represented as a grey
line.
Finally, in Fig. 5 (c), we show a comparison of the expected FWHM of the
central spectral peak for AERIS and the modified version with respect to the
error amplitude. The modified version recovers a FWHM close to the minimum
possible given the nuclear $T_{2}^{*}$ for the considered error range.
|
Chun-Wei Ho1, Chao-Han Huck Yang1, Sabato Marco Siniscalchi1,2,3
# Differentially Private Adapters for Parameter Efficient Acoustic Modeling
###### Abstract
In this work, we devise a parameter-efficient solution to bring differential
privacy (DP) guarantees into adaptation of a cross-lingual speech classifier.
We investigate a new frozen pre-trained adaptation framework for DP-preserving
speech modeling without full model fine-tuning. First, we introduce a noisy
teacher-student ensemble into a conventional adaptation scheme leveraging a
frozen pre-trained acoustic model and attain superior performance than DP-
based stochastic gradient descent (DPSGD). Next, we insert residual adapters
(RA) between layers of the frozen pre-trained acoustic model. The RAs reduce
training cost and time significantly with a negligible performance drop.
Evaluated on the open-access Multilingual Spoken Words (MLSW) dataset, our
solution reduces the number of trainable parameters by 97.5% using the RAs
with only a 4% performance drop with respect to fine-tuning the cross-lingual
speech classifier while preserving DP guarantees.
Index Terms: speech classification, differential privacy, domain adaptation,
parameter efficient tuning
## 1 Introduction
With the rapid growth of the computation ability and commercial datasets, more
and more personal data are collected, which poses the issue of protecting
sensitive data. The United States Census Bureau, for instance, announced a new
security standard [1] based on Differential Privacy (DP) [2]. The
$(\epsilon,\delta)$-DP mechanism allows us to measure the security of
algorithms and provides a guarantee based on a privacy budget. However,
ensuring differential privacy degrades the system's performance [3] because it
restricts access to the data. In addition, training a large model with DP is
not only time-consuming but also leads to a more severe drop in performance.
[3].
Nonetheless, there are many benefits associated with the use of large-scale
datasets and large models. For example, large-scale datasets are fundamental
to deploying well-trained deep neural networks (DNNs) [4, 5]; moreover, if the
size of the DNN is large enough, it can reach the global minima from any
initialization with the gradient descent algorithm [6]. Although the global
optimality was only proven in tensor factorization, [6] shows the benefits
associated with large connectionist models. Indeed, there exist several large
pre-trained models that have been proven vital for different downstream tasks
[7, 8, 9, 10, 11, 12, 13, 14] after fine-tuning - in this work, we will use
the term fine-tuning and adaptation interchangeably.
Unfortunately, fine-tuning a pre-trained larger model, in addition to being a
time-intensive procedure, can also distort the pre-trained features and
underperform out-of-distribution [15]. Training large models with differential
privacy is even harder because DP-related perturbations are introduced into
the training process. Therefore, a feasible solution to estimate and exploit a
representation of a large pre-trained model is becoming a pressing issue to be
tackled.
This work aims at investigating the benefits of leveraging model adaptation
and parameter efficient techniques in the context of differential privacy. In
particular, we propose a cross-domain differential private fine-tuning
framework 111GitHub Link: https://github.com/Chun-wei-Ho/Private-Speech-
Adapter. leveraging a deep frozen model pre-trained on public source data, and
private target data. We consider the case when there is a domain mismatch
between source and target domains. In the proposed framework the frozen pre-
trained model doesn't guarantee privacy but provides information from non-
sensitive source data. We also use additional parameters (weights) to serve as
a domain adaptor, which provides information from the target data and
introduces DP guarantees. In particular, DP stochastic gradient descent
(DPSGD) [16, 17, 18], and Private Aggregation of Teacher Ensembles (PATE) [19,
20] are used to attain DP guarantees.
For DPSGD, we follow what was proposed by Da et al.in [21]. Since the
experimental evidence demonstrated poor results with DPSGD, we devised a PATE-
based solution, which led to a substantial performance improvement. Figure 1
shows the proposed PATE-based solution to perform model adaptation (fine-
tuning) while attaining DP-privacy guarantees. The additional weights shown in
the figure are trained on different disjoint chunks of the sensitive data.
Those weights are then inserted into the frozen pre-trained large models using
the solutions discussed in [21]. The obtained frozen pre-trained model is
aggregated with different weights together based on PATE's algorithm. Finally,
the student model queries from the aggregated teacher model using non-
sensitive target domain data and learns only from non-sensitive data to
preserve privacy. To the best of the authors' knowledge, our work is the first
to propose cross-domain DP-based acoustic modeling adaptation. The overall
solution does not only guarantee DP, but it also is parameter efficient.
Figure 1: Proposed private aggregation of teacher ensembles [19] (PATE)-based
adapter for parameter efficient fine-tuning on acoustic and speech processing.
## 2 Related Works
### 2.1 Differential Privacy in a Nutshell
The DP mechanism [2] is established to evaluate the security of an algorithm.
DP is parameterized by the privacy budget variable $\epsilon$, and $\delta$
defined as follows:
Definition 1 An algorithm $\mathcal{A}$ is said to be $(\epsilon,\delta)$-DP
if for all adjacent datasets $D$ and $D^{\prime}$, and for any possible event
$S$, the algorithm satisfies:
$\text{Pr}[\mathcal{A}(D)\in S]\leq
e^{\epsilon}\text{Pr}[\mathcal{A}(D^{\prime})\in S]+\delta$ (1)
The above equation, in some sense, guarantees that the outcomes of the
algorithm with $D$ and $D^{\prime}$ are indistinguishable.
There are several methods to achieve $(\epsilon$, $\delta)$-DP, and most of
them require some DP-oriented perturbation. The perturbation guarantees
$(\epsilon,\delta)$-DP by making the output of the algorithm, $\mathcal{A}(D)$
and $\mathcal{A}(D^{\prime})$, indistinguishable. The simplest method to
guarantee DP is to introduce the Laplace perturbation to the output of
$\mathcal{A}$. It has been shown that we can achieve pure DP ($\delta=0$) with
Laplace perturbation added [22].
Although there exist several ways to estimate the privacy budget $\epsilon$,
one of the most convenient methods is Renyi Differential Privacy (RDP) [23],
which is based on the Renyi Divergence by (2), which is similar to the
Kullback-Leibler Divergence:
$D_{\alpha}(P\|Q)=\frac{1}{\alpha-1}E_{X\sim
Q}\log\left(\frac{P(x)}{Q(x)}\right)^{\alpha}$ (2)
The RDP is defined as follows:
Definition 2 An algorithm $\mathcal{A}$ is said to be $\alpha,\epsilon$-RDP if
for all adjacent datasets $D$ and $D^{\prime}$, the algorithm satisfies:
$D_{\alpha}(f(D)\|f(D^{\prime}))\leq\epsilon$ (3)
It has been proven in [23] that if any algorithm satisfies
$\alpha,\epsilon$-RDP, it's also an
$\left(\epsilon+\frac{\log(1/\delta)}{\alpha-1},\delta\right)$-DP algorithm.
We use RDP to evaluate DP in this study.
### 2.2 Privacy Preserving in Machine Learning
A common method to preserve privacy is by DP-based perturbations. However,
perturbations also degrade the system's performance. Finding a trade-off
between performance and privacy has become an important topic worth
investigating. Two popular algorithms have been designed to preserve privacy
in machine learning. The first is DP stochastic gradient descent (DPSGD) [16,
17, 18], in which the effect of single data is restricted by per-utterance
gradient clipping, and the noises are added to satisfy a certain privacy
budget $\epsilon$. The second method is PATE [19], which is based on three
stages: First, several teacher models are trained on disjoint chunks of
sensitive data. Then, the outputs of the teacher models $T_{i}(x,\theta_{i})$
are aggregated using a private aggregation algorithm (4). Finally, the student
model is trained on some public data and the output of the teacher models,
defined as $T(x,\theta)$ in (4). PATE models achieve $(\epsilon,\delta)$-DP by
introducing noises in the aggregation phase and by hiding sensitive data from
the student model. The amount of noise is determined by the ``smooth
sensitivity'' [24] of the teacher models, which is also called data-dependent
privacy. By reducing the required DP-oriented perturbation while aggregating,
PATE has been tested as the state-of-the-art results in different
applications, e.g., [25, 26].
$T(x,\theta)=\text{argmax}\\{T_{i}(x,\theta_{i})+\text{Lap}_{i.i.d}(\lambda)\\}$
(4)
### 2.3 Parameter Efficiency & Differential Privacy
Training a huge deep model taking into account DP requirements can be
troublesome because we have to restrict the information extracted from the
data. Furthermore, the perturbation introduces randomness into the learning
phase. The amount of perturbation required under the same privacy budget is
depended on the model size. The larger the model is, the more perturbation we
need to preserve privacy. For example, the perturbation added to the gradients
is proportional to the square root of the number of trainable parameters in
DPSGD. That in turn leads to a trade-off between the model capacity, and DP
guarantees. In many DP setups [19, 27], smaller and simpler model
architectures end up providing superior performance. Nonetheless, Da et
al.[21] proposed to use parameter efficient methods to deal with the noise
injection while training large models with DPSGD. In their study, it has been
experimentally proven that larger models with parameter efficiency lead to
better results when used in combination with DPSGD. We posit that parameter
efficiency serves as a conduit between large models and privacy budgets. To
this end, we investigate a first attempt to advance parameter-efficient
learning with PATE, which has been demonstrated to have wide-ranging
applications for performance-driven tasks.
### 2.4 Parameter Efficient Algorithms
In this study, we mainly focus on two parameter efficient algorithms. Linear
Probing (LP) [15] prevents distortions by freezing the entire encoder while
training the linear head222The last linear layer is referred to as “head”
only. By reusing the pre-trained weights completely, Linear Probing is
effective when the source domain and the target domain are similar to each
other.
Adapters [28] modifies the feature extractors by inserting some adapting
layers without changing the pre-trained weights. More specifically, the
relationship between the output of the $i^{th}$ layers
$\hat{\mathcal{F}}_{\theta}^{i}(x)$ and the output of the $(i-1)^{th}$ layers
$\hat{\mathcal{F}}_{\theta}^{i-1}(x)$ are described in (5), where $\Theta$
denotes non-trainable parameters, and $\theta$ denotes trainable parameters.
$\mathcal{A}_{\theta}$ denotes some non-linear function parameterized by
$\theta$. The hat notation, $\hat{\cdot}$, indicates the functions whose
inputs are the model input, $x$, instead of the output of the previous layer.
$\displaystyle\theta^{*}=\arg\min_{\theta}\left\\{\mathcal{L}_{\text{error}}(\sigma(\hat{\mathcal{F}}_{\theta}^{N}(x)),\hat{y})\right\\}$
(5)
$\displaystyle\text{where}~{}~{}~{}\begin{cases}\hat{\mathcal{A}}_{\theta}^{i}(x)=&\mathcal{A}_{\theta}^{i}(\hat{\mathcal{F}}_{\theta}^{i-1}(x))\\\
\hat{\mathcal{F}}_{\theta}^{i}(x)=&\underbrace{\mathcal{F}_{\Theta}^{i}(\hat{\mathcal{F}}_{\theta}^{i-1}(x))}_{\text{original
encoder
(frozen)}}+\underbrace{\hat{\mathcal{A}}_{\theta}^{i}(x)}_{\text{Adapter
output}}\end{cases}$
DNN Residule Adapter ($\text{RA}_{\text{DNN}}$) [29], a common adapter uses a
simple up-projector and a simple down-projector along with a residual path to
define the non-linear function $\mathcal{A}_{\theta}$, which modifies the
input feature, $\hat{\mathcal{F}}_{\theta}^{i-1}(x)$, by a limited matrix
rank. It has been experimentally proven that $\text{RA}_{\text{DNN}}$ can
attain comparable performance results to those obtained through a fine-tuning
of the whole model parameters but using only up to 2 % of parameters [30].
## 3 Proposed DP based Parameter Efficient Adaptation for Acoustic Modeling
In this study, two of the most popular privacy-preserving algorithms, DPSGD
and PATE were investigated. For DPSGD, we used the same setup in [21], where
only the $\text{RA}_{\text{DNN}}$s are updated during training. Figure 1 shows
instead the proposed PATE-based solution, where $N$ different additional
weights are trained on the different disjoint chunks from the sensitive
dataset. The weights are then inserted into the global teacher model and are
aggregated together using the private aggregation algorithm proposed in [19].
The student model, on the other hand, learns from the public data queried from
the private teacher models. Therefore, the student can learn from private data
without direct access to it. As explained in Section 2.2, the amount of
required DP-oriented perturbation is determined by the sensitivity of the
teacher models. Therefore, by applying data-dependant privacy and domain
adaptation, we were able to successfully reduce the amount of DP-oriented
perturbation required to preserve privacy.
### 3.1 DNN Residual Adapters Connection
As discussed in Section 2.4, $\text{RA}_{\text{DNN}}$ is one of the common
parameter-efficient adapters. In this study, we also investigated different
non-linear functions, $\mathcal{\hat{A}}_{\theta}(x)$. Inspired by [31, 32],
we try to connect the $\text{RA}_{\text{DNN}}$s using some skip connections.
Instead of just performing neighboring connections, we tried to connect the
$\text{RA}_{\text{DNN}}$s in three different ways and investigate their
effects. The three connection ways are summarized in (6). The connections are
inspired by Unet [33] and DenseNet [34].
$\displaystyle\text{Neighboring:}\quad\hat{\mathcal{A}}_{\theta}^{i}(x)=\mathcal{A}_{\theta}^{i}(\hat{\mathcal{F}}_{\theta}^{i-1}(x)+\hat{\mathcal{A}}_{\theta}^{i-1}(x))$
(6) $\displaystyle\text{Unet-alike
\cite[cite]{[\@@bibref{}{ronneberger2015u}{}{}]}:}\quad\hat{\mathcal{A}}_{\theta}^{i}(x)=\mathcal{A}_{\theta}^{i}(\hat{\mathcal{F}}_{\theta}^{i-1}(x)+\hat{\mathcal{A}}_{\theta}^{N-i}(x))\
\forall i>\frac{N}{2}$ $\displaystyle\text{DenseNet-alike
\cite[cite]{[\@@bibref{}{huang2017densely}{}{}]}:}\quad\hat{\mathcal{A}}_{\theta}^{i}(x)=\mathcal{A}_{\theta}^{i}(\hat{\mathcal{F}}_{\theta}^{i-1}(x)+\sum_{k=1}^{i-1}\hat{\mathcal{A}}_{\theta}^{k}(x))$
As defined in (6), the neighboring connections connect the output of the
previous layer. In the Unet-alike connection, the last $i$ layers are
connected to the first $i$ layers. And in the DenseNet-alike connection, every
layer is connected to every preceding layer.
### 3.2 Evaluation of Utility
We leveraged Eric Hulburd's work [35] to assess the quality of the proposed
approach and used the utility defined in (7) that takes both parameter
efficiency and performance:
$\text{Utility}=\frac{\text{Accuracy}-50}{\log(\text{Number of trainable
parameters})}$ (7)
## 4 Experiments & Results
### 4.1 Experimental Setup
We assessed our framework on a keyword classification task. Specifically, we
used the English Google Speech Command V2 (EGSP-V2) [36] as source domain, and
the Multilingual Spoken Words [37] as the target domain. We took into account
only four languages, namely English, German, French, and Russian, and
generated smaller subsets from them, referred to as MLSW-mini 333The list of
train/test split is reported on https://github.com/Chun-wei-Ho/Private-Speech-
Adapter., to simulate low-resource conditions. MLSW-mini configuration is
shown in Table 1. And the EGSP-V2 was used to pre-train the deep classifier.
Then, we adapted the model to MLSW-mini with DP. For DPSGD, we used MLSW-mini-
train and half of MLSW-mini-test to train the model. For PATE, we trained the
teacher models on MLSW-mini-train. Then we trained the student model on half
of the MLSW-mini-test. The remaining data in MLSW-mini-test was used for
evaluation. The proposed setup follows the standard PATE setup [19]. The
privacy budget $\epsilon$ is 8.0 444We follow a common privacy budget
($\epsilon$=8) based on [21] and Apple’s official document in
https://www.apple.com/privacy/docs/Differential_Privacy_Overview.pdf for
French, German, and English, and 11.6 for Russian.
Table 1: MLSW-mini dataset. "# Words" indicates the number of unique words in
the language. The sample rate of the waveforms is 16 kHz. Each waveform is
roughly 1 second long.
Language | # Words | # Samples/word | Total Train Audio Time
---|---|---|---
en (Germanic) | 18 | 4501-4927 | 23 hours 34 mins
de (Germanic) | 15 | 4011-4910 | 18 hours 14 mins
fr (Romance) | 13 | 4081-4988 | 16 hours 01 mins
ru (Slavic) | 23 | 1002-4758 | 11 hours 00 mins
The deep architecture used as a pre-trained model is the Keyword Transformer
(KWT) [38]. KWT first performs a time-distributed linear project of the mel-
spectrogram; it then concatenates the features with token embeddings. Next,
the concatenated features are fed into 12 layers of transformer blocks with
dimension 192 and classified using a linear head. The setups are similar to
[38] with the only difference being that 12 trainable
$\text{RA}_{\text{DNN}}$s (with different dimensions) were inserted between
the transformer blocks in the fine-tuning phase.
The Mel-spectrogram generated with a 30 ms analysis window, a 10 ms frame
shift, and 40-points DFT is the input feature used in both pre-training, and
fine-tuning. For optimization, we used AdamW except for DPSGD. The number of
epochs was set to 200. All the other setups are the same as those in [38].
### 4.2 Cross-lingual Adaptation Results
In this section, we investigate the effect of domain adaption with cross-
lingual data and compare the two introduced DP algorithms, DPSGD and PATE,
with and without $\text{RA}_{\text{DNN}}$s.
In Table 2, the baseline method, i.e., adapting the whole KWT network
parameters without DP guarantees, attains a classification accuracy equal to
96.49 with a utility of 3.0 on the France language. By comparing the results
with or without DP in Table 2, we can see that both utility and accuracy drop
when DP constraints are imposed. In particular, the accuracy drops from 96.49
to 53.40 when DPSGD, the fourth row, is used, and the utility drops from 3.0
to 0.22. PATE, in the fifth row, instead can limit the drop in accuracy and
utility.
Furthermore, by comparing the results of training from scratch (fs) and fine-
tuning (FT), we conclude that domain adaptation is required to successfully
train a model with DP. But differ from what was reported on language modeling
in [21], DPSGD is not effective in cross-lingual acoustic adaptation but PATE
is. We argue the difference is mainly because the domain mismatch is larger in
cross-lingual tasks, and the mechanism of data dependant privacy in PATE
reduces the amount of perturbation needed to be added under the same level of
privacy budget. We also evaluate all the selected languages listed in Table 1,
reporting an overall average results in the last two rows in Table 2. The
results indicate that our method works not only on French but also in multi-
lingual scenario.
Table 2: Comparison between DPSGD and PATE with and without
$\text{RA}_{\text{DNN}}$s on MLSW-mini. The All language results are the
weighted average accuracy based on the number of utterances in the four
selected languages.
lang | Method | DP | # Train Para. | Utility | Acc. (%)
---|---|---|---|---|---
fr | from Scratch (fS) | | 5.4 M (100 %) | 2.88 | 94.58
| fS w/ PATE | | 5.4 M (100 %) | 2.66 | 91.23
en $\to$ fr | Fine-tune (FT) | | 5.4 M (100 %) | 3.00 | 96.49
| FT w/ DPSGD | | 5.4 M (100 %) | 0.22 | 53.40
| FT w/ PATE | | 5.4 M (100 %) | 2.72 | 92.10
| LP w/ PATE | | 21.7 K (0.4 %) | 1.13 | 61.13
| $\text{RA}_{\text{DNN}}$ w/ DPSGD | | 0.9 M (14.6 %) | 0.77 | 60.69
| $\text{RA}_{\text{DNN}}$ w/ PATE | | 0.9 M (14.6 %) | 3.05 | 91.82
all | fS | | 5.4 M (100 %) | 2.71 | 92.03
| fS w/ PATE | | 5.4 M (100 %) | 2.32 | 85.97
en $\to$ all | FT | | 5.4 M (100 %) | 2.99 | 96.38
| FT w/ PATE | | 5.4 M (100 %) | 2.49 | 88.51
| $\text{RA}_{\text{DNN}}$ w/ PATE | | 0.9 M (17 %) | 2.75 | 87.72
### 4.3 Residual Adapter Size Effect on Fine-tuning
In this section, the effects of $\text{RA}_{\text{DNN}}$s are discussed. The
MLSW-mini French in Table 1 is used for our experiment. We performed the
experiments with a privacy budget $\epsilon=7.96$ using a PATE [19] based KWT
[38] model. We also used use $\text{RA}_{\text{DNN-d}}$ to denote that the
down-projection dimension of the $\text{RA}_{\text{DNN}}$s is $d$. We tried
several $d$ values ranging from 3 to 288, where 288 is twice the dimension of
the original feature dimension, and 3 is instead 64 times smaller than the
original feature dimension. As summarized in Table 3,
$\text{RA}_{\text{DNN-24}}$ attains the best utility, with an 88.08 % accuracy
training 2.46 % of parameters only. In addition, by appropriately choosing the
size of $\text{RA}_{\text{DNN}}$s, $\text{RA}_{\text{DNN-288}}$ provides a
result that is comparable with that of the fully fine-tuned model.
Table 3: Results with $\text{RA}_{\text{DNN}}$ for PATE with different
$\text{RA}_{\text{DNN}}$ dimension and a privacy budget $\epsilon=7.96$ on
MLSW-mini French. $\text{RA}_{\text{DNN-d}}$ means the down-projection
dimension is $d$.
Method | DP | # Train Para. | Utility | Acc. (%)
---|---|---|---|---
FT | | 5.4 M (100%) | 3.00 | 96.49
FT w/ PATE | | 5.4 M (100%) | 2.72 | 92.10
LP | | 21.7 K (0.4%) | 1.13 | 61.13
$\text{RA}_{\text{DNN-24}}$ | | 0.1 M (2.5%) | 3.22 | 88.08
$\text{RA}_{\text{DNN-288}}$ | | 1.4 M (20.2%) | 2.98 | 92.07
The effect of trainable parameters is also investigated in Figure 2. First of
all, as the number of trainable parameters increases, the model accuracy
increases. However, it saturates at the fine-tuned accuracy when the number of
parameters exceeds 20% of the total model parameters. The latter means that we
only have to train 20% of the parameters to reach the best performance, and
increasing the number of trainable parameters does not help. In addition, the
best utility occurs when adapting $2.46\%$ of parameters. Reducing it does not
lead to any beneficial effect, and the accuracy begins to degrade rapidly.
Increasing the $\text{RA}_{\text{DNN}}$ size improves the overall accuracy,
but the utility drops because the number of trainable parameters increases
accordingly.
(a) Accuracy
(b) Utility
Figure 2: Accuracy and utility of PATE-$\text{RA}_{\text{DNN}}$ architecture
with different $\text{RA}_{\text{DNN}}$ sizes. (a) The model performance
converges to the fine-tuning result when 20 % of the parameters are adapted.
(b) Our method achieves the best utility when 2.46 % of parameters are
adapted.
### 4.4 Different Connections of Residual Adapters
We now investigate into the different $\text{RA}_{\text{DNN}}$ connections
described in Section 3.1. As shown in Table 4, connecting the
$\text{RA}_{\text{DNN}}$s in our task isn't necessarily helpful. We believe
the reason is that the additional information from the other
$\text{RA}_{\text{DNN}}$s is too noisy for a few-shot domain adaptation. We
can validate the hypothesis from the fact that the DenseNet-alike connections
provide the worst performance albeit it's more complicated than the other
structures. And the results, same as our other experiments, lead to a
conclusion that the simpler, the more promising.
Table 4: Experiments of PATE with different connections from EGSP-V2 to MLSW-mini French with $\epsilon=8.0$. Model structure | Connection type | Acc. (%)
---|---|---
$\text{RA}_{\text{DNN-24}}$ | No connection | 88.08
| Neighboring [31] | 86.91
| Unet-alike | 87.61
| DenseNet-alike | 86.72
$\text{RA}_{\text{DNN-288}}$ | No connection | 92.07
| Neighboring [31] | 91.49
| Unet-alike | 91.77
| DenseNet-alike | 91.13
## 5 Conclusion
In this work, we tackled the problem of preserving privacy in a cross-lingual
speech classification task. First, we tried to port what done on language
modeling by [21] using DPSGD, but we observed a significant performance drop
using their method. Thus, we proposed a novel PATE-based solution, which,
differently from DPSGD, led to a small drop in performance while still
preserving DP guarantees.
Furthermore, to reduce the computational burden while fine-tuning with DP, we
tested LP and $\text{RA}_{\text{DNN}}$. LP was not effective; whereas,
$\text{RA}_{\text{DNN}}$ allows a reduction of 97.5% of the parameters to be
adapted while keeping a comparable performance of the PATE model. We also
performed an ablation study to verify skip connection strategies on
$\text{RA}_{\text{DNN}}$. Although skip-connection does not give any
performance improvement, the exploring of different parameter-efficient
architectures leveraging PATE is useful for future studies.
Acknowledgments The authors would like to express their gratitude to Prof.
Chin-Hui Lee from Georgia Tech for providing helpful insights and suggestions.
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|
Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2023.0322000
This research was sponsored by the Defense Threat Reduction Agency (DTRA) and
the DEVCOM Army Research Laboratory (ARL) under Grant No. W911NF2120076.
Corresponding author: Yi-Ting Shen (e-mail: ytshen@umd.edu).
# Archangel: A Hybrid UAV-based Human Detection Benchmark with Position and
Pose Metadata
YI-TING SHEN1 YAESOP LEE1 HEESUNG KWON2 DAMON M. CONOVER2 SHUVRA S.
BHATTACHARYYA1 NIKOLAS VALE2 JOSHUA D. GRAY3 G. JEREMY LEONG4 KENNETH
EVENSEN5 AND FRANK SKIRLO5 University of Maryland, ECE Department and
UMIACS, College Park, MD, USA DEVCOM Army Research Laboratory (ARL), Adelphi,
MD, USA Fibertek Inc., Herndon, VA, USA Department of Energy, Washington,
DC, USA Defense Threat Reduction Agency (DTRA), Fort Belvoir, VA, USA
###### Abstract
Learning to detect objects, such as humans, in imagery captured by an unmanned
aerial vehicle (UAV) usually suffers from tremendous variations caused by the
UAV’s position towards the objects. In addition, existing UAV-based benchmark
datasets do not provide adequate dataset metadata, which is essential for
precise model diagnosis and learning features invariant to those variations.
In this paper, we introduce Archangel, the first UAV-based object detection
dataset composed of real and synthetic subsets captured with similar imagining
conditions and UAV position and object pose metadata. A series of experiments
are carefully designed with a state-of-the-art object detector to demonstrate
the benefits of leveraging the metadata during model evaluation. Moreover,
several crucial insights involving both real and synthetic data during model
optimization are presented. In the end, we discuss the advantages,
limitations, and future directions regarding Archangel to highlight its
distinct value for the broader machine learning community.
###### Index Terms:
UAV-based object detection, human detection, UAV-based benchmark dataset,
position metadata, synthetic data, model optimization.
=-21pt
## I Introduction
With the recent rapid advancement in edge computing technology coupled with
resource-constrained mobile platforms, particularly unmanned aerial vehicles
(UAVs) with electro-optical (EO) sensor payloads, a wide range of UAV-enabled
applications have been more prevalent. Notable examples include UAV-enabled
search and rescue in disaster management [1], aerial surveillance and
reconnaissance for civilian and military purposes [2], precision agriculture
[3], traffic analysis [4], and intelligent transportation applications [5].
Lately, owing to the remarkable progress of artificial intelligence and
machine learning technology, tailored to distinct constraints of small UAV
platforms, these UAV-based applications have been frequently providing
promising solutions and successfully achieving the goals and operational
requirements of their corresponding applications. Central to the above UAV-
based applications are streamlined plug-ins that can effectively interrogate
real-time imagery, captured with UAVs, and provide image/video analytics
relevant to UAV-based scene understanding, particularly object detection and
recognition.
Recently, extensive efforts in object detection and recognition have led to
extraordinary advances in the perception accuracy in various challenges
associated with large-scale object detection benchmarks that were captured
primarily with ground-based cameras [6, 7, 8]. Compared to ground-based object
detection and recognition, UAV-based object detection poses unique and severe
challenges, as UAV flight inevitably results in a wider range of variations in
the conditions for capturing images, including the altitudes and viewing
angles of cameras, system turbulence, and weather events. These variations
lead to more drastic variations in object appearances/attributes, and thus
pose additional challenges to onboard detection models in the location and
recognition of objects of interest. In general, changes in object
appearances/attributes, caused by varying the image collection conditions,
entail three major dependencies: (1) pose dependency caused by changes in the
UAV position or camera viewing angle, (2) scale dependency owing to the
distance between the UAV and object, and (3) image quality dependency due to
UAV turbulence or various weather conditions.
We argue that developing object detection models which can adequately learn
features invariant to these dependencies is key to substantially enhancing
object detection accuracy in UAV-based perception. This requires curating UAV-
based datasets that include images and metadata that carefully depict the full
spectrum of correlations between the target scene on the ground and the camera
on the UAV, whose imaging conditions constantly change as the UAV navigates
the entire range of given operational requirements. Therefore, it is
imperative to have a UAV-based object detection benchmark carefully curated
with object poses, UAV positions, and weather information in the form of
metadata for accurate model validation and verification.
Existing UAV-based benchmarks, such as VisDrone [9], UAVDT [10], Okutama-
Action [11], and Standford Drone Dataset [12], provide limited metadata.
Despite containing a wide variety of scenes captured using UAVs under
different circumstances with various types of objects of interest, they do not
provide a complete set of metadata, such as object poses and UAV positions,
for each image in the datasets. This significant lack of information about how
objects on the ground are projected through the camera lens as a function of
UAV positions can lead to considerable limitations in learning about the
objects in UAV-based images, as the appearances/attributes of the objects are
subject to large variations.
Figure 1: Examples of images in Archangel-Synthetic (top), Archangel-Mannequin
(middle), and Archangel-Real (bottom). Each image is labeled with its UAV
position [Height, Radius]: [20m, 20m] (left), [35m, 20m] (middle), and [50m,
20m] (right). Each instance is labelled with its object pose: stand (green),
squat or kneel (red), and prone (cyan).
To overcome the limitation resulting from lack of metadata, we introduce a
large-scale UAV-based dataset, called Archangel, collected with comprehensive
position and pose information by the DEVCOM Army Research Laboratory (ARL)
(Fig. 1). Archangel comprises three sub-datasets: Archangel-Real [13],
Archangel-Mannequin [13] and Archangel-Synthetic [14]. Archangel-Real
comprises video sequences captured by a UAV flying at various altitudes and
radii of rotation circles. It sets a group of real humans as targets, and each
human is in one of three possible poses (i.e., stand, kneel or squat111The
terms kneel and squat are used interchangeably throughout this paper., and
prone) (Fig. 2). Similarly, Archangel-Mannequin sets a group of mannequins and
different types of vehicles as targets. The imaging conditions for these two
sub-datasets, such as UAV altitudes, ranges to targets, and object poses, are
the same. Unlike Archangel-Real and Archangel-Mannequin, which were collected
in real-world environments, Archangel-Synthetic was generated using the Unity
game engine [15]. It includes a number of different virtual characters, who
are in the same poses as described above and rendered with diverse
illumination conditions. Archangel-Synthetic is designed for augmenting the
other two real sub-datasets and for studying various issues tied to optimizing
machine learning (ML) models using synthetic data, such as synthetic data
augmentation and domain adaptation [16], with respect to UAV-based scene
understanding.
Figure 2: Examples of the three poses in Archangel: stand (left), squat or
kneel (middle), and prone (right).
In addition to collecting a new dataset, we further characterize Archangel
using state-of-the-art (SoTA) object detection models, specifically the YOLOv5
family [17] which has five different levels of architectural complexity and is
pre-trained on MS-COCO [8], a large-scale ground-based object detection
dataset. In this paper, we focus on the YOLOv5 models with lower complexity
(i.e., $YOLOv5n6$, $YOLOv5s6$, $YOLOv5m6$) since they are able to run on
computing resource typically available on small UAV platforms (Tab. II). For
each model, we evaluate its human detection performance using Archangel-Real.
Since we programmed our UAVs to circle around the human targets at various
altitudes and radii during data collection (Fig. 3), the detection accuracy
can be compared across the whole range of UAV positions.
Furthermore, we optimize the pre-trained YOLOv5 models with different fine-
tuning strategies using various hybrid combinations of subsets from Archangel-
Mannequin and Archangel-Synthetic. These fine-tuning strategies have been
designed to provide valuable guidelines for leveraging synthetic data in
training ML models to boost their performance. A comprehensive performance
comparison between the baseline and the optimized YOLOv5 models is presented
to demonstrate how incorporating a combination of real and synthetic data can
enhance detection performance across varying UAV positions (Sec. V and VI).
One of the critical findings from the comparative performance analysis
indicates that if the real and synthetic data used for fine-tuning is balanced
with respect to the amount of data, a significant performance boost can be
achieved even with a low-complexity model, such as $YOLOv5n6$. Furthermore,
the optimization based on the real and synthetic data is much more effective
on infrequent object poses that are rarely seen in the original dataset for
pre-training. For example, prone from Archangel is not often seen in MS-COCO
[8], so the performance improvement on this pose is more evident.
This paper is extended from our previous preliminary studies [14, 13], where
we mainly focused on introducing new datasets separately without extensive
data analysis and characterization. As a result, the major scope of this paper
is to extensively study the three sub-datasets jointly as a unified UAV-based
benchmark with metadata for human detection. In summary, the contributions of
this paper are as follows:
1. 1.
We present a unified Archangel dataset222Archangel is available for access
through: https://a2i2-archangel.vision after substantially restructuring the
three sub-datasets since our conference publications [13, 14], including
additional labeling Archangel-Real and an expansion of the range of Archangel-
Synthetic. Note that Archangel was not previously available for access due to
incomplete labeling and restructuring. To the best of our knowledge, Archangel
is the first UAV-based object detection dataset which contains real and
synthetic sub-datasets captured with similar imaging conditions and includes
an extensive set of metadata (e.g., object poses and UAV positions).
2. 2.
We conduct extensive data analysis on Archangel by jointly analyzing its three
sub-datasets (Archangel-Synthetic, Archangel-Mannequin and Archangel-Real). In
particular, we provide several important guidelines on exploiting real and
synthetic data together to improve UAV-based object detectors.
## II Related Work
### II-A UAV-based Object Detection Datasets
There is an increasing number of large-scale benchmarks for object detection,
utilizing images captured with some fixed or moving cameras on the ground [8,
7, 18, 19, 20, 21, 22], yet relatively few datasets have been collected with
UAVs. Moreover, these UAV-based object detection datasets all have their own
limitations. For instance, VisDrone [9] is one of the major datasets for UAV-
based object detection. It consists of images captured with UAVs in dozens of
different scenarios, in which ten categories of objects were selected and
carefully labelled. While it does have an advantage in data diversity,
VisDrone does not provide any metadata, such as UAV positions. In contrast to
VisDrone, UAVDT [10], another well-known benchmark for UAV-based detection and
tracking, provides three different kinds of UAV-specific metadata (i.e.,
weather condition, flying altitude and viewing angle) and a few object
attributes such as vehicle category. However, the annotations for the metadata
are coarse (i.e., 3 categories for each). Also, the dataset does not contain
the human category. Unlike UAVDT, Okutama-Action [11] is composed of images
from aerial views and contains humans in different human poses. However, it
provides limited metadata regarding UAV altitudes and camera viewing angles.
A212-Haze [23], which is the first real haze and object detection dataset with
in-situ smoke measurements aligned to aerial imagery and is used in the recent
$UG^{2}$\+ Challenge [24], provides UAV position metadata. However, A212-Haze
does not include a synthetic subset, limiting its usage for facilitating
studies on how to improve UAV-based object detectors by using synthetic data.
More recently, DGTA [25] generates synthetic datasets associated with existing
UAV-based object detection datasets and provides UAV position metadata for the
generated datasets. Nevertheless, the existing UAV-based datasets they use,
such as VisDrone [9], are still lack of metadata, limiting DGTA’s usage for
precise model diagnosis.
There are two other types of datasets which are closely related to UAV-based
object detection. First, datasets such as DOTA [26] include aerial images
collected from satellites or aircraft. Although they are usually curated for
remote sensing applications, detecting objects in such datasets also suffers
from severe variations in the scale and orientation of the objects. Second,
some UAV-based datasets are designed for certain vision tasks strongly
associated with object detection. For instance, CARPK [27] is a large-scale
car parking lot dataset designed for object counting. MOR-UAV [28] is a large-
scale moving object recognition dataset comprising videos captured by a UAV in
various environments, such as urban areas and highways. Stanford Drone Dataset
[12] is used for analyzing various object trajectories in the real world from
the top-view. UAV123 [29] is used for low altitude UAV-based object tracking.
Similar to Archangel, UAV123 also includes synthetic data generated by a
photo-realistic simulator.
A comprehensive investigation of recent UAV-based datasets is shown in Tab. I.
Note that Archangel is the first ever UAV-based dataset collection not only
containing both real and synthetic data but also providing an extensive set of
metadata, including object poses and UAV positions.
TABLE I: Comparison of recent UAV-based datasets. (1k = 1000)
General Information | Metadata
---|---
Name | Tasks | Year | #Clips | #Images | Resolution | Syn/Real | Human | Obj. Poses | UAV Pos. | Lighting Cond.
Stanford Drone Dataset [12] | TF | 2016 | 60 | 929.5k | 1400$\times$1904 | R | ✓ | - | - | -
UAV123 [29] | OT | 2016 | 123 | 112.6k | 1280$\times$720 | S+R | ✓ | - | - | -
Okutama-Action [11] | OD, AR | 2017 | 43 | 77.4k | 3840$\times$2160 | R | ✓ | ✓ | - | -
CARPK [27] | OC | 2017 | - | 1.4k | 1280$\times$720 | R | - | - | - | -
UAVDT [10] | OD, OT | 2018 | 100 | 80k | 1024$\times$540 | R | - | - | ✓ | ✓
VisDrone [9] | OD, OT | 2018 | 263 | 179.3k, static: 10k | various | R | ✓ | - | - | -
DroneSURF [30] | FRD | 2019 | 200 | 411.5k | 1280$\times$720 | R | ✓ | - | - | -
AU-AIR [31] | OD | 2020 | 8 | 32.8k | 1920$\times$1080 | R | ✓ | - | ✓ | -
MOR-UAV [28] | MOR | 2020 | 30 | 10.9k | various | R | - | - | - | -
DOTA [26] | OD | 2021 | - | 11.3k | various | R | - | - | - | -
UAV-Human [32] | AR, PE, PR, ATR | 2021 | AR: 67.4k | PE: 22.5k, PR: 41.3k, ATR: 22.3k | 1920$\times$1080 | R | ✓ | ✓ | - | -
SeaDroneSee [33] | OD, OT | 2022 | SOT: 208, MOT: 22 | OD: 5.6k, MOT: 54.1k | various | R | ✓ | - | ✓ | -
A212-Haze [23] | IER, OD | 2022 | - | 1k | 1845$\times$1500 | R | ✓ | - | ✓ | -
DGTA [25] | OD | 2022 | - | VisDrone: 50k, SeaDroneSee: 100k, Cattle: 50k | 3840$\times$2160 | S | ✓ | - | ✓ | -
Archangel-Real | OD | 2022 | 69 | 41.4k | 1304$\times$978 | R | ✓ | ✓ | ✓ | -
Archangel-Mannequin | OD | 2022 | 598 | 178.8k | 1920$\times$1080 | R | ✓ | ✓ | ✓ | -
Archangel-Synthetic | OD | 2022 | - | 4423.7k | 512$\times$512 | S | ✓ | ✓ | ✓ | ✓
Notation | Description | Notation | Description | Notation | Description | Notation | Description
---|---|---|---|---|---|---|---
TF | Trajectory Forecasting | OT | Object Tracking | OD | Object Detection | AR | Action Recognition
OC | Object Counting | FRD | Face Recognition & Detection | PE | Pose Estimation | MOR | Moving Object Recognition
PR | Person Re-identification | ATR | Attribute Recognition | SOT/MOT | Single/Multiple Object Tracking | IER | Image Enhancement & Restoration
### II-B UAV-based Object Detection Methods
With the rapid development of generic object detection methods [34] and the
aforementioned UAV-based object detection benchmarks (Tab. I), the detection
accuracy of UAV-based object detectors has improved significantly over the
past few years. In addition to common issues for generic object detection,
UAV-based object detection has its own unique challenges [35]. In general, all
of the challenges can be roughly divided into three categories. First, objects
in UAV-based images are usually much smaller [36]. Therefore, many solutions
have been proposed to address this problem to date. For example, Liu et al.
[37] proposed HRDNet that fused information from both high- and low-resolution
inputs to simultaneously preserve features of small objects and maintain
computational costs. Similarly, Liu et al. [38] introduced a multi-branch and
parallel structure (MPFPN) to extract more powerful features for tiny object
detection. Besides the scale of an object, target objects in UAV-based
datasets are usually crowded and sparsely distributed, reducing both the
accuracy and efficiency of an object detector. Thus, Yang et al. [39] proposed
ClusDet that performed object cluster proposal first before detecting objects.
Finally, UAV-based datasets contain many UAV-specific nuisances [40], such as
varying UAV altitudes and viewing angles. These nuisances cause tremendous
variations in object appearances, causing degraded detection performance. To
address this issue, Wu et al. [40] proposed to adopt adversarial training to
learn domain-robust features from UAV-specific nuisances coarsely annotated by
the authors. In this paper, we posit that UAV-based object detection can be
further enhanced by providing UAV-based benchmarks with a set of fine-grained
metadata, such as that contained in Archangel.
In addition to improving the detection accuracy, reducing the computational
cost to achieve real-time on-board processing is also very important for UAV-
based object detection approaches. One way to improve latency is to skip
unnecessary computation. For instance, Ammour et al. [41] proposed to extract
candidate regions of target objects first via over-segmentation. After that,
only windows around the candidate regions were sent to the pre-trained CNN and
linear SVM for feature extraction and classification. Another way of reducing
computational overhead is to use more efficient one-stage object detectors,
such as YOLO [42, 17], RetinaNet [43], CenterNet [44], and EfficientDet [45].
These one-stage object detectors directly classify and locate objects without
generating region proposals, resulting in improved latency. As an example, Liu
et al. [46] adapted the original network architecture of YOLO by making it
more suitable for UAV-based object detection. In this paper, we also utilize
YOLOv5 [17] for all the experiments due to the advantage of its low complexity
(Tab. II).
TABLE II: Complexity of the YOLOv5 models [17] used in this study. Model | #Parameters (M) | FLOPs (G)
---|---|---
YOLOv5n6 | 3.1 | 4.3
YOLOv5s6 | 12.3 | 16.2
YOLOv5m6 | 35.3 | 49.1
## III The Archangel Dataset
The data collection process for Archangel is illustrated in Fig. 3 and a brief
comparison of the three sub-datasets is provided in Tab. III. In the
following, we will go through the data collection process of each sub-dataset
in detail.
Figure 3: Illustration of the data collection process for Archangel. For each
data collection, a number of objects (real people, mannequins or virtual
characters) on the ground were captured by a camera mounted on a UAV (real or
simulated). Each of the objects was in one of the three defined poses (stand,
kneel, and prone). The UAV circled around the objects at a predefined
altitudes and radii of rotation circles. TABLE III: Comparison of the three
sub-datasets comprising Archangel.
Archangel | Image Size | Targets | Altitude (m) | Radius (m) | Field of View | Camera Pitch Angle
---|---|---|---|---|---|---
Synthetic | 512$\times$512 | Virtual characters | [5-80] increment by 5 | [5-80] increment by 5 | 22.5$\degree$ | various
Mannequin | 1920$\times$1080 | Mannequins, vehicles | [15-50] increment by 5 | [15-50] increment by 5 | 120$\degree$ | 45$\degree$
Real | 1304$\times$978 | Real people | [15-50] increment by 5 | [20-50] increment by 5 | 45$\degree$ | 22.5$\degree$, 45$\degree$, 67.5$\degree$
### III-A Archangel-Mannequin
Target Objects. During this data collection, a group of mannequins were used
as human surrogates primarily due to the safety guidelines of the test
facility and the difficulty of asking humans to maintain certain strenuous
poses, such as prone and squat, for a long period of time. The mannequins were
dressed in casual outfits and positioned in three different poses (i.e.,
stand, kneel, and prone). Hence, the distribution gap between human attributes
and those of mannequins’ is not noticeably large. Additionally, the dataset
also includes a small group of various types of civilian vehicles as targets,
such as sports utility vehicles (SUVs), minivans, and sedans. Each target in
the dataset was labeled as mannequin-standing, -kneeling, -prone or civilian
vehicles.
Data Collection. The imagery was captured using a contractor-built UAV
equipped with an onboard electro-optical (EO) camera (ELP-USBFHD01M-L21) with
a 1920$\times$1080 pixel array and a lens with approximately 120° field-of-
view (FOV). The UAV camera was pitched forward by 45° relative to level
flight. During the course of multiple UAV flights, the UAV operated over a
wide range of altitudes and radii of rotation circles while keeping the camera
pointed inward toward the targets and circling a central point. Both the
altitude and radius of the rotation circle were varied from 15-50 meters in
5-meter increments. Since the target objects were stationary and the camera
pitch angle was constant, the target objects were spread across different
regions of the camera’s FOV, resulting in different view angles.
### III-B Archangel-Synthetic
Motivation. While Archangel-Mannequin provides valuable aerial imagery with
pose and position metadata well suited for UAV-based human detection, the
imaging conditions of this data collection were limited. Furthermore,
Archangel-Mannequin did not incorporate some important factors, such as
various human appearances/attributes, extended ranges of UAV altitudes and
radii of rotation circles, and different illumination conditions. To overcome
these restrictions, a large-scale synthetic imagery (i.e., Archangel-
Synthetic) dataset containing multiple virtual humans in the same poses as
that used in Archangel-Mannequin was generated using the Unity game engine
[15] to augment Archangel.
Data Generation and Labeling. In the Unity-based simulation, a 3D scene is
constructed using a terrain asset (i.e., background) and one or more target
assets (i.e., virtual characters in different outfits and poses). For the
current version of Archangel-Synthetic, we use only a simple terrain model
(i.e., desert), but we plan to integrate more complex terrain models in future
work.
For each target asset, we first created a Unity project and configured the
lighting and camera parameters in a virtual 3D environment. A Unity terrain
asset (i.e., the desert background) and the target asset were then added to
the 3D environment. After the 3D environment was configured as above, a C#
script was then used to control the position and viewing angle of the camera
as it circled around the target. At each step, the camera was pointed at the
center of the target. The script was iterated to encompass the whole range of
UAV camera altitudes, the radii of the circles, and the camera viewing angles
relative to the target, thus producing imagery captured at various camera-to-
target distances and camera pitch angles. Additionally, the sun angle was
varied to generate synthetic images captured at different times of a day with
corresponding illumination conditions. This resulted in large-scale synthetic
imagery with significant target pose, scale, and illumination variations.
To synthesize images and annotations from the virtual 3D environment
constructed above, we used an open source software asset, Image Synthesis for
Machine Learning [47]. Specifically, the software produced an image
segmentation mask where each target object in a synthetic image was assigned a
unique scalar value. To generate the bounding box annotations for each target,
a Python script was used to parse each segmentation mask, identify each target
object, and measure the center, width, and height of the tightest bounding box
encompassing the target. Additionally, the target category, the camera
position, the target orientation relative to the camera, the camera-to-target
distance, the camera pitch angle, and the number of pixels inside the
segmentation mask were recorded in a single JavaScript Object Notation (JSON)
file for each trial.
Properties. Archangel-Synthetic includes mountainous desert terrain and eight
different virtual characters, each in three different poses (i.e., stand,
squat, and prone). In a single trial of synthetic data generation (i.e., a
virtual character with a certain pose), both the altitude of the camera and
the radius of the rotation circle were varied from 5-80 meters in 5-meter
increments. Additionally, the camera viewing angle relative to the character
was varied from 0°-358° in 2° increments and four different sun angles were
simulated. This resulted in over 4.4M images included in Archangel-Synthetic.
Each image contains 512$\times$512 pixels with horizontal and vertical fields-
of-view of 22.5°.
### III-C Archangel-Real
System Design. The dataset was collected with an ARL-designed UAV platform
called the Dawn Dove (D2). The D2 is a re-configurable UAV, with the ability
to shift the center of gravity by adjusting arm angles, arm placement,
battery, sensor payload, and on-board processor location. It is composed of a
combination of 3D printed polyethylene terephthalate glycol (PETG) and carbon
fiber infused nylon, and traditional carbon fiber parts. It can carry various
types of sensor payloads and on-board processors. It has an approximate
payload capacity of 1.5 lbs and an approximate flight time of 8 minutes. For
this data collection, the sensor payload consisted of a UI-3250ML-C-HQ EO
camera with an Edmund Optics 6mm/F1.4 lens and a FLIR Boson 640 8.7 mm IR
camera. The cameras were co-located on the front of the D2 and the EO camera’s
image was cropped to match the FOV of the FLIR Boson (50° HFOV). The on-board
processor was an NVIDIA Xavier NX with a 1 TB NVMe SSD for additional data
storage.
Data Collection. In this dataset, the targets consisted of real people wearing
civilian clothing in three different poses: stand, kneel, and prone. The data
collection process involved having the D2 fly circles at radii ranging from
20-50 meters, at intervals of 5 meters, and at altitudes ranging from 15-50
meters, at intervals of 5 meters. The camera angle relative to level flight
was manually adjusted between flights from -22.5°, -45°, and -67.5°to ensure
the targets remained within the FOV of the cameras. In total, 52 circles were
flown around the targets. To fly the circles, custom Robot Operating System
(ROS) based autonomy code was used, along with a custom Python-based graphical
user interface (GUI), which communicated with the UAV. From the ground control
station (GCS), the target GPS location, circle radius, altitude, maximum
velocity, and file name were entered into the GUI. Once the circle parameters
were entered, the D2 was manually armed, launched, and switched over to
“offboard mode” which passed control of the UAV to the GCS. The GCS then
commanded the UAV to perform the autonomous circle. Once complete, the next
circle’s parameters were entered into the GUI and sent to the UAV while still
in the air. This was repeated each flight until the UAV had to be brought back
down to replace the battery. In addition to stationary targets, a few circles
were also flown where the people walked, jogged, crawled, and waved. Note that
Archangel-Real involves human subjects as UAV-based detection instances.
However, an Institutional Review Board (IRB) approval was exempted since one
cannot identify individuals in the dataset.
### III-D Importance of the Camera Parameters
Before moving on to the data analysis section, we want to highlight the
importance of revealing the camera parameters used in the data collection.
Note that the scale of human instances in UAV-based object detection datasets,
such as Archangel, is strongly influenced by the camera parameters used in the
data collection, including FOV, pixel-array size, and pitch angles, in
addition to the UAV altitude and radius of rotation circle. Hence, the
detection results can vary greatly when using different camera parameters.
However, all the conclusions derived from the following data analysis of
Archangel can still be applied to other UAV-based datasets using different
camera parameters through extrapolation. That is, the performance gap can be
easily calibrated by adjusting the scale of human instances if the camera
parameters and the original object size are known a priori.
## IV Experimental Setup
Overview. In this paper, we designed a series of experiments based on the flow
shown in Fig. 4. In brief, for each experiment, we selected one of the three
pre-trained YOLOv5 models (Tab. II) and fine-tuned the model on varying
amounts of UAV-based real (i.e., Archangel-Mannequin) and synthetic (i.e,
Archangel-Synthetic) data. We then evaluated the model on a sequestered UAV-
based dataset (i.e., Archangel-Real). Based on the results, the designed
experimental flow can provide valuable insights into optimizing UAV-based
object detectors with hybrid sets of real and synthetic data.
Figure 4: Overview of the designed experimental flow. The pre-trained YOLOv5
models with various complexity were fine-tuned on different hybrid sets of
UAV-based real and synthetic data and evaluated on a hold-out UAV-based
dataset.
Datasets. Note that acquiring the best performance for an UAV-based object
detector is not the primary purpose of this study. Thus, we subsampled each of
the three sub-datasets of Archangel to explore optimal strategies for fine-
tuning or evaluating models:
1. 1.
Archangel-Mannequin: The dataset consists of video clips collected in 11 UAV
flight trials. In this paper, we carefully split the dataset into two subsets
so that each covered the whole range of the UAV positions covered during the
entire data collection. The video clips collected in 6 of the 11 trials (i.e.,
Trial-5, 6, 8, 9, 10, 11) were used for evaluating models. The rest (i.e.,
Trial-1, 2, 3, 4, 7) were used for fine-tuning models. All the video clips
were uniformly subsampled at 3 fps. This resulted in two sets of frames, Arch-
Mann-FT37, containing 6.7k frames for fine-tuning models, and Arch-Mann-Eval,
containing 11.2k frames for evaluating models. Arch-Mann-FT37 is named based
on the amount of data it has compared to the entire Archangel-Mannequin in
terms of percentage (i.e., 37%).
2. 2.
Archangel-Real: Similarly, we uniformly subsampled the video clips in
Archangel-Real at 1 fps. This resulted in a set of frames, named as Arch-Real-
Eval, containing 4.1k frames for evaluating models.
3. 3.
Archangel-Synthetic: Only one virtual character in all of the three poses was
used. For each UAV position, only one of the four sun angles was randomly
selected. Additionally, instead of using all the UAV positions, we uniformly
sampled images across each rotation circle in 60° increments. This resulted in
a set of images, named as Arch-Syn-FT, containing 4.6k images for fine-tuning
models.
Each of the three sub-datasets has its own unique usage in this study. In
general, Archangel-Real serves as the primary UAV-based benchmark for
measuring detection accuracy. Archangel-Mannequin can be viewed as the real
UAV-based fine-tuning dataset for adapting the detection models to the target
UAV-based domain. Although it includes mannequins instead of real humans,
fine-tuning on this dataset is shown to be effective in the following
sections. Archangel-Synthetic, on the other hand, is used as the synthetic
version of the UAV-based fine-tuning dataset, which can be combined with
Archangel-Mannequin to further optimize the models.
Evaluation. We utilized standard AP50, the average precision with an IOU
(Intersection of Union) threshold of 0.5, as the metric to measure the
performance of each object detector. Moreover, AP50 was computed for each pose
respectively. As more than one pose may exist in a single image, to obtain the
performance for only one certain pose, the other two poses were ignored during
the evaluation process.
Implementation Details. The official repository of YOLOv5 [17] was used for
both fine-tuning and evaluating models. If not specified otherwise, the
default hyperparameters were adopted. The input images were rescaled (i.e.,
imgsz=1280) first before being fed into all the models. During fine-tuning,
the backbone for each model was frozen (i.e., freeze=10) to prevent the model
from easily over-fitting. We fine-tuned each model for 20 epochs with a batch
size of 16 on a server with 4 NVIDIA GeForce RTX 2080 TI GPUs. During the
evaluation, we set the confidence threshold to be 0.05.
## V Results
The Performance of Pre-trained Models. To begin with, we evaluated the three
pre-trained YOLOv5 models on Arch-Mann-Eval and Arch-Real-Eval. The models
were pre-trained on MS-COCO, a representative ground-based dataset. The
results are shown in Fig. 5. From the results, we can gain several useful
insights on UAV-based object detection. First, larger pre-trained models
achieved better accuracy across all the evaluation datasets and poses. One
possible reason for this is that larger models, compared with smaller ones,
can explore better and find more powerful features for classification and
detection [48]. Although it implies that we can get higher accuracy by using
larger pre-trained models, such larger models may not fit well on small UAV
platforms with computational constraints.
Another trend we can observe is that the pre-trained models had much better
accuracy on stand. It is mainly because that the dataset used for pre-training
models (i.e., MS-COCO) contains significantly more human instances in stand
(i.e., 84.53%) [49]. In other words, it is impractical to directly use
detectors pre-trained on standard datasets, especially in unusual scenarios
where we need to detect people in uncommon positions, such as search and
rescue in disaster relief. It is worth mentioning that the pre-trained YOLOv5
models performed much worse on Arch-Mann-Eval than on Arch-Real-Eval. That is
mainly because Arch-Mann-Eval contains some other objects, such as traffic
cones and fiducials, which are easily misclassified as humans when captured by
the pre-trained detectors at high altitudes [13].
Figure 5: AP50 of the pre-trained YOLOv5 models evaluated on Arch-Mann-Eval
and Arch-Real-Eval.
Fine-tuning Models on a Real UAV-based Dataset: Arch-Mann-FT37. As we have
discussed, most ground-based datasets used to fine-tune models usually lack
UAV-specific samples for the models to learn from, such as human instances in
non-standing positions captured from various camera viewing angles and
altitudes. Therefore, we allowed the pre-trained YOLOv5 models to acquire such
knowledge by fine-tuning the models on Arch-Mann-FT37. Moreover, given the
significant challenges of collecting and annotating UAV-based datasets [9] and
the lack of existing large-scale UAV-based object detection benchmarks, we
explored the idea of fine-tuning models in the small-data regime [50]. More
precisely, we subsampled the original fine-tuning dataset and created several
smaller subsets for fine-tuning, which contained much less data (i.e., Arch-
Mann-FT20, Arch-Mann-FT10, Arch-Mann-FT5 and Arch-Mann-FT2). We followed the
same naming strategy as Arch-Mann-FT37 for the extra fine-tuning datasets.
The results are presented in Fig. 6. We would like to highlight the importance
of having an evaluation dataset with different characteristics from the fine-
tuning dataset. As we fine-tuned the models on data from Arch-Mann-FT37, we
could improve their detection accuracy on a similar evaluation dataset such as
Arch-Mann-Eval. However, the models fine-tuned on too much data from Arch-
Mann-FT37 tended to perform worse on Arch-Real-Eval. We argue that this was
because the models started to learn certain dataset-specific features from the
fine-tuning dataset, adversely affecting the models’ generalization capability
to unseen datasets, such as the evaluation dataset in our case. In practice,
ML models embedded into UAVs are usually deployed to new environments unseen
during training. Thus, in the following experiments, we chose to fine-tune
models on data from Arch-Mann-FT37 and Arch-Syn-FT but evaluate them on Arch-
Real-Eval.
Figure 6: AP50 of the YOLOv5 models fine-tuned on the different subsets of
Arch-Mann-FT37. Each model was evaluated on Arch-Mann-Eval (top) and Arch-
Real-Eval (bottom).
Fine-tuning Models on Both Real and Synthetic Datasets: Arch-Mann-FT37 and
Arch-Syn-FT. One of the significant advantages of Archangel is that it
contains both real and synthetic subsets acquired from similar imaging
conditions in data collections and synthetic rendering, respectively. Hence,
investigating the effect of augmenting the original UAV-based fine-tuning
datasets with UAV-based synthetic data is another major topic for this study.
To do so, we fine-tuned the pre-trained YOLOv5 models on Arch-Syn-FT.
Additionally, a well-known concern about learning from synthetic data is that,
compared with real data, synthetic data usually contains much less variations
in appearances/attributes of objects or structures of scenes [51]. Therefore,
instead of fine-tuning models only on Arch-Syn-FT, we also explored the idea
of multi-source learning [52], constructing multiple hybrid fine-tuning
datasets by directly merging Arch-Syn-FT with all the subsets of Arch-Mann-
FT37 used in the previous experiment.
The results are shown in Fig. 7. For prone, a rarely seen pose in the pre-
training dataset, the detection accuracy of the pre-trained models was very
low. After fine-tuning the models on the various hybrid subsets of Arch-Syn-FT
and Arch-Mann-FT37, the detection accuracy continually increased with few
exceptions. For stand, the pre-trained models performed much better than they
did for prone as expected, but fine-tuning on the hybrid sets of the real and
synthetic data still provided much improvement over the pre-trained models, as
clearly observed from the detection accuracy of YOLOv5n6. Similar observations
can be made for kneel.
We now compare the results shown in Fig. 6 with the ones shown in Fig. 7 to
highlight the effect of adding the synthetic data to the fine-tuning dataset
based only on the subsets of Arch-Mann-FT37. The results are shown in Fig. 8.
For prone, we can observe significant performance improvement across all the
models and hybrid subsets for fine-tuning after introducing the synthetic data
into the fine-tuning datasets, compared with the results of fine-tuning on
data from Arch-Mann-FT37 only. One explanation for the above finding is that
most of the human instances in the pre-training dataset are in certain upright
positions, including stand and kneel. Therefore, adding synthetic characters
in prone captured from various camera viewing angles and UAV altitudes to the
fine-tuning dataset can effectively aid the models to learn how to detect
humans in prone. More in-depth analysis of this issue will be discussed in the
ablation study (Sec. VI).
Additionally, Arch-Syn-FT, which includes only one virtual character and one
type of background, might be too simple for larger models, such as YOLOv5s6
and YOLOv5m6 in our case, to learn from, causing them to overfit to the fine-
tuning dataset. A proof of this assumption was that using Arch-Syn-FT along
with the subsets of Arch-Mann-FT37 to fine-tune YOLOv5s6 and YOLOv5m6
significantly decreased their performance on stand and kneel, especially when
we included fewer data from Archangel-Mann-FT37 (Fig. 8). On the other hand,
fine-tuning YOLOv5n6 on Arch-Syn-FT along with the subsets of Arch-Mann-FT37
did not have such a negative effect. As a result, in the following ablation
study, we focused on analyzing the results of YOLOv5n6 once we included Arch-
Syn-FT in the fine-tuning dataset.
Figure 7: AP50 of the YOLOv5 models fine-tuned on the various hybrid sets
constructed by Arch-Syn-FT and the different subsets of Arch-Mann-FT37. Figure
8: AP50 improvement of the YOLOv5 models after adding Arch-Syn-FT to the
original fine-tuning datasets based on the different subsets of Arch-Mann-
FT37.
## VI Ablation Study
Adjusting the Size of the Synthetic Dataset: Arch-Syn-FT. We have demonstrated
the effects of directly combining the various subsets of Arch-Mann-FT37 with
the same set of synthetic data (i.e., Arch-Syn-FT) for fine-tuning models
(Fig. 7 and 8). In this section, we are interested in further exploring the
outcome of using different amounts of synthetic data to fine-tune models.
Notably, we aim to investigate whether a ”balanced” fine-tuning dataset, which
contains the same amount of real and synthetic data, is better than its
”unbalanced” counterpart, in which the ratio of the synthetic data to the real
data varies due to the use of a fixed set of the synthetic data across all the
hybrid fine-tuning datasets.
To achieve this goal, for each real fine-tuning dataset (i.e., Arch-Mann-FT37,
Arch-Mann-FT20, Arch-Mann-FT10, Arch-Mann-FT5, and Arch-Mann-FT2), the
corresponding amount of data was randomly sampled from Arch-Syn-FT to match
the real fine-tuning dataset. Namely, if the real fine-tuning dataset was
smaller than Arch-Syn-FT, a subset of Arch-Syn-FT was randomly selected and
combined with the real fine-tuning dataset to form a balanced fine-tuning
dataset. Similarly, if the real fine-tuning dataset was larger, a random
subset of Arch-Syn-FT was duplicated before the combination. We denote the
synthetic fine-tuning dataset as Arch-Syn-FT-B if the aforementioned data
balancing procedure has been conducted.
The results are shown in Fig. 9, 10 and 11. Comparing Fig. 8 with Fig. 10, we
can observe that the negative effect of fine-tuning larger models (i.e.,
YOLOv5s6 and YOLOv5m6) on hybrid fine-tuning sets largely decreases or even
diminishes. Moreover, the improvement becomes more significant, especially for
YOLOv5n6. From Fig. 11, it is shown that fine-tuning YOLOv5n6 on the balanced
hybrid sets of the real and synthetic data can always be at least on par with
the best performance setting for fine-tuning. These findings indicate that the
proportion of each data to the whole fine-tuning dataset significantly impacts
the model’s fine-tuning performance.
Figure 9: AP50 of the YOLOv5 models fine-tuned on the various balanced hybrid
sets constructed by Arch-Syn-FT-B and the different subsets of Arch-Mann-FT37.
Figure 10: AP50 improvement of the YOLOv5 models after adding Arch-Syn-FT-B to
the original fine-tuning datasets based on the different subsets of Arch-Mann-
FT37. Figure 11: AP50 comparison of YOLOv5n6 fine-tuned on the different
hybrid sets of Arch-Syn-FT, Arch-Syn-FT-B, and the different subsets of Arch-
Mann-FT37.
Leaving One Pose Out from the Synthetic Dataset: Arch-Syn-FT. We have claimed
that fine-tuning models with synthetic human instances are particularly
important for prone, a pose rarely seen in the original training data. In this
section, we would like to provide more evidence to support this statement.
Specifically, we used YOLOv5n6 for this set of experiments since it showed
less sign of overfitting when fine-tuning on Arch-Syn-FT (Fig. 7).
Additionally, the technique of dataset balancing was adopted due to its
positive effect on fine-tuning models (Fig. 9 and 10). We followed a similar
procedure to generate each hybrid fine-tuning dataset, except that each time
one of the poses, stand, kneel, or prone, was excluded in advance from Arch-
Syn-FT, resulting in a ”leave-one-pose-out” fine-tuning dataset, Arch-Syn-FT-
NoSt, Arch-Syn-FT-NoKn, or Arch-Syn-FT-NoPr, respectively.
The results are presented in Fig. 12. In general, the detection accuracy of
stand and kneel did not obviously change when we removed any one of the poses
from the synthetic fine-tuning dataset. However, the performance of prone
degraded drastically when we excluded the virtual characters in prone from the
fine-tuning dataset, which strongly supports our earlier claim.
Figure 12: AP50 of YOLOv5n6 fine-tuned on the various balanced hybrid sets
constructed by the three leave-one-pose-out synthetic fine-tuning datasets
(i.e., Arch-Syn-FT-NoSt-B, Arch-Syn-FT-NoKn-B and Arch-Syn-FT-NoPr-B) and the
different subsets of Arch-Mann-FT37.
Precise Model Diagnosis: Performance Comparison on the Altitude/Radius Grid.
So far, we have demonstrated that the detection accuracy of the pre-trained
YOLOv5 models on the UAV-based evaluation dataset can be considerably boosted
by fine-tuning the models on the hybrid sets of UAV-based real and synthetic
data. For example, we have shown that the AP50 of the pre-trained YOLOv5n6 can
be increased by about 30 in AP value by fine-tuning it on a joint set of Arch-
Mann-FT2 and Arch-Syn-FT-B (Fig. 11). In this section, we further analyze this
particular example with the complete information about the UAV positions over
the altitude/radius grid as shown in Fig. 13. Our goal is to give an idea of
how to utilize the metadata provided by Archangel to diagnose problems with a
UAV-based object detection model.
The results are presented in Fig. 13. In this figure, we can clearly observe
how the pre-trained YOLOv5n6’s performance gradually progresses with the
different fine-tuning datasets, from the real-data-only dataset to the
unbalanced hybrid dataset to the balanced hybrid dataset. Initially, the pre-
trained YOLOv5n6 performs fairly well at the low altitudes but fails at the
high altitudes due to the curse of the pre-training dataset, which is composed
mostly of ground-based human instances. After being fine-tuned on Arch-Mann-
FT2, the fine-tuned YOLOv5n6 gets much better at detecting human instances
from a relatively higher altitude or larger circle radii. However, the
detection accuracy of the human instances at close range decreases
considerably. We argue that this is mainly because the image resolution of
Arch-Mann-FT2 (i.e., 1920x1080) is much larger than that of the evaluation
dataset (i.e., 1304x978) and the bounding boxes contained in Arch-Mann-FT2 are
generally small. In other words, fine-tuning YOLOv5n6 on Arch-Mann-FT2
inevitably causes a bias on the model toward detecting tiny objects. In
contrast, this negative effect does not occur when we fine-tune the model on
the hybrid set of Arch-Mann-FT2 and Arch-Syn-FT (or Arch-Syn-FT-B). We believe
that this is because our synthetic dataset covers an extended range of UAV
positions over the grid so as to cause less bias on the fine-tuning process.
Based on the results shown in Fig. 13, to further improve the model’s
detection accuracy, we can focus on improving the model’s performance at high
altitudes and large circle radii, or the performance of non-standing
positions, such as kneel and prone. Notably, we find that the performance
improvement of kneel is surprisingly small to high altitudes and large circle
radii, which requires a deeper investigation in future work.
Figure 13: AP50 comparison on the altitude/radius grid of YOLOv5n6 fine-tuned
on the different hybrid sets of Arch-Mann-FT2, Arch-Syn-FT and Arch-Syn-FT-B.
## VII Discussion and Future Direction
With the series of experiments presented in Sec. V and VI, we have
demonstrated the distinctive value of the Archangel dataset. Particularly, we
have clearly illustrated how to utilize the dataset’s metadata to evaluate and
diagnose the UAV-based object detectors on the altitude/radius grid. Moreover,
we have systematically analyzed how to involve both real and synthetic data
within the UAV-based fine-tuning process. Such fundamental studies of UAV-
based perception had not been achieved until we curated Archangel.
Despite having all these merits, Archangel is still in its early stage and has
much room for improvement and development. In the following, we suggest a few
possible future directions regarding Archangel:
Direct Extension of this Study. Many results shown in Sec. V and VI imply that
increasing the data diversity of Archangel is one of the most promising future
research directions. For instance, since we have demonstrated that fine-tuning
models with virtual characters in unusual poses is particularly effective, we
can include more atypical poses into Archangel to further explore this
phenomenon, which is crucial especially in search and rescue scenarios where
finding people in severe physiological states is the priority. Additionally,
we can diversify the appearances/attributes of either the real or synthetic
human instances in the dataset, investigating whether this will resolve the
issue of overfitting as we have discussed earlier. For the same purpose, we
can increase the diversity of components beyond the foreground objects, such
as the real and synthetic backgrounds included in Archangel. Finally, we can
extend Archangel to include more object categories, such as various types of
vehicles, which frequently exist in other UAV-based object detection datasets
(Tab. I) so that Archangel can be used in conjunction with those datasets.
Next, exploring more sophisticated fine-tuning strategies is another potential
direction to extend this work. In this study, we have demonstrated that the
performance of the pre-trained SoTA object detector can be boosted
considerably by fine-tuning the model on a balanced UAV-based fine-tuning
dataset constructed by directly merging a real subset and a synthetic subset.
Nevertheless, it is worth exploring if there is a better strategy for sampling
each subset or merging the two subsets. For instance, to build up a balanced
fine-tuning dataset, instead of randomly selected samples from the synthetic
fine-tuning dataset, we may do the sampling based on certain distance
measurements.
UAV-based Visual Representation Learning with Metadata. In this study, we used
the position and pose metadata provided by Archangel only for accurate model
evaluation and diagnosis. However, we believe that there is a huge potential
to utilize such metadata during training for better visual representation
learning. A notable example for this is NDFT [40], where the authors exploited
adversarial training with the coarse metadata labeled by themselves to enhance
the robustness of the learned features for UAV-based object detection. We
expect that such a framework will benefit greatly from the extensive metadata
provided by Archangel. Beyond UAV-based perception, in medical [53] and
underwater [54] imaging, it has also been shown that dataset metadata is
useful for learning visual representation with self-supervised learning or
contrastive learning.
UAV-based Synthetic Data Generation and Augmentation. We have found that fine-
tuning larger models with the synthetic data that we generated often causes
the issue of overfitting. This issue might be mitigated by directly
synthesizing more diverse images. However, as we have discussed, there is
usually a huge domain gap between the synthetic data and real data in terms of
object appearances/attributes and scene structures, which may not be solved by
simply increasing the number of synthetic images. Hence, addressing this
domain gap issue within the scope of UAV-based perception is another important
future research direction for Archangel. Possible solutions include: (1)
jointly training an object detector with a generative model which transforms
synthetic images to be more visually realistic [55, 56], and (2) formulating
the process of synthetic data generation as a learning problem to synthesize
scene structures better matching real-world scene distributions [57].
## VIII Conclusion
In this paper, we introduce a unique UAV-based object detection dataset,
Archangel, to encourage the community to continue developing more effective
UAV-based object detection approaches with dataset metadata and synthetic
data. A comprehensive study is carefully designed to show how to utilize
Archangel to fully optimize a state-of-the-art object detector with a hybrid
fine-tuning dataset comprising both real and synthetic data. Additionally, we
also demonstrate the huge benefit of leveraging the dataset metadata during
model evaluation by comparing the performance of the model across the
different object poses and UAV positions. As we have discussed, although there
is still much room for improvement, we hope that Archangel is useful for the
broader machine learning community and can lead to future advances in the area
of UAV-based perception.
## Acknowledgment
This research was sponsored by the Defense Threat Reduction Agency (DTRA) and
the DEVCOM Army Research Laboratory (ARL) under Grant No. W911NF2120076. The
authors would like to thank the US Army Artificial Innovation Institute (A2I2)
for supporting the annotation of Archangel.
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| Yi-Ting Shen received the B.S. degree in Electrical Engineering from
National Taiwan University in 2016 and the M.S. degree in Electronics
Engineering from National Taiwan University in 2019. He joined the Department
of Electrical and Computer Engineering at the University of Maryland, College
Park as a Ph.D. student in 2020. His research interests include computer
vision, machine learning and hardware/software co-design.
---|---
| Yaesop Lee received the bachelor’s degree in Electrical Engineering from
Sogang University, South Korea, and master’s degree in ENTS from University of
Maryland, College Park. She is currently pursuing Ph.D. degree in the
Department of Electrical and Computer Engineering at the University of
Maryland, College Park. Her research interests include Embedded Computer
Vision and Real-time Image/Signal Processing.
---|---
| Heesung Kwon is Senior Researcher and Team Lead at the DEVCOM Army Research
Laboratory (ARL). He received the B.Sc. degree in Electronic Engineering from
Sogang University, Seoul, Korea, in 1984, and the MS and Ph.D. degrees in
Electrical Engineering from the State University of New York at Buffalo in
1995 and 1999, respectively. Dr. Kwon served as one of the government leads of
an ARL collaborative research program–the Internet of Battlefield Things
(IoBT). Dr. Kwon also served as Associate Editor of IEEE Trans. on Aerospace
and Electronic Systems. He has published over 150 journal papers, book
chapters, and conference papers on various topics. Dr. Kwon is a co-recipient
of the best paper award at the Army Science Conference in 2004 and the best
paper runner-up award at the IEEE International Conference on Biometrics:
Theory, Applications, and Systems (BTAS 2016). He has been on Technical
Program Committee for various conferences and workshops relevant to
image/video analytics and machine learning.
---|---
| Damon M. Conover is an electtrical engineer in the Intelligent Perception
Branch at the DEVCOM Army Research Laboratory (ARL). His research focuses on
3D geospatial data processing and visualization, simulation and synthetic data
generation, and robotic systems. He is particularly interested in the
intersection of geospatial information and robotics for high-level planning.
Dr. Conover earned his Ph.D. from George Washington University in 2015.
---|---
| Shuvra S. Bhattacharyya is a Professor in the Department of Electrical and
Computer Engineering at the University of Maryland, College Park. He holds a
joint appointment in the University of Maryland Institute for Advanced
Computer Studies (UMIACS), and is affiliated with the Maryland Crime Research
and Innovation Center (MCRIC). He also holds a part-time visiting position as
Chair of Excellence in Design Methodologies and Tools at Institut National Des
Sciences Appliquées (INSA) in Rennes, France. He received the Ph.D. degree
from the University of California at Berkeley. He has held industrial
positions as a Researcher at the Hitachi America Semiconductor Research
Laboratory, and Compiler Developer at Kuck & Associates. From 2015 through
2018, he was a part-time visiting professor in the Department of Pervasive
Computing at the Tampere University of Technology (now Tampere University),
Finland, as part of the Finland Distinguished Professor Programme. He has also
held visiting research positions with the U.S. Air Force Research Laboratory
(AFRL). He is a Fellow of the IEEE.
---|---
| Nikolas Vale is a mechanical engineer at the DEVCOM Army Research
Laboratory (ARL) under the Autonomous Sensing & Integration Branch. His
focuses are on designing, constructing, programming and operating unmanned
aerial vehicle (UAV) research platforms. He is particularly interested in
developing modular, reconfigurable hardware and software systems for the
purposes of data collection, experimentation and prototyping.
---|---
| Joshua D. Gray is a mechanical engineer at the DEVCOM Army Research
Laboratory (ARL) under the Autonomous Sensing & Integration Branch. His
focuses are on designing, constructing, and operating unmanned aerial vehicle
(UAV) research platforms. He is particularly interested in developing small,
multi-rotor UAVs for experimentation and data collection.
---|---
| G. Jeremy Leong serves as a Program Manager within DTRA’s Counter-WMD
Advanced Research Division. Prior to joining DTRA, Jeremy spent 13 years in
energy research; most recently 6 years at the U.S. Department of Energy where
he oversaw several programs in fuels and chemicals, advanced materials, as
well as modeling & simulation/high performance computing for multiple
directorates. In the past 8 years, he managed a total Federal applied R&D
budget of over $180M with numerous recognitions for successful technology
transitions and deployments. Dr. Leong earned his Ph.D. in Applied Chemistry
(Catalyst Science and Engineering) at the Colorado School of Mines. Since
2009, he has authored more than 30 peer reviewed manuscripts, internal
assessments, and congressional reports. He has also served on numerous
technical panels and presented more than two dozen conference talks on
modeling & simulation, software development, as well as the chemistry of
materials and their practical applications.
---|---
| Kenneth Evensen served twenty-seven years in various levels of command in
the United States Army. Notably he served as the Program Manager for
developing the next generation of tactical networking for the Department of
Defense within the Joint Tactical Radio System Program Executive Office. He
retired as an active member of the Army Acquisition Corps and upon retirement
he named as the Director of the United States Army International Technology
Center - Pacific from late 2012 thru early 2018 and was responsible for
technology search and coordinating cooperation opportunities in science and
technology throughout the Pacific Region on behalf of the US Army. He
currently serves as the Division Chief for the Advanced Research Division
within the Counter WMD Technologies Department, Research & Development
Directorate, Defense Threat Reduction Agency.
---|---
| Frank Skirlo presently supports DTRA’s Advanced Research Division, Counter-
WMD Department, Research & Development Directorate as a Technical Program
Analyst. He served as a U.S. Army Signal Officer for over 24 years of active
service, and retired at the rank of Colonel in December 2011, after completing
a variety of worldwide assignments, including command and staff positions in
Germany, Korea, and Hawaii. His last assignment was with the Joint Staff J-6,
overseeing C4 transport programs and requirements. After retiring from the
military, Dr. Skirlo supported the U.S. Army C5ISR Center’s Night Vision and
Electronics Sensors (now Research and Technology Integration) Directorate’s
Integrated Sensor Architecture (ISA) project. Dr. Skirlo earned his Ph.D. in
Electrical Engineering from George Mason University, with his thesis focusing
on image processing and UAS-based aided target detection. He also has a Master
of Strategic Studies from the U.S. Army War College, Master of Science degree
in Electrical Engineering from the University of Colorado at Colorado Springs,
and a Bachelor of Science degree from the University of Florida.
---|---
|
# Lipschitz Interpolation: Non-parametric Convergence under Bounded Stochastic
Noise
Julien Walden Huang<EMAIL_ADDRESS>
University of Oxford Stephen Roberts<EMAIL_ADDRESS>
University of Oxford Jan-Peter Calliess<EMAIL_ADDRESS>
University of Oxford
###### Abstract
This paper examines the asymptotic convergence properties of Lipschitz
interpolation methods within the context of bounded stochastic noise. In the
first part of the paper, we establish probabilistic consistency guarantees of
the classical approach in a general setting and derive upper bounds on the
uniform convergence rates. These bounds align with well-established optimal
rates of non-parametric regression obtained in related settings and provide
new precise upper bounds on the non-parametric regression problem under
bounded noise assumptions. Practically, they can serve as a theoretical tool
for comparing Lipschitz interpolation to alternative non-parametric regression
methods, providing a condition on the behaviour of the noise at the boundary
of its support which indicates when Lipschitz interpolation should be expected
to asymptotically outperform or underperform other approaches. In the second
part, we expand upon these results to include asymptotic guarantees for online
learning of dynamics in discrete-time stochastic systems and illustrate their
utility in deriving closed-loop stability guarantees of a simple controller.
We also explore applications where the main assumption of prior knowledge of
the Lipschitz constant is removed by adopting the LACKI framework (Calliess et
al. (2020)) and deriving general asymptotic consistency.
Keywords: Non-parametrics, System Identification, Lipschitz Interpolation,
Non-linear systems, Convergence Rates, Asymptotics
## 1 Introduction
Non-parametric regression methods are a flexible class of machine learning
algorithms that often enjoy strong estimation success even when little is
known about the underlying target function or data-generating distribution.
Generally, theoretical guarantees on their convergence have been well-
researched with classical optimal convergence rates obtained in seminal work
by Stone (1982) in the case where the noise is assumed to be essentially
Gaussian; although various extensions relax this assumption, and a variety of
general consistency results can be derived (Györfi et al. (2002)). In this
paper, we will extend the literature on the non-parametric estimation problem
by considering the setting of bounded stochastic noise and a non-parametric
regression framework known as Lipschitz interpolation111Also referred to as
non-linear set membership (Milanese and Novara (2004)) or kinky inference
(Calliess et al. (2020)) (Beliakov (2006). This theoretical context is of
particular interest to the field of control where Lipschitz interpolation has
been used in various predictive control applications and bounded noise
assumptions are commonly made.
More generally, the popularity of data-driven adaptive control frameworks has
increased significantly in both industry and academia over the past two
decades. These frameworks assume that the underlying system dynamics are
partially or completely unknown and need to be identified through
computational machine learning-based approaches. Traditionally, linear
parametric regression methods have been extensively studied and utilized for
this purpose (Ljung (2010)). However, with recent advancements, the research
focus has expanded to encompass non-linear approaches, reflecting the growing
interest in more expressive modelling techniques (Schoukens and Ljung (2019)).
Non-linear non-parametric regression methods such as the Nadaraya-Watson
estimator (Nadaraya (1964)), Polynomial Spline Interpolation (Stone (1994)),
general Lipschitz Interpolation or Gaussian process regression (Williams and
Rasmussen (2006)) which offer flexible predictors capable of modelling complex
dynamics with minimal hyperparameter tuning have emerged as natural tools in
this context. In particular, Gaussian process regression and Lipschitz
interpolation frameworks have been increasingly studied due to their inherent
ability to provide uncertainty quantification and worst-case error guarantees
which enable the design of robust controllers that ensure system safety and
meet desired performance criterias (e.g. see Hewing et al. (2019), Canale et
al. (2007)).
In contrast to the probabilistic nature of the uncertainty characterisation
provided by Gaussian processes, Lipschitz interpolation-based techniques offer
deterministic guarantees in the form of feasible systems sets (Milanese and
Novara (2004)) and worst-case error bounds (Calliess et al. (2020)). This has
been particularly useful in the context of safety-critical model predictive
control (MPC) where a number of associated non-linear controllers have been
designed and are being researched (see Canale et al. (2014), Manzano et al.
(2020), Manzano et al. (2021) for a selection). At the same time, this
popularity has led to several recent extensions of the original Lipschitz
interpolation framework: (Calliess et al. (2020)) relaxes the assumption of
prior knowledge of the Lipschitz constant in favour of a fully data-driven
approach by incorporating a Lipschitz constant estimation procedure,
(Maddalena and Jones (2020)) proposes an equivalent smooth formulation which
is more suited for controllers that rely on gradient computations, (Blaas et
al. (2019)) extends the framework by incorporating localised Lipschitz
constants and (Manzano et al. (2022)) proposes a computationally more
efficient approach that retains key properties of the original Lipschitz
interpolation framework.
Given the growing use of Lipschitz interpolation frameworks in control,
obtaining a strong theoretical understanding of this method is essential.
While several finite sample guarantees and worst-case error bounds already
exist (see in particular Milanese and Novara (2004), Calliess et al. (2020)),
few consistency results have been derived, to the best of our knowledge, none
under stochastic noise. By contrast, numerous asymptotic guarantees and
convergence rates have been obtained for other popular non-parametric methods.
In particular, for alternative safe-learning frameworks based on Gaussian
processes, both the pointwise convergence of the posterior mean function
(Seeger et al. (2008), Yang et al. (2017), Wynne et al. (2021)) and the
contraction rate of the posterior distribution, which provide a measure of
uncertainty quantification (van der Vaart and van Zanten (2008), Van Der Vaart
and Van Zanten (2011)), of the fitted Gaussian processes have been derived.
These types of asymptotic properties are crucial for adaptive control
applications as they guarantee that the learned dynamics and error bounds
accurately converge to the true underlying system dynamics while also
providing a characterisation of the long-run performance of the regression
method. This in turn ensures that the controllers built on these data-driven
frameworks become increasingly more successful the longer the interaction with
the underlying plant progresses. Considering the computational advantages of
Lipschitz interpolation over Gaussian process regression (Calliess et al.
(2020)), deriving analogous asymptotic guarantees for Lipschitz interpolation
is therefore strongly desirable and constitutes the main motivation of this
paper. Specifically, the following contributions to the literature are made:
* •
(Main Result) In the case of independent input sampling, general consistency
and upper bounds on the asymptotic convergence rates are obtained for both the
prediction function (Theorem 7) and the worst-case error bounds (Corollary 8)
of the general Lipschitz constant interpolation framework. While convergence
lower bounds do not exist for the exact setting considered in this paper
signifying that the optimality of our bounds is not (yet) established, the
obtained rates are consistent with the optimal convergence rates for non-
parametric regression in related settings; e.g. with the classical convergence
rate results derived by (Stone (1982)).
* •
In the case of discrete-time non-linear and noisy dynamical systems, we show
that the Lipschitz interpolation framework and worst-case bounds converge
point-wise in moments (Corollary 9 and ensuing discussion) and that, under an
additional sampling assumption, the convergence rates match the ones derived
in the first part of the paper. The first result can be directly applied in
the context of the existing non-linear controllers discussed above (e.g.
Canale et al. (2014), Manzano et al. (2020)) and we provide a simple
illustration in the context of online learning-based trajectory tracking
control (see Section 6).
* •
In a general sampling setting, probabilistic consistency is shown (Theorem 15)
for the fully data-driven LACKI (Lazily Adapted Constant Kinky Inference)
estimator (Calliess et al. (2020)) that extends the general Lipschitz
interpolation framework by removing the key assumption of prior knowledge of
the Lipschitz constant. This result generalises on Theorem 16 of (Calliess et
al. (2020)) which derives the consistency of the LACKI estimator in the noise-
free setting.
We note that with the goal of obtaining a precise characterisation of the
convergence rates of Lipschitz interpolation methods, we make a non-standard
noise assumption (Assumption 2) utilising the concept of ”non-regular” noise
(Ibragimov and Has’ Minskii (2013)) which describes the behaviour of the tails
of the noise distribution in proximity of assumed noise bounds. This type of
assumption has been used in recent research on non-parametric boundary
regression (see Hall and Van Keilegom (2009), Jirak et al. (2014) and ensuing
works) and allows for a better comparison between the convergence rates of
Lipschitz interpolation derived in this paper and the ones known for Gaussian
process regression and other kernel methods. In fact, the convergence rate
bounds obtained in this paper provide an explicit condition on the tail
behaviour of the noise that indicates when the Lipschitz interpolation should
be expected to asymptotically outperform or underperform other non-parametric
regression paradigms.
## 2 Lipschitz Interpolation: Set-up & Assumptions
Given an input space $\mathcal{X}\subset\mathbb{R}^{d}$ endowed with a metric
$\,\mathfrak{d}:\mathcal{X}^{2}\to\mathbb{R}_{\geq 0}$ and an output
space222Here, it would be possible to extend the analysis done in this paper
to a vector-valued output space, i.e. $\mathcal{Y}\subset\mathbb{R}^{m}$ for
$m\in\mathbb{N}$, by applying the obtained results in a component-wise
fashion. $\mathcal{Y}\subset\mathbb{R}$ endowed with a metric
$\,\mathfrak{d}_{\mathcal{Y}}:\mathcal{Y}^{2}\to\mathbb{R}_{\geq 0}$, the goal
of non-parametric regression is to learn an unknown target function
$f:\mathcal{X}\to\mathcal{Y}$. In this paper, we will assume that $f$ belongs
to the class of $L$-Lipschitz (continuous) functions (with respect to
$\,\mathfrak{d}_{\mathcal{X}}$, $\,\mathfrak{d}_{\mathcal{Y}}$ and some
$L\in\mathbb{R}_{+}$) which we formally define as:
$Lip(L,\,\mathfrak{d}):=\\{h:\mathcal{X}\to\mathcal{Y}|\,\mathfrak{d}_{\mathcal{Y}}(h(x),h(x^{\prime}))\leq
L\,\mathfrak{d}(x,x^{\prime}),\forall x,x^{\prime}\in\mathcal{X}\\}.$
The smallest non-negative number $L^{*}$ for which $f$ is $L^{*}$-Lipschitz is
called the _best_ Lipschitz constant of $f$, i.e.
$L^{*}=\min\\{L\in\mathbb{R}_{\geq 0}|f\in Lip(L,\,\mathfrak{d})\\}$. The
functional assumption that the target function $f$ is $L$-Lipschitz continuous
for some $L\in\mathbb{R}_{+}$ is essential for the application of Lipschitz
interpolation frameworks and is standard in the relevant literature (Milanese
and Novara (2004), Beliakov (2006), Calliess et al. (2020)).
In order to learn $f$, we assume that a sequence of sample sets
$(\mathcal{D}_{n})_{n\in\mathbb{N}}:=(G^{\mathcal{X}}_{n},G^{\mathcal{Y}}_{n})_{n\in\mathbb{N}}$
defined such that $\mathcal{D}_{n}\subset\mathcal{D}_{n+1}$ for
$n\in\mathbb{N}$ is available, where
$G_{n}^{\mathcal{X}}:=\\{s_{i}|i=1,...,N_{n}\\}\subset\mathcal{X}$ represents
a set of sample inputs that can be either deterministically or randomly
generated and
$G^{\mathcal{Y}}_{n}:=\\{\tilde{f}_{i}|i=1,...,N_{n}\\}\subset\mathcal{Y}$
denotes the set of noise-corrupted values of the target function $f$
associated with the inputs in $G_{n}^{\mathcal{X}}$. Unless stated otherwise,
we will also assume that elements of $G^{\mathcal{Y}}_{n}$ are of the form
$\tilde{f}_{k}=f(s_{k})+e_{k}$ where $(e_{k})_{k\in\mathbb{N}}$ is a
collection of random variables denoting the additive observational noise.
In this paper, we will make the following assumption on the noise:
###### Assumption 1
(General noise assumptions) The noise variables $(e_{k})_{k\in\mathbb{N}}$ are
assumed to be independent and identically333The identically distributed
assumption is made to alleviate notation and is not technically needed in our
derivations. distributed random variables with compact support:
$\exists\bar{\mathfrak{e}}>0$ such that $\forall k\in\mathbb{N}:$
$\mathbb{P}\left(e_{k}\in[-\bar{\mathfrak{e}},\bar{\mathfrak{e}}]\right)=1$.
Furthermore, we assume that the bounds of the support are tight in the
following sense:
There exists $\bar{\epsilon}>0$, $\forall
k\in\mathbb{N},\epsilon\in(0,\bar{\epsilon})$:
$\mathbb{P}(e_{k}>\bar{\mathfrak{e}}-\epsilon)>0\quad\text{ and
}\quad\mathbb{P}(e_{k}<-\bar{\mathfrak{e}}+\epsilon)>0$
In order to derive precise upper bounds on the convergence rates, we will
sometimes make an additional noise assumption which describes the behaviour of
the noise at the boundary of its support. This assumption is given formally as
follows:
###### Assumption 2
(Assumptions on the boundary behaviour of the noise) Assume that Assumption 2
holds. We assume that the behaviour of the noise near the bounds of the
support can be characterised in the following sense: There exists
$\bar{\epsilon},\gamma,\eta>0,\forall
k\in\mathbb{N},\epsilon\in(0,\bar{\epsilon})$:
$\mathbb{P}(e_{k}>\bar{\mathfrak{e}}-\epsilon)>\gamma\epsilon^{\eta}\quad\text{
and
}\quad\mathbb{P}(e_{k}<-\bar{\mathfrak{e}}+\epsilon)>\gamma\epsilon^{\eta}.$
(a) $\eta=0.5$
(b) $\eta=1$
(c) $\eta=2$
(d) $\eta=3$
Figure 1: Illustration of Assumption 2 for various $\eta$. The target
function is given by $f(x)=-\sin(3x)x^{2}+5$ and the noise distribution is
defined as a mixture of two truncated Weibull distributions. The solid lines
define the error bounds of the observed data, i.e. (with abuse of notation)
$f\pm\bar{\mathfrak{e}}$.
###### Example 1
(Noise distributions) For $\eta=1$, commonly used noise distributions with
bounded support such as the uniform or the truncated Gaussian distributions
satisfy Assumption 2. More generally, any noise distribution for which the
density can be bounded away from zero on a bounded symmetric support satisfies
the assumption with $\eta=1$.
The assumption of boundedness of the support of the noise distribution given
in Assumption 1 is standard in the Lipschitz interpolation literature (e.g.
see Milanese and Novara (2004), Calliess et al. (2020)) as it ensures that the
functions $\mathfrak{u}_{n},\mathfrak{l}_{n}$ defined in Definition 1 are
generally well-behaved. By contrast, as noted in the introduction of this
paper, the assumption on the tail of the noise distribution stated in
Assumption 2 is non-standard444But as shown in Example 1, many standard noise
distributions arise as special cases. in the literature. While this assumption
will be not needed to ensure the asymptotic consistency of Lipschitz
interpolation frameworks, the precise characterisation of the bounded tail of
the noise distribution as a function of $\gamma$ and $\eta$ given in
Assumption 2 makes it possible to derive a more refined convergence rate
result that depends on $\eta$.
The general Lipschitz interpolation framework considered in this paper is
defined as follows:
###### Definition 1
(Lipschitz interpolation) Using the set-up defined above, we define the
sequence of predictors $(\hat{f}_{n})_{n\in\mathbb{N}}$,
$\hat{f}_{n}:\mathcal{X}\to\mathcal{Y}$ associated to
$(\mathcal{D}_{n})_{n\in\mathbb{N}}$, as
$\,\mathfrak{\hat{f}_{n}}\bigl{(}x):=\frac{1}{2}\mathfrak{u}_{n}(x)+\frac{1}{2}\mathfrak{l}_{n}(x),$
(1)
where $\mathfrak{u}_{n},\mathfrak{l}_{n}:\mathcal{X}\to\mathcal{Y}$ are
defined as
$\displaystyle\mathfrak{u}_{n}(x)=\min_{i=1,...,N_{n}}\tilde{f}_{i}+L\,\mathfrak{d}(x,s_{i})$
$\displaystyle\mathfrak{l}_{n}(x)=\max_{i=1,...,N_{n}}\tilde{f}_{i}-L\,\mathfrak{d}(x,s_{i})$
and $L\in\mathbb{R}_{\geq 0}$ is a selected hyper-parameter.
Ideally, the hyper-parameter $L\in\mathbb{R}_{\geq 0}$ can be set to be larger
than the best Lipschitz constant $L^{*}$ of the unknown target function. In
this case, existing finite sample and worst-case guarantees can be utilised.
###### Remark 2
(Alternative Formulation) In some works (see in particular Milanese and Novara
(2004), Calliess et al. (2020)) an alternative formulation is given for the
Lispchitz interpolation predictors. In this case, the bounds
($\bar{\mathfrak{e}}$) on the noise distribution are assumed known and are
explicitly used in the formulation of
$\tilde{}\mathfrak{u}_{n},\tilde{}\mathfrak{l}_{n}:\mathcal{X}\to\mathcal{Y}$:
$\displaystyle\tilde{}\mathfrak{u}_{n}(x)=\min_{i=1,...,N_{n}}\tilde{f}_{i}+L\,\mathfrak{d}(x,s_{i})+\bar{\mathfrak{e}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle\tilde{}\mathfrak{l}_{n}(x)=\max_{i=1,...,N_{n}}\tilde{f}_{i}-L\,\mathfrak{d}(x,s_{i})-\bar{\mathfrak{e}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
This formulation is useful for computing tight worst-case upper and lower
bound guarantees in practice and can be used in the context of this paper to
weaken Assumption 1 by considering asymmetric noise bounds, i.e.
$e\in[\bar{\mathfrak{e}}_{1},\bar{\mathfrak{e}}_{2}]$ with probability $1$
where $\bar{\mathfrak{e}}_{1}<0<\bar{\mathfrak{e}}_{2}\in\mathbb{R}$. All
results derived in this paper can be shown to hold for the alternative
Lipschitz interpolation formulation.
Figure 2: Illustration of of the consistency of Lipschitz interpolation for
the target function: $f(x)=\sqrt{x}\sin(2x^{2})+0.5x$ on the input space
$\mathcal{X}=[0,2]$, with uniform sampling on $\mathcal{X}$ and with
independent uniform noise: $U([-0.5,0.5])$ on the observations ($\eta=1$). The
Lipschitz interpolation plotted in the lefthand figure utilised 100 samples
and assumed that a bound of ($\bar{\mathfrak{e}}^{\prime}=0.7$) on the noise
bound ($\bar{\mathfrak{e}}=0.5$) was known in order to compute the lower and
upper bounds (the LI predictors can be constructed without knowledge of
$\bar{\mathfrak{e}}^{\prime}$). The convergence rate and standard deviation
plotted on the righthand figure were obtained by running the experiment
independently 20 times. Both plots assumed that access to a bound on the best
Lipschitz constant was known in order to apply the Lipschitz interpolation
framework.
As the description of the input and output metrics has been general so far, we
make the following simplifying assumption on the output metric in order to
obtain our theoretical results.
###### Assumption 3
(Assumption on $\,\mathfrak{d}_{\mathcal{Y}}$). In this paper we will restrict
ourselves to the case,
$\,\mathfrak{d}_{\mathcal{Y}}(y,y^{\prime})=\left\|y-y^{\prime}\right\|_{\mathcal{Y}}$,
$\forall y,y^{\prime}\in\mathcal{Y}$ where $\left\|.\right\|_{\mathcal{Y}}$ is
a norm on $\mathcal{Y}$. It will therefore be sufficient to derive our
asymptotic results for the case: $\left\|.\right\|_{\mathcal{Y}}=|.|$ as
discussed below.
As the norms on $\mathcal{Y}\subset\mathbb{R}$ are of the form
$\left\|y-y^{\prime}\right\|=c|y-y^{\prime}|$ $\forall
y,y^{\prime}\in\mathcal{Y}$ for some $c>0$, it is sufficient to consider the
case $\left\|y-y^{\prime}\right\|_{\mathcal{Y}}=|y-y^{\prime}|$, $\forall
y,y^{\prime}\in\mathcal{Y}$ in order to achieve our theoretical results.
Assumption 3 is necessary in order to ensure that for arbitrary
$x,x^{\prime}\in\mathcal{X}$, the relations: $f(x)\leq
f(x^{\prime})+\frac{L}{c}\,\mathfrak{d}(x,x^{\prime})$ and
$f(x)-\frac{L}{c}\,\mathfrak{d}(x,x^{\prime})\leq f(x^{\prime})$ hold. In
particular, for any sub-linear metric $\,\mathfrak{d}_{\mathcal{Y}}$, these
inequalities no longer hold. We note however that no restrictions are made on
the input metric $\,\mathfrak{d}$.
## 3 Asymptotic Consistency and Convergence Rates
In order for our consistency results to hold for both random and deterministic
sampling approaches, we recall Definition 8: ”Becoming dense, rates,
$\stackrel{{\scriptstyle r}}{{\longrightarrow}},\stackrel{{\scriptstyle
r}}{{\leadsto}},\stackrel{{\scriptstyle r}}{{\twoheadrightarrow}}$” of
(Calliess et al. (2020)) to define general sampling conditions for
$(G_{n})_{n\in\mathbb{N}}$.
###### Definition 3
(Uniformly dense sampling) We say that the sequence of sets of sample inputs
$(G_{n})_{n\in\mathbb{N}}$ becomes uniformly dense relative to $\mathcal{X}$
at a rate $r$ (denoted by $(G_{n})\stackrel{{\scriptstyle
r}}{{\twoheadrightarrow}}\mathcal{X}$) if $\exists
r:\mathbb{N}\to\mathbb{R}_{+}$ such that $\lim_{n\to\infty}r(n)=0$ and
$\forall n\in\mathbb{N}$, $\sup_{x\in\mathcal{X}}\inf_{s_{n}\in
G_{n}}\,\mathfrak{d}(s_{n},x)\leq r(n)$.
Using this definition, we can provide the following asymptotic guarantee for
the general Lipschitz interpolation method.
###### Theorem 4
Suppose Assumptions 1 and 3 hold, $\mathcal{X}$ is bounded and the target
function $f\in Lip(L^{*},\,\mathfrak{d})$555unless specified otherwise,
Lipschitz continuity will be assumed to be w.r.t. the metrics
$\,\mathfrak{d}$,$\,\mathfrak{d}_{\mathcal{Y}}$ on the spaces
$\mathcal{X},\mathcal{Y}$. with best Lipschitz constant
$L^{*}\in\mathbb{R}_{+}$. If the sampling set sequence
$(\mathcal{D}_{n})_{n\in\mathbb{N}}$ has sample inputs
$(G_{n}^{\mathcal{X}})_{n\in\mathbb{N}}$ such that $\exists r\in
o(1):(G_{n}^{\mathcal{X}})\stackrel{{\scriptstyle
r}}{{\twoheadrightarrow}}\mathcal{X}$ and the sequence of predictors
$(\hat{f}_{n})_{n\in\mathbb{N}}$ are computed by a general Lipschitz
interpolation framework with a hyperparameter $L\in\mathbb{R}_{\geq 0}$ set
such that $L\geq L^{*}$ then we have:
$\forall\epsilon>0\text{,
}\lim_{n\to\infty}\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))>\epsilon\right)=0.$
Before providing the proof of Theorem 4, we recall the notion of
$\epsilon$-covering that will be used in multiple proofs of this paper.
###### Definition 5
($\epsilon$-Cover) Let $d\in\mathbb{N}$, $\epsilon>0$ and consider a set
$\mathcal{X}\subset\mathbb{R}^{d}$ and a metric $\,\mathfrak{d}$ on
$\mathbb{R}^{d}$. Denoting $B_{\epsilon}(x)$ the ball of radius $\epsilon$
centred in $x\in\mathcal{X}$ with respect to $\,\mathfrak{d}$, we define an
$\epsilon$-cover of $\mathcal{X}$ as a subset
$Cov(\epsilon)\subset\mathbb{R}^{d}$ such that
$\mathcal{X}\subset\bigcup_{x\in Cov(\epsilon)}B_{\epsilon}(x)$ and the
associated set of balls as $\mathcal{B}:=\\{B_{\epsilon}(x)|x\in
Cov(\epsilon)\\}$. We say furthermore that $Cov(\epsilon)$ is a
$\epsilon$-minimal cover of $\mathcal{X}$ if
$|Cov(\epsilon)|=min\\{n:\exists\epsilon\text{-covering over $\mathcal{X}$of
size n}\\}.$
Proof We begin by establishing a general bound on
$\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))$, $\forall n\in\mathbb{N}$,
$\forall x\in\mathcal{X}$. For any $x\in\mathcal{X}$ we have:
$\displaystyle\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))=|\hat{f}_{n}(x)-f(x)|$
$\displaystyle=\left|\frac{1}{2}\min_{i=1,...,N_{n}}\\{\tilde{f}_{i}+L\,\mathfrak{d}(x,s_{i})\\}+\frac{1}{2}\max_{i=1,...,N_{n}}\\{\tilde{f}_{i}-L\,\mathfrak{d}(x,s_{i})\\}-f(x)\right|$
$\displaystyle=\left|\frac{1}{2}\min_{i=1,...,N_{n}}\\{\tilde{f}_{i}-f(x)+L\,\mathfrak{d}(x,s_{i})\\}+\frac{1}{2}\max_{i=1,...,N_{n}}\\{\tilde{f}_{i}-f(x)-L\,\mathfrak{d}(x,s_{i})\\}\right|.$
Using the Lipschitz continuity of $f$ , we obtain the following set of
inequality relations for the two terms stated above:
$\displaystyle(1)\min_{i=1,...,N_{n}}\\{e_{i}\\}\leq\min_{i=1,...,N_{n}}\left\\{\tilde{f}_{i}-f(x)+L\,\mathfrak{d}(x,s_{i})\right\\}\leq\min_{i=1,...,N_{n}}\\{e_{i}+(L^{*}+L)\,\mathfrak{d}(x,s_{i})\\}.$
$\displaystyle(2)\max_{i=1,...,N_{n}}\left\\{e_{i}-(L^{*}+L)\,\mathfrak{d}(x,s_{i})\right\\}\leq\max_{i=1,...,N_{n}}\left\\{\tilde{f}_{i}-f(x)-L\,\mathfrak{d}(x,s_{i})\right\\}\leq\max_{i=1,...,N_{n}}\\{e_{i}\\}.$
In combination, we see that
$\displaystyle|\hat{f}_{n}(x)-f(x)|$
$\displaystyle\leq\frac{1}{2}\max\Big{\\{}\underbrace{\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{i=1,...,N_{n}}\left\\{e_{i}+(L^{*}+L)\,\mathfrak{d}(x,s_{i})\right\\}}_{(I)},$
$\displaystyle\underbrace{-\min_{i=1,...,N_{n}}\\{e_{i}\\}-\max\limits_{i=1,...,N_{n}}\left\\{e_{i}-(L^{*}+L)\,\mathfrak{d}(x,s_{i})\right\\}}_{(II)}\Big{\\}}.$
$(I),(II)$ can then be bounded using the assumption of uniform convergence of
the grid (see Definition 3). Define $R:=\frac{\epsilon}{4(L^{*}+L)}$ and
consider the minimal covering of $\mathcal{X}$ of radius $R$ with respect to
$\,\mathfrak{d}$ that we denote $Cov(R)$ and the associated set of hyperballs
$\mathcal{B}$. By uniform convergence of the sample inputs, there exists
$M\in\mathbb{N}$ such that $\forall n>M$: $\forall B\in\mathcal{B}$, $|B\cap
G_{n}^{\mathcal{X}}|>0$. Then, the following upper bound holds for $(I)$ with
$n>M$ ($(II)$ can be bounded in a similar way):
$\displaystyle\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{i=1,...,N_{n}}\\{e_{i}+(L^{*}+L)\,\mathfrak{d}(x,s_{i})\\}$
$\displaystyle\leq$
$\displaystyle\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\\{e(s_{i})+(L^{*}+L)\,\mathfrak{d}(x,s_{i})\\}$
$\displaystyle\leq$
$\displaystyle\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\left\\{e(s_{i})+2(L^{*}+L)R\right\\}$
where with abuse of notation, $e(s_{i})$ denotes the noise term associated
with the input $s_{i}$ and $B^{x}$ denotes a hyperball $B\in\mathcal{B}$ such
that $x\in B$. Similarly for $(II)$, we obtain
$(II)\leq\max_{i=1,...,N_{n}}\\{-e_{i}\\}+\min_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\\{-e(s_{i})+2(L^{*}+L)R\\}.$
Let $\epsilon>0$. Utilising these bounds, $\forall n>M$, we obtain
$\displaystyle\mathbb{P}(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))>\epsilon)$
$\displaystyle\leq\mathbb{P}\Big{(}2(L^{*}+L)R+\frac{1}{2}\sup_{x\in\mathcal{X}}\max\Big{\\{}\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in
B^{x}\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\},$
$\displaystyle-\min_{i=1,...,N_{n}}\\{e_{i}\\}-\max_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\Big{\\}}>\epsilon\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\frac{1}{2}\max_{B\in\mathcal{B}}\max\Big{\\{}\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\},$
$\displaystyle-\min_{i=1,...,N_{n}}\\{e_{i}\\}-\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\Big{\\}}>\epsilon-2(L^{*}+L)R\Big{)}$
$\displaystyle\leq\mathbb{P}\left(\max_{B\in\mathcal{B}}\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}>\frac{\epsilon}{2}\right)$
$\displaystyle+\mathbb{P}\left(\max_{B\in\mathcal{B}}\max_{i=1,...,N_{n}}\\{-e_{i}\\}+\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{-e(s_{i})\\}\\}>\frac{\epsilon}{2}\right)$
where the last inequality follows by definition of $R$. Both probability terms
stated above can be shown to converge to $0$ as follows:
$\mathbb{P}\left(\max_{B\in\mathcal{B}}\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}>\frac{\epsilon}{2}\right)$
$=1-\mathbb{P}\left(\max_{B\in\mathcal{B}}\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\leq\frac{\epsilon}{2}\right)$
$=1-\mathbb{P}\left(\forall
B\in\mathcal{B},\max_{i=1,...,N_{n}}\\{e_{i}\\}+\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\leq\frac{\epsilon}{2}\right)$ $\leq
1-\mathbb{P}\left(\forall B\in\mathcal{B}:\max_{i=1,...,N_{n}}\\{e_{i}\\}\in
I_{1},\min_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\in
I_{2}\right).$
Here $I_{1}:=[\bar{\mathfrak{e}}-\frac{\epsilon}{4},\bar{\mathfrak{e}}]$ and
$I_{2}:=[-\bar{\mathfrak{e}},-\bar{\mathfrak{e}}+\frac{\epsilon}{4}]$.
Applying a similar argument to the one given in $(\star\star)$ in the proof of
Theorem 7, we have that the last term is upper bounded by
$1-\prod_{B\in\mathcal{B}}\mathbb{P}\left(\max_{i=1,...,N_{n}}\\{e_{i}\\}\in
I_{1},\min_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}\\{e(s_{i})\\}\in I_{2}\right)$
$\leq 1-\mathbb{P}\left(\max_{i=1,...,N_{n}}\\{e_{i}\\}\in
I_{1},\min_{i=1,...,L_{n}}\\{e_{i}\\}\in I_{2}\right)^{|\mathcal{B}|}$
where $L_{n}:=\min_{B\in\mathcal{B}}|B\cap G_{n}^{\mathcal{X}}|$ and $|.|$ is
used to denote the cardinality operator for finite sets. By the uniformity of
the convergence of the sample inputs, we have that
$\lim_{n\to\infty}L_{n}=\lim_{n\to\infty}N_{n}=+\infty$. Using basic
identities of probability theory and applying Assumption 1, we have that
$\lim_{n\to\infty}\mathbb{P}\left(\max_{i=1,...,N_{n}}\\{e_{i}\\}\in
I_{1},\min_{i=1,...,L_{n}}\\{e_{i}\\}\in I_{2}\right)=1$
which implies that
$\lim_{n\to\infty}\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))>\epsilon\right)=0$
and concludes the proof.
Theorem 4 ensures that the classical Lipschitz interpolation method is
asymptotically consistent for a general selection of input metrics.
Furthermore, a similar result for Lipschitz interpolation with a multi-
dimensional output setting $\mathcal{Y}\subset R^{m}$ for $m\in\mathbb{N}$
follows naturally by applying Theorem 4 to each output component function (the
noise assumption would need to be modified in this case; e.g. see Assumption
7).
In general, we are mostly interested in simple metric choices for
$\,\mathfrak{d}$. In this case with additional assumptions on $\mathcal{X}$
and $\mathcal{Y}$, we can extend the result obtained in Theorem 4 by deriving
asymptotic rates of convergence for the general Lipschitz interpolation
method. More precisely, we have the following definition (Györfi et al.
(2002)):
###### Definition 6
Consider a sequence of non-parametric predictors
$(\hat{f}_{n})_{n\in\mathbb{N}}$ and a class of functions $\mathcal{C}$
endowed with a norm $\left\|.\right\|$. Let $(a_{n})_{n\in\mathbb{N}}$ be a
sequence of positive constants in $\mathbb{R}$. We define
$(a_{n})_{n\in\mathbb{N}}$ as the rate of convergence of
$(\hat{f}_{n})_{n\in\mathbb{N}}$ on $\mathcal{C}$ with respect to
$\left\|.\right\|$ if there exists $c>0$ such that
$\limsup_{n\to\infty}\sup_{f\in\mathcal{C}}\mathbb{E}\left[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|\right]=c<\infty.$
In order to avoid extreme cases of compact spaces, the following general
assumption provides a light geometric assumption on $\mathcal{X}$.
###### Assumption 4
(Geometric Assumption on $\mathcal{X}$) Let $\mathcal{X}\subset\mathbb{R}^{d}$
be compact and convex. There exist two constants $r_{0}>0,\theta\in(0,1]$ such
that $\forall x\in\mathcal{X}$, $r\in\left(0,r_{0}\right):\newline
\operatorname{vol}\left(B_{r}(x)\cap\mathcal{X}\right)\geq\theta\operatorname{vol}\left(B_{r}(x)\right)$.
Assumption 4 has been used in the learning theory literature (e.g. see Hu et
al. (2020) Bachoc et al. (2021)) and ensures that for all $x\in\mathcal{X}$, a
constant fraction of ball with a sufficiently small radius and centred in $x$
is contained in $\mathcal{X}$. For example, if $\mathcal{X}$ is a the unit
hypercube then Assumption 4 holds with $r_{0}=1,\theta=2^{-d}$.
The additional assumptions on the sampling of the sample inputs
$(D_{n})_{n\in\mathbb{N}}$ and metric of the input space $\,\mathfrak{d}$ are
relatively standard and are given as follows:
###### Assumption 5
(Assumption on Sampling) $(G_{n}^{\mathcal{X}})_{n\in\mathbb{N}}$ is a
randomly sampled sequence on $\mathcal{X}$ with a sampling distribution
density that is bounded away from zero on $\mathcal{X}$.
###### Assumption 6
(Hölder Condition) We restrict the input space metrics under consideration to
be of the form $\,\mathfrak{d}(x,y)=\left\|x-y\right\|_{p}^{\alpha}$ where
$\alpha\in(0,1]$ and $\left\|.\right\|_{p}$ denotes the usual $p$-norm on
$\mathbb{R}^{d}$ with $p\in\mathbb{N}\cup\\{+\infty\\}$.
###### Theorem 7
Consider an input space $\mathcal{X}\subset\mathbb{R}^{d}$ that satisfies
Assumption 4, an output space $\mathcal{Y}\subset\mathbb{R}$ and the function
space $\mathcal{C}=Lip(L^{*},\,\mathfrak{d})$ with $L^{*}\in\mathbb{R}_{\geq
0}$ endowed with the supremum-norm:
$\left\|h\right\|_{\infty}=\sup_{x\in\mathcal{X}}\left\|h(x)\right\|_{\mathcal{Y}}$.
Assume that Assumptions 1, 2, 5, 6 with $\alpha\in(0,1]$, $p\in\mathbb{N}$
hold. Then, any sequence of predictors $(\hat{f}_{n})_{n\in\mathbb{N}}$
generated by the general Lipschitz interpolation framework set with a
hyperparameter $L\geq L^{*}$ achieves a rate of convergence of at least
$(a_{n})_{n\in\mathbb{N}}:=\left((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}}\right)_{n\in\mathbb{N}}$
with respect to $\left\|.\right\|_{\infty}$, i.e.
$\limsup_{n\to\infty}\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\mathbb{E}\left[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|_{\infty}\right]<\infty.$
Proof See appendix B.
Convergence lower bounds do not exist for the exact setting considered in this
paper signifying that we cannot directly compare the rates stated in Theorem 7
to a theoretically optimal convergence rate. Instead, we can note that the
convergence rate of Lipschitz interpolation is in line with several known
optimal rates in related settings (see Table 1), i.e. non-parametric
regression on the Lipschitz continuous function space endowed with an $L_{2}$
or $L_{\infty}$ norm. In particular, we note that the exponent of the
convergence rate derived for Lipschitz interpolation exactly matches the
exponent of the convergence rate derived in Tsybakov (2004) in the case where
the noise distribution is assumed to be uniform (i.e. $\eta=1$). Our
convergence rate is however larger by a log-factor due to a difference in
norm.
Furthermore, by varying $\eta$ in Assumption 2, we can compare our rate of
convergence:
O$\left((n^{-1}log(n)\right)^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$
to classical non-parametric convergence rates. More precisely, we observe the
following:
* •
For $\eta<2$: the derived convergence rates for Lipschitz interpolation are
better than the known optimal convergence rates obtained under a Gaussian tail
noise assumption on the noise distribution:
$(n^{-1}log(n))^{\frac{\alpha}{2\alpha+d}}$ (Stone (1982)) which are
attained666Note that these methods can be shown to converge at this rate under
the simple assumption of bounded variance (Györfi et al. (2002)). by Gaussian
process regression (Yang et al. (2017)) and other kernel-based non-parametric
methods such as local polynomial regression (Stone (1982)) or the Nadaraja-
Watson estimator (Tsybakov (2004), Müller and Wefelmeyer (2010)).
* •
For $\eta>2$: the opposite becomes true and these alternative non-parametric
methods can be expected to converge quicker asymptotically than the Lipschitz
interpolation framework.
Algorithm/Type | Convergence Rate | Noise Assumption | Norm
---|---|---|---
LI (Upper Bound) | O$\left(n^{-1}log(n)\right)^{\frac{\alpha}{d+\eta\alpha}}$ | Bounded | $L_{\infty}$.
Optimal (Tsybakov (2004)) | $\Theta\left(n^{-1}\right)^{\frac{\alpha}{d+\alpha}}$ | Uniform ($\eta=1$) | $L_{2}$
Optimal (Stone (1982)) | $\Theta\left(n^{-1}log(n)\right)^{\frac{\alpha}{d+2\alpha}}$ | Gaussian 444Various generalisations of this noise assumption exist, see Stone (1982). | $L_{\infty}$
Optimal (Stone (1982)) | $\Theta\left(n^{-1}\right)^{\frac{\alpha}{d+2\alpha}}$ | Gaussian44footnotemark: 4 | $L_{2}$
Optimal (Jirak et al. (2014)) | $\Theta\left(n^{-1}\right)^{\frac{\alpha}{1+\eta\alpha}}$ (d=1) | Boundary Regr. | $L_{2}$
Upper Bound (Selk et al. (2022)) | O$\left(n^{-1}log(n)\right)^{\frac{\alpha}{d+\eta\alpha}}$ | Boundary Regr. | $L_{\infty}$
Table 1: Comparison of the convergence rate derived in Theorem 7 with optimal
rates of convergence rates in similar settings and discussion given in this
section.
This ”$\eta$-condition” provides a theoretical tool for comparing the expected
long-run performance of Lipschitz interpolation relative to alternative non-
parametric methods and can help guide the choice of the system identification
approach if information on the non-regularity of the noise distribution is
obtainable. We note that the convergence rates of the kernel-based non-
parametric methods stated in Table 1 hold under general noise assumptions (see
footnotes 4 and 6 below) and that, aside from the Nadaraja-Watson estimator,
no formal derivation of improved convergence rates in the bounded noise
setting considered in this paper currently exists777To the extent of our
knowledge. for these methods. As these kernel-based non-parametric frameworks
generally rely on local averaging of the noise in order to prove convergence,
it is expected that their convergence rates do not improve with respect to
their classical convergence rates (stated in Table 1) under Assumption 1 and
Assumption 2. This has been formally shown to be true for the Nadaraja-Watson
estimator by Müller and Wefelmeyer (2010) and a more general discussion on the
topic can be found in Meister and Reiß (2013).
Figure 3: Illustration of the behaviour of the convergence rates derived in
Theorem 7 for various values of $(d,\alpha,\eta)$.
As discussed above, the convergence rates obtained in Theorem 7 under the
bounded noise assumptions are better than the classical optimal convergence
rates derived by Stone (1982). This is possible as the lower bounds of these
optimal convergence rates are generally derived under the condition that the
noise has a positive density with respect to the Lebesgue measure on
$\mathbb{R}$ which does not hold for the noise assumptions of this paper. As a
consequence, $O\left(n^{-1}log(n)\right)^{\frac{\alpha}{d+\eta\alpha}}$
provides a new general upper bound on the non-parametric regression problem in
the bounded noise setting and future work can be done on deriving lower bounds
to match these results. We expect the lower bounds to be tight given recent
results by Jirak et al. (2014) on the optimal convergence rates of the related
non-parametric boundary regression problem (see below for a more detailed
discussion).
The optimality results of Theorem 2 of Milanese and Novara (2004) show that
$(\tilde{}\mathfrak{u}_{n})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n})_{n\in\mathbb{N}}$ are exactly equal to the upper
and lower bounds of the feasible systems set, i.e. the set of all data-
consistent Lipschitz continuous systems and therefore provide worst-case error
prediction bounds. With little modification to the proof of Theorem 7, both
error bounds can be shown to converge to $f$ at the same rate as
$(\hat{f}_{n})_{n\in\mathbb{N}}$ as stated in the following Corollary.
###### Corollary 8
Assume that the setting and assumptions of Theorem 7 holds. The worst-case
prediction guarantees $(\tilde{}\mathfrak{u}_{n})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n})_{n\in\mathbb{N}}$ defined in Remark 2 with second
hyperparameter: $\bar{\mathfrak{e}}^{\prime}=\bar{\mathfrak{e}}$, converge
uniformly to any target function $f\in Lip(L^{*},\,\mathfrak{d})$ at a rate of
at least $((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$.
Proof Follows from the proof of Theorem 7.
A connection between our convergence results and succeeding work on non-
parametric boundary regression (see Hall and Van Keilegom (2009) and ensuing
works) can be made. More precisely, consider the predictive functions
$(\tilde{}\mathfrak{u}_{n}^{\prime})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n}^{\prime})_{n\in\mathbb{N}}$ defined for all
$n\in\mathbb{N}$ as
$\tilde{}\mathfrak{u}_{n}^{\prime}:\mathcal{X}\to\mathcal{Y}$,
$x\mapsto\mathfrak{u}_{n}(x)+\bar{\mathfrak{e}}_{1}+\bar{\mathfrak{e}}_{2}$
and $\tilde{}\mathfrak{l}_{n}^{\prime}:\mathcal{X}\to\mathcal{Y}$,
$x\mapsto\mathfrak{l}_{n}(x)-\bar{\mathfrak{e}}_{1}-\bar{\mathfrak{e}}_{2}$
where $\bar{\mathfrak{e}}_{1},\bar{\mathfrak{e}}_{2}$ are tight asymmetric
bounds on the noise. $(\tilde{}\mathfrak{u}_{n}^{\prime})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n}^{\prime})_{n\in\mathbb{N}}$ can be interpreted as
conservative non-parametric boundary regression methods. Therefore, in the
context of bounded noise, the two problems are equivalent and we can again
slightly modify the proof of Theorem 7 to obtain the same uniform asymptotic
convergence rates of $O(n^{-1}log(n))^{\frac{\alpha}{\eta d+\alpha}})$ as
$(\hat{f}_{n})_{n\in\mathbb{N}}$. These rates exactly match the recently
derived best convergence rates in the multivariate boundary regression problem
(Selk et al. (2022)) and have the same exponent888They differ by a log-factor
which is usual when considering the $L_{\infty}$ norm instead of the $L_{2}$
norm. as the optimal rates derived with respect to the $L_{2}$ norm (Jirak et
al. (2014)). In order to properly define
$(\tilde{}\mathfrak{u}_{n}^{\prime})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n}^{\prime})_{n\in\mathbb{N}}$, prior knowledge of an
upper bound on the Lipschitz constant ($L\geq L^{*}$) as well as the Hölder
exponent ($\alpha$) and of tight bounds of the noise
($\bar{\mathfrak{e}}_{1},\bar{\mathfrak{e}}_{2}$) are needed. However in
contrast to the proposed ”best” non-parametric estimators that attain the
optimal rates, we do not require prior knowledge of the degree of ”non-
regularity” of the noise ($\eta$, defined in Assumption 2) which is usually
required in order to define an optimal bandwidth hyperparameter (Drees et al.
(2019), Selk et al. (2022)). In the bounded noise setting, our assumption is
therefore arguably more natural and simpler to verify in practice as ($\eta$)
is generally hard to determine precisely.
## 4 Online Learning: Asymptotics
A set-up not yet explicitly considered in this paper but relevant to control
applications is when the output variables can be used as input variables. More
specifically, we consider the case where $f$ models the dynamics of a non-
linear autoregressive stochastic system with exogenous control variables:
$y_{n}=f(x_{n})+e_{n}$
where $x_{n}=(y_{n-d_{y}},...,y_{n-1},u_{n-d_{u}},...,u_{n})$ with
$y_{i}\in\mathcal{Y}\subset\mathbb{R}$ and
$u_{i}\in\mathcal{U}\subset\mathbb{R}^{s}$ for $d_{y},d_{u},s\in\mathbb{N}$
and $e_{n}\in\mathbb{R}$ is a noise variable that satisfies Assumption 1.
Here, $y_{i}$ denotes the autoregressive inputs and the $u_{i}$ denote vectors
of past and current control inputs. In this setting, we will therefore
consider
$\mathcal{X}=\mathbb{R}^{d_{y}}\times\mathcal{U}^{d_{u}+1}\subset\mathbb{R}^{d_{y}+(d_{u}+1)(s)},\mathcal{Y}=\mathbb{R}$.
If the dynamics and control inputs are such that the underlying dynamical
system is ergodic then Theorem 4 can be applied and a weaker version of
Theorem 7 can be derived. However, in general, this cannot be guaranteed and
the following result on the asymptotic point-wise convergence of the general
Lipschitz interpolation framework is needed.
###### Corollary 9
Consider
$\mathcal{X},\mathcal{Y},(x_{n})_{n\in\mathbb{N}},(u_{n})_{n\in\mathbb{N}},(y_{n})_{n\in\mathbb{N}}$
as defined above, $L^{*}\geq 0$ and $(\hat{f}_{n})_{n\in\mathbb{N}}$ as
defined in Definition 1 with $L\geq L^{*}$ and
$(\mathcal{D}_{n})_{n\in\mathbb{N}}=(x_{n},y_{n})_{n\in\mathbb{N}}$. Suppose
that Assumptions 1, 3 hold. Assume furthermore that
$\mathcal{U}\subset\mathbb{R}^{s}$ is bounded. Then $\forall p\in\mathbb{N}$,
$M^{*}\in\mathbb{R}^{+}$
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}\left[\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{\mathcal{Y}}^{p}\right]\to
0$
where $\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ denotes the set containing
all functions in $Lip(L^{*},\,\mathfrak{d})$ that are bounded by $M^{*}$, i.e.
$\left\|f\right\|_{\infty}\leq M^{*}$.
Proof As in the proof of Theorem 7, we have that for all $n\geq 1$ and any
sampling procedure $\mathcal{D}_{n}$,
$\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\left\|\hat{f}_{n}-f\right\|_{\infty}$
is uniformly bounded with probability $1$. This follows from (1) the existence
of a bounded set $\tilde{}\mathcal{X}\subset\mathcal{X}$ such that
$(x_{n})_{n\in\mathbb{N}}\subset\tilde{}\mathcal{X}$ (with probability 1)
which is due to the boundedness of
$\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$, the compactness of $\mathcal{U}$
and Assumption 1 (which implies that the noise is bounded), (2) $f\in
Lip(L^{*},\,\mathfrak{d})$ and (3) by construction of the Lipschitz
interpolation framework. More precisely, we have $\forall n\in\mathbb{N}$,
$\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\left\|\hat{f}_{n}-f\right\|_{\infty}\leq
2\bar{\mathfrak{e}}+2L\delta_{\,\mathfrak{d}}(\tilde{}\mathcal{X})$ where
$\delta_{\,\mathfrak{d}}(\tilde{}\mathcal{X}):=\sup_{x,y\in\tilde{}\mathcal{X}}\,\mathfrak{d}(x,y)$.
Using Lemma 25, it is therefore sufficient to show convergence in probability,
i.e.
$\forall\epsilon>0:\text{
}\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{P}(|\hat{f}_{n}(x_{n+1})-f(x_{n+1})|>\epsilon)=0$
which can be done through a modified proof of Theorem 4 as follows. Fix
$\epsilon>0$ and consider the minimal covering of $\bar{}\mathcal{X}$ by balls
of radius $r<\frac{\epsilon}{4(L^{*}+L)}$ which we denote $cov(r)$ and the
associated set of hyperballs $\mathcal{B}$ (the existence of a finite covering
is guaranteed by the boundedness of $\tilde{}\mathcal{X}$). There exists
$N_{1}\in\mathbb{N}$ such that for all $B\in\mathcal{B}$; $(x_{n})_{n\geq
N_{1}}\cap B\in\\{0,+\infty\\}$. Denoting by
$\mathcal{\tilde{B}}\subset\mathcal{B}$ the subset of $\mathcal{B}$ consisting
of hyperballs that contain an infinite number of elements of $(x_{n})_{n\geq
N_{1}}$, we can proceed as in the proof of Theorem 4.
Let $f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ be arbitrary. For
$n>N_{1}$ sufficiently large (such that there is at least one sample input in
each hyperball of $\mathcal{\tilde{B}}$), applying the same arguments as in
the proof of Theorem 4:
$\mathbb{P}\left(\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x_{n+1}),f(x_{n+1}))>\epsilon\right)$
$\leq\mathbb{P}\left(\max_{B\in\mathcal{\tilde{B}}}\left|\max_{i=1,...,n}\\{e_{i}\\}+\min_{x_{i}\in
B}\\{e(x_{i})\\}\right|>\frac{\epsilon}{2}\right)+\mathbb{P}\left(\max_{B\in\mathcal{\tilde{B}}}\left|\min_{i=1,...,n}\\{e_{i}\\}+\max_{x_{i}\in
B}\\{e(x_{i})\\}\right|>\frac{\epsilon}{2}\right).$
As the choice of $f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ was
arbitrary and the upper bound expressed above does not depend on $f$, we have
that both terms of this upper bound can be treated with the same approach as
the one used to conclude the proof of Theorem 4. This implies
$\displaystyle\quad\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{P}\left(|\hat{f}_{n}(x_{n+1})-f(x_{n+1})|>\epsilon\right)$
$\displaystyle\leq\lim_{n\to\infty}\mathbb{P}\left(\max_{B\in\mathcal{\tilde{B}}}|\max_{i=1,...,n}\\{e_{i}\\}+\min_{x_{i}\in
B}\\{e(x_{i})\\}|>\frac{\epsilon}{2}\right)$
$\displaystyle+\lim_{n\to\infty}\mathbb{P}\left(\max_{B\in\mathcal{\tilde{B}}}|\min_{i=1,...,n}\\{e_{i}\\}+\max_{x_{i}\in
B}\\{e(x_{i})\\}|>\frac{\epsilon}{2}\right)$ $\displaystyle=0$
which concludes the proof.
The setting considered in Corollary 9 is the same as the one considered in
Milanese and Novara (2004) and in ensuing applications of the Lipschitz
interpolation framework in the context of MPC (see Canale et al. (2014),
Manzano et al. (2020)). As in Corollary 8, the worst-case prediction
guarantees $(\tilde{}\mathfrak{u}_{n})_{n\in\mathbb{N}}$,
$(\tilde{}\mathfrak{l}_{n})_{n\in\mathbb{N}}$ can be shown to provide similar
guarantees to the one proposed Corollary 9 which provides a theoretical
guarantee that even conservative adaptive controllers relying on worst-case
bounds of Lipschitz interpolation methods will consider the true underlying
dynamics in the long run. In Section 6, a slight modification of Corollary 18
that considers dynamics with multidimensional outputs
$(y_{n})_{n\in\mathbb{N}}$ is given. This extension is then applied in the
context of tracking control in order to obtain closed-loop stability
guarantees for a simple online-learning based controller
To conclude this section, we remark that if an additional assumption is made
on the sequence of inputs $(x_{n})_{n\in\mathbb{N}}$, then the convergence
rate derived in Theorem 7 holds in the online learning setting. This
assumption is given using the following definition on the ”regularity of the
sampling” of $(x_{n})_{n\in\mathbb{N}}$.
###### Definition 10
(Regularity Assumption for $(x_{n})_{n\in\mathbb{N}}$) We say that
$(x_{n})_{n\in\mathbb{N}}$ is regularly sampled on a set
$\bar{}\mathcal{X}\subset\mathcal{X}$ if $\exists N\in\mathbb{N}$,
$(x_{n})_{n\in\mathbb{N}_{\geq N}}\subset\bar{}\mathcal{X}$ and $\exists
M\in\mathbb{N}$ such that $\forall n>N$ and $\forall
A\subset\bar{}\mathcal{X}$ ,
$\mathbb{P}(x_{n+M}\in A|x_{n})>C\mu(A)$
where $\mu(A)$ denotes the Lebesgue measure of $A$ and $C>0$ is an arbitrary
constant.
In essence, Definition 1 states that $(x_{n})_{n\in\mathbb{N}}$ is regularly
sampled on a given set $\bar{}\mathcal{X}\subset\mathcal{X}$ if
$(x_{n})_{n\in\mathbb{N}}$ will eventually be contained in $\bar{}\mathcal{X}$
and will continue to visit all of $\bar{}\mathcal{X}$ with non-zero
probability. The existence of such a set depends implicitly on the target
function and the defined control inputs.
###### Corollary 11
Assume that the setting and assumptions of Corollary 9 hold and consider
$f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$. If Assumption 2 holds and
the stochastic control law $u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ is
defined such that $(x_{n})_{n\in\mathbb{N}}$ is regularly sampled on a bounded
set $\bar{}\mathcal{X}\subset\mathcal{X}$ that satisfies Assumption 4, then
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{\mathcal{Y}}]<\infty.$
where
$(a_{n})_{n\in\mathbb{N}}:=((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$.
###### Remark 12
From the proof of Corollary 9, we have that if $\mathcal{U}$ is bounded and
$f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$, then there exists a bounded
$\tilde{}\mathcal{X}\subset\mathcal{X}$ that contains
$(x_{n})_{n\in\mathbb{N}}$ with probability $1$. Therefore, only the second
part of Definition 10 and the geometric shape of $\bar{}\mathcal{X}$ need to
be checked in order for Corollary 11 to hold.
Proof The proof of Corollary 11 follows from Theorem 7. More precisely:
By assumption, we have that there exists $M,N\in\mathbb{N}$ and a bounded set
$\bar{}\mathcal{X}\subset\mathcal{X}$ such that Definition 10 and Assumption 4
hold. Consider the sequence $(x_{n})_{n\in\mathbb{N}{\geq
N}}\subset\bar{}\mathcal{X}$ and the subsequence
$(\tilde{x}_{n})_{n\in\mathbb{N}}\subset(x_{n})_{n\in\mathbb{N}_{\geq N}}$
defined such that $\tilde{x}_{n}=x_{Mn+N}$ for all $n\in\mathbb{N}$. From
Definition 10, we have that for all $n\in\mathbb{N}$, $\tilde{x}_{n}$ is
sampled on $\bar{}\mathcal{X}$ with a probability distribution whose density
is bounded away from zero on all of $\bar{}\mathcal{X}$.
Then, defining $(\hat{f}_{n}^{M})_{n\in\mathbb{N}}$ as the predictors of the
Lipschitz interpolation framework with hyperparameter $L$ and sample inputs
$(\tilde{x}_{n})_{n\in\mathbb{N}}$, we can apply Theorem 7 to
$(\hat{f}_{n}^{M})_{n\in\mathbb{N}}$. This implies that
$(\hat{f}_{n}^{M})_{n\in\mathbb{N}}$ converges uniformly on
$\bar{}\mathcal{X}$ to $f$ at a rate that is upper bounded by
$(a_{\left\lfloor\frac{n}{M}\right\rfloor})_{n\in\mathbb{N}}=\tilde{c}(a_{n})_{n\in\mathbb{N}}$
for some $\tilde{c}>$ that depends on $M$ and where $n\in\mathbb{N}$ denotes
the index of the original sequence: $(x_{n})_{n\in\mathbb{N}}$. As the
asymptotic convergence rate of $(\hat{f}_{n})_{n\in\mathbb{N}}$ is at least as
fast as the convergence rate of $(\hat{f}_{n}^{M})_{n\in\mathbb{N}}$ due to
the fact that the input samples utilised by
$(\hat{f}_{n}^{M})_{n\in\mathbb{N}}$ are also utilised by
$(\hat{f}_{n})_{n\in\mathbb{N}}$, we have that
$(\hat{f}_{n})_{n\in\mathbb{N}}$ achieves the same uniform convergence rate on
$\bar{}\mathcal{X}$. Finally, as $(x_{n})_{n\in\mathbb{N}_{\geq
N}}\subset\bar{}\mathcal{X}$, the same converge rate holds for the pointwise
asymptotic convergence of $(x_{n})_{n\in\mathbb{N}}$, i.e.
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{\mathcal{Y}}]$
with
$(a_{n})_{n\in\mathbb{N}}:=((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$.
While Corollary 11 provides an interesting extension to Theorem 7, the
characterisation of the regularly sampling set $\bar{X}$ and the necessity of
ensuring that Assumption 4 holds for $\bar{}\mathcal{X}$ can be difficult to
do in practice. Therefore, in comparison to Corollary 9 which can be directly
utilised in various control applications, Corollary 11 is essentially a
theoretical result.
## 5 Removing the Lipschitz Constant Assumption
The main difficulty of the Lipschitz interpolation framework is obtaining a
suitable hyper-parameter that properly estimates the Lipschitz constant of the
unknown target function. In cases where prior knowledge of the Lipschitz
constant of $f$ is not obtainable, an additional step is therefore needed.
While one solution would be to compute this estimate offline beforehand, this
approach is problematic when considering a stream of data. Instead, one can
consider the approach developed by Novara et al. (2013) and applied in the
context of Lipschitz interpolation by Calliess et al. (2020) which utilises a
modified version of Strongin’s Lipschitz constant estimator (Strongin (1973))
to $(\mathcal{D}_{n})_{n\in\mathbb{N}}$ to obtain a sequence
$(L(n))_{n\in\mathbb{N}}$ of approximations of $L^{*}$. These estimates can be
continuously updated with the arrival of new data and are defined formally in
the following definition.
###### Definition 13
(LACKI rule) The Lazily Adapted Lipschitz Constant Kinky Inference (LACKI)
rule computes a Lipschitz interpolation predictor $\,\mathfrak{\hat{f}_{n}}$
as per Definition 1, but where L depends on
$(\mathcal{D}_{n})_{n\in\mathbb{N}}$ and is computed as follows:
$L(n):=\max\Bigl{\\{}0,\max_{(s,s^{\prime})\in
U_{n}}\frac{\,\mathfrak{d}_{\mathcal{Y}}(\tilde{f}(s),\tilde{f}(s^{\prime}))-\lambda}{\,\mathfrak{d}(s,s^{\prime})}\Bigr{\\}},$
(2)
where $U_{n}=\\{(g_{1},g_{2})\in G_{n}^{\mathcal{X}}\times
G_{n}^{\mathcal{X}}|\,\mathfrak{d}(g_{1},g_{2})>0\\}$ and $\lambda$ is a
hyperparameter.
The noise bounds can be correctly estimated if the $\lambda$ hyper-parameter
of the LACKI rule is set to $2\bar{\mathfrak{e}}$. Calliess et al. (2020)
provides worst-case prediction bounds even when the noise bounds are not
correctly estimated. In this paper, we focus on the case where the noise
bounds are known and $\lambda$ can be correctly specified. We note that the
Lipschitz estimator $L(n)$ given by LACKI is the smallest Lipschitz constant
that is consistent with the data. In other words, it reduces the hypothesis
space of Lipschitz continuous functions $Lip(L(n),\,\mathfrak{d})$ that the
target function f could belong to.
We start by showing that the LACKI rule proposed in Definition 13 converges
asymptotically to the best Lipschitz constant of the unknown target function.
###### Lemma 14
If the assumptions of Theorem 4 hold, then :
$\forall\epsilon>0,\lim_{n\to\infty}\mathbb{P}(|L(n)-L^{*}|>\epsilon)=0$
Proof Fix an arbitrary $\epsilon>0$. We start by defining an auxiliary
function F:
$\displaystyle F:Dom(F):=\mathcal{X}\times\mathcal{X}-\\{(x$
$\displaystyle,x)|x\in\mathcal{X}\\}\longrightarrow\mathbb{R}_{\leq 0}$
$\displaystyle(x,y)$
$\displaystyle\longmapsto\frac{\,\mathfrak{d}_{\mathcal{Y}}{(f(x),f(y))}}{\,\mathfrak{d}(x,y)}$
By construction, $L^{*}=\sup_{(x,y)\in Dom(F)}F(x,y)$ and there exists
$(x_{1},x_{2})\in Dom(F)$ such that $L^{*}-\frac{\epsilon}{2}\leq
F(x_{1},x_{2})\leq L^{*}$. Hence,
$\mathbb{P}\left(|L(n)-L^{*}|>\epsilon\right)$
$\leq\mathbb{P}\left(|L(n)-F(x_{1},x_{2})|+|F(x_{1},x_{2})-L^{*}|>\epsilon\right)$
$=\mathbb{P}\left(|F(x_{1},x_{2})-L(n)|>\frac{\epsilon}{2}\right)=\mathbb{P}\left(F(x_{1},x_{2})-L(n)>\frac{\epsilon}{2}\right).$
Since F is continuous on its domain, we have that $\exists\delta_{1}>0$ such
that $\forall(x,y)\in B_{\delta_{1}}((x_{1},x_{2}))$999Here,
$B_{\delta}((x_{1},x_{2}))$ denotes the ball centered in $(x_{1},x_{2})$ of
radius $\delta_{1}$ with respect to
$\,\mathfrak{d}_{\mathcal{X}\times\mathcal{X}}$ defined such that
$\,\mathfrak{d}_{\mathcal{X}\times\mathcal{X}}((x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime}))=\,\mathfrak{d}(x_{1},x_{1}^{\prime})+\,\mathfrak{d}(x_{2},x_{2}^{\prime})$
$\cap$ $Dom(F)$, $|F(x_{1},x_{2})-F(x,y)|<\frac{\epsilon}{2}$. Defining
$0<\delta_{2}<\min\\{\frac{\delta_{1}}{2},\frac{\,\mathfrak{d}(x_{1},x_{2})}{2}\\}$,
we consider the two hyperballs $B_{1}:=B_{\delta_{2}}(x_{1})$,
$B_{2}:=B_{\delta_{2}}(x_{1})$. Then
$\displaystyle F(x_{1},x_{2})$ $\displaystyle-L(n)$
$\displaystyle=F(x_{1},x_{2})$ $\displaystyle-\max_{(s,s^{\prime})\in
U_{n}}\frac{|\tilde{f}(s)-\tilde{f}(s^{\prime})|-\lambda}{\,\mathfrak{d}(s,s^{\prime})}$
$\displaystyle\leq F(x_{1},x_{2})$ $\displaystyle-\max_{s_{i}\in
B_{1},s_{j}\in
B_{2}}\frac{|\tilde{f}(s_{1})-\tilde{f}(s_{j})|-\lambda}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle\leq F(x_{1},x_{2})$
$\displaystyle-\max_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{|f(s_{i})-f(s_{j})|+|e(s_{i})-e(s_{j})|-\lambda}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle\leq F(x_{1},x_{2})$
$\displaystyle-\min_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{|f(s_{i})-f(s_{j})|}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle-\max_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{|e(s_{i})-e(s_{j})|-\lambda}{\,\mathfrak{d}(s_{i},s_{j})}.$
where
$cond(x,y):=\left\\{sgn\left(f(s_{i})-f(s_{j})\right)=sgn\left(e(s_{i})-e(s_{j})\right)\right\\}$
and with abuse of notation, $\tilde{f}(s_{i})$ $e(s_{i})$ denote the noise
term associated with the input $s_{i}$. By definition of $B_{1}$, $B_{2}$, we
have
$\displaystyle F(x_{1},x_{2})-\min_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in
B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{|f(s_{i})-f(s_{j})|}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle=F(x_{1},x_{2})-\min_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in
B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}F(s_{i},s_{j})\leq\frac{\epsilon}{4}.$
Substituting this value into the initial expression, we can obtain the upper
bound
$\displaystyle\frac{\epsilon}{4}-\max_{\begin{subarray}{c}s_{i}\in
B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{|e(s_{i})-e(s_{j})|-\lambda}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{4}+\min_{\begin{subarray}{c}s_{i}\in
B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{\lambda-|e(s_{i})-e(s_{j})|}{\,\mathfrak{d}(s_{i},s_{j})}$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{4}+\min_{\begin{subarray}{c}s_{i}\in
B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{\lambda-|e(s_{i})-e(s_{j})|}{\,\mathfrak{d}(x_{1},x_{2})-2\delta_{2}}.$
By the assumption of uniformly dense sampling, there exists $M\in\mathbb{N}$
such that $r(M)<{\delta_{2}}$. Therefore, for $n>M$,
$\mathbb{P}\left(F(x_{1},x_{2})-L(n)>\frac{\epsilon}{2}\right)$
$\leq\mathbb{P}\left(\min_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\frac{\lambda-|e(s_{i})-e(s_{j})|}{\,\mathfrak{d}(x_{1},x_{2})-2\delta_{2}}>\frac{\epsilon}{4}\right)$
$\leq\mathbb{P}\left(\min_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}\left\\{\lambda-|e(s_{i})-e(s_{j})|\right\\}>\frac{\epsilon}{4}(\,\mathfrak{d}(x_{1},x_{2})-2\delta_{2})\right)$
$=\mathbb{P}\left(\max_{\begin{subarray}{c}s_{i}\in B_{1},s_{j}\in B_{2}\\\
cond(s_{i},s_{j})\end{subarray}}|e(s_{i})-e(s_{j})|<\lambda-\frac{\epsilon}{4}(\,\mathfrak{d}(x_{1},x_{2})-2\delta_{2})\right).$
As $\lambda=2\bar{\mathfrak{e}}$ and
$\,\mathfrak{d}(x_{1},x_{2})>2\delta_{2}$, the last expression can be shown to
converge to 0 as $n$ goes to $\infty$ by a similar argument to the one used in
the proof of Theorem 4.
Lemma 14 proves that the modified version of Strongin’s estimate defined in
Definition 13 is a consistent Lipschitz constant estimator under bounded
noise. It is therefore of interest for applications outside the one considered
in this paper, e.g. see in particular global optimisation methods that depend
explicitly on the Lipschitz constant (see for example Malherbe and Vayatis
(2017)). One main drawback however is that none of the finite sample estimates
generated by the LACKI rule upper bound the true Lipschitz constant. This is
discussed in more detail after Theorem 15.
Using Theorem 4 and Lemma 14, we can now show that the sequence of LACKI
predictors $(\hat{f}_{n})_{n\in\mathbb{N}}$ converges uniformly and in
probability to the target function $f$.
###### Theorem 15
If the assumptions of Theorem 4 hold, then the sequence of LACKI predictors
$(\hat{f}_{n})_{n\in\mathbb{N}}$ with $\lambda=2\bar{\mathfrak{e}}$ converges
to f uniformly and in probability:
$\forall\epsilon>0,\lim_{n\to\infty}\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))>\epsilon\right)=0$
Proof The proof of Theorem 15 follows from Theorem 4 and Lemma 14. Fix an
arbitrary $\epsilon>0$, we have
$\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),f(x))>\epsilon\right)$
$\leq\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}(x),\hat{f}_{n}^{*}(x))>\frac{\epsilon}{2}\right)+\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}^{*}(x),f(x))>\frac{\epsilon}{2}\right)$
where $(\hat{f}_{n}^{*})_{n\in\mathbb{N}}$ denotes the general Lipschitz
interpolation framework with a hyperparameter equal to the best Lipschitz
constant $L^{*}$ of $f$. The second term of the upper bound given above
converges to 0 as $n\to\infty$ by Theorem 4. For the first term, we have
$\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}_{\mathcal{Y}}(\hat{f}_{n}^{*}(x),\hat{f}_{n}(x))>\frac{\epsilon}{2}\right)$
$\leq\mathbb{P}\left(\sup_{x\in\mathcal{X}}\frac{1}{2}|\min_{i=1,...,N_{n}}\\{\tilde{f}_{i}+L(n)\,\mathfrak{d}(x,s_{i})\\}-\min_{i=1,...,N_{n}}\\{\tilde{f}_{i}+L^{*}\,\mathfrak{d}(x,s_{i})\\}|>\frac{\epsilon}{4}\right)$
$+\mathbb{P}\left(\sup_{x\in\mathcal{X}}\frac{1}{2}|\max_{i=1,...,N_{n}}\\{\tilde{f}_{i}-L(n)\,\mathfrak{d}(x,s_{i})\\}-\max_{i=1,...,N_{n}}\\{\tilde{f}_{i}-L^{*}\,\mathfrak{d}(x,s_{i})\\}|>\frac{\epsilon}{4}\right)$
$\leq\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}(x,s^{*}_{i})|L^{*}-L(n)|>\frac{\epsilon}{4}\right)+\mathbb{P}\left(\sup_{x\in\mathcal{X}}\,\mathfrak{d}(x,s^{*}_{k})|L^{*}-L(n)|>\frac{\epsilon}{4}\right)$
$\leq
2\mathbb{P}\left(\delta_{\,\mathfrak{d}}(\mathcal{X})|L^{*}-L(n)|>\frac{\epsilon}{4}\right)$
(3)
where
$s^{*}_{i}:=\text{argmin}_{i=1,...,N_{n}}\\{\tilde{f}_{i}+L(n)\,\mathfrak{d}(x,s_{i})\\}$
and
$s^{*}_{k}:=\text{argmax}_{i=1,...,N_{n}}\\{\tilde{f}_{i}-L(n)\,\mathfrak{d}(x,s_{i})\\}$.
As $\delta_{\,\mathfrak{d}}(\mathcal{X})$ is finite by assumption, Lemma 14
can be applied to show that
$\mathbb{P}(\delta_{\,\mathfrak{d}}(\mathcal{X})|L^{*}-L(n)|>\frac{\epsilon}{4})$
converges to 0.
In general, it suffices for the sequence of Lipschitz constant estimates to
converge to a value that is bigger than the best Lipschitz constant in order
for the consistency guarantees given in Theorem 15 to hold. This follows from
the fact that Lemma 14 holds for any Lipschitz interpolation framework with
$L\geq L^{*}$. Furthermore, if the Lipschitz constant estimate can be
guaranteed to be feasible101010i.e. $\hat{L}(n)\geq L^{*}$. in a finite number
of queries and is asymptotically bounded, then the rate of convergence of the
adaptive Lipschitz interpolation method matches the one derived in Theorem 7.
Unfortunately, as remarked above, the LACKI rule proposed in Definition 13 is
not feasible for any finite number of sample points but converges only
asymptotically to the true best Lipschitz constant. One approach to remedying
this problem would be to include a multiplicative factor $\kappa\geq 1$
(similar to the original approach proposed by Strongin (1973) in the noiseless
sampling setting) in the LACKI rule. However, developing a principled approach
to setting $\kappa$ is non-trivial and depends on second order partial
derivatives of the unknown target function.
Furthermore, in contrast to the general Lipschitz interpolation approach, the
LACKI estimator is also not necessarily asymptotically consistent in the
setting of a non-linear discrete-time dynamic system. This is due to the fact
that dependent on the sampling sequence, the LACKI rule may never become large
enough to ensure that the relations (1) and (2) derived in the proof of Lemma
14 hold. This issue could potentially be fixed by including a ”memory hyper-
parameter” that limits the number of past observations considered in the
$\mathfrak{u}_{n}$, $\mathfrak{l}_{n}$ functions. This extension will be
investigated in future work.
In essence, while the general Lipschitz interpolation framework can be shown
to perform well as a non-parametric estimation method, the additional
difficulty of Lipschitz constant estimation implies that many of the desirable
asymptotic properties become difficult to obtain for a fully adaptive version
of the framework. A detailed discussion on this issue can be found in Huang et
al. (2023) where optimal convergence rates are given for the Lipschitz
constant estimation problem and a feasible asymptotically consistent
estimation method is developed.
## 6 Connections to Online Learning and Control
We conclude this paper by providing a simple illustration of the potential
applicability of our results to learning-based control. More precisely, we
slightly modify the online consistency results of the general Lipschitz
interpolation stated in Section 4 in order to obtain closed-loop stability of
a class of online learning-based trajectory tracking controllers discussed in
Sanner and Slotine (1991), Åström and Wittenmark (2013), Chowdhary et al.
(2013), Calliess et al. (2020).
We briefly recall the setting of the trajectory tracking control problem
considered by Calliess et al. (2020). The goal is to ensure that a sequence of
states $(y_{n})_{n\in\mathbb{N}}$ follows a given reference trajectory
$(\xi_{n})_{n\in\mathbb{N}}$. In order to do so, it is assumed that the states
$(y_{n})_{n\in\mathbb{N}}$ satisfy a multivariate recurrence relation
described as follows:
$y_{n}=f(x_{n})$
where $x_{n}=(y_{n-d_{y}},...,y_{n-1},u_{n-d_{u}},...,u_{n})$ with
$y_{i}\in\mathcal{Y}\subset\mathbb{R}^{l}$ denoting the past autoregressive
inputs, $u_{i}\in\mathcal{U}\subset\mathbb{R}^{s}$ denoting a vector of past
or current control inputs for $d_{y},d_{u},s,l\in\mathbb{N}$. In this setting,
we will therefore consider
$\mathcal{X}=\mathbb{R}^{(l)(d_{y})}\times\mathcal{U}^{d_{u}+1}\subset\mathbb{R}^{(l)(d_{y})+(s)(d_{u}+1)}$,
$\mathcal{Y}=\mathbb{R}^{l}$. Note that in contrast to the setting considered
in Section 4 the noise does not impact the state and will only be assumed to
be observational: we assume that the Lipschitz interpolation framework has
access to noisy samples of function values $f(x_{i})$ at each time step $i<n$:
$\mathcal{D}_{n}=\\{(x_{i},\tilde{f}_{i})|i<n\\}$.
Under this assumption on the system dynamics, the problem becomes equivalent
to defining a control law that ensures that the tracking error
$(\zeta_{n})_{n\in\mathbb{N}}$, $\zeta_{n}=\xi_{n}-y_{n}$ becomes stable:
obtaining, in an ideal scenario, a closed-loop recurrence relation
$\zeta_{n+1}=\phi(\zeta_{n})$
where $\phi$ is a contraction with a desirable fixed point $\zeta_{*}$,
typically $\zeta_{*}=0$.
This type of stability is well-known to be achievable when the dynamics of the
states $(y_{n})_{n\in\mathbb{N}}$ are known and sufficiently well-behaved
(Åström and Wittenmark (2013)) or when $f$ is assumed unknown but well
approximated by linear learning-based methods (Limanond and Tsakllis (2000)).
Obtaining such guarantees in the setting where $f$ is assumed both unknown and
non-linear is less straightforward although significant research has been
conducted with the use of non-parametric regression methods (Sanner and
Slotine (1991), Chowdhary et al. (2013), Calliess et al. (2020)).
Under a general assumption on the control law, the online-learning guarantees
of the Lipschitz interpolation method (Corollary 9 and Lemma 18) derived in
this paper can be shown to directly imply the convergence of the tracking
error to a fixed point, therefore ensuring the asymptotic stability of the
controller.
To do so, we begin by formally extending the online guarantees of the
Lipschitz interpolation stated in Corollary 9 to the multi-dimensional online
setting described above. In this case, the Lipschitz interpolation framework
is applied component-wise as follows:
###### Definition 16
(Multi-dimensional Lipschitz interpolation) Let $L\in\mathbb{R}_{\geq 0}$ be a
selected hyper-parameter. Using the set-up defined above, we define the
sequence of predictors $(\hat{f}_{n})_{n\in\mathbb{N}}$,
$\hat{f}_{n}:\mathcal{X}\to\mathcal{Y}$ associated to
$(\mathcal{D}_{n})_{n\in\mathbb{N}}$, as
$\forall
j\in\\{1,...,l\\},\quad\,\mathfrak{\hat{f}_{n}}^{i}\bigl{(}x):=\frac{1}{2}\mathfrak{u}_{n}^{j}(x)+\frac{1}{2}\mathfrak{l}_{n}^{j}(x),$
where $\mathfrak{u}_{n}^{j},\mathfrak{l}_{n}^{j}:\mathcal{X}\to\mathbb{R}$ are
defined as
$\displaystyle\mathfrak{u}_{n}^{j}(x)=\min_{i=1,...,N_{n}}{\tilde{f}_{n,i}^{j}}+L\,\mathfrak{d}(x,s_{i})$
$\displaystyle\mathfrak{l}_{n}^{j}(x)=\max_{i=1,...,N_{n}}{\tilde{f}_{n,i}^{j}}-L\,\mathfrak{d}(x,s_{i})$
for all $j\in\\{1,...,l\\}$.
We note that under Assumption 8 provided below, it is relatively
straightforward to observe that each component of the target function is also
Lipschitz continuous with the same Lipschitz constant. This implies that the
properties utilised in the previous sections hold component-wise for the
multi-dimensional Lipschitz interpolation framework.
In order to derive the desired online guarantee for the Lipschitz
interpolation framework described in Definition 16, we first extend the
assumptions of the previous sections to the multi-dimensional output setting.
###### Assumption 7
(Assumption on multi-dimensional noise) The noise variables
$(e_{k})_{k\in\mathbb{N}}$, $e_{k}\in\mathbb{R}^{d}$ are assumed to be
independent and identically111111The identically distributed assumption is
made to alleviate notation and is not technically needed in our derivations.
distributed random variables such that
$\exists\bar{\mathfrak{e}}\in\mathbb{R}_{+}^{d}$, such that $\forall
k\in\mathbb{N}$ $j\in\\{1,...,d\\}$,
$e_{k}^{j}\in[-\bar{\mathfrak{e}}^{j},\bar{\mathfrak{e}}^{j}]$ with
probability $1$. We assume further that the bounds of the support are tight,
i.e. $\forall\epsilon>0$, $\forall j\in\\{1,...,d\\}$,
$\mathbb{P}(e_{k}^{j}\in[-\bar{\mathfrak{e}}^{j}+\epsilon,\bar{\mathfrak{e}}^{j}]),\mathbb{P}(e_{k}^{j}\in[-\bar{\mathfrak{e}}^{j},\bar{\mathfrak{e}}^{j}-\epsilon])>0.$
###### Assumption 8
(Assumption on $\,\mathfrak{d}_{\mathcal{Y}}$). In this section, we will
restrict ourselves to the case,
$\,\mathfrak{d}_{\mathcal{Y}}(y,y^{\prime})=\left\|y-y^{\prime}\right\|_{1}$,
$\forall y,y^{\prime}\in\mathcal{Y}$ where $\left\|.\right\|_{1}$ denotes the
usual 1-norm.
###### Remark 17
By the strong equivalence of norms on $\mathbb{R}^{l}$, it is sufficient to
show the results of this section for
$\left\|.\right\|_{\mathcal{Y}}=\left\|.\right\|_{1}$. Additionally, we note
that if a Lipschitz constant of the target function is known for a given norm
on $\mathbb{R}^{l}$, then it is straightforward to compute a feasible
Lipschitz constant for any other norm on $\mathbb{R}^{l}$.
###### Corollary 18
(Multidimensional Online Learning) Consider the multidimensional setting
described above121212With the same arguments, one can show that the same
result holds for the multidimensional version of the dynamical system
described in Section 4., $L^{*},M^{*}\in\mathbb{R}^{+}$ and
$(\hat{f}_{n})_{n\in\mathbb{N}}$ as defined in Definition 16 with $L\geq
L^{*}$ and
$(\mathcal{D}_{n})_{n\in\mathbb{N}}=(x_{n},y_{n})_{n\in\mathbb{N}}$. Assume
that Assumptions 7 and 8 hold. Assume furthermore that $\mathcal{U}$ is
compact. Then
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}[\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{\mathcal{Y}}]\to
0$
where we recall that $\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ denotes the
set containing all functions in $Lip(L^{*},\,\mathfrak{d})$ that are bounded
by $M^{*}$, i.e. $\left\|f(x)\right\|_{\mathcal{Y}}\leq M^{*}$.
Proof By the strong equivalence of norms on $\mathbb{R}^{l}$, it is
sufficient to show Lemma 18 for
$\left\|.\right\|_{\mathcal{Y}}=\left\|.\right\|_{1}$:
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}[\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{1}]\to
0.$
This is implied if $\forall j\in\\{1,...,l\\}$:
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}[|f^{j}(x_{n+1})-\hat{f}^{j}_{n}(x_{n+1})|]\to
0$
where $f^{j}_{n},\hat{f}^{j}_{n}$ denote the j-th component functions of
$f_{n},\hat{f}_{n}$. This statement can be derived using the same arguments as
the ones given in the proof of Corollary 9 as, under Assumption 8, $f$ is
component-wise Lipschitz continuous with Lipschitz constant $L^{*}$ and by
construction, the multi-dimensional Lipschitz interpolation framework can be
considered component-wise.
Utilising Lemma 18, we can now state the closed-loop guarantees of an online
controller based on the Lipschitz interpolation framework.
###### Theorem 19
Assume the settings described above. Assume that reference trajectory
$(\xi_{n})_{n\in\mathbb{N}}$ is bounded and that the recursive plant dynamics
satisfy: $f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ for some
$L^{*},M^{*}>0$. Let $(\hat{f}_{n})_{n\in\mathbb{N}}$ be the predictors
generated by the Lipschitz interpolation framework with hyperparameter $L\geq
L^{*}$ and
$(\mathcal{D}_{n})_{n\in\mathbb{N}}=(x_{n},\tilde{f}_{i})_{n\in\mathbb{N}}$.
If there exists a bounded control law
$u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ such that the closed loop
dynamics are given by:
$\zeta_{n+1}=\phi(\zeta_{n})+d_{n}$
where $d_{n}:=f(x_{n})-\hat{f}(x_{n})$ is the one-step prediction error and
$\phi$ is a contraction with a fixed point $\zeta_{*}\in\mathcal{X}$ and
Lipschitz constant $\lambda_{\phi}\in[0,1)$, then we have
$\lim_{n\to\infty}\mathbb{E}[\left\|\zeta_{n}-\zeta_{*}\right\|_{\mathcal{Y}}]=0.$
Proof The proof follows a modified version of the proof Theorem 17 of
Calliess et al. (2020) and an application of Corollary 19.
Define the _nominal reference error_ $(\bar{\zeta}_{n})_{n\in\mathbb{N}}$,
$\bar{\zeta}_{0}=\zeta_{0}$, $\bar{\zeta}_{n+1}=\phi(\bar{\zeta}_{n})$ for
$n\in\mathbb{N}$. Fix an arbitrary $\epsilon>0$. The proof of Theorem 19
follows from the following sequence of steps.
By the Banach fixed point Theorem, $\exists n_{0}\in\mathbb{N}$ such that
$\forall n\geq n_{0}$,
$\left\|\bar{\zeta}_{n}-\zeta_{*}\right\|_{\mathcal{Y}}<\frac{\epsilon}{3}$.
Inductively, one can show that $\forall n,k\in\mathbb{N}$,
$\mathbb{E}[\left\|\zeta_{k}-\bar{\zeta}_{k}\right\|_{\mathcal{Y}}]$
$\leq\lambda_{\phi}^{n}\mathbb{E}[\left\|\zeta_{k}-\bar{\zeta}_{k}\right\|_{\mathcal{Y}}]+\sum^{n-1}_{i=0}\lambda_{\phi}^{n-1-i}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]$
$\leq\lambda_{\phi}^{n}\mathbb{E}[\left\|\zeta_{k}-\bar{\zeta}_{k}\right\|_{\mathcal{Y}}]+\frac{1}{1-\lambda_{\phi}}\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}].$
By Lemma 18, we have that $\mathbb{E}[\left\|d_{n}\right\|_{\mathcal{Y}}]$
converges to 0 as n goes to infinity. Therefore, $\exists k_{0}\in\mathbb{N}$
such that $\forall k\geq k_{0}$,
$\frac{1}{1-\lambda_{\phi}}\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]\leq\frac{\epsilon}{3}.$
Let $m_{0}:=\max\\{n_{0},k_{0}\\}$. Since $\lambda_{\phi}^{q_{0}}<1$, there
exists $q_{0}\in\mathbb{N}$ such that:
$\lambda_{\phi}^{q_{0}}\mathbb{E}[\left\|\zeta_{m_{0}}-\bar{\zeta}_{m_{0}}\right\|_{\mathcal{Y}}]<\frac{\epsilon}{3}.$
Let $M:=m_{0}+q_{0}$. Combining the above steps, we have for any $m>M$ there
exists $n\geq q_{0}$ such that $m=m_{0}+n$. This implies
$\mathbb{E}[\left\|\zeta_{m}-\zeta_{*}\right\|_{\mathcal{Y}}]\leq\left\|\zeta_{*}-\bar{\zeta}_{m}\right\|_{\mathcal{Y}}+\mathbb{E}[\left\|\zeta_{m}-\bar{\zeta}_{m}\right\|_{\mathcal{Y}}]$
$\leq\frac{\epsilon}{3}+\lambda_{\phi}^{q_{0}}\mathbb{E}[\left\|\zeta_{m_{0}}-\bar{\zeta}_{m_{0}}\right\|_{\mathcal{Y}}]+\frac{1}{1-\lambda_{\phi}}\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{m_{0}+i}\right\|_{\mathcal{Y}}]$
$\leq\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon.$
As the choice of $\epsilon$ was arbitrary, this concludes the proof.
Theorem 19 provides a theoretical result on the global stability in
expectation of a general class of control problems where both the system
dynamics and the error dynamics are assumed to be both unknown and non-linear.
An extension of Theorem 19 which states that the convergence rates of the
Lipschitz constant estimator derived in Corollary 11 hold for the tracking
error $(\zeta_{n})_{n\in\mathbb{N}}$ can also be obtained. However, this
result would be contingent on the difficult-to-verify ”regularity of the
sampling” assumption of $(x_{n})_{n\in\mathbb{N}}$ (as defined in Definition
10) and is therefore of limited interest. We provide it in the appendix for
completeness.
Unfortunately, for numerous applications, the contraction assumption on
dynamics of the tracking error $\phi$ is too stringent to be achieved in
practice. To alleviate this issue, Theorem 19 can be extended to consider the
more general assumption that $\phi$ is an eventually contracting function if
$\phi$ is also assumed to be a linear. More formally, we define an eventually
contracting function as follows.
###### Definition 20
(Eventually Contracting Function) Let $l\in\mathbb{N}$. A continuous function
$h:\mathbb{R}^{l}\to\mathbb{R}^{l}$ is said to be eventually contracting if
there exists $N\in\mathbb{N}$ and $\lambda\in[0,1)$ such that $\forall
x,y\in\mathbb{R}^{l}$:
$\,\mathfrak{d}(h^{N}(x),h^{N}(y))\leq\lambda\,\mathfrak{d}(x,y).$
As with the contracting functions considered above, eventually contracting
functions can be shown to admit a unique fixed point $\xi^{*}$. Additionally,
it is well-known that a linear function $h:\mathbb{R}^{l}\to\mathbb{R}^{l}$
defined as $h(x)=Mx$ for some matrix $M\in\mathbb{R}^{l\times l}$ is
eventually contracting if and only if the spectral radius of $M$ is strictly
smaller than 1: $\rho(M)<1$.
The assumption of the existence of a control law
$u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ such that the closed loop
dynamics are given by: $\zeta_{n+1}=\phi(\zeta_{n})+d_{n}$and $\phi$ is
eventually contracting can be observed in applications such as the removal of
wing rock during the landing of modern fighter aircraft (Monahemi and Krstic
(1996), Chowdhary et al. (2013)). Therefore, if Theorem 19 can be extended to
hold under these assumptions, then, conditional on the existence of feasible
Lipschitz constant estimate, the online learning-based reference tracking
controllers utilising a Lipschitz interpolation can be ensured to be globally
asymptotically stable in expectation in these settings. This alternative
result is stated in the following corollary.
###### Corollary 21
Assume the setting and initial assumptions of Theorem 19. If there exists a
bounded control law $u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ such that
the closed loop dynamics are given by:
$\zeta_{n+1}=\phi(\zeta_{n})+d_{n}$
where $d_{n}:=f(x_{n})-\hat{f}(x_{n})$ is the one-step prediction error and
$\phi:\mathbb{R}^{l}\to\mathbb{R}^{l}$, $\phi(\zeta)=M\zeta$ for a matrix
$M\in\mathbb{R}^{l\times l}$ that is a stable, i.e. $\rho(M)<1$. Then
$\lim_{n\to\infty}\mathbb{E}[\left\|\zeta_{n}\right\|_{\mathcal{Y}}]=0.$
Proof The proof of Corollary 21 is similar to the on given for Theorem 19.
Define the nominal reference error $(\bar{\zeta}_{n})_{n\in\mathbb{N}}$,
$\bar{\zeta}_{0}=\zeta_{0}$, $\bar{\zeta}_{n+1}=\phi(\bar{\zeta}_{n})$ for
$n\in\mathbb{N}$. Fix an arbitrary $\epsilon>0$.
As $\rho(M)<1$, we have that $\lim_{n\to\infty}\bar{\zeta}_{n}=0$ (Hasselblatt
and Katok (2003)). This implies that $\exists n_{0}\in\mathbb{N}$ such that
$\forall n\geq n_{0}$,
$\left\|\bar{\zeta}_{n}\right\|_{\mathcal{Y}}<\frac{\epsilon}{3}$.
Inductively, one can show that $\forall n,k\in\mathbb{N}$,
$\mathbb{E}[\left\|\zeta_{n+k}-\bar{\zeta}_{n+k}\right\|_{\mathcal{Y}}]$
$\leq\left\|M^{n}\right\|_{\mathcal{Y}}\mathbb{E}\left[\left\|(\zeta_{k}-\bar{\zeta}_{k})\right\|_{\mathcal{Y}}\right]+\sum^{n-1}_{i=0}\left\|M^{n-1-i}\right\|_{\mathcal{Y}}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}].$
By Gelfand’s formula we have
$\lim_{n\to\infty}\left\|M^{k}\right\|^{\frac{1}{k}}=\rho(M)<1$ for any matrix
norm $\left\|.\right\|$. This implies that there exists $n_{1}$ such that for
all $n\geq n_{1}$:
$\left\|M^{n}\right\|_{\mathcal{Y}}\leq\left\|M^{n}\right\|_{\mathcal{Y}}^{\frac{1}{n}}<\lambda_{\phi}<1$
for some $\lambda_{\phi}\in(0,1)$. Utilising this relation and matrix-vector
inequalities, we obtain the following inequalities: let $n\geq n_{1}$, there
exists $n_{2}\in\mathbb{N}\cup\\{0\\}$ such that
$n=n_{1}\left\lfloor\frac{n}{n_{1}}\right\rfloor+n_{2}$:
$\left\|M^{n}\right\|_{\mathcal{Y}}=\left\|M^{n_{1}(\left\lfloor\frac{n}{n_{1}}\right\rfloor-1)}M^{n_{1}+n_{2}}\right\|_{\mathcal{Y}}\leq\left\|M^{n_{1}}\right\|_{\mathcal{Y}}^{(\left\lfloor\frac{n}{n_{1}}\right\rfloor-1)}\left\|M^{n_{1}+n_{2}}\right\|_{\mathcal{Y}}\leq\lambda_{\phi}^{(\left\lfloor\frac{n}{n_{1}}\right\rfloor-1)}\lambda_{\phi}=\lambda_{\phi}^{\left\lfloor\frac{n}{n_{1}}\right\rfloor}.$
Substituting this inequality into the bound given above, we obtain:
$\left\|M^{n}\right\|_{\mathcal{Y}}\mathbb{E}\left[\left\|(\zeta_{k}-\bar{\zeta}_{k})\right\|_{\mathcal{Y}}\right]+\sum^{n-1}_{i=0}\left\|M^{n-1-i}\right\|_{\mathcal{Y}}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]$
$\leq\lambda_{\phi}^{\left\lfloor\frac{n}{n_{1}}\right\rfloor}\mathbb{E}[\left\|(\zeta_{k}-\bar{\zeta}_{k})\right\|_{\mathcal{Y}}]+\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]\left(K_{n_{1}}+\sum^{\left\lceil\frac{n-1}{n_{1}}\right\rceil}_{i=1}n_{1}\lambda_{\phi}^{i}\right)$
$\leq\lambda_{\phi}^{\left\lfloor\frac{n}{n_{1}}\right\rfloor}\mathbb{E}[\left\|(\zeta_{k}-\bar{\zeta}_{k})\right\|_{\mathcal{Y}}]+\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]\left(K_{n_{1}}+\frac{n_{1}}{1-\lambda_{\phi}}\right)$
where $K_{n_{1}}:=\sum^{n_{1}-1}_{i=0}\left\|M^{i}\right\|_{\mathcal{Y}}$ By
Lemma 18, we have that $\mathbb{E}[\left\|d_{n}\right\|_{\mathcal{Y}}]$
converges to 0 as n goes to infinity. Therefore, $\exists k_{0}\in\mathbb{N}$
such that $\forall k\geq k_{0}$,
$\left(K_{n_{1}}+\frac{n_{1}}{1-\lambda_{\phi}}\right)\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{k+i}\right\|_{\mathcal{Y}}]\leq\frac{\epsilon}{3}.$
Let $m_{0}:=\max\\{n_{0},k_{0}\\}$. There exists $q_{0}\in\mathbb{N}$ such
that
$\lambda_{\phi}^{\left\lfloor\frac{q_{0}}{n_{1}}\right\rfloor}\mathbb{E}[\left\|\zeta_{m_{0}}-\bar{\zeta}_{m_{0}}\right\|_{\mathcal{Y}}]<\frac{\epsilon}{3}.$
Let $M:=m_{0}+q_{0}$. Combining the above steps, we have that for all $m>M$,
there exists $n\geq q_{0}$ such that $m=m_{0}+n$. This implies
$\mathbb{E}[\left\|\zeta_{m}\right\|_{\mathcal{Y}}]\leq\left\|\bar{\zeta}_{m}\right\|_{\mathcal{Y}}+\mathbb{E}[\left\|\zeta_{m}-\bar{\zeta}_{m}\right\|_{\mathcal{Y}}]$
$\leq\frac{\epsilon}{3}+\lambda_{\phi}^{\left\lfloor\frac{q_{0}}{n_{1}}\right\rfloor}\mathbb{E}[\left\|\zeta_{m_{0}}-\bar{\zeta}_{m_{0}}\right\|_{\mathcal{Y}}]+\left(K_{n_{1}}+\frac{n_{1}}{1-\lambda_{\phi}}\right)\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{m_{0}+i}\right\|_{\mathcal{Y}}]$
$\leq\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon.$
As the choice of $\epsilon$ was arbitrary, this concludes the proof.
### 6.1 Example - model-reference adaptive control of a single pendulum
As a simple illustration, we replicate a modification of the model-reference
adaptive control example in Calliess et al. (2020). Here, we control an Euler
discretisation of a torque-actuated single pendulum in set-point control mode:
We consider a torque controlled pendulum where forces $u$ can be applied to
the joint of the pendulum. The angle of the pendulum is called $q$. We define
a state $x=[q\dot{q}]$. In continuous-time, it’s dynamics are given by the ODE
$\ddot{q}=f(x)+u$ where $f(x)=-\sin(q)-\dot{q}$ may be uncertain a priori and
hence, needs to be learned online while we control.
Figure 4: Illustration of the pendulum control example. A single run is
depicted in the leftmost figure showing how the controller learns to drive the
state to the set-point in spite of the noise and initially uncertain dynamics.
Note, “Time” is simulation time $t=\Delta n$ [sec.] for discrete time steps
$n=0,1,...$. The second figure shows how the mean trajectories, averaged over
30 repetitions (each with new draws from the noise distribution), converge to
the set-point. An illustration of our theory, predicting vanishing tracking
errors in the mean, is depicted in the rightmost figure: For each repetition
of the experiment, the colored lines show the error trajectories
$(\left\|\xi({\Delta n})-x(\Delta n)\right\|)_{n=0,1,2,\dots}$ as well as
their empirical mean (black dashed line).
As explained in Section 4 of Calliess et al. (2020), using online learning of
noisy measurements of angular accelerations (but assuming full state
observability) we can use Lipschitz interpolation to learn a model
$\hat{f}_{n}$ at time step $n$ define a control law
$u(x)=-\hat{f}_{n}(x)-K_{1}x_{1}-K_{2}x_{2}$ with gains $K_{1},K_{2}>0$ such
that the closed-loop error dynamics becomes:
$\displaystyle\zeta_{n+1}$ $\displaystyle=\phi(\zeta_{n})+\Delta d_{n}$ (4)
$\displaystyle=\underbrace{M\zeta_{n}}_{=:\phi(\zeta_{n})}+\Delta d_{n}$ (5)
where $\Delta=0.1$ is a sampling period for the time discretisation,
$d_{n}=f(x_{n})-\hat{f}_{n}(x_{n})$ is the tracking error and
$M=\left(\begin{array}[h]{cc}1&\,\Delta\\\ -\Delta K_{1}&1-\Delta
K_{2}\end{array}\right)$ is a matrix, where the gain parameters
$K_{1},K_{2}=1$ were chosen to render $M$ stable (i.e. such that its spectral
radius $\rho(M)<1$). This renders the closed-loop tracking error dynamics
consistent with the one considered in Corollary 21 and as previously
discussed, implies that the error dynamics are an eventual contraction.
For this experiment, we chose Lipschitz interpolation with a fixed Lipschitz
constant $L=11$ which is a Lipschitz constant of the true dynamics. To learn
$f$ the Lipschitz interpolator had access to acceleration corrupted by
uniformly distributed noise drawn i.i.d. from the interval [-2,2] given a
performance example in a relatively low signal-to-noise ration setting.
Starting in initial state $x_{0}=[-2.,-1.]$, the controller was given a set-
point reference $\xi_{n}=[2\pi,0],\forall n\in\mathbb{N}$. An example run of
the controller as well as empirical measurements of the tracking dynamics
across 30 trials of the experiments for different noise realisations are given
in Figure 4. Note, consistent with our theory, the plots show how the tracking
error appears to vanish in the mean (and in fact for all realisations).
Before we conclude, we will need to point out some limitations to the online
control application given in this section:
###### Remark 22
Firstly, our example assumed knowledge of the Lipschitz constant. While not an
uncommon assumption, we recognise that having to know the true Lipschitz
constant is a practical limitation. Therefore methods such as LACKI (Calliess
et al. (2020)) or POKI (Calliess (2017)) could also be employed to incorporate
full black-box learning. Our example however, merely serves as a simple
illustration of our currently developed theory. Secondly, we assumed that
states were observable but that only accelerations were noisy.
Extending our results to learning-based control settings that involve
Lipschitz constant parameter estimation and extending the theory to realistic
settings of noisy state observations would, in our opinion, be an interesting
direction to investigate in future work.
## 7 Conclusion
In conclusion, this paper has provided a comprehensive investigation into the
asymptotic convergence properties of general Lipschitz interpolation methods
in the presence of bounded stochastic noise. Through our analysis, we have
established probabilistic consistency guarantees for the classical approach
within a broad context. Furthermore, by deriving upper bounds on the uniform
convergence rates, we have aligned these bounds with the well-established
optimal rates of non-parametric regression observed in similar settings. These
rates provide a precise characterisation of the impact of the behaviour of the
noise at the boundary of its support on the non-parametric uniform convergence
rates and are, as far as the authors of this paper are aware, novel to the
literature.
These established bounds can also serve as useful tools for the comparative
asymptotic assessment of Lipschitz interpolation against alternative non-
parametric regression techniques determining the circumstances under which
Lipschitz interpolation frameworks can be anticipated to be asymptotically
better or worse. In particular, an explicit condition on the noise’s behaviour
at the boundary of its support can be utilised to predict this out-or under-
performance.
Extending our work, we have expanded our asymptotic results to consider online
learning in discrete-time stochastic systems. The additional consistency
guarantees we provide in this context carry practical significance, as we show
how they can be utilised to establish closed-loop stability assurances for a
simple online learning-based controller in the setting of model reference
adaptive control. We note that these asymptotic results also hold for the
worst-case upper and lower bounds provided by the ceiling and floor predictors
of the Lipschitz interpolation framework. This implies that even the most
conservative adaptive controllers relying on worst-case bounds of Lipschitz
interpolation methods will consider the true underlying dynamics in the long
run.
Finally, we have provided a brief theoretical study of the fully data-driven
LACKI framework (Calliess et al. (2020)) which extends classical Lipschitz
interpolation by incorporating a Lipschitz constant estimation mechanism into
the algorithm. We show asymptotic consistency of both the Lipschiz constant
estimation method and the extended framework which can serve to further
theoretically motivate the use of the LACKI in practice.
Two research avenues are of interest with respect to the theoretical results
derived in this paper. The first would be the derivation of lower bounds on
the non-parametric convergence rates under the same settings and assumptions
as the ones utilised in Theorem 7. These would, ideally but not unexpectedly,
given existing results on the optimal convergence rates of non-parametric
boundary regression by Jirak et al. (2014), demonstrate the optimality of the
upper bounds on the non-parametric convergence rates developed in this paper.
The second potential research direction would be to extend the convergence
rate upper bounds stated in Theorem 7 such that they hold for the practical
and fully data-driven extensions of Lipschitz interpolation such as LACKI
(Calliess et al. (2020)) or POKI Calliess (2017).
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## A Additional Results (Convergence rate of tracking error)
We provide the theoretical convergence rates obtained for the tracking error
in the application to online learning-based control. As noted in Section 6,
this result is not generally applicable as verification of the ”regular
sampling” condition defined in Definition 10 is difficult to do in practice.
###### Corollary 23
Assume that the setting and assumptions of Corollary 18 hold. Assume
furthermore that the stochastic control law
$u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ is defined such that
$(x_{n})_{n\in\mathbb{N}}$ is regularly sampled on a set
$\bar{}\mathcal{X}\subset\mathcal{X}$ that satisfies Assumption 4 and that the
noise vectors $(\epsilon_{n})_{n\in\mathbb{N}}$ are component-wise
independent. Then,
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}\left\|f(x_{n+1})-\hat{f}_{n}(x_{n+1})\right\|_{\mathcal{Y}}]<\infty.$
where
$(a_{n})_{n\in\mathbb{N}}:=((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$.
Proof (sketch) Applying the same arguments as in the proof of Lemma 18, it is
sufficient to consider: $\forall j\in\\{1,...,l\\}$
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}|f^{j}(x_{n+1})-\hat{f}^{j}_{n}(x_{n+1})|].$
Since the noise is component-wise independent, we can apply the arguments
utilised in the proof of Corollary 11 for all $j\in\\{1,...,l\\}$ and conclude
the proof.
###### Theorem 24
Assume the settings and assumptions of Theorem 19 hold. If the stochastic
control law $u_{n+1}:=u(x_{n},\hat{f}_{n},\mathcal{D}_{n})$ is such that
$(x_{n})_{n\in\mathbb{N}}$ is regularly sampled on a set
$\bar{}\mathcal{X}\subset\mathcal{X}$ that satisfies Assumption 4 and the
noise vectors $(\epsilon_{n})_{n\in\mathbb{N}}$ are component-wise
independent. Then,
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}\left\|e_{n}-e_{*}\right\|_{\mathcal{Y}}]<\infty$
where
$(a_{n})_{n\in\mathbb{N}}:=((n^{-1}log(n))^{\frac{\alpha}{d+\eta\alpha}})_{n\in\mathbb{N}}$.
Proof (sketch) Follows from applying the proof of Theorem 19 and noting that
the slowest converging term at the end of the proof is given by
$\frac{1}{1-\lambda_{\phi}}\max_{i=0,...,n-1}\mathbb{E}[\left\|d_{m_{0}+i}\right\|_{\mathcal{Y}}].$
This term can be upper bounded by applying Corollary 23 which therefore
provides the convergence rate and concludes the proof.
## B Proof of Theorem 7
Proof From the proof of Theorem 4, we have $\forall f\in
Lip(L^{*},\,\mathfrak{d})$, $x\in\mathcal{X}$,
$|\hat{f}_{n}(x)-f(x)|$
$\leq\max\Big{\\{}\min_{i=1,...,N_{n}}\\{\frac{e_{i}}{2}+\frac{L^{*}+L}{2}\,\mathfrak{d}(x,s_{i})\\}+\max_{i=1,...,N_{n}}\\{\frac{e_{i}}{2}\\},$
$-\min_{i=1,...,N_{n}}\\{\frac{e_{i}}{2}\\}-\max\limits_{i=1,...,N_{n}}\\{\frac{e_{i}}{2}-\frac{L^{*}+L}{2}\,\mathfrak{d}(x,s_{i})\\}\Big{\\}}.$
Consider the minimal covering of $\mathcal{X}$ of radius
$R_{n}:=a_{n}=\left(n^{-1}log(n)\right)^{\frac{\alpha}{d+\eta\alpha}}$ with
respect to $\,\mathfrak{d}$ denoted $Cov(R_{n})$ and the associated set of
hyperballs; $\mathcal{B}_{n}$. Assuming that $n$ is large enough such that
every hyperball in $\mathcal{B}_{n}$ contains at least one input of
$G_{n}^{\mathcal{X}}$, we have that the following upper bound holds:
$|\hat{f}_{n}(x)-f(x)|$ $\leq\max\left\\{\min_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\\{\frac{e(s_{i})}{2}\\},\min_{s_{i}\in B^{x}\cap
G_{n}^{\mathcal{X}}}\\{-\frac{e(s_{i})}{2}\\}\right\\}+\frac{\bar{\mathfrak{e}}}{2}+(L^{*}+L)R_{n}$
where with abuse of notation, $e(s_{i})$ denotes the noise variable associated
with the input $s_{i}$ and $B^{x}\in\mathcal{B}_{n}$ such that $x\in B$. For
all $n\in\mathbb{N}$, we define the following random variable and event
$\displaystyle A_{n}$
$\displaystyle:=\max_{B\in\mathcal{B}_{n}}\max\left\\{\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}\\{\frac{e(s_{i})}{2}\\},-\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}\\{\frac{e(s_{i})}{2}\\}\right\\}+\frac{\bar{\mathfrak{e}}}{2}$
$\displaystyle E_{n}$ $\displaystyle:=\left\\{\forall
B\in\mathcal{B}_{n},|\\{i\in[n]\\},s_{i}\in B\\}|>0\right\\}.$
Then, from ($\star$) given at the end of the proof we have that it is
sufficient to consider
$\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\mathbb{E}\left[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|_{\infty}\Big{|}E_{n}\right]\mathbb{P}(E_{n})$
in order for Theorem 7 to hold. For $n\in\mathbb{N}$ sufficiently large such
that $\mathbb{P}(E_{n})>0$ (see ($\star$)), we can apply the upper bound on
$|\hat{f}_{n}(x)-f(x)|$ derived above:
$\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\mathbb{E}\left[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|_{\infty}\Big{|}E_{n}\right]=\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\mathbb{E}\left[a_{n}^{-1}\sup_{x\in\mathcal{X}}|f_{n}(x)-f(x)|\Big{|}E_{n}\right]$
$\leq a_{n}^{-1}\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\mathbb{E}\left[\sup_{x\in\mathcal{X}}\max\left\\{\min_{s_{i}\in
B^{x}\cap G_{n}^{\mathcal{X}}}\\{\frac{e(s_{i})}{2}\\},\min_{s_{i}\in
B^{x}\cap
G_{n}^{\mathcal{X}}}\\{-\frac{e(s_{i})}{2}\\}\right\\}+\frac{\bar{\mathfrak{e}}}{2}+(L^{*}+L)R_{n}\Big{|}E_{n}\right]$
$=a_{n}^{-1}(L^{*}+L)R_{n}+\mathbb{E}\left[a_{n}^{-1}\max_{B\in\mathcal{B}_{n}}\max\left\\{\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}\\{\frac{e(s_{i})}{2}\\},\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}\\{-\frac{e(s_{i})}{2}\\}\right\\}+\frac{\bar{\mathfrak{e}}}{2}\Big{|}E_{n}\right]$
$=(L^{*}+L)a_{n}^{-1}R_{n}+\mathbb{E}\left[a_{n}^{-1}A_{n}\Big{|}E_{n}\right].$
By definition of $R_{n}$, the first term: $(L^{*}+L)a_{n}^{-1}R_{n}=(L^{*}+L)$
is bounded for all $n\in\mathbb{N}$. We can therefore focus on upper bounding
the second term:
$\mathbb{E}\left[a_{n}^{-1}A_{n}\Big{|}E_{n}\right]\mathbb{P}(E_{n})=\mathbb{E}\left[a_{n}^{-1}A_{n}\right].$
Using $0\leq A_{n}\leq 2\bar{\mathfrak{e}}$ with probability 1, we have
$\forall C_{0}>0$,
$a_{n}^{-1}A_{n}\leq C_{0}1_{\\{a_{n}^{-1}A_{n}\leq
C_{0}\\}}+2\bar{\mathfrak{e}}a_{n}^{-1}1_{\\{a_{n}^{-1}A_{n}>C_{0}\\}}$
with probability $1$. This implies that
$\mathbb{E}[a_{n}^{-1}A_{n}]\leq
C_{0}+2\bar{\mathfrak{e}}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n}).$
It is therefore sufficient to show that $\exists C_{0}>0$ such that
$\limsup_{n\to\infty}\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n})<\infty.$
We have
$\mathbb{P}(A_{n}>C_{0}a_{n})=1-\mathbb{P}(\forall
B\in\mathcal{B}_{n}:\min_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}e(s_{i})\in
I_{1},\max_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}e(s_{i})\in I_{2})$
$\stackrel{{\scriptstyle(\star\star)}}{{\leq}}1-\prod_{B\in\mathcal{B}_{n}}\mathbb{P}\left(\min_{s_{i}\in
B\cap G_{n}^{\mathcal{X}}}e(s_{i})\in I_{1},\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e(s_{i})\in I_{2}\right)$ $\leq
1-\mathbb{P}\left(\min_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in
I_{1},\max_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in
I_{2}\right)^{|\mathcal{B}_{n}|}$
where $I_{1}:=[-\bar{\mathfrak{e}},-\bar{\mathfrak{e}}+2C_{0}a_{n})$,
$I_{2}:=(\bar{\mathfrak{e}}-2C_{0}a_{n},\bar{\mathfrak{e}}]$ and
$N_{\mathcal{B}_{n}}:=\min_{B\in\mathcal{B}_{n}}|B\cap G_{n}^{\mathcal{X}}|$.
The second to last inequality follows from arguments given in $(\star\star)$
provided at the end of the proof. For $n$ large enough such that
$2C_{0}a_{n}<\bar{\epsilon}$, we can apply Assumption 2 to simplify the left
hand expression as follows;
$\mathbb{P}\left(\min_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in
I_{1},\max_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in I_{2}\right)$
$\geq\mathbb{P}\left(\min_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in
I_{1}\right)\cdot\mathbb{P}\left(\max_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in
I_{2}|\min_{i\in 1,...,N_{\mathcal{B}_{n}}}e_{i}\in I_{1}\right)$
$\geq\left(1-\left(1-\gamma(2C_{0}a_{n})^{\eta}\right)^{N_{\mathcal{B}_{n}}}\right)\left(1-\left(1-\gamma(2C_{0}a_{n})^{\eta}\right)^{N_{\mathcal{B}_{n}}-1}\right)$
$\geq\left(1-2\left(1-\gamma(2C_{0}a_{n})^{\eta}\right)^{N_{\mathcal{B}_{n}}-1}\right)^{2}.$
Therefore, we have
$\sup_{f\in Lip(L^{*},\,\mathfrak{d})}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n})$
$\leq
a_{n}^{-1}\left(1-\left(1-2(1-\gamma(2C_{0}a_{n})^{\eta})^{N_{\mathcal{B}_{n}}-1}\right)^{2|\mathcal{B}_{n}|}\right).$
$\leq
a_{n}^{-1}\left(1-\left(1-2(1-\gamma(2C_{0}a_{n})^{\eta})^{N_{\mathcal{B}_{n}}-1}\right)^{\frac{C_{1}}{{a_{n}}^{\frac{d}{\alpha}}}}\right).$
where we used the fact that there exists a constant $C_{1}>0$ (that can depend
on $d$) such that
$2|\mathcal{B}_{n}|\leq\frac{C_{1}}{{R_{n}}^{\frac{d}{\alpha}}}=\frac{C_{1}}{{a_{n}}^{\frac{d}{\alpha}}}$
which is a modification of (Wu (2017), Theorem 14.2) that follows from the
assumed convexity of $\mathcal{X}$. By Lemma 26, in order for the above
expression to be bounded, it is sufficient that
$2(1-\gamma(2C_{0}a_{n})^{\eta})^{N_{\mathcal{B}_{n}}-1}$ behaves like
$C_{2}^{\prime}{a_{n}}^{({\frac{d}{\alpha}}+1)}$ for an arbitrary
$C_{2}^{\prime}>0$ as $n$ goes to infinity. More precisely, let
$C_{2}^{\prime}=1$, it is sufficient to have:
$2(1-\gamma(2C_{0}a_{n})^{\eta})^{N_{\mathcal{B}_{n}}-1}\leq{a_{n}}^{({\frac{d}{\alpha}}+1)}$
$\iff N_{\mathcal{B}_{n}}\geq
1+({\frac{d}{\alpha}}+1)\frac{\log\left(a_{n}\right)}{\log\left(1-\gamma(2C_{0}a_{n})^{\eta}\right)}$
as $n$ goes to infinity. The right-hand expression can be re-expressed as the
series expansion:
$\frac{d+\alpha}{\alpha\gamma(2C_{0})^{\eta}}\frac{1}{{a_{n}}^{\eta}}\log(\frac{1}{a_{n}})+O(\log(\frac{1}{a_{n}}))$
as $a_{n}$ goes to $0$. Therefore, for any
$C_{2}>\frac{d+\alpha}{\alpha\gamma(2C_{0})^{\eta}}$ and $n>0$ large enough,
we have
$1+({\frac{d}{\alpha}}+1)\frac{\log(a_{n})}{\log(1-\gamma(2C_{0}a_{n})^{\eta})}<C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$
and it suffices to have
$N_{\mathcal{B}_{n}}\geq C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$
as $n$ goes to infinity in order for $\lim_{n\to\infty}\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n})$ to be
bounded:
If $N_{\mathcal{B}_{n}}\geq
C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$,
$\limsup_{n\to\infty}\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n})\leq
a_{n}^{-1}\left(1-\left(1-{a_{n}}^{({\frac{d}{\alpha}}+1)}\right)^{\frac{C_{1}}{{a_{n}}^{\frac{d}{\alpha}}}}\right)\leq
2C_{1}$
where the last inequality follows from Lemma 26.
Therefore, the final step of the proof is to show that
$N_{\mathcal{B}_{n}}\geq C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$
occurs with a probability that converges to $1$ at a rate of $a_{n}$ as n goes
to infinity.
More precisely, let $n\in\mathbb{N}$ and fix an arbitrary constant
$C_{2}>\frac{d+\alpha}{\alpha\gamma(2C_{0})^{\eta}}$ based on the condition
given above (note that $C_{0}>0$ can be set arbitrarily large).
$S_{n}:=\left\\{\forall B\in\mathcal{B}_{n}:\text{ }|\\{i\in[n]\\},s_{i}\in
B\\}|>C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}\right\\}$
i.e. the event that there is more than
$C_{2}\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$ queries in each hyperball
in $\mathcal{B}_{n}$. Utilising the asymptotic bound developed in the first
part of the proof, we have by the law of total probability:
$\limsup_{n\to\infty}\mathbb{E}[a_{n}^{-1}A_{n}]\leq
C_{0}+2\bar{\mathfrak{e}}\limsup_{n\to\infty}a_{n}^{-1}\mathbb{P}(A_{n}>C_{0}a_{n})$
$\leq
C_{0}+2\bar{\mathfrak{e}}\limsup_{n\to\infty}a_{n}^{-1}\left(\mathbb{P}(A_{n}>C_{0}a_{n}|S_{n})+\mathbb{P}(S_{n}^{c})\right)\leq
C_{0}+4\bar{\mathfrak{e}}C_{1}+\limsup_{n\to\infty}a_{n}^{-1}\mathbb{P}(S_{n}^{c})$
where the last inequality can be obtained by applying Lemma 26.
To conclude the proof, we need to show that
$\limsup_{n\to\infty}a_{n}^{-1}\mathbb{P}(S_{n}^{c})$ is bounded. We have
(denoting $b_{n}:=\log(\frac{1}{a_{n}})\frac{1}{{a_{n}}^{\eta}}$ to alleviate
notation):
$\mathbb{P}(S_{n})=\mathbb{P}\left(\forall
B\in\mathcal{B}_{n}:|\\{i\in[n]\\},s_{i}\in B\\}|>C_{2}b_{n}\right)$
$\geq\prod_{B\in\mathcal{B}_{n}}\mathbb{P}\left(|\\{i\in[\left\lfloor\frac{n}{2}\right\rfloor]\\},s_{i}\in
B\\}|>C_{2}b_{n}\right)$
$\geq\prod_{B\in\mathcal{B}_{n}}\left(1-\mathbb{P}\left(|\\{i\in[\left\lfloor\frac{n}{2}\right\rfloor]\\},s_{i}\in
B\\}|\leq C_{2}b_{n}\right)\right)$
where the first inequality stated above follows from $(\star\star\star)$
(shown at the end of the proof) for $n\in\mathbb{N}$ if $C_{2}$ satisfies the
condition: $C_{2}\leq\frac{d+\eta\alpha}{2C_{1}\alpha}$ where
$C_{1},d,\alpha,\eta$ are constants.
Then, Assumption 4 on $\mathcal{X}$ implies that there
$r_{0}>0,\theta\in(0,1]$ such that $\forall x\in\mathcal{X}$,
$r\in\left(0,r_{0}\right),\operatorname{vol}\left(B_{r}(x)\cap\mathcal{X}\right)\geq\theta\operatorname{vol}\left(B_{r}(x)\right)$.
Therefore, for all $n\in\mathbb{N}$ such that $R_{n}<r_{0}$, we have that
Assumption 4 can be applied to $B\in\mathcal{B}_{n}$. Using Assumption 5, we
have that the random variable defined by
$|\\{i\in[\left\lfloor\frac{n}{2}\right\rfloor]\\},s_{i}\in B\\}|$ follows a
binomial distribution with a success probability $p$ that can be lower bounded
by
$C_{3}^{\prime}\frac{vol(B)}{vol(\mathcal{X})}=C_{3}^{\prime\prime}a_{n}^{\frac{d}{\alpha}}$
for $C_{3}^{\prime},C_{3}^{\prime\prime}>0$ and with expectation:
$\mathbb{E}\left[|\\{i\in[\left\lfloor\frac{n}{2}\right\rfloor]\\},s_{i}\in
B\\}|\right]=\left\lfloor\frac{n}{2}\right\rfloor
p\geq\frac{C_{3}^{\prime\prime}}{3}n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}=C_{3}n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}$
where $C_{3}:=\frac{C_{3}^{\prime\prime}}{3}$. (Note: the ”$3$” denominator
was arbitrarily selected in order to remove the ceiling operator in the above
equation).
As the only condition on $C_{2}$ is given by the bound
$C_{2}>\frac{d+\alpha\xi}{\alpha\gamma(2C_{0})^{\eta}}$, $C_{2}$ can be set
arbitrarily small as $C_{0}$ can be set arbitrarily large. Therefore, there
exists $C_{0}>0$, $C_{2}>0$ such that
$C_{2}\leq\min\\{\frac{d+\eta\alpha}{2C_{1}\alpha},\frac{C_{3}(d+\eta\alpha)}{\alpha}\\}$
which implies that $(\star\star\star)$ holds and that:
$C_{2}b_{n}=C_{2}\left(\frac{\alpha}{d+\eta\alpha}\right)\log\left(\frac{n}{\log{(n)}}\right)\left(\frac{n}{\log(n)}\right)^{\frac{\eta\alpha}{\eta\alpha+d}}$
$\leq
C_{2}\left(\frac{\alpha}{d+\eta\alpha}\right)n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}\leq
C_{3}n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}\leq\frac{\mathbb{E}[|\\{i\in[n]\\},s_{i}\in
B\\}|]}{2}.$
This last relation implies that we can apply Lemma 1 of Stone (1982) to obtain
the upper bound:
$\mathbb{P}\left(|\\{i\in[n]\\},s_{i}\in B\\}|\leq
C_{2}b_{n}\right)\leq\mathbb{P}\left(|\\{i\in[n]\\},s_{i}\in
B\\}|\leq\frac{\mathbb{E}\left[|\\{i\in[n]\\},s_{i}\in
B\\}|\right]}{2}\right)$
$\leq\left(\frac{2}{e}\right)^{\frac{\mathbb{E}[|\\{i\in[n]\\},s_{i}\in
B\\}|]}{2}}$
which in turn implies
$\left(1-\mathbb{P}\left(|\\{i\in[n]\\},s_{i}\in B\\}|\leq
C_{2}b_{n}\right)\right)^{|\mathcal{B}_{n}|}\geq\left(1-\left(\frac{2}{e}\right)^{\frac{\mathbb{E}[|\\{i\in[n]\\},s_{i}\in
B\\}|]}{2}}\right)^{|\mathcal{B}_{n}|}.$
Plugging this expression back into
$\limsup_{n\to\infty}a_{n}^{-1}\mathbb{P}(S_{n}^{c})$, we obtain
$a_{n}^{-1}\mathbb{P}\left(S_{n}^{c}\right)\leq
a_{n}^{-1}\left(1-\left(1-(\frac{2}{e})^{\frac{\mathbb{E}[|\\{i\in[n]\\},s_{i}\in
B\\}|]}{2}}\right)^{|\mathcal{B}_{n}|}\right).$ $\leq
a_{n}^{-1}\left(1-\left(1-(\frac{2}{e})^{\frac{C_{3}}{2}n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}}\right)^{C_{1}{a_{n}}^{\frac{d}{\alpha}}}\right)$
which converges to $0$ as $n$ goes infinity and concludes the proof (follows
from the exponential speed of convergence of
$(\frac{2}{e})^{\frac{C_{3}}{2}n^{\frac{\eta\alpha}{\eta\alpha+d}}\log(n)^{\frac{d}{d+\eta\alpha}}}$).
($\star$) For completeness we revisit the assumption made in the proof. Recall
for all $n\in\mathbb{N}$,
$E_{n}:=\left\\{\forall B\in\mathcal{B}_{n},|\\{i\in[n]\\},s_{i}\in
B\\}|>0\right\\}.$
Then, by law of total expectation, we have $\forall f\in
Lip(L^{*},\,\mathfrak{d})$, $n\in\mathbb{N}$ sufficiently large such that
$\mathbb{P}(E_{n})>0$ (which exists since
$\mathbb{P}(E_{n})>\mathbb{P}(S_{n})\overset{n\to\infty}{\rightarrow}1$ );
$\mathbb{E}[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|_{\infty}]$
$=a_{n}^{-1}(\mathbb{E}\left[\left\|\hat{f}_{n}-f\right\|_{\infty}\Big{|}E_{n}\right]\mathbb{P}(E_{n})+\mathbb{E}\left[\left\|\hat{f}_{n}-f\right\|_{\infty}\Big{|}E_{n}^{c}\right]\mathbb{P}(E_{n}^{c})).$
For all $n\geq 1$, $f\in Lip(L^{*},\,\mathfrak{d})$ and any sampling procedure
$\mathcal{D}_{n}$, $\left\|\hat{f}_{n}-f\right\|_{\infty}$ is uniformly
bounded with probability $1$. More precisely, we have $\sup_{f\in
Lip(L^{*},\,\mathfrak{d})}\left\|\hat{f}_{n}-f\right\|_{\infty}$ $\leq
2\bar{\mathfrak{e}}+2L\delta_{\,\mathfrak{d}}(\mathcal{X})$ where
$\delta_{\,\mathfrak{d}}(\mathcal{X}):=\sup_{x,x^{\prime}\in\mathcal{X}}\,\mathfrak{d}(x,x^{\prime})$
with probability $1$ which follows from $f\in Lip(L^{*},\,\mathfrak{d})$ and
by the design of the Lipschitz interpolation framework. This bound is finite
by the assumed compactness of $\mathcal{X}$. Therefore, denoting
$K:=2\bar{\mathfrak{e}}+2L\delta_{\,\mathfrak{d}}(\mathcal{X})$, we have that
the above statement is upper bounded by
$\mathbb{E}\left[a_{n}^{-1}\left\|\hat{f}_{n}-f\right\|_{\infty}\Big{|}E_{n}\right]\mathbb{P}(E_{n})+a_{n}^{-1}K\mathbb{P}(E_{n}^{c}).$
The first term is equal to the simplified expression assumed earlier in the
proof and the second term converges to 0 since
$\mathbb{P}(E_{n}^{c})\leq\mathbb{P}(S_{n}^{c})$ and
$\lim_{n\to\infty}a_{n}^{-1}\mathbb{P}(S_{n}^{c})=0$ as shown above.
($\star\star$) For all $B\in\mathcal{B}_{n}$, let $E(B)$ denote the event
$E(B):=\left\\{\min_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}e_{i}\in
I_{1},\max_{s_{i}\in B\cap G_{n}^{\mathcal{X}}}e_{i}\in I_{2}\right\\}.$
Then, imposing an arbitrary ordering of the hyperballs in $\mathcal{B}_{n}$,
we have
$\mathbb{P}\left(\forall B\in\mathcal{B}_{n},\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{1},\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{2}\right)$
$=\mathbb{P}\left(E(B_{1})\right)\prod_{i=2}^{|\mathcal{B}_{n}|}\mathbb{P}\left(E(B_{i})|E(B_{i-1}),...,E(B_{1})\right).$
For all $i\in\\{1,...,|\mathcal{B}_{n}|\\}$, we observe that either there
exists $j\in\\{1,...,i-1\\}\text{ such that }B_{i}\cap B_{j}\cap
G_{n}^{\mathcal{X}}\neq 0$ in which case
$\mathbb{P}\left(E(B_{i})|E(B_{i-1}),...,E(B_{1})\right)>\mathbb{P}(E(B_{i}))$
or no such $j$ exists, in which case
$\mathbb{P}(E(B_{i})|E(B_{i-1}),...,E(B_{1}))=\mathbb{P}(E(B_{i})).$
Therefore, we have that
$\mathbb{P}(\forall B\in\mathcal{B}_{n},\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{1},\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{2})$
$\geq\prod_{B\in\mathcal{B}_{n}}\mathbb{P}(\min_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{1},\max_{s_{i}\in B\cap
G_{n}^{\mathcal{X}}}e_{i}\in I_{2}).$
$(\star\star\star)$ In order to alleviate notation, for all
$B\in\mathcal{B}_{n}$, we define the following event:
$\mathcal{E}_{B}(n):=\left\\{|\\{i\in[n]\\},s_{i}\in
B\\}|>C_{2}b_{n}\right\\}.$
It is trivial to see that for all $B\in\mathcal{B}_{n}$, $\mathcal{E}_{B}(n)$
is increasing in $n$ (when $b_{n}$ is kept fixed). Utilising the arbitrary
numbering of $\mathcal{B}_{n}$ defined above in $(\star\star)$, we have
$\mathbb{P}(S_{n})=\mathbb{P}(\forall
B\in\mathcal{B}_{n}:\mathcal{E}_{B}(n))=\mathbb{P}\left(\mathcal{E}_{B_{2}}(n)\right)\prod_{i=1}^{|\mathcal{B}_{n}|}\mathbb{P}\left(\mathcal{E}_{B_{i}}(n)|\mathcal{E}_{B_{i-1}}(n),...,\mathcal{E}_{B_{1}}(n)\right)$
$\geq\prod_{i=1}^{|\mathcal{B}_{n}|}\mathbb{P}\left(\mathcal{E}_{B_{i}}(n-(i-1)C_{2}b_{n}\right)\geq\prod_{B\in\mathcal{B}_{n}}\mathbb{P}\left(\mathcal{E}_{B}(n-|\mathcal{B}_{n}|C_{2}b_{n}\right)$
where the second to last inequality holds due to the independence of the input
sampling. Computing $|\mathcal{B}_{n}|C_{2}b_{n}$, we obtain
$|\mathcal{B}_{n}|C_{2}b_{n}\leq
C_{2}\frac{C_{1}}{a_{n}^{\frac{d}{\alpha}}}\log(\frac{1}{a_{n}})\frac{1}{a_{n}^{\eta}}=C_{1}C_{2}\frac{\log(\frac{1}{a_{n}})}{a_{n}^{\frac{d+\eta\alpha}{\alpha}}}$
$=nC_{1}C_{2}\frac{\alpha}{d+\eta\alpha}(1-\frac{\log(\log(n))}{\log(n)})\leq
nC_{1}C_{2}\frac{\alpha}{d+\eta\alpha}.$
Therefore, setting the condition $C_{2}\leq\frac{d+\eta\alpha}{2C_{1}\alpha}$,
we have
$\mathbb{P}(S_{n})\geq\prod_{B\in\mathcal{B}_{n}}\mathbb{P}\left(\mathcal{E}_{B}\left(n(1-C_{1}C_{2}\frac{\alpha}{d+\eta\alpha})\right)\right)\geq\prod_{B\in\mathcal{B}_{n}}\mathbb{P}\left(\mathcal{E}_{B}(\frac{n}{2})\right).$
## C Technical Lemmas
###### Lemma 25
Assume that the settings and assumptions of Theorem 7 hold. Let
$(g_{n})_{n\in\mathbb{N}}$ denote a sequence of non-parametric predictors and
$(b_{n})_{n\in\mathbb{N}}$ denote a convergence rate sequence (that converges
to 0). If $\exists K>0$ such that
$\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\left\|\hat{g}_{n}-f\right\|_{\infty}<K$
$\forall n\in\mathbb{N}$ with probability 1, then
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}[(f(x_{n+1})-\hat{g}_{n}(x_{n+1}))^{p}]\to
0$ (6)
if and only if $\forall\epsilon>0$
$\lim_{n\to\infty}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{P}(|f(x_{n+1})-\hat{g}_{n}(x_{n+1})|>\epsilon)=0$
(7)
where $\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$ is as defined in Corollary
9.
Proof ”$\implies$” can be trivially obtained by applying Markov’s inequality.
We show the ”$\impliedby$” statement. Fix $\epsilon>0$ and consider an
arbitrary $f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})$. Define
$A_{n}:=|f(x_{n+1})-\hat{g}_{n}(x_{n+1})|>\epsilon$, we have
$(f(x_{n+1})-\hat{g}_{n}(x_{n+1}))^{p}\leq\epsilon^{p}1_{A_{n}^{c}}+K^{p}1_{A_{n}}$
with probability $1$. This implies that
$\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{E}[(f(x_{n+1})-\hat{g}_{n}(x_{n+1}))^{p}]$
$\leq\epsilon^{p}+K^{p}\sup_{f\in\overline{Lip}(L^{*},\,\mathfrak{d},M^{*})}\mathbb{P}(|f(x_{n+1})-\hat{g}_{n}(x_{n+1})|>\epsilon)$
$\stackrel{{\scriptstyle n\to 0}}{{\leq}}\epsilon^{p}.$
As the choice of $\epsilon$ was arbitrary, $(\ref{equ:conv moment})$ holds.
###### Lemma 26
$\forall p,c>0$, we have
$\limsup_{x\to\infty}x\left(1-(1-\frac{1}{x^{p+1}})^{cx^{p}}\right)\leq 2c$
Proof Lemma 26 can be shown as follows.
$x\left(1-(1-\frac{1}{x^{p+1}})^{cx^{p}}\right)=x\left(1-e^{cx^{p}\log(1-\frac{1}{x^{p+1}})}\right).$
Expanding the exponent based on the power series expression of $\log(1+x)$, we
obtain
$cx^{p}\log(1-\frac{1}{x^{p+1}})=-cx^{p}\sum_{m=1}^{\infty}\frac{1}{mx^{m(p+1)}}=-\frac{cx^{p}}{x^{p+1}}\sum_{m=0}^{\infty}\frac{1}{(m+1)x^{m(p+1)}}$
$\geq-\frac{c}{x}\sum_{m=0}^{\infty}\frac{1}{x^{m(p+1)}}=-\frac{c}{x}\frac{x^{(p+1)}}{x^{(p+1)}-1}\geq-\frac{2c}{x}$
for sufficiently large $x$. Substituting this equation back into the initial
bound, we obtain:
$\limsup_{x\to\infty}x\left(1-e^{cx^{p}\log(1-\frac{1}{x^{p+1}})}\right)\leq\limsup_{x\to\infty}x\left(1-e^{-\frac{2c}{x}}\right)\stackrel{{\scriptstyle
x\to\infty}}{{\rightarrow}}2c.$
|
# Species Distribution Modeling with Expert Elicitation and Bayesian
Calibration
Karel Kaurila Sanna Kuningas Antti Lappalainen Jarno Vanhatalo Department
of Mathematics and Statistics, University of Helsinki Natural Resources
Institute Finland Organismal and Evolutionary Biology Research Programme,
University of Helsinki
(2022)
###### Abstract
Species distribution models (SDMs) are key tools in ecology, conservation and
management of natural resources. They are commonly trained by scientific
survey data but, since surveys are expensive, there is a need for
complementary sources of information to train them. To this end, several
authors have proposed to use expert elicitation since local citizen and
substance area experts can hold valuable information on species distributions.
Expert knowledge has been incorporated within SDMs, for example, through
informative priors. However, existing approaches pose challenges related to
assessment of the reliability of the experts. Since expert knowledge is
inherently subjective and prone to biases, we should optimally calibrate
experts’ assessments and make inference on their reliability. Moreover,
demonstrated examples of improved species distribution predictions using
expert elicitation compared to using only survey data are few as well. In this
work, we propose a novel approach to use expert knowledge on species
distribution within SDMs and demonstrate that it leads to significantly better
predictions. First, we propose expert elicitation process where experts
summarize their belief on a species occurrence proability with maps. Second,
we collect survey data to calibrate the expert assessments. Third, we propose
a hierarchical Bayesian model that combines the two information sources and
can be used to make predictions over the study area. We apply our methods to
study the distribution of spring spawning pikeperch larvae in a coastal area
of the Gulf of Finland. According to our results, the expert information
significantly improves species distribution predictions compared to
predictions conditioned on survey data only. However, experts’ reliability
also varies considerably, and even generally reliable experts had spatially
structured biases in their assessments. This suggests that expert elicitation
can be an efficient tool, for example, in natural resources management and
conservation area planning, but expert information should be analyzed with
care and preferably calibrated.
expert opinion,
Supra Bayes,
hierarchical models,
Gaussian process,
random effects,
bias correction,
fisheries,
62F15, 62P12;,
60G15,
###### keywords:
###### keywords:
[class=MSC]
††volume: TBA††issue: TBA
, , , and
## 1 Introduction
Species distribution models (SDMs) are key tools in ecology, conservation and
management of natural resources. They are used to study, for example, species
habitat preferences (Kallasvuo et al., 2017; Elith and Leathwich, 2009),
interspecific interactions (Vanhatalo et al., 2020; Ovaskainen and Abrego,
2020) and to build species distribution maps by predicting species presence
and abundance over extended regions where species data have not been collected
(Kotta et al., 2019; Mäkinen and Vanhatalo, 2018; Gelfand et al., 2006). SDMs
are commonly trained by controlled survey data but, since organizing
scientific surveys is expensive and human labor intensive, there is also a
need for complementary sources of information on species distributions. One
such source is expert knowledge which has been used to inform species
distribution predictions either independently or together with survey data
(Murray et al., 2009; Pearman-Gillman et al., 2020; Crawford et al., 2020).
However, expert knowledge is inherently subjective posing challenges related
to model calibration (Murray et al., 2009; Di Febbraro et al., 2018).
Moreover, it is not clear what type of expert knowledge should be collected
and who are the experts to trust (Pearce et al., 2001; Di Febbraro et al.,
2018).
In this work, we propose a novel approach to elicit expert knowledge on
species distribution and to calibrate it using survey data within hierarchical
Bayesian framework. Expert elicitation refers to a process during which a
statistician extracts probability distributions, or point estimates, for a
parameter or a variable of interest from one or more people who are believed
to have valuable information about them. Comprehensive, general, treatments of
modern approaches to expert elicitation are given by O’Hagan et al. (2006) and
Dias et al. (2018). Expert elicitation is extensively used in science,
management and other applications (e.g., Burgman et al., 2011; Vanhatalo et
al., 2014; Nevalainen et al., 2018; O’Hagan, 2019; Perälä et al., 2020; LaMere
et al., 2020). In the context of species distribution modeling, expert
elicitation has been used especially in conservation and management
applications. For example, Crawford et al. (2020) used expert opinion to
inform habitat suitability models for concervation planning in US and Di
Febbraro et al. (2018) assessed the feasibility of monitoring habitat quality
for bird communities in Central Italy by using survey data and expert driven
models. Pearman-Gillman et al. (2020) used expert elicitation to model the
distribution of several species in New England. They used a web questionnaire
to elicit species presence probabilities and information on the impact of
covariates on the species occurency. When fitting the SDM, the assessments
were pooled together by weighting them based on the experts’ self-assessment
on their confidence. Murray et al. (2009) elicited informative priors from
experts for a species distribution model, where the presence or absence of a
threatened species was modelled with logistic regression. They provided
experts with an interactive tool to tune the parameters of the beta
distribution to give their assessment of species presence in various
environments.
We will move forward from the above mentioned approaches by explicitly
modeling and correcting for the systematic biases in the expert assessments.
Correcting for possible biases is crucial for reliable use of expert
information in inference since humans are prone to psychological
idiosyncrasies and subjective biases (Dias et al., 2018; Burgman et al., 2011;
Tversky and Kahneman, 1974). We will follow the so called _supra Bayesian_
approach, where expert information is treated as observations which are linked
to model parameters through a conditional probability distribution (likelihood
function; Genest and Schervish, 1985; Lindley and Singpurwalla, 1986; French,
2011; Albert et al., 2012). Our approach is based on three components: i)
expert elicitation process which summarizes experts’ own assessment of their
region of expertize and belief on species occurrence proability within that
area, ii) data from carefully designed survey within the study region that can
be used to calibrate the expert assessments, and iii) a hierarchical Bayesian
model that combines the two information sources.
We motivate our approach with a real wold case study where we map the
distribution of newly hatched pikeperch (_Sander lucioperca_) larvae within
the Porvoo-Sipoo fisheries region (a public corporation whose purpose is to
develop fishery in their region) in the Gulf of Finland (see Figure 1).
Pikeperch is a top predator of Baltic Sea coastal areas and central species in
the coastal ecosystem of the Baltic Sea. The coastal pikeperch population
forms a commercially important fish stock and pikeperch are also highly sought
after by recreational fishers. Hence, knowledge about its spawning areas
(i.e., areas with newly hatched larvae) has both economical and conservation
importance. Pikeperch is of fresh water origin and has specific habitat
requirements for their reproduction, selecting shallow ($<$10 m deep),
vegetated, and sheltered bays that warm up early in the spring (Veneranta et
al., 2011; Kallasvuo et al., 2017). Information on pikeperch spawning areas is
important for implementation of conservation measures to protect the spawning,
the larvae and male who are securing the development of their offspring. Maps
of the spawning areas can be used to guide local fisheries management and
coastal area planning (Kallasvuo et al., 2017) and our aim is to improve these
maps by combining survey data with expert information.
The rest of the paper is organized as follows. We first summarize the data
collection and expert elicitation process used in our motivating case study.
After that we propose a hierarchical Bayesian model for these data and methods
to conduct model comparison. We then present the results from our case study,
discuss them and provide some concluding remarks.
## 2 Data collection and expert elicitation
Figure 1: The study area in Porvoo-Sipoo archipelago, in the Baltic Sea. The
dots show the locations from where the survey data was collected.
### 2.1 Survey data and environmental covariates
The study area consists of coastal environment types ranging from open water
to sheltered bays. In our analysis, we used three environmental covariates to
characterize the environment: _depth_ (m), _distance to deep_ water ($>$10 m)
and _shoreline density_ (km/m2). The latter two covariates are proxies for how
close to shoreline and how sheltered a location is. Each of the covariates
were available throughout the study area as raster maps with 50 m resolution.
More detailed description of the covariates and how they were constructed is
provided by Kallasvuo et al. (2017).
To collect survey data on the distribution of the larvae of pikeperch from the
study area we conducted a field survey of the surface water layer in June 2017
with paired Gulf ichthyoplankton samplers. These are small nets that are
pulled by a small boat over a transect of 500 meters and have been used to
quantitatively monitor the abundance and spatial occurrence of pikeperch
larvae also in earlier studies (see Veneranta et al., 2011; Kallasvuo et al.,
2017; Långnabba et al., 2019, for a more detailed description). We sampled in
total 92 sites (Figure 1) which were dispersed over the entire study area
covering all main habitat types. The sampling was scheduled to the _a priori_
estimated peak larval season (Liu and Vanhatalo, 2020) in June. From each
sample, we counted early-stage larvae (size range of 3–17 mm) and recorded the
effort; that is, the volume of water sampled, which equals the length of the
transect multiplied by the size (area) of the opening of the Gulf
ichthyoplankton sampler. The Gulf ichthyoplankton samplers are accurate method
for larval sampling and the sampling times were set so that the variability in
catchability due to weather conditions was minimized (Långnabba et al., 2019).
Hence, data collected with the Gulf ichthyoplankton samplers will be treated
as the ground truth in this study.
### 2.2 Expert elicitation
We elicited information concerning the pikeperch spawning grounds from ten
active fishermen (to be called experts hereafter) whose fishing areas fall
inside our study area. The experts were suggested by the executive director of
the local fisheries region. Before the elication, the executive director also
confirmed by a phone call to all experts that the experts were well motivated
to participate in the elicitation process. We sent to each expert by post a
printed map of the study area, crayons with three different colors and filling
instructions. First the experts were asked to draw the borders of their
assessment regions to the map; assessment regions were defined to be areas
within which an expert was confident to state his/her beliefs. After this,
each expert was asked to color the areas, within their assessment region,
where they believed pikeperch did or did not spawn. The experts were asked to
describe the strength of their belief using four categories that were
described as follows (translated from the original Finnish and Swedish
versions):
* •
(Colour 1) A generally known, locally or regionally important pikeperch
spawning area (the probability that pikeperch spawns in the area is over 90%)
* •
(Colour 2) Another area where pikeperch most likely spawns (the probability
that pikeperch spawns in the area is 50 – 90 %)
* •
(Colour 3) An area where pikeperch might spawn (the probability that pikeperch
spawns in the area is 10 – 50%)
* •
(Uncoloured) The areas that are not marked to belong to any of the above three
categories, but are inside an assessment region, are considered areas where
pikeperch does not spawn according to the general knowledge (the probability
that pikeperch spawns in the area is less than 10%)
Sea areas outside an expert’s assessment regions were considered as missing
information from that expert. Figure A.1 shows an example of an expert
elicitation form and a map drawn by an expert. The elicitation process and
elicitation questions were tested before the actual elicition with the
executive director of the local fisheries region. We then revised the
questions and the elicitation process according to her feedback. The written
descriptions of the colour categories were particularly carefully planned
together so that they would be correctly understood by the fishermen. The
expert elicitation was organized in spring 2018 after which we digitized the
expert drawn maps by scanning them and storing them as raster maps. We aligned
the expert assessment raster maps with the raster maps of the environmental
covariates (Section 2.1) using the ArcGIS software so that each expert’s
answers formed one raster map layer whose lattice match that of the
environmental covariates. The grid cells of the expert assessment maps were
labeled by one of the above four categories or NA (see Figure A.2). The latter
denotes an area outside an expert’s assessment region. There are no
significant errors from the digitalization since the questionnaire maps were
printed in the same coordinate system as the raster maps of the environmental
covariates.
The motivation behind asking directly about spawning areas and the reasoning
for using the four above classes to describe experts’ knowledge were the
following. Earlier works have indicated that expert information is most
accurate within areas from where experts have personal experience (Pearce et
al., 2001; Di Febbraro et al., 2018; Crawford et al., 2020). It has also been
shown that experts are typically better in assessing their belief on real
world variables, which they can observe, than on parameters of statistical
models (O’Hagan et al., 2006). The latter are mathematical abstractions whose
interpretation requires expertise in modeling. Hence, we wanted the experts to
directly assess their beliefs concerning a variable that has a clear
undebatable real world meaning instead of asking them to provide prior
information for some parameters of our species distribution model (Section 3).
The definition of pikeperch spawning area is clear and it is understood in
uniform manner by both the fishermen and researchers. Even though the
fishermen cannot directly observe pikeperch spawning they are assumed to be
skilled to assess spawning areas based on their personal experience on where
pikeperch concentrate during spawning season. We also wanted to allow experts
to express the uncertainty in their knowledge since this provides a more
complete picture of their beliefs than hard cut division to spawning and no-
spawning areas (see also O’Hagan et al., 2006). However, since filling in the
questionnaire maps was rather tedious we wanted to restrict the level of
detail. The four categories described above provide a compromise between these
two targets.
## 3 Species distribution models
### 3.1 Model for survey data
#### 3.1.1 Model for larval counts
We followed Liu and Vanhatalo (2020) and Kallasvuo et al. (2017) and modeled
the distribution of pikeperch larvae with a log Gaussian Cox process (LGCP)
with intensity function
$\lambda(\operatorname{\mathbf{s}},\operatorname{\mathbf{x}}(\operatorname{\mathbf{s}}))$
where $\operatorname{\mathbf{s}}$ denotes a location inside the study area and
$\operatorname{\mathbf{x}}(\operatorname{\mathbf{s}})$ is a vector of
environmental covariates at that location. We modeled the log intensity with a
linear function of the covariates and a spatial random effect:
$\log\lambda(\textbf{s}_{i},\textbf{x}_{i})=\alpha+\boldsymbol{\beta}^{T}\textbf{x}_{i}+\phi(\textbf{s}_{i}),$
(3.1)
where the intercept $\alpha$ and linear weights, $\boldsymbol{\beta}$ were
given a zero mean Gaussian prior with variance 100. The spatial random effect
$\phi(\textbf{s})$ followed the Barrier model of Bakka et al. (2019), which is
a non-stationary Gaussian process whose covariance does not travel through
physical barriers – islands in our application.
The Barrier model is defined as a Stochastic Partial Differential Equation
(SPDE) which leads to a Gaussian process with a non-stationary covariance
function. For a stationary covariance function the covariance between two
points $\operatorname{\mathbf{s}}$ and $\operatorname{\mathbf{s}}^{\prime}$
depends only on the distance
$d(\operatorname{\mathbf{s}},\operatorname{\mathbf{s}}^{\prime})$. In the
Barrier model the covariance travels only through water, while land areas are
considered as barriers through which the covariance does not travel. This
property is achieved by defining the spatial random effect, $\phi(s)$, as the
solution to the following SPDE:
$\displaystyle\phi(s)-\nabla\frac{r^{2}}{8}\phi(s)$
$\displaystyle=r\sqrt{\frac{\pi}{2}}\sigma_{\phi}\mathcal{W}(s),\text{ for
}s\in\Omega_{w}$ $\displaystyle\phi(s)-\nabla\frac{r^{2}_{b}}{8}\phi(s)$
$\displaystyle=r_{b}\sqrt{\frac{\pi}{2}}\sigma_{\phi}\mathcal{W}(s),\text{ for
}s\in\Omega_{l},$ (3.2)
where $\Omega_{w}$ is water, $\Omega_{l}$ is land, $\mathcal{W}(s)$ is white
noise, $\sigma_{\phi}$ is the marginal standard deviation of the process and
$r$ is the range parameter governing how fast the correlation decreases with
distance through water. The parameter $r_{b}$ is the correlation range on land
and it is set to be a fraction of $r$ to remove the correlation there (Bakka
et al., 2019). Here, we use the default value $r_{b}=0.2r$. The details on
implementing the barrier model are given in Appendix B.1. The parameters of
the barrier model follow a Penalized Complexity (PC) prior (Simpson et al.,
2017; Bakka et al., 2019). These priors shrink the effect towards a base
model, where $\sigma\rightarrow 0$ and $l\rightarrow\infty$. We defined the
prior through probabilities $p(\sigma>1)=0.01$ and $p(l<0.5km)=0.01$.
As detailed in Section 2.1 the transect observations correspond to the number
of pikeperch larvae caught with a net that samples a volume of water over a
transect. We denote by $\operatorname{\mathbf{s}}_{i}$ the coordinates of the
middle point of the $i$th transect, by $y_{i}$ the number of larvae caught
over the transect and by
$\operatorname{\mathbf{x}}_{i}=\operatorname{\mathbf{x}}(\operatorname{\mathbf{s}}_{i})$
the environmental covariates at the middle of the transect. Since survey
transects were short compared to the resolution of the mesh used to implement
the model (see Appendix B.1), we treated the larvae density as a constant over
each transect. Hence, the observation model for the survey data
$\operatorname{\mathbf{y}}=[y_{1},\dots,y_{n}]^{T}$ is
$p(\operatorname{\mathbf{y}}|\boldsymbol{\lambda},\boldsymbol{\epsilon})=\prod_{i=1}^{n}\mathrm{Poisson}(V_{i}\lambda_{i}\epsilon_{i})$
where $V_{i}$ is the volume of water sampled,
$\lambda_{i}=\lambda(\operatorname{\mathbf{s}}_{i},\operatorname{\mathbf{x}}_{i})$
is the intensity of the log Gaussian Cox process and
$\boldsymbol{\epsilon}=[\epsilon_{1},\dots,\epsilon_{n}]^{T}$ is a vector of
independent random effects capturing overdispersion in larval counts arising
from local phenomena and stochasticity in sampling. Note though that, in the
survey data set, each point has the same sampling volume $V_{i}=V$ (28.35 m3)
so we gave Gamma prior for the random effects,
$\epsilon_{i}\sim\mathrm{Gamma}(r,1/r)$, which leads to negative binomial
observation model for the larval counts
$p(\operatorname{\mathbf{y}}|\lambda,r)=\prod_{i=1}^{n}\mathrm{Negative-
Binomial}(V_{i}\lambda_{i},r)$ (3.3)
so that the observed larval counts are actually modeled to arise from an
overdispersed log Gaussian Cox process which approaches the Poisson process as
the overdispersion parameter $r$ increases (see also, Liu and Vanhatalo,
2020). We gave $\text{Gamma}(\sqrt{10},1/\sqrt{10})$ prior for the
overdispersion parameter $r$.
#### 3.1.2 Model for larval presence
As an alternative to larval abundance modeling we considered also an
occurrence model where the occurrence of larvae in a survey sample was modeled
with
$\mathbbm{1}(y_{i}>0)\sim\text{Bernoulli}\left(\pi(\textbf{s}_{i},\textbf{x}_{i})\right)$
(3.4)
where $\mathbbm{1}(y_{i}>0)$ is an indicator function returning one if
$y_{i}>0$ and zero otherwise and
$\pi(\textbf{s}_{i},\textbf{x}_{i})=\text{logit}^{-1}\left(\alpha+\boldsymbol{\beta}^{T}\textbf{x}_{i}+\phi(\textbf{s}_{i})\right)$
denotes the probability of presence for pikeperch larvae. The priors for model
parameters $\alpha$, $\beta$ and $\phi$ were the same as in the abundance
model.
### 3.2 Joint model for expert assessments and survey data
#### 3.2.1 Model for expert opinions
We constructed the model for expert assessments by first building a model for
experts’ subjective probabilities on the occurrence of larvae. For this, we
denote by $\bar{\pi}_{ji}\in[0,1]$ the $j$th expert’s subjective probability
that larvae are present at location $s_{i}$ and model this probability with
Beta distribution
$\bar{\pi}_{ji}\sim Beta(\bar{\mu}_{ji},\bar{s}_{j}),$ (3.5)
where $\bar{\mu}_{ji}=E[\bar{\pi}_{ji}]$, and $\bar{s}_{j}$ is the (prior
sample size) parameter governing the spread of the Beta distribution. We fixed
$\bar{s}_{j}=2$ to encode vague prior predictive distribution for expert
opinions. We then model the expected value of an expert’s subjective
probability with logistic regression so that it is a function of the log of
the true larval density or the logit of the true probability of presence of
larvae, so that
$\bar{\mu}_{ji}=\mathrm{logit}^{-1}(\bar{\alpha}_{j}+\bar{c}_{j}(\boldsymbol{\beta}^{T}\textbf{x}_{i}+\phi(\textbf{s}_{i}))+\bar{\varphi}_{j}(\textbf{s}_{i})).$
(3.6)
Here, the parameter $\bar{\alpha}_{j}$ is an intercept, $\bar{c}_{j}$ is a
parameter for the expert’s skill and $\bar{\varphi}_{j}(\textbf{s}_{i})$ is a
residual error. Note that, for example,
$\log\lambda_{i}-\alpha=\boldsymbol{\beta}^{T}\textbf{x}_{i}+\phi(\textbf{s}_{i})$,
so $\text{logit}(\mu_{ji})$ is proportional to $\log\lambda_{i}$ (or
$\mathrm{logit}(\pi_{i})$) but we have subtracted $\alpha$ from (3.6) in order
to improve the identifiability of the parameters. The parameter $\bar{c}_{j}$,
thus, describes how strongly an expert’s assessment follows the true
underlying larvae intensity or probability of presence, so that a positive
$\bar{c}_{j}$ indicates that an expert has information about larvae
distribution. The error terms $\bar{\varphi}_{j}(\textbf{s}_{i})$ correct for
spatially correlated local biases in the expert assessment that can result
from actual bias in an expert’s opinion but also from inaccuracies in
expressing them. For example, it is likely that the experts coloured the maps
with smaller resolution than the true resolution in the larvae
presence/absence pattern, so $\bar{\varphi}_{j}(\textbf{s}_{i})$ also accounts
for spatial correlation resulting from this.
In case of a conflict between expert assessment and survey data, we want the
model to explain the expert assessment with $\bar{\varphi}$ instead of
$\bar{c}_{j}(\boldsymbol{\beta}^{T}\textbf{x}_{i}+\phi(\textbf{s}_{i}))$ so
that an expert’s assessment does not provide information on larvae intensity
or occurrence probability. To attain this behaviour we give relatively
stricter prior for $\bar{c}_{j}$ than to $\bar{\varphi}$. Hence, priors for
the parameters were $\bar{\alpha}_{j}\sim N(0,2^{2})$ and $\bar{c}_{j}\sim
N(0,0.5^{2})$. The latter implies 95% prior probability that $\bar{c}_{j}$ is
less than one. We then modeled the expert bias function
$\bar{\varphi}_{j}(\textbf{s}_{i})$, with Besag-York-Mollié (BYM) -model
(Besag et al., 1991), which induces spatial correlation among neighboring
pixels in an expert assessment graph (see Appendix B.1 for details). In the
BYM-model $\bar{\varphi}_{j}\sim N(0,Q^{-1})$, where the precision matrix
$Q=\tau_{u}R+\tau_{v}I$, $I$ is the identity matrix corresponding to the
i.i.d. random effect, and
$R_{kl}=\begin{cases}n_{k},&k=l\\\ -\mathbbm{1}\\{k\sim l\\},&k\neq
l\end{cases}$ (3.7)
where $n_{k}$ is the number of neighbors of vertex $k$, and $k\sim l$
indicates that vertices $k$ and $l$ are neighbors, i.e. they are connected by
an edge. The hyperparameters of the BYM model are precision for the spatially
correlated effect $\tau_{u}$ and precision for the i.i.d. effect $\tau_{v}$.
We gave both of them gamma prior $\Gamma(2,8)$. When marginalizing over the
precision this induces a heavy tailed scaled Student-$t$ distribution with
$\nu=4$ and $s=2$ for $\bar{\varphi}$.
#### 3.2.2 Likelihood functions for the expert assessments
As explained in Section 2.2 the expert assessments were coded as raster maps
so that the assessment of the $j$’th expert for grid cell $i$ at location
$\operatorname{\mathbf{s}}_{i}$ is a categorical variable
$z_{ij}\in\\{1,2,3,4\\}$ if the grid cell belongs to the expert’s assessment
region (Figure A.2). If the grid cell is outside expert’s assessment area
$z_{ij}=\text{NA}$, which means that the corresponding grid cell is excluded
from the likelihood function of the expert’s assessments to be detailed below.
We ordered the categories so that $z_{ij}=1$ corresponds to the lowest
($<10\%$) and $z_{ij}=4$ corresponds to the highest ($>90\%$) subjective
probability of presence of an expert.
As the first model for the expert assessments we treated them as binary
statements about presence of pikeperch larvae. In this model, a location
within an expert’s assessment region was considered to be labeled as presence
if the expert had given it over 50% probability for larvae presence (i.e.
$z_{ij}\in\\{3,4\\})$) and otherwise it was considered as an absence. The
observation model for the binary absence assessment is then
$\displaystyle\text{Pr}(z_{ij}\in\\{1,2\\})$
$\displaystyle=\text{Pr}(\bar{\pi}_{ji}\leq
0.5)=F_{\text{Beta}}(0.5|\bar{\mu}_{ji},\bar{s}_{j})$ (3.8)
where $F_{\text{Beta}}(0.5|\mu_{ji},s_{j})$ is the cumulative distribution
function of the Beta distribution. Similarly for the binary presence
assessment we have
$\text{Pr}(z_{ij}\in\\{3,4\\})=1-F_{\text{Beta}}(0.5|\bar{\mu}_{ji},\bar{s}_{j})$
This observation model induces a likelihood function for
$\text{logit}(\bar{\mu}_{ji})$ that resembles closely the logit-function. We
can similarly form the observation model for all expert assessment categories
$\displaystyle\text{Pr}(z_{ij}|\mu_{ji},s_{j})$
$\displaystyle=\begin{cases}F_{\text{Beta}}(0.1|\mu_{ji},s_{j}),&\text{ if
}z_{ij}=1\\\
F_{\text{Beta}}(0.5|\mu_{ji},s_{j})-F_{\text{Beta}}(0.1|\mu_{ji},s_{j}),&\text{
if }z_{ij}=2\\\
F_{\text{Beta}}(0.9|\mu_{ji},s_{j})-F_{\text{Beta}}(0.5|\mu_{ji},s_{j}),&\text{
if }z_{ij}=3\\\ 1-F_{\text{Beta}}(0.9|\mu_{ji},s_{j}),&\text{ if
}z_{ij}=4,\end{cases}$ (3.9)
which again induces likelihood functions for $\text{logit}(\mu_{ji})$ and
through that to the model parameters.
We made a further assumption that, conditionally on the model parameters, the
expert assessments are mutually independent. Hence, the joint distribution for
all expert assessments over the whole study region can be written as
$p\left(\operatorname{\mathbf{z}}_{1},\dots,\operatorname{\mathbf{z}}_{J}|\bar{\alpha},\bar{c},\boldsymbol{\beta},\varphi,\bar{\varphi}\right)=\prod_{j=1}^{J}\prod_{i=1}^{n_{j}}p(z_{ij}|\bar{\alpha}_{j},\bar{c}_{j},\boldsymbol{\beta},\varphi,\bar{\varphi}_{j})$
(3.10)
where $J$ is the number of experts, $n_{j}$ is the number of grid cells inside
the assessment area of the $j$th expert,
$\bar{\alpha}=\\{\bar{\alpha}_{1},\dots,\bar{\alpha}_{J}\\}$,
$\bar{c}=\\{\bar{c}_{1},\dots,\bar{c}_{J}\\}$ and
$\bar{\varphi}=\\{\bar{\varphi}_{1},\dots,\bar{\varphi}_{J}\\}$. When we
combine the two data sets, $\operatorname{\mathbf{y}}$ (the survey
observations) and $\operatorname{\mathbf{z}}_{j},j=1,\dots,J$ (the expert
observations), the observations are independent given the latent function and
model parameters so that, in case of count observations from surveys, the full
observation model is
$p(\operatorname{\mathbf{y}},\operatorname{\mathbf{z}}_{1},\dots,\operatorname{\mathbf{z}}_{J}|\bar{\alpha},\bar{c},\bar{\varphi},\alpha,\boldsymbol{\beta},\varphi,r)=\left[\prod_{i=1}^{n}p(y_{i}|\lambda_{i},r)\right]\prod_{j=1}^{J}\prod_{k=1}^{n_{j}}p(z_{kj}|\bar{\alpha}_{j},\bar{c}_{j},\bar{\varphi}_{j},\boldsymbol{\beta},\varphi).$
(3.11)
The full model with the presence-absence model for the survey observations is
constructed analogously.
## 4 Posterior inference and model comparison
We implemented all the models and conducted the posterior inference using the
Integrated Nested Laplace Approximation (INLA) R package (R-INLA, Rue et al.,
2009). The technical implementation of the barrier model requires a triangular
mesh over the study area on which the model is constructed (Bakka et al.,
2019). The survey observations and expert assessments are then mapped to the
mesh with projection matrices. The details on how we constructed these are
given in the Appendix B.1. R-INLA does not support the exact likelihood
functions derived in Section 3.2, so we searched for the closest match to them
from among the Binomial likelihood functions, which are readily available in
R-INLA. The details of the matching process and the resulting likelihood
functions are given in the Appendix C. After setting up the models, we
constructed the INLA approximation for the model parameters and used the model
to predict the larvae intensity over the whole study area.
Since the survey data were collected with a standardized and extensively
tested sampling method (see Veneranta et al., 2011; Kallasvuo et al., 2017,
and their references) and the sampling time was matched with the peak larval
time (Liu and Vanhatalo, 2020), we considered them to accurately reflect the
larval areas of pikeperch. On the other hand, we had less evidence on the
expert elicitation process and the reliability of the experts. Hence, we
considered the survey data to be more reliable source of information on larvae
distribution than the expert assessments, and compared the alternative models’
based on how well they predict survey data that have been left out from the
training data. The INLA approximation gives access to approximate leave-one-
out (LOO) predictive densities for all observations. These are also called
Conditional Predictive Ordinate (CPO) values and, for the $i$’th survey
observation, it is defined as
$\mathrm{CPO}_{i}=p(y_{i}|\mathbf{x}_{i},\mathbf{s}_{i},D_{\setminus i})$
(4.1)
where $D_{\setminus i}$ includes all the other data except
$\\{y_{i},\mathbf{x}_{i},\mathbf{s}_{i}\\}$. The CPO values can be used to
calculate the widely used average LOO cross-validation log predictive density
$\mathrm{lpd}=\frac{1}{n_{\mathrm{survey}}}\sum_{i=1}^{n_{\mathrm{survey}}}\log\mathrm{CPO}_{i}.$
(4.2)
With models where survey data are modeled as presence-absence data, we can
calculate three additional model comparison metrics. First, we calculated
classification accuracy,
$\mathrm{ACC}=\frac{1}{n_{\mathrm{survey}}}\sum_{i=1}^{n_{\mathrm{survey}}}\mathbbm{1}(\mathrm{CPO}_{i}\geq
0.5)$, i.e. the ratio of correctly classified observations. Because the
observations are skewed towards absences, we also calculated the balanced
classification accuracy $\mathrm{bACC}=\frac{\mathrm{TPR}+\mathrm{TNR}}{2}$,
which is the average of true positive rate (TPR) and true negative rate (TNR)
that are calculated similarly to ACC. Log predictive density is known to be
sensitive to outliers (i.e., observations for which the predictive density is
much lower than in average) for which reason Gneiting et al. (2007) proposed
to use the Continuous Ranked Probability Score (CRPS) as an alternative metric
for comparing probabilistic predictions. The CRPS for a single prediction is
defined as
$\text{CRPS}(F_{i},y_{i})=E_{F_{i}}|Y_{i}-y_{i}|-\frac{1}{2}E_{F_{i}}|Y_{i}-Y_{i}^{\prime}|,$
(4.3)
where $F_{i}$ is the posterior predictive distribution given by the model
being evaluated, $Y_{i}$ and $Y_{i}^{\prime}$ are independent and identical
random variables that follow $F_{i}$ and $y_{i}$ is an observation. In the
case of Bernoulli likelihood for survey data, the LOO posterior predictive
distribution for the $i$’th observation is $Y_{i}\sim
F=\text{Bernoulli}(\hat{\pi}_{i})$ where
$\hat{\pi}_{i}=\text{Pr}(Y_{i}=1|\mathbf{x}_{i},\mathbf{s}_{i},D_{\setminus
i})$. Hence, we can simplify the CRPS for the LOO posterior predictive
distributions of these models to the square of the posterior predictive
probability of the incorrect class:
$\displaystyle\text{CRPS}(F_{i},y_{i})$
$\displaystyle=\hat{\pi}_{i}(1-y_{i})+\left(1-\hat{\pi}_{i}\right)y_{i}-\frac{1}{2}\left(2\hat{\pi}_{i}\left(1-\hat{\pi}_{i}\right)\right)$
$\displaystyle=\begin{cases}\hat{\pi}_{i}^{2}&\text{ if }y_{i}=0\\\
(1-\hat{\pi}_{i})^{2}&\text{ if }y_{i}=1\end{cases}$
$\displaystyle=(1-\mathrm{CPO}_{i})^{2}.$
## 5 Results
We received in total 11 expert assessments out of which ten were used for the
analyses. The assessed areal coverage of experts varied considerably. Some
experts expressed their views for very small regions whereas some of them
covered almost the entire study area (see Figure A.2). One expert assessment
was excluded, as the area assessed by the expert was considered too small for
the analysis. In some cases, the request of first drawing the expert’s own
assessment area in the map was poorly understood and needed to be clarified
later on. Some experts were more uncertain in their statements (they used only
categories 2 or 3) than others (who used all four categories). In general the
digitalization of the expert assessment maps worked reasonably well but in few
cases borders between two colors were hard to distinguish and needed some
interpretation by us.
In general, the models integrating survey data and expert assessments
(hereafter survey+expert models) outperformed the survey only model in their
predictive performance (table 1). In occurrence predictions, that is survey
data were modeled as occurrence data (Bernoulli likelihood (3.4)), the best
model was the one where expert assessments were modeled as presence-absence
statements (3.8). However, the survey+expert models outperformed the survey
only model in terms of classification accuracy (ACC and bACC) and CRPS
statistics but not in LPD statistics (see table 1). Since LPD is more
sensitive to outlying observations than CRPS (Gneiting et al., 2007), this
indicates that survey+expert models made more over-confident false presence-
absence predictions than the survey only models. In count predictions, that is
survey data were modeled as count data (Negative-Binomial likelihood (3.3)),
the survey+expert models had clearly better LPD statistics than the survey
only model (table 1). The differences in LPD statistics between alternative
larval abundance models were rather large and the best model was the
survey+expert model where we used all four expert assessment categories,
observation model (3.9).
Table 1: Posterior predictive model comparison for alternative models. First two columns specify the observation model for survey data and expert elicited maps; ”–” in the latter denotes a survey data only model, ”p/a” denotes the presence absence model ((3.4) for survey data and (3.8) for expert assessments), and ”abu” denotes abundance model (3.3) for survey data and four categories model (3.9) for expert assessments. Other abbreviations are lpd for leave-one out cross validation log predictive density, ACC for classification accuracy, bACC for balanced classification accuracy and CRPS for conditional predictive ordinate. Observation models | lpd | ACC | bACC | CRPS
---|---|---|---|---
Survey data | Expert maps | | | |
p/a | – | -83.6 | 0.718 | 0.683 | 0.538
p/a | p/a | -95.2 | 0.737 | 0.724 | 0.436
p/a | abu | -87.3 | 0.737 | 0.721 | 0.465
abu | – | -452 | - | - | -
abu | p/a | -410 | - | - | -
abu | abu | -294 | | |
All the models produced posterior predictive larval distribution maps that
were similar in their overall pattern; that is, the larvae density and
occurrence probability is the highest in the sheltered bays in the northern
parts of the study area and smallest in open sea areas in the south (see
figures 2 and B.4a). However, there are clear differences in the finer scale
patterns of the predictions. For example, the survey+expert models predict
consistently lower densities and occurrence probabilities in southern areas
than the survey only models. The survey+expert models predicted also somewhat
higher larval densities and occurrence probability than the survey only model
in some of the bays in the north whereas in the north-easternmost bay the
survey only models tended to predict higher densities and occurrence
probabilities than the survey+expert model. In general, the survey+expert
models reduce the uncertainty compared to survey only models almost everywhere
in the study area. Only in the south-eastern and south-western areas, the
survey+expert models had larger posterior predictive standard deviation for
the log intensity and logit probability than the survey only models (see
figures 3 and B.4b).
Figure 2: Posterior predictions for larval distribution in the study area with
survey only and survey+experts models and their difference ([survey only] -
[survey+experts]). On top row, the survey data are modeled as occurrence data
and the maps show the posterior predictive mean of the log odds ratio for
larval occurrence and their difference. On bottom row, survey data are modeled
as count data and the maps show the posterior predictive mean of the log
larval density and their difference. In both rows, the expert assessments
included all four categories.
Figure 3: Posterior predictive uncertainty for larval distribution in the
study area with survey only and survey+experts models and their difference
([survey only] - [survey+experts]). On top row, the survey data are modeled as
occurrence data and the maps show the posterior predictive standard deviation
of the log odds ratio for larval occurrence and their difference. On bottom
row, survey data are modeled as count data and the maps show the posterior
predictive standard deviation of the log larval density and their difference.
In both rows, the expert assessments included all four categories.
Experts’ reliability, when measured by the $\bar{c}$ parameters, varied
considerably (see figures 4(b) and B.2). Experts 2-5 were the reliable ones
having significantly positive $\bar{c}$ parameters whereas all the other
experts were unreliable in the sense that the posterior distributions their
$\bar{c}$ parameters did not differ from zero significantly. The third expert
showed some local bias in his/her assessment in a couple of inner bayes in the
north but otherwise the spatial bias terms of the reliable experts were small
(see Figure B.3). The posterior distributions of the parameters of the survey
data model were considerably narrower in the survey+expert models than in the
survey only models (figures 4(a) and B.1). Log larval density and log odds
ratio of larval occurrence responded negatively to depth and openness
(lined3km) and positively to distance to 10m depth curve. The overdispersion
in survey data was significant since the posterior distribution of
overdispersion parameter, $r$, was concentrated in small values. The barrier
model had more significant effect on log larval density and log odds-ratio of
occurrence in survey+expert models than in the survey only models
(a) Survey model parameters
(b) Expert coefficients
Figure 4: Posterior distributions for model parameters in the survey+experts
model with Negative-Binomial observation model for survey data and Beta
observation model for expert assessments.
We also conducted sensitivity analyses for the prior distributions. With too
relaxed barrier model priors, INLA sometimes ran into numerical problems, with
the latent variable $\eta$ evaluating to `NAN`\- or `INF`-values at several
points in the mesh. Numerical problems did not arise when we defined the PC-
priors to correspond the general knowledge on likely scale and spatial
correlation distance in log density and log odds ratio (the priors reported in
Section 2.1). When the BYM-model was left out, or when we restricted its
variance parameters too heavily towards zero by prior distributions, the
survey+expert models effectively ignored the survey data information and
fitted the log abundance and log odds ratio based on experts information only.
However, when expert bias terms (BYM-model components) were _a priori_ allowed
to vary more than the log density or log odds ratio, conflict between survey
data and expert assessment was explained by the bias term, and the expert
assessment did not contribute to the posterior distribution of log density and
log odds ratio in practice. In addition to prior distributions, the resolution
of expert-mesh had an impact on how much weight we gave to expert assessments
so that larger grid cells in the expert-mesh implied reduced the effect of
expert assessments compared to fine grid cells.
## 6 Discussion
Our results show clear differences between the survey only and survey+expert
models both in predictions and parameter inference. In general, the
survey+expert models outperformed the survey only models in their predictive
performance, indicating that adding expert information to SDMs can be
benefitial. The general pattern of the predicted species distributions was
similar across different models, but there were clear differences in some
areas where expert assessments were available (Figure 2). The biggest
differences between the alternative models being in the south. This is
reasonable since all experts, who had marked the southern areas into their
assessment regions, had given the smallest probability for larval occurrence
there (see Figure A.2) whereas there were only a few survey observations from
the southern part of the study region. Hence, the expert assessments have the
largest impact in the south. In addition to changing the predictive mean of
the larvae abundance and occurrence probability, including expert assessments
into the models decreased their predictive uncertainty as well. Similarly, the
posterior distributions of the parameters of the larval density and occurrence
probability models were considerably narrower in the survey+expert models than
in the survey only models (figures 4(a) and B.1). All these results show that
the expert assessments provided significant information on factors affecting
larvae distribution.
These results suggest that expert elicitation can be a viable solution in
studies concerning distribution of species in space and time when resources
for scientific sampling are limited. However, our results demonstrate also
that expert elicitation should be applied with care and optimally combined
with more objective information to reduce the effect of subjective biases.
These findings align well with earlier studies that have shown that the skill
of experts in assessing species distributions varies and depends, for example,
on study area, species and background of an expert (Pearce et al., 2001; Di
Febbraro et al., 2018; Crawford et al., 2020). Bias correction and weighing
expert assessments based on experts’ skills have been emphasized also in many
other expert assessment tasks (O’Hagan et al., 2006; Burgman et al., 2011;
Dias et al., 2018; Perälä et al., 2020).
The expert elicitation process turned out well-functioning. Important first
step was to confirm the commitment of the experts to the process. As a result,
responses were gotten from all candidate experts. For successful
implementation of an expert elicitation by post, it is important that
instructions for filling questionnaire are clear and that the level of details
are kept meaningful. Another, but more consuming option, would be to run a
face-to-face meeting or an expert elicitation workshop to collect the expert’s
views.
The technical implementation of our models was done using the R-INLA software
(Rue et al., 2009) since it allows straightforward implementation of the
barrier model (3.1.1). The barrier model is justified in our application area
since it is scattered by islands, peninsulas and other physical obstacles for
marine species that make traditional stationary spatial random effect an
unrealistic assumption (Bakka et al., 2019). This choice did not come without
price, though, since we were restricted to the built-in likelihood functions
in INLA which affected our practical choice for the expert assessment models
(see Section 3.2 and Appendix C). We believe this choice had only minor effect
on the results though. More fundamental technical consideration is related to
the prior distributions. Firstly, the priors for the hyperparameters of
barrier model had to be chosen carefully to keep the R-INLA calculations
numerically stable. Secondly, to prevent expert assessments from overruling
the survey observations, the mesh resolution and the prior distributions for
the hyperparameters of the expert bias terms (BYM-model parameters) had to be
chosen reasonably. Too small mesh size and too narrow prior for the BYM-model
prevents the model from rejecting biased expert assessments. This is
understandable because we included the expert assessments into our models as
point-wise observations at expert-mesh nodes. Hence, if expert bias terms
(BYM-model components) were restricted to near zero, the combined likelihood
of expert assessments would outweight the likelihood of survey data in
information concerning the log larvae density or log odds ratio of larvae
occurrence probability (there are far more expert-mesh nodes than survey data
observation locations). For this reason, the expert-mesh resolution should be
chosen so that it corresponds to the resolution at which the experts can draw
their maps. Similarly, the variance parameters of the BYM-model should be
given wide priors.
In summary, our results are encouraging for further development of expert
elicitation methods in species distribution modeling. They suggest also that
the methods proposed in this work could also be scaled to larger scale
applications; such as environmental accounting and environmenal management.
Biodiversity loss has become an equally important challenge for humanity and
wellbeing of our planet as climate change. Today, confrontation between nature
and economic wellbeing has started to disappear and general demand for
introducing natural capital into national accounting systems to appear
(Dasgupta, 2021). This demand parallels other calls for biodiversity
preserving measures (e.g., the System of Environmental Economic Accounting
United Nations et al., 2021) in that they all rely on monitoring of species,
biodiversity and ecosystem processes. Current technology does not, however,
allow continuous measuring of biota over space and time so predictive models
are needed to fill in the gaps. Species distribution models are routinely used
for this task so that they are trained with observational data to make
predictions on species occurrence and abundance at spatiotemporal locations
not covered by observations. However, survey data can be expensive and
logistically challenging to collect whereas expert information can often be
collected with relatively inexpensively. For this reason expert elicitation is
a tempting method to collect compelementary information to survey data
(Pearman-Gillman et al., 2020; Murray et al., 2009). The other benefit from
expert elicitation in the context of environmental accounting could be
stakeholder engagement since management and policy decisions are typically
better received by stakeholders and other interest groups if they have been
heard and their knowledge incorporated in the decision making process (La Mere
et al., 2020).
## 7 Conclusions
We have presented a formal Bayesian approach to include expert information
into species distribution modeling. Our approach effectively integrates
spatial expert information with point-wise survey data and allows expert
calibration and assessment of experts’ reliability. Our results show that
expert information can significantly improve species distribution predictions
compared to predictions conditioned on survey data only. This suggests that
expert elicitation can be an efficient tool, for example, in natural resources
management, conservation area planning and in other applications that require
knowledge on distributions of species and other biotic variables. However, our
results also demonstrate that experts’ reliability may considerably vary, and
that even generally reliable experts can have spatially structured biases in
their beliefs. Hence, expert information should be analyzed with care and
preferably calibrated with more objective information sources such as survey
data.
## Acknowledgements
This work has been funded by the Academy of Finland (grant 317255, KK and JV)
and the Finnish Operational Program for the European Maritime and Fisheries
Fund (EMFF, AL and SK).
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The analogy is further limited by the fact that graph embeddings of transformers must have some (possibly arbitrary) node identifier as input in order to tokenize a graph using the node/edge encoding without losing the ability to associate each node with its incident edges.
However, the juxtaposition of the 1-WL and -based limitations on the abilities of GNNs to solve connectivity-based tasks suggests a fundamental gap in capabilities between models that is apparent in multiple theoretical lenses.
§ EXPERIMENTAL DETAILS
§.§ Datasets
We evaluate our model on the diverse graph reasoning tasks presented in GraphQA <cit.>. We used the public code of the dataset available at <https://github.com/google-research/google-research/tree/master/graphqa>. The code to generate the datasets is licensed under the Apache License, Version 2.0. The tasks in the dataset range in difficulty and encompass the following categories:
* Graph-level:
node counting (counting the number of nodes in a graph), edge counting (counting the number of edges in a graph),
cycle check (determining whether a graph contains a cycle), and
triangle counting (counting the number of triangles in a graph).
* Node-level:
node degree (calculating the degree of a given node in a graph).
* Edge-level:
connectivity (finding if there is a path from one node to another),
edge existence (whether a given edge exists in a graph, and
shortest path (finding the length of the shortest path from one node to another).
The graphs used in the experiments in this paper and the corresponding graph reasoning tasks are taken from <cit.>.
There are $1,000$ graphs in the original train set, $500$ graphs in the dev set, and $500$ graphs in the test set. The graphs are generated randomly using Erdős-Rényi (ER) random graph model <cit.>. Graph size ranges from 5 to 20 nodes.
Train set statistics. Average number of nodes: 11.90; average number of edges: 37.01; average node degree: 5.43.
Test set statistics. Average number of nodes: 12.37; average number of edges: 39.79; average node degree: 5.70.
Histogram of minimum cycle lengths for cycle check instances.
While random instances of graph reasoning tasks provide a valuable assessment of the task complexity on realistic graphs, they do not necessarily reflect the “worst case” graph inputs that convey negative results like <Ref> and <Ref>.
For example, the reduction that establishes that cycle check is “as hard as” graph connectivity and the consequential logarithmic-depth hardness results hinge on the consideration of graph instances with $n$ nodes and polynomial cycle length.
However, as witnessed by <Ref>, the shortest cycles observed in 1000 instances of GraphQA cycle check is almost always of length three, and only 3.2% of instances are larger.
As a consequence, identifying the existence of a cycle on the GraphQA dataset is inherently local, which is reflected by a strong performance by heuristic-based GNN solutions (<Ref>)—despite the fact that efficient GNNs for worst-case cycle check do not exist (<Ref>).
For our experiments on the effect of the scale of the number of training data points in the final results we obtain, we use the open-source code of GraphQA available to generate a larger training dataset of 100K examples. We follow the original instructions and parameters to create this larger training dataset.
§.§ Implementation Details
Model Hyperparameters.
We fixed the number of iterations as 1,000,000 and train standard decoder-only transformers with $L = 12$ layers, $m = 768$ embedding dimension, $H = 12$ heads, learning rate $5 \cdot 10^{-4}$, and dropout $0.1$.
These models have an approximate parameter count of 60,000,000.
We used random search <cit.> over the following set of hyperparameters to select a universal architecture for all tasks: The range provided for the learning rate and dropout rate are $[10^{-4}, 10^{-1}]$ and $[0, 0.5]$. The number of layers $L$ and embedding dimension $m$ is selected from $L \in \set{4, 6, 8, 10, 12, 14, 16}$ and $m \in \set{192, 384, 576, 768, 960, 1152, 1344, 1536}$. We
employed the GLU <cit.> activation as a non-linearity.
Model Selection.
We implemented our model in JAX <cit.> and used AdamW <cit.> as the optimizer. Optimal hyperparameters for each task and model were determined by training on the GraphQA$_{Train}$ dataset and evaluating performance on the GraphQA$_{Dev}$ dataset. The results presented in the paper are based on the held-out GraphQA$_{Test}$ dataset.
Hardware Acceleration. All experiments were conducted using Google's TPUv3 and TPUv5e accelerators <cit.>.
§.§ Baseline Results
To rigorously evaluate the performance of transformers on graph reasoning tasks, we compare them against three established categories of baselines:
* Prompting-based methods. These methods provide the LLM with a textual descriptions of the graph and question within the prompt. We consider the following variations and copy the results from the original papers:
* zero-shot. In this approach, the model is given a task description and immediately asked to produce the desired output. No additional examples or demonstrations are provided.
* few-shot. This approach provides the model with a few examples of the task and their desired outputs <cit.>. Unlike traditional training, these examples are included directly in the prompt, allowing the model to learn and adapt during the inference.
* CoT. Chain-of-thought (CoT) prompting <cit.> provides examples each showing step-by-step reasoning, teaching the LLM to generate its own thought processes for tackling new tasks.
* zero-cot. Zero-shot CoT <cit.> builds upon Chain-of-Thought (CoT) prompting by eliminating the need for training examples. The LLM generates its own step-by-step reasoning process using a simple trigger phrase like “Let's think step by step”.
* cot-bag. BAG prompting <cit.> extends cot to improve the performance of LLMs on graph-related tasks by appending “Let's construct a graph with the nodes and edges first” to the prompt.
* Graph-based methods. These models are specifically designed to process graphs as input and are trained task-specific. They leverage the connections between nodes to learn patterns and make predictions, making them ideal for tasks where a graph is involved. We use GCN <cit.>, MPNN <cit.>, and GIN <cit.> from this category. GraphToken <cit.> is a GNN-based model that processes the graph and feed the output of the GNN as soft-tokens to an LLM.
* Transformer models (Ours). The last class of model are task-specific vanilla transformer models <cit.>. The 60M transformer-1K model is the one described above trained on $1,000$ training examples from the GraphQA training set.
To investigate the impact of training data scale, we generated a larger dataset containing $100,000$ examples, ensuring the same distribution as the original training set by using the official GraphQA code and trained 60M transformer-100K on that. The 11B transformer (FT)-1K is a vanilla transformer model that is started with a pre-trained checkpoint of T5 <cit.> and is fine-tuned on the 1K training dataset.
We also include two fine-tuned PaLM 2 <cit.> transformers of size XXS and XS.
Similar to prompting baselines, this model receives a textual description of the graph as input to leverage its textual reasoning capabilities.
The results for zero-shot, zero-cot, few-shot, cot, and cot-bag are taken from <cit.>. Results for soft-prompt and GraphToken are sourced from <cit.>.
We independently evaluated GCN, MPNN, and GIN models on these tasks. We used the original architectures proposed in their respective papers and performed hyperparameter tuning on the GraphQA$_{Dev}$ dataset.
§.§ Further Experimental results
Comparison of train and test scaling on all tasks.
<Ref> presents a comprehensive comparison of the graph reasoning capabilities across various baseline models and our proposed transformer architectures. The results highlight several key findings, which we summarize below:
Transformers Exhibit Strong Performance on Graph-based Reasoning Problems.
While transformers are not explicitly designed for graph reasoning tasks like graph-based models, they demonstrate surprisingly strong performance in this domain. The results of this study indicate that transformers, despite their versatility as a general architecture, can often match or even surpass specialized graph models on a variety of graph reasoning benchmarks.
Transformers Excel at Retrieval Tasks. As proved in <Ref>, retrieval tasks can be solved by transformers. The obtained results confirm that such tasks are relatively easy for transformers as they obtained the full accuracy on most of such tasks. One exception here is the node degree task that GNNs outperform transformers but still transformers perform relatively well. We discuss why GNNs outperform transformers well for this task.
Larger Transformers Excel at Solving Search Tasks. As discussed in <Ref>, transformers are effective for search tasks, albeit requiring a larger number of parameters compared to retrieval tasks. This is empirically evident in the comparison between Transformer-1K and Transformer-1K (pretrained). It's worth noting that the pretrained transformer used here has 11 billion parameters, a significant increase from the 1 million parameters in Transformer-1K.
Transformers Excel at Capturing Global Patterns.
An interesting observation here is the performance gap between Transformers and GNNs across tasks with varying emphasis on local versus global graph structure. Notably:
* Local Structure: The node degree task, which relies heavily on local node information, is best handled by MPNN, a GNN-based model.
* Global Structure: In contrast, tasks like connectivity, triangle counting and shortest path, which require understanding global graph patterns, are dominated by Transformer models.
Notably, vanilla Transformers achieve a remarkable 45% relative improvement over even much larger LLMs augmented with GNN-generated soft prompts (GraphToken) on the shortest path task. This showcases the exceptional ability of Transformers to capture long-range dependencies, a critical factor in understanding global graph structure.
§.§.§ Sample complexity ablations
To develop a more comprehensive understanding of graph reasoning tasks learnability by small transformers, we train a variety of transformers for 1,000,000 steps on each task on a range of sample sizes.
In <Ref>, we demonstrate how the model performance improves as a function of sample complexity.
By doing so, we witness the relative hardness of each task in terms of the marginal benefit of new samples.
* Edge count and node count are “easy” retrieval tasks that can be solved perfectly with as few as 100 samples.
* Edge existence attains near-perfect train and test classification accuracy as the number of training samples approaches 100,000.
* Connectivity, shortest path,
and node degree demonstrate a sharp improvement in evaluation error as a function of the sample size. These models perfectly fit the training set in most sample size regimes, but yield a closer correspondence between training and testing error when trained on 100,000 samples.
* Cycle check and triangle count have persistent gaps between training and testing error and overfit even in the large sample setting.
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22 2118
Towards Syntactic Epistemic Logic
# Towards Syntactic Epistemic Logic
Sergei Artemov
The Graduate Center Address for correspondence: The Graduate Center, The City
University of New York, 365 Fifth Avenue, New York City, NY 10016, USA. The
City University of New York
365 Fifth Avenue New York City NY 10016 USA
<EMAIL_ADDRESS>
###### Abstract
Traditionally, Epistemic Logic represents epistemic scenarios using a single
model. This, however, covers only complete descriptions that specify truth
values of all assertions. Indeed, many—and perhaps most—epistemic descriptions
are not complete. Syntactic Epistemic Logic, SEL, suggests viewing an
epistemic situation as a set of syntactic conditions rather than as a model.
This allows us to naturally capture incomplete descriptions; we discuss a case
study in which our proposal is successful. In Epistemic Game Theory, this
closes the conceptual and technical gap, identified by R. Aumann, between the
syntactic character of game-descriptions and semantic representations of
games.
††volume: 186††issue: 1-4
## 1 Introduction
In this paper, we argue for a paradigm shift in the way that logic and
epistemic-related applications – in particular, game theory – specify
epistemic scenarios.111The preliminary version of this paper was delivered as
an invited talk at the 15th LMPS Congress in 2015 [3]. Given a verbal
description of a situation, a typical epistemic user cherrypicks a “natural
model” (Kripke or Aumann’s) and then regards it as a formalization of the
original description. This approach carries with it two fundamental
deficiencies:
> I. It covers only complete descriptions, whereas many (intuitively most)
> epistemic situations are partially described and cannot be adequately
> specified by a single model.222Epistemic logicians have been mostly aware of
> (I) but this did not stop the wide spread culture of identifying an
> epistemic scenario with a single Kripke model (or Aumann structure in Game
> Theory).
>
> II. The traditional epistemic reading of Kripke/Aumann models requires
> common knowledge of the model which restricts their generality and utility
> even further.
### 1.1 Overspecification
A typical case of (I) is the overspecification problem. Consider the following
description:
A tossed coin lands heads up. Alice sees the coin, Bob does not. (1)
Students in an epistemic logic class normally produce a Kripke S5-model of
this situation as in Figure 1.
$\textstyle{2}$$\textstyle{1}$$\textstyle{h}$$\textstyle{\neg
h}$$\textstyle{R_{B}}$$\textstyle{R_{A,B}}$$\textstyle{R_{A,B}}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 1: Model $\mathcal{M}_{1}$.
In this model, there are two possible worlds 1 and 2, arrows represent
indistinguishability relations $R_{A}$ and $R_{B}$ between worlds, $h$ is a
propositional letter for “heads,” and node 1 represents the real world at
which $h$ holds.
$\mathcal{M}_{1}$ is a model of (1) which, however, overspecifies (1): in this
model there are propositions which are true but do not follow from (1), e.g.,
* •
$\mathbf{K}_{A}\neg\mathbf{K}_{B}h$ \- Alice knows that Bob does not know
$h$;333$\mathbf{K}_{A}$ and $\mathbf{K}_{B}$ are knowledge modalities for
Alice and Bob.
* •
$\mathbf{K}_{B}(\mathbf{K}_{A}h\vee\mathbf{K}_{A}\neg h)$ \- Bob knows that
Alice knows whether $h$;
* •
etc.
We will see in Section 4 that scenario (1) “as is” does not have a single-
model specification at all.
In a situation in which an epistemic scenario is described syntactically but
formalized as a model, a completeness analysis relating these two modes is
required. For example, the Muddy Children puzzle is given syntactically but
then presented as a model tacitly presumed to be commonly known (cf. [14, 16,
17, 18, 19]). In Section 5, we show that this choice of a specifying model can
be justified. However, the Muddy Children case is a fortuitous exception: see
Sections 5 and 6 for more epistemic scenarios without single model
specifications.
Existing approaches to mitigate overspecification include
* •
Supervaluations: given a syntactically defined situation $\cal S$, assume
“$F$ holds in $\cal S$” iff “$F$ is true in all models of $\cal S$.”
This approach has been dominant in mathematical logic with formal theories as
“situations,” and it manifests itself in Gödel’s Completeness Theorem.
* •
Non-standard truth values: Kleene three-valued logic or other, more exotic
ways of defining truth values. This approach has generated mathematically
attractive models, but it has neither dethroned the supervaluation tradition
in mathematical logic, nor changed the ill-founded “natural model culture” in
epistemology.
Here we explore the supervaluation approach in epistemology by representing
epistemic scenarios in a logical language syntactically and considering the
whole class of the corresponding models, not just one cherrypicked model. This
also eliminates problem (II).
## 2 What is Syntactic Epistemic Logic?
The name Syntactic Epistemic Logic was suggested by Robert Aumann (cf. [9])
who identified the conceptual and technical gap between the syntactic
character of game descriptions and the predominantly semantic way of analyzing
games via relational/partition models.
Suppose the initial description ${\cal I}$ of an epistemic situation is
syntactic in a natural language. The long-standing tradition in epistemic
logic and game theory is then to proceed to a specific epistemic model ${\cal
M}_{\cal I}$, and take the latter as a mathematical definition of ${\cal I}$:
$\mbox{\it informal description $\cal I$}\ \ \Rightarrow\ \mbox{\it``natural
model'' ${\cal M}_{\cal I}$.}$ (2)
Hidden dangers lurk within this process: a syntactic description $\cal I$ may
have multiple models and picking one of them (especially declaring it common
knowledge) is not generally sound. Furthermore, if we seek an exact
specification, then only deductively complete scenarios can be represented
(cf. Theorem 4.3). Epistemic scenarios outside this group, which include
situations with asymmetric and less-than-common knowledge (e.g., mutual
knowledge) of conditions, do not have single-model presentations, but can be
specified and handled syntactically.
Through the framework of Syntactic Epistemic Logic, SEL, we suggest making the
syntactic formalization ${\cal S}_{\cal I}$ a formal definition of the
situation described by $\cal I$:
$\mbox{\it description $\cal I$}\ \Rightarrow\ \mbox{\it syntactic
formalization ${\cal S}_{\cal I}$}\Rightarrow\mbox{\it all of ${\cal S}_{\cal
I}$'s models.}$ (3)
The first step from $\cal I$ to ${\cal S}_{\cal I}$ is formalization and it
has its own subtleties which we will not analyze here.
The SEL approach (3), we argue, encompasses a broader class of epistemic
scenarios than a semantic approach (2). In this paper, we provide motivations
and sketch basic ideas of Syntactic Epistemic Logic. Specific suggestions of
general purpose formal systems is a work in progress, cf. [4].
SEL provides a more balanced view of the epistemic universe as being comprised
of two inseparable entities, syntactic and semantic. Such a dual view of
objects is well-established in mathematical logic where the syntactic notion
of a formal theory is supplemented by the notion of a class of all its models.
One could expect equally productive interactions between syntax and semantics
in epistemology as well.
The definition of a game with epistemic conditions, cf. [6, 7], was originally
semantic in a single-model format. In more recent papers (cf. [1, 9]), Aumann
acknowledges the deficiencies of purely semantic formalizations and asks for
some kind of “syntactic epistemic logic” to bridge a gap between the syntactic
character of game descriptions and the semantic way of analyzing games.
In this paper, we look at extensive games; the syntactic epistemic approach to
strategic games has been tried in [2]. However, neither of these papers
considers Epistemic Game Theory in its entirety, including probabilistic
belief models, cf. [12]; we leave this for future studies.
## 3 Logical postulates and derivations
We consider the language of classical propositional logic augmented by
modalities $\mathbf{K}_{i}$, for agent $i$’s knowledge, $i=1,2,\ldots,n$. For
the purposes of this paper, we consider the usual “knowledge postulates” (cf.
[10, 13, 14, 16, 19]) corresponding to the multi-agent modal logic ${\sf
S5}_{n}$:444The same approach works for other epistemic modal logics.
* •
classical logic postulates and rule Modus Ponens $A,A\\!\rightarrow\\!B\vdash
B$;
* •
distributivity:
$\mathbf{K}_{i}(A\\!\rightarrow\\!B)\\!\rightarrow\\!(\mathbf{K}_{i}A\\!\rightarrow\\!\mathbf{K}_{i}B)$;
* •
reflection: $\mathbf{K}_{i}A\\!\rightarrow\\!A$;
* •
positive introspection:
$\mathbf{K}_{i}A\\!\rightarrow\\!\mathbf{K}_{i}\mathbf{K}_{i}A$;
* •
negative introspection:
$\neg\mathbf{K}_{i}A\\!\rightarrow\\!\mathbf{K}_{i}\neg\mathbf{K}_{i}A$;
* •
necessitation rule: $\vdash A\ \ \Rightarrow\ \ \vdash\mathbf{K}_{i}A$.
A derivation in ${\sf S5}_{n}$ is a derivation from ${\sf S5}_{n}$-axioms by
${\sf S5}_{n}$-rules (Modus Ponens and necessitation). The notation
$\vdash A$ (4)
is used to represent the fact that $A$ is derivable in ${\sf S5}_{n}$.
### 3.1 Derivations from hypotheses
For a given set of formulas $\Gamma$ (here called “hypotheses” or
“assumptions”) we consider derivations from $\Gamma$: assume all ${\sf
S5}_{n}$-theorems, $\Gamma$, and use classical reasoning (rule Modus Ponens).
The notation
$\Gamma\vdash A$ (5)
represents $A$ is derivable from $\Gamma$.
It is important to distinguish the role of necessitation in reasoning without
assumptions (4) and in reasoning from a nonempty set of assumptions (5). In
(4), necessitation can be used freely: what is derived from logical postulates
($\vdash A$) is known ($\vdash\mathbf{K}_{i}A$). In (5), the rule of
necessitation is not postulated: if $A$ follows from a set of assumptions
$\Gamma$, we cannot conclude that $A$ is known, since $\Gamma$ itself can be
unknown. However, for some “good” sets of assumptions $\Gamma$, necessitation
is a valid rule (cf. $\Gamma_{3}$ from Example 4.2, ${\sf MC}_{n}$ from
Section 5).
###### Example 3.1
If we want to describe a situation in which proposition $m$ is known to agent
1, we consider the set of assumptions $\Gamma$:
$\Gamma=\\{\mathbf{K}_{1}m\\}.$
From this $\Gamma$, by reflection principle
$\mathbf{K}_{1}m\\!\rightarrow\\!m$ from ${\sf S5}_{n}$, we can derive $m$,
$\Gamma\vdash m.$
Likewise, we can conclude ‘1 knows that 1 knows $m$’ by using positive
introspection:
$\Gamma\vdash\mathbf{K}_{1}\mathbf{K}_{1}m.$
However, we cannot conclude that agent 2 knows $m$:
$\Gamma\not\vdash\mathbf{K}_{2}m.$
This is rather clear intuitively since when assuming ‘1 knows $m$,’ we do not
settle the question of whether ‘2 knows $m$.’555A rigorous proof of this non-
derivability can be made by providing a counter-model. Therefore, there is no
necessitation in this $\Gamma$, since we have $\Gamma\vdash m$ but
$\Gamma\not\vdash\mathbf{K}_{2}m$.
### 3.2 Common knowledge and necessitation
We will also use abbreviations: for “everybody’s knowledge”
${\bf E}X={\mathbf{K}_{1}}X\wedge\ldots\wedge{\mathbf{K}_{n}}X,$
and “common knowledge”
${\bf C}X=\\{X,\ {\bf E}X,\ {\bf E}^{2}X,\ {\bf E}^{3}X,\ \ldots\\}.$
As one can see, ${\bf C}X$ is an infinite set of formulas. Since modalities
$\mathbf{K}_{i}$ commute with the conjunction, ${\bf C}X$ is provably
equivalent to the set of all formulas which are $X$ prefixed by iterated
knowledge modalities:
${\bf C}X=\\{P_{1}P_{2}\ldots P_{k}X\mid k=0,1,2,\ldots,\ \
P_{i}\in\\{\mathbf{K}_{1},\ldots,\mathbf{K}_{n}\\}\\}.$
Naturally,
${\bf C}\Gamma=\bigcup\\{{\bf C}F\mid F\in\Gamma\\}$
that states “$\Gamma$ is common knowledge.”
> The set of formulas ${\bf C}X$ emulates common knowledge of $X$ using the
> conventional modalities $\\{\mathbf{K}_{1},\ldots,\mathbf{K}_{n}\\}$. This
> allows us to speak, to the extent we need here, about common knowledge
> without introducing a special modality and new principles.
The following proposition states that the rule of necessitation corresponds to
common knowledge of all assumptions. If $\Gamma,\Delta$ are sets of formulas,
then $\Gamma\vdash\Delta$ means $\Gamma\vdash X$ for each $X\in\Delta$.
###### Proposition 3.2
A set of formulas $\Gamma$ is closed under necessitation if and only if
$\Gamma\vdash{\bf C}\Gamma$, i.e., that $\Gamma$ proves its own common
knowledge.
###### Proof 3.3
Direction ‘if.’ Assume $\Gamma\vdash{\bf C}\Gamma$ and prove by induction on
derivations that $\Gamma\vdash X$ yields $\Gamma\vdash\mathbf{K}_{i}X$. For
$X$ being a theorem of ${\sf S5}_{n}$, this follows from the rule of
necessitation in ${\sf S5}_{n}$. For $X\in\Gamma$, it follows from the
assumption that $\Gamma\vdash{\bf C}X$, hence $\Gamma\vdash\mathbf{K}_{i}X$.
If $X$ is obtained from Modus Ponens, $\Gamma\vdash Y\\!\rightarrow\\!X$ and
$\Gamma\vdash Y$. By the induction hypothesis,
$\Gamma\vdash\mathbf{K}_{i}(Y\\!\rightarrow\\!X)$ and
$\Gamma\vdash\mathbf{K}_{i}Y$. By the distributivity principle of ${\sf
S5}_{n}$, $\Gamma\vdash\mathbf{K}_{i}X$.
For ‘only if,’ suppose that $\Gamma$ is closed under necessitation and
$X\in\Gamma$, hence $\Gamma\vdash X$. Using appropriate instances of the
necessitation rule in $\Gamma$ we can derive $P_{1}P_{2}P_{3},\ldots,P_{k}X$
for each prefix $P_{1}P_{2}P_{3},\ldots,P_{k}$ with $P_{i}$ is one of
$\mathbf{K}_{1},\mathbf{K}_{2},\ldots,\mathbf{K}_{n}$. Therefore,
$\Gamma\vdash{\bf C}X$ and $\Gamma\vdash{\bf C}\Gamma$.
## 4 Kripke structures and models
A Kripke structure is a convenient vehicle for specifying epistemic assertions
via truth values of atomic propositions and the combinatorial structure of the
set of global states of the system. A Kripke structure
${\cal M}=\langle W,R_{1},R_{2},\ldots,\\!\Vdash\\!\rangle$
consists of a non-empty set $W$ of possible worlds, “indistinguishability”
equivalence relations $R_{1},R_{2},\ldots$ for each agent, and truth
assignment ‘$\ \\!\Vdash\\!\ $’ of atoms at each world. The predicate ‘$F$
holds at $u$’ ($u\\!\Vdash\\!F$) respects Booleans and reads epistemic
assertions as
$\mbox{\em$u\\!\Vdash\\!\mathbf{K}_{i}F\ \ \ $ {\em iff} $\ \ \ $ for each
state $v\in W$ with $uR_{i}v$, $v\\!\Vdash\\!F$ holds}.$
Conceptually, ‘agent $i$ at state $u$ knows $F$’
$(u\\!\Vdash\\!\mathbf{K}_{i}F)$ encodes the situation in which $F$ holds at
each state indistinguishable from $u$ for agent $i$.
A model of a set of formulas $\Gamma$ is a pair $({\cal M},u)$ of a Kripke
structure ${\cal M}$ and a state $u$ such that all formulas from $\Gamma$ hold
at $u$:
${\cal M},u\\!\Vdash\\!F\ \ \mbox{\em for all $F\in\Gamma$.}$
A pair $({\cal M},u)$ is an exact model of $\Gamma$ if
$\Gamma\vdash F\ \ \Leftrightarrow\ \ {\cal M},u\\!\Vdash\\!F.$
An epistemic scenario (a set of ${\sf S5}_{n}$-formulas) $\Gamma$ admits a
semantic definition iff $\Gamma$ has an exact model.
There is a simple criterion to determine whether $\Gamma$ admits semantic
definitions (Theorem 4.3) and we argue that “most” epistemic scenarios lack
semantic definitions. These observations provide a justification for Syntactic
Epistemic Logic with its syntactic approach to epistemic scenarios.
A formula $F$ follows semantically from $\Gamma$,
$\Gamma\models F,$
if $F$ holds in each model $({\cal M},u)$ of $\Gamma$. A well-known fact
connecting syntactic derivability from $\Gamma$ and semantic consequence is
given by the Completeness Theorem777There are many sources in which the proof
of this theorem can be found, e.g., [10, 11, 13, 14, 15, 16, 19].:
$\Gamma\vdash F\ \ \Leftrightarrow\ \ \Gamma\models F.$
This fact has been used to claim the equivalence of the syntactic and semantic
approaches and to define epistemic scenarios semantically by a model. However,
the semantic part of the Completeness Theorem
$\Gamma\models F$
refers to the validity of $F$ in all models of $\Gamma$, not in an arbitrary
single model.
We challenge the model theoretical doctrine in epistemology and show the
limitations of single-model semantic specifications, cf. Theorem 4.3.
### 4.1 Canonical model
The Completeness Theorem claims that if $\Gamma$ does not derive $F$, then
there is a model $({\cal M},u)$ of $\Gamma$ in which $F$ is false. Where does
this model come from?
The standard answer is given by the canonical model construction. In any model
$({\cal M},u)$ of $\Gamma$, the set of truths $\cal T$ contains $\Gamma$ and
is maximal, i.e., for each formula $F$,
$F\in{\cal T}\ \ \ \mbox{ or }\ \ \ \neg F\in{\cal T}.$
This observation suggests the notion of a possible world as a maximal set of
formulas $\Gamma$ which is consistent, i.e., $\Gamma\not\vdash\bot$.
A canonical model ${\cal M}({\sf S5}_{n})$ of ${\sf S5}_{n}$ (cf. [10, 11, 13,
14, 15, 16]) consists of all possible worlds over ${\sf S5}_{n}$.
Accessibility relations are defined on the basis of what is known at each
world: for maximal consistent sets $\alpha$ and $\beta$,
$\alpha R_{i}\beta\ \ \ $ iff $\ \ \ \alpha_{\mathbf{K}_{i}}\subseteq\beta$
where
$\alpha_{\mathbf{K}_{i}}=\\{F\mid\mathbf{K}_{i}F\in\alpha\\},$
i.e.,
$\mbox{\em all facts that are known at $\alpha$ are true at $\beta$}.$
Evaluations of atomic propositions are defined accordingly:
$\alpha\\!\Vdash\\!p_{i}\ \ \mbox{ iff }\ \ p_{i}\in\alpha.$
The standard Truth Lemma shows that Kripkean truth values in the canonical
model agree with possible worlds: for each formula $F$,
$\alpha\\!\Vdash\\!F\ \ \mbox{ iff }\ \ F\in\alpha.$
The canonical model ${\cal M}({\sf S5}_{n})$ of ${\sf S5}_{n}$ serves as a
parametrized universal model for each consistent epistemic scenario $\Gamma$.
Given $\Gamma$, by the well-known Lindenbaum construction, extend $\Gamma$ to
a maximal consistent set $\alpha$. By definition, $\alpha$ is a possible world
in ${\cal M}({\sf S5}_{n})$. By the Truth Lemma, all formulas from $\Gamma$
hold in $\alpha$:
${\cal M}({\sf S5}_{n}),\alpha\ \\!\Vdash\\!\ \Gamma.$
### 4.2 Deductive completeness
###### Definition 4.1
A set of ${\sf S5}_{n}$-formulas $\Gamma$ is deductively complete if
$\Gamma\vdash F\ \ \mbox{ or }\ \ \Gamma\vdash\neg F.$
###### Example 4.2
Consider examples in the language of the two-agent epistemic logic ${\sf
S5}_{2}$ with one propositional variable $m$ and knowledge modalities
$\mathbf{K}_{1}$ and $\mathbf{K}_{2}$.
1\. $\Gamma_{1}=\\{m\\}$, where $m$ is a propositional letter. Neither
$\mathbf{K}_{1}m$ nor $\neg\mathbf{K}_{1}m$ is derivable in $\Gamma_{1}$ and
this can be easily shown on corresponding models. Hence $\Gamma_{1}$ is not
deductively complete.888In classical logic without epistemic modalities,
$\Gamma_{1}$ is deductively complete: for each modal-free formula $F$ of one
variable $m$, either $\Gamma_{1}\vdash F$ or $\Gamma_{1}\vdash\neg F$.
2\. $\Gamma_{2}=\\{{\bf E}m\\}$, i.e., both agents have first-order knowledge
of $m$. However, the second-order knowledge assertions, e.g.,
$\mathbf{K}_{2}\mathbf{K}_{1}m$, are independent,999Again, there are easy
countermodels.
$\Gamma_{2}\not\vdash\mathbf{K}_{2}\mathbf{K}_{1}m\ \ \mbox{\ and }\ \
\Gamma_{2}\not\vdash\neg\mathbf{K}_{2}\mathbf{K}_{1}m.$
This makes $\Gamma_{2}$ deductively incomplete.
3\. $\Gamma_{3}={\bf C}m$, i.e., it is common knowledge that $m$. This set is
deductively complete. Indeed, first note that, by Proposition 3.2,
$\Gamma_{3}$ admits necessitation:101010which is not the case for $\Gamma_{1}$
and $\Gamma_{2}$.
$\Gamma_{3}\vdash F\ \ \Rightarrow\ \ \Gamma_{3}\vdash\mathbf{K}_{i}F,\ \ \
i=1,2.$
To establish the completeness property: for each formula $F$,
$\Gamma_{3}\vdash F\ \ \mbox{ or }\ \ \Gamma_{3}\vdash\neg F,$
run induction on $F$. The base case when $F$ is $m$ is covered, since
$\Gamma_{3}\vdash m$. The Boolean cases are straightforward. Case
$F=\mathbf{K}_{i}X$. If $\Gamma_{3}\vdash X$, then, by necessitation,
$\Gamma_{3}\vdash\mathbf{K}_{i}X$. If $\Gamma_{3}\vdash\neg X$, then, since S5
proves $\neg X\\!\rightarrow\\!\neg\mathbf{K}_{i}X$,
$\Gamma_{3}\vdash\neg\mathbf{K}_{i}X$.
### 4.3 Semantic definitions and complete scenarios
The following observation provides a necessary and sufficient condition for
semantic definability. Let $\Gamma$ be a consistent set of formulas in the
language of ${\sf S5}_{n}$.111111The same criteria hold for any other normal
modal logic which has a canonical model in the usual sense.
###### Theorem 4.3
$\Gamma$ is semantically definable if and only if it is deductively complete.
###### Proof 4.4
The ‘only if’ direction. Suppose $\Gamma$ has an exact model $({\cal M},u)$,
i.e.,
$\Gamma\vdash F\ \ \ \Leftrightarrow\ \ \ {\cal M},u\\!\Vdash\\!F.$
The set of true formulas in $({\cal M},u)$ is maximal: for each formula $F$,
${\cal M},u\\!\Vdash\\!F\ \ \mbox{ or }\ \ {\cal M},u\\!\Vdash\\!\neg F,$
hence $\Gamma$ is deductively complete: for each $F$,
$\Gamma\vdash F\ \ \mbox{ or }\ \ \Gamma\vdash\neg F.$
The ‘if’ direction. Suppose $\Gamma$ is consistent and deductively complete.
Then the deductive closure $\widetilde{\Gamma}$ of $\Gamma$
$\widetilde{\Gamma}=\\{F\mid\Gamma\vdash F\\},$
is a maximal consistent set, hence an element of the canonical model ${\cal
M}({\sf S5}_{n})$. We claim that $({\cal M}({\sf S5}_{n}),\widetilde{\Gamma})$
is an exact model of $\Gamma$, i.e., for each $F$,
$\Gamma\vdash F\ \ \ \Leftrightarrow\ \ \ {\cal M}({\sf
S5}_{n}),\widetilde{\Gamma}\\!\Vdash\\!F.$
Indeed, if $\Gamma\vdash F$, then $F\in\widetilde{\Gamma}$ by the definition
of $\widetilde{\Gamma}$. By the Truth Lemma in ${\cal M}({\sf S5}_{n})$, $F$
holds at the world $\widetilde{\Gamma}$. If $\Gamma\not\vdash F$, then, by
deductive completeness of $\Gamma$, $\Gamma\vdash\neg F$, hence, as before,
$\neg F$ holds at $\widetilde{\Gamma}$, i.e., ${\cal M}({\sf
S5}_{n}),\widetilde{\Gamma}\not\\!\Vdash\\!F$.
Theorem 4.3 shows serious limitations of semantic definitions. Since,
intuitively, deductively complete scenarios $\Gamma$ are exceptions, “most”
epistemic situations cannot be defined semantically.
In Section 5.4, we provide a yet another example of an incomplete but
meaningful epistemic scenario, a natural variant of the Muddy Children puzzle,
which, by Theorem 4.3 does not have a semantic definition, but can
nevertheless be easily specified and analyzed syntactically.
In Section 6, we consider an example of an extensive game with incomplete
epistemic description which cannot be defined semantically, but admits an easy
syntactic analysis.
## 5 The Muddy Children puzzle
Consider the standard Muddy Children puzzle, which is formulated
syntactically.
> A group of $n$ children meet their father after playing in the mud. Their
> father notices that $k>0$ of the children have mud on their foreheads. The
> children see everybody else’s foreheads, but not their own. The father says:
> “some of you are muddy,” then adds: “Do any of you know that you have mud on
> your forehead? If you do, raise your hand now.” No one raises a hand. The
> father repeats the question, and again no one moves. After exactly $k$
> repetitions, all children with muddy foreheads raise their hands
> simultaneously. Why?
### 5.1 Standard syntactic formalization
This can be described in ${\sf S5}_{n}$ with modalities
$\mathbf{K}_{1},\mathbf{K}_{2},\ldots,\mathbf{K}_{n}$ for the children’s
knowledge and atomic propositions $m_{1},m_{2},\ldots,m_{n}$ with $m_{i}$
stating “child $i$ is muddy.” The initial configuration, which we call ${\sf
MC}_{n}$, includes common knowledge assertions of the following assumptions:
1\. Knowing about the others:
$\bigwedge_{i\neq j}[\mathbf{K}_{i}(m_{j})\vee\mathbf{K}_{i}(\neg m_{j})].$
2\. Not knowing about themselves:
$\bigwedge_{i=1,\ldots,n}[\neg\mathbf{K}_{i}(m_{i})\wedge\neg\mathbf{K}_{i}(\neg
m_{i})].$
Transition from the verbal description of the situation to ${\sf MC}_{n}$ is a
straightforward formalization of a given syntactic description to another,
logic friendly syntactic form.
### 5.2 Semantic solution
In the standard semantic solution, the set of assumptions ${\sf MC}_{n}$ is
replaced by a Kripke model: $n$-dimensional cube $Q_{n}$ ([14, 16, 17, 18,
19]). To keep things simple, we consider the case $n=k=2$.
$\textstyle{1,0}$$\textstyle{1,1}$$\textstyle{0,0}$$\textstyle{0,1}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$
Figure 2: Model $Q_{2}$.
Logical possibilities for the truth value combinations121212$1$ standing for
‘true’ and $0$ for ‘false’ of $(m_{1},m_{2})$, namely (0,0), (0,1), (1,0), and
(1,1) are declared possible worlds. There are two indistinguishability
relations denoted by solid arrows (for 1) and dotted arrows (for 2). It is
easy to check that conditions 1 (knowing about the others) and 2 (not knowing
about themselves) hold at each node of this model. Furthermore, $Q_{2}$ is
assumed to be commonly known.
$\textstyle{1,0}$$\textstyle{1,1}$$\textstyle{0,1}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{1,1}$$\textstyle{\bullet}$
Figure 3: Models $\mathcal{M}_{2}$ and $\mathcal{M}_{3}$.
After the father publicly announces $m_{1}\vee m_{2}$, node $(0,0)$ is no
longer possible and model $\mathcal{M}_{2}$ now becomes common knowledge. Both
children realize that in $(1,0)$, child 2 would know whether (s)he is muddy
(no other 2-indistinguishable worlds), and in $(0,1)$, child 1 would know
whether (s)he is muddy. After both children answer “no” to whether they know
what is on their foreheads, worlds $(1,0)$ and $(0,1)$ are no longer possible,
and each child eliminates them. The only remaining logical possibility here is
model ${\cal M}_{3}$. Now both children know that their foreheads are muddy.
### 5.3 Justifying the model
The semantic solution in Section 5.2 adopts $Q_{n}$ as a semantic equivalent
of a theory ${\sf MC}_{n}$. Can this choice of the model be justified? In the
case of Muddy Children, the answer is ‘yes.’
Let $u$ be a node at $Q_{n}$. i.e., $u$ is an $n$-tuple of $0$’s and $1$’s and
$u\\!\Vdash\\!m_{i}$ iff $i$’s projection of $u$ is $1$. Naturally, $u$ is
represented by a formula $\pi(u)$:
$\pi(u)=\bigwedge\\{m_{i}\mid u\\!\Vdash\\!m_{i}\\}\wedge\bigwedge\\{\neg
m_{i}\mid u\\!\Vdash\\!\neg m_{i}\\}.$
It is obvious that $v\\!\Vdash\\!\pi(u)$ iff $v=u$.
By ${\sf MC}_{n}(u)$ we understand the Muddy Children scenario with specific
distribution of truth values of $m_{i}$’s corresponding to $u$:
${\sf MC}_{n}(u)={\sf MC}_{n}\cup\\{\pi(u)\\}.$
So, each specific instance of Muddy Children is formalized by an appropriate
${\sf MC}_{n}(u)$.
###### Theorem 5.1
Each instance ${\sf MC}_{n}(u)$ of Muddy Children is deductively complete and
$(Q_{n},u)$ is its exact model
${\sf MC}_{n}(u)\vdash F\ \ \ \mbox{ iff }\ \ \ \ Q_{n},u\\!\Vdash\\!F.$
Proof:131313We have chosen to present a syntactic proof of Theorem 5.1. A
semantic proof that makes use of bi-simulations can also be given. The
direction ‘only if’ claims that $(Q_{n},u)$ is a model for ${\sf MC}_{n}(u)$
is straightforward. First, $Q_{n}$ is an ${\sf S5}_{n}$-model and all
principles of ${\sf S5}_{n}$ hold everywhere in $Q_{n}$. It is easy to see
that principles ‘knowing about the others’ and ‘not knowing about himself’
hold at each node. Furthermore, as $\pi(u)$ holds at $u$, everything that can
be derived from ${\sf MC}_{n}(u)$ holds at $u$.
To establish the ‘if’ direction, we first note that, by Proposition 3.2,
necessitation is admissible in ${\sf MC}_{n}$: for each $F$,
${\sf MC}_{n}\vdash F\ \ \Rightarrow\ \ \ {\sf MC}_{n}\vdash\mathbf{K}_{i}F.$
The theorem now follows from the statement ${\cal S}(F)$:
> for all nodes $u\in Q_{n}$,
>
> $Q_{n},u\\!\Vdash\\!F\ \ \ \ \Rightarrow\ \ \ {\sf
> MC}_{n}\vdash\pi(u)\\!\rightarrow\\!F$
>
> and
>
> $Q_{n},u\\!\Vdash\\!\neg F\ \ \ \Rightarrow\ \ {\sf
> MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\neg F.$
We prove that ${\cal S}(F)$ holds for all $F$ by induction on $F$.
The case $F$ is one of the atomic propositions $m_{1},m_{2},\ldots,m_{n}$ is
trivial since ${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!m_{i}$, if
$u\\!\Vdash\\!m_{i}$ and ${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\neg
m_{i}$, if $u\\!\Vdash\\!\neg m_{i}$. The Boolean cases are also
straightforward.
The case $F=\mathbf{K}_{i}X$. Consider the node $u^{i}$ which differs from $u$
only at the $i$-coordinate. Without a loss of generality, we may assume that
$u\\!\Vdash\\!m_{i}$ and $u^{i}\\!\Vdash\\!\neg m_{i}$; the alternative
$u\\!\Vdash\\!\neg m_{i}$ and $u^{i}\\!\Vdash\\!m_{i}$ is similar.
Suppose $Q_{n},u\\!\Vdash\\!\mathbf{K}_{i}X$. Then $Q_{n},u\\!\Vdash\\!X$ and
$Q_{n},u^{i}\\!\Vdash\\!X$. By the induction hypothesis,
${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!X\ \ $ and $\ \ {\sf
MC}_{n}\vdash\pi(u^{i})\\!\rightarrow\\!X$.
By the rules of logic (splitting premises)
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!(m_{i}\\!\rightarrow\\!X)\ \ $
and $\ \ {\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!(\neg
m_{i}\\!\rightarrow\\!X)$,
where $\pi(v)_{-i}$ is $\pi(v)$ without its $i$-th coordinate141414Formally,
$\pi(v)_{-i}=\bigwedge\\{m_{j}\mid v\\!\Vdash\\!m_{j},\ j\neq
i\\}\wedge\bigwedge\\{\neg m_{j}\mid v\\!\Vdash\\!\neg m_{j},\ j\neq i\\}$. .
By further reasoning,
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!X$.
By necessitation in ${\sf MC}_{n}$, and distributivity,
${\sf MC}_{n}\vdash\mathbf{K}_{i}\pi(u)_{-i}\\!\rightarrow\\!\mathbf{K}_{i}X$.
By ‘knowing about the others’ principle, and since $\pi(u)_{-i}$ contains only
atoms other them $m_{i}$,
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!\mathbf{K}_{i}\pi(u)_{-i}$,
hence
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!\mathbf{K}_{i}X$,
and
${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\mathbf{K}_{i}X$.
Now suppose $Q_{n},u\\!\Vdash\\!\neg\mathbf{K}_{i}X$. Then
$Q_{n},u\\!\Vdash\\!\neg X$ or $Q_{n},u^{i}\\!\Vdash\\!\neg X$. By the
induction hypothesis,
${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\neg X\ \ $ or $\ \ {\sf
MC}_{n}\vdash\pi(u^{i})\\!\rightarrow\\!\neg X$.
In the former case we immediately get ${\sf
MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\neg\mathbf{K}_{i}X$, by reflection $\neg
X\\!\rightarrow\\!\neg\mathbf{K}_{i}X$. So, consider the latter, i.e., ${\sf
MC}_{n}\vdash\pi(u^{i})\\!\rightarrow\\!\neg X$. As before,
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!(\neg
m_{i}\\!\rightarrow\\!\neg X).$
By contrapositive,
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!(X\\!\rightarrow\\!m_{i}).$
By necessitation and distribution,
${\sf
MC}_{n}\vdash\mathbf{K}_{i}\pi(u)_{-i}\\!\rightarrow\\!(\mathbf{K}_{i}X\\!\rightarrow\\!\mathbf{K}_{i}m_{i}).$
By ‘knowing about others,’ as before,
${\sf
MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!(\mathbf{K}_{i}X\\!\rightarrow\\!\mathbf{K}_{i}m_{i}).$
By ‘not knowing about himself,’ ${\sf MC}_{n}\vdash\neg\mathbf{K}_{i}m_{i}$,
hence
${\sf MC}_{n}\vdash\pi(u)_{-i}\\!\rightarrow\\!\neg\mathbf{K}_{i}X,$
and
${\sf MC}_{n}\vdash\pi(u)\\!\rightarrow\\!\neg\mathbf{K}_{i}X.\vskip 3.0pt
plus 1.0pt minus 1.0pt$
As we see, in the case of Muddy Children given by a syntactic description,
${\sf MC}_{n}(u)$, picking one “natural model” $(Q_{n},u)$ could be justified.
However, in a general setting, the approach
given a syntactic description, pick a “natural model”
is intrinsically flawed: by Theorem 4.3, in many (intuitively, most) cases,
there is no model description at all. Furthermore, if there is a “natural
model,” a completeness analysis in the style of what we did for ${\sf MC}_{n}$
in Theorem 5.1 is required.
### 5.4 Incomplete scenario: Muddy Children Explicit
Here is a natural modification, ${\sf MCE}_{n,k}$, of the standard Muddy
Children.
> A group of $n$ children meet their father after playing in the mud. Each
> child sees everybody else’s foreheads. The father says: “$k$ of you are
> muddy” after which it becomes common knowledge that all children know
> whether they are muddy. Why?
This description does not specify whether children initially know if they are
muddy; hence the initial description of ${\sf MCE}_{n,k}$ is, generally
speaking, not complete151515In particular, prior to father’s announcement
${\sf MCE}_{2,2}$ does not specify whether $\mathbf{K}_{1}m_{1}$ holds or
not.. By Theorem 4.3, the initial ${\sf MCE}_{2,2}$ is not semantically
definable. Therefore, ${\sf MCE}_{2,2}$ cannot be treated by “natural model”
methods.
However, here is a syntactic analysis of ${\sf MCE}_{n,k}$ which can be shaped
as a formal logical reasoning within an appropriate extension of ${\sf
S5}_{n}$.
> After father’s announcement, each child knows that if she sees $k$ muddy
> foreheads, then she is not muddy, and if she sees $k\\!-\\!1$ muddy
> foreheads, she is muddy: this secures that each child knows whether she is
> muddy. Moreover, everybody can reflect on this reasoning and this makes it
> common knowledge that each child knows whether she is muddy.
### 5.5 Some additional observations
If we want to go beyond complete epistemic scenarios, we need a mathematical
apparatus to handle classes of models, and not just single models. The format
of syntactic specifications in some version of the modal epistemic language is
a viable candidate for such an apparatus.
The traditional model solution of ${\sf MC}_{n}$ without completeness analysis
uses a strong additional assumption – common knowledge of a specific model
$Q_{n}$ and hence, strictly speaking, does not resolve the original Muddy
Children puzzle; it rather corresponds to a different scenario of a more
tightly controlled epistemic states of agents, e.g.,
> A group of robots programmed to reason about model $Q_{n}$ meet their
> programmer after playing in the mud. …
One could argue that the given model solution of ${\sf MC}_{n}$ actually
codifies some deductive solution in the same way that geometric reasoning is
merely a visualization of a rigorous derivation in some sort of axiom system
for geometry. This is a valid point which can be made scientific within the
framework of Syntactic Epistemic Logic.
## 6 Syntactic Epistemic Logic and games
Consider a variant Centipede Lite, CL, of the well-known Centipede game (cf.
[17]) with risk-averse rational players Alice and Bob. No cross-knowledge of
rationality, let alone common knowledge, is assumed!
$\textstyle{2,1}$$\textstyle{\bullet}$$\textstyle{1(A)}$$\textstyle{1,4}$$\textstyle{\bullet}$$\textstyle{2(B)}$$\textstyle{4,3}$$\textstyle{\bullet}$$\textstyle{3(A)}$$\textstyle{3,6}$
Figure 4: Centipede game tree
CL admits the following rigorous analysis.
> At 3, Alice plays down. At 2, Bob plays down because he is risk-averse and
> cannot rule out that Alice plays down at 3 (since it is true). At 1, Alice
> plays down because she cannot rule out Bob’s playing down at 2. So, CL has
> the so-called Backward Induction solution “down at each node.”
CL is not complete (epistemic assumptions, such as Bob knows that Alice plays
“across” at 3, are not specified), hence CL cannot be defined by a single
Kripke/Aumann model.
## 7 Incomplete and complete scenarios
How typical are deductively incomplete epistemic scenarios? We argue that this
is the rule rather than the exception. Epistemic conditions more flexible than
common knowledge of the game and rationality (mutual knowledge of rationality,
asymmetric epistemic assumptions when one player knows more than the other,
etc.) lead to semantic undefinability.
Semantically non-definable scenarios are the “dark matter” of the epistemic
universe: they are everywhere, but cannot be visualized as a single model. The
semantic approach does not recognize these “dark matter” scenarios; SEL deals
with them syntactically.
The question remains: how manageable are semantic definitions of deductively
complete scenarios?
### 7.1 Cardinality and knowability issue
Models of complete $\Gamma$’s provided by Theorem 4.3 are instances of the
canonical model ${\cal M}({\sf S5}_{n})$ at nodes $\widetilde{\Gamma}$
corresponding to $\Gamma$. This generic solution is, however, not satisfactory
because of the highly nonconstructive nature of the canonical model ${\cal
M}({\sf S5}_{n})$.
As was shown in [8], the canonical model ${\cal M}({\sf S5}_{n})$ for any
$n\geq 1$ has continuum-many possible worlds even with just one propositional
letter. This alone renders models $({\cal M}({\sf
S5}_{n}),\widetilde{\Gamma})$ not knowable under any reasonable meaning of
“known.” The canonical model for ${\sf S5}_{n}$ is just too large to be
considered known and hence does not a priori satisfy the knowability of the
model requirement II from Section 1.
This observation suggests that the question about existence of an
epistemically acceptable (“known”) model for a given deductively complete set
$\Gamma$ requires a case-by-case consideration.
### 7.2 Complexity considerations
Epistemic models of even simple and complete scenarios can be prohibitively
large compared to their syntactic descriptions. For example, the Muddy
Children model $Q_{n}$ is exponential in $n$ whereas its syntactic description
${\sf MC}_{n}$ is quadratic in $n$.
Consider a real-life epistemic situation after the cards have been initially
dealt in the game of poker. One can show that for each distribution of cards,
its natural syntactic description in epistemic logic is deductively complete
([5]) and hence admits a model characterization. Moreover, it has a natural
finite model of the type given in [14] with hands as possible worlds and with
straightforward knowledge relations. However, with 52 cards and 4 players
there are over $10^{24}$ different combinations of hands. This yields that
explicit formalization of the model not practical. Players reason using
concise syntactic descriptions of the rules of poker and of its “large” model
in the natural language, which can also be syntactically formalized in some
kind of extension of epistemic logic.
In this and some other real life situations, models are prohibitively large
whereas appropriate syntactic descriptions can be quite manageable.
## 8 Further observations
An interesting question is why the traditional semantic approach, despite its
aforementioned shortcomings, produces correct answers in many situations. One
of possible reasons for this is pragmatic self-limitation.
Given a syntactic description $\cal D$, we intuitively seek a solution that
logically follows from $\cal D$. Even if we reason on a “natural model” of
$\cal D$, normally overspecified, we try not to use features of the model that
are not supported by $\cal D$. If we conclude a property $P$ by such self-
restricted reasoning about the model, then $P$ indeed logically follows from
$\cal D$.
> This situation resembles Geometry, in which we reason about “models”, i.e.,
> combinations of triangles, circles, etc., but have a rigorous system of
> postulates in the background. We are trained not to venture beyond given
> postulates even in informal reasoning.
Such an ad hoc pragmatic approach needs a scientific foundation, which could
be provided within the framework of Syntactic Epistemic Logic.
## 9 Syntactic Epistemic Logic suggestions
The Syntactic Epistemic Logic suggestion, in brief, is to make an appropriate
syntactic formalization of an epistemic scenario its formal specification.
This extends the scope of scientific epistemology and offers a remedy for two
principal weaknesses of the traditional semantic approach. The reader will
recall that those weaknesses were the restricting single model requirement and
a hidden assumption of the common knowledge of this model.
SEL suggests a way to handle incomplete scenarios which have rigorous
syntactic descriptions (cf. Muddy Children Explicit, Centipede Lite, etc.).
SEL offers a scientific framework for resolving the tension, identified by R.
Aumann [9], between a syntactic description and its hand-picked model. If,
given a syntactic description $\Gamma$ we prefer to reason on a model $\cal
M$, we have to establish completeness of $\Gamma$ with respect to $\cal M$.
Appropriate syntactic specifications could also help to handle situations for
which natural models exist but are too large for explicit presentations.
SEL can help to extend Epistemic Game Theory to less restrictive epistemic
conditions. A broad class of epistemic scenarios does not define higher-order
epistemic assertions and rather addresses individual knowledge, mutual and
limited-depth knowledge, asymmetric knowledge, etc. and hence is deductively
incomplete and has no exact single model characterizations. However, if such a
scenario allows a syntactic formulation, it can be handled scientifically by a
variety of mathematical tools, including reasoning about its models.
Since the basic object in SEL is a syntactic description of an epistemic
scenario rather than a specific model, there is room for a new syntactic
theory of updates and belief revision.
### Acknowledgements
The author is grateful to Adam Brandenburger, Alexandru Baltag, Johan van
Benthem, Robert Constable, Melvin Fitting, Vladimir Krupski, Anil Nerode,
Elena Nogina, Eoin Moore, Vincent Peluce, Tudor Protopopescu, Bryan Renne,
Richard Shore, and Cagil Tasdemir for useful discussions. Special thanks to
Karen Kletter for editing early versions of this text.
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|
# Large scale compressible turbulence in the ism of CSWA13, a star-Forming
Lensed Galaxy at z = 1.87 with outflowing wind
Itzhak Goldman 1,2 1Physics Department Afeka College, Tel Aviv 6998812, Israel
2 Astrophysics Department, Tel Aviv University, Tel Aviv 6997801 , Israel
###### Abstract
Recently, Keerthi Vasan G. et al. (2024)presented spatially resolved
observations of a wind outflow in CSWA13, a gravitationally lensed Star-
Forming galaxy at $z=1.87$. The gravitational lensing allowed for a
substantially improved spatial and kinematic resolution of the wind and of the
nebular gas. In this paper we take advantage of the resolved data to test for
the existence of turbulence and to study its nature. We derive the spatial
structure functions of the residual nebular and wind velocities along the
major axis of the galaxy. The structure functions, of both velocity fields,
reveal a supersonic compressible large scale turbulence. The turbulent
timescale corresponding to the largest scale is about 200 Myr, an order of
magnitude larger than the estimated age of the wind and of the young stars.
This implies that the turbulence in the ism formed well before the wind and
the young stars. Given the large spatial scale of the turbulence, it is
plausible that the source of the turbulence is a large scale one e.g. a merger
or tidal event that triggered the formation of molecular clouds, in the cores
of which the young stars formed. A steepening of the structure functions on
the smaller scales provides an estimate of the effective depth along the line
of sight of the turbulent layer. The latter turns out to be $\sim 2kpc$.
###### keywords:
galaxy outflows , galaxy evolution, interstellar turbulence
## 1 introduction
High redshift galaxies are characterized by high star formation rates as well
as outflowing winds, generated by the young stars, e.g. (Bournaud et al.,
2009; Hoffmann et al., 2022; Rizzo et al., 2021; Sanders et al., 2023; Shah et
al., 2022) The high rate of star formation is attributed to the assembly
process of the galaxy, involving gas inflow from the circum galactic medium
(CGM) and also by more violent events such as mergers that lead to formation
of molecular clouds and eventually to star formation. Such mergers could
result in large scale shocks and also in large scale turbulence, created by
the shocks and by the instabilities caused by the mergers.
Observations of high redshift galaxies indeed display velocity dispersions
that are usually interpreted as manifestation of turbulence. It has been
argued that accretion onto disk galaxies can generate large scale turbulence,
in particular at the disk outskirts, e. g. (Forbes et al., 2023; Goldman &
Fleck, 2023). Turbulence can be generated also by mergers and tidal
interactions. To establish the existence of turbulence and moreover, to
understand its nature, a power spectrum or structure function of the velocity
field are needed. This in turn demands observations with high enough spatial
resolution which, for galaxies at high redshifts, are challenging.
Gravitational lensing can help with this regard. A recent paper (Keerthi Vasan
G. et al., 2024) presented a study of a wind outflow in CSWA13, which is a
gravitationally lensed star-forming galaxy at $z=1.87$. The gravitational
lensing allowed for a substantially improved spatial and kinematic resolution.
The authors obtained, among other results, two velocity fields along the major
axis of the galaxy: the nebular gas velocity traced by the $C_{|||]}$ emission
line. that represents also the velocity of the young stars embedded in the
nebular gas and the wind velocity traced by the $Si_{||}^{*}$ florescent
emission line. Each of these velocity fields , exhibits a large scale shear.
In the present paper we set to check wether these velocity fields can be used
to test for the existence of turbulence, and if so to obtain its
characteristics.
The two residual velocity fields are obtained in section 2. The
autocorrelation functions are presented in section 3. In section 4 we obtain
the structure functions. In the appendix, the theoretical structure function
of a quantity that is the result of integration along the line of sight
direction, is derived. This provides an estimate of the depth of the turbulent
layer. Discussion is presented in section 5.
## 2 The residual velocity fields
We digitized the velocity curves of Fig. 8 in Keerthi Vasan G. et al. (2024)
and obtained the nebular and wind velocity as functions of position along the
galactic major axis.The nebular velocity and the wind velocity exhibit each a
large scale shear. We subtracted from each velocity field the corresponding
large scale shear, and then removed the remaining mean value. Doing so
resulted in two residual velocity fields along the major galactic axis. We
derived the autocorrelation function and the structure function for each.
Structure functions rather as function of$x/D$.than power spectra were
employed since the former are more detailed on the smaller and medium spatial
scales. They are also more reliable at treating data at the borders of the
data domain (Nestingen-Palm et al., 2017).
### 2.1 The residual nebular velocity along the major galaxy axis
The velocity posses a large scale shear of $96.2km\ s^{-1}/(6.43kpc)$. After
subtracting the shear, and the remaining mean value , the residual velocity is
obtained and is displayed in Fig. 1.
Figure 1: The residual nebular velocity in units of $km/s$ as function of
position along the major axis, in units of $207.5pc$.
### 2.2 The residual wind velocity along the major galaxy axis
The velocity posses a large scale shear of $152.2km\ s^{-1}/(6.43kpc)$. After
subtracting the shear, and the remaining mean value , the residual velocity is
obtained and is displayed in Fig. 2.
Figure 2: The residual wind velocity in units of $km/s$ as function of
position along the major axis, in units of $207.5pc$.
## 3 autocorrelation functions
The one-dimensional autocorrelation function, $C(x)$, of a residual velocity
$v(x)$ (with a zero mean value) is
$C(x)=<v(x+x^{\prime})v(x^{\prime})>,$ (1)
The brackets indicate ensemble average which by invoking the ergodic principle
can be replaced by an average over $x^{\prime}$. Here $x$ denotes the spatial
lag between two positions along the major galaxy axis.
### 3.1 The autocorrelation function of the residual nebular velocity
The computed observational normalized autocorrelation function of the residual
nebular velocity is displayed in Fig, 3. It implies that the residual values
of the nebular velocity are in fact correlated over large spatial range,
comparable to the size of the major axis.
Figure 3: The normalized observational autocorrelation function of the
residual nebular velocity as function of the spatial lag, in units $207.5pc$.
### 3.2 The autocorrelation function of the residual wind velocity
Fig.4. presents the normalized autocorrelation function of the wind velocity.
It exhibits long range correlation similar to that of the nebular velocity.
Figure 4: The normalized observational autocorrelation function of the
residual wind velocity as function of the spatial lag, in units of $207.5pc$.
## 4 structure functions
Long range autocorrelation could be a signature of turbulence. In order to
test for the existence of turbulence and understand its nature, we evaluate
the structure functions for the two residual velocity fields.
The one-dimensional structure function of a quantity $f(x)$ defined along a
straight line is
$S_{f}(x)=<\left(f(x^{\prime}+x)-f(x^{\prime})\right)^{2}>=2C_{f}(0)-2C_{f}(x),$
(2)
with the lag $x$ being the difference between two positions. $C_{f}(x)$ is the
auto correlation function of $f(x)$ defined as
$C_{f}(x)=<f(x^{\prime}+x)f(x)>.$ (3)
In the following, the computed structure functions of the observational
residual nebular velocity, and of the residual wind velocity are presented,
### 4.1 The observational structure function of the residual nebular velocity
Fig.5 displays the structure function of the observational residual nebular
velocity. The blue line has a logarithmic slope of 1. The orange line has a
logarithmic slope of 2. In the appendix it is shown that the structure
function, evaluated along a lateral line, of a quantity that is an integral
over the line of sight direction increases the logarithmic slope by 1 when the
lateral lag is smaller than the effective depth of the turbulent layer.
The structure function at the largest lag is $2<C(0)^{2}=391(km/s)^{2}$. Thus,
the one dimensional rms turbulent velocity is $13.9km/s$. Assuming isotropic
turbulence, the three-dimensional turbulent velocity is $24km/s$
Figure 5: The observational structure function of the residual nebular
velocity in units of $(km/s)^{2}$ as function of the spatial lag, in units of
$207.5pc$. The asymptotes have logarithmic slopes of 1 and 2.
### 4.2 The observational structure functions of the residual wind velocity
Fig.6 displays the structure function of the residual wind velocity. The blue
line has a logarithmic slope of 1. The orange line has a logarithmic slope of
2. This behavior is similar to that of the structure function of the residual
nebular velocity.
The structure function at the largest lag is $2<C(0)^{2}=291(km/s)^{2}$,
implying a one-dimensional rms turbulent velocity of $12.1km/s$. Assuming
isotropic turbulence, the three-dimensional turbulent velocity is $21km/s$
Figure 6: The observational structure function of the residual wind velocity,
in units of $(km/s)^{2}$ as function of the spatial lag in units of $207.5pc$.
The asymptotes have logarithmic slopes of 1 and 2.
## 5 discussion
### 5.1 The nature of the turbulence
The observational structure functions of the two residual velocity fields
have, each, a logarithmic slope equaling 1 on the large spatial scales and a
logarithmic slope of 2 on the small spatial scales. This dependence
characterizes compressible turbulence with a one-dimensional power spectrum
$\propto k^{-2}$ with $k$ denoting the one -dimensional spatial wave number.
This power spectrum is steeper than the Kolmogorov power spectrum, which
corresponds to subsonic incompressible turbulence with a one-dimensional power
law with exponent of $-5/3$ and structure function with logarithmic slope of
2/3 and 5/3 for the large and small scales, respectively.
Such a power spectrum was derived by Burgers (1948) describing a hierarchy of
shocks in compressible gas. Compressible turbulence power spectra were
observed in HI intensity maps in the Milky Way (MW) galaxy (Green, 1993)and in
the SMC (Stanimirovic et al., 1999). This power spectrum has been observed
also in molecular clouds (Larson, 1981; Leung et al., 1982; Dame et al.,
1986),in the HII region Sharpless 142 (Roy & Joncas, 1985). It has been found
in a shocked cloud near the Milky Way galaxy center (Contini & Goldman, 2011),
and recently in the Gamma ray emission from the large Magellanic Cloud.
(Besserglik & Goldman, 2021). It has been also obtained in numerical
simulations e.g. (Passot et al., 1988; Vázquez-Semadeni et al., 1997; Kritsuk
et al., 2007; Federrath et al., 2021).
The steeper slope signals that (unlike in the Kolmogorov spectrum) the rate of
energy transfer in the turbulence cascade is not constant but decreases with
increasing wavenumber. This is expected in a compressible turbulence since
part of the energy at a given wavenumber in the cascade, is diverted to
compression of the gas. Indeed, a theoretical derivation of the compressible
turbulence power spectrum based on this argument has been obtained (Goldman,
2021a).
The three-dimensional rms turbulent velocities estimated in section 4.are
supersonic, in line with the shape of the structure functions.
The turbulence timescale of the largest scale eddies is $\sim l_{0}/v_{0}\sim
200$ Myr where $l_{0}$ is the largest spatial scale and $v_{0}$ is the
turbulent velocity on this scale. This timescale represents the eddy
correlation time on the largest spatial scale and therefore a lower bound on
the time span over which the turbulence was generated. This time span is an
order of magnitude larger than the age of the young stars and the outflowing
wind. Thus, the turbulence is older than the young stars and the wind that was
created by the latter. The timescales of the large scale shears are about 20
Myr, so they formed at the same time as the wind. The emerging picture is that
the young stars as well as the wind and the shear were formed on the
background of the turbulent interstellar gas. The generating source of this
large scale turbulence, must itself be correlated over the largest scale of
the turbulence, rather than a collection of smaller scale sources. The
probable source is a merger or a close tidal interaction with a smaller
galaxy.
### 5.2 The effective depth of the turbulent region
. The emitted photons that determine the velocities originate from different
depths along the line of sight. The issue of power spectra and structure
functions of quantities which are the result of integration along the line-of-
sight has been addressed by e.g. (Stutzki et al., 1998; Goldman, 2000, 2021b;
Lazarian & Pogosyan, 2000; Miville-Deschênes et al., 2003a). These authors
concluded that when the lateral spatial scale is smaller than the depth of the
layer, the logarithmic slope of the power spectrum steepens by $-1$ compared
to its value when the lateral scale is large compared to the depth. The
logarithmic slope of the structure function increases by 1. This behavior was
indeed revealed in observational power spectra of Galactic and extra Galactic
turbulence ( e.g. Elmegreen et al. (2001),Block et al. (2010), Miville-
Deschênes et al. (2003b) ) and in solar photospheric turbulence (Abramenko &
Yurchyshyn, 2020)).
In the appendix, the theoretical structure function of a quantity that is the
result of integration along the line of sight, is obtained. For the specific
case of compressible turbulence we found that $D=1.83x_{tr}$, where $D$ is the
effective depth of the turbulent layer and $x_{tr}$ is the observational lag
marking the slopes transition. The effective depth is the depth of a layer
with depth independent of the lateral coordinate, that would yield the
observational structure function.
From Fig.5 the observational transition lag for the nebular velocity $(1.31\pm
0.04)kpc$ yielding an effective depth of $(2.4\pm 0.07)kpc$. The observational
transition lag of the wind velocity structure function, from Fig.6, is
$(1.06\pm 0.04)kpc$ implying an effective depth of $(1.9.\pm 0.07)kpc$.
## Appendix A The theoretical structure function of a quantity that is the
result of integration along the line of sight
Consider a function $f(x)$ where $x$ is a straight line in a lateral
direction. and is an integral along the line of sight:
$f(x)=\int_{0}^{D}g(x,z)dz.$ (4)
Here, $z$ is the line of sight coordinate and $D$ the depth. A plane parallel
geometry is assumed for simplicity.
The autocorrelation function of $f(x$ is:
$\displaystyle C_{f}(x)=<f(x+x^{\prime})f(x)>=$ (5)
$\displaystyle\int_{0}^{D}\int_{0}^{D}<g(x^{\prime},z)g(x+x^{\prime},z^{\prime})>dzdz^{\prime}=$
$\displaystyle\int_{0}^{D}\int_{0}^{D}C_{g}(x,z-z^{\prime})dzdz^{\prime}.$
,
The autocorrelation function $C_{g}(x,z-z^{\prime})$ can be expressed by the
two-dimensional power spectrum, $P_{2}(k_{x},k_{z})$,
$\displaystyle\hskip-42.67912ptC_{g}(x,z-z^{\prime})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(k_{x}x+k_{z}(z-z^{\prime}))}$
(6) $\displaystyle P_{2}(k_{x},k_{z})dk_{x}dk_{z}.$
leading to
$C_{f}(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{ik_{x}x}P_{2}(k_{x},k_{z})\frac{\sin^{2}\left(k_{z}D/2\right)}{\left(k_{z}D/2\right)^{2}}dk_{x}dk_{z}.$
(7)
from equations (2) and (A4) follows the expression for the structure function
$\displaystyle
S_{f}(x,D)\propto\int_{0}^{\infty}\int_{0}^{\infty}\sin^{2}(k_{x}x/2)\frac{\sin^{2}\left(k_{z}D/2\right)}{\left(k_{z}D/2\right)^{2}}$
(8) $\displaystyle P_{2}(k_{x},k_{z})dk_{x}dk_{z}.$
In the case of a turbulence with a one-dimensional power spectrum which is a
power law with index $-m$, the two dimensional power spectrum is
$P_{2}(k_{x},k_{z})\propto\left(k_{x}^{2}+k_{z}^{2}\right)^{-(m+1)/2.}\ \ \ \
\ \ $ (9)
resulting in
$\displaystyle
S_{f}(x,D)\propto\int_{0}^{\infty}\int_{0}^{\infty}\sin^{2}(k_{x}x/2)\frac{\sin^{2}\left(k_{z}D/2\right)}{\left(k_{z}D/2\right)^{2}}$
(10) $\displaystyle\left(k_{x}^{2}+k_{z}^{2}\right)^{-(m+1)/2}dk_{x}dk_{z}.$
It is convenient to define the dimensionless variables
$\eta=k_{z}D/2.\ \ \ \ \ \ \ ;\ \ \ \mu=k_{x}D/2.$
Figure 7: Theoretical structure function, which is an integral along the line
of sight, as function of$x/D$. The straight lines have logarithmic slopes
equaling 1 and 2. Figure 8: Theoretical structure function of a quantity,
which is an integral along the line of sight, as function of$x/D$. The
straight line has a logarithmic slope of 1.5.
The structure function of equation (A7) can be expressed as
$S_{f}(x,D)\propto\int_{0}^{\infty}I(\mu)\sin^{2}\left(\mu x/D\right)d\mu.$
(11)
where $I(\mu)$ is
$I(\mu)=\int_{0}^{\infty}\left(\mu^{2}+\eta^{2}\right)^{-(m+1)/2}\frac{\sin^{2}\eta}{\eta^{2}}d\eta.$
Equation (A9) implies that the structure function argument is $x/D$. Also,
inspection of equations (A9) and (A10) reveals that for $x<<D$ the structure
function is proportional to $x^{m}$ while for $x>>D$ it is proportional to
$x^{m-1}$ .
A numerical solution for the case of $m=2$ is presented in Fig.7together power
laws with exponents 1 and 2. In order to find the value of $x_{tr}/D$ where
$x_{tr}$ denotes the transition lag between the two slopes a power law with
exponent of 1.5 is plotted in Fig.8 together with the structure function. The
value of $x_{tr}/D$ is taken to be the value for which the logarithmic slope
of the structure function equals 1.5, i.e. where the straight line is tangent
to the structure function. The result is $x_{tr}/D\sim 0.547$ thus $D\sim
1.83x_{tr}$.
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|
# Socioeconomic agents as active matter in nonequilibrium Sakoda-Schelling
models
Ruben Zakine<EMAIL_ADDRESS>Chair of Econophysics and Complex
Systems, École polytechnique, 91128 Palaiseau Cedex, France LadHyX, CNRS,
École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Jérôme Garnier-Brun Chair of Econophysics and Complex Systems, École
polytechnique, 91128 Palaiseau Cedex, France LadHyX, CNRS, École
polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Antoine-Cyrus Becharat Chair of Econophysics and Complex Systems, École
polytechnique, 91128 Palaiseau Cedex, France LadHyX, CNRS, École
polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Michael Benzaquen Chair of Econophysics and Complex Systems, École
polytechnique, 91128 Palaiseau Cedex, France LadHyX, CNRS, École
polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Capital Fund Management, 23 Rue de l’Université, 75007 Paris, France
(September 3, 2024)
###### Abstract
How robust are socioeconomic agent-based models with respect to the details of
the agents’ decision rule? We tackle this question by considering an
occupation model in the spirit of the Sakoda-Schelling model, historically
introduced to shed light on segregation dynamics among human groups. For a
large class of utility functions and decision rules, we pinpoint the
nonequilibrium nature of the agent dynamics, while recovering the equilibrium-
like phase separation phenomenology. Within the mean-field approximation we
show how the model can be mapped, to some extent, onto an active matter field
description. Finally, we consider non-reciprocal interactions between two
populations, and show how they can lead to non-steady macroscopic behavior. We
believe our approach provides a unifying framework to further study geography-
dependent agent-based models, notably paving the way for joint consideration
of population and price dynamics within a field theoretic approach.
## I Introduction
Will a collective system in which individuals share the same common goal ever
reach an optimal state? This nontrivial question is at the very core of strong
debates among economists, notably because the notion of “optimal state” is
intrinsically political and most often ill-defined. Despite the common idea
that a system made of agents individualistically improving their outcome will
spontaneously converge by the action of the “invisible hand” to an optimal
collective state, simple models have been shown to contradict this belief [1,
2, 3]. A well documented example of such system is the celebrated Schelling
model [4]. The latter can be considered to be a variant of the model
previously111To be perfectly precise, the first mention of Sakoda’s model can
be traced back to his unpublished PhD thesis completed in 1946, while
Schelling’s work can be found in a 1969 working paper [5]. In any case, there
is no reason to think either author took inspiration from the other, the
objective of the papers being clearly quite different. introduced by Sakoda
[6], and will thus be referred henceforth as the Sakoda-Schelling model. To
understand some aspects of urban segregation in post-WWII American cities, and
more widely of urban and social dynamics, both authors proposed simple lattice
models of idealized cities. Each site, representing an accommodation, can be
empty or occupied by an agent belonging to one of two sub-populations in the
system. Interestingly, Schelling observed that when introducing a slight
preference for agents to be surrounded by neighbors of their own group, the
system evolves towards configurations with completely segregated regions.
While in fact not very well suited to explain urban segregation, which is
intimately related to past and present public policies rather than self-
organization [7, 8], the model illustrates how the micromotives of the agents
may lead to unanticipated macrobehavior [9].
Along the years, the Sakoda-Schelling model has attracted further attention of
statistical physicists [10, 11, 12, 13], due to its simple microscopic rules,
its paradoxical macroscopic consequences and its unconventional non-local
particle moves. To the usual end of bridging the gap from _micro to macro_ ,
mappings onto equilibrium systems were suggested [14], but with limited
analytical results. To gain a more in-depth understanding of the mechanism
through which individual choices may lead to sub-optimal collective outcomes,
Grauwin et al. introduced a modified version of the Schelling model with a
single type of agent occupying a lattice divided in pre-defined neighborhoods,
or blocks [15]. In this occupation model, the agents now base their decisions
on the neighborhood density, which is identical for all the agents in a given
block. This fixed neighborhood structure then allows to describe analytically
the steady state as the minimizer of a free energy, and to recover a
nontrivial phase with suboptimal jam-packed neighborhoods. Subsequent works
have then explored variations of these different models focusing on the effect
of altruistic agents [16], dynamics close to criticality [17, 18, 19] or habit
formation [20].
Even in the seemingly simpler occupation problem of Grauwin et al. [15],
several questions persist, both from the socioeconomic and statistical physics
perspectives. In particular, the role of the specific decision rule and the
precise nature of neighborhoods on the phenomenology of the model remain
unclear. Indeed, to allow for the standard techniques of statistical mechanics
to be applicable, the choice of the neighborhoods and the dynamics is very
constrained, see [21]. As will be discussed in detail, most non-trivial
decision rules lead the system out of thermodynamic equilibrium, requiring
calculations that are not always readily tractable. As it is extremely
difficult to empirically determine how economic agents actually make
decisions, the physics-inspired theoretical analysis of toy models has a
significant part to play, in particular to determine the robustness of
qualitative findings to specific modeling choices. Besides, as argued in [21]
and by some of us in [22], the intrinsically individualistic nature of agent-
specific moves in socioeconomic models means that the description of
collective behaviors as the minimization of some global energy is often not
possible. Understanding simple out-of-equilibrium dynamics as those that arise
from the decision rules presented here is therefore also necessary from the
methodological point of view.
The purpose of this paper is to assess, within a general Sakoda-Schelling like
occupation model, whether and how the sub-optimal concentration of agents in
overly dense regions still occurs out of equilibrium. Most importantly, we
relax the assumption of taking a specific decision rule, and no longer require
pre-defined block neighborhoods as in [15]. The resulting heterogeneity of
interactions in our model then requires the use of out-of-equilibrium
statistical mechanics techniques, the progress of which in the last decade can
be credited to active matter theory. Overall, we find that the phenomenology
of the model is largely unaffected by its nonequilibrium nature, suggesting
that the tendency of agents to aggregate sub-optimally is robust to large
classes of decision rules. This being said, our analysis highlights
interesting theoretical subtleties, notably related to the non-monotonicity of
the utility functions considered, that may, in turn, contribute to the
understanding of other complex physical systems.
The paper is organized as follows. In Sec. II we introduce a Schelling-like
occupation model, in which we keep the utility function and decision rule as
general as possible to allow for nonequilibrium dynamics. We the perform a
numerical analysis of the model. In Sec. III we present a mean-field
description of the dynamics, and determine the region in parameter space where
condensation necessarily occurs. In Sec. IV we show how the dynamics can be
mapped on the Active Model B [23], which is considered to be the natural
nonequilibrium extension of the Cahn-Hilliard field relaxation [24]. This
mapping notably allows to compute the phase densities of the concentrated
states. In Sec. V we propose some relevant generalizations of the model,
namely with two different populations and a housing market. Finally, in Sec.
VI we discuss the implications of our study and conclude.
## II A Sakoda-Schelling occupation model
### II.1 Setup
Consider a city structured as a two-dimensional rectangular lattice composed
of $M=L_{x}\times L_{y}$ sites (or houses). Each site can be occupied by at
most one of the $N(\leq M)$ agents living in this city. On each site of
coordinate $\bm{r}=(i,j)$, the occupation field $n$ takes the value
$n(\bm{r})=1$ if the site is occupied, $n(\bm{r})=0$ if it is vacant. It is
assumed that each agent $k$ wants to maximize their own utility $u_{k}$, which
depends on the local density of agents around them. Typically, it is natural
to think that people like to gather in relatively dense areas to benefit from
the city life, but not too dense as crowding might degrade the quality of
life. Agents estimate the local density by averaging the occupation field with
a probability-density-function kernel $G_{\sigma}$, where $\sigma$ stands for
the interaction range. The kernel is assumed to be isotropic and identical for
all agents. The smoothed occupation field $\tilde{n}$ at site $\bm{r}$ is thus
given by the discrete convolution
$\displaystyle\tilde{n}(\bm{r})=\sum_{\bm{r}^{\prime}}G_{\sigma}(\bm{r}-\bm{r}^{\prime})n(\bm{r}^{\prime}).$
(1)
At each time step, an agent $k$ can decide to move out from their occupied
site $\bm{r}_{k}$ and to settle on a new, randomly chosen, empty site
$\bm{r}_{k}^{\prime}$ where the utility $u[\tilde{n}(\bm{r}_{k}^{\prime})]$ –
quantifying the agent’s satisfaction – might exceed their previous utility
$u[\tilde{n}(\bm{r}_{k})]$. We assume that the decision to move to the new
site is a function of the utility difference $\Delta u_{k}\equiv
u[\tilde{n}(\bm{r}_{k}^{\prime})]-u[\tilde{n}(\bm{r}_{k})]$. While the very
existence of the utility function is debatable from a behavioural standpoint
[25], classical economics has traditionally taken agents to be strict utility
maximizers, meaning the move will be accepted if $\Delta u_{k}>0$ and rejected
otherwise. In order to mitigate this assumption, a common approach is to
introduce a stochastic decision rule of the form
$\displaystyle\mathbb{P}(\bm{r}_{k}\to\bm{r}_{k}^{\prime})=f_{\Gamma}(\Delta
u_{k}),$ (2)
where the function $f_{\Gamma}$ is larger than $\frac{1}{2}$ whenever $\Delta
u_{k}>0$. Typically, $f_{\Gamma}$ is a positive and monotonic function of the
utility difference, with $\lim_{x\to-\infty}f_{\Gamma}(x)=0$ and
$\lim_{x\to+\infty}f_{\Gamma}(x)=1$ [26]. The parameter $\Gamma\geq 0$, known
as the intensity of choice, or simply the rationality, quantifies the
propensity of agents to go for strict utility maximizing. In particular,
$\Gamma\to 0$ corresponds to random decision making, while $\Gamma\to\infty$
means perfectly rational agents.
In reality, the specific shape of the function $f_{\Gamma}$ is unknown. In the
socio-economics literature, it is most of the time taken as the logistic
function
$\displaystyle f_{\Gamma}(x)=\frac{1}{1+e^{-\Gamma x}},$ (3)
defining the so-called _logit rule_ [27, 26]. The various reasons and
justifications of this decision rule are discussed and summarized in [21]. In
a nutshell, it can be motivated axiomatically [27], or by the fact that
$f_{\Gamma}$ is a maximum entropy distribution and therefore optimizes an
exploration-exploitation tradeoff when the cost associated with information
scales as $1/\Gamma$ [28, 29]. As empirical evidence supporting this choice
remains extremely scarce, its popularity is in reality largely motivated by
convenience [25]. Indeed, many calculations are made possible thanks to the
fact that it preserves detailed balance with respect to the Gibbs-Boltzmann
measure in the particular case where agents’ utility change also coincides
with a global utility difference [30]. In this context, $T\equiv 1/\Gamma$ can
naturally be interpreted as the temperature, or “social temperature”, of the
system. In the following, the function $f_{\Gamma}$ will be left unspecified,
unless stated otherwise. In the Monte Carlo simulations we will notably use
the logit rule for simplicity.
The last ingredient to specify is the utility function $u$ of the agents. As
stated above, we assume that the utility depends on the locally smoothed
occupation $\tilde{n}$ only, and that it is non monotonic. As in Ref. [15], we
assume that the utility is maximal for some density
$\rho^{\star}\geq\frac{1}{2}$. We specifically choose for the simulations
$\displaystyle u(x)=-\left|x-\rho^{\star}\right|^{\alpha},$ (4)
with $\alpha>0$, see Fig. 1(a), but theoretical computations below will keep
$u$ unspecified.
Figure 1: (a) Utility function $u(\rho)=-|\rho-\rho^{\star}|^{\alpha}$ for
$\rho^{\star}=0.5$, $\alpha=\\{0.5,1,2\\}$. Panels (b), (c) and (d) show
snapshots of the stationary state for these different utility functions,
starting from the same homogeneous profile at $\rho_{0}=0.5$. Here
$\Gamma=100$, $\sigma=3$ and $L_{x}=L_{y}=100$. The stationary density
$\rho_{d}$ in the dense phase is $\rho_{d}=0.575(5)$ for $\alpha=0.5$ in (b),
$\rho_{d}=0.575(5)$ for $\alpha=1$ in (c), and $\rho_{d}=0.585(5)$ for
$\alpha=2$ in (d). These bulk densities are all significantly higher than the
density $\rho^{\star}$ for which agents maximize their utility. Note the
accumulation of agents at the edge of the empty domain in (b) and (c), see
Sec. IV.
### II.2 In or out of equilibrium?
As mentioned above, an often unspoken motivation for the use of the logit rule
in the modeling of socioeconomic systems is that it may satisfy detailed
balance. Indeed, as described by Grauwin et al. [15, 31] or in [21] in a more
general setting, if one manages to find a system-wide energy-like function
$\mathcal{H}$ such that
$\displaystyle\begin{split}\Delta
u_{k}&=\mathcal{H}([\\{n(\bm{r})\\},n(\bm{r}_{k}^{\prime})=1,n(\bm{r}_{k})=0])\\\
&\quad-\mathcal{H}([\\{n(\bm{r})\\},n(\bm{r}_{k}^{\prime})=0,n(\bm{r}_{k})=1]),\end{split}$
(5)
then the usual tools of equilibrium statistical mechanics can be used. The
steady-state distribution of agents is notably identified as the minimum of
the free energy, which is a Lyapunov function of the dynamics prescribed by
the logit rule.
At the agent level, the existence of such a global quantity is usually the
symptom of either altruistic individuals (that voluntarily maximize some
collective satisfaction) or of a central planner (that constructs individual
rewards towards a collective objective). Outside of these two cases, the
existence of a free energy when agents are individualistic is in fact
restricted to a limited number of carefully chosen models (see [22] for a
related discussion in the context of microeconomics). In the literature of
Schelling-like models, taking a city divided in neighborhoods or blocks [15],
where agents share the same utility, yields such a free energy description
(which is importantly not a simple aggregation of individual utilities). In
our model, however, this is no longer true.
Figure 2: Loop of four configurations with $N=3$ agents on $M=5$ sites
breaking Kolmogorov’s criterion when the utility is non-linear and is a
function of an individual perceived density. Shaded and unshaded nodes
correspond to occupied and empty sites respectively. The dashed line indicates
a possible segmentation of the system into two distinct neighborhoods.
To explicitly show that the dynamics breaks detailed balance despite the logit
rule, one may consider a small system and find a specific cycle breaking
Kolmogorov’s criterion [32]. Such a cycle between four consecutive states with
$N=3$ agents placed on a one-dimensional “city” with $M=5$ sites is
illustrated in Fig. 2. The ratio of transition rates between states $X$ and
$Y$, that differ by an agent located on sites $i$ in $X$, versus $j$ in state
$Y$, is given by
$\frac{W_{X\to Y}}{W_{Y\to
X}}=\frac{1+\mathrm{e}^{-\Gamma[u(\tilde{n}_{i}^{X})-u(\tilde{n}_{j}^{Y})]}}{1+\mathrm{e}^{-\Gamma[u(\tilde{n}_{j}^{Y})-u(\tilde{n}_{i}^{X})]}}=\mathrm{e}^{\Gamma[u(\tilde{n}_{j}^{Y})-u(\tilde{n}_{i}^{X})]}.$
(6)
As a result, the ratio between the product of forward rates, $W_{+}$, and the
product of backwards rates, $W_{-}$, in the cycle shown in Fig. 2, is given by
$\displaystyle\frac{W_{+}}{W_{-}}=\mathrm{e}^{\Gamma[u(\tilde{n}_{5}^{B})-u(\tilde{n}_{3}^{A})-u(\tilde{n}_{2}^{B})+u(\tilde{n}_{3}^{D})+u(\tilde{n}_{2}^{A})-u(\tilde{n}_{5}^{D})]}.$
(7)
For a generic non-linear utility function, $W_{+}\neq W_{-}$, which is a
signature of nonequilibrium dynamics. For a linear utility function on the
other hand, considering that the convolution kernel $G_{\sigma}$ is isotropic,
all terms in the exponential cancel out, leading to $W_{+}=W_{-}$ (which would
be also satisfied for any other cycle). In this situation, the utility
difference can simply be interpreted as an energy difference, where the kernel
$G_{\sigma}$ plays the role of a pairwise interaction potential between the
agents. Interestingly, this small cycle also illustrates how the introduction
of neighborhoods can salvage the equilibrium description for a generic
utility. Splitting the lattice in two neighborhoods along the dashed line
shown in Fig. 2 and taking an identical value of $\tilde{n}$ for all agents on
each neighborhood, the terms in the exponential in Eq. (7) indeed cancel out
for any utility function since $\tilde{n}_{5}^{B}=\tilde{n}_{5}^{D}$,
$\tilde{n}_{3}^{A}=\tilde{n}_{2}^{A}$ and
$\tilde{n}_{2}^{B}=\tilde{n}_{3}^{D}$.
Figure 3: (a) Typical dense domain size $L_{d}(t)$ during coarsening as a
function of time $t$. A unit of time is defined as $N$ Monte Carlo steps,
where $N$ is the number of agents. $L_{d}(t)$ is averaged over 5 independent
simulations. (b), (c), (d) and (e) show snapshots at different times. Starting
from a disordered configuration, we quench the system at low temperature, or
high rationality $\Gamma$, corresponding to $T\simeq T_{c}/6$. Parameters:
$L_{x}=L_{y}=600$, $\rho_{0}=0.3$, $\sigma=1$, $\alpha=3/2$, $T=0.01$.
### II.3 Microscopic simulations
Having established the out-of-equilibrium nature of our model, we start by
performing numerical simulations to assess whether the concentration of agents
in overly dense regions is generic and robust to different shapes of the
utility function. Here, all numerical simulations are performed on a two-
dimensional grid with periodic boundary conditions. The utility is maximal for
$\rho^{\star}=1/2$. For the sake a simplicity, here we use the logit decision
rule and a truncated Gaussian kernel
$G_{\sigma}(\bm{r})=\begin{cases}\frac{1}{N_{\sigma}}\mathrm{e}^{-\frac{1}{2\sigma^{2}}\|\bm{r}\|^{2}},\quad\text{
if $\|\bm{r}\|$ }\leq 4\sigma,\\\ 0,\quad\text{otherwise},\end{cases}$ (8)
where $N_{\sigma}$ enforces the normalization of the kernel.
#### II.3.1 Phase separation
For large system size $L_{x},L_{y}\gg\sigma$, we explore the behavior for
different global densities $\rho_{0}=N/(L_{x}L_{y})$ and for various
rationality parameters $\Gamma$. Numerical results are qualitatively similar
for all the values of $\alpha$ we tested, ranging from $\alpha=0.5$ to
$\alpha=2$, see Fig. 1. The phenomenology can be summarized as follows. When
rationality is low ($\Gamma\to 0$, $T\to\infty$), the stationary state remains
homogeneous because agents settle at random. When rationality is high, agents
may aggregate in dense clusters, which can surprisingly be more crowded than
what agents’ utilities prescribe. This was already discussed in [15] where the
authors point out that the homogeneous state is actually an unstable Nash
equilibrium, even though all agents maximize their utility. The
destabilization occurs as one agent randomly moves to another region (with no
regard to the effect it may have on the other agents utilities), which
decreases the average density at their original site and increases the average
density where they settle. Agents in the lower-density region will eventually
move to gain the utility they lost when their neighbors moved out. This
dynamics will eventually empty some regions, in which agent’s return becomes
statistically less and less probable. The final state, where a dense phase and
an empty phase coexist, is a stable Nash equilibrium.
One can quantify the condensation dynamics when starting from the homogeneous
state and taking high rationality. The system undergoes a spinodal
decomposition where dense clusters grow and merge until there is one large
dense cluster only, as shown in Fig. 3. The final cluster topology ultimately
depends on noise realization and on the box dimensions. We measure the cluster
size $L_{d}(t)$ as a function of time $t$ using the radial structure factor
(see App. A). We find $L_{d}(t)\sim t^{1/z}$, with the dynamical exponent
$z\in[2,3]$, reminiscent of the coarsening exponent observed in a 2D Ising
system with long-range Kawasaki dynamics [33, 34, 35, 36]. Interestingly, and
consistent with the findings of [36] in the low temperature region, our
results suggest an exponent closer to the local Kawasaki dynamics result $z=3$
(see Fig. 3(a)), despite long-range particle displacements.
#### II.3.2 Critical point and critical exponents
The complete phase separation that occurs when rationality is high indicates
the use of the order parameter $m\equiv\rho_{d}-\rho_{g}$, where
$\rho_{d},\rho_{g}$ are the average densities of the dense and “gas” (dilute)
phases, respectively. At the critical point $(\rho_{c},T_{c})$, we expect a
second-order phase transition where $m$ goes to $0$ with power-law scaling
$\displaystyle m\underset{\tau\to 0^{+}}{\sim}\tau^{\beta},$ (9)
where $\tau=(T_{c}-T)/T_{c}>0$ defines the rescaled temperature difference,
and $\beta$ is the order-parameter critical exponent. Measuring the critical
exponents allows one to determine to which universality class the system
belongs to, providing precious information on the system behavior at large
scales. Since simulations are carried out in finite systems, measuring the
critical point with precision requires numerical ruse. We follow the approach
that has been extensively used to measure critical exponents in systems
undergoing a Motility-Induced Phase Separation (referred to as MIPS) [37, 38,
39], see App. B.
Simulations are performed in a rectangular domain of size $L_{x}\times L_{y}$,
with $L_{x}=3L_{y}$, with periodic boundary conditions to keep flat interfaces
between a stripe of liquid (dense phase) and a stripe of gas (dilute phase).
Starting with the dense phase in the center of the system, we track the center
of mass such that we always compute the densities in the bulk of each phase.
To compute the local density inside the bulk of each phase, we consider square
boxes of size $\ell=L_{y}/2$, centered either in $0$ in the gas bulk or
centered in $L_{x}/2$ in the dense bulk (Fig. 8). The local density in each
box fluctuates and it is given by $\rho_{b}=N_{b}/\ell^{2}$ with $N_{b}$ the
number of agents in the box $b$ in a given realization of the system. The
distribution of the density in the system is thus bimodal for $T<T_{c}$ and
unimodal when the system is homogeneous. Defining
$\displaystyle\Delta\rho=\frac{N_{b}-\langle N_{b}\rangle}{\ell^{2}},$ (10)
where the $\langle\cdot\rangle$ stands for averaging on the four boxes and on
independent realizations of the simulation, we compute the celebrated Binder
cumulant [40, *rovere_simulation_1993, 42]
$\displaystyle
Q_{\ell}(\Delta\rho,T)=\frac{\langle(\Delta\rho)^{2}\rangle}{\langle(\Delta\rho)^{4}\rangle},$
(11)
for a given box size $\ell$ and a given temperature $T$. For $\ell$ large
enough, the curves $Q_{\ell}(T)$ all intersect in $T=T_{c}$ where the behavior
of the system is universal. It is important to mention that the critical
density is not known _a priori_. It has to be assessed beforehand to ensure
that the system, as $T$ changes, goes through the critical point, where the
phase transition is of second-order type. To locate $\rho_{c}$, we compute the
Binder cumulant at fixed temperature, close to the estimated critical point,
for various densities $\rho_{0}$. The critical density then corresponds to the
maximal fluctuations of $\Delta\rho$, translated in a peak of the Binder
cumulant, see Fig. 4(a). Once the critical point is precisely located,
additional critical exponents can be measured. Notably, defining the
susceptibility $\chi$ as
$\displaystyle\chi\equiv\frac{\langle(N_{b}-\langle
N_{b}\rangle)^{2}\rangle}{\langle
N_{b}\rangle}=\frac{\langle(\Delta\rho)^{2}\rangle}{\langle
N_{b}\rangle}\ell^{4},$ (12)
one obtains
$\displaystyle\chi\sim\ell^{\gamma/\nu},\quad\frac{dQ_{\ell}}{d\tau}\Big{|}_{\tau=0}\sim\ell^{1/\nu},$
(13)
at the critical point.
Figure 4: Numerical experiments for $\sigma=1$, $\alpha=3/2$. (a) Binodal
densities measured for $L_{x}=200$ and $L_{y}=66$ ($\ell=33$), inset showing
the Binder cumulant as a function of the density and fitted (continuous line)
to determine the critical density. (b), (c) and (d) show the numerical
measurements of the critical exponents close to the critical point
$(\rho_{c},T_{c})=(0.271,0.0620)$ determined using various system sizes
ranging from $\ell=20$ to $\ell=40$.
We report in Fig. 4 the various results on the critical point and on the
critical exponent for $\sigma=1$ and $\alpha=3/2$. Using the Binder cumulant,
one identifies the critical point at $\rho_{c}=0.271(5)$ and
$T_{c}=0.0620(2)$, where the uncertainty on the last digit appears in the
parentheses. The phase diagram in space $(\rho,T)$ is shown in Fig. 4(a), the
black star indicates the critical point and the circular markers show the
densities of the coexisting phases: they define the _binodal_ frontier. The
exponent $\beta$ is directly measured from the order parameter $m$ as function
of reduced temperature $\tau$, at a fixed system size $L_{x}=220$ [see
Fig.9(c)]. From the Ising-2D ansatz, we check that $\nu=1$ yields a neat
collapse of the Binder cumulant, see Fig. 9(b). The exponent $\gamma$ is
obtained by varying the system size at the critical temperature $T_{c}$ and
assuming $\nu=1$ [see Fig.9(d)]. We report in Table 1 the values found for the
critical exponents in the cases $\alpha=3/2$ (Fig. 4) and $\alpha=1/2$ (not
shown here).
Model | $\rho_{c}$ | $T_{c}$ | $\beta$ | $\gamma$
---|---|---|---|---
Ising 2D (exact) | 0.5 | | 0.125 | 1.75
$\alpha=1/2$, $\sigma=1$ | 0.309(5) | 0.0983(5) | 0.120(8) | 1.71(5)
$\alpha=3/2$, $\sigma=1$ | 0.271(5) | 0.0620(2) | 0.119(5) | 1.74(5)
Table 1: Critical density and exponents for nonequilibrium Sakoda-Schelling
model for $\alpha=1/2$ and $\alpha=3/2$.
They differ by less than 5% from the 2D Ising static exponents. These results
enjoin us to assert with a high degree of confidence that the model considered
here belongs to the 2D-Ising universality class. Since the system is out of
equilibrium and particle displacements can be of infinite range, recovering
the Ising universality class is a priori nontrivial. However, finding other
critical exponents would have been surprising since the ingredients at play
are the ones of the Ising model, namely, short-range and isotropic
interactions, a homogeneous medium and a two state degree of freedom (sites
are empty or occupied). This result must also be put into perspective with the
recent debate on the universality class(es) of systems undergoing MIPS [37,
39, 38, 43, 44], and their associated active field theories [45, 46]. Notably
here, our interaction kernel $G_{\sigma}$ provides a so-called _quorum-
sensing_ interaction, like that found in assemblies of bacteria [47]. The
particle dynamics is however quite different for bacteria and for our agents.
The remaining of the paper shall be devoted to establishing a quantitative
relation between our Sakoda-Schelling occupation model and the field-theory
descriptions of MIPS. The first step along this path is to formulate a mean-
field approximation of our model.
## III Field theory and the local-move approximation
### III.1 General description
The computation starts by writing the expectation of the occupation number
$n_{\bm{r},s+1}\equiv n(\bm{r},s+1)$ of site $\bm{r}$ at time $s+1$,
conditioned on the previous configuration $\\{n_{\bm{r},s}\\}$. Averaging over
multiple realizations of noise and using a mean-field approximation in which
all correlation functions factorize, one obtains
$\displaystyle\begin{split}\langle n_{\bm{r},s+1}\rangle-\langle
n_{\bm{r},s}\rangle&=(1-\langle
n_{\bm{r},s}\rangle)\sum_{\bm{r}^{\prime}\neq\bm{r}}\langle
n_{\bm{r}^{\prime},s}\rangle f_{\Gamma}(\Delta
u^{s}_{\bm{r}^{\prime}\to\bm{r}})\\\ &-\langle
n_{\bm{r},s}\rangle\sum_{\bm{r}^{\prime}\neq\bm{r}}(1-\langle
n_{\bm{r}^{\prime},s}\rangle)f_{\Gamma}(\Delta
u^{s}_{\bm{r}\to\bm{r}^{\prime}}),\end{split}$ (14)
where $\Delta u^{s}_{\bm{r}\to\bm{r}^{\prime}}\equiv
u(\langle\tilde{n}_{\bm{r}^{\prime},s}\rangle)-u(\langle\tilde{n}_{\bm{r},s}\rangle)$.
For convenience, we take the continuous time and continuous space limit,
following the common procedure to obtain a mean-field description of exclusion
processes on lattices (see e.g. [48]). The average occupation number $\langle
n\rangle$ is now described by the density $\rho$, while the spatially smoothed
average occupation number $\langle\tilde{n}\rangle$ is described by the field
$\phi\equiv G_{\sigma}*\rho$. The master equation for the occupation
probability then takes the form of a noiseless hydrodynamic equation, in our
case:
$\displaystyle\partial_{t}\rho(x,t)=$
$\displaystyle[1-\rho(x,t)]\int\mathrm{d}y\,\rho(y,t)w_{\Gamma}([\phi],y,x,t)$
(15)
$\displaystyle-\rho(x,t)\int\mathrm{d}y\,[1-\rho(y,t)]w_{\Gamma}([\phi],x,y,t),$
with the transition rate from $y$ to $x$ explicitly given by
$\displaystyle w_{\Gamma}([\phi],y,x,t)=\omega
f_{\Gamma}\left(u(\phi(x,t))-u(\phi(y,t))\right),$ (16)
where $\omega$ is homogeneous to an inverse time scale and $f_{\Gamma}$ is
left unspecified. Equation (15) is valid in any dimension, but, for
simplicity, we will work out the mean-field computations in dimension 1 in
space. This can be justified _a posteriori_ when we compare the mean-field
(MF) to the Monte Carlo (MC) simulations. Let us also mention that the
dimension does not play a role in determining the phase densities in the
steady state of coarse-grained field theories (Allen-Cahn [49], Cahn-Hilliard
[50], etc.).
Integrating Eq. (15) over space, one immediately sees that the total density
$\int\rho$ is conserved. One can also check that in the very specific case
where $u(\phi)$ is linear in $\phi$, one can build a free-energy functional
that is a Lyapunov function of the non-local MF dynamics, ensuring a
convergence towards local minima and preventing limit cycles and oscillatory
dynamics. This is a natural consequence of the fact that detailed balance is
satisfied at microscopic level. In App. C, we construct this free energy and
show that the dynamics is relaxational.
### III.2 Linear stability analysis
In the general case, we would like to understand how the homogeneous state
becomes unstable. To do so, we consider a small perturbation around the
homogeneous state: $\rho(x,t)=\rho_{0}+\rho_{1}(x,t)$, with $\rho_{1}$ the
perturbation. By linearity of the convolution, one has
$\phi(x,t)=\rho_{0}+\phi_{1}(x,t)$, with $\phi_{1}\equiv G_{\sigma}*\rho_{1}$.
A Taylor expansion of Eq. (15) combined with mass conservation (i.e
$\int_{D}\rho_{1}=\int_{D}\phi_{1}=0$, where $D$ is the full domain), finally
yields:
$\displaystyle\begin{split}\partial_{t}\rho_{1}(x,t)=&\
2\Omega\rho_{0}(1-\rho_{0})f^{\prime}_{\Gamma}(0)u^{\prime}(\rho_{0})\phi_{1}(x,t)\\\
&-\Omega f_{\Gamma}(0)\rho_{1}(x,t),\end{split}$ (17)
with $\Omega$ the full domain size. Defining the Fourier transform for any
field $h$ as $\hat{h}(k)=\int\mathrm{d}x\,e^{-ikx}h(x)$, one obtains
$\displaystyle\partial_{t}\hat{\rho}_{1}(k,t)=\Lambda(k)\hat{\rho}_{1}(k,t),$
(18) $\displaystyle\Lambda(k)=\Omega
f_{\Gamma}(0)\left(2\rho_{0}(1-\rho_{0})\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}u^{\prime}(\rho_{0})\hat{G}_{\sigma}(k)-1\right).$
(19)
This last equation shows that the homogeneous state is unstable if there
exists a mode $k^{\star}$ such that
$\displaystyle
2\rho_{0}(1-\rho_{0})\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}u^{\prime}(\rho_{0})\hat{G}_{\sigma}(k^{\star})>1.$
(20)
The manifold for which the inequality becomes an equality defines the spinodal
in the phase diagram $(\rho_{0},\Gamma)$. In particular, for any monotonically
decreasing kernel $G_{\sigma}(|x|)\in L_{2}(\mathbb{R})$, one has
$\hat{G}_{\sigma}(0)>|\hat{G}_{\sigma}(k)|$, such that for large system size,
the stability of the homogeneous state is given by the stability of modes
$k\to 0$, and the spinodal is thus defined by the equation
$\displaystyle
2\rho_{0}(1-\rho_{0})\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}u^{\prime}(\rho_{0})=1.$
(21)
Note that this criterion is generic as it only depends on the decision rule
through $f_{\Gamma}(0)$ and $f^{\prime}_{\Gamma}(0)$. The simulations also
reveal the existence of a bistable region in the vicinity of this spinodal.
This is the binodal region, where hysteresis and bistability can notably be
observed, and which can be fully characterized in the case of an equilibrium
system [12]. Here however, there is a priori no free energy one can rely on to
describe the nucleation scenario and to obtain the densities of the phase-
separated state.
Figure 5: Comparison between Monte Carlo simulations and mean-field results
for $\alpha=3/2$, $\sigma=7$ and $L_{x}=200$, $L_{y}=66$ ($\ell=33$). (a)
Phase diagram in the $(\rho_{0},T)$ plane. The mean-field binodal (continuous
black line) is given by measuring the densities of the bulk of each plateau in
a phase separated state. The circles are the bulk averaged densities in Monte
Carlo (MC) simulations. The dashed black line represent the mean-field
spinodal, which is obtained analytically from the linear stability analysis
(see Eq. (20)), with $\hat{G}_{\sigma}(k_{1})=e^{-\sigma^{2}k_{1}^{2}/2}$ and
$k_{1}=2\pi/L_{x}$. The diamonds indicate the loss of stability of the
homogeneous state in the MC simulations. The black square is the critical
point for $\sigma/L\to 0$. (b) Averaged density profile $\rho(x)$ from MC
simulations (continuous green line) for $\rho_{0}=0.35$, $T=0.05$. The dashed
black line is the stationary solution of the mean-field equation Eq. (15) for
the same parameters, solved on a grid of step size 1 with a Euler explicit
scheme.
### III.3 Comparison to microscopic simulations
The MF prediction is expected to be accurate for systems with high
connectivity, which here corresponds to large $\sigma$. In the following, we
shall take the limit $L\to\infty$, $\sigma\to\infty$ with $\sigma/L\to 0$ to
obtain mean-field predictions that are independent of both $\sigma$ and $L$,
and perform numerical simulations as close as possible to this scaling regime.
The first analytical prediction of the MF description is the spinodal, that
determines the onset of instability of the homogeneous state, see Eq. (21).
The spinodal is the dashed line in the $(\rho,T)$ phase diagram in Fig. 5(a).
To check the prediction, we start in the MC simulations from a uniformly
distributed configuration of agents for three different values of temperature,
$T=0.04$, $0.08$, $0.11$, and we detect the frontier across which the
homogeneous profile either coarsens, or needs a nucleation event to converge
to the separated state. This frontier is marked with the diamonds, which
agrees with the MF prediction.
Second, the MF dynamical Eq. (15) can be solved numerically with an Euler
explicit scheme. From the numerical solution, one obtains the densities of the
bulk of each phase when a phase separation occurs: these densities define the
binodal, the continuous line in Fig. 5(a). These MF phase densities are
perfectly recovered by the MC simulations (circles). In addition, one can
compare the steady-state average density profile from MC simulations to the
mean-field stationary density, which superimpose almost exactly, see Fig.
5(b).
As previously stated, the MF predictions fail for small values of $\sigma$.
The phase diagram in Fig. 4(a) is for instance obtained for $\sigma=1$, and
indeed strongly differs from the MF solution. For $\sigma=1$, we notably
identify the critical point at $(\rho_{c},T_{c})=(0.271,0.0620)$, whereas the
MF predicts $(\rho_{c},T_{c})_{\mathrm{MF}}=(0.2763,0.1418)$, where, as
expected, $T_{c}^{\sigma=1}<T_{c}^{\mathrm{MF}}$.
### III.4 Local-move approximation
To make progress into the identification of a possible effective free energy
functional, it may be convenient to consider slightly modified dynamics where
jumps are now only authorized in the direct neighborhood of the agents.
Indeed, considering an evolution enforcing a local mass conservation will
allow for more familiar partial differential equations (PDEs) and field
theoretic approaches on conserved scalar fields. Here, the absence of
macroscopic density currents in the steady state, both in MC simulations and
in the MF solution suggests that the system generically converges to a
stationary stable fixed point, where the details of the dynamics become
inconsequential. In addition, when the majority of agents have aggregated in a
single dense cluster in the steady state, it is unlikely that they would
perform moves outside of the bulk, in low-density regions, since the utility
there is minimal. The local-move approximation, as it strongly simplifies the
description, thus appears natural.222Dynamically, the coarsening exponent
$z\simeq 3$ displayed in Fig. 3(a), and which is also observed in a Cahn-
Hilliard relaxation dynamics can also be invoked to support the idea of local
moves.
Following the Taylor expansion outlined in App. D, the local mean-field
dynamics is given by
$\displaystyle\partial_{t}\rho$
$\displaystyle=f_{\Gamma}(0)\partial_{x}^{2}\rho-2f_{\Gamma}^{\prime}(0)\partial_{x}[\rho(1-\rho)\partial_{x}u],$
(22)
which can be rewritten as the canonical equation for the mass-conserving
dynamics
$\partial_{t}\rho=\partial_{x}[M[\rho]\partial_{x}\mu([\rho],x)],$ (23)
with the mobility operator $M[\rho]=\rho(1-\rho)$, stemming from the non-
overlaping nature of the agents, and with the chemical potential
$\mu=\mu_{\mathrm{ent.}}+\mu_{\mathrm{util.}}$ where
$\displaystyle\mu_{\mathrm{ent.}}$
$\displaystyle=f_{\Gamma}(0)\log\left(\frac{\rho}{1-\rho}\right)$ (24)
$\displaystyle\mu_{\mathrm{util.}}$
$\displaystyle=-2f_{\Gamma}^{\prime}(0)u[\phi(x)].$ (25)
The first contribution to the chemical potential $\mu_{\mathrm{ent.}}$ is
purely local and accounts for entropy in the system where agents cannot
overlap. The second contribution $\mu_{\mathrm{util.}}$ encodes the drive from
agents’ utility. This term exhibits non-locality with respect to the field
$\rho$, and as a consequence, cannot be expressed as a functional derivative
of any free energy, in general [51, 52, 53]. However, in the particular case
of a linear utility in $\phi$, one again recovers that
$\mu_{\mathrm{util.}}+\mu_{\mathrm{ent.}}$ can be written as the functional
derivative of the free energy $\mathcal{F}$ given in App. C and, as a
consequence, the dynamics (23) becomes a gradient descent [54]. Let us
emphasize that, here again, the decision rule is kept general, and that the
entire local dynamics only depend on it through $f_{\Gamma}(0)$ and
$f^{\prime}_{\Gamma}(0)$.
Performing the linear stability analysis on the dynamics with local moves (see
App. D), we find that the criterion for the homogeneous solution to be
unstable is identical to that given in Eq. (21), when moves are global. Also,
the stationary density profiles computed either with the local, or with the
non-local MF PDEs for the same parameters are identical, as shown in App. E.
Both these observations therefore allow us to confirm the relevance of the
local-move approximation to characterize the system in the long-time limit.
Finally, note that the local hydrodynamic equations can also be obtained using
the path integral approach on a lattice [55], which, in passing, provides the
fluctuating hydrodynamics:
$\displaystyle\partial_{t}\rho=\partial_{x}\left[\rho(1-\rho)\partial_{x}\mu([\rho],x)+\sqrt{\rho(1-\rho)}\xi\right],$
(26)
where $\xi(x,t)$ is a Gaussian white noise with zero mean and with
$\langle\xi(x,t)\xi(x^{\prime},t^{\prime})\rangle=2f_{\Gamma}(0)\delta(t-t^{\prime})\delta(x-x^{\prime})$.
We then remark that when the utility is linear, the stochastic field evolution
describes a complete equilibrium dynamics, irrespective of the choice of the
decision rule: A rule that breaks detailed balance at the microscopic level
can still lead to an equilibrium field theory after coarse graining. Similar
findings had been pinpointed in active matter models [56, 52]. While not
central to the present work, the fluctuations can be studied in more detail,
providing information on the nucleation scenarii and on transition paths
between macroscopic states for instance [57, 58, 59, 60]. The study of the
associated functional Fokker-Planck equation using the tools described in [61,
62] may also be an interesting perspective for future works. In the case of
non-local moves, the formalism from [55] cannot be straightforwardly adapted,
since the local gradient expansion of the jump rates in the action breaks
down. Establishing an appropriate fluctuating hydrodynamic description in the
case of non-local dynamics is therefore an open problem.
## IV Generalized-thermodynamic construction
Since the previous section has shown that the phase separation is well
described by the local-move approximation, we can now use the machinery of
field theory for scalar active matter (e.g. Active Model B), as developed in
[23, 63, 64]. This mapping will notably allow us to obtain the binodal
densities for some class of utility functions detailed below.
### IV.1 The generalized-thermodynamic expansion
Even though $\mu$ in Eq. (23) cannot be written as the functional derivative
[52, 51, 53], the dynamics can be analyzed by resorting to a gradient
expansion. Indeed, expanding the chemical potential up to
$O(\nabla^{4},\rho^{2})$ terms yields
$\displaystyle\mu[\rho]=g_{0}(\rho)+\lambda(\rho)(\nabla\rho)^{2}-\kappa(\rho)\nabla^{2}\rho+O(\nabla^{4},\rho^{2}),$
(27)
with $g_{0}$, $\lambda$, $\kappa$ local function of the field $\rho$, and a
generalized thermodynamic mapping [63, 65] can yield the prediction of the
binodal densities.
For simplicity, we will now assume that the smoothing kernel is a Gaussian
distribution of zero mean and variance $\sigma^{2}$. In Fourier space, the
smoothed field is given by
$\hat{\phi}(k)=\hat{\rho}_{k}\exp({-{\sigma^{2}k^{2}}/{2}})$, which can be
truncated to leading order:
$\displaystyle\hat{\phi}_{k}\simeq\hat{\rho}_{k}\left(1-\frac{k^{2}\sigma^{2}}{2}+O(\sigma^{4}|k|^{4})\right).$
(28)
In real space, this translates into
$\phi=\rho+\frac{\sigma^{2}}{2}\nabla^{2}\rho+O(\nabla^{4},\rho)$. This allows
us to further expand the $\mu_{\mathrm{util.}}$ given in Eq. (25). To leading
order in the $O(\nabla,\rho)$ expansion, one has
$\displaystyle\mu_{\mathrm{util.}}=-2f^{\prime}_{\Gamma}(0)\left[u(\rho)+\frac{\sigma^{2}}{2}u^{\prime}(\rho)\partial_{x}^{2}\rho+O(\partial_{x}^{4},\rho)\right].$
(29)
Combining this expansion of $\mu_{\mathrm{util.}}$ with the entropic
contribution $\mu_{\mathrm{ent.}}$, it is now possible to identify the
different terms in Eq. (27), namely:
$\displaystyle
g_{0}(\rho)=-2f^{\prime}_{\Gamma}(0)u(\rho)+f_{\Gamma}(0)\log\left(\frac{\rho}{1-\rho}\right);$
(30)
$\displaystyle\lambda(\rho)=0;\quad\kappa(\rho)=f^{\prime}_{\Gamma}(0)\sigma^{2}u^{\prime}(\rho).$
(31)
This identification enables us to follow up to the next step, which is finding
the proper function $R(\rho)$ and the generalized functional $\mathcal{G}[R]$
by means of which the dynamics will be given by
$\displaystyle\partial_{t}\rho(x,t)=\partial_{x}\cdot\left[M[\rho]\partial_{x}\frac{\delta\mathcal{G}}{\delta
R(x,t)}\Big{|}_{R(\rho)}\right].$ (32)
A double-tangent construction on $\mathcal{G}[R]$ then provides the binodal
densities [63]. Since $\lambda(\rho)=0$, the differential equation that the
function $R$ must satisfy (see [63, 65]) is
$\displaystyle\kappa(\rho)R^{\prime\prime}(\rho)=-\kappa^{\prime}(\rho)R^{\prime}(\rho),$
(33)
which simplifies into $(\kappa R^{\prime})^{\prime}=0$, where the ′ denotes
the derivative with respect to $\rho$.
### IV.2 Implications of non-monotonous utilities
The previous equation suggests $R^{\prime}(\rho)=C/\kappa(\rho)$, with $C$
some constant. However, one has to be careful at this stage. In the case
considered here, where the utility of agents reaches its maximum for some
density $\rho^{\star}$, it is clear that $\kappa(\rho)$ undergoes a sign
change at $\rho^{\star}$, and more precisely since $f^{\prime}_{\Gamma}(0)>0$,
we have $\mathrm{sign}[\kappa(\rho)]=\mathrm{sign}[u^{\prime}(\rho)]$. To our
knowledge, the fact that $\kappa(\rho)$ may not remain strictly positive has
never been considered in the active matter literature, even though it bears
important physical meaning. Consider a system of quorum-sensing moving
bacteria whose microscopic velocity $v(\rho)$ is density dependent [47].
Coarse-graining typically yields $\kappa(\rho)\simeq-v^{\prime}(\rho)/v(\rho)$
[66, 65], and one obtains $\kappa(\rho)>0$ when the velocity of the particles
is a decreasing function of local density, eventually leading to bacteria
condensation, i.e. MIPS [56, 67]. A positive $\kappa>0$ is thus naturally
interpreted as an “effective surface tension” in this framework. On the other
hand, a negative $\kappa(\rho)$ would be the reflect of an increasing motility
with increasing bacterial density, which is also biologically relevant if one
considers that bacteria are likely to avoid competition for resources in
crowded areas. Yet, and this is a key remark here, it does not necessarily
mean that the phase separation is arrested or that the system undergoes a
microphase separation when $\kappa<0$, notably because higher order gradient
terms that were discarded in the field expansion then become relevant and may
stabilize the interfaces. More specifically here, for $\rho>\rho^{\star}$,
$u^{\prime}(\rho)<0$ such that the term $O(\partial_{x}^{2}\rho)$ destabilizes
the interfaces but the term $O(\partial_{x}^{4}\rho)$ prevents the gradient
from diverging, with a role similar to the positive bending stiffness in the
Helfrich Hamiltonian of membranes [68, 69]. This leads to density overshoots
and spatial oscillations at the edges of the densely populated domain, but the
dense phase ultimately reaches a plateau, as can be observed in snapshots
(Figs. 1, 5 and 10). Conversely, a strictly positive $\kappa(\rho)$ suppresses
the density overshoots, as illustrated in Fig. 11(b). More generally, we
believe that such a macroscopic feature, namely, a spatially oscillating
density profile could interestingly be exploited in experimental systems to
provide important clues on the microscopic properties of the constituents
under study.
Coming back to $(\kappa R^{\prime})^{\prime}=0$ with $\kappa(\rho^{\star})=0$
and assuming that $R(\rho)$ simply needs to be bijective and continuous, one
deduces that different constants are now a priori needed on each interval
$(0,{\rho^{\star}})$ and $({\rho^{\star}},1)$ to compute $R(\rho)$. For the
specific class of utility function $u(\rho)=-|\rho-{\rho^{\star}}|^{\alpha}$,
$\alpha\neq 2$, one obtains
$\displaystyle
R^{\prime}(\rho)=\begin{cases}\frac{C_{1}}{\sigma^{2}f^{\prime}_{\Gamma}(0)\alpha({\rho^{\star}}-\rho)^{(\alpha-1)}}\text{
if $\rho<{\rho^{\star}}$,}\\\
-\frac{C_{2}}{\sigma^{2}f^{\prime}_{\Gamma}(0)\alpha(\rho-{\rho^{\star}})^{(\alpha-1)}}\text{
if $\rho>{\rho^{\star}}$}.\end{cases}$ (34)
For $\alpha\neq 2$, the function $R$ is given by
$\displaystyle
R(\rho)=\begin{cases}\displaystyle\frac{C_{1}({\rho^{\star}}-\rho)^{2-\alpha}}{\sigma^{2}f^{\prime}_{\Gamma}(0)\alpha(\alpha-2)}+C_{3}\text{
if $\rho<{\rho^{\star}}$,}\\\
\displaystyle\frac{C_{2}(\rho-{\rho^{\star}})^{2-\alpha}}{\sigma^{2}f^{\prime}_{\Gamma}(0)\alpha(\alpha-2)}+C_{4}\text{
if $\rho>{\rho^{\star}}$},\end{cases}$ (35)
with $C_{i}$ the interval dependent constants. These equations show that the
case $\alpha<2$ may admit an acceptable change of variable. For $\alpha\geq
2$, the function $R$ displays a divergence at $\rho={\rho^{\star}}$, which
makes impossible to recover an homeomorphism $R:\rho\mapsto R(\rho)$ on the
whole domain $(0,1)$.
In the next paragraph, we detail the procedure introduced in [63, 65] to
recover the binodal densities. We show that we can extend the procedure to
negative $\kappa(\rho)$ – assuming that higher order gradient terms stabilize
the interface. This is one of the central results of this paper.
The function $R$ is bijective, and can thus be inverted to get $\rho(R)$, and
this allows us to define a new chemical potential $g[R]\equiv\mu[\rho(R)]$.
The functional $\mathcal{G}[R]$ is then obtained by integrating $g[R]$ on each
domain and by gluing together the two integrated parts at ${\rho^{\star}}$.
Explicitly using the notations introduced in [65], we have
$\displaystyle g=\frac{\delta\mathcal{G}}{\delta
R},\qquad\mathcal{G}=\int\mathrm{d}x\left[\Phi(R)+\frac{\kappa}{2R^{\prime}}(\partial_{x}R)^{2}\right],$
(36)
where $\Phi(R)$ defines a generalized free energy density verifying
$\displaystyle\frac{\mathrm{d}\Phi}{\mathrm{d}R}=g_{0}[\rho(R)],$ (37)
with $g_{0}$ defined in Eq. (30). The double-tangent construction on $\Phi(R)$
then yields the binodal densities. To be more explicit, from Eq. (35), we
assume that $R$ is simply given by
$\displaystyle R(\rho)=\begin{cases}-(\rho^{\star}-\rho)^{2-\alpha}\text{ if
$\rho<{\rho^{\star}}$,}\\\ (\rho-\rho^{\star})^{2-\alpha}\text{ if
$\rho>{\rho^{\star}}$},\end{cases}$ (38)
which can be inverted into
$\rho(R)=\rho^{\star}+\operatorname*{sgn}(R)|R|^{\frac{1}{2-\alpha}}$.
Injecting $\rho(R)$ in Eq. (30), one obtains
$\displaystyle\begin{split}g_{0}(R)=&\,2f^{\prime}_{\Gamma}(0)|R|^{\frac{\alpha}{2-\alpha}}\\\
&+f_{\Gamma}(0)\log\left[\frac{\rho^{\star}+\operatorname*{sgn}(R)|R|^{\frac{1}{2-\alpha}}}{1-\rho^{\star}-\operatorname*{sgn}(R)|R|^{\frac{1}{2-\alpha}}}\right].\end{split}$
(39)
The explicit formula for $\Phi(R)$ is more involved and we choose to display
$R(\rho)$ in Fig. 6(a) for $\alpha=3/2$ and $\rho^{\star}=1/2$. To obtain the
binodal densities, one either performs the double-tangent construction on
$\Phi$ (Fig. 6(b), inset) or the Maxwell construction, the latter being easier
to handle numerically [65]. We provide details on these two constructions in
Appendix G. In a nutshell, one obtains the coexistence densities of the dilute
and the dense phases from the constraints of the steady state. Indeed, in the
steady state, the interface between the phases does not move and each phase
has a fixed density, which translates into equality across phases of pressure
and chemical potential, respectively. We then compare the predictions to the
phase densities measured in MC simulations in Fig. 6(b), and report an
excellent match. Note that the interaction range $\sigma$, that appears in
$\kappa$ but not in $\Phi$, does not play a role in the predicted coexistence
densities of the infinite size system, confirming that the sub-optimal
aggregation of agents in a dense cluster is not limited to finite size
lattices.
Figure 6: (a) Utility $u(\rho)$ (solid line), and change of variable $R(\rho)$
(dashed line) for $\alpha=3/2$ and $\rho^{\star}=1/2$. (b) Comparison between
the semi-analytical prediction (dark line) and the binodal densities both
obtained via the Monte Carlo simulations (green circles) and solving
numerically the mean-field Eq. (15) (green dot-dashed line). The decision rule
here is the logit function, whose values in $0$ are $f_{\Gamma}(0)=1/2$ and
$f^{\prime}_{\Gamma}(0)=\Gamma/4$. Inset: double-tangent construction on
$\Phi(R)$ for $T=0.083$. The dense phase is given by $R_{d}=0.170$, yielding
$\rho_{d}=0.53$, and the gaseous phase is given by $R_{g}=-0.69$, yielding
$\rho_{g}=0.022$.
As a final remark, and to provide some validity criterion for the extended
change variable, we should mention that $R(\rho)$ taken alone does not contain
information on the sign of $\kappa(\rho)$. This is why we claim that, if the
system indeed undergoes a true phase separation and that $\nabla\rho$ has
finite left and right limits at each point of the domain, then the sign of
$\kappa$ does not matter and a computation with negative $\kappa$ still
predicts the correct binodal densities. In other words, it is not the negative
sign of $\kappa$ _per se_ that leads to the failure of the gradient expansion,
but rather the fact that there exists some points $x_{c}$ in the domain such
that $|\partial_{x}\rho\small|_{x_{c}^{\pm}}|=+\infty$. The fact that
$\partial_{x}\rho$ may not be continuous but keeps finite values at the right
and at the left of each point of the domain does not seem to be an issue when
performing the gradient expansion, probably because the set of points where
the derivative is higher than some arbitrary value is a set of zero measure.
The breakdown of the gradient expansion for a positive and monotonic
$\kappa(\rho)$ is illustrated in App. F.
## V Further socioeconomic considerations
### V.1 Two populations
A natural extension of the problem is to restore some diversity among agents,
as initially considered by both Sakoda and Schelling. Here we consider two
types of interacting agents (say $A$ and $B$), with possibly different utility
functions, which could for example represent higher and lower revenue
individuals, or city dwellers and business professionals, etc.333The recent
work [70] on urban segregation in the United States has brought to our
attention the existence of surveys [71, 72, 73] confirming the idea that
different sub-populations may require markedly different utility functions. A
central question in this case is whether the system reaches fixed points, or
if more complicated dynamics can persist in the long time limit, especially if
the two populations have competing interests. Recent work has been devoted to
studying nonreciprocal interactions between different kinds of particles,
exhibiting the wealth of possible dynamical behavior when particle
displacements are local [74, 75]. An interesting question in our setup is for
instance: do propagating waves (or frustrated states) survive when nonlocal
moves are allowed? Indeed, one may expect that enforcing local displacement
constitutes a dynamical constraint that drives the system in a particular way.
Allowing for nonlocal moves may change the dynamics of how the frustrated
states are resolved. One may think of three major types of interactions:
* •
First, a cooperative interaction where agents $A$ and agents $B$ may maximize
their utility when agents of opposite type are found in their neighborhood.
This kind of interaction will typically lead to homogeneous well-mixed
systems, or to some condensation into a dense phase where agents are well-
mixed, but since frustration is not implemented in the microscopic rules, we
reasonably expect stationary states.
* •
Second, each agent type may decide to settle among peers and/or avoid agents
of the other type in their surroundings. One should then expect a complete
phase separation into two domains, one displaying a majority of $A$s and, the
other, a majority of $B$s. Whether the $A-B$ phase separation additionally
displays some condensation depends on the self-affinity of each agent type.
* •
Third, frustrated situations in which $A$ settles with $A$ but wants to avoid
$B$ agents, while $B$ agents would like to gather and settle close to $A$. In
this situation, we may expect non stationary patterns, stemming from the fact
that all agents cannot be satisfied at the same time.
With this last situation in mind, we have considered the following utility
functions ($u_{A}$ for $A$ agents and $u_{B}$ for $B$ agents):
$\displaystyle u_{A}(x,[\phi_{A,B}])$
$\displaystyle=-|\phi_{A}(x)-\rho^{\star}|^{2}+c_{1}\phi_{B}(x)$ (40)
$\displaystyle u_{B}(x,[\phi_{A,B}])$
$\displaystyle=-|\phi_{A}(x)-\rho^{\star}|^{2}+c_{2}\phi_{B}(x),$ (41)
where $c_{1}<0$ translates the fact that $A$s are fleeing from $B$, and
$c_{2}>0$ translates the fact that $B$s have a tendency to gather with $B$s.
The term $-|\phi_{A}-\rho^{\star}|^{2}$ enjoins both populations to settle
among $A$ populated areas. Of course, the specific shape of utilities taken
here may be restrictive and can be easily generalized.
The extension of the mean-field dynamics to this two population problem is
rather straightforward. Writing $\rho_{A}(x,t)$ (resp. $\rho_{B}(x,t)$) the
density of agents $A$ (resp. $B$) at location $x$ and time $t$, and denoting
the total density by $\rho(x,t)\equiv\rho_{A}(x,t)+\rho_{B}(x,t)$, we now have
an evolution equation of the form
$\displaystyle\partial_{t}\rho_{A}(x,t)$
$\displaystyle=[1-\rho(x,t)]\int\rho_{A}(y,t)w_{\Gamma_{A}}([\phi_{A,B}],y,x,t)\,\mathrm{d}y$
(42)
$\displaystyle-\rho_{A}(x,t)\int[1-\rho(y,t)]w_{\Gamma_{A}}([\phi_{A,B}],x,y,t)\,\mathrm{d}y,$
and, by symmetry, a similar equation for $B$. The transition rates depend on
the utility function of each agent type and are a priori agent specific.
Denoting $u_{Z}(x)\equiv u_{Z}(x,[\phi_{A,B}])$ (with $Z=A$ or $B$), we set
$\displaystyle
w_{\Gamma_{Z}}([\phi_{A,B}],y,x,t)=\omega_{Z}f_{\Gamma_{Z}}[u_{Z}(x)-u_{Z}(y)],$
(43)
where $\omega_{Z}$ and $\Gamma_{Z}$ can be agent dependent.
Figure 7: Snapshots of the system for two frustrated interaction parameter
choices. (a) Stationary demixing in a region where LSA presents complex
eigenvalues. The agent $A$ phase still contains some $B$ agents. Parameters:
$c_{1}=-2$, $c_{2}=1$, $\sigma=3$, $\bar{\rho}_{A}=0.2$, $\bar{\rho}_{B}=0.5$,
$\Gamma=10$. (b) Chaotic propagation of polarized blobs in a region where LSA
presents pure real eigenvalues (null imaginary part). Parameters: $c_{1}=-2$,
$c_{2}=0.5$, $\sigma=7$, $\bar{\rho}_{A}=0.6$, $\bar{\rho}_{B}=0.2$,
$\Gamma=100$. For both (a) and (b), $L_{x}=L_{y}=300$. Movies are available
online [76].
In App. H, we perform the linear stability analysis of the homogeneous state.
As expected, in the frustrated two-population system, unstable modes can
display temporal oscillations. However, these oscillations may stop when
nonlinear terms become relevant, and the system may end up in a stationary
phase separation (similar to classical demixing in equilibrium systems), as
displayed in Fig. 7(a). Reciprocally, non-oscillating growing modes at the
linear level may give rise to propagating structures and waves when
nonlinearities become important, as shown in Fig. 7(b) (see Supplementary
Material in [76]). In our system, and at odds with recent work [74, 75], the
oscillatory nature of the non-homogeneous steady state cannot be predicted
from a simple linear stability analysis about the homogeneous solution.
A thorough analysis of the emerging behaviors in the multi-population system
would require more work, beyond the scope of the present paper. Still, it is
remarkable that, here as well, the linear stability analysis in the case of
local jumps yields exactly the same instability conditions as the nonlocal
dynamics ones (see results of Appendices H and I). As a consequence, we expect
that some results of the recent works [74, 75] should be relevant, to some
extent, to describe our multi-population system.
### V.2 Housing market
A common and reasonable criticism of the kind of model developed here is that,
while the perceived density may be a significant factor in the decision making
process of agents, the price of a house should also necessarily be taken into
account. Indeed, in classical economics, the market is usually considered to
be the mechanism through which the optimal allocation of goods and assets
occurs (despite some contradicting empirical evidence e.g. [77]). As a result,
one could rightfully argue that a housing market is necessary to ensure that
agents eventually reach a steady state where their utility is maximal, at odds
with what we have observed in the condensed phase.
Incorporating pricing in the model is not trivial, however, and there are a
number of ways in which this could be done. A common approach in the modeling
of socioeconomic systems is to introduce an agent-dependent budget and to
constrain agents’ moves based on such budget, as done in [78] for example.
Realistically, this budget should then be heterogeneous among the agents (e.g.
Pareto distributed for instance). While relevant and interesting, this agent-
specific dependence as well as its formulation as a hard constraint would
require a different modeling approach, and is unlikely to be tractable
analytically. The alternative that we take here is to consider that if a given
move leads to excess costs for an agent, its utility would decrease. We may
then conveniently stay within the modeling framework of our model and assume
that such a price field $\psi$ is an increasing function of the smoothed
density field $\phi$, such that houses are more expensive in dense
neighborhoods– or in other words: prices are driven by demand only. In its
most general form, we propose the price-adjusted utility
$\bar{u}(\phi)=u(\phi)-u_{\mathrm{p}}[\psi(\phi)],$ (44)
where $u_{\mathrm{p}}$ is the price penalty, assumed to be an increasing
function of the price, and therefore, of the density $\phi$. This penalty is
then, by construction, expected to drive the system away from condensation.
For concreteness, one can consider the example of a linear penalty term on the
utility function , and that the price grows linearly with the local (smoothed)
density, such that:
$\bar{u}(\phi)=u(\phi)-\gamma\phi,$ (45)
with $\gamma>0$. Interestingly, introducing such a coupling boils down to
introducing a pair-wise repulsive interaction between agents. The condition to
observe condensation is then shifted by $\gamma$, and reads explicitly
$2\rho_{0}(1-\rho_{0})\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}(u^{\prime}(\rho_{0})-\gamma)>1.$
(46)
Clearly, condensation cannot occur anymore if the price penalty on the utility
is too important.
This example therefore illustrates that it is indeed possible for an
appropriate housing market to destroy the condensation observed in our
continuous model. However, it is important to note that the outcome (without
the price penalty) for agents is not necessarily improved by this brutal
homogenization through the price field. Besides, it can be argued that the
effect of price should not be as trivial as a utility penalty. In fact, other
models have used price as a proxy for social status, in which case agents are
actually more attracted by the most expensive site they can afford [78]. In
App. J we also consider the case of a more complex price field dependence.
Ongoing research is devoted to performing a comprehensive study of housing
price correlations in empirical data and the careful construction of an
adequate pricing dynamics [79].
## VI Discussion
Let us summarize what we have achieved in this paper. We have introduced a
neighborhood-less extension of the Sakoda-Schelling model for the occupation
of a lattice representing a city. In this version of the model, the agents
attempt to maximize a utility that is a function of their perceived local
density, and are most satisfied when they are located in a site where such
density is of an intermediate value, i.e. neither empty nor too crowded.
Having that agents only consider their own site dependent utility, and that
their utility is nonlinear, drives the system out of equilibrium. As a result,
the system can no longer be studied by constructing a free energy directly
from an aggregate system-wide utility function, as was done by Grauwin et al.
[15] for agents inhabiting predefined neighborhoods or blocks in which the
utility is identical for all. Using numerical simulations as well as a mean-
field description of the nonequilibrium dynamics, we have established that the
apparent disparity between micromotives and macrobehaviours initially observed
by Schelling [4] is robust to the absence of neighborhoods and to the out-of-
equilibrium nature of our extension. Interestingly, we find that the
transition between the fully homogeneous state and the phase-separated one
likely belongs to the 2D Ising universality class, a debated topic in the
active matter literature regarding the very similar Motility Induced Phase
Separation (MIPS) phenomenon. Taking advantage of the similarity between our
problem and the Active Model B (describing MIPS), we predict the local density
in the bulk of the concentrated phase, confirming that the majority of agents
find themselves in a sub-optimal situations with a perceived density exceeding
the ideal value.
While seemingly technical, the fact that the original observations of
Schelling is robust to out-of-equilibrium dynamics actually carries far
reaching consequences, in our opinion. Indeed, as discussed in Sec. II.2,
equilibrium descriptions of socioeconomic problems require the decision rule
$f_{\Gamma}(x)$ to be the “logit” function. This very specific choice is a
common source of criticism, as any results are then a priori uncertain to hold
for other decision rules. Here, on the other hand, our out-of-equilibrium
description presents no such restriction, as all calculations have been
written as generally as possible and, interestingly, turn out to only depend
on $f_{\Gamma}(0)$ and $f^{\prime}_{\Gamma}(0)$. While the numerical results
presented here are those using the classical choice of the logit function for
lack of a more plausible decision rule, one could readily adapt the outcomes
following behavioral evidence that another function is more appropriate, and
we expect the entire phenomenology of our model to hold for any other sigmoid
function. More generally, we believe that this robustness to other decision
rules holds for a large number of socioeconomic models that have been
described using the methods of statistical physics [22, 80, 81, 82, 83]. Of
course, subtleties around the dynamics, such as the relaxation time towards
the steady state or the coarsening dynamics discussed here, will inherently be
affected by the specific choice that governs transition rates. This being
said, we have observed a remarkable similarity in the local and non-local
versions of our model for which the dynamics are yet qualitatively very
different. It is therefore possible that there also exists some degree of
universality in the dynamical behavior of different decision rules, at least
at the mean-field level.
Going back to the Sakoda-Schelling model, we have also introduced some
generalizations that we believe are natural and relevant. First, the
introduction of different sub-populations is interesting, as the exhibited
dynamical patterns are impossible to observe in an equilibrium version of the
model. Second, we have seen that introducing a linear dependence of the price
on the density has the effect of delaying the transition, eventually killing
it off completely if the price penalty in the utility function is strong
enough. As previously stated, however, this mechanism remains very simple. In
order to determine more plausible effects of a housing market, a thorough
analysis of real estate transactions appears to be a promising avenue, in
particular given the increasing availability of open datasets in this area in
major European cities. An extensive study of French data is currently
underway, hopefully allowing us to couple this continuous Sakoda-Schelling
model with a plausible housing market model in the near future [79]. Note
finally that the recent preprint [70] revisits the problem of urban
segregation in the United States and proposes a two-population model very
similar to ours. Studying the validity of hydrodynamic descriptions of the
two-population problem using the census data brought forth in this paper
appears to be an important perspective. Such a comparison could notably be
necessary to highlight the role of ingredients not taken into account in
existing models – such as public policy and economic inequalities – in the
emergence of strong geographic disparities.
## Acknowledgements
We warmly thank Jean-Philippe Bouchaud for his numerous insights on this
study, as well as Claire Alais and Noé Beserman who participated in the early
stages of this project. We also thank Eric Vanden-Eijnden for fruitful
discussions, as well as Eric Bertin for useful comments on the manuscript. R.
Z. also thanks Jérémy O’Byrne for precious discussions along the years on
thermodynamic mappings. J. G.-B. finally thanks Samy Lakhal for fruitful
discussions on the linear utility case. This research was conducted within the
Econophysics & Complex Systems Research Chair, under the aegis of the
Fondation du Risque, the Fondation de l’École polytechnique, the École
polytechnique and Capital Fund Management.
## Appendix A Coarsening exponent
We start from a homogeneous system of size $L_{x}\times L_{y}$ with
$L_{x}=L_{y}$, and we quench it below the critical temperature in the spinodal
region. The system undergoes a spinodal decomposition where dense domains
coarsen until forming one single large cluster. The typical size of the
domains, denoted $L_{d}$ grows with time as $\sim t^{1/z}$, where the growth
exponent $z$ indicates the physics at play. To measure the typical domain
size, we compute first the structure factor, given by
$\displaystyle
S(\bm{k},t)=\Big{|}\sum_{\bm{r}}e^{-i\bm{k}\cdot\bm{r}}\phi(\bm{r},t)\Big{|}^{2}.$
(47)
Using isotropy of the system, we average the structure factor over given
shells $q=(k_{x}^{2}+k_{y}^{2})^{\frac{1}{2}}$ and we obtain the radial
structure factor $s(q,t)=\int_{[0,2\pi]}S(q,\theta,t)\mathrm{d}\theta$. The
typical domain size is given by
$\displaystyle
L_{d}(t)=2\pi\dfrac{\int_{k_{1}}^{\Lambda}s(q,t)\mathrm{d}q}{\int_{k_{1}}^{\Lambda}q\,s(q,t)\mathrm{d}q},$
(48)
with $\Lambda$ the ultraviolet cutoff and $k_{1}=2\pi/L_{x}$ the infrared
cutoff. On our finite grid, the integral takes the form of a discrete sum, the
wavenumber $q$ ranges from $2\pi/N_{x}$ to $2\pi(N_{x}-1)/N_{x}$, and the
increment $\mathrm{d}q$ is replaced by $2\pi/N_{x}$.
Figure 8: Snapshot of a Monte Carlo simulation. Green sites are occupied,
black sites are empty. We draw the boxes of size $\ell=L_{x}/6$ that are used
to measure the liquid and the gas densities. Here, the system size is
$L_{x}=200$, $L_{y}=66$. Parameters: $\alpha=3/2$, $\rho_{0}=0.35$, $T=0.05$.
Figure 9: Binder cumulant, order parameter and compressibility close to the
critical point $(\rho_{c},T_{c})=(0.271,0.0620)$ computed for $\alpha=3/2$ and
$\sigma=1$, as a function of the temperature $T$.
## Appendix B Monte Carlo simulations for 2nd-order phase transitions
The Monte Carlo simulation setup that we used is shown in Fig. 8. As
introduced in recent works on MIPS, we study the phase transition using four
boxes of size $\ell\times\ell$ located in the bulks of the dense and “gas”
phases. The initial condition for the simulations is a fully separated state,
where a slab of density 1 coexists with a slab of density 0. We measure the
system decorrelation time $\tau_{d}$ and we start recording data after $\sim
1.5\tau_{d}$. Each simulation is run for a time $\geq 5\tau_{d}$, and each
symbol in Fig. 9 aggregates the data of $80$ independent simulations.
The collapse of the different observable with the 2D Ising critical exponents
is displayed in Fig. 9.
## Appendix C Lyapunov function for non-local moves
We show below that, in the mean-field limit, with the logit decision rule and
a linear utility function, the hydrodynamic evolution has a Lyapunov function.
Starting from the pairwise Hamiltonian
$\mathcal{H}=-\frac{\nu}{2}\sum_{\bm{r},\bm{r}^{\prime}}n(\bm{r})G_{\sigma}(\bm{r}-\bm{r}^{\prime})n(\bm{r}^{\prime}),$
(49)
which can be shown to satisfy Eq. (5) when $u(\phi)=\nu\phi$, we take the
continuous-space limit and we account for the entropic contribution to find
the free energy functional
$\displaystyle\mathcal{F}[\rho]=-\frac{\nu}{2}\int\mathrm{d}x\,\mathrm{d}y\,\rho(x)G_{\sigma}(x-y)\rho(y)+TS[\rho],$
(50)
with the entropy $S[\rho]=\int[\rho\log(\rho)+(1-\rho)\log(1-\rho)]$. Before
addressing the dynamics of non-local moves, let us address the one of local
moves. It turns out that when moves are local, the use of a linear utility and
the logit rule implies that the mean-field dynamics is of gradient type,
analogous to a Wasserstein gradient flow but with $M[\rho]=\rho(1-\rho)$
instead of $M[\rho]=\rho$ [54], with $\mathcal{F}$ playing the role of the
free energy potential. Naturally, the stochastic dynamics is in detailed-
balance with the Gibbs-Boltzmann measure $e^{-\Gamma\mathcal{F}}$. When the
moves are no longer local, the gradient structure at the mean-field level is
difficult to unveil, even though the stochastic dynamics remains in detailed-
balance. The functional $\mathcal{F}$ remains a Lyapunov function of the
dynamics, i.e. it is non-increasing as the dynamics evolves. This is what we
show explicitly in the next paragraph.
For simplicity, we define the chemical potential
$\displaystyle\mu(x)\equiv\frac{\delta\mathcal{F}}{\delta\rho(x)}=-\nu\phi(x)+T\log\Big{(}\frac{\rho(x)}{1-\rho(x)}\Big{)},$
(51)
such that inverting the equation yields
$\displaystyle\rho(x)=\frac{e^{\frac{\Gamma}{2}(\mu(x)+\nu\phi(x))}}{D(x)},$
(52)
with $D(x)\equiv 2\cosh[\frac{\Gamma}{2}(\mu(x)+\nu\phi(x))]$. Then, computing
the total time derivative on the functional yields (we omit the time
dependence of the fields to alleviate notations):
$\displaystyle\begin{split}\frac{d\mathcal{F}}{dt}=&\int_{x}\frac{\delta\mathcal{F}}{\delta\rho(x)}\partial_{t}\rho(x,t)\mathrm{d}x\\\
=&\int_{x}\mu(x)(1-\rho(x,t))\int_{y}\rho(y,t)w_{\Gamma}([\phi],y,x,t)\mathrm{d}y\mathrm{d}x\\\
&-\int_{x}\mu(x)\rho(x,t)\int_{y}(1-\rho(y,t))w_{\Gamma}([\phi],x,y,t)\mathrm{d}y\mathrm{d}x\\\
=&-\iint\mathrm{d}x\mathrm{d}y\,\frac{Z(x,y)}{4D(x)D(y)\cosh[\frac{\Gamma}{2}(\phi(x)-\phi(y))]},\end{split}$
(53)
where we have replaced $w_{\Gamma}$ by the logit decision rate [see Eq. (16)]
and we have symmetrized and simplified the second line to obtain the third
line, and where we define
$Z(x,y)\equiv[\mu(y)-\mu(x)](e^{\frac{\Gamma}{2}(\mu(y)-\mu(x))}-e^{\frac{\Gamma}{2}(\mu(x)-\mu(y))})\geq
0$, $\forall x,y$. We thus conclude that $\frac{d\mathcal{F}}{dt}\leq 0$, i.e.
that $\mathcal{F}$ is a Lyapunov function of the hydrodynamic evolution when
the utility is linear. Now, more than that, the function is always strictly
decreasing unless it starts from a fixed point, which forbids limit cycles.
Indeed, one notices that the integrand in Eq. (53) is always positive, unless
$\mu(x,t)=\mu(y,t)$, for all $x,y$. Setting $C(t)=\mu(x,t)$ on the manifold
where $\mathcal{F}$ is constant, we have
$\displaystyle 1-\rho(x,t)=\rho(x,t)e^{\Gamma C(t)}e^{\Gamma\nu\phi(x,t)}.$
(54)
If we inject this relation in the non-local mean-field evolution Eq. (15),
then we obtain $\partial_{t}\rho(x,t)=0$, indicating that $\rho(x,t)$ must be
a stationary fixed point when the Lyapunov function is constant.
## Appendix D Local mean field description and LSA
In this section, we consider a modified dynamics where agents are allowed to
relocate on neighboring site only. For simplicity, we also consider that the
system is one dimensional. It is thus possible to perform a Taylor expansion
of the different fields assuming that all fields are smooth in the mean-field
limit. The jump probability between two neighboring sites becomes
$\displaystyle
f_{\Gamma}[u(x+a)-u(x)]=f_{\Gamma}\left(a\partial_{x}u+\frac{a^{2}}{2}\partial_{x}^{2}u\right),$
(55)
where $a$ is the lattice size, and $u$ is the utility on position $x$. The
evolution of the density (for non-overlapping agents) is thus given by
$\displaystyle\begin{split}\partial_{t}\rho=&\rho(x+a)[1-\rho(x)]f_{\Gamma}(-a\partial_{x}u-\frac{a^{2}}{2}\partial_{x}^{2}u)\\\
&+\rho(x-a)[1-\rho(x)]f_{\Gamma}(a\partial_{x}u-\frac{a^{2}}{2}\partial_{x}^{2}u)\\\
&-\rho(x)[1-\rho(x+a)]f_{\Gamma}(a\partial_{x}u+\frac{a^{2}}{2}\partial_{x}^{2}u)\\\
&-\rho(x)[1-\rho(x-a)]f_{\Gamma}(-a\partial_{x}u+\frac{a^{2}}{2}\partial_{x}^{2}u).\end{split}$
(56)
After Taylor expansion up to $O(a^{2})$ and time rescaling, it turns out that
the evolution equation simplifies into
$\displaystyle\partial_{t}\rho$
$\displaystyle=f_{\Gamma}(0)\partial_{x}^{2}\rho-2f_{\Gamma}^{\prime}(0)\partial_{x}[\rho(1-\rho)\partial_{x}u].$
(57)
Then, expanding around an homogeneous state, we write
$\rho=\rho_{0}+\rho_{1}(x,t)$, $\phi=\rho_{0}+\phi_{1}(x,t)$, and we obtain to
leading order in the perturbation:
$\partial_{t}\rho_{1}=f_{\Gamma}(0)\partial_{x}^{2}\rho_{1}-2f_{\Gamma}^{\prime}(0)\rho_{0}(1-\rho_{0})u^{\prime}(\rho_{0})\partial_{x}^{2}\phi_{1}.$
(58)
In Fourier space the evolution of the mode $k$ is given by
$\partial_{t}\hat{\rho}_{1}=\Lambda(k)\hat{\rho}_{1}$, with
$\Lambda(k)=-k^{2}f_{\Gamma}(0)\left(1-2\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}\rho_{0}(1-\rho_{0})u^{\prime}(\rho_{0})\hat{G}_{\sigma}(k)\right).$
(59)
From this, we deduce that the homogeneous system is unstable if there exists a
mode $k^{\star}$ such that
$\displaystyle
1<2\frac{f^{\prime}_{\Gamma}(0)}{f_{\Gamma}(0)}\rho_{0}(1-\rho_{0})u^{\prime}(\rho_{0})\hat{G}_{\sigma}(k^{\star}).$
(60)
This criterion is exactly the same as the one found for the non-local move
dynamics.
## Appendix E Local versus non-local PDEs
To illustrate the effectiveness of our local-move approximation in the
description of the steady state of the system, we have solved numerically both
the local and the non-local mean-field PDEs for the same parameters. The
resulting density profiles, displayed in Fig. 10, appear to be strictly
identical up to numerical errors.
Figure 10: 1D steady-state profiles of the density $\rho$ computed by solving
the mean-field dynamics with local moves (solid line), or non-local moves
(dashed line). The two density profiles superimpose almost exactly,
independently of the various parameter values. It was checked that different
initial configurations lead to the same final state. a) Parameters: utility
exponent $\alpha=1$, $\rho_{0}=0.4$, $\sigma=12$, $L=300$, $\Gamma=9$. b)
$\alpha=3/2$, $\rho_{0}=0.3$, $\sigma=10$, $L=300$, $\Gamma=15$.
## Appendix F Breakdown of the gradient expansion
To illustrate the possible breakdown of the gradient expansion even for
$\kappa(\rho)>0$, we consider a strictly increasing and continuous utility
with a $|\rho-\rho^{\star}|^{-1/2}$ divergence. As shown in the comparison
with the mean-field PDE solved numerically in Fig. 11, the generalized
thermodynamics fails at predicting the bulk densities in this pathological
case.
Figure 11: a) Monotonic utility
$u(\rho)=\operatorname*{sgn}(\rho-\rho^{\star})|\rho-\rho^{\star}|^{1/2}$
(solid line), and bijective change of variable
$R(\rho)=\operatorname*{sgn}(\rho-\rho^{\star})|\rho-\rho^{\star}|^{3/2}$
(dashed line) for $\rho^{\star}=1/2$. b) Density profile when phase separation
occurs, for $\Gamma=2.5$, $\sigma=10$, $L_{x}=300$. The plateaus of the liquid
and the gas phase do not match the plateaus predicted by the generalized
thermodynamic mapping because the gradient expansion is no longer valid close
to $\rho=0.5$.
## Appendix G Details on the double-tangent and the Maxwell constructions
For completeness, we provide a summary of the two approaches that yields the
binodal densities, as they are presented in [65]. As stated the main text,
when a phase separation occurs in equilibrium, the density profile in the
stationary state is no longer evolving, i.e. the free energy, constrained by
the fact that the total mass of the field is fixed, has reached a minimum.
Since interfaces have a sub-extensive contribution in the thermodynamics
limit, one can work with the free energy density $f(\rho)$. The chemical
potential it thus $\mu(\rho)=\frac{\mathrm{d}f}{\mathrm{d}\rho}$. In the
steady state, one has equality of the chemical potential in the liquid and in
the gas, i.e
$\displaystyle\mu(\rho_{\ell})=\mu(\rho_{g})=\bar{\mu},$ (61)
where $\rho_{\ell}$ and $\rho_{g}$ denote the densities in the liquid and in
the gas, respectively. Since the interface does not move, one also has
equality of pressure across the interface, i.e
$\displaystyle P(\rho_{\ell})=P(\rho_{g})=\bar{P},$ (62)
with $P(\rho)=\rho\mu(\rho)-f(\rho)$. We are thus looking for the two
densities such that the tangent to the free energy density $f$ in
$\rho_{\ell}$ and $\rho_{g}$ is the same, with the slope given by
$\displaystyle\bar{\mu}=\frac{f(\rho_{\ell})-f(\rho_{g})}{\rho_{\ell}-\rho_{g}}.$
(63)
Equivalently, the coexisting densities can be obtained via the Maxwell equal-
area construction that imposes
$\displaystyle\int_{v_{\ell}}^{v_{g}}[P(v)-\bar{P}]\mathrm{d}v=0,$ (64)
where $v\equiv 1/\rho$ is the volume per particle, and
$v_{g/\ell}=1/\rho_{g/\ell}$.
Figure 12: (a) Double-tangent construction (in red) on the function $\Phi(R)$
and (b) Maxwell equal-area construction on $h_{0}(w)$, with $h_{0}(w)=\bar{h}$
in red. Shaded areas in (b) are equal. Here we have plotted these functions
for $\alpha=3/2$, $\rho^{\star}=1/2$, $f_{\Gamma}(0)=1/2$,
$f^{\prime}_{\Gamma}(0)=\Gamma/4$, and $\Gamma=10$. We find $\bar{h}=0.0212$,
$R_{\ell}=1.092$ and $R_{g}=0.327$, (or $w_{\ell}=0.915$ and $w_{g}=3.058$),
translating into $\rho_{\ell}=0.508$ and $\rho_{g}=0.047$.
Here, even though we lie out of equilibrium, we have shown how to find a
function $\Phi(R)$ that plays a role similar to a free energy density by means
of the change of variable $\rho(R)$. This function is always convex for $T$
above the mean-field critical temperature $T_{c}^{\mathrm{MF}}$ and display a
non-convex region for $T<T_{c}^{\mathrm{MF}}$. The function
$g_{0}(R)=\frac{\mathrm{d}\Phi}{\mathrm{d}R}$ is now the chemical potential,
and the double construction on $\Phi$ imposes
$\displaystyle
g_{0}(R_{\ell})=g_{0}(R_{g})=\frac{\Phi(R_{\ell})-\Phi(R_{g})}{R_{\ell}-R_{g}}.$
(65)
The equal-area Maxwell construction involves the so-called generalized
pressure $h_{0}$:
$\displaystyle h_{0}(R)=R\frac{\mathrm{d}\phi(R)}{\mathrm{d}R}-\Phi(R),$ (66)
and setting $w=1/R$, we find $w_{\ell}$ and $w_{g}$ such that
$\displaystyle\int_{w_{\ell}}^{w_{g}}[h_{0}(w)-\bar{h}]\mathrm{d}w=0.$ (67)
In practice, the function $h_{0}$ can be obtained numerically and the volume
$w_{\ell}$ and $w_{g}$ can be obtained with a numerical solver using Eq. (67),
more easily than solving the double-tangent condition. We show both
constructions in Fig. 12. Note again that $R$ has no clear physical meaning
and is an intermediary variable of computations. As such, one can always
choose integration constants in Eq. (35) such that $R(\rho)\geq R(0)>0$ to
ensure that $w=1/R$ is correctly defined. In the main text though, we have
chosen $R(\rho)$ centered in $0$ for simplicity, and computed $g_{0}(R)$ and
$\Phi(R)$ accordingly. If we enforce below $R(0)=1-\rho^{\star}{}^{2-\alpha}$,
then taking $\alpha=3/2$, $\rho^{\star}=1/2$, $f_{\Gamma}(0)=1/2$ and
$f^{\prime}_{\Gamma}(0)=\Gamma/4$, we can obtain an analytic expression for
$\Phi(R)$
$\displaystyle\begin{split}\Phi(R)=&\frac{\sqrt{2}}{2}\left(2\Theta(R-1)-1\right)\left[\frac{R-1}{\sqrt{2}}\log\left(\frac{1+2(R-1)^{2}}{1-2(R-1)^{2}}\right)+\arctan\left((R-1)\sqrt{2}\right)-\operatorname{arctanh}\left((R-1)\sqrt{2}\right)\right]\\\
&+\frac{\Gamma}{8}|R-1|(R-1)^{3}.\end{split}$ (68)
$\displaystyle\begin{split}\Phi(R)=\sqrt{2}(2\Theta(R-1)-1)\left[\frac{R-1}{2\sqrt{2}}\log\left(\frac{2+(R-1)^{2}}{2-(R-1)^{2}}\right)+\arctan\left(\frac{R-1}{\sqrt{2}}\right)-\operatorname{arctanh}\left(\frac{R-1}{\sqrt{2}}\right)\right]+\frac{\Gamma}{64}|R-1|(R-1)^{3}.\end{split}$
(69)
$\displaystyle\begin{split}\Phi(R)=&(2\Theta(R)-1)\left[\frac{R}{2}\log\left(\frac{2+R^{2}}{2-R^{2}}\right)+\sqrt{2}\arctan\left(\frac{R}{\sqrt{2}}\right)-\sqrt{2}\operatorname{arctanh}\left(\frac{R}{\sqrt{2}}\right)\right]+\frac{1}{64}\Gamma|R|R^{3}\end{split}$
(70)
## Appendix H Linear stability for two coupled populations
We consider the evolution of a perturbation of the homogeneous state in Eq.
(42) (and in its coupled analogue for the field $\rho_{B}$). Close to the
homogeneous state $\rho_{A}(x)\equiv\bar{\rho}_{A}$,
$\rho_{B}(x)\equiv\bar{\rho}_{B}$, with
$\rho_{0}=\bar{\rho}_{A}+\bar{\rho}_{B}$, we expand the fields
$\rho_{Z}(x,t)=\bar{\rho}_{Z}+\rho_{Z,1}(x,t)$, with $Z=A$ or $B$, and the
perturbation fields are denoted with index $1$. One also has
$\rho(x,t)=\rho_{0}+\rho_{1}(x,t)$, and
$\phi_{Z}(x,t)=\bar{\rho}_{Z}+\phi_{Z,1}(x,t)$. Keeping leading order terms in
Eq. (42) yields
$\displaystyle\partial_{t}\rho_{A,1}=\Omega\omega_{A}\left[-\bar{\rho}_{A}f_{\Gamma_{A}}(0)\rho_{1}(x,t)+2(1-\rho_{0})\bar{\rho}_{A}f^{\prime}_{\Gamma_{A}}(0)[\phi_{A,1}(x,t)\partial_{1}u_{A}+\phi_{B,1}(x,t)\partial_{2}u_{A}]-(1-\rho_{0})f_{\Gamma_{A}}(0)\rho_{A,1}(x,t)\right],$
(71)
where $\partial_{1}u_{A}$ is a shorthand notation for $\frac{\partial
u_{A}}{\partial\bar{\rho}_{A}}[\bar{\rho}_{A},\bar{\rho}_{B}]$. Taking the
logistic function $f_{\Gamma_{A}}(0)=\frac{1}{2}$,
$f^{\prime}_{\Gamma_{A}}(0)=\frac{\Gamma_{A}}{4}$, the linear evolution
simplifies into
$\displaystyle\partial_{t}\rho_{A,1}(x,t)=\frac{\Omega\omega_{A}}{2}\left[-\bar{\rho}_{A}\rho_{1}(x,t)+(1-\rho_{0})\bar{\rho}_{A}\Gamma_{A}[\phi_{A,1}(x,t)\partial_{1}u_{A}+\phi_{B,1}(x,t)\partial_{2}u_{A}]-(1-\rho_{0})\rho_{A,1}(x,t)\right].$
(72)
Similarly, we obtain for the evolution of $B$:
$\displaystyle\partial_{t}\rho_{B,1}(x,t)=\frac{\Omega\omega_{B}}{2}\left[-\bar{\rho}_{B}\rho_{1}(x,t)+(1-\rho_{0})\bar{\rho}_{B}\Gamma_{B}[\phi_{A,1}(x,t)\partial_{1}u_{B}+\phi_{B,1}(x,t)\partial_{2}u_{B}]-(1-\rho_{0})\rho_{B,1}(x,t)\right].$
(73)
Denoting $\hat{\rho}_{Z}(k,t)$ the Fourier transform of $\rho_{Z,1}(x,t)$, the
evolution equation can be cast in Fourier space into
$\displaystyle\partial_{t}\begin{pmatrix}\hat{\rho}_{A}(k,t)\\\
\hat{\rho}_{B}(k,t)\end{pmatrix}=L\begin{pmatrix}\hat{\rho}_{A}(k,t)\\\
\hat{\rho}_{B}(k,t)\end{pmatrix},$ (74)
with
$\displaystyle
L=\frac{\Omega}{2}\begin{pmatrix}\omega_{A}(\bar{\rho}_{B}-1+(1-\rho_{0})\bar{\rho}_{A}\Gamma_{A}\hat{G}_{\sigma}(k)\partial_{1}u_{A})&\omega_{A}(-\bar{\rho}_{A}+(1-\rho_{0})\bar{\rho}_{A}\Gamma_{A}\hat{G}_{\sigma}(k)\partial_{2}u_{A})\\\
\omega_{B}(-\bar{\rho}_{B}+(1-\rho_{0})\bar{\rho}_{B}\Gamma_{B}\hat{G}_{\sigma}(k)\partial_{1}u_{B})&\omega_{B}(\bar{\rho}_{A}-1+(1-\rho_{0})\bar{\rho}_{B}\Gamma_{B}\hat{G}_{\sigma}(k)\partial_{2}u_{B})\end{pmatrix}.$
(75)
For simplicity, we will consider that agents are equally rational
($\Gamma_{A}=\Gamma_{B}=\Gamma$) and that their moving rates are also
identical ($\omega_{A}=\omega_{B}=\omega$).
We are looking for conditions to observe dynamical patterns and/or static
phase separation. Notably, the homogeneous state is linearly unstable if one
eigenvalue of $L$ has a positive real part. It is important to stress that the
linear stability analysis is unable to predict the dynamic behavior when
nonlinear terms become relevant. Whether the eigenvalues display an imaginary
part or not _does not_ bring any information on the final dynamics of the
system. For the sake of completeness, we explicitate the criteria to have
eigenvalues with positive real part and zero imaginary part, referred to as
case (i), and eigenvalues with positive real part and nonzero imaginary part,
referred to as case (ii). We lie in case (i) if
$\displaystyle\begin{cases}\operatorname*{tr}L>0\\\
(\operatorname*{tr}L)^{2}-4\det L>0,\end{cases}\text{ or
}\begin{cases}\operatorname*{tr}L<0\\\ \det L<0.\end{cases}$ (76)
Case (ii) is obtained if
$\displaystyle\begin{cases}\operatorname*{tr}L>0\\\
(\operatorname*{tr}L)^{2}-4\det L<0.\end{cases}$ (77)
The criterion $\operatorname*{tr}L>0$ notably simplifies into
$\displaystyle\bar{\rho}_{A}\partial_{1}u_{A}+\bar{\rho}_{B}\partial_{2}u_{B}>\frac{1}{\Gamma\hat{G}_{\sigma}(k)}\left(\frac{2-\rho_{0}}{1-\rho_{0}}\right).$
(78)
In the main text we have come up with utility functions that lead to
eigenvalues with positive real parts and non zero imaginary parts, thus
suggesting chasing instability. In some cases, oscillations were observed
close to the homogeneous state but they eventually vanished at late times.
Whether or not the chasing instability or oscillations are sustained cannot be
predicted from the simple linear stability analysis but would require to
perform a weakly non-linear analysis which is beyond the scope of this present
paper.
## Appendix I LSA for two populations with local moves
We start from the local jump approximation of the mean-field equation for the
coupled fields. We find that the dynamics can be cast into
$\partial_{t}\rho_{A}=\partial_{x}[\rho_{A}(1-\rho_{A}-\rho_{B})\partial_{x}\mu([\rho_{A,B}],x)],$
(79)
with $\mu=\mu_{\mathrm{ent.}}+\mu_{\mathrm{util.}}$,
$\mu_{\mathrm{ent.}}=w_{\Gamma_{A}}(0)\log\left(\frac{\rho_{A}}{1-\rho_{A}-\rho_{B}}\right),$
(80) $\mu_{\mathrm{util.}}=-2w_{\Gamma_{A}}^{\prime}(0)u_{A}([\rho],x),$ (81)
and likewise for $\rho_{B}$. One can look into the stability of an homogeneous
state with densities $\bar{\rho}_{A}$ and $\bar{\rho}_{B}$, expanding around
this state with a utility $u(\phi_{A},\phi_{B})$ for agents $A$ and
$v(\phi_{A},\phi_{B})$ for agents $B$. For convenience, we will take
$w_{\Gamma_{A}}(0)=w_{\Gamma_{B}}(0)=\omega f_{\Gamma}(0)=\omega/2$ and
$w_{\Gamma_{A}}^{\prime}(0)=w_{\Gamma_{B}}^{\prime}(0)=\omega
f_{\Gamma}^{\prime}(0)=\omega\Gamma/4$. Expanding around the homogeneous state
$(\bar{\rho}_{A}$ , $\bar{\rho}_{B})$ leads to
$\begin{cases}\partial_{t}\rho_{A,1}=\frac{\omega}{2}\left[(1-\bar{\rho}_{B})\partial_{x}^{2}\rho_{A,1}+\bar{\rho}_{A}\partial_{x}^{2}\rho_{B,1}-\partial_{x}(\Gamma\bar{\rho}_{A}(1-\rho_{0})\partial_{x}\phi_{A,1}\partial_{1}u+\partial_{x}\phi_{B,1}\partial_{2}u)\right]\\\
\partial_{t}\rho_{B,1}=\frac{\omega}{2}\left[(1-\bar{\rho}_{A})\partial_{x}^{2}\rho_{B,1}+\bar{\rho}_{B}\partial_{x}^{2}\rho_{A,1}-\partial_{x}(\Gamma\bar{\rho}_{B}(1-\rho_{0})\partial_{x}\phi_{A,1}\partial_{1}v+\partial_{x}\phi_{B,1}\partial_{2}v)\right],\end{cases}$
(82)
Hence, in Fourier space, the linear system can be cast into
$\displaystyle\partial_{t}\begin{pmatrix}\hat{\rho}_{A}(k,t)\\\
\hat{\rho}_{B}(k,t)\end{pmatrix}=K\begin{pmatrix}\hat{\rho}_{A}(k,t)\\\
\hat{\rho}_{B}(k,t)\end{pmatrix},$ (83)
with
$\displaystyle K=\frac{\omega
k^{2}}{2}\begin{pmatrix}\bar{\rho}_{B}-1+\Gamma\bar{\rho}_{A}(1-\rho_{0})\hat{G}_{\sigma}(k)\partial_{1}u&-\bar{\rho}_{A}+\Gamma\bar{\rho}_{A}(1-\rho_{0})\hat{G}_{\sigma}(k)\partial_{2}u\\\
-\bar{\rho}_{B}+(1-\rho_{0})\bar{\rho}_{B}\Gamma\hat{G}_{\sigma}(k)\partial_{1}v&\bar{\rho}_{A}-1+(1-\rho_{0})\bar{\rho}_{B}\Gamma\hat{G}_{\sigma}(k)\partial_{2}v\end{pmatrix}.$
(84)
It is interesting to note that the evolution matrix $K$ is directly
proportional to $L$ and, as a consequence, the stability criterion of the
homogeneous state with local moves is exactly the same as the one found for
non-local moves.
## Appendix J Introducing a non-linear price field
Figure 13: a) Non-monotonic price field $\psi(\phi)$ as a function of the
locally perceived density $\phi$, given by Eq. (85). Parameters:
$\rho_{\mathrm{p}}^{\star}=0.25$ and ${\alpha_{\mathrm{p}}}=2$. b) Spinodal
curves for parameters $\lambda=5,{\alpha_{\mathrm{p}}}=2,\alpha=3$ and
densities [$\rho^{\star}=0.2,\rho_{\mathrm{p}}^{\star}=0.4$] (solid line),
[$\rho^{\star}=0.45,\rho_{\mathrm{p}}^{\star}=0.2$] (dotted line) and
[$\rho^{\star}=0.5,\rho_{\mathrm{p}}^{\star}=0.1$] (dashed line).
For completeness, we can also consider a non-monotonic price field (see Fig.
13(a))
$\psi=\rho_{\mathrm{p}}^{\star}-\lvert\phi-\rho_{\mathrm{p}}^{\star}\rvert^{{\alpha_{\mathrm{p}}}}.$
(85)
The intuition behind this relation is that prices can be lower in overcrowded
neighborhoods as well as in empty neighborhoods, and are maximized for a given
density $\rho_{\mathrm{p}}^{\star}$. Assuming again that the price-adjusted
utility has the form $\bar{u}(\phi)=u(\phi)-u_{\mathrm{p}}[\psi(\phi)]$, with
$u_{\mathrm{p}}[\psi]=\lambda\psi$ and $\lambda>0$, the total utility for the
agents is then given by
$\bar{u}(\phi)=-\lvert\phi-\rho^{\star}\rvert^{\alpha}+\lambda\lvert\phi-\rho_{\mathrm{p}}^{\star}\rvert^{\alpha_{\mathrm{p}}}+cst,$
(86)
We can inject this expression into the linear-stability condition, see Eq.
(21), to pinpoint the condensation. In Fig. 13(b) we take $\alpha=3$,
$\lambda=5,{\alpha_{\mathrm{p}}}=2$, and we interestingly observe that some
spinodal curves display several re-entrance points.
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|
# Head-Free Lightweight Semantic Segmentation with Linear Transformer
Bo Dong 1, Pichao Wang 1 , Fan Wang 2 Work done during an internship at
Alibaba Group.Corresponding author; work done at Alibaba Group, and now
affiliated with Amazon Prime Video.
###### Abstract
Existing semantic segmentation works have been mainly focused on designing
effective decoders; however, the computational load introduced by the overall
structure has long been ignored, which hinders their applications on resource-
constrained hardwares. In this paper, we propose a head-free lightweight
architecture specifically for semantic segmentation, named Adaptive Frequency
Transformer (AFFormer). AFFormer adopts a parallel architecture to leverage
prototype representations as specific learnable local descriptions which
replaces the decoder and preserves the rich image semantics on high-resolution
features. Although removing the decoder compresses most of the computation,
the accuracy of the parallel structure is still limited by low computational
resources. Therefore, we employ heterogeneous operators (CNN and Vision
Transformer) for pixel embedding and prototype representations to further save
computational costs. Moreover, it is very difficult to linearize the
complexity of the vision Transformer from the perspective of spatial domain.
Due to the fact that semantic segmentation is very sensitive to frequency
information, we construct a lightweight prototype learning block with adaptive
frequency filter of complexity $O(n)$ to replace standard self attention with
$O(n^{2})$. Extensive experiments on widely adopted datasets demonstrate that
AFFormer achieves superior accuracy while retaining only 3M parameters. On the
ADE20K dataset, AFFormer achieves 41.8 mIoU and 4.6 GFLOPs, which is 4.4 mIoU
higher than Segformer, with 45% less GFLOPs. On the Cityscapes dataset,
AFFormer achieves 78.7 mIoU and 34.4 GFLOPs, which is 2.5 mIoU higher than
Segformer with 72.5% less GFLOPs. Code is available at
https://github.com/dongbo811/AFFormer.
## Introduction
\begin{overpic}[width=433.62pt]{Imgs/s3.pdf} \end{overpic} Figure 1: Left:
Computational complexity under different input scales. Segformer (Xie et al.
2021) significantly reduces the computational complexity compared to
traditional methods, such as PSPNet (Zhao et al. 2017) and DeepLabV3+ (Chen et
al. 2018) which have mobilenetV2 (Sandler et al. 2018) as backbone. However,
Segformer still has a huge computational burden for higher resolutions. Right:
AFFormer achieves better accuracy on ADE20K and Cityscapes datasets with
significantly lower FLOPs.
Semantic segmentation aims to partition an image into sub-regions (collections
of pixels) and is defined as a pixel-level classification task (Long,
Shelhamer, and Darrell 2015; Xie et al. 2021; Zhao et al. 2017; Chen et al.
2018; Strudel et al. 2021; Cheng, Schwing, and Kirillov 2021) since Fully
Convolutional Networks (FCN) (Long, Shelhamer, and Darrell 2015). It has two
unique characteristics compared to image classification: pixel-wise dense
prediction and multi-class representation, which is usually built upon high-
resolution features and requires a global inductive capability of image
semantics, respectively. Previous semantic segmentation methods (Zhao et al.
2017; Chen et al. 2018; Strudel et al. 2021; Xie et al. 2021; Cheng, Schwing,
and Kirillov 2021; Yuan et al. 2021b) focus on using the classification
network as backbone to extract multi-scale features, and designing a
complicated decoder head to establish the relationship between multi-scale
features. However, these improvements come at the expense of large model size
and high computational cost. For instance, the well-known PSPNet (Zhao et al.
2017) using light-weight MobilenetV2 (Sandler et al. 2018) as backbone
contains 13.7M parameters and 52.2 GFLOPs with the input scale of $512\times
512$. The widely-used DeepLabV3+ (Chen et al. 2018) with the same backbone
requires 15.4M parameters and 25.8 GFLOPs. The inherent design manner limits
the development of this field and hinders many real-world applications. Thus,
we raise the following question: can semantic segmentation be as simple as
image classification?
Recently vision Transformers (ViTs) (Liu et al. 2021; Lee et al. 2022; Xie et
al. 2021; Strudel et al. 2021; Cheng, Schwing, and Kirillov 2021; Xu et al.
2021; Lee et al. 2022) have shown great potential in semantic segmentation,
however, they face the challenges of balancing performance and memory usage
when deployed on ultra-low computing power devices. Standard Transformers has
computational complexity of $O(n^{2})$ in the spatial domain, where $n$ is the
input resolution. Existing methods alleviate this situation by reducing the
number of tokens (Xie et al. 2021; Wang et al. 2021; Liang et al. 2022; Ren et
al. 2022) or sliding windows (Liu et al. 2021; Yuan et al. 2021a), but they
introduce limited reduction on computational complexity and even compromise
global or local semantics for the segmentation task. Meanwhile, semantic
segmentation as a fundamental research field, has extensive application
scenarios and needs to process images with various resolutions. As shown in
Figure 1, although the well-known efficient Segformer (Xie et al. 2021)
achieves a great breakthrough compared to PSPNet and DeepLabV3+, it still
faces a huge computational burden for higher resolutions. At the scale of
$512\times 512$, although Segformer is very light compared to PSPNet and
DeepLabV3+, it is almost twice as expensive as ours (8.4 GFLOPs vs 4.6
GFLOPs); at the scale of $2048\times 2048$, even 5x GFLOPs is required (384.3
GFLOPs vs 73.2 GFLOPs). Thus, we raise another question: can we design an
efficient and lightweight Transformer network for semantic segmentation in
ultra-low computational scenarios?
The answers to above two questions are affirmative. To this end, we propose a
head-free lightweight semantic segmentation specific architecture, named
Adaptive Frequency Transformer (AFFormer). Inspired by the properties that ViT
maintains a single high-resolution feature map to keep details (Dosovitskiy et
al. 2021) and the pyramid structure reduces the resolution to explore
semantics and reduce computational cost (He et al. 2016; Wang et al. 2021; Liu
et al. 2021), AFFormer adopts a parallel architecture to leverage the
prototype representations as specific learnable local descriptions which
replace the decoder and preserves the rich image semantics on high-resolution
features. The parallel structure compresses the majority of the computation by
removing the decoder, but it is still not enough for ultra-low computational
resources. Moreover, we employ heterogeneous operators for pixel embedding
features and local description features to save more computational costs. A
Transformer-based module named prototype learning (PL) is used to learn the
prototype representations, while a convolution-based module called pixel
descriptor (PD) takes pixel embedding features and the learned prototype
representations as inputs, transforming them back into the full pixel
embedding space to preserve high-resolution semantics.
However, it is still very difficult to linearize the complexity of the vision
Transformer from the perspective of spatial domain. Inspired by the effects of
frequency on classification tasks (Rao et al. 2021; Wang et al. 2020), we find
that semantic segmentation is also very sensitive to frequency information.
Thus, we construct a lightweight adaptive frequency filter of complexity
$O(n)$ as prototype learning to replace the standard self attention with
$O(n^{2})$. The core of this module is composed of frequency similarity
kernel, dynamic low-pass and high-pass filters, which capture frequency
information that is beneficial to semantic segmentation from the perspectives
of emphasizing important frequency components and dynamically filtering
frequency, respectively. Finally, the computational cost is further reduced by
sharing weights in high and low frequency extraction and enhancement modules.
We also embed a simplified depthwise convolutional layer in the feed-forward
network (FFN) layer to enhance the fusion effect, reducing the size of the two
matrix transformations.
With the help of parallel heterogeneous architecture and adaptive frequency
filter, we use only one convolutional layer as classification layer (CLS) for
single-scale feature, achieving the best performance and making semantic
segmentation as simple as image classification. We demonstrate the advantages
of the proposed AFFormer on three widely-used datasets: ADE20K, Cityscapes and
COCO-stuff. With only 3M parameters, AFFormer significantly outperforms the
state-of-the-art lightweight methods. On ADE20K, AFFormer achieves 41.8 mIoU
with 4.6 GFLOPs, outperforming Segformer by 4.4 mIoU, while reducing GFLOPs by
45%. On Cityscapes, AFFormer achieves 78.7 mIoU and 34.4 GFLOPs, which is 2.5
mIoU higher than Segformer, with 72.5% less GFLOPs. Extensive experimental
results demonstrate that it is possible to apply our model in computationally
constrained scenarios, which still maintaining the high performance and
robustness across different datasets.
\begin{overpic}[width=433.62pt]{Imgs/3.pdf} \end{overpic} Figure 2: An
Overview of Adaptive Frequency Transformer (AFFormer). We first displays the
overall structure of parallel heterogeneous network. Specifically, the feature
$\boldsymbol{F}$ after patch embedding is first clustered to obtain the
prototype feature $\boldsymbol{G}$, so as to construct a parallel network
structure, which includes two heterogeneous operators. A Transformer-based
module as prototype learning to capture favorable frequency components in
$\boldsymbol{G}$, resulting prototype representation
$\boldsymbol{G}^{\prime}$. Finally $\boldsymbol{G}^{\prime}$ is restored by a
CNN-based pixel descriptor, resulting $\boldsymbol{F}^{\prime}$ for the next
stage.
## Related Work
### Semantic Segmentation
Semantic segmentation is regarded as a pixel classification task (Strudel et
al. 2021; Xu et al. 2017; Xie et al. 2021). In the last two years, new
paradigms based on visual Transformers have emerged, which enable mask
classification via queries or dynamic kernels (Zhang et al. 2021; Li et al.
2022; Cheng, Schwing, and Kirillov 2021; Cheng et al. 2022). For instance,
Maskformer (Cheng, Schwing, and Kirillov 2021) learns an object query and
converts it into an embedding of masks. Mask2former (Cheng et al. 2022)
enhances the query learning with a powerful multi-scale masked Transformer
(Zhu et al. 2021). K-Net (Zhang et al. 2021) adopts dynamic kernels for masks
generation. MaskDINO (Li et al. 2022) brings object detection to semantic
segmentation, further improving query capabilities. However, all above methods
are not suitable for low computing power scene due to the high computational
cost of learning efficient queries and dynamic kernels. We argue that the
essence of these paradigms is to update pixel semantics by replacing the whole
with individual representations. Therefore, we leverage pixel embeddings as a
specific learnable local description that extracts image and pixel semantics
and allows semantic interaction.
### Efficient Vision Transformers
The lightweight solution of vision Transformer mainly focuses on the
optimization of self attention, including following ways: reducing the token
length (Wang et al. 2021; Xie et al. 2021; Wang et al. 2022) and using local
windows (Liu et al. 2021; Yuan et al. 2021a). PVT (Wang et al. 2021) performs
spatial compression on keys and values through spatial reduction, and PVTv2
(Wang et al. 2022) further replaces the spatial reduction by pooling
operation, but many details are lost in this way. Swin (Liu et al. 2021; Yuan
et al. 2021a) significantly reduce the length of the token by restricting self
attention to local windows, while these against the global nature of
Transformer and restrict the global receptive field. At the same time, many
lightweight designs (Chen et al. 2022; Mehta and Rastegari 2022) introduce
Transformers in MobileNet to obtain more global semantics, but these methods
still suffer from the square-level computational complexity of conventional
Transformers. Mobile-Former (Chen et al. 2022) combines the parallel design of
MobileNet (Sandler et al. 2018) and Transformer (Dosovitskiy et al. 2021),
which can achieve bidirectional fusion performance of local and global
features far beyond lightweight networks such as MobileNetV3. However, it only
uses a very small number of tokens, which is not conducive to semantic
segmentation tasks.
## Method
In this section, we introduce the lightweight parallel heterogeneous network
for semantic segmentation. The basic information is first provivided on the
replacement of semantic decoder by parallel heterogeneous network. Then, we
introduce the modeling of pixel descriptions and semantic frequencies.
Finally, the specific details and the computational overhead of parallel
architectures are discussed.
### Parallel Heterogeneous Architecture
The semantic decoder propagates the image semantics obtained by the encoder to
each pixel and restores the lost details in downsampling. A straightforward
alternative is to extract image semantics in high resolution features, but it
introduces a huge amount of computation, especially for vision Transformers.
In contrast, we propose a novel strategy to describe pixel semantic
information with prototype semantics. For each stage, given a feature
$\boldsymbol{F}\in\mathbb{R}^{H\times W\times C}$, we first initial a grid
$\boldsymbol{G}\in\mathbb{R}^{h\times w\times C}$ as a prototype of the image,
where each point in $\boldsymbol{G}$ acts as a local cluster center, and the
initial state simply contains information about the surrounding area. Here we
use a $1\times C$ vector to represent the local semantic information of each
point. For each specific pixel, because the semantics of the surrounding
pixels are not consistent, there are overlap semantics between each cluster
centers. The cluster centers are weighted initialized in its corresponding
area $\alpha^{2}$, and the initialization of each cluster center is expressed
as:
$\boldsymbol{G}(s)=\sum_{i=0}^{n}w_{i}x_{i}$ (1)
where $n=\alpha\times\alpha$, $w_{i}$ denotes the weight of $x_{i}$, and
$\alpha$ is set to 3. Our purpose is to update each cluster center $s$ in the
grid $G$ instead of updating the feature $\boldsymbol{F}$ directly. As
$h\times w\ll H\times W$, it greatly simplifies the computation.
Here, we use a Transformer-based module as prototype learning to update each
cluster center, which contains $L$ layers in total, and the updated center is
denoted as $\boldsymbol{G}^{\prime}(s)$. For each updated cluster center, we
recover it by a pixel descriptor. Let $F_{i}^{\prime}$ denote the recovered
feature, which contains not only the rich pixel semantics from $F$, but also
the prototype semantics collected by the cluster centers
$\boldsymbol{G}^{\prime}(s)$. Since the cluster centers aggregate the
semantics of surrounding pixels, resulting in the loss of local details, PD
first models local details in $F$ with pixel semantics. Specifically, $F$ is
projected to a low-dimensional space, establishing local relationships between
pixels such that each local patch keeps a distinct boundary. Then
$\boldsymbol{G}^{\prime}(s)$ is embedded into $F$ to restore to the original
space feature $F^{\prime}$ through bilinear interpolation. Finally, they are
integrated through a linear projection layer.
\begin{overpic}[width=390.25534pt]{Imgs/2.pdf} \end{overpic} Figure 3: The
effect of different frequency components on semantic segmentation. We use the
cut-edge method Segformer (Xie et al. 2021) to evaluate the impact of
frequency components on semantic segmentation on the widely used ADE20K
dataset (Zhou et al. 2017). The image is transformed into the frequency domain
by a fast Fourier transform (Heideman, Johnson, and Burrus 1984), and high-
frequency information is filtered out using a low-pass operator with a radius.
Removing high-frequency components at different levels results the prediction
performance drops significantly.
### Prototype Learning by Adaptive Frequency Filter
#### Motivation
Semantic segmentation is an extremely complex pixel-level classification task
that is prone to category confusion. The frequency representation can be used
as a new paradigm of learning difference between categories, which can
excavate the information ignored by human vision (Zhong et al. 2022; Qian et
al. 2020). As shown in Figure 3, humans are robust to frequency information
removal unless the vast majority of frequency components are filtered out.
However, the model is extremely sensitive to frequency information removal,
and even removing a small amount would result in significant performance
degradation. It shows that for the model, mining more frequency information
can enhance the difference between categories and make the boundary between
each category more clear, thereby improving the effect of semantic
segmentation.
\begin{overpic}[width=390.25534pt]{Imgs/s4.pdf} \end{overpic} Figure 4:
Structure of the adaptive frequency filter in prototype learning. The
prototype as learnable local description utilizes frequency component
similarity kernel to enhance different components while combining efficient
and dynamic low-pass and high-pass filters to capture more frequency
information.
Since feature $\boldsymbol{F}$ contains rich frequency features, each cluster
center in the grid $\boldsymbol{G}$ also collects these frequency information.
Motivated by the above analysis, extracting more beneficial frequencies in
grid $\boldsymbol{G}$ helps to discriminate the attributes of each cluster. To
extract different frequency features, the straightforward way is to transform
the spatial domain features into spectral features through Fourier transform,
and use a simple mask filter in the frequency domain to enhance or attenuate
the intensity of each frequency component of the spectrum. Then the extracted
frequency features are converted to the spatial domain by inverse Fourier
transform. However, Fourier transform and inverse transform bring in
additional computational expenses, and such operators are not supported on
many hardwares. Thus, we design an adaptive frequency filter block based on
the vanilla vision Transformer from the perspective of spectral correlation to
capture important high frequency and low frequency features directly in the
spatial domain. The core components are shown in Figure 4 and the formula is
defined as:
$\boldsymbol{AFF}(X)=\underbrace{||\boldsymbol{D}_{h}^{fc}(X)||_{H}}_{corr.}+\underbrace{||\boldsymbol{D}_{m}^{lf}(X)||_{M}+||\boldsymbol{D}_{n}^{hf}(X)||_{N}}_{dynamic~{}filters},$
(2)
where $\boldsymbol{D}_{h}^{fc}$, $\boldsymbol{D}_{m}^{lf}(X)$ and
$\boldsymbol{D}_{n}^{hf}(X)$ denote the frequency similarity kernel with ${H}$
groups to achieve frequency component correlation enhancement, dynamical low-
pass filters with ${M}$ groups and dynamical high-pass filters with ${N}$
groups, respectively. $||\cdot||$ denotes concatenation. It is worth noting
that these operators adopt a parallel structure to further reduce the
computational cost by sharing weights.
#### Frequency Similarity Kernel (FSK)
Different frequency components distribute over in $\boldsymbol{G}$, and our
purpose is to select and enhance the important components that helps semantic
parsing. To this end, we design a frequency similarity kernel module.
Generally, this module is implemented by the vision Transformer. Given a
feature $\boldsymbol{X}\in\mathbb{R}^{(hw)\times C}$, with relative position
encoding on $\boldsymbol{G}$ through a convolution layer (Wu et al. 2021). We
first use a fixed-size similarity kernel
$\boldsymbol{A}\in\mathbb{R}^{C/H\times C/H}$ to represent the correspondence
between different frequency components, and select the important frequency
components by querying the similarity kernel. We treat it as a function
transfer that computes the keys $\boldsymbol{K}$ and values $\boldsymbol{V}$
of frequency components through a linear layer, and normalizes the keys across
frequency components by a Softmax operation. Each component integrates a
similarity kernel $\boldsymbol{A}_{i,j}$, which is computed as:
$\boldsymbol{A}_{i,j}={e^{\boldsymbol{k}_{i}\boldsymbol{v}_{j}^{\top}}}/{\sum\limits_{j=1}^{n}e^{\boldsymbol{k}_{i}}},$
(3)
where $\boldsymbol{k}_{i}$ represents the $i$-th frequency component in
$\boldsymbol{{K}}$, $\boldsymbol{v}_{j}$ represents the $j$-th frequency
component in $\boldsymbol{V}$. We also transform the input $\boldsymbol{X}$
into the query $\boldsymbol{Q}$ through a linear layer, and obtain the
component-enhanced output through interactions on the fixed-size similarity
kernel.
#### Dynamic Low-Pass Filters (DLF)
Low-frequency components occupy most of the energy in the absolute image and
represent most of the semantic information. A low-pass filter allows signals
below the cutoff frequency to pass, while signals above the cutoff frequency
are obstructed. Thus, we employ typical average pooling as a low-pass filter.
However, the cutoff frequencies of different images are different. To this
end, we control different kernels and strides in multi-groups to generate
dynamic low-pass filters. For $m$-th group, we have:
$\boldsymbol{D}_{m}^{lf}(\boldsymbol{\boldsymbol{v}}^{m}))=\boldsymbol{B}(\Gamma_{s\times
s}(\boldsymbol{\boldsymbol{v}}^{m})),$ (4)
where $\boldsymbol{B}(\cdot)$ represents bilinear interpolation and
$\Gamma_{s\times s}$ denotes the adaptive average pooling with the output size
of $s\times s$.
#### Dynamic High-Pass Filters (DHF)
High-frequency information is crucial to preserve details in segmentation. As
a typical high-pass operator, convolution can filter out irrelevant low-
frequency redundant components to retain favorable high-frequency components.
The high-frequency components determine the image quality and the cutoff
frequency of the high-pass for each image is different. Thus, we divide the
value $\boldsymbol{V}$ into $\boldsymbol{N}$ groups, resulting
$\boldsymbol{\boldsymbol{v}}^{n}$. For each group, we use a convolution layer
with different kernels to simulate the cutoff frequencies in different high-
pass filters. For the $n$-th group, we have:
$\boldsymbol{D}_{n}^{hf}(\boldsymbol{\boldsymbol{v}}^{n}))=\Lambda_{k\times
k}(\boldsymbol{v}^{n}),$ (5)
where $\Lambda_{k\times k}$ denotes the depthwise convolution layer with
kernel size of $k\times k$. In addition, we use the Hadamard product of query
and high-frequency features to suppress high frequencies inside objects, which
are noise for segmentation.
FFN helps to fuse the captured frequency information, but owns a large amount
of calculation, which is often ignored in lightweight designs. Here we reduce
the dimension of the hidden layer by introducing a convolution layer to make
up for the missing capability due to dimension compression.
#### Discuss
For the frequency similarity kernel, the computational complexity is
$\mathcal{O}(hwC^{2})$. The computational complexity of each dynamic high-pass
filter is $\mathcal{O}(hwCk^{2})$, which is much smaller than that of
frequency similarity kernel. Since the dynamic low-pass filter is implemented
by adaptive mean pooling of each group, its computational complexity is about
$\mathcal{O}(hwC)$. Therefore, the computational complexity of a module is
linear with the resolution, which is advantageous for high resolution in
semantic segmentation.
## Experiments
### Implementation Details
We validate the proposed AFFormer on three publicly datasets: ADE20K (Zhou et
al. 2017), Cityscapes (Cordts et al. 2016) and COCO-stuff (Caesar, Uijlings,
and Ferrari 2018). We implement our AFFormer with the PyTorch framework base
on MMSegmentation toolbox (Contributors 2020). Follow previous works (Cheng,
Schwing, and Kirillov 2021; Zhao et al. 2017), we use ImageNet-1k to pretrain
our model. During semantic segmentation training, we employ the widely used
AdamW optimizer for all datasets to update the model parameters. For fair
comparisons, our training parameters mainly follow the previous work (Xie et
al. 2021). For the ADE20K and Cityscapes datasets, we adopt the default
training iterations 160K in Segformer, where mini-batchsize is set to 16 and
8, respectively. For the COCO-stuff dataset, we set the training iterations to
80K and the minibatch to 16. In addition, we implement data augmentation
during training for ADE20K, Cityscapes, COCO-stuff by random horizontal
flipping, random resizing with a ratio of 0.5-2.0, and random cropping to
$512\times 512$, $1024\times 1024$, $512\times 512$, respectively. We evaluate
the results with mean Intersection over Union (mIoU) metric.
Table 1: Comparison to state of the art methods on ADE20K with resolution at $512\times 512$. Here we use the Segformer as the baseline and report the percentage growth. MV2=MobileNetV2, EN=EfficientNet, SV2=ShuffleNetV2. Model | #Param. | FLOPs | mIoU
---|---|---|---
FCN-8s | 9.8M | 39.6G | 19.7
PSPNet (MV2) | 13.7M | 52.2G | 29.6
DeepLabV3+ (MV2) | 15.4M | 25.8G | 38.1
DeepLabV3+ (EN) | 17.1M | 26.9G | 36.2
DeepLabV3+ (SV2) | 16.9M | 15.3G | 37.6
Lite-ASPP | 2.9M | 4.4G | 36.6
R-ASPP | 2.2M | 2.8G | 32.0
LR-ASPP | 3.2M | 2.0G | 33.1
HRNet-W18-Small | 4.0M | 10.2G | 33.4
HR-NAS-A | 2.5M | 1.4G | 33.2
HR-NAS-B | 3.9M | 2.2G | 34.9
PVT-v2-B0 | 7.6M | 25.0G | 37.2
TopFormer | 5.1M | 1.8G | 37.8
EdgeViT-XXS | 7.9M | 24.4G | 39.7
Segformer (LVT) | 3.9M | 10.6G | 39.3
Swin-tiny | 31.9M | 46G | 41.5
Xcit-T12/16 | 8.4M | 21.5G | 38.1
ViT | 10.2M | 24.6G | 37.4
PVT-tiny | 17.0M | 33G | 36.6
Segformer | 3.8M | 8.4G | 37.4
AFFormer-tiny | 1.6M(-58%) | 2.8G(-67%) | 38.7(+1.3)
AFFormer-small | 2.3M(-41%) | 3.6G(-61%) | 40.2(+2.8)
AFFormer-base | 3.0M(-21%) | 4.6G(-45%) | 41.8(+4.4)
Table 2: Comparison to state of the art methods on Cityscapes val set. The FLOPs are test on the resolution of $1024\times 2048$. Meanwhile, we also report the percentage increase compared to Segformer. Model | #Param. | FLOPs | mIoU
---|---|---|---
FCN | 9.8M | 317G | 61.5
PSPNet (MV2) | 13.7M | 423G | 70.2
DeepLabV3+ (MV2) | 15.4M | 555G | 75.2
SwiftNetRN | 11.8M | 104G | 75.5
EncNet | 55.1M | 1748G | 76.9
Segformer | 3.8M | 125G | 76.2
AFFormer-tiny | 1.6M(-58%) | 23.0G(-82%) | 76.5(+0.3)
AFFormer-small | 2.3M(-41%) | 26.2G(-79%) | 77.6(+1.4)
AFFormer-base | 3.0M(-21%) | 34.4G(-73%) | 78.7(+2.5)
Table 3: Speed-accuracy tradeoffs at different scales on Cityscapes. Model | size | FLOPs | mIoU
---|---|---|---
Segformer (3.8M) | $512\times 1024$ | 17.7G | 71.9
AFFormer-base (3.0M) | $512\times 1024$ | 8.6G(-51.4%) | 73.5(+1.6)
Segformer (3.8M) | $640\times 1280$ | 31.5G | 73.7
AFFormer-base (3.0M) | $640\times 1280$ | 13.4G(-57.5%) | 75.6(+1.9)
Segformer (3.8M) | $768\times 1536$ | 51.7G | 75.3
AFFormer-base (3.0M) | $768\times 1536$ | 19.4G(-62.5%) | 76.5(+1.2)
Segformer (3.8M) | $1024\times 2048$ | 125G | 76.2
AFFormer-base (3.0M) | $1024\times 2048$ | 34.4G(-72.5%) | 78.7(+2.5)
Table 4: Comparison to state of the art methods on COCO-stuff. We use a single-scale results at the input resolution of $512\times 512$. MV3=MobileNetV3 Model | #Param. | FLOPs | mIoU
---|---|---|---
PSPNet (MV2) | 13.7M | 52.9G | 30.1
DeepLabV3+ (MV2) | 15.4M | 25.9G | 29.9
DeepLabV3+ (EN) | 17.1M | 27.1G | 31.5
LR-ASPP (MV3) | – | 2.37G | 25.2
AFFormer-base | 3.0M | 4.6G | 35.1
### Comparisons with Existing Works
#### Results on ADE20K Dataset.
We compare our AFFormer with top-ranking semantic segmentation methods,
including CNN-based and vision Transformer-based models. Following the
inference settings in (Xie et al. 2021), we test FLOPs at $512\times 512$
resolution and show the single scale results in Table 1. Our model AFFormer-
base improves by 5.2 mIoU under the same computing power consumption as Lite-
ASPP, reaching 41.8 mIoU. At the same time, by reducing the number of layers
and channels, we obtain AFFormer-tiny and AFFormer-small versions to adapt to
different computing power scenarios. For the lightweight and efficient
Segformer (8.4 GFLOPs),our base version (4.6 GFLOPs) also gain 4.4 mIoU using
half the computing power and the tiny version (2.4 GFLOPs) with only 1/4 the
computing power improving 1.3 mIoU. Only 1.8 GFLOPs are needed for the lighter
topformer, but our base version has 2.1M less parameters (5.1M vs 3M) with 4.0
higher mIoU.
Table 5: Ablation studies on the parallel structure. Setting | #Param. | FLOPs | mIoU
---|---|---|---
w/o PD | 2.78G | 2.98G | 39.2
w/o PL | 0.42G | 1.65G | 19.5
Parallel | 3.0G | 4.6G | 41.8
Table 6: Ablation studies on heterogeneous architecture. Setting | #Param. | FLOPs | mIoU
---|---|---|---
All PD | 0.6M | 1.85G | 27.4
All PL | 3.6M | 7.0G | 41.6
Heterogeneous | 3.0M | 4.6G | 41.8
#### Results on Cityscapes Dataset.
Table 2 shows the results of our model and the cutting-edge methods on
Cityscapes. Although the Segformer is efficient enough, due to its square-
level complexity, we only use 30% of the computational cost to reach 78.7
mIoU, which is 2.5 mIoU improvement with a 70% reduction in FLOPs. Meanwhile,
we report the results at different high resolutions in Table 4. At the short
side of {512, 640, 768, 1024}, the computational cost of our model is 51.4%,
57.5%, 62.5% and 72.5% of that of Segformer, respectively. Meanwhile, the mIoU
are improved by 1.6, 1.9, 1.2 and 2.5, respectively. The higher the input
resolution, the more advantageous of our model in both computational cost and
accuracy.
#### Results on COCO-stuff Dataset.
COCO-stuff dataset contains a large number of difficult samples that collected
in COCO. As show in Table 4, although complex decoders (_e.g._ , PSPNet,
DeepLabV3+) can achieve better results than LR-ASPP (MV3), they bring a lot of
computational cost. Our model achieves an accuracy of 35.1 mIoU while only
taking 4.5 GFLOPs, achieving the best trade-off.
### Ablation Studies
All the ablation studies are conducted on ADE20K dataset with AFFormer-base
unless otherwise specified.
#### Rationalization of Parallel Structures.
Parallel architecture is the key to removing the decoder head and ensuring
accuracy and efficiency. We first adjust the proposed structure to a naive
pyramid architecture (denoted as “w/o PD”) and a ViT architecture (denoted as
“w/o PL”) to illustrate the advantages of the parallel architecture.
Specifically, the “w/o PD” means removing PD module and keeping only PL
module, while the “w/o PL” does the opposite. As shown in Table 5, the setting
“w/o PD” reduces 2.6 mIoU due to the lack of high-resolution pixel semantic
information. The “w/o PL” structure without the pyramid structure has a
significant reduction in accuracy due to few parameters and lack of rich image
semantic information. It also demonstrates that our parallel architecture can
effectively combine the advantages of both architectures.
#### Advantages of Heterogeneous Structure.
The purpose of the heterogeneous approach is to further reduce the
computational overhead. The PL module is adopted to learn the prototype
representation in the clustered features, and then use PD to combine the
original features for restoration, which avoids direct calculation on the
high-resolution original features and reduce the computational cost. It can be
seen from Table 6 that when the parallel branch is adjusted to the pixel
description module (denote as “All PD”), which means that the prototype
representation is learned by PD module. The model size is only 0.6M, and the
FLOPs are reduced by 2.5G, but the accuracy is reduced by 14.3 mIoU. This is
due to the PD lacks the ability to learn great prototype representations. In
contrast, after we replace the PD module with the PL module (denote as “All
PL”), the FLOPs are increased by 2.4G, but there is almost no difference in
accuracy. We believe that the PD module is actually only a simple way to
restore the learned prototype, and the relatively complex PL module saturates
the model capacity.
#### Advantages of Adaptive Frequency Filter.
We use two datasets with large differences, including ADE20K and Cityscapes,
to explore the core components in adaptive frequency filter module. The main
reason is that the upper limit of the ADE20K dataset is only 40 mIoU, while
the upper limit of the Cityscapes is 80 mIoU. The two datasets have different
degrees of sensitivity to different frequencies. We report the benefits of
each internal component in the Table 7. We find that DHF alone outperforms
DLF, especially on the Cityscapes dataset by 2.6 mIoU, while FSK is
significantly higher than DLF and DHF on ADE20K. This shows that ADE20K may be
more inclined to an intermediate state between high frequency and low
frequency, while Cityscapes needs more high frequency information. The
combined experiments show that the combination of the advantages of each
component can stably improve the results of ADE20K and Cityscapes.
#### Frequency Statistics Visualization.
We first count the characteristic frequency distribution of different stages,
as shown in Figure 5. It can be found that the curves of $G_{2}$ and $F_{2}$
almost overlap, indicating that the frequencies after clustering are very
similar to those in the original features. The same goes for $G_{3}$ and
$F_{3}$. Whereas, the learned prototype representation after frequency
adaptive filtering significantly improves the contained frequency information.
After PD restoration, different frequency components can be emphasized in
different stages. As shwon in Figure 6, we also analyze the frequency effects
of the core components in the AFF module. As expected, DLF and DHF show strong
low-pass and high-pass capabilities, respectively, as FSK does. At the same
time, we also found that the important frequency components screened and
enhanced by FSK are mainly concentrated in the high frequency part, but the
frequency signal is more saturated than that of DHF. This also shows that the
high-frequency component part is particularly important in the semantic
segmentation task, because it emphasizes more on the boundary details and
texture differences between objects. Meanwhile, according to the analysis in
Table 7 (the effects of ADE20K and Cityscapes have been steadily improved),
each core component has its own advantages, and the AFF module shows strong
robustness in various types and complex scenes.
#### Speed and Memory Costs.
Meanwhile, we report the speed on the Cityscapes dataset in Table 8. We can
find that the proposed model improves by 10 FPS and performs much better than
Segformer on such high-resolution Cityscapes images.
Table 7: Ablation studies on frequency aware statistics. Setting | #Param. | FLOPs | ADE20K | Cityscapes
---|---|---|---|---
DLF | 2.4M | 3.6G | 38.7 | 75.7
DHF | 2.6M | 3.9G | 39.3 | 78.3
FSK | 2.9M | 4.2G | 40.5 | 75.3
DLF + DHF | 2.7M | 3.9G | 41.1 | 77.8
DLF + FSK | 2.8M | 4.2G | 40.0 | 76.2
DHF + FSK | 2.9M | 4.3G | 41.2 | 77.3
Whole | 3.0M | 4.6G | 41.8 | 78.7
Table 8: The FPS is tested on a V100 NVIDIA GPU with a batch size of 1 on the resolution of 1024x2048. Model | FPS | mIoU
---|---|---
Segformer | 12 | 76.2
AFFormer | 22 | 78.7
\begin{overpic}[width=411.93767pt]{Imgs/4.pdf} \par\end{overpic} Figure 5:
Frequency analysis of stage-2 (left) and stage-3 (right).
\begin{overpic}[width=411.93767pt]{Imgs/5.pdf} \end{overpic} Figure 6:
Frequency analysis of the core components in PL module.
## Conclusion
In this paper, we propose AFFormer, a head-free lightweight semantic
segmentation specific architecture. The core is to learn the local description
representation of the clustered prototypes from the frequency perspective,
instead of directly learning all the pixel embedding features. It removes the
complicated decoder while having linear complexity Transformer and realizes
semantic segmentation as simple as regular classification. The various
experiments demonstrate that the AFFormer owns powerful accuracy and great
stability and robustness at low computational cost.
## Acknowledgements
This work was supported by Alibaba Group through Alibaba Research Intern
Program.
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# Distributed no-regret edge resource allocation with limited communication
Saad Kriouile, Dimitrios Tsilimantos, and Theodoros Giannakas
Huawei Paris Research Centre, France<EMAIL_ADDRESS>This work was
supported by the CHIST-ERA LeadingEdge project, grant CHIST-ERA-18-SDCDN-004
through ANR grant number ANR-19-CHR3-0007-06. The three authors contributed
equally; the order is random.
###### Abstract
To accommodate low latency and computation-intensive services, such as the
Internet-of-Things (IoT), 5G networks are expected to have cloud and edge
computing capabilities. To this end, we consider a generic network setup where
devices, performing analytics-related tasks, can partially process a task and
offload its remainder to base stations, which can then reroute it to cloud
and/or to edge servers. To account for the potentially unpredictable traffic
demands and edge network dynamics, we formulate the resource allocation as an
online convex optimization problem with service violation constraints and
allow limited communication between neighboring nodes. To address the problem,
we propose an online distributed (across the nodes) primal-dual algorithm and
prove that it achieves sublinear regret and violation; in fact, the achieved
bound is of the same order as the best known centralized alternative. Our
results are further supported using the publicly available Milano dataset.
###### Index Terms:
Online convex optimization, edge computing, resource allocation, distributed
algorithms.
## I Introduction
### I-A Motivation
It is envisioned that globally more than 29.3 billion networked devices will
be connected to the Internet of Things (IoT) by 2023 [1], offering automation
and real-time monitoring of machine and human-driven processes. A main
challenge in IoT deployment lies with the massive amount of connected devices;
and in particular with the device heterogeneity (e.g., different computational
capabilities) and the diverse and potentially stringent service (task)
requirements [2]. To host the unprecedented IoT data traffic, the edge
computing paradigm has recently gained a lot of momentum as complementary to
that of the cloud and it has been deemed as a key enabler to what ultimately
will be the so-called “Cloud-to-Things Continuum” [3, 4].
In that framework, a plethora of spatially distributed devices collect data
from sensors and perform low-latency impromptu computation (e.g., machine-
learning inference) using energy-limited resources. The envisaged “Cloud-to-
Things Continuum” allows flexible task offloading _from_ an IoT device, via
base stations, _towards_ more computationally powerful edge servers and, if
needed, to the cloud. Although this architecture is promising, allocating
resources for IoT computations has two distinct fundamental challenges: (a)
the resources are allocated in the presence of highly unpredictable and non-
stationary request patterns (demands) and network conditions; (b) the network
nodes handling those tasks, namely devices, base stations and servers, are
distributed and should act in the absence of a centralized entity with full
observability. Naturally, the following question arises:
“Can we offer an efficient distributed data-driven algorithm for resource
allocation in the IoT context?”
In order to address this question, in this paper we consider a distributed
setting with nodes of different capabilities and employ Online Convex
Optimization (OCO) [5]. The use of OCO is suitable for problems that are
difficult to model due to unpredictable dynamics and provides concrete
theoretical guarantees for such problems even in the worst-case.
### I-B Related Work
The related literature can be split into two categories. The first corresponds
to studies of _IoT network optimization_ in the edge-cloud setting, using
similar system model and assumptions to ours. An offline version of the
problem is formulated and then decomposed across its different domains (fog
and cloud) in [6], resulting in convex subproblems. In [7], the authors
consider the latency minimization and develop an algorithm based on the online
secretary framework—however, no constraints are used in their formulation.
Closer to our work, [2, 4] formulate the resource allocation as an OCO
problem, and model the service violations as long-term constraints. The
learning rate is adapted to the different IoT tasks in [2]; and further
extending the notion of constraint violation (see [8]), the number of
violations is also considered in [4]. Unlike our work, both approaches are
centralized.
The second one deals with works on _OCO with constraints in generic settings_
, such as [9, 10, 11]. Although generic, these works do not apply to our
problem as they develop centralized algorithms. Delayed feedback on the cost
and constraint functions arrive to a centralized agent in [12]; our approach
adopts a different feedback model based on limited exchange of information
between nodes. Finally, a distributed OCO algorithm in an environment with
time-varying constraints is presented and analyzed in [13], but, unlike our
approach, the nodes/actors use synchronous information and make consensus
steps.
### I-C Contributions and Structure
In this work, we approach the resource allocation in an edge-cloud scenario in
a distributed way; our main contributions can be summarized as follows.
(C.1) We model the resource allocation as a distributed constrained OCO
problem. The network nodes (devices, base stations, edge servers) are cast as
individual OCO agents that must collectively optimize a given network
performance metric and satisfy service requirement guarantees (modeled as
long-term constraints). To this end, we define a limited communication model
between neighboring agents that allows them to exchange crucial information,
related to their coupling constraints.
(C.2) We propose an online primal-dual algorithm, based on projected gradient
descent, with a sub-linear regret bound $\mathcal{O}(T^{1/2})$ and a sub-
linear constraint violation bound $\mathcal{O}(T^{3/4})$. These bounds are
equivalent to a centralized approach besides a multiplicative factor. We
validate our theoretical results with numerical simulations on a commonly used
dataset, and compare the performance of our algorithm to benchmarks.
The remainder of this paper is organized as follows. We summarize our system
model and assumptions in Section II. Then, we introduce the OCO formulation of
the problem and present our proposed algorithm and its theoretical guarantees
in Section III. We show our numerical results in Section IV and conclude the
paper in Section V.
## II Problem Setup
In this section we present our network model and main assumptions. In
particular, we start by the edge computing components that are available in
our IoT application; then we present the control (optimization) variables, and
finally we discuss our performance objectives and system constraints.
In what follows, we use bold fonts for vectors and matrices, and $\mathcal{A}$
for a set with $|\mathcal{A}|=A$ elements.
### II-A Topology and Computational Requests
We consider a layer of IoT sensors, which receive computational requests
(e.g., analytics tasks) that need to be executed, similarly to [2, 4, 14, 6,
15, 7]. Time is slotted and at every timeslot $t$ those requests arrive to a
set of devices $\mathcal{D}$. We denote the vector of requests as
$\mathbf{r}^{t}\in\mathbb{R}_{+}^{D}$. Before $\mathbf{r}^{t}$ is revealed,
the Network Operator (NO) has to reserve resources across its network
infrastructure in order to accommodate them.
We assume that the network consists of the following nodes, as shown in Fig.
1.
* •
Devices at the edge, denoted by $\mathcal{D}$;
* •
Base Stations (BSs) at the edge, denoted by $\mathcal{B}$;
* •
Servers at the edge, denoted by $\mathcal{S}$;
* •
A cloud server at the core network, denoted by $C$.
Throughout the rest of this work, we will refer to the above entities (except
for the cloud) as “nodes” or “agents”. We denote by $\mathcal{N}$ the set of
all nodes across the network with $N=D+B+S$.
Each device can process locally part of the computation and can also offload
tasks via a wireless channel to the BSs. Then, the BSs can forward an incoming
task either to the edge servers (wirelessly) or to the cloud. Finally, each
edge server can process the received tasks locally, reroute to other edge
servers or forward to the cloud; the latter only executes tasks. For
simplicity, we assume full connectivity between nodes, but our methodology
applies to any connectivity graph.
Figure 1: Topology of our edge computing setting
### II-B Distributed Control Variables
The NO wishes to optimize a set of performance metrics in a _distributed_
manner. Therefore, we wish to design a system where each agent decides its own
actions. At every $t$, the control variables for every device
$d\in\mathcal{D}$, BS $b\in\mathcal{B}$ and server $s\in\mathcal{S}$ are
$\displaystyle\mathbf{x}_{d}^{t}$
$\displaystyle=[w^{t}_{d0},w^{t}_{d1},\dots,w^{t}_{dB},p^{t}_{d1},\dots,p^{t}_{dB}]\in\mathbb{R}_{+}^{2B+1}$
(1) $\displaystyle\mathbf{x}_{b}^{t}$
$\displaystyle=[y^{t}_{bC},y^{t}_{b1},\dots,y^{t}_{bS},q^{t}_{b1},\dots,q^{t}_{bS}]\in\mathbb{R}_{+}^{2S+1}$
(2) $\displaystyle\mathbf{x}_{s}^{t}$
$\displaystyle=[z^{t}_{sC},z^{t}_{s1},\dots,z^{t}_{sS}]\in\mathbb{R}_{+}^{S+1}.$
(3)
For each device $d$, $w^{t}_{d0}$ is the locally executed tasks and
$w^{t}_{db}$, $p^{t}_{db}$ are the offloaded tasks and the respective
transmission power to each available BS. For each BS $b$, $y_{b,C}^{t}$
denotes the tasks offloaded to the cloud and $y_{bs}^{t},q_{bs}^{t}$ are the
offloaded tasks and the respective transmission power to each available server
$s\in\mathcal{S}$. Finally, for each server $s$, $z^{t}_{sC}$ is the offloaded
tasks to the cloud and $z^{t}_{ss^{\prime}}$ is the offloaded tasks to each
available server $s^{\prime}\in\mathcal{S}$, including $z^{t}_{ss}$ that
denotes the locally processed tasks.
For every $t$, it must hold
$\mathbf{x}_{d}^{t}\in\Omega_{d},\mathbf{x}_{b}^{t}\in\Omega_{b},\mathbf{x}_{s}^{t}\in\Omega_{s}$,
where $\Omega_{d},\Omega_{b},\Omega_{s}$ are time-invariant box constraints of
the form $\\{\mathbf{x}:\mathbf{0}\leq\mathbf{x}\leq\mathbf{\bar{x}}\\}$. Note
that these constraints are local, meaning that an agent needs no external
information in order to satisfy them. To denote the _collective_ control
variable of all nodes, we use the following notation:
$\mathbf{x}^{t}_{\mathcal{D}}=\\{\mathbf{x}^{t}_{d}\\}_{\forall
d\in\mathcal{D}},\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{x}^{t}_{\mathcal{B}}=\\{\mathbf{x}^{t}_{b}\\}_{\forall
b\in\mathcal{B}},\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{x}^{t}_{\mathcal{S}}=\\{\mathbf{x}^{t}_{s}\\}_{\forall
s\in\mathcal{S}}.$ (4)
Finally, we use
$\mathbf{x}^{t}=\\{\mathbf{x}^{t}_{\mathcal{D}},\mathbf{x}^{t}_{\mathcal{B}},\mathbf{x}^{t}_{\mathcal{S}}\\}\in\Omega\subset\mathbb{R}_{+}^{V}$
to denote all the variables across the network, with
$V=D(2B+1)+B(2S+1)+S(S+1)$.
### II-C Performance Objectives
Our main target is to choose the resources $\mathbf{x}^{t}$, such that a
cumulative cost for the total delay and transmit power across all nodes is
minimized. The cost function at each node depends on its control variables and
it is considered time-varying in order to capture the unpredictable network
dynamics at every timeslot, e.g. the randomness of the wireless channels or
the network congestion levels. More precisely, we have the following cost
functions per node:
$\displaystyle f_{d}^{t}(\mathbf{x}_{d}^{t})$
$\displaystyle=c^{t}_{d}(w_{d0}^{t})+\sum_{b\in
B}c^{t}_{db}(w_{db}^{t},p_{db}^{t})$ (5) $\displaystyle
f_{b}^{t}(\mathbf{x}_{b}^{t})$ $\displaystyle=\sum_{s\in
S}c^{t}_{bs}(y_{bs}^{t},q_{bs}^{t})+c^{t}_{bC}(y_{bC}^{t})$ (6) $\displaystyle
f^{t}_{s}(\mathbf{x}_{s}^{t})$ $\displaystyle=\sum_{s^{\prime}\in
S}c^{t}_{ss^{\prime}}(z_{ss^{\prime}}^{t})+c^{t}_{sC}(z_{sC}^{t}).$ (7)
The functions $c^{t}_{d}(.),c^{t}_{ss}(.)$ represent local processing delay
cost, $c^{t}_{db}(.),c^{t}_{bs}(.)$ capture both delay and power cost for the
wireless links between nodes, $c^{t}_{bC}(.),c^{t}_{sC}(.)$ are used for the
offloading delay cost to the cloud and $c^{t}_{ss^{\prime}}(.)$ introduces the
delay cost for the wired links between servers. At every timeslot $t$, the
total cost is expressed as
$f^{t}(\mathbf{x}^{t})=\sum_{d\in\mathcal{D}}f^{t}(\mathbf{x}^{t}_{d})+\sum_{b\in\mathcal{B}}f^{t}(\mathbf{x}^{t}_{b})+\sum_{s\in\mathcal{S}}f^{t}(\mathbf{x}^{t}_{s}).$
(8)
Similar to our model, most related works, e.g., [2, 4, 7], assume that local
processing delay and communication delay, often specified via standard queuing
models, are expressed by functions of only local control variables. As we will
see next, this is not the case for the problem constraints, where there exist
constraints that couple agents to preserve the flow of tasks inside the
network.
### II-D Constraints
We now focus on the constraints imposed by our application. In practice, a
good decision must first ensure that the incoming tasks are either processed
locally or offloaded to other nodes (_flow conservation constraint_).
Moreover, the transmission data rate of a wireless link must be sufficient for
the assigned offloaded tasks (_rate constraint_). We model these rates using
the well known Shannon capacity. All constraint functions are considered time-
varying in order to model the unknown dynamics of incoming tasks and channel
gains. Given that the agents first reserve resources and then the tasks are
revealed, it is possible to have service violations, i.e. the provisioned
resources are not adequate or cannot be realized.
For each device $d\in\mathcal{D}$, the constraint functions are
$\displaystyle g^{t}_{d0}(\mathbf{x}_{d}^{t})$
$\displaystyle=r^{t}_{d}-w^{t}_{d0}-\sum_{b\in\mathcal{B}}w^{t}_{db}$ (9)
$\displaystyle g_{db}^{t}(\mathbf{x}_{d}^{t})$
$\displaystyle=w^{t}_{db}-b_{w}\log_{2}\big{(}1+\alpha_{db}^{t}p_{db}^{t}),\forall\leavevmode\nobreak\
b\in\mathcal{B},$ (10)
where $b_{w}$ is a constant for the transmission bandwidth and
$\alpha_{db}^{t}$ is an unknown variable for the channel gain, including the
effect of interference and noise. In a similar way, for each BS
$b\in\mathcal{B}$ we have
$\displaystyle g^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}_{\mathcal{D}}^{t})$
$\displaystyle=\sum_{d\in\mathcal{D}}w^{t}_{db}-y^{t}_{bC}-\sum_{s\in\mathcal{S}}y^{t}_{bs}$
(11) $\displaystyle g_{bs}^{t}(\mathbf{x}_{b}^{t})$
$\displaystyle=y^{t}_{bs}-b_{w}\log_{2}\big{(}1+\alpha_{bs}^{t}q_{bs}^{t}),\forall\leavevmode\nobreak\
s\in\mathcal{S},$ (12)
where $\alpha_{bs}^{t}$ is defined as $\alpha_{db}^{t}$ above. Notice that
(11) is a _coupling constraint_ , i.e. to evaluate the function at BS $b$, we
need to know the external variables $\\{w^{t}_{db}\\}_{d\in\mathcal{D}}$ of
devices. Conventionally, we denote this dependency with conditional arguments
to distinguish from locally available variables, as denoted by
$g^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}_{\mathcal{D}}^{t})$. Finally, for
each server $s\in\mathcal{S}$, we have only the following flow conservation
constraint function
$\displaystyle
g_{s0}^{t}(\mathbf{x}_{s}^{t};\mathbf{x}_{\mathcal{B}}^{t},\mathbf{x}_{\mathcal{S}_{-s}}^{t})\\!=\\!\sum_{b\in\mathcal{B}}y^{t}_{bs}+\\!\\!\\!\sum_{s^{\prime}\in\mathcal{S}_{-s}}\\!\\!\\!z^{t}_{s^{\prime}s}-\\!\\!\sum_{s^{\prime}\in\mathcal{S}}\\!z^{t}_{ss^{\prime}}\\!-\\!z^{t}_{sC}$
(13)
where $\mathcal{S}_{-s}$ is used to denote the set of edge servers
$\mathcal{S}$ excluding $s$. (13) is also a coupling constraint, since server
$s$ needs to know the external variables $\\{y^{t}_{bs}\\}_{b\in\mathcal{B}}$
of BSs and $\\{z^{t}_{s^{\prime}s}\\}_{s^{\prime}\in\mathcal{S}_{-s}}$ of
other servers.
To denote the collective set of constraints per device $d$ and per BS $b$, we
use the notation
$\displaystyle\mathbf{g}_{d}^{t}(\mathbf{x}_{d}^{t})$
$\displaystyle=g^{t}_{d0}(\mathbf{x}_{d}^{t})\cup\\{g^{t}_{db}(\mathbf{x}_{d}^{t})\\}_{\forall
b\in\mathcal{B}}$ (14)
$\displaystyle\mathbf{g}_{b}^{t}(\mathbf{x}_{b}^{t};\mathbf{x}_{\mathcal{D}}^{t})$
$\displaystyle=g^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}_{\mathcal{D}}^{t})\cup\\{g^{t}_{bs}(\mathbf{x}_{b}^{t})\\}_{\forall
s\in\mathcal{S}}$ (15)
and write
$\mathbf{g}^{t}(\mathbf{x}^{t})=\\{\\{\mathbf{g}^{t}_{d}\\}_{d\in\mathcal{D}},\\{\mathbf{g}^{t}_{b}\\}_{d\in\mathcal{B}},\\{g^{t}_{s0}\\}_{s\in\mathcal{S}}\\}:\mathbb{R}_{+}^{V}\to\mathbb{R}^{M}$
to denote all constraints across the network, where $M=B(D+S)+N$.
### II-E Exchange of Information
Due to the coupling constraints, a fully distributed solution is no longer
possible. To circumvent this, we allow each node to exchange information with
other nodes, so that it can take into account the coupling constraints in its
local decisions. The exact messages to exchange is part of the algorithm
design, which ideally should achieve a performance close to a centralized
approach with a minimal exchange of information. To limit the communication
overhead, we further consider that each node can send information to other
nodes only once during a timeslot, i.e. at the beginning of a timeslot. As we
will see in the next section, this practical assumption introduces delayed
feedback between nodes, as not all of the required information is available
for a single transmission step within a timeslot.
## III OCO Formulation and Decentralized Algorithm
The goal of this section is to formulate the resource allocation, described in
the previous section, as an Online Convex Optimization (OCO) problem [5] and
propose an algorithm to solve it. Importantly, we explain how we adapt a
centralized algorithm to transcend towards one which can run in a distributed
fashion; finally, we present the performance guarantees of our solution.
### III-A OCO Preliminaries
We start with the basics of OCO in a centralized algorithm for two reasons:
(a) it helps to introduce useful concepts and metrics; and (b) we benchmark
our distributed algorithm with respect to a centralized one. To formulate the
resource allocation as an OCO problem, we first need to define the sequence of
events taking place for the central agent during every timeslot $t$.
1. 1.
The agent implements an action $\mathbf{x}^{t}$.
2. 2.
The environment reveals all the unknown variables, e.g. computation requests
$\mathbf{r}^{t}$ and channel gains $\alpha_{ij}^{t}$, which in the OCO
framework can be random variables or even controlled by an adversary. Using
these, the functions $f^{t}(\mathbf{.})$ and $\mathbf{g}^{t}(\mathbf{.})$
become known.
3. 3.
The agent receives or evaluates cost and constraint violations, i.e. the
values $f^{t}(\mathbf{x}^{t})$ and $\mathbf{g}^{t}(\mathbf{x}^{t})$.
4. 4.
The agent updates its action to $\mathbf{x}^{t+1}$.
Below, we define the benchmark actions and the metrics to evaluate an
algorithm that produces a sequence of actions
$\\{\mathbf{x}^{t}\\}_{t=1,\dots,T}$.
###### Definition 1 (Static Regret).
The fixed optimal action $\mathbf{x}_{*}$ and the static regret are defined as
$\displaystyle\mathbf{x}_{*}=\underset{x\in\Omega}{\mathrm{argmin}}\sum_{t=1}^{T}f^{t}(\mathbf{x}),\emph{s.t.}\leavevmode\nobreak\
\mathbf{g}^{t}(\mathbf{x})\leq 0,\leavevmode\nobreak\ t=1,\dots,T$ (16)
$\displaystyle\emph{Reg}_{S}(T)=\sum_{t=1}^{T}f^{t}(\mathbf{x}^{t})-\sum_{t=1}^{T}f^{t}(\mathbf{x}_{*}).$
(17)
###### Definition 2 (Dynamic Regret).
The per slot optimal action $\\{\mathbf{x}_{*}^{t}\\}_{t=1,\dots,T}$ and the
dynamic regret are defined as
$\displaystyle\mathbf{x}_{*}^{t}=\underset{x\in\Omega}{\mathrm{argmin}}f^{t}(\mathbf{x}),\emph{s.t.}\leavevmode\nobreak\
\mathbf{g}^{t}(\mathbf{x})\leq 0$ (18)
$\displaystyle\emph{Reg}_{D}(T)=\sum_{t=1}^{T}f^{t}(\mathbf{x}^{t})-\sum_{t=1}^{T}f^{t}(\mathbf{x}_{*}^{t}).$
(19)
###### Definition 3 (Fit).
Using the clipped constraint function
$h_{m}^{t}(\mathbf{x}^{t}):=[g_{m}^{t}(\mathbf{x}^{t})]^{+}=\max\\{0,g_{m}^{t}(\mathbf{x}^{t})\\}$,
the fit is defined as
$\emph{Fit}(T)=\sum_{t=1}^{T}\sum_{m=1}^{M}h_{m}^{t}(\mathbf{x}^{t}).$ (20)
The static regret is a standard metric for evaluating OCO-based algorithms;
however, there is also a growing interest recently for the dynamic regret
[16], which is in principle a much harder metric. For the fit we use a clipped
version of the constraint, i.e. we do not allow negative and positive values
of the constraints to average out in the long run. This is a suitable modeling
approach, as it would be unrealistic to assume that overprovisioning in
certain timeslots can compensate for missing resources or channel rate
violations in other timeslots. Finally, similarly to [8], we define the
gradient of $h_{m}(\mathbf{x})$ as
$\displaystyle\nabla
h_{m}(\mathbf{x})=\nabla[g_{m}(\mathbf{x})]^{+}=\begin{cases}\mathbf{0}&\text{if
$g_{m}(\mathbf{x})\leq 0$}\\\ \nabla g_{m}(\mathbf{x})&\text{if
$g_{m}(\mathbf{x})>0$}\end{cases}$
In the remainder, we use $\mathbf{h}^{t}$ for our analysis with subscripts
that have the same meaning as for $\mathbf{g}^{t}$ in (9)-(15).
Overall, the objective of an algorithm is to establish that Reg${}_{S}(T)$,
Reg${}_{D}(T)$ and Fit$(T)$ are all sublinear in the time horizon $T$ [5].
### III-B Decomposition Across the Agents
A centralized algorithm can use the Lagrange function
$\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})=f^{t}(\mathbf{x}^{t})+\sum_{m=1}^{M}h_{m}^{t}(\mathbf{x})\lambda^{t}_{m}$
(21)
and apply the primal-dual updates [8] as follows
$\displaystyle\boldsymbol{\lambda}^{t}=\frac{\mathbf{h}^{t}({\mathbf{x}}^{t})}{\eta\sigma},\leavevmode\nobreak\
\mathbf{x}^{t+1}=\mathcal{P}_{\Omega}\left({\mathbf{x}}^{t}-\eta{\nabla}_{\mathbf{x}}\mathcal{L}^{t}({\mathbf{x}}^{t},\boldsymbol{\lambda}^{t})\right),$
(22)
where $\lambda^{t}_{m}$ is the Lagrange multiplier for constraint $m$, $\eta$
is the gradient step size, $\sigma$ is a constant and
$\mathcal{P}_{\Omega}(\mathbf{x})$ is the projection of $\mathbf{x}$ onto
$\Omega$. Ideally, we want to perform these updates in a distributed way so
that they are as close as possible to the updates of the centralized
algorithm. For any node $n\in\mathcal{N}$ (device, BS or server), we have
$\displaystyle\boldsymbol{\lambda}^{t}_{n}$
$\displaystyle=\frac{\mathbf{h}_{n}^{t}(\mathbf{x}_{n}^{t};\mathbf{x}_{\mathcal{E}_{n}}^{t})}{\eta\sigma}$
(23) $\displaystyle\mathbf{x}_{n}^{t+1}$
$\displaystyle=\mathcal{P}_{\Omega_{n}}\left(\mathbf{x}_{n}^{t}-\eta\nabla_{\mathbf{x}_{n}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})\right),$
(24)
where $\mathcal{E}_{n}$ is the set of nodes with variables required for node
$n$, i.e. $\mathcal{E}_{d}=\emptyset$, $\mathcal{E}_{b}=\mathcal{D}$ (11),
$\mathcal{E}_{s}=\mathcal{B}\cup\mathcal{S}_{-s}$ (13), and
$\boldsymbol{\lambda}^{t}$ uses the same subscripts for constraints as
$\mathbf{g}^{t},\mathbf{h}^{t}$.
We now focus on the gradients in (24) and write
$\nabla_{\mathbf{x}_{n}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})=\mathbf{H}^{t}_{nL}+\mathbf{H}^{t}_{nE}$
(25)
where $\mathbf{H}^{t}_{nL}$ denotes the part that depends only on local cost
and constraint functions at node $n$,
$\mathbf{H}^{t}_{nL}=\nabla_{\mathbf{x}_{n}}f_{n}^{t}(\mathbf{x}_{n}^{t})+\boldsymbol{\lambda}_{n}^{t\top}\nabla_{\mathbf{x}_{n}}\mathbf{h}^{t}_{n}(\mathbf{x}_{n}^{t};\mathbf{x}_{\mathcal{E}_{n}}^{t}),$
(26)
and $\mathbf{H}^{t}_{nE}$ describes the part that depends on external
constraints from other nodes. For clarity, we provide the explicit expression
for each type of node:
$\displaystyle\mathbf{H}^{t}_{dE}$
$\displaystyle=\sum_{b\in\mathcal{B}}\lambda^{t}_{b0}\nabla_{\mathbf{x}_{d}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}_{\mathcal{D}}^{t})$
(27) $\displaystyle\mathbf{H}^{t}_{bE}$
$\displaystyle=\sum_{s\in\mathcal{S}}\lambda_{s0}^{t}\nabla_{\mathbf{x}_{b}}h^{t}_{s0}(\mathbf{x}_{s}^{t};\mathbf{x}_{\mathcal{B}}^{t},\mathbf{x}_{\mathcal{S}_{-s}}^{t})$
(28) $\displaystyle\mathbf{H}^{t}_{sE}$
$\displaystyle=\sum_{s^{\prime}\in\mathcal{S}_{-s}}\\!\\!\lambda^{t}_{s^{\prime}0}\nabla_{\mathbf{x}_{s}}h_{s^{\prime}0}^{t}(\mathbf{x}_{s^{\prime}}^{t};\mathbf{x}_{\mathcal{B}}^{t},\mathbf{x}_{\mathcal{S}_{-s^{\prime}}}^{t}).$
(29)
Notice that in (27)-(29), only one dimension of the gradients can be non-zero,
as each node has only a single variable involved in coupling constraints of
another node, i.e. $w_{db},y_{bs}$ and $z_{ss^{\prime}}$ for devices, BSs and
servers.
###### Remark 1.
According to the definition of $\mathbf{h}^{t}(.)$ and the flow constraints,
the summation terms in (27)-(29) simplify to
$\lambda^{t}_{n0}\mathbf{1}_{h^{t}_{n0}(\mathbf{x}_{n}^{t};\mathbf{x}_{\mathcal{E}_{n}}^{t})>0}$,
where $\mathbf{1}_{(.)}$ is the indicator function.
### III-C Distributed Algorithm
As one can already notice, a distributed version of the algorithm requires
exchange of information at two different steps. Specifically, a node $n$ needs
to send its variables to nodes that need them in their coupling constraints
and then, receive the gradient-related feedback $\mathbf{H}^{t}_{nE}$. We now
turn our focus on the adopted communication model and examine how this message
exchange can be implemented.
(a) Ideal case
(b) Proposed approach
Figure 2: Exchanged messages during time slot $t$ between device $d$ and BS
$b$.
An ideal scenario is shown in Fig. 2(a), where nodes can send messages at any
moment during a time slot, focusing for simplicity in the link between a
device and a BS. As we can see, the BS $b$ needs to collect
$\mathbf{x}^{t}_{\mathcal{E}_{b}}=\\{w^{t}_{db}\\}_{d\in\mathcal{D}}$ and
perform few processing steps before it can send its feedback to the device,
which can then perform the primal update $\mathbf{x}_{d}^{t+1}$. In practice,
this is not possible in our model as nodes are allowed to send messages only
once at the beginning of a time slot. We address this limitation by allowing
nodes to send their feedback at the next time slot, as shown in Fig. 2(b). As
a result, the device uses the outdated feedback $\mathbf{H}^{t-1}_{dE}$ for
its updates.
Overall, we define the following messages between nodes $n,v$, where
$n\in\mathcal{E}_{v}$, and summarize them in Table I.
1. 1.
$m^{t}_{1,n\to v}:=\mathbf{x}^{t}_{n}\in\mathbf{x}^{t}_{\mathcal{E}_{v}}$;
required to evaluate the coupling constraint
$h^{t}_{v0}(\mathbf{x}^{t}_{v};\mathbf{x}^{t}_{\mathcal{E}_{v}})$ at node $v$,
which is then used for the dual update of $\lambda^{t}_{v0}$ in (23) and the
local term $\mathbf{H}^{t}_{vL}$.
2. 2.
$m^{t}_{2,v\to
n}:=\lambda^{t-1}_{v0}\mathbf{1}_{h^{t-1}_{v0}(\mathbf{x}_{v}^{t-1};\mathbf{x}_{\mathcal{E}_{v}}^{t-1})>0}$;
feedback required to evaluate $\mathbf{H}^{t-1}_{nE}$ for the primal update at
node $n$.
TABLE I: Messages $(m^{t}_{1},m^{t}_{2})$ between nodes From$\backslash$To | Device $d^{\prime}$ | BS $b^{\prime}$ | Server $s^{\prime}$
---|---|---|---
Device $d$ | $-$ | $w^{t}_{db^{\prime}}$ | $-$
BS $b$ | $\lambda^{t-1}_{b0}\mathbf{1}_{h^{t-1}_{b0}>0}$ | $-$ | $y^{t}_{bs^{\prime}}$
Server $s$ | $-$ | $\lambda^{t-1}_{s0}\mathbf{1}_{h^{t-1}_{s0}>0}$ | $z^{t}_{ss^{\prime}}$, $\lambda^{t-1}_{s0}\mathbf{1}_{h^{t-1}_{s0}>0}$
Let us now consider the primal/dual updates of the proposed approach. First,
the dual update (23) remains the same and thus, identical to the centralized
case. Then, for the primal update (24), only the gradient term is modified and
approximated by
$\nabla_{\mathbf{x}_{n}}\hat{\mathcal{L}}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})=\mathbf{H}^{t}_{nL}+\mathbf{H}^{t-1}_{nE}.$
(30)
The steps of our proposed distributed algorithm are presented in Algorithm 1
for any node $n$.
Algorithm 1 Distributed OCO for node $n$
1:Initialize: parameters $\sigma$, $\eta$
2:Set first action $\mathbf{x}^{1}_{n}\in\Omega_{n}$ and define
$\boldsymbol{\lambda}^{0}_{n}=\mathbf{0}$.
3:for $t=1,\dots,T$ do
4: Play action $\boldsymbol{x}^{t}_{n}$
5: Send messages $m^{t}_{1,n\to v}$ to nodes $v:n\in\mathcal{E}_{v}$
6: Receive from environment functions $f^{t}_{n}(.)$ and
$\boldsymbol{g}^{t}_{n}(.)$
7: Receive feedback messages $m^{t}_{2,v\to n}$ from nodes $v$
8: Compute $\boldsymbol{\lambda}_{n}^{t}$ with (23) $\triangleright$ Dual
update
9: Update $\boldsymbol{x}_{n}^{t+1}$ with (24),(30) $\triangleright$ Primal
update
10:end for
### III-D Performance Guarantees
We make the following standard assumptions, widely used in online learning
literature (e.g., see [8, 11]), and then formally state our main theorem.
###### Assumption 1.
* •
(i) Set $\Omega_{n}$ is bounded and convex; specifically it holds that
$\left\lVert\mathbf{x}_{n}-\mathbf{y}_{n}\right\rVert\leq R$,
$\forall\mathbf{x}_{n},\mathbf{y}_{n}\in\Omega_{n}$, for
$n\in\mathcal{D,B,S}$.
* •
(ii) For $t=1,\dots,T$, functions $f^{t}_{n}$ and $g^{t}_{n,i}$ are convex and
Lipschitz with $\left\lVert\nabla_{\mathbf{x}_{n}}f^{t}_{n}\right\rVert\leq
F^{\prime}$ and $\left\lVert\nabla_{\mathbf{x}_{n}}g_{n,i}^{t}\right\rVert\leq
G^{\prime}$, for $n\in\mathcal{D,B,S}$ and
$\mathbf{g}_{n}^{t}=\\{g^{t}_{n,i}\\}_{i=1,\dots,M_{n}}$ (with $M_{n}$ the
number of constraints at node $n$).
Below we write a list of implications that we use for the proof of our
theorem. First, $f^{t}_{n}$ and $g^{t}_{n,i}$ are both bounded, i.e.,
$|f^{t}_{n}|\leq F$, $|g^{t}_{n,i}|\leq G^{\prime\prime}$. Second, since
$\left\lVert\nabla g_{n,i}^{t}\right\rVert\leq G^{\prime}$, then
$\left\lVert\nabla h_{n,i}^{t}\right\rVert\leq G^{\prime}$ (comes from the
definition of gradient of $h$). Third, since $g^{t}_{n,i}$ is bounded,
$h^{t}_{n,i}$ is also bounded by definition; hence $|h^{t}_{n,i}|\leq
G^{\prime\prime}$ and $\left\lVert\mathbf{h}^{t}\right\rVert\leq G$. For
simplicity we write that $|f^{t}_{n}|,\left\lVert\nabla
f^{t}_{n}\right\rVert\leq F$ and $|h_{n,i}^{t}|,\left\lVert\nabla
h_{n,i}^{t}\right\rVert,\left\lVert\mathbf{h}_{n}^{t}\right\rVert\leq G$.
Fourth, since $g_{n,i}^{t}$ is convex, then $h_{n,i}^{t}$ is as well.
A proof of the first and fourth implications can be found in Appendix B.
###### Theorem 1.
Given Assumption 1, and $\sigma>3KG^{2}$, Algorithm 1 guarantees that
$\displaystyle{\text{Reg}_{S}}(T)$
$\displaystyle\leq\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T\triangleq U_{sr},$ (31) $\displaystyle{\text{Reg}_{D}}(T)$
$\displaystyle\leq U_{sr}+\frac{R}{\eta}V(\boldsymbol{x}_{*}^{1:T}),$ (32)
$\displaystyle{\text{Fit}}(T)$
$\displaystyle\leq\sqrt{\frac{\eta\sigma}{\beta}MT(U_{sr}+2NFT)},$ (33)
where $E$ is the number of edges in the network topology,
$K=2D(3B+1)+2B(3S+1)+2S(2S-1)$, $\beta=1-\frac{3KG^{2}}{\sigma}$ and
$V(\mathbf{x}_{*}^{1:T})=\sum_{t=1}^{T}\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}_{*}^{t-1}\right\rVert$.
###### Proof.
The proof can be found in Appendix A. ∎
An immediate implication of Theorem 1 is that, for step size
$\eta=\mathcal{O}(T^{-1/2})$, we have
* •
$\textnormal{Reg}_{S}(T)=\mathcal{O}(T^{1/2})$
* •
$\textnormal{Reg}_{D}(T)=\mathcal{O}(\max\\{T^{1/2},T^{1/2}V(\mathbf{x}_{*}^{1:T})\\})$
* •
$\textnormal{Fit}(T)=\mathcal{O}(T^{3/4})$
We conclude that our distributed algorithm, although using outdated Lagrange
multipliers, achieves sublinear static regret and fit; if additionally,
$V(\mathbf{x}_{*}^{1:T})=o(T^{1/2})$, then dynamic regret is also sublinear.
In fact, we achieve the same order of bounds as the centralized algorithm in
[4] (in the case where the authors ignore the outages), which tackles the same
setting. Note that choosing $\eta=\mathcal{O}(T^{-1/2})$ yields the minimum
regret while preserving a sublinear fit.
## IV Performance Evaluation
### IV-A Simulation Setup
Topology and box constraints. We assume a fully connected setting with nodes
$D,B,S=2$. Moreover, the upper bounds of the control variables are as follows:
for every device $d$, $\overline{w}_{d0}=2$, $\overline{w}_{db}=25$, and
$\overline{p}_{db}=25$, for every BS $b$, $\overline{y}_{bC}=30$,
$\overline{y}_{bs}=25$, and $\overline{q}_{bs}=25$ and for every server $s$,
$\overline{z}_{sC}=50$, $\overline{z}_{ss}=15$ and
$\overline{z}_{ss^{\prime}}=10$.
Costs. We model the cost functions using expressions for delay (from
$M/M/1$111To avoid numerical instabilities, e.g., $M/M/1$ delay becoming
infinite, we use standard convex extensions [4].) and power [7]. The local
processing delay of node $n$ for $x$ tasks is $c_{n}(x)=1/(\overline{x}-x)$,
where $\overline{x}$ denotes the capacity of node $n$; this cost is used to
model $c^{t}_{d}(w^{t}_{d0})$ and $c^{t}_{ss}(z^{t}_{ss})$. Then, the cost
related to wireless offloading, i.e., $c^{t}_{db}(w^{t}_{db},p^{t}_{db})$ and
$c^{t}_{bs}(y^{t}_{bs},q^{t}_{bs})$, is modeled as
$c^{t}_{nn^{\prime}}(x,y)=1/(R^{t}_{nn^{\prime}}(y)-x)+\frac{1}{2}y^{2}$,
where
$R^{t}_{nn^{\prime}}(y)=b_{w}\log_{2}\big{(}1+\alpha_{nn^{\prime}}^{t}y)$ is
the channel rate. Finally, the delays for offloading to the cloud, i.e.
$c^{t}_{bC}(y^{t}_{bC})$ and $c^{t}_{sC}(z^{t}_{sC})$, are modeled as
$c_{nC}^{t}(x)=d_{nC}^{t}x$, where $d_{nC}^{t}$ is a time-varying unknown
environment parameter.
Unknown variables. For each time slot $t$ we need: (a) channel gains
$\alpha_{db}^{t},\alpha_{bs}^{t}$, (b) cloud delay costs
$d_{bC}^{t},d_{sC}^{t}$, and (c) traffic requests $\mathbf{r}^{t}$. We model
(a) and (b) as random variables sampled from $\mathcal{U}(8,15)$ and
$\mathcal{U}(3,10)$ respectively. For (c), we mainly use the publicly
available Milano dataset [17]; in particular, we extract the aggregate
Internet traffic arrivals measured in MBs to $D=2$ BSs (devices in our model).
We also provide results using synthetic demands, drawn from
$\mathcal{U}(1,10)$.
Metrics and Baselines. The performance of an online algorithm is evaluated
using the static and dynamic regrets (the respective benchmarks are found
using CVXPY [18]), and the fit. We plot these metrics for proposed Algorithm
1, which we refer to as Cooperative algorithm, and for two more baselines.
First, the _Selfish_ is a distributed algorithm without information exchange
between nodes; i.e. $\mathbf{H}_{nE}^{t}=\mathbf{0}$ in (25). Second, the
_Centralized_ algorithm assumes a controller with access to all necessary
information in order to do the updates optimally. This is essentially the
algorithm described in [8], adapted to our setting with time-varying
constraints $\mathbf{h}^{t}$.
### IV-B Simulation Results
In all our plots, the $x$-axis represents the horizon length $T$, which we
vary from $0$ to $300$ time slots. For each algorithm we plot the average
value across $4$ independent runs and with shade the corresponding standard
deviation. Notice that all metrics are normalized by the horizon $T$.
Our setup is challenging for a distributed algorithm, as the flow conservation
constraints _couple_ different nodes. To this end, we first investigate the
Fit$(T)$ for the Milano and the synthetic datasets in Figs. 3(a), 3(b). The
fit of the _Centralized_ algorithm quickly converges to zero, suggesting that
it learns to play actions that respect most of the time-varying constraints;
the reason is that it performs the best possible primal and dual updates with
the freshest information. The fit of the proposed _Cooperative_ algorithm
converges almost together with the _Centralized_ for the Milano demands (Fig.
3(a)) and slightly slower for the synthetic ones (Fig. 3(b)). Therefore, the
modified gradients proposed in our algorithm suffice in order to satisfy the
constraints in the long run. The _Selfish_ baseline exemplifies the necessity
of at least some information exchange between the nodes; we can see in both
figures the fit increasing. This behavior is expected, as the Fit$(T)$ of
_Centralized_ and _Collaborative_ is sublinear, whereas the one of _Selfish_
can be shown to be linear as its updates totally ignore the coupling (flow
conservation) constraints.
Having discussed the “feasibility” aspect of the algorithms (i.e., how they
perform in terms of constraints) we now focus on the objective function and in
particular on the regrets in Figs. 3(c), 3(d). We plot these metrics only for
the Milano dataset; but the respective plots for the synthetic one are
similar. A first observation is that the regrets of the _Selfish_ algorithm
are the lowest among all three methods. This should not come as a surprise,
since by construction, the algorithm solves a more relaxed version of the
problem (ignores flow constraints) and therefore can achieve better cost
values. The _Centralized_ algorithm has slightly higher regrets, which is
justified by its effort to also satisfy the fit. Finally, our _Collaborative_
algorithm has regrets that also go to zero and are very close to the ones of
the _Centralized_ solution.
(a) Fit - Milano
(b) Fit - Synthetic
(c) Static regret - Milano
(d) Dynamic Regret - Milano
Figure 3: Performance metrics vs horizon length $T$
Finally, we comment on the jump at $T\approx 110$ in Fig. 3(c), which is not
present in Fig. 3(d). The difference of the two plots lies with the
benchmarks; in Fig. 4, the $y$-axis is in fact the cost gap between them, i.e.
$\frac{1}{T}\sum_{t=1}^{T}\Big{(}f^{t}(x_{*})-f^{t}(x^{t}_{*})\Big{)}$; and in
that plot we obviously see that same jump. This behavior can be explained by
the change of demand in Fig. 4. Essentially, the static benchmark, for
$T>110$, solves an optimization problem by finding a feasible $x_{*}$ for that
extreme demand (at $T\approx 110$), and will be then _more constrained_
compared to the dynamic benchmark, which solves the problem for each $t$
individually.
Figure 4: Milano dataset: (a) Difference of regrets (and benchmarks) vs
horizon $T$; (b) Demands of the devices over time.
## V Conclusions
We revisit the problem of resource allocation in an IoT network, where devices
can process part of the traffic requests, and/or offload the rest to more
powerful computational entities (cloud, edge servers). The distributed nature
of the setting, as well as the unpredictable environment, motivated us to
model the network nodes (devices, BSs, servers) as distributed OCO actors.
However, the nodes of the network are naturally coupled by flow conservation
constraints at each node, which we model as long-term constraints, and as a
result a fully decentralized algorithm is no longer possible.
In order to address this challenge, we propose a distributed OCO algorithm
with limited communication between nodes, which practically leads to partially
outdated gradient updates. Nevertheless, we show theoretically that our
algorithm achieves sub-linear regret bound $\mathcal{O}(T^{1/2})$ and sub-
linear constraint violation bound $\mathcal{O}(T^{3/4})$, which is the same
order of bounds as a centralized algorithm for this setting. Numerical results
based on real data traces confirm our theoretical findings.
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## Appendix A Proof of Theorem 1
In this section, we provide the upper bounds of the static regret, the dynamic
regret and the fit as a function of $T$. To that extent, we remind that:
$\displaystyle f^{t}(\mathbf{x}^{t})$ $\displaystyle=\sum_{d\in
D}f_{d}^{t}(\mathbf{x}_{d}^{t})+\sum_{b\in
B}f_{b}^{t}(\mathbf{x}_{b}^{t})+\sum_{s\in S}f_{s}^{t}(\mathbf{x}_{s}^{t})$
$\displaystyle\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})$
$\displaystyle=f^{t}(\mathbf{x}^{t})+[\mathbf{h}^{t}(\mathbf{x}^{t})]^{\top}\boldsymbol{\lambda}^{t},\quad\text{with}\;\;\boldsymbol{\lambda}^{t}=\left((\boldsymbol{\lambda}_{d}^{t})_{d\in
D},(\boldsymbol{\lambda}^{t}_{b})_{b\in
B},(\boldsymbol{\lambda}_{s}^{t})_{s\in S}\right).$
For ease of understanding, we also recall that:
* •
$\eta$ is the step-size of the decentralized algorithm.
* •
$|f^{t}_{n}|,\left\lVert\nabla f^{t}_{n}\right\rVert\leq F$; and
$\left\lVert\nabla
g^{t}_{n,i}\right\rVert,\left\lVert\mathbf{g}^{t}_{n}\right\rVert\leq G$ for
$n\in\mathcal{D,B,S}$.
* •
$R$ is the radius of the space set $\Omega$.
* •
$\sigma$ is a constant that we will define later.
* •
$D,B,S$ is the number of devices, BSs and servers respectively.
* •
$N$ is the number of nodes, i.e. $N=D+B+S$.
* •
$M$ is the number of the constraint functions, with $M=D(B+1)+B(S+1)+S$.
* •
$E$ is the total number of edges, i.e. $E=DB+BS+S(S-1)$.
In order to prove the present theorem, let us first introduce the following
Lemma.
###### Lemma 1.
For any node $n$, the next inequality holds:
$\displaystyle(\mathbf{x}_{n}^{t}-\mathbf{x}_{n})^{\top}\nabla_{\mathbf{x}_{n}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})$
$\displaystyle\leq\frac{1}{2\eta}\left(\left\lVert\mathbf{x}_{n}-\mathbf{x}_{n}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{n}-\mathbf{x}_{n}^{t+1}\right\rVert^{2}\right)$
$\displaystyle\quad+2\eta
F^{2}+2\eta(c_{1,n}+c_{2,n})G^{2}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+2\eta
c_{2,n}G^{2}\left(\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}\right)$
$\displaystyle\quad+\frac{3}{2}\eta\left(F^{2}+c_{1,n}G^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}+c_{2,n}G^{2}\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2}\right)+\frac{\eta}{2}c_{2,n}G^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}$
$\displaystyle\quad+(\mathbf{x}_{n}^{t}-\mathbf{x}_{n})^{\top}\mathbf{H}^{t}_{nE}-(\mathbf{x}_{n}^{t-1}-\mathbf{x}_{n})^{\top}\mathbf{H}^{t-1}_{nE}$
$\displaystyle=\frac{1}{2\eta}\left(\left\lVert\mathbf{x}_{n}-\mathbf{x}_{n}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{n}-\mathbf{x}_{n}^{t+1}\right\rVert^{2}\right)+\frac{7}{2}\eta
F^{2}$ $\displaystyle\quad+2(c_{1,n}+2c_{2,n})\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+\frac{1}{2}(3c_{1,n}+5c_{2,n})\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}+\frac{3}{2}c_{2,n}\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2}$
$\displaystyle\quad+(\mathbf{x}_{n}^{t}-\mathbf{x}_{n})^{\top}\mathbf{H}^{t}_{nE}-(\mathbf{x}_{n}^{t-1}-\mathbf{x}_{n})^{\top}\mathbf{H}^{t-1}_{nE}$
(34)
where $\mathbf{H}^{t}_{nE}$ is defined in (27)-(29), $c_{1,n}$ is the number
of local constraints at node $k$ and $c_{2,n}$ is the number of flow
constraints from other nodes that include an optimization variable of node
$k$. Specifically:
$c_{1,n}=\begin{cases}B+1&\text{for $n=d$}\\\ N+1&\text{for $n=b$}\\\
1&\text{for $n=s$}\end{cases},\quad c_{2,n}=\begin{cases}B&\text{for $n=d$}\\\
N&\text{for $n=b$}\\\ N-1&\text{for $n=s$}\end{cases}.$
###### Proof.
We prove in detail only the expression for the devices, as the respective
expressions for BSs and servers is straight-forward following exactly the same
steps. Thus, we have
$\displaystyle\left\lVert\mathbf{x}_{d}-\mathbf{x}_{d}^{t+1}\right\rVert^{2}$
$\displaystyle\overset{(a)}{\leq}\left\lVert\mathbf{x}_{d}-\Bigg{(}\mathbf{x}_{d}^{t}-\eta\bigg{(}\nabla_{\mathbf{x}_{d}}f_{d}^{t}(\mathbf{x}_{d}^{t})+\nabla_{\mathbf{x}_{d}}^{\top}[\mathbf{h}_{d}^{t}(\mathbf{x}_{d}^{t})]\boldsymbol{\lambda}^{t}_{d}+\sum_{b\in\mathcal{B}}\nabla_{\mathbf{x}_{d}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda^{t-1}_{b0}\bigg{)}\Bigg{)}\right\rVert^{2}$
$\displaystyle\overset{(b)}{=}\left\lVert\big{(}\mathbf{x}_{d}-\mathbf{x}_{d}^{t}\big{)}+\eta\Bigg{[}\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})+\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}\Big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\Big{]}\bigg{]}\Bigg{]}\right\rVert^{2}$
$\displaystyle\overset{(c)}{\leq}\left\lVert\mathbf{x}_{d}-\mathbf{x}_{d}^{t}\right\rVert^{2}+2\eta^{2}\left\lVert\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})\right\rVert^{2}+2\eta^{2}\left\lVert\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}\Big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\Big{]}\bigg{]}\right\rVert^{2}$
$\displaystyle\quad+2\eta(\mathbf{x}_{d}-\mathbf{x}^{t}_{d})^{\top}\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})+2\eta(\mathbf{x}_{d}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}\Big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\Big{]}\bigg{]},$
(35)
where:
* •
(a) comes from the update rule of $\mathbf{x}_{d}^{t+1}$ and Assumption 1 on
the projection non-expansiveness.
* •
(b) uses the expression of
$\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})$
from (25).
* •
(c) uses
$\left\lVert\mathbf{x}+\mathbf{y}\right\rVert^{2}=\left\lVert\mathbf{x}\right\rVert^{2}+\left\lVert\mathbf{y}\right\rVert^{2}+2\mathbf{x}^{\top}\mathbf{y}$
and then $\left\lVert\mathbf{x}+\mathbf{y}\right\rVert^{2}\leq
2(\left\lVert\mathbf{x}\right\rVert^{2}+\left\lVert\mathbf{y}\right\rVert^{2})$.
Therefore, we have:
$\displaystyle(\mathbf{x}_{d}^{t}-\mathbf{x}_{d})^{\top}\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})$
$\displaystyle\leq\frac{1}{2\eta}\bigg{(}\left\lVert\mathbf{x}_{d}-\mathbf{x}_{d}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{d}-\mathbf{x}_{d}^{t+1}\right\rVert^{2}\bigg{)}+\eta\left\lVert\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})\right\rVert^{2}$
$\displaystyle\quad+\eta\left\lVert\nabla_{\mathbf{x}_{d}}\sum\limits_{b\in\mathcal{B}}\big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\big{]}\right\rVert^{2}$
$\displaystyle\quad+(\mathbf{x}_{d}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}\big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\big{]}\bigg{]}.$
(36)
We continue by analyzing each term of (A). Specifically, we have:
$\displaystyle\eta\left\lVert\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})\right\rVert^{2}$
$\displaystyle=\eta\left\lVert\nabla_{\mathbf{x}_{d}}f_{d}^{t}(\mathbf{x}_{d}^{t})+\nabla_{\mathbf{x}_{d}}^{\top}[\mathbf{h}_{d}^{t}(\mathbf{x}_{d}^{t})]\boldsymbol{\lambda}^{t}_{d}+\sum_{b\in\mathcal{B}}\nabla_{\mathbf{x}_{d}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda^{t}_{b0}\right\rVert^{2}$
$\displaystyle\overset{(a)}{\leq}\eta\Bigg{(}F+\left\lVert\nabla_{\mathbf{x}_{d}}^{\top}[\mathbf{h}_{d}^{t}(\mathbf{x}_{d}^{t})]\boldsymbol{\lambda}^{t}_{d}+\sum_{b\in\mathcal{B}}\nabla_{\mathbf{x}_{d}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda^{t}_{b0}\right\rVert\Bigg{)}^{2}$
$\displaystyle\overset{(b)}{\leq}2\eta
F^{2}+2\eta\bigg{(}\left\lVert\nabla_{\mathbf{x}_{d}}^{\top}[\mathbf{h}_{d}^{t}(\mathbf{x}_{d}^{t})]\boldsymbol{\lambda}^{t}_{d}\right\rVert+\sum_{b\in\mathcal{B}}\left\lVert\nabla_{\mathbf{x}_{d}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda^{t}_{b0}\right\rVert\bigg{)}^{2}$
$\displaystyle\overset{(c)}{\leq}2\eta F^{2}+2\eta
G^{2}\big{(}|\lambda^{t}_{d0}|+\sum\limits_{b\in\mathcal{B}}|\lambda_{db}^{t}|+\sum\limits_{b\in\mathcal{B}}|\lambda_{b0}^{t}|\big{)}^{2}$
$\displaystyle\overset{(d)}{\leq}2\eta
F^{2}+2\eta(2B+1)G^{2}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}$
(37)
where:
* •
(a) uses
$\left\lVert\mathbf{x}+\mathbf{y}\right\rVert\leq\left\lVert\mathbf{x}\right\rVert+\left\lVert\mathbf{y}\right\rVert$
and then $\left\lVert\nabla{f^{t}_{d}(\mathbf{x}^{t})}\right\rVert\leq F$.
* •
(b) uses first $\left\lVert\mathbf{x}+\mathbf{y}\right\rVert^{2}\leq
2(\left\lVert\mathbf{x}\right\rVert^{2}+\left\lVert\mathbf{y}\right\rVert^{2})$
and then
$\left\lVert\mathbf{x}+\mathbf{y}\right\rVert\leq\left\lVert\mathbf{x}\right\rVert+\left\lVert\mathbf{y}\right\rVert$.
* •
(c) follows from $\left\lVert\nabla{h^{t}_{i}(\mathbf{x}^{t})}\right\rVert\leq
G\;\forall i\in[1,M]$.
* •
(d) uses $(\sum\limits_{i=1}^{K}a_{i})^{2}\leq
K\sum\limits_{i=1}^{K}a_{i}^{2}$ and the fact that $2B+1\leq
M\Rightarrow|{\lambda}_{d0}^{t}|^{2}+\sum\limits_{b\in\mathcal{B}}|\lambda_{db}^{t}|^{2}+\sum\limits_{b\in\mathcal{B}}|\lambda_{b0}^{t}|^{2}\leq\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}$.
Following the same procedure, i.e. steps (b)-(d), for the second term of (A),
we get:
$\displaystyle\eta\left\lVert\nabla_{\mathbf{x}_{d}}\sum\limits_{b\in\mathcal{B}}\big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b}^{t}\big{]}\right\rVert^{2}$
$\displaystyle\leq 2\eta
G^{2}\left(\Big{(}\sum\limits_{b\in\mathcal{B}}|\lambda_{b0}^{t-1}|\Big{)}^{2}+\Big{(}\sum\limits_{b\in\mathcal{B}}|\lambda_{b0}^{t}|\Big{)}^{2}\right)$
$\displaystyle\leq 2\eta
BG^{2}\Big{(}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}\Big{)}$
(38)
For the last term of (A), if we add and subtract the term
$(\mathbf{x}_{d}^{t-1})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}\bigg{]}$
we get:
$\displaystyle(\mathbf{x}_{d}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}\big{[}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}-h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\big{]}\bigg{]}=(\mathbf{x}_{d}-\mathbf{x}_{d}^{t-1})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}\bigg{]}$
$\displaystyle\quad-(\mathbf{x}_{d}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\bigg{]}+(\mathbf{x}_{d}^{t-1}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}\bigg{]}.$
(39)
For the last term of (A) we have:
$\displaystyle(\mathbf{x}_{d}^{t-1}-\mathbf{x}_{d}^{t})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}\bigg{]}\overset{(a)}{\leq}\frac{1}{2\eta}\left\lVert\mathbf{x}_{d}^{t}-\mathbf{x}_{d}^{t-1}\right\rVert^{2}+\frac{\eta}{2}\left\lVert\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t-1}_{b0}(\mathbf{x}_{b}^{t-1};\mathbf{x}^{t-1}_{\mathcal{D}})\lambda_{b0}^{t-1}\bigg{]}\right\rVert^{2}$
$\displaystyle\quad\overset{(b)}{\leq}\frac{\eta}{2}\left\lVert\nabla_{\mathbf{x}_{d}}f_{d}^{t-1}(\mathbf{x}_{d}^{t-1})+\nabla^{\top}_{\mathbf{x}_{d}}[\mathbf{h}^{t-1}_{d}(\mathbf{x}_{d}^{t-1})]\boldsymbol{\lambda}_{d}^{t-1}+\sum\limits_{b\in\mathcal{B}}\nabla_{\mathbf{x}_{d}}[h^{t-2}_{b0}(\mathbf{x}_{b}^{t-2};\mathbf{x}^{t-2}_{\mathcal{D}})]\lambda_{b0}^{t-2}\right\rVert^{2}+\frac{\eta}{2}BG^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}$
$\displaystyle\quad\overset{(c)}{\leq}\frac{3\eta}{2}\left(F^{2}+(B+1)G^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}+BG^{2}\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2}\right)+\frac{\eta}{2}BG^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2},$
(40)
where:
* •
(a) uses
$\mathbf{x}^{\top}\mathbf{y}\leq\frac{1}{2\eta}\left(\left\lVert\mathbf{x}\right\rVert^{2}+\eta^{2}\left\lVert\mathbf{y}\right\rVert^{2}\right)$
* •
(b) comes from the update rule of $\mathbf{x}_{d}^{t}$ and Assumption 1.
* •
(c) uses $\left\lVert\mathbf{x}+\mathbf{y}+\mathbf{z}\right\rVert^{2}\leq
3(\left\lVert\mathbf{x}\right\rVert^{2}+\left\lVert\mathbf{y}\right\rVert^{2}+\left\lVert\mathbf{z}\right\rVert^{2})$
Combining (A)-(A) we directly obtain (1). Following the same steps for BSs and
servers completes the proof of Lemma 1. ∎
The next step to prove the theorem is to combine the results of Lemma 1 for
all nodes. According to Assumption 1,
$\mathcal{L}^{t}(.,\boldsymbol{\lambda})$ is a convex function with
$\mathbf{x}$ and thus, we have
$\displaystyle\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})-\mathcal{L}^{t}(\mathbf{x},\boldsymbol{\lambda}^{t})$
$\displaystyle\leq(\mathbf{x}^{t}-\mathbf{x})^{\top}\nabla_{x}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})$
$\displaystyle=\sum\limits_{d\in\mathcal{D}}(\mathbf{x}_{d}^{t}-\mathbf{x}_{d})^{\top}\nabla_{\mathbf{x}_{d}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})+\sum\limits_{b\in\mathcal{B}}(\mathbf{x}_{b}^{t}-\mathbf{x}_{b})^{\top}\nabla_{\mathbf{x}_{b}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})+\sum\limits_{s\in\mathcal{S}}(\mathbf{x}_{s}^{t}-\mathbf{x}_{s})^{\top}\nabla_{\mathbf{x}_{s}}\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t}).$
(41)
Then, from Lemma 1 we get:
$\displaystyle\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})-\mathcal{L}^{t}(\mathbf{x},\boldsymbol{\lambda}^{t})$
$\displaystyle\leq\frac{1}{2\eta}\Big{(}\left\lVert\mathbf{x}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}-\mathbf{x}^{t+1}\right\rVert^{2}\Big{)}+\frac{7}{2}\eta
NF^{2}$ $\displaystyle\quad+\Big{(}(6DB+2D)+(6BS+2B)+(4S^{2}-2S)\Big{)}\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}$
$\displaystyle\quad+\Big{(}(4DB+\frac{3}{2}D)+(4BS+\frac{3}{2}B)+(\frac{5}{2}S^{2}-S)\Big{)}\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}$
$\displaystyle\quad+\frac{3}{2}\Big{(}DB+BS+S(S-1)\Big{)}\eta
G^{2}\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2}+Q^{t}(\mathbf{x})-Q^{t-1}(\mathbf{x})$
(42)
where
$\displaystyle Q^{t}(\mathbf{x})$
$\displaystyle:=\sum\limits_{d\in\mathcal{D}}\Bigg{[}(\mathbf{x}_{d}^{t}-\mathbf{x}_{d})^{\top}\nabla_{\mathbf{x}_{d}}\bigg{[}\sum\limits_{b\in\mathcal{B}}h^{t}_{b0}(\mathbf{x}_{b}^{t};\mathbf{x}^{t}_{\mathcal{D}})\lambda_{b0}^{t}\bigg{]}\Bigg{]}+\sum\limits_{b\in\mathcal{B}}\Bigg{[}(\mathbf{x}_{b}^{t}-\mathbf{x}_{b})^{\top}\nabla_{\mathbf{x}_{b}}\bigg{[}\sum\limits_{s\in\mathcal{S}}h^{t}_{s0}(\mathbf{x}^{t}_{s};\mathbf{x}^{t}_{\mathcal{B}},\mathbf{x}^{t}_{\mathcal{S}_{-s}})\lambda_{s0}^{t}\bigg{]}\Bigg{]}$
$\displaystyle\quad+\sum\limits_{s\in\mathcal{S}}\Bigg{[}(\mathbf{x}_{s}^{t}-\mathbf{x}_{s})^{\top}\nabla_{\mathbf{x}_{s}}\bigg{[}\sum\limits_{s^{\prime}\in\mathcal{S}_{-s}}h_{s^{\prime}0}^{t}(\mathbf{x}_{s^{\prime}}^{t};\mathbf{x}_{\mathcal{B}}^{t},\mathbf{x}^{t}_{\mathcal{S}_{-s^{\prime}}})\lambda_{s^{\prime}0}^{t}\bigg{]}\Bigg{]}$
(43)
To upper-bound the multiplicative terms in front of the multipliers in (A), we
define
$\displaystyle K=2D(3B+1)+2B(3S+1)+2S(2S-1),$ (44)
and get
$\displaystyle\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})-\mathcal{L}^{t}(\mathbf{x},\boldsymbol{\lambda}^{t})$
$\displaystyle\leq\frac{1}{2\eta}\underbrace{\Big{(}\left\lVert\mathbf{x}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}-\mathbf{x}^{t+1}\right\rVert^{2}\Big{)}}_{\text{(a)}}+\underbrace{Q^{t}(\mathbf{x})-Q^{t-1}(\mathbf{x})}_{\text{(b)}}+\frac{7}{2}\eta
NF^{2}$ $\displaystyle\quad+\eta
KG^{2}\underbrace{\Big{(}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2}\Big{)}}_{\text{(c)}}.$
(45)
We then sum from $t=1\to T$ and upper-bound the (a)-(c) terms of (A) using the
following:
* •
(a)
$\sum\limits_{t=1}^{T}(\left\lVert\mathbf{x}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}-\mathbf{x}^{t+1}\right\rVert^{2})=\left\lVert\mathbf{x}-\mathbf{x}^{1}\right\rVert^{2}-\left\lVert\mathbf{x}-\mathbf{x}^{T+1}\right\rVert^{2}\leq\left\lVert\mathbf{x}^{1}-\mathbf{x}\right\rVert^{2}\leq
R^{2}$;
* •
(b)
$\sum\limits_{t=1}^{T}(Q^{t}(\mathbf{x})-Q^{t-1}(\mathbf{x}))=Q^{T}(\mathbf{x})-Q^{0}(\mathbf{x})\leq\left\lVert
Q^{T}(\mathbf{x})\right\rVert+\left\lVert
Q^{0}(\mathbf{x})\right\rVert\leq\frac{2REG^{2}}{\eta\sigma}$, where in the
last step we use $\left\lVert
Q^{t}(\mathbf{x})\right\rVert\leq\left(DB+BS+S(S-1)\right)RG\left\lVert\boldsymbol{\lambda}^{t}\right\rVert$
according to (A) and the update rule
$\boldsymbol{\lambda}^{t}=\frac{\mathbf{h}^{t}({\mathbf{x}}^{t})}{\eta\sigma}$;
* •
(c)
$\sum\limits_{t=1}^{T}(\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-1}\right\rVert^{2}+\left\lVert\boldsymbol{\lambda}^{t-2}\right\rVert^{2})=3\sum\limits_{t=1}^{T}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}-2\left\lVert\boldsymbol{\lambda}^{T}\right\rVert^{2}-\left\lVert\boldsymbol{\lambda}^{T-1}\right\rVert^{2}$,
assuming $\left\lVert\boldsymbol{\lambda}^{0}\right\rVert=0$ and
$\left\lVert\boldsymbol{\lambda}^{-1}\right\rVert=0$.
Putting everything together we have:
$\sum\limits_{t=1}^{T}\left(\mathcal{L}^{t}(\mathbf{x}^{t},\boldsymbol{\lambda}^{t})-\mathcal{L}^{t}(\mathbf{x},\boldsymbol{\lambda}^{t})\right)\leq\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T+3\eta
KG^{2}\sum\limits_{t=1}^{T}\left\lVert\boldsymbol{\lambda}^{t}\right\rVert^{2}\\\
$ (46)
### A-A Static Regret
If we set $x=x_{*}$ for which $\mathbf{h}^{t}(\mathbf{x}_{*})=0$ and use
$\boldsymbol{\lambda}^{t}=\frac{\mathbf{h}^{t}({\mathbf{x}}^{t})}{\eta\sigma}$,
we get
$\displaystyle\sum\limits_{t=1}^{T}\Big{(}f^{t}(\mathbf{x}^{t})+\frac{\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert^{2}}{\eta\sigma}-f^{t}(\mathbf{x}_{*})\Big{)}\leq\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T+3KG^{2}\sum\limits_{t=1}^{T}\frac{\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert^{2}}{\eta\sigma^{2}},$
(47)
that yields
$\displaystyle\sum\limits_{t=1}^{T}\Big{(}f^{t}(\mathbf{x}^{t})-f^{t}(\mathbf{x}_{*})\Big{)}+\frac{1}{\eta\sigma}\sum\limits_{t=1}^{T}\left\lVert\mathbf{h}^{t}({\mathbf{x}^{t}})\right\rVert^{2}(1-\frac{3KG^{2}}{\sigma})\leq\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T.$ (48)
For $\sigma>3KG^{2}$, the upper bound $U_{sr}$ of the static regret is found
by
$\sum\limits_{t=1}^{T}\Big{(}f^{t}(\mathbf{x}^{t})-f^{t}(\mathbf{x}_{*})\Big{)}\leq\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T\triangleq U_{sr}.$ (49)
### A-B Dynamic Regret
Choosing again $\sigma>3KG^{2}$ and plugging the instantaneous optimal
solution $x=x^{t}_{*}$ in (A), we follow the procedure we used for the static
regret and obtain:
$\sum\limits_{t=1}^{T}\Big{(}f^{t}(\mathbf{x}^{t})-f^{t}(\mathbf{x}^{t}_{*})\Big{)}\leq\frac{1}{2\eta}\sum\limits_{t=1}^{T}(\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t+1}\right\rVert^{2})+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T.$ (50)
The only term that needs different handling is
$\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t+1}\right\rVert^{2}$,
for which we can write
$\displaystyle\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t+1}\right\rVert^{2}=\underbrace{\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t-1}-\mathbf{x}^{t}\right\rVert^{2}}_{\text{(a)}}+\underbrace{\left\lVert\mathbf{x}_{*}^{t-1}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t+1}\right\rVert^{2}}_{\text{(b)}}.$
(51)
We are interested in the telescopic sums of (a, b). In particular, for (a) we
use
$\mathbf{x}^{2}-\mathbf{y}^{2}=(\mathbf{x}-\mathbf{y})^{\top}(\mathbf{x}+\mathbf{y})$
and get
$\displaystyle\sum\limits_{t=1}^{T}\left(\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t-1}-\mathbf{x}^{t}\right\rVert^{2}\right)$
$\displaystyle=\sum\limits_{t=1}^{T}(\mathbf{x}_{*}^{t}-\mathbf{x}_{*}^{t-1})^{\top}(\mathbf{x}_{*}^{t}+\mathbf{x}_{*}^{t-1}-2\mathbf{x}^{t})\leq\sum\limits_{t=1}^{T}\left\lVert\mathbf{x}_{*}^{t}+\mathbf{x}_{*}^{t}-2\mathbf{x}^{t}\right\rVert\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}_{*}^{t-1}\right\rVert$
$\displaystyle\leq
2R\sum\limits_{t=1}^{T}\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}_{*}^{t-1}\right\rVert=2RV(\mathbf{x}_{*}^{1:T})$
(52)
where $V(\mathbf{x}_{*}^{1:T})$ is the sum of distances of consecutive optimal
solutions. Moreover, for term (b) we have
$\sum\limits_{t=1}^{T}(\left\lVert\mathbf{x}_{*}^{t-1}-\mathbf{x}^{t}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{t}-\mathbf{x}^{t+1}\right\rVert^{2})=\left\lVert\mathbf{x}_{*}^{0}-\mathbf{x}^{1}\right\rVert^{2}-\left\lVert\mathbf{x}_{*}^{T}-\mathbf{x}^{T+1}\right\rVert^{2}\leq
R^{2}.$ (53)
Finally, the upper bound $U_{dr}$ of the dynamic regret is
$\sum\limits_{t=1}^{T}\Big{(}f^{t}(\mathbf{x}^{t})-f^{t}(\mathbf{x}^{t}_{*})\Big{)}\leq\frac{R}{\eta}V(\mathbf{x}_{*}^{1:T})+\frac{R^{2}}{2\eta}+\frac{2REG^{2}}{\eta\sigma}+\frac{7}{2}\eta
NF^{2}T\triangleq U_{dr}.$ (54)
By combining (49),(54), the relation to the bound of the static regret is
$U_{dr}=U_{sr}+\frac{R}{\eta}V(\mathbf{x}_{*}^{1:T})$.
### A-C Fit
In order to bound the fit, we choose $\sigma>3KG^{2}$ in (48) and use
$\sum\limits_{t=1}^{T}\left(f^{t}(\mathbf{x}_{*})-f^{t}(\mathbf{x}^{t})\right)\leq
2NFT$, which gives
$\sum\limits_{t=1}^{T}\left\lVert\mathbf{h}^{t}({\mathbf{x}^{t}})\right\rVert^{2}\leq\frac{1}{\beta}(\frac{\sigma
R^{2}}{2}+2REG^{2}+\frac{7}{2}\eta^{2}\sigma NF^{2}T+2\eta\sigma NFT),$ (55)
where $\beta=1-\frac{3KG^{2}}{\sigma}$. By further using the property
$\Big{(}\sum\limits_{i=1}^{n}a_{i}\Big{)}^{2}\leq
n\sum\limits_{i=1}^{n}a_{i}^{2}$, we obtain
$\displaystyle\Big{(}\sum\limits_{t=1}^{T}\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert\Big{)}^{2}\leq
T\sum\limits_{t=1}^{T}\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert^{2}$
$\displaystyle\leq\frac{T}{\beta}(\frac{\sigma
R^{2}}{2}+2REG^{2}+\frac{7}{2}\eta^{2}\sigma NF^{2}T+2\eta\sigma NFT).$ (56)
Taking the square root on both sides of (56)
$\displaystyle\sum\limits_{t=1}^{T}\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert\leq\Big{(}\frac{T}{\beta}(\frac{\sigma
R^{2}}{2}+2REG^{2}+\frac{7}{2}\eta^{2}\sigma NF^{2}T+2\eta\sigma
NFT)\Big{)}^{1/2}$ (57)
Finally,
$\sum_{m=1}^{M}h_{m}^{t}(\mathbf{x}^{t})\leq\sqrt{M\sum_{m=1}^{M}(h_{m}^{t}(\mathbf{x}^{t}))^{2}}=\sqrt{M}\left\lVert\mathbf{h}^{t}(\mathbf{x}^{t})\right\rVert$
which gives us the upper bound $U_{f}$ on the fit
$\displaystyle\sum\limits_{t=1}^{T}\sum\limits_{m=1}^{M}h_{m}^{t}(\mathbf{x}^{t})\leq\Big{(}\frac{MT}{\beta}(\frac{\sigma
R^{2}}{2}+2REG^{2}+\frac{7}{2}\eta^{2}\sigma NF^{2}T+2\eta\sigma
NFT)\Big{)}^{1/2}.$ (58)
By combining (49),(58), the relation to the bound of the static regret is
$U_{f}=\sqrt{\frac{\eta\sigma}{\beta}MT(U_{sr}+2NFT)}$.
## Appendix B Proof for implications of Assumption 1
Here we show formally two of the implications of Assumption 1, namely (A) that
$f(\mathbf{x})$ is bounded and (B) that clipped function $h(\mathbf{x})$ is
convex. Below, for simplicity we drop the node $n$ and timestep $t$ sub and
superscripts.
### B-A Function $f(\mathbf{x})$ is bounded
We want to show that for a finite $f$ for which Assumption 1 holds, there
exists $F$ such that $|f|\leq F$ for all $\mathbf{x}\in\Omega$. From
$\left\lVert\nabla_{\boldsymbol{x}}f\right\rVert\leq F^{\prime}$, it is
implied that for any $\boldsymbol{x},\boldsymbol{y}\in\Omega$, we have
$|f(\boldsymbol{x})-f(\boldsymbol{y})|\leq
F^{\prime}\underbrace{\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert}_{\leq
R}\leq F^{\prime}R.$
Furthermore, using that
$|f(\boldsymbol{x})|-|f(\boldsymbol{y})|\leq|f(\boldsymbol{x})-f(\boldsymbol{y})|\leq
F^{\prime}R$, we get
$|f(\boldsymbol{x})|\leq F^{\prime}R+|f(\boldsymbol{y})|=F,$
where we do not know $|f(\boldsymbol{y})|$, but we know that by definition it
is finite. Since Assumption 1 holds for $g$, the exact same steps can be
followed in order to show that $g$ is bounded.
### B-B Convexity of $h(\mathbf{x})$
Recall the definitions of $h(\mathbf{x})$ and its gradient
$\displaystyle h(\mathbf{x})=[g(\mathbf{x})]^{+}=\begin{cases}0&\text{if
$g(\mathbf{x})\leq 0$}\\\ g(\mathbf{x})&\text{if
$g(\mathbf{x})>0$}\end{cases}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \nabla
h(\mathbf{x})=\nabla[g(\mathbf{x})]^{+}=\begin{cases}\mathbf{0}&\text{if
$g(\mathbf{x})\leq 0$}\\\ \nabla g(\mathbf{x})&\text{if
$g(\mathbf{x})>0$}\end{cases}$
We want to show that if $g(\mathbf{x})$ is convex, that is,
$g(\mathbf{x})-g(\mathbf{y})\leq(\mathbf{x}-\mathbf{y})^{\top}\nabla
g(\mathbf{x})$, then for any $\mathbf{x},\mathbf{y}\in\Omega$, we have
$h(\mathbf{x})-h(\mathbf{y})\leq(\mathbf{x}-\mathbf{y})^{\top}\nabla
h(\mathbf{x}).$
In what follows, we only use the above definitions of $h$ and its gradient,
and the fact that $g$ is convex. The four possible cases are the following.
(a): If $g(\mathbf{x})>0$ and $g(\mathbf{y})>0$, then
$h(\mathbf{x})-h(\mathbf{y})=g(\mathbf{x})-g(\mathbf{y})\leq(\mathbf{x}-\mathbf{y})^{\top}\nabla
g(\mathbf{x})=(\mathbf{x}-\mathbf{y})^{\top}\nabla h(\mathbf{x}).$
(b): If $g(\mathbf{x})\leq 0$ and $g(\mathbf{y})\leq 0$, then
$h(\mathbf{x})-h(\mathbf{y})=0-0\leq 0=(\mathbf{x}-\mathbf{y})^{\top}\nabla
h(\mathbf{x})$.
(c): If $g(\mathbf{x})>0$ and $g(\mathbf{y})\leq 0$, then
$h(\mathbf{x})-h(\mathbf{y})=h(\mathbf{x})=g(\mathbf{x})\leq
g(\mathbf{x})-g(\mathbf{y})\leq(\mathbf{x}-\mathbf{y})^{\top}\nabla
g(\mathbf{x})=(\mathbf{x}-\mathbf{y})^{\top}\nabla h(\mathbf{x}).$
(d): If $g(\mathbf{x})\leq 0$ and $g(\mathbf{y})>0$, then
$h(\mathbf{x})-h(\mathbf{y})=-h(\mathbf{y})=-g(\mathbf{y})\leq
0=(\mathbf{x}-\mathbf{y})^{\top}\nabla h(\mathbf{x}).$
|
# From Paper to Platform: Evolution of a Novel Learning Environment for
Tabletop Exercises
Valdemar Švábenský 0000-0001-8546-280X Masaryk UniversityFaculty of
InformaticsBrnoCzech Republic<EMAIL_ADDRESS>, Jan Vykopal
0000-0002-3425-0951 Masaryk UniversityFaculty of InformaticsBrnoCzech Republic
<EMAIL_ADDRESS>, Martin Horák 0000-0002-1835-6465 Masaryk
UniversityFaculty of InformaticsBrnoCzech Republic<EMAIL_ADDRESS>,
Martin Hofbauer 0009-0005-3998-9164 Masaryk UniversityFaculty of
InformaticsBrnoCzech Republic<EMAIL_ADDRESS>and Pavel Čeleda
0000-0002-3338-2856 Masaryk UniversityFaculty of InformaticsBrnoCzech Republic
<EMAIL_ADDRESS>
(2024)
###### Abstract.
For undergraduate students of computing, learning to solve complex practical
problems in a team is an essential skill for their future careers. This skill
is needed in various fields, such as in cybersecurity and IT governance.
Tabletop exercises are an innovative teaching method used in practice for
training teams in incident response and evaluation of contingency plans.
However, tabletop exercises are not yet widely established in university
education. This paper presents data and teaching experience from a
cybersecurity course that introduces tabletop exercises in classrooms using a
novel technology: INJECT Exercise Platform (IXP), a web-based learning
environment for delivering and evaluating the exercises. This technology
substantially improves the prior practice, since tabletop exercises worldwide
have usually been conducted using pen and paper. Unlike in traditional
tabletop exercises, which are difficult to evaluate manually, IXP provides
insights into students’ behavior and learning based on automated analysis of
interaction data. We demonstrate IXP’s capabilities and evolution by comparing
exercise sessions hosted throughout three years at different stages of the
platform’s readiness. The analysis of student data is supplemented by the
discussion of the lessons learned from employing IXP in computing education
contexts. The data analytics enabled a detailed comparison of the teams’
performance and behavior. Instructors who consider innovating their classes
with tabletop exercises may use IXP and benefit from the insights in this
paper.
tabletop exercise, incident response, team collaboration, cybersecurity,
hands-on training, learning analytics, INJECT
††journalyear: 2024††copyright: rightsretained††conference: Proceedings of the
2024 Innovation and Technology in Computer Science Education V. 1; July 8–10,
2024; Milan, Italy††booktitle: Proceedings of the 2024 Innovation and
Technology in Computer Science Education V. 1 (ITiCSE 2024), July 8–10, 2024,
Milan, Italy††doi: 10.1145/3649217.3653639††isbn:
979-8-4007-0600-4/24/07††ccs: Social and professional topics Computing
education
## 1\. Introduction
Collaborative problem-solving of complex issues (CPSCI), such as the
resolution of incidents in large organizations that critically rely on
information technology (IT), is a core competency for the twenty-first-century
workforce (Fiore et al., 2018; CC2020 Task Force, 2020). However, many
university graduates lack necessary skills in these areas (Fiore et al.,
2018).
University students of applied computing (a target student demographic of this
paper) learn CPSCI in cybersecurity and IT governance courses, among others.
These courses cover topics like cyber incident response, emergency readiness,
information sharing, or contingency plan validation when managing an IT
infrastructure.
Computing educators found it difficult to provide students with practical
learning experience in such courses (Ottis, 2014). Thus, researchers and
instructors have been exploring innovative ways to teach these
interdisciplinary topics, which connect technological and human aspects of
computing, in an immersive and meaningful way (Ottis, 2014).
### 1.1. What Are Tabletop Exercises?
A tabletop exercise (TTX) is a type of a teaching activity designed to train
professional teams in incident response to a crisis situation (Grance et al.,
2006). The simulated crisis happens in the context of business operations in
an organization, e.g., a phishing attack on employees or malware infecting the
company infrastructure. The team members (exercise participants) hold various
roles in the organization, e.g., manager or cybersecurity incident responder
(Angafor et al., 2023). During the exercise, the team discusses which actions
to take to effectively respond to the emergency while following proper
protocols and regulations. The team discussions are facilitated by
instructors, who also present an exercise debriefing at the end.
TTXs are an effective educational tool, which enhance incident preparedness of
individuals, particularly their communication, coordination, and collaboration
(Angafor et al., 2023). In computing education, TTXs provide students with
realistic incident response experience and deepen their understanding of
related processes. TTXs are especially relevant for cybersecurity and
information security management courses and align with broader IT governance
courses (European Union Agency for Cybersecurity (2010), ENISA).
### 1.2. Problem Statement and Innovation
TTXs differ from technical hands-on exercises in an emulated computer
infrastructure, such as in a cyber range (Yamin et al., 2020) or a locally
virtualized learning environment (Vykopal et al., 2021). Instead, TTXs are
much more lightweight and do not dive into technical matters deeply. They are
traditionally conducted using pen and paper or simple online office
applications, such as Google Forms, to collect participant responses. The
advantage of this approach is its low cost and low barrier to entry. On the
other hand, the assessment of the participating teams has to be done manually
by the instructors, which is highly time-consuming. It takes days or even
weeks until the trainees can receive educational feedback, which diminishes
its effectiveness and decreases learning gains from the TTX.
We aim to transition the TTX format from this low-tech approach into INJECT
Exercise Platform (IXP): a novel, lightweight, open-source environment for
supporting the deployment and evaluation of TTXs. This represents a major
innovation that automates repetitive tasks for instructors, leaving them more
room for teaching. Since IXP automatically collects exercise data, it can
deliver pedagogical insights and feedback using the methods of learning
analytics. This paper shares our experience in deploying IXP in computing
classes and analyzes student data from these classes.
### 1.3. Goals and Scope of This Paper
We developed a novel TTX, which we deployed on three occasions (“runs”) with
three groups of learners. In the first run, we used only online Microsoft
Office (Microsoft, 2023) applications. In the second run, we used a simple
prototype of the TTX platform. Finally, the third run demonstrated a more
developed version of IXP. Our goal is to compare the student data and teacher
perspective on facilitating the TTX in these three different versions of the
learning environment. Specifically, this paper explores the following research
questions:
1. (1)
What types of insights about the student behavior and learning can the
platform deliver to instructors?
2. (2)
What is the instructors’ teaching experience when comparing the three exercise
runs?
## 2\. Related Work
As this paper focuses on transitioning TTXs from pen-and-paper format to a
software platform, we reviewed literature covering all three stages of the
transition: in-person pen-and-paper format, online pen-and-paper format, and
online platform for TTX delivery. We also review analytics of data from TTXs.
Based on our review of related work, we summarize the unique contributions of
our paper.
### 2.1. Pen-and-Paper Tabletop Exercises
Ottis (2014) described how to create lightweight TTXs for cybersecurity
education. The TTX detailed in the paper is in-person with two groups of
participants: “red” and “blue”. Red teams are in charge of creating the
exercise scenario from the attackers’ perspective. (A TTX scenario is an
outline of the sequence of events that drive the exercise and guide
participant discussion (U.S. Cybersecurity & Infrastructure Security Agency,
CISA).) Blue teams are responsible for handling the attack events. The paper
presents observations from eight such TTXs with 250 students in total.
Angafor et al. (2023) used Microsoft Teams to conduct an online pen-and-paper-
like TTX for 20 participants from an unnamed company. After the exercise, the
participants answered a survey about their awareness of attack mitigation
controls, as well as feedback on the completeness of controls currently used
in the company. The authors used descriptive statistics to analyze the survey.
However, the publication does not include any learning analytics.
Brilingaitė et al. (2017) conducted a TTX with students of IT and social
sciences. A custom software that hosted the exercise also logged data about
user actions. These actions include the number of messages and the number of
exercise events in different states of progress, which do not offer extensive
possibilities for analysis. The option of further analysis of exercise logs is
mentioned, but neither these logs nor the analysis are available.
### 2.2. Software for Tabletop Exercises
TTXs in the cybersecurity context are quite prominent (European Union Agency
for Cybersecurity (2012), ENISA; European Union Agency for Cybersecurity
(2015), ENISA), which leads to research and development of software solutions.
While there are companies, such as Privasec (Global, 2023) or Red Goat (Goat,
2023), that provide paid software for TTXs, open-source solutions exist as
well.
We discovered 16 open-source projects for TTXs (Vykopal et al., 2024) on
GitHub, out of which 11 contained software solutions. Most of them are simple
and specifically tailored for delivering just one specific exercise scenario.
An example is an application (Lewis, 2022) that presented the scenario in a
few sentences via a command-line interface and asked the participants to
discuss the possible solutions.
While some software solutions are more advanced than just presentations of
scenarios, they focus on the cybersecurity aspects of the exercise, as opposed
to more discussion-based problems. These solutions are also more technical. An
example is Ransomware Simulator (McKeown, 2022), which works as a reporting
tool for incidents, asking for the event ID, owner, summary, and response to
the event. The instructors can add the option of a simulated ransomware attack
launched at a specified time, locking participants out of the tool.
The open-source software solution that we consider to have the most features
is OpenEx (Filigran, 2023). This solution is not specifically tailored for
cybersecurity TTXs, and it allows to create and execute different scenarios.
Unlike other software we found, OpenEx records logs of participant
interactions within the scenario. This enables analyzing the data gathered
during the exercise; however, OpenEx does not implement such analysis. Another
downside of OpenEx is using real email infrastructure for participant
communication, which can lead to delays due to antispam or system outages.
### 2.3. Data Analytics in Tabletop Exercises
As of October 2023, searching “tabletop exercise” in the multi-faceted
citation database Scopus (Elsevier, 2023) returned 418 papers. While this
amount is non-negligible, the number of publications with learning analytics
of data from TTXs is low.
Mareš et al. (2023) conducted a TTX for 33 experts in cybersecurity and
related fields, such as law. The authors analyzed data from two surveys about
the participants’ behavior, performance, and workload handling. The first
survey was carried out immediately after the exercise, and the following
survey two weeks later. Higher performance of participants was significantly
correlated with their lower levels of perceived stress ($r=0.30$ to $0.37$,
$p=0.039$). However, this study does not include data about the participants’
learning.
Hsieh et al. (2023) compared fire safety knowledge acquisition between drill-
based and game-based learning. Although this study did not use the TTX format
as defined above, the game-based learning was carried out as a tabletop game.
The authors used t-tests to measure the knowledge gain of both groups. The
knowledge gain of the game-based group ($t=12.58$) was substantially larger
than for the other group ($t=6.14$), with $p<0.001$ for both statistics.
We did not find any other publication containing learning analytics of TTX
data. The only data currently being collected from TTXs are not focused on the
educational process, but on feedback on the exercise itself and its perceived
usefulness for the participants (Ottis, 2014; Kopustinskas et al., 2020; Ota
et al., 2022). These data are valuable for the exercise creators but do not
provide deeper insight into the TTX participant behavior.
### 2.4. Novel Contributions of This Paper
TTXs are suited for computing education, and some software solutions for
conducting TTXs exist. However, the existing research does not focus on TTX
participant learning behavior. The software solutions do not implement
analytics, other than descriptive. So, our work contributes to educators and
researchers with these inputs:
* •
We propose an innovated TTX format (Section 3.1).
* •
Since TTXs rarely use dedicated software for evaluation and in-depth analysis,
we develop a new learning environment that provides these functionalities
(Section 3.2).
* •
Unlike traditional TTXs, in which the instructors analyze the exercise data
manually, we demonstrate the platform’s capabilities in automated data
collection and analysis. The data come from three runs of a novel TTX deployed
in an authentic teaching context (Section 4).
* •
For instructors and practitioners, we share the practical lessons learned from
using the platform (Section 5).
* •
Artifacts associated with this work are available (Section 6).
## 3\. Tabletop Exercise Delivery
We now define the key features of TTXs. The purpose of this section is
twofold: (1) to provide the background for our research study, which is
detailed in Section 4, and (2) to represent a contribution on its own by
defining the innovated TTX format and its properties.
### 3.1. Proposed Exercise Format
#### 3.1.1. Participant Roles
Human participants in a TTX can have one of three roles. Designers prepare the
exercise and its scenario. Instructors facilitate the exercise by guiding the
participants and evaluate the exercise at its end. They may or may not be
different from designers. Trainees attend the exercise to improve their
skills. Trainees are grouped into teams that are independent of each other
(i.e., each team completes the same tasks in parallel). Each person may have a
different role in the team.
#### 3.1.2. Components of the Exercise
An inject is a pre-scripted message, such as an email, provided to trainees
during the TTX. Its purpose is to move the scenario forward and prompt
additional actions. For example, it can inform the trainees about a data
breach in their company, requiring them to respond accordingly (Grance et al.,
2006).
A tool is a simplified simulated version of a real-world computer
application/service. Its purpose is to allow trainees to perform actions to
respond to injects. For example, trainees in a TTX do not use an actual email
client, but an “email” tool to send and receive in-exercise messages. Another
example is a “browser”, a tool that returns an in-exercise website based on
the provided exercise URL.
A milestone is a true/false condition that denotes whether a specific team
reached a certain important situation in the TTX scenario. Its purpose is to
track each team’s progress through the TTX. For example, it can mark that a
team used an email tool to respond to a query from a manager. The exercise
milestones can be completed in any order, and there is rarely a single correct
solution. These properties make the team assessment challenging.
#### 3.1.3. Exercise Workflow
The exercise is driven by injects, which are either provided by the
instructor, or triggered automatically based on time (e.g., an inject is sent
after 20 minutes of the exercise) or based on milestone (in)completion. Some
injects may be provided to all teams at once; others are conditioned by the
team’s progress. This allows each team to progress through the scenario
independently of other teams. As a result, each team can progress at their own
pace, which removes the need for them to wait idle.
When a new inject arrives, trainees in a team discuss the situation to agree
on which action to take (e.g., which tool to use). Effective communication
under time pressure is a crucial component of TTXs. Trainees do not choose an
inject response from pre-defined options, but have to think of a unique open-
ended solution. If a team gets stuck, the instructor should help them by
asking guiding questions or providing a gentle hint via responding to the
team’s emails.
### 3.2. Exercise Platform
Figure 1. Trainees’ view of the INJECT Exercise Platform. In the left sidebar
(A), trainees see injects or email conversations. The middle pane (B) shows
injects, emails, and outcomes of tools, depending on the view chosen in the
left bar. In the right sidebar (C), trainees see all available tools. After
clicking on the tool, a dropdown menu (D) for the tool’s arguments appears.
Screenshot of the frontend of the Tabletop Exercise Platform.
We developed a novel learning environment called INJECT Exercise Platform
(IXP), which is an interactive web application for supporting the delivery and
evaluation of TTXs.
Designers can use the platform to instantiate an exercise definition, which
prescribes the exercise story, injects, available tools, and milestones. An
exercise definition is implemented as a set of structured text-based files (in
YAML format) that are both human- and machine-readable. It automates a
substantial portion of the TTX, since it provides trainees with tools and
inject response templates.
Instructors deploy an exercise definition in IXP when they want to host an
exercise. The definition is created only once, but thanks to the platform’s
automation capabilities, it can be deployed repeatedly under the same
conditions for different trainees. Compared to manually-hosted exercises, IXP
significantly reduces the workload and personnel requirements for TTX
delivery.
Trainees interact with the scenario through automated tools during the
exercise. This interaction moves the scenario forward and impacts the
simulated environment. Figure 1 shows the trainees’ view of the IXP to
demonstrate some of the interaction options.
### 3.3. Exercise Content Example
To illustrate the exercise format and the components of the platform, we now
describe an exercise that we developed for the IXP. This TTX was also selected
for our research study described in Section 4.
#### 3.3.1. Learning Objectives
The TTX was based on a real cyber attack that happened at our university, in
order to provide the trainees with an authentic learning experience. The
learning objectives of the TTX are: (a) to perform cyber incident triage, (b)
to coordinate and execute incident response, and (c) to mitigate the impacts
of an incident on a large organization with multiple involved parties.
#### 3.3.2. Description of the Story
In the TTX scenario, the trainees assume the role of the members of a Computer
Security Incident Response Team (CSIRT) of the university. At the beginning,
the trainees receive an initial inject: a report of a phishing email targeting
the university employees. Several employees have fallen victims to the attack
and submitted their login credentials to a fraudulent phishing website. As a
result, a malicious actor accessed sensitive information on the employees’
internal project server and also deleted important files, making the project
website unavailable. As the time progresses, the trainees receive more and
more injects in the form of emails from the affected employees, asking the
team to take swift action in response to the ongoing emergency.
#### 3.3.3. Available Tools
The trainees can take numerous actions at each point of the TTX. They can
apply technical solutions (e.g., inspect network data or block traffic to/from
certain IP addresses) as well as take managerial/governance steps (e.g.,
notify the responsible persons according to the data protection law). Each
action (e.g., using a specific tool) or inaction (e.g., not responding to a
query quickly enough) may trigger another inject to propel the scenario
forward and place the trainees in the midst of another time-critical issue. To
support discussion in a team, only one person from the team can interact with
the tools in IXP at any given time, after the members consult and mutually
agree on their progress.
## 4\. Research Methods
We delivered the exercise described in Section 3 on three occasions,
throughout three years with the total of 91 university students of computing.
This section describes the design of our research study. The research
questions were posed in Section 1.3.
### 4.1. Course Context
The TTX is the culmination of the course titled Cybersecurity in an
Organization, which is taught at the Faculty of Informatics, Masaryk
University: a large, public university in Central Europe.
#### 4.1.1. Learning Outcomes
The course graduates should understand the role and services of a CSIRT in an
organization. Specifically, the course covers knowledge and skills required
for the work role of Cyber Defense Incident Responder as defined by the NICE
Cybersecurity Workforce Framework (for Cybersecurity Careers and, NICCS).
#### 4.1.2. Teaching Format
The course is offered once per academic year and spans a standard 13-week Fall
semester. It is taught in-person using a combination of flipped classroom
sessions, discussion, and homework assignments. The TTX at the course’s end
provides a hands-on learning experience with the knowledge and skills studied
throughout the semester. All teaching materials are written in English, but
the language of instruction is local (Czech).
#### 4.1.3. Student Population
All students enrolled in the course were students of Faculty of Informatics,
and the vast majority pursued their degree in cybersecurity. The class size
was up to 42 students. Most students were undergraduates. In the latest
semester, we had 34 bachelor-level students (31 of which in the cybersecurity
degree program) and 7 master-level students.
### 4.2. Field Studies Setup and Participants
This paper analyzes data and experience from three groups of trainees who
completed the TTX. Table 1 summarizes the three training sessions as the IXP
readiness increased over three years.
Table 1. Information about the three TTX runs. Run | Date | Students (team division) | Platform
---|---|---|---
#1 | Nov 25, 2021 | 19 (5 teams of 3–4 people) | Documents
#2 | Nov 23, 2022 | 36 (9 teams of 4 people) | Prototype
#3 | Nov 22, 2023 | 36 (12 teams of 3 people) | IXP
To ensure a fair comparison, we attempted to keep the three TTX runs as
consistent and similar as possible. All three runs took place within the same
course at the same stage of the semester, on the same exercise, and with the
same core instructors (though with slightly different teaching assistants).
The modality of all runs was fully in-person. The duration of the exercise was
80–90 minutes.
Before each exercise run, the TTX was thoroughly tested during a dry run with
our colleagues and senior graduate students (not students of the course). The
purpose of the test run was to verify that the platform is ready for practical
usage and that the exercise can be meaningfully completed, and to fix any
errors or issues that could negatively impact the learning experience of
trainees.
The only substantial difference between the three runs is the subject of
examination in this paper – the TTX platform readiness.
* •
Run 1 was an imitation of a pen-and-paper exercise using shared text-based
documents on Microsoft SharePoint.
* •
Run 2 was deployed in the first prototype of the dedicated IXP developed as a
master’s thesis (Urban, 2023).
* •
Run 3 featured the latest version of IXP, which was substantially improved by
a dedicated development team, mainly including bug fixes and a better user
experience.
### 4.3. Research Ethics and Data Privacy
This research did not require an approval from the university’s institutional
review board. All trainees receive their course points simply for active
participation. IXP does not store any personal information that could reveal
the trainees’ identity. The data exported for analysis are anonymous and
cannot be linked to specific individuals. The trainees were informed that
their anonymized exercise activity data may be used for educational research
purposes. Lastly, post-exercise surveys were voluntary and anonymous.
### 4.4. Exercise Data Collection
Since the Run 2, IXP provides transparent, automated collection of exercise
metadata and actions of trainees. The log records are stored in the standard
JSONL format (Ward, 2023), and each record has a uniform timestamp with
microsecond precision. Per each team, IXP gathers and categorizes the records
into four log files:
* •
Which injects did the team receive (inject_categories.jsonl).
* •
In-exercise email communication (emails.jsonl).
* •
Actions performed using the in-exercise tools (action_logs.jsonl).
* •
Reached milestones (milestones.jsonl).
The format of the logs was improved for the latest version of IXP deployed for
Run 3. To enable uniform data analysis, we automatically converted the logs
from Run 2 to this new format.
### 4.5. Exercise Data Analysis
To analyze the exercise data, we used a combination of a learning analytics
dashboard built into IXP and dedicated Python scripts. The scripts process the
logs after the TTX ends in order to provide additional analytics for assessing
the team performance more deeply, such as correct/incorrect tool usage. The
scripts also enable to evaluate the TTX as a whole, by looking at metrics like
time needed to reach individual milestones.
## 5\. Results and Their Discussion
We present and compare the results of data analyses from Run 2 and 3. Since
Run 1 was executed using text-based documents, it did not yield logs in the
above-described format. All observations are tied to implications for
computing educators.
### 5.1. Analysis of the TTX Data of Trainees
#### 5.1.1. Run 2
The TTX had 14 defined milestones, which captured actions such as blocking
traffic from certain sources, communicating with the affected users, and
notifying responsible parties. The teams reached between 5 and 12 milestones,
with an average of 10 (71%). Only 2 out of the 9 teams scored below average,
indicating they may have benefited from an intervention by an instructor.
The TTX provided the teams with 7 possible tools. The team that reached the
fewest milestones also used the least amount of tools (6 times in total
compared to the overall team average of 20 occurrences of tool usage).
However, the team that reached the second smallest number of milestones had
the second largest number of tool usages. While other explanations are
possible, this may indicate that rare tool usage is associated with low
exercise completion, but frequent tool usage does not necessarily imply
success (the tools might not be used efficiently).
#### 5.1.2. Run 3
In order to improve the granularity of capturing teams’ actions, we added 8
additional milestones to the TTX. When looking just at the 14 original
milestones, the teams reached between 4 and 11 of them, with an average of 8
(57%). Compared to Run 2, this lower ratio indicates that if an instructor
provided a post-exercise debrief, it might be a valuable learning experience
for all trainees. This debrief would inform the trainees about the additional
actions that could have been made but were missed.
Overall, the first reached milestone was to visit the compromised website. The
first team that reached this milestone did so in around 8 minutes from the TTX
start. This milestone was also the fastest to reach across all teams, in 15.5
minutes on average. However, the slowest team to achieve this milestone took
30 minutes. Instead, this team focused on four other milestones before,
prioritizing different aspects of the TTX compared to the vast majority of
teams. This can be an interesting observation for the instructor, showing
possibly alternative approaches to solving the in-exercise problems.
Regardless, this team took rather long to reach their first milestone, which
suggests they may have benefited from a hint or intervention.
The milestones that took the longest to complete encompassed the communication
with the simulated employees. Four teams took a little more than an hour to
address their stakeholders, and this step was completely overlooked by five
teams. Although this skill is non-technical, it is still an important part of
the cyber incident responders’ work role. Therefore, instructors can use these
insights from the TTX data to remind the learners about this responsibility or
revise the course content in this aspect.
The TTX provided the teams with 11 possible tools (additional 4 compared to
Run 2). A team used a tool between 10 and 46 times throughout the entire TTX,
with 31 uses on average (including repeated uses of the same tool). The
approaches of individual teams differed vastly: different teams used certain
tools more often and (almost) ignored other tools. For example, all teams used
the tool to block traffic incoming from a certain IP address, but only two-
thirds of the teams blocked traffic outgoing to the compromised website.
Finally, the improvement of IXP for Run 3 enabled to evaluate the syntactical
correctness of tool usage. When looking at these data, all tools have much
more correct rather than incorrect invocations, showing that all trainees
understood the tools’ interfaces. However, the DNS lookup tool has
substantially higher percentage of erroneous applications compared to other
tools. Within the errors that were not simply typos, there might have been
confusion among the trainees that could be addressed by the instructor.
Looking at the teams’ written communication, they engaged in 6 email threads
on average. The team that communicated the most (9 threads) reached the most
milestones, and vice versa, the team that communicated the least (3 threads)
reached almost the least number of milestones. This provides a teaching
opportunity if the instructor compares the differences between the teams,
showing that active communication is crucial while resolving a crisis.
#### 5.1.3. Summary
The automatic collection and analysis of data provided by the IXP equips
instructors and researchers with valuable insights that would be difficult to
obtain otherwise, especially in the traditional pen-and-paper TTX format. By
adding more granularity to the milestones and enhancing the platform’s logging
capabilities, we were able to observe deeper insights in Run 3 compared to Run
2. These include difficult milestones and errors in tool usage.
### 5.2. Trainees’ Learning Experience in IXP
To complement the analysis of exercise logs, we present the results of a post-
exercise survey administered to all trainees after Run 3.
In the overall evaluation, 35 out of 36 learners considered the TTX scenario
realistic. A majority, 29 out of 36, found the TTX beneficial for practical
applications because it improved their understanding of incident handling. One
participant stated, “At the beginning, we were quite lost. There is just so
much difference between having a specified incident handling task and having
to figure out everything by yourself.” Additionally, 31 participants expressed
satisfaction with the ease of use of IXP to facilitate the exercise.
Our survey unveiled three pivotal insights for refining future exercises.
Primarily, we encountered challenges in effectively communicating which in-
exercise email addresses are trustworthy. Consequently, specific teams
refrained from accessing some exercise emails, deeming them potentially
malicious.
Secondly, trainees would like IXP’s email feature to resemble familiar
interfaces more. The current version can send and receive emails, but the
trainees expected more features, like auto-saving drafts or showing emails in
threads. The absence of such features led to communication delays, influencing
the learning experience.
Finally, some teams got stuck in various stages of the scenario. Instructors
were briefed to assist by sending exercise emails to guide these teams.
However, this approach proved challenging as instructors struggled to identify
the right moments for intervention. Given the continuous team discussions,
instructors found it hard to determine when it was appropriate to influence
the discussion.
### 5.3. Instructors’ Teaching Experience in IXP
During a focus group discussion hosted with the instructors after Run 3, the
instructors observed that enhancing IXP led to improving the following two key
aspects in the teaching practice:
* •
Reliability: When using shared documents in Run 1, students sometimes
accidentally rewrote their past conversation, and instructors got confused
when working with multiple teams. A dedicated learning environment eliminates
these errors.
* •
Involvement: Run 1 and 2 had fewer teams, almost exclusively with 4 people.
With the improvement of IXP for Run 3, we were able to have more teams, almost
exclusively with 3 people. This means that a single student got more
opportunities to speak in the team and to be actively involved in the
decision-making, improving their individual experience.
## 6\. Conclusion
Tabletop exercises are a promising method for innovating computing courses.
They enable students to exercise collaborative problem-solving in the context
of cybersecurity, IT governance, and other domains of applied informatics.
Introducing the INJECT Exercise Platform, a dedicated learning environment for
TTXs, alleviates many challenges that instructors face. For example, having a
platform to automate repetitive tasks, such as providing injects or outputs of
tools, enables instructors to focus on the exercise facilitation. The
automation capabilities of the learning environment also enable further
educational research.
We release the IXP as open-source software with an example exercise at
https://inject.muni.cz. The research data, Python scripts for data processing,
and complete results are also available at
https://gitlab.fi.muni.cz/inject/papers/2024-iticse-from-paper-to-platform.
### 6.1. Open Research Challenges
Currently, it is difficult to identify the right moments for intervention
during TTXs. A key challenge is how to determine when a team would benefit
from a hint, using insights from exercise data. For example, measuring
expected time to reach a milestone can imply how long the instructor should
wait before giving a team a hint. This would help teams to navigate through
scenario challenges.
Another limitation is that IXP does not yet support instructors in quickly
reacting to expected trainee responses. Therefore, future work can explore
machine learning and natural language processing techniques to evaluate the
similarity in the responses to injects between different teams. Then, the
platform can provide instructors with pre-defined responses that would suit
the trainees’ inputs.
Finally, future work should use the data to measure team performance (Amon et
al., 2019), evaluate students’ achievement of learning objectives, and
experimentally determine the effect of IXP on skill acquisition.
###### Acknowledgements.
This research was supported by the Open Calls for Security Research 2023–2029
(OPSEC) program granted by the Ministry of the Interior of the Czech Republic
under No. VK01030007 – Intelligent Tools for Planning, Conducting, and
Evaluating Tabletop Exercises.
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|
# Interference Exploitation in Full Duplex Communications: Trading
Interference Power for Both Uplink and Downlink Power Savings
Mahmoud T. Kabir, Muhammad R. A. Khandaker, and Christos Masouros
###### Abstract
This paper considers a multiuser full-duplex (FD) wireless communication
system, where a FD radio base station (BS) serves multiple single-antenna
half-duplex (HD) uplink and downlink users simultaneously. Unlike conventional
interference mitigation approaches, we propose to use the knowledge of the
data symbols and the channel state information (CSI) at the FD radio BS to
exploit the multi-user interference constructively rather than to suppress it.
We propose a multi-objective optimisation problem (MOOP) via the weighted
Tchebycheff method to study the trade-off between the two desirable system
design objectives namely the total downlink transmit power minimisation and
the total uplink transmit power minimisation problems at the same time
ensuring the required quality-of-service (QoS) for all users. In the proposed
MOOP, we adapt the QoS constraints for the downlink users to accommodate
constructive interference (CI) for both generic phase shift keying (PSK)
modulated signals as well as for quadrature amplitude modulated (QAM) signals.
We also extended our work to a robust design to study the system with
imperfect uplink, downlink and self-interference CSI. Simulation results and
analysis show that, significant power savings can be obtained. More
importantly, however, the MOOP approach here allows for the power saved to be
traded off for both uplink and downlink power savings, leading to an overall
energy efficiency improvement in the wireless link.
###### Index Terms:
full-duplex, multi-objective optimization, constructive interference, power
minimization, robust design.
## I Introduction
The ever-increasing need for improved spectrum-efficiency in wireless links
has brought FD at the forefront of research attention. By allowing
simultaneous transmission and reception, FD since it has the potential to
drastically improve the spectral efficiency of the HD communication networks
[1, 2, 3, 4, 5, 6]. One major hurdle with the FD communication systems is the
self-interference (SI) from the transmit antennas to the receive antennas of
the wireless transceiver. This interference raises the noise floor and it
becomes a dominant factor in the performance of the FD system. However, major
breakthroughs have been made in practical FD system setups [1] and [2] that
show that the SI can be partially cancelled to within a few dB of the noise
floor. While others focused on resource management, in [3], the authors
investigated the spectral efficiency of FD small cell wireless systems by
considering a joint beamformer design to maximize the spectral efficiency
subject to power constraints. In [4], the authors discussed the resource
allocation problems in FD-MIMO, FD-Relay, FD-OFDMA and FD-HetNet systems
including power control, interference-aware beamforming, e.t.c. Also, resource
allocation and scheduling in FD-MIMO-OFDMA relaying systems was studied in
[5]. In [6], the authors used massive arrays at the FD relay station to cancel
out loop interference and as a result increase the sum spectral efficiency of
the system.
Many of the above FD solutions build upon existing beamforming solutions in
the literature, that have been extensively developed for the downlink channel,
moving from the sophisticated but capacity achieving non-linear beamforming
techniques [7, 8, 9, 10, 11] to the less complex linear beamforming techniques
[12, 13, 14, 15, 16]. Several optimization based schemes that provide optimal
solutions subject to required quality of service (QoS) constraints have been
proposed for multi-input single-output (MISO) systems in [17, 18, 19, 20]. In
[21, 22], the authors addressed the problem of robust designs in downlink
multiuser MISO systems with respect to erroneous channel state information
(CSI). The work in [23] focused on addressing both max-min signal-to-
interference (SINR) balancing problem and power minimisation problem with SINR
constraints. More recently, it has been shown in [13, 14, 24, 25] that with
the knowledge of the users’ data symbols and the CSI, the interference can be
classified into constructive and destructive interference. And further
findings in [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] show that
tremendous gains can be achieved by exploiting the constructive interference
based on symbol level optimization for both PSK and QAM modulations. However,
these findings are all based on MISO HD systems.
Our work extends the above interference exploitation concept to the FD
transmission, by employing multi-objective optimization, as most recently
studied for FD in [38, 39, 40]. The authors in [38] investigated the power
efficient resource allocation for a MU-MIMO FD system. They proposed a multi-
objective optimisation problem (MOOP) to study the total uplink and downlink
transmit power minimization problems jointly via the weighed Tchebycheff
method. They extended their work to a robust and secure FD systems model in
the presence of roaming users (eavesdroppers) in [39]. Similarly, in [40] the
authors used a similar model to investigate the resource allocation for FD
simultaneous wireless information and power transfer (SWIPT) systems.
Accordingly, in this work we aim to further reduce the power consumption in FD
MU-MIMO wireless communication systems by adopting the concept of constructive
interference in the literature to the downlink channel for both PSK and QAM
modulation. By exploiting interference constructively, useful signal power
from interference, we can provide a truly power efficient resource allocation
for a FD MU-MIMO system. The interference exploitation concept is yet to be
explored in the realm of FD transmission, where FD offers the unique
opportunity to trade-off the harvested interference power for both uplink and
downlink power savings through the MOOP designs. Against the state-of-the-art,
we summarize our contributions below:
1. 1.
We first formulate the FD beamforming design problem that minimizes (a)the
total downlink transmit power and, (b)the total uplink transmit power problem,
for PSK and QAM modulation separately. Both problems are subject to downlink
users SINR requirement based on the constructive interference regions and
uplink users SINR requirement. Unlike conventional FD beamformers, we show
that the proposed optimizations are convex and can be easily solved by
conventional solvers.
2. 2.
Building on the above single-objective problems, we then formulate a multi-
objective problem to study the trade-off between the total uplink and downlink
transmit power minimization problems jointly via the weighed Tchebycheff
method. Again, unlike the conventional FD beamformers, we show that the
proposed optimization is convex.
3. 3.
We further derive robust MOOP for both the conventional and the proposed
interference exploitation approach by recasting the MOOP into a virtual
multicast problem for erroneous downlink, uplink and SI CSI with bounded
errors.
The rest of the paper is organised as follows. Section II introduces the
system model that is considered in this paper. Section III describes the two
conventional power minimisation problems of interest to the system operator
and then briefly describes the MOOP formulation based on the two problems. In
Section IV, the proposed power minimization optimisation problems based on
constructive interference regions are presented for PSK and QAM modulations.
Then in Section V, we present the robust version of the optimisation problem
presented in Section IV. In Section VI, we provide a computational complexity
analysis of the MOOP formulations. Section VII illustrates the important
results and discussions. And finally we conclude in Section VIII.
## II System Model
We consider a FD multiuser communication system as shown in Fig. 1. The system
consists of a FD radio BS with N antennas serving K HD downlink users and J HD
uplink users. Each user is equipped with a single antenna to reduce hardware
complexity. Let ${\textbf{h}_{i}}\in\mathbb{C}^{N\times 1}$ be the channel
vector between the FD radio BS and the i-th downlink user, and
${\textbf{f}_{j}}\in\mathbb{C}^{N\times 1}$ be the channel vector between the
FD radio BS and the j-th uplink user. We denote the transmit signal vector
from the FD radio BS to the i-th downlink user as
$\displaystyle{\textbf{t}_{i}}$ $\displaystyle={\textbf{w}_{i}}d_{i}$ (1)
where ${\textbf{w}_{i}}\in\mathbb{C}^{N\times 1}$ and $d_{i}$ denote the
beamforming vector and the unit data symbol for the i-th downlink user. The
received signal at the i-th downlink user is:
$\displaystyle y_{i}$ $\displaystyle=\underset{\textrm{desired
signal}}{\underbrace{{\textbf{h}_{i}^{H}}{\textbf{t}_{i}}}}+\underset{\textrm{interference
plus noise}}{\underbrace{\sum_{k\neq
i}^{K}{\textbf{h}_{i}^{H}}{\textbf{t}_{k}}+n_{i}}}$ (2)
where $n_{i}\sim{\mathcal{CN}}\left(0,\sigma_{i}^{2}\right)$ represents the
additive white Gaussian noise AWGN at the i-th downlink user. For each time
slot the FD radio BS transmits K independent unit data symbols d
simultaneously at the same frequency to the K downlink users. The first term
in (2) represents the desired signal while the second term is the multiuser
interference signal. The received signal from the J uplink users at the FD
radio BS is:
Figure 1: System model with a FD radio BS with N antennas, K HD downlink users
and J HD uplink users.
$\displaystyle{\textbf{y}^{BS}}$
$\displaystyle=\sum_{j=1}^{J}{\sqrt{P}_{j}}{\textbf{f}_{j}}x_{j}+\underset{\textrm{residual
self-
interference}}{\underbrace{{\textbf{G}}\sum_{k=1}^{K}{\textbf{t}_{k}}}}+{\textbf{z}}$
(3)
where ${\textit{P}_{j}}$ and ${\textit{x}_{j}}$ denotes the uplink transmit
power and the data symbol from the j-th uplink user respectively. The vector
${\textbf{z}}\sim{\mathcal{CN}}(0,{\sigma}_{N}^{2})$ represents the additive
white Gaussian noise AWGN at the FD radio BS. The matrix
${\textbf{G}}\in\mathbb{C}^{N\times N}$ denotes the self-interference (SI)
channel at the FD radio BS. In the literature, different SI mitigation
techniques have been proposed [41, 42] to reduce the effect of self-
interference. In order to isolate our proposed scheme from the specific
implementation of a SI mitigation technique, since the SI cannot be cancelled
perfectly in FD systems due to limited dynamic range at the receiver even if
the SI channel is known perfectly [39, 42], we model the residual SI after
cancellation as $\left({\textbf{G}}\sum_{k=1}^{K}{\textbf{t}_{k}}\right)$ as
in [38, 39]. Accordingly, the first term of (3) represents the desired signal
from the j-th uplink user and the second term represents the residual SI.
Before we formulate the problem, we first define the signal-to-interference
ratio (SINR) at the i-th downlink user and at the FD radio BS respectively as
$\displaystyle SINR_{i}^{DL}$
$\displaystyle=\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}$ (4)
$\displaystyle SINR_{j}^{UL}$
$\displaystyle=\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}}\mid^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}$
(5)
where ${\textbf{u}_{j}}\in^{N\times 1}$ is the receive beamforming vector for
detecting the receivied symbol from the j-th uplink user. To reduce
complexity, we assume a zero-forcing receiver at the BS. Hence, the receive
beamformer for the j-th uplink user is given as
$\displaystyle{\textbf{u}_{j}}$
$\displaystyle=({\textbf{r}_{j}}{\textbf{F}^{\dagger}})^{H}$ (6)
where
${\textbf{r}_{j}}=[\underset{j-1}{\underbrace{0,\ldots,0,}}1,\underset{J-j}{\underbrace{0,\ldots,0}}]$,
${\textbf{F}^{\dagger}}=({\textbf{F}}^{H}{\textbf{F}})^{-1}{\textbf{F}}^{H},^{\dagger}$
denotes the pseudo-inverse operation and
${\textbf{F}}=[{\textbf{f}}_{1},\dots,{\textbf{f}}_{J}]$.
## III Conventional Power Minimization Problem
In this section, we study the conventional power minimization (PM) problem
where all the interferences are treated as undesired signals. We first
formulate the downlink and uplink power minimization problems, which aim to
minimize the total average downlink and uplink transmit power, respectively,
subject to the downlink users SINR and uplink users SINR. Then we formulate a
multi-objective PM problem that aims to investigate the two system’s
objectives (downlink and uplink) jointly.
### Problem 1: Total Downlink Transmit PM Problem
The downlink PM problem for FD optimisation is typically formulated as [38,
39]:
$\begin{split}\mathcal{P}1:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{i=1}^{K}{\left\|{\textbf{w}_{i}}\right\|}^{2}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (7)
where, $I_{j}=\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}}\mid^{2}$,
we define $\Gamma_{i}^{DL}$ and $\Gamma_{j}^{UL}$ as the minimum required
SINRs for the i-th downlink user and the j-th uplink user, respectively. This
problem aims to minimize the total downlink transmit power with no regards to
the consumed uplink transmit power. This problem is non-convex and it is
commonly solved via semidefinite relaxation as in [38, 39].
### Problem 2: Total Uplink Transmit PM Problem
The uplink PM problem for FD optimisation is typically formulated as [38, 39]:
$\begin{split}\mathcal{P}2:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (8)
where, $\Gamma_{i}^{DL}$ and $\Gamma_{j}^{UL}$ are the minimum required SINRs
for the i-th downlink user and the j-th uplink user, respectively. This
problem unlike problem $\mathcal{P}1$ aims to minimize the total uplink
transmit power with no regards to the consumed downlink transmit power.
Problem $\mathcal{P}2$ is non-convex and it is commonly solved via
semidefinite relaxation as in [38, 39].
### Problem 3: Multi-objective PM Problem
This formulation combines the two objectives of problem $\mathcal{P}1$ and
$\mathcal{P}2$ since both objectives are very important to both the users and
system operator. The multi-objective optimization is employed when there is
need to study jointly the trade-off between two desirable objectives via the
concept of Pareto optimality. A point is said to be Pareto optimal if there is
no other point that improves any of the objectives without decreasing the
others [43]. [43] did a survey of multi-objective optimization methods in
engineering. By using the weighted Tchebycheff method [43] which can acheive
the complete Pareto optimal set with lower computational complexity, the
multi-objective PM problem for FD optimisation is typically formulated as [38,
39],
$\begin{split}\mathcal{P}3:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\underset{a=1,2}{\text{max}}\left\\{{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\right\\}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (9)
where $R_{a}$ and $R_{a}^{*}$ denote the objective value and the optimal
objective value of the a-th optimisation problem, respectively. The variable
${\lambda_{a}}\geq 0$, $\sum{\lambda_{a}}=1$, specifies the priority given to
the a-th objective i.e. for a given ${\lambda_{1}}=0.8$ means $80\%$ priority
is given to the objective of problem $\mathcal{P}1$ and $20\%$ priority to the
objective of problem $\mathcal{P}2$. By varying ${\lambda_{a}}$ we can obtain
the complete Pareto optimal set. Problem $\mathcal{P}3$ is a non-convex
problem due to the SINR constraints A1 and A2, and it is commonly solved via
semidefinite relaxation as in [38, 39].
$\begin{split}\mathcal{P}4:\quad\underset{{\textbf{w}_{k}},{P_{j}}}{\text{min}}\quad&{\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}\right\|}^{2}\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{e^{j({\phi}_{k}-{\phi}_{1})}}}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (16)
## IV Power Minimization Problem based on Constructive Interference
In this section, we study the PM optimization problems based on constructive
interference. With prior knowledge of the CSI and users’ data symbols for the
downlink users, the instantaneous interference can be exploited rather than
suppressed [32]. To be precise, constructive interference is the interference
that pushes the received signal further into the detection region of the
constellation and away from the detection threshold [32]. This concept has
been thoroughly studied in the literature for both PSK and QAM modulation. We
refer the reader to [26, 27, 28, 29, 30, 31, 32, 33] for further details of
this topic. Motivated by this idea, here, we apply this concept to the PM
problems in Section III for both PSK and QAM modulations. We note that
constructive interference is only applied to the downlink users and not the
uplink users following that only the prior knowledge of the CSI and users’
data symbols for the downlink users are available at the BS. Nevertheless, we
show in the following that power savings can be obtained for both uplink and
downlink transmission, by means of the MOOP design.
### IV-A Constructive Interference for PSK modulation
To illustrate this concept, we provide a geometric illustration of the
constructive interference regions for a QPSK constellation in Fig. 2. We can
define the total transmit signal vector as
$\displaystyle\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}=\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}d_{i}$
(10)
where $d_{i}=de^{{\phi}_{i}}$ is the desired symbol for the i-th downlink
user. Therefore, the received signal (2) at the i-th downlink user can be
redefined as
$\displaystyle y_{i}$
$\displaystyle={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}+m_{i}$
(11)
$\displaystyle={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}d_{i}+m_{i}$
(12)
Accordingly, since the interference contributes constructively to the received
signal, it has been shown in [14] that the downlink SNR at the i-th downlink
user (4) can be rewritten as
$\displaystyle SNR_{i}^{DL}$
$\displaystyle=\frac{{\left|{\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right|}^{2}}{\sigma_{i}^{2}}$
(13)
Without loss of generality, by taking user 1 as reference the instantaneous
transmit power for a unit symbol is
$\displaystyle
P_{total}={\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}\right\|}^{2}$
(14)
As detailed in [32], the shaded area in Fig. 2 is the region of constructive
interference. If the recieved signal $y_{i}$ falls within this region, then
interference has been exploited constructively. The angle
$\theta=\pm\frac{\pi}{M}$ determines the maximum angle shift of the
constructive interference region for a modulation order $M$, $a_{I}$ and
$a_{R}$ are the imaginary and real parts of the received signal $y_{i}$
without the noise, respectively. The detection threshold is determined by
$\gamma=\sqrt{\Gamma_{i}^{DL}\sigma_{i}}$.
Therefore, by applying these definitions and basic geometry from Fig. 2 it can
be seen that for the received signal to fall in the constructive region of the
constellation we need to have $a_{I}\leq(a_{R}-\gamma)\tan\theta$.
Accordingly, we can define the downlink SINR constraint that guarantees
constructive interference at the i-th downlink user by
$\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\leq\\\
\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta$
(15)
Figure 2: Constructive interference region for a QPSK constellation point
#### IV-A1 Total Downlink Transmit PM Problem
Based on the analysis above, we can modify the SINR constraints for the
downlink users to accommodate CI. The optimisation problem for the total
downlink transmit PM is expressed in $\mathcal{P}4$. The total minimum
downlink transmit power is minimised subject to constraint B1, which
guarantees constructive interference for the downlink users for minimum
required SINR $\Gamma_{i}^{DL}$ while the constraint B2 guarantees that the
uplink users their minimum required SINR $\Gamma_{j}^{UL}$. Unlike its
conventional counterpart $\mathcal{P}1$, it can be seen that $\mathcal{P}4$ is
convex due to the substitution of the conventional downlink SINR constraint
with the CI SNR constraints and can be tackled with standard solvers.
#### IV-A2 Total Uplink Transmit PM Problem
On the other hand, we formulate the uplink transmit PM problem by minimising
the total uplink transmit power with no regards to the downlink transmit
power.
$\begin{split}\mathcal{P}5:\,\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\,&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\\\
&\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{I^{PSK}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (17)
where, $I^{PSK}_{j}=\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{e^{j({\phi}_{k}-{\phi}_{1})}}}\right|^{2}$.
Again, it can be seen that the above problem is convex and can be tackled with
standard solvers.
#### IV-A3 Multi-objective PM Problem
By adapting the downlink SINR constraints in $\mathcal{P}2$, we can further
obtain the MOOP for interference exploitation in the FD scenario under study
as
$\begin{split}\mathcal{P}6:\,\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\\\
&\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{I^{PSK}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\
&B3:{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\end{split}$ (18)
where $t$ is an auxiliary variable.
It can be observed that, due to the substitution of the conventional downlink
SINR constraint with the CI SNR constraints, this formulation unlike the
conventional problem in $\mathcal{P}3$ is convex and thus can be optimally
solved using standard convex softwares like CVX [44].
### IV-B Constructive Interference for QAM modulation
Figure 3: Schematic representation of 16QAM constellation points
To illustrate the concept of constructive interference for QAM modulation we
provide a schematic representation for 16QAM constellation points in Fig. 3.
Based on [31], to guarantee constructive interference for the constellation
points, we rewrite the SINR constraints for the downlink users to exploit the
specific detection regions for each group of constellation points separately
as detailed below. First, we redefine the received signal without noise (12)
and the instantaneous transmit power (14) in terms of amplitude not phase as
$\displaystyle
y_{i}={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k},\forall i$ (19)
and,
$\displaystyle
P_{total}={\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right\|}^{2}$ (20)
From Fig. 3, to ensure constructive interference is achieved and the
constellation points are received in the correct detection region for the
downlink users, the following constraints are adopted. Note that the dotted
lines in Fig. 3 represent the decision boundaries.
* •
For the group of constellation points in the box labelled ”1” in Fig. 3, since
they are all surrounded by the decision boundaries, the constraints should
guarantee that the received signals achieve the exact constellation point so
as not to exceed the decision boundaries. The constraints are
$\displaystyle C1:$ $\displaystyle
Re(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle C2:$
$\displaystyle Im(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$
* •
For the group of constellation points labelled ”2” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries, which is the real axis. The constraints are
$\displaystyle C1:$ $\displaystyle
Re(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle C2:$
$\displaystyle Im(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$
* •
For the group of constellation points labelled ”3” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries, which is the imaginary axis. The constraints are
$\displaystyle C1:$ $\displaystyle
Re(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle
C2:$ $\displaystyle Im(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$
* •
For the group of constellation points labelled ”4” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries. Here, the constellation points are not
surrounded by the decision boundaries and therefore have a larger detection
region that extend infinitely. The constraints are
$\displaystyle C1:$ $\displaystyle
Re(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle
C2:$ $\displaystyle
Im(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$
#### IV-B1 Total Downlink Transmit PM Problem
By adopting the required constraints C1 and C2 for the corresponding group
constellation points, the total downlink transmit PM optimisation problem is
expressed as
$\begin{split}\mathcal{P}7:\,\underset{{\textbf{w}_{k}},P_{j}}{\text{min}}\quad&{\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right\|}^{2}\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j.\end{split}$ (21)
where $I^{QAM}_{j}=\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{d_{k}}}\mid^{2}$.
#### IV-B2 Total Uplink Transmit PM Problem
Similarly, the uplink PM problem can be written for the case of QAM as
$\begin{split}\mathcal{P}8:\,\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j.\end{split}$ (22)
#### IV-B3 Multi-objective PM Problem
Finally, we can design the MOOP for the case of QAM by employing the above
constraints C1 and C2 as
$\begin{split}\mathcal{P}9:\,\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\
&C4:{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\end{split}$ (23)
Again, it can be observed that unlike their conventional counterparts, all
three optimizations above are convex and can be optimally solved using
standard convex softwares like CVX [44].
## V Multi-objective Optimization Problem with Imperfect CSI
### V-A Conventional Robust MOOP
In this section we study the robustness of the system when the downlink, the
uplink and the SI CSI is not perfectly known. For each channel, the actual CSI
is modeled as
$\displaystyle{\textbf{h}}_{i}={\check{\textbf{h}}}_{i}+{\Delta{\textbf{h}}}_{i},\forall
i,$ (24)
$\displaystyle{\textbf{f}}_{j}={\check{\textbf{f}}}_{j}+{\Delta{\textbf{f}}}_{j},\forall
j,$ (25)
and,
$\displaystyle{\textbf{G}}={\check{\textbf{G}}}+{\Delta{\textbf{G}}}.$ (26)
where ${\check{\textbf{h}}}_{i},{\check{\textbf{f}}}_{j}$ and
${\check{\textbf{G}}}$ denote the downlink, the uplink and the SI CSI
estimates known to the FD BS, respectively. And
${\Delta{\textbf{h}}}_{i},\forall i{\Delta{\textbf{f}}}_{j},\forall j$ and
${\Delta{\textbf{G}}}$ represent the downlink, the uplink and the SI CSI
uncertainties, respectively, which are assumed to be bounded such that
$\displaystyle\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\,\textrm{for
some}\,\epsilon_{h,i}\geq 0,$ (27)
$\displaystyle\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\,\textrm{for
some}\,\epsilon_{f,i}\geq 0,$ (28)
$\displaystyle\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\,\textrm{for
some}\,\epsilon_{G}\geq 0.$ (29)
We assume that the FD BS has no knowledge of
${\Delta{\textbf{h}}}_{i},{\Delta{\textbf{f}}}_{j}$ and ${\Delta{\textbf{G}}}$
except for the error bounds, hence, we take the worst-case approach for our
problem design.
Henceforth, we focus on the multi-objective problem formulation since it is a
generalisation of both the downlink and uplink optimisation problems.
Therefore, the multi-object formulation of problem $\mathcal{P}3$ for
imperfect CSI is
$\begin{split}\mathcal{P}10:\quad\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\frac{{\left|\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{w}_{i}}\right|}^{2}}{\sum_{k\neq
i}^{K}{\left|\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{w}_{k}}\right|}^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\frac{{P_{j}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|^{2}+C_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (30)
where
$C_{j}=\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\textbf{w}_{k}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}$
In the downlink and uplink SINR constraints, there are infinitely many
inequalities which make the worst-case design particularly challenging. To
make $\mathcal{P}10$ more tractable, we formulate the problem as a
semidefinite program (SDP) then transform the constraints into linear matrix
inequalities (LMI), which can be efficiently solved by existing solvers like
CVX [44]. The SDP transformation of problem $\mathcal{P}10$ is
$\begin{split}&\underset{{\textbf{W}_{i}},P_{j},t}{\text{min}}\quad t\\\
\text{s.t.}\quad&\widetilde{\textrm{D1}}:\frac{\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{W}_{i}}\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)}{\sum_{k\neq
i}^{K}\left((\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i})^{H}{\textbf{W}_{k}}(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i})\right)+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\widetilde{\textrm{D2}}:\frac{P_{j}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{U}_{j}}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)}{\sum_{n\neq
j}^{J}P_{n}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{U}_{j}}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)+\widetilde{C}_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &\widetilde{\textrm{D3}}:{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\\\
&\widetilde{\textrm{D4}}:{\textbf{W}_{i}}\succeq 0,\forall i.\end{split}$ (31)
where,
$\widetilde{C}_{j}=\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}$
and we define $\textbf{W}_{i}=\textbf{w}_{i}\textbf{w}_{i}^{H}$ and
$\textbf{U}_{j}=\textbf{u}_{j}\textbf{u}_{j}^{H}$.
By applying the S-procedure as in [45] we can convert these constraints into
LMIs. First, we can rearrange constraint $\widetilde{\textrm{D1}}$ into
$\displaystyle\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{Q}_{i}}\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)-\Gamma_{i}^{DL}\sigma_{i}^{2}\geq
0,\forall i,$ (32)
where, we introduce
$\displaystyle{\textbf{Q}_{i}}\triangleq{\textbf{W}_{i}}-\Gamma_{i}^{DL}\sum_{k\neq
i}^{K}{\textbf{W}_{k}},\forall i$
and then for constraint $\widetilde{\textrm{D2}}$, let’s define two vectors
$\widetilde{\textbf{f}}$ and $\widetilde{\Delta{\textbf{f}}}$ as
$\displaystyle\widetilde{\textbf{f}}={\begin{bmatrix}\check{\textbf{f}}_{j}\\\
\vdots\\\ \check{\textbf{f}}_{J}\end{bmatrix}}\in\mathbb{C}^{NJ\times
1},\widetilde{\Delta{\textbf{f}}}={\begin{bmatrix}{\Delta{\textbf{f}}}_{j}\\\
\vdots\\\ {\Delta{\textbf{f}}}_{J}\end{bmatrix}}\in\mathbb{C}^{NJ\times 1}$
(33)
hence, we can define any
$\check{\textbf{f}}_{j}=\textbf{B}_{j}\widetilde{\textbf{f}}$ and
${\Delta{\textbf{f}}}_{j}=\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}},$ for
$j=1,\ldots,J,$ with $\textbf{B}_{j}\in\mathbb{R}^{N\times NJ}$ defined as
$\textbf{B}_{j}={\begin{bmatrix}\textbf{B}_{j,1},\ldots,\textbf{B}_{j,J}\end{bmatrix}},$
where $\textbf{B}_{j,j}=\textbf{I}_{N}$ and $\textbf{B}_{j,n}=\textbf{0}_{N},$
for $n=1,\ldots,J,n\neq j.$ We have $\textbf{I}_{N}$ and $\textbf{0}_{N}$ to
be an $N\times N$ identity matrix and zero matrix, respectively.
Now constraint $\widetilde{\textrm{D2}}$ can be rewritten as
$\displaystyle\frac{P_{j}\left((\textbf{B}_{j}\widetilde{\textbf{f}}+\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}})^{H}{\textbf{U}_{j}}(\textbf{B}_{j}\widetilde{\textbf{f}}+\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}})\right)}{\sum_{n\neq
j}^{J}P_{n}\left((\textbf{B}_{n}\widetilde{\textbf{f}}+\textbf{B}_{n}\widetilde{\Delta{\textbf{f}}})^{H}{\textbf{U}_{j}}(\textbf{B}_{n}\widetilde{\textbf{f}}+\textbf{B}_{n}\widetilde{\Delta{\textbf{f}}})\right)+\widetilde{C}_{j}}\geq\Gamma_{j}^{UL},\forall
j$ (34)
and can be simplified to give
$\displaystyle\frac{\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)}{\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}}\geq\Gamma_{j}^{UL}$
(35)
where we introduce
$\displaystyle{\textbf{Z}_{j}}\triangleq
P_{j}\textbf{B}_{j}^{T}{\textbf{U}_{j}}\textbf{B}_{j}-\Gamma_{j}^{UL}\sum_{n\neq
j}^{J}P_{n}\textbf{B}_{n}^{T}{\textbf{U}_{j}}\textbf{B}_{n},\forall j$
we further simplify (35) by introducing slack variables $s_{j}>0,\forall{j}$
[45], such that (35) can be written as the following two constraints
$\displaystyle\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)\geq
s_{j}\Gamma_{j}^{UL},\forall{j},$ (36)
$\displaystyle\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}\leq
s_{j},\forall j.$ (37)
Next, we review the definitions of the S-procedure.
Lemma 1. (S-procedure [45]): Let $g_{l}(\textbf{x}),$ $l=1,2,$ be defined as
$\displaystyle
g_{l}(\textbf{x})=\textbf{x}^{H}\textbf{A}_{l}\textbf{x}+2Re\left\\{\textbf{b}_{l}^{H}\textbf{x}\right\\}+c_{l}$
where $\textbf{A}_{l}\in\mathbb{C}^{n\times
n},\textbf{b}_{l}\in\mathbb{C}^{n}$ and $c_{l}\in\mathbb{R}$. Then, the
implication of $g_{1}(\textbf{x})\geq 0\Rightarrow g_{2}(\textbf{x})\geq 0$
holds if and only if there exists a $\lambda\geq 0$ such that
$\displaystyle\delta{\begin{bmatrix}\textbf{A}_{1}&\textbf{b}_{1}\\\
\textbf{b}_{1}^{H}&c_{1}\end{bmatrix}}-{\begin{bmatrix}\textbf{A}_{2}&\textbf{b}_{2}\\\
\textbf{b}_{2}^{H}&c_{2}\end{bmatrix}}\succeq 0$
provided there exists a point $\hat{\textbf{x}}$ with
$g_{1}(\hat{\textbf{x}})>0.$
Following Lemma 1 and using the fact that
$\textrm{Tr}\left\\{\textbf{ABCD}\right\\}=\textrm{vec}\left(\textbf{A}^{H}\right)^{H}\left(\textbf{D}^{H}\otimes\textbf{B}\right)\textrm{vec}\left(\textbf{C}\right)$,
constraints (32), (36) and (37) can be expanded as
$\displaystyle{\Delta{\textbf{h}}}_{i}^{H}{\textbf{Q}_{i}}{\Delta{\textbf{h}}}_{i}+2Re\left\\{\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}{\Delta{\textbf{h}}}_{i}\right\\}+\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}\check{\textbf{h}}_{i}-\Gamma_{i}^{DL}\sigma_{i}^{2}\geq
0,\forall i$ (38)
$\displaystyle\widetilde{\Delta{\textbf{f}}}^{H}{\textbf{Z}_{j}}\widetilde{\Delta{\textbf{f}}}+2Re\left\\{\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\Delta{\textbf{f}}}\right\\}+\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-s_{j}\Gamma_{j}^{UL}\geq
0,\forall j,$ (39)
${\Delta{\textbf{g}}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right){\Delta{\textbf{g}}}+2Re\left\\{\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right){\Delta{\textbf{g}}}\right\\}\\\
+\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}-s_{j}\leq
0,\forall j$ (40)
$\begin{split}{\mathcal{P}11}:\quad\underset{{\textbf{W}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&{\begin{bmatrix}{\delta_{i}{\textbf{I}}+\textbf{Q}_{i}}&{\textbf{Q}_{i}}\check{\textbf{h}}_{i}\\\
\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}&\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}\check{\textbf{h}}_{i}-\Gamma_{i}^{DL}\sigma_{i}^{2}-\delta_{i}\epsilon_{h,i}^{2}\end{bmatrix}}\succeq
0,\forall i,\\\
&{\begin{bmatrix}{\mu_{j}{\textbf{I}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-s_{j}\Gamma_{j}^{UL}-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\succeq
0,\forall j,\\\
&{\begin{bmatrix}{\rho{\textbf{I}}-\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)}&-\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}\\\
-\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)&s_{j}-\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}-\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}-\rho\epsilon_{G}^{2}\end{bmatrix}}\succeq
0,\forall j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\},\\\ &{\textbf{W}_{i}}\succeq 0,\delta_{i}\geq
0,\mu_{j}\geq 0,\rho\geq 0,s_{j}>0,\forall i,j.\end{split}$ (41)
we define
${\check{\textbf{g}}}=\textrm{vec}\left({\check{\textbf{G}}}^{H}\right)$ and
${\Delta{\textbf{g}}}=\textrm{vec}\left({\Delta{\textbf{G}}}^{H}\right)$
where, $\textrm{vec}\left(\cdot\right)$ stacks the columns of a matrix into a
vector and $\otimes$ stands for Kronecker product.
Hence, by exploiting the S-procedure in Lemma 1, (38), (39) and (40) can be
formulated as LMIs and the conventional robust optimisation problem
$\mathcal{P}10$ can be reformulated as shown in (41).
The problem ${\mathcal{P}11}$ is convex, and can be efficiently solved using
CVX [44]. The resulting optimal values obtained from ${\mathcal{P}11}$ provide
a lower bound for the conventional power minimisation problem.
Note that the problem ${\mathcal{P}11}$ is a relaxed form of
${\mathcal{P}10}$. When the relaxation in ${\mathcal{P}11}$ is tight, i.e.
${\mathcal{P}11}$ returns all rank-one solutions $({\textbf{W}_{i}})$, then
the optimal solution $({\textbf{w}_{i}})$ to solve ${\mathcal{P}10}$ can be
obtained by matrix decomposition or randomisation as in [46], such that
${\textbf{W}_{i}}=\textbf{w}_{i}\textbf{w}_{i}^{H},\forall i.$ Otherwise, the
required power in original problem ${\mathcal{P}10}$ is always higher than
that in ${\mathcal{P}11}$.
### V-B Robust MOOP based on Constructive Interference
To study the robustness of the proposed system for the case of constructive
interference, we first formulate $\mathcal{P}6$ as a virtual multicast problem
[47]. To facilitate this, we simply incorporate each user’s channel with its
respective data symbol i.e.
${\widetilde{\textbf{h}}_{i}}={\textbf{h}_{i}}{e^{j({\phi}_{1}-{\phi}_{i})}}$
and let
$\textbf{w}=\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}$.
Following this the multicast formulation of problem $\mathcal{P}6$ can be
written as
$\begin{split}{\mathcal{P}12}:\quad&\underset{\textbf{w},P_{j},t}{\text{min}}\quad
t\\\
\text{s.t.}\quad&\left|Im\left({{\widetilde{\textbf{h}}_{i}}^{H}}\textbf{w}\right)\right|\leq\left(Re\left({{\widetilde{\textbf{h}}_{i}}^{H}}\textbf{w}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\left|{\textbf{u}_{j}^{H}}{\textbf{G}}\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (42)
Based on the multicast formulation ${\mathcal{P}12}$, for the worst-case
design we model the imperfect CSI as
$\displaystyle\widetilde{\textbf{h}}_{i}={\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i},\forall
i$ (43)
where ${\widetilde{\textbf{h}}}_{i}$ denotes the downlink CSI estimate known
to the FD BS. And ${\Delta\widetilde{\textbf{h}}}_{i}$ is the downlink CSI
uncertainty which is bounded such that
$\left\|{\Delta\widetilde{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2}$.
And we model the uplink and the SI CSI as in Section V-A, respectively. The
robust formulation of problem ${\mathcal{P}12}$ is
$\begin{split}{\mathcal{P}13}:\,\underset{\textbf{w},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\left|Im\left(({\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i})^{H}\textbf{w}\right)\right|\\\
&\leq\left(Re\left(({\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i})^{H}\textbf{w}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\frac{{P_{j}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|^{2}+I_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (44)
where
$I_{j}=\left|{\textbf{u}_{j}^{H}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}$.
First, let’s consider the downlink SINR constraint. For convenience we
separate the real and imaginary part of the complex notations and represent
them as real valued numbers. Let
$\displaystyle{\underline{\textbf{w}}}$
$\displaystyle\triangleq{\begin{bmatrix}Re({\textbf{w}})\\\
Im({\textbf{w}})\end{bmatrix}},$ (45)
$\displaystyle{\underline{\widetilde{\textbf{h}}}}_{i}$
$\displaystyle\triangleq{\begin{bmatrix}Im({\widetilde{\textbf{h}}_{i}})^{H}&Re({\widetilde{\textbf{h}}_{i}})^{H}\end{bmatrix}},$
(46) $\displaystyle{\Delta\underline{\widetilde{\textbf{h}}}}_{i}$
$\displaystyle\triangleq{\begin{bmatrix}Im(\Delta{\widetilde{\textbf{h}}_{i}})^{H}&Re(\Delta{\widetilde{\textbf{h}}_{i}})^{H}\end{bmatrix}},$
(47) $\displaystyle{\boldsymbol{\Pi}}$
$\displaystyle\triangleq{\begin{bmatrix}{\textbf{0}}_{N}&-{\textbf{I}}_{N}\\\
{\textbf{I}}_{N}&{\textbf{0}}_{N}\end{bmatrix}}.$ (48)
Where, ${\textbf{0}}_{N}$ and ${\textbf{I}}_{N}$ denote $N$ x $N$ all-zero
matrix and identity matrix, respectively.
With the new notations we can express the real and imaginary terms of downlink
SINR constraint in ${\mathcal{P}13}$ as:
$\displaystyle\textrm{Im}({\widetilde{\textbf{h}}_{i}^{H}}{\textbf{w}})=({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}){\underline{\textbf{w}}},$
$\displaystyle\textrm{Re}({\widetilde{\textbf{h}}_{i}^{H}}{\textbf{w}})=({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}){\boldsymbol{\Pi}}{\underline{\textbf{w}}}$
(49)
From the definition of the error bound, we have
$\left\|{\Delta\widetilde{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2}$,
the downlink SINR constraint can be guaranteed by the following constraint
$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad\left|\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\underline{\textbf{w}}}\right|\\\
-\left(\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta\leq
0,\forall i$ (50)
Hence, by considering the absolute value term, (50) is equivalent to the
following two constraints
$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (51)
$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\Delta\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (52)
Therefore, the the robust formulations of (51) and (52) are given by
${\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (53)
$-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{-\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (54)
Next, we consider the uplink SINR constraint in problem (44). Following
equations (33) and (34) in Section V-A, the uplink SINR constraint can be
rewritten as
$\frac{\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)}{\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}+{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\\\
\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j.$ (55)
and we note that (55) can be guaranteed by the following constraints
$\underset{\lVert{\Delta\widetilde{\textbf{f}}}_{j}\rVert^{2}\leq\epsilon_{f,j}^{2}}{\text{max}}\quad\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)-\Gamma_{j}^{UL}\left({c}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}\right)\\\
\geq 0,\forall j$ (56)
$\displaystyle\underset{\lVert{\Delta\widetilde{\textbf{G}}}\rVert^{2}\leq\epsilon_{G}^{2}}{\text{max}}\quad\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}+{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|^{2}\leq
c_{j},\forall j$ (57)
where $c_{j}>0,\forall{j}$ are introduced as slack variables [45].
Similar procedure as in Section V-A can be applied to (56). By exploiting the
S-procedure in Lemma 1, (56) can be expanded and converted into a LMI as shown
below
${\begin{bmatrix}{\mu_{j}{\textbf{I}_{N}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-\Gamma_{j}^{UL}c_{j}-\Gamma_{j}^{UL}\sigma_{N}^{2}\textrm{Tr}({\textbf{U}_{j}})-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\\\
\succeq 0,\forall j,$ (58)
We note that by using the fact that
$\left\|x+y\right\|^{2}\leq\left(\left\|x\right\|+\left\|y\right\|\right)^{2}$,
(57) can always be guaranteed by the following constraint
$\displaystyle\underset{\lVert{\Delta\widetilde{\textbf{G}}}\rVert^{2}\leq\epsilon_{G}^{2}}{\text{max}}\quad\left(\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}\right|+\left|{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|\right)^{2}\leq
c_{j},\forall{j}$ (59)
whose robust formulation is given by
$\displaystyle\left(\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}\right|+\epsilon_{G}\left|{\textbf{u}_{j}^{H}}\textbf{w}\right|\right)^{2}\leq
c_{j},\forall{j}$ (60)
Futhermore, we define
${\underline{\textbf{Y}}}_{j}\triangleq{\begin{bmatrix}Re({\textbf{u}_{j}^{H}}{\textbf{G}})&-Im({\textbf{u}_{j}^{H}}{\textbf{G}})\\\
Im({\textbf{u}_{j}^{H}}{\textbf{G}})&Re({\textbf{u}_{j}^{H}}{\textbf{G}})\end{bmatrix}}$
and
${\underline{\textbf{U}}}_{j}\triangleq{\begin{bmatrix}Re({\textbf{u}_{j}^{H}})&-Im({\textbf{u}_{j}^{H}})\\\
Im({\textbf{u}_{j}^{H}})&Re({\textbf{u}_{j}^{H}})\end{bmatrix}}$, therefore,
the constraint (60) can be written in terms of real valued numbers as
$\displaystyle\left(\left|{\underline{\textbf{Y}}}_{j}{\underline{\textbf{w}}}\right|+\epsilon_{G}\left|{\underline{\textbf{U}}}_{j}{\underline{\textbf{w}}}\right|\right)^{2}\leq
c_{j},\forall{j}$ (61)
Therefore, the robust optimisation problem based on CI is
$\begin{split}{\mathcal{P}14}:&\,\underset{{\underline{\textbf{w}}},P_{j},t}{\text{min}}\quad
t\\\ \text{s.t.}\\\
&{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta,\forall
i\\\
&-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{-\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta,\forall
i,\\\
&{\begin{bmatrix}{\mu_{j}{\textbf{I}_{N}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-\Gamma_{j}^{UL}c_{j}-\Gamma_{j}^{UL}\sigma_{N}^{2}\textrm{Tr}({\textbf{U}_{j}})-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\succeq 0,\forall j,\\\
&\left(\left|{\underline{\textbf{Y}}}_{j}{\underline{\textbf{w}}}\right|+\epsilon_{G}\left|{\underline{\textbf{U}}}_{j}{\underline{\textbf{w}}}\right|\right)^{2}\leq
c_{j},\forall{j},\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\},\\\ &\mu_{j}\geq 0,\,c_{j}>0,\forall j.\end{split}$
(62)
Note that problem ${\mathcal{P}14}$ is a convex problem and thus can be
optimally solved using standard convex softwares like CVX [44]. After we
obtain the optimal ${\underline{\textbf{w}}}^{*}$ and $P_{j}^{*}$, the robust
solution ${\textbf{w}}^{*}$ can be obtained from the relation in (45).
## VI Computational Complexity Analysis
In this Section, we mathematically characterize the computational complexity
of the conventional and proposed schemes based on MOOP formulations.
### VI-A Transmit Complexity
We note that the convex MOOP formulations
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ involve
only LMI and second-order cone (SOC) constraints. As such, the problems can be
sovled by a standard interior-point method (IPM) [48]. Therefore we can use
the worst-case runtime to analyse the complexity of the conventional and the
proposed CI schemes.
Following [49] and [50], the complexity of a generic IPM for solving problems
like ${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$
involve the computation of a per-optimization cost. In each iteration, the
computation cost is dominated by (i) the formation of the coefficient matrix
of the linear system, and (ii) the factorization of the coefficient matrix.
The cost of formation of the coefficient ($C_{form}$) matrix is on the order
of
$\displaystyle C_{form}=\underset{\textrm{due to the
LMI}}{\underbrace{n\sum_{a=1}^{A}k_{a}^{3}+n^{2}\sum_{a=1}^{A}k_{a}^{2}}}+\underset{\textrm{due
to the SOC}}{\underbrace{n\sum_{a=A+1}^{B}k_{a}^{2}}}$
while the cost of factorizing ($C_{fact}$) is on the order of $C_{fact}=n^{3}$
($n=$ number of decision variables). Hence, the total computation cost per
optimization is on the order of $C_{form}+C_{fact}$ [49]. We assume for the
sake of simplicity that the decision variables in
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ are
real-valued.
Hence, using these concepts, we now analyse the comutational complexity of
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$. First
we consider SDP formulation of ${\mathcal{P}3}$, which has $K$ LMI (trace)
constraints of size 1, $J$ LMI (trace) constraints of size 1, $K$ SOC
constraints of size $N$, $J$ LMI (trace) constraints of size 1 and $K$ LMI
(trace) constraints of size $N$. Therefore, the complexity of the SDP
formulation of ${\mathcal{P}3}$ is on the order shown in the first row of
Table I. Similarly, we can determine the complexity order of the formulations
${\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ as shown in Table I,
respectively. From Table I, we can show that the proposed MOOP formulation
${\mathcal{P}6}$ has lower complexity than the SDP formulation of
${\mathcal{P}3}$ since it has lower order of variables to compute i.e lower
cost of factorization ($C_{fact}$). Also, we can straightforwardly show that
for the robust MOOP, the proposed formulation ${\mathcal{P}14}$ has a lower
complexity than the conventional formulation ${\mathcal{P}11}$ since
${\mathcal{P}11}$ involves a more complicated set of constraints (5 LMI
constraints and 1 SOC constrsint). This is also consistent with our simulation
results in the following Section.
Table I: Complexity Analysis of the MOOP Formulations MOOP | Complexity Order
---|---
${\mathcal{P}3}$(SDP) | $\mathcal{O}((KN^{2}+J)[K(1+N^{3})+2J+(KN^{2}+J)(K(1+N^{2})$
| $+2J)+KN^{2}+(KN^{2}+J)^{2}])$
${\mathcal{P}6}$ | $\mathcal{O}((KN+J)[2J(1+(KN+J))+2KN^{2}+(KN+J)^{2}])$
${\mathcal{P}11}$ | $\mathcal{O}((KN^{2}+J)[K(N+1)^{2}+J(NJ+1)^{3}+J(N^{2}+1)^{3}$
| $+J+KN^{3}+(KN^{2}+J)(K(N+1)^{2}+J(NJ+1)^{2}$
| $+J(N^{2}+1)^{2}+J+KN^{2})+(KN^{2}+J)(KN^{2})$
| $+(KN^{2}+J)^{2}])$
${\mathcal{P}14}$ | $\mathcal{O}((2N+J)[J(NJ+1)^{3}+J(N+1)^{3}+J$
| $+(2N+J)(J(NJ+1)^{2}+J+12N^{2})+(2N+J)^{2})])$
At this point, we emphasize that as the MOOP formulations in ${\mathcal{P}3}$
and ${\mathcal{P}11}$ are data independent, they only need to be applied once
during each channel coherence time. While as the proposed MOOP formulations in
${\mathcal{P}6}$ and ${\mathcal{P}14}$ are data dependent, they need to be run
on a symbol by symbol basis. In the following section we compare the resulting
transmit complexity of conventional and proposed MOOP approaches for both slow
and fast fading scenarios, and show that the average execution time per
downlink frames is comparable for both techniques.
### VI-B Receiver Complexity
At the receiver side, for the case of the conventional beamforming, the
downlink users in our FD system scenario need to equalize the composite
channel ${\textbf{h}_{i}^{H}}{\textbf{w}_{i}}^{*}$ to recover their data
symbols, where $\left\\{{\textbf{w}_{i}}^{*}\right\\}_{i=1}^{K}$ is the
optimal solution of ${\mathcal{P}3}$. For the case of the proposed CI scheme,
since the received symbols already lie in the constructive region of the
constellation as shown in Fig. 2 and Fig. 3, equalization is not required by
the downlink users. This automatically translates to reduced complexity at the
receiver. Accordingly, this implies that CSI is not required for detection at
the downlink users for the proposed CI scheme. Thus, depending on the
signaling and pilots already involved for the SINR estimation, the proposed CI
scheme may lead to further savings in training time and overhead. Most
importantly, this makes the proposed scheme resistant to any quantization
errors from the CSI acquisition at the receiver.
## VII Results
In this section, we investigate the performance of our proposed system through
simulations. We model all channels as independent and identically distributed
Rayleigh fading for both the perfect and imperfect CSI cases. Systems with
QPSK and 16QAM modulation are considered while it is clear that the benefit
extends to any lower or higher order modulation. For comparison in every
scenario, we compare the proposed technique, constructive interference (CI)
with the conventional case i.e. when all interference is treated as harmful
signal [38, 39]. We use $N\times K\times J$ to denote an FD radio BS with $N$
antennas, $K$ downlink users and $J$ uplink users, respectively.
Figure 4: Average system object trade-off region achieved by the proposed
scheme versus the conventional scheme N = 9, K = 6, J = 3.
### VII-A Uplink-Downlink Power Trade-off
In Fig. 4, we investigate the trade-off between the downlink and uplink total
transmit power for the case of $N=9,K=6,J=3$ antennas. The trade-off region is
obtained by solving problem ${\mathcal{P}3}$ and ${\mathcal{P}6}$ for the
conventional and CI case, respectively, for $0\leq{\lambda_{a}}\leq
1,a\in(1,2)$ with a step size of 0.1. Note that ${\lambda_{a}}$ determines the
priority of the a-th objective. We assume the same required SINR for all
downlink users to be $\Gamma_{i}^{DL}=10dB$ and $\Gamma_{i}^{UL}=0dB$ for all
uplink users. It can be seen from the plot that there is a trade-off between
the two objectives (downlink and uplink) i.e. an increase in one leads to a
decrease in the other and vice versa.
Figure 5: Average system object trade-off region achieved by the proposed
scheme versus the conventional scheme N = 8, K = 6, J = 3. Figure 6: Average
system object trade-off region achieved by the proposed scheme versus the
conventional scheme N = 6, K = 6, J = 6.
We compare the trade-off plot for the conventional scheme and the CI schemes
when applied to QPSK and 16QAM modulations. We study the trade-off plots when
the total number of antennas at the users is equal to the number of antennas
at the FD radio BS. Thus, we can observe that the CI scheme has power savings
of about 7dB for the uplink users in both QPSK and 16QAM modulations and about
2dB and 1.2dB power savings for the downlink users in QPSK and 16QAM
modulations, respectively. Note that the proposed scheme is only outperformed
by the conventonal beamforming for the case $\lambda_{1}=0,\lambda_{2}=1$,
where all priority is given to the uplink PM problem, where interference
exploitation does not apply.
In Fig. 5, we plot the case when we have $N=8,K=6,J=3$. The same trend can be
seen with Fig. 4, where we have for QPSK modulation power savings of about 6dB
and 2dB for the uplink and downlink users, respectively. And for 16QAM
modulation, we have power savings of about 6dB and 1.8dB for the uplink and
downlink users, respectively. This two scenarios $N=9,K=6,J=3$ and
$N=8,K=6,J=3$, show a practical perspective in the sense that there is usually
more antennas at the FD radio BS than the number of antennas at the users and
the optimisation problems are always feasible.
In Fig. 6, we show a scenario where we have equal number of antennas at the FD
radio BS and at the users $N=K=J=6$. With this setup we can see for QPSK
modulation uplink and downlink user power savings of about 12dB and 4dB,
respectively, and about 10dB and 2dB, respectively, for 16QAM modulation. The
reason is because for $N=K=J=6$ the problem is more restricted in the
optimisation variable dimensions and the conventional scheme in this scenario
leads to greatly increased uplink and downlink powers while for the CI scheme
this scenario can be accommodated and has higher feasibility so consumes lower
power. These results highlight a key advantage of the proposed scheme over the
conventional approaches.
Figure 7: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.9,\lambda_{2}=0.1$ and $\Gamma^{UL}=0$dB for QPSK modulation
Figure 8: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.1,\lambda_{2}=0.9$ and $\Gamma^{UL}=0$dB for QPSK modulation
Figure 9: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.1$ for QPSK modulation
### VII-B Average Transmit Power versus Minimum Required SINR
In Fig. 7 and Fig. 8, we investigate the power consumption of the downlink and
uplink users for different minimum required downlink SINR ($\Gamma_{i}^{DL}$).
For both plots we assume a minimum required uplink SINR $\Gamma_{j}^{UL}=0dB$
for all uplink users. In Fig. 7, we select $\lambda_{1}=0.9$ and
$\lambda_{2}=0.1.$ which gives higher priority to the total downlink transmit
power minimisation problem. It can be observed that both the uplink and
downlink power consumption increases with increase in $\Gamma_{i}^{DL}$. This
is because an increase in the downlink SINR requirment translates to increace
in downlink transmit power and hence increase in the SI power. Therefore, the
uplink users have to transmit with a higher power to meet their QoS
requirement ($\Gamma_{j}^{UL}$). However, we can still see power savings of
12dB and 5dB for the uplink and downlink users, respectively, for the CI
scheme compared to the conventional scheme. Also, we note that while CI is
applied to only the downlink users, more power is saved for the uplink users
than the downlink users. This is because with CI the total downlink transmit
power is reduced and this directly reduces the residual SI power at the FD BS.
Accordingly, the constructive interference power has been traded off for both
uplink and downlink power savings. The same trend can be seen in the Fig. 8,
where $\lambda_{1}=0.1$ and $\lambda_{2}=0.9$. It can be observed that in this
scenario since we give higher priority to the uplink power minimisation
problem, we have higher power savings for the uplink users and lower power
savings for the downlink users compared to the Fig. 7.
### VII-C MOOP with Imperfect CSI
In Fig. 9 and 10, we investigate the performance of the proposed CSI-robust CI
scheme for $N=K=J=6$, we select $\lambda_{1}=0.9$ and $\lambda_{2}=0.1$.
Figure 10: Average power consumption versus error bounds when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0dB$ and $\Gamma^{DL}=10dB$ for
QPSK modulation Figure 11: Average execution time per optimisation versus
number of downlink users with $N=J=6$ when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01$
Fig. 9 shows the Average power consumption for the uplink and downlink users
when the error bounds $\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.1$. It can be
seen that the CI scheme shows better performance than the conventional scheme
with power savings of 6dB and 4dB for the uplink and downlink users,
respectively. In addition, for the conventional cases, feasible solutions can
only be found for minimum required downlink SINR $\Gamma_{i}^{DL}\leq 20$dB.
This indicates that the channel error tolerance of the conventional scheme is
much lower than that of the proposed CI scheme. This is also shown in Fig. 10,
which shows the average power consumption with increasing error bounds. It can
be seen that feasible solutions can only be found for
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}\leq 0.2$. Besides, even if feasible
results could be found, significant amount of power will be consumed as can be
seen for error bound values between 0.15 and 0.2 for both uplink and downlink
users.
### VII-D Complexity
In Fig. 11, we compare the Average execution time per optimisation of the
conventional scheme and the proposed CI scheme for different number of
downlink users ($K$) with $N=J=6$. We fixed
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01.$ It can be seen that for the
perfect CSI case, the proposed CI scheme takes 83% of time taken by the
conventional scheme. While for the imperfect CSI case, the proposed CI scheme
takes about 28% of the time taken by the conventional scheme. This is because
the conventional approach involves a more complicated set of constraints,
hence, more computational cost as shown in Section VI-A above. Besides, the
proposed MOOP ${\mathcal{P}14}$ formulation involves a multicast approach
which reduces the number variables to compute.
As we have noted above however, the proposed data dependent optimization needs
to be run on a symbol-by-symbol basis. To obtain a fairer comparison, we plot
in Fig. 12 the average execution time per frame versus the number of downlink
users for slow and fast fading channels. Here, we assume the LTE Type 2 TDD
frame structure [51], where each frame is subdivided to 10 subframes each with
a duration $1ms$ and containing 14 symbol-time slots. Accordingly, we assume
that for fast fading the channel is constant for the duration of a subframe
with a number of symbols per coherence time $N_{coh}=14$, while for slow
fading we assume a coherence time equal to 5 subframes with $N_{coh}=70$ [51].
The results for both slow and fast fading channels show the end complexity of
the proposed CI approaches are comparable to those with the conventional
approaches. Accordingly, and in conjunction with the performance improvements
shown in the previous results, it can be seen that the proposed schemes
provide a much more favorable performance complexity trade-off w.r.t.
conventional interference mitigation.
Figure 12: Average execution time versus number of downlink users for
slow/fast fading channels with $N=J=6$ when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01.$
## VIII Conclusion
In this paper we studied the application of the interference exploitation
concept to a MU-MIMO system with a FD radio BS. The optimisation problem was
formulated as a convex Multi-Objective Optimisation problem (MOOP) via the
weighted Tchebycheff method. The MOOP was formulated for both PSK and QAM
modulations by adapting the decision thresholds in both cases to accommodate
for constructive interference. The CI scheme was also extended to robust
designs for imperfect downlink, uplink and SI CSI with bounded CSI errors.
Simulation results proved the significant power savings of the CI scheme over
the conventional scheme in every scenario. More importantly, we have shown
that through the FD MOOP formulation, constructive interference power can be
traded off for both uplink and downlink power savings.
## Acknowledgment
The author would like to thank the Federal Republic of Nigeria and the
Petroleum Technology Development Fund (PTDF) for funding his PhD.
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|
# Learning Generalized Causal Structure in Time-series
Aditi Kathpalia
Department of Complex Systems Institute of Computer Science of the Czech
Academy of Sciences
Prague Czech Republic
<EMAIL_ADDRESS>
Keerti Panchakshari Charantimath
Department of Mathematics Indian Institute of Technology Kharagpur
West Bengal India
<EMAIL_ADDRESS>
Nithin Nagaraj
Consciousness Studies Programme National Institute of Advanced Studies
Indian Institute of Science Campus Bengaluru India
<EMAIL_ADDRESS>
###### Abstract
The science of causality explains/determines ‘cause-effect’ relationship
between the entities of a system by providing mathematical tools for the
purpose. In spite of all the success and widespread applications of machine-
learning (ML) algorithms, these algorithms are based on statistical learning
alone. Currently, they are nowhere close to ’human-like’ intelligence as they
fail to answer and learn based on the important “Why?” questions. Hence,
researchers are attempting to integrate ML with the science of causality.
Among the many causal learning issues encountered by ML, one is that these
algorithms are dumb to the temporal order or structure in data. In this work
we develop a machine learning pipeline based on a recently proposed
‘neurochaos’ feature learning technique (_ChaosFEX_ feature extractor), that
helps us to learn generalized causal-structure in given time-series data.
## 1 Introduction
Current machine learning (ML) algorithms are based on statistical learning,
that is, these algorithms learn mostly by identifying associations in the
given data. As the popularity and range of ML algorithms have increased to
deal with specific tasks of classification and prediction, researchers focus
on the task of bringing these ML algorithms closer to human-like intelligence
so that these algorithms can be used to deal with _higher-level_ problems. One
of the major ways in which human intelligence differs from machine
intelligence is that humans learn through ‘cause-effect’ reasoning. Put more
specifically, humans have the capability to answer _‘what if’_ kind of
questions. _What if I intervene and do something to a given system? What if I
had acted in a different way?_ Machine intelligence is still far away from
answering these kind of questions [1, 2].
Let alone learning based on a causal understanding, machines are not even
capable of making sense of temporal or sequential order in the data. Further,
they lack the ability of _generalization_ which involves transfer of learning
from one problem to another and learning features that categorize together
datasets that are more alike than different. This lack of generalization stems
mainly from an inability of these algorithms to learn causal structures [2].
While a lot of research is being currently pursued at the intersection of
causality and machine learning, we focus on causal learning based on time
series data. Time series data are real valued data points arranged in a
chronological order. Such data is often collected as measurements in different
fields such as finance, medicine, climate science etc. Currently, there are
two main tasks for which causal inference is done using time series data. One
is studying the effect of treatment or interventions on certain variable of a
system. This treatment may be provided as a discrete event or may be
continuously given over time. This task is generally referred to as _causal
treatment effect estimation_ and finds applications in estimating the effects
of policies implemented so as to increase/ reduce the sale of certain goods or
in estimating the effect of drugs given to patients [3, 4, 5]. The second one
is discovering causal relationships between different variables of a system,
for which temporal evolution data is available. This task is generally
referred to as _causal network discovery_ and is useful in domains where we
wish to study the interaction between different variables such as the role of
human and natural factors in climate change [6, 7, 5].
For our work, we take one step backward and ask the question of whether ML
algorithms can even identify whether given time series has an inherent causal
structure associated with it, that is, whether the past values of a time
series effect its present values or if that is not the case. Learning this
structure for time series is essential for making sense of any ordered data
given as input to ML algorithms and is also essential to meaningfully answer
the two causality tasks discussed above. There are of course mathematical
techniques that rely on fitting autoregressive models and testing for time
dependence by estimating serial correlation in given time series. The latter
techniques include the Durbin–Watson test [8], Ljung–Box test [9] and
Breusch–Godfrey test [10]. However, these methods have their limitations: they
are parametric, requiring the choice of maximum order; do not work with
overlapping data or in the case of heteroskedasticity in the error process.
Further, since they are purely statistical, based on autocorrelation, they
cannot handle data with intervention(s)/ perturbation(s) where the cause-
effect relation in time series builds up as a result of external event(s). We
are interested in developing a non-parametric way of identifying causal
structure in time series data that works irrespective of the underlying model
and learns _generalized_ causal-structure in data, unaffected by distribution
or model shifts. For this task, we rely on extracting strong features which
can identify whether or not there is an underlying causal-structure and later
using a simple ML algorithm, classify given time-series data based on having a
causal-structure or not. We have used time series values directly, frequency
domain characteristics of the process, and ‘neurochaos’ based features
extracted from the frequency representation of the process to train the ML
classifiers in separate models. _Neurochaos_ inspired feature learning [11]
has been proposed based on a _ChaosNet_ neural network architecture [12] that
is composed of neurons, which individually mimic chaotic firing properties of
the human brain neurons. We compare the performance of these models and show
that neurochaos based learning is able to extract meaningful features for
learning generalized causal-structure in time series data.
This paper is organized as follows: Section 2 describes the time series data
simulated for the study. Section 3 describes the methods, that is the features
and the classifiers used for distinguishing between causal and non-causal
structure in temporal data. Results are given in Section 4 and we conclude
with Section 5, discussing the results, and hinting at future research
directions.
## 2 Datasets
The following time-series datasets were simulated and used in this work: Time-
series having a causal-structure or causal time-series were generated using
the following models:
* •
Autoregressive (AR) processes
AR processes are random processes in which value of the time-series at any
time depends linearly on its past values and on a noise term. The general
equation governing an AR ($p$) process, $X$, is given as:
$X(t)=c+\sum_{i=1}^{p}{a_{i}X(t-i)}+\varepsilon_{t},$ (1)
where, $t$ denotes time, $c$ is a constant, $p$ is the order of the AR
process, $a_{i}$ is the coefficient at lag $i$ time step(s) and
$\varepsilon_{t}$ is the noise term at time $t$. Order of an AR process is the
maximum past lag on which the value of the process at any time depends.
AR processes are used to model several processes occurring in nature,
economics etc. [13, 14]
* •
Autoregressive Moving Average (ARMA) processes
These processes have an AR part and an MA part. Based on the AR part, each
term in the time-series is regressed on its past values and based on the MA
part, the noise term at any time point is modelled as a linear combination of
instantaneous noise term (at that time point) and noise terms at past time
points. In mathematical form, an ARMA($p,q$) process $X$ can be expressed as:
$X(t)=c+\sum_{i=1}^{p}{a_{i}X(t-i)}+\sum_{i=0}^{q}{b_{i}\varepsilon(t-i)},$
(2)
where, $t$ denotes time, $c$ is a constant, $p$ is the order of the AR part
and $q$ is the order of the MA part, $a_{i}$ is the AR coefficient at lag $i$
time step(s) and $b_{i}$ is the MA coefficient at lag $i$ time step(s),
$\varepsilon(t)$ is the noise term at time $t$.
ARMA processes are widely used in the modelling of financial and geological
processes [15, 13].
* •
Autoregressive Fractionally Integrated Moving Average (ARFIMA) These processes
are used to model long term memory processes. An ARMA($p^{\prime},q$) process
when expressed in terms of the lag or backshift operator can be written as:
$(1-\sum_{i=1}^{p^{\prime}}{a_{i}B^{i}})X(t)=(1+\sum_{i=1}^{q}{b_{i}}B^{i})\varepsilon(t),$
(3)
Now, if the polynomial $(1-\sum_{i=1}^{p^{\prime}}{a_{i}B^{i}})$ has a unit
root of multiplicity $d$, then:
$(1-\sum_{i=1}^{p^{\prime}}{a_{i}B^{i}})=(1-\sum_{i=1}^{p^{\prime}-d}{a_{i}B^{i}})(1-B)^{d},$
(4)
This is the property expressed by Autoregressive Integrated Moving Average or
ARIMA($p,d,q$) processes with $p=p^{\prime}-d$. For the case of ARFIMA
processes, the difference parameter $d$ is allowed to take non-integer values.
They are thus generally expressed as:
$(1-\sum_{i=1}^{p^{\prime}}{a_{i}B^{i}})(1-B)^{d}X(t)=(1+\sum_{i=1}^{q}{b_{i}}B^{i})\varepsilon(t),$
(5)
ARFIMA models are also widely used, for example in the modelling of economic,
geological and physiological time series [16, 17].
Time-series without a causal structure or non-causal time-series were
generated as having time-series value at each time point, as independently and
randomly chosen from a normal distribution, $\mathcal{N}(\mu,\sigma^{2})$
(where $\mu$ and $\sigma$ denote the mean and standard deviation of the
distribution) or uniform distribution, $U(b_{u},b_{l})$ (where $b_{u}$ and
$b_{l}$ denote the upper bound and lower bound of the distribution).
## 3 Methods
### 3.1 Using time-series values directly
In this case, time-series values were directly passed through ML classifiers
to check if they could classify time series with a causal structure and
without one appropriately. For this purpose, Logistic Regression (LR) [18] and
Long-short term memory (LSTM) [19] classifiers were used. LSTM classifiers
have a recurrent neural network architecture and include not only feedforward
but also feedback connections. Along with being applied to single data points,
they have also been shown to work for large sequential datasets such as speech
and text data.
### 3.2 Using frequency domain characteristics of time-series
Any periodic structure in a time series is reflected in its frequency domain
amplitude. Since continuous causal effect within a time series (where its past
values affect present values), can be thought of as imbuing some periodic
nature to the time series, we consider it better to analyze the signals in
frequency domain. The fast fourier transform algorithm was used to obtain the
discrete fourier transform of simulated signals. Frequency domain amplitudes
of the signal were then directly passed through ML classifiers: LR and LSTM.
### 3.3 Using ChaosFEX features of the frequency domain signal
_ChaosNet_ is an artificial neural network inspired by the chaotic firing of
neurons in the human brain [12]. Based on ChaosNet, a hybrid learning
architecture was proposed later that uses ‘neurochaos’ or ‘ChaosFEX’ 111The
code for ChaosFEX feature extraction is available at
https://github.com/pranaysy/ChaosFEX and this code was used in our work. This
code is available open source under the license: Apache License, Version 2.0.
Consent of the authors was taken to use the code. features combined with
traditional machine learning approaches for the purpose of classification
[11]. The basis of ChaosNet or the extraction of ChaosFEX features is the
_topological transitivity_ property of Chaos. The ChaosNet architecture is
composed of 1-D chaotic neurons called as Generalized Luröth Series (GLS)
maps. The number of neurons in the architecture depend upon the number of
inputs/features provided to the architecture, with each neuron receiving a
single value as input for each data instance. These neurons are set to have an
initial activity of $q$ units. Input data is first normalized (to lie between
$0$ and $1$) and then passed to these GLS neurons. Each GLS neuron starts
firing and keeps on firing until it reaches the epsilon neighborhood of the
input value (also called stimulus) provided to it. The epsilon neighborhood of
a stimulus $y$ is defined as the interval $(y-\varepsilon,y+\varepsilon)$,
where $\varepsilon>0$.
In this work, we use a GLS neuron $T:[0,1)\rightarrow[0,1)$ defined by the
following equation:
$T(y)=\begin{cases}\frac{y}{b},&0\leq y<b,\\\ \frac{1-y}{1-b},&b\leq
y<1,\end{cases}$ (6)
where, $y\in[0,1)$ and $0<b<1$. Let the trajectory of a GLS neuron for a
stimulus $y$ be given by $A=[q\rightarrow T(q)\rightarrow
T^{2}(q)\ldots\rightarrow T^{N}(q)]$. Thus, it can be seen that the neuron
takes $N$ time steps to reach the epsilon neighborhood of the stimulus. $N$ is
referred to as the firing time. The fraction of the time when the trajectory
is above the discrimination threshold ($b$) is referred to as the _Topological
Transitivity-Symbolic Sequence (TTSS) Feature_ for the stimulus $y$.
To classify time-series as having a causal structure or not, normalized
frequency domain amplitudes of given time series were passed as input to the
ChaosFEX feature extractor and the TTSS features extracted from the same
(there will be one _TTSS_ feature at each frequency) were then passed as
inputs to the ML classifier, LR. This scheme, thus exploited the neurochaos
based hybrid learning architecture for our task. Such an architecture has been
shown to learn from finite sample datasets and can help to use chaos as a
kernel trick which can be useful to make given data linearly separable.
## 4 Results
The results obtained by using each of the methods discussed above are detailed
in the subsections below. We term each of our methods as models, as what the
methods are essentially trying to do is distinguish between a causal and a
non-causal underlying structure. So, even though we are not strictly trying to
fit a model, we are trying to learn a generalized characteristic of the
temporal order in given data.
Each of the models were trained and tested using the AR training and testing
set which consisted of AR series as the causal time-series and time series
with independent entries, randomly chosen from normal distribution,
$\mathcal{N}(0,0.01)$, as the non-causal time series. We term time-series
generated in the latter way as random time-series. 1250 AR and 1250 random
time-series, each having a length of 2000 time points were simulated. Each AR
series was of the form, $X(t)=a_{k}X(t-k)+\varepsilon_{t}$, with the order
$k$, being randomly chosen between 1 and 20. These series were initialized to
random values. The noise term followed the distribution $\mathcal{N}(0,0.01)$
and the AR coefficient for each simulation was randomly chosen such that
$a_{k}\in U(0.8,0.9)$. The training to testing split for this dataset was
70:30.
For further testing of the models, the following datasets were generated.
These testing sets had a shift in the probability distribution of time-series
when compared to the training set. We call them Distribution shift testing set
I and Distribution shift testing set II. For both of these, causal time-series
were generated in exactly the same manner as for the AR training and testing
set. The non-causal time series in Distribution shift testing set I were
generated using $\mathcal{N}(0,0.09)$ and in Distribution shift testing set II
were generated using $U(-0.6,0.6)$. The number of non-causal time series in
both datasets were 1250 and each time-series was simulated with 2000 time
points. For these datasets, models were trained using the AR testing set
described in the previous paragraph and tested based on these datasets.
### 4.1 Time-series values model
Performance metrics for the time-series values model using both LR and LSTM on
the simulated training and testing sets are shown in Table 1. The non-causal
time series were labelled as Class-0 and the causal time series as Class-1.
The Precision, Recall and F1-Score columns are given in the format
$(\cdot,\cdot)$, with the first value being for Class-0 and the second value
for Class-1.
LR was implemented using the _Scikit-Learn_ [20] package and LSTM was
implemented using the _Keras_ [21] framework in Python. For LR, all parameters
were kept as default. For LSTM, the layers were stacked in the following
order: LSTM layer with 10 outputs, dropout layer with a parameter value of
0.5, dense hidden layer with 10 outputs and relu activation function and
finally, a dense output layer with two outputs and soft-max activation
function. This classification model used categorical cross-entropy as the loss
function and adam optimizer for optimization. The optimizer was run for 50
epochs with a batch size of 1250. This classification model was run 15 times
to avoid local minima and the run which gave the best testing accuracy was
used.
Table 1: Prediction for time-series values model. LR: Logistic Regression, LSTM: Long Short-Term Memory. Scores are given in the format $(\cdot,\cdot)$, with the first value being for Class-0 and the second value for Class-1. Dataset | Classifier | Precision | Recall | $\mathbf{F_{1}Score}$ | Accuracy
---|---|---|---|---|---
AR training set | LR | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
| LSTM | (0.98, 1.00) | (1.00, 0.98) | (0.99, 0.99) | 99%
AR testing set | LR | (0.53, 0.54) | (0.70, 0.37) | (0.60, 0.44) | 54%
| LSTM | (0.98, 0.99) | (0.99, 0.98) | (0.99, 0.99) | 99%
Distribution shift | LR | (0.49, 0.48) | (0.58, 0.38) | (0.53, 0.43) | 48%
testing set I | LSTM | (0.15, 0.49) | (0.01, 0.97) | (0.01, 0.65) | 49%
Distribution shift | LR | (0.48, 0.47) | (0.56, 0.38) | (0.52, 0.42) | 47%
testing set II | LSTM | (0.00, 0.49) | (0.00, 0.97) | (0.00, 0.65) | 48%
It can be seen from Table 1 that the LR classifier works well for the AR
training set but fails for the AR testing set, indicating that it is
overfitting the training set. Tuning the hyperparameters for the classifier
did not help to improve its performance. LSTM gives good performance for both
AR training as well as testing set but fails when there is a distribution
shift in the random time-series. This is probably because the LSTM is
recognizing only the random time-series used for training as non-causal and
anything other than that as causal. Thus, it fails to learn the causal
structure in time-series.
### 4.2 Frequency-domain representation model
Performance metrics for the frequency-domain representation model using both
LR and LSTM are shown in Table 2 The Precision, Recall and F1-Score columns
are given in the format $(\cdot,\cdot)$, with the first value being for
Class-0 (non-causal time series) and the second value for Class-1 (causal time
series). In order to observe the characteristics of frequency amplitude of the
signals, it would be essential to demean the time-series so as to remove any
DC component that is present. However, since all our simulated datasets have a
zero mean, we skipped this step.
The hyperparameters and architecture used for LR and LSTM as well as the
packages used for the implementation remained the same as in Section 4.1.
Table 2: Prediction for frequency domain representation model. LR: Logistic Regression, LSTM: Long Short-Term Memory. Scores are given in the format $(\cdot,\cdot)$, with the first value being for Class-0 and the second value for Class-1. Dataset | Classifier | Precision | Recall | $\mathbf{F_{1}Score}$ | Accuracy
---|---|---|---|---|---
AR training set | LR | (0.99, 1.00) | (1.00, 0.99) | (1.00, 1.00) | 100%
| LSTM | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
AR testing set | LR | (0.99, 1.00) | (1.00, 0.99) | (0.99, 0.99) | 99%
| LSTM | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
Distribution shift | LR | (0.00, 0.50) | (0.00, 0.99) | (0.00, 0.66) | 50%
testing set I | LSTM | (0.00, 0.50) | (0.00, 1.00) | (0.00, 0.67) | 50%
Distribution shift | LR | (0.00, 0.50) | (0.00, 0.99) | (0.00, 0.66) | 50%
testing set II | LSTM | (0.00, 0.50) | (0.00, 1.00) | (0.00, 0.67) | 50%
Figure 1 shows the amplitude spectrum of the fourier transformed signal for a
realization of AR(15) process. Figures 2(a) and 2(b) show the same for
realizations of random time-series generated using distributions
$\mathcal{N}(0,0.01)$ and $U(-0.6,0.6)$ respectively. It can be seen from
these figures that the frequency domain characteristics of causal and non-
causal time series used are quite different, yet the implemented ML
classifiers are unable to distinguish between the two.
Figure 1: Frequency amplitude spectrum for a realization of AR(15) process
(a) (b)
Figure 2: Frequency amplitude spectrum for a realization of random time-series
generated using $\mathcal{N}(0,0.01)$ (left) and $U(-0.6,0.6)$ (right).
It can be seen from Table 2 that both LR and LSTM accurately classify the AR
training and testing sets, however, fail for testing sets with distribution
shifts in non-causal time-series. This again seems to be because the
classifiers are recognizing only the random time-series used for training as
non-causal and anything other than that as causal and have failed to learn any
causal-structure from the data.
### 4.3 ChaosFEX feature representation model (FT+ChaosFEX)
Performance metrics when TTSS features of the amplitudes at different
frequencies were passed as inputs to the ML classifier logistic regression are
shown in Table 3. The Precision, Recall and F1-Score columns are given in the
format $(\cdot,\cdot)$, with the first value being for Class-0 (non-causal
time series) and the second value for Class-1 (causal time series).
Table 3: Prediction for FT+ChaosFEX model with Logistic Regression classifier. Scores are given in the format $(\cdot,\cdot)$, with the first value being for Class-0 and the second value for Class-1. Dataset | Precision | Recall | $\mathbf{F_{1}Score}$ | Accuracy
---|---|---|---|---
AR training set | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
AR testing set | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
Distribution shift testing set I | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
Distribution shift testing set II | (1.00, 1.00) | (1.00, 1.00) | (1.00, 1.00) | 100%
AR(100) testing set | (NA, 1.00) | (NA, 0.99) | (NA, 1.00) | 99%
ARMA testing set | (NA, 1.00) | (NA, 1.00) | (NA, 1.00) | 100%
ARFIMA testing set | (NA, 1.00) | (NA, 1.00) | (NA, 1.00) | 100%
It can be seen that this model classifies accurately not just the AR training
and testing sets but also both the testing sets with distribution shifts in
the non-causal time series. To further check the robustness of the model, it
was tested on more testing sets in which there was a shift in the distribution
of causal time-series. Three testing sets generated in this manner included
AR(100), ARMA and ARFIMA as causal time-series and no non-causal time series.
Since these datasets did not include any non-causal time series, the first
value for precision, recall and $F_{1}$ score are marked as NA (Not-
Applicable) in Table 3.
The specific details of the three new testing sets mentioned above are as
follows. Each of these datasets consisted of 1250 causal time-series with 2000
time points each. Each time series was initialized to random values and had an
instantaneous noise term, $\varepsilon_{t}\in\mathcal{N}(0,0.01)$. AR(100)
processes followed the form $X(t)=a_{100}X(t-100)+\varepsilon_{t}$, where, for
each realization, $a_{100}\in U(0.8,0.9)$. For each realization in the set of
ARMA and ARFIMA processes, the AR and MA orders were randomly chosen between 1
and 20 and the AR and MA coefficients were randomly chosen from $U(0.8,0.9)$.
The difference parameter $d$ was randomly chosen from $U(-0.5,0.5)$ for ARFIMA
processes.
It can be seen from the performance metrics in Table 3 that the model worked
extremely well for accurate classification of above discussed causal time-
series with distribution shifts and with the presence of causal influence that
was occurring at very different scales as compared to the influence in the
causal-time series used in the training set.
We also plot figures to illustrate the difference in TTSS features of the
amplitudes at different frequencies for causal and non-causal time series used
in this section. Figures 3(a), 3(b), 3(c) and 3(d) show the figures for causal
time series, AR(15), AR(100), ARMA and ARFIMA respectively. Figures 4(a) and
4(b) show the figures for non-causal time series generated using
$\mathcal{N}(0,0.01)$ and $U(-0.6,0.6)$ respectively. Clearly the number of
peaks and valleys in TTSS feature plots of causal time-series are much lower
than those in TTSS feature plots of non-causal time series.
(a) (b)
(c) (d)
Figure 3: TTSS feature representation for the amplitude at each frequency for
causal time-series: (a)AR(15), (b)AR(100), (c)ARMA and (d)ARFIMA
(a) (b)
Figure 4: TTSS feature representation for the amplitude at each frequency for
non-causal time-series generated from: (a) $\mathcal{N}(0,0.01)$ and (b)
$U(-0.6,0.6)$.
Hyperparameters used for ChaosFEX feature extraction for all datasets in this
section are $q=0.33$, $b=0.499$, $\varepsilon=0.01$ and maximum trajectory
length of firing for each GLS neuron was set to 1000. This means that the GLS
neuron would stop firing after a maximum trajectory length of 1000 if it did
not reach the $\varepsilon$ neighborhood of the stimulus. In [22] and [23],
the authors have found the hyperparameters tuned to $q=0.34$, $b=0.499$ and
$\varepsilon$ in the range $0.005$ to $0.3$, to be the most effective for the
classification of sequential data such as genome sequences and spoken digits.
Thus we fine tuned the hyperparameters by limiting the exploration of the
parameters around this range. Hyperparameter tuning for the LR classifier was
also done in this case. All parameters were set to default values other than
_‘ $max\\_iter$’, ‘tol’_ and _‘C’_ , which were set to $1000,0.001$ and
$0.001$ respectively.
## 5 Discussion, concluding remarks and future research directions
We show that ML classifiers (Logistic regression and LSTM), when used by
themselves directly on time-series measurements are dumb to the temporal/
causal-structure in the data. This fact has also been discussed in existing
literature [1, 2]. When time-series values were directly passed to the
classifiers, LR and LSTM, they failed to learn any causal-structure
characteristics from the data. Even though LSTM has been developed for the
classification of sequential data, it seemed to be learning some statistical
features from the data, giving high classification accuracy when the testing
set followed the same distribution as the training set and failing when there
was a distribution shift in the non-causal time series structure.
Even though characteristics of the fourier transformed time series of causal
and non-causal type are quite different, classifiers LR and LSTM, when
directly used on the frequency domain amplitudes fail to learn the causal
structure. Though they could accurately classify testing set which was exactly
the same as training set, the classification accuracy dropped to about $50\%$,
when there was a change in the distribution of non-causal time series. Almost
all of these time series were being wrongly classified as causal.
TTSS feature representation of the amplitude values at different frequencies
proved to be robust in learning the causal structure. We show that it works in
case of distribution shifts in both causal and non-causal time series and
could learn a generalized structure, identifying the processes correctly
irrespective of the scale at which causal-influence existed in the processes.
Good performance of the developed pipeline on our simple problem illustrates
the power of chaos and of ‘neurochaos’ based hybrid learning architectures for
developing more sophisticated causality based ML architectures as well as in
using ML for causal inference tasks. Since, even a simple linear classifier,
LR, is able to distinguish between TTSS features from causal and non-causal
time series, the strength of this feature seems to lie in the ability to
transform the given data to make it linearly separable. Hence, the developed
pipeline seems to be helpful in strong feature extraction for causal-structure
learning purposes.
Future work would involve more rigorous testing of the TTSS based pipeline
with other datasets having different levels of noise, strength of causal
coefficient and different causal/ non-causal structures. Simulated data with
‘interventional perturbations’ will also be provided to the algorithms to test
their performance on such data. Understanding and classification of these kind
of cause-effect treatment datasets will benefit the most from the application
of the proposed approach. Using classifiers other than LR, for example Support
Vector Machine with a linear kernel, to classify based on the TTSS model will
also be done. Other features based on the chaotic firing trajectory of GLS
neurons in the Chaosnet architecture have also been utilized in hybrid ML
based learning tasks [11, 22]. We would also like to check the performance of
the ChaosFEX based model using features other than the TTSS feature (or GLS
firing rate) for classification of causal and non-causal time-series.
Experiments and analysis will be done to get a better theoretical
understanding of how and why the ChaosFEX model works well for causal-
structure learning tasks. Finally, we would like to apply the developed
technique to recognize real time-series data with causal structure. We would
like to check the performance of the method for cause-effect treatment
estimation tasks for real data, both when the treatment/intervention being
provided is a discrete event or a continuous event. There are still not many
well developed techniques available for the latter purpose [5] and the
proposed pipeline seems like a promising approach.
## Acknowledgment
The authors are thankful to Harikrishnan N.B., National Institute of Advanced
Studies, for providing help with the use of ChaosFEX toolbox. N. Nagaraj
gratefully acknowledges the financial support of Tata Trusts and Dept. of
Science and Tech., Govt. of India (grant no. DST/CSRI/2017/54). A. Kathpalia
acknowledges the financial support of the Czech Science Foundation, Project
No. GA19-16066S and the Czech Academy of Sciences, Praemium Academiae awarded
to M. Paluš.
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|
# Frequency Limited $\mathcal{H}_{2}$ Optimal Model Reduction of Large-Scale
Sparse Dynamical Systems
Xin Du School of Mechatronic Engineering and Automation, Shanghai University,
Shanghai-200072, China and Key Laboratory of Modern Power System Simulation
and Control & Renewable Energy Technology, Ministry of Education(Northeast
Electric Power University), Jilin-132012, China<EMAIL_ADDRESS>M. Monir
Uddin Department of Mathematics and Physics, North south University,
Dhaka-1229, Bangladesh<EMAIL_ADDRESS>A. Mostakim Fony
Department of Mathematics, Chittagong University, Chittagong, Bangladesh,
<EMAIL_ADDRESS>Md. Tanzim Hossain Department of Electrical and
Computer Engineering, North South University, Dhaka-1229, Bangladesh,
<EMAIL_ADDRESS>Mohammaed Sahadat-Hossain Department of
Mathematics and Physics, North south University, Dhaka-1229, Bangladesh,
<EMAIL_ADDRESS>
###### Abstract
We mainly consider the frequency limited $\mathcal{H}_{2}$ optimal model order
reduction of large-scale sparse generalized systems. For this purpose we need
to solve two Sylvester equations. This paper proposes efficient algorithm to
solve them efficiently. The ideas are also generalized to index-1 descriptor
systems. Numerical experiments are carried out using Python Programming
Language and the results are presented to demonstrate the approximation
accuracy and computational efficiency of the proposed techniques.
keywords Frequency limited model reduction, $\mathcal{H}_{2}$ optimal
condition, frequency limited Graminas, Sylvester and Lyapunov equations
## 1 Introduction
Model order reduction (MOR) is a process to approximate a high-order dynamical
system by a substantially low-order system with a maximum accuracy. This tool
is now widely used in different disciplines of science, engineering and
technology to reduce the complexity of the model. In general, the reduced
order models are used in controller design, simulation and optimization. For
motivation, applications and techniques of MOR see, e.g,. [1, 2].
The commonly useful methods for the model reduction of large-scale linear time
invariant dynamical systems are the balanced truncation and $\mathcal{H}_{2}$
optimal model reduction [1]. Both the methods are well established and
successfully investigated to find the model reduction of large-scale sparse
dynamical systems. In the recent times frequency and time limited model
reduction methods have taken a lot of attentions due to its demand in real-
life applications. In many applications, a specific frequency interval is more
important, i.e., the ROM should maintain a superior accuracy within that
desired frequency interval. Balanced truncation based frequency limited model
reduction was discussed by Gawronski and Juang in [3]. The computational
techniques for the time and frequency limited balanced truncation were
discussed in [4].
The optimal $\mathcal{H}_{2}$ model reduction methods have been studied and
investigated in [5, 6, 7, 8, 9]. See, also the references cited therein. In
all these papers the proposed technique is based on either Gramian assistance
first-order optimality conditions [5] or the tangential interpolaton [6] of
the transfer function. In fact both the conditions are coincided which is
shown in [9]. These papers only discuss the model reduction on the infinite
frequency interval. For the time limited case the we refer the readers to
[10]. Although, in [11] authors briefly introduced optimal $\mathcal{H}_{2}$
model reduction problem of standard state space systems considering a
restricted frequency interval, there the implementation details were not
given.
This paper focuses on the computational techniques of the frequency limited
optimal $\mathcal{H}_{2}$ model reduction method of large-scale sparse
systems. We mainly generalized the idea as in [12, 9] in which the proposed
algorithm is called two sided iteration algorithm (TSIA). Moreover, to
implement the frequency limited TSIA we need to solve two frequency limited
Sylvester equations. This paper also discusses how to solve the Sylvester
equations efficiently by preserving the sparsity of the system. Besides the
generalized systems the idea is also extended for index-1 descriptor systems.
The benefits of the algorithmic improvements presented in this paper are
illustrated by several numerical examples. We have generated the the results
by using Python Programming Language. Rest of this paper is organized as
follows.
Section 2 overview the TSIA and the optimal $\mathcal{H}_{2}$ model reduction
of generalized system. Then the ideas of this section are discussed in the
next sections for the frequency-limited model order reduction. Section 5
presents the algorithm for solving frequency limited Sylvesters equations
which provides projectors to carry out the FLMOR. The results of the numerical
experiments are depicted in Section 6 which show the efficiency and capability
of the proposed methods.
## 2 The TSIA and $\mathcal{H}_{2}$ optimal model order reduction of
generalized Systems
The goal of this section is to review the basic idea of $\mathcal{H}_{2}$
optimal model order reduction of generalized Systems. Let us consider a linear
time invariant continuous-time system of the form
$\displaystyle E\dot{x}(t)$ $\displaystyle=Ax(t)+Bu(t),$ (1) $\displaystyle
y(t)$ $\displaystyle=Cx(t)+Du(t),$
where $E\in\mathbb{R}^{n\times n}$ is non-singular, and
$A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times p}$,
$C\in\mathbb{R}^{m\times n}$ and ${D}\in\mathbb{R}^{m\times p}$. The transfer-
function matrix of this system is defined by $\text{G}(s)=C(sE-A)^{-1}B+D$,
where $s\in\mathbb{C}$. The controllability and the observability Gramians of
the system on the infinite frequency range can be defined as [13]
$\displaystyle P$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}(\imath\omega
E-A)^{-1}BB^{T}(\imath\omega E^{T}-A^{T})\mathop{d}\omega\quad\text{and}$
$\displaystyle Q$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}(\imath\omega
E^{T}-A^{T})^{-1}C^{T}C(\imath\omega E-A)\mathop{d}\omega,$
and they are the solutions of the continuous-time algebraic Lyapunov equations
$\displaystyle APE^{T}+EPA^{T}+BB^{T}=0\quad\text{and}$ (2) $\displaystyle
A^{T}QE+E^{T}QA+C^{T}C=0,$
respectively. We want to construct a substantially reduced-order model
$\displaystyle\dot{\hat{x}}(t)$ $\displaystyle=\hat{A}\hat{x}(t)+\hat{B}u(t),$
(3) $\displaystyle\hat{y}(t)$ $\displaystyle=\hat{C}\hat{x}(t)+\hat{D}u(t),$
where $\hat{A}\in\mathbb{R}^{r\times r}$, $\hat{B}\in\mathbb{R}^{r\times p}$,
$\hat{C}\in\mathbb{R}^{m\times r}$ and $\hat{D}\in\mathbb{R}^{m\times p}$. The
goal is to minimize the error
$\displaystyle\Xi=\|\text{G}-\hat{\text{G}}\|_{\mathcal{H}_{2}},$ (4)
where $\|.\|_{\mathcal{H}_{2}}$ denotes the system’s $\mathcal{H}_{2}$-norm
[1] and $\hat{G}(s)=\hat{C}(sI-\hat{A})^{-1}\hat{B}+\hat{D}$ is the transfer-
function matrix of the reduced system. The $\mathcal{H}_{2}$-norm of the error
system as defined in (4) can be measured by
$\displaystyle\Xi=:\sqrt{\mathrm{Tr}\left(C_{\Xi}P_{\Xi}C_{\Xi}^{T}\right)}\quad\text{or}\quad\sqrt{\mathrm{Tr}\left(B_{\Xi}^{T}Q_{\Xi}B_{\Xi}\right)},$
(5)
where $P_{\Xi}$ and $Q_{\Xi}$ are the solutions of the Lyapunov equations
$\displaystyle
A_{\Xi}P_{\Xi}E_{\Xi}^{T}+E_{\Xi}P_{\Xi}A_{\Xi}^{T}+B_{\Xi}B_{\Xi}^{T}=0\quad\text{and}$
(6a) $\displaystyle
A_{\Xi}^{T}Q_{\Xi}E_{\Xi}+E_{\Xi}^{T}Q_{\Xi}A_{\Xi}+C_{\Xi}^{T}C_{\Xi}=0,$
(6b)
in which
$\displaystyle E_{\Xi}=\begin{bmatrix}E&\\\ &\hat{I}\end{bmatrix},\quad
A_{\Xi}=\begin{bmatrix}A&\\\ &\hat{A}\end{bmatrix},\quad
B_{\Xi}=\begin{bmatrix}B\\\ \hat{B}\end{bmatrix}\quad\text{and}\quad
C_{\Xi}=\begin{bmatrix}C&-\hat{C}\end{bmatrix}.$
Now, partitioning $P_{\Xi}$ and $Q_{\Xi}$ as
$\displaystyle P_{\Xi}=\begin{bmatrix}P&M\\\
M^{T}&\hat{P}\end{bmatrix}\quad\text{and}\quad Q_{\Xi}=\begin{bmatrix}Q&N\\\
N^{T}&\hat{Q}\end{bmatrix},\,\text{respectively}$
and plugging into (6) we obtain the following algebraic matrix equations
$\displaystyle APE^{T}+EPA^{T}+BB^{T}$ $\displaystyle=0,$ (7a)
$\displaystyle\hat{A}\hat{P}+\hat{P}\hat{A}^{T}+\hat{B}\hat{B}^{T}$
$\displaystyle=0,$ (7b) $\displaystyle AM+EM\hat{A}^{T}+B\hat{B}^{T}$
$\displaystyle=0,$ (7c) $\displaystyle A^{T}QE+E^{T}QA+C^{T}C$
$\displaystyle=0,$ (7d)
$\displaystyle\hat{A}^{T}\hat{Q}+\hat{Q}\hat{A}+\hat{C}^{T}\hat{C}$
$\displaystyle=0,$ (7e) $\displaystyle A^{T}N+E^{T}N\hat{A}-C^{T}\hat{C}$
$\displaystyle=0,$ (7f)
where $\hat{P}$ and $\hat{Q}$ are respectively known as the controllability
and observability Gramians of the reduced systems. Therefore, the
$\mathcal{H}_{2}$ norm of the error system in (5) can be measured by
$\Xi=:\begin{cases}\sqrt{{\mathrm{Tr}\left(CPC^{T}\right)}+{\mathrm{Tr}\left(\hat{C}\hat{P}\hat{C}^{T}\right)}+2{\mathrm{Tr}\left(CM\hat{C}^{T}\right)}}\\\
\quad\text{or}\quad\\\
\sqrt{{\mathrm{Tr}\left(B^{T}QB\right)}+{\mathrm{Tr}\left(\hat{B}^{T}\hat{Q}\hat{B}\right)}+2{\mathrm{Tr}\left(B^{T}N\hat{B}\right)}}.\end{cases}$
(8)
The first-order optimality conditions for the optimal $\mathcal{H}_{2}$ model
reduction was given in [14], which is known as Wilson conditions. Wilson
conditions are based on the first derivatives of (4) with respect to
$\hat{A}$, $\hat{B}$ and $\hat{C}$ as follows:
$\displaystyle\nabla\Xi_{\hat{A}}=2(\hat{Q}\hat{P}+W^{T}EV),\,\nabla\Xi_{\hat{B}}=2(\hat{Q}\hat{B}+W^{T}E^{-1}B),\,\nabla\Xi_{\hat{C}}=2(\hat{C}\hat{P}-CV).$
Setting these three derivatives to zero leads to the _Wilson conditions_ ,
$\displaystyle\hat{Q}\hat{P}+N^{T}EM=0,$ (9)
$\displaystyle\hat{Q}\hat{B}+N^{T}E^{-1}B=0,$ (10)
$\displaystyle\hat{C}\hat{P}-CM=0.$ (11)
These three conditions in fact yield the left and and right projection
matrices to compute an optimal reduced order system (3) and in the optimal
reduced order system the reduced matrices are formed as
$\displaystyle\hat{A}=W^{T}E^{-1}AV,\quad\hat{B}=W^{T}B,\quad\,\text{and}\quad\hat{C}=CV,$
(12)
where $V=M\hat{P}^{-1}$ and $W^{T}=-\hat{Q}^{-1}N^{T}$ and hence it can be
proved that $W^{T}EV=I$. However, we can not guarantee that $\hat{P}$ and
$\hat{Q}$ are invertible, since to assure this the reduced model should be
completely controllable and observable [1]. In the case that they are
invertible, the multiplication from the right is only a transformation of
bases and does not change the subspace. The idea by Xu and Zeng [9] was to
satisfy the Wilson conditions by setting
$\displaystyle W=N\quad\text{and}\quad V=M.$ (13)
Note that $V$ and $W$ can be computed by solving the matrix equations (7c) and
(7f), respectively which are known as Sylvesters equations. Another important
observation is that if we want to compute the optimal projection subspace we
already need the optimal solution $\hat{A}$, $\hat{B}$ and $\hat{C}$. However,
this is not known prior. A possible solution is to start with a reduced model,
which emerged from an arbitrary projection of the original model, solve matrix
equations (7c) and (7c), compute the projectors, and restart the process with
the newly obtained reduced model until we are satisfied. In this way we get a
kind of a fixed point iteration. This procedure is called two sided iteration
algorithm (TSIA) by Xu and Zeng in [9].
The Wilson conditions are Gramian-based conditions since they are related to
Gramians of the system. The Hyland-Bernstein conditions [7] are another
gramian-based first-order optimal conditions, which were shown to be
equivalent to the Wilson conditions [15]. Van Dooren et al., were
characterized the tangential interpolation based $\mathcal{H}_{2}$ optimal
conditions in [12]. One drawback of interpolation based model reduction is to
selection of interpolation points. However in [15] authors proposed the
Iterative Rational Krylov Algorithms (IRKA) to resolve this problem. On the
other hand Xu and Zeng showed in [9] that both the Gramian and interpolation
based optimality conditions are the same. In [9] authors also presented the
two sided iterative algorithm (TSIA) for a standard system which is slightly
modified as in [12]. For our convenient the TSIA for the $\mathcal{H}_{2}$
optimal model reduction for the generalized system (1) is summarized in
Algorithm 1
Input : $E,A,B,C,D$.
Output : $\hat{A},\hat{B},\hat{C}$, $\hat{D}:=D$.
1 Choose matrices $W_{0}\in\mathbb{R}^{n\times r}$ and
$V_{0}\in\mathbb{R}^{n\times r}$ such that $W_{0}^{T}V_{0}=I$.
2 Construct the reduced-order matrices:
$\hat{A}=W_{0}^{T}E^{-1}AV_{0},\,\hat{B}=W_{0}^{T}E^{-1}B\quad\text{and}\quad\hat{C}=CV_{0}$
.
3 while _$i\leq N-1$_ do
4 Compute $V_{i}$ and $W_{i}$ by solving Sylvesters $\displaystyle
AV+EV\hat{A}^{T}+B\hat{B}^{T}=0$ (14a) $\displaystyle
A^{T}W+E^{T}W\hat{A}-C^{T}\hat{C}=0,$ (14b)
5 Compute $W_{i+1}=W_{i}(V_{i}^{T}W_{i})^{-1}$ and $V_{i+1}=V_{i}$
6 Construct the reduced-order matrices
$\hat{A}=W_{i+1}^{T}E^{-1}AV_{i+1},\hat{B}=W_{i+1}^{T}E^{-1}B$ and
$\hat{C}=CV_{i+1}$.
7 $i=i+1$.
8 end while
Algorithm 1 Two-sided iteration algorithm (TSIA).
## 3 FL $\mathcal{H}_{2}$ optimal MOR of generalized systems
This section moves to the frequency limited $\mathcal{H}_{2}$ optimal model
reduction of the system (1). For this purpose we first define frequency
limited Gramians. In the previous section the system Gramians have been
defined on the infinite frequency interval. If we replace the infinite
interval (-$\infty$, $\infty$) into a finite interval $\omega=[\omega_{1}$,
$\omega_{2}]$) then the controlability and the observablity Gramians can be
defined as
$\displaystyle P_{\omega}$
$\displaystyle=\frac{1}{2\pi}\int_{\omega_{1}}^{\omega_{2}}(\imath\nu
E-A)^{-1}BB^{T}(\imath\nu E^{T}-A^{T})\mathop{d}\nu$ (15) $\displaystyle
Q_{\omega}$
$\displaystyle=\frac{1}{2\pi}\int_{\omega_{1}}^{\omega_{2}}(\imath\nu
E^{T}-A^{T})^{-1}C^{T}C(\imath\nu E-A)\mathop{d}\nu,$ (16)
which satisfies the frequency limited Lyapunov equations
$\displaystyle
AP_{\omega}E^{T}+EP_{\omega}A^{T}+B_{\omega}BB^{T}+BB^{T}B_{\omega}^{\ast}=0,$
(17) $\displaystyle
A^{T}Q_{\omega}E+E^{T}Q_{\omega}A+C_{\omega}^{\ast}C^{T}C+C^{T}CC_{\omega}=0,$
with
$\displaystyle
B_{\omega}=\frac{\imath}{2\pi}\ln(A+\imath\omega_{2}E)(A+\imath\omega_{1}E)^{-1}\,\,\text{and}$
$\displaystyle
C_{\omega}=\frac{\imath}{2\pi}\ln(A+\imath\omega_{1}E)^{-1}(A+\imath\omega_{2}E).$
The goal in this paper is to constructed a reduced model
$\hat{G}=(\hat{A},\hat{B},\hat{C})$ from a given model $G=(E,A,B,C)$ that
minimizes the error
$\displaystyle\Xi_{\omega}=\|\text{G}-\hat{\text{G}}\|_{\mathcal{H}_{2,\omega}},$
(18)
where $\|.\|_{\mathcal{H}_{2,\omega}}$ denotes the $\mathcal{H}_{2}$-norm on
the prescribed frequency range $\omega$. The transfer-function matrix of the
reduced system is The $\mathcal{H}_{2}$-norm of the error system as defined in
(4) can be measured efficiently by
$\displaystyle\Xi_{\omega}=:\sqrt{\mathrm{Tr}\left(C_{\Xi}P_{\Xi,\omega}C_{\Xi}^{T}\right)}\quad\text{or}\quad\sqrt{\mathrm{Tr}\left(B_{\Xi}^{T}Q_{\Xi,\omega}B_{\Xi}\right)},$
(19)
where $P_{\Xi,\omega}$ and $Q_{\Xi,\omega}$ are the solutions of the Lyapunov
equations
$\displaystyle
A_{\Xi}P_{\Xi,\omega}E_{\Xi}^{T}+E_{\Xi}P_{\Xi,\omega}A_{\Xi}^{T}+B_{\Xi,\omega}B_{\Xi}B_{\Xi}^{T}+B_{\Xi}B_{\Xi}^{T}B_{\Xi,\omega}^{\ast}=0\quad\text{and}$
(20a) $\displaystyle
A_{\Xi}^{T}Q_{\Xi,\omega}E_{\Xi}+E_{\Xi}^{T}Q_{\Xi,\omega}A_{\Xi}+C_{\Xi,\omega}^{\ast}C_{\Xi}^{T}C_{\Xi}+C_{\Xi}^{T}C_{\Xi}C_{\Xi,\omega}=0,$
(20b)
where
$\displaystyle
B_{\Xi,\omega}=\frac{\imath}{2\pi}\ln(A_{\Xi}+\imath\omega_{2}E_{\Xi})(A_{\Xi}+\imath\omega_{1}E_{\Xi})^{-1}\,\,\text{and}$
$\displaystyle
C_{\Xi,\omega}=\frac{\imath}{2\pi}\ln(A_{\Xi}+\imath\omega_{1}E_{\Xi})^{-1}(A_{\Xi}+\imath\omega_{2}E_{\Xi}).$
Due to the structure of $E_{\Xi}$, $A_{\Xi}$, $B_{\Xi}$ and $C_{\Xi}$ we can
Partition $P_{\Xi,\omega}$, $Q_{\Xi,\omega}$, $B_{\Xi,\omega}$ and
$C_{\Xi,\omega}$ as follows
$\displaystyle P_{\Xi,\omega}=\begin{bmatrix}P_{\omega}&M_{\omega}\\\
M_{\omega}^{T}&\hat{P}_{\omega}\end{bmatrix},\quad
Q_{\Xi,\omega}=\begin{bmatrix}Q_{\omega}&N_{\omega}\\\
N_{\omega}^{T}&\hat{Q}_{\omega}\end{bmatrix},$ $\displaystyle
B_{\Xi,\omega}=\begin{bmatrix}B_{\omega}&0\\\
0&\hat{B}_{\omega}\end{bmatrix},\qquad
C_{\Xi,\omega}=\begin{bmatrix}C_{\omega}&0\\\
0&\hat{C}_{\omega}\end{bmatrix}.$
Therefore (20) yields
$\displaystyle
AP_{\omega}E^{T}+EP_{\omega}A^{T}+B_{\omega}BB^{T}+BB^{T}B_{\omega}^{\ast}=0,$
(21a)
$\displaystyle\hat{A}\hat{P}_{\omega}+\hat{P}_{\omega}\hat{A}^{T}+\hat{B}_{\omega}\hat{B}\hat{B}^{T}+\hat{B}\hat{B}^{T}\hat{B}_{\omega}^{\ast}=0,$
(21b) $\displaystyle
AM_{\omega}+EM_{\omega}\hat{A}^{T}+B_{\omega}B\hat{B}^{T}+B\hat{B}^{T}\hat{B}_{\omega}^{\ast}=0,$
(21c) $\displaystyle
AQ_{\omega}E^{T}+EQ_{\omega}A^{T}+C_{\omega}^{\ast}C^{T}C+C^{T}CC_{\omega}=0,$
(21d)
$\displaystyle\hat{A}\hat{Q}_{\omega}+\hat{Q}_{\omega}\hat{A}^{T}+\hat{C}_{\omega}^{\ast}\hat{C}^{T}\hat{C}+\hat{C}^{T}\hat{C}\hat{C}_{\omega}=0,$
(21e) $\displaystyle
AN_{\omega}+EN_{\omega}\hat{A}^{T}+C_{\omega}^{\ast}C^{T}\hat{C}-C^{T}\hat{C}\hat{C}_{\omega}=0,$
(21f)
with
$\displaystyle\hat{B}_{\omega}=\frac{\imath}{2\pi}\ln(\hat{A}+\imath\omega_{2}I)(\hat{A}+\imath\omega_{1}I)^{-1}\,\,\text{and}$
$\displaystyle\hat{C}_{\omega}=\frac{\imath}{2\pi}\ln(\hat{A}+\imath\omega_{1}I)^{-1}(\hat{A}+\imath\omega_{2}I).$
Following the discussion in the above section here we also construct reduced
order model by constructing the reduced matrices as in (12). We solve the the
Sylvesters equations (21c) and (21f) to construct $V=M_{\omega}$ and
$W=N_{\omega}$. The constructed reduced order model is $\mathcal{H}_{2}$
optimal on the limited frequency range and satisfies Wilson’s first-order
optimality conditions. The whole procedure is summarized in Algorithm 2. The
main computation tasks in this algorithm is to solve sparse-dense Sylvesters
equations (22a) and (22b). Following section will presents how to solve them
efficiently.
Input : $E,A,B,C$.
Output : $\hat{A},\hat{B},\hat{C}$, $\hat{D}:=D$.
1 Choose matrices $W_{0}\in\mathbb{R}^{n\times r}$ and
$V_{0}\in\mathbb{R}^{n\times r}$ such that $W_{0}^{T}V_{0}=I$.
2 Construct the reduced-order matrices
$\hat{A}=W_{0}^{T}E^{-1}AV_{0},\hat{B}=W_{0}^{T}E^{-1}B$ and $\hat{C}=CV_{0}$.
3 while _( $i\leq N-1$)_ do
4 Compute $V_{i}=M_{\omega}$ and $W_{i}=N_{\omega}$ by solving the Sylvester
$\displaystyle
AM_{\omega}+EM_{\omega}\hat{A}^{T}+B_{\omega}B\hat{B}^{T}+B\hat{B}^{T}\hat{B}_{\omega}^{\ast}=0,$
(22a) $\displaystyle
AN_{\omega}+EN_{\omega}\hat{A}^{T}+C_{\omega}^{\ast}C^{T}\hat{C}-C^{T}\hat{C}\hat{C}_{\omega}=0,$
(22b) Compute $W_{i+1}=W_{i}(V_{i}^{T}W_{i})^{-1}$ and $V_{i+1}=V_{i}$.
5 Construct the reduced-order matrices
$\hat{A}=W_{i+1}^{T}E^{-1}AV_{i+1},\hat{B}=W_{i+1}^{T}E^{-1}B$ and
$\hat{C}=CV_{i+1}$.
6 $i=i+1$.
7 end while
Algorithm 2 Two-sided iteration algorithm (TSIA).
## 4 Solution of semi-generalized Sylvester equations
Above section shows that to perform the frequency limited model reduction of
system (1) we need to solve two frequency limited matrix equations namely
Sylvester equations (22a) and (22b). This section discusses how to solve them
efficiently. Since the Sylvester equations (22a) and (22b) are duel of each
other we only interested to elaborate the solution of (22a). Another one can
be solved by applying the same procedure. For our convenient we rewrite the
equation (22a) as
$\displaystyle AX+EX\hat{A}^{T}+F=0,$ (23)
where $F=B_{\omega}BB^{T}+BB^{T}B_{\omega}^{\ast}$ and $X=M_{\omega}$. The
technique that we have followed here was presented in [16] where $E=I$ is an
identity matrix. In [17] authors generalized the idea of [16] for the equation
like (23) where $F=B\hat{B}$.
Considering the _Schur decomposition_ of $\hat{A}$ as $QSQ^{\ast}$ such that
$QQ^{\ast}=Q^{\ast}Q=I$ and inserting this into (23) we get
$\displaystyle AX+EXQSQ^{\ast}+F=0.$ (24)
By multiplying this from the right with $Q$ we obtain
$\displaystyle
A\underbrace{XQ}_{\tilde{X}}+E\underbrace{XQ}_{\tilde{X}}S+\underbrace{FQ}_{\tilde{F}}=0,$
(25)
Observing that $S$ is a upper triangular matrices leads to a formula for the
first column of $\tilde{X}$:
$\displaystyle A\tilde{X}_{1}+E\tilde{X}_{1}S_{1,1}+\tilde{F}_{1}=0,$ (26)
$\displaystyle\Leftrightarrow\quad$
$\displaystyle(A+S_{11}E)\tilde{X}_{1}=-\tilde{F}_{1}.$ (27)
For all other columns we have to take care of the linear combination of $E$
matrix. If we consider the second column of the solution
$\displaystyle(A+S_{22}E)\tilde{X}_{2}=-\tilde{F}_{2}-S_{12}E\tilde{X}_{1}.$
(28)
In this way the arbitrary column $j$ of $\tilde{X}$ we find
$\displaystyle(A+S_{jj}E)\tilde{X}_{j}=-\tilde{F}_{j}-E\sum_{i=1}^{j-1}S_{ij}\tilde{X}_{i}.$
(29)
To obtain the solution of the original system we multiply $\tilde{X}$ by
$U^{\ast}$ from the right.
Input : $E,A,\hat{A},F$ from (23).
Output : $X\in\mathbb{R}^{n\times r}$ solution of (23).
1 Compute the Schur decomposition $\hat{A}=QSQ^{\ast}$ and Define
$\tilde{F}=FQ$
2 for _$i=1,\cdots,r$_ do
3 Compute $\hat{F}=-\tilde{F}_{j}-E\sum_{i=1}^{j-1}S_{i,j}\tilde{X}_{i}$
4 Solve $(A+S_{jj}E)\tilde{X}_{j}=\hat{F}$
5 end for
$X=\tilde{X}Z^{\ast}$.
Algorithm 3 Solution of semi-generalized Sylvester equations.
## 5 FLMOR of structured index-1 systems
The index 1 descriptor system that we consider in the section has the
following form.
$\begin{array}[]{rcl}E_{1}\dot{x}(t)&=J_{1}x(t)+J_{2}z(t)+B_{1}u(t)\\\
0&=J_{3}x(t)+J_{4}z(t)+B_{2}u(t)\\\
y(t)&=C_{1}x(t)+C_{2}z(t)+D_{a}u(t),\end{array}$ (30)
Where $x(t)\in\mathbb{R}^{n_{1}}$ is the vector with differential variables
and $z(t)\in\mathbb{R}^{n_{2}}$ is the vector with algebraic variables. Model
reduction of such descriptor system has been discussed in a couple of previous
research articles, e.g., [18, 19, 20, 21, 22, 20, 23, 24] on an unrestricted
frequency limit. In [18] author uses spectral projector to split the system
into finite and infinite parts and balancing based method was applied to the
finite part. Other papers implemented MOR without computing the spectral
projector, rather eliminating the algebraic part the system was converted into
an ODE system. However, practical implementation was carried out without
computing the ODE system explicitly. This paper generalizes this idea for of
the FLMOR discussed in the above section.
By eliminating the algebraic variables i.e., $z(t)\in\mathbb{R}^{n_{2}}$ of
the system we obtain a generalized system (1) where the coefficients matrices
are defined as
$\displaystyle E:=E_{1},\quad A:=J_{1}-J_{2}{J_{4}}^{-1}J_{3},\quad
B:=B_{1}-J_{2}{J_{4}}^{-1}B_{2},$ (31) $\displaystyle
C:=C_{1}-C_{2}{J_{4}}^{-1}J_{3},\quad D:=D_{a}-C_{2}{J_{4}}^{-1}B_{2}.$
The index-1 and generalized systems are equivalent since the responses of the
systems are same and their finite eigenvalues are coincided. Such structured
system are arising in power system model [25]. For the FLMOR of index-1 system
we define $V$ and $W$ by solving the corresponding Sylvesters equations of the
generalized system as discussed in Section 3. Now, applying these
transformations the reduced system matrices can be constructed as:
$\displaystyle\hat{A}:=\hat{J}_{1}-\hat{J}_{2}{J}_{4}^{-1}\hat{J}_{3},\quad\hat{B}:=\hat{B}_{1}-\hat{J}_{2}\hat{J}_{4}^{-1}B_{2},$
(32)
$\displaystyle\hat{C}:=\hat{C}_{1}-C_{2}J_{4}^{-1}\hat{J}_{3},\quad\hat{D}:=D_{a}-C_{2}{J}_{4}^{-1}B_{2},$
where $\hat{J}_{1}=W^{T}E_{1}^{-1}J_{1}V$, $\hat{J}_{2}=W^{T}J_{2}$,
$\hat{J}_{3}=J_{3}V$, $\hat{B}_{1}=W^{T}E_{1}^{-1}B_{1}$ and
$\hat{C}_{1}=C_{1}V$. To compute the the projectors $V$ and $W$ by solving the
corresponding Sylvesters equations is a challenging task since the input
matrices in (32) are highly dense. In the following text we discuss how to
solve the Sylvesters equations related to the index-1 system (30) efficiently.
To solve the Sylvesters equations we can use Algorithm 3. In this algorithm
the main expensive task is to solve a linear system at each iteration step. At
Step 4 of the algorithm we need to solve the linear system
$\displaystyle(A+S_{ii}E)\tilde{X}_{j}=\hat{F}.$
Plugging $A$ and $E$ from (31) we obtain
$\displaystyle(J_{1}-J_{2}J_{4}^{-1}J_{3}+S_{ii}E_{1})\tilde{X}_{j}=\hat{F},$
which can be rewritten as
$\displaystyle(J_{1}+S_{ii}E_{1}-J_{2}J_{4}^{-1}J_{3})\tilde{X}_{j}=\hat{F}.$
(33)
A close observation reveal that instead of this we can solve the following
linear system
$\displaystyle\begin{bmatrix}J_{1}+S_{ii}E_{1}&J_{2}\\\
J_{3}&J_{4}\end{bmatrix}\begin{bmatrix}\tilde{X}_{j}\\\
\Gamma\end{bmatrix}=\begin{bmatrix}\hat{F}\\\ 0\end{bmatrix}$ (34)
for $\tilde{X}_{j}$. Although, linear system in (34) is larger than the system
in (33) it is sparse and hence can be solved by any sparse solvers (e.g.,
direct [26, 27] or iterative e.g., [28, 29]) efficiently.
Input : $E_{1},J_{1},J_{2},J_{3},J_{4},B_{1},B_{2},\hat{A}$.
Output : $X\in\mathbb{R}^{n\times r}$.
1 Form $F=B_{\omega}B\hat{B}^{T}+B\hat{B}^{T}\hat{B}_{\omega}^{\ast}$, where
$\displaystyle
B_{\omega}=\frac{\imath}{2\pi}\ln(J_{1}+\imath\omega_{2}E_{1}-J_{2}J_{4}^{-1}J_{3})(J_{1}+\imath\omega_{1}E_{1}-J_{2}J_{4}^{-1}J_{3})^{-1},$
$\displaystyle\hat{B}_{\omega}=\frac{\imath}{2\pi}\ln(\hat{A}+\imath\omega_{1}I)^{-1}(\hat{A}+\imath\omega_{2}I),\quad\text{and}\quad
B=B_{1}-J_{2}J_{4}^{-1}B_{2}$ Compute the Schur decomposition
$\hat{A}=QSQ^{\ast}$ and Define $\tilde{F}=FQ$
2 for _$i=1,\cdots,r$_ do
3 Compute $\hat{F}=-\tilde{F}_{j}-E_{1}\sum_{i=1}^{j-1}S_{i,j}\tilde{X}_{i}$
4 Solve $\displaystyle\begin{bmatrix}J_{1}+S_{ii}E_{1}&J_{2}\\\
J_{3}&J_{4}\end{bmatrix}\begin{bmatrix}\tilde{X}_{j}\\\
\Gamma\end{bmatrix}=\begin{bmatrix}\hat{F}\\\ 0\end{bmatrix}$ for
$\tilde{X}_{j}$
5 end for
$X=\tilde{X}Z^{\ast}$.
Algorithm 4 Sylvester equation for index-1 system.
## 6 Numerical results
To asses the efficiency of our proposed techniques in this section we discuss
the numerical results. For our convenient we have splitted the section into
several subsections.
### 6.1 Model examples
We consider the following model examples for the numerical experiments.
###### Example 6.1 (International Space Station (ISS)).
This is a model of stage 1R (Russian Service Module) of the ISS. It has n=270
states, p=3 inputs and m=3 outputs. The details of the model can be obtain in
[30].
###### Example 6.2 (Clamped beam model (CBM)).
This structural model was obtained by spatial discretization of an appropriate
partial differential equation (see [31]). The dimension of the model is n=348
and it is a single input single output i.e., SISO system. The input represents
the force applied to the structure at the free end, and the output is the
resulting displacement.
###### Example 6.3 (Triple chain oscillator (TCO) model).
Although this example was originated in [32], the setup was described in [33]
which resulted in second-order system. We convert into firs-order form in
which the dimension of the system is n=10000. It is also an SISO system and
input, output matrices are transpose of each other.
###### Example 6.4 (Power system model).
We consider several Brazilian Power System (BPS) models from [21] which are in
index-1 descriptor form. Table 1 shows number of differential ($n_{1}$) and
algebraic $n_{2}$ variables and inputs/outputs of several models which are
used for the numerical experiments here.
model name | $n_{1}$ | $n_{2}$ | $m/p$
---|---|---|---
BPS-606 | 606 | 1142 |
BPS-1142 | 1142 | 8593 | 4/4
BPS-1450 | 1450 | 9855 |
BPS-1693 | 1693 | b11582 |
Table 1: Dimension of differential and algebraic variables of different
Brazilian power system models.
### 6.2 Setup Hardware and Software
The experiments are carried out with Python 3.7.9 on a board with AMD
$\text{Ryzen}^{\text{TM}}$ $\text{Threadripper}^{\text{TM}}$ 1920X 12-Core
Processor with a 3.5 GHz clock speed and 128 GB RAM.
### 6.3 Error analysis of reduce-order model
The proposed techniques was applied to the all model examples mentioned above.
For the ISS, CBM, TCO we apply Algorithm 2 to obtain frequency limited reduced
order model. On the other for the BPS models we have applied the techniques
discussed in Section 5. We have computed different dimensional reduced order
models for different model examples which are mentioned in Table 2. The table
also shows $\mathcal{H}_{2}$ norm of the error systems in both frequency
restricted and unrestricted cases. For all the models frequency restricted
reduced order model show much more better accuracy than the frequency
frequency underused ones within the assigned frequency intervals.
Model | $r$ | $\Xi_{\omega}$ | $\Xi$
---|---|---|---
ISS | 30 | 7.3244$\times 10^{-8}$ | 2.8000$\times 10^{-3}$
TCO | 30 | 3.4850$\times 10^{-4}$ | 9.6000$\times 10^{-3}$
BPS-606 | 30 | 7.8992$\times 10^{-4}$ | 1.0600$\times 10^{-2}$
BPS-1142 | 35 | 3.4421$\times 10^{-5}$ | 5.3000$\times 10^{-3}$
BPS-1450 | 35 | 9.3818$\times 10^{-8}$ | 1.5063$\times 10^{-5}$
BPS-1693 | 45 | 4.2995$\times 10^{-6}$ | 1.3000 $\times 10^{-3}$
Table 2: Dimension of reduced order models and $\mathcal{H}_{2}$ norms of the
error systems with and without limited frequency intervals.
We also have investigated the frequency domain analysis including the errors
of the original and reduced order models by using sigma plots. Exemplary, we
have depicted the frequency responses of TCO and BPS models only. Figures 1
and 2 show comparisons of the frequency responses of the of the original and
the reduced order models of TCO and BPS models, respectively . In both the
figures from absolute and the relative errors we observe that frequency
restricted reduced order models approximate with the original models with
higher accuracy within on the prescribed frequency ranges.
$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$\displaystyle\omega$$\sigma{}_{\text{max}}\text{(G(j}\omega\text{))}$Full
modelFrequency unrestrictedFrequency restricted (a) Sigma plot
$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$10^{-4}$$10^{-1}$$10^{2}$$\displaystyle\omega$$\sigma{}_{\text{max}}\text{(G(j}\omega\text{)-}\hat{\text{G}}\text{(j}\omega\text{))}$
(b) Absolute error
$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$10^{-6}$$10^{-4}$$10^{-2}$$10^{0}$$\displaystyle\omega$$\frac{\sigma{}_{\text{max}}\text{(G(j}\omega\text{)-}{\hat{\text{G}}}\text{(j}\omega\text{))}}{\sigma{}_{\text{max}}\text{(G(j}\omega\text{))}}$
(c) Relative error
Figure 1: Comparison of original and 30 dimensional reduced systems on the
frequency range [1,2] for TCO.
$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$10^{1.5}$$10^{2}$$\displaystyle\omega$$\sigma{}_{\text{max}}\text{(G(j}\omega\text{))}$Full
modelFrequency unrestrictedFrequency restricted (a) Sigma plot
$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$\displaystyle\omega$$\sigma{}_{\text{max}}\text{(G(j}\omega\text{)-}\hat{\text{G}}\text{(j}\omega\text{))}$
(b) Absolute error
$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$10^{-6}$$10^{-4}$$10^{-2}$$10^{0}$$\displaystyle\omega$$\frac{\sigma{}_{\text{max}}\text{(G(j}\omega\text{)-}{\hat{\text{G}}}\text{(j}\omega\text{))}}{\sigma{}_{\text{max}}\text{(G(j}\omega\text{))}}$
(c) Relative error
Figure 2: Comparison of original and 45 dimensional reduced systems on the
frequency range [6,10] for BPS-1693.
### 6.4 Comparison of sparse and dense system
We already mentioned that in the TSIA the main computational task is solving
two Sylvester equations. This paper presents Algorithm 4 to solve the
Sylvester equations efficiently for index-1 system. We know that by converting
index-1 system into a generalized system we can solve the Sylvester equation
using Algorithm 3. Figure 3 shows the time comparisons of Algorithms 3 and 4
for solving Sylvester equations of index-1 system. We see that if the
dimension of the system is increased then the computational times for
Algorithms 3 is increased rapidly. On the other hand the computational time
with Algorithm 4 is nominal in compare to Algorithms 3.
BPS-606BPS-1142BPS-1450BPS-1693$0$$1$$2$$3$$4$$5$$6$$7$$8$0.621.943.197.740.530.730.871.16time(sec)Algorithm
3Algorithm 4 Figure 3: Comparison of computational time to solve the Sylvester
equation for different dimensional BPS model using Algorithms 3 and 4.
## 7 Conclusions
In this paper we have discussed the frequency limited $\mathcal{H}_{2}$
optimal model order reduction of large-scale sparse dynamical systems. For
this purpose we have solved two mixed (Sparse and dense) Sylvester equations
which are dual of each other. We have shown that how to solve the Sylvester
equations efficiently without loosing the sparsity of large sparse system
matrices. The ideas are also generalized to index-1 descriptor systems.
Index-1 system can be converted into a generalized system by eliminating the
algebraic equations, which however convert the system from sparse to dense. We
have discussed how to perform the model reduction without converting the dense
system explicitly. Numerical experiments has been carried out to demonstrate
the approximation accuracy and computational efficiency of the proposed
algorithm using Python Programming Language.
#### Acknowledgments.
This research work was funded by NSU-CTRG research grant under the project
No.: CTRG-19/SEPS/05. It was also supported by National Natural Science
Foundation of China under Grant No. (61873336, 61873335), the Fundamental
Research Funds for the Central Universities under Grant (FRF-BD-19-002A), and
the High-end foreign expert program of Shanghai University,
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# Coloring hypergraphs with excluded minors
Raphael Steiner
###### Abstract.
Hadwiger’s conjecture, among the most famous open problems in graph theory,
states that every graph that does not contain $K_{t}$ as a minor is properly
$(t-1)$-colorable.
The purpose of this work is to demonstrate that a natural extension of
Hadwiger’s problem to hypergraph coloring exists, and to derive some first
partial results and applications.
Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph
$H_{1}$ is a minor of a hypergraph $H_{2}$, if a hypergraph isomorphic to
$H_{1}$ can be obtained from $H_{2}$ via a finite sequence of the following
operations:
* •
deleting vertices and hyperedges,
* •
contracting a hyperedge (i.e., merging the vertices of the hyperedge into a
single vertex).
First we show that a weak extension of Hadwiger’s conjecture to hypergraphs
holds true: For every $t\geq 1$, there exists a finite (smallest) integer
$h(t)$ such that every hypergraph with no $K_{t}$-minor is $h(t)$-colorable,
and we prove
$\left\lceil\frac{3}{2}(t-1)\right\rceil\leq h(t)\leq 2g(t)$
where $g(t)$ denotes the maximum chromatic number of graphs with no
$K_{t}$-minor. Using the recent result by Delcourt and Postle that
$g(t)=O(t\log\log t)$, this yields $h(t)=O(t\log\log t)$.
We further conjecture that $h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$,
i.e., that every hypergraph with no $K_{t}$-minor is
$\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable for all $t\geq 1$, and
prove this conjecture for all hypergraphs with independence number at most
$2$.
By considering special classes of hypergraphs, the above additionally has some
interesting applications for ordinary graph coloring, such as:
* •
every graph $G$ is $O(kt\log\log t)$-colorable or contains a $K_{t}$-minor
model all whose branch-sets are $k$-edge-connected,
* •
every graph $G$ is $O(qt\log\log t)$-colorable or contains a $K_{t}$-minor
model all whose branch-sets are modulo-$q$-connected (i.e., every pair of
vertices in the same branch-set has a connecting path of prescribed length
modulo $q$),
* •
by considering cycle hypergraphs of digraphs, we obtain known results on
strong minors in digraphs with large dichromatic number as special cases. We
also construct digraphs with dichromatic number
$\left\lceil\frac{3}{2}(t-1)\right\rceil$ not containing the complete digraph
on $t$ vertices as a strong minor, thus answering a question by Mészáros and
the author in the negative.
Department of Computer Science, Institute of Theoretical Computer Science, ETH
Zürich, Switzerland<EMAIL_ADDRESS>The author was
supported by an ETH Zurich Postdoctoral Fellowship.
## 1\. Preliminaries
In this short first section we introduce essential terminology and notation
used throughout the paper. The reader familiar with hypergraphs may want to
skip this technical section and consult it at a later stage should there be
any unclarities.
A _hypergraph_ is a tuple $(V,E)$ consisting of a finite set $V$ of vertices
and a set $E\subseteq 2^{V}\setminus\\{\emptyset\\}$ of hyperedges. Throughout
this paper, all hypergraphs and graphs considered are assumed to have only
hyperedges of size at least $2$, i.e., they are loopless. We refer to a
hyperedge $e\in E$ as a _graph edge_ if it is of size $2$.
We denote $V(H):=V$, $E(H):=E$ for the vertex- and edge-set of a hypergraph.
Given two hypergraphs $H_{1}$ and $H_{2}$, we say that $H_{1}$ is a
_subhypergraph of $H_{2}$_, in symbols, $H_{1}\subseteq H_{2}$, if
$V(H_{1})\subseteq V(H_{2})$ and $E(H_{1})\subseteq E(H_{2})$. We further say
that $H_{1}$ is a _proper_ subhypergraph if at least one of the two inclusions
above is strict.
Given a hypergraph $H$ and a set $W\subseteq V(H)$, we denote by $H[W]$ the
_subhypergraph of $H$ induced by $W$_, whose vertex-set is $W$ and whose edge-
set is $\\{e\in E(H)|e\subseteq W\\}$. For a hypergraph $H$ and an edge $e\in
E(H)$ we write $H-e:=(V(H),E(H)\setminus\\{e\\})$ for the subhypergraph
obtained by deleting $e$. Similarly, for a vertex $v\in V(H)$ we denote by
$H-v:=H[V(H)\setminus\\{v\\}]$ the induced subhypergraph obtained by deleting
$v$ and all the hyperedges containing $v$. For deleting sets $W\subseteq V(H)$
of vertices or $F\subseteq E(H)$ of hyperedges we use the analogous notation
$H-W:=H[V(H)\setminus W],H-F:=(V(H),E(H)\setminus F)$.
We say that a vertex subset $W$ is _independent_ in $H$, if $H[W]$ contains no
hyperedges, and we denote by $\alpha(H)$ the maximum size of an independent
set in $H$.
A hypergraph $H$ is _connected_ if for every partition of its vertex-set into
non-empty subsets $X$ and $Y$, there exists a hyperedge $e$ in $H$ with $e\cap
X\neq\emptyset\neq e\cap Y$. Equivalently, $H$ is connected if for every pair
$x,y$ of distinct vertices there exists $k\geq 2$, a sequence of vertices
$x_{1},\ldots,x_{k}$ in $H$ and a sequence $e_{1},\ldots,e_{k-1}$ of
hyperedges in $H$ such that $x=x_{1},y=x_{k}$ and $x_{i},x_{i+1}\in e_{i}$ for
all $i\in[k-1]$. We say that a subset $W$ of $V(H)$ is _connected_ in $H$ if
the induced subhypergraph $H[W]$ is connected.
In this manuscript, as usual, a _proper coloring_ of a hypergraph with a
color-set $S$ is defined as a mapping $c:V(H)\rightarrow S$ such that for any
$s\in S$ the color-class $c^{-1}(s)$ is independent in $H$. In other words, we
assign colors from $S$ to the vertices of $H$ such that no hyperedge is
monochromatic. The _chromatic number_ $\chi(H)$ is the minimum possible size
of a color-set that can be used for a proper coloring of $H$. Note that
$\chi(H)$ coincides with the ordinary chromatic number when $H$ is a graph.
For an integer $k$, a hypergraph $H$ is called _$k$ -color-critical_ if
$\chi(H)=k$ but $\chi(H^{\prime})<\chi(H)$ for every proper subhypergraph
$H^{\prime}$ of $H$.
A _fractional coloring_ of a hypergraph $H$ is an assignment of real-valued
weights $w(I)\geq 0$ to all independent vertex sets $I$ in $H$ such that
$\sum_{v\in I}{w(I)}\geq 1$ holds for every $v\in V(H)$. The _weight_ of a
fractional coloring is defined to be the total weight $\sum{w(I)}$, where the
sum is taken over all independent sets in $H$. Finally, the _fractional
chromatic number_ of a hypergraph $H$, denoted by $\chi_{f}(H)$, is the
smallest total weight a fractional coloring of $H$ can achieve (this infimum
is always attained). We refer to [2] for an example of previous usage of the
fractional chromatic number of hypergraphs.
A hypergraph is called _Sperner_ if it does not contain distinct hyperedges
$e,f$ such that $e\subset f$. Given a hypergraph $H$, we denote by
$\text{min}(H)$ the subhypergraph on the same vertex-set as $H$, whose
hyperedges are exactly the inclusion-wise minimal hyperedges of $H$, i.e.,
$e\in E(\text{min}(H))$ if and only if $e\in E(H)$ and there exists no
hyperedge $f\in E(H)\setminus\\{e\\}$ with $f\subset e$. It is easy to check
that a set of vertices is independent in $H$ if and only if it is independent
in $\text{min}(H)$. Also, it is clear by definition that $\text{min}(H)$ is
Sperner. This immediately implies the following.
###### Observation 1.
For every hypergraph $H$, the subhypergraph $\text{min}(H)\subseteq H$ is
Sperner and satisfies $\alpha(H)=\alpha(\text{min}(H))$,
$\chi(H)=\chi(\text{min}(H))$ and $\chi_{f}(H)=\chi_{f}(\text{min}(H))$.
## 2\. Introduction
Recall that given a graph $G_{1}$, another graph $G_{2}$ is a _minor_ of
$G_{1}$ if $G_{2}$ is isomorphic to at least one graph that can be obtained
from $G_{1}$ through a finite sequence of the following operations:
* •
moving to a subgraph,
* •
contracting an edge (i.e., identifying its endpoints into a common vertex, and
identifying parallel edges created by this process afterwards).
Maybe the most important open problem in graph coloring remains Hadwiger’s
conjecture, claiming the following relationship between minor-containment and
the chromatic number:
###### Conjecture 1 (Hadwiger 1943 [14]).
For every integer $t\geq 2$, if $G$ is a graph which does not contain $K_{t}$
as a minor, then $\chi(G)\leq t-1$.
In this paper, we develop and study a hypergraph analogue of Hadwiger’s
conjecture. For this purpose, we generalize the definition of a graph minor to
hypergraph minors in (what seems to the author) the most straightforward way
as follows:
###### Definition 1.
Let $H_{1}$ and $H_{2}$ be hypergraphs. We say that $H_{2}$ is a _minor_ of
$H_{1}$ if $H_{2}$ is isomorphic to at least one hypergraph that can be
obtained from $H_{1}$ via a finite sequence of the following operations:
* •
moving to a subhypergraph,
* •
contracting a hyperedge (i.e., identifying all vertices contained in the
hyperedge and removing loops or parallel hyperedges created by this process
afterwards).
To avoid confusion, let us give a more formal description of a hyperedge-
contraction in the following: Let a hypergraph $H$ and $e\in E(H)$ be given.
Let $v\notin V(H)$ be a “new” vertex, which will serve as the contraction
vertex for $e$. Define a map $\phi:V(H)\rightarrow(V(H)\setminus
e)\cup\\{v\\}$ as follows: $\phi(x):=x$ for every $x\in V(H)\setminus e$, and
$\phi(x):=v$ for every $x\in e$. We may now define the new hypergraph $H/e$
obtained by contracting $e$ as having vertex-set $V(H/e):=(V(H)\setminus
e)\cup\\{v\\}$ and edge-set $E(H/e):=\\{\phi(f)|f\in E(H),f\not\subseteq
e\\}$. Note that, if we start from a hypergraph $H$ which is Sperner, the
contracted hypergraph $H/e$ may no longer be Sperner.
We remark that various other definitions of minors for hypergraphs and
simplicial complexes have been considered in the literature that are not
addressed here, see for instance [1, 6, 17, 26].
For some of our proofs, it will be convenient to take a slightly different
view of graph and hypergraph minors using so-called _branch-sets_ and _minor
models_. We extend these notions (which are well-established in graph minor
theory) to hypergraphs as follows:
###### Definition 2.
Let $H_{1}$ and $H_{2}$ be hypergraphs. An _$H_{2}$ -minor model_ in $H_{1}$
is a collection $(B_{h})_{h\in V(H_{2})}$ of disjoint non-empty sets of
vertices in $H_{1}$, such that:
* •
for every $h\in V(H_{2})$, the set $B_{h}$ is connected in $H_{1}$, and
* •
for every hyperedge $f\in E(H_{2})$, there is a hyperedge $e\in E(H_{1})$ such
that $e\subseteq\bigcup_{h\in f}{B_{h}}$ and $e\cap B_{h}\neq\emptyset$ for
every $h\in f$.
The sets $B_{h},h\in V(H_{2})$ are called the _branch-sets_ of the
$H_{2}$-minor model.
It is easy to see from the definition that if a hypergraph $H_{1}$ contains an
$H_{2}$-minor model, then by contracting in $H_{1}$ all the edges within the
connected branch-sets (in arbitrary order) and potentially deleting some
superfluous edges and vertices afterwards, we obtain $H_{2}$ as a minor of
$H_{1}$. Similarly, it is not hard to see and left for the reader to verify
that if $H_{1}$ contains $H_{2}$ as a minor, then $H_{1}$ contains an
$H_{2}$-minor model.
#### New results.
The main insight of this paper is that the above notion of hypergraph minors
harmonizes well with the established notion of the hypergraph chromatic number
$\chi(H)$, and allows for extending known partial results for Hadwiger’s
conjecture to hypergraph coloring. To discuss these statements more formally,
we use the following notation: Given an integer $t\geq 2$, we denote by $g(t)$
the maximum chromatic number of a graph not containing $K_{t}$ as a minor, and
similarly we denote by $h(t)$ the largest chromatic number of a hypergraph not
containing $K_{t}$ as a minor. Note that at first glance it is far from
obvious why the function $h(t)$ should be well-defined for every value of $t$.
In our first main result below, we show that $h(t)$ exists for every $t\geq 2$
and, quite surprisingly, is tied to the graph function $g(t)$ by a factor of
at most $2$.
###### Theorem 1.
Every $K_{t}$-minor free hypergraph is $2g(t)$-colorable. Equivalently,
$h(t)\leq 2g(t)$.
The same proof idea as for Theorem 1 also works for fractional coloring,
yielding the following analogous result.
###### Proposition 1.
Let $t\geq 2$ be an integer, and let $g_{f}(t)$ be the supremum of the
fractional chromatic numbers taken over all $K_{t}$-minor free graphs. Then
every $K_{t}$-minor free hypergraph $H$ satisfies $\chi_{f}(H)\leq 2g_{f}(t)$.
Hadwiger’s conjecture has been proved for $t\leq 6$, the case $t=6$ was
settled by Robertson, Seymour and Thomas [21]. Thus, $g(t)=t-1$ for $t\leq 6$.
Asymptotically, the best known bound is that $g(t)=O(t\log\log t)$ as proved
by Delcourt and Postle [9]. Finally, it was proved by Reed and Seymour [20]
that $g_{f}(t)\leq 2(t-1)$. Combining these known partial results for
Hadwiger’s conjecture with Theorem 1 and Proposition 1, we get the following
set of immediate consequences.
###### Corollary 2.
Let $t\geq 2$ be an integer, and let $H$ be a hypergraph without a
$K_{t}$-minor. Then the following hold.
* •
if $t\in\\{2,3,4,5,6\\}$, then $\chi(H)\leq 2t-2$.
* •
for $t\geq 3$, we have $\chi(H)\leq Ct\log\log t$, where $C>0$ is some
absolute constant.
* •
$\chi_{f}(H)\leq 4t-4$.
Given that $h(t)$ exists for every integer $t\geq 2$, it is tempting to hope
or conjecture that Hadwiger’s conjecture generalizes one-to-one to
hypergraphs, in the sense that $h(t)=t-1$ for every $t\geq 2$. However, a more
careful thought shows that this is not the case, and that in fact
$h(t)\geq\left\lceil\frac{3}{2}(t-1)\right\rceil$ for every integer $t\geq 2$.
###### Observation 2.
For $t\geq 2$ the complete $3$-uniform hypergraph $H=K_{3(t-1)}^{(3)}$ does
not contain $K_{t}$ as a minor and has chromatic number
$\chi(H)=\left\lceil\frac{3}{2}(t-1)\right\rceil$. Thus,
$h(t)\geq\left\lceil\frac{3}{2}(t-1)\right\rceil$.
###### Proof.
A set of vertices is independent in $H$ if and only if it has size at most
$2$, which immediately yields
$\chi(H)=\left\lceil\frac{3}{2}(t-1)\right\rceil$. Next suppose towards a
contradiction that $H$ contains $K_{t}$ as a minor. Then $H$ must contain a
$K_{t}$-minor model consisting of $t$ disjoint and connected branch-sets
$(B_{i})_{i=1}^{t}$, such that for every distinct $i,j\in[t]$ there is a
hyperedge $e$ of $H$ contained in $B_{i}\cup B_{j}$ with $e\cap
B_{i}\neq\emptyset\neq e\cap B_{j}$. Note that no branch-set $B_{i}$ can be of
size $2$, since no such set is connected in $H$. Furthermore, there can be at
most one branch-set of size $1$: If there were distinct $i,j$ with
$|B_{i}|=|B_{j}|=1$, then $B_{i}\cup B_{j}$ would be too small to host a
hyperedge of $H$. Consequently
$3(t-1)=|V(H)|\geq\sum_{i=1}^{t}{|B_{i}|}\geq 3(t-1)+1,$
a contradiction, and this concludes the proof. ∎
Despite a significant effort, we were not able to find examples providing
better lower bounds on $h(t)$ than the one given by the previous proposition.
This finally leads us towards a hypergraph analogue of Hadwiger’s conjecture
as follows:
###### Conjecture 2.
For every integer $t\geq 2$ we have
$h(t)=\left\lceil\frac{3}{2}(t-1)\right\rceil$. In other words, if a
hypergraph $H$ does not contain $K_{t}$ as a minor, then
$\chi(H)\leq\left\lceil\frac{3}{2}(t-1)\right\rceil$.
Intriguingly,Conjecture 2 remains open even for $K_{3}$-minor-free
hypergraphs. The best upper bound on the chromatic number of $K_{3}$-minor
free hypergraphs we are aware of is $4$, provided by Corollary 2.
###### Problem 1.
Is every $K_{3}$-minor-free hypergraph $3$-colorable?
We remark that independently of our work, van der Zypen [24] has stated an
extension of Hadwiger’s conjecture to hypergraphs. However, his alternative
conjecture uses a much less restricted version of hypergraph minors, hence
leading to a more restricted class of hypergraphs with no $K_{t}$-minor. For
instance, according to the definition in [24], every hypergraph consisting of
a single hyperedge on $t$ vertices would contain $K_{t}$ as a minor. An
argument outlined in [25] shows that for _finite_ values of $t$ van der
Zypen’s conjecture is a consequence of the ordinary Hadwiger’s conjecture for
graphs. Since our focus in this paper is on finite hypergraphs, we will not
discuss this variant in more detail here.
As additional evidence for our Conjecture 2 we verify it for all hypergraphs
without $3$-vertex independent sets. This is a natural special case to look
at, due to the considerable attention that the special case of Hadwiger’s
conjecture for graphs with independence number $2$ has received in the past,
see e.g. [4, 5, 8, 10, 11, 15, 19].
###### Theorem 3.
Let $t\geq 2$ be an integer, and let $H$ be a hypergraph such that
$\alpha(H)\leq 2$. If $H$ does not contain $K_{t}$ as a minor, then
$\chi(H)\leq\left\lceil\frac{3}{2}(t-1)\right\rceil$.
#### Applications.
We present three simple examples of how our bounds for coloring $K_{t}$-minor
free hypergraphs can be applied to produce new or recover known results for
coloring graphs and directed graphs with excluded minor-like substructures. We
believe that more applications of a similar flavour will be found in the
future.
Our first two applications address the following natural question. While
Hadwiger’s conjecture guarantees the existence of a $K_{t}$-minor model with
connected branch-sets in graphs with high chromatic number, in principle these
are not very dense structures, as every branch-set could span a tree only.
However, intuitively, graphs of high chromatic number should also contain
$K_{t}$-minor models with “richer” branch-sets (i.e., denser, of higher
connectivity, etc). Our first result in this direction confirms this intuition
by proving that a very moderate bound on the chromatic number suffices to
guarantee high edge-connectivity in the branch-sets.
###### Theorem 4.
There exists an absolute constant $C>0$ such that for every pair of integers
$k\geq 1,t\geq 3$ every graph $G$ satisfying $\chi(G)>Ckt\log\log t$ contains
a $K_{t}$-minor model with branch-sets $(B_{i})_{i=1}^{t}$, such that for each
$i\in[t]$ the induced subgraph $G[B_{i}]$ is $k$-edge-connected. Furthermore,
for any distinct $i,j\in[t]$ there are $k$ distinct edges in $G$ connecting
$B_{i}$ and $B_{j}$.
In our next application of Theorem 1 we consider another way of strengthening
the connectivity requirement for the branch-sets of a complete minor, by
requiring that distinct vertices in the same branch-set can be connected by a
path of any desired length modulo $q$. For $q=2$, this is somewhat reminiscent
of (but not equivalent to) the recently popular notion of _odd minors_. The
interested reader may consult e.g. [12] for a definition and background.
###### Definition 3.
Let $q\geq 2$ be an integer and $G$ a graph. We say that $G$ is _modulo
$q$-connected_ if for every pair of distinct vertices $u,v\in V(G)$ and every
residue $r\in\\{0,1,\ldots,q-1\\}$ there exists a path $P$ in $G$ with
endpoints $u,v$ and length $\ell(P)\equiv_{q}r$.
###### Theorem 5.
There exists an absolute constant $c>0$ such that for every pair of integers
$q\geq 2$, $t\geq 3$ every graph $G$ satisfying $\chi(G)>Cqt\log\log t$
contains a $K_{t}$-minor model with branch-sets $(B_{i})_{i=1}^{t}$, such that
for each $i\in\\{1,\ldots,t\\}$ the induced subgraph $G[B_{i}]$ is
modulo-$q$-connected.
In our third and last application, we give a short reproof of the main result
from [16] on coloring digraphs with excluded strong complete minors. The
research on this topic was initiated by Axenovich, Girão, Snyder and Weber
[3], and then further addressed by Mészáros and the author [16]. To state the
result, we adopt the following terminology from [3, 16]:
For digraphs $D$ and $F$, we say that _$D$ contains $F$ as a strong minor_, if
there exist disjoint non-empty subsets $(B_{f})_{f\in V(F)}$ of $V(D)$ such
that $B_{f}$ induces a strongly connected subdigraph of $D$ for every $f\in
V(F)$ and for every arc $(f_{1},f_{2})$ in $F$, there is an arc from
$B_{f_{1}}$ to $B_{f_{2}}$ in $D$. The _dichromatic number_ $\vec{\chi}(D)$ of
a digraph $D$ is the smallest number of colors that can be used to color the
vertices of $D$ such that every color class spans an acyclic subdigraph of
$D$.
The following is a main result from [16]. By considering _cycle hypergraphs_
of digraphs, we will show that it is an immediate consequence of Theorem 1.
For $t\geq 1$ we denote by
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$ the complete digraph of
order $t$ (containing all possible $t(t-1)$ arcs).
###### Theorem 6.
If $D$ is a digraph, $t\geq 2$ an integer, and $D$ does not contain
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$ as a strong minor, then
$D$ has dichromatic number at most $h(t)\leq 2g(t)$.
It was raised as an open problem in [16] whether or not every digraph $D$ with
no $\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$ strong minor has
dichromatic number at most $t$. Inspired by our lower bound on $h(t)$ from
Proposition 2, we will actually show that this is _not_ the case:
###### Proposition 2.
For every $t\geq 2$ there is a digraph $D$ with no strong
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$-minor and dichromatic
number at least $\left\lceil\frac{3}{2}(t-1)\right\rceil$.
#### Structure of the paper.
In Section 3 we give the proofs of our main results, including Theorem 1,
Proposition 1 and Theorem 3. In Section 4 we give the formal proofs of the
three applications of these main results to graph and hypergraph coloring,
including the proofs of Theorem 4, Theorem 5, Theorem 6 and Proposition 2.
## 3\. Proofs of Theorem 1, Proposition 1 and Theorem 3
We start by giving the (joint111We decided to merge the proofs of these
results into one, since they are based on largely the same idea.) proof of
Theorem 1 and Proposition 1. The idea of the proof is to decompose the
hypergraph into connected and $2$-colorable pieces and derive a graph from
that partition.
###### Proof of Theorem 1 and Proposition 1.
Let $t\geq 2$ be an integer, and $H$ be a hypergraph with no $K_{t}$-minor.
Our goal is to show that $\chi(H)\leq 2g(t)$, and $\chi_{f}(H)\leq 2g_{f}(t)$.
W.l.o.g. we may assume that $H$ is Sperner, for if not then by Observation 1
the subhypergraph $\text{min}(H)$ is Sperner, also contains no $K_{t}$-minor
and has the same chromatic and fractional chromatic number as $H$.
Let us inductively construct a partition of the vertex-set $V(H)$ into non-
empty sets $(X_{i})_{i=1}^{n}$ for some number $n$ as follows: For $i\geq 1$,
as long as $X_{1}\cup\cdots\cup X_{i-1}\neq V(H)$, pick $X_{i}$ as a subset of
$V(H)\setminus(X_{1}\cup\cdots\cup X_{i-1})$, such that $X_{i}$ is inclusion-
wise maximal with respect to the following two properties:
* •
$H[X_{i}]$ is connected, and
* •
$\chi(H[X_{i}])\leq 2$.
Note that such a set always exists, since any singleton set in
$V(H)\setminus(X_{1}\cup\cdots\cup X_{i-1})$ satisfies both of the above
properties. We now define a graph $G$ by $V(G):=[n]$ and
$E(G):=\Set{\\{i,j\\}\in\binom{[n]}{2}\>}{\>\text{there exists }e\in
E(H)\text{ with }e\subseteq X_{i}\cup X_{j}\text{ and }e\cap
X_{i}\neq\emptyset\neq e\cap X_{j}}.$
Since $H[X_{i}]$ is connected for $i=1,\ldots,n$, by contracting (in arbitrary
order) all hyperedges of $H$ contained in $H[X_{i}]$ for some $i\in[n]$, we
obtain a hypergraph $\tilde{H}$ on $n$ vertices. It now follows directly from
the definition of $G$ that for every edge $\\{i,j\\}\in E(G)$, the pair
consisting of the two contraction vertices of $X_{i}$ and $X_{j}$ forms a
hyperedge in $\tilde{H}$. Hence, after deleting all edges in $\tilde{H}$ which
are not such pairs, we obtain a graph isomorphic to $G$. Thus, $G$ is a minor
of $H$. Since $H$ does not contain $K_{t}$ as a minor, the same must be true
for $G$, and it follows that $\chi(G)\leq g(t)$ and $\chi_{f}(G)\leq
g_{f}(t)$.
Claim 1. Let $e\in E(H)$. If $I(e):=\\{i\in[n]|e\cap X_{i}\neq\emptyset\\}$
forms an independent set in $G$, then there exists $j\in[n]$ such that
$e\subseteq X_{j}$.
###### Proof.
Suppose towards a contradiction that the claim is wrong, and let $e\in E(H)$
be chosen such that $|I(e)|$ is minimized among all hyperedges for which the
claim fails.
Let $j:=\min I(e)$ be the (index-wise) smallest element of $I(e)$. Since $e$
does not satisfy the claim, we have $e\not\subseteq X_{j}$ and hence
$Y:=e\setminus X_{j}\neq\emptyset$. By minimality of $j$ we have $Y\subseteq
X_{j+1}\cup\cdots\cup X_{n}$. If $I(e)=\\{j,j^{\prime}\\}$ for some
$j^{\prime}>j$, then by definition of $I(e)$ we have $e\subseteq X_{j}\cup
X_{j^{\prime}}$ and $e\cap X_{j}\neq\emptyset\neq e\cap X_{j^{\prime}}$. The
definition of $G$ now yields $\\{j,j^{\prime}\\}\in E(G)$, contradicting the
fact that $I(e)$ is an independent set in $G$. Moving on, we may therefore
assume that $|I(e)|\geq 3$. Let $I_{1},I_{2}$ be two non-empty disjoint sets
such that $I(e)\setminus\\{j\\}=I_{1}\cup I_{2}$. By definition, $X_{j}$ is
inclusion-wise maximal among the subsets $X$ of
$V(H)\setminus(X_{1}\cup\cdots\cup X_{j-1})$ for which $H[X]$ is connected and
$\chi(H[X])\leq 2$. We will now show that the strict superset $X_{j}\cup Y$ of
$X_{j}$ also satisfies these two properties, a contradiction which then
concludes the proof of Claim 1.
It remains to show that $H[X_{j}\cup Y]$ is also connected and $2$-colorable.
The connectivity follows directly from the facts that $H[X_{j}]$ is connected,
$Y=e\setminus X_{j}$ and $e\cap X_{j}\neq\emptyset$. Let now
$c:X_{j}\rightarrow\\{1,2\\}$ be a proper $2$-coloring of $H[X_{j}]$, and
extend $c$ to a $2$-coloring $c^{\prime}$ of $H[X_{j}\cup Y]$ by putting
$c^{\prime}(y):=1$ for every $y\in Y$ such that $y\in X_{i}$ for some $i\in
I_{1}$, and $c^{\prime}(y):=2$ for every $y\in Y$ such that $y\in X_{i}$ for
some $i\in I_{2}$. Towards a contradiction, suppose that there is a
monochromatic hyperedge $f$ in $H[X_{j}\cup Y]$ with respect to $c^{\prime}$.
Then since $f$ is monochromatic and $f\subseteq X_{j}\cup Y$, we conclude that
$I(f)\subseteq\\{j\\}\cup I_{1}$ or $I(f)\subseteq\\{j\\}\cup I_{2}$. In each
case, $I(f)\subsetneq I(e)$, and thus $I(f)$ is independent in $G$ and
strictly smaller than $I(e)$. Now our initial assumption on the minimality of
$|I(e)|$ tells us that $f$ satisfies the statement of Claim 1. We know however
that this is not the case: $f\subseteq X_{j}$ would imply that $f$ is a
monochromatic edge in the proper $2$-coloring $c$ of $H[X_{j}]$, which is
impossible, and similarly $f\subseteq X_{i}$ for some
$i\in[n]\setminus\\{j\\}$ would imply that $f\subseteq Y=e\setminus X_{j}$, a
contradiction since $H$ is Sperner. All in all, this concludes the argument
that $H[X_{j}\cup Y]$ is both connected and $2$-colorable, concluding the
proof of Claim 1. ∎
We can now argue that $\chi(H)\leq 2\chi(G)\leq 2g(t)$ and $\chi_{f}(H)\leq
2\chi_{f}(G)\leq 2g_{f}(t)$, completing the proof of Theorem 1.
To do so, for every $i\in[n]$, fix a proper $2$-coloring
$c_{i}:X_{i}\rightarrow\\{1,2\\}$ of $H[X_{i}]$.
To verify $\chi(H)\leq 2\chi(G)$, consider a proper coloring
$c_{G}:V(G)\rightarrow[\chi(G)]$ of $G$. Let
$c_{H}:V(H)\rightarrow[\chi(G)]\times\\{1,2\\}$ be defined as
$c_{H}(x):=(c_{G}(i),c_{i}(x))$ for every $x\in X_{i}$, $i\in[n]$. We claim
that $c_{H}$ defines a proper coloring of $H$. Suppose not, then there exists
$e\in E(H)$ (w.l.o.g. inclusion-wise minimal) such that all vertices in $e$
receive the same color under $c_{H}$, say $c_{H}(x)=(\alpha,\beta)$ for every
$x\in e$. This shows that for every $i\in[n]$ such that $e\cap
X_{i}\neq\emptyset$, we have $c_{G}(i)=\alpha$. Since $c_{G}$ is a proper
coloring of $G$, this means that (with the notation from Claim 1) the set
$I(e):=\\{i\in[n]|e\cap X_{i}\neq\emptyset\\}$ is independent in $G$. Hence,
we may apply Claim 1 to $e$, which yields that there exists $j\in[n]$ such
that $e\subseteq X_{j}$. We then have $c_{j}(x)=\beta$ for every $x\in e$,
contradicting the facts that $e$ is a hyperedge of $H[X_{j}]$ and that $c_{j}$
was chosen as a proper coloring of $H[X_{j}]$. This contradiction shows that
$c_{H}$ is indeed a proper coloring of $H$, yielding $\chi(H)\leq 2\chi(G)\leq
2g(t)$, as claimed.
To verify $\chi_{f}(H)\leq 2\chi_{f}(H)$, we proceed similarly. Denote by
$\mathcal{I}(G)$, $\mathcal{I}(H)$ the collections of independent sets in $G$
and $H$, and let $(w_{G}(I))_{I\in\mathcal{I}(G)}$ be a non-negative weight
assignment such that $\sum_{i\in I\in\mathcal{I}(G)}{w_{G}(I)}\geq 1$ for
every $i\in[n]$ and $\sum_{I\in\mathcal{I}(G)}{w_{G}(I)}=\chi_{f}(G)$. Note
that for every $i\in[n]$, the subsets $c_{i}^{-1}(1),c_{i}^{-1}(2)$ of $X_{i}$
are independent in $H$. Using Claim 1 this implies that for every
$I\in\mathcal{I}(G)$, and any string $s\in\\{1,2\\}^{I}$, the set
$J(I,s):=\bigcup_{i\in I}{c_{i}^{-1}(s_{i})}$
is independent in $H$.
We now define a weight assignment $w_{H}$ for $\mathcal{I}(H)$ by putting
$w_{H}(J(I,s)):=2^{1-|I|}w_{G}(I)$ for every $I\in\mathcal{I}(G)$ and
$s\in\\{1,2\\}^{I}$, and assigning weight $0$ to all remaining independent
sets in $H$. We then have for every $i\in[n]$ and $v\in X_{i}$:
$\sum_{v\in J\in\mathcal{I}(H)}{w_{H}(J)}=\sum_{i\in
I\in\mathcal{I}(G)}\sum_{\begin{subarray}{c}s\in\\{1,2\\}^{I}:\\\
s_{i}=c_{i}(v)\end{subarray}}{2^{1-|I|}w_{G}(I)}$ $=\sum_{i\in
I\in\mathcal{I}(G)}{2^{|I|-1}(2^{1-|I|}w_{G}(I))}=\sum_{i\in
I\in\mathcal{I}(G)}{w_{G}(I)}\geq 1.$
Therefore, $w_{H}$ is a legal weight assignment for $H$, and it follows that
$\chi_{f}(H)\leq\sum_{J\in\mathcal{I}(H)}{w_{H}(J)}=\sum_{I\in\mathcal{I}(G)}{2^{|I|}(2^{1-|I|}w_{G}(I))}=2\sum_{I\in\mathcal{I}(G)}{w_{G}(I)}=2\chi_{f}(G),$
as desired. ∎
We proceed by preparing the proof of Theorem 3. A crucial ingredient to that
proof is the following lemma. We note that its statement restricted to graphs
is well-known (compare [10]), and that our proof follows similar lines as the
one in the case of graphs.
###### Lemma 7.
Let $H$ be a hypergraph on $n$ vertices satisfying $\alpha(H)\leq 2$. Then $H$
contains $K_{\lceil n/3\rceil}$ as a minor.
###### Proof.
For every hypergraph $H$ with $\alpha(H)\leq 2$ we may instead consider
$\text{min}(H)$. By Observation 1 this subhypergraph satisfies
$\alpha(\text{min}(H))=\alpha(H)\leq 2$, is Sperner, and if it contains
$K_{\lceil n/3\rceil}$ as a minor then the same is true for $H$. So in the
remainder, we assume w.l.o.g. that $H$ is Sperner. Note that this implies that
$H$ contains no $r$-edges for any $r>3$, since if it were to contain an
$r$-edge $e$, removing any vertex $x\in e$ would result in the independent set
$e\setminus\\{x\\}$ of size $r-1>2$ in $H$, which is impossible.
In the following let us call a subset $X\subseteq V(H)$ of exactly $3$
vertices _nice_ if the induced subhypergraph $H[X]$ contains as hyperedges
either exactly one hyperedge of size $3$ or exactly two graph edges. Let
$\mathcal{X}$ be a maximal collection of pairwise disjoint nice sets in $H$
(here we allow $\mathcal{X}=\emptyset$). Let $U:=\bigcup\mathcal{X}$ denote
the set of vertices contained in a member of $\mathcal{X}$. Then by maximality
$V(H)\setminus U$ does not contain a nice set. This in particular means that
$H-U$ does not contain any hyperedges of size more than $2$, so it is a graph,
and also no induced $3$-vertex path. It is well-known (and not hard to verify)
that the only graphs with no induced $3$-vertex paths are disjoint unions of
complete graphs. Since $\alpha(H-U)\leq\alpha(H)\leq 2$, this means that $H-U$
is the disjoint union of at most $2$ complete graphs. Let us write $n=a+b$,
where $a:=|U|$ and $b:=|V(H)\setminus U|$. From the above we find that there
exists a subset $C\subseteq V(H)\setminus U$ of size
$|C|\geq\frac{b}{2}\geq\frac{b}{3}$ such that $H[C]$ is a complete graph. We
now claim that the members of $\mathcal{X}$ together with the singletons in
$C$ form the branch-sets of a minor model for a $K_{t}$ of order
$t:=|\mathcal{X}|+|C|\geq\frac{a}{3}+\frac{b}{3}=\frac{n}{3}$ in $H$, which
will conclude the proof.
To see this, note that every member of $\mathcal{X}$ (and trivially every
singleton in $C$) induces a connected subhypergraph of $H$. Furthermore, for
every nice set $X\in\mathcal{X}$ and every vertex $y$ of $H$ outside $X$,
there exists an edge $e\in E(H)$ with $e\subseteq X\cup\\{y\\}$ and $y\in e$.
Indeed, if such an edge would not exist, then it is easy to see that an
independent set of size $2$ in $H[X]$ (which exists since $X$ is nice) joined
by $y$ would form an independent set of size $3$ in $H$, a contradiction.
The above implies that for any two distinct members $X,Y\in\mathcal{X}$ there
exists a hyperedge in their union intersecting both $X$ and $Y$, and also that
for every $X\in\mathcal{X}$ and $c\in C$ there is a hyperedge contained in
$X\cup\\{c\\}$ intersecting both $X$ and $\\{c\\}$. Since additionally $C$ is
a clique, this confirms our above claim that $\mathcal{X}$ together with
$\\{c\\}_{c\in C}$ yields a complete minor in $H$.
∎
To prove Theorem 3, we additionally need a structural result about small
color-critical hypergraphs established by Stiebitz, Storch and Toft in [22].
We adopt the following terminology from their paper:
###### Definition 4.
Given two hypergraphs $H_{1}=(V_{1},E_{1})$ and $H_{2}=(V_{2},E_{2})$ with
disjoint vertex-sets $V_{1}$ and $V_{2}$, their _Dirac-sum_ $H_{1}+H_{2}$ is
the hypergraph with the vertex-set $V=V_{1}\cup V_{2}$ and the edge-set
$E=E_{1}\cup E_{2}\cup\\{\\{x,y\\}|x\in V_{1},y\in V_{2}\\}$.
We may now state their result as follows.
###### Theorem 8 (Stiebitz, Storch and Toft [22], Theorem 1).
Let $H$ be a $k$-color critical hypergraph on at most $2k-2$ vertices. Then
there exist two non-empty disjoint subhypergraphs $H_{1}$ and $H_{2}$ of $H$
such that $H=H_{1}+H_{2}$.
We remark the following simple property of the Dirac sum for later use:
###### Observation 3.
$\chi(H_{1}+H_{2})=\chi(H_{1})+\chi(H_{2})$ for all hypergraphs $H_{1},H_{2}$
on disjoint vertex-sets.
We are now sufficiently prepared for the proof of Theorem 3.
###### Proof of Theorem 3.
Towards a contradiction, suppose the claim is false and let $t\geq 2$ be the
smallest integer for which there exists a counterexample. Let $H$ be a non-
trivial hypergraph with $|V(H)|+|E(H)|$ minimum such that:
* •
$\alpha(H)\leq 2$
* •
$H$ does not contain $K_{t}$ as a minor, and
* •
$\chi(H)>\left\lceil\frac{3}{2}(t-1)\right\rceil$.
We start with a few observations about $H$.
Claim 1. $H$ is Sperner.
###### Subproof..
If $H$ is not Sperner, then $\text{min}(H)$ is a proper subhypergraph of $H$,
which by Observation 1 has the same chromatic and independence number as $H$,
and clearly also does not contain $K_{t}$ as a minor. This is a contradiction
to our minimality assumption on $H$. ∎
Claim 2. $H$ is $k$-color-critical, with
$k:=\left\lceil\frac{3}{2}(t-1)\right\rceil+1$.
###### Subproof.
By our assumptions on $H$ we have $\chi(H)\geq k$. Thus the claim follows if
we can show that for every proper subhypergraph $H^{\prime}\subsetneq H$ we
have $\chi(H^{\prime})\leq k-1=\left\lceil\frac{3}{2}(t-1)\right\rceil$.
Clearly, this is equivalent to showing that $H-v$ and $H-e$ for every choice
of $v\in V(H)$ and $e\in E(H)$ are
$\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable. Note for every $v\in
V(H)$ that $H-v$ is an induced subhypergraph of $H$, which directly implies
that $\alpha(H-v)\leq 2$. Trivially, also $H-v$ does not contain $K_{t}$ as a
minor, which means that we must have
$\chi(H-v)\leq\left\lceil\frac{3}{2}(t-1)\right\rceil$, for otherwise $H-v$
would be a counterexample to the claim of the theorem with
$|V(H-v)|+|E(H-v)|<|V(H)|+|E(H)|$, contradicting the minimality of $H$.
Let now $e\in E(H)$ be arbitrary. To argue that $H-e$ is
$\left\lceil\frac{3}{2}(t-1)\right\rceil$-colorable, we have to be a bit more
careful: Deleting the hyperedge $e$ could increase the independence number,
such that $H-e$ may fall out of the class of hypergraphs with independence
number $2$. We avoid this obstacle by instead considering the hypergraph
$H^{\prime}:=H/e$, obtained from $H$ by contracting $e$. Then it is not hard
to see that $\alpha(H^{\prime})\leq\alpha(H)=2$. Further note that
$H^{\prime}$ as a minor of $H$ cannot contain $K_{t}$ as a minor. Now the
minimality of $H$ as a counterexample and the fact
$|V(H^{\prime})|+|E(H^{\prime})|<|V(H)|+|E(H)|$ imply that
$\chi(H^{\prime})\leq\left\lceil\frac{3}{2}(t-1)\right\rceil$. Thus there
exists a proper coloring
$c^{\prime}:V(H^{\prime})\rightarrow\\{1,\ldots,\left\lceil\frac{3}{2}(t-1)\right\rceil\\}$
of $H^{\prime}$. Let
$c:V(H)\rightarrow\\{1,\ldots,\left\lceil\frac{3}{2}(t-1)\right\rceil\\}$ be
the coloring defined by $c(x):=c^{\prime}(\phi(x))$ for every $x\in V(H)$,
where $\phi:V(H)\rightarrow V(H/e)$ denotes the “contraction map” as defined
in the preliminaries. We claim that $c$ is a proper coloring of the hypergraph
$H\setminus e$. Indeed, consider any edge $f\in E(H)\setminus\\{e\\}$. Then by
Claim 1, we cannot have $f\subseteq e$, and hence the image $\phi(f)$ (by
definition of $H/e$) forms a hyperedge in $H/e$. If $f$ were to be
monochromatic with respect to the coloring $c$, then its image $\phi(f)$ would
be monochromatic with respect to the coloring $c^{\prime}$ of $H^{\prime}$.
Since this is impossible, we conclude that indeed $H\setminus e$ is properly
colored by $c$, proving that $\chi(H\setminus
e)\leq\left\lceil\frac{3}{2}(t-1)\right\rceil=k-1$ for every $e\in E(H)$. This
concludes the proof of Claim 2. ∎
In the following denote $n:=|V(H)|$. By Lemma 7 applied to $H$, we find that
$H$ contains $K_{\lceil n/3\rceil}$ as a minor, which by our assumptions on
$H$ implies that $t\geq\left\lceil\frac{n}{3}\right\rceil+1$. This yields
$n\leq 3(t-1)=2\cdot\frac{3}{2}(t-1)\leq 2(k-1)=2k-2$. Since by Claim 2 $H$ is
$k$-color-critical, we may apply Theorem 8 to $H$. This yields the existence
of a pair $(H_{1},H_{2})$ of non-empty disjoint subhypergraphs of $H$ such
that $H=H_{1}+H_{2}$.
In the following, let us denote by $t_{1}$ and $t_{2}$ the largest integers
such that $H_{1}$ contains $K_{t_{1}}$ as a minor, and such that $H_{2}$
contains $K_{t_{2}}$ as a minor, respectively. It is easy to see that any
sequence of deletions and contractions resulting in the minor of $K_{t_{1}}$
of $H_{1}$ and $K_{t_{2}}$ of $H_{2}$ can be concatenated to yield a sequence
of deletions and contractions in $H$. Performing this sequence of operations
is then easily seen to result in the minor
$K_{t_{1}}+K_{t_{2}}=K_{t_{1}+t_{2}}$ of $H$. By our assumptions on $H$, this
implies that $t_{1}+t_{2}\leq t-1$.
Since $H_{1}$ and $H_{2}$ are induced subhypergraphs of $H$, they satisfy
$\alpha(H_{1}),\alpha(H_{2})\leq\alpha(H)\leq 2$.
Hence, by our minimality assumption on $t$ at the beginning of this proof,
since $t_{i}+1\leq t_{1}+t_{2}<t$ for $i=1,2$, and since $H_{i}$ does not
contain $K_{t_{i}+1}$ as a minor for $i=1,2$, we find that
$\chi(H_{i})\leq\left\lceil\frac{3((t_{i}+1)-1)}{2}\right\rceil=\left\lceil\frac{3t_{i}}{2}\right\rceil$
for $i=1,2$.
Now pick (arbitrarily) a pair of vertices $x\in V(H_{1})$ and $y\in V(H_{2})$.
We distinguish two cases depending on the behaviour of the largest clique
minors in $H_{1}-x$ and $H_{2}-y$.
#### Case 1.
$H_{1}-x$ contains $K_{t_{1}}$ as a minor, and $H_{2}-y$ contains $K_{t_{2}}$
as a minor.
Then, repeating the argument from above, we find that
$(H_{1}-x)+(H_{2}-y)=H-\\{x,y\\}$ contains
$K_{t_{1}}+K_{t_{2}}=K_{t_{1}+t_{2}}$ as a minor. Also note that every vertex
in $H$ is connected by a graph edge to either $x$ or $y$. Therefore,
contracting the edge $\\{x,y\\}$, and then successively performing the
deletions and contractions that transform $H-\\{x,y\\}$ into
$K_{t_{1}+t_{2}}$, yields the complete graph $K_{t_{1}+t_{2}+1}$ as a minor of
$H$. Since we assumed that $H$ does not contain $K_{t}$ as a minor, this
implies that $t_{1}+t_{2}+1\leq t-1$. We may now use this to bound the
chromatic number of $H$ as follows:
$\chi(H)=\chi(H_{1})+\chi(H_{2})\leq\left\lceil\frac{3t_{1}}{2}\right\rceil+\left\lceil\frac{3t_{2}}{2}\right\rceil\leq\left(\frac{3t_{1}}{2}+\frac{1}{2}\right)+\left(\frac{3t_{2}}{2}+\frac{1}{2}\right)$
$=\frac{3(t_{1}+t_{2}+1)-1}{2}<\frac{3(t_{1}+t_{2}+1)}{2}\leq\frac{3(t-1)}{2}\leq\left\lceil\frac{3(t-1)}{2}\right\rceil,$
which is a contradiction to our initial assumption
$\chi(H)>\left\lceil\frac{3(t-1)}{2}\right\rceil$ on $H$. Hence, we obtain the
desired contradiction, which concludes the proof of the theorem in Case 1.
#### Case 2.
$H_{1}-x$ does not contain $K_{t_{1}}$ as a minor, or $H_{2}-y$ does not
contain $K_{t_{2}}$ as a minor. Due to symmetry, in the following we may
assume without loss of generality that $H_{1}-x$ does not contain $K_{t_{1}}$
as a minor.
Then we can obtain a better estimate on the chromatic number of $H_{1}$: Since
$H_{1}-x$ does not contain $K_{t_{1}}$ as a minor and since
$\alpha(H_{1}-x)\leq\alpha(H_{1})\leq 2$, by minimality of $t$ we obtain
$\chi(H_{1}-x)\leq\left\lceil\frac{3(t_{1}-1)}{2}\right\rceil$. Since the
deletion of a vertex can lower the chromatic number of a hypergraph by at most
one, we conclude:
$\chi(H)=\chi(H_{1})+\chi(H_{2})\leq(\chi(H_{1}-x)+1)+\chi(H_{2})$
$\leq\left\lceil\frac{3(t_{1}-1)}{2}\right\rceil+\left\lceil\frac{3t_{2}}{2}\right\rceil+1$
Consider first the case that both $t_{1}$ and $t_{2}$ are odd numbers, meaning
that both $3t_{1}$ and $3t_{2}$ are odd, and both $3(t_{1}-1)$ and
$3(t_{2}-1)$ are even. Then we may estimate, using the above:
$\chi(H)\leq\frac{3(t_{1}-1)}{2}+\frac{3t_{2}+1}{2}+1$
$=\frac{3(t_{1}+t_{2})}{2}\leq\frac{3(t-1)}{2}\leq\left\lceil\frac{3(t-1)}{2}\right\rceil.$
Again, this yields a contradiction to the assumption
$\chi(H)>\left\lceil\frac{3(t-1)}{2}\right\rceil$ and concludes the proof in
this case.
Finally, assume that at least one of $t_{1}$ and $t_{2}$ is even. Then we may
estimate:
$\chi(H)=\chi(H_{1})+\chi(H_{2})\leq\left\lceil\frac{3t_{1}}{2}\right\rceil+\left\lceil\frac{3t_{2}}{2}\right\rceil\leq\frac{3t_{1}}{2}+\frac{3t_{2}}{2}+\frac{1}{2}=\frac{3(t_{1}+t_{2})}{2}+\frac{1}{2}$
$\leq\frac{3(t-1)}{2}+\frac{1}{2}<\frac{3(t-1)}{2}+1\leq\left\lceil\frac{3(t-1)}{2}\right\rceil+1.$
This is a contradiction, since our initial assumption on $H$ means that
$\chi(H)\geq\left\lceil\frac{3(t-1)}{2}\right\rceil+1$.
This concludes the proof of the theorem also in Case 2. ∎
## 4\. Applications
### 4.1. Complete minors with branch-sets of high edge-connectivity
As a first simple application of Theorem 1 (or, rather Corollary 2), we want
to use it to give a short proof of Theorem 4. To reach this goal, we associate
with every graph $G$ and every integer $k\geq 1$ a hypergraph
$H_{\text{conn}}(G,k)$ as follows: Its vertex-set is $V(G)$, and a subset of
vertices $e\subseteq V(G)$ of size at least $2$ forms a hyperedge of
$H_{\text{conn}}(G,k)$ if and only if the induced subgraph $G[e]$ is $k$-edge-
connected. The next lemma collects simple but important properties of this
hypergraph.
###### Lemma 9.
Let $G$ be a graph and $k\geq 1$. Then
1. (1)
If $W\subseteq V(G)$ is connected in $H_{\text{conn}}(G,k)$, then $G[W]$ is
$k$-edge-connected.
2. (2)
If $\chi(G)>k$ then $H_{\text{conn}}(G,k)$ contains at least one hyperedge.
3. (3)
$\chi(G)\leq k\chi(H_{\text{conn}}(G,k))$.
###### Proof.
For the proof, we abbreviate $H:=H_{\text{conn}}(G,k)$.
1. (1)
Let $W\subseteq V(G)$ be such that $H[W]$ is connected. Suppose towards a
contradiction that $G[W]$ is not $k$-edge connected. Then there exists a set
$F$ of at most $k-1$ edges in $G[W]$ such that $G[W]-F$ is disconnected. Let
$X$ denote the vertex-set of a connected component in $G[W]-F$ and let
$Y:=W\setminus X$. Then both $X$ and $Y$ are non-empty, and there are at most
$k-1$ edges in $G[W]$ with endpoints in $X$ and $Y$. Since $H[W]$ is
connected, there has to exist a hyperedge $e$ in $H$ with $e\subseteq W=X\cup
Y$ such that $e\cap X\neq\emptyset\neq e\cap Y$. Then $G[e]$ is a $k$-edge
connected induced subgraph of $G[W]$ containing vertices of both $X$ and $Y$.
However, since $X$ and $Y$ can be separated in $G[W]$ by deleting at most
$k-1$ edges, the same is true for $G[e]$, a contradiction to the fact that
$G[e]$ is $k$-edge connected. This contradiction proves the claim, $G[W]$ must
indeed also be $k$-edge connected.
2. (2)
Suppose $\chi(G)>k$. Let $G^{\prime}\subseteq G$ be a minimal subgraph of $G$
with chromatic number greater than $k$. Then clearly $G^{\prime}$ is
$(k+1)$-color critical. It is a well-known result, compare e.g. [23], that
every $(k+1)$-color-critical graph is $k$-edge-connected. Hence, $G^{\prime}$
is $k$-edge-connected. Denoting by $e:=V(G^{\prime})$ its vertex-set, we
easily see that also $G[e]$ must be $k$-edge-connected, that is, $e$ forms a
hyperedge of $H$.
3. (3)
Let $c:V(H)\rightarrow S$ be a proper coloring of $H$ for some color-set $S$
of size $\chi(H)$. Then for every $s\in S$ the color-class $c^{-1}(s)$ forms
an independent set in $H$. In other words,
$H[c^{-1}(s)]=H_{\text{conn}}(G[c^{-1}(s)],k)$ contains no hyperedges. Using
item (2), this implies that $\chi(G[c^{-1}(s)])\leq k$ for every $s\in S$.
Thus,
$\chi(G)\leq\sum_{s\in S}{\chi(G[c^{-1}(s)])}\leq k|S|=k\chi(H),$
as desired.
∎
Using Lemma 9, Theorem 4 now falls out as an immediate consequence of our
hypergraph result Corollary 2.
###### Proof of Theorem 4.
By Corollary 2 there exists a constant $c>0$ such that $h(t)\leq ct\log\log t$
for $t\geq 3$. Let now $G$ be any given graph, $k\geq 1$ and $t\geq 3$
integers.
Let $C:=c$ and suppose $\chi(G)>Ckt\log\log t$. Then by Lemma 9
$\chi(H_{\text{conn}}(G,k))\geq\frac{1}{k}\chi(G)>ct\log\log t\geq h(t)$. This
means that $H_{\text{conn}}(G,k)$ contains $K_{t}$ as a minor, i.e., there is
a $K_{t}$-minor model in $H_{\text{conn}}(G,k)$ with branch-sets
$(B_{i})_{i=1}^{t}$. Then for every $i\in[t]$, the set $B_{i}$ is connected in
$H_{\text{conn}}(G,k)$ and thus $G[B_{i}]$ is $k$-edge-connected by Lemma 9,
(1). Now consider distinct $i,j\in[k]$. Then by definition of a minor model
there exists a hyperedge $e\subseteq B_{i}\cup B_{j}$ of
$H_{\text{conn}}(G,k)$ such that $e\cap B_{i}\neq\emptyset\neq e\cap B_{j}$.
But then, since $G[e]$ is a $k$-edge-connected graph, at least $k$ edges must
be spanned between $B_{i}$ and $B_{j}$ in $G$, and this proves the claim of
the theorem. ∎
### 4.2. Complete minors with modulo $q$-connected branch-sets
To derive Theorem 5, we use the following definition of a hypergraph. For any
graph $G$ and any number $q\geq 2$, we denote by $H_{\text{mod}}(G,q)$ the
hypergraph with vertex-set $V(G)$ and in which a subset $e\subseteq V(G)$ of
size at least $2$ is a hyperedge if and only if $G[e]$ is a modulo
$q$-connected graph. Similar to Lemma 9 from the previous section, the
following lemma relates the coloring and connectivity properties of
$H_{\text{mod}}(G,q)$ and $G$.
###### Lemma 10.
Let $G$ be a graph and $q\geq 2$. Then
1. (1)
If $W\subseteq V(G)$ is connected in $H_{\text{mod}}(G,q)$, then $G[W]$ is
modulo $q$-connected.
2. (2)
If $\chi(G)>315q$ then $H_{\text{mod}}(G,q)$ contains at least one hyperedge.
3. (3)
$\chi(G)\leq 315q\cdot\chi(H_{\text{conn}}(G))$.
In the proof of the above Lemma, we need two statements from the literature,
the first relating chromatic number and vertex-connectivity, and the second
relating vertex-connectivity and modulo $q$-connectivity of graphs.
###### Theorem 11 (Girão and Narayanan [13]).
Let $k\geq 1$ be an integer. Every graph $G$ with $\chi(G)>7k$ contains a
subgraph $G^{\prime}$ such that $\chi(G^{\prime})\geq k$ and $G^{\prime}$ is
$k$-vertex-connected.
We remark that the constant in the previous theorem was recently improved from
$7$ to $3+\frac{1}{16}$ by Nguyen [18]. For simplicity, we stick with the
constant $7$ here.
###### Theorem 12 (Chen, Chen, Gao and Hu [7]).
Let $k$ and $m_{1},\ldots,m_{k}$ be positive integers, and let $G$ be a
$45(m_{1}+\ldots+m_{k})$-vertex-connected graph. Then at least one of the
following holds:
* •
There exists a vertex-set $X\subseteq V(G)$ of size $|X|\leq 4k-2$ such that
$G-X$ is bipartite, or
* •
for every choice of pairwise distinct vertices
$x_{1},\ldots,x_{k},y_{1},\ldots,y_{k}$ in $G$ and for every choice of
integers $d_{1},\ldots,d_{k}$ there exist $k$ disjoint paths
$P_{1},\ldots,P_{k}$ in $G$ such that $P_{i}$ has endpoints $x_{i}$ and
$y_{i}$, and its length is congruent to $d_{i}$ modulo $2m_{i}$.
Setting $k:=1$ in Theorem 12, we obtain the following special case.
###### Corollary 13.
Let $q\geq 2$ be an integer. Then every $45q$-vertex-connected graph $G$
contains a set $X$ of at most $2$ vertices such that $G-X$ is bipartite, or
$G$ is modulo $2q$-connected (and thus also modulo $q$-connected).
We are now prepared for the proof of Lemma 10.
###### Proof.
During the proof we use the abbreviation $H=H_{\text{mod}}(G,q)$.
1. (1)
Let $W\subseteq V(G)$ be connected in $H_{\text{mod}}(H,q)$. Let a pair of
distinct vertices $x,y\in W$ as well as a residue $r\in\\{0,1,\ldots,q-1\\}$
be given to us. We have to show that there exists a path in $G[W]$ with
endpoints $x,y$ and whose length is congruent to $r$ modulo $q$. Since $H[W]$
is connected, for some $k\geq 2$ there exists a sequence
$x=x_{1},x_{2},\ldots,x_{k}=y$ of vertices in $W$ and hyperedges
$e_{1},\ldots,e_{k-1}$ of $H[W]$ such that $x_{i},x_{i+1}\in e_{i}$ for all
$i\in[k-1]$. We can further assume that $k\geq 2$ is chosen minimal such that
a sequence as above exists. If $k=2$, then $x$ and $y$ are distinct vertices
in the modulo $q$-connected induced subgraph $G[e_{1}]$ of $G[W]$, and thus
can be connected by a path of length $r$ modulo $q$, as desired. So assume
$k\geq 3$. Then our minimality assumption on $k$ implies that $x_{1}\notin
e_{2}\cup\cdots\cup e_{k-1}$. Note that $H[e_{2}\cup\cdots\cup e_{k-1}]$ forms
a connected subhypergraph of $H$, and hence we may apply Lemma 9, (1), to find
that $G[e_{2}\cup\cdots\cup e_{k-1}]$ is a connected subgraph of $G[W]$. Let
$Q$ be a shortest path in $G[e_{2}\cup\cdots\cup e_{k-1}]$ that connects
$y=x_{k}$ to a vertex in $e_{1}$ (such a path exists, since $x_{2}\in
e_{1}\cap e_{2}$). Let $z$ denote the other endpoint of $Q$. Then we have
$V(Q)\cap e_{1}=\\{z\\}$ since $Q$ is shortest. Furthermore, $x\neq z$, since
$x\notin e_{2}\cup\cdots\cup e_{k-1}$. Let $r^{\prime}\in\\{0,1,\ldots,q-1\\}$
be such that $r^{\prime}+\ell(Q)\equiv_{q}r$. Then we find a path $R$ in
$G[e_{1}]$ connecting $x$ and $z$ of length congruent to $r^{\prime}$ modulo
$q$. It is now apparent that $R\cup Q$ forms a path in $G[W]$ connecting $x$
and $y$ of length $\ell(R)+\ell(Q)\equiv_{q}r^{\prime}+\ell(Q)\equiv_{q}r$.
This concludes the proof that $G[W]$ is modulo $q$-connected.
2. (2)
Since $\chi(G)>315q=7\cdot 45q$, we may apply Theorem 11 and find a
$45q$-vertex-connected subgraph $G^{\prime}$ of $G$ with $\chi(G^{\prime})\geq
45q$. W.l.o.g. we may assume that $G^{\prime}$ is an induced subgraph of $G$.
Applying Corollary 13, we find that there is a set $X$ of at most $2$ vertices
such that $G^{\prime}-X$ is bipartite, or $G^{\prime}$ is modulo
$q$-connected. However, the first option is not feasible, since
$\chi(G^{\prime}-X)\geq\chi(G^{\prime})-|X|\geq 45q-2>2$ for every set
$X\subseteq V(G^{\prime})$ of size $2$. Hence, $V(G^{\prime})$ forms a
hyperedge in $H_{\text{mod}}(G,q)$, proving the assertion.
3. (3)
The proof is analogous to the proof of item (3) of Lemma 9, and is thus left
out.
∎
Similar as in the previous subsection, we can now derive Theorem 5 as a direct
consequence of Corollary 2.
###### Proof of Theorem 5.
Let $c>0$ be as guaranteed by Corollary 2 such that $h(t)\leq ct\log\log t$
for $t\geq 3$. Let $G$ be any given graph, $q\geq 2$ and $t\geq 3$ integers.
Define $C:=315c$ and suppose that $\chi(G)>Cqt\log\log t$. Then by Lemma 9
$\chi(H_{\text{mod}}(G,q))\geq\frac{1}{315q}\chi(G)>ct\log\log t\geq h(t)$.
Therefore $H_{\text{mod}}(G,q)$ contains $K_{t}$ as a minor. Let
$(B_{i})_{i=1}^{t}$ be corresponding branch-sets of a minor model, then for
every $i\in[t]$ the set $B_{i}$ is connected in $H_{\text{mod}}(G,q)$ and thus
$G[B_{i}]$ is modulo $q$-connected by Lemma 10, (1). For distinct $i,j\in[k]$
we have by definition of a minor model that there is a hyperedge $e\subseteq
B_{i}\cup B_{j}$ with $e\cap B_{i}\neq\emptyset\neq e\cap B_{j}$. Since $G[e]$
is connected, this means there is at least one edge in $G$ between $B_{i}$ and
$B_{j}$, and hence the set collection $(B_{i})_{i=1}^{t}$ indeed describes a
$K_{t}$-minor model in $G$, as desired. ∎
### 4.3. Strong complete minors in digraphs
To prove Theorem 6, it will be crucial to relate strong minors in digraphs
with the complete minors in their _cycle hypergraphs_. Given a digraph $D$,
its _cycle hypergraph_ $\mathcal{C}(D)$ has the same vertex-set $V(D)$ as $D$,
and a set $e\subseteq V(D)$ forms a hyperedge of $\mathcal{C}(D)$ if and only
if there exists a directed cycle $C$ in $D$ whose vertex-set equals $e$. It is
clear by definition that a set $W$ of vertices is independent in
$\mathcal{C}(D)$ if and only if $D[W]$ is acyclic. This directly yields that
$\vec{\chi}(D)=\chi(\mathcal{C}(D))$ for every digraph $D$. The next lemma
shows that complete minors in $\mathcal{C}(D)$ induce strong complete minors
in $D$.
###### Lemma 14.
Let $t\geq 1$ be an integer. If $\mathcal{C}(D)$ contains $K_{t}$ as a minor,
then $D$ contains $\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$ as a
strong minor.
###### Proof.
Let $(B_{i})_{i=1}^{t}$ be the branch-sets of a $K_{t}$-minor model in
$\mathcal{C}(D)$. Then for every $i\in[t]$, the subhypergraph
$\mathcal{C}(D)[B_{i}]$ is connected. We claim this means that $D[B_{i}]$ is
strongly connected. Indeed, if not then there is a partition of $B_{i}$ into
non-empty sets $X,Y$ such that no arc in $D$ starts in $X$ and ends in $Y$.
But then no directed cycle in $D[B_{i}]$ can intersect both $X$ and $Y$,
contradicting the connectivity of $\mathcal{C}(D)[B_{i}]$.
Next we claim that for every distinct $i,j\in[t]$ there is an arc from $B_{i}$
to $B_{j}$, and an arc from $B_{j}$ to $B_{i}$ in $D$. Indeed, there exists a
hyperedge $e$ in $\mathcal{C}(D)$ such that $e\subseteq B_{i}\cup B_{j}$ and
$e\cap B_{i}\neq\emptyset\neq e\cap B_{j}$. Then there is a directed cycle $C$
in $D$ with $V(C)=e$, and hence $C$ must traverse an arc starting in $B_{i}$
and ending in $B_{j}$, and vice versa.
With the above observations we see that $(B_{i})_{i=1}^{t}$ certify the
existence of a strong
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$-minor of $D$, as desired.
∎
The proof of Theorem 6 is now immediate using Theorem 1.
###### Proof of Theorem 6.
Let $D$ be a digraph with no strong
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$-minor. By Lemma 14 the
cycle hypergraph $\mathcal{C}(D)$ is $K_{t}$-minor free, and Theorem 1 implies
$\vec{\chi}(D)=\chi(\mathcal{C}(D))\leq h(t)\leq 2g(t)$. ∎
We conclude this section with the lower-bound construction for the dichromatic
number of digraphs with no strong
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$-minor.
###### Proof of Proposition 2.
Let $t\geq 2$ be given. Let $A,B,C$ be $3$ vertex-disjoint sets of size $t-1$,
and let $D$ be the digraph on the vertex-set $A\cup B\cup C$ with the
following arcs: For every pair of distinct vertices $u,v$ that are contained
together in one of $A$, $B$ or $C$, we add both the arcs $(u,v)$ and $(v,u)$.
Finally, we add all arcs from $A$ to $B$, from $B$ to $C$ and from $C$ to $A$.
For $\vec{\chi}(D)\geq\left\lceil\frac{3}{2}(t-1)\right\rceil$ it suffices to
note that every set of more than $2$ vertices spans a directed cycle in $D$,
and hence at least $\frac{3(t-1)}{2}$ colors are needed in any coloring with
acyclic color classes.
Next, suppose towards a contradiction that $D$ contains
$\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}$ as a strong minor. Then
there are disjoint non-empty sets $(B_{i})_{i=1}^{t}$ such that each of
$D[B_{i}],i\in[t]$ is strongly connected and such between any two sets
$B_{i},B_{j}$ there exist arcs in both directions. These conditions imply that
each $B_{i}$ is either contained in one of $A,B,C$ or intersects each of $A,B$
and $C$ in at least one vertex (and thus has size at least $3$ in this case).
Since $D$ contains $3(t-1)<3t$ vertices, at least one of the sets has to be
fully contained in either $A,B$ or $C$. W.l.o.g. assume that $B_{1}\subseteq
A$. Since $|A|=t-1$, there must exist $i\geq 2$ such that $B_{i}\cap
A=\emptyset$. Then by the above we have $B_{i}\subseteq B$ or $B_{i}\subseteq
C$. But then $B_{1}$ and $B_{i}$ are not connected by arcs in both directions,
a contradiction. This concludes the proof. ∎
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# BERT Embeddings for Automatic Readability Assessment
Joseph Marvin Imperial
National University
Manila, Philippines
<EMAIL_ADDRESS>
###### Abstract
Automatic readability assessment (ARA) is the task of evaluating the level of
ease or difficulty of text documents for a target audience. For researchers,
one of the many open problems in the field is to make such models trained for
the task show efficacy even for low-resource languages. In this study, we
propose an alternative way of utilizing the information-rich embeddings of
BERT models with handcrafted linguistic features through a combined method for
readability assessment. Results show that the proposed method outperforms
classical approaches in readability assessment using English and Filipino
datasets—obtaining as high as 12.4% increase in F1 performance. We also show
that the general information encoded in BERT embeddings can be used as a
substitute feature set for low-resource languages like Filipino with limited
semantic and syntactic NLP tools to explicitly extract feature values for the
task.
## 1 Introduction
Automatic readability assessment is the task of evaluating the level of ease
or difficulty of text documents such as web articles, story and picture books,
test materials, and medical prescriptions. Often readability levels can be
expressed in many forms: discrete values with grade and age levels such as in
the Common European Framework of Reference for Languages
(CEFR)111https://www.cambridgeenglish.org/exams-and-tests/cefr/, or with
continuous values from a given range such as in the famous Lexile Reading
Framework222https://lexile.com/. In machine learning setting, this task is
most often viewed as a classification task where an annotated set of corpora
is trained with its corresponding gold-standard labels evaluated by an expert
as mostly done in previous works Vajjala (2021); Chatzipanagiotidis et al.
(2021); Weiß and Meurers (2018); Xia et al. (2016); Reynolds (2016); Hancke et
al. (2012); Vajjala and Meurers (2012). Recent works have tried testing
unexplored resources by utilizing large pre-trained language models such as
Bidirectional Encoder Representations or BERT Devlin et al. (2019) which is
based on the attention-driven Transformer architecture by Vaswani et al.
(2017) by (a) directly processing the data to the network Martinc et al.
(2021); Tseng et al. (2019) or by (b) using the discrete output of the network
via transfer learning Deutsch et al. (2020) as an additional feature. For
these methods, however, evidence of efficacy are only seen in high-resources
readability datasets in English. Thus, we propose an alternative way of
incorporating the knowledge of large language models such as BERT by combining
its information-rich sentence embeddings as a separate feature set for
traditional machine learning algorithms with handcrafted linguistic features.
We argue that this method is not only low-resource friendly but also preserves
the semantic and syntactic information encoded by the attention heads of BERT
since the embeddings itself will be used. We show that such information can
act as a substitute for languages with limited tools for explicitly extracting
semantic and syntactic features where results describe non-significance in
difference of performances between models using semantic and syntactic
features versus models using BERT embeddings.
## 2 Previous Work
The first generation of readability formulas and indices date as early as
1920-1940s with the works of Thorndike (1921), Dale and Chall (1948), and
Flesch (1948) primarily using surface-based variables such as raw frequencies
and average values of sentences and words per document. The process for using
such indices requires manual computation and plugging of values to formulas
which can be tedious as the length of a document increases. Likewise, experts
argue that considering narrow, surface-based features do not entirely capture
the linguistic complexity of a given text Macahilig (2015); Collins-Thompson
and Callan (2004); Si and Callan (2001). Thus, incorporation of deeper,
linguistic variables such as a language’s semantics, syntax, morphology, and
discourse properties are imperative and worth exploring for the task. To
answer this call, the use of handcrafted linguistic features remained the most
popular type of input for training readability assessment models through the
years. Handcrafted linguistic features are often represented as real-valued
numbers serving as potential predictors of the difficulty of reading
materials. These features span on a wide range of linguistically motivated
factors that base on syntax, semantics, morphology, cohesion, and cognition to
name a few. These features also serve as the input in the form of vectors for
conventional readability assessment setups using traditional classification-
based algorithms. To note, not all linguistic features can be applied or
extracted for all languages as some have limited NLP tools suitable for use
especially for low-resource languages. Notable works in various languages such
as Greek Chatzipanagiotidis et al. (2021), German Weiss and Meurers (2019);
Weiß and Meurers (2018); Hancke et al. (2012), Bangla Sinha et al. (2012), and
Filipino Imperial and Ong (2021a, 2020) have used this approach in combination
with traditional machine learning algorithms such as Logistic Regression and
Support Vector Machines. Likewise, another reason why studies have resorted to
the classical approach of model building is that deep neural models are not
practical for the task without a large amount of training data.
The advent of large and complex pre-trained language models such as BERT and
its variations spawned a handful of studies on how these models fare with the
readability assessment tasks. The work of Martinc et al. (2021) on the
supervised experiment setup explored directly using English benchmark corpus
such as Weebit and OneStopEnglish as input for BERT via transfer learning
while Deutsch et al. (2020) explored using the final discrete output of BERT
as a feature for the same datasets. Results from both studies show
effectiveness of BERT for English data as direct input while no significant
improvement is seen when the discrete output itself is used as a feature.
While these results are remarkable, BERT’s effectiveness remain a gray area
for low-resource languages.
## 3 Task Definition
We define our task at hand as a supervised learning setup. Given a text
document $d$ where a feature vector $x=[x_{1},x_{2}\ldots,x_{n}]$ is
extracted, a model $M$ is trained using said collection of features $X$ along
with the gold label $Y$ or expert-identified readability level. The label is
relative in form (discrete or continuous) based on how readability levels are
categorized for each corpus.
Data | Doc Count | Sent Count | Vocab
---|---|---|---
OSE | 567 | 4,890 | 17,818
CCE | 168 | 20,945 | 78,965
Adarna House | 265 | 10,018 | 16,058
Table 1: Data distribution for English and Filipino corpus.
## 4 Corpus
We describe each corpus used in the study below as well as the statistics and
breakdown in Table 1
OneStopEnglish. The OSE corpus is a collection of 567 texts in three different
reading levels (beginner, intermediate, and advanced) for adult ESL learners
from the MacMillan Education website333https://www.onestopenglish.com/. This
corpus was first used in the work of Vajjala and Lučić (2018) and has become
one of the most-used benchmark datasets for readability assessment and text
simplification in English.
Common Core Exemplars. The CCE dataset contains 168 prose texts from the
Appendix B of the Common Core State Standards Initiative
(CCSS)444http://www.corestandards.org/assets/Appendix_B.pdf for English
Language studies and first used by Flor et al. (2013) for readability
assessment. The initiative was a project of the National Governors Association
and the Council of Chief State School Officers in USA555http://www.ccsso.org.
The dataset is divided into three age-range categories: 2-5, 6-7, and 9-12.
Adarna House. The Adarna House corpus is a collection of 265 story books for
grades 1-3 from Adarna House Inc.666https://adarna.com.ph/, the largest
children’s literature publisher in the Philippines. This corpus has been used
by Imperial et al. (2019); Imperial and Ong (2020, 2021a) for readability
assessment in Filipino777Filipino is considered as a low-resource language
Cruz et al. (2020a, b).
Figure 1: The proposed combined training approach using sentence embeddings
from BERT model and extracted handcrafted linguistic feature sets.
## 5 BERT Embeddings + Handcrafted Linguistic Features
BERT’s efficacy on a wide range of NLP tasks stems from its implicit
capability to encode linguistic knowledge such as hierarchical parse trees
Hewitt and Manning (2019), parts of speech and syntactic chunks Liu et al.
(2019); Tenney et al. (2019), semantic roles Ettinger (2019) as well as entity
types and relations Tenney et al. (2019) to name a few. In view with this, we
find such amount of knowledge an extremely valuable resource which can
potentially improve performances of readability assessment models especially
for low-resource languages if used correctly. Thus, to maximize the potential
of BERT for low-resource readability assessment, we propose a combined
training of its raw embeddings with handcrafted linguistic feature sets
through a concatenation process and feeding them to traditional machine
learning algorithms. The embeddings of BERT generated by the multi-head
attention layers are information-rich, specifically on semantic and syntactic
knowledge Rogers et al. (2020), due to the nature of its training. We describe
our proposed architecture in Figure 1 with a sample Filipino sentence for
context.
## 6 Experiment Setup
For the OSE and CCE corpus in English, we extracted over 155 linguistic
features covering lexical diversity and density features, syntactic features
based on parse trees, morphosyntactic properties of lemmas, and word-level
psycholinguistic features. For the Adarna House corpus in Filipino, we
extracted over 54 linguistic features covering traditional surface-based
features, lexical features based on POS tags, language model features,
morphology based on verb inflection, and orthographic features based on
syllable pattern. The size of the BERT embeddings for all datasets remain
equal with a fixed dimension of $H$ = 768 since the base version of BERT for
English Devlin et al. (2019) and Filipino Cruz et al. (2020c); Cruz and Cheng
(2020, 2019) were used. The embeddings and extracted linguistic feature sets
were concatenated, for a total of 923 dimensions for combined features for
both English datasets and 823 for the Filipino dataset. Recipes for feature
extraction were obtained from the studies of Vajjala and Meurers (2016, 2014)
for English and Imperial and Ong (2020, 2021a, 2021b) for Filipino. We used
the sentence-transformers library by Reimers and Gurevych (2019) with mean
pooling option to extract BERT embedding representations for the readability
corpora888We release the script for extracting BERT embeddings at
https://github.com/imperialite/BERT-Embeddings-For-ARA.
For the traditional machine learning algorithms, we used three of the commonly
utilized in previous works: Logistic Regression, Support Vector Machines, and
Random Forest. Models for each dataset were trained on a 5-fold cross
validation procedure. We used weighted F1 as the overall metric for
performance evaluation.
## 7 Results
### 7.1 Ablation
We compared performances of models on three different setups, (a) linguistic
features only, (b) BERT sentence embeddings only, and (c) combined training of
the two feature embeddings to gauge the efficacy of the proposed framework.
Method | OSE | CCE | Adarna
---|---|---|---
Linguistic Features | 0.676 | 0.774 | 0.389
BERT Embeddings | 0.620 | 0.747 | 0.505
Combined Features (Ling + BERT) | 0.732 | 0.778 | 0.554
(a) Logistic Regression
Method | OSE | CCE | Adarna
---|---|---|---
Linguistic Features | 0.691 | 0.732 | 0.414
BERT Embeddings | 0.611 | 0.826 | 0.487
Combined Features (Ling + BERT) | 0.704 | 0.893 | 0.571
(b) Support Vector Machines
Method | OSE | CCE | Adarna
---|---|---|---
Linguistic Features | 0.683 | 0.842 | 0.423
BERT Embeddings | 0.439 | 0.770 | 0.504
Combined Features (Ling + BERT) | 0.690 | 0.861 | 0.467
(c) Random Forest
Table 2: F1 performance via training with (a) Logistic Regression, (b) Support
Vector Machines, and (c) Random Forest using handcrafted linguistic features,
BERT sentence embeddings, and combined training of both.
As described in Table 2, generally speaking, models trained using the proposed
combined training of handcrafted linguistic feature sets with contexual BERT
embeddings outperform both performances of only using each exclusively on
English and Filipino datasets. On average, we note an increase of performance
of 2.63% for OSE, 6.23% for CCE, and 12.4% in weighted F1 score for Adarna
House across all algorithms. From this, we infer that extracting and
incorporating the information-rich embeddings of any readability dataset using
BERT to commonly-used linguistic feature sets can substantially improve model
performance.
Interestingly, there are a few notable cases reported in Table 2 where BERT
embeddings alone outperformed the traditional method of using handcrafted
linguistic feature sets as primary input. These cases are evident in the all
models utilizing the Adarna House dataset in Filipino with an average increase
of 9.5% weighted F1 scores. From this, we infer that the general semantic and
syntactic knowledge implicitly encoded in BERT embeddings as detailed in
probing tasks from previous works Rogers et al. (2020); Hewitt and Manning
(2019); Liu et al. (2019); Tenney et al. (2019) may be significantly more
informative than the traditional handcrafted linguistic features for
discriminating reading difficulty. Consequently, this poses as a probable and
alternative solution for low-resource languages with little to no NLP tools
such as a good part-of-speech tagger, stemmer, syntactic parse tree extractor,
and morphological analyzer to name a few for manually extracting linguistic
information from documents. Since BERT models are trained in an self-
supervised manner, the overhead of developing these tools from scratch can be
disregarded, at least for readability assessment. We discuss further
experiments on this inference in the next section.
Model w/ Removed Features | OSE | CCE | Adarna
---|---|---|---
Logistic Regression | 0.744 | 0.865 | 0.492
Support Vector Machines | 0.615 | 0.869 | 0.507
Random Forest | 0.669 | 0.791 | 0.431
Full Model (Ling + BERT) | 0.732 | 0.893 | 0.571
Table 3: Performances of models via F1 score after retraining with semantic
and syntactic handcrafed linguistic features removed to test if information-
rich BERT embeddings can act as substitution for such features. Best
performing model utilizing combined features from Table 2 appended for
comparison.
Figure 2: Decomposing large feature sets on a 25%, 50%, 75%, 95%, and 100%
(full) variance percentages using PCA for Logistic Regression, Support Vector
Machines, and Random Forest (left to right).
### 7.2 Substituting Semantic and Syntactic Features for BERT Embeddings
To empirically test if BERT embeddings can act as substitute for semantic and
syntactic linguistic features for readability assessment, we removed features
from the three datasets that assume semantic and syntactic knowledge. For OSE
and CCE, we removed 56 features covering part-of-speech densities, lexical
richness, type-token densities, and general parse-tree based features. For
Adarna, we removed 22 features covering part-of-speech densities, type-token
densities, and verb inflection densities. There are no parse-tree based
features for Adarna House as there are currently no NLP tools for extracting
such feature set for Filipino. The rest of the linguistic features from the
datasets denoting other aspects of reading difficulty measurement such as
frequency-based features and syllable patterns remain unchanged. Models were
retrained using the three selected machine learning algorithms for comparison.
Results of substitution experiments can be found in Table 3. Generally
speaking, it is evident that models trained using the combined method still
outperforms models using the reduced feature set on the account of CCE and
Adarna data. However, we note the 1.2% increase in F1 score on the OSE data.
Stemming from this observation, we also note small differences in performances
of using the combined features against decreased features. In the CCE corpus,
the highest performing model using decreased features obtained 86.9% F1 score
which is less 2.4% than using the model with combined features. For the Adarna
data, the difference is 6.4%.
To identify if such difference is significant, we used a two-tailed test of
difference via Mann-Whitney U using the performance scores of models with
combined features and models with decreased features for all datasets. We
arrived at a $p$-value of 0.522 ($p>$0.5), meaning that the difference of the
scores between two groups is not significant999The distribution of the two
groups are of equal variances with $p$-value of 0.619.. Thus, we conclude that
BERT embeddings can be fully used as a substitute for semantic and syntactic
features if such information cannot be explicitly extracted from readability
data due to the lack of NLP tools and low-resourceness of other languages. To
add, since BERT models are trained in an self-supervised manner and there are
over 3,000 pretrained models from online
repositories101010https://huggingface.co/models, these resources and the
proposed combined training method as a viable option.
### 7.3 Feature Decomposition for Performance Boost
In extending the effort to improve the performance of BERT-enriched
readability assessment models and reduce feature size or dimensionality, we
resorted to the use of feature decomposition to the large feature vector sizes
(BERT + linguistic features) via Principal Components Analysis (PCA). PCA
works by projecting the overall feature set (often large) to a lower
dimensional property while preserving quality and information of features
Hotelling (1933). We experimented on differing values of variance percentages:
25, 50, 75, 95, and 100 (full, no feature removed). Results of feature
decomposition via PCA for each machine learning model are described in Figure
2.
For SVM and Random Forest, all datasets have the highest performances if all
features are retained (100 variance percentage). While for Logistic
Regression, 75 variance percentage obtained the highest performance with 82.7
F1 score for OSE, 95 variance percentage obtained the highest performance with
83.8% F1 score for CCE, and 100% or full features for Adarna. Thus, we infer
that there is no need to perform feature decomposition to find the principal
components as the highest-performing models for OSE, CCE, and Adarna use 100%
of the combined feature set (BERT + linguistic features).
## 8 Conclusion
In this study, we proposed an alternative way of combining information-rich
BERT embeddings with handcrafted linguistic features for the readability
assessment task. Results from our experiments showed that the method
outperforms classical, vanilla approaches in readability assessment using
English (OSE and CCE) and Filipino (Adarna) datasets in various machine
learning algorithms such as Logistic Regression, Support Vector Machines, and
Random Forest. We also demonstrated that the knowledge implicitly encoded in
BERT embeddings (semantic and syntactic information) can be used as a full
substitute feature set for low-resource languages like Filipino with limited
NLP tools to explicitly extract feature values for the task. We are looking
forward to the application of our proposed method to other languages
struggling with the extraction deep linguistic features to trained readability
assessment models. Future directions of the study include deeper exploration
of BERT such as isolating extracted embeddings for each of the twelve
attention layers.
## 9 Acknowledgments
We would like to thank Dr. Ani Almario from Adarna House, Dr. Sowmya Vajjala
from the National Research Council of Canada, and Dr. Michael Flor from ETS
for providing the Adarna, OSE, and CCE datasets respectively.
## 10 Ethical Considerations
We report that there are no major ethical concerns in the study as it involves
no human subjects nor discriminate any identifiable group of people. As for
the dataset, for Adarna House, permission was obtained from the publishing
house while for the OSE and CCE datasets, it remains open-sourced. As for
energy consumption, the study only uses pre-trained BERT models and the
authors did not actually perform the pre-training phase itself.
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|
Candes et al., (2006) for recovering traffic patterns using GPS traces. My
approach can provide accurate recovery when tested using the ground-truth
traffic data from loop detectors, and is among few that have exploited the
effectiveness of Compressed Sensing on traffic pattern processing Lin et al.,
(2019).
In order to further improve the estimation accuracy of traffic conditions for
spatial data interpolation, I proposed an iterative approach, which embeds
map-matching and travel-time estimation as its sub-routines. This process is
further improved by the statistical modeling and learning of traffic
conditions of a road segment. The results of my approach are accurate
estimations of traffic conditions in areas with GPS data coverage—achieving up
to 97% relative improvement in estimation accuracy over previous
techniques—and coarse estimations of traffic conditions in areas without GPS
data coverage (of a city).
For achieving accurate estimations of traffic conditions in data-deficient
areas, using the abovementioned results, I presented a method to dynamically
interpolate spatial missing traffic data. In particular, I have leveraged
traffic simulation to ensure the consistency of traffic flows on the
boundaries of areas with and without GPS data coverage. A metamodel-based
simulation optimization is further developed to save the computational cost of
using traffic simulation in optimization. Compared to the simulation-only
approach, my technique has achieved on average a 7% error rate and up to 90
times speedup. My approach is the first dynamical and efficient method for
interpolating large-scale traffic data while ensuring the flow consistency on
city-scale boundaries.
After fully reconstructing spatial-temporal traffic at a city scale, I
visualized the reconstructed traffic in various forms such as 2D flow map, 2D
animation, and 3D animations. These visual representations can be adopted to
improve many ITS applications including the analysis of traffic patterns at
street level, region level, and the city level, and enrich virtual environment
applications such as virtual tourism and the training of general driving
behaviors or autonomous driving.
Regarding autonomous driving, I presented ADAPS, a framework that consists of
two simulation platforms and a hierarchical control policy. ADAPS can be used
to simulate, analyze various traffic scenarios, especially accidents, and
automatically produce labeled training data. In addition, ADAPS represents a
more efficient online learning mechanism compared to previous techniques,
attributing to the switch from the reset modeling approach to the generative
modeling approach. Using the hierarchical control policy and the efficient
online learning mechanism of ADAPS, robust control policies for autonomous
driving can be learned and applied to obtain normal driving and safe
navigation in dangerous situations including accidents.
### Future Work
There are many future development and research directions can stem from this
dissertation, at the macroscopic level of traffic (i.e., _city-scale
traffic_), the microscopic level of traffic (i.e., _autonomous driving_), the
connection between the two levels, and beyond. I will discuss a few of them in
each category in the following.
#### Macroscopic Level
On the “city-scale traffic” side, first of all, it would be useful to develop
an interactive simulation platform. Using the platform, policy makers, city
planners, and other users can easily edit a road network and alter a transport
policy in order to test the effectiveness of these changes via observing the
response of simulated traffic flows propagating in a city. Building such a
platform would require several elements: a 3D virtual environment with a user
interface, a road network construction mechanism Wilkie et al., (2012);
Musialski et al., (2013), a road network editing mechanism Chen et al.,
(2008), and a real-time traffic simulation technique Sewall et al., 2011b ;
Wilkie et al., (2013); Garcia-Dorado et al., (2014). The 3D virtual
environment can be built using a game engine such as Unity
(https://unity.com/) or Unreal (https://www.unrealengine.com). The rest of the
elements have been explored to various degrees in existing studies. Unifying
these elements would be an interesting topic.
Secondly, simulating city-scale traffic, depending on the levels of detail,
can be computationally prohibitive. A scalable approach that can combine
modern machine learning techniques and traffic flow models is highly
desirable. Such a approach can especially benefit applications with highly
interactive and real-time demands such as the simulation platform mentioned
above. The metamodel-based simulation optimization presented in Chapter 3 is
an example work in this direction. However, the functional component of the
metamodel is currently chosen to be quadratic polynomial, which offers limited
expressiveness. With the emergence of deep learning LeCun et al., (2015), it
would be interesting to replace the quadratic polynomial with a deep neural
network or an LSTM network for potential improvements on the traffic
reconstruction accuracy.
Thirdly, traffic participants are not limited to cars, mixed traffic involving
cars, pedestrians, cyclists, and other motorists are commonly seen in many
regions across the globe. A simulation model that can encompass all these
different traffic modalities can enrich various real-world and virtual-world
applications mentioned in the previous chapters. In order to achieve this
goal, knowledge from traffic engineering literature can be exploited. Mixture
models, although not necessarily built for simulation, have been explored and
developed Faghri and Egyháziová, (1999); Laxman et al., (2010). Examining the
possibility of extending these models for mixture traffic simulation is a
promising research direction.
#### Microscopic Level
On the “autonomous driving” side, the system presented in Chapter 4 is an end-
to-end system, which means a single model is trained to map the sensor input
directly to the control command output. Such an approach is straightforward
and usually results in a more compact model as it does not contain
intermediate steps. However, an end-to-end system based on deep learning can
be hard to interpret. Also, a large number of training examples are often
needed to train an end-to-end model.
In contrast, the traditional engineering pipeline, which consists of several
modules, can be adopted for autonomous driving. In this approach, the sensor
input will get processed and passed to its subsequent modules such as
detection, tracking, and planning, before the final control command is
produced. This conventional approach has both advantages and disadvantages.
Advantages include, since there are multiple modules in the pipeline, less
training examples are needed to learn a model; the prior knowledge can be
incorporated into the problem; the explainability is improved as the final
control command is produced by a planning algorithm rather than directly from
the raw sensor input. Disadvantages include, the uncertainty and errors of
each module are difficult to propagate backwards to its preceding modules,
thus causing the system to suffer potential compounding errors; the
computation is not shared between modules: each module is trained
independently for a different objective; human experts are usually needed to
tailor each module so that the system can achieve maximum performance.
While both end-to-end and traditional engineering approaches have their own
characteristics, given that the safety is of the leading concerns these days
regarding autonomous driving, the traditional engineering approach is likely
to prevail in the near future due to its superior explainability and
controllability. Hence, it would be interesting to develop the “traditional
engineering” version of ADAPS. The only element needs to be modified is the
hierarchical control policy, which currently is represented by a single model
with three neural networks. The other elements such as the simulation
platforms and online learning mechanism remain applicable.
Another aspect can be improved in ADAPS is the generation of accident data.
Currently, the accident simulation is simple and arbitrary. However, in
traffic engineering, there exists rich literature on accident analysis and
prevention. By exploring which, a systematically way of simulating accidents
can be developed, which can bring further justifications to ADAPS on
autonomous driving training and testing. One imminent research direction is to
incorporate the pre-crash scenarios published by the National Highway Traffic
Safety Administration of the United States Najm et al., (2013) into our
simulation platform, and then develop a sampling mechanism to produce accident
data for learning a control policy.
Beyond the abovemetioned immediate research topics that can be built on top of
ADAPS, there are many other interesting research directions. In general, the
safety, control, and coordination aspects of autonomous driving all need
further exploration and development. One future research direction can be
exploring the possibility of using simulations to assist sample-efficient
learning for a control policy. Another direction, inspired by the observation
that the training of autonomous driving is largely context-dependent, is to
develop theory and practice in transferring the learned behaviors of an
autonomous vehicle from one environment to other environments. This
generalization ability, observed in humans, is largely missing in autonomous
driving at the moment.
#### Connection Between The Two Levels
Although this dissertation has been addressing the macroscopic level and the
microscopic level of traffic as two separate topics, the two aspects have
tight connection, where many applications and developments can be drawn. From
the macro-to-micro perspective, the estimated city-scale traffic conditions
can be immediately adopted for better routing and planning of AVs. The
reconstructed traffic can be incorporated into virtual environments to provide
rich traffic semantics for training the navigation and decision-making of
autonomous driving. From the micro-to-macro perspective, AVs can be treated as
probe vehicles to gather traffic information in a city so that traffic
reconstruction can be achieved with higher accuracy. The AV can also be
dispatched to multiple road users as a sharing transportation tool. This way
not only the number of vehicles on the road is reduced, which can assist in
alleviating traffic jams, but also less space is needed to physically
accommodate the large number of vehicles, which implies additional socio-
economic benefits. Back to the macro-to-micro perspective, an efficient
traffic reconstruction technique can contribute to the design of the
dispatching algorithm for AVs to maximize their sharing functionality.
In conclusion, now and into the near future, AVs will be operating not only in
traffic but also along with human-driven vehicles. This assembly brings many
challenges as well as research opportunities. It would be interesting to
develop simulation models for the mixture of human-driven and autonomous
vehicles, with the flexibility to choose the percentage of each type of the
vehicle. As AVs can be considered part of the overall cyber-physical system,
they can serve as additional “degrees of freedom” to the traffic system, which
can be potentially “tuned” to regulate traffic flows Wu et al., (2018). The
applications range from alleviating traffic congestion to assisting flow
distribution in social gatherings or evacuation situations. Lastly, it would
be imperative to consider human factors in addition to technology development,
given essentially autonomous and intelligent systems are designed and built to
improve people’s life. As technology advances, developing collaborative rather
than competitive relationships between autonomous systems and humans is the
challenge that scientists and engineers will be facing. My future efforts will
be centered around this challenge, with the focus on building an cooperative
mixed traffic system.
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The Chisholm rational approximant is a natural generalization to two variables of the well-known single variable approximant, and has the advantage of reducing to the latter when one of the variables is set equals to 0. We present, to our knowledge, the first automated Mathematica package to evaluate diagonal Chisholm approximants of two variable series. For the moment, the package can only be used to evaluate diagonal approximants i.e. the maximum powers of both the variables, in both the numerator and the denominator, is equal to some integer $M$. We further modify the original method so as to allow us to evaluate the approximants around some general point $(x,y)$ not necessarily $(0,0)$. Using the approximants around general point $(x,y)$, allows us to get a better estimate of the result when the point of evaluation is far from $(0,0)$. Several examples of the elementary functions have been studied which shows that the approximants can be useful for analytic continuation and convergence acceleration purposes. We continue our study using various examples of two variable hypergeometric series, $\mathrm{Li}_{2,2}(x,y)$ etc that arise in particle physics and in the study of critical phenomena in condensed matter physics. The demonstration of the package is discussed in detail and the Mathematica package is provided as an ancillary file.
Program summary:
* Program Title: .
* Licensing provisions: GNU General Public License v3.0.
* Programming language: Wolfram Mathematica version 13.2.0 and beyond.
* Nature of problem: To find the diagonal rational approximant of two variable series, analogous to approximant of one variable series.
* Solution method: implementation of Chisholm's method to find the diagonal approximant of the two variable series.
§ INTRODUCTION
In practical problems it is possible that only the lower order terms of a series are known [1, 2, 3] and it is desirable to estimate the higher order terms of the series using this information. For a given series in one variable, truncated up to certain order $M$ in its variable, one can estimate the higher order terms by using rational approximants. Rational approximants are approximations of a given truncated series (which is actually infinite series but is known only up to certain order) using a rational function. A popular and widespread rational approximant for series in one variable is the approximants [4, 5, 6].
Though in physics applications the two-variable series also appear quite frequently, Appell $F_{1}$ hypergeometric series[7, 8, 9], $\text{Li}_{2,2}(x,y)$[10, 11, 12, 13, 14] series to name a few. Thus, it is desirable to have approximants analogous to approximants for the multi-variable case. There are various ways to form such multivariate approximations [15, 16, 17, 18, 19, 20, 21, 22, 23]. Though, in the present work, we are interested in the construction and study of bi-variate rational approximants. The generalisation to the bi-variate case is not straightforward, as the correct number of linear equations to determine the coefficients in the approximation cannot be formed, unlike the case of approximants. Chisholm later proposed a way to obtain the correct number of linear equations [15]. This bi-variate generalization
of approximant is the Chisholm approximant(CA), which shares some desirable properties similar to the approximant. Apart from other properties, it is also reducible to the case of approximant when one of the variables is set to 0.
With this motivation, we use the method presented in [15] to construct approximants of series in two variables. We will also modify the method so as to obtain the CA around any given point $(x,y)$. The method is further implemented in a package (Chisholm Diagonal). We focus only on the construction of the diagonal approximant, which implies that the maximum powers of both the variables, in both the numerator and denominator, is equal to some integer $M$. Though we would like to mention that there are ways to construct off-diagonal CA for both the two variable cases [17, 20]. For the case of one variable the approximant has also been used for the purpose of numerical analytic continuation [1, 24]. Motivated by this we further study applications of these approximants for the purpose of analytic continuation and convergence acceleration using examples of multivariable hypergeometric series and generalized multiple polylogarithms(MPLs). It is sometimes possible that some of the coefficients in the general series of a two-variable function are zero and hence the CA approximant does not exist. We discuss how using certain transformations we can obtain the Ca for these series. Some examples of the series with these properties are the ones that arise in the study of critical phenomena in condensed matter physics systems [25, 26, 27]. This further shows the application of these approximants in theoretical physics.
The article is organized as follows. We give a brief review of approximants in section <ref> discussing its properties with examples. In section <ref> we will discuss the method to construct the CA given in [15]. A description of the package is given in section <ref>. We then discuss various examples of elementary functions and applications of these approximants in section <ref> and <ref> respectively. This will be followed by a summary and discussion in section <ref>.
§ PADÉ APPROXIMANT
In this section, we review approximants [4, 5, 6] and discuss some of their application using examples. Padé approximant of a given series is an approximation using a rational function of a given order. Consider a series $f(x)$ around $x=0$
\begin{equation}
f(x) = \sum_{n=0}^{\infty} a_{n} x^{n}
\end{equation}
The Padé approximant of $f(x)$, denoted by $R_{M/N} \equiv [M/N]$[When the pade approximant is evaluated around point $x=a$ then we would denote it as $[M/N]_{a}$ and $[M/N]_{0}$ is denoted as $[M/N]$. ], is given by
\begin{equation} \label{eqn:pade0}
R_{M/N} = \frac{\sum_{i=0}^{M} p_{i} x^{i}}{1+\sum_{j=1}^{N} q_{j} x^{j}} = \frac{P(x)}{Q(x)}
\end{equation}
where $P(x)$ and $Q(x)$ are polynomials of degree $M$ and $N$ respectively. When the degree of the polynomial in numerator and denominator is the same then the approximant, $[M/M]$, is called the diagonal approximant.
The coefficients $p_{i}$ and $q_{j}$ can be obtained by setting
\begin{align}
f(x) = \frac{P(x)}{Q(x)}
\end{align}
\begin{align}
\left[1+\sum_{j=1}^{N} q_{j} x^{j}\right] \left[\sum_{n=0}^{\infty} a_{n} x^{n}\right] = \sum_{i=0}^{M} p_{i} x^{i}
\end{align}
By collecting different powers of $x$, one can find a set of equations, which is needed to be solved. We, hereon, specialise for the case of diagonal approximants i.e. $M=N$, and we get following set of equations
\begin{align}
p_{0} &= a_{0} \nonumber \\
p_{1} &= a_{1} + a_{0}q_{1} \nonumber \\
\vdots \nonumber \\
p_{M} &= a_{M} + a_{M-1} q_{1} + \cdots + a_{0} q_{M} \nonumber \\
0 &= a_{M+1}+ a_{M} q_{1}+ \cdots + a_{1}q_{M} \nonumber \\ \vdots \nonumber \\
0 &= a_{2M}+ a_{2M-1}q_{1} + \cdots + a_{M}q_{M}
\end{align}
Further, without loss of generality, one can normalize the series: $p_{0}= a_{0} =1$. With this we have $2M$ unknown coefficients and $2M$ linear equations to solve.
There exist various algorithms for efficient calculation of these coefficients [28, 29]. When the solution of the set of linear equations exists, Padé approximant is unique for a given $M$ (and $N$ in general).
(a) Comparison of the percentage error of Taylor series truncated at $\mathcal{O}(x^{10})$ and Padé approximants $[5/5]$ around point $x= 0$, in red and blue respectively. (b) Comparison of the percentage error of Padé approximants $[5/5]$ evaluated around point $x= 0$ and $x= 1.5$, in red and blue respectively. The maximum % error for the case of $[5/5]_{1.5}$ is of the order of $10^{-7}$ for the range of $x$ shown in the plot.
An important feature of the approximant is that it often gives a better approximation of the series than the corresponding truncated Taylor series it is constructed from. To show this, we take the series of $e^{x}$, and compare the result of the Taylor series truncated at $\mathcal{O}(x^{10})$ with that of approximant $[5/5]$ of $e^{x}$. In Fig.(<ref>) we plot the corresponding percentage error for a range of values of $x$. We clearly see that the results obtained using approximant agree better with the exact function for a larger range of $x$ as compared to the truncated Taylor series.
The approximant obtained using the Eq.(<ref>) is obtained around the point $x=0$. Due to this the approximant defined using Eq.(<ref>) tends to deviate from the exact result as we move further away from $x=0$. We can generalize this process and evaluate the approximant around any given point $x= a$ and obtain the approximant $[M/N]_{a}$. This allows us to get better results farther away from $x=0$ and around our point of interest, $x=a$. In Fig.(<ref>) we show the comparison of two approximants obtained around $x=0$ and $x=1.5$ by plotting the percentage error using the two approximants in red and blue respectively. We see from the plot that if our point of interest is $x=2$ then $[5/5]_{1.5}$ agrees better with the exact $e^{x}$, as compared to $[5/5]$.
Another interesting feature of approximant is that it may sometimes provide results outside the domain of convergence of the corresponding Taylor series it is constructed from. As an example, consider the Gauss hypergeometric $_2F_1$ series
\begin{align}
_2F_1(a,b,c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}
\end{align}
which is valid in $|z|<1$. Clearly, the points $z= \frac{1}{2} \left(1\pm i \sqrt{3}\right)$ do not belong to the defining domain of convergence of the $_2F_1$ series. It turns out that, the well-known analytic continuations of $_2F_1(\dots; z)$ around $z=1$ or $z=\infty$ can not be used to find the value of the function at these points. A special treatment [30] is required to evaluate the value of the Gauss $_2F_1$ at these special points.
In the following, we find the value of $_2F_{1}(1/2,1/3,1/5;z)$ at $z= \frac{1}{2} \left(1- i \sqrt{3}\right)$ using the $[10/10]$ approximants and compare with the result obtained using the inbuilt command .
We first store terms up to $\mathcal{O}(x^{20})$ of the Gauss $_2F_1(1/2,1/3,1/5;z)$ series in the variable .
ser = Normal[Series[Hypergeometric2F1[1/2,1/3,1/5,z],{z,0,20}]];
Next, we find the approximation using the inbuilt command at $x=0$ of order $10$.
approx = PadeApproximant[ser,{z,0,10}];
Finally, we evaluate the at the point $z = \frac{1}{2} \left(1- i \sqrt{3}\right)$ and compare with implementation of
{N[approx /.x->(1-I3)2,10], N[Hypergeometric2F1[12,13,15,(1-I3)2],10]}
{0.7062090573-0.8072538749 I,0.7062090573-0.8072538748 I}
We see that the result obtained using the approximant is quite accurate and matches well with the implementation in . We also see that we just need 20 terms in the series of $_2F_1$ to obtain the $[10/10]$ approximant.
§ CHISHOLM APPROXIMANT
In this section, we briefly outline the procedure to construct Chisholm approximant (CA) for two variable series following [15]. Consider a bi-variate series of the following form
\begin{equation}\label{eq:genseries}
= \sum_{m,n=0}^{\infty} c_{mn}x^{m}y^{n}
\end{equation}
We seek rational diagonal approximants of the series above. Similar to the case of approximant we denote the CA of order $M$ as $[M/M]$. If on the other hand, the CA is obtained around the point $(a,b)$ then it is denoted as $[M/M]_{(a,b)}$. Thus for $f_{M,M}(x,y) \equiv [M/M]$ approximants we have
\begin{equation}
f_{M,M}(x,y) = \frac{\sum_{p,q=0}^{M} a_{pq}x^{m}y^{n}}{\sum_{r,s=0}^{M} b_{rs}x^{r}y^{s}}
\end{equation}
where $a_{p q}$ and $b_{r s}$ are coefficients to be determined.
Without loss of generality, we can assume that $c_{00}=1$. This will allow us to choose
\begin{equation*}
a_{00}= b_{00}=1
\end{equation*}
The total number of coefficients to be determined for $f_{M,M}$ is
\begin{equation}
2[(M+1)^{2}-1] = 2M^{2}+4M
\end{equation}
Thus, we need the same number of equations to solve the unknown coefficients. To demonstrate how one can construct the required number of equations, we consider the $f_{1,1}$ approximant of the general series given by Eq.(<ref>).
\begin{equation}\label{eq:Chisholm1}
f_{1,1}(x,y) = \frac{1+ a_{10}x +a_{01}y+a_{11}x y}{1+ b_{10}x +b_{01}y+b_{11}x y}
\end{equation}
The coefficients of the approximation can be found by setting
\begin{align}
f(x,y) = f_{1,1} (x,y)
\end{align}
\begin{equation}
f(x,y) \left[1+ b_{10}x +b_{01}y+b_{11}x y\right]= 1+ a_{10}x +a_{01}y+a_{11}x y
\end{equation}
To get the right number of consistency equations to solve for coefficients we need the following expansion of $f(x,y)$
\begin{equation}\label{eq:func11}
f(x,y) = 1+ c_{10}x + c_{01}y + c_{11}xy + c_{20}x^{2}+c_{02}y^{2}+c_{21}x^{2}y + c_{12}xy^{2}
\end{equation}
Using Eq. (<ref>) and Eq. (<ref>) we now form the following two sets of equations
* By comparing coefficients of $x, y, xy, x^{2}$ and $y^{2}$, we obtain
\begin{align}\label{eq:consistent1}
b_{10}+c_{10} &= a_{10} \nonumber \\
b_{01}+c_{01} &= a_{01} \nonumber \\
(b_{11}+c_{11})+(b_{10}c_{01} +b_{01}c_{10}) &= a_{11}\nonumber \\
c_{20}+c_{10}b_{10}&=0 \nonumber \\
c_{02}+c_{01}b_{01}&=0 \nonumber \\
\end{align}
* From above we see that we have already obtained 5 equations and we have a total of 6 unknowns so we need one more equation. We form it by adding the coefficients of $x^{2}y$ and $xy^{2}$. We thus obtain the following equation
\begin{align}\label{eq:consistent2}
(c_{20}+c_{11})b_{01}+(c_{11}+c_{02})b_{10}+ (c_{10}+c_{01})b_{11}+c_{21}+c_{12} &=0
\end{align}
The above strategy, shown specifically for $[1/1]$ CA is a special case of the general procedure to find $[M/M]$ CA for any $ M$. The approximants thus obtained, have the following properties
* The approximants are symmetrical in the variables $x$ and $y$.
* The approximants when they exist are unique.
* If $x=0$ or $y=0$ then the approximants become the diagonal approximants in the other variable.
* The approximants are invariant under all transformations of the group
\begin{equation}
x = \frac{A u}{1- B u}, \quad y = \frac{A v}{1- B v} \quad (A \neq 0).
\end{equation}
* Consider the reciprocal of the series Eq. (<ref>) which is given by
\begin{equation}\label{eq:genseriesreci}
\frac{1}{f(x,y)}= \sum_{m,n=0}^{\infty} d_{mn}x^{m}y^{n}
\end{equation}
then the reciprocal of the approximant defined from Eq.(<ref>) is equal to the corresponding approximant defined from
It has been also shown in [15] that these are the only approximants satisfying all the above properties.
The approximants formed from Eq.(<ref>) are constructed around the point $(x,y) = (0,0)$. Similar to the case of the approximants we can modify the method to obtain the CA around any point $(a,b)$. To do this we need the series of the following form
\begin{equation}
\sum_{m,n=0}^{\infty} c'_{mn}(x-a)^{m}(y-b)^{n}
\end{equation}
Analogous to series (<ref>), we assume the series is of the following form
\begin{equation}\label{eq:genseriesab}
\sum_{m,n=0}^{\infty} c'_{mn}X^{m}Y^{n}
\end{equation}
where $X=x-a$ and $Y=y-b$.
With the series given in Eq. (<ref>), we now repeat the procedure discussed above and obtain approximant in the new variables $X$ and $Y$. Finally, in the approximant obtained we substitute back $X=x-a$ and $Y=y-b$, to obtain the CA around point $(a,b)$. In a later section <ref> we use such a procedure to find CA of series with $X$ and $Y$ as general functions of variables $x$ and $y$.
§ DESCRIPTION OF
The method presented in section
<ref> has been automatized in the accompanying package . We demonstrate the usage of the package below.
After downloading the package and putting it in the same directory as the notebook we can call the package as follows:
ChrisholmD.wl 1.0
Authors : Souvik Bera Tanay Pathak
The only command of the package is , which can be called as follows
The various elements of the input are given below.
* : This is the series for which we want to determine its CA. The series is always given around $(0,0)$ even for the cases when the approximant is to be determined around point $(a,b)$.
* : It is a list containing three elements. and refer to the point around which the approximant is to be determined and refers to the required order of the approximant.
* : It is a list containing two entries. These are the variables of the series and also the resulting approximant.
The output of the above command is the CA of the of order around the point ().
Let us illustrate the usage of the command by a simple example of double variable series of
$\exp\left(x+y\right)$. To obtain its CA around $(x,y)=(0,0)$ of order $1$, we can use the following command, where the series of $\exp\left(x+y\right)$ is stored in the variable
Note that, the output expression is symmetric in $x$ and $y$. Substituting $y=0$, we obtain
\begin{equation*}
\frac{1+\frac{x}{2}}{1-\frac{x}{2}}.
\end{equation*}
This expression is the well-known $[1/1]$ approximant of $e^{x}$ and can be easily verified in .
To form the correct number of consistency equations for evaluation of $[M/M]$ approximant(analogous to Eqn.(<ref>) and (<ref>)) we need to provide all the terms of the series of the form $x^{\alpha}y^{\beta}$ such that $\alpha+\beta \leq 2M+1, \{ \alpha,\beta\} \neq 2M+1$. Giving extra terms won't affect the computation and the result, but if the correct terms are not present in the series then there would be an error message displayed saying : .
§ EXAMPLES OF TWO-VARIABLE SERIES
In this section, we numerically study the CAs of some elementary series and compare their result with the exact series.
We will take the examples of elementary functions: $\exp(\frac{x+y}{2}),\, \sin\left(\frac{x+y}{2}\right),\, \sinh\left(\frac{x+y}{2}\right)$ and $\log(1+x+y)$.
§.§ $\exp\left(\frac{x+y}{2}\right)$
We can obtain the CA for $\exp(\frac{x+y}{2})$ around $(0,0)$ using the following command
The CA of the same function around $(3,6)$ can be obtained in a similar way
In Table (<ref>) and (<ref>) we compare the values obtained using the CA and the values obtained using the in-built functions. We also provide the percentage error in the evaluation of values using the CA. The Table (<ref>) corresponds to the values obtained using the CA around $(0,0)$ while the Table (<ref>) corresponds to the values obtained using the CA around $(3,6)$. We observe from Table (<ref>) that the error is less when the chosen points are closer to $(0,0)$ [A point $(x,y)$ is closer to $(0,0)$ if its Euclidean distance from $(0,0)$ is less as compared to others.] and worsen as we move away from the $(0,0)$. A similar pattern in the numerical values is observed in the Table (<ref>) also. Thus, for computation purposes if, for example, the point of evaluation is $(5,5)$ then it is better to use $[10/10]_{(3,6)}$ approximant than $[10/10]$ approximant.
Table of values of $\exp(\frac{x+y}{2})$
{x,y} CA Function % Error
{0,0} 1.000000000 1.000000000 0
{0,3} 4.481689070 4.481689070 $2.2\times10^{-14}$
{0,6} 20.08553692 20.08553692 $3.2\times10^{-9 }$
{0,9} 90.01712793 90.01713130 $3.7\times10^{-6 }$
{3,0} 4.481689070 4.481689070 $2.2\times10^{-14}$
{3,3} 20.08553692 20.08553692 $4.5\times10^{-14}$
{3,6} 90.01713130 90.01713130 $3.2\times10^{-9 }$
{3,9} 403.4287784 403.4287935 $3.7\times10^{-6 }$
{6,0} 20.08553692 20.08553692 $3.2\times10^{-9}$
{6,3} 90.01713130 90.01713130 $3.2\times10^{-9 }$
{6,6} 403.4287935 403.4287935 $6.4\times10^{-9}$
{6,9} 1808.042347 1808.042414 $3.8\times10^{-6}$
{9,0} 90.01712793 90.01713130 $3.7\times10^{-6 }$
{9,3} 403.4287784 403.4287935 $3.7\times10^{-6}$
{9,6} 1808.042347 1808.042414 $3.8\times10^{-6}$
{9,9} 8103.083320 8103.083928 $7.5\times10^{-6}$
Table of values obtained using CA around $(0,0)$.
{x,y} CA Function % Error
{0,0} 1.000000000 1.000000000 $1.2\times10^{-13}$
{0,3} 4.481689070 4.481689070 $1.1\times10^{-19} $
{0,6} 20.08553692 20.08553692 $5.4\times10^{-20}$
{0,9} 90.01713130 90.01713130 $1.5\times10^{-33} $
{3,0} 4.481689070 4.481689070 $1.2\times10^{-13} $
{3,3} 20.08553692 20.08553692 $5.4\times10^{-20}$
{3,6} 90.01713130 90.01713130 $7.3\times10^{-42}$
{3,9} 403.4287935 403.4287935 $5.4\times10^{-20} $
{6,0} 20.08553692 20.08553692 $1.2\times10^{-13} $
{6,3} 90.01713130 90.01713130 $2.8\times10^{-35 }$
{6,6} 403.4287935 403.4287935 $5.4\times10^{-20}$
{6,9} 1808.042414 1808.042414 $1.1\times10^{-19} $
{9,0} 90.01713130 90.01713130 $4.6\times10^{-29}$
{9,3} 403.4287935 403.4287935 $1.2\times10^{-13}$
{9,6} 1808.042414 1808.042414 $1.2\times10^{-13}$
{9,9} 8103.083928 8103.083928 $1.2\times10^{-13}$
Table of values obtained using CA around $(3,6)$.
§.§ $\sin\left(\frac{x+y}{2}\right)$
We obtain the $[10/10]$ CA for $\sin\left(\frac{x+y}{2}\right)$ around $(0,0)$ as follows
Similarly, We also obtain the $[10/10]$ CA for $\sin\left(\frac{x+y}{2}\right)$ around $(1.6,1.6)$
We compare the values obtained using the CA and the values obtained using the in-built function in Table (<ref>) and (<ref>). We observe that unlike the case of $\exp\left(\frac{x+y}{2}\right)$ the agreement between the CA and the exact function worsens quickly.
Table of values of $\sin\left(\frac{x+y}{2}\right)$
{x,y} CA Function % Error
{0.1,0.1} 0.09983341665 0.09983341665 $2.4\times10^{-41}$
{0.1,1.6} 0.7512804051 0.7512804051 $2.7\times10^{-20}$
{0.1,3.1} 0.9995736030 0.9995736030 $2.0\times10^{-14}$
{0.1,4.6} 0.7114733528 0.7114733528 $1.0\times10^{-10}$
{1.6,0.1} 0.7512804051 0.7512804051 $2.7\times10^{-20}$
{1.6,1.6} 0.9995736030 0.9995736030 $1.2\times10^{-14}$
{1.6,3.1} 0.7114733528 0.7114733528 $7.3\times10^{-11}$
{1.6,4.6} 0.04158066227 0.04158066243 $4.0\times10^{-7}$
{3.1,0.1} 0.9995736030 0.9995736030 $2.0\times10^{-14}$
{3.1,1.6} 0.7114733528 0.7114733528 $7.3\times10^{-11}$
{3.1,3.1} 0.04158066201 0.04158066243 $1.0\times10^{-6}$
{3.1,4.6} -0.6506251887 -0.6506251371 $7.9\times10^{-6}$
{4.6,0.1} 0.7114733528 0.7114733528 $1.0\times10^{-10}$
{4.6,1.6} 0.04158066227 0.04158066243 $4.0\times10^{-7}$
{4.6,3.1} -0.6506251887 -0.6506251371 $7.9\times10^{-6}$
{4.6,4.6} -0.9936946941 -0.9936910036 0.00037
Table of values obtained using CA around $(0,0)$.
{x,y} CA Function % Error
{0.1,0.1} 0.09983341665 0.09983341665 $4.1\times10^{-11}$
{0.1,1.6} 0.7512804051 0.7512804051 $3.8\times10^{-20}$
{0.1,3.1} 0.9995736030 0.9995736030 $1.9\times10^{-14}$
{0.1,4.6} 0.7114733528 0.7114733528 $1.1\times10^{-10}$
{1.6,0.1} 0.7512804051 0.7512804051 $3.8\times10^{-20}$
{1.6,1.6} 0.9995736030 0.9995736030 $2.4\times10^{-20}$
{1.6,3.1} 0.7114733528 0.7114733528 $1.8\times10^{-14}$
{1.6,4.6} 0.04158066243 0.04158066243 $1.1\times10^{-9}$
{3.1,0.1} 0.9995736030 0.9995736030 $1.9\times10^{-14}$
{3.1,1.6} 0.7114733528 0.7114733528 $1.8\times10^{-14}$
{3.1,3.1} 0.04158066243 0.04158066243 $1.0\times10^{-10}$
{3.1,4.6} -0.6506251369 -0.6506251371 $2.3\times10^{-8}$
{4.6,0.1} 0.7114733528 0.7114733528 $1.1\times10^{-10}$
{4.6,1.6} 0.04158066243 0.04158066243 $1.1\times10^{-9}$
{4.6,3.1} -0.6506251369 -0.6506251371 $2.3\times10^{-8}$
{4.6,4.6} -0.9936908505 -0.9936910036 0.000015
Table of values obtained using CA around $(1.6,1.6)$.
§.§ $\sinh(\frac{x+y}{2})$
Next, we consider the hyperbolic function
$\sinh\left(\frac{x+y}{2}\right)$ and find its $[10/10]$ CA around $(0,0)$
Analogously, the $[10/10]$ CA around $(1.6,1.6)$ is obtained as
We compare the values obtained using the CA and the values obtained using the in-built function in Table (<ref>) and (<ref>). The behaviour of CA for $\sinh(\frac{x+y}{2})$ is similar to that of $\sin(\frac{x+y}{2})$
Table of values of $\sinh\left(\frac{x+y}{2}\right)$.
{x,y} CA Function % Error
{0.1,0.1} 0.1001667500 0.1001667500 $2.4\times10^{-41}$
{0.1,1.6} 0.9561159600 0.9561159600 $2.2\times10^{-20}$
{0.1,3.1} 2.375567953 2.375567953 $1.0\times10^{-14}$
{0.1,4.6} 5.195100281 5.195100281 $2.1\times10^{-11}$
{1.6,0.1} 0.9561159600 0.9561159600 $2.2\times10^{-20}$
{1.6,1.6} 2.375567953 2.375567953 $5.3\times10^{-15}$
{1.6,3.1} 5.195100281 5.195100281 $1.2\times10^{-11}$
{1.6,4.6} 11.07645104 11.07645104 $2.3\times10^{-9}$
{3.1,0.1} 2.375567953 2.375567953 $1.0\times10^{-14}$
{3.1,1.6} 5.195100281 5.195100281 $1.2\times10^{-11}$
{3.1,3.1} 11.07645104 11.07645104 $5.3\times10^{-9}$
{3.1,4.6} 23.48589183 23.48589175 $3.6\times10^{-7}$
{4.6,0.1} 5.195100281 5.195100281 $2.1\times10^{-11 }$
{4.6,1.6} 11.07645104 11.07645104 $2.3\times10^{-9}$
{4.6,3.1} 23.48589183 23.48589175 $3.6\times10^{-7}$
{4.6,4.6} 49.73713860 49.73713190 0.000013
Table of values obtained using CA around the point $(0,0)$.
{x,y} CA Function % Error
{0.1,0.1} 0.1001667500 0.1001667500 $6.1\times10^{-13}$
{0.1,1.6} 0.9561159600 0.9561159600 $5.6\times10^{-21}$
{0.1,3.1} 2.375567953 2.375567953 $8.3\times10^{-15 }$
{0.1,4.6} 5.195100281 5.195100281 $1.5\times10^{-11}$
{1.6,0.1} 0.9561159600 0.9561159600 $5.6\times10^{-21}$
{1.6,1.6} 2.375567953 2.375567953 $2.1\times10^{-20}$
{1.6,3.1} 5.195100281 5.195100281 $5.3\times10^{-15}$
{1.6,4.6} 11.07645104 11.07645104 $1.0\times10^{-11}$
{3.1,0.1} 2.375567953 2.375567953 $8.3\times10^{-15}$
{3.1,1.6} 5.195100281 5.195100281 $5.3\times10^{-15}$
{3.1,3.1} 11.07645104 11.07645104 $9.2\times10^{-16}$
{3.1,4.6} 23.48589175 23.48589175 $1.3\times10^{-11}$
{4.6,0.1} 5.195100281 5.195100281 $1.5\times10^{-11}$
{4.6,1.6} 11.07645104 11.07645104 $1.0\times10^{-11}$
{4.6,3.1} 23.48589175 23.48589175 $1.3\times10^{-11}$
{4.6,4.6} 49.73713191 49.73713190 $1.0\times10^{-8}$
Table of values obtained using CA obtained around the point $(1.6,1.6)$.
§.§ $\log(1+x+y)$
The series of $\log(1+x+y)$ is given by
\begin{equation}\label{eq:log00}
\log(1+x+y) = \sum_{m=1}^{\infty} \frac{(-1)^{m+1}(x+y)^{m}}{m}
\end{equation}
which converges for $|x+y|< 1$. To satisfy the assumption that $a_{00}=1$ is we artificially add 1 to the above series of $\log(1+x+y)$ so as to obtain its CA. To obtain the CA for $\log\left(1+x+y\right)$ around $(0,0)$ we use following command
{x,y} CA Function % Error
{0.1,0.1} 0.1823215568 0.1823215568 $5.8\times10^{-31}$
{0.1,1.1} 0.7884573604 0.7884573604 $4.6\times10^{-14}$
{0.1,2.1} 1.163150810 1.163150810 $2.4\times10^{-10}$
{0.1,3.1} 1.435084525 1.435084525 $1.8\times10^{-8}$
{1.1,0.1} 0.7884573604 0.7884573604 $4.6\times10^{-14}$
{1.1,1.1} 1.163150810 1.163150810 $3.5\times10^{-12}$
{1.1,2.1} 1.435084525 1.435084525 $2.3\times10^{-10}$
{1.1,3.1} 1.648658626 1.648658626 $5.2\times10^{-9}$
{2.1,0.1} 1.163150810 1.163150810 $2.4\times10^{-10}$
{2.1,1.1} 1.435084525 1.435084525 $2.3\times10^{-10}$
{2.1,2.1} 1.648658619 1.648658626 $3.8\times10^{-7}$
{2.1,3.1} 1.824549134 1.824549292 $8.7\times10^{-6 }$
{3.1,0.1} 1.435084525 1.435084525 $1.8\times10^{-8}$
{3.1,1.1} 1.648658626 1.648658626 $5.2\times10^{-9}$
{3.1,2.1} 1.824549134 1.824549292 $8.7\times10^{-6}$
{3.1,3.1} 1.974099414 1.974081026 0.00093
Table of values of $\log(1+x+y)$.
In Table (<ref>), apart from the first entry all the other points lie outside the region of convergence of the series given by Eq. (<ref>).
We thus observe from Table (<ref>), that the CA formed using the series (<ref>), is also valid where the series is not and also matches well with the values obtained using the in-built function (which automatically uses the suitable analytic continuations of $\log(1+x+y)$). The matching worsens as the chosen point moves away from the point $(0,0)$ as is evident from the table. Though with increasing the order of the CA we can obtain better agreement. It is also important to note that the CA obtained using Eq.(<ref>) cannot be used for evaluating $\log(1+x+y)$ when the point $(x,y)$ lies on the cut. This is due to the fact that such an approximant does not contain any information of the cut structure of $\log(1+x+y)$ and its suitable analytic continuation should be used to construct CA that would give the correct values on the cut too.
§ APPLICATIONS
Analogous to the Padé approximants we study the use of Chisholm approximants for the analytic continuation purposes. As an application of numerical analytic continuation, we consider Appell $F_{1}$[7, 8], Appell $F_{2}$ [7, 31, 32] and $\text{Li}_{2,2}(x,y)$[10, 11, 12, 13, 14]. However, it is to be noted that the order to which the approximation is taken, affects the numerical value. We show that the values obtained using approximation are in good agreement with the values obtained by the numerical evaluation of known analytic continuations.
§.§ Appell $F_1$
We consider double variable Appell $F_1$ series. The analytic continuations of $F_1$ have been previously derived in [8]. Appell $F_{1}$ is defined as follows [9, 8]
\begin{align}\label{f1definition}
F_{1}(a,b_{1},b_{2},c,x,y)=\sum_{m, n=0}^{\infty} \frac{(a)_{m+n}\left(b_1\right)_m\left(b_2\right)_n}{(c)_{m+n} m ! n !} x^m y^n
\end{align}
with region of convergence : $ |x|< 1 \wedge |y| < 1 $. We discuss various properties of CA obtained for this series below.
We form $[10/10]$ approximant of the above series by taking terms from $m,n= 0 \cdots 20$[We remark that such a sum is taken for the convenient purposes. For $[10/10]$ approximant only 251 terms are required.]. The comparison of values obtained using $[10/10]$ CA and values obtained by summing the series ( Eq.(<ref>)) from 0 to 100 in each of the summation indices, are shown in Table (<ref>). Furthermore, we have the following transformation formula of $F_{1}$
\begin{equation}\label{f1ac}
F_{2}(a,b_{1},b_{2},c,x,y)= (1-x)^{b_{1}} (1-y)^{b_2} F_{2}\left(c-a,b_{1},b_{2},c,\frac{x}{x-1},\frac{y}{y-1}\right)
\end{equation}
where the RHS converges for: $\dfrac{x}{x-1}< 1 \wedge \dfrac{y}{y-1}< 1$. This relation provides the analytic continuation of $F_{1}$ and covers the whole third quadrant of the real $x-y$ plane.
ROC of AC of $F_2$ given by Eq.(<ref>).
In Table(<ref>) we compare the values obtained using $[10/10]$ CA of Eq.(<ref>) and values obtained by summing the series in Eq.(<ref>) from 0 to 100 in each of the summation indices. We see that the values obtained using CA matches even outside the ROC of Eq.(<ref>).
Comparison of values obtained using CA and the function $F_{1}\left(a,b_{1},b_{2},c,x,y\right)$ , Eq.(<ref>).
{x,y} CA Function % Error
{0.1,0.1} 1.207502432 1.207502432 $6.1 \times 10^{-24}$
{0.1,0.34} 1.471358983 1.471358983 $2.3\times 10^{-15}$
{0.1,0.58} 1.948702367 1.948702367 $1.1\times 10^{-10}$
{0.1,0.82} 3.245962293 3.245962139 $4.7\times 10^{-6}$
{0.34,0.1} 1.659460805 1.659460805 $1.8\times 10^{-15}$
{0.34,0.34} 1.961119271 1.961119271 $5.3\times 10^{-11}$
{0.34,0.58} 2.502849864 2.502849840 $9.9\times 10^{-7}$
{0.34,0.82} 3.962087028 3.961749783 0.0085
{0.58,0.1} 2.530523511 2.530523511 $7.9\times 10^{-11}$
{0.58,0.34} 2.900515131 2.900515140 $3.4\times 10^{-7}$
{0.58,0.58} 3.557119523 3.557105989 0.00038
{0.58,0.82} 5.269593568 5.298264613 0.54
{0.82,0.1} 5.155409814 5.155410035 $4.3\times 10^{-6}$
{0.82,0.34} 5.715281448 5.715371394 0.0016
{0.82,0.58} 6.686208931 6.684782078 0.021
Table of values obtained using CA of Eq.(<ref>).
{x,y} CA Eq.(<ref>) % Error
{-1.,-1.} 0.07863469382 0.07863466908 0.000031
{-1.,-1.5} 0.009242093487 0.009242025192 0.00074
{-1.,-2.} -0.04009680166 -0.04009609854 0.0018
{-1.,-2.5} -0.07715771667 -0.07715280459 0.0064
{-1.,-3.} -0.1061062366 -0.1060897986 0.015
{-1.5,-1.} -0.03466561327 -0.03466663472 0.0029
{-1.5,-1.5} -0.09715488084 -0.09716351628 0.0089
{-1.5,-2.} -0.1413096277 -0.1413367821 0.019
{-1.5,-2.5} -0.1742764748 -0.1743217876 0.026
{-1.5,-3.} -0.1998862081 -0.1999311654 0.022
{-2.,-1.} -0.1120107537 -0.1120225723 0.011
{-2.,-1.5} -0.1693922192 -0.1695202070 0.076
{-2.,-2.} -0.2095106666 -0.2099700949 0.22
{-2.,-2.5} -0.2392184278 -0.2400338776 0.34
{-2.,-3.} -0.2622159926 -0.2632658502 0.40
Comparison of values obtained using CA of Eq.(<ref>) and the values obtained using AC, Eq.(<ref>).
§.§ Appell $F_2$
Appell $F_{2}$ is defined as [9, 31, 32]
\begin{equation}\label{appellf2}
F_{2}(a,b_{1},b_{2},c_{1},c_{2},x,y)= \sum_{m, n=0}^{\infty} \frac{(a)_{m+n}\left(b_1\right)_m\left(b_2\right)_n}{(c_1)_{m}(c_2)_{n} m ! n !} x^m y^n
\end{equation}
which converges for $|x|+|y| <1$.
We compute the $[10/10]$ CA of Appell $F_{2}$ with the pochhammer parameters: $a=\frac{3}{10}, b_1=\frac{4}{10}, b_2=\frac{3}{17}, c_1=\frac{1}{5}, c_2=\frac{1}{7}$. To find the CA we take series of $F_{2}$ from $m,n =0, \cdots, 20$. We tabulate the comparison of values obtained using the CA around $(0,0)$ and using the series, Eq.<ref> in table (<ref>). In table (<ref>) we compare values obtained using the CA around $(x,y)$(the point at which the result is required) with the values obtained using the package . The values from the package as well as the values obtained by summing Eq.(<ref>) are obtained in both cases by summing the series from 0 to 150 in each of the summation indices. We observe from the tables (<ref>) and (<ref>) that the agreement between the values using CA and values obtained using is better when the points lie in the first and third quadrant.
Table of values of Appell $F_{2}$
{x,y} CA Eq.(<ref>) Error
{-0.6,-0.2} 0.7373422441 0.7364876835 0.12
{-0.6,0.2} 0.7596772196 0.7594320572 0.032
{-0.2,-0.6} 0.7889825335 0.7891964716 0.027
{-0.2,-0.2} 0.8549285608 0.8549285555 $6.2\times 10^{-7}$
{-0.2,0.2} 0.9431865106 0.9431860672 0.000047
{-0.2,0.6} 1.063513188 1.059374908 0.39
{0.2,-0.6} 0.8959261238 0.8960573837 0.015
{0.2,-0.2} 1.035523120 1.035523421 0.000029
{0.2,0.2} 1.298900246 1.298899102 0.000088
{0.6,-0.2} 1.355309400 1.356644887 0.098
{0.6,0.2} -2.473368787 2.533662025 $2.0\times 10^{2}$
Table of values obtained using CA around $(0,0)$ of Eq.(<ref>).
{x,y} CA Eq.(<ref>) Error
{-0.6,-0.2} 0.7364873706 0.7364876835 0.000042
{-0.6,0.2} 0.7597201054 0.7594320572 0.038
{-0.2,-0.6} 0.7891961013 0.7891964716 0.000047
{-0.2,-0.2} 0.8549285555 0.8549285555 $1.0\times 10^{-14}$
{-0.2,0.2} 0.9431860672 0.9431860672 $4.1\times 10^{-12}$
{-0.2,0.6} 1.059376291 1.059374908 0.00013
{0.2,-0.6} 0.8962813844 0.8960573837 0.025
{0.2,-0.2} 1.035523421 1.035523421 $2.9\times 10^{-12}$
{0.2,0.2} 1.298899102 1.298899102 $8.7\times 10^{-12}$
{0.6,-0.2} 1.356646128 1.356644887 0.000091
{0.6,0.2} 2.531769261 2.533662025 0.075
Table of values obtained CA around $(x,y)$ of Eq.(<ref>).
Outside its region of convergence, $F_2$ can be evaluated using the analytic continuations(ACs) evaluated in [32]. To illustrate the use of the package and to obtain the CA when the series is not of the form given by Eq.(<ref>), we take the following AC of Appell $F_2$ [32]
\begin{align}\label{eq:f2ac}
& F_{2}(a,b_{1},b_{2},c_{1},c_{2},x,y) = \frac{\Gamma \left(c_2\right)\Gamma \left(b_2-a\right)}{\Gamma \left(b_2\right) \Gamma \left(c_2-a\right)} (-y)^{-a} \sum_{m,n=0}^{\infty}\frac{\left(b_1\right)_m (a)_{m+n} \left(a-c_2+1\right)_{m+n}}{m! n! \left(c_1\right)_m \left(a-b_2+1\right)_{m+n}} \left(-\frac{x}{y}\right)^{m}\left(\frac{1}{y}\right)^{n} + \nonumber \\
&\frac{\Gamma \left(c_1\right) \Gamma \left(c_2\right) \Gamma \left(a-b_1-b_2\right)(-x)^{-b_1} (-y)^{-b_2}}{\Gamma (a) \Gamma \left(c_1-b_1\right) \Gamma \left(c_2-b_2\right)} \sum_{m,n=0}^{\infty} \frac{\left(b_1\right)_m \left(b_2\right)_n \left(b_1-c_1+1\right)_m \left(b_2-c_2+1\right)_n}{\left(-a+b_1+b_2+1\right)_{m+n} m! n!} \left(\frac{1}{x}\right)^{m}\left(\frac{1}{y}\right)^{n}+ (-x)^{b_2-a}\nonumber \\
& \frac{ \Gamma (c_1) \Gamma(c_2)\Gamma(a-b_2) \Gamma(-a+b_1+b_2)(-y)^{-b_2}}{\Gamma (a) \Gamma(b_1) \Gamma(c_2-b_2) \Gamma(-a+b_2+c_1)} \sum_{m,n=0}^{\infty}\frac{(b_2)_n (b_2-c_2+1)_n (a-b_2)_{m-n} (a-b_2-c_1+1)_{m-n}}{(a-b_1-b_2+1)_{m-n} m! n!} \left(\frac{1}{x}\right)^{m}\left(\frac{x}{y}\right)^{n}
\end{align}
which converges for : $\frac{1}{| x| }<1\land \left| \frac{x}{y}\right| <1\land \left| \frac{x}{y}\right| < \left|\frac{ x }{ x +1}\right|\land \frac{1}{| y| }<1\land \left| \frac{x}{y}\right| +\frac{1}{| y| }<1$.
It is to be noted that the series in Eq.(<ref>) is not of the form given by Eq.(<ref>). To find the CA for these it is first desirable to convert the series into the form given by Eq.(<ref>). We simply do this by considering the series of the following form
\begin{equation*}
\sum_{m,n=0}^{\infty} c_{mn}X^{m}Y^{n}
\end{equation*}
Here $X$ and $Y$ are function of $x$ and $y$. As an example the first series in Eq.(<ref>) has:
\begin{equation*}
c_{mn}= \frac{\left(b_1\right)_m (a)_{m+n} \left(a-c_2+1\right)_{m+n}}{m! n! \left(c_1\right)_m \left(a-b_2+1\right)_{m+n}} \quad X= -\frac{x}{y} \quad Y= \frac{1}{y}
\end{equation*}
We then obtain its CA using the following command
here, denotes the first series in Eq.(<ref>). In the CA thus obtained, as a function of $X$ and $Y$, we then substitute $X= -\frac{x}{y}$ and $Y= \frac{1}{y}$ so as to obtain the CA for the first series in Eq.(<ref>). Similarly, the procedure can be repeated for other series in Eq.(<ref>).
In Table (<ref>) we present the values obtained using the CA of Eq.(<ref>) and . The first column denotes whether the point at which Appell $F_{2}$ is evaluated lies inside the ROC of Eq.(<ref>) or not by means of True and False respectively.
ROC {x,y} CA o/p % Error
False {5.,5.} c]<EMAIL_ADDRESS>+0.08296505721 I c]<EMAIL_ADDRESS>+0.08296374276 I 0.0015
True {5.,15.} c]<EMAIL_ADDRESS>+0.03474758530 I c]<EMAIL_ADDRESS>+0.03474758527 I $8.5\times 10^{-8}$
False {15.,5.} c]<EMAIL_ADDRESS>+0.03402313454 I c]<EMAIL_ADDRESS>+0.03401893668 I 0.012
False {15.,15.} c]<EMAIL_ADDRESS>+0.01889956754 I c]<EMAIL_ADDRESS>+0.01889956359 I 0.000021
False {-15.,5.} c]<EMAIL_ADDRESS>-0.2380483933 I c]<EMAIL_ADDRESS>-0.05575437776 I $8.3\times 10^{2}$
False {-15.,15.} c]<EMAIL_ADDRESS>-0.5199484573 I c]<EMAIL_ADDRESS>-0.1778996098 I $2.2\times 10^{2}$
False {-5.,5.} c]<EMAIL_ADDRESS>-0.6223061875 I c]<EMAIL_ADDRESS>-0.4558571901 I $1.1\times 10^{2}$
True {-5.,15.} c]<EMAIL_ADDRESS>-0.01973305678 I c]<EMAIL_ADDRESS>-0.01973305706 I $2.8\times 10^{-7}$
False {-15.,-15.} 0.02446537613 0.02446537612 $3.4\times 10^{-8}$
False {-15.,-5.} 0.04005221085 0.04005451644 0.0058
True {-5.,-15.} 0.04088637870 0.04088637870 $5.4\times 10^{-9}$
False {-5.,-5.} 0.08293654594 0.08293657495 0.000035
True {5.,-15.} c]<EMAIL_ADDRESS>-0.06470315121 I c]<EMAIL_ADDRESS>-0.06470315236 I $1.1\times 10^{-6}$
False {5.,-5.} c]<EMAIL_ADDRESS>-0.3574591665 I c]<EMAIL_ADDRESS>-0.5418921340 I 33.
False {15.,-15.} c]<EMAIL_ADDRESS>-0.1974597530 I c]<EMAIL_ADDRESS>-0.1974707931 I 0.0069
False {15.,-5.} c]<EMAIL_ADDRESS>-0.7270318659 I c]<EMAIL_ADDRESS>-0.01090328199 I $8.1\times 10^{2}$
Table for values using CA of Eq.(<ref>) and using the
We observe from the above that even when the point is outside the ROC of Eq.(<ref>) the values obtained using the CA are in good agreement with the value obtained using the package . More specifically this happens when the points lie either in the first quadrant or the third quadrant and we have a mismatch for the values lying in second or fourth quadrant.
§.§ Application in condensed matter- 1
In [27, 26] it is required to obtain the two variable approximants of the zero-field susceptibility of the three-dimensional Ising model. The double series expansion of the susceptibility can be written as follows
\begin{equation}\label{eqn:condeg1}
f(z_{1},z_{2})= 1+ \sum_{l} P_{l}(z_{1})z_{2}^{l}
\end{equation}
where $z_{1}$ and $z_{2}$ are functions of coupling constants and temperature, which we omit for the present purpose. $P_{l}(z_{1})$ is a polynomial of degree $l$ in $z_{1}$. In [33] the polynomial $P_{l}(z_{1})$ has been evaluated for various systems. For the illustration purposes of the package, we will take $P_{l}(z_{1})$ to be Legendre polynomials. Though, it is not related to any physical system still it would illustrate the procedure well without loss of its features. It would also be advantageous as we would be able to calculate higher order CA in contrast to [33] where such polynomials are not known up to very high order. We present the result in the accompanying file in the section .
We now describe some of the features of this study. Firstly, we note that for the series given by Eq. (<ref>) the CA does not exist as the correct number of consistency equations cannot be formed for a given order. To remove this problem we would instead consider the following transformation of the variables
\begin{equation*}
z_{1} \to x-y \quad z_{2} \to x+y.
\end{equation*}
After the above transformation, we further need to add $x+y$ to the resulting series to as to obtain CA. Finally from the CA thus obtained we would subtract $x+y$. It can be done using the following command
where $f(x,y)$ is the Eq.(<ref>) after transformation. In the above obtain CA then do the reverse transformation:
\begin{equation*}
x \to \frac{z_{1}+z_{2}}{2}, \quad y \to \frac{z_{2}- z_{1}}{2}
\end{equation*}
to obtain the approximants in terms of $z_{1}$ and $z_{2}$.
In Table (<ref>) we compare the values obtained using the CA thus obtained and summing the series given by Eq.(<ref>) from $l=0 \cdots 100$.
{x,y} CA Eq.(<ref>) % Error
{0.01,0.01} 1.000050004 1.000050004 $7.7\times 01^{-45}$
{0.01,0.21} 0.9806278224 0.9806278224 $1.7\times 01^{-17}$
{0.01,0.41} 0.9285167140 0.9285167140 $1.0\times 01^{-11}$
{0.01,0.61} 0.8575244528 0.8575244529 $1.6\times 01^{-8}$
{0.01,0.81} 0.7808926017 0.7808926175 $2.0\times 01^{-6}$
{0.21,0.01} 1.002056325 1.002056325 $6.9\times 01^{-21}$
{0.21,0.21} 1.022807183 1.022807183 $7.4\times 01^{-18}$
{0.21,0.41} 1.002056325 1.002056325 $5.2\times 01^{-15}$
{0.21,0.61} 0.9466454705 0.9466454705 $7.2\times 01^{-12}$
{0.21,0.81} 0.8717431760 0.8717431762 $3.1\times 01^{-8}$
{0.41,0.01} 1.004074771 1.004074771 $6.9\times 01^{-16}$
{0.41,0.21} 1.070943751 1.070943751 $2.5\times 01^{-12}$
{0.41,0.41} 1.096388415 1.096388415 $9.7\times 01^{-12}$
{0.41,0.61} 1.070943751 1.070943751 $1.3\times 01^{-11}$
{0.41,0.81} 1.004074771 1.004074771 $2.2\times 01^{-9}$
{0.61,0.01} 1.006105463 1.006105463 $3.5\times 01^{-13}$
{0.61,0.21} 1.126586259 1.126586259 $2.0\times 01^{-9 }$
{0.61,0.41} 1.223613551 1.223613552 $5.2\times 01^{-8 }$
{0.61,0.61} 1.261986642 1.261986643 $9.9\times 01^{-8 }$
{0.61,0.81} 1.223613551 1.223613552 $9.9\times 01^{-8 }$
{0.81,0.01} 1.008148527 1.008148527 $2.5\times 01^{-11}$
{0.81,0.21} 1.191912890 1.191912892 $2.1\times 01^{-7}$
{0.81,0.41} 1.408729931 1.408730186 0.000018
{0.81,0.61} 1.613950569 1.613953225 0.00016
{0.81,0.81} 1.705228240 1.705233720 0.00032
Table of values obtained using CA of Eq.(<ref>) and Eq.(<ref>)
§.§ Application in condensed matter- 2
Another example where such approximants have been useful has been in the study of critical phenomena in [25, 27]. The function that is of interest in these studies is the following
\begin{equation}\label{eqn:condeg2}
f(z_{1},z_{2}) = \frac{1}{e^{z_{1} z_{2}}-z_{2}}
\end{equation}
We again observe that similar to the example in sub-section (<ref>), CA cannot be obtained using the series of the right-hand side of Eq.(<ref>). Repeating the same procedure as described in sub-section (<ref>) we obtain the CA for Eq.(<ref>). The comparison of values thus obtained are presented in Table (<ref>). The value of the function is obtained using the in-built function.
{x,y} CA Eq.(<ref>) % Error
{0.1,0.1} 1.098840521 1.098840521 $1.3\times 01^{-28}$
{0.1,1.1} 61.43234252 61.43234252 $3.0\times 01^{-10}$
{0.1,2.1} -1.154305335 -1.154305292 $3.7\times 01^{-6}$
{0.4,0.1} 1.062912997 1.062912997 $2.9\times 01^{-20}$
{0.4,1.1} 2.208933189 2.208933189 $1.7\times 01^{-9 }$
{0.4,2.1} 4.621756970 4.621777384 0.00044
{0.7,0.1} 1.028268985 1.028268985 $5.8\times 01^{-16}$
{0.7,1.1} 0.9436043051 0.9436043056 $4.9\times 01^{-8}$
{0.7,2.1} 0.4445907354 0.4445955791 0.0011
{1.,0.1} 0.9948556828 0.9948556828 $4.7\times 01^{-13}$
{1.,1.1} 0.5251642849 0.5251642910 $1.2\times 01^{-6}$
{1.,2.1} 0.1648319239 0.1648486631 0.010
{1.3,0.1} 0.9626229087 0.9626229087 $7.7\times 01^{-11}$
{1.3,1.1} 0.3248124384 0.3248125061 0.000021
{1.3,2.1} 0.07541934296 0.07556929931 0.20
{1.6,0.1} 0.9315229375 0.9315229375 $4.7\times 01^{-9}$
{1.6,1.1} 0.2122038089 0.2122044106 0.00028
{1.6,2.1} 0.03776965663 0.03746835206 0.80
{1.9,0.1} 0.9015103550 0.9015103563 $1.5\times 01^{-7}$
{1.9,1.1} 0.1431607395 0.1431656615 0.0034
{1.9,2.1} 0.01957320339 0.01924746664 1.7
{0.81,0.21} 1.191912890 1.191912892 $2.1\times 01^{-7}$
{0.81,0.41} 1.408729931 1.408730186 0.000018
{0.81,0.61} 1.613950569 1.613953225 0.00016
{0.81,0.81} 1.705228240 1.705233720 0.00032
Table of values obtained using CA of Eq.(<ref>)
and Eq.(<ref>)
§.§ $\mathrm{Li}_{2,2}(x, y)$
The classical polylogarithms are defined by iterative integrals
\begin{align}
\text{Li}_{n+1} (x) = \int_0^x \frac{\text{Li}_n(t)}{t} dt
\end{align}
It can also be represented as infinite sums
\begin{align}
\text{Li}_n (x) = \sum_{i=1}^\infty \frac{x^i}{i^n}
\end{align}
which is valid for $|x|<1$. The value of the polylogarithms can be found for $|x|\geq 1$ by performing analytic continuations.
Similarly, $\text{Li}_{2,2}(x,y)$ is defined as
\begin{align}
\text{Li}_{2,2}(x,y) = \sum_{i>j>0}^\infty \frac{x^i y^j}{i^2 j^2} = \sum_{i=1,j=1}^\infty \frac{x ^i (x y)^j}{(i+j)^2 j^2} \label{eqn:Li_def}
\end{align}
which is valid in the domain $|x|\leq1 \wedge |x y| \leq 1$. Note that, this order of argument in the above definition is same as that of [34, 35], but revered compared to the definitions in [13, 36].
The classical and multiple polylogarithms frequently appear in the Feynman integral calculus. There exist computer programs, that can handle the manipulation and evaluations of MPLs [37, 38, 35, 39, 40, 34].
In [41], it is conjectured that all the MPLs up to weight 4 can be expressed in terms of classical polylogarithms with weight up to 4 and $\text{Li}_{2,2}(x,y)$, which is later proved in [34]. Furthermore, the authors of the later paper provides an algorithm to evaluate the double variable series $\text{Li}_{2,2}(x,y)$. In the remainder of the section we study the CA of of the series of $\text{Li}_{2,2}(x,y)$.
It is worth pointing out that, the bivariate MPLs can be written in terms of Kampé de Fériet functions as follows
\begin{align}
\text{Li}_{p,q}(x,y) = \frac{x^2 y}{2^p}\text{KdF}^{p:1;q+1}_{p:0;q}
\left[
\setlength{\arraycolsep}{0pt}% local assignment
\begin{array}{c@{{}:{}}c@{;{}}c} 2,\dots,2& 1 & 1,\dots, 1
\\[1ex]
3, \dots, 3& \linefill & 2, \dots ,2
\end{array}
\;\middle|\;
x,x y
\right]
\end{align}
The following relations are well known from the literature which can be used to find the numerical value of $\text{Li}_{2,2}(x,y)$ beyond its defining region of convergence (ROC),
\begin{align}
\mathrm{Li}_{2,2}(x, y) &=-\mathrm{Li}_{2,2}(y, x)-\mathrm{Li}_4(x y)+\mathrm{Li}_2(x) \mathrm{Li}_2(y) \label{eqn:stuffle}\\
\mathrm{Li}_{2,2}(x, y) & =\mathrm{Li}_{2,2}\left(\frac{1}{x}, \frac{1}{y}\right)-\mathrm{Li}_4(x y)+3\left(\mathrm{Li}_4\left(\frac{1}{x}\right)+\mathrm{Li}_4(y)\right)+2\left(\mathrm{Li}_3\left(\frac{1}{x}\right)-\mathrm{Li}_3(y)\right) \log (-x y) \nonumber\\
& +\mathrm{Li}_2\left(\frac{1}{x}\right)\left(\frac{\pi^2}{6}+\frac{\log ^2(-x y)}{2}\right)+\frac{1}{2} \mathrm{Li}_2(y)\left(\log ^2(-x y)-\log ^2(-x)\right) \label{eqn:inversion}
\end{align}
The first of the two relations (i.e., Eq. (<ref>)) is known as the stuffle relation, and the second relation is known as the inversion relation.
Another relation can be obtained by applying the stuffle relation on the $\mathrm{Li}_{2,2}\left(\frac{1}{x}, \frac{1}{y}\right)$ appearing on the RHS of Eq. (<ref>),
\begin{align}
\mathrm{Li}_{2,2}(x, y) & =-\mathrm{Li}_{2,2}\left(\frac{1}{y}, \frac{1}{x}\right)-\mathrm{Li}_4\left(\frac{1}{x y}\right)+\mathrm{Li}_2\left(\frac{1}{x})\right) \mathrm{Li}_2\left(\frac{1}{y}\right)-\mathrm{Li}_4(x y)+3\left(\mathrm{Li}_4\left(\frac{1}{x}\right)+\mathrm{Li}_4(y)\right)\\
&+2\left(\mathrm{Li}_3\left(\frac{1}{x}\right)-\mathrm{Li}_3(y)\right) \log (-x y) +\mathrm{Li}_2\left(\frac{1}{x}\right)\left(\frac{\pi^2}{6}+\frac{\log ^2(-x y)}{2}\right)+\frac{1}{2} \mathrm{Li}_2(y)\left(\log ^2(-x y)-\log ^2(-x)\right) \label{eqn:stuffleinversion}
\end{align}
These relations are valid for
\begin{equation}
\label{eq:rocLi}
\begin{aligned}
\text{Eq.} \eqref{eqn:Li_def} &: \hspace{1cm}|x|<1 \wedge |x y| <1\\
\text{Eq.} \eqref{eqn:stuffle} &: \hspace{1cm}|y| < 1 \wedge |x y| < 1 \\
\text{Eq.} \eqref{eqn:inversion} &: \hspace{1cm} \left| x \right| > 1\land \left| x y \right| > 1\\
\text{Eq.} \eqref{eqn:stuffleinversion} &: \hspace{1cm} \left| x \right| < 1\land \left| x y \right| > 1
\end{aligned}
\end{equation}
Note that the first few terms of the $\mathrm{Li}_{2,2}(x, y)$ series are
\begin{align*}
\mathrm{Li}_{2,2}(x, y) = \frac{x^2 y}{4} + \frac{x^3 y}{9} + \frac{x^3 y^2}{36} + \frac{x^4 y^2}{64} + \dots
\end{align*}
To fulfil the requirement, as discussed in previous examples, we consider the function
\begin{align}
f(x,y) = 1+ x+y +\mathrm{Li}_{2,2}(x-y, x+y)
\end{align}
and find the CA of $f(x,y)$ of order 10. Later it is massaged to yield the CA of $\mathrm{Li}_{2,2}(x, y) $.
In the Table (<ref>), we compute the CAs of the
$\mathrm{Li}_{2,2}(x, y)$ series appearing in the RHS of Eq. (<ref>), Eq. (<ref>) and Eq. (<ref>) (denoted as CA1, CA2 and CA3 respectively) and compare with the result of the [35] implementation through the
package [37]. Here, CA0 denotes the CA of the defining series (i.e., Eq. (<ref>)).
It is to be noted that, the $\text{Li}_n$ series appearing in the expression of the analytic continuations of the $\mathrm{Li}_{2,2}(x, y)$ can be approximated using the Padé approximation. However, those are evaluated with the inbuilt commands.
We take values of $(|x|,|y|)$ from $(0.1,0.1)$ to $(2.1,2.1)$ in the interval of 0.5 for each of the variables and compute the CAs at these points. A small negative imaginary part is added to both the variables in order to avoid the branch cut issues of polylogarithms, which is not shown explicitly in Table (<ref>). The first column of the Table indicates which region of convergences of the series $\mathrm{Li}_{2,2}(x, y)$ (Eq. (<ref>)) the selected point belongs to. We observe that, in some instances, even if the chosen point does not belong to the region of convergence of the series, still its CA produces the correct result when compared to the implementaion.
Table of values of $\text{Li}_{2,2}(x,y)$
ROCs $\{|x|,|y|\}$ CA0 CA1 CA2 CA3 o/p
False,False} {0.1,0.1} c]<EMAIL_ADDRESS>$-7.99326\times 10^{-13}$ I c]<EMAIL_ADDRESS>$-7.99326\times10^{-13} $I c]<EMAIL_ADDRESS>+21.689 I c]<EMAIL_ADDRESS>-21.689 I c]<EMAIL_ADDRESS>$+0. \times10^{-13}$ I
False,False} {0.1,0.6} c]<EMAIL_ADDRESS>$-3.51482\times10^{-12}$ I c]<EMAIL_ADDRESS>$-3.51482\times10^{-12}$ I c]<EMAIL_ADDRESS>+2.56458 I c]<EMAIL_ADDRESS>+9.35645 I c]<EMAIL_ADDRESS>$+0.\times10^{-12}$ I
False,False} {0.1,1.1} c]<EMAIL_ADDRESS>$-6.27839\times10^{-12}$ I c]<EMAIL_ADDRESS>-0.0307264 I c]<EMAIL_ADDRESS>-0.0954793 I c]<EMAIL_ADDRESS>+3.84641 I c]<EMAIL_ADDRESS>$+0.\times10^{-12}$ I
False,False} {0.1,1.6} c]<EMAIL_ADDRESS>$-9.09157\times10^{-12}$ I c]<EMAIL_ADDRESS>-0.151521 I c]<EMAIL_ADDRESS>-0.995102 I c]<EMAIL_ADDRESS>+1.32557 I c]<EMAIL_ADDRESS>$+0.\times10^{-12}$ I
False,False} {0.1,2.1} c]<EMAIL_ADDRESS>$-1.19236\times10^{-11}$ I c]<EMAIL_ADDRESS>-0.239188 I c]<EMAIL_ADDRESS>-1.35315 I c]<EMAIL_ADDRESS>+0.632078 I c]<EMAIL_ADDRESS>$+0.\times10^{-11}$ I
False,False} {0.6,0.1} c]<EMAIL_ADDRESS>$-1.82808\times10^{-11}$ I c]<EMAIL_ADDRESS>$-1.82808\times10^{-11}$ I c]<EMAIL_ADDRESS>-9.35645 I c]<EMAIL_ADDRESS>-2.56458 I c]<EMAIL_ADDRESS>$+0.\times10^{-11}$ I
False,False} {0.6,0.6} c]<EMAIL_ADDRESS>$-4.80714\times10^{-11}$ I c]<EMAIL_ADDRESS>$-4.80714\times10^{-11}$ I c]<EMAIL_ADDRESS>-3.62341 I c]<EMAIL_ADDRESS>+3.62341 I c]<EMAIL_ADDRESS>$+0.\times10^{-11}$ I
False,False} {0.6,1.1} c]<EMAIL_ADDRESS>$-8.29067\times10^{-11}$ I c]<EMAIL_ADDRESS>-0.217858 I c]<EMAIL_ADDRESS>-1.96513 I c]<EMAIL_ADDRESS>+0.12099 I c]<EMAIL_ADDRESS>$+0.\times10^{-11}$ I
False,False} {0.6,1.6} c]<EMAIL_ADDRESS>$-1.26853\times10^{-10}$ I c]<EMAIL_ADDRESS>-1.07432 I c]<EMAIL_ADDRESS>-1.22965 I c]<EMAIL_ADDRESS>+0.0000562957 I c]<EMAIL_ADDRESS>$+0.\times10^{-10}$ I
False,True} {0.6,2.1} c]<EMAIL_ADDRESS>$-1.66348\times10^{-10}$ I c]<EMAIL_ADDRESS>-1.68944 I c]<EMAIL_ADDRESS>-0.889454 I c]<EMAIL_ADDRESS>-0.00749342 I c]<EMAIL_ADDRESS>-0.00749 I
False,False} {1.1,0.1} c]<EMAIL_ADDRESS>$-1.65742\times10^{-10}$ I c]<EMAIL_ADDRESS>-0.0307264 I c]<EMAIL_ADDRESS>-3.87714 I c]<EMAIL_ADDRESS>+0.0647529 I c]<EMAIL_ADDRESS>-0.030726 I
False,False} {1.1,0.6} c]<EMAIL_ADDRESS>$-7.06191\times10^{-10}$ I c]<EMAIL_ADDRESS>-0.217858 I c]<EMAIL_ADDRESS>-0.338848 I c]<EMAIL_ADDRESS>+1.74727 I c]<EMAIL_ADDRESS>-0.21786 I
True,False} {1.1,1.1} c]<EMAIL_ADDRESS>$-1.60629\times10^{-9}$ I c]<EMAIL_ADDRESS>-1.17132 I c]<EMAIL_ADDRESS>-0.58566 I c]<EMAIL_ADDRESS>-0.58566 I c]<EMAIL_ADDRESS>-0.58566 I
True,False} {1.1,1.6} c]<EMAIL_ADDRESS>$-1.38643\times10^{-9}$ I c]<EMAIL_ADDRESS>-3.52497 I c]<EMAIL_ADDRESS>-1.23401 I c]<EMAIL_ADDRESS>-1.23401 I c]<EMAIL_ADDRESS>-1.2340 I
True,False} {1.1,2.1} c]<EMAIL_ADDRESS>$-2.17691\times10^{-9}$ I c]<EMAIL_ADDRESS>-5.00396 I c]<EMAIL_ADDRESS>-1.899 I c]<EMAIL_ADDRESS>-1.899 I c]<EMAIL_ADDRESS>-1.8990 I
False,False} {1.6,0.1} c]<EMAIL_ADDRESS>$+3.02096\times10^{-8}$ I c]<EMAIL_ADDRESS>-0.151521 I c]<EMAIL_ADDRESS>-1.47709 I c]<EMAIL_ADDRESS>+0.843581 I c]<EMAIL_ADDRESS>-0.15152 I
False,False} {1.6,0.6} c]<EMAIL_ADDRESS>$-1.37615\times10^{-9}$ I c]<EMAIL_ADDRESS>-1.07432 I c]<EMAIL_ADDRESS>-1.07438 I c]<EMAIL_ADDRESS>+0.155323 I c]<EMAIL_ADDRESS>-1.07432 I
True,False} {1.6,1.1} c]<EMAIL_ADDRESS>$-6.56091\times10^{-8}$ I c]<EMAIL_ADDRESS>-3.52497 I c]<EMAIL_ADDRESS>-2.29096 I c]<EMAIL_ADDRESS>-2.29096 I c]<EMAIL_ADDRESS>-2.2910 I
True,False} {1.6,1.6} c]<EMAIL_ADDRESS>$-3.05954\times10^{-8}$ I c]<EMAIL_ADDRESS>-6.69136 I c]<EMAIL_ADDRESS>-3.34568 I c]<EMAIL_ADDRESS>-3.34568 I c]<EMAIL_ADDRESS>-3.3457 I
True,False} {1.6,2.1} c]<EMAIL_ADDRESS>$-1.23116\times10^{-9}$ I c]<EMAIL_ADDRESS>-8.33243 I c]<EMAIL_ADDRESS>-4.20211 I c]<EMAIL_ADDRESS>-4.20211 I c]<EMAIL_ADDRESS>-4.2021 I
False,False} {2.1,0.1} c]<EMAIL_ADDRESS>$+2.0216\times10^{-10}$ I c]<EMAIL_ADDRESS>-0.239188 I c]<EMAIL_ADDRESS>-0.871266 I c]<EMAIL_ADDRESS>+1.11396 I c]<EMAIL_ADDRESS>-0.23919 I
True,False} {2.1,0.6} c]<EMAIL_ADDRESS>$-1.30701\times10^{-10}$ I c]<EMAIL_ADDRESS>-1.68944 I c]<EMAIL_ADDRESS>-1.68195 I c]<EMAIL_ADDRESS>-0.799988 I c]<EMAIL_ADDRESS>-1.6819 I
True,False} {2.1,1.1} c]<EMAIL_ADDRESS>$-1.0927\times10^{-8}$ I c]<EMAIL_ADDRESS>-5.00396 I c]<EMAIL_ADDRESS>-3.10496 I c]<EMAIL_ADDRESS>-3.10496 I c]<EMAIL_ADDRESS>-3.1050 I
True,False} {2.1,1.6} c]<EMAIL_ADDRESS>$-2.38178\times10^{-9}$ I c]<EMAIL_ADDRESS>-8.33243 I c]<EMAIL_ADDRESS>-4.13032 I c]<EMAIL_ADDRESS>-4.13032 I c]<EMAIL_ADDRESS>-4.1303 I
True,False} {2.1,2.1} c]<EMAIL_ADDRESS>$-3.54955\times10^{-9}$ I c]<EMAIL_ADDRESS>-9.78067 I c]<EMAIL_ADDRESS>-4.89033 I c]<EMAIL_ADDRESS>-4.89033 I c]<EMAIL_ADDRESS>-4.8903 I
The defining series of $\mathrm{Li}_{2,2}(x, y)$ (i.e., Eq. (<ref>)) is very slowly convergent around the point $\left(|x|, |y|\right) = (1,1)$, where the presented acceleration technique is found to be useful. In the Table <ref>, we show a comparison of the values obtained from summing the series defined in Eq. (<ref>) and its CA with order $(o)$ varying from 5 to 20. In the right most column, the number of terms used for the summation is indicated, which we choose for convenience to be $(2o+1)^2$ for a given order $o$. We clearly see that the value obtained using CA is more accurate compared to the value obtained by summing the series. Note that
\begin{align}
\mathrm{Li}_{2,2}(1, 1) = \frac{\pi^4}{120} = 0.811742425283354
\end{align}
Higher order CA is needed to obtain more accurate result.
Table of values of
$\mathrm{Li}_{2,2}(x, y)$ at $(1,1)$
Order value from CA value of $\mathrm{Li}_{2,2}(x, y)$ number of terms
from series in the summation
5 0.726068215009552 0.690568727620971 121
6 0.743812703465901 0.706590246937065 169
7 0.763247794268115 0.718828257083165 225
8 0.771956662975054 0.728488465125224 289
9 0.780513212719172 0.736311803998949 361
10 0.785440863070842 0.742779189825413 441
11 0.789944433469386 0.748216626989709 529
12 0.793057205586444 0.752853064361067 625
13 0.795665279868944 0.756854084618387 729
14 0.797835077625447 0.760342450837344 841
15 0.799406544586245 0.763411133663296 961
16 0.801148912443423 0.766131848591896 1089
17 0.802067561873525 0.768560812629968 1225
18 0.806191214220949 0.770742723657730 1369
19 0.803711956950532 0.772713572001726 1521
20 0.804066077181726 0.774502665799193 1681
§ SUMMARY AND DISCUSSION
We present an automation to evaluate Chisholm approximants [15] for two variable series, in . The CA are a natural generalization of the well-known approximants for the one variable case. They have the advantage of reducing to the latter when one of the variables in the approximants is set to 0. They also have various other symmetric and group properties. For the moment, we just focus on the diagonal approximants. We present several examples to demonstrate the usage of the package using some elementary functions such as $\exp(\frac{x+y}{2}), \sin(\frac{x+y}{2}), \sinh(\frac{x+y}{2})$ and $\log(1+x+y)$. For the case of $\log(1+x+y)$ we see that the CA is also valid where the Taylor series, using which the approximants have been constructed from is not valid. This shows that the CA also performs the analytic continuation in some cases. Furthermore, we show some applications of these approximants in physics. We consider examples of hypergeometric functions such as Appell $F_1$, $F_2$ and $\mathrm{Li}_{2,2}(x,y)$ show the utility of these approximants for their evaluation purposes. As has been shown for the case of $\mathrm{Li}_{2,2}(x,y)$ the
CA can also be used to accelerate the convergence of the double series. We further present the application of these approximants in the study of critical phenomena in condensed matter physics.
We emphasise that the method presented in this paper is not the only way to obtain the two-variable approximant. Other methods presented in [42, 18, 43, 44] can also be used to find the two-variable approximants. As a future problem, it would be interesting to study and compare the efficiency of these methods and the results obtained using them. We also notice that the present implementation of CA is symmetric and one can also look for possibilities of CA which break the symmetry in one of the other ways as has been already mentioned in [15]. A simple way to break the symmetry of the CA obtained is to consider the simple off-diagonal Chisholm approximants [17] analogous to approximant case. Another exciting direction of study would be to develop a package for $N-$variable approximant[16], using the simple generalization of Chisholm's method for the two-variable case. Similar to the two variable case there are various methods to form the rational approximants for the $N-$variables [20, 45, 21, 22, 46, 23] case which can be studied and compared.
§ ACKNOWLEDGEMENT
We would like to thank B. Ananthanarayan for the useful suggestions and comments on the manuscript. We would also like to thank Sudeepan Datta for stimulating discussions. This work is part of TP's doctoral thesis.
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|
# StoC-ToT: Stochastic Tree-of-Thought with Constrained Decoding for Complex
Reasoning in Multi-Hop Question Answering
Zhenyu Bi
Virginia Tech
<EMAIL_ADDRESS>
&Daniel Hajialigol
Virginia Tech
<EMAIL_ADDRESS>
&Zhongkai Sun
Amazon Alexa AI
<EMAIL_ADDRESS>
Jie Hao
Amazon Alexa AI
<EMAIL_ADDRESS>
&Xuan Wang
Virginia Tech
<EMAIL_ADDRESS>
###### Abstract
Multi-hop question answering (MHQA) requires a model to retrieve and integrate
information from multiple passages to answer a complex question. Recent
systems leverage the power of large language models and integrate evidence
retrieval with reasoning prompts (e.g., chain-of-thought reasoning) for the
MHQA task. However, the complexities in the question types (bridge v.s.
comparison questions) and the reasoning types (sequential v.s. parallel
reasonings) require more novel and fine-grained prompting methods to enhance
the performance of MHQA under the zero-shot setting. In this paper, we propose
StoC-ToT, a stochastic tree-of-thought reasoning prompting method with
constrained decoding for MHQA and conduct a detailed comparison with other
reasoning prompts on different question types and reasoning types.
Specifically, we construct a tree-like reasoning structure by prompting the
model to break down the original question into smaller sub-questions to form
different reasoning paths. In addition, we prompt the model to provide a
probability estimation for each reasoning path at each reasoning step. At
answer time, we conduct constrained decoding on the model to generate more
grounded answers and reduce hallucination. Experiments comparing StoC-ToT with
on two MHQA datasets and five large language models showed that StoC-ToT
outperforms other reasoning prompts by a significant margin.
## 1 Introduction
Question answering (QA) is a fundamental task in natural language processing
(NLP) that involves designing systems capable of understanding human language
questions and providing accurate and relevant answers. With the recent
advancement of large language models (LLMs) that demonstrated superior
reasoning ability Brown et al. (2020), researchers have been focusing more on
complex QA tasks, such as multi-hop question answering (MHQA). MHQA is more
challenging as it requires models to understand complicated questions, perform
multiple reasoning steps, and gather evidence across documents. Figure 1 shows
an example of a two-hop MHQA question. To answer that question in Figure 1,
the QA model needs to first figure out who is the actor that received the 2016
Academy Honorary Award. Then based on the answer to the previous question, the
QA model needs to further answer a second question about which movie the actor
co-starred with Chris Tucker.
State-of-the-art methods for MHQA are fully-supervised methods that often
follow a retrieve-and-read framework, including a passage retrieving module
that gathers relative evidence from documents and a reading comprehension
module to reason about the evidence Zhu et al. (2021); Li et al. (2022). Other
methods include beam-search Zhang et al. (2023) and label-smoothing Yin et al.
(2023). However, these methods often require extensive pre-training or fine-
tuning and do not generalize well to other datasets.
Figure 1: An example of the MHQA question. This question has two hops that
require the model to reason about before answering the final question.
Large language models (LLMs), on the other hand, show remarkable reasoning
ability and rich knowledge of general-domain questions. Many LLMs can answer
simple and straightforward questions that do not require complex reasoning
without any supervision involved but often fail to deal with complex questions
requiring multiple reasoning steps. To tackle the problem, researchers have
developed many prompting techniques to improve LLM’s reasoning ability, such
as chain-of-thought (CoT) Wei et al. (2022), self-consistency CoT (Sc-CoT)
Wang et al. (2023), and tree-of-thought (ToT) prompting Yao et al. (2023a).
CoT has been shown effective across tasks requiring extensive, step-by-step
reasoning, such as math calculation and reading comprehension. However, there
could be various possible reasoning paths for many complex multi-hop
questions, and CoT models cannot "turn back" when they have made a mistake
along their reasoning paths. Sc-CoT further improves on CoT by proposing
different chains of thought, thus expanding the reasoning space. However,
there is no local reasoning expansion within each chain, and the "majority
voting" strategy often fails in open-domain tasks where the output space is
unlimited. ToT, designed to maintain different reasoning paths along its
reasoning process, is more suitable for dealing with complex question types.
However, the intermediate reasoning steps in NLP generation tasks are much
less constrained and require more than a simple rule-based evaluation. The
complexities in the question types (bridge v.s. comparison questions in Table
1), as well as the reasoning types (sequential v.s. parallel reasonings in
Table 2), require more novel and fine-grained prompting methods to enhance the
reasoning ability of LLMs.
Figure 2: Overview of our framework, with the example in Figure 1. The top-
right Corner shows the overall structure of the constructed tree, with each
node’s label on the left. Darker green in the nodes means a higher evaluated
probability of the reasoning path. The original Question is colored in blue.
We chose the first round of our tree-building process as an example in the
purple block.
To tackle the challenges and design a more reliable reasoning method for open-
domain NLP tasks, we propose StoC-ToT, a stochastic ToT-based framework that
instructs the model to generate different reasoning paths from the same
question and assign probability scores to reasoning paths to effectively avoid
reasoning dead-ends. To the best of our knowledge, our work is the first to
adapt the tree-of-thought reasoning prompting to natural language tasks that
require complex reasoning, such as MHQA. We provide an example overview of our
framework in Figure 2. Specifically, we construct a tree-like reasoning
structure by prompting the model to break down the original question into
smaller sub-questions to form different reasoning paths. We evaluate the
validity of each reasoning path on three levels of aspects and arrive at a
model-given probability score. At answer time, we innovatively propose to use
constrained decoding in the answering process to reduce hallucination by
forcing the model to generate grounded answers from evidence and letting
models give concise and exact answers. Ultimately, we arrive at the best
answer by choosing the path with the highest aggregated probability score.
Experiments on two benchmarking MHQA datasets demonstrate that StoC-ToT
significantly improves the reasoning ability of LLMs in complex reasoning
scenarios, especially with GPT-4, improving Exact Match accuracy by 7%, and F1
score by 7.8 points on the HotpotQA dataset over the original tree-of-thought
prompting. Our contributions are as follows:
* •
We propose StoC-ToT, which constructs a stochastic reasoning tree in complex
reasoning scenarios. We introduce stochastic estimations on different
reasoning paths, which helps the model have a more reliable reasoning process
than previous reasoning prompting methods.
* •
We innovatively propose to use constrained decoding in the answering process.
This step reduces model hallucination by forcing the model to generate
grounded answers from evidence and letting models give concise and exact
answers.
* •
We evaluate the effectiveness of StoC-ToT by conducting experiments on two
MHQA datasets. We observe substantial improvements over other reasoning
prompting methods, with StoC-ToT surpassing all other selected reasoning
prompting baselines on 5 tested models.
## 2 Related Work
#### Multi-Hop Question Answering
Multi-hop Question Answering (MHQA) is a challenging task requiring models to
reason over different evidence across documents to answer a complex multi-hop
question. Many high-quality MHQA datasets have been developed, including
HotpotQA Yang et al. (2018), WikiHop Welbl et al. (2018), MuSiQue Trivedi et
al. (2022), and others. Among these, HotpotQA is the task’s most
representative and widely used dataset. Previous state-of-the-art MHQA models
often follow a two-stage pipeline: a retriever that extracts evidence from the
documents, and a reader that reasons about the evidence to arrive at an answer
Zhu et al. (2021); Li et al. (2022). Other methods include beam-search Zhang
et al. (2023) and label-smoothing Yin et al. (2023). Some LLM-based frameworks
Yao et al. (2023b); Gou et al. (2024) were also evaluated on the task of MHQA,
but their performance fell short compared with supervised methods.
#### Reasoning Prompting of LLMs
Various prompt engineering methods have been developed Wei et al. (2022); Wang
et al. (2023); Yao et al. (2023a); Besta et al. (2024); Sel et al. (2024);
Chen et al. (2023), aiming to improve large language models’ reasoning ability
across various tasks and domains. Chain-of-thought (CoT) prompting Wei et al.
(2022) prompts the large language models (LLMs) to divide their reasoning
process into smaller steps when solving a question, forming a chain of
thoughts. Chain-of-thought self-consistency prompting Wang et al. (2023)
improves on the CoT method by proposing different reasoning chains and
ensembles on the final result. Tree-of-thought (ToT) prompting method Yao et
al. (2023a) actively maintains a tree of thoughts, where each thought is a
coherent language sequence that serves as an intermediate step toward problem-
solving. Graph-of-thought Besta et al. (2024) further improves ToT by
constructing a Directed Graph instead of a tree. LLMs can loop over a thought
to refine it and aggregate thoughts or chains.
#### Constrained Decoding
Constrained decoding is the technique that asks the models to generate outputs
following a given set of rules. The most common way of conducting constrained
generation uses beam search Och and Ney (2004) in decoding time. Before the
LLM era, works on constrained decoding focused on task-specific sequence-to-
sequence models that span across many fields, such as machine translation
Hokamp and Liu (2017); Post and Vilar (2018), named entity recognition Lester
et al. (2020), and dialogue generation Balakrishnan et al. (2019). Recently,
Microsoft introduced Guidance 111https://github.com/guidance-ai/guidance,
which allows users of various large language models to control their outputs
given a human-defined vocabulary or rules.
## 3 Method
### 3.1 Task Formation
Given a multi-hop question $Q$ and background corpus of evidence $P$, the goal
of our framework is to output the answer $A$ to question $Q$, drawing its
reasoning with the support of multiple evidence passages $p_{1},p_{2},...$
retrieved from corpus $P$.
### 3.2 StoC-ToT Framework
For each of the questions $Q$, multiple reasoning lines and, thus, multiple
ways of breaking down the question could exist. However, not every reasoning
line would lead us to the right answer, and they take us to dead ends. To
avoid such reasoning dead-ends, we build a stochastic reasoning tree to
represent the possible reasoning lines and the probability of each reasoning
line taking us to the right answer. We achieve this by proposing a self-
interactive framework that automatically builds the reasoning tree given a
multi-hop question. Figure 2 shows our framework with an example question.
In our reasoning process, we first prompt the model to propose different
possible sub-questions to solve at each reasoning step. Each sub-question
corresponds to one possible reasoning path and is presented as a node in the
tree. We then ask the model to answer the generated sub-questions. To prevent
hallucination and make the model more focused on the given question and
evidence, we build a vocabulary bank using words from the evidence list and
the original question and instruct the model to do constrained decoding from
the vocabulary bank when generating its answers. After answering every sub-
question generated from the same question in the previous reasoning level, we
prompt the model to evaluate each reasoning path and estimate how likely the
reasoning path would lead us to the right answer. This probability estimation
would be assigned to the corresponding node in the tree. After the reasoning
process finishes, each reasoning path would have an aggregated probability
calculated from nodes along the path.
Formally, given a question $Q$, we instruct the model to generate sub-
questions $q_{1},q_{2},...,q_{n}$, and build a tree structure with the
original question $Q$ as the root node and each question $q_{i}$ as subsequent
nodes. The tree would expand as each sub-question $q_{i}$ has its sub-question
$q_{j}$, and the reasoning paths are thus represented as branches in the tree
structure. From the original question $Q$ and the evidence list
$E=e_{1},e_{2},...,e_{n}$, we build a vocabulary bank
$V=[w_{1},w_{2},...,w_{n}],w_{i}\in{Q},w_{j}\in{E}$. We then prompt the model
to generate their answer $a_{1},a_{2},...,a_{n}$ using only $w_{i}\in{V}$. We
describe the details of our framework below.
#### Example-Based Sub-Question Generation
Our framework starts with the sub-question generation module, which generates
sub-questions $q_{1},q_{2},...,q_{n}$ using the question $Q_{g}$ from the
previous reasoning level. The sub-questions are generated based on both the
model’s reasoning ability and the model’s semantic understanding of the
question $Q_{g}$. An example is given in Figure 2, where the sub-questions
from nodes 2 and 3 were generated using the question from node 1. However, we
cannot guarantee that each sub-question asked is a good sub-question, and
sometimes, the generated sub-question merely repeats the previous question. We
introduce the paraphrase detection module and pass on the generated sub-
questions to reduce redundancy and improve question quality.
#### Paraphrase Detection
Answering repetitive questions often leads to low-quality answers and time-
consuming steps. Following the sub-question generation module, we introduce
the paraphrase detection module to reduce redundancy and improve question
quality. In this module, we prompt the model and ask it to distinguish
informative questions from questions that merely repeat what is already stated
at the previous reasoning level. If a sub-question is a paraphrase, we
instruct the model to stop generating sub-questions from the current question.
In other words, we prune the low-quality sub-branch of the tree that could
otherwise be generated. By pruning these branches, we effectively improve the
efficiency of our framework.
#### Evidence Retrieval and Answering
We then move on to answering the question after our paraphrase detection
module. Our evidence retrieval and answering module focuses on retrieving
evidence and generating answers to the given sub-question. We also pass in the
full evidence list provided and prompt the model to give out an answer to the
given sub-question. The evidence retrieval and answering module selects
relative evidence from an evidence pool for each sub-question and uses words
only from the vocabulary bank to generate its final answer. We will discuss
details of constrained decoding in Section 3.3. The generated sub-answer and
the answered sub-question are then passed on to the sub-question generation
module at the next level to continue the reasoning process.
#### Validity Estimation
Not each sub-question asked is a good sub-question, and not each reasoning
path is reasonable. After every sub-question $q_{i}$ generated from the same
question $Q_{g}$ has been answered, we prompt the model to provide a
probability estimation $p_{i}$ for each $(q_{i},a_{i})$ pair. This probability
is the model’s evaluation of going down the correct reasoning path.
Specifically, this probability is obtained by prompting the model to consider
the following three aspects:
* •
Question Level: Is the question semantically clear and answerable?
* •
Reasoning Level: Is the reasoning line coherent when considering previous
levels?
* •
Answer Level: Does the evidence fully support the answer to the question?
As shown in Figure 2, we conduct validity estimation for sub-questions and
sub-answers in nodes 2 and 3 since the sub-questions were generated from the
same question in node 1.
At the leaf node of our tree, we would have a final question $q_{f}$. along
with a final answer $A$ to the original question $Q$, and also an aggregated
probability $p_{final}=\prod_{i}p_{i}$, with each $p_{i}$ being the
probability of the nodes along the reasoning path. We assign $p_{final}$ to
the leaf node, representing the aggregated probability of answer $A$ being the
correct answer to $Q$.
### 3.3 Constrained Decoding
One challenge for generative LLMs in the task of question answering is
hallucination. LLMs often fail to pay attention to the golden evidence and
hallucinate their own reference even when large amounts of evidence exist. To
alleviate the problem of LLM halluscination during evidence selection and
answer generation, we innovatively propose to use constrained decoding in the
answering process to reduce hallucination by forcing the model to generate
grounded answers from evidence and let models give concise and exact answers.
As shown in Figure 2, we conduct constrained decoding by asking the model to
generate words from the vocabulary bank, consisting of words taken only from
the original question and the evidence list provided. More formally, we
construct a vocabulary bank $V={w_{1},w_{2},...,w_{i}}$ from all words in the
provided evidence sentences. We conduct a simple filtering by removing common
English stop words. We then instruct the model’s evidence retrieval and
answering module to construct its answers using words only from the given
vocabulary $V$.
#### Code-based Constrained Decoding
For open-source LLMs (e.g., Llama), we build our logit processor at the
decoding time. Specifically, for every word $w_{j}\notin{V}$, we manually set
the score to negative infinity to prevent the model from generating them.
Thus, every answer generated will only use words from the evidence list.
#### Prompt-based Constrained Decoding
For closed-source LLMs (e.g., GPT models), since we do not have access to
their decoding function, we had to instruct the GPT models using prompts to do
constrained decoding. We provide our prompt template used in Appendix A.
## 4 Experimental Setup
Table 1: Performance comparison of StoC-ToT and baseline methods on the
HotpotQA dataset.
Prompting Method | GPT3.5 | GPT4 | LLaMa2(13B) | LLaMa2(70B) | LLaMa3(8B)
---|---|---|---|---|---
EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1
Zero-Shot Vanilla | 34.0 | 45.0 | 51.0 | 65.0 | 25.5 | 36.5 | 30.5 | 41.0 | 27.5 | 40.7
Chain-of-Thought | 35.5 | 47.3 | 52.0 | 66.8 | 30.5 | 42.5 | 33.5 | 45.0 | 32.5 | 44.6
Tree-of-Thought | 36.5 | 49.5 | 55.0 | 68.5 | 29.5 | 41.3 | 35.5 | 47.3 | 30.5 | 37.5
StoC-ToT | 45.5 | 56.2 | 62.0 | 76.3 | 31.0 | 43.0 | 43.0 | 56.3 | 33.0 | 44.5
w/o constrained decoding | 40.5 | 53.5 | 59.5 | 73.0 | 31.0 | 43.0 | 40.5 | 53.5 | 32.0 | 44.3
Table 2: Performance comparison of StoC-ToT and baseline methods on the MusiQue dataset. Prompting Method | GPT3.5 | GPT4 | LLaMa2(13B) | LLaMa3(8B)
---|---|---|---|---
EM | F1 | EM | F1 | EM | F1 | EM | F1
Zero-Shot Vanilla | 17.0 | 28.8 | 31.5 | 41.2 | 9.5 | 16.0 | 12.0 | 19.2
Chain-of-Thought | 18.0 | 29.7 | 32.5 | 44.2 | 11.0 | 17.5 | 12.5 | 21.6
Tree-of-Thought | 20.5 | 32.0 | 35.0 | 47.3 | 11.0 | 17.2 | 12.0 | 20.6
StoC-ToT | 26.5 | 38.0 | 42.0 | 55.3 | 11.5 | 18.0 | 14.5 | 22.0
w/o constrained decoding | 24.0 | 35.5 | 38.5 | 51.0 | 11.5 | 18.0 | 14.0 | 22.0
#### Dataset
We compare StoC-ToT with baseline methods on the HotpotQA dataset Yang et al.
(2018) and the MuSiQue dataset Trivedi et al. (2022), both of which are widely
used MHQA datasets across state-of-the-art MHQA baselines. The experiments are
conducted under the distractor setting, where we provide the model with an
evidence pool containing both golden and irrelevant evidence. The model needs
to find the golden evidence to answer the question correctly. We randomly
selected 200 examples from each dataset as our evaluation set.
#### Baselines
We included three baselines:
* •
Vanilla Prompting with no examples provided. We only provide the model with
questions and evidence and instruct it to output the answer.
* •
Chain-of-Thought (CoT) prompting Wei et al. (2022) with a standard input-
output (IO) prompt. We design the prompt with one in-context example, which
presents the whole reasoning chain, including all intermediate steps.
* •
Tree-of-Thought prompting Yao et al. (2023a) with slight modifications to
adapt to the MHQA task. We largely followed the original framework and used
majority voting on the reasoning lines to decide the final answer.
We recognize that there are LLM-based retrieval augmented generation
frameworks Yao et al. (2023b); Gou et al. (2024) that were also evaluated on
HotpotQA. However, we excluded them from our baselines as they used outside
knowledge bases, which are under a different testing scenario.
### 4.1 Implementation
We experiment with the baselines and our model utilizing five LLMs:
GPT-3.5-turbo Brown et al. (2020) and GPT-4OpenAI et al. (2024) from OpenAI,
LLaMa 2-13B Touvron et al. (2023), LLaMa 2-70B, and LLaMa 3-8B from MetaAI.
Due to the lengthy running time, LLaMa 2-70B was not tested on the MusiQue
dataset. For all models, We set the temperature to 0.5, $top_{k}$ to 1.0, and
maximum number of iterations to 5.
### 4.2 Evaluation Metric
Following the metrics in Yang et al. (2018), we use Exact Match and F1 score
as two evaluation metric. For an answer $a$ given by our framework, the Exact
Match score equals 1 if the answer span matches the golden answer exactly and
0 otherwise. The F1 metric measures the average overlap between the prediction
and ground truth answers.
## 5 Results
### 5.1 Overall Results
We compare StoC-ToT with LLM baselines on the HotpotQA dataset and the MusiQue
dataset and present our results in Tables 1 and 2. The backbone LLMs in our
experiments include GPT3.5, GPT4, Llama2-13B, Llama2-70B, and Llama3-8B. Due
to time constraints, we only tested with Llama2-70B on the HotpotQA dataset.
On the HotpotQA dataset, StoC-ToT attains an on-average increase in
performance of over 6 % compared with vanilla prompting on GPT models, and the
improvement goes up to 11% when we further implement StoC-ToT with constrained
decoding. On the more challenging MusiQue dataset, we still see an increase in
performance of StoC-ToT compared with the other baselines, most notably on
GPT4, where we observe an 11.5% EM improvement (from 31.50 to 42.0).
#### Comparison with Tree-of-Thought
StoC-ToT surpasses the original Tree-of-Thought prompting by 7% with the GPT4
model on both tested datasets. For LLMs with inferior reasoning ability, such
as LLaMa2-8B, we still observe a performance improvement, even on the harder
MusiQue dataset. These results suggest that StoC-ToT is more effective at
forming and selecting reliable reasoning paths under complex reasoning
scenarios.
#### Constrained Decoding
Even though the LLM’s reasoning ability can be improved by reasoning
prompting, such techniques have little help in preventing hallucination.
However, StoC-ToT implements constrained decoding, which makes the model much
more grounded to evidence when answering the question, effectively addressing
hallucination issues and improving the overall performance of our framework.
### 5.2 Ablation Study
#### Sensitivity to Demonstration Question Type
Table 3: Performance of StoC-ToT with different prompt types on the HotpotQA
dataset in terms of EM score. “Com" represents comparison questions, and “Bri"
represents bridge questions.
Model Variant | GPT3.5 | GPT4 | LLaMa2(13B) | LLaMa2(70B) | LLaMa3(8B)
---|---|---|---|---|---
Prompt/Question Type | Com | Bri | Com | Bri | Com | Bri | Com | Bri | Com | Bri
Prompt: Comparison | 58.8 | 41.0 | 76.5 | 57.2 | 38.2 | 31.9 | 58.8 | 41.0 | 44.1 | 33.7
Prompt: Bridge | 55.9 | 43.4 | 73.5 | 59.0 | 35.3 | 32.5 | 55.9 | 42.2 | 41.2 | 34.9
We study the effect on StoC-ToT performance when different types of
demonstration questions are provided in the prompt template. The HotPotQA
dataset specified two types of questions. The "Bridge" question contains a
"bridge entity” that connects the question and the final answer. In contrast,
the "Comparison" question requires the model to compare two entities of the
same type. Of the 200 questions in our evaluation set, 34 are comparison
questions, and 166 are bridge questions. Examples of bridge and comparison
questions are in Table 4.
We examined StoC-ToT performance under the two different question types, each
with a different prompt template: one containing only a comparison question as
an example and the other containing only a bridge question as an example. We
provide the content of our templates in Appendix A. Results are shown in Table
3. We observe that the difference in prompt templates influences the
performance of our framework under different question types by a small margin.
The comparison questions are generally easier to solve, and StoC-ToT performs
better on comparison questions than on bridge questions. StoC-ToT will handle
comparison questions better if the prompt template contains comparison
questions and vice versa.
#### Question and Reasoning Types
(a) Question Type
(b) Reasoning Type
Figure 3: Performace comparison of Chain-of-Thought, Tree-of-Thought, and
StoC-ToT on questions of different question types (Left) and reasoning types
(Right). Experiments were done on the HotpotQA dataset. Table 4: Question Type
Examples. On the left side, the bridging entity is highlighted in red, and the
final question is highlighted in orange. On the right side, entities that are
being compared are highlighted in blue.
Bridge Question | Comparison Question
---|---
What distinction is held by the former NBA player who was a member of the Charlotte Hornets during their 1992-93 season and was head coach for the WNBA team Charlotte Sting? | Were Scott Derrickson and Ed Wood of the same nationality?
Table 5: Reasoning Type Examples. On the left side, the entity in red needs to
be found before solving the question in orange. On the right side, questions
with parallel reasoning contain parts (highlighted in blue) that can be solved
in arbitrary order.
Sequential Reasoning | Parallel Reasoning
---|---
The football manager who recruited David Beckham managed Manchester United during what timeframe? | What distinction is held by the former NBA player who was a member of the Charlotte Hornets during their 1992-93 season and was head coach for the WNBA team Charlotte Sting?
We examine StoC-ToT, Tree-of-Thought prompting, and Chain-of-Thought prompting
by comparing their performance under different question-type settings.
Detailed results are shown in Figure 3(a). StoC-ToT performs better at both
Bridge Questions and Sequential Questions, suggesting that StoC-ToT can avoid
reasoning dead-ends and is better at forming intermediate reasoning lines.
We also conduct an in-depth analysis of the reasoning types in the existing
MHQA datasets by randomly selecting 100 questions from our testing set. The
questions are roughly divided into two categories: 1) tree-like parallel
reasoning and 2) chain-like sequential reasoning. Questions with parallel
reasoning contain two or more reasoning paths that can be solved arbitrarily.
Questions with sequential reasoning follow a strict reasoning chain, and all
the sub-questions must be solved to form the correct reasoning process. All
comparison questions are parallel reasoning, but some bridge questions contain
parallel reasoning. Examples of sequential and parallel reasoning questions
are in Table 5. Out of the selected 100 questions, 59 questions were
Sequential and 41 questions were Parallel. Results are shown in Figure 3(b).
StoC-ToT performs better on both reasoning types, especially on questions
containing parallel reasoning. This suggests that StoC-ToT’s stochastic way of
forming the tree is very effective when solving questions containing multiple
reasoning paths.
#### Performance and Hops
Figure 4: Performance comparison of CoT, ToT, and StoC-ToT on different number
of hops in the question. Experiments done in the MusiQue dataset.
As the number of hops increases in a question, the reasoning line gets more
complex and varied. Figure 4 shows the performances of different prompting
techniques on questions in the MusiQue dataset with different numbers of hops.
StoC-ToT performs best in all categories, demonstrating our framework’s
superior ability to deal with complex reasoning scenarios. This ablation study
was conducted only on GPT4, as other models performed poorly on 3-hop and
4-hop scenarios, regardless of the reasoning prompting technique used.
#### Error Analysis
•Semantically Correct•Wrong Answer•Intermediate Answer•No Answer
Figure 5: Ratio of different categories in error cases, on the HotpotQA
dataset.
We conduct a detailed analysis of the errors made by our framework on GPT3 and
GPT4, and present our results in Figure 5. We categorize the errors into four
types: (1) No Answer: our framework did not come up with an answer for the
question due to not finishing the reasoning process; (2) Intermediate Answer:
our framework came up with an answer for one of the intermediate hops instead
of for the final question; (3) Wrong Answer: our framework came up with an
answer that is neither the final answer nor one of the intermediate answers;
(4) Semantically Correct: our framework came up with the right answer, but did
not have an exact match with the final answer. Appendix B shows examples of
each error category. Large amounts of error cases were correct answers with
extra wording or hallucination errors, signaling potential improvements over
our constrained decoding scheme. Reasoning process errors, including no answer
and intermediate answer, make up only 25% of the total error cases. This
result shows that our framework is capable of building a robust reasoning
process for complex questions.
## 6 Conclusion
This paper proposes StoC-ToT, a stochastic tree-of-thought reasoning framework
with constrained generation for multi-hop question answering. StoC-ToT is
specialized in dealing with complex reasoning scenarios in natural language
tasks. Experiments on two benchmark datasets show that our framework
outperforms previous reasoning prompting techniques with multiple Large
Language Models. Detailed analysis shows that our framework is capable of
building a robust reasoning process given different types of questions.
Further research can aim to enhance the reliability of our framework by
proposing better validity evaluation schemes and more effective methods for
improving groundedness and preventing hallucination.
## Limitations
Our framework relies on initiating multiple model instances and requires
multiple prompts per round. The repetitive callings impose heavy time costs
for our framework, even after implementing our paraphrase module. Another
limitation comes from how we generated sub-questions. Currently, we directly
prompt the model to generate sub-questions. A more complex standard can be
used to increase the quality of the sub-questions generated. Also, more
extensive experiments should be provided, including experimenting on other
different datasets and case studies.
## Ethics Statement
This research adhered to the ethical standards and best practices outlined in
the ACL Code of Ethics. Language Models can sometimes produce illogical or
inaccurate reasoning paths, so their outputs should be cautiously used. The
outputs are only examined to understand how a model arrives at its answers and
investigate why it makes certain errors. All experiments used publicly
available datasets from previously published works and did not involve ethical
or privacy issues.
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* Yin et al. (2023) Zhangyue Yin, Yuxin Wang, Xiannian Hu, Yiguang Wu, Hang Yan, Xinyu Zhang, Zhao Cao, Xuanjing Huang, and Xipeng Qiu. 2023. Rethinking label smoothing on multi-hop question answering. In _Chinese Computational Linguistics - 22nd China National Conference, CCL 2023, Harbin, China, August 3-5, 2023, Proceedings_ , volume 14232 of _Lecture Notes in Computer Science_ , pages 72–87. Springer.
* Zhang et al. (2023) Jiahao Zhang, Haiyang Zhang, Dongmei Zhang, Yong Liu, and Shen Huang. 2023. Beam retrieval: General end-to-end retrieval for multi-hop question answering. _CoRR_ , abs/2308.08973.
* Zhu et al. (2021) Fengbin Zhu, Wenqiang Lei, Chao Wang, Jianming Zheng, Soujanya Poria, and Tat-Seng Chua. 2021. Retrieving and reading: A comprehensive survey on open-domain question answering. _CoRR_ , abs/2101.00774.
## Appendix A Prompt Templates
#### Sub Question Generation Template
The prompt template containing one comparison question and one bridge question
is given below:
prompt: Break a question into high-quality sub-questions that are easier to
answer. Here are two examples as guidelines:
"Question: Are Tokyo and Busan in the same country? Thought 1: I could either
find which country Tokyo is located in, or which country Busan is located in.
Sub Question 1-1: Which country is Tokyo located in? Sub Question 1-2: Which
country is Busan located in?"
"Question: Tokyo is located in the country that has what colors present on its
national flag? Thought 1: I need to first find out which country Tokyo is
located in. Sub Question 1-1: Which country is Tokyo located in?"
Only give out your thought process and current-level sub-questions. Do not
give out answers to your questions. Question: Given Question. Thought 1:
#### Prompt-based Constrained Generation Template
The prompt template at answering time is given below:
prompt: Given a question and a list of evidence that may of help, give your
answer directly, using words only from the vocabulary bank, without any
explanations.
Question: Given Question. Evidence as reference: Given Evidence. Vocabulary
Bank: Given Vocabulary. Answer:
## Appendix B Examples of the Error Cases
•Type-2: Intermediate Answer
Question:
Where does the hotel and casino located in which Bill Cosby’s third album was
recorded?
Answer given by StoC-ToT on GPT4:
Las Vegas.
Golden Answer:
Las Vegas Strip in Paradise.
•Type-3: Wrong Answer
Question:
Aside from the Apple Remote, what other device can control the program Apple
Remote was originally designed to interact with?
Answer given by StoC-ToT on GPT4:
siri remote and devices with netsupport manager software
Golden Answer:
keyboard function keys
•Type-4: Semantically Correct
Question:
Roger O. Egeberg was Assistant Secretary for Health and Scientific Affairs
during the administration of a president that served during what years?
Answer given by StoC-ToT on GPT4:
1969 to 1974
Golden Answer:
1969 until 1974
|
# Liouville results for semilinear integral equations with conical diffusion
Isabeau Birindelli, Lele Du and Giulio Galise Dipartimento di Matematica
Guido Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185,
Roma, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
Nonexistence results for positive supersolutions of the equation
$-Lu=u^{p}\quad\text{in $\mathbb{R}^{N}_{+}$}$
are obtained, $-L$ being any symmetric and stable linear operator, positively
homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely
continuous and positive only in a relative open set of the unit sphere of
$\mathbb{R}^{N}$. The results are sharp: $u\equiv 0$ is the only nonnegative
supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while
nontrivial supersolutions exist, at least for some specific $-L$, as soon as
$p>\frac{N+s}{N-s}$.
The arguments used rely on a rescaled test function’s method, suitably adapted
to such nonlocal setting with weak diffusion; they are quite general and also
employed to obtain Liouville type results in the whole space.
## 1\. Introduction
In this paper we obtain Liouville theorems for supersolutions of semilinear
integral equations in the half-space
$\mathbb{R}^{N}_{+}:=\left\\{x\in\mathbb{R}^{N}:\,x_{N}>0\right\\}$ with the
following class of nonlocal operators
(1.1)
$Lu\left(x\right)=\left(1-s\right)\int_{\mathbb{R}^{N}}\frac{u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy,$
where $0<s<1$ and $a\left(\theta\right)\in
L^{\infty}\left(\mathbb{S}^{N-1}\right)$ is a nonnegative and even function on
the unit sphere $\mathbb{S}^{N-1}$ of $\mathbb{R}^{N}$. We will be more
precise later, but let us emphasize immediately that the function
$a\left(\theta\right)$ can be chosen so that the operator diffuses only along
a cone.
The main result will be about nonexistence of classical solutions, besides the
trivial one, $u\in C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$ of
the problem
(1.2) $\displaystyle\left\\{\begin{aligned} -Lu&\geq
u^{p},&&x\in\mathbb{R}_{+}^{N},\\\ u&\geq
0,&&x\in\mathbb{R}^{N},\end{aligned}\right.$
where $\mathcal{L}_{s}$ is the natural functional space required that will be
defined precisely later.
Nonexistence of entire bounded harmonic functions, goes back to Cauchy (1844)
and Liouville (1850), but since the acclaimed works of Gidas and Spruck [24]
much attention has been given to so called semilinear Liouville theorems in
the whole space and in half-spaces. Indeed these nonexistence results in
unbounded domains correspond to existence results in bounded domains thanks to
the priori bounds they imply and the use of Leray-Schauder degree theory.
A particular role has been given in these decades to determine the range of
existence, or better lack of it, when instead of considering solutions, one
concentrates on supersolutions. This has been done for very different cases:
the case of cone-like domains for linear and quasilinear operators appeared
for instance in [1, 6, 5, 23, 26], existence and nonexistence for positive
supersolutions in the whole space, half-spaces and exterior domains for fully
nonlinear inequalities can be found e.g. in [2, 17, 27], systems and sub-
Laplacians have been respectively addressed in [10, 29] and [4, 7], while as
far as nonlinear degenerate elliptic operators are concerned we refer to [3,
8].
We cannot recall all the results in this area but the interested readers can
refer to the book [30] and the references therein. We just remind that, for
the Laplacian, if
$1\leq p\leq\frac{N+1}{N-1}$
then the inequality
$\Delta u+u^{p}\leq 0\quad\text{in $\mathbb{R}^{N}_{+}$}$
admits only the trivial solution (in fact such result is still true for
$p\geq-1$, but in this paper we only consider the superlinear case $p\geq 1$).
Furthermore, the bound $\frac{N+1}{N-1}$ is sharp in the sense that for
$p>\frac{N+1}{N-1}$ it is possible to construct explicit positive
supersolutions.
In the realm of nonlocal operators, the results are more recent but
nonetheless really numerous, see e.g. [11, 13, 19, 35] concerning entire
$s$-harmonic functions and [14, 18, 21, 34] for positive solutions and
supersolutions of Lane-Emden type equations/systems in $\mathbb{R}^{N}$ or in
exterior domains. Since we mainly concern here with supersolutions of
semilinear nonlocal operators in half-spaces, we will recall results directly
related with this cases. In [15], using the method of moving planes in
integral forms, Chen, Fang and Yang have proved that for
$1<p\leq\frac{N+2s}{N-2s}$ any nonnegative and locally bounded distributional
solution of
$\displaystyle\left\\{\begin{aligned}
(-\Delta)^{s}u&=u^{p},&&x\in\mathbb{R}_{+}^{N},\\\
u&=0,&&x\in\overline{\mathbb{R}_{-}^{N}},\end{aligned}\right.$
where $\mathbb{R}^{N}_{-}:=\left\\{x\in\mathbb{R}^{N}:\,x_{N}<0\right\\}$, is
identically zero. The result has been extended in [16] for a larger class of
operators.
Instead, for supersolutions, the proofs and the range for which the validity
of the Liouville theorem holds are different and very few results are
available in the half-spaces. We wish to mention the result for nonlocal fully
nonlinear operators by Nornberg, dos Prazeres and Quaas [28] where they study
the existence of fundamental solutions in conical domains for fully nonlinear
integral operators of Isaacs type and, as application, they obtain Liouville
type results in subcritical regimes. In the special case where the cone is the
half-space $\mathbb{R}^{N}_{+}$ and the diffusion operator is $(-\Delta)^{s}$
they find in particular that for
$1\leq p\leq\frac{N+s}{N-s}$
the only nonnegative (in $\mathbb{R}^{N}$) viscosity supersolutions of the
problem
$(-\Delta)^{s}u\geq u^{p}\quad\text{in $\mathbb{R}^{N}_{+}$}$
is the trivial one (even in this case the results is still valid for
$p\geq-1$).
In the works mentioned above, even in the nonlinear case, the kernels of
operators considered are, up to some constant, bounded above and below by that
of the fractional Laplacian. Here instead, the operators $L$ we consider are
part of the larger class of infinitesimal generators of stable Lévy processes
(1.3)
$Lu\left(x\right)=(1-s)\int_{\mathbb{S}^{N-1}}\int_{0}^{+\infty}\frac{u\left(x+t\theta\right)+u\left(x-t\theta\right)-2u\left(x\right)}{t^{1+2s}}dtd\mu,$
where the spectral measure $\mu=\mu\left(\theta\right)$ on $\mathbb{S}^{N-1}$
satisfies the ellipticity conditions
$\displaystyle
0<\lambda\leq\inf_{\nu\in\mathbb{S}^{N-1}}\int_{\mathbb{S}^{N-1}}\left|\nu\cdot\theta\right|^{2s}d\mu\quad\text{and}\quad\int_{\mathbb{S}^{N-1}}d\mu\leq\Lambda<\infty.$
When $\mu$ is absolutely continuous and $d\mu=a\left(\theta\right)d\theta$,
then (1.1) and (1.3) coincide.
At this stage it is worth to point out that the operator (1.1) converges, as
$s\to 1^{-}$, to a second order linear uniformly elliptic operators with
constant coefficients, in the sense that for any $u\in
C^{2}_{0}(\mathbb{R}^{N})$, then
$\lim_{s\to
1^{-}}(1-s)Lu(x)=\sum_{i,j=1}^{N}a_{ij}\partial^{2}_{ij}u(x)\qquad\forall
x\in\mathbb{R}^{N},$
where the constant coefficients $a_{ij}$ depend on the function $a(\theta)$.
Moreover, let us point out that, since the normalizing constant $1-s$ in (1.1)
is irrelevant for the issues that we address in the present paper, henceforth
it will be omitted.
We refer to the previous works of Ros-Oton and Serra [32] and in particular to
the very interesting paper [33] where, in order to give boundary estimates,
some linear Liouville theorems in half-spaces are proved under the further
assumption that the function $a(\theta)$ is positive bounded away from zero
everywhere.
Contrarily to the cases where $a(\theta)$ is positive bounded away from zero
everywhere, here we only suppose that $a(\theta)$ is positive in some relative
open set on $\mathbb{S}^{N-1}$. Clearly if $a(\theta)$ is bounded and bounded
away from zero then the kernel of the operator (1.1) can be compared from
above and below by that of the fractional Laplacian, which corresponds to the
case $a(\theta)$ is a constant function. Other results concerning
classifications of entire solutions and Hölder regularity for nonlocal
operators with anisotropic diffusion can be found respectively in [20, 25].
We can now give the precise conditions on the function $a(\theta)$. Let us fix
the following notation: for any vector $\nu\in\mathbb{S}^{N-1}$ and
$0<\tau\leq 1$, we define the closed two fold cone
$\Sigma_{\nu,\tau}\left(x\right)$ of aperture
$\arccos(1-\tau)\in\left(0,\frac{\pi}{2}\right]$, with vertex
$x\in\mathbb{R}^{N}$ and axis $\nu$, by
(1.4)
$\displaystyle\Sigma_{\nu,\tau}\left(x\right):=\left\\{y\in\mathbb{R}^{N}:\;\left|\left(y-x\right)\cdot\nu\right|\geq\left(1-\tau\right)\left|y-x\right|\right\\}.$
In all the paper, we assume that for some constants $0<d<D$,
(1.5) $\displaystyle 0\leq a(\theta)\leq D\quad\text{in $\mathbb{S}^{N-1}$}$
and that there exist $\nu_{0}\in\mathbb{S}^{N-1}$ and $0<\tau_{0}\leq 1$ such
that
(1.6) $\displaystyle a(\theta)\geq d>0\quad\text{in
$\Sigma_{\nu_{0},\tau_{0}}\left(0\right)\cap\mathbb{S}^{N-1}$}.$
Hence in all the proofs we can only use that the operator (1.1) diffuses along
a fixed cone. This feature will induce some delicate geometric constructions.
Let
$\displaystyle\mathcal{L}_{s}=\left\\{u\in
L^{1}_{loc}\left(\mathbb{R}^{N}\right):\,\limsup_{\left|x\right|\rightarrow+\infty}\frac{\left|u\right|}{\left|x\right|^{2s-\delta}}<+\infty\text{\;
for some\,}\delta\in(0,2s]\right\\}.$
Our main result, is the following
###### Theorem 1.1.
Let $s\in(0,1)$ and let $L$ be any operator of the form (1.1) satisfying
(1.5)-(1.6). If $1\leq p\leq\frac{N+s}{N-s}$ and $u\in
C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$ is a solution of
(1.2), then $u\equiv 0$.
We were inspired by the pioneering proof used by Berestycki, Capuzzo Dolcetta
and Nirenberg in [5] for supersolutions of semilinear Liouville Theorem in
cones. But it is important to notice that the key ingredients they use need to
be completely reconsidered due both to the nonlocal character of the operators
we consider and their weak diffusion. Even the simple integrations by parts
formula, since the test functions we shall use in the proof of Theorem 1.1 are
not smooth on $\partial\mathbb{R}^{N}_{+}$, needs to be proved (see
Proposition 2.2). The other novelty is in the ad hoc construction of the test
functions. Indeed we construct a sequence of nonnegative compactly supported
test functions that converge to 1 in the whole half space, but their supports
depend on the cone $\Sigma_{\nu_{0},\tau_{0}}\left(0\right)$ where the
function $a(\theta)$ is supported.
Interestingly the bound on $p$, does not depend on the size of the cone.
Furthermore, at least in the case of the fractional Laplacian, the bound is
optimal: for any $p>\frac{N+s}{N-s}$ the problem
$\displaystyle\left\\{\begin{aligned} \left(-\Delta\right)^{s}u&\geq
u^{p},&&x\in\mathbb{R}_{+}^{N},\\\
u&=0,&&x\in\mathbb{R}_{-}^{N}\end{aligned}\right.$
admits positive solutions $u\in
C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$. This is proved in
Theorem 4.1.
The arguments used in the proof of Theorem 1.1 are quite flexible and they
also apply to get the equivalent Liouville result in the whole space
$\mathbb{R}^{N}$, in a simplified form due to the absence of the boundary of
the domain which, instead, poses additional difficulties in the case of half-
spaces. We obtain nonexistence of positive solutions provided $1\leq
p\leq\frac{N}{N-2s}$, see Theorem 5.1. This range is known to be optimal for
the fractional Laplacian and even more, as showed by Felmer and Quaas in [22],
for Pucci’s nonlinear operators ${\mathcal{M}}^{\pm}$, which are extremal
operators among the class of linear integral operators whose kernels,
differently to the cases treated in the present paper, are comparable to that
of the fractional Laplacian. The critical exponents associated to Pucci’s
operators are $p=\frac{N^{\pm}}{N^{\pm}-2s}$, where $N^{\pm}$ are dimensional-
like numbers which reduce to the dimension $N$ when
$-{\mathcal{M}}^{\pm}=(-\Delta)^{s}$.
The paper is organized as follows. Section 2 contains several technical
results that will be used throughout the paper. Section 3 is devoted to the
proof of Theorem 1.1 and Section 4 is concerned with the optimality of the
exponent $p=\frac{N+s}{N-s}$. In the last Section 5, we prove a Liouville
theorem in the whole space. Let us mention that the case $N=1$ is much simpler
and some remarks concerning the results in that case are scattered along the
paper (See e.g. Remarks 4.2 and 5.2).
## 2\. preliminary
In this section we prove several technical results that will be used in the
proof of Theorem 1.1. We start by the following simple lemma, where it is
proved that the nonnegativity assumption of $u$ in
$\overline{\mathbb{R}^{N}_{-}}$ can be, in fact, replaced by the simplest one
$u=0$ in $\overline{\mathbb{R}_{-}^{N}}$. This reduction is useful in the
integration by parts formula of Proposition 2.2.
Henceforth we denote by $e_{N}=(0,\ldots,0,1)$ and we use the notation
$C=C(\cdot)$ to denote positive constants depending on given quantities.
###### Lemma 2.1.
For any nonnegative classical solution $u$ of (1.2), the translation-
truncation function (see Figure 1)
(2.1) $\displaystyle\widetilde{u}\left(x\right)=\left\\{\begin{aligned}
&u\left(x+e_{N}\right),&&x\in\mathbb{R}_{+}^{N},\\\
&0,&&x\in\overline{\mathbb{R}_{-}^{N}}\end{aligned}\right.$
is a nonnegative classical solution of the problem
(2.2) $\displaystyle\left\\{\begin{aligned} -Lu&\geq
u^{p},&&x\in\mathbb{R}_{+}^{N},\\\
u&=0,&&x\in\overline{\mathbb{R}_{-}^{N}}.\end{aligned}\right.$
Figure 1. The graph of $\widetilde{u}$.
###### Proof.
For $x\in\mathbb{R}_{+}^{N}$ and any $y\in\mathbb{R}^{N}$ it holds that
$\displaystyle\widetilde{u}\left(x\pm y\right)\leq u\left(x+e_{N}+y\right).$
Hence
$\displaystyle-L\widetilde{u}\left(x\right)$
$\displaystyle=-\int_{\mathbb{R}^{N}}\frac{\widetilde{u}\left(x+y\right)+\widetilde{u}\left(x-y\right)-2\widetilde{u}\left(x\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\geq-\int_{\mathbb{R}^{N}}\frac{u\left(x+e_{N}+y\right)+u\left(x+e_{N}-y\right)-2u\left(x+e_{N}\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=-Lu\left(x+e_{N}\right).$
Since $u$ is a solution of (1.2), we have
$\displaystyle-L\widetilde{u}\left(x\right)\geq-Lu\left(x+e_{N}\right)\geq
u^{p}\left(x+e_{N}\right)=\widetilde{u}^{p}\left(x\right).$
∎
For any $\widetilde{u}$ as in Lemma 2.1, we find that $\widetilde{u}$
satisfies the following regularity condition up to the boundary of
$\mathbb{R}^{N}_{+}$:
(2.3)
$\displaystyle\left\|u\right\|_{C^{2}\left(K\cap\overline{\mathbb{R}^{N}_{+}}\right)}<+\infty,\quad\forall\text{
compact }K\subseteq\overline{\mathbb{R}_{+}^{N}}.$
It is well known that $L$ is a self adjoint operator but we need to take care
of the “integration by part” since the test function is not smooth. This is
the object of the next Proposition.
###### Proposition 2.2.
Let $(2s-1)_{+}<\alpha<2s$, $\psi\in C^{2}_{0}\left(\mathbb{R}^{N}\right)$ and
$v_{\alpha}\left(x\right)=\left(x_{N}\right)_{+}^{\alpha}\psi\left(x\right)$.
Then for any $u\in C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$
satisfying (2.3) and $u=0$ in $\overline{\mathbb{R}_{-}^{N}}$, we have
$\displaystyle\int_{\mathbb{R}^{N}}uLv_{\alpha}dx=\int_{\mathbb{R}^{N}}v_{\alpha}Ludx.$
###### Proof.
Assume that supp $\psi\subseteq B_{R}$, $R>1$. Let us define two maps
$\displaystyle
F_{1},F_{2}:\mathbb{R}^{N}\times\left(\mathbb{R}^{N}\backslash\left\\{0\right\\}\right)\rightarrow\mathbb{R}$
by
$\displaystyle
F_{1}\left(x,y\right):=\frac{v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)u\left(x\right)$
and
$\displaystyle
F_{2}\left(x,y\right):=\frac{u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)v_{\alpha}\left(x\right).$
What needs to be proved in order to get (2.3) is
(2.4)
$\displaystyle\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left|F_{1}\right|dy\right)dx<+\infty$
and
(2.5)
$\displaystyle\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left|F_{2}\right|dy\right)dx<+\infty.$
Once (2.4) and (2.5) are proved i.e. that $F_{1},F_{2}\in
L^{1}\left(\mathbb{R}^{N}\times\mathbb{R}^{N}\right)$, we may apply Fubini-
Tonelli’s theorem and it is classical that
$\displaystyle\int_{\mathbb{R}^{N}}uLv_{\alpha}dx=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}F_{1}dy\right)dx=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}F_{1}dx\right)dy$
$\displaystyle=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left[v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right]u\left(x\right)dx\right)\frac{1}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left[u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right]v_{\alpha}\left(x\right)dx\right)\frac{1}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}F_{2}dx\right)dy=\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}F_{2}dy\right)dx=\int_{\mathbb{R}^{N}}v_{\alpha}Ludx.$
We first prove (2.4). Note that
(2.6) $\displaystyle
F_{1}\left(x,y\right)=0\quad\forall\left(x,y\right)\in\overline{\mathbb{R}_{-}^{N}}\times\left(\mathbb{R}^{N}\backslash\left\\{0\right\\}\right),$
we only consider
$\left(x,y\right)\in\mathbb{R}_{+}^{N}\times\left(\mathbb{R}^{N}\backslash\left\\{0\right\\}\right)$.
For any $x\in\mathbb{R}^{N}_{+}\cap\left\\{\left|x\right|\geq 2R\right\\}$,
there is $v_{\alpha}\left(x\right)=0$. Moreover, if $\left|y\pm x\right|\geq
R$, then $v_{\alpha}\left(x\pm y\right)=0$, while if $\left|y\pm x\right|\leq
R$, we have
$\displaystyle\left|y\right|\geq\left|x\right|-\left|y-x\right|\geq\frac{\left|x\right|}{2}.$
Therefore, for any $x\in\mathbb{R}^{N}_{+}\cap\left\\{\left|x\right|\geq
2R\right\\}$,
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=\int_{B_{R}\left(x\right)}\frac{\left|v_{\alpha}\left(x-y\right)\right|}{\left|y\right|^{N+2s}}dy+\int_{B_{R}\left(-x\right)}\frac{\left|v_{\alpha}\left(x+y\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=\int_{B_{R}\left(x\right)}\frac{\left|\left(x_{N}-y_{N}\right)_{+}^{\alpha}\psi\left(x-y\right)\right|}{\left|y\right|^{N+2s}}dy+\int_{B_{R}\left(-x\right)}\frac{\left|\left(x_{N}+y_{N}\right)_{+}^{\alpha}\psi\left(x+y\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle\leq
2^{N+2s}R^{\alpha}\left\|\psi\right\|_{L^{\infty}\left(\mathbb{R}^{N}\right)}\left(\left|B_{R}\left(x\right)\right|+\left|B_{R}\left(-x\right)\right|\right)\frac{1}{\left|x\right|^{N+2s}}$
$\displaystyle=C\left(N,s,\alpha,R,\left\|\psi\right\|_{L^{\infty}\left(\mathbb{R}^{N}\right)}\right)\frac{1}{\left|x\right|^{N+2s}}.$
For any $x\in\mathbb{R}^{N}_{+}\cap B_{2R}$, letting $z=\frac{y}{x_{N}}$, we
have
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\left|\left(x_{N}+y_{N}\right)_{+}^{\alpha}\psi\left(x+y\right)+\left(x_{N}-y_{N}\right)_{+}^{\alpha}\psi\left(x-y\right)-2x_{N}^{\alpha}\psi\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=x_{N}^{\alpha}\int_{\mathbb{R}^{N}}\frac{\left|\left(1+\frac{y_{N}}{x_{N}}\right)_{+}^{\alpha}\psi\left(x+y\right)+\left(1-\frac{y_{N}}{x_{N}}\right)_{+}^{\alpha}\psi\left(x-y\right)-2\psi\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=\left(\int_{\mathbb{R}^{N}}\frac{\left|A\left(x,z\right)\right|}{\left|z\right|^{N+2s}}dz\right)x_{N}^{\alpha-2s},$
where
$\displaystyle
A\left(x,z\right)=\left(1+z_{N}\right)_{+}^{\alpha}\psi\left(x+x_{N}z\right)+\left(1-z_{N}\right)_{+}^{\alpha}\psi\left(x-x_{N}z\right)-2\psi\left(x\right).$
Since $A$ satisfies
$\displaystyle\left\\{\begin{aligned} &A\left(x,0\right)=0&&\forall
x\in\mathbb{R},\\\
&A\left(x,z\right)=A\left(x,-z\right)&&\forall\left(x,z\right)\in\mathbb{R}^{N}\times\mathbb{R}^{N},\\\
&A\in
C^{2}\left(\mathbb{R}^{N}\times\overline{B_{\frac{1}{2}}}\right),\end{aligned}\right.$
and $\alpha>0$, then for any $x\in\mathbb{R}^{N}_{+}\cap B_{2R}$ a second
order Taylor expansion yields
(2.7) $\displaystyle\left|A\right|\leq
C\left(\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&\left|z\right|^{2},&&\left|z\right|<\frac{1}{2},\\\
&\left|z\right|^{\alpha},&&\left|z\right|\geq\frac{1}{2}.\end{aligned}\right.$
Therefore, for $x\in\mathbb{R}^{N}_{+}\cap B_{2R}$, by the condition
$\alpha<2s$,
(2.8)
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|A\right|}{\left|z\right|^{N+2s}}dz\leq
C\left(N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right).$
We conclude that
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
(2.9) $\displaystyle\leq
C\left(N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&\frac{1}{x_{N}^{2s-\alpha}},&&x\in\mathbb{R}^{N}_{+}\cap B_{2R},\\\
&\frac{1}{\left|x\right|^{N+2s}},&&x\in\mathbb{R}^{N}_{+}\cap\left\\{\left|x\right|\geq
2R\right\\}.\end{aligned}\right.$
Let us notice that by the assumption $u\in\mathcal{L}_{s}$, $u$ satisfies
(2.3) and $u=0$ in $\overline{\mathbb{R}_{-}^{N}}$, we deduce that there
exists $\beta=\beta\left(u\right)>0$ such that
(2.10)
$\displaystyle\left|u\right|\leq\beta\left(1+\left|x\right|^{2s-\delta}\right)\quad\forall
x\in\mathbb{R}^{N}.$
Since $\alpha>2s-1$, by (1.5), (2.6), (2) and (2.10),
$\displaystyle\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left|F_{1}\right|dy\right)dx$
$\displaystyle=\int_{\mathbb{R}_{+}^{N}}\left|u\right|\left(\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}\left|a\left(\frac{y}{\left|y\right|}\right)\right|dy\right)dx$
$\displaystyle\leq
D\int_{\mathbb{R}_{+}^{N}}\left|u\right|\left(\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy\right)dx$
$\displaystyle\leq
C\left(D,N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\left[\int_{\mathbb{R}^{N}_{+}\cap
B_{2R}}\frac{\left|u\right|}{x_{N}^{2s-\alpha}}dx+\int_{\mathbb{R}^{N}_{+}\cap\left\\{\left|x\right|\geq
2R\right\\}}\frac{\left|u\right|}{\left|x\right|^{N+2s}}dx\right]$
$\displaystyle\leq
C\left(D,N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\left[\left\|u\right\|_{L^{\infty}\left(B_{2R}\right)}\int_{\mathbb{R}^{N}_{+}\cap
B_{2R}}\frac{1}{x_{N}^{2s-\alpha}}dx\right.$
$\displaystyle\quad\left.+\beta\int_{\mathbb{R}^{N}_{+}\cap\left\\{\left|x\right|\geq
2R\right\\}}\frac{1+\left|x\right|^{2s-\delta}}{\left|x\right|^{N+2s}}dx\right]<+\infty.$
Next, we prove (2.5). Since
(2.11) $\displaystyle
F_{2}\left(x,y\right)=0\quad\forall\left(x,y\right)\in\left(\overline{\mathbb{R}^{N}_{-}}\cup\left\\{\left|x\right|\geq
R\right\\}\right)\times\left(\mathbb{R}^{N}\backslash\left\\{0\right\\}\right),$
we need to focus on $\left(x,y\right)\in\left(\mathbb{R}_{+}^{N}\cap
B_{R}\right)\times\left(\mathbb{R}^{N}\backslash\left\\{0\right\\}\right)$.
For any $x\in\mathbb{R}^{N}_{+}\cap B_{R}$, by (2.10),
(2.12) $\displaystyle\left|u\left(x\pm
y\right)\right|\leq\beta\left(1+\left|x\pm y\right|^{2s-\delta}\right)\leq
C\left(s,\delta,\beta,R\right)\left(1+\left|y\right|^{2s-\delta}\right)\quad\forall
y\in\mathbb{R}^{N}.$
Moreover, by (2.3), we obtain $u\in
C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap B_{2R}\right)$ and together with
(2.12), we infer that
$\displaystyle\frac{\left|u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right|}{\left|y\right|^{N+2s}}\leq\left\\{\begin{aligned}
&\left\|u\right\|_{C^{2}\left(B_{x_{N}}\left(x\right)\right)}\frac{1}{\left|y\right|^{N+2s-2}},&&\left|y\right|<x_{N},\\\
&C\left(s,\delta,\beta,R\right)\frac{1+\left|y\right|^{2s-\delta}}{\left|y\right|^{N+2s}},&&\left|y\right|\geq
x_{N}.\end{aligned}\right.$
Consequently, for any $x\in\mathbb{R}^{N}_{+}\cap B_{R}$, we get
$\left\|u\right\|_{C^{2}\left(B_{x_{N}}\left(x\right)\right)}\leq\left\|u\right\|_{C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap
B_{2R}\right)}$ and then
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle\leq\left\|u\right\|_{C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap
B_{2R}\right)}\int_{\left|y\right|<x_{N}}\frac{1}{\left|y\right|^{N+2s-2}}dy+C\left(s,\delta,\beta,R\right)\int_{\left|y\right|\geq
x_{N}}\frac{1+\left|y\right|^{2s-\delta}}{\left|y\right|^{N+2s}}dy$
$\displaystyle\leq
C\left(s,\delta,\beta,R,\left\|u\right\|_{C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap
B_{2R}\right)}\right)\left(x_{N}^{2-2s}+\frac{1}{x_{N}^{2s}}+\frac{1}{x_{N}^{\delta}}\right)$
(2.13) $\displaystyle\leq
C\left(s,\delta,\beta,R,\left\|u\right\|_{C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap
B_{2R}\right)}\right)\frac{1}{x_{N}^{2s}}.$
Reasoning that $\alpha>2s-1$, by (1.5), (2.11) and (2),
$\displaystyle\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^{N}}\left|F_{2}\right|dy\right)dx$
$\displaystyle=\int_{\mathbb{R}^{N}_{+}\cap
B_{R}}x_{N}^{\alpha}\left|\psi\right|\left(\int_{\mathbb{R}^{N}}\frac{\left|u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right|}{\left|y\right|^{N+2s}}\left|a\left(\frac{y}{\left|y\right|}\right)\right|dy\right)dx$
$\displaystyle\leq
D\left\|\psi\right\|_{L^{\infty}\left(\mathbb{R}^{N}\right)}\int_{\mathbb{R}^{N}_{+}\cap
B_{R}}x_{N}^{\alpha}\left(\int_{\mathbb{R}^{N}}\frac{\left|u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)\right|}{\left|y\right|^{N+2s}}dy\right)dx$
$\displaystyle\leq
C\left(D,s,\delta,\beta,R,\left\|\psi\right\|_{L^{\infty}\left(\mathbb{R}^{N}\right)},\left\|u\right\|_{C^{2}\left(\overline{\mathbb{R}^{N}_{+}}\cap
B_{2R}\right)}\right)\int_{\mathbb{R}^{N}_{+}\cap
B_{R}}\frac{1}{x_{N}^{2s-\alpha}}dx<+\infty.$
∎
###### Remark 2.3.
For the sake of completeness we point out that the assumption $\alpha<2s$ in
Proposition 2.2 can be in fact removed. Such condition has been used in
(2.7)-(2.8). On the other hand, since the function $\psi$ is compactly
supported in the ball $B_{R}$, a more precise estimate of (2.7) can be
obtained. Indeed, since for any $x\in\mathbb{R}^{N}_{+}\cap B_{2R}$ and
$|z|\geq\frac{3R}{x_{N}}$ we have that $\psi(x\pm x_{N}z)=0$, then
$\displaystyle\left|A\right|\leq
C\left(\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&\left|z\right|^{2},&&\left|z\right|<\frac{1}{2},\\\
&\left|z\right|^{\alpha},&&\frac{1}{2}\leq\left|z\right|<\frac{3R}{x_{N}},\\\
&1,&&\left|z\right|\geq\frac{3R}{x_{N}},\end{aligned}\right.$
and
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|A\right|}{\left|z\right|^{N+2s}}dz\leq
C\left(N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&1+\frac{1}{x_{N}^{\alpha-2s}},&&\alpha>2s,\\\
&1+\left|\log{x_{N}}\right|,&&\alpha=2s,\\\
&1,&&\alpha<2s.\end{aligned}\right.$
Therefore for any $x\in\mathbb{R}^{N}_{+}\cap B_{2R}$,
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy$
$\displaystyle=\left(\int_{\mathbb{R}^{N}}\frac{\left|A\right|}{\left|z\right|^{N+2s}}dz\right)x_{N}^{\alpha-2s}$
$\displaystyle\leq
C\left(N,s,\alpha,R,\left\|\psi\right\|_{C^{2}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&1,&&\alpha>2s,\\\ &1+\left|\log{x_{N}}\right|,&&\alpha=2s,\\\
&\frac{1}{x_{N}^{2s-\alpha}},&&\alpha<2s\end{aligned}\right.$
and as a result
$\displaystyle\int_{\mathbb{R}^{N}_{+}\cap
B_{2R}}\left(\int_{\mathbb{R}^{N}}\left|F_{1}\right|dy\right)dx$
$\displaystyle\leq
D\left\|u\right\|_{L^{\infty}\left(B_{2R}\right)}\int_{\mathbb{R}^{N}_{+}\cap
B_{2R}}\left(\int_{\mathbb{R}^{N}}\frac{\left|v_{\alpha}\left(x+y\right)+v_{\alpha}\left(x-y\right)-2v_{\alpha}\left(x\right)\right|}{\left|y\right|^{N+2s}}dy\right)dx<+\infty.$
We are now concerned with the computation of the operator (1.1) acting on
barrier type functions. The result of the next Lemma is partially known, in
particular the fact that for $\alpha=s$, $\left(x_{N}\right)_{+}^{\alpha}$ is
harmonic see e.g. [12, 33]. But a complete result for $\alpha\in(0,2s)$
doesn’t seem to be available, so we decided to put here the statement and a
short proof.
###### Lemma 2.4.
Let $0<\alpha<2s$ and
$w_{\alpha}\left(x\right)=\left(x_{N}\right)_{+}^{\alpha}$. For any
$x\in\mathbb{R}^{N}_{+}$,
$\displaystyle Lw_{\alpha}\left(x\right)=C_{\alpha}x_{N}^{\alpha-2s},$
where
$\displaystyle C_{\alpha}\left\\{\begin{aligned} &<0,&&0<\alpha<s,\\\
&=0,&&\alpha=s,\\\ &>0,&&s<\alpha<2s.\end{aligned}\right.$
###### Proof.
For any $x\in\mathbb{R}^{N}_{+}$,
$\displaystyle Lw_{\alpha}\left(x\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\left(x_{N}+y_{N}\right)_{+}^{\alpha}+\left(x_{N}-y_{N}\right)_{+}^{\alpha}-2x_{N}^{\alpha}}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=\int_{\mathbb{S}^{N-1}}\left(\int_{0}^{+\infty}\frac{\left(x_{N}+r\theta_{N}\right)_{+}^{\alpha}+\left(x_{N}-r\theta_{N}\right)_{+}^{\alpha}-2x_{N}^{\alpha}}{r^{1+2s}}dr\right)a\left(\theta\right)d\theta$
$\displaystyle=x_{N}^{\alpha}\int_{\mathbb{S}^{N-1}}\left(\int_{0}^{+\infty}\frac{\left(1+\frac{\theta_{N}}{x_{N}}r\right)_{+}^{\alpha}+\left(1-\frac{\theta_{N}}{x_{N}}r\right)_{+}^{\alpha}-2}{r^{1+2s}}dr\right)a\left(\theta\right)d\theta.$
By the change of variable $t=\frac{\left|\theta_{N}\right|}{x_{N}}r$, with
$\theta_{N}\neq 0$, we obtain
$\displaystyle\int_{0}^{+\infty}\frac{\left(1+\frac{\theta_{N}}{x_{N}}r\right)_{+}^{\alpha}+\left(1-\frac{\theta_{N}}{x_{N}}r\right)_{+}^{\alpha}-2}{r^{1+2s}}dr$
$\displaystyle=\left(\frac{\left|\theta_{N}\right|}{x_{N}}\right)^{2s}\int_{0}^{+\infty}\frac{\left(1+t\right)_{+}^{\alpha}+\left(1-t\right)_{+}^{\alpha}-2}{t^{1+2s}}dt$
$\displaystyle=:c_{\alpha}\left|\theta_{N}\right|^{2s}x_{N}^{-2s}.$
Moreover, the above equalities are still true for $\theta_{N}=0$. Thus we have
$\displaystyle Lw_{\alpha}\left(x\right)=C_{\alpha}x_{N}^{\alpha-2s}$
with
$C_{\alpha}:=c_{\alpha}\left(\int_{\mathbb{S}^{N-1}}\left|\theta_{N}\right|^{2s}a\left(\theta\right)d\theta\right).$
By the assumption (1.6) it turns out that
$\int_{\mathbb{S}^{N-1}}\left|\theta_{N}\right|^{2s}a\left(\theta\right)d\theta>0.$
Hence the sign of $C_{\alpha}$ is given by $c_{\alpha}$. For this, by the same
arguments of [12, Lemma 2.4] concerning the case $\alpha=s$ (see also [9,
Lemma 2.3]), we get that
$\displaystyle c_{\alpha}\left\\{\begin{aligned} &<0,&&0<\alpha<s,\\\
&=0,&&\alpha=s,\\\ &>0,&&s<\alpha<2s.\end{aligned}\right.$
Hence the result follows. ∎
###### Lemma 2.5.
Let $u,g,h\in C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$.
* (1)
For any $R>0$ and $u_{R}=u\left(\frac{x}{R}\right)$,
$\displaystyle
Lu_{R}\left(x\right)=R^{-2s}Lu\left(\frac{x}{R}\right)\quad\forall
x\in\mathbb{R}_{+}^{N}.$
* (2)
For $u=gh$,
$\displaystyle
Lu\left(x\right)=g\left(x\right)Lh\left(x\right)+h\left(x\right)Lg\left(x\right)+l\left[g,h\right]\left(x\right)\quad\forall
x\in\mathbb{R}_{+}^{N},$
where
$\displaystyle l\left[g,h\right]\left(x\right)=$
$\displaystyle\int_{\mathbb{R}^{N}}\left\\{\frac{\left[g\left(x+y\right)-g\left(x\right)\right]\left[h\left(x+y\right)-h\left(x\right)\right]}{\left|y\right|^{N+2s}}\right.$
$\displaystyle\left.+\,\frac{\left[g\left(x-y\right)-g\left(x\right)\right]\left[h\left(x-y\right)-h\left(x\right)\right]}{\left|y\right|^{N+2s}}\right\\}a\left(\frac{y}{\left|y\right|}\right)dy.$
###### Proof.
For any $R>0$,
$\displaystyle Lu_{R}\left(x\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{u\left(\frac{x+y}{R}\right)+u\left(\frac{x-y}{R}\right)-2u\left(\frac{x}{R}\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=R^{-2s}\int_{\mathbb{R}^{N}}\frac{u\left(\frac{x}{R}+y\right)+u\left(\frac{x}{R}-y\right)-2u\left(\frac{x}{R}\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle=R^{-2s}Lu\left(\frac{x}{R}\right),$
hence we get Lemma 2.5-(1). Next, for $x\in\mathbb{R}_{+}^{N}$ and any
$y\in\mathbb{R}^{N}$,
$\displaystyle g$
$\displaystyle\left(x+y\right)h\left(x+y\right)+g\left(x-y\right)h\left(x-y\right)-2g\left(x\right)h\left(x\right)$
$\displaystyle=\left[g\left(x+y\right)+g\left(x-y\right)-2g\left(x\right)\right]h\left(x\right)+\left[h\left(x+y\right)+h\left(x-y\right)-2h\left(x\right)\right]g\left(x\right)$
$\displaystyle\quad+\left[g\left(x+y\right)-g\left(x\right)\right]\left[h\left(x+y\right)-h\left(x\right)\right]+\left[g\left(x-y\right)-g\left(x\right)\right]\left[h\left(x-y\right)-h\left(x\right)\right].$
Lemma 2.5-(2) is a direct result of the above identity. ∎
We conclude this section with a continuity property of the function $l$
introduced in Lemma 2.5-(2) that will be used in Theorem 1.1.
###### Lemma 2.6.
Let $(2s-1)_{+}<\alpha<\min\left\\{1,2s\right\\}$, $\varphi\in
C^{1}_{0}\left(\mathbb{R}^{N}\right)$ and
$w_{\alpha}\left(x\right)=\left(x_{N}\right)_{+}^{\alpha}$. For any
$\left\\{x_{n}\right\\}\subset\mathbb{R}_{+}^{N}$ and $x_{n}\rightarrow
x_{0}\in\overline{\mathbb{R}_{+}^{N}}$ as $n\rightarrow+\infty$, we have
$\displaystyle\lim_{n\rightarrow+\infty}l\left[w_{\alpha},\varphi\right]\left(x_{n}\right)=l\left[w_{\alpha},\varphi\right]\left(x_{0}\right).$
###### Proof.
By Lemma 2.5-(2),
$\displaystyle l\left[w_{\alpha},\varphi\right]\left(x_{n}\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\left\\{\frac{\left[\left(\left(x_{n}\right)_{N}+y_{N}\right)_{+}^{\alpha}-\left(x_{n}\right)_{N}^{\alpha}\right]\left[\varphi\left(x_{n}+y\right)-\varphi\left(x_{n}\right)\right]}{\left|y\right|^{N+2s}}\right.$
$\displaystyle\quad\left.+\,\frac{\left[\left(\left(x_{n}\right)_{N}-y_{N}\right)_{+}^{\alpha}-\left(x_{n}\right)_{N}^{\alpha}\right]\left[\varphi\left(x_{n}-y\right)-\varphi\left(x_{n}\right)\right]}{\left|y\right|^{N+2s}}\right\\}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle:=\int_{\mathbb{R}^{N}}H_{n}\left(y\right)dy.$
Since $0<\alpha<1$, for any $n\in\mathbb{N}$ and $y\in\mathbb{R}^{N}$ we have
$\displaystyle\left|\left(\left(x_{n}\right)_{N}\pm
y_{N}\right)_{+}^{\alpha}-\left(x_{n}\right)_{N}^{\alpha}\right|\leq\left|y_{N}\right|^{\alpha}.$
Moreover,
$\displaystyle\left|\varphi\left(x_{n}\pm
y\right)-\varphi\left(x_{n}\right)\right|\leq
C\left(\left\|\varphi\right\|_{C^{1}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&|y|,&&\left|y\right|<1,\\\ &1,&&\left|y\right|\geq 1.\end{aligned}\right.$
Hence we get
$\displaystyle\left|H_{n}\left(y\right)\right|\leq
H\left(y\right)=C\left(D,\left\|\varphi\right\|_{C^{1}\left(\mathbb{R}^{N}\right)}\right)\times\left\\{\begin{aligned}
&\frac{1}{\left|y\right|^{N+2s-\alpha-1}},&&\left|y\right|<1,\\\
&\frac{1}{\left|y\right|^{N+2s-\alpha}},&&\left|y\right|\geq
1.\end{aligned}\right.$
By the assumption $2s-1<\alpha<2s$, we have $H\in
L^{1}\left(\mathbb{R}^{N}\right)$. By Lebesgue’s dominated convergence
theorem, we reach the conclusion. ∎
## 3\. Nonexistence in the half space $\mathbb{R}_{+}^{N}$
This section is devoted to the proof of Theorem 1.1. Observe that Theorem 1.1
is true if $N=1$, in this case the operators (1.1) satisfying (1.5)-(1.6)
reduce, up to a positive multiplicative constant, to the $1$-dimensional
fractional Laplacian. The proof can be done in a similar way to the case
$N\geq 2$ but every step is much simpler, details are left to the reader. So
from now on we suppose that $N\geq 2$ unless otherwise specified.
We start with some geometric considerations concerning the cones
$\Sigma_{\nu,\tau}\left(x\right)$, see (1.4) for their definition. Next lemma
is basic, we report a proof for the reader’s convenience.
###### Lemma 3.1.
Given $\nu\in{\mathbb{S}}^{N-1}$ and $\tau\in(0,1]$, then for any
$x\in\partial B_{1}$ it holds
$\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap B_{1}\right|>0.$
###### Proof.
If $x\cdot\nu\neq 0$, there exists $t_{0}\in\mathbb{R}$ and
$\varepsilon_{0}>0$ such that
$y_{0}=x+t_{0}\nu\in\Sigma_{\nu,\tau}\left(x\right)\cap B_{1}$ and
$B_{\varepsilon_{0}}\left(y_{0}\right)\subset\left(\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\right)$, hence
$\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\right|>\left|B_{\varepsilon_{0}}\left(y_{0}\right)\right|>0.$
If $x\cdot\nu=0$, then we can pick $\widetilde{\nu}\in\mathbb{S}^{N-1}$ such
that $x\cdot\widetilde{\nu}\neq 0$ and
$0<\left|\widetilde{\nu}-\nu\right|\leq\frac{\tau}{2}$. For any
$y\in\Sigma_{\widetilde{\nu},\frac{\tau}{2}}\left(x\right)$ we have
$\displaystyle\left|\left(y-x\right)\cdot\nu\right|$
$\displaystyle\geq\left|\left(y-x\right)\cdot\widetilde{\nu}\right|-\left|\left(y-x\right)\cdot\left(\widetilde{\nu}-\nu\right)\right|$
$\displaystyle\geq\left(1-\frac{\tau}{2}\right)\left|y-x\right|-\frac{\tau}{2}\left|y-x\right|=\left(1-\tau\right)\left|y-x\right|.$
Then
$\Sigma_{\widetilde{\nu},\frac{\tau}{2}}\left(x\right)\subseteq\Sigma_{\nu,\tau}\left(x\right)$
and by the first part of the proof we conclude that
$\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\right|\geq\left|\Sigma_{\widetilde{\nu},\frac{\tau}{2}}\left(x\right)\cap
B_{1}\right|>0.$
∎
###### Remark 3.2.
In the whole space $\mathbb{R}^{N}$, for any $x\in\partial B_{1}$, it is
obvious that
$\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\cap\mathbb{R}^{N}\right|=\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\right|>0$
as showed in Lemma 3.1, see also Figure 2.
Figure 2. The blue area $\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\right|>0$.
However, in the half space $\mathbb{R}_{+}^{N}$, there might be that for some
$x\in\partial\mathbb{R}_{+}^{N}\cap\partial B_{1}$
$\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\cap\mathbb{R}^{N}_{+}\right|=0,$
see Figure 3.
Our strategy is therefore the following: since, as a consequence of Lemma 3.1,
$\left|\Sigma_{\nu,\tau}\left(0\right)\cap B_{1}\left(e_{N}\right)\right|>0,$
we can then slightly move the center of the ball $B_{1}\left(e_{N}\right)$
from $e_{N}=(0,\ldots,0,1)$ to
$\left(1-\gamma\right)e_{N}=\left(0,...,0,1-\gamma\right)$, with $\gamma>0$
sufficiently small, in order to guarantee that
$\displaystyle\left|B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap{\mathbb{R}^{N}_{-}}\right|>0$
and that for any $x\in\partial\mathbb{R}_{+}^{N}\cap\partial
B_{1}\left(\left(1-\gamma\right)e_{N}\right)$
(3.1) $\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\mathbb{R}^{N}_{+}\right|>0,$
see Figure 4. In fact, condition (3.1) is still true for any $x\in\partial
B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\overline{\mathbb{R}^{N}_{+}}$.
Figure 3. $\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\cap\mathbb{R}^{N}_{+}\right|=0$. Figure 4. The blue area
$\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\mathbb{R}^{N}_{+}\right|>0$.
The above considerations are summarized in the following
###### Corollary 3.3.
Given $\nu\in{\mathbb{S}}^{N-1}$ and $\tau\in(0,1]$, then there exist
$\gamma=\gamma\left(\nu,\tau\right)\in\left(0,1\right)$ such that for any
$x\in\partial
B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\overline{\mathbb{R}^{N}_{+}}$,
$\displaystyle\left|\Sigma_{\nu,\tau}\left(x\right)\cap
B_{1}\left(\left(1-\gamma\right)e_{N}\right)\cap\mathbb{R}^{N}_{+}\right|>0.$
Next theorem is the main result from which Theorem 1.1 easily follows, as
proved at the end of this section.
###### Theorem 3.4.
Assume $1\leq p\leq\frac{N+s}{N-s}$. If $u\in
C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}$ is a nonnegative
solution of (2.2) satisfying (2.3), then $u\equiv 0$.
###### Proof.
For the $\nu_{0},\tau_{0}$ given in (1.6), by Corollary 3.3, there exists
$0<\gamma_{0}<1$ such that for any $x\in\partial
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)\cap\overline{\mathbb{R}^{N}_{+}}$,
(3.2) $\displaystyle\left|\Sigma_{\nu_{0},\tau_{0}}\left(x\right)\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)\cap\mathbb{R}^{N}_{+}\right|>0.$
We choose $\varphi\in C^{\infty}_{0}\left(\mathbb{R}^{N}\right)$ such that
$0<\varphi\leq 1$ in $B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$ and
$\displaystyle\varphi\left(x\right)=\left\\{\begin{aligned} &1,&&x\in
B_{1-\frac{\gamma_{0}}{2}}\left(\left(1-\gamma_{0}\right)e_{N}\right),\\\
&0,&&x\notin
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right),\end{aligned}\right.$
see Figure 5.
Figure 5. The graph of $\varphi$.
For any $s\leq\alpha<\min\left\\{1,2s\right\\}$, we define
$\displaystyle\phi_{\alpha}\left(x\right):=w_{\alpha}(x)\varphi\left(x\right):=\left(x_{N}\right)_{+}^{\alpha}\varphi\left(x\right).$
Step 1. For any given $\alpha_{0}\in\left(s,\min\left\\{1,2s\right\\}\right)$,
we show that there exists $M>0$ such that
(3.3)
$\displaystyle-L\phi_{\alpha_{0}}\left(x\right)-L\phi_{s}\left(x\right)\leq
M\phi_{s}\left(x\right)\quad\forall x\in\mathbb{R}_{+}^{N}.$
For this we first note that for any
$x\in\mathbb{R}_{+}^{N}\cap\left\\{\left|x-\left(1-\gamma_{0}\right)e_{N}\right|\geq
1\right\\}$, by the assumption (1.5) we infer that
$\displaystyle-L\phi_{\alpha_{0}}\left(x\right)-L\phi_{s}\left(x\right)=$
$\displaystyle-\int_{\mathbb{R}^{N}}\frac{\phi_{\alpha_{0}}\left(x+y\right)+\phi_{\alpha_{0}}\left(x-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle-\int_{\mathbb{R}^{N}}\frac{\phi_{s}\left(x+y\right)+\phi_{s}\left(x-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy\leq
0=M\phi_{s}\left(x\right).$
Hence we only need to show that
(3.4) $\displaystyle\inf_{x\in\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)}\frac{L\phi_{\alpha_{0}}\left(x\right)+L\phi_{s}\left(x\right)}{\phi_{s}\left(x\right)}>-\infty.$
Assuming that (3.4) is not true, then there exists a convergent sequence
$\left\\{x_{n}\right\\}_{n}\subset\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$ such that
(3.5)
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)}{\phi_{s}\left(x_{n}\right)}=-\infty.$
Let $x_{n}\rightarrow x_{\infty}\in\overline{\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)}$ as $n\rightarrow+\infty$.
We distinguish different cases, each of them producing a contradiction to
(3.5).
Case 1: $x_{\infty}\in\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$. We obtain
$\displaystyle\lim_{n\rightarrow+\infty}L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)=L\phi_{\alpha_{0}}\left(x_{\infty}\right)+L\phi_{s}\left(x_{\infty}\right)$
and
$\displaystyle
0<\lim_{n\rightarrow+\infty}\phi_{s}\left(x_{n}\right)=\phi_{s}\left(x_{\infty}\right)<+\infty,$
then
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)}{\phi_{s}\left(x_{n}\right)}=\frac{L\phi_{\alpha_{0}}\left(x_{\infty}\right)+L\phi_{s}\left(x_{\infty}\right)}{\phi_{s}\left(x_{\infty}\right)}.$
Case 2: $x_{\infty}\in\mathbb{R}_{+}^{N}\cap\partial
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$. Recalling that $a\geq 0$,
by (1.6) and (3.2) we have
$\displaystyle\lim_{n\rightarrow+\infty}L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\phi_{\alpha_{0}}\left(x_{\infty}+y\right)+\phi_{\alpha_{0}}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\quad+\int_{\mathbb{R}^{N}}\frac{\phi_{s}\left(x_{\infty}+y\right)+\phi_{s}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\geq
d\int_{\Sigma_{\nu_{0},\tau_{0}}\left(0\right)}\frac{\phi_{\alpha_{0}}\left(x_{\infty}+y\right)+\phi_{\alpha_{0}}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}dy$
$\displaystyle\quad+d\int_{\Sigma_{\nu_{0},\tau_{0}}\left(0\right)}\frac{\phi_{s}\left(x_{\infty}+y\right)+\phi_{s}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}dy>0.$
Moreover, $\phi_{s}\left(x_{n}\right)>0$ for any $n$ and
$\displaystyle\lim_{n\rightarrow+\infty}\phi_{s}\left(x_{n}\right)=0,$
thus
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)}{\phi_{s}\left(x_{n}\right)}=+\infty.$
Case 3: $x_{\infty}\in\partial\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$. By Lemmas 2.4 and 2.5-(2),
for any $x\in\mathbb{R}^{N}_{+}$
(3.6)
$\begin{split}L\phi_{\alpha_{0}}\left(x\right)+L\phi_{s}\left(x\right)&=C_{\alpha_{0}}\varphi\left(x\right)x^{\alpha_{0}-2s}_{N}+x_{N}^{\alpha_{0}}L\varphi\left(x\right)+x_{N}^{s}L\varphi\left(x\right)\\\
&\quad+l\left[w_{\alpha_{0}},\varphi\right]\left(x\right)+l\left[w_{s},\varphi\right]\left(x\right),\end{split}$
where $C_{\alpha_{0}}>0$ and
$\displaystyle
l\left[w_{\alpha_{0}},\varphi\right]\left(x\right)+l\left[w_{s},\varphi\right]\left(x\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\left\\{\frac{\left[\varphi\left(x+y\right)-\varphi\left(x\right)\right]\left[\left(x_{N}+y_{N}\right)_{+}^{\alpha_{0}}+\left(x_{N}+y_{N}\right)_{+}^{s}-x_{N}^{\alpha_{0}}-x_{N}^{s}\right]}{\left|y\right|^{N+2s}}\right.dy$
$\displaystyle\quad\left.+\frac{\left[\varphi\left(x-y\right)-\varphi\left(x\right)\right]\left[\left(x_{N}-y_{N}\right)_{+}^{\alpha_{0}}+\left(x_{N}-y_{N}\right)_{+}^{s}-x_{N}^{\alpha_{0}}-x_{N}^{s}\right]}{\left|y\right|^{N+2s}}\right\\}a\left(\frac{y}{\left|y\right|}\right)dy.$
We see that
$\displaystyle\lim_{n\rightarrow+\infty}C_{\alpha_{0}}\varphi\left(x_{n}\right)\cdot\left(x_{n}\right)_{N}^{\alpha_{0}-2s}=+\infty$
and
$\displaystyle\lim_{n\rightarrow+\infty}\left(x_{n}\right)_{N}^{\alpha_{0}}L\varphi\left(x_{n}\right)+\left(x_{n}\right)_{N}^{s}L\varphi\left(x_{n}\right)=0.$
Moreover Lemma 2.6 yields
$\displaystyle\lim_{n\rightarrow+\infty}l\left[w_{\alpha_{0}},\varphi\right]\left(x_{n}\right)+l\left[w_{s},\varphi\right]\left(x_{n}\right)=l\left[w_{\alpha_{0}},\varphi\right]\left(x_{\infty}\right)+l\left[w_{s},\varphi\right]\left(x_{\infty}\right),$
hence by (3.6),
$\displaystyle\lim_{n\rightarrow+\infty}L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)=+\infty.$
By the fact
$\displaystyle\lim_{n\rightarrow+\infty}\phi_{s}\left(x_{n}\right)=0,$
and $\phi_{s}\left(x_{n}\right)>0$ for any $n$, we get
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)}{\phi_{s}\left(x_{n}\right)}=+\infty.$
Case 4: $x_{\infty}\in\partial\mathbb{R}_{+}^{N}\cap\partial
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)$. Notice that by (3.6), since
$\varphi(x_{n})>0$ for any $n$,
(3.7)
$\begin{split}L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)&\geq\left(x_{n}\right)_{N}^{\alpha_{0}}L\varphi\left(x_{n}\right)+\left(x_{n}\right)_{N}^{s}L\varphi\left(x_{n}\right)\\\
&\quad+l\left[w_{\alpha_{0}},\varphi\right]\left(x_{n}\right)+l\left[w_{s},\varphi\right]\left(x_{n}\right).\end{split}$
It is obvious that
(3.8)
$\displaystyle\lim_{n\rightarrow+\infty}\left(x_{n}\right)_{N}^{\alpha_{0}}L\varphi\left(x_{n}\right)+\left(x_{n}\right)_{N}^{s}L\varphi\left(x_{n}\right)=0.$
Since $a\geq 0$, by (1.6), Lemma 2.6 and (3.2) we have
$\displaystyle\lim_{n\rightarrow+\infty}l\left[w_{\alpha_{0}},\varphi\right]\left(x_{n}\right)+l\left[w_{s},\varphi\right]\left(x_{n}\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\phi_{\alpha_{0}}\left(x_{\infty}+y\right)+\phi_{\alpha_{0}}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\quad+\int_{\mathbb{R}^{N}}\frac{\phi_{s}\left(x_{\infty}+y\right)+\phi_{s}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\geq
d\int_{\Sigma_{\nu_{0},\tau_{0}}\left(0\right)}\frac{\phi_{\alpha_{0}}\left(x_{\infty}+y\right)+\phi_{\alpha_{0}}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}dy$
(3.9)
$\displaystyle\quad+d\int_{\Sigma_{\nu_{0},\tau_{0}}\left(0\right)}\frac{\phi_{s}\left(x_{\infty}+y\right)+\phi_{s}\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}dy>0.$
Combining (3.7), (3.8) and (3) with the fact
$\displaystyle\lim_{n\rightarrow+\infty}\phi_{s}\left(x_{n}\right)=0$
and $\phi_{s}\left(x_{n}\right)>0$ for any $n$, even in this case we get :
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\phi_{\alpha_{0}}\left(x_{n}\right)+L\phi_{s}\left(x_{n}\right)}{\phi_{s}\left(x_{n}\right)}=+\infty.$
In conclusion, we have reached a contradiction to (3.5) for any
$x_{\infty}\in\overline{\mathbb{R}_{+}^{N}\cap
B_{1}\left(\left(1-\gamma_{0}\right)e_{N}\right)}$, then inequality (3.3)
holds.
Step 2. For any $R>0$, we make the following rescaling
$\displaystyle\varphi_{R}\left(x\right)=\varphi\left(\frac{x}{R}\right)=\left\\{\begin{aligned}
&1,&&x\in
B_{R\left(1-\frac{\gamma_{0}}{2}\right)}\left(R\left(1-\gamma_{0}\right)e_{N}\right),\\\
&0,&&x\notin
B_{R}\left(R\left(1-\gamma_{0}\right)e_{N}\right).\end{aligned}\right.$
For any $s\leq\alpha<\min\left\\{1,2s\right\\}$, we define
$\displaystyle\phi_{\alpha,R}\left(x\right)=\left(x_{N}\right)_{+}^{\alpha}\varphi_{R}\left(x\right).$
Since $u$ is solution of (2.2), by Proposition 2.2 and Lemma 2.5-(1) we obtain
$\displaystyle\int_{\mathbb{R}_{+}^{N}}u^{p}\phi_{s,R}dx$
$\displaystyle\leq-R^{s}\int_{\mathbb{R}^{N}}\left(\frac{x_{N}}{R}\right)_{+}^{s}\varphi_{R}Ludx$
$\displaystyle\leq-R^{s}\int_{\mathbb{R}^{N}}\left(\frac{x_{N}}{R}\right)_{+}^{s}\varphi_{R}Ludx-R^{s}\int_{\mathbb{R}^{N}}\left(\frac{x_{N}}{R}\right)_{+}^{\alpha_{0}}\varphi_{R}Ludx$
$\displaystyle=-R^{-s}\int_{\mathbb{R}^{N}}uL\phi_{s}\left(\frac{x}{R}\right)dx-R^{-s}\int_{\mathbb{R}^{N}}uL\phi_{\alpha_{0}}\left(\frac{x}{R}\right)dx$
$\displaystyle=R^{-s}\int_{\mathbb{R}_{+}^{N}}u\left[-L\phi_{\alpha_{0}}\left(\frac{x}{R}\right)-L\phi_{s}\left(\frac{x}{R}\right)\right]dx.$
Now we use (3.3) to infer that
$\displaystyle\int_{\mathbb{R}_{+}^{N}}u\left[-L\phi_{\alpha_{0}}\left(\frac{x}{R}\right)-L\phi_{s}\left(\frac{x}{R}\right)\right]dx\leq
M\int_{\mathbb{R}_{+}^{N}}u\phi_{s}\left(\frac{x}{R}\right)dx=MR^{-s}\int_{\mathbb{R}_{+}^{N}}u\phi_{s,R}dx$
and that
(3.10) $\displaystyle\int_{\mathbb{R}_{+}^{N}}u^{p}\phi_{s,R}dx\leq
MR^{-2s}\int_{\mathbb{R}_{+}^{N}}u\phi_{s,R}dx.$
Since $0\leq\phi_{s,R}\leq(x_{N})_{+}^{s}$ and
$B_{R}\left(R\left(1-\gamma_{0}\right)e_{N}\right)\subset B_{2R}$, we may
apply the Hölder inequality to get, for any $p>1$,
$\displaystyle\int_{\mathbb{R}_{+}^{N}}u\phi_{s,R}dx$
$\displaystyle\leq\left(\int_{\mathbb{R}_{+}^{N}\cap
B_{R}\left(R\left(1-\gamma_{0}\right)e_{N}\right)}u^{p}\phi_{s,R}dx\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}_{+}^{N}\cap
B_{R}\left(R\left(1-\gamma_{0}\right)e_{N}\right)}\phi_{s,R}dx\right)^{\frac{p-1}{p}}$
$\displaystyle\leq
C\left(N,p\right)R^{\frac{\left(N+s\right)\left(p-1\right)}{p}}\left(\int_{\mathbb{R}_{+}^{N}}u^{p}\phi_{s,R}dx\right)^{\frac{1}{p}}.$
Therefore, by (3.10),
(3.11) $\displaystyle\int_{\mathbb{R}_{+}^{N}}u^{p}\phi_{s,R}dx\leq
C\left(N,M,p\right)R^{N+s-\frac{2sp}{p-1}}.$
Step 3. In the case $p=1$, we immediately obtain $u\equiv 0$ by letting
$R\rightarrow+\infty$ in (3.10).
We now consider $p>1$. Since $\varphi_{R}\rightarrow\varphi_{\infty}\equiv 1$
in $\mathbb{R}^{N}_{+}$ and $\phi_{s,R}\rightarrow\phi_{s,\infty}=x_{N}^{s}$
as $R\rightarrow+\infty$, if we let $R\rightarrow+\infty$ in (3.11), we have
$u\equiv 0$ provided $p<\frac{N+s}{N-s}$.
If instead $p=\frac{N+s}{N-s}$, then we infer that
(3.12) $\displaystyle\int_{\mathbb{R}_{+}^{N}}x^{s}_{N}u^{p}dx<+\infty.$
We can rewrite
$\displaystyle\int_{\mathbb{R}_{+}^{N}}u\phi_{s,R}dx=\int_{\mathbb{R}_{+}^{N}\cap
B_{\sqrt{R}}}u\phi_{s,R}dx+\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}u\phi_{s,R}dx.$
Using once again the fact $0\leq\phi_{s,R}\leq(x_{N})_{+}^{s}$, by the Hölder
inequality we obtain
$\displaystyle\int_{\mathbb{R}_{+}^{N}\cap B_{\sqrt{R}}}u\phi_{s,R}dx$
$\displaystyle\leq\left(\int_{\mathbb{R}_{+}^{N}\cap
B_{\sqrt{R}}}x_{N}^{s}u^{p}dx\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}_{+}^{N}\cap
B_{\sqrt{R}}}x_{N}^{s}dx\right)^{\frac{p-1}{p}}$ $\displaystyle\leq
C\left(N,p\right)R^{\frac{\left(N+s\right)\left(p-1\right)}{2p}}\left(\int_{\mathbb{R}_{+}^{N}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}$
$\displaystyle=C\left(N,p\right)R^{s}\left(\int_{\mathbb{R}_{+}^{N}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}$
and
$\displaystyle\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}u\phi_{s,R}dx$
$\displaystyle\leq\left(\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}x_{N}^{s}u^{p}dx\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}x_{N}^{s}dx\right)^{\frac{p-1}{p}}$ $\displaystyle\leq
C\left(N,p\right)R^{\frac{\left(N+s\right)\left(p-1\right)}{p}}\left(\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}$
$\displaystyle=C\left(N,p\right)R^{2s}\left(\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}.$
By (3.10), we get
$\displaystyle\int_{\mathbb{R}_{+}^{N}}u^{p}\phi_{s,R}dx\leq
C\left(N,M,p\right)\left[R^{-s}\left(\int_{\mathbb{R}_{+}^{N}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}+\left(\int_{\mathbb{R}_{+}^{N}\cap\left\\{\sqrt{R}\leq\left|x\right|\leq
2R\right\\}}x^{s}_{N}u^{p}dx\right)^{\frac{1}{p}}\right].$
Therefore, by (3.12) the right-hand side of the above inequality goes to $0$
as $R\rightarrow+\infty$ and we again conclude that $u\equiv 0$. ∎
Now we are ready to prove Theorem 1.1.
###### Proof of Theorem 1.1.
Let $u$ be any nonnegative classical solution of (1.2). By Lemma 2.1, the
function $\widetilde{u}$, defined in (2.1), is a nonnegative classical
solution of (2.2). Theorem 3.4 yields $\widetilde{u}\equiv 0$ in
$\mathbb{R}^{N}$, hence
$\displaystyle u\left\\{\begin{aligned} &\geq 0,&&x_{N}<1,\\\ &=0,&&x_{N}\geq
1.\end{aligned}\right.$
For any $\overline{x}$ with $\overline{x}_{N}=1$, we have
$\displaystyle 0=u^{p}\left(\overline{x}\right)\leq-
Lu\left(\overline{x}\right)=-\int_{\mathbb{R}^{N}}\frac{u\left(\overline{x}+y\right)+u\left(\overline{x}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy\leq
0$
and $Lu\left(\overline{x}\right)=0$. Since $u\in
C^{2}\left(\mathbb{R}_{+}^{N}\right)$, then $u\equiv 0$ in
$\Sigma_{\nu_{0},\tau_{0}}\left(\overline{x}\right)\cap\mathbb{R}^{N}_{+}$.
The arbitrariness of $\overline{x}$ implies $u\equiv 0$ in
$\mathbb{R}^{N}_{+}$ and moreover $u=0$ a.e. in
$\overline{\mathbb{R}_{-}^{N}}$, see Figure 6.
Figure 6.
∎
## 4\. Optimality of the critical exponent $\frac{N+s}{N-s}$
We prove that the exponent $p=\frac{N+s}{N-s}$ in Theorem 1.1 is sharp, in the
sense that there is at least some operator in the class (1.1) for which
problem (1.2) has nontrivial solutions as soon as $p>\frac{N+s}{N-s}$. In
fact, this occurs whenever $a(\theta)$ is any positive constant function, in
which case $-L$ coincides, up to a multiplicative constant, with
$(-\Delta)^{s}$.
###### Theorem 4.1.
For any $p>\frac{N+s}{N-s}$ and $N\geq 2$, the problem
(4.1) $\displaystyle\left\\{\begin{aligned} \left(-\Delta\right)^{s}u&\geq
u^{p},&&x\in\mathbb{R}_{+}^{N},\\\
u&=0,&&x\in\mathbb{R}_{-}^{N}\end{aligned}\right.$
admits positive classical solutions.
###### Proof.
Let
$\displaystyle w_{\alpha}\left(x\right)=\left(x_{N}\right)^{\alpha}_{+},\quad
0<\alpha<s.$
Consider the Kelvin transform of $w_{\alpha}$, given by
$\displaystyle\overline{w}_{\alpha}\left(x\right):=\frac{1}{\left|x\right|^{N-2s}}w_{\alpha}\left(\frac{x}{\left|x\right|^{2}}\right)=\frac{\left(x_{N}\right)^{\alpha}_{+}}{\left|x\right|^{N-2s+2\alpha}}\in
C^{2}\left(\mathbb{R}_{+}^{N}\right)\cap\mathcal{L}_{s}.$
By the general property of the Kelvin transformation
$\left(-\Delta\right)^{s}\overline{w}_{\alpha}(x)=\frac{1}{\left|x\right|^{N+2s}}\left(-\Delta\right)^{s}w_{\alpha}\left(\frac{x}{\left|x\right|^{2}}\right),$
see e.g. [31, Proposition A.1], from Lemma 2.4 we infer that for any
$x\in\mathbb{R}^{N}_{+}$,
$-\left(-\Delta\right)^{s}\overline{w}_{\alpha}\left(x\right)=C_{\alpha}\frac{1}{\left|x\right|^{N+2s}}\left(\frac{x_{N}}{\left|x\right|^{2}}\right)^{\alpha-2s}=C_{\alpha}\frac{x_{N}^{\alpha-2s}}{\left|x\right|^{N-2s+2\alpha}},$
where $C_{\alpha}<0$.
For $0<\varepsilon\leq\left(-C_{\alpha}\right)^{\frac{1}{p-1}}$, we define the
function
$\displaystyle
u\left(x\right)=\varepsilon\overline{w}_{\alpha}\left(x\right).$
For any $x\in\mathbb{R}_{+}^{N}$, then we have
$\displaystyle-\left(-\Delta\right)^{s}u\left(x\right)+u^{p}\left(x\right)$
$\displaystyle=\varepsilon
C_{\alpha}\frac{x_{N}^{\alpha-2s}}{\left|x\right|^{N-2s+2\alpha}}+\varepsilon^{p}\frac{x_{N}^{\alpha
p}}{\left|x\right|^{\left(N-2s+2\alpha\right)p}}$ (4.2)
$\displaystyle=\varepsilon\frac{x_{N}^{\alpha-2s}}{\left|x\right|^{N-2s+2\alpha}}\left(C_{\alpha}+\varepsilon^{p-1}\frac{x_{N}^{\alpha
p-\alpha+2s}}{\left|x\right|^{\left(N-2s+2\alpha\right)\left(p-1\right)}}\right).$
When $\frac{N+s}{N-s}<p<\frac{N}{N-2s}$, we can choose
$\alpha\in\left(0,s\right)$ in such a way
$\displaystyle\alpha p-\alpha+2s=\left(N-2s+2\alpha\right)\left(p-1\right),$
i.e. $\alpha=\frac{N-\left(N-2s\right)p}{p-1}$. Then, from (4), for any
$x\in\mathbb{R}_{+}^{N}$ we have
(4.3)
$\displaystyle-\left(-\Delta\right)^{s}u\left(x\right)+u^{p}\left(x\right)\leq\varepsilon\frac{x_{N}^{\alpha-2s}}{\left|x\right|^{N-2s+2\alpha}}\left(C_{\alpha}+\varepsilon^{p-1}\right)\leq
0.$
Thus $u$ is a classical solution of (4.1) when
$\frac{N+s}{N-s}<p<\frac{N}{N-2s}$.
If $p\geq\frac{N}{N-2s}$, then for any $\alpha\in\left(0,s\right)$ one has
$\displaystyle\alpha p-\alpha+2s<\left(N-2s+2\alpha\right)\left(p-1\right).$
In view (4), $u$ satisfies (4.3) for any $x_{N}\geq 1$. As a consequence, the
function
$\displaystyle\widetilde{u}\left(x\right)=\left\\{\begin{aligned}
&u\left(x+e_{N}\right),&&x\in\mathbb{R}_{+}^{N},\\\
&0,&&x\in\overline{\mathbb{R}_{-}^{N}}\end{aligned}\right.$
is in turn a solution of (4.1). ∎
###### Remark 4.2.
The existence result of Theorem 4.1 remains valid even if $N=1$. We briefly
discuss this aspect by distinguishing the cases $0<s<\frac{1}{2}$ and
$\frac{1}{2}\leq s<1$.
If $0<s<\frac{1}{2}$, using the fact that the function $|x|^{-(1-2s)}$ is
$s$-harmonic in $\mathbb{R}\backslash\left\\{0\right\\}$, one can argue as in
the proof of Theorem 4.1 with $N$ replaced by $1$.
If instead $\frac{1}{2}\leq s<1$, the function $|x|^{2s-1}$ is still
$s$-harmonic in $\mathbb{R}\backslash\left\\{0\right\\}$ (if $s=\frac{1}{2}$
we simply mean the constant $1$) but unbounded at infinity if $s>\frac{1}{2}$.
The equivalent formulation of (4) for the function
$u(x)=\varepsilon\frac{(x)_{+}^{\alpha}}{|x|^{2\alpha-2s+1}}$
is
$\displaystyle-\left(-\Delta\right)^{s}u\left(x\right)+u^{p}\left(x\right)$
$\displaystyle=\frac{\varepsilon
C_{\alpha}}{x^{1+\alpha}}+\frac{\varepsilon^{p}}{x^{\left(1+\alpha-2s\right)p}}$
$\displaystyle=\frac{\varepsilon}{x^{1+\alpha}}\left(C_{\alpha}+\frac{\varepsilon^{p-1}}{x^{\left(1+\alpha-2s\right)p-1-\alpha}}\right)\qquad\forall
x\in\mathbb{R}_{+},$
where $C_{\alpha}<0$. In this case we can choose
$\alpha=\frac{1+(2s-1)p}{p-1}$ is such a way
$(1+\alpha-2s)p=\alpha+1$
and, differently from the previous cases $N=1$ and $0<s<\frac{1}{2}$ or $N\geq
2$, the exponent $\alpha$ is now always positive since, for any
$p>\frac{1+s}{1-s}$, it turns out that
$\frac{1+\left(2s-1\right)p}{p-1}\in\left(2s-1,s\right)$. Hence
$-\left(-\Delta\right)^{s}u\left(x\right)+u^{p}\left(x\right)\leq
0\quad\forall x\in\mathbb{R}_{+}.$
## 5\. Nonexistence in the whole space $\mathbb{R}^{N}$
We prove in this section that below the nonlocal Serrin exponent, the only
entire non negative supersolution in $\mathbb{R}^{N}$ is the trivial one. The
proof is similar to the case of the half space but simpler and it is given for
completeness and clarity sake.
###### Theorem 5.1.
Let $N\geq 2$ and $1\leq p\leq\frac{N}{N-2s}$. If $u\in
C^{2}\left(\mathbb{R}^{N}\right)\cap\mathcal{L}_{s}$ is a nonnegative solution
of
(5.1) $\displaystyle-Lu\geq u^{p}\quad\text{in $\mathbb{R}^{N}$},$
then $u\equiv 0$.
###### Proof.
We choose $\varphi\in C^{\infty}_{0}\left(\mathbb{R}^{N}\right)$ such that
$0<\varphi\leq 1$ in $B_{2}$ and
$\displaystyle\varphi\left(x\right)=\left\\{\begin{aligned} &1,&&x\in
B_{1},\\\ &0,&&x\notin B_{2}.\end{aligned}\right.$
We claim that there exists $M>0$ such that
(5.2) $\displaystyle-L\varphi\left(x\right)\leq
M\varphi\left(x\right)\quad\forall x\in\mathbb{R}^{N}.$
By the assumption (1.5), for any $\left|x\right|\geq 2$
$\displaystyle-L\varphi\left(x\right)=-\int_{\mathbb{R}^{N}}\frac{\varphi\left(x+y\right)+\varphi\left(x-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy\leq
0=M\varphi\left(x\right).$
Hence (5.2) is equivalent to
(5.3) $\displaystyle\inf_{x\in
B_{2}}\frac{L\varphi\left(x\right)}{\varphi\left(x\right)}>-\infty.$
Assume (5.3) does not hold, then there exists a convergent sequence
$\left\\{x_{n}\right\\}_{n}\subset B_{2}$ such that
(5.4)
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\varphi\left(x_{n}\right)}{\varphi\left(x_{n}\right)}=-\infty.$
Let $x_{n}\rightarrow x_{\infty}\in\overline{B_{2}}$ as $n\rightarrow+\infty$.
We distinguish two cases.
Case 1: $\left|x_{\infty}\right|<2$. In this case
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\varphi\left(x_{n}\right)}{\varphi\left(x_{n}\right)}=\frac{L\varphi\left(x_{\infty}\right)}{\varphi\left(x_{\infty}\right)}$
which is a finite quantity since $\varphi\in
C^{\infty}_{0}\left(\mathbb{R}^{N}\right)$ and
$\varphi\left(x_{\infty}\right)>0$.
Case 2: $\left|x_{\infty}\right|=2$. By the assumptions (1.5)-(1.6) and using
Lemma 3.1, we have
$\displaystyle\lim_{n\rightarrow+\infty}L\varphi\left(x_{n}\right)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\varphi\left(x_{\infty}+y\right)+\varphi\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}a\left(\frac{y}{\left|y\right|}\right)dy$
$\displaystyle\geq
d\int_{\Sigma_{\nu_{0},\tau_{0}}(0)}\frac{\varphi\left(x_{\infty}+y\right)+\varphi\left(x_{\infty}-y\right)}{\left|y\right|^{N+2s}}dy>0.$
Thus
$\displaystyle\lim_{n\rightarrow+\infty}\frac{L\varphi\left(x_{n}\right)}{\varphi\left(x_{n}\right)}=+\infty.$
Therefore, the assumption (5.4) can not occur, so that (5.2) holds.
For any $R>0$, we consider the rescaled test-function
$\displaystyle\varphi_{R}\left(x\right)=\varphi\left(\frac{x}{R}\right).$
Multiplying (5.1) by $\varphi_{R}$, integrating by parts, then using Lemma
2.5-(1) and (5.2) we have
(5.5)
$\displaystyle\int_{\mathbb{R}^{N}}u^{p}\varphi_{R}dx\leq-\int_{\mathbb{R}^{N}}uL\varphi_{R}dx\leq
MR^{-2s}\int_{\mathbb{R}^{N}}u\varphi_{R}dx.$
By the Hölder inequality,
$\displaystyle\int_{\mathbb{R}^{N}}u\varphi_{R}dx$
$\displaystyle\leq\left(\int_{B_{2R}}u^{p}\varphi_{R}dx\right)^{\frac{1}{p}}\left(\int_{B_{2R}}\varphi_{R}dx\right)^{\frac{p-1}{p}}$
$\displaystyle\leq
C\left(N,p\right)R^{\frac{N\left(p-1\right)}{p}}\left(\int_{\mathbb{R}^{N}}u^{p}\varphi_{R}dx\right)^{\frac{1}{p}}.$
Thus, by (5.5), we obtain that for any $p>1$
(5.6) $\displaystyle\int_{\mathbb{R}^{N}}u^{p}\varphi_{R}dx\leq
C\left(N,M,p\right)R^{N-\frac{2sp}{p-1}}.$
In the case $p=1$, we let $R\rightarrow+\infty$ in (5.5) to get $u\equiv 0$.
When $1<p<\frac{N}{N-2s}$, letting $R\rightarrow+\infty$ in (5.6), we infer
that $u\equiv 0$. If $p=\frac{N}{N-2s}$, then (5.6) yields, in the limit as
$R\rightarrow+\infty$, that
(5.7) $\displaystyle\int_{\mathbb{R}^{N}}u^{p}dx<+\infty.$
Moreover, in this case, we can rewrite
$\displaystyle\int_{\mathbb{R}^{N}}u\varphi_{R}dx=\int_{\left|x\right|\leq\sqrt{R}}u\varphi_{R}dx+\int_{\sqrt{R}\leq\left|x\right|\leq
2R}u\varphi_{R}dx.$
By the Hölder inequality
$\displaystyle\int_{\left|x\right|\leq\sqrt{R}}u\varphi_{R}dx$
$\displaystyle\leq
C\left(N,p\right)R^{s}\left(\int_{\mathbb{R}^{N}}u^{p}dx\right)^{\frac{1}{p}}$
and
$\displaystyle\int_{\sqrt{R}\leq\left|x\right|\leq 2R}u\varphi_{R}dx$
$\displaystyle\leq
C\left(N,p\right)R^{2s}\left(\int_{\sqrt{R}\leq\left|x\right|\leq
2R}u^{p}dx\right)^{\frac{1}{p}}.$
Therefore, by (5.5), it follows that
$\displaystyle\int_{\mathbb{R}^{N}}u^{p}\varphi_{R}dx\leq
C\left(N,M,p\right)\left[R^{-s}\left(\int_{\mathbb{R}^{N}}u^{p}dx\right)^{\frac{1}{p}}+\left(\int_{\sqrt{R}\leq\left|x\right|\leq
2R}u^{p}dx\right)^{\frac{1}{p}}\right].$
In view of (5.7), we again conclude that $u\equiv 0$ by letting
$R\rightarrow+\infty$. ∎
###### Remark 5.2.
If $N=1$, Theorem 5.1 reads as follows: if $s\geq\frac{1}{2}$, then $u\equiv
0$ is the only nonnegative solution of (5.1) for any $p\geq 1$; if instead
$s<\frac{1}{2}$, then the same conclusion holds provided $1\leq
p\leq\frac{1}{1-2s}$.
## Acknowledgements
I.B and G.G. are partially supported by INdAM-GNAMPA. The authors wish to
thank Sapienza University of Rome for the financial support from the research
project Ateneo 2022 “At the edge of reaction-diffusion equations: from
population dynamics to geometric analysis, a qualitative approach”,
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|
# Entanglement entropy in conformal quantum mechanics
Michele Arzano<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”,
Università di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy Alessandra D’Alise
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università di
Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy Domenico Frattulillo
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università
di Napoli Federico II, I-80125 Napoli, Italy
INFN, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy
###### Abstract
We consider sets of states in conformal quantum mechanics associated to
generators of time evolution whose orbits cover different regions of the time
domain. States labelled by a continuous global time variable define the two-
point correlation functions of the theory seen as a one-dimensional conformal
field theory. Such states exhibit the structure of a thermofield double built
on bipartite eigenstates of generators of non-global time evolution. In terms
of the correspondence between radial conformal symmetries in Minkowski
spacetime and time evolution in conformal quantum mechanics proposed in Arzano
(2020, 2021) such generators coincide with conformal Killing vectors tangent
to worldlines of Milne and diamond observers at constant radius. The
temperature of the thermofield double states in conformal quantum mechanics
reproduces the temperatures perceived by such diamond and Milne observers. We
calculate the entanglement entropy associated to the thermofield double states
and obtain a UV divergent logarithmic behaviour analogous to known results in
two-dimensional conformal field theory in which the entangling boundary is
point-like.
## I Introduction
Entanglement entropy plays a central role in our understanding of the
interplay between the quantum realm and the geometry of spacetime. In quantum
field theory, it can characterize the quantum correlations between degrees of
freedom inside and outside a given region of spacetime. Due to the presence of
short scale correlations such geometric entropy is generally UV divergent but
has the notable feature that the leading divergent term is proportional to the
area of the entangling surface. Such property suggests a link between
entanglement entropy and thermodynamic properties of black holes. Indeed,
according to the celebrated entropy-area relation Bekenstein (1973), black
holes are characterized by an entropy proportional to their horizon surface
area. While there is to date no general consensus on the origin of the degrees
of freedom responsible for such entropy Jacobson (1999); Carlip (2007), its
area scaling suggests that black hole entropy is intimately related to quantum
correlations across the horizon Solodukhin (2011).
Entanglement entropy is notoriously hard to compute in quantum field theory
(see Rangamani and Takayanagi (2017) for a review of the various techniques).
One notable exception where analytical results exist is two-dimensional
conformal field theory (CFT2) Holzhey et al. (1994); Calabrese and Cardy
(2004). In this letter we show that $0+1$-dimensional conformal field theory,
i.e. conformal quantum mechanics, provides yet another model in which
entanglement entropy associated to correlations of the two-point function can
be calculated analytically in a rather straightforward way.
Our analysis is motivated by the correspondence between radial conformal
Killing vectors in Minkowski space-time and the generators of conformal
transformations of the real line which describe alternative time evolutions in
conformal quantum mechanics. In Arzano (2020, 2021) it was shown that such
alternative time evolutions coincide with the tangent vectors to worldlines of
observers sitting at the origin within a causal diamond and in Milne space-
time. The definition of positive and negative frequency modes for a
conformally invariant field for such observers are different than the one of
inertial observers and lead to the construction of different Fock spaces with
different vacuum states Olson and Ralph (2011); Su and Ralph (2016); Higuchi
et al. (2017); Wald (2019). The inertial vacuum density matrix is an entangled
state between the modes associated to such observers and the ones defined in
the space-time region from which they cannot exchange signals with. Tracing
over the inaccessible degrees of freedom the vacuum density matrix becomes a
thermal state at a characteristic diamond or Milne temperature. In conformal
quantum mechanics one can identify states which have a similar structure
associated to time domains covered by orbits of different generators of time
evolution and find analogous temperatures Arzano (2020, 2021). In particular
states labelled by a continuous time variable are used to build a two-point
function which corresponds the restriction to wordlines of observers sitting
at the origin of the two-point function of a massless scalar field in
Minkwoski space-time defined on the inertial vacuum state. Such states exhibit
the structure of a thermofield double in terms of excitations of the
Hamiltonian which generates time translations in time domains with boundaries.
Here we show how it is possible to quantify the entanglement of such state in
terms of the Von Neumann entropy of the associated reduced density matrix
obtained by tracing over one set of degrees of freedom of the thermofield
double. The result diverges logarithmically when the UV regulator is sent to
zero as in two-dimensional conformal field theory, a result expected in models
where the entangling boundary is point-like.
In the next section we start by recalling the correspondence between radial
conformal Killing vectors in 3+1-dimensional Minkowski space time and
conformal transformations of the real line with particular focus on generators
whose orbits exhibit boundaries. In Section 3 we examine the role of such
generators in conformal quantum mechanics, we introduce the sets of states
naturally associated to them and spell out their significance in the
definition of the two-point function of the model seen as a one-dimensional
conformal field theory. In Section 4 we show how states labelled by a
continuous time variable, whose inner product gives the two-point function of
the theory, exhibit the structure of a thermofield double state in terms of
the excitations of Hamiltonians whose orbits exhibit boundaries on the time
domain. We point out how, after tracing over one set of degrees of freedom of
the thermofield double, one obtains thermal density matrices at the Milne and
diamond temperature. In Section 5 we proceed to evaluate the entanglement
entropy of the reduced vacuum density matrix. This requires an appropriate
regularization of the vacuum state in order to control its non-normalizability
which is directly related to the divergence of the two-point function at
coincident points. We conclude in Section 6 with a summary and comments on the
results obtained.
## II Radial conformal Killing vectors in Minkowski spacetime and conformal
transformations of the real line
Let us consider the Minkowski metric in spherical coordinates
$ds^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,.$ (1)
A vector field $\xi$ is a conformal Killing vector if
$\mathcal{L}_{\xi}g_{\mu\nu}\propto g_{\mu\nu}$ (2)
where $g_{\mu\nu}$ is a generic metric and $\mathcal{L}_{\xi}$ denotes the Lie
derivative. For Minkowski spacetime all radial conformal Killing vectors were
classified in Herrero and Morales (1999) and they have the general form
$\xi=\left(a(t^{2}+r^{2})+bt+c\right)\,\partial_{t}+r(2at+b)\,\partial_{r}$
(3)
with $a,b,c$ real constants. Central to the present work is the fact that such
vectors can be written as
$\xi=aK_{0}+bD_{0}+cP_{0}\,,$ (4)
with $P_{0}$, $D_{0}$ and $K_{0}$ generating, respectively, time translations,
dilations and special conformal transformations
$P_{0}=\partial_{t}\,,\qquad D_{0}=r\,\partial_{r}+t\,\partial_{t}\,,\qquad
K_{0}=2tr\,\partial_{r}+(t^{2}+r^{2})\,\partial_{t}\,,$ (5)
whose commutators close the $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra
$[P_{0},D_{0}]=P_{0}\,,\qquad[K_{0},D_{0}]=-K_{0}\,,\qquad[P_{0},K_{0}]=2D_{0}\,.$
(6)
The causal character of different conformal Killing vectors changes according
to the values of the real constants $a,b$ and $c$ Herrero and Morales (1999).
For example, when $a=0$ and $b=0$ the conformal Killing vectors are everywhere
time-like. This is the case of the vector $P_{0}$ which generates evolution in
inertial time and whose integral lines are worldlines of static inertial
observers i.e. straight infinite lines with $r=\mathrm{const}$. In general,
however, such conformal Killing vectors are not everywhere time-like. For
example generators for which $a=0$ and $b\neq 0$ are null on the light-cone
emanating from the point $(t=-c/b,r=0)$, time-like inside such light-cone and
space-like outside. The generator of dilations $D_{0}$ is one of such vector
whose integral lines are straight lines emanating from the origin in the
$t$-$r$ plane and within the future oriented light-cone describe worldlines of
comoving observers in a expanding Milne universe Ling (2020) (contracting in
the past cone). The Milne universe is a flat FRLW space-time with scale factor
linear in time
$ds^{2}=-d\bar{t}^{\,2}+\bar{t}^{\,2}\left(d\chi^{2}+\sinh\chi^{2}d\Omega^{2}\right)\,.$
(7)
Such metric is simply the Minkowski space-time metric restricted to the future
cone and rewritten in hyperbolic slicing coordinates $t=\bar{t}\,\cosh\chi$
and $r=\bar{t}\,\sinh\chi$. Worldlines of comoving Milne observers are
straight lines with $\chi=\mathrm{const.}$. These observers move with radial
Minkowskian velocity $r/t=\tanh\chi$ which approaches the value of 1 as
$\chi\rightarrow\infty$ i.e. as the worldline of the comoving observer
approaches the light-cone. Notice how the time evolution of Milne comoving
observers is not eternal, albeit being still infinite, since integral lines of
$D_{0}$ have a beginning (and an end for the past cone) at the origin
$(t=0,r=0)$. Finally notice that the conformal Killing $D_{0}$ vector becomes
null on the light-cone which thus represents a conformal Killing horizon Dyer
and Honig (1979).
Another conformal Killing vector which will be relevant for our discussion is
the following combination of translations and special conformal
transformations
$S_{0}=\frac{1}{2}\left(\alpha P_{0}-\frac{K_{0}}{\alpha}\right)\ .$ (8)
Such conformal Killing vector exhibits a more articulated causal structure
determined by the intersection of two light-cones emanating from the points
$(t=\pm\alpha,r=0)$: it is null on these light-cones, time-like inside or
outside both light-cones and space-like elsewhere. Thus conformal
transformations generated by the vector map the intersection of the two light-
cones into itself Jacobson (2016), such region is a causal diamond of radius
$\alpha$. The integral lines of $S_{0}$ within the diamond are worldlines of
accelerated observers Herrero and Morales (1999) with a finite life-time
(since they are restricted to the causal diamond) and take the form
$t^{2}-\left(r-\alpha\omega\right)^{2}=\alpha^{2}\left(1-\omega^{2}\right)$
(9)
parametrized by a dimensionless parameter $\omega\neq 0$. Notice how the
generators of boosts, which describe time evolution in Rindler space-time, are
not included in the above classification since they are not radial (written in
spherical coordinates they are not independent of angular variables).
The connection with conformal quantum mechanics stems from the observation
that along $r=\mathrm{const}$ worldlines (but also on light-cones
$u=t-r=\mathrm{const}$, $v=t+r=\mathrm{const}$) the conformal Killing vector
(3) takes the form
$\xi=\left(a\,t^{2}+b\,t+c\,\right)\partial_{t}\,.$ (10)
This is the general expression for a generator of conformal transformations of
the real (time) line. In particular $P_{0}=\partial_{t}$ generates
translations in “inertial time” $t$ covering the entire time line. This time
variable can be identified with the proper time of static inertial observers
in Minkowski space-time (observers with four-velocity parallel to the
(conformal) Killing vector $P_{0}$). The dilation Killing vector
$D_{0}=t\partial_{t}$ generates translation in “Milne time” $\tau$ defined by
$D_{0}=\alpha\partial_{\tau}\,.$ (11)
The only worldline at fixed radius in the Milne universe is the one at $r=0$
for which $\tau$ corresponds to the conformal time. One can easily show that
$t=\pm\alpha\,\exp(\frac{\tau}{\alpha})$ (12)
and thus the Milne time coordinate covers only half (the regions $t>0$ or
$t<0$) of the whole time domain. Finally the conformal Killing vector
$S_{0}=\frac{1}{2\alpha}\left(\alpha^{2}-t^{2}\right)\partial_{t}$ generates
translation in “diamond time” $\sigma$
$S_{0}=\alpha\partial_{\sigma}\,.$ (13)
In Minkowski space-time this vector corresponds to the restriction to the
worldline of a static diamond observer at $r=0$ of the conformal Killing
vector (8). On the time line this time variable is related to inertial time by
the transformation
$t=\alpha\,\tanh{\left(\frac{\sigma}{2\alpha}\right)}$ (14)
and thus it covers only the segment $|t|<\alpha$ of the time domain.
## III Time evolution in conformal quantum mechanics and the CFT1 two-point
function
The quantum mechanical counterpart of the generator (10) is
$G=i\xi=aK+bD+cH\,,$ (15)
where $K=iK_{0}$, $D=iD_{0}$ and $H=iP_{0}$, corresponds to the most general
Hamiltonian of conformal quantum mechanics, a quantum mechanical model first
studied in de Alfaro et al. (1976) characterized by an inverse square
potential Lagrangian invariant under the one-dimensional group of conformal
transformations of the time axis $\mathrm{SL}(2,\mathbb{R})$. Such model can
be understood as a one-dimensional conformal field theory Chamon et al.
(2011); Jackiw and Pi (2012), as we briefly recall below.
In de Alfaro et al. (1976) two sets of states were constructed: the
eigenstates $|n\rangle$ of the elliptic operator $R$ which has discrete
spectrum and are normalizable, and states labelled by inertial time
$|t\rangle$ on which the operator generating inertial time translations $H$
acts as a derivative
$H|t\rangle=-i\derivative{t}|t\rangle\,.$ (16)
Such states are labelled by the continuous parameter $t$ and are non-
normalizable as we will show below. The action of the remaining
$\mathrm{SL}(2,\mathbb{R})$ generators is given by
$\displaystyle D\ket{t}$
$\displaystyle=-i\left(t\derivative{t}+r_{0}\right)\ket{t}$ (17)
$\displaystyle K\ket{t}$
$\displaystyle=-i\left(t^{2}\derivative{t}+2r_{0}t\right)\ket{t}\,.$ (18)
Introducing ladder operators
$L_{0}=R=\frac{1}{2}\left(\frac{K}{\alpha}+\alpha H\right)\qquad L_{\pm}=S\pm
iD=\frac{1}{2}\left(\frac{K}{\alpha}-\alpha H\right)\pm iD\,,$ (19)
whose commutators are
$\commutator{L_{0}}{L_{\pm}}=\pm
L_{\pm},\qquad\commutator{L_{-}}{L_{+}}=2L_{0}\,,$ (20)
we define sates $|n\rangle$ labelled by positive integers $n=0,1,...$ on which
the action of the operators (19) is given by
$\displaystyle L_{0}\ket{n}$ $\displaystyle=(n+r_{0})\ket{n}$ (21)
$\displaystyle L_{\pm}\ket{n}$ $\displaystyle=\sqrt{((n+r_{0})(n+r_{0}\pm
1))-r_{0}(r_{0}-1))}\ket{n\pm 1}\,,$ (22)
with the orthonormality relation
$\langle n|n^{\prime}\rangle=\delta_{nn^{\prime}}\,.$ (23)
The constant $r_{0}$, the eigenvalue of the ground state $|n=0\rangle$, is a
positive real number Jackiw and Pi (2012) and it is related to the eigenvalue
of the Casimir operator
$\mathcal{C}=R^{2}-S^{2}-D^{2}=\frac{1}{2}\left(HK+KH\right)-D^{2}\,,$ (24)
given by
$\mathcal{C}\ket{n}=r_{0}(r_{0}-1)\ket{n}\,.$ (25)
For $r_{0}$ integer and half integer with $r_{0}\geq 1$ the set of states
$\ket{n}$ provides an irreducible representation of the Lie algebra
$\mathfrak{sl}(2,\mathbb{R})$ belonging to the so-called discrete series Sun
(2021).
In what follows we focus on the case $r_{0}=1$ as in Arzano (2021, 2020) since
this is the case in which the two-point function conformal quantum mechanics,
seen as a one dimensional conformal field theory, is equivalent to the
restriction of the two-point function of a massless scalar field along the
worldline of observers sitting at $r=0$ in Minkowski space-time.
Wavefunctions representing the states $|n\rangle$ as functions of the inertial
time coordinate $t$ can be obtained by considering the action of the $R$
operator on the $\langle t|$ states obtaining the differential equation
$\matrixelement{t}{R}{n}=\frac{i}{2}\left[\left(\alpha+\frac{t^{2}}{\alpha}\derivative{t}\right)+2\frac{t}{\alpha}\right]\innerproduct{t}{n}=(n+1)\innerproduct{t}{n}$
(26)
whose solution is given by
$\innerproduct{t}{n}=-\frac{\alpha^{2}c_{n}e^{2i(n+1)\tan^{-1}\left(\frac{\alpha}{t}\right)}}{\alpha^{2}+t^{2}}\
.$ (27)
One can determine the normalization constants by iteration (see Appendix A)
obtaining
$c_{n}=\sqrt{\frac{\Gamma(2+n)}{n!}}\ .$ (28)
Plugging this expression in (27) we can obtain the following equation relating
$|t\rangle$ and $|n\rangle$ states111Where we used the relation
$2i\tan^{-1}{x}=\log{\frac{1+ix}{1-ix}}$ in (27) to write (29).
$\ket{t}=\left(\frac{\frac{\alpha+it}{\alpha-it}+1}{2}\right)^{2}\
\sum_{n}(-1)^{n}\left(\frac{\alpha+it}{\alpha-it}\right)^{n}\
\sqrt{\frac{\Gamma(2+n)}{n!}}\ket{n}\ .$ (29)
These states can be written in terms of the action of the creation operator
$L_{+}$ on the ground state $\ket{n=0}$
$\ket{t}=\left(\frac{\frac{\alpha+it}{\alpha-it}+1}{2}\right)^{2}\
\exp{-\left(\frac{\alpha+it}{\alpha-it}\right)\,L_{+}}\ket{n=0}\,,$ (30)
and in particular
$\ket{t=0}=\exp\left(-L_{+}\right)\ket{n=0}\,.$ (31)
As discussed in Chamon et al. (2011); Jackiw and Pi (2012) the inner product
between the $|t\rangle$ states can be interpreted as the two-point function of
a one-dimensional CFT. One can explicitly evaluate such two-point function
using (29)
$G(t_{1},t_{2})\equiv\innerproduct{t_{1}}{t_{2}}=-\frac{\alpha^{2}}{4\
(t_{1}-t_{2})^{2}}\,.$ (32)
Notice how such expression matches, modulo a constant factor, that of the two-
point function of a free massless scalar field in Minkowski space-time along
the trajectory of a static inertial observer sitting at $r=0$ Arzano (2020).
Moreover, since the Hamiltonian $H$ generates the time evolution, the
$t$-state can actually be obtained with a time translation from the $t=0$
vacuum
$\ket{t}=e^{iHt}\ket{t=0}=\frac{1}{4}e^{(\alpha+it)H}\ket{n=0}$ (33)
and thus the two-point function (32) can be written as
$G(t_{1},t_{2})=\innerproduct{t_{1}}{t_{2}}=\bra{t=0}e^{-iHt_{1}}\,e^{iHt_{2}}\ket{t=0}\,.$
(34)
It is instructive to look at the two-point function (34) in terms of
eigenstates of the generator $H$. These eigenstates $\ket{E}$, first
introduced in de Alfaro et al. (1976), are defined by
$H\ket{E}=E\ket{E}$ (35)
and satisfy the conditions
$\innerproduct{E}{E^{\prime}}=\delta(E-E^{\prime})\mbox{\quad
and\quad}\int_{0}^{+\infty}\ \differential E\ket{E}\bra{E}=\mathbb{1}\ .$ (36)
Following de Alfaro et al. (1976) we can write $\ket{t}$ as
$\ket{t}=e^{iHt}\ket{t=0}=\int_{0}^{\infty}\differential E\
\frac{\alpha\sqrt{E}}{2}e^{iEt}\ket{E}$ (37)
and obtain the overlap between $\ket{E}$ and $\ket{t}$
$\innerproduct{t}{E}=\frac{\alpha\sqrt{E}}{2}e^{-iEt}\ .$ (38)
Therefore, the states $\ket{E}$ are similar in spirit to the momentum
eigenstates $\ket{\mathbf{p}}$ that one introduces in QFT, in terms of which
the action of a field operator $\phi(\mathbf{x})$ on the vacuum state is
$\phi(\mathbf{x})\ket{0}=\int\frac{\differential^{3}p}{(2\pi)^{3}}\frac{1}{2E_{p}}e^{-i\mathbf{p}\cdot\mathbf{x}}\ket{\mathbf{p}}\
,$ (39)
where
$\innerproduct{\mathbf{p}}{\mathbf{p}^{\prime}}=2E_{\mathbf{p}}(2\pi)^{3}\
\delta^{(3)}(\mathbf{p}-\mathbf{p}^{\prime})\ .$ (40)
The analogy between (37) and (39) clearly suggests that the state $\ket{t=0}$
plays a role analogous to the inertial vacuum for quantum fields in Minkowski
space-time which is the averaging state on which one builds the two-point
function.
## IV “Vacuum” states and horizon temperature
We have seen that conformal quantum mechanics can be interpreted as a $0+1$
dimensional field theory in which any generator of conformal transformations
of the time axis can be used to define time evolution. Such time evolution can
be mapped to motions along $r=\text{const.}$ orbits of time-like conformal
Killing vectors in Minkowski space-time. For massless fields in Minkowski
space-time one can construct a Fock space using any conformal Killing vector
in the domain where the latter is time-like. As in the more familiar case of
the Unruh effect Unruh (1976), where boost Killing vectors in the Rindler
wedge are used to define positive frequency field modes, Fock spaces
constructed using mode decompositions based on different conformal Killing
vectors will lead to different Fock spaces and thus to different notions of
particles and vacuum states. For the Milne cone and causal diamond in
Minkowski space-time these different quantizations were explored in Higuchi et
al. (2017); Wald (2019); Olson and Ralph (2011); Su and Ralph (2016). These
works suggest that, in analogy with the Unruh effect, for both Milne and
diamond observers the vacuum state for inertial observers appears as a thermal
state.
Let us go back to the correspondence between generators of time evolution in
conformal quantum mechanics and time-like conformal Killing vectors
determining the wordlines of Milne and diamond observers. In light of what we
recalled above, we do expect that in conformal quantum mechanics one should be
able to identify an inertial “vacuum” state which is thermally populated by
excitations of the Hamiltonian describing the conformal quantum mechanics
counterparts of the Milne and diamond time evolutions. This is indeed the case
as first shown in Arzano (2021). To see this, let us recall that the
$\mathfrak{sl}(2,\mathbb{R})$ Lie algebra (19) can be realized in terms of two
sets of creation and annihilation operators
$a^{\dagger}_{L},a^{\dagger}_{R},a_{L},a_{R}$
$L_{0}=\frac{1}{2}\left(a^{\dagger}_{L}a_{L}+a^{\dagger}_{R}a_{R}+1\right)\
,\quad L_{+}=a^{\dagger}_{L}a^{\dagger}_{R}\mbox{\quad
and\quad}L_{-}=a_{L}a_{R}\,.$ (41)
This shows that the ground state of the $R$-operator has a bipartite structure
$\ket{n=0}=\ket{0}_{L}\otimes\ket{0}_{R}\ ,$ (42)
and that the $\ket{t=0}$ state in (31) can be written as
$\ket{t=0}=e^{-a^{\dagger}_{L}a^{\dagger}_{R}}\ket{0}_{L}\ket{0}_{R}=\sum_{n}\
(-1)^{n}\ket{n}_{L}\ket{n}_{R}=-i\sum_{n}\ e^{i\pi
L_{0}}\ket{n}_{L}\ket{n}_{R}\,.$ (43)
From the last equality it is clear that the $\ket{t=0}$ state exhibits a
structure similar to that of a thermofield double state for a harmonic
oscillator (see e.g. Lykken (2021) for a pedagogical review). Such state can
be built by “doubling” the oscillator’s degrees of freedom and is defined by
the superposition
$|TFD\rangle=\frac{1}{Z(\beta)}\sum^{\infty}_{n=0}e^{-\beta
E_{n}/2}|n\rangle_{L}\otimes|n\rangle_{R}\,,$ (44)
where $Z(\beta)=\sum^{\infty}_{n=0}e^{-\beta E_{n}}$ is the partition function
at inverse temperature $\beta$. The state (44) is highly entangled and,
tracing over the degrees of freedom of one copy of the system, we obtain a
thermal density matrix
$Tr_{L}\\{|TFD\rangle\langle TFD|\\}=\frac{e^{-\beta H}}{Z(\beta)}$ (45)
at a temperature $T=1/\beta$. The Hamiltonian $H$ is known as modular
Hamiltonian. For a quantum field in Minkowski space-time the inertial vacuum
state can be seen as a thermofield double state built on two copies of the
Rindler Hilbert space Valdivia-Mera (2020). Tracing over the degrees of
freedom of one copy (i.e. looking at the state from the point of view of
Rindler observers whose worldlines are restricted to a space-like wedge) one
obtains a thermal state at the Unruh temperature. The modular Hamiltonian in
this case is the generator of boosts which can be identified, modulo a factor
with dimensions of inverse length related to the magnitude of the
acceleration, with the generator of time evolution for Rindler observers.
We see that, setting aside normalization issues which will be the focus of the
next section, in conformal quantum mechanics we are dealing with a similar
scenario. Indeed we see that tracing over one set of degrees of freedom the
reduced density matrix associated to the state (43) has the form of a thermal
density matrix for the modular Hamiltonian $-iL_{0}$ at a temperature
$T=1/2\pi$. Since $L_{0}$ can be identified with the elliptic generator $R$ of
the $SO(2)$ compact subgroup of $SL(2,\mathbb{R})$ its “Wick rotated”
counterpart $iL_{0}$ will generate non compact transformations. Indeed one
finds Arzano (2021) that the generators of the Lie algebra (20), besides the
identification (17), have two alternative realizations in terms of the
generators $H,D$ and $K$ given by
$L_{0}=iS\ ,\qquad L_{+}=i(D-R)\ ,\qquad L_{-}=-i(D+R)$ (46)
and
$L_{0}=iD\ ,\qquad L_{+}=-i\alpha H\ ,\qquad L_{-}=-i\frac{K}{\alpha}$ (47)
in which the the modular Hamiltonian $-iL_{0}$ coincides with the generators
$S$ and $D$. From our discussion in Section 2 we see that, when divided by the
constant $\alpha$ with dimensions of length, these two Hamiltonians generate,
respectively, translations on diamond and Milne times.
In the case of the diamond Hamiltonian we notice that the identification (46)
can be obtained by “Wick rotating” the length parameter $\alpha\rightarrow
i\alpha$. Under this map the generator $R$ turns into $iS$ and the
wavefunctions (27) into eigenfunctions of the operator $L_{0}=iS$. Following
steps analogous to the ones leading to (43) we find that the state
$|t=0\rangle$ has the structure of a thermofield double state for the modular
Hamiltonian $S/\alpha$ at a temperature $T=1/(2\pi\alpha)$.
For the Milne Hamiltonian the picture is less straightforward since the
generator $D$ is ill defined at $t=0$. One can solve the eigenvalue equation
$(n+1)\,{}_{D}\langle
t|n\rangle=\matrixelement{t}{iD}{n}=-\left[t\derivative{t}+1\right]\,{}_{D}\langle
t|n\rangle$ (48)
to obtain the eigenstates (see Appendix B)
${}_{D}\langle t|n\rangle=\frac{(-1)^{n}}{2}\
\alpha^{n+2}\sqrt{\frac{\Gamma(2+n)}{n!}}\,t^{-n-2}\ .$ (49)
Since the conformal transformation
$t^{\prime}=\frac{\alpha(t-\alpha)}{\alpha+t}$ (50)
maps the generator $S$ written as a differential operator in terms of the time
variable $t^{\prime}$ into the generator $D$ as a differential operator in the
variable $t$, one can obtain the eigenfunctions (49) starting from the
eigenfunctions of the $R$ operator (27), performing the Wick rotation
$\alpha\rightarrow i\alpha$ and then the conformal transformation (50) on the
time variable. The states $|t\rangle_{D}$ can be now written in terms of
eigenstates of the $L_{0}=iD$ operator as
$\begin{split}\ket{t}_{D}&=\frac{1}{2}\left(\frac{\alpha}{t}\right)^{2}\sum_{n}(-1)^{n}\left(\frac{\alpha}{t}\right)^{n}\sqrt{\frac{\Gamma(2+n)}{n!}}\ket{n}=\frac{1}{2}\left(\frac{\alpha}{t}\right)^{2}e^{-\frac{\alpha}{t}L_{+}}\ket{n=0}\
.\end{split}$ (51)
We see that now the state $t=\alpha$ exhibits the structure of a thermofield
double state for the modular Hamiltonian $D/\alpha$ at the temperature
$T=1/(2\pi\alpha)$. The point $t=\alpha$ is the image of the origin under the
conformal mapping (50) and it corresponds to the origin of the conformal time
$\tau$ variable defined by $t=\alpha\ e^{\frac{\tau}{\alpha}}$.
The state $\ket{t=0}$, as evidenced by eq. (43), is an entangled state with
respect to the bi-partition of the Hilbert space in terms of $L$ and $R$
degrees of freedom. In analogy with the case of the inertial vacuum written as
a thermofield double state over the excitations of the left and right Rindler
modes (the two complementary domains of the evolution of the boost modular
Hamiltonian), we can think of the two sets of degrees of freedom in (43) as
belonging to the domain of diamond and Milne time evolution and their
complements (the restriction to $r=0$ wordlines of the entanglement considered
in Higuchi et al. (2017) and Olson and Ralph (2011)).
We can quantify this entanglement by calculating the Von Neumann entropy of
the reduced density matrix obtained by tracing over one set of degrees of
freedom in the density matrix associated to the inertial vacuum $\ket{t=0}$.
Such entanglement entropy can be seen as the $0+1$-dimensional analogue of the
entanglement entropy a quantum field across space-time regions. Unlike its
higher dimensional counterparts, the simple structure of the state $\ket{t=0}$
makes it rather straightforward to calculate the entanglement entropy
associated to the diamond and Milne time domains.
## V Entanglement entropy
In order to derive the entanglement entropy associated to the partition of the
$\ket{t=0}$ state we first notice that such state is non-normalizable. This is
to be expected given the correspondence between the inner product
$\innerproduct{t_{1}}{t_{2}}$ and the restriction of the two-point function of
a massless field to the $r=0$ worldline since it reflects the UV divergence of
the latter for coincident points.
We can regularize the state $\ket{t=0}$ via an infinitesimal translation in
imaginary time. We consider a state at time $t=i\epsilon$
$\ket{t=i\epsilon}=\left(\frac{\frac{\alpha-\epsilon}{\alpha+\epsilon}+1}{2}\right)^{2}\
e^{-\frac{\alpha-\epsilon}{\alpha+\epsilon}\ L_{+}}\ket{n=0}$ (52)
where $\epsilon$ can be interpreted as a short-distance cut-off scale. We have
$\ket{t=i\epsilon}=\left(\frac{\alpha}{\alpha+\epsilon}\right)^{2}\sum_{n=0}^{\infty}(-1)^{n}\left(\frac{\alpha-\epsilon}{\alpha+\epsilon}\right)^{n}\ket{n}_{L}\ket{n}_{R}$
(53)
so that
$\bra{t=i\epsilon}\ket{t=i\epsilon}=\frac{\alpha}{4\epsilon}\left(\frac{\alpha}{\alpha+\epsilon}\right)^{2}\equiv\frac{1}{\mathcal{N}^{2}}\
.$ (54)
In order to derive the reduced density matrix we normalize the
$\ket{t=i\epsilon}$ and introduce a state $\ket{\delta}$ with unitary norm
$\ket{\delta}\equiv\mathcal{N}\ket{i\epsilon}\ .$ (55)
Let us now consider the density matrix
$\rho_{RL}=\ket{\delta}\bra{\delta}$ (56)
and compute the reduce density matrix $\rho_{L}$ explicitly by tracing over
the R degrees of freedom
$\rho_{L}=\Tr_{R}{\rho_{LR}}=\mathcal{N}^{2}\left(\frac{\alpha}{\alpha+\epsilon}\right)^{4}\sum_{n=0}^{\infty}\left(\frac{\alpha-\epsilon}{\alpha+\epsilon}\right)^{2n}\ket{n}_{L}\bra{n}_{L}\
.$ (57)
The Von Neumann entropy of the reduced density matrix is then given by
$S=-\Tr{\rho_{L}\log\rho_{L}}=-\frac{(\alpha-\epsilon)^{2}\log\left(\frac{(\alpha-\epsilon)^{2}}{(\alpha+\epsilon)^{2}}\right)}{4\alpha\epsilon}-\log\left(\frac{4\alpha\epsilon}{(\alpha+\epsilon)^{2}}\right)\,,$
(58)
which considering the limit $\epsilon\rightarrow 0$ leads to
$S=\log\left(\frac{\alpha}{\epsilon}\right)+\text{const}+\order{\epsilon^{2}}\,.$
(59)
We see that the result obtained exhibits a logarithmic divergence when the UV
cut-off scale $\epsilon$ is sent to zero. Let us recall that in a
d-dimensional free quantum field theory the entanglement entropy associated to
a spatial region $\mathcal{A}$ is UV divergent with a leading divergent term
proportional to
$\frac{\text{Area}(\partial\mathcal{A})}{\epsilon^{d-2}}\ ,$ (60)
where $\text{Area}(\partial\mathcal{A})$ is the area of the boundary of the
region (entangling surface) and $\epsilon$ is a UV cut-off Rangamani and
Takayanagi (2017). Two-dimensional conformal theories $\mathrm{CFT_{2}}$ are a
special case since they predict a logarithmic divergence and therefore the
entanglement entropy fails to follow an area law. For example the entanglement
entropy between a shorter line-segment with length $\alpha$ and a longer one
with length $L$ containing it in the limit $\frac{\alpha}{L}\ll 1$ reads
Calabrese and Cardy (2004); Saravani et al. (2014); Rangamani and Takayanagi
(2017)
$S=\frac{c}{3}\log{\frac{\alpha}{\epsilon}}+\text{const}\ ,$ (61)
where $c$ is the central charge of the CFT which is equal to $1$ in a quantum
field theory of a massless bosonic field. This peculiar behaviour can be
understood heuristically by arguing that the logarithm arises as a limiting
case of a power law divergence and it is consistent with the entangling
surface comprising of a set of disconnected points.
We observe that the analytical behaviour of our result (59) for the
entanglement entropy in $\mathrm{CFT_{1}}$ is the same as in
$\mathrm{CFT_{2}}$ and in particular it shows the same logarithmic divergence.
This is in line with the point-like nature of the entangling surface.
## VI Conclusions
Conformal quantum mechanics is a simple one-dimensional model which, as we
have seen, possesses enough structure to mimic the non-trivial vacuum
structure of higher dimensional free quantum field theories. Such non-trivial
structure is due to the existence of excitations in the theory which are
associated to Hamiltonians whose orbits posses a boundary, the one dimensional
counterparts of horizons in higher dimensions. As in quantum field theories in
higher dimensions we have seen that one can associate temperatures to these
horizons. Moreover, we evaluated the entanglement entropy associated to the
bipartite decomposition of the states on which one builds the two-point
function of the theory into modes of Hamiltonians whose orbits do not cover
the entire time domain. The possibility of entanglement between time domains
in Minkowski space-time has been considered before Olson and Ralph (2011);
Higuchi et al. (2017). Here we provide an explicit calculation of entanglement
entropy in conformal quantum mechanics for partitions of states in terms of
modular Hamiltonians which define evolution in time domains with boundaries.
These domains can be embedded in Minkowski space-time as wordlines of diamond
and Milne observers at the origin and, thus, one could interpret our result as
quantifying a worldline entanglement entropy (see Anninos et al. (2012);
Nakayama (2012) for a similar interpretation of conformal quantum mechanics as
worldline quantum mechanics for static patch observers in de Sitter space-
time) for observers who cannot have access to the past beyond a certain point
(the initial “singularity” for Milne observers) or to the future and the past
beyond two points (the future and past tips of the diamond for diamond
observers). The worldline boundaries are point-like and the entanglement
entropy exhibits a logarithmic divergence similar to that found in two-
dimensional CFTs, where the boundaries of the spatial region considered are
also point-like. It is tempting to speculate that such worldline entropy could
play a role as a tool for facilitating the calculation of entanglement entropy
for conformally invariant quantum field theories in higher space-time
dimensions. We leave this task for future investigations.
## Acknowledgements
We acknowledge support from the INFN Iniziativa Specifica QUAGRAP. This
research was carried out in the frame of Programme STAR Plus, financially
supported by the University of Napoli Federico II and Compagnia di San Paolo.
This work also falls within the scopes of the European Union COST Action
CA18108 Quantum gravity phenomenology in the multi-messenger approach.
## Appendix A Determining the coefficients $c_{n}$
In order to find the coefficients $c_{n}$ we act with the ladder operators. We
start with the action of $L_{+}$
$\displaystyle\sqrt{(n+1)(n+2)}\innerproduct{t}{n+1}$
$\displaystyle=\matrixelement{t}{L_{+}}{n}=\left[\left(i\frac{t}{\alpha}-1\right)+\left(i\frac{t^{2}}{2\alpha}-i\frac{\alpha}{2}-t\right)\derivative{t}\right]\innerproduct{t}{n}$
(A.62)
which gives
$\frac{c_{n}}{c_{n+1}}=\frac{(n+1)}{\sqrt{(n+1)(n+2)}}$ (A.63)
while acting with $L_{-}$
$\displaystyle\sqrt{n(n+1)}\innerproduct{t}{n-1}$
$\displaystyle=\matrixelement{t}{L_{-}}{n}=\left[\left(i\frac{t}{\alpha}+1\right)+\left(i\frac{t^{2}}{2\alpha}-i\frac{\alpha}{2}+t\right)\derivative{t}\right]\innerproduct{t}{n}$
(A.64)
we arrive at
$\frac{c_{n-1}}{c_{n}}=\frac{n}{\sqrt{n(n+1)}}\ ,$ (A.65)
hence we conclude that
$c_{n}=\sqrt{\frac{\Gamma(2+n)}{\Gamma(n+1)}}=\sqrt{\frac{\Gamma(2+n)}{n!}}\
.$ (A.66)
## Appendix B Eigenstates of the $iD$ generator
The differential equation
$(n+1)\innerproduct{t}{n}=\matrixelement{t}{iD}{n}=-\left[t\derivative{t}+1\right]\innerproduct{t}{n}$
(B.67)
admits solutions
$\innerproduct{t}{n}=c_{n}\ t^{-n-2}\ .$ (B.68)
To determine the coefficients $c_{n}$ we act with $L_{+}$ on these functions
$\displaystyle\sqrt{(n+1)(n+2)}\innerproduct{t}{n+1}$
$\displaystyle=\matrixelement{t}{L_{+}}{n}=\alpha\derivative{t}\innerproduct{t}{n}$
(B.69)
and obtain
$\sqrt{(n+1)(n+2)}\ c_{n+1}=-\alpha\ c_{n}(2+n)\ .$ (B.70)
The action with $L_{-}$ gives
$\displaystyle\sqrt{n(n+1)}\innerproduct{t}{n-1}$
$\displaystyle=\matrixelement{t}{L_{-}}{n}=\frac{1}{\alpha}\left[t^{2}\derivative{t}+2t\right]\innerproduct{t}{n}$
(B.71)
whose solution is
$-\alpha\sqrt{n(n+1)}c_{n-1}=n\ c_{n}\ .$ (B.72)
By combining the two results obtained we arrive at the following expression
for the coefficients $c_{n}$
$c_{n}=\frac{(-1)^{n}}{2}\ \alpha^{n+2}\sqrt{\frac{\Gamma(2+n)}{n!}}\ .$
(B.73)
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# Study of water Cherenkov detector design for ground-based gamma-ray
experiments
F. Bisconti11footnotetext: Corresponding author. and A. Chiavassa
###### Abstract
In the framework of the development of the SWGO experiment we have performed a
detailed study of the single unit of an extensive air shower observatory based
on an array of water Cherenkov detectors. Indeed, one of the possible water
Cherenkov detector unit configurations for SWGO consists of tanks, and to
reach a high detection efficiency and discrimination capability between gamma-
ray and hadronic air showers, different tank designs are under investigation.
In this study, we considered double-layer tanks with several sizes, shapes and
number of photo-multiplier tubes (PMTs). Muons, electrons, and gamma-rays with
energies typical of secondary particles in extensive air showers have been
simulated entering the tanks with zenith angles from 0 to 60 degrees. The tank
response was evaluated considering the number of photoelectrons produced by
the PMTs, the detection efficiency, and the time resolution of the measurement
of the first photon. This analysis allowed to compare the performance of tanks
with different size, configuration of PMTs, and with circular, hexagonal and
square geometry. The method used and the results will be discussed in this
paper.
## 1 Introduction
Wide field of view gamma-ray observatories can be realized by an array of
water Cherenkov detectors, covering areas ranging from $10^{4}$ to $10^{6}$
square meter, usually located in desertic areas. Secondary particles produced
in extensive air showers induced by astrophysical gamma-rays or hadrons (that
represent a background source), can be detected measuring the Cherenkov light
produced when they cross the detectors filled with clean water. A next-
generation gamma-ray observatory is the SWGO experiment [1, 2], which will be
realized at high altitude in the Southern Hemisphere, to be complementary to
other gamma-ray experiments in the Northern Hemisphere, like HAWC [3] and
LHAASO [4], for the observation of the entire sky. It will operate with close
to 100% duty cycle and order steradian field of view. The site has to be at
high altitude (above $4\,400$ m a.s.l.), in order to be closer to the maximum
of the extensive air showers induced by astrophysical gamma-rays with primary
energy in the range of interest (between 100 GeV and a few PeV). The SWGO
design will be primarily based on water Cherenkov detectors, and the final
array and detector unit configurations are still to be defined [1]. One
configuration under study consists of and array of (surface) water Cherenkov
tanks arranged in a high fill-factor core (with area considerably larger than
HAWC) and a low density outer array.
To study the single tank behaviour, we performed simulations of particles
crossing tanks with different size and configuration of PMTs. We simulated
double-layer tanks [5], in which the lower layer helps in the gamma/hadron
discrimination, as muons are more abundant in hadronic showers and they can
cross the upper layer reaching the lower layer where they are measured. We
considered tanks of different shape, with circular (Circular-DLT), hexagonal
(Hexagonal-DLT) and square (Square-DLT) base.
To simulate the particles crossing the tanks and their response we used the
HAWCSim framework [6], which makes use of GEANT4 [7] to simulate the
interaction of the particle with the tank itself and the water volume,
including the production of the Cherenkov photons that can be detected by the
PMTs inside the tank.
## 2 Simulations
### 2.1 Particles
In this analysis we considered the most abundant particles contained in an
extensive air shower generated by 400 GeV protons and 200 GeV photons at an
observation level of $4\,500-5\,000$ m a.s.l.. Therefore, we performed
simulations of electrons, gamma-rays and muons with fixed energies: 10 MeV,
100 MeV and 1 GeV electrons and photons, and 1 GeV and 10 GeV muons. To define
the directions, we used azimuth angles $\phi$ uniformly distributed in the
range $0-360\deg$ and zenith angles $\theta$ in the range $0-60\deg$ sampled
on a $\cos^{2}\theta$ distribution.
The particles were generated on a large circular surface 10 cm above the tank
and centered with it. The size of the generation area is such that even the
most inclined particles could enter the tank from the lateral walls of the
upper layer, to avoid the detection of particles entering in the tank directly
from the lower layer that would affect the overall performance of a single
tank study. This has to be considered in the context of a sparse array of
tanks, while in a dense array the nearby tanks contribute to the detection
capability of the large scale experiment. Therefore, for each tank design,
particle type and energy, 10000 particles entering the upper layer of the
tanks have been analyzed.
### 2.2 Specifications of the tanks
#### 2.2.1 Shapes and dimensions of the tanks
In this analysis, Circular-DLTs, Hexagonal-DLTs and Square-DLTs were
considered. In Fig. 1 examples of the Geant4 visualization of the three tank
designs crossed by a muon are shown.
Figure 1: Geant4 visualization of a Circular-DLT, an Hexagonal-DLT and a
Square-DLT, crossed by a 1 GeV vertical muon. All tanks have widths of 3 m
(diameter for Circular-DLT, two times the side for Hexagonal-DLT, and side for
Square-DLT) and lower layers 1 m high. The upper layers were simulated with
non-reflective walls, while the lower layer with reflective walls. The green
line represents the simulated muon and the red lines a sample of Cherenkov
photons.
The height of the upper layer was chosen allowing the Cherenkov photons to
reach any PMT at the base of the upper layer. Assuming a vertical particle
entering the tank from the center of the roof, the Cherenkov photons should be
able to reach the lateral walls of a Circular-DLT or the corners of an
Hexagonal-DLT and a Square-DLT at the base of the upper layer. For Circular-
DLTs the height $h$ and radius $r$ follow the relation $h=r/\tan{\theta_{C}}$,
where $\theta_{C}=41.2\deg$ is the emission angle of the Cherenkov photons
with respect to the trajectory of the particle crossing the water. Similarly,
for Hexagonal-DLTs with side $L$, the height is $h=L/\tan{\theta_{C}}$, and
for Square-DLTs with half side $l$, $h=\sqrt{2}l/\tan{\theta_{C}}$. To the
height calculated with previous formulas, 1 m of water is added to have 90%
probability that gamma-rays interact by pair production. The lower layer, with
height independent of the radius, is dedicated to muon measurements, allowing
for the gamma/hadron discrimination and the separation of mass groups of
charged primaries (from 2 to 4). For the lower layer, we chose heights of 0.5
m, 0.75 m and 1 m. The dimensions of the tanks are collected in Tab. 1.
Tank | Width (m) | Cir.&Hex. Height u.l. (m) | Sqr. Height u.l. (m) | Height l.l. (m)
---|---|---|---|---
T1 | 3 | 2.7 | 3.4 | 0.5, 0.75, 1
T2 | 3.5 | 3.0 | 3.8 | 0.5, 0.75, 1
T3 | 4 | 3.3 | 4.2 | 0.5, 0.75, 1
T4 | 4.5 | 3.6 | 4.6 | 0.5, 0.75, 1
T5 | 5 | 3.9 | 5.0 | 0.5, 0.75, 1
T6 | 5.5 | 4.2 | 5.4 | 0.5, 0.75, 1
Table 1: Size of the tanks. “Width” is the diameter of Circular-DLT, the side
of Square-DLT and two times the side of Hexagonal-DLT; “Cyl.&Hex. Height u.l.”
is the height of the upper layer of Circular-DLT and Hexagonal-DLT; “Sqr.
Height u.l.” is the height of the upper layer of Square-DLT; “Height l.l.” is
the height of the lower layer.
#### 2.2.2 Properties of the inner walls
For the inner walls of the upper layers, we used both reflective (Tyvek) and
non-reflective (Polypropylene) materials. The reflectivity of the materials
depends on the wavelength of the incident photons. Tyvek has a reflectivity of
$0.63-0.92$ in the wavelength range $250-650$ nm; polypropylene has a
reflectivity of 0.10 over the same wavelength range. Reflective walls allow
for a better detection capability, but might extend the detection time, due to
possible consecutive reflections of photons on the walls before they reach the
PMTs. This results in a higher detection efficiency as photons that would not
be detected with non-reflective walls are instead detected with reflective
walls, but also widen the time resolution for the detection of the first
photon. For the lower layer we used reflective walls, as the priority was
given to the detection efficiency of particles entering the lower layer rather
than the timing.
#### 2.2.3 PMTs
In the upper layer we used two configurations of PMTs looking upwards: one
central 10" PMT or four peripheral 5" PMTs placed at half radius in Circular-
DLTs, half the apothem in Hexagonal-DLTs, and half diagonal in the Square-DLT.
Signals of the peripheral PMTs are summed in one unique output. In the lower
layer we used one central 10" PMT or 5" PMT looking downwards. In each layer,
the two PMT configurations have to be considered independently. In the
simulations, we used two models of PMTs from Hamamatsu: the 10" R7081HQE PMT,
and the 8" R5912 PMT, then re-scaled to a 5" PMT during the analysis phase.
## 3 Analysis
For the evaluation of the tank response, the parameters taken into account
are:
((a))
((b))
((c))
((d))
((e))
((f))
((g))
((h))
((i))
Figure 2: Distributions used in the analysis of the tank performance: number
of PEs (a-c), arrival time of the first photon (d-f), and arrival time of any
photon (g-i). These plots refer to simulations of 1 GeV electrons, gamma-rays
and muons crossing a Circular-DLT with non-reflective walls in the upper layer
and reflective walls in the lower layer. The statistics boxes shown on the
right-side of the plots can be used to analyze the results for the different
configurations of PMTs. For example, the “Mean” value in (a-c) can be used to
estimate the average number of PEs, while the width of the distribution “Std
Dev” in (d-f) can be used to evaluate the time resolution of the measurement
of the first photon.
* •
The number of photoelectrons (PEs) produced in both layers. In the upper layer
we considered separately the configuration with one central 10" PMT or four
peripheral 5" PMTs. For the lower layer we considered individually a central
10" PMT or 5" PMT. This allowed to understand which of the two configurations
in the upper layer gives the higher detection efficiency and the better time
resolution of the first detected photon and, for the lower layer, how a
different size of the PMT changes the performance.
* •
The time resolution of the measurement of the first photon in the upper layer,
evaluated as the standard deviation of the distribution of the first photon
arrival time.
* •
The detection efficiency of both layers. The efficiency is calculated as the
number of detected particles (events) divided by the number of particles
entering the upper layer of the tank (10000). The latter is based on simple
geometrical considerations, based on the height of the entrance point of the
particles. Due to the random direction of particles, a fraction of the
particles that enter the upper layer does not enter the lower layer. We tried
to evaluate the number of particles actually entering the lower layer based on
the initial direction of the particles, but this was not possible due to non-
tracked deflections of the particle trajectories occurring while they cross
the tank. We performed a set of simulations where only 10 GeV vertical muons
were thrown through a Circular-DLT, and in this case the detection efficiency
was $\sim$1 for both the upper and lower layer. This demonstrated that the
inefficiency of the lower layer is only due to geometrical constraints, and
would be effectively reduced considering a joint detection of inclined
particles by neighbouring tanks in a dense array. Therefore, also in the
calculation of the detection efficiency of the lower level we used as
reference the number of particles entering the upper layer of the tank. This
effect underestimates the detection efficiency of the lower layer for any type
of tank and particle, but the comparison between the different configurations
remains valid. For the upper layer, we considered as threshold both 1 PE and
the coincidence of 2 PEs produced within 30 ns by the central 10" PMT or by
the four peripheral 5" PMTs, while for the lower layer the threshold was only
of 1 PE.
In Fig. 2 sample distributions of the number of PEs (a-c), the arrival time of
the first photon (d-f), and the arrival time of photons (g-i) of are shown.
They refer to simulations of 1 GeV electrons, gamma-rays and muons crossing a
Circular-DLT with non-reflective walls in the upper layer and reflective walls
in the lower layer. The distribution of the number of PEs in the lower layer
has a higher average for muons than for electrons and gamma-rays, as the
latter are absorbed by the water of the upper layer. The PE distributions also
show that particles generate in average more PEs on the central 10" PMT than
on the four peripheral 5" PMTs. The distributions of the arrival time of the
first photon are similar. In the sample of distributions reported, the timing
resolution for the four peripheral PMTs is slightly narrower than for the
central PMT.
The time distributions of photons are also presented to show the effect of the
reflective walls in the lower layer. Especially for muons that reach easily
the lower layer, a long tail and a bump after the main peak are due to the
consecutive reflections on the walls before photons reach the PMT. This effect
is more visible when reflective covers are used also for the upper layer, due
to the higher statistics of PEs generated in the upper layer. Depending on the
distributions of the impact point on the tank and the direction of the
particles, a sequence of bumps might appear instead of the tail.
## 4 Results
### 4.1 Comparison of tanks with different size, reflective properties of the
inner walls and PMT configuration
In this section, plots are referred to Circular-DLTs, but the results are
similar for all tank geometries. In Fig. 3–5 the performance of the upper
layer is shown, considering a central 10" PMT or four peripheral 5" PMTs.
Panels (a) are relative to non-reflecting walls, while panels (b) are for
reflecting walls. Similarly, in Fig. 6–7 the performance of the lower layer,
which has always reflective walls, is shown considering a 10" PMT or a 5" PMT.
The number of detected PEs in the upper layers (see Fig. 3) and consequently
the detection efficiency (see Fig. 4) decrease while increasing the size of
the tank. This is due to the decrease of the ratio between the area of the PMT
and that of the base of the tank. To verify this, we made some test
simulations using different tank widths and rescaling the PMT size in order to
have a constant ratio between the area of the PMT and the base of the tank,
and the detection efficiency remained almost constant. With 1 PE threshold,
the detection efficiency of the upper layers considering one central 10" PMT
or four peripheral 5" PMTs are comparable, although more PEs are produced in
the central PMT. The sensitive area based on the size of the PMTs is similar
for the two configurations. Therefore, the difference is related to the
position of the PMTs. The efficiency for 10 MeV and 100 MeV particles is a few
ten percent, while it is above 70% for higher energies. With 2 PEs threshold
(plots not shown in this work), the efficiency is reduced by a few ten percent
for 10 MeV and 100 MeV particles, while it is similar for particles with
higher energy. With reflective walls in the upper layers, the number of PEs
and the detection efficiency are higher than those for non-reflective walls,
even for low energy particles (compare Fig. 3 (a) with (b) and Fig. 4 (a) with
(b)). Using 2 PEs threshold, the effect is the same as that for non-reflecting
walls.
((a))
((b))
Figure 3: Number of PEs detected in the upper layer with non-reflecting walls
(a) and reflective walls (b) in Circular-DLT, note the different scales on the
y-axis.
((a))
((b))
Figure 4: Detection efficiency of the upper layer with non-reflecting walls
(a) and reflective walls (b) in Circular-DLT.
The time resolution of the measurement of the first photon worsen, i.e. the
standard deviation of the distribution gets larger, with the increase of the
size of the tank (see Fig. 5). In average, using non-reflecting walls it
ranges from $\sim$2.5 ns in small tanks to $\sim$3.5 ns in large tanks.
Considering 2 PEs threshold, it has smaller values, between $\sim$1 ns and
$\sim$2 ns. Using reflective walls, it slightly increases for 100 MeV and 1
GeV particles, and rises up to $\sim$18 ns for particles of 10 MeV, because
the time distribution of the first photon shows a long tail for these
particles. It has similar values for the central 10" PMT and the four
peripheral 5" PMTs. Considering 2 PEs threshold, it has slightly lower values.
((a))
((b))
Figure 5: Time resolution of the measurement of the first photon in the upper
layer with non-reflective walls (a) and reflecting walls (b) in Circular-DLT.
Like in the upper layer, the number of detected PEs (see Fig. 6) and the
detection efficiency (see Fig. 7) in the lower layers decrease with the size
of the tank. Electrons and gamma-rays of 10 MeV and 100 MeV are rarely
detected in lower layers. Considering a 5" PMT instead of a 10" PMT, the
efficiency slightly decreases although the number of produced PEs is the 25%,
proportional to the area of the photocathode. Both kinds of PMTs are placed at
the center of the ceiling of the lower layer, so there is no effect due to
different positioning like it happens in the upper layer. The height of the
lower layer influences the number of PEs, which is lower for 0.5 m and
comparable for 0.75 m and 1 m, but does not affect the detection efficiency
(plots not shown in this work). In all the configurations, the detection
efficiency of the lower layer is underestimated in the same way due to
geometrical constraints (see details about the calculation of the detection
efficiency in Section 3).
((a)) Figure 6: Number of PEs detected in the lower layer with reflective
walls. ((a)) Figure 7: Detection efficiency of the lower layer with reflective
walls. In all the configurations, it is underestimated due to geometrical
constraints.
### 4.2 Comparison of tank shapes
The analysis described in the previous section was performed for the three
tank shapes of interest: Circular-DLTs, Square-DLTs and Hexagonal-DLTs. In
addition to tanks with circular base, which is the most commonly used shape
for the construction of ground based astrophysical experiments due to their
stronger strain resistance to the water pressure, ease of construction, and
lower cost, tanks with square and the hexagonal bases have been considered
because they would offer a higher fill factor, particularly important when it
is necessary to cover areas with high density of tanks, like the inner array
of the SWGO experiment.
The plots shown in this section represent the aforementioned parameters in
function of the size of the tanks with different geometries and were made for
1 GeV particles, in order to compare the tank performance in the same
conditions. The size of the tanks is represented by their width, i.e. diameter
for Circular-DLT, two times the side for Hexagonal-DLT, and side for Square-
DLT. In Fig. 8–13 the performance of the upper layers of tanks with different
shapes are shown, considering a central 10" PMT or four peripheral 5" PMTs.
Panels (a) are relative to non-reflecting walls, while panels (b) are for
reflecting walls. Similarly, in Fig. 14–17 the performance of the lower layer
is shown considering a 10" PMT or a 5" PMT.
In general the Circular-DLT and Hexagonal-DLT produce more PEs than the
Square-DLT.
((a))
((b))
Figure 8: Comparison of the number of PEs detected in the upper layer with
non-reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the central 10"PMTs.
((a))
((b))
Figure 9: Comparison of the number of PEs detected in the upper layer with
non-reflective walls (a), and reflective walls (b) for 1 GeV particles, for
different geometries and considering only the peripheral 5"PMTs.
With reflective walls, the number of PEs is about three times that obtained
with non-reflective walls, in any kind of tank (compare panels (a) with panels
(b) in Fig. 8 and Fig. 9). Comparing the response of the central 10" PMT and
the four peripheral 5" PMTs in the upper layer, the number of PEs for the
first configuration is higher than the number of PEs for the latter
configuration for all tank geometries (compare panel (a) and panel (b) in Fig.
8 with the same panels in Fig. 9). The detection efficiency of the upper layer
is similar for Circular-DLTs and Hexagonal-DLTs, while it is slightly lower
for Square-DLTs. As already shown, it is higher using reflective walls
(compare panels (a) with panels (b) in Fig. 10 and Fig. 11). Although the
number of PEs is higher for the central 10" PMT, the detection efficiency for
particles of 1 GeV is similar for both configurations of PMTs (compare panels
(a) and (b) in Fig. 10 with those in Fig. 11).
((a))
((b))
Figure 10: Comparison of detection efficiency of the upper layer with non-
reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the central 10"PMTs.
((a))
((b))
Figure 11: Comparison of the detection efficiency of the upper layer with non-
reflective walls (a) and reflective walls (b), for 1 GeV particles, for
different geometries and considering only the peripheral 5"PMTs.
The temporal resolution of the measurement of the first photon considering the
central 10" PMT is $\sim$0.5 ns larger than that of the four peripheral 5"
PMTs (compare panels (a) and panels (b) in Fig. 12 with those in Fig. 13).
However, such values are slightly higher for Square-DLT than the others. In
general, with reflective walls the temporal resolution of the measurement of
the first photon in upper layers is about $1-2$ ns larger than with non-
reflective walls, in all kinds of tanks (compare panels (a) with panels (b) in
Fig. 12 and Fig. 13).
((a))
((b))
Figure 12: Comparison of the time resolution of the measurement of the first
photon in case of non-reflecting walls (a) and reflective walls (b) in the
upper layer, for 1 GeV particles, for different geometries and considering
only the peripheral 10" PMTs in the upper layer.
((a))
((b))
Figure 13: Comparison of the time resolution of the measurement of the first
photon in case of non-reflecting walls (a) and reflective walls (b) in the
upper layer, for 1 GeV particles, for different geometries and considering
only the peripheral 5" PMTs in the upper layer.
In the lower layer of the three kinds of tanks, the number of PEs produced by
the 10" PMT is four times that produced by the 5" PMT, simply because of the
ratio between the active areas of the sensors (see Fig. 14 and Fig. 15). The
height of the lower layer influences the number of PEs, which is lower for 0.5
m but comparable for 0.75 m and 1 m.
((a)) Figure 14: Comparison of the number of PEs in the lower layer with
reflective walls, for 1 GeV particles, for different geometries and
considering the 10" PMT. ((a)) Figure 15: Comparison of the number of PEs in
the lower layer with reflective walls, for 1 GeV particles, for different
geometries and considering the 5" PMT.
Despite the difference in the number of PEs detected with the two PMT
configurations, the detection efficiency of the lower layer does not vary (see
Fig. 16 and Fig. 17). It is similar for Circular-DLT and Hexagonal-DLT and
higher than for Square-DLT, and it is similar also for the three different
heights considered. In all the configurations, the detection efficiency of the
lower layer is underestimated in the same way due to geometrical constraints
(see details about the calculation of the detection efficiency in Section 3).
((a)) Figure 16: Comparison of the detection efficiency of the lower layer
with reflective walls, for 1 GeV particles, for different geometries and
considering the 10" PMT. In all the configurations, it is underestimated due
to geometrical constraints. ((a)) Figure 17: Comparison of the detection
efficiency of the lower layer with reflective walls, for 1 GeV particles, for
different geometries and considering the 5" PMT. In all the configurations, it
is underestimated due to geometrical constraints.
## 5 Conclusion
This study allowed to compare the response of double-layer tanks with
different shapes, i.e. with circular, hexagonal and square base of different
size, to the passage of single particles of different type and energy.
Moreover, it offered the possibility to compare tanks with reflective and non-
reflective walls in the upper layer.
We found that regardless of the tank design and the reflective properties of
the walls in the upper layer, the performance of the tanks worsen while
increasing the width of the tank, because the “sensitive area”, i.e. the area
covered by the PMTs, decreases with respect to that of the base of the tank.
By using reflective walls instead of non-reflective walls in the upper layers,
the detection efficiency increases, but the time resolution of the measurement
of the first photon widen, in particular for particle with low energy.
In lower layers, electrons and gamma-rays of 10 MeV and 100 MeV are rarely
detected. The height of the lower layer influences the number of PEs, which is
lower for 0.5 m but comparable for 0.75 m and 1 m. Howewer, the detection
efficiency of the lower layer does not vary much with its height.
Comparing the performance of the central 10" PMT and the four peripheral 5"
PMTs in the upper layer, the first configuration produces more PEs than the
second one, although both sensitive areas are similar, but the detection
efficiencies are comparable. In a similar comparison for the lower layer, the
10" PMT produces more PEs than the 5" PMT due to the larger area of the
photocathode, but the detection efficiencies are similar.
The comparison of the performance of tanks with different geometries revealed
that the Circular-DLTs and Hexagonal-DLTs have similar performance, which is
better than that of the Square-DLTs. Nevertheless, for the final design of
ground based astrophysical observatories like the SWGO array, it should be
taken into account also that with Hexagonal-DLT and Square-DLT it is possible
to achieve a higher fill factor, although they are potentially more expensive
solutions.
## Acknowledgments
We thank our colleagues within the SWGO Collaboration for the discussions and
the software framework used in this work. We thank the HAWC Collaboration for
providing the AERIE software.
## References
* [1] U. Barres de Almeida (for the SWGO Collaboration), The Southern Wide-Field Gamma-ray Observatory (SWGO), arXiv e-prints: arXiv:2012.13740 (2020)
* [2] J. Hinton et al. (for the SWGO Collaboration), The Southern Wide-field Gamma-ray Observatory: Status and Prospects, in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)023 (2021)
* [3] T. DeYoung (for the HAWC Collaboration), The HAWC observatory, Nuclear Instruments and Methods in Physics Research A, 692, 72 (2012)
* [4] X. Bai et al., The Large High Altitude Air Shower Observatory (LHAASO) Science White Paper, arXiv e-prints: arXiv:1905.02773 (2019)
* [5] S. Kunwar et al. (for the SWGO Collaboration), Double-layered Water Cherenkov Detector for the Southern Wide-field-of-view Gamma-ray Observatory (SWGO), in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)902 (2021)
* [6] H. Schoorlemmer et al. (for the SWGO Collaboration), Simulating the performance of the Southern Wide-view Gamma-ray Observatory, in Proceedings of the 37th International Cosmic Ray Conference, Berlin, Germany - Online, PoS(ICRC2021)903 (2021)
* [7] J. Allison et al., Geant4 developments and applications, IEEE Transactions on Nuclear Science 53 No. 1 (2006) 270-278
|
# 1D Global Bosonization of Quantum Gravity
L. A. Glinka111E-mail to<EMAIL_ADDRESS><EMAIL_ADDRESS>
_Bogoliubov Laboratory of Theoretical Physics_ ,
_Joint Institute for Nuclear Research_ ,
_Joliot–Curie 6, 141980 Dubna, Moscow Region, Russia_
###### Abstract
Reduction of the Wheeler–DeWitt equation to the Klein–Gordon–Fock evolution
for bosonic field by using of global bosonization to one-dimensional is
proposed. The second quantization of the theory is carried out, and the
Quantum Gravity is constructed in terms of the Fock–Bogoliubov–Heisenberg
initial data operator basis. It is shown that this leads to understanding of
mass of the bosonic field as a scaled initial data mass by one-point
correlations of two bosonic fields.
## 1 Introduction: Unsolved Quantum Gravity
The Einstein–Hilbert field equations of General Relativity [1, 2] 222We use
the system of units $c=\hbar=k_{B}=8\pi G/3=1$.
$R_{\mu\nu}-\dfrac{R[g]}{2}g_{\mu\nu}+\Lambda
g_{\mu\nu}=3T_{\mu\nu},\leavevmode\nobreak\ \leavevmode\nobreak\
R[g]=g^{\kappa\lambda}R_{\kappa\lambda}$ (1)
where $g_{\mu\nu}$ is a non-degenerate and symmetric
$\left(\\!\\!\\!\begin{array}[]{c}0\vspace*{-4pt}\\\
2\end{array}\\!\\!\\!\right)$-tensor field, $R_{\mu\nu},\Lambda,T_{\mu\nu}$
are the metric-contracted Riemann curvature tensor, cosmological constant, and
stress-energy tensor, and $R[g]$ is the Ricci scalar curvature of a pseudo-
Riemannian manifold $(M,g)$ [3, 4], arise due to the Palatini principle [5]
$\dfrac{\delta S[g]}{\delta g_{\mu\nu}}=0,$ (2)
used to the Einstein–Hilbert action modified by a boundary term
$S[g]=-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K[h]+\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}R[g]+\dfrac{\Lambda}{3}+\mathcal{L}\right\\},$
(3)
springs from allowing variations for which the normal derivatives on $\partial
M$ are non-zero, in order to cancel surface terms. Here $K[h]$ is the
extrinsic curvature of an induced three-dimensional spacelike boundary
$(\partial M,h)$, and $\mathcal{L}$ is the Matter fields Lagrangian provoking
the stress-energy tensor $T_{\mu\nu}$
$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta
g^{\mu\nu}}.$ (4)
Stationarity of the Matter fields results in existence of a global timelike
Killing vector field for a metric field $g_{\mu\nu}$. A coordinate system can
be chose such that the Killing vector field equals $\dfrac{\partial}{\partial
t}$ and the foliation $t=constant$ is spacelike. Then a metric field depends
at most on a spatial coordinates $x^{i}$, so the $t$ can be treated globally
[6], and $3+1$ decomposition of a metric
$\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
g^{\mu\nu}=\left[\begin{array}[]{cc}-\dfrac{1}{N^{2}}&\dfrac{N^{j}}{N^{2}}\vspace*{5pt}\\\
\dfrac{N^{i}}{N^{2}}&h^{ij}-\dfrac{N^{i}N^{j}}{N^{2}}\end{array}\right],$ (9)
$\displaystyle h_{ik}h^{kj}=\delta_{i}^{j},\leavevmode\nobreak\
\leavevmode\nobreak\ N^{i}=h^{ij}N_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ g=N^{2}h,$ (10)
has also a global sense. In this case the action (3) becomes
$\displaystyle S[g]\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int dt\leavevmode\nobreak\
L(\pi,\pi^{i},\pi^{ij},N,N_{i},h_{ij}),$ (11) $\displaystyle
L(\pi,\pi^{i},\pi^{ij},N,N_{i},h_{ij})\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\int_{\partial
M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}-NH-
N_{i}H^{i}\right\\},$ (12)
where
$\displaystyle\dot{h}_{ij}$ $\displaystyle=$ $\displaystyle\dfrac{\partial
h_{ij}}{\partial t}=N_{i|j}+N_{j|i}-2NK_{ij},$ (13) $\displaystyle H$
$\displaystyle=$
$\displaystyle\sqrt{h}\left\\{K^{2}-K_{ij}K^{ij}+R[h]-2\Lambda-6T_{nn}\right\\},$
(14) $\displaystyle H^{i}$ $\displaystyle=$
$\displaystyle-2\pi^{ij}_{\leavevmode\nobreak\
;j}=-2\pi^{ij}_{\leavevmode\nobreak\
,j}-h^{il}\left(2h_{jl,k}-h_{jk,l}\right)\pi^{jk},$ (15)
where the second formula follows from the Gauss-Codazzi equations [7]. Here
$K_{ij}$ is the extrinsic-curvature tensor ($K=K^{i}_{i}$), and $\pi^{ij}$ is
the canonical conjugate momentum field to the field $h_{ij}$
$\pi^{ij}=\dfrac{\delta
L}{\delta\dot{h}_{ij}}=-\sqrt{h}\left(K^{ij}-h^{ij}K\right).$ (16)
Time-preservation requirement [8] of the primary constraints [9] for (11)
$\pi=\dfrac{\delta L}{\delta\dot{N}}\approx 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \pi^{i}=\dfrac{\delta L}{\delta\dot{N_{i}}}\approx 0,$
(17)
leads to the secondary constraints
$H\approx 0,\leavevmode\nobreak\ \leavevmode\nobreak\ H^{i}\approx 0,$ (18)
called the Hamiltonian constraint and the diffeomorphism constraint,
respectively. The diffeomorphism constraint merely reflects spatial
diffeoinvariance, and the Hamiltonian constraint gives the dynamics. By (16)
the Hamiltonian constraint becomes the Einstein–Hamilton–Jacobi equation [10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]
$G_{ijkl}\pi^{ij}\pi^{kl}+\sqrt{h}\left(R[h]-2\Lambda-6T_{nn}\right)=0,$ (19)
where $G_{ijkl}$ is called the Wheeler superspace metric
$G_{ijkl}=\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right).$
(20)
Canonical quantization [45] of (19) by the commutation relations [46]
$\displaystyle i\left[\pi^{ij}(x),h_{kl}(y)\right]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(21) $\displaystyle i\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y),$ (22) $\displaystyle
i\left[\pi(x),N(y)\right]$ $\displaystyle=$ $\displaystyle\delta^{(3)}(x,y),$
(23)
leads to the Wheeler–DeWitt equation [47, 9]
$\left\\{-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+h^{1/2}\left(R[h]-2\Lambda-6T_{nn}\right)\right\\}\Psi[h]=0,$ (24)
and the other first class constraints are conditions on the wave function
$\Psi[h]$
$\pi\Psi[h]=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\pi^{i}\Psi[h]=0,\leavevmode\nobreak\ \leavevmode\nobreak\ H^{i}\Psi[h]=0.$
(25)
Furthermore, the canonical commutation relations hold
$\left[\pi(x),\pi^{i}(y)\right]=\left[\pi(x),H^{i}(y)\right]=\left[\pi^{i}(x),H^{j}(y)\right]=\left[\pi^{i}(x),H(y)\right]=0,$
(26)
and in consequence $H_{i}$ are generators of diffeomorphisms
$\widetilde{x}^{i}=x^{i}+\delta x^{i}$ [9]
$\displaystyle\left[h_{ij},i\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-h_{ij,k}\delta x^{k}-h_{kj}\delta
x^{k}_{\leavevmode\nobreak\ ,i}-h_{ik}\delta x^{k}_{\leavevmode\nobreak\
,j}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (27)
$\displaystyle\left[\pi_{ij},i\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-\left(\pi_{ij}\delta
x^{k}\right)_{,k}+\pi_{kj}\delta x^{i}_{\leavevmode\nobreak\
,k}+\pi_{ik}\delta x^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (28)
or in more conventional form
$\displaystyle i\left[H_{i}(x),H_{j}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{\partial M}H_{a}c^{a}_{ij}d^{3}z,$ (29)
$\displaystyle i\left[H(x),H_{i}(y)\right]\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!H\delta^{(3)}_{,i}(x,y),$ (30) $\displaystyle
i\left[\int_{\partial M}H\delta x_{1}d^{3}x,\int_{\partial M}H\delta
x_{2}d^{3}x\right]\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\int_{\partial
M}H^{a}\left(\delta x_{1,a}\delta x_{2}-\delta x_{1}\delta
x_{2,a}\right)d^{3}x,$ (31)
where $H_{i}=h_{ij}H^{j}$, and
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-\delta^{a}_{j}\delta^{b}_{i}\delta^{(3)}_{,b}(y,z)\delta^{(3)}(x,z)\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (32)
are structure constants of diffeomorphism group. Commutators (29-31) show the
first-class constrained system property.
The Wheeler–DeWitt equation (24) has been studied intensively since 30 years
[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. In fact, this is an equation on
superspace [78], defined as a space of all equivalence class of metrics
related by the action of the diffeomorphism group of a compact, connected,
orientable, Hausdorff, $C^{\infty}$ 3-dimensional spacelike manifold without
boundary $\partial M$. If $Riem(\partial M)$ consists all $C^{\infty}$
Riemannian metrics on $\partial M$, and $Dif\\!f(\partial M)$ is a group of
all $C^{\infty}$ diffeomorphisms of $\partial M$ that preserve orientation,
then the Wheeler superspace $S(\partial M)$ is the space of all orbits of
$Dif\\!f(\partial M)$, _i.e._ $S(\partial M)=Riem(\partial M)/Dif\\!f(\partial
M)$. $S(\partial M)$ is a connected, second-countable, metrizeable space. All
geometries with the same kind of symmetry are manifold in $S(\partial M)$ \-
they have homeomorphic neighbouhoods. However, symmetric geometries
neighbourhoods are not homeomorphic to nonsymmetric geometries ones, and by
this $S(\partial M)$ is not manifold. Superspace can be decomposed by its
subspaces on a countable, partially-ordered,
$C^{\infty}$-Fr$\mathrm{\acute{e}}$chet manifold partition, that is an
inverted stratification indexed by the symmetry type - geometries with a given
symmetry are completely contained within the boundary of less symmetric
geometries. The minisuperspace models, _i.e._ Quantum Cosmology [79, 80, 81,
82, 83, 84, 85], study certain strata of superspace. Fischer [78] proved by
suitable choice of a subgroup of $Dif\\!f(\partial M)$ and by action of this
subgroup on $Riem(\partial M)$, for n-dimensional $\partial M$ the superspace
$S(\partial M)$ can be extended to a manifold $S_{e}(\partial M)$ such that
$\dim S_{e}(\partial M)/S(\partial M)=n(n+1)$.
## 2 Global Bosonization to One Dimension
The superspace has no physical consequences [86] and is the main structural
problem of the theory. In this section we will construct linearization of the
Quantum Gravity, global bosonization to one dimension.
### 2.1 Reduction Problem
Let us consider the standard relation of General Relativity [87] between
variations of a metric field determinant and a metric field
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}=g\left(g^{00}\delta g_{00}+g^{ij}\delta
g_{ij}+g^{0j}\delta g_{0j}+g^{i0}\delta g_{i0}\right).$ (33)
The $3+1$ parametrization (9) allows determine the partial variations
$\displaystyle\delta g_{00}$ $\displaystyle=$ $\displaystyle-\delta
N^{2}+N^{i}N^{j}\delta h_{ij}+h_{ij}N^{i}\delta N^{j}+h_{ij}N^{j}\delta
N^{i},$ (34) $\displaystyle\delta g_{ij}$ $\displaystyle=$
$\displaystyle\delta h_{ij},$ (35) $\displaystyle\delta g_{0j}$
$\displaystyle=$ $\displaystyle h_{ij}\delta N^{i}+N^{i}\delta h_{ij},$ (36)
$\displaystyle\delta g_{i0}$ $\displaystyle=$ $\displaystyle h_{ij}\delta
N^{j}+N^{j}\delta h_{ij},$ (37)
as well as the total variation
$\displaystyle\delta g=N^{2}\delta h+h\delta N^{2}.$ (38)
Taking a contravariant metric field components of (9) we obtain from (33)
$\displaystyle N^{2}\delta h=N^{2}hh^{ij}\delta h_{ij},$ (39)
so that the global relation between first functional derivatives is
established
$\dfrac{\delta}{\delta h_{ij}}=hh^{ij}\dfrac{\delta}{\delta h}.$ (40)
The global reduction (40) has deep sense - the first functional derivative
operator $\dfrac{\delta}{\delta h_{ij}}$ is an object from a vector space
spanned by the contravariant 3-space metric $h^{ij}$. Therefore, as the
consequence of (40) one can determine the Wheeler–DeWitt second derivative
functional operator (24)
$\displaystyle-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}},$ (41)
where was used the obvious identity
$\displaystyle\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)h^{ij}h^{kl}=\delta_{i}^{l}\delta^{i}_{l}+\delta^{j}_{l}\delta^{l}_{j}-\delta^{i}_{i}\delta^{k}_{k}=-3.$
(42)
Hence the Wheeler–DeWitt equation (24) becomes the one-dimensional
Klein–Gordon–Fock type evolution
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+m^{2}\right)\Psi[h]=0,$ (43)
where
$m^{2}\equiv m^{2}[h]=\dfrac{2}{3h}\left(R[h]-2\Lambda-6T_{nn}\right),$ (44)
is the square of mass of the bosonic field $\Psi[h]$. By using of the notation
$\Phi=\left[\begin{array}[]{c}\Psi\\\
\Pi_{\Psi}\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\vec{\partial}=\left[\begin{array}[]{c}\dfrac{\delta}{\delta h}\\\
0\end{array}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{M}=\left[\begin{array}[]{cc}0&1\\\ -m^{2}&0\end{array}\right]\geq 0,$
(45)
the second order scalar equation (43) becomes the first order vector equation
$\left(i\mathbf{\Gamma}\vec{\partial}-\mathbb{M}\right)\Phi[h]=0,$ (46)
where $\Gamma$ matrices obey the relations
$\mathbf{\Gamma}=\left[-i\mathbf{1},\mathbf{0}\right],\leavevmode\nobreak\
\leavevmode\nobreak\
\left\\{\mathbf{\Gamma}^{a},\mathbf{\Gamma}^{b}\right\\}=2\eta^{ab}\mathbf{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \eta^{ab}=\left[\begin{array}[]{cc}-1&0\\\
0&0\end{array}\right],$ (47)
where $\mathbf{1}$ and $\mathbf{0}$ are unit and null two-dimensional
matrices.
We have seen that application of the global reduction (40) to the
Wheeler–DeWitt equation (24), that has also a global nature by a character of
the decomposition (9), results in the bosonic quantum mechanics (43). This
scalar-type second order functional evolution was reduced directly to the
vector-type first order functional equation (46) with some two-component field
$\Phi[h]$ as a solution. In the equation (43) as well as in its the reduced
form (46) the superspace metric is completely absent. By this reason the most
mysterious element of the Wheeler Quantum Gravity’s logics was formally
excluded from considerations – the notion of superspace as well as its
mathematical properties are not need to further analysis. In further
developments of this paper we will concentrate on canonical quantization in
the bosonic Fock space of the reduced equation (46).
### 2.2 Fock–Bogoliubov–Heisenberg initial data basis
Next step of the bosonization is the field quantization of the equation (46)
$\Phi[h]\rightarrow\mathbf{\Phi}[h]\Rightarrow\left(i\mathbf{\Gamma}\vec{\partial}-\mathbb{M}\right)\mathbf{\Phi}[h]=0,$
(48)
according to canonical commutation relations proper for the Bose statistics
[88, 89, 90]
$\displaystyle\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle-i\delta(h^{\prime}-h),$ (49)
$\displaystyle\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Pi}_{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle 0,$ (50)
$\displaystyle\left[\mathbf{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]$
$\displaystyle=$ $\displaystyle 0.$ (51)
By using of the second quantization method [91, 92, 93], from the equation
(43) spring that the field operator $\mathbf{\Phi}[h]$ of the reduced equation
(46) can be represent in the Fock space of annihilation and creation
functional operators
$\mathbf{\Phi}[h]=\mathbb{Q}[h]\mathfrak{B}[h],$ (52)
where $\mathfrak{B}[h]$ is a dynamical basis in the Fock space
$\mathfrak{B}[h]=\left\\{\left[\begin{array}[]{c}\mathsf{G}[h]\\\
\mathsf{G}^{\dagger}[h]\end{array}\right]:\left[\mathsf{G}[h^{\prime}],\mathsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}[h^{\prime}],\mathsf{G}[h]\right]=0\right\\},$
(53)
and $\mathbb{Q}[h]$ is the second quantization matrix
$\mathbb{Q}[h]=\left[\begin{array}[]{cc}\dfrac{1}{\sqrt{2|m[h]|}}&\dfrac{1}{\sqrt{2|m[h]|}}\\\
-i\sqrt{\dfrac{|m[h]|}{2}}&i\sqrt{\dfrac{|m[h]|}{2}}\end{array}\right].$ (54)
In this way the operator equation (48) becomes the equation for a basis
$\mathfrak{B}[h]$
$\dfrac{\delta\mathfrak{B}[h]}{\delta
h}=\left[\begin{array}[]{cc}-im[h]&\dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}\\\ \dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}&im[h]\end{array}\right]\mathfrak{B}[h].$ (55)
Actually, there is a nonlinearity given by coupling between annihilation and
creation operators present as nondiagonal terms in (55), so the equation (55)
can not be solved standardly. In order to solving, let us suppose that in the
Fock space exists a new basis $\mathfrak{B}^{\prime}[h]$
$\mathfrak{B}^{\prime}[h]=\left\\{\left[\begin{array}[]{c}\mathsf{G}^{\prime}[h]\\\
\mathsf{G}^{\prime\dagger}[h]\end{array}\right]:\left[\mathsf{G}^{\prime}[h^{\prime}],\mathsf{G}^{\prime\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}^{\prime}[h^{\prime}],\mathsf{G}^{\prime}[h]\right]=0\right\\},$
(56)
for which the the Bogoliubov transformation
$\mathfrak{B}^{\prime}[h]=\left[\begin{array}[]{cc}u[h]&v[h]\\\
v^{\ast}[h]&u^{\ast}[h]\end{array}\right]\mathfrak{B}[h],\leavevmode\nobreak\
\leavevmode\nobreak\ |u[h]|^{2}-|v[h]|^{2}=1,$ (57)
and the Heisenberg evolution
$\dfrac{\delta\mathfrak{B}^{\prime}[h]}{\delta
h}=\left[\begin{array}[]{cc}-i\lambda[h]&0\\\
0&i\lambda[h]\end{array}\right]\mathfrak{B}^{\prime}[h],$ (58)
are supposed to hold together. We will call briefly this special basis as the
Fock–Bogoliubov–Heisenberg (FBH) operator basis. The diagonalization procedure
(56)-(58) converts the operator basis evolution (55) onto the Bogoliubov
coefficients one
$\dfrac{\delta}{\delta h}\left[\begin{array}[]{c}u[h]\\\
v[h]\end{array}\right]=\left[\begin{array}[]{cc}-im[h]&\dfrac{1}{2m[h]}\dfrac{\delta
m[h]}{\delta h}\\\ \dfrac{1}{2m[h]}\dfrac{\delta m[h]}{\delta
h}&im[h]\end{array}\right]\left[\begin{array}[]{c}u[h]\\\
v[h]\end{array}\right],$ (59)
and the basis $\mathfrak{B}^{\prime}[h]$ takes a meaning of static operator
basis associated with initial data
$\mathfrak{B}^{\prime}[h]\equiv\mathfrak{B}_{I}=\left\\{\left[\begin{array}[]{c}\mathsf{G}_{I}\\\
\mathsf{G}^{\dagger}_{I}\end{array}\right]:\left[\mathsf{G}_{I},\mathsf{G}^{\dagger}_{I}\right]=1,\left[\mathsf{G}_{I},\mathsf{G}_{I}\right]=0\right\\},$
(60)
within the vacuum state can be correctly defined
$|0\rangle_{I}=\left\\{|0\rangle_{I}:\mathsf{G}_{I}|0\rangle_{I}=0,\leavevmode\nobreak\
0={{}_{I}}\langle 0|\mathsf{G}_{I}^{\dagger}\right\\}.$ (61)
In the other words, the integrability problem consists in the equations (59).
However, the Bogoliubov coefficients are additionally constrained by the
hyperbolic rotation condition (56). By this it is useful to apply the
superfluid parametrization, for which the solutions are
$\displaystyle u[h]$ $\displaystyle=$
$\displaystyle\dfrac{1+\mu[h]}{2\sqrt{\mu[h]}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\},$ (62) $\displaystyle v[h]$
$\displaystyle=$
$\displaystyle\dfrac{1-\mu[h]}{2\sqrt{\mu[h]}}\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\},$ (63)
where $\mu[h]$ is a mass scale
$\mu[h]=\dfrac{m_{I}}{m[h]}.$ (64)
This establish the relation between a dynamical basis $\mathfrak{B}[h]$ and
the initial data FBH basis $\mathfrak{B}_{I}$ as follows
$\mathfrak{B}[h]=\mathbb{G}[h]\mathfrak{B}_{I},$ (65)
where the transformation matrix $\mathbb{G}[h]$ is
$\displaystyle\mathbb{G}[h]=\left[\begin{array}[]{cc}\dfrac{\mu[h]+1}{2\sqrt{\mu[h]}}e^{-i\theta[h]}\vspace*{10pt}&\dfrac{\mu[h]-1}{2\sqrt{\mu[h]}}e^{i\theta[h]}\\\
\dfrac{\mu[h]-1}{2\sqrt{\mu[h]}}e^{-i\theta[h]}&\dfrac{\mu[h]+1}{2\sqrt{\mu[h]}}e^{i\theta[h]}\end{array}\right],$
(68)
where $i\theta[h]$ is given by a phase of (62). By this reason, the solution
of the equation (48) can be expressed in the initial data basis as
$\mathbf{\Phi}[h]=\mathbb{Q}[h]\mathbb{G}[h]\mathfrak{B}_{I}.$ (69)
### 2.3 One-point correlations
The second quantized equation (43), _i.e._
$\left(\mu^{2}[h]\dfrac{\delta^{2}}{\delta
h^{2}}+m^{2}_{I}\right)\mathbf{\Psi}[h]=0,$ (70)
has a solution
$\displaystyle\mathbf{\Psi}[h]=\frac{\mu[h]}{2\sqrt{2m_{I}}}\left(\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\}\mathsf{G}_{I}+\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\mu[h^{\prime}]}\right\\}\mathsf{G}_{I}^{\dagger}\right),$ (71)
that is a direct conclusion of the relation (69). This field acts on the
initial data vacuum state as follows
$\displaystyle\mathbf{\Psi}[h]|0\rangle_{I}$ $\displaystyle=$
$\displaystyle\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{i\theta[h]}\mathsf{G}^{\dagger}_{I}|0\rangle_{I},$
(72) $\displaystyle{{}_{I}}\langle 0|\mathbf{\Psi}^{\dagger}[h]$
$\displaystyle=$ $\displaystyle{{}_{I}}\langle
0|\mathsf{G}_{I}\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{-i\theta[h]}.$ (73)
By this reason, one can consider the following many-field quantum states
$\displaystyle|h,n\rangle$ $\displaystyle\equiv$
$\displaystyle\left(\mathbf{\Psi}[h]\right)^{n}|0\rangle_{I}=\left(\frac{\mu[h]}{2\sqrt{2m_{I}}}e^{i\theta[h]}\right)^{n}\mathsf{G}^{\dagger
n}_{I}|0\rangle_{I},$ (74) $\displaystyle\langle n^{\prime},h^{\prime}|$
$\displaystyle\equiv$ $\displaystyle{{}_{I}}\langle
0|\left(\mathbf{\Psi}^{\dagger}[h^{\prime}]\right)^{n^{\prime}}={{}_{I}}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\left(\frac{\mu[h^{\prime}]}{2\sqrt{2m_{I}}}e^{-i\theta[h^{\prime}]}\right)^{n^{\prime}},$
(75)
and determine the two-point quantum correlator of two many-field states
$\displaystyle\langle
n^{\prime},h^{\prime}|h,n\rangle=\dfrac{\mu^{n^{\prime}}[h^{\prime}]\mu^{n}[h]}{\left(8m_{I}\right)^{(n^{\prime}+n)/2}}e^{-im_{I}\theta_{n^{\prime},n}[h^{\prime},h]}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger n}_{I}|0\rangle_{I},$ (76)
where
$\theta_{n^{\prime},n}[h^{\prime},h]=n^{\prime}\int_{h_{I}}^{h^{\prime}}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}-n\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}.$ (77)
Application of the normalization
$\langle 1,h_{I}|h_{I},1\rangle=\dfrac{1}{8m_{I}}{{}_{I}}\langle
0|0\rangle_{I}\equiv 1\Longrightarrow{{}_{I}}\langle 0|0\rangle_{I}=8m_{I},$
(78)
allows define the following correlators
$\displaystyle\langle n^{\prime},h|h,n\rangle$ $\displaystyle=$
$\displaystyle\left(\dfrac{\langle 1,h|h,1\rangle}{{{}_{I}}\langle
0|0\rangle_{I}}\right)^{(n^{\prime}+n)/2}e^{-i(n^{\prime}-n)\theta[h]}{{}_{I}}\langle
0|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger n}_{I}|0\rangle_{I},$ (79)
$\displaystyle\dfrac{\langle n,h^{\prime}|h,n\rangle}{{{}_{I}}\langle
0|0\rangle_{I}}$ $\displaystyle=$ $\displaystyle\left(\dfrac{\langle
1,h^{\prime}|h,1\rangle}{{{}_{I}}\langle 0|0\rangle_{I}}\right)^{n},$ (80)
where
$\displaystyle\langle 1,h^{\prime}|h,1\rangle$ $\displaystyle=$
$\displaystyle\mu[h^{\prime}]\mu[h]\exp\left\\{im_{I}\int_{h^{\prime}}^{h}\dfrac{\delta
h^{\prime\prime}}{\mu[h^{\prime\prime}]}\right\\},$ (81) $\displaystyle\langle
1,h|h,1\rangle$ $\displaystyle=$ $\displaystyle\mu^{2}[h].$ (82)
The last formula (82) together with the definition (64) leads to the relation
between the mass of the bosonic field $\mathbf{\Psi}[h]$ and the initial data
mass $m_{I}$
$m[h]=\lambda[h]m_{I},\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda[h]=\dfrac{1}{\sqrt{\langle 1,h|h,1\rangle}},$ (83)
that means the arbitrary mass $m[h]$ is only rescaled the initial data mass
$m_{I}$, and the scale $\lambda$ is directly related to one-point correlations
of the quantum bosonic field $\mathbf{\Psi}[h]$. Therefore, actually the mass
$m[h]$ for arbitrary $h$ is given by correlations of two bosonic fields
$\mathbf{\Psi}$ in the point $h$. Finally note that the two-point correlator
(81), that can be rewritten in the power series form
$\langle
1,h^{\prime}|h,1\rangle=\mu[h^{\prime}]\mu[h]\prod_{n=0}^{\infty}\sum_{p=0}^{\infty}a_{pn}[h,h^{\prime}|h_{I}]\left(\dfrac{\delta^{n}}{\delta
h^{n}}\mu^{2}[h]\Biggr{|}_{h_{I}}\right)^{p},$ (84)
with a coefficients
$a_{pn}[h,h^{\prime}|h_{I}]=\dfrac{1}{p!}\left[im_{I}\dfrac{(2n-3)!}{2^{2n-1}(n-1)!}\sum_{k=0}^{n+1}\dfrac{(-1)^{k}}{k!(n-k+1)!}(h_{I})^{n-k+1}\left(h^{k}-h^{\prime
k}\right)\right]^{p}.$ (85)
The series gives an opportunity to study perturbationally the two-point
correlations around the initial data point $h=h_{I}$.
## 3 Summary
In spite of a work in the Hamiltonian approach to General Relativity and the
primary quantization, the method of global bosonization to one $h$-dimension
of the Wheeler–DeWitt Quantum Gravity and its second quantization in the
Fock–Bogoliubov–Heisenberg initial data basis, which was presented in details
in this paper differs seriously from the previous authors considerations. The
main difference is a quantum field theory formulation of the Quantum Gravity,
that leads to the FBH initial data basis and considering the theory in terms
of the quantum bosonic field $\mathbf{\Psi}[h]$ associated with a
3-dimensional induced spacelike geometry $(\partial M,h)$. The proposed
approach is not the so called third quantization [94, 95, 96, 97, 98, 99,
100], where the Fock operator bases formalism is not applied. The main goal of
the presented linearization is a canceling of the Wheeler’s superspace notion
from considerations, and formulation of the Quantum Gravity in terms of the
Klein–Gordon–Fock operator evolution and the one-point correlations, that
results in the mass scale of the system.
The author benefited discussions from A.B. Arbuzov, B.M. Barbashov, V.N.
Pervushin, V.B. Priezzhev, D.V. Shirkov, and A.N. Sissakian. Special thanks
are directed to Profs. I.Ya. Aref’eva, G. ’t Hooft, and B.G. Sidharth for
interesting and critical remarks.
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|
# Bistability and oscillations in cooperative microtubule and kinetochore
dynamics in the mitotic spindle
Felix Schwietert and Jan Kierfeld Physics Department, TU Dortmund University,
44221 Dortmund, Germany<EMAIL_ADDRESS>
###### Abstract
In the mitotic spindle microtubules attach to kinetochores via catch bonds
during metaphase, and microtubule depolymerization forces give rise to
stochastic chromosome oscillations. We investigate the cooperative stochastic
microtubule dynamics in spindle models consisting of ensembles of parallel
microtubules, which attach to a kinetochore via elastic linkers. We include
the dynamic instability of microtubules and forces on microtubules and
kinetochores from elastic linkers. A one-sided model, where an external force
acts on the kinetochore is solved analytically employing a mean-field approach
based on Fokker-Planck equations. The solution establishes a bistable force-
velocity relation of the microtubule ensemble in agreement with stochastic
simulations. We derive constraints on linker stiffness and microtubule number
for bistability. The bistable force-velocity relation of the one-sided spindle
model gives rise to oscillations in the two-sided model, which can explain
stochastic chromosome oscillations in metaphase (directional instability). We
derive constraints on linker stiffness and microtubule number for metaphase
chromosome oscillations. Including poleward microtubule flux into the model we
can provide an explanation for the experimentally observed suppression of
chromosome oscillations in cells with high poleward flux velocities.
Chromosome oscillations persist in the presence of polar ejection forces,
however, with a reduced amplitude and a phase shift between sister
kinetochores. Moreover, polar ejection forces are necessary to align the
chromosomes at the spindle equator and stabilize an alternating oscillation
pattern of the two kinetochores. Finally, we modify the model such that
microtubules can only exert tensile forces on the kinetochore resulting in a
tug-of-war between the two microtubule ensembles. Then, induced microtubule
catastrophes after reaching the kinetochore are necessary to stimulate
oscillations. The model can reproduce experimental results for kinetochore
oscillations in PtK1 cells quantitatively.
††: New J. Phys.
* March 2020
Keywords: mitotic spindle, directional instability, microtubule dynamics,
kinetochore oscillations, bistability, stochastic simulation
## 1 Introduction
Proper separation of chromosomes during mitosis is essential for the
maintenance of life and achieved by the mitotic spindle, which is composed of
two microtubule (MT) asters anchored at the spindle poles. The spindle
contains three types of MTs classified according to their function [1]: astral
MTs interact with the cell membrane to position the spindle poles, interpolar
MTs interact with MTs from the opposite pole to maintain spindle length, and,
finally, kinetochore MTs link to the chromosomes via the kinetochores at the
centromere and can apply pulling forces via the linkage. The MT-kinetochore
bond is a catch bond [2], i.e., tightening under tension but the molecular
nature of the MT-kinetochore link is not exactly known and a complete
mechanistic understanding of the catch bond is missing [3, 4] but probably
involves Aurora B [5]; the Ndc80 complexes and Dam1 (in yeast) are believed to
play a major role in the MT-kinetochore link. One function of the spindle is
to align the chromosomes in the metaphase plate at the spindle equator. It has
been observed in several vertebrate cells that chromosomes do not rest during
metaphase but exhibit oscillations along the pole to pole axis known as
directional instability [6, 7, 8, 9, 10, 11, 12], whereas in Drosophila
embryos and Xenopus eggs a directional instability does not occur [13, 14]. If
present, these oscillations are stochastic and on the time scale of minutes,
i.e., on a much larger time scale than the dynamic instability of single MTs.
Both single kinetochores and the inter-kinetochore distance oscillate; inter-
kinetochore or breathing oscillations occur with twice the frequency of single
kinetochore oscillations [11].
A quantitative understanding of the underlying mechanics of the MT-
kinetochore-chromosome system is still lacking. In the past, several
theoretical models have been proposed that reproduce chromosome oscillations
[15, 16, 17, 18, 19, 20, 21]. (see table 1 and reviews [22, 23]) These models
have in common that they simplify to a quasi-one-dimensional geometry and
contain two ensembles of MTs growing from the two spindle poles that connect
to one chromosome that is represented by two kinetochores connected by a
spring (the cohesin bond). Kinetochores follow overdamped motion [16, 17, 18,
19, 20] or are assumed to reach force balance instantaneously under the
influence of MT depolymerization and polymerization forces (because the
friction force is small) [15, 21].
Several MT depolymerization and polymerization forces are included into the
models. The models neglect explicit spindle pole dynamics but possibly include
poleward MT flux [16, 20], which describes a constant flux of tubulin from the
plus-ends towards the spindle pole and is probably driven by plus-end directed
kinesin-5 motors pushing overlapping antiparallel interpolar MTs apart and
kinesin-13 proteins that depolymerize MTs at the centrosome [24]. The main
poleward forces on kinetochores are generated by depolymerization of MTs which
builds up and transmits a poleward force via the MT-kinetochore link. Only in
the model of Civelekoglu-Scholey et al. [16] the main poleward force is
generated by MT depolymerization motors at the spindle poles. In order to be
able to exert poleward pulling forces the MT-kinetochore bond needs to remain
intact during depolymerization and “slide” with the depolymerizing MT plus
end. The force that can be exerted depends on the nature of this bond and is
high if it is a catch bond that tightens under tension [2]. All models include
switching between polymerizing and depolymerizing MT states; in most models
this switching is caused by catastrophe and rescue events (dynamic instability
[25]), only Shtylla and Keener [17, 18] do not introduce explicit MT
catastrophes but catastrophe-like events are triggered by a chemical feedback
loop if MTs approach the kinetochore.
The two ensembles of MTs are engaged in a kind of tug-of-war and exert
antagonistic forces via the spring connecting kinetochores: poleward (P)
depolymerization forces of one ensemble generate an antipoleward (AP) force on
the other kinetochore. Experiments suggest that kinetochore MTs can only exert
P-directed pulling forces by depolymerization but are not able to directly
exert AP-directed pushing forces on the kinetochore during polymerization [7,
26]. During directional instability, the spindle switches between the left and
the right ensemble pulling actively in P-direction by depolymerization while
the respective other ensemble is passively following in AP-direction by
polymerization without actively pushing. Nevertheless, some models have
included AP-directed MT pushing forces [16, 17, 18, 20, 21]. Antagonistic AP-
directed forces on the kinetochores can also be generated by polar ejection
forces (PEFs); they originate from non-kinetochore MTs interacting with the
chromosome arms via collisions or chromokinesins belonging to the kinesin-4
and kinesin-10 families [27] and pushing them thereby towards the spindle
equator. The action of different P- and AP-directed forces can move
kinetochores back and forth and also tense and relax the inter-kinetochore
cohesin bond.
Models differ in their assumptions about the MT-kinetochore link and the
mechanism how MT dynamics is directed by mechanical forces to give rise to
kinetochore and inter-kinetochore distance oscillations. The model by Joglekar
and Hunt [15] uses the thermodynamic Hill sleeve model [28] for the MT-
kinetochore connection, which assumes equally spaced rigid linkers that
diffuse on the discrete MT lattice. Shtylla and Keener [17, 18] combine a
continuous version of the Hill sleeve model with a negative chemical feedback
between the force at the MT-kinetochore interface and the depolymerization
rate. In Hill sleeve models there is no effect of MT insertion and, thus,
force onto the MT dynamic instability, i.e., on catastrophe and rescue rates.
The Hill sleeve can transmit pulling forces onto the kinetochore up to a
critical force above which MTs pull out of the sleeve [15], and there is
evidence that the Hill sleeve exhibits catch-bond-like behavior [29]. More
recent studies show that the kinetochore is not rigid, as supposed in the Hill
sleeve model, but should be viewed as a flexible framework [30]. Moreover,
Ndc80 fibrils have been suggested as main force transmitter [31, 4, 32], which
motivated Keener and Shtylla to modify their Hill sleeve model by replacing
the rigid attachment sites with elastic linkers and allowing for a force
feedback onto MT depolymerization [33]. However, sleeve models remain
speculative as electron microscopy has not yet revealed appropriate structures
[34, 35]. Civelekoglu-Scholey et al. [16] proposed a model in which MTs and
kinetochores are linked by motor proteins (dyneins) walking towards the MT
minus end; these links have no catch-bond-like behavior. The links are assumed
to be able to transmit tension onto MTs that promotes MT rescue. In [21] no
explicit linkers are introduced but permanent MT-kinetochore links are assumed
that can transmit both pulling and pushing forces onto MTs. As the exact
nature of MT-kinetochore linking structures is not known, a model of the MT-
kinetochore linkage as a generic elastic structure seems reasonable, as in
recent models where the MTs are linked to the kinetochore via (visco-)elastic
springs [19, 20]. The MT-kinetochore bond can be modeled as a catch bond, and
the elastic linkers also transmit forces back onto the MT allowing for a force
feedback onto MT dynamics as it has been measured in [2].
In the model of Shtylla and Keener [17], MTs that are attached to the same
kinetochore share the force from the cohesin bond equally and exhibit
synchronous dynamics. The last assumption is contradictory to the experimental
observation that one kinetochore MT ensemble does not coherently
(de)polymerize but always consists of a mixture of both states [36, 37]. Klemm
et al. [21] take into account this observation by dividing each MT ensemble
into a growing and a shrinking sub-ensemble, but also make the strong
assumption of equal force sharing between the MTs within each sub-ensemble.
All other models allow for individual MT dynamics and for different forces
between MTs depending on the distances of MTs to the kinetochore.
The main mechanism for oscillations differs between models depending on the
main antagonistic AP-directed force that switches a depolymerizing P-directed
ensemble back into AP-directed polymerization. Switching can be triggered by
the AP-directed force that the other ensemble can exert via the cohesin spring
during depolymerization and by AP-directed PEFs if MT catastrophes are
suppressed or rescues promoted under tension. In the model by Joglekar and
Hunt [15] AP-directed PEFs are essential for switching. Civelekoglu-Scholey et
al. [16] proposed a model in which force is transmitted by motor proteins. By
variation of the model parameters they were able to reproduce a wide range of
chromosome behavior observed in different cell types. In this model, a
depolymerizing P-directed ensemble switches because tension in the cohesin
spring and PEFs promote rescue events. A modified model [19] uses viscoelastic
catch bonds and accounts for the observation that in PtK1 cells only
chromosomes in the center of the metaphase plate exhibit directional
instability [11]. They explain this dichotomy with different distributions of
PEFs at the center and the periphery of the metaphase plate. In the model by
Shtylla and Keener [17, 18] MT catastrophe-like events are only triggered by a
chemical feedback such that kinetochore oscillations become coupled to
oscillations of the chemical negative feedback system: AP-directed MT
polymerization exerts pushing forces onto the kinetochore but triggers
switching into a depolymerizing state, and MT depolymerization exerts
P-directed pulling forces and triggers switching back into a polymerizing
state.
Whereas in [15, 16, 19] AP-directed PEFs are present and in the model by
Joglekar and Hunt [15] also essential for realistic kinetochore oscillations,
Banigan et al. [20] presented a minimal model with simple elastic linkers and
neglecting PEFs. Referring to experiments with budding yeast kinetochores [2],
they modeled MT dynamics with force-dependent velocities, catastrophe and
rescue rates. In this model, kinetochore oscillations arise solely from the
collective behavior of attached MTs under force and the resulting interplay
between P-directed depolymerization forces and AP-directed polymerization
forces of the opposing MT ensembles. Force-dependent velocities, catastrophe
and rescue rates are essential to trigger switching of kinetochore motion and
oscillations in this model. MTs can exert pushing forces such that it is
unclear to what extent the oscillation mechanism remains functional if pushing
forces are absent as suggested experimentally. Also the recent model by Klemm
et al. [21], which aims to describe kinetochore dynamics in fission yeast,
does not rely on PEFs. It uses a permanent MT-kinetochore bond and
oscillations result from the interplay between MT depolymerization and
polymerization forces via force-dependence in MT dynamics; also in this model
MTs can exert pushing forces. Moreover, the model makes the strong assumption
of equal force sharing between all growing or shrinking MTs attached to a
kinetochore. The model also includes kinesin-8 motors that enhance the
catastrophe rate and have a centering effect on the chromosome positions.
Table 1: Overview of assumptions of models exhibiting stochastic chromosome oscillations. In the referred sections we discuss how poleward flux, PEFs and the absence of pushing forces affect kinetochore dynamics in the model used for this work. | linker | catch | equal | force-dep. | MT | | pole-
---|---|---|---|---|---|---|---
Ref. (year) | model | bonds | force | MT | pushing | PEFs | ward
| | | sharing | rescue/cat. | forces | | MT flux
Joglekar [15] (2002) | Hill sleeve | | no | no | no | yes | no
Civelekoglu [16] (2006) | motor | no | no | yes | yes | yes | yes
Civelekoglu [19] (2013) | viscoelastic | yes | no | yes | no | yes | no
Shtylla [17, 18] (2010) | Hill sleeve | | yes | no | yes | yes | no
Banigan [20] (2015) | elastic | yes | no | yes | yes | no | yes
Klemm [21] (2018) | permanent | | yes | yes | yes | no | no
this work | elastic | yes | no | yes | sec. 8 | sec. 7 | sec. 6
In all Refs. [15, 16, 17, 18, 19, 20, 21] a sufficient set of ingredients is
given for the respective model to exhibit oscillations including a specific
choice of parameter values. It is much harder to give necessary conditions and
parameter ranges for oscillations, which means to obtain quantitative bounds
on model parameters, than to give a sufficient set of conditions. This is the
aim of the present paper within a model that starts from the minimal model by
Banigan et al. and generalizes this model in several respects in later
sections, see table 1. In this way we discuss the complete inventory of
possible forces acting on the kinetochore and its influence on oscillations.
It is also difficult to trace the actual mechanism leading to oscillations. An
essential part in our quantitative analysis is a mean-field solution of the
one-sided minimal model of Banigan et al. [20], where a single kinetochore
under force is connected to one or several MTs that experience length-
dependent individual loads and feature force-dependent dynamics. Force-
velocity relations for a single kinetochore, which is connected to one or
several MTs have been investigated previously based on a sleeve-like MT-
kinetochore interface [17, 18, 33, 29]. Here, we can derive an analytical
solution of the one-sided minimal model from a novel mean-field approach. For
this purpose, we start from the Fokker-Planck equations for the length
distribution of the MT-kinetochore linkers. The only mean-field approximation
is to neglect stochastic velocity fluctuations of the attached kinetochore.
Our solution clearly shows that the force feedback of linkers onto the MT
depolymerization dynamics via catch (or permanent) bonds is essential for a
bistable force-velocity relation within the minimal model. Moreover, the
stationary state solution allows us to quantify the parameter range for a
bistability in the parameter plane of MT-kinetochore linker stiffness and MT
numbers.
By interpreting the force-velocity relation as phase space diagram for the
two-sided model as in [20], we show that bistability in the one-sided model is
a necessary condition for kinetochore oscillations in the two-sided model.
Beyond that, we are able (1) to quantify an oscillatory regime, in which
kinetochores exhibit directional instability, in the parameter plane of linker
stiffness and MT numbers predicting that linkers have to be sufficiently
stiff; (2) to describe kinetochore motion in this oscillatory regime,
calculate frequencies which agree with in vivo measurements [11] and to
explain frequency doubling of breathing compared to single kinetochore
oscillations; (3) to describe kinetochore motion in the non-oscillatory regime
as fluctuations around a fixed point; (4) to show that high poleward flux
velocities move the system out of the oscillatory regime and thereby explain
why directional instability has been observed in mitotic vertebrate cells but
not in Drosophila embryos and Xenopus eggs; (5) to show that polar ejection
forces reduce the amplitude of oscillations, induce a phase shift between
sister kinetochores and are necessary to align the chromosome at the spindle
equator; (6) to derive as necessary condition for oscillations that either MTs
must be able to apply pushing forces on the kinetochore or a catastrophe has
to be induced with increased catastrophe rate when a MT reaches the
kinetochore. All these results are validated by stochastic simulations; (7) to
provide a set of model parameters that reproduce experimental results for
kinetochore oscillations in PtK1 cells quantitatively.
In particular, we quantify lower bounds for linker stiffnesses that allow
oscillations, whose value depends on the behavior of MTs growing against the
kinetochore. If kinetochore MTs can exert pushing forces, we find oscillations
for linker stiffnesses
$>$16\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$; also
if MT catastrophes are induced upon reaching the kinetochore, we find
oscillations in a similar range of linker stiffnesses. These constraints
provide useful additional information on MT-kinetochore linkers whose
molecular nature is not completely unraveled up to now.
## 2 Mitotic spindle model
We use a one-dimensional model of the mitotic spindle (figure 1(a)), similar
to the model from [20]. The $x$-coordinate specifies positions along the one-
dimensional model, and we choose $x=0$ to be the spindle equator. The spindle
model contains a single chromosome represented by two kinetochores, which are
linked by a spring with stiffness $c_{\mathrm{k}}$ and rest length $d_{0}$.
Two centrosomes margin the spindle at $\pm x_{\mathrm{c}}$. From each
centrosome a constant number $M$ of MTs emerges with their plus ends directed
towards the spindle equator. Each MT exhibits dynamic instability [25] and
attaches to and detaches from the corresponding kinetochore stochastically.
Attached MTs are connected to the kinetochore by a linker, which we model as
Hookean polymeric spring with stiffness $c$ and zero rest length. This spring
exerts a force $F_{\mathrm{mk}}=-c(x_{\mathrm{m}}-X_{\mathrm{k}})$ on each MT,
and each MT exerts a counter force $-F_{\mathrm{mk}}$ on the kinetochore,
where $X_{\mathrm{k}}$ and $x_{\mathrm{m}}$ are kinetochore and MT position.
Figure 1: One-dimensional model of the mitotic spindle. (a) Two-sided model:
$M$ MTs arise from each centrosome and can attach to / detach from the
corresponding kinetochore. (b) One-sided model: Left half of two-sided model.
The cohesin bond is replaced by the external force $F_{\mathrm{ext}}$. MTs are
not confined by a centrosome and permanently attached to the kinetochore. MT-
kinetochore distances $x_{i}=x_{\mathrm{m,}i}-X_{\mathrm{k}}$ are the only
relevant coordinates.
In the following we define all MT parameters for MTs in the left half of the
spindle model; for MTs in the right half position velocities $v$ and forces
$F$ have opposite signs. In the left half, tensile forces on the MT-
kinetochore link arise for $X_{\mathrm{k}}>x_{\mathrm{m}}$ and pull the MT in
the positive $x$-direction, $F_{\mathrm{mk}}>0$. In [2], the velocities of MT
growth $v_{\mathrm{m}+}$ and shrinkage $v_{\mathrm{m}-}$ as well as the rates
of catastrophe $\omega_{\mathrm{c}}$, rescue $\omega_{\mathrm{r}}$ and
detachment $\omega_{\mathrm{d}\pm}$ have been measured while MTs were attached
to reconstituted yeast kinetochores. They can all be described by an
exponential dependence on the force $F_{\mathrm{mk}}$ that acts on the MT plus
end:
$\displaystyle
v_{\mathrm{m}\pm}=v^{0}_{\pm}\exp\left(\frac{F_{\mathrm{mk}}}{F_{\pm}}\right),\qquad\omega_{i}=\omega^{0}_{i}\exp\left(\frac{F_{\mathrm{mk}}}{F_{i}}\right),$
(1)
(for $i=\mathrm{r},\mathrm{c},\mathrm{d}+,\mathrm{d}-$) with
$F_{+},~{}F_{\mathrm{r}},~{}F_{\mathrm{d+}}>0$ and
$F_{-},~{}F_{\mathrm{c}},~{}F_{\mathrm{d-}}<0$ for the characteristic forces,
because tension ($F_{\mathrm{mk}}>0$) enhances growth velocity, rescue and
detachment of a growing MT, while it suppresses shrinking velocity,
catastrophe and detachment of a shrinking MT (note that we use signed
velocities throughout the paper, i.e., $v_{\mathrm{m}-}<0$ and
$v_{\mathrm{m}+}>0$). Suppression of detachment of shrinking MTs is the catch
bond property of the MT-kinetochore link. The attachment rate is assumed to
follow a Boltzmann distribution,
$\displaystyle\omega_{\mathrm{a}}=\omega^{0}_{\mathrm{a}}\exp\left(\frac{c(X_{\mathrm{k}}-x_{\mathrm{m}})^{2}}{2k_{\mathrm{B}}T}\right),$
(2)
according to the MT-kinetochore linker spring energy.
The kinetochore motion is described by an overdamped dynamics,
$\displaystyle
v_{\mathrm{k}}\equiv\dot{X}_{\mathrm{k}}=\frac{1}{\gamma}\left(F_{\mathrm{kk}}+F_{\mathrm{km}}\right),$
(3)
with the friction coefficient $\gamma$, and the forces $F_{\mathrm{kk}}$ and
$F_{\mathrm{km}}=-\sum_{\rm att.~{}MTs}F_{\mathrm{mk}}$ originating from the
cohesin bond between kinetochores and the MT-kinetochore linkers of all
attached MTs, respectively.
We perform simulations of the model by integrating the deterministic equations
of motion for MTs ($\dot{x}_{\mathrm{m,i}}=v_{\mathrm{m}\pm,i}$ for
$i=1,...,M$) and kinetochores (eq. (3)) and include stochastic switching
events between growth and shrinking as well as for attachment and detachment
to the kinetochore for each MT. For integration we employ an Euler algorithm
with a fixed time step $\Delta t\leq 10^{-3}\,{\rm s}$ which is small enough
to ensure $\omega_{i}\Delta t\ll 1$ for all stochastic switching events (see
table 2). The algorithm is described in the supplementary material in more
detail. We use parameter values from experiments as listed in table 2.
Table 2: Model parameters.
* Transition parameters | $\omega_{i}$ | $\omega_{i}^{0}$ ($\mathrm{s}^{-1}$) | $F_{i}$ ($\mathrm{p}\mathrm{N}$) |
---|---|---|---|---
catastrophe | $\omega_{\mathrm{c}}$ | $0.0019$ | $-2.3$ | [2]
rescue | $\omega_{\mathrm{r}}$ | $0.024$ | -$6.4$ | [2]
detachment in growing state | $\omega_{\mathrm{d}+}$ | $0.000\,11$ | -$3.8$ | [2]
detachment in shrinking state | $\omega_{\mathrm{d}-}$ | $0.035$ | $-4.0$ | [2]
attachment rate | $\omega_{\mathrm{a}}$ | $1.0$ | estimated
Velocity parameters | $v_{\mathrm{m}\pm}$ | $v_{\pm}^{0}$ ($\mathrm{n}\mathrm{m}\mathrm{s}^{-1}$) | $F_{\pm}$ ($\mathrm{p}\mathrm{N}$) |
growth | $v_{\mathrm{m}+}$ | -00$5.2$ | -$8.7$ | [2]
shrinking | $v_{\mathrm{m}-}$ | $-210.0$ | $-3.2$ | [2]
Other parameters | Symbol | Value | |
cohesin bond stiffness | $c_{\mathrm{k}}$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ | estimated
cohesin bond rest length | $d_{0}$ | $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [7]
centrosome position | $x_{\mathrm{c}}$ | $6.8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ | [10]
friction coefficient | $\gamma$ | $1\text{\,}\mathrm{pN}\text{\,}\mathrm{s}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ | estimated
thermal energy | $k_{\mathrm{B}}T$ | $4\text{\,}\mathrm{pN}\text{\,}\mathrm{nm}$ | estimated
We start with the investigation of the minimal model, i.e. neglecting poleward
flux and PEFs and using the same simple spring model for the MT-kinetochore
linker as Banigan et al. where the MT plus ends are able to “overtake” the
kinetochore ($x_{\mathrm{m}}>X_{\mathrm{k}}$, again for MTs in the left half
of the spindle) and thereby exert pushing forces $F_{\mathrm{km}}>0$ on the
kinetochore (which could be interpreted as a compression of the MT-kinetochore
linker). Later, we will generalize the minimal model as described in the
introduction, see table 1. In a first step, we add poleward MT flux, which
describes a constant flux of tubulin from the plus-ends towards the spindle
pole [24], by shifting the MT velocities $v_{\mathrm{m}\pm}$. PEFs, which push
the kinetochore away from the pole [27], will be included in a second step as
external forces, which depend on the absolute positions of the kinetochores.
Finally, we will take account of the hypothesis that MTs are not able to apply
pushing forces on the kinetochore [7, 26] by modifying the model such that the
growth of a MT is stalled or that the MT undergoes a catastrophe when it
reaches the kinetochore.
At the centrosome, MTs are confined: It is reasonable to assume that they
undergo a forced rescue and detach from the kinetochore if they shrink to zero
length. If the mean distance of MTs from the spindle equator is sufficiently
small, $|\langle x_{\mathrm{m}}\rangle|\ll|x_{\mathrm{c}}|$, we can also
consider the MTs as unconfined ($|x_{\mathrm{c}}|\rightarrow\infty$). Then
both MT and kinetochore dynamics solely depend on their relative distances and
not on absolute positions, which simplifies the analysis.
## 3 Mean-field theory for bistability in the one-sided model
We first examine the one-sided model of Banigan et al. [20], which only
consists of the left half of the two-sided model with an external force
applied to the kinetochore (figure 1(b)). In simulations of this one-sided
spindle model, kinetochore movement exhibits bistable behavior as a function
of the applied force, i.e., within a certain force range there are two
metastable states for the same external force: In one state the MTs
predominantly grow and the kinetochore velocity is positive while in the other
state the kinetochore has a negative velocity as a consequence of mainly
shrinking MTs. It depends on the history which of these two states is assumed:
When the system enters the bistable area in consequence of a force change, the
kinetochore velocity will maintain its direction (following its current
metastable branch) until the force is sufficiently large that the system
leaves the bistable area again (the current metastable branch becomes
unstable). Later we will show that this hysteresis of the one-sided model can
explain stochastic chromosome oscillations in metaphase if two one-sided
models are coupled in the full two-sided model.
In the following, we present a Fokker-Planck mean-field approach that lets us
derive bistability analytically and obtain constraints for the occurrence of
bistability. We obtain a system of Fokker-Planck equations (FPEs) for the $M$
MT-kinetochore distances $x_{i}\equiv x_{\mathrm{m,}i}-X_{\mathrm{k}}$
($i=1,...,M$) and decouple the MT dynamics in a mean-field approximation,
which neglects kinetochore velocity fluctuations.
We make two assumptions. First we assume that all $M$ MTs are always attached
to the kinetochore. Because the MT-kinetochore links are catch bonds this
assumption is equivalent to assuming that these links are predominantly under
tension. We will check below by comparison with numerical simulations to what
extent this assumption can be justified. Secondly, we neglect that MTs are
confined by a centrosome. Then, as mentioned above, the only relevant
coordinates are the relative MT-kinetochore distances $x_{i}$, which measure
the extension of the $i$-th linker.
The MTs are coupled because they attach to the same kinetochore: each MT
experiences a force $F_{\mathrm{mk},i}=-cx_{i}$ from the elastic linker to the
kinetochore, which is under tension (compression) for $x_{i}<0$ ($x_{i}>0$);
the kinetochore is subject to the total counter force
$F_{\mathrm{km}}=c\sum_{i}x_{i}$. Therefore, the kinetochore velocity
$v_{\mathrm{k}}$ is a stochastic variable depending on all distances $x_{i}$,
on the one hand, but determines the velocities
$\dot{x}_{i}=v_{\text{m}\pm}(x_{i})-v_{\text{k}}$ of MTs relative to the
kinetochores, on the other hand. The equations can be decoupled to a good
approximation because the one-sided system assumes a steady state with an
approximately stationary kinetochore velocity $v_{\mathrm{k}}$ after a short
time (rather than, for example, a cooperative oscillation as for an MT
ensemble pushing against an elastic barrier [38]). In our mean-field
approximation we then assume a constant kinetochore velocity
$v_{\mathrm{k}}\equiv\langle v_{\mathrm{k}}\rangle$ and neglect all stochastic
fluctuations around this mean. This mean value is determined by the mean
linker extension $\langle x\rangle$ via
$v_{\mathrm{k}}=\frac{1}{\gamma}\left(F_{\mathrm{ext}}+cM\langle
x\rangle\right).$ (4)
Fluctuations around the mean value are caused by fluctuations of the force
$F_{\mathrm{km}}=c\sum_{i}x_{i}$ around its mean $\langle
F_{\mathrm{km}}\rangle=Mc\langle x\rangle$, which become small for large $M$
(following the central limit theorem).
If $v_{\mathrm{k}}$ is no longer a stochastic variable, the dynamics of the
MT-kinetochore extensions $x_{i}$ decouple. Then, the probability distribution
for the $M$ extensions $x_{i}$ factorizes into $M$ identical factors
$p_{\pm}(x_{i},t)$, where $p_{\pm}(x,t)$ are the probabilities to find one
particular MT in the growing ($+$) or shrinking ($-$) state with a MT-
kinetochore linker extensions $x$. We can derive two FPEs for the dynamical
evolution of $p_{\pm}(x,t)$,
$\displaystyle\partial_{t}p_{+}(x,t)$
$\displaystyle=-\omega_{\mathrm{c}}(x)p_{+}(x,t)+\omega_{\mathrm{r}}(x)p_{-}(x,t)-\partial_{x}\big{(}v_{+}(x)p_{+}(x,t)\big{)},$
(5) $\displaystyle\partial_{t}p_{-}(x,t)$
$\displaystyle=\phantom{-}\omega_{\mathrm{c}}(x)p_{+}(x,t)-\omega_{\mathrm{r}}(x)p_{-}(x,t)-\partial_{x}\big{(}v_{-}(x)p_{-}(x,t)\big{)},$
(6)
where $v_{\pm}(x)$ denotes the relative velocity of MT and kinetochore,
$\displaystyle v_{\pm}(x)\equiv
v_{\text{m}\pm}(x)-v_{\text{k}}=v_{\pm}^{0}\exp\left(-\frac{cx}{F_{\pm}}\right)-v_{\text{k}},$
(7)
and
$\omega_{\mathrm{c},r}(x)=\omega_{\mathrm{c},r}^{0}\exp\left(-{cx}/{F_{\mathrm{c},r}}\right)$.
The velocity $v_{\mathrm{k}}$ is no longer stochastic but self-consistently
determined by (4). We note that these FPEs are equivalent to single MT FPEs
with position-dependent velocities, catastrophe and rescue rates [39, 40, 41,
42].
We will now obtain the force-velocity relation of the whole MT ensemble by
first solving the FPEs (5) and (6) in the stationary state
$\partial_{t}p_{\pm}(x,t)=0$ and then calculating the mean linker extension
$\langle x\rangle$ for given kinetochore velocity $v_{\mathrm{k}}$ using the
stationary distribution $p_{\pm}(x)$. The external force that is necessary to
move the kinetochore with velocity $v_{\mathrm{k}}$ then follows from (4),
$F_{\mathrm{ext}}=\gamma v_{\text{k}}-cM\langle x\rangle(v_{\mathrm{k}}).$ (8)
The MT-kinetochore distance $x$ is limited to a maximal or a minimal value
$x_{\mathrm{max}}$ or $x_{\mathrm{min}}$ for a given kinetochore velocity
$v_{\mathrm{k}}>0$ or $<0$, respectively, see table 3. These limiting values
are reached if the relative MT-kinetochore velocities vanish after the linker
extension $x$ has adjusted: First we consider $v_{k}<0$ and a shrinking MT. If
we start with a compressed linker ($x>0$), the MT starts to shrink fast, the
compression is reduced and the linker may get under tension ($x<0$) because
the relative velocity is negative, $\dot{x}=v_{-}(x)<0$. The MT-kinetochore
distance $x$ continues to decrease until $\dot{x}=v_{-}(x_{\mathrm{min}})=0$
in (7), where the shrinking velocity of the MTs is the same as the prescribed
kinetochore velocity ($v_{\mathrm{m},-}=v_{\mathrm{k}}$). Further shrinking to
$x<x_{\mathrm{min}}$ is not possible but distances $x>x_{\mathrm{min}}$ can
always be reached if MTs are rescued. If $v_{k}<0$ and the MT grows, on the
other hand, there is no upper bound on $x$, as the relative velocity
$\dot{x}=v_{+}(x)$ is always positive; $x$ starts to grow into the compressive
regime $x>0$ and continues to grow without upper bound (for very large
compressive linker extensions, MT growth is suppressed, but the kinetochore
still moves such that $v_{+}(x)\approx-v_{\mathrm{k}}>0$). Analogously, if
$v_{k}>0$ and MTs grow, $x$ grows until $\dot{x}=v_{+}(x_{\mathrm{max}})=0$,
and smaller distances can be reached by catastrophe but there is no lower
bound on $x$ for shrinking MTs. Linker extensions $x_{\mathrm{max}}$
($x_{\mathrm{min}}$) are reached as stationary states if catastrophes
(rescues) are suppressed (for example, because of large forces), such that MTs
can grow (shrink) for sufficiently long times. If the external force
$F_{\mathrm{ext}}$ is prescribed rather than a kinetochore velocity, all MTs
reach a stationary state with the same velocity $\tilde{v}_{\pm}$ given by
(8), $F_{\mathrm{ext}}=\gamma\tilde{v}_{\pm}-cMx_{\mathrm{max,min}}$. In this
stationary state, both MT-tips and kinetochore move with the same velocity
$\displaystyle\tilde{v}_{\pm}\equiv\frac{MF_{\pm}}{\gamma}\,W\left(\frac{\gamma
v^{0}_{\pm}}{MF_{\pm}}\exp\left(\frac{F_{\mathrm{ext}}}{MF_{\pm}}\right)\right),$
(9)
where $W()$ denotes the Lambert-W function (defined by $x=W(x)e^{W(x)}$).
Table 3: Maximal or a minimal value $x_{\mathrm{max}}$ or $x_{\mathrm{min}}$
of the stationary linker extension distribution $p(x)$ from conditions
$v_{-}(x_{\mathrm{min}})=0$ and $v_{+}(x_{\mathrm{max}})=0$. The distance
$x_{\mathrm{min}}$ ($x_{\mathrm{max}}$) is a function of the prescribed
kinetochore velocity $v_{\mathrm{k}}$ and has to be specified separately
depending on the direction of $v_{\mathrm{k}}$; $x_{\mathrm{min}}$
($x_{\mathrm{max}}$) is approached if the MTs shrink (grow) for a sufficiently
long time.
* | MT shrinks | MT grows
---|---|---
$v_{\mathrm{k}}>0$ | $v_{-}(x)<-v_{\mathrm{k}}\;{\rm always}$ | $v_{+}(x)>0\;\text{for}\;x<x_{\mathrm{max}}$
| $x_{\mathrm{min}}=-\infty$ | $x_{\mathrm{max}}=({F_{+}}/{c})\ln\left({v^{0}_{+}}/{v_{\mathrm{k}}}\right)$
$v_{\mathrm{k}}<0$ | $v_{-}(x)<0\;\text{for}\;x>x_{\mathrm{min}}$ | $v_{+}(x)>v_{\mathrm{k}}\;{\rm always}$
| $x_{\mathrm{min}}=({F_{-}}/{c})\ln\left({v^{0}_{-}}/{v_{\mathrm{k}}}\right)$ | $x_{\mathrm{max}}=\infty$
$v_{\mathrm{k}}=0$ | $v_{-}(x)<0\;{\rm always}$ | $v_{+}(x)>0\;{\rm always}$
| $x_{\mathrm{min}}=-\infty$ | $x_{\mathrm{max}}=\infty$
In the complete absence of stochastic switching between growth and shrinking
by catastrophes and rescues, the MT ensemble reaches stationary states with
peaked distributions $p_{+}(x)\propto\delta(x_{\mathrm{max}}-x)$ and
$p_{-}(x)\propto\delta(x-x_{\mathrm{min}})$. Stochastic switching shifts and
broadens these peaks, and the FPEs (5) and (6) lead to a distribution
$p_{\pm}(x,t)$ of linker extensions $x$ in the growing and shrinking states
with statistical weight $p_{\pm}(x,t)>0$ in the whole interval
$x_{\mathrm{min}}\leq x\leq x_{\mathrm{max}}$. At the boundaries
$x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ of this interval, the probability
current density
$\displaystyle j(x,t)\equiv v_{+}(x,t)p_{+}(x,t)+v_{-}(x,t)p_{-}(x,t)$ (10)
has to vanish. Furthermore, in any stationary state
($\partial_{t}p_{\pm}(x,t)=0$) the current density is homogeneous, as can be
seen by summing up the FPEs (5) and (6):
$\displaystyle
0=\partial_{x}(v_{+}(x)p_{+}(x)+v_{-}(x)p_{-}(x))=\partial_{x}j(x).$ (11)
Together with $j=0$ at the boundaries this implies that $j=0$ everywhere in a
steady state. The resulting relation $v_{+}(x)p_{+}(x)=-v_{-}(x)p_{-}(x)$ can
be used to reduce the stationary FPEs to a single ordinary differential
equation with the solution [41]
$\displaystyle
p_{\pm}(x)=\frac{\pm\mathcal{N}}{v_{\pm}(x)}\exp\left(-\int\left(\frac{\omega_{\text{c}}(x)}{v_{+}(x)}+\frac{\omega_{\text{r}}(x)}{v_{-}(x)}\right)\mathrm{d}x\right)$
(12)
for the stationary distribution of linker extensions $x$ in the growing and
shrinking states. The normalization constant $\mathcal{N}$ must be chosen so
that the overall probability density $p(x)\equiv p_{+}(x)+p_{-}(x)$ satisfies
$\int_{x_{\mathrm{min}}}^{x_{\mathrm{max}}}p(x)\mathrm{d}x=1$. Obviously,
$p_{\pm}(x)=0$ for $x>x_{\mathrm{max}}$ and $x<x_{\mathrm{min}}$. The
stationary probability densities $p_{\pm}(x)$ from (12) can then be used to
calculate the mean distance $\langle x\rangle$ as a function of the
kinetochore velocity $v_{k}$, which enters through (7) for $v_{\pm}(x)$. The
integral in the exponent in (12) as well as the normalization can be evaluated
numerically to obtain an explicit $\langle x\rangle(v_{\mathrm{k}})$-relation,
which is shown in figure 2(a).
Figure 2: Mean-field results compared to stochastic simulations of the one-
sided model. (a) The master curve $\langle x\rangle(v_{\mathrm{k}})$ from the
mean-field approach (red line) agrees with simulation results for different
MT-numbers $M=5,20,50,200$. The dashed lines mark $x_{\rm
min,max}(v_{\mathrm{k}})$ from table 3. We run simulations with constant
external forces and average over 80 simulations for each force. Initially, the
MT-kinetochore distance is either $x_{\mathrm{min}}$ or $x_{\mathrm{max}}$
while all MTs grow or shrink with velocity $\tilde{v}_{\pm}$, respectively.
The system then enters a (meta-)stable state, in which we measure the mean
kinetochore velocity and MT-kinetochore distances. The marker size depicts the
time the system rests in this state on average, which is a measure for its
stability (maximum marker size corresponds to
$t_{\mathrm{rest}}\geq$1000\text{\,}\mathrm{s}$$). As predicted, the mean-
field approach turns out to be correct in the limit of many MTs, and in this
limit the $\langle x\rangle(v_{\mathrm{k}})$-relation is independent of the
MT-number $M$. (b) Resulting force-velocity relations for different MT-numbers
$M=5,20,50,200$. The dashed lines show the large velocity limit
$v_{\mathrm{k}}\approx\tilde{v}_{\pm}$ given by (9). We used a linker
stiffness of $c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ both in (a) and (b).
At this point it should be noticed that in the mean-field theory the $\langle
x\rangle(v_{\mathrm{k}})$-relation is independent of the MT number $M$.
Therefore, we call it master curve henceforth. In figure 2(a) we compare the
mean-field theory result to stochastic simulations and find that the mean-
field approach becomes exact in the limit of large $M$, where fluctuations in
the kinetochore velocity around its mean in (4) can be neglected.
The master curve is a central result and will be the basis for all further
discussion. Together with the force-balance (8) on the kinetochore, the master
curve will give the force-velocity relation for the MT-kinetochore system. A
positive slope of the master curve, as it can occur for small
$v_{\mathrm{k}}\approx 0$ (see figure 2(a)), gives rise to an instability of
the MT-kinetochore system: Then, a positive kinetochore velocity fluctuation
$\delta v_{\mathrm{k}}>0$ leads to a MT-kinetochore linker compression
$\delta\langle x\rangle>0$. According to the force-balance (8), a compression
$\delta\langle x\rangle>0$ puts additional forward-force on the kinetochore
leading to a positive feedback and further increase $\delta v_{\mathrm{k}}>0$
of the kinetochore velocity. This results in an instability, which will
prevent the system to assume mean linker extensions $\langle x\rangle$ in this
unstable regime. This is confirmed by stochastic simulation results in figure
2(a), which show that the unstable states are only assumed transiently for
very short times. Therefore, occurrence of a positive slope in the master
curve in figure 2(a) is the essential feature that will give rise to
bistability in the one-sided model and, finally, to oscillations in the full
two-sided model.
Now we want to trace the origin of this instability for small
$v_{\mathrm{k}}\approx 0$. If the MTs are growing (shrinking) for a long time,
all linker extensions assume the stationary values $x\approx
x_{\mathrm{max}}(v_{\mathrm{k}})$ ($x\approx
x_{\mathrm{min}}(v_{\mathrm{k}})$) from table 3, where the MT-velocity adjusts
to the kinetochore velocity, $v_{\mathrm{k}}\approx v_{\text{m}\pm}(x)$. If
the kinetochore velocity increases in these states by a fluctuation (i.e.,
$\delta v_{\mathrm{k}}>0$), the MT-kinetochore linkers are stretched (i.e.,
$\delta x<0$), which slows the kinetochore down again resulting in a stable
response. This is reflected in the negative slopes of both $x_{\rm
max}(v_{\mathrm{k}})$ (for $v_{\mathrm{k}}>0$) and $x_{\rm
min}(v_{\mathrm{k}})$ (for $v_{\mathrm{k}}<0$). Because of constant stochastic
switching between catastrophes and rescues the mean linker extension exhibits
fluctuations about $x_{\rm max}$ and $x_{\rm min}$, but we expect also the
master curve $\langle x\rangle(v_{\mathrm{k}})$ to have a negative slope for a
wide range of velocities $v_{\mathrm{k}}$. Figure 2(a) shows that this is
actually the case for kinetochore velocities $v_{\mathrm{k}}$ around the
force-free growth or shrinking velocities $v^{0}_{\pm}$ of the MTs, i.e., if
the imposed kinetochore velocity $v_{\mathrm{k}}$ roughly “matches” the force-
free growing or shrinking MT velocity. Then a small mismatch can be
accommodated by small linker extensions $x$, which do not dramatically
increase fluctuations by triggering catastrophe or rescue events.
The situation changes for small negative or small positive values of the
kinetochore velocity around $v_{\mathrm{k}}\approx 0$. For
$v_{\mathrm{k}}\lesssim 0$, MT-kinetochore linkers develop logarithmically
growing large negative extensions $x_{\rm min}$ (see table 3) corresponding to
a slow kinetochore trailing fast shrinking MTs that strongly stretch the
linker. Likewise, for $v_{\mathrm{k}}\gtrsim 0$, MT-kinetochore linkers
develop logarithmically growing large positive extensions $x_{\rm max}$
corresponding to a slow kinetochore trailing fast growing MTs that strongly
compress the linker. Around $v_{\mathrm{k}}\approx 0$, the system has to
switch from large negative $x$ to large positive $x$ because the resulting
tensile force $F_{\mathrm{mk}}=-cx$ on the shrinking MT will destabilize the
shrinking state and give rise to MT rescue at least for $x<-F_{r}/c$.
Therefore, also the mean value $\langle x\rangle$ switches from negative to
positive values resulting in a positive slope of the master curve if the
stationary distributions $p_{-}(x)$ and $p_{+}(x)$ remain sufficiently peaked
around the linker extensions $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$, also
in the presence of fluctuations by catastrophes and rescues. In the
supplementary material, we show that the stationary distributions assume a
power-law behavior $p_{+}(x)\propto(x_{\mathrm{max}}-x)^{\alpha_{+}}$
[$p_{-}(x)\propto(x-x_{\mathrm{min}})^{\alpha_{-}}$] around $x_{\mathrm{max}}$
[$x_{\mathrm{min}}$] for $v_{\mathrm{k}}>0$ [$v_{\mathrm{k}}<0$] with
exponents $\alpha_{\pm}$ that depend on the MT-kinetochore stiffness $c$ as
$\alpha_{\pm}+1\propto 1/c$ in the presence of fluctuations. It follows that
distributions are peaked (i.e., have a large kurtosis) and bistability emerges
if the MT-kinetochore linker stiffness $c$ is sufficiently large such that
deviations of the MT velocity from the kinetochore velocity become suppressed
by strong spring forces. This is one of our main results. We also find that
$\alpha_{\pm}+1\propto(|v_{\mathrm{k}}/v_{\pm}^{0}|)^{-1-|F_{\pm}/F_{\mathrm{c,r}}|}$
such that the distributions become also peaked around $x_{\mathrm{min,max}}$
in the limit of large velocities $|v_{\mathrm{k}}|$. Then the velocity
approaches $v_{\mathrm{k}}\approx\tilde{v}_{\pm}(F_{\mathrm{ext}})$ for a
prescribed external force such that $\tilde{v}_{\pm}$ from (9) represents the
large velocity and large force limit of the force-velocity relation of the
kinetochore (see figure 2(b)).
In the unstable regime around $v_{\mathrm{k}}\approx 0$, the linker length
distribution $p(x)$ is typically broad without pronounced peaks and has a
minimal kurtosis (as a function of $v_{\mathrm{k}}$) in the presence of
catastrophe and rescue fluctuations. In this regime the system assumes a state
with a heterogeneous stationary distribution of growing and shrinking MTs,
i.e., the total probabilities to grow or shrink become comparable, $\int
p_{+}(x)\mathrm{d}x\sim\int p_{-}(x)\mathrm{d}x$. If the kinetochore velocity
is increased, $\delta v_{\mathrm{k}}>0$, the system does not react by $\delta
x<0$, i.e., by increasing the average tension in the linkers in order to pull
MTs forward, but by switching MTs from the shrinking to the growing state (on
average), which then even allows to relax the average linker tension.
Using the force-balance (8) on the kinetochore, the master curve is converted
to a force-velocity relation for the MT-kinetochore system; the results are
shown in figure 2(b). The bistability in the master curve directly translates
to a bistability in the force-velocity relation of the MT ensemble, and we
obtain a regime with three branches of possible velocities for the same
external force. The upper and the lower branches agree with our simulation
results and previous simulation results in [20], and our mean-field results
become exact in the limit of large $M$, see figure 2(b). These branches
correspond to the two stable parts of the master curve with negative slope,
that are found for kinetochore velocities $v_{\mathrm{k}}$ roughly matching
the force-free growth or shrinking velocities $v^{0}_{\pm}$ of the MTs. The
mid branch corresponds to the part of the master curve with positive slope,
where the system is unstable. Also figure 2(b) demonstrates that this
instability is confirmed by stochastic simulations results.
Finally, we note that a simpler theoretical approach, where it is assumed that
all linkers assume identical extensions $x_{i}\approx x$ and all attached MTs
are in the same state (growing or shrinking), is exact for a single MT ($M=1$)
by definition but not sufficient to obtain a bistable force-velocity relation
for MT ensembles ($M>1$) (see supplementary material). The same assumption of
identical MT positions has already been used to study an ensemble of MTs that
are connected to the same kinetochore via Hill sleeve like linkers [17, 29].
The model of Klemm et al. [21] divides each MT ensemble into a growing and a
shrinking sub-ensemble, and assumes equal load sharing only between MTs within
each sub-ensemble. We can show that, together with a force-sensitive rescue
force, this is sufficient to obtain a bistable force-velocity relation in a
corresponding one-sided model.
## 4 Bistability gives rise to oscillations in the two-sided model
As already worked out by Banigan et al. [20], the bistability in the force-
velocity relation of the one-sided MT ensemble can be considered to be the
cause for stochastic oscillations in the two-sided model. Each ensemble can be
either on the lower branch of the force-velocity relation, where it mainly
depolymerizes and exerts a P-directed pulling force ($v_{\mathrm{k}}<0$) or on
the upper branch where it mainly polymerizes and exerts an AP-directed pushing
force ($v_{\mathrm{k}}>0$). The external force in the one-sided model is a
substitute for the spring force
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
of the cohesin bond in the full model with a stiffness $c_{\mathrm{k}}$ and
rest length $d_{0}$, see table 2. Since the cohesin force is a linear function
of the inter-kinetochore distance, the force-velocity relation can be treated
as distance-velocity (phase space) diagram for the two kinetochores (see
figure 3(a)), where both kinetochores move as points on the force-velocity
relation. The cohesin bond always affects the two kinetochores in the same way
because action equals reaction: if the cohesin spring is stretched, both
kinetochores are pulled away from their pole (AP), if it is compressed, both
kinetochores are pushed polewards (P). Thus, the kinetochores always have the
same position on the $F_{\mathrm{kk}}$-axis in the
$F_{\mathrm{kk}}$-$v_{\mathrm{k}}$ diagram in figure 3(a), if
$F_{\mathrm{kk}}$ on the horizontal axis is defined as the force on the
kinetochore in AP-direction (i.e., $F_{\mathrm{kk,l}}\equiv F_{\mathrm{kk}}$
and $F_{\mathrm{kk,r}}\equiv-F_{\mathrm{kk}}$ for the left/right kinetochore).
Likewise, we define $v_{\mathrm{k}}$ on the vertical axis as the velocity in
AP-direction (i.e., $v_{\mathrm{k,l}}\equiv\dot{X}_{\mathrm{k,l}}$ and
$v_{\mathrm{k,r}}\equiv-\dot{X}_{\mathrm{k,r}}$ for the left/right
kinetochore). The upper/lower stable branch of the force-velocity relation is
denoted by $v^{\pm}_{\mathrm{k}}(F_{\mathrm{kk}})$. Typically, a kinetochore
on the upper (lower) branch has $v^{+}_{\mathrm{k}}>0$
($v^{-}_{\mathrm{k}}<0$) and, thus moves in AP-(P-)direction. Using
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
for the spring force, we find
$\dot{F}_{\mathrm{kk}}=-c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)$,
i.e., kinetochores move with the sum of their AP-velocities along the force-
velocity curve in the $F_{\mathrm{kk}}$-$v_{\mathrm{k}}$ diagram.
Figure 3: Bistability gives rise to oscillations in the two-sided model.
(a,b) Different states of sister kinetochore motion can be deduced from the
bistability of the force-velocity relation: either both kinetochores are in
the upper branch (0) or one is in the upper and the other one in the lower
branch (2, $2^{\prime}$). In the first case, both kinetochores move away from
their pole (AP) towards each other. Thus, the spring force $F_{\text{kk}}$
decreases until it reaches $F_{\mathrm{min}}$. Since the upper branch is not
stable anymore below $F_{\mathrm{min}}$, either the left (1) or the right
($1^{\prime}$) kinetochore switches to the lower branch and changes direction
to poleward movement (P). The system is then in state 2 or $2^{\prime}$, where
both kinetochores move into the same direction: the leading kinetochore P, the
trailing kinetochore AP. As P- is much faster than AP-movement (MT shrinking
is much faster than growth), the inter-kinetochore distance and the spring
force are increasing. Above $F_{\mathrm{max}}$ only AP-movement is stable,
which is why the leading kinetochore changes direction (3, $3^{\prime}$) and
the system switches to state 0 again. (c) Solution of the equations of motion
(19) for $c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ and $M=25$ with an imposed periodic order of states
($0-2-0-2^{\prime}-0-...$). The initial condition is
$F_{\text{kk}}=F_{\mathrm{max}}$ (both kinetochores at the right end of the
upper branch). For an animated version see video S1 in the supplementary
material.
Oscillations arise from the two kinetochores moving through the hysteresis
loop of the bistable force-velocity relation as described in figure 3(a).
Three states are possible (see figure 3(b)). In state $0$, both kinetochores
move in AP-direction (i.e., in opposite directions) relaxing the
$F_{\mathrm{kk}}$-force from the cohesin bond, i.e., on the upper branch and
to the left in the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram with velocity
$\dot{F}_{\mathrm{kk}}=-2c_{\text{k}}v^{+}_{\mathrm{k}}<0$. After reaching the
lower critical force $F_{\mathrm{min}}$ of the hysteresis loop, one of the two
kinetochores reverses its direction and switches to the lower branch resulting
into states $2$ or $2^{\prime}$ where one kinetochore continues in AP-
direction with $v^{+}_{\mathrm{k}}>0$ while the other is moving in P-direction
with $v^{-}_{\mathrm{k}}<0$ (i.e., both move in the same direction). In the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram, this results in a motion to the
right with velocity
$\dot{F}_{\mathrm{kk}}=c_{\text{k}}(-v^{-}_{\mathrm{k}}-v^{+}_{\mathrm{k}})>0$
because MTs typically shrink much faster than they grow ($-v^{0}_{-}\gg
v^{0}_{+}$, see table 2). Moving on opposite P- and AP-branches increases the
kinetochore distance and builds up $F_{\mathrm{kk}}$-force in the cohesin
bond. After reaching the upper critical force $F_{\mathrm{max}}$ of the
hysteresis loop, it is always the kinetochore on the lower branch moving in
P-direction which switches back and state $0$ is reached again. This behavior
is in agreement with experimental results [11]. The system oscillates by
alternating between state $0$ and one of the states $2$ or $2^{\prime}$ (which
is selected randomly with equal probability).
For each of the states 0, 2 and $2^{\prime}$ depicted in figure 3(ab) the two
branches $v^{\pm}_{\mathrm{k}}=v^{\pm}_{\mathrm{k}}[F_{\mathrm{kk}}]$ provide
deterministic equations of motion for the kinetochore positions. Inserting
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
we obtain both kinetochore velocities as functions of the kinetochore
positions and find
$\displaystyle\eqalign{\text{state
0:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
\text{state
2:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{-}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
\text{state
$2^{\prime}$:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{-}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0.}$
(19)
Solving these equations gives idealized deterministic trajectories of the
sister kinetochores, when we also assume that the left and the right
kinetochore pass the lower branch alternately such that the order of states is
a periodic sequence $0-2-0-2^{\prime}-0-...$ as shown in the example in figure
3(c). Then single kinetochores oscillate with half the frequency of inter-
kinetochore (breathing) oscillations, just as observed in PtK1 cells [11].
Moreover, we can obtain numerical values of the frequencies directly from the
trajectories. For a MT-kinetochore linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
20–25 MTs per kinetochore, which is a realistic number for mammalian cells
[43], we get periods of 206–$258\text{\,}\mathrm{s}$ and
103–$129\text{\,}\mathrm{s}$ for kinetochore and breathing oscillations,
respectively. These values coincide with experimental results of
$239\text{\,}\mathrm{s}$ and $121\text{\,}\mathrm{s}$ measured in PtK1 cells
[11].
The calculated trajectories are idealized since they neglect stochastic
fluctuations that occur in simulations of the two-sided model and have two
main effects on the kinetochore dynamics which already arise in simulations
that comply with the assumptions behind the mean-field theory (no confinement
($x_{\mathrm{c}}\to\infty$) and permanent bonds ($\omega_{\mathrm{d}}=0$)):
Firstly, the sister kinetochores do not pass the lower branch alternately but
in random order. Therefore, we observe phases where one kinetochore moves in
AP-direction for several periods, while the other one changes its direction
periodically but moves polewards on average (figure 4(a)). Since this does not
influence the trajectory of the inter-kinetochore distance, breathing
oscillations still occur in a more or less regular manner, which allows us to
measure their frequencies by Fourier analysis. We will show below that
additional polar ejection forces suppress this random behavior and force the
kinetochores to pass the lower branch alternately. As a second effect of the
stochastic character of the simulation, kinetochores do not change the branch
instantaneously after crossing the critical forces $F_{\mathrm{max}}$ or
$F_{\mathrm{min}}$. Instead, they tend to maintain their primary state for a
while (figure 4(b)) and follow the metastable states that we also observe in
the one-sided model (figure 2(b)). Hence, the frequencies we measure in the
simulations are smaller than those we calculate from the Fokker-Planck mean-
field approach (figure 4(c)). The latter effect vanishes in the limit of many
MTs (large $M$): the switching points approach the theoretical values
$F_{\mathrm{max}}$ and $F_{\mathrm{min}}$, and the simulated breathing
frequencies converge to our mean-field predictions.
Figure 4: Oscillations in stochastic simulations of the unconfined model
compared to mean-field results. (a) Kinetochore trajectories and breathing
oscillations in the two-sided model without confinement
($x_{\mathrm{c}}\to\infty$) and detachment ($\omega_{\mathrm{d}}=0$). The
kinetochores behave as described in figure 3 with a random order of states
$2/2^{\prime}$. The breathing oscillations are regular enough to assign a
frequency by Fourier analysis, see (d). With less MTs oscillations are more
fluctuative. (b) Kinetochore velocity against cohesin force in simulations of
the unconfined two-sided model without detachment (green). For many MTs the
velocity follows very precisely the predicted hysteresis from the mean-field
approach (red). For animated versions see videos S2 ($M=25$) and S3 ($M=500$)
in the supplementary material. (c) Double-logarithmic plot of frequencies of
breathing oscillations as a function of MT number $M$: calculated from the
mean-field approach according to figure 3 (red) and measured in simulations of
the unconfined (green diamonds) as well as the confined model with detachable
catch bonds (blue circles) and with permanent attachment (orange triangles).
Confinement becomes relevant for large MT numbers. In the presence of
detachable catch bonds only $75\text{\,}\mathrm{\char 37\relax}$ of the MTs
are attached on average, which corresponds to a simple shift of the curve to
lower MT numbers. (d) Trajectories from (a) in Fourier space. While
$\tilde{X}_{\mathrm{k,r}}$ has its maximum at $f=0$ due to the random order of
states in figure 3, $\Delta\tilde{X}_{\mathrm{k}}$ has a distinct peak that
becomes sharper for large $M$ indicating regular breathing oscillations. For
all simulations the MT-kinetochore linker stiffness was
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.
So far we have demonstrated that the mean field theory correctly describes
kinetochore dynamics in simulations of the unconfined model where we suppress
detachment in order to prevent unattached MTs from shrinking towards infinity.
As shown in figure 5(ab), kinetochore oscillations also survive in simulations
of the confined model independently of whether the MTs are able to detach from
the kinetochore, i.e., to rupture the catch bond. However, confinement by the
centrosome influences the kinetochore dynamics in the limit of large $M$:
since more MTs exert a higher force on the kinetochore, it is possible that
one of the two sisters gets stuck at the centrosome for a while (see figure
5(ab)). Hence, the frequencies measured in the confined two-sided model
deviate from the frequencies in the unconfined case above $M\approx 200$.
Figure 5: Dynamics in the confined model with detachable MTs. (a) Kinetochore
positions $X_{\mathrm{k}}$ and inter-kinetochore distance $\Delta
X_{\mathrm{k}}$ over time in simulations with a total number of $M=25$ and
$M=100$ MTs per spindle pole. Oscillations as described in figure 3 are
recognizable. With 100 MTs one kinetochore can get stuck to the centrosome for
a while. (b) Distribution of kinetochore positions. The kinetochores are not
aligned to the spindle equator and for $M=100$ they are most likely to be
found near the centrosomes. (c) Number of attached MTs $M^{\mathrm{att}}$ over
time. MTs are more likely to be attached when the correspondent kinetochore is
near the centrosome since the free MTs can reattach to the kinetochore faster
in that case. (d) Distribution of $M^{\mathrm{att}}$. On average
$75\text{\,}\mathrm{\char 37\relax}$ of the MTs are attached independently of
the total MT number $M$.
If we enable detachment in our simulations we find that the number of attached
MTs correlates with the kinetochore position (see figure 5(c)) since due to
the exponential distribution of free MTs and the distance dependent attachment
rate (2) detached MTs are more likely to reattach to the kinetochore the
closer it is to the centrosome. Moreover, on average, about
$75\text{\,}\mathrm{\char 37\relax}$ of the MTs are attached independently of
the total MT number (see figure 5(cd)). Therefore, the catch bond nature of
the link leads to an effective behavior similar to a system without detachment
but with less MTs, which explains the difference in frequencies between the
confined models with and without detachment in figure 4(c). We conclude that
detachment does not play a major role for the occurrence of kinetochore
oscillations in cells with many MTs as despite detachment there are always
enough MTs attached to justify our mean-field approximation. Hence, (periodic)
changes in the number of attached MTs as they can be seen in figure 5(c) are
rather a passive consequence than an active source of kinetochore
oscillations. This argumentation may not be tenable, if just a few MTs are
attached to a kinetochore, so that even detachment of a single MT effects the
total force acting on the kinetochore significantly. Then, detachment can be
the primary cause of directional instability as worked out by Gay et al. [44],
who modeled the mitotic spindle of fission yeast.
Taking into account the results of the last paragraph, we will mainly
investigate the unconfined model with permanently attached MTs in the
following sections. This procedure is reasonable as we do not lose any
qualitative key features of kinetochore dynamics on the one hand, and, on the
other hand, gain a much better comparability of our mean field theory with the
appropriate stochastic simulations.
We finally note that in all cases we examined (confined / unconfined system,
permanent / detachable bonds) the kinetochore oscillations become more
fluctuative if less MTs are attached. This leads to the conclusion that
kinetochore oscillations are a result of the collective dynamics of an
ensemble of MTs that exhibit a force-dependent dynamic instability
individually. Such a behavior can not be described correctly based on the
simple assumption that all linkers have the same extension, i.e., that MTs
share the load equally and all attached MTs are in the same state (growing or
shrinking), (see supplementary material). Therefore, the model of Shtylla and
Keener [17] which does assume equal load sharing and synchronous MT dynamics
requires a chemical feedback as an additional mechanism in order to obtain
kinetochore oscillations. The model of Klemm et al. [21] divides each MT
ensemble into a growing and a shrinking sub-ensemble, and assumes equal load
sharing only between MTs within each sub-ensemble. Together with a force-
sensitive rescue force, this is sufficient to obtain oscillations.
## 5 Constraints on linker stiffness and MT number for bistability and
oscillations
### 5.1 Constraints for bistability in the one-sided model
We already argued above in Sec. 3 that bistability (and thus oscillations) can
only emerge if the MT-kinetochore linker is sufficiently stiff. To analyze the
influence of the linker stiffness $c$ and the MT number $M$ on bistability
quantitatively, the transformation from the master curve to the force-velocity
relation is visualized in figure 6(a) as search for the intersections of the
master curve with linear functions
$\displaystyle\langle x\rangle=\frac{1}{cM}(\gamma
v_{\mathrm{k}}-F_{\mathrm{ext}}).$ (20)
In the limit of large $M$ these linear functions have zero slope. Bistable
force-velocity relations with three intersection points are only possible if
the master curve has positive slope for intermediate $v_{\mathrm{k}}$
resulting in a maximum and minimum. The extrema of the master curve vanish,
however, in a saddle-node bifurcation if the linker stiffness drops below
$c_{\mathrm{bist}}=$7.737\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, which is, therefore, a lower bound for the occurrence of
bistability. In the case of finite MT numbers $M$, bistable force-velocity
relations can only be found if the slope in the inflection point of the master
curve exceeds $\gamma/cM$ (the slope of the linear function (20)). This allows
us to quantify a bistable regime in the parameter plane of linker stiffness
$c$ and MT number $M$ as shown in figure 6(b).
Figure 6: Constraints for bistability in the one-sided model. (a) Master
curves for different linker stiffnesses $c$ and linear functions according to
(20). In the limit of large $M$ the linear functions have zero slope and
bistability occurs if the master curve has two extrema, which is the case for
$c>c_{\mathrm{bist}}$. For finite $M$ bistable solutions are possible if the
linear functions have a smaller slope than the inflection point of the master
curve. (b) Resulting bistable regime in the parameter plane of linker
stiffness $c$ and MT number $M$.
### 5.2 Constraints for oscillations in the two-sided model
We showed in Sec. 4 that bistability of the one-sided model is a necessary
condition for oscillations in the two-sided model. Now we show that
bistability in the one-sided model is, however, not sufficient for
oscillations in the full model. If the force-velocity relation is interpreted
as phase space diagram for the two kinetochores, kinetochores only switch
branches in the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram if their velocity
changes its sign at the turning points $F_{\mathrm{min}}$ and
$F_{\mathrm{max}}$. If this is not the case and one of the two branches
crosses $v_{\mathrm{k}}=0$ (e.g. the right branch for
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ in
figure 6(a), which transforms to the upper branch of the force-velocity
relation), the intersection point is a stable fixed point in the phase space
diagram (see figure 7(a)). At this fixed point kinetochore motion will relax
to zero velocity and just exhibit fluctuations around an equilibrium distance
instead of oscillations.
Figure 7: Kinetochore dynamics in the non-oscillatory regime. (a) Schematic
explanation of kinetochore motion in the non-oscillatory regime based on the
force-velocity relation. Where the upper branch crosses zero velocity, inter-
kinetochore distance has a fixed point, around which it fluctuates. With
higher linker stiffnesses $c$ the fixed point comes closer to the left turning
point $F_{\mathrm{min}}$. When $c$ is just slightly smaller than
$c_{\mathrm{osc}}$, fluctuations can be large enough for the kinetochore
distance to leave the upper stable branch. Then, one of the two sister
kinetochores passes once through the lower branch. (b,c) This behavior can be
observed in simulations. While at
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
kinetochores just fluctuate around the fixed point, at
$c=$14\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ the
kinetochores occasionally pass through the hysteresis loop. Simulations were
performed with an unconfined system and 100 MTs on each side.
As a sufficient condition for oscillations we have to require – besides
bistability – a strictly positive velocity in the upper and a strictly
negative velocity in the lower branch in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram. Based on this condition we
quantify an oscillatory regime in the parameter plane of linker stiffness $c$
and MT number $M$ in figure 8(a). In the limit of many MTs the sufficient
condition for oscillations can be formulated in terms of the master curve: the
maximum of the master curve has to be located at a positive and the minimum at
a negative velocity. This is the case for
$c>c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, which is, therefore, a lower bound for the occurrence of
oscillations. This constraint on the linker stiffness for metaphase chromosome
oscillations provides additional information on MT-kinetochore linkers whose
molecular nature is not known up to now.
Figure 8: Constraints for oscillations in the two-sided model. (a)
Oscillatory regime in the parameter plane of linker stiffness $c$ and MT
number $M$. (b) Mean inter-kinetochore distance according to (22) (red) and
measured in simulations (blue) with $M=100$. Below
$c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ (dashed line) both results match, whereas in the oscillatory regime
mean inter-kinetochore distance diverges from the fixed point, and its
standard deviation increases notably.
Because of stochastic fluctuations, the transition between oscillatory and
non-oscillatory regime is not sharp in our simulations. In the non-oscillatory
regime kinetochores fluctuate around a fixed point of inter-kinetochore
distance, where the upper branch crosses $v_{\mathrm{k}}=0$. However, these
fluctuations can be large enough for the inter-kinetochore distance to shrink
and leave the upper branch on the left side, especially for stiffnesses $c$
slightly below $c_{\mathrm{osc}}$. If that happens, one kinetochore passes
once through the lower branch of the force-velocity relation just as in an
oscillation. The difference to genuine oscillations is that these are randomly
occurring single events (resulting in a Poisson process). Randomly occurring
oscillations are visualized in figure 7 for $c<c_{\mathrm{osc}}$ and
$c\lesssim c_{\mathrm{osc}}$. Moreover, the force-velocity relations as well
as the kinetochore trajectories measured in corresponding simulations are
shown.
In the non-oscillatory regime, the fixed point should determine the mean
inter-kinetochore distance $\langle\Delta X_{\mathrm{k}}\rangle=\langle
X_{\text{k,r}}-X_{\text{k,l}}\rangle$. Solving the FPEs for $v_{\text{k}}=0$,
we compute the (external) force $F_{0}$ that has to be applied to one
kinetochore to stall its motion:
$\displaystyle F_{0}=\gamma v_{\text{k}}-cM\langle x\rangle=-cM\langle
x\rangle(v_{\text{k}}=0).$ (21)
In the two-sided model this force is applied to the kinetochores by the
cohesin bond at the fixed point. With $F_{\text{kk}}=c_{\text{k}}(\Delta
X_{\mathrm{k}}-d_{0})$ we compute the corresponding mean inter-kinetochore
distance:
$\displaystyle\langle\Delta
X_{\mathrm{k}}\rangle=\frac{F_{0}}{c_{\text{k}}}+d_{0}=-\frac{cM}{c_{\text{k}}}\langle
x\rangle(v_{\text{k}}=0)+d_{0}.$ (22)
Figure 8(b) shows that simulations agree with this result in the non-
oscillatory regime. At $c_{\mathrm{osc}}$ the transition to the oscillatory
regime can be recognized, where the mean inter-kinetochore distance deviates
from the fixed point (22). Moreover, the variance of $\Delta X_{\mathrm{k}}$
increases significantly at $c_{\mathrm{osc}}$ due to the transition to the
oscillatory regime.
In order to provide an overview and to make orientation easier for the reader,
we summarize in figure 9 where the stochastic simulations from the last three
sections and the master curves in figure 6(a) are located in the parameter
plane of linker stiffness $c$ and MT number $M$, and which regime they are
part of.
Figure 9: Locations in $c$-$M$-parameter plane of the master curves from
figure 6(a) and the simulations from figures 2, 4, 5, 7 and 8.
## 6 Poleward microtubule flux suppresses oscillations
An effect we have not included so far is poleward MT flux, which was observed
in several metazoan cells (table 4). It describes the constant flux of tubulin
from the plus-ends towards the spindle pole and is probably driven by plus-end
directed kinesin-5 motors pushing overlapping antiparallel MTs apart as well
as kinesin-13 proteins that are located at the centrosome and depolymerize the
MTs at their minus-ends [24]. During metaphase, spindle and MT length can be
maintained by simultaneous polymerization at the plus-ends [45], which results
in a behavior similar to treadmilling of MTs [46].
Table 4: Metaphase poleward flux velocities $v_{\mathrm{f}}$ and occurrence
of directional instability. For a more detailed review of poleward flux
measurements see [45]
* Cell type | $v_{\mathrm{f}}($\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$) | Directional instability
---|---|---
LLC-PK1 (porcine) | $8.3$ [8] | yes [8]
PtK1 (rat kangaroo) | $7.7$ [47] | yes [11]
PtK2 (rat kangaroo) | $10$ [8] | yes [12]
Newt lung | $9.0$ [48] | yes [6]
U2OS (human) | $8.8$ [9] | yes [9]
Drosophila embryo | $32$ [49] | no [13]
Xenopus egg | $37$ [50] | no [14]
Poleward flux can be easily included in our model by subtracting a constant
flux velocity $v_{\mathrm{f}}$ from the MT velocity. Then, the relative MT-
kinetochore velocities (7) become
$\displaystyle
v_{\pm}(x)=v_{\pm}^{0}\exp\left(-\frac{cx}{F_{\pm}}\right)-v_{\mathrm{f}}-v_{\mathrm{k}}.$
(23)
Hence, the flux velocity can be treated as an offset to the constant
kinetochore velocity in the solution of the stationary FPEs. The final effect
is a shift of both the master curves and the force-velocity relations by
$v_{\mathrm{f}}$ towards smaller kinetochore velocities $v_{\mathrm{k}}$ as
shown in figure 10(a). If the shift is so large that the left turning point
$F_{\mathrm{min}}$ of the force-velocity hysteresis is located at a negative
velocity, poleward flux suppresses directional instability because a fixed
point emerges, and we expect similar behavior as for intermediate linker
stiffnesses in the previous section (see figure 7). In the limit of many MTs,
the maximum flux velocity that still allows directional instability is given
by the velocity in the maximum of the master curve, which provides the
boundary of the oscillatory regime in the parameter plane of linker stiffness
$c$ and poleward flux velocity $v_{\mathrm{f}}$ (figure 10(b)). Phase space
diagrams (figure 10(c)) and kinetochore trajectories (figure 10(d)) from
simulations with appropriate flux velocities confirm our arguments exhibiting
similar behavior as for intermediate linker stiffnesses in figure 7. For small
flux velocities the boundary of the oscillatory regime in figure 10(b)
approaches our above result
$c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. For increasing flux velocities the oscillatory regime shrinks, and
its boundary has a maximum at
$c\approx$50\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ with
$v_{\mathrm{f}}\approx$3.11\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$.
We conclude that kinetochore oscillations can be suppressed by moderate flux
velocities independently of the linker stiffness.
Figure 10: Poleward flux suppresses oscillations. (a) Due to (23), the force-
velocity relation is shifted by the amount of the flux velocity
$v_{\mathrm{f}}$ towards smaller kinetochore velocities. If the flux is slower
than the kinetochore velocity $v_{\mathrm{min}}$ in the left turning point
$F_{\mathrm{min}}$, the kinetochores still oscillate. For larger flux
velocities, a fixed point arises on the upper branch and the kinetochores
behave as described in figure 7. (b) Oscillatory regime in the parameter plane
of $c$ and $v_{\mathrm{f}}$ in the limit of many MTs. Fast poleward flux
suppresses kinetochore oscillations for arbitrary linker stiffnesses $c$.
(b,c) Phase space diagrams and MT trajectories from simulations of the
unconfined two-sided model with
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=100$. While at
$v_{\mathrm{f}}=$2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$ the system
is still in the oscillatory regime, where hysteresis is recognizable in phase
space, at $v_{\mathrm{f}}=$4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$
kinetochores show fluctuative motion as described in figure 7.
Our theory also agrees with and explains simulation results in [20], where,
for large flux velocities, suppression of kinetochore oscillations were
observed but at the same time maintenance of bistability. Moreover, our
results explain the experimentally observed correlation between flux velocity
and directional instability. Kinetochore oscillations have been observed in
the mitotic vertebrate cells listed in table 4 (LLC-PK1, PtK1/2, newt lung,
U2OS) which have poleward flux velocities not exceeding
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$, whereas in the mitosis of
a Drosophila embryo as well as in meiosis of a Xenopus egg, where flux
velocities are three to four times higher, chromosomes do not exhibit
directional instability.
## 7 Polar ejection forces provide an alternating oscillation pattern and
chromosome alignment at the spindle equator
So far, we have not included polar ejection forces (PEFs). They originate from
non-kinetochore MTs interacting with the chromosome arms and pushing them
thereby towards the spindle equator, either through collisions with the
chromosome arms or via chromokinesins [27], and provide additional pushing
forces on kinetochores. Therefore, they can be included into the model by
adding forces $F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}})$ and
$F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}})$ acting on kinetochores, which depend on
the absolute position of the kinetochores [19]. Due to the exponential length
distribution of free MTs as well as the spherical geometry of the MT asters,
the density of non-kinetochore MTs decreases monotonically with the distance
from the spindle pole. Therefore, we assume that PEFs reach their maximum at
the centrosome and vanish at the spindle equator (located at $x=0$), where
opposite PEFs compensate each other. This assumption is supported by the
monotonic PEF distribution that has been measured in vivo by Ke et al. [51].
Here, we will only discuss linearized PEFs
$\displaystyle F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}})=-kX_{\mathrm{k,l}},\qquad
F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}})$ $\displaystyle=kX_{\mathrm{k,r}},$ (24)
where the spring constant $k$ defines the strength of the forces, and the
signs are chosen so that a positive force acts in AP-direction. We show in
figure S3 in the supplementary material that other force distributions do not
differ qualitatively in their influence on the kinetochore dynamics.
To determine kinetochore trajectories of the two-sided model in the presence
of PEFs, we can start from the same force-velocity relations as for the basic
one-sided model. In the presence of PEFs, the total forces $F_{\mathrm{k,l}}$
and $F_{\mathrm{k,r}}$ that act on the left and the right kinetochore in AP-
direction depend on the absolute kinetochore positions $X_{\mathrm{k,l}}$ and
$X_{\mathrm{k,r}}$:
$\displaystyle F_{\mathrm{k,l}}$ $\displaystyle=F_{\mathrm{kk}}(\Delta
X_{\mathrm{k}})+F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}}),$ (25) $\displaystyle
F_{\mathrm{k,r}}$ $\displaystyle=F_{\mathrm{kk}}(\Delta
X_{\mathrm{k}})+F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}}).$ (26)
We can investigate the motion of kinetochores in the full two-sided model
again by using a phase space diagram; in the presence of PEFs we use a
$v_{\mathrm{k}}$-$F_{\mathrm{k}}$-diagram with the total force
$F_{\mathrm{k}}$ in AP-direction on the horizontal axis and the velocity
$v_{\mathrm{k}}$ in AP-direction on the vertical axis. Because the total
forces contain the external PEFs they are no longer related by action and
reaction and, thus, the two kinetochores no longer have the same position on
the $F_{\mathrm{k}}$-axis, but they still remain close to each other on the
$F_{\mathrm{k}}$-axis as long as the cohesin bond is strong enough.
A kinetochore on the upper/lower branch moves in AP-/P-direction with
$v^{\pm}_{\mathrm{k}}(F_{\mathrm{k}})$ if $v^{+}_{\mathrm{k}}>0$
($v^{-}_{\mathrm{k}}<0$). A kinetochore on the upper AP-directed branch will
relax its AP-directed PEFs, while a kinetochore on the lower P-directed branch
will build up AP-directed PEFs. After a time of equilibration the kinetochores
behave as described in figure 11. When one kinetochore changes its direction
from P to AP (switches to the upper branch) the sister kinetochore, which was
on the upper branch before, becomes the leading kinetochore (here, “leading”
refers to the position in the force velocity phase space). Therefore, the
kinetochores do not reach the left turning point $F_{\mathrm{min}}$ at the
same time so that it is always the leading kinetochore that switches to the
lower branch. Since in general the absolute P-velocity is much larger than the
AP-velocity ($-v_{-}$ for the lower branch is much larger than $+v_{+}$ for
the upper branch), the AP-directed PEF contribution to the total force
increases faster on the lower branch than on the upper one. As a result, the
P-moving kinetochore overtakes its sister on the $F_{\mathrm{k}}$-axis before
switching back to the upper branch such that the leading kinetochore
automatically becomes the trailing kinetochore in the next oscillation period
(again, “leading” and “trailing” in terms of phase space positions). This
periodic change of kinetochore positions in the force-velocity diagram leads
to both regular breathing and regular single kinetochore oscillations, as the
kinetochores alternately pass the lower branch. Solving appropriate equations
of motions similar to (19) for each of the states depicted in figure 11(ab),
we determine the deterministic trajectories in figure 11(c) confirming this
regular alternating oscillation pattern.
Figure 11: Kinetochore motion in the presence of PEFs. (a,b) At the beginning
of state 1 the left kinetochore (green) has just switched from P- to AP-
movement, so that both kinetochores are on the upper branch. Both kinetochores
move in AP-direction, which means that both the cohesin force and the PEFs
decrease and both kinetochores move left in the force-velocity diagram. Due to
different PEFs, the right kinetochore (red) reaches the left turning point
$F_{\mathrm{min}}$ first and switches to the lower branch, which marks the
start of state 2. This state is dominated by the fast P-movement of the right
kinetochore, which causes a steep increase of both $F_{\mathrm{kk}}$ and
$F_{\mathrm{PEF,r}}$. Therefore, the right kinetochore moves to the right in
the force-velocity diagram. Meanwhile, the left sister still moves in AP-
direction and $F_{\mathrm{k,l}}$ increases slightly as the increase of
$F_{\mathrm{kk}}$ is larger than the decrease of $F_{\mathrm{PEF,l}}$. Since
$\dot{F}_{\mathrm{k,r}}>\dot{F}_{\mathrm{k,l}}$, the right kinetochore
overtakes its sister on the $F_{\mathrm{k}}$-axis before it reaches the right
turning point and switches to the upper branch. The then following states
$1^{\prime}$ and $2^{\prime}$ are the exact opposite to 1 and 2 with swapped
kinetochores. (c) Solution of the corresponding equations of motion for
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=25$. For an animated version see video S4 in the supplementary material.
The alternating oscillation pattern robustly survives in stochastic
simulations in the presence of moderate PEFs
($k\sim$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
as we demonstrate in figure 12(a) by means of the kinetochore trajectories in
real space. In figure 12(b), emergence of regular oscillations is illustrated
in Fourier space: Whereas for rather small values of $k$ single kinetochore
oscillations are still irregular resulting in a nearly monotonic decreasing
Fourier transform, for
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
single kinetochore motion has a distinct peak in the Fourier space indicating
a regular shape of oscillations in real space. Moreover, frequency doubling of
breathing compared to single kinetochore oscillations can directly be
recognized by comparing the corresponding Fourier transforms. As a consequence
of regular oscillations, the kinetochores stay near the spindle equator and
can not get stuck to one of the centrosomes as in the basic model, see
histograms of kinetochore positions in figure 12(c). We conclude that PEFs are
necessary to assure proper chromosome alignment in the metaphase plate at the
spindle equator. This is consistent with an experiment by Levesque and Compton
[52], who observed mitosis of vertebrate cells after suppressing the activity
of chromokinesins and, thus PEFs. This resulted in $17.5\text{\,}\mathrm{\char
37\relax}$ of the cells in at least one chromosome not aligning at the
equator, but locating near a spindle pole.
Figure 12: Kinetochore dynamics under the influence of PEFs. (a) Kinetochore
trajectories with different PEF constants $k$ from simulations with $M=100$,
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
without confinement at the spindle poles. The PEFs force the kinetochores to
oscillate regularly and to stay near the spindle equator. For
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
kinetochores oscillate as described in figure 11. Since with strong PEFs
kinetochores tend to switch to the lower branch simultaneously when reaching
$F_{\mathrm{min}}$ in the phase space at the same time, for
$k=$1000\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
oscillations are in antiphase due to symmetric initial conditions before the
system equilibrates at $t\approx$1500\text{\,}\mathrm{s}$$. After
equilibration, periods of antiphase oscillations reappear over and over again
due to fluctuations. Stronger PEFs cause a more fluctuative kinetochore
motion. Especially for moderate MT numbers, this can lead to suppression of
kinetochore oscillations. For animated versions of phase space trajectories
see videos S5 ($k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$) and S6
($k=$1000\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
in the supplementary material. (b) Single (right) kinetochore and breathing
oscillations in Fourier space. For weak PEFs
($k=$1\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
single kinetochore oscillations are still irregular and
$\tilde{X}_{\mathrm{k,r}}$ has its maximum at $f=0$. If
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
$\tilde{X}_{\mathrm{k,r}}$ has a distinct peak at half the breathing
frequency, indicating regular oscillations as described in figure 11 and
frequency doubling of breathing compared to single kinetochore oscillations.
With sufficiently strong PEFs
($k\gtrsim$100\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$) frequency doubling is lost as a consequence of antiphase
oscillations and the peaks of $\tilde{X}_{\mathrm{k,r}}$ and
$\Delta\tilde{X}_{\mathrm{k}}$ coincide with each other. (c) Histograms of
kinetochore positions and inter-kinetochore distances for the realistic case
of $M=25$. Chromosomes are aligned at the spindle equator despite missing
confinement at the centrosome. The range of kinetochore positions is narrower
and the distances smaller if PEFs are stronger.
Moreover, PEFs reduce the amplitude and increase the frequency of
oscillations. The amplitude decreases for increasing PEF strength $k$ as the
kinetochores have to cover a smaller distance between the turning points at
$F_{\mathrm{min}}$ and $F_{\mathrm{max}}$. The increase of the frequency is
linear in $k$, which can be deduced from the linear increase of
$|\dot{F}_{\mathrm{k}}|$:
$\displaystyle|\dot{F}_{\mathrm{k,l}}|$
$\displaystyle=\left|c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)+kv_{\mathrm{k,l}}\right|,$
(27) $\displaystyle|\dot{F}_{\mathrm{k,r}}|$
$\displaystyle=\left|c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)+kv_{\mathrm{k,r}}\right|$
(28)
(defining $v_{\mathrm{k,l}}\equiv\dot{X}_{\mathrm{k,l}}$ and
$v_{\mathrm{k,r}}\equiv-\dot{X}_{\mathrm{k,r}}$ as the velocities in AP-
direction as before).
Since PEFs do not have any influence on the underlying master curves and
force-velocity relations, they do not affect the kinetochore velocities
$v_{\mathrm{k}}$ and never completely suppress kinetochore oscillations in the
deterministic Fokker-Planck model, but only reduce their amplitude and
increase their frequency. For strong PEFs, however, this gives rise to
kinetochore motion with a fluctuative character, see figure 12 (see also video
S6 in the supplementary material). The same observation was made in the model
of Civelekoglu-Scholey et al. [19]. Additionally, we detect sister kinetochore
oscillations being in antiphase if PEFs are strong enough
($k\gtrsim$100\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$), see figure 12(a). This follows from the phase space velocities
$\dot{F}_{\mathrm{k}}$ being dominated by the strong PEFs compared to inter-
kinetochore tension: Imagine, both kinetochores are in the upper branch of the
phase space and reach the turning point $F_{\mathrm{min}}$ at nearly the same
time. When now one of the two kinetochores switches to the lower branch and
starts moving polewards, its sister does not change its direction in phase
space as in state $2/2^{\prime}$ in figure 11(a) but continues moving left
since the decrease of PEFs due to its poleward motion can not be compensated
by the increasing AP-directed cohesin tension if $k\gg c_{\mathrm{k}}$. As a
consequence, the kinetochore will switch to the lower branch just after its
sister and both kinetochores pass the lower branch simultaneously, i.e. move
apart from each other, finally resulting in antiphase oscillations. While the
antiphase behavior vanishes after a certain time of equilibration in the
deterministic model, in stochastic simulations periods of antiphase
oscillations can be observed over and over again regardless of whether the
system has been equilibrated before. A characteristic of antiphase
oscillations is the loss of frequency doubling which also appears in the
Fourier space where the peaks of single kinetochore and breathing motion
coincide with each other if PEFs are strong, see figure 12(b). Since antiphase
kinetochore oscillations have not been observed experimentally, we conclude
that in vivo PEFs are weak compared to the inter-kinetochore tension but
strong enough to assure chromosome alignment at the spindle equator. Compared
to experimental results [6, 7, 10, 11, 12, 19], in our model,
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
seems a reasonable choice as it assures regular oscillations with frequency
doubling, keeps the inter-kinetochore distance within a suitable range of
$1.2\pm 0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and aligns kinetochores in
a realistic maximum distance of $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from
the spindle equator with a standard deviation of
$0.88\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the lifelike case of $M=25$.
## 8 Catastrophe promotion at the kinetochore is required to stimulate
directional instability if microtubules can not exert pushing forces
So far, we assumed that MTs are also able to exert pushing forces on the
kinetochore. During oscillations we find, on average, slightly less (48%) MT-
kinetochore links under tension, while a substantial part of linkers also
exerts pushing forces. Two experimental results suggest, however, that MTs do
not directly exert pushing forces on the kinetochore: In [7], it was shown
that the link between chromosomes is always under tension; the experiments in
[26] demonstrated that, after removal of the cohesin bond, AP-moving
kinetochores immediately stop indicating that kinetochore MTs can not exert
pushing forces, while P-moving kinetochores continue moving due to MT pulling
forces.
In view of these experimental results and in order to answer the question
whether MT pushing forces are essential for bistability and oscillations, we
analyze variants of our basic model, where MT growth is confined at the
kinetochore, i.e., where the relative coordinate
$x=x_{\mathrm{m}}-X_{\mathrm{k}}$ is limited to $x\leq 0$ such that MTs can
only exert tensile forces on the kinetochore. This implies that the
kinetochore undergoes a catastrophe if it reaches the kinetochore, i.e., if
the relative coordinate reaches $x=0$ from below in the one-sided model.
Different choices for the corresponding catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ at $x=0$ are possible: (i) A reflecting
boundary, i.e., $\omega_{\mathrm{c}}^{\mathrm{kin}}=\infty$, where a
catastrophe is immediately triggered if the MT plus-end reaches the
kinetochore. (ii) A “waiting” boundary condition, where the relative velocity
$v_{+}=v_{\mathrm{m}+}-v_{\mathrm{k}}=0$ stalls if the MT reaches $x=0$ (in
the simulation, we set the MT velocity to $v_{\mathrm{m}+}=v_{\mathrm{k}}$).
In contrast to the reflecting boundary condition, the catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ at the kinetochore is finite such that
the MT waits at the kinetochore until it undergoes a catastrophe for a mean
waiting time $1/\omega_{\mathrm{c}}^{\mathrm{kin}}$, as similarly observed in
metaphase of PtK1 cells [36]. Because $x=0$ also results in
$F_{\mathrm{mk}}=0$, the force-free catastrophe rate seems a natural choice,
$\omega_{\mathrm{c}}^{\mathrm{kin}}=\omega^{0}_{\mathrm{c}}$ [see (1)], which
should be realized in the absence of any additional catastrophe regulating
proteins at the centromere. (iii) If catastrophes are promoted by regulating
proteins, but not immediately as for (i), we obtain intermediate cases of
waiting boundary conditions with
$\omega^{0}_{\mathrm{c}}<\omega_{\mathrm{c}}^{\mathrm{kin}}<\infty$. In
mammalian cells, such regulating mechanisms could be provided by the kinesin
MCAK, which is localized at the centromere during metaphase [53] and has been
reported to increase the catastrophe rate of MTs roughly 7-fold [54].
Therefore, waiting boundary conditions with an increased catastrophe rate
appear to be the most realistic scenario. We introduce a numerical catastrophe
enhancement factor $n\geq 1$ characterizing the increased catastrophe rate,
$\omega_{\mathrm{c}}^{\mathrm{kin}}=n\omega^{0}_{\mathrm{c}}$. Within this
general scenario reflecting boundary conditions (i) are recovered for
$n=\infty$ and (ii) waiting boundary conditions with the zero force
catastrophe rate for $n=1$. We will discuss the general case (iii) in the
following.
In our basic model, where MTs can exert pushing forces on kinetochores, the
pushing phases where $x>0$ can also be interpreted as a an effective waiting
phase at the kinetochore with a catastrophe rate, which is effectively
increased by the pushing forces. Therefore, the behavior of our basic model
resembles a model with waiting boundary conditions with an increased
catastrophe rate $n>1$ at the kinetochore. MT pushing forces are not essential
for bistability and oscillations and have a similar effect as an increased
catastrophe rate at the kinetochore as our detailed analysis will show.
In the Fokker-Planck solution for the one-sided model, all confining boundary
conditions limit the maximum MT-kinetochore distance $x_{\mathrm{max}}$ to
zero, where it is positive in the basic model. When $x_{\mathrm{max}}$ is
negative in the basic model (for $v_{\mathrm{k}}>v^{0}_{+}$, see table 3),
confining boundary conditions do not modify the basic model, since the MTs are
not able to reach the fast kinetochore. For negative kinetochore velocities
$v_{\mathrm{k}}<v^{0}_{-}$, the minimum distance $x_{\mathrm{min}}$ becomes
positive while $x_{\mathrm{max}}$ is zero. Then, all confining boundary
conditions fix the MT tips to the kinetochore position as they do not shrink
fast enough to move away from the poleward-moving kinetochore after a
catastrophe resulting in $\langle x\rangle=0$ and $F_{\mathrm{ext}}=\gamma
v_{\mathrm{k}}$. All in all, confinement leads to the following maximal and
minimal values for the MT-kinetochore distance $x$ modifying table 3:
$\displaystyle
x^{\mathrm{conf}}_{\mathrm{max}}=\cases{0,&$v_{\mathrm{k}}<v^{0}_{+}$\\\
x_{\mathrm{max}},&$v_{\mathrm{k}}\geq v^{0}_{+}$,}\qquad
x^{\mathrm{conf}}_{\mathrm{min}}=\cases{0,&$v_{\mathrm{k}}<v^{0}_{-}$\\\
x_{\mathrm{min}},&$v_{\mathrm{k}}\geq v^{0}_{-}$.}$ (29)
We calculate the master curves $\langle x\rangle(v_{\mathrm{k}})$ for all
three types of confining boundary conditions (see figure 13(a)). Because
$x^{\mathrm{conf}}_{\mathrm{max}}\leq 0$ for any confining boundary condition,
also $\langle x\rangle<0$, i.e., the complete master curves lie in the regime
of tensile MT-kinetochore linker forces reflecting the fact that pushing
forces are strictly suppressed. Therefore, the MT-kinetochore catch bond is on
average under tension establishing a more firm MT-kinetochore connection
during the stochastic chromosome oscillations in metaphase. Oscillations then
become a tug-of-war, in which both sets of MTs only exert pulling forces onto
each other.
Figure 13: Microtubule confinement at the kinetochore. (a) Master curves of a
system with a waiting boundary condition for various
$\omega_{\mathrm{c}}^{\mathrm{kin}}=n\,\omega_{\mathrm{c}}^{0}$ and
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$. (b)
Regimes in the parameter plane of $c$ and $\omega_{\mathrm{c}}^{\mathrm{kin}}$
in the limit of many MTs. Outside the blue region, the master curve is
bistable. In the orange region, the left branch of the master curve and,
therefore, the lower branch of the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram
cross $v_{\mathrm{k}}=0$, which leads to a fixed point suppressing
oscillations (see text), whereas in the red region oscillations are possible.
In stochastic simulations, kinetochores already oscillate at much smaller
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ than predicted by the master curves.
Additionally, a new kind of fixed point, which is depicted in (c), emerges in
the shaded region. (c,d) Phase space diagrams and kinetochore trajectories
from simulations of the unconfined two-sided model with
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=100$. The blue dots mark the new kind of fixed point, where the leading
kinetochore in the lower branch moves with the same velocity as the trailing
kinetochore in the upper branch. Then the inter-kinetochore distance remains
constant, while the center of mass moves with a constant velocity as in (d)
for $\omega_{\mathrm{c}}^{\mathrm{kin}}=20\,\omega_{\mathrm{c}}^{0}$ at
$t\approx$25\,000\text{\,}\mathrm{s}$$. In the presence of PEFs, these fixed
points are absent and the shaded region in (b) does not apply.
With a waiting boundary condition at the kinetochore, the probability
densities $p_{\pm}(x,t)$ have to be supplemented with the probability $Q(t)$
to find a MT at the kinetochore ($x=0$). Besides the FPEs (5) and (6) for the
probability densities, we also have to solve the equation for the time
evolution of $Q(t)$:
$\displaystyle\partial_{t}Q(t)=v_{+}(0)p_{+}(0,t)-\omega_{\mathrm{c}}^{\mathrm{kin}}Q(t).$
(30)
The analogous model for a free MT that grows against a rigid wall has already
been solved in [55, 41]. In the stationary state, (30) leads to
$Q=p_{+}(0){v_{+}(0)}/{\omega_{\mathrm{c}}^{\mathrm{kin}}}$. For the
probability densities $p_{\pm}(x)$ we get the same solution as for the basic
model without confinement, except for the normalization constant. The overall
probability density can then be written as $p(x)=p_{+}(x)+p_{-}(x)+Q\delta(x)$
and has to satisfy
$\int_{x^{\mathrm{conf}}_{\mathrm{min}}}^{x^{\mathrm{conf}}_{\mathrm{max}}}p(x)\mathrm{d}x=1$.
From the overall probability density $p(x)$ we obtain the master curves, which
we show in figure 13(a) for $n=1,5,20,50,200,\infty$ and a linker stiffness of
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.
Again we can analyze the master curves for extrema to obtain constraints on
linker stiffness $c$ and catastrophe enhancement factor
$n=\omega_{\mathrm{c}}^{\mathrm{kin}}/\omega^{0}_{\mathrm{c}}$ for the
occurrence of bistability and oscillations. The results of this analysis are
shown in figure 13(b) as colored regions. It turns out that extrema in the
master curve and, thus, bistability occur if the linker stiffness is
sufficiently high $c>c_{\mathrm{bist}}$. For the zero force catastrophe rate
$n=1$ we find a high threshold value
$c_{\mathrm{bist}}=$178\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, in the limit of a reflecting boundary $n=\infty$ a very low
threshold
$c_{\mathrm{bist}}=$1.218\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$.
We remind that a sufficient condition for oscillations is the absence of a
stable fixed point, where one of the two branches in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram crosses $v_{\mathrm{k}}=0$. In
contrast to the basic model, the maxima of the master curve are now located at
a positive velocity for $n>1$. Therefore, oscillations are suppressed by a
fixed point $v^{-}_{\mathrm{k}}=0$ on the lower branch in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram, which occurs if the velocity is
positive in the minimum of the master curve. In general, oscillations occur if
the linker stiffness is sufficiently high $c>c_{\mathrm{osc}}$. Again we find
a high threshold value
$c_{\mathrm{osc}}=$280\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ for $n=1$ and a low threshold
$c_{\mathrm{osc}}=$1.237\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ for a reflecting boundary condition ($n=\infty$).
For $n<10$ the threshold values remain high. Moreover, at such high linker
stiffnesses and for for small $n$, the simulations of the two-sided model do
not show the expected behavior. For $n=1$ and high linker stiffnesses in the
oscillatory regime the kinetochore trajectories do not exhibit regular
oscillations. Naively, one could argue that kinetochore oscillations are
suppressed due to the lack of a pushing force and can be restored by
additional PEFs. However, this is not the case, since, as stated above, PEFs
do not affect the master curve that determines the regime of kinetochore
motion. One reason for the absence of oscillations is that, for the zero force
catastrophe rate ($n=1$) the waiting time
$1/\omega_{\mathrm{c}}^{\mathrm{kin}}\sim 500{\rm s}$ (see table 2) at the
kinetochore is large compared to the typical oscillation periods, which are in
the range of $100-200{\rm s}$.
Figure 13(b) also shows that oscillations require increased catastrophe rates
with $n\gtrsim 20$ over a wide range of linker stiffnesses from
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ to
$c=$200\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.
For $n>1$, at the boundary between bistable and oscillatory regime in figure
13(b), a fixed point $v^{-}_{\mathrm{k}}=0$ on the lower branch of the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$ phase space diagrams appears, which can
suppress oscillations. This fixed point is, however, less relevant because the
kinetochores will only occasionally pass the lower branch simultaneously,
which is necessary to reach this fixed point. Furthermore, this fixed point is
located near the right turning point $F_{\mathrm{max}}$ so that the
kinetochores can easily leave the fixed point by a stochastic fluctuation (as
in figure 7). For these two reasons, in stochastic simulations, oscillations
already occur for $n\gtrsim 5$, that is at a much lower $n$ than the
deterministically predicted $n\gtrsim 20$, but not for $n=1$, i.e., in the
absence of a catastrophe promoting mechanism.
The fixed point analysis of the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$ phase space
diagrams reveals that also a new type of fixed point corresponding to a non-
oscillatory motion emerges for $n\lesssim 100$ in the shaded regions in figure
13(b). In this new type of fixed point, the leading P-moving kinetochore in
the lower branch of the master curve has the same velocity as the trailing AP-
moving kinetochore in the upper branch (see figure 13(c)) so that
$\dot{F}_{\mathrm{kk}}=-c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)=0$,
and the inter-kinetochore distance remains constant, while the center of mass
moves with a constant velocity (see figure 13(d)). In the presence of PEFs,
however, this new type of fixed point does not survive because for the P-
moving kinetochore the AP-directed PEFs increase, whereas they decrease for an
AP-moving kinetochore. Then the upper blue dot in figure 13(c) moves to the
left, while the lower blue point moves to the right such that this new type of
fixed point is unstable in the presence of PEFs. Therefore, in the entire
shaded region in figure 13(b) PEFs are essential to re-establish oscillations.
We conclude that both the linker stiffness
$c>$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
the catastrophe rate $\omega_{\mathrm{c}}^{\mathrm{kin}}$ at the kinetochore
($n\gtrsim 20$ or $n\gtrsim 5$ in the presence of stochastic fluctuations)
have to be sufficiently large to obtain bistability and oscillations. Because
additional catastrophe promoting proteins are necessary to increase the
catastrophe rate at the kinetochore, the lowest values of $n$, which still
enable oscillations, might be advantageous in the cellular system. We note
that poleward flux can influence existence and positions of fixed points: An
intermediate flow velocity can eliminate a fixed point on the lower branch by
moving it into the unstable area of the phase space diagram. If flux is
sufficiently large it can establish additional fixed points on the upper
branch of the phase space diagrams, which suppress oscillations as in the
basic model.
Moreover, the linker stiffness has to be sufficiently high to give linker
extensions compatible with experimental results. An important part of the MT-
kinetochore linkage is Ndc80, which is a rod-like fibril of total length
around $60\,{\rm nm}$ [56, 57] consisting of two coiled-coil regions with a
flexible hinge that can adopt bending angles up to $120^{\circ}$ with a broad
distribution [57]. This bending corresponds to linker length changes of
$|x|\sim$50\text{\,}\mathrm{nm}$$. Moreover, fluorescent labeling showed total
intra-kinetochore stretches around $100\text{\,}\mathrm{nm}$ [58] or
$50\text{\,}\mathrm{nm}$ [12]. Therefore, we regard linker extensions
$x\lesssim$100\text{\,}\mathrm{nm}$$ as realistic values. For large $n\gg 20$
only a small linker stiffness is necessary to enable oscillations. At the
small threshold stiffness, the average linker length $|\langle x\rangle|$ is
typically $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in this regime. Increasing
the linker stiffness leads to a decreasing linker length $|\langle x\rangle|$.
We conclude that, for $n\gg 20$, experimental observations of linker
extensions $|x|\lesssim$100\text{\,}\mathrm{nm}$$ put a stronger constraint on
linker stiffness than the experimental observations of oscillations. Linker
stiffnesses significantly above
$5\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ and,
thus, far above $c_{\mathrm{osc}}$ are necessary to obtain a realistic linker
length.
For $n\sim 10-20$, which is compatible with the experimental result $n\sim 7$
for the catastrophe promoter MCAK [54], and a linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$, the
increased catastrophe rate at the kinetochore leads to a realistic behavior
with linker extensions $x\sim$100\text{\,}\mathrm{nm}$$, which are also
compatible with the experimental results [56, 57, 58, 12] (see figure 13(a)).
This parameter regime is within the shaded regions in figure 13(b) and PEFs
are necessary to establish oscillations. The linker extension is independent
of PEFs.
For an increased catastrophe rate around $n\sim 10-20$ and a linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$, the
more realistic model with waiting boundary conditions at the kinetochore
exhibits a similar behavior as our basic model because pushing phases where
$x>0$ in the basic model have a similar duration as waiting times at the
kinetochore in the more realistic model.
## 9 Model parameters can be adjusted to reproduce kinetochore oscillations
in PtK1 cells
So far, we took the experimentally measured parameters for MT transitions and
velocities from table 2 for granted in order to analyze the effects of
poleward flux, PEFs and confinement at the kinetochore by means of our mean-
field theory. These values stem from experiments with yeast kinetochores [2],
which can only bind one MT [59], whereas the mean-field theory is only correct
if the kinetochores are attached to multiple MTs as in metazoan cells.
Moreover, in budding yeast, the Ndc80 fibrils are connected to MTs via ring-
like Dam1 complexes, which do not appear in metazoan cells [60]. In this
section, we demonstrate that by adjusting the parameters of MT dynamics our
model can reproduce experimental data of metazoan spindles using the example
of PtK1 cells.
Our model exhibits a large difference of P versus AP-velocity ($\sim 100$ vs.
$\sim$4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, see figure 8) which
is the origin of frequency doubling and also appears in PtK1 cells but not in
this extent ($\sim 19$ vs.
$\sim$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$) [11]. As a
consequence, in our model both kinetochores move towards each other in AP-
direction (state 0 in figure 3) most of the time, whereas in the experiment,
mostly one kinetochore moves in P- while the trailing sister is moving in AP-
direction (state $2/2^{\prime}$ in figure 3). In a first step we will respect
these results by adjusting the master curve (or force velocity relation) in a
way that the two stable branches fit the experimentally measured velocities.
This objective will be achieved by modifying the force-free MT velocities
$v_{\pm}^{0}$ (shifting the upper / lower branch up- or downwards) and the
corresponding characteristic forces $F_{\pm}$ (altering the slope of the upper
/ lower branch). Moreover, as a last parameter of MT dynamics, we will change
the rescue rate $\omega_{\mathrm{r}}^{0}$ in order to adjust the MT-
kinetochore distance to a realistic value. In a second step we will fit the
measured frequencies and amplitudes by varying the parameters that do not
affect the master curves ($c_{\mathrm{k}}$, $k$).
Using the model with confinement at the kinetochore, we assume a ten times
increased catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}=10\omega_{\mathrm{c}}^{0}$ according to
experimental results [54]. We set the linker stiffness to
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
keep it unchanged henceforth since this value results in strongly bistable
master curves and the manifold consequences that a further modification of $c$
has on kinetochore dynamics are hard to handle. The flux velocity is
$v_{\mathrm{f}}=$8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$ (see table
4). The force-free MT growth velocity $v_{+}^{0}$ has to be greater than
$v_{\mathrm{f}}$ for two reasons: Firstly, detached MTs would not have a
chance to reach the kinetochore again, otherwise. Secondly, this choice
prevents a fixed point at the upper branch, as the left turning point in phase
space (maximum of the master curve) is located at $v_{+}^{0}-v_{\mathrm{f}}$,
when the MTs are confined at the kinetochore. We increase the force-free
growth velocity roughly four-fold to
$v_{+}^{0}=$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, so that the
minimum AP-velocity
$v_{+}^{0}-v_{\mathrm{f}}=$12\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$
in the left turning point $F_{\mathrm{min}}$ lies below the observed mean
velocity of $\sim$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$. In order
to adjust the maximum AP-velocity, we reduce the characteristic force in MT
growth to $F_{+}=$5\text{\,}\mathrm{p}\mathrm{N}$$, which leads to a steeper
upper branch in the phase space diagram. The force-free shrinking velocity
$v_{-}^{0}$ should be smaller than the observed P-velocity since the lower,
P-directed branch always lies above it. Analogously to the upper branch and
$F_{+}$, also the slope of the lower branch can be adjusted by varying the
characteristic force $F_{-}$: An increase of $F_{-}$, i.e. a decrease of its
absolute value, steepens the lower branch and thereby slows down the poleward
motion. It turns out that it is a good choice to keep the values for
$v_{-}^{0}$ and $F_{-}$ from table 2 unchanged. Finally, we reduce the rescue
rate $\omega_{\mathrm{r}}^{0}$, which lets MTs shrink to smaller lengths
$x_{\mathrm{m}}$ (the minimum of the master curve is shifted downwards) and
increases the MT-kinetochore distance $|x|=|X_{\mathrm{k}}-x_{\mathrm{m}}|$ to
a realistic value.
Since we enable detachment in this section, we set $M=35$ as it results in a
mean number of $\sim 20$ attached MTs. Finally, we adjust the strength of PEFs
$k$ and the cohesin bond stiffness $c_{\mathrm{k}}$ to the following
conditions: Firstly, the PEFs have to be strong enough to assure proper
chromosome alignment at the equator as well as a regular oscillation pattern,
but should not dominate compared to the inter-kinetochore tension in order to
prevent antiphase oscillations. Secondly, $k$ and $c_{\mathrm{k}}$ affect the
amplitude and the frequency of kinetochore oscillations which should resemble
experimental results in the same manner: An increase of both $k$ and
$c_{\mathrm{k}}$ decreases the amplitude and increases the frequency. We find
that $k=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
and
$c_{\mathrm{k}}=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ fulfill both conditions. In table 5, we list all parameters that we
have changed compared to table 2.
Table 5: Parameters to reproduce of kinetochore oscillations in PtK1 cells.
Parameters not listed here have been unchanged compared to table 2.
* Description | Symbol | Value
---|---|---
zero force rescue rate | $\omega_{\mathrm{r}}^{0}$ | $0.012\text{\,}{\mathrm{s}}^{-1}$
zero force MT growth velocity | $v_{+}^{0}$ | $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$
characteristic force of MT growth | $F_{+}$ | $5\text{\,}\mathrm{p}\mathrm{N}$
catastrophe rate at the kinetochore | $\omega_{\mathrm{c}}^{\mathrm{kin}}$ | $0.019\text{\,}{\mathrm{s}}^{-1}$
MT flux velocity | $v_{\mathrm{f}}$ | $8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$
PEF coefficient | $k$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
cohesin bond stiffness | $c_{\mathrm{k}}$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
MT-kinetochore linker stiffness | $c$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
number of MTs | $M$ | 35
The resulting kinetochore dynamics is shown in figure 14. The simulated
kinetochore trajectories in figure 14(a) are very similar to the experimental
results in [11, 19] as they exhibit frequency doubling of breathing compared
to single kinetochore oscillations and move predominantly in phase, i.e. there
is a leading P- and a trailing AP-kinetochore (state $2/2^{\prime}$ in figure
3). The motion of the inter-kinetochore distance is rather fluctuative,
resulting in a broad Fourier transform, in which the maximum at the breathing
frequency is hardly recognizable, see figure 14(d). This is the only
significant difference to the real kinetochore motion. The distributions of
kinetochore positions as well as inter-kinetochore and MT-kinetochore
distances (figure 14(e-g)) are in good agreement with experimental results
[19].
Figure 14: Reproduction of kinetochore oscillations in PtK1 cells. (a)
Kinetochore positions and inter-kinetochore distance over time. Although the
breathing oscillations are rather fluctuative, frequency doubling is
recognizable. (b) Number of attached MTs over time. (c) Kinetochore motion in
phase space (green) compared to the mean-field force-velocity relation (red,
calculated with the mean number of attached MTs). For an animated version see
video S7 in the supplementary material. (d) Position of the right kinetochore
and inter-kinetochore distance in Fourier space. Fluctuative breathing
oscillations lead to a Fourier transform with broad maxima, which are almost
only recognizable in the smoothed curve (dark blue). (e-h) Distributions of
kinetochore positions $X_{\mathrm{k}}$, inter-kinetochore distance $\Delta
X_{\mathrm{k}}$, MT-kinetochore distance $|x|$, and the number of attached MTs
$M^{\mathrm{att}}$.
In table 6, we list several characteristic quantities of kinetochore
oscillations that have also been determined experimentally for PtK1 cells.
Comparison with our model results shows quantitative agreement. In particular,
the large discrepancy in the P- and AP-velocities is eliminated.
Table 6: Characteristic quantities of model kinetochore oscillations compared
to experimental results in PtK1 cells.
* Description | Model | Experiment |
---|---|---|---
mean P velocity | $21.5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | $19.0\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | [11]
mean AP velocity | $15.7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | $15.7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | [11]
single kinetochore frequency | $4.27\text{\,}\mathrm{mHz}$ | 4.14–$4.23\text{\,}\mathrm{mHz}$ | [11]
breathing frequency | $\sim$8.6\text{\,}\mathrm{mHz}$$ | $8.25\text{\,}\mathrm{mHz}$ | [11]
mean inter-kinetochore distance | $1.83\pm 0.42\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | $1.90\pm 0.44\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
mean MT-kinetochore distance | $0.081\pm 0.042\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | $0.11\pm 0.04\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
standard deviation of kinetochore position | $0.76\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | 0.5–$1.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
mean number of attached MTs | 21.4 | 20–25 | [43]
## 10 Discussion
We provided an analytical mean-field solution of the one-sided spindle model
introduced by Banigan et al. [20], which becomes exact in the limit of large
MT numbers. The mean-field solution is based on the calculation of the mean
linker extension $\langle x\rangle$ as a function of a constant kinetochore
velocity $v_{\mathrm{k}}$ (the master curve). Together with the equation of
motion of the kinetochore we obtained the force-velocity relation of the one-
sided model from the master curve. Our solution clearly shows that the force
feedback of linkers onto the MT depolymerization dynamics is essential for a
bistable force-velocity relation within the minimal model. The shape of the
distribution $p_{\pm}(x)$ of linker lengths (12) is governed by this force
feedback, and we traced the bistability to the peakedness (kurtosis) of this
distribution.
Bistability of the force-velocity relation in the one-sided model is a
necessary (but not sufficient) condition for oscillations in the two-sided
model. Interpreting the bistable force-velocity relation as phase space
diagram, we mathematically described kinetochore oscillations as an emergent
result of collective dynamics of coupled MTs that exhibit dynamic instability
individually. Our theory becomes exact in the limit of large MT numbers $M$.
This interpretation of oscillations is underpinned by the experimental
observations that kinetochore oscillations in budding yeast [61, 62, 63],
where each kinetochore is attached to one MT [59], as well as in fission yeast
[64, 21], where two to four MTs interact with the same kinetochore [65], have
a considerably more fluctuative character than the regular oscillations in
vertebrate cells [6, 7, 8, 9, 10, 11, 12] with $\sim 20$ MTs per kinetochore
[43, 66]. Moreover, we were able to deduce idealized kinetochore oscillations,
whose periods conform with experimental results [11]. For a MT-kinetochore
linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
20–25 MTs per kinetochore, we get periods of 206–$258\text{\,}\mathrm{s}$ and
103–$129\text{\,}\mathrm{s}$ for kinetochore and breathing oscillations,
respectively. Our approach reproduced the frequency doubling of breathing
compared to single kinetochore oscillations, observed in the experiment [11].
Both in the model and in the experiment this doubling originates from the
different velocities of AP- and P-moving kinetochores, which ensure that a
P-to-AP switch ($3/3^{\prime}$ in figure 3) always follows an AP-to-P switch
of the same kinetochore ($1/1^{\prime}$ in figure 3). In the model the
velocity difference is, however, much larger. As a consequence, in our model
with 20–25 MTs an AP-to-P switch follows 96–$119\text{\,}\mathrm{s}$ after a
P-to-AP switch of the sister kinetochore, which is $93\text{\,}\mathrm{\char
37\relax}$ of a breathing period, whereas in PtK2 cells a mean interval of
merely $6\text{\,}\mathrm{s}$ has been measured [12]. In other words, in our
model, most of the time both kinetochores move towards each other in AP-
direction (state 0 in figure 3), whereas in the experiment, mostly one
kinetochore moves in P- while the trailing sister is moving in AP-direction
(state $2/2^{\prime}$ in figure 3). In our model, different AP- and
P-velocities are based on the fact that the MT shrinkage is much faster than
growth. The model parameters for MT dynamics were taken from experimental
measurements with yeast kinetochores [2], which, however, are distinct from
metazoan kinetochores in two main points: firstly, they can only attach to one
MT [59]; secondly, the Ndc80 fibrils are connected to MTs via ring-like Dam1
complexes, which do not appear in metazoan cells [60]. We show in section 9
that this discrepancy can be eliminated by adjusting some MT parameters and,
moreover, the model can reproduce kinetochore oscillations in PtK1 cells
quantitatively.
In experiments with HeLa cells Jaqaman et al. [67] observed an increase of
oscillation amplitudes and periods when they weakened the cohesin bond. In our
model, a smaller cohesin stiffness $c_{\mathrm{k}}$ has the same two effects
as the inter-kinetochore distance has to be larger to reach the turning points
$F_{\mathrm{min}}$ and $F_{\mathrm{max}}$ of the hysteresis loop, and the
phase space velocity
$\dot{F}_{\mathrm{kk}}=c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)$
and, therefore, the frequencies are proportional to $c_{\mathrm{k}}$.
Our analytical approach also allowed us to go beyond the results of [20] and
quantify constraints on the linker stiffness $c$ and the MT number for
occurrence of bistability in the one-sided model and for the occurrence of
oscillations in the full model. We found that bistability requires linker
stiffnesses above
$c_{\mathrm{bist}}\simeq$8\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. Bistability is, however, not sufficient for oscillations. Our
phase space interpretation showed that bistability only leads to directional
instability if the two branches of the force-velocity relation are also
separated by the zero velocity line. This condition quantifies the oscillatory
regime in the parameter plane of $c$ and $M$. We predict that oscillations
should only be observable if the MT-kinetochore linker stiffness is above
$c_{\mathrm{osc}}\simeq$16\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. Our model can thus provide additional information on the MT-
kinetochore linkers whose molecular nature is unknown up to now. Several Ndc80
fibrils, which cooperatively bind to the MT, are an important part of the MT-
kinetochore link and the stiffness of this Ndc80 link has been determined
recently using optical trap measurements [68]. These experiments found
stiffnesses above
$\sim$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
which are compatible with our bounds. Moreover, they found a stiffening of the
link under force, which could be included in our model in future work.
The derivation of the lower bound for the stiffness for the occurrence of
oscillations is based on the occurrence of a new zero AP-velocity fixed point
in the force-velocity diagram of the kinetochores, which suppresses
oscillations upon decreasing the stiffness. Also the influence of poleward
flux to the system could be analyzed by a fixed point analysis of the force-
velocity diagram. Since poleward MT flux shifts the force-velocity towards
smaller AP-velocities of the kinetochore, the upper branch may cross zero
velocity establishing again a zero velocity fixed point suppressing
oscillations. This explains why high flux velocities suppress directional
instability and rationalizes the correlation between kinetochore oscillations
and poleward flux observed in several cells (table 4). It has been observed in
newt lung cells that oscillations are occasionally ($11\text{\,}\mathrm{\char
37\relax}$ of time) interrupted by phases in which the kinetochores pause
their motion [6] analogously to resting in the fixed point in our model. This
indicates that the spindle of newt lung cells operates near the boundary
between the oscillatory and the non-oscillatory regime.
Also experimental results in [69, 70, 71, 72] on the effects of
phosphorylation of Hec1, which is part of mammalian Ndc80 complex, onto
kinetochore dynamics can be rationalized by our force-velocity diagram of the
kinetochores. Dephosphorylation leads to hyper-stable MT-kinetochore
attachments, increases the inter-kinetochore distance, damps or completely
suppresses oscillations, and lets the kinetochores more often be found in a
“paused state”. The increase of the inter-kinetochore distance can be
explained with the hyper-stable MT-kinetochore attachments: in the oscillatory
regime, the bistable area of the force-velocity relation increases if more MTs
are attached to the kinetochore (figure 2(b)); in the non-oscillatory regime,
the mean distance $\langle\Delta X_{\mathrm{k}}\rangle$ is a linear function |
$e^{K}_{G}(V_{j}(1))=(1-\tau^{-j}\xi^{-1})^{a_{j}}$, where $\xi$ denote the
standard $1$-dimensional representation of $S^{1}$. Hence
$K^{*}_{G}(\mathbb{P}(V_{j}))\cong R(G)[\xi,\xi^{-1}]/(e^{K}_{G}(V_{j}(1)))$.
Hence
$S^{-1}K^{*}_{G}(\mathbb{P}(V_{j}))\cong\frac{\mathbb{Q}(\omega)[\xi,\xi^{-1}]}{(1-\tau^{-j}\xi^{-1})^{a_{j}})}.$
Let $\mathbb{C}_{j}$ denote the $1$-dimensional representation of $G$ where
$g$ acts as multiplication by $\omega^{j}$. Then $N_{j}$ is a direct sum of
line bundles of the form $\mathbb{C}_{i}(1)$ where $i\neq j$. So it suffices
to show that $e^{K}_{G}(\mathbb{C}_{i}(1))=1-\omega^{-i}\xi^{-1}$ is
invertible. We have
$\displaystyle 1-\omega^{-i}\xi^{-1}$
$\displaystyle=1-\omega^{-(i-j)}+\omega^{-(i-j)}(1-\omega^{-j}\xi^{-1})$
$\displaystyle=(1-\omega^{-(i-j)})(1-u)$
where $u=(1-\omega^{i-j})^{-1}(1-\omega^{-j}\xi^{-1})$. Then $u^{a_{j}}=0$ in
$S^{-1}K^{*}_{G}(\mathbb{P}(V_{j}))$ because
$0=e^{K}_{G}(V_{j}(1))=(1-\omega^{-j}\xi^{-1})^{a_{j}}$. Therefore
$(1-\omega^{-i}\xi^{-1})^{-1}=(1-\omega^{-(i-j)})^{-1}(1+u+u^{2}+\cdots),$
where there are only finitely many terms on the right since $u$ is nilpotent.
∎
Equipped now with the $K$-theoretic equivalents of Lemmas 6.2 and 6.4 and
using the localisation theorem in $K$-theory, we obtain the $K$-theoretic
equivalent of Theorem 6.5:
###### Theorem 6.7.
Let $f$ be a $G$-monopole map. Assume that $\mathbb{R}\oplus H^{+}$ can be
given a $G$-invariant complex structure which we use to $K$-orient $H^{+}$.
Then
$\chi(SW^{K,\phi}_{G,f}(\theta),g)=\chi(e^{K}_{G}(H^{+}/(H^{+})^{g}),g)\sum_{j=0}^{n-1}\chi(SW^{K,\phi}_{G,f^{s_{j}}}(e^{K}_{G_{\mathfrak{s}}}(D/D_{j})^{-1}\theta,g).$
### 6.3. Free actions revisited
In this section we will apply Theorem 6.7 to the case of free actions. Let $X$
be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let $G$ be a
group which acts smoothly, orientation preservingly and freely on $X$. Assume
that $b_{+}(X)^{G}=b_{+}(X/G)>0$. Let $\mathfrak{s}$ be a $G$-invariant spinc-
structure and assume that $d(X,\mathfrak{s})=0$. Since
$d(X,\mathfrak{s})=2d-b_{+}(X)-1=0$, this implies that $b_{+}(X)$ is odd.
In the case of free actions with $d(X,\mathfrak{s})=0$, we have a method of
$K$-orienting $\mathbb{R}\oplus H^{+}(X/H)$ for every subgroup $H$. First,
from the proof of Proposition 5.13 we see that $\mathbb{R}\oplus H^{+}(X)$ is
isomorphic to several copies of the real regular representation. In fact,
since $d(X,\mathfrak{s})=0$, there are $2d_{0}$ copies, where $d_{0}=d/|G|$.
Therefore, $\mathbb{R}\oplus H^{+}(X)$ can be (non-canonically) equipped with
a $G$-invariant complex structure $I$ for which $\mathbb{R}\oplus H^{+}(X)$ is
isomorphic as a complex representation to $d_{0}$ copies of the complex
regular representation. Fix a choice of such a complex structure. We use this
to define a $K$-orientation on $\mathbb{R}\oplus H^{+}(X)$. Then for any
$H\subseteq G$, we have an isomorphism $\mathbb{R}\oplus
H^{+}(X/H)\cong(\mathbb{R}\oplus H^{+}(X))^{H}$. Moreover, $(\mathbb{R}\oplus
H^{+}(X))^{H}$ is a complex subspace of $\mathbb{R}\oplus H^{+}(X)$ and hence
it has an induced complex structure and a $K$-orientation.
###### Theorem 6.8.
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let $G$
be a finite group which acts smoothly and freely on $X$. Let $\mathfrak{s}$ be
a spinc-structure whose isomorphism class is fixed by $G$ and suppose that
$d(X,\mathfrak{s})\geq 0$. Assume that $b_{+}(X)^{G}>0$ and if
$b_{+}(X)^{G}=1$ then fix a chamber $\phi$.
Fix a choice of $G$-invariant complex structure on $\mathbb{R}\oplus H^{+}(X)$
for which $\mathbb{R}\oplus H^{+}(X)$ is isomorphic to a sum of copies of the
complex regular represention and use this to $K$-orient $\mathbb{R}\oplus
H^{+}(X/H)$ for every $H\subseteq G$, as described above. Then we have
$\sum_{g\in G}\sum_{s^{\prime}}SW(X/\langle
g\rangle,\mathfrak{s}^{\prime})=0\;({\rm mod}\;|G|)$
where the second sum is over spinc-structures on $X/\langle g\rangle$ whose
pullback to $X$ is isomorphic to $\mathfrak{s}$. Equivalently, we have
$\sum_{(H)\text{
cyclic}}\phi(|H|)|G/N_{G}H|\sum_{\mathfrak{s}^{\prime}|q_{H}^{*}(\mathfrak{s}^{\prime})\cong\mathfrak{s}}SW(X/H,\mathfrak{s}^{\prime})=0\;({\rm
mod}\;|G|)$
where the first sum is over conjugacy classes of cyclic subgroups of $G$ and
the second sum is over spinc-structures on $X/H$ whose pullback to $X$ is
isomorphic to $\mathfrak{s}$. If $b_{+}(X/H)=1$ then
$SW(X/H,\mathfrak{s}^{\prime})$ is defined using the chamber $\phi$.
###### Proof.
Let $f$ be the $G$-equivariant Bauer–Furuta monopole map of
$(X,\mathfrak{s})$. Let $M=SW^{K}_{G,f}(1)\in R(G)$. The restriction of $M$ to
$R(1)\cong\mathbb{Z}$, which is the rank of $M$, is just the ordinary
($K$-theoretic) Seiberg–Witten invariant of $(X,\mathfrak{s})$. Since
$d(X,\mathfrak{s})=0$, it follows that this equals the usual Seiberg–Witten
invariant $SW(X,\mathfrak{s})$ (see [4, §6]). Thus
$\chi(M,1)=SW(X,\mathfrak{s})$.
Let $g\in G$ have order $n$. Set $H=\langle g\rangle$. Since $H$ is cyclic, we
can choose a splitting $H_{\mathfrak{s}}\cong S^{1}\times H$. Then as in
Section 6.2, let $s_{j}$ denote the splitting given by
$s_{j}(g)=(\omega^{-j},g)$, $\omega=e^{2\pi i/n}$. Let $\mathbb{C}_{j}$ be the
$1$-dimensional representation of $H$ where $g$ acts as multiplication by
$\omega^{j}$. Use the splitting $s_{0}$ to regard $D$ as a representation of
$H$. Since the action is free, we have
(6.1) $D\cong\bigoplus_{j=0}^{n-1}\mathbb{C}_{j}^{d/n}.$
Recall that we have fixed a complex structure on $\mathbb{R}\oplus H^{+}(X)$
for which $\mathbb{R}\oplus H^{+}(X)$ is isomorphic to a sum of copies of the
regular representation of $G$. Restricting to $H$, we therefore have that
$\mathbb{R}\oplus H^{+}(X)$ as a representation of $H$ is isomorphic to a sum
of copies of the regular representation. Hence
(6.2) $H^{+}/(H^{+})^{H}\cong\bigoplus_{j=1}^{n-1}\mathbb{C}_{j}^{d/n}.$
Consider the reduced monopole map $f^{s_{j}}$. Its $H$-equivariant
$K$-theoretic Seiberg–Witten invariant takes the form of a map of
$R(H)$-modules:
$SW^{K}_{H,f^{s_{j}}}:R(H)[\xi,\xi^{-1}]\to R(H).$
We have $R(H)\cong\mathbb{Z}[t]/(t^{n}-1)$, where $t=[\mathbb{C}_{1}]$. By
Proposition 6.1, it follows that
$SW^{K}_{H,f^{s_{j}}}(1)=SW(X/H,\mathfrak{s}_{s_{j}})$. Under the splitting
$s_{j}$, $s_{j}^{*}(\xi)=t^{-j}$, hence $s_{j}^{*}(t^{j}\xi)=1$. This implies
that $SW^{K}_{H,f^{s_{j}}}(t^{j}\xi)=SW(X/H,\mathfrak{s}_{s_{j}})$, hence
$SW^{K}_{H,f^{s_{j}}}(\xi)=t^{-j}SW(X/H,\mathfrak{s}_{s_{j}})$ and more
generally $SW^{K}_{H,f^{s_{j}}}(\xi^{m})=t^{-mj}SW(X/H,\mathfrak{s}_{s_{j}})$
for any $m$. Therefore,
$\chi(SW^{K}_{H,f^{s_{j}}}(\xi^{m}),g)=\omega^{-mj}SW(X/H,\mathfrak{s}_{s_{j}}).$
Theorem 6.7 gives
(6.3)
$\chi(M,g)=\chi(e^{K}_{H}(H^{+}/(H^{+})^{H}),g)\sum_{j=0}^{n-1}\chi(SW^{K}_{H,f^{s_{j}}}(e^{K}_{H_{\mathfrak{s}}}(D/D_{j})^{-1}),g).$
From Equation (6.1) we have
$e^{K}_{H_{\mathfrak{s}}}(D/D_{j})=\prod_{i\neq
j}(1-\omega^{-i}\xi^{-1})^{d/n}$
and hence
$\displaystyle\chi(SW^{K}_{H,f^{s_{j}}}(e^{K}_{H_{\mathfrak{s}}}(D/D_{j})^{-1},g)$
$\displaystyle=\left(\prod_{i\neq
j}(1-\omega^{j-i})^{-d/n}\right)SW(X/H,\mathfrak{s}_{s_{j}})$
$\displaystyle=\left(\prod_{i=1}^{n-1}(1-\omega^{j})\right)^{-d/n}SW(X/H,\mathfrak{s}_{s_{j}}).$
From Equation (6.2) we have
$e^{K}_{H}(H^{+}/(H^{+})^{H})=\prod_{i=1}^{n-1}(1-t^{-i})^{d/n},$
hence
$\chi(e^{K}_{H}(H^{+}/(H^{+})^{H}),g)=\left(\prod_{i=1}^{n-1}(1-\omega^{-i})\right)^{d/n}.$
Substituting into Equation (6.3) gives
$\chi(M,g)=\sum_{j=0}^{n-1}SW(X/H,\mathfrak{s}_{s_{j}}).$
Now since $\sum_{g\in G}\chi(M,g)=0\;({\rm mod}\;|G|)$, we have shown that
$\sum_{g\in G}\sum_{s^{\prime}}SW(X/\langle
g\rangle,\mathfrak{s}^{\prime})=0\;({\rm mod}\;|G|).$
∎
### 6.4. A constraint on smooth group actions
###### Theorem 6.9.
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let $G$
be a finite group acting on $X$ by orientation preserving diffeomorphisms. Let
$\mathfrak{s}$ be a $G$-invariant spinc-structure. Assume that
$b_{+}(X)^{G}>0$ and that $d(X,\mathfrak{s})=0$, hence $b_{+}(X)$ is odd.
Suppose also that $\mathbb{R}\oplus H^{+}(X)$ can be equipped with a
$G$-invariant complex structure. Suppose that $SW(X,\mathfrak{s})\neq 0\;({\rm
mod}\;|G|)$. Then for some non-trivial cyclic subgroup $\\{1\\}\neq H\subseteq
G$ and some splitting $s:H\to G_{\mathfrak{s}}$, we have
$2\,dim_{\mathbb{C}}(D^{sH})>dim_{\mathbb{R}}(H^{+}(X)^{H})$.
###### Proof.
Let $f$ be the $G$-equivariant Bauer–Furuta monopole map of
$(X,\mathfrak{s})$. Let $M=SW^{K}_{G,f}(1)\in R(G)$. Then
$\chi(M,1)=SW(X,\mathfrak{s})$. The complex structure on $\mathbb{R}\oplus
H^{+}(X)$ yields a complex structure on $\mathbb{R}\oplus H^{+}(X)^{H}$ for
every $H\subseteq G$ and hence a $K$-orientation.
Let $g\in G$ have order $n$. Theorem 6.7 gives
$\chi(M,g)=\chi(e^{K}_{H}(H^{+}/(H^{+})^{H}),g)\sum_{s}\chi(SW^{K,\phi}_{H,f^{s}}(e^{K}_{G_{\mathfrak{s}}}(D/D^{sH})^{-1}\theta,g)$
where the sum is over splittings $s:H\to H_{\mathfrak{s}}$. Now since $s(H)$
acts trivially on the domain and codomain of $f^{s}$, there are no
obstructions to achieving equivariant transversality. The expected dimension
of the moduli space of $sH$-invariant solutions to the Seiberg–Witten
equations is $2\,dim_{\mathbb{C}}(D^{sH})-dim_{\mathbb{R}}(H^{+}(X)^{H})-1$.
If this is negative then $SW^{K}_{H,f^{s}}=0$.
Now we re-write the congruence $\sum_{g\in G}\chi(M,g)=0\;({\rm mod}\;|G|)$ as
$SW(X,\mathfrak{s})=-\sum_{g\neq 1}\chi(M,g)\;({\rm mod}\;|G|).$
Hence if $SW(X,\mathfrak{s})\neq 0\;({\rm mod}\;|G|)$ then $\chi(M,g)\neq 0$
for some $g\in G\setminus\\{1\\}$ and thus
$2\,dim_{\mathbb{C}}(D^{sH})>dim_{\mathbb{R}}(H^{+}(X)^{H})$ for some
splitting $s$ of $H$. ∎
### 6.5. Divisibility conditions
Let $f:S^{V,U}\to S^{V^{\prime},U^{\prime}}$ be an ordinary monopole map over
a point, $V=\mathbb{C}^{a},V^{\prime}=\mathbb{C}^{a^{\prime}}$,
$U=\mathbb{R}^{b},U^{\prime}=\mathbb{R}^{b^{\prime}}$, $d=a-a^{\prime}$,
$b_{+}=b^{\prime}-b$. Assume that $b_{+}\geq 1$ and that $2d-b_{+}-1=2m$ is
even and non-negative. Choose an orientation on $H^{+}$ and choose a chamber
if $b_{+}=1$. Then the abstract Seiberg–Witten invariant of $f$ is defined and
given by $SW_{f}(x^{m})\in H^{0}(pt;\mathbb{Z})=\mathbb{Z}$. Let us denote it
by $SW(f)$ for simplicity. If $f$ is the Bauer–Furuta monopole map of
$(X,\mathfrak{s})$, then $SW(f)=SW(X,\mathfrak{s})$ is the usual
Seiberg–Witten invariant.
###### Theorem 6.10.
Let $f:S^{V,U}\to S^{V^{\prime},U^{\prime}}$ be an ordinary monopole map over
a point and let $p$ be a prime. If $SW(f)\neq 0\;({\rm mod}\;p)$, then
$(b_{+}-1)/2$ is divisible by $p^{e+1}$ whenever
$p^{e}\leq\left\lfloor\frac{m}{p-1}\right\rfloor$, where $2d-b_{+}-1=2m$. In
particular, if $SW(f)\neq 0\;({\rm mod}\;p)$ and $b_{+}\neq 1\;({\rm
mod}\;2p)$ then $m\leq p-2$.
###### Proof.
We give the proof in the case $p$ is odd. The case $p=2$ is similar. We will
make use of the Steenrod powers $P^{i}$. Using the Borel model, the Steenrod
powers can be defined on $S^{1}$-equivariant cohomology with
$\mathbb{Z}_{p}$-coefficients. Let $W$ be an $S^{1}$-equivariant vector bundle
on $B$ with Thom class $\tau_{B}$. Then
(6.4) $P^{j}(\tau_{W})=q_{j}(W)\tau_{W}$
for some characteristic class $q_{j}(W)\in
H^{2(p-1)j}_{S^{1}}(B;\mathbb{Z}_{p})$. Setting $q_{t}=q_{0}+tq_{1}+\cdots$,
one finds [20, Chapter 19] that if $W$ is complex with Chern roots
$\\{a_{i}\\}$ then:
$q_{t}(W)=\prod_{i}(1+ta_{i}^{p-1}).$
Setting $P_{t}=P^{0}+tP^{1}+\cdots$, we have
$P_{t}(\tau_{W})=q_{t}(W)\tau_{W}$.
Considering $\mathbb{C}$ with the standard $S^{1}$-action and $\mathbb{R}$
with the trivial $S^{1}$-action as $S^{1}$-equivariant vector bundles over
$B=pt$, we have
$q_{t}(\mathbb{C})=1+tx^{p-1},\quad q_{t}(\mathbb{R})=1.$
Since $deg(x)=2$, we have $P^{1}(x)=x^{p}$ and $P^{j}(x)=0$ for $j>1$. Hence
$P_{t}(x)=(1+tx^{p-1})x$. Consider
$f^{*}:H^{*}_{S^{1}}(S^{V^{\prime},U^{\prime}};S^{U};\mathbb{Z}_{p})\to
H^{*}_{S^{1}}(S^{V,U};S^{U};\mathbb{Z}_{p})$. Suspending if necessary, we can
assume $U$ is even-dimensional. Then
$f^{*}(\tau^{\phi}_{V^{\prime},U^{\prime}})=\delta\tau_{U}\eta^{\phi}$, where
$\eta^{\phi}=SW(f)x^{a-1-m}$. Applying $P_{t}$ to both sides of
$SW(f)\delta\tau_{U}x^{a-1-m}=f^{*}(\tau^{\phi}_{V^{\prime},U^{\prime}})$
gives
$\displaystyle(1+tx^{p-1})^{a-1-m}SW(f)\delta\tau_{U}x^{a-1-m}$
$\displaystyle=(1+tx^{p-1})^{a^{\prime}}f^{*}(\tau^{\phi}_{V^{\prime},U^{\prime}})$
$\displaystyle=(1+tx^{p-1})^{a^{\prime}}SW(f)\delta\tau_{U}x^{a-1-m}.$
(To prove this we used that Equation (6.4) also holds for the refined Thom
class $\tau^{\phi}_{V^{\prime},U^{\prime}}$. This follows by a straightforward
extension of the usual proof. We also used that $P_{t}$ commutes with the
coboundary operator $\delta$).
If $SW(f)\neq 0\;({\rm mod}\;p)$, then the above equation reduces to
$(1+tx^{p-1})^{a-1-m}x^{a-1-m}=(1+tx^{p-1})^{a^{\prime}}x^{a-1-m}.$
This is an equality in
$H^{*}(\mathbb{P}(V);\mathbb{Z}_{p})[[t]]\cong\mathbb{Z}_{p}[x][[t]]/(x^{a})$.
Multiplying both sides by
$(1+tx^{p-1})^{-a^{\prime}}=1-a^{\prime}tx^{p-1}+\cdots$ gives
$(1+tx^{p-1})^{d-1-m}x^{a-1-m}=x^{a-1-m}\text{ in
}\mathbb{Z}_{p}[x][[t]]/(x^{a}).$
Expanding on the left, we get
$\binom{d-1-m}{j}=0\;({\rm mod}\;p)\text{ whenever }1\leq j\leq m/(p-1).$
Let $p^{e}\leq\lfloor m/(p-1)\rfloor$. Setting $j=p^{u}$ with $u=0,1,\dots,e$
we get
$\binom{d-1-m}{p^{u}}=0\;({\rm mod}\;p)\text{ for }0\leq u\leq e.$
This implies that $d-1-m$ is divisible by $p^{e+1}$. Noting that
$2m=2d-b_{+}-1$, we have $d-m-1=(b_{+}-1)/2$, and hence $(b_{+}-1)/2$ is
divisible by $p^{e+1}$. ∎
###### Remark 6.11.
The conclusion that $m\leq p-2$ when $SW(f)\neq 0\;({\rm mod}\;p)$ and
$b_{+}\neq 1\;({\rm mod}\;2p)$ was also shown in [17, Corollary 1.5].
### 6.6. $\mathbb{Z}_{p}$-actions
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let
$G=\mathbb{Z}_{p}=\langle g\rangle$ act on $X$ by orientation preserving
diffeomorphism, where $p$ is a prime. For convenience we will work with
$\mathbb{Z}_{p}$-coefficients. For odd $p$ we have
$H^{*}_{\mathbb{Z}_{p}}(pt;\mathbb{Z}_{p})\cong\mathbb{Z}_{p}[u,v]/(u^{2})$
where $deg(u)=1$, $deg(v)=2$. For $p=2$ we have
$H^{*}_{\mathbb{Z}_{2}}(pt;\mathbb{Z}_{2})\cong\mathbb{Z}_{2}[u]$ where
$deg(u)=2$. In this case we set $v=u^{2}$ and sometimes write $u=v^{1/2}$.
Suppose that $H^{+}(X)^{G}\neq 0$ and let $\phi$ be a chamber. Let
$\mathfrak{s}$ be a $G$-invariant spinc-structure. Fix a trivialisation
$G_{\mathfrak{s}}\cong S^{1}\times G$. This amounts to choosing a lift of
$\mathbb{Z}_{p}$ to the spinor bundles associated to $\mathfrak{s}$. Then the
equivariant Seiberg–Witten invariants
$SW_{\mathbb{Z}_{p},X,\mathfrak{s}}^{\phi}:H^{*}_{\mathbb{Z}_{p}}(pt;\mathbb{Z}_{p})[x]\to
H^{*-d(X,\mathfrak{s})}_{\mathbb{Z}_{p}}(pt;\mathbb{Z}_{p})$ are defined. Let
$\mathbb{C}_{j}$ be the $1$-dimensional complex representation of $G$ where
$g$ acts as multiplication by $\omega^{j}$, $\omega=e^{2\pi i/p}$. Then
$D=\bigoplus_{j=0}^{p-1}\mathbb{C}_{j}^{d_{j}}$ for some
$d_{0},\dots,d_{p-1}$. The weights $d_{j}$ can be computed using the $G$-spin
theorem. Let $b_{0}$ be the dimension of $H^{+}(X)^{G}$. To each
$j\in\mathbb{Z}_{p}$ we obtain a splitting $s_{j}:\mathbb{Z}_{p}\to
S^{1}\times\mathbb{Z}_{p}$ given by $s_{j}(g)=(\omega^{-j},g)$. Let $f$ denote
the $G$-monopole map associated to $(X,\mathfrak{s})$ and $f^{s_{j}}$ the
reduced monopole map. The expected dimension of the moduli space for
$f^{s_{j}}$ is $2d_{j}-b_{0}-1$. Let $\delta_{j}=d_{j}-(b_{0}+1)/2$. If this
is integral and non-negative then we obtain a reduced Seiberg–Witten invariant
$\overline{SW}^{s_{j},\phi}_{G,X,\mathfrak{s}}=SW_{f^{s_{j}}}^{\phi}(x^{\delta_{j}})\in\mathbb{Z}$.
If $\delta_{j}$ is non-integral or negative, then we set
$\overline{SW}^{s_{j},\phi}_{G,X,\mathfrak{s}}=0$. Recall that each splitting
$s_{j}$ defines an isomorphism
$\psi_{s_{j}}:H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z}_{p})\to
H^{*}_{\mathbb{Z}_{p}}(pt;\mathbb{Z})[x]$. If we use $s_{0}$ to identify
$H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z}_{p})$ with
$H^{*}_{\mathbb{Z}_{p}}(pt;\mathbb{Z})[x]$ then $\psi_{s_{j}}$ is given by
$\psi_{s_{j}}(x)=x-jv$, $\psi_{s_{j}}(v)=v$. Since $s_{j}(G)$ acts trivially
on the domain and codomain of $f_{s_{j}}$, we have that
$SW^{\phi}_{G,f^{s_{j}}}(\psi_{s_{j}}^{-1}(x^{m}))=\overline{SW}^{s_{j},\phi}_{G,X,\mathfrak{s}}$
if $m=\delta_{j}$ and is zero otherwise. Then since
$\psi_{s_{j}}^{-1}(x)=x+jv$, it follows that for any
$\theta\in\mathbb{Z}_{p}[x,v]$,
$SW_{G,f^{s_{j}}}^{\phi}(\theta)=c_{j}\overline{SW}^{s_{j},\phi}_{G,X,\mathfrak{s}}$,
where $c_{j}$ is the coefficient of $(x+jv)^{\delta_{j}}$ when $\theta$ is
written as a polynomial in $x+jv$ with coefficients in $\mathbb{Z}_{p}[v]$.
Let $k=\mathbb{Z}_{p}(v)$ be the ring of rational functions in $v$ with
coefficients in $\mathbb{Z}_{p}$. Let $k(x)$ be the ring of rational functions
in $x$ with coefficients in $k$. For any $a\in k$ we have a natural
homomorphism $k(x)\to k((x-a))$ from $k(x)$ into the ring $k((x-a))$ of formal
Laurent series in $x-a$ with coefficients in $k$ and with finite polar part.
If $f\in k(x)$ then the image of $f$ in $k((x-a))$ can be uniquely written as
$f=\sum_{j=-n}^{\infty}c_{j}(x-a)^{j}$, $c_{j}\in k$. We refer to $c_{j}$ as
the coefficient of $(x-a)^{j}$ in the Laurent expansion of $f$ at $a$. For any
$j\in\mathbb{Z}_{p}$ and integers $n,n_{0},n_{1},\dots,n_{p-1}$, define
$c_{j}(n;n_{0},\dots,n_{p-1})$ to be the coefficient of $(x+jv)^{n}$ in the
Laurent expansion of $\prod_{i=0}^{p-1}(x+iv)^{n_{i}}$. It follows that
$c_{j}(n;n_{0},\dots,n_{p-1})=\left(\sum_{k_{i}}\prod_{i|i\neq
j}\binom{n_{i}}{k_{i}}(i-j)^{n_{i}-k_{i}}\right)v^{\sum_{i}n_{i}-n}$
where the sum is over non-negative integers
$k_{0},\dots,\hat{k}_{j},\dots,k_{p-1}$ such that
$k_{0}+\cdots+\hat{k}_{j}+\cdots+k_{p-1}=n-n_{j}$. We will extend the
definition of $c_{j}(n;n_{0},\dots,n_{p-1})$ to non-integer values of $n$ by
setting $c_{j}(n;n_{0},\dots,n_{p-1})=0$ if $n$ is non-integral.
###### Theorem 6.12.
For any non-negative integers $m_{0},\dots,m_{p-1}$, we have the following
equality in $H^{*}(\mathbb{Z}_{p};\mathbb{Z}_{p})$:
$\displaystyle
SW^{\phi}_{\mathbb{Z}_{p},X,\mathfrak{s}}(x^{m_{0}}(x+v)^{m_{1}}\cdots(x+(p-1)v)^{m_{p-1}})$
$\displaystyle\quad\quad=e_{\mathbb{Z}_{p}}(H^{+}/(H^{+})^{g})\sum_{j=0}^{p-1}c_{j}\left(-\dfrac{(b_{0}+1)}{2};m_{0}-d_{0},\dots,m_{p-1}-d_{p-1}\right)\overline{SW}^{s_{j},\phi}_{G,X,\mathfrak{s}}.$
###### Proof.
By Theorem 6.5, we have
$SW^{\phi}_{G,f}(\theta)=e_{G}(H^{+}/(H^{+})^{g})\sum_{j=0}^{p-1}SW^{\phi}_{G,f^{s_{j}}}(e_{G_{\mathfrak{s}}}(D/D_{j})^{-1}\theta).$
If $\theta=\prod_{i=0}^{p-1}(x+iv)^{m_{i}}$, then
$e_{G_{\mathfrak{s}}}(D/D_{j})^{-1}\theta=\prod_{i=0}^{p-1}(x+iv)^{m_{i}}\prod_{i|i\neq
j}(x+jv)^{-d_{i}}.$
Hence $SW^{\phi}_{G,f^{s_{j}}}(e_{G_{\mathfrak{s}}}(D/D_{j})^{-1}\theta)$
equals $SW^{G,s_{j},\phi}_{X,\mathfrak{s}}$ times the coefficient of
$(x+jv)^{\delta_{j}}$ in $\prod_{i=0}^{p-1}(x+iv)^{m_{i}}\prod_{i|i\neq
j}(x+jv)^{-d_{i}}$. But $\delta_{j}=d_{j}-(b_{0}+1)/2$, so this is
$\alpha_{j}$ times the coefficient of $(x+jv)^{-(b_{0}+1)/2}$ in
$\prod_{i=0}^{p-1}(x+iv)^{m_{i}-d_{i}}$, which is
$c_{j}(-(b_{0}+1)/2;m_{0}-d_{0},\dots,m_{p-1}-d_{p-1})$. ∎
For instance, when $p=2$, we get:
$SW_{\mathbb{Z}_{2},X,\mathfrak{s}}^{\phi}(x^{m_{0}}(x+v)^{m_{1}})=\left(\binom{m_{1}-d_{1}}{\delta_{0}-m_{0}}\overline{SW}^{s_{0},\phi}_{G,X,\mathfrak{s}}+\binom{m_{0}-d_{0}}{\delta_{1}-m_{1}}\overline{SW}^{s_{1},\phi}_{G,X,\mathfrak{s}}\right)v^{m_{0}+m_{1}-\delta}$
where we set $\binom{a}{b}=0$ if $b$ is non-integral or negative.
## 7\. Kähler actions
Suppose $(X,I)$ is a compact complex surface with $b_{1}(X)=0$ and suppose
that $G$ is a finite group which acts on $X$ by biholomorphism. Since
$b_{1}(X)=0$, $X$ admits a Kähler metric. Furthermore, the set of Kähler
metrics is convex so by averaging we can find a $G$-invariant Kähler metric
$g$ with associated Kähler form $\omega$. The Hodge decomposition yields an
equality $H^{+}(X)=\mathbb{R}[\omega]\oplus Re(H^{2,0}(X))$ and hence
$H^{+}(X)\cong\mathbb{R}\oplus H^{0}(X,K_{X})$ where $K_{X}$ denotes the
canonical bundle of $X$. Since $H^{0}(X,K_{X})$ is a complex vector space,
this provides a distinguished orientation on $H^{+}(X)$ given by
$\\{\omega,e_{1},Ie_{1},\dots,e_{n},Ie_{n}\\}$, where $e_{1},\dots,e_{n}$ is a
complex basis for $H^{0}(X,K_{X})$. We also have a distinguished chamber given
by setting $\phi=[\omega]$. We call this the Kähler chamber and we call
$-\phi$ the anti-Kähler chamber. Since
$H^{+}(X)^{G}\cong\mathbb{R}[\omega]\oplus H^{0}(X,K_{X})^{G}$, we see that
$dim(H^{+}(X)^{G})\geq 1$ and equals $1$ only if $H^{0}(X,K_{X})^{G}=0$.
The complex structure on $X$ defines a spinc-structure $\mathfrak{s}_{can}$,
the canonical spinc-structure with spinor bundles
$S^{\pm}=\wedge^{0,ev/odd}T^{*}X$. Any other spinc-structure is of the form
$\mathfrak{s}_{L}=L\otimes\mathfrak{s}_{can}$ for some complex line bundle $L$
and we have $c(\mathfrak{s}_{L})=2c_{1}(L)-c_{1}(K_{X})$. The spinor bundles
for $\mathfrak{s}_{L}$ are given by $S^{\pm}_{L}=L\otimes S^{\pm}$. The charge
conjugate of $\mathfrak{s}_{L}$ is $\mathfrak{s}_{K_{X}^{*}L}$. A spinc-
structure $\mathfrak{s}_{L}$ is preserved by $G$ if and only $G$ preserves the
isomorphism class of $L$. Furthermore, since $G$ has a canonical lift to
$S^{\pm}$ we see that lifts of $G$ to $S^{\pm}_{L}$ correspond to lifts of $G$
to $L$.
Let $L$ be a complex line bundle whose isomorphism class is preserved by $G$.
To compute the equivariant Seiberg–Witten invariants of
$(X,\mathfrak{s}_{L})$, we first consider the Seiberg–Witten equations with
respect to a $2$-form perturbation of the form $\eta=i\lambda\omega$, where
$\lambda$ is a sufficiently large positive real number. Note that it suffices
to consider only the Kähler chamber since charge conjugation exchanges the
Kähler and anti-Kähler chambers. We follow the approach of [27, Chapter 12].
Let $\mathcal{M}=\mathcal{M}(X,L,g,\eta)$ denote the moduli space of solutions
to the $\eta$-perturbed Seiberg–Witten equations on $X$ with respect to the
metric $g$ and spinc-structure $\mathfrak{s}_{L}$. By [27, Proposition 12.23],
we have a natural bijection between $\mathcal{M}$ and the space of effective
divisors on $X$ representing $c_{1}(L)$. If the image of $c_{1}(L)$ in
$H^{2}(X;\mathbb{R})$ is not of type $(1,1)$, then there are no divisors
representing $c_{1}(L)$ and hence $\mathcal{M}$ is empty. On the other hand if
$c_{1}(L)$ is of type $(1,1)$, then $L$ admits a holomorphic structure. Since
$b_{1}(X)=0$, the holomorphic structure is unique up to isomorphism and so we
can regard $L$ as a fixed holomorphic line bundle. Let $V^{i}=H^{i}(X,L)$
denote the cohomology groups of $L$ and let $h^{i}(L)$ denote the dimension of
$V^{i}$. The $V^{i}$ are representations of $G_{\mathfrak{s}_{L}}$ where the
$S^{1}$-subgroup acts by scalar multiplication. Effective divisors
representing $c_{1}(L)$ are in bijection with non-zero holomorphic sections of
$L$ up to scale, so we have an identification
$\mathcal{M}\cong\mathbb{P}(V^{0})$. Under this identification the action of
$G$ on $\mathcal{M}$ corresponds to the action on $\mathbb{P}(V^{0})$ induced
by the action of $G_{\mathfrak{s}_{L}}$ on $V^{0}$.
Although the moduli space $\mathcal{M}$ is a smooth manifold, the perturbation
$\eta=i\lambda\omega$ is typically not a regular perturbation. That is, the
moduli space $\mathcal{M}$ is typically not cut out transversally and its
dimension, $2(h^{0}(L)-1)$, is typically larger than the expected dimension
which is $2(h^{0}(L)-1)-2(h^{1}(L)-h^{2}(L)+h^{2,0}(X))$. To compute the
equivariant Seiberg–Witten equations in this setting we will make use of the
technique of obstruction bundles [13], [27, §12.9]. The failure of
$\mathcal{M}$ to be cut out transversally is measured by a bundle
$Obs\to\mathcal{M}$ called the obstruction bundle. The fibres of $Obs$ are the
cokernels of the linearisation of the Seiberg–Witten equations. As shown in
[27, §12.9], we have an exact sequence of bundles on $\mathcal{M}$:
(7.1) $0\to\widetilde{V}^{1}\to Obs\to
H^{2}(X,\mathcal{O})\to\widetilde{V}^{2}\to 0,$
where $\widetilde{V}^{i}$ denotes the vector bundle over
$\mathbb{P}(V^{0})=S(V^{0})/S^{1}$ given by
$\widetilde{V}^{i}=V^{i}\times_{S^{1}}S(V^{0})$ and $H^{2}(X,\mathcal{O})$ is
to be regarded as a trivial vector bundle on $\mathbb{P}(V^{0})$. In the
presence of a $G$-action, the obstruction bundle $Obs$ is a $G$-equivariant
vector bundle and the above sequence is an exact sequence of $G$-equivariant
vector bundles on $\mathcal{M}$.
###### Lemma 7.1.
Assume that $G_{\mathfrak{s}}$ is split. Then for any $\theta\in
H^{*}_{G_{\mathfrak{s}_{\scalebox{0.7}{$\scriptscriptstyle
L$}}}}\\!(pt;\mathbb{Z})$, we have
$SW_{G}^{\omega}(\theta)=(\pi_{\mathbb{P}(V^{0})})_{*}(e_{G}(Obs)\theta)$
where $\theta$ is regarded as an element of
$H^{*}_{G}(\mathbb{P}(V^{0});\mathbb{Z})$ by pulling it back to $S(V^{0})$ and
using $H^{*}_{G_{\mathfrak{s}_{\scalebox{0.7}{$\scriptscriptstyle
L$}}}}\\!(S(V^{0});\mathbb{Z})\cong H^{*}_{G}(\mathbb{P}(V^{0});\mathbb{Z})$.
###### Proof.
In the non-equivariant setting, this follows from the technique of obstruction
bundles. We need to extend the result to the equivariant setting. If $Obs$
admits a $G$-invariant section whhich is tranverse to the zero section, then
the usual argument can be carried out equivariantly. However in general an
equivariant vector bundle need not admit invariant sections which are
transverse to the zero section. To get around this problem we will work with
families instead of equivariantly.
For any compact smooth manifold $B$ and any principal $G$-bundle $P\to B$, we
have an associated family $E=P\times_{G}X$. Since the Kähler structure of $X$
is preserved by $G$, the family $E$ is a Kähler family in the sense that the
fibres are equipped with a smoothly varying Kähler structure. We will use this
structure to evaluate the families Seiberg–Witten invariants
$SW_{G,E,\mathfrak{s}_{L}}^{\omega}(\theta)$. The families moduli space is
$\mathcal{M}_{E}=P\times_{G}\mathcal{M}$. It is not cut out transversally, but
the obstruction bundle technique can be applied. The obstruction bundle
$Obs_{E}$ for this family is just the associated vector bundle
$Obs_{E}=P\times_{G}Obs$ and therefore we have
$SW_{G,E,\mathfrak{s}_{L}}^{\omega}(\theta)=(\pi_{*})(e(Obs_{E})\theta)$
where $\pi$ is the projection map $\pi:\mathcal{M}_{E}\to B$. It follows
immediately that
$SW_{G,E,\mathfrak{s}_{L}}^{\omega}(\theta)=(f_{P})^{*}((\pi_{\mathbb{P}(V^{0})})_{*}(e_{G}(Obs)\theta))$
where $f_{P}:B\to BG$ is the classifying map for $P$. Then by Theorem 4.2, it
follows that
$SW_{G}^{\omega}(\theta)=(\pi_{\mathbb{P}(V^{0})})_{*}(e_{G}(Obs)\theta)$. ∎
We turn to the computation of $e_{G}(Obs)$. Since $Obs$ is a complex vector
bundle of rank $r=h^{1}(L)-h^{2}(L)+h^{2,0}(X)$, we have
$e_{G}(Obs)=c_{r,G}(Obs)$, where $c_{r,G}$ denotes the $r$-th equivariant
Chern class. Let $c_{G}=1+c_{1,G}+\cdots$ denote the total equivariant Chern
class and $s_{G}=1+s_{1,G}+\cdots$ the total equivariant Segre class. From
(7.1) we have
$c_{G}(Obs)=c_{G}(\widetilde{V}^{1})s_{G}(\widetilde{V}^{2})c_{G}(H^{2}(X,\mathcal{O}))$
and hence
$e_{G}(Obs)=\sum_{\begin{subarray}{c}i+j+k=r\\\ i,j,k\geq
0\end{subarray}}c_{i,G}(\widetilde{V}^{1})s_{j,G}(\widetilde{V}^{2})c_{k,G}(H^{2}(X,\mathcal{O})).$
Assume now that $G_{\mathfrak{s}_{\scalebox{0.7}{$\scriptscriptstyle L$}}}\\!$
is a split extension. Fix a splitting
$G_{\mathfrak{s}_{\scalebox{0.7}{$\scriptscriptstyle L$}}}\\!\cong S^{1}\times
G$. A choice of splitting amounts to a choice of lift of the $G$-action to
$L$. This makes $V^{i}$ into representations of $G$ and then
$\widetilde{V}^{i}\cong V^{i}\otimes\mathcal{O}_{V^{0}}(1)$. The equivariant
Chern and Segre classes of $\widetilde{V}^{i}$ can now be expressed in terms
of the Chern and Segre classes of $V^{i}$ and
$x=c_{1,G}(\mathcal{O}_{V^{0}}(1))$ using the following identities for a
vector bundle $E$ of rank $r$ and a line bundle $N$:
$c_{j}(E\otimes N)=\sum_{l=0}^{j}c_{l}(E)c_{1}(N)^{j-l}\binom{r-l}{j-l},\quad
s_{j}(E\otimes N)=\sum_{l=0}^{j}s_{l}(E)c_{1}(N)^{j-l}\binom{-r-l}{j-l}.$
The same result also applies more generally when $E$ is a virtual vector
bundle. We can simplify the expression for $e_{G}(Obs)$ by writing it in terms
of the virtual bundle $\widetilde{V}^{1}-\widetilde{V}^{2}$, namely
$e_{G}(Obs)=\sum_{\begin{subarray}{c}i+j=r\\\ i,j\geq
0\end{subarray}}c_{i,G}(\widetilde{V}^{1}-\widetilde{V}^{2})c_{j,G}(H^{2}(X,\mathcal{O})).$
We also have $H^{*}_{G_{\mathfrak{s}_{\scalebox{0.7}{$\scriptscriptstyle
L$}}}}\\!(pt;\mathbb{Z})\cong H^{*}_{G}(pt;\mathbb{Z})[x]$ and so it suffices
to compute $SW_{G}^{\omega}(x^{m})$ for each $m\geq 0$. We also have that
$(\pi_{\mathbb{P}(V^{0})})_{*}(x^{j})=s_{j-(d-1),G}(V^{0})$ where
$d=h^{0}(L)-h^{1}(L)+h^{2}(L)$. Putting it all together, we have
$\displaystyle
SW_{G}^{\omega}(x^{m})=(\pi_{\mathbb{P}(V^{0})})_{*}\left(\sum_{\begin{subarray}{c}i+j=r\\\
i,j\geq
0\end{subarray}}c_{i,G}(\widetilde{V}^{1}-\widetilde{V}^{2})c_{j,G}(H^{2}(X,\mathcal{O}))x^{m}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=r\\\ i,j\geq
0\end{subarray}}\sum_{l=0}^{i}\binom{h^{1}(L)-h^{2}(L)-l}{i-l}c_{l,G}(V^{1}-V^{2})c_{j,G}(H^{2}(X,\mathcal{O}))(\pi_{\mathbb{P}(V^{0})})_{*}(x^{m+i-l})$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=r\\\ i,j\geq
0\end{subarray}}\sum_{l=0}^{i}\binom{h^{1}(L)-h^{2}(L)-l}{i-l}s_{i-l+m-(h^{0}(L)-1),G}(V^{0})c_{l,G}(V^{1}-V^{2})c_{j,G}(H^{2}(X,\mathcal{O})).$
###### Theorem 7.2.
Let $X$ be a compact complex surface with $b_{1}(X)=0$. Let $G$ be a finite
group that acts on $X$ by biholomorphisms. Let $L$ be a $G$-equivariant line
bundle. If $L$ is not holomorphic, or if $L$ is holomorphic and $h^{0}(L)=0$,
then $SW^{\omega}_{G,X,\mathfrak{s}_{L}}=0$. If $L$ is holomorphic and
$h^{0}(L)>0$, then
$\displaystyle SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=r\\\ i,j\geq
0\end{subarray}}\sum_{l=0}^{i}\binom{h^{1}(L)-h^{2}(L)-l}{i-l}s_{i-l+m-(h^{0}(L)-1),G}(V^{0})c_{l,G}(V^{1}-V^{2})c_{j,G}(H^{2}(X,\mathcal{O}))$
where $V^{i}=H^{i}(X,L)$, $d=h^{0}(L)-h^{1}(L)+h^{2}(L)$,
$r=h^{1}(L)-h^{2}(L)+h^{2,0}(X)$.
The expression for $SW_{G,X,\mathfrak{s}_{L}}^{\omega}$ given by Theorem 7.2
gives a formula for $SW_{G,X,\mathfrak{s}_{L}}^{\omega}$ purely in terms of
the representations $V^{0},V^{1},V^{2}$ and $H^{2}(X,\mathcal{O})$ and hence
purely in terms of the complex geometry of $X$. Next, we restrict to the case
$b_{+}(X)=1$.
###### Theorem 7.3.
Let $X$ be a compact complex surface with $b_{1}(X)=0$ and $b_{+}(X)=1$. Let
$G$ be a finite group that acts on $X$ by biholomorphisms. Let $L$ be a
$G$-equivariant line bundle. Let $D=V^{0}-V^{1}+V^{2}$ and
$d=h^{0}(L)-h^{1}(L)+h^{2}(L)$.
* (1)
If $h^{0}(L)>0$, then
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=s_{m-(d-1)}(D),\quad
SW_{G,X,\mathfrak{s}_{L}}^{-\omega}(x^{m})=0.$
* (2)
If $h^{2}(L)>0$, then
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=0,\quad
SW_{G,X,\mathfrak{s}_{L}}^{-\omega}(x^{m})=-s_{m-(d-1)}(D).$
* (3)
If $h^{0}(L)=h^{2}(L)=0$, then $SW_{G,X,\mathfrak{s}_{L}}^{\pm\omega}=0$.
###### Proof.
Part (3) is clear, so it remains to prove (1) and (2). Since $b_{+}(X)=1$, we
have $H^{2}(X,\mathcal{O})\cong H^{0}(X,K_{X})^{*}=0$. If
$h^{0}(L),h^{2}(L)>0$, then there are non-zero holomorphic sections
$\alpha,\beta$ of $L$ and $K^{*}_{X}L$. But then $\alpha\beta$ is a non-zero
section of $K_{X}$, which is impossible. This means that at most one of
$h^{0}(L),h^{2}(L)$ is non-zero. Consider the case that $h^{0}(L)>0$,
$h^{2}(L)=0$. The case $h^{0}(L)=0$, $h^{2}(L)>0$ will follow from charge
conjugation symmetry. If $h^{0}(L)>0$, then
$SW_{G,X,\mathfrak{s}_{L}}^{-\omega}=0$ and hence the wall-crossing formula
gives $SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=s_{m-(d-1)}(D)$. ∎
Next, we consider the case that $X$ is a K3 surface:
###### Theorem 7.4.
Let $X$ be a complex K3 surface and let $G$ be a finite group which acts on
$X$ by biholomorphisms. Let $L$ be a $G$-equivariant line bundle.
* (1)
If $c_{1}(L)$ is not $(1,1)$ then $SW_{G,X,\mathfrak{s}_{L}}^{\pm\omega}=0$.
* (2)
If $c_{1}(L)$ is $(1,1)$, $c_{1}(L)\neq 0$ and $G$ acts trivially on $K_{X}$,
then there is only one chamber and $SW_{G,X,\mathfrak{s}_{L}}=0$.
* (3)
If $c_{1}(L)=0$ and $G$ acts trivially on $K_{X}$, then there is only one
chamber and $SW_{G,X,\mathfrak{s}_{L}}(x^{m})=s_{1,G}(L)^{m}$.
* (4)
If $c_{1}(L)=0$ and $G$ acts non-trivially on $K_{X}$, then
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=s_{1,G}(L)^{m}$,
$SW_{G,X,\mathfrak{s}_{L}}^{-\omega}=(s_{1,G}(L)-s_{1,G}(K_{X}))^{m}$.
* (5)
If $c_{1}(L)$ is $(1,1)$, $c_{1}(L)\neq 0$ and $G$ acts non-trivially on
$K_{X}$, then there are two chambers. If $h^{0}(L)>0$, then
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=e_{G}(H^{0}(X,K_{X}))s_{m-(d-1)}(D),\quad
SW_{G,X,\mathfrak{s}_{L}}^{-\omega}(x^{m})=0.$
If $h^{0}(L)=0$, then
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=0,\quad
SW_{G,X,\mathfrak{s}_{L}}^{-\omega}(x^{m})=-e_{G}(H^{0}(X,K_{X}))s_{m-(d-1)}(D).$
###### Proof.
Case (1) is clear, so we can assume that $c_{1}(L)$ is $(1,1)$.
Suppose that $h^{0}(L)$ and $h^{2}(L)$ are non-zero. If $\alpha\in
H^{0}(X,L),\beta\in H^{0}(X,L^{*})$ are non-zero, then $\alpha\beta\in
H^{0}(X,\mathcal{O})$ is non-zero and hence non-vanishing. This means that
$\alpha,\beta$ are non-vanishing and hence $c_{1}(L)=0$. This means that if
$c_{1}(L)\neq 0$ then $h^{0}(L)$ or $h^{2}(L)$ is zero. Now if the action of
$G$ on $K_{X}$ is trivial, then $H^{+}(X)^{G}=H^{+}(X)$ and there is only one
chamber. Then it follows that $SW_{G,X,\mathfrak{s}_{L}}=0$ since $h^{0}(L)=0$
or $h^{2}(L)=0$. This proves (2). If $c_{1}(L)\neq 0$ and $G$ acts non-
trivially on $K_{X}$ then there are two chambers, but we have that either
$h^{0}(L)=0$, in which case $SW_{G,X,\mathfrak{s}_{L}}^{\omega}=0$, or
$h^{2}(L)=0$, in which case $SW_{G,X,\mathfrak{s}_{L}}^{-\omega}=0$. Then the
invariants for the other chamber are given by the wall-crossing formula. This
proves (5).
It remains to prove (3) and (4). Hence we assume that $c_{1}(L)=0$. This means
that $h^{0}(L)=h^{2}(L)=1$, $h^{1}(L)=0$, $d=2$, $r=0$. So Theorem 7.2
simplifies to
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}(x^{m})=\binom{-1}{0}s_{m,G}(V^{0})=s_{m,G}(V^{0}).$
But $V^{0}=H^{0}(X,L)\cong L$ is $1$-dimensional, hence
$s_{m,G}(V^{0})=s_{1,G}(L)^{m}$. This proves (3). Finally in case (4) the
formula for $SW_{X,G,\mathfrak{s}_{L}}^{-\omega}$ follows from the formula for
$SW_{G,X,\mathfrak{s}_{L}}^{\omega}$ and charge conjugation, or alternatively
from the wall-crossing formula. ∎
## 8\. Gluing formulas
In this section we prove gluing formulas for the equivariant Seiberg–Witten
invariants of equivariant connected sums.
Let $f:S^{V,U}\to S^{V^{\prime},U^{\prime}}$ be a $G$-monopole map. Define the
$G_{\mathfrak{s}}$-equivariant degree $deg_{G_{\mathfrak{s}}}(f)\in
H^{b_{+}-2d}_{G_{\mathfrak{s}}}(pt;\mathbb{Z}_{w})$ by
$f^{*}(\tau_{V^{\prime},U^{\prime}})=deg_{G_{\mathfrak{s}}}(f)\tau_{V,U}$. The
following result is an extension of [5, Theorem 6.1] to the case where
$G_{\mathfrak{s}}$ is not necessarily split.
###### Proposition 8.1.
Let $f:S^{V,U}\to S^{V^{\prime},U^{\prime}}$ be a $G$-monopole map. If
$e(H^{+})=0$, then $deg_{G_{\mathfrak{s}}}(f)=0$. If $e(H^{+})\neq 0$, then
$d\leq 0$ and $deg_{G_{\mathfrak{s}}}(f)=e(H^{+})s_{G_{\mathfrak{s}},-d}(D)$.
Furthermore in this case we have that
$e_{G}(H^{+})s_{G_{\mathfrak{s}},j}(D)=0$ for $j>-d$.
###### Proof.
Consider the commutative square
$\textstyle{S^{V,U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S^{V^{\prime},U^{\prime}}}$$\textstyle{S^{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\iota}$$\textstyle{S^{U^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$
where $\iota:S^{U}\to S^{U^{\prime}}$ is given by inclusion. We have
$i^{*}f^{*}(\tau_{V^{\prime},U^{\prime}})=i^{*}(\tau_{V,U}deg_{G_{\mathfrak{s}}}(f))=\tau_{U}e_{G_{\mathfrak{s}}}(V)deg_{G_{\mathfrak{s}}}(f).$
On the other hand,
$\iota^{*}i^{*}(\tau_{V^{\prime},U^{\prime}})=\iota^{*}(\tau_{U^{\prime}}e_{G_{\mathfrak{s}}}(V^{\prime}))=\tau_{U}e(H^{+})e_{G_{\mathfrak{s}}}(V^{\prime}).$
Since $f\circ i=i\circ\iota$, we may equate these two expressions, giving
$e_{G_{\mathfrak{s}}}(V)deg_{G_{\mathfrak{s}}}(f)=e(H^{+})e_{G_{\mathfrak{s}}}(V^{\prime}).$
Using a spectral sequence argument it can be shown that
$e_{G_{\mathfrak{s}}}(V)$ is not a zero divisor and so the above equation
uniquely determines $deg_{G_{\mathfrak{s}}}(f)$. However, it seems difficult
to obtain an explicit formula for $deg_{G_{\mathfrak{s}}}(f)$ from this
equation. We can remedy this by considering the larger group
$\widehat{G}_{\mathfrak{s}}=S^{1}\times G_{\mathfrak{s}}$. We let
$\widehat{G}_{\mathfrak{s}}$ act on $S^{V,U}$ and $S^{V^{\prime},U^{\prime}}$
through the homomorphism $S^{1}\times G_{\mathfrak{s}}\to G_{\mathfrak{s}}$
given by $(u,g)\mapsto ug$. Then $f$ can be regarded as a
$\widehat{G}_{\mathfrak{s}}$-equivariant map. Repeating the above computation
for this larger group gives
(8.1)
$e_{\widehat{G}_{\mathfrak{s}}}(V)deg_{\widehat{G}_{\mathfrak{s}}}(f)=e(H^{+})e_{\widehat{G}_{\mathfrak{s}}}(V^{\prime}).$
Let $\mathbb{C}_{1}$ be the $1$-dimensional representation of $S^{1}\times
G_{\mathfrak{s}}$ where the first factor acts by scalar multiplication and the
second factor acts trivially. Then
$H^{*}_{\widehat{G}_{\mathfrak{s}}}(pt;\mathbb{Z}_{w})\cong
H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z}_{w})[y]$ where
$y=c_{1}(\mathbb{C}_{1})$. Equation (8.1) can be expanded as
$(y^{a}+c_{G,1}(V)y^{a-1}+\cdots+c_{G,a}(V))deg_{\widehat{G}_{\mathfrak{s}}}(f)=e(H^{+})(y^{a^{\prime}}+c_{G,1}(V^{\prime})y^{a-1}+\cdots+c_{G,a^{\prime}}(V^{\prime})).$
If $e(H^{+})=0$, then it follows that $deg_{\widehat{G}_{\mathfrak{s}}}(f)=0$.
Suppose instead that $e(H^{+})\neq 0$. Then we can write
$deg_{\widehat{G}_{\mathfrak{s}}}(f)=f_{0}y^{k}+f_{1}y^{k-1}+\cdots+f_{k}$ for
some $k\geq 0$, where $f_{0}\neq 0$. Expanding the left hand side of the above
equation we see that $a^{\prime}=k+a$, hence $d=a-a^{\prime}=-k\leq 0$. Now
consider the above equation in the group of formal Laurent series of the form
$\sum_{j=-\infty}^{n}c_{n}y^{n}$ where $n$ is any integer and $c_{j}\in
H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z}_{w})$. This group is a module over the
ring $H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z})[[y^{-1}]]$ of formal power
series in $y$ with coefficients in $H^{*}_{G_{\mathfrak{s}}}(pt;\mathbb{Z})$.
Multiplying both sides by $1+y^{-1}s_{G,1}(V)+y^{-2}s_{G,2}(V)+\cdots$ we
obtain
$y^{a}(f_{0}y^{-d}+f_{1}y^{-d-1}+\cdots+f_{k})=e(H^{+})(y^{a^{\prime}}+s_{G,1}(D)y^{a^{\prime}-1}+s_{G,2}(D)y^{a^{\prime}-2}+\cdots).$
Equating coefficients we see that $f_{j}=e(H^{+})s_{G,j}(D)$ for $j\leq-d$ and
$e(H^{+})s_{G,j}(D)=0$ for $j>-d$. Thus
$deg_{\widehat{G}_{\mathfrak{s}}}(f)=e(H^{+})(y^{-d}+s_{G,1}(D)y^{-d-1}+\cdots+s_{G,-d}(D)).$
Restricting to $G_{\mathfrak{s}}\subset\widehat{G}_{\mathfrak{s}}$ amounts to
setting $y=0$ in the above equality, giving
$deg_{G_{\mathfrak{s}}}(f)=e(H^{+})s_{G,-d}(D).$
∎
### 8.1. Abstract gluing formula
We first prove a general gluing formula for the abtract Seiberg–Witten
invariants of a smash product of monopole maps. Let $G$ be a finite group and
$G_{\mathfrak{s}}$ an $S^{1}$-central extension of $G$. Suppose that we have
two $G$-monopole maps
$f_{i}:S^{V_{i},U_{i}}\to S^{V^{\prime}_{i},U^{\prime}_{i}},\quad i=1,2$
over $B=pt$ and assume $f_{1},f_{2}$ are $G_{\mathfrak{s}}$-equivariant for
the same central extension of $G$. Then the smash product
$f=f_{1}\wedge f_{2}:S^{V,U}\to S^{V^{\prime},U^{\prime}}$
is again $G_{\mathfrak{s}}$-equivariant, where $V=V_{1}\oplus V_{2}$,
$U=U_{1}\oplus U_{2}$, $V^{\prime}=V^{\prime}_{1}\oplus V^{\prime}_{2}$,
$U^{\prime}=U^{\prime}_{1}\oplus U^{\prime}_{2}$. As usual we write
$D=V-V^{\prime}$, $H^{+}=U^{\prime}-U$. Similarly define $D_{i},H^{+}_{i}$ for
$i=1,2$. Since $(H^{+})^{G}=(H^{+}_{1})^{G}\oplus(H^{+}_{2})^{G}$, we see that
$f$ admits a chamber if and only if at least one of $f_{1}$ and $f_{2}$ admits
a chamber. Suppose that this is the case and let $\phi=(\phi_{1},\phi_{2})$ be
a chamber. Thus either $\phi_{1}\neq 0$ or $\phi_{2}\neq 0$.
Our orientation conventions are as follows. We orient $H^{+}$ according to
$H^{+}=H^{+}_{1}\oplus H^{+}_{2}$, orient $U$ according to $U=U_{1}\oplus
U_{2}$ and orient $U^{\prime}$ according to $U^{\prime}=U\oplus H^{+}$. Note
that this is different from taking $U^{\prime}$ to be $U^{\prime}_{1}\oplus
U^{\prime}_{2}$ by a factor of $(-1)^{b_{2}(b_{+,1})}$, where $b_{i}$ denotes
the rank of $U_{i}$ and $b_{+,i}$ the rank of $H^{+}_{i}$.
###### Theorem 8.2.
If $(H^{+}_{1})^{G},(H^{+}_{2})^{G}$ are both non-zero, then the
Seiberg–Witten invariants of $f$ are all zero. If $(H^{+}_{1})^{G}\neq 0$,
$(H^{+}_{2})^{G}=0$, then
$SW_{G,f}^{\phi}(\theta)=SW_{G,f_{1}}^{\phi_{1}}(e_{G}(H^{+}_{2})s_{G_{\mathfrak{s}},-d_{2}}(D_{2})\theta).$
where $s_{G_{\mathfrak{s}},-d_{2}}$ is taken to be zero if $d_{2}>0$. In
particular, if $G_{\mathfrak{s}}\cong S^{1}\times G$ is the trivial extension
then
$SW_{G,f}^{\phi}(\theta)=\sum_{k+l=-d_{2}}s_{G,l}(D_{2})SW_{G,f_{1}}^{\phi_{1}}(x^{k}e_{G}(H^{+}_{2})\theta).$
###### Proof.
Without loss of generality we can assume $(H^{+}_{1})^{G}\neq 0$. If we also
have $(H^{+}_{2})^{G}\neq 0$, then $dim((H^{+})^{G})>1$ and so
$SW_{G,f}^{\phi}$ is independent of the choice of chamber. Hence without loss
of generality we can assume that $\phi=(\phi_{1},0)$, where $\phi_{1}\neq 0$.
We have that
$SW_{G,f}^{\phi}(\theta)=(\pi_{\mathbb{P}(V)})_{*}(\eta^{\phi}\theta)$ where
$f^{*}(\tau^{\phi}_{V^{\prime},U^{\prime}})=(-1)^{b}\delta\tau_{U}\eta^{\phi}$.
Similarly
$SW_{G,f_{1}}^{\phi_{1}}(\theta)=(\pi_{\mathbb{P}(V_{1})})_{*}(\eta^{\phi_{1}}_{1}\theta)$
where
$f^{*}(\tau^{\phi_{1}}_{V^{\prime}_{1},U^{\prime}_{1}})=(-1)^{b_{1}}\delta\tau_{U_{1}}\eta^{\phi_{1}}_{1}$.
Recall that the equivariant degree $deg_{G_{\mathfrak{s}}}(f_{2})$ of $f_{2}$
is defined by
$f_{2}^{*}(\tau_{V^{\prime}_{2},U^{\prime}_{2}})=\tau_{V_{2},U_{2}}deg_{G_{\mathfrak{s}}}(f_{2})$.
Consider the external cup product
$H^{i}_{G_{\mathfrak{s}}}(S^{V^{\prime}_{1},U^{\prime}_{1}},S^{U_{1}};\mathbb{Z})\times
H^{j}_{G_{\mathfrak{s}}}(S^{V^{\prime}_{2},U^{\prime}_{2}},pt;\mathbb{Z})\to
H^{i+j}_{G_{\mathfrak{s}}}(S^{V^{\prime},U^{\prime}},S^{U};\mathbb{Z}).$
Using this cup product we can write
$\tau^{\phi}_{V^{\prime},U^{\prime}}=(-1)^{b_{+,1}b_{2}}\tau^{\phi_{1}}_{V^{\prime}_{1},U^{\prime}_{2}}\tau_{V^{\prime}_{2},U^{\prime}_{2}}.$
The sign factor $(-1)^{b_{+,1}b_{2}}$ arises because our orientation
conventions described above. From this, we find
$\displaystyle(-1)^{b}\delta\tau_{U}\eta^{\phi}$
$\displaystyle=f^{*}(\tau^{\phi}_{V^{\prime},U^{\prime}})$
$\displaystyle=(-1)^{b_{+,1}b_{2}}(f_{1}\wedge
f_{2})^{*}(\tau^{\phi_{1}}_{V^{\prime}_{1},U^{\prime}_{1}}\tau_{V^{\prime}_{2},U^{\prime}_{2}})$
$\displaystyle=(-1)^{b_{+,1}b_{2}}f_{1}^{*}(\tau^{\phi_{1}}_{V^{\prime}_{1},U^{\prime}_{1}})f_{2}^{*}(\tau_{V^{\prime}_{2},U^{\prime}_{2}})$
$\displaystyle=(-1)^{b_{+,1}b_{2}+b_{1}}\delta\tau_{U_{1}}\eta^{\phi_{1}}_{1}\tau_{V_{2},U_{2}}deg_{G_{\mathfrak{s}}}(f_{2})$
$\displaystyle=(-1)^{b_{+,1}b_{2}+b_{1}+(b_{+,1}+1)b_{2}}\delta\tau_{U}e_{G_{\mathfrak{s}}}(V_{2})\eta^{\phi_{1}}_{1}deg_{G_{\mathfrak{s}}}(f_{2})$
$\displaystyle=(-1)^{b}\delta\tau_{U}e_{G_{\mathfrak{s}}}(V_{2})\eta^{\phi_{1}}_{1}deg(f_{2}).$
Hence
(8.2)
$\eta^{\phi}=e_{G_{\mathfrak{s}}}(V_{2})\eta^{\phi_{1}}_{1}deg_{G_{\mathfrak{s}}}(f_{2}).$
Let $\iota_{1}:\mathbb{P}(V_{1})\to\mathbb{P}(V_{2})$ be the inclusion map.
Since $(\iota_{1})_{*}(1)=e_{G_{\mathfrak{s}}}(V_{2})$, we have
$\displaystyle SW_{G,f}^{\phi}(\theta)$
$\displaystyle=(\pi_{\mathbb{P}(V)})_{*}(\eta^{\phi}\theta)$
$\displaystyle=(\pi_{\mathbb{P}(V)})_{*}(e_{G_{\mathfrak{s}}}(V_{2})\eta^{\phi_{1}}_{1}deg_{G_{\mathfrak{s}}}(f_{2})\theta)$
$\displaystyle=(\pi_{\mathbb{P}(V)})_{*}(\iota_{1})_{*}(\eta^{\phi_{1}}_{1}deg_{G_{\mathfrak{s}}}(f_{2})\theta)$
$\displaystyle=(\pi_{\mathbb{P}(V_{1})})_{*}(\eta^{\phi_{1}}_{1}deg_{G_{\mathfrak{s}}}(f_{2})\theta)$
$\displaystyle=SW^{\phi_{1}}_{G,f_{1}}(deg_{G_{\mathfrak{s}}}(f_{2})\theta).$
Now if $(H^{+}_{2})^{G}\neq 0$, then $e(H^{+}_{2})=0$ and
$deg_{G_{\mathfrak{s}}}(f_{2})=0$ by Proposition 8.1. Hence in this case
$SW_{G,f}^{\phi}$ vanishes. If $(H^{+}_{2})^{G}=0$, then $d_{2}\leq 0$ and
Proposition 8.1 gives
$deg_{G_{\mathfrak{s}}}(f_{2})=e(H^{+}_{2})s_{G_{\mathfrak{s}},-d_{2}}(D_{2})$,
which gives
$SW_{G,f}^{\phi}(\theta)=SW_{G,f_{1}}^{\phi_{1}}(e_{G}(H^{+}_{2})s_{G_{\mathfrak{s}},-d_{2}}(D_{2})\theta).$
Lastly if $G_{\mathfrak{s}}\cong S^{1}\times G$ is the trivial extension, then
$s_{G_{\mathfrak{s}},-d_{2}}(D_{2})=\sum_{k=0}^{-d_{2}}x^{k}s_{G,-d_{2}-k}(D_{2})$
and hence
$SW_{G,f}^{\phi}(\theta)=\sum_{k+l=-d_{2}}s_{G,l}(D_{2})SW_{G,f_{1}}^{\phi_{1}}(x^{k}e_{G}(H^{+}_{2})\theta).$
∎
### 8.2. Equivariant connected sum
Let $X,Y$ be compact, oriented, smooth $4$-manifolds. Let $G$ be a finite
group acting smoothly and orientation preservingly on $X$. Let $H$ be a
subgroup of $G$ and assume that $H$ acts smoothly and orientation preservingly
on $Y$. We will construct an action of $G$ on the connected sum of $X$ with
$|G/H|$ copies of $Y$. For this we need to assume that $X$ and $Y$ contain
points $x,y$ whose stabiliser groups are $H$. We also need that
$T_{x}X,T_{y}Y$ are isomorphic as real representations of $H$ via an
orientation reversing isomorphism. Choose such an isomorphism $\psi:T_{x}X\to
T_{y}Y$. Then we can form the $G$-equivariant connected sum
$X\\#ind^{G}_{H}(Y)$ as follows. Here $ind^{G}_{H}(Y)$ is the disjoint union
of $G/H$ copies of $Y$ which is made into a $G$-space by taking
$ind^{G}_{H}(Y)=G\times_{H}Y$. We remove from $X$ and $ind^{G}_{H}(Y)$
equivariant tubular neighbourhoods $\nu(Gx),\nu(Gy)$ of the orbits $Gx,Gy$ and
identify the boundaries of $X\setminus\nu(Gx)$,
$ind^{G}_{H}(Y)\setminus\nu(Gy)$ by the orientation reversing $G$-equivariant
isomorphism $\partial\nu(Gx)\to\partial\nu(Gy)$ determined by $\psi$. We note
that the construction of $X\\#ind^{G}_{H}(Y)$ depends on the choice of points
$x,y$ and the isomorphism $\psi$. The underlying smooth manifold of
$X\\#ind^{G}_{H}(Y)$ is $X\\#|G/H|Y$, the connected sum of $X$ with $|G/H|$
copies of $Y$. Note also that if $b_{1}(X)=b_{1}(Y)=0$, then
$b_{1}(X\\#ind^{G}_{H}(Y))=0$.
Suppose that $\mathfrak{s}_{X}$ is a $G$-invariant spinc-structure on $X$ and
$\mathfrak{s}_{Y}$ is a $H$-invariant spinc-structure on $Y$. Let $S^{\pm}$
denote the spinor bundles corresponding to $\mathfrak{s}_{Y}$. Since $H$ fixes
$y$, we see that $H_{\mathfrak{s}_{Y}}$ lifts to the fibres $S^{\pm}_{y}$.
Hence the extension class of $H_{\mathfrak{s}_{Y}}$ is given by
$Sq_{3}^{\mathbb{Z}}(T_{y}Y)\in H^{3}_{H}(pt;\mathbb{Z})$, the third integral
Stiefel–Whitney class of $T_{y}Y$. Similarly, since $H$ fixes $x$, we see that
the restriction $G_{\mathfrak{s}_{X}}|_{H}$ of the extension
$G_{\mathfrak{s}_{X}}$ to $H$ has extension class
$Sq_{3}^{\mathbb{Z}}(T_{x}X)$. Since $T_{x}X$ and $T_{y}Y$ are orientation
reversingly isomorphic, it follows that the extension classes agree (reversing
orientation has the effect of exchanging the roles of positve and negative
spinors). Therefore $H_{\mathfrak{s}_{Y}}$ and $G_{\mathfrak{s}_{X}}|_{H}$ are
isomorphic as extensions of $H$. The isomorphism $\psi:T_{x}X\to T_{y}Y$
determines the isomorphism $G_{\mathfrak{s}_{X}}|_{H}\to
H_{\mathfrak{s}_{Y}}$. We can induce a $G$-invariant spinc-structure
$ind^{G}_{H}(\mathfrak{s}_{Y})$ on $ind^{G}_{H}(Y)$ as follows. We take the
spinor bundles of $ind^{G}_{H}(\mathfrak{s}_{Y})$ to be given by
$G_{\mathfrak{s}_{X}}\times_{H_{\mathfrak{s}_{Y}}}S^{\pm}$. The underlying
isomorphism class of $ind^{G}_{H}(\mathfrak{s}_{Y})$ is simply given by
equipping each connected component of $ind^{G}_{H}(Y)$ with a copy of
$\mathfrak{s}_{Y}$. However our construction makes it clear that the extension
of $G$ determined by $ind^{G}_{H}(\mathfrak{s}_{Y})$ is isomorphic to
$G_{\mathfrak{s}_{X}}$. It follows that the spinc-structures
$\mathfrak{s}_{X}$ and $ind^{G}_{H}(\mathfrak{s}_{Y})$ can be identified
equivariantly on the boundaries of $X\setminus\nu(Gx)$ and
$ind^{G}_{H}(Y)\setminus\nu(Gx)$ to form a spinc-structure
$\mathfrak{s}_{X}\\#ind^{G}_{H}(\mathfrak{s}_{Y})$ on
$X\\#ind^{G}_{H}(\mathfrak{s}_{Y})$.
To summarise, given a $G$-invariant spinc-structure $\mathfrak{s}_{X}$ on $X$
and a $H$-invariant spinc-structure $\mathfrak{s}_{Y}$ on $Y$, the choice of
an orientation reversing isomorphism $\psi:T_{x}X\to T_{y}Y$ of
representations of $H$ allows us to construct a $G$-invariant spinc-structure
$\mathfrak{s}=\mathfrak{s}_{X}\\#ind^{H}_{G}(\mathfrak{s}_{Y})$ on
$X\\#ind^{G}_{H}(Y)$ and moreover the extensions $G_{\mathfrak{s}}$ and
$G_{\mathfrak{s}_{X}}$ are isomorphic.
Suppose $f:S^{V,U}\to S^{V^{\prime},U^{\prime}}$ is an $H$-equivariant
monopole map with respect to some extension $H_{\mathfrak{s}}$. Suppose
$H\subseteq G$ and $H_{\mathfrak{s}}=G_{\mathfrak{s}}|_{H}$ for some extension
$G_{\mathfrak{s}}$. We will define an induced monopole map
$ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(f):S^{ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(V),ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(U)}\to
S^{ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(V^{\prime}),ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(U^{\prime})}$
as follows. Choose representatives $g_{1},\dots,g_{n}$ for the left cosets of
$H$ in $G$ and choose lifts $\tilde{g}_{i}$ of $g_{i}$ to $G_{\mathfrak{s}}$.
Then $\tilde{g}_{1},\dots,\tilde{g}_{n}$ are representatives for the left
cosets of $H_{\mathfrak{s}}$ in $G_{\mathfrak{s}}$. For any representation $W$
of $H_{\mathfrak{s}}$,
$ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(W)=\mathbb{R}[G_{\mathfrak{s}}]\otimes_{\mathbb{R}[H_{\mathfrak{s}}]}W$.
Regard $S^{W}$ as $W\cup\\{\infty\\}$ and similarly
$S^{ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(W)}=ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(W)\cup\\{\infty\\}$.
Then $ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(f)$ is defined by
$ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(f)(\sum_{i}g_{i}\otimes
w_{i})=\begin{cases}\sum_{i}g_{i}\otimes f(w_{i})&\text{if
}f(w_{i})\neq\infty\text{ for all }i,\\\ \infty&\text{otherwise}.\end{cases}$
This is well-defined because $f$ is $H_{\mathfrak{s}}$-equivariant.
###### Theorem 8.3.
Let $X,Y$ be compact, oriented, smooth $4$-manifolds with
$b_{1}(X)=b_{1}(Y)=0$. Let $G$ be a finite group acting smoothly and
orientation preservingly on $X$. Let $H$ be a subgroup of $G$ and suppose that
$H$ acts smoothly and orientation preservingly on $Y$. Suppose that there is
an $x\in X$ and $y\in Y$ with stabiliser group $H$ and an orientation
reversing isomorphism $\psi:T_{x}X\to T_{y}Y$ of representations of $H$.
Suppose that $\mathfrak{s}_{X}$ is a $G$-invariant spinc-structure on $X$ and
$\mathfrak{s}_{Y}$ is a $H$-invariant spinc-structure on $Y$. Set
$Z=X\\#ind^{G}_{H}(Y)$,
$\mathfrak{s}=\mathfrak{s}_{X}\\#ind^{G}_{H}(\mathfrak{s}_{Y})$. Then:
* (1)
If $H^{+}(X)^{G},H^{+}(Y)^{H}$ are both non-zero then the equivariant
Seiberg–Witten invariants of $Z$ vanish.
* (2)
If $H^{+}(X)^{G}\neq 0$ and $H^{+}(Y)^{H}=0$, then for any chamber $\phi\in
H^{+}(X)^{G}\setminus\\{0\\}$ we have
$SW^{\phi}_{G,Z,\mathfrak{s}}(\theta)=SW^{\phi}_{G,X,\mathfrak{s}_{X}}(e(ind^{G}_{H}(H^{+}(Y)))s_{G_{\mathfrak{s}},-d_{Y}}(ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(D_{Y}))\theta).$
###### Proof.
Let $f_{X},f_{Y},f_{Z}$ denote the equivariant monopole maps for $X,Y$ and
$Z$. Bauer’s connected sum formula [9] extends easily to the $G$-equivariant
setting. Then by a straightforward extension of [29, Theorem 3.1], we see that
$f_{Z}=f_{X}\wedge ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(f_{Y})$.
Set $f=f_{Z}$, $f_{1}=f_{X}$,
$f_{2}=ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(f_{Y})$. We use the notation
$H^{+},D,H^{+}_{i},D_{i}$ as in Section 8.1. Then $H^{+}_{1}=H^{+}(X)$,
$H^{+}_{2}=ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(H^{+}(Y))$,
$(H^{+}_{1})^{G}=H^{+}(X)^{G}$, $(H^{+}_{2})^{G}=H^{+}(Y)^{H}$. Hence if
$H^{+}(X)^{G},H^{+}(Y)^{H}$ are both non-zero, then the Seiberg–Witten
invariants of $Z$ vanish by Theorem 8.2.
Now suppose $H^{+}(X)^{G}\neq 0$ and $H^{+}(Y)^{H}=0$. Theorem 8.2 gives
$SW^{\phi}_{G,Z,\mathfrak{s}}(\theta)=SW^{\phi}_{G,X,\mathfrak{s}}(e(H^{+}_{2})s_{G_{\mathfrak{s}},-d_{Y}}(D_{2})\theta).$
The result follows, since
$H^{+}_{2}=ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(H^{+}(Y))$ and
$D_{2}=ind^{G_{\mathfrak{s}}}_{H_{\mathfrak{s}}}(D_{Y})$. ∎
We consider now the case where $H^{+}(X)^{G}=0$ and $H^{+}(Y)^{H}\neq 0$. Here
it is more difficult to obtain a general formula for the cohomological
invariants so we restrict to the case $G=\mathbb{Z}_{p}$ and $H=1$. Let
$\mathbb{C}_{j}$ denote the $1$-dimensional complex representation of
$\mathbb{Z}_{p}=\langle g\rangle$ where $g$ acts as multiplication by
$\omega^{j}$, $\omega=e^{2\pi i/p}$. When $p$ is odd we may choose the
orientation on $ind^{G}_{1}(H^{+}(Y))$ such that
$ind^{G}_{1}(H^{+}(Y))/H^{+}(Y)\cong\bigoplus_{j=1}^{(p-1)/2}\mathbb{C}_{j}^{b_{+}(Y)}$.
Recall that $H^{*}_{\mathbb{Z}_{p}}(pt;\mathbb{Z})\cong\mathbb{Z}[v]/(pv)$,
$deg(v)=2$. When $p=2$ to avoid orientability issues we use
$\mathbb{Z}_{2}$-coefficients. We have
$H^{*}_{\mathbb{Z}_{2}}(pt;\mathbb{Z}_{2})\cong\mathbb{Z}_{2}[u]$, $deg(u)=1$.
In this case we set $v=u^{2}$ and we will also denote $u$ by $v^{1/2}$.
###### Theorem 8.4.
Let $X,Y$ be compact, oriented, smooth $4$-manifolds with
$b_{1}(X)=b_{1}(Y)=0$. Let $G=\mathbb{Z}_{p}$ where $p$ is prime act smoothly
and orientation preservingly on $X$. Suppose that $\mathfrak{s}_{X}$ is a
$G$-invariant spinc-structure on $X$ and $\mathfrak{s}_{Y}$ is a spinc-
structure on $Y$ with $d(Y,\mathfrak{s}_{Y})=2d_{Y}-b_{+}(Y)-1=0$. Set
$Z=X\\#ind^{G}_{1}(Y)$,
$\mathfrak{s}=\mathfrak{s}_{X}\\#ind^{G}_{1}(\mathfrak{s}_{Y})$. Suppose that
$H^{+}(X)^{G}=0$ and $b_{+}(Y)>0$. Then
$SW_{G,Z,\mathfrak{s}}^{\phi}(x^{m})=(-1)^{d_{Y}+1}h{\sum_{l}}^{\prime}e(H^{+}(X))s_{-d_{X}-l}(D_{X})SW(Y,\mathfrak{s}_{Y},\phi_{Y})v^{m+l-(p-1)/2}$
where the sum ${\sum_{l}}^{\prime}$ is over $l$ such that $0\leq l\leq-d_{X}$,
$m+l>0$, $m+l=0\;({\rm mod}\;p-1)$ and $h=\prod_{j=1}^{(p-1)/2}j^{b_{+}(Y)}$
for $p\neq 2$, $h=1$ for $p=2$.
###### Proof.
We give the proof for $p\neq 2$. The case $p=2$ is similar. Let
$f_{X},f_{Y},f_{Z}$ denote the monopole maps for $X,Y,Z$. We have that
$f_{Z}=ind^{G}_{1}(f_{Y})\wedge f_{X}$. Theorem 8.2 gives
$SW_{G,Z,\mathfrak{s}}^{\phi}(x^{m})=SW_{G,ind^{G}_{1}(f_{Y})}^{\phi_{Y}}(e_{G}(H^{+}(X))s_{G_{\mathfrak{s}},-d_{X}}(D_{X})x^{m}).$
Next we will compute $SW_{G,ind^{G}_{1}(f_{Y})}^{\phi_{Y}}$ using Theorem 6.5.
It is easy to see that for each splitting $s_{j}:\mathbb{Z}_{p}\to
S^{1}\times\mathbb{Z}_{p}$, we have $(ind^{G}_{1}(f_{Y}))^{s_{j}}=f_{Y}$.
Furthermore,
$ind^{G}_{1}(D_{Y})\cong\bigoplus_{j=0}^{p-1}\mathbb{C}_{j}^{d_{Y}}$. As
explained above we can orient $ind^{G}_{1}(H^{+}(Y))$ in such a way that
$ind^{G}_{1}(H^{+}(Y))/H^{+}(Y)\cong\bigoplus_{j=1}^{(p-1)/2}\mathbb{C}_{j}^{b_{+}(Y)}$.
Then
$\displaystyle SW^{\phi_{Y}}_{G,ind^{G}_{1}(f_{Y})}(x^{m})$
$\displaystyle=e_{G}(ind^{G}_{1}(H^{+}(Y))/H^{+}(Y))\sum_{j=0}^{p-1}SW^{\phi_{Y}}_{G,f_{Y}}(e_{S^{1}\times
G}(D/D_{j})^{-1}(x+jv)^{m})$
$\displaystyle=\prod_{r=1}^{(p-1)/2}(rv)^{b_{+}(Y)}\sum_{j=0}^{p-1}SW^{\phi_{Y}}_{G,f_{Y}}\left(\psi_{s_{j}}^{-1}\left(\prod_{k=1}^{p-1}(x+kv)^{-d_{Y}}(x-jv)^{m}\right)\right)$
where $\psi_{s_{j}}$ was defined in Section 6.6. Since
$d(Y,\mathfrak{s}_{Y})=0$, we have
$SW^{\phi_{Y}}_{G,f_{Y}}(1)=SW(Y,\mathfrak{s}_{Y},\phi_{Y})$ and
$SW^{\phi_{Y}}_{G,f_{Y}}(\psi_{s_{j}}^{-1}(x^{m}))=0$ for $m>0$. Hence
$\displaystyle SW^{\phi_{Y}}_{G,ind^{G}_{1}(f_{Y})}(x^{m})$
$\displaystyle=(-1)^{d_{Y}}\prod_{r=1}^{(p-1)/2}(rv)^{b_{+}(Y)}SW(Y,\mathfrak{s}_{Y},\phi_{Y})v^{m-d_{Y}(p-1)}\sum_{j=0}^{p-1}(-j)^{m}$
$\displaystyle=(-1)^{d_{Y}}h\,SW(Y,\mathfrak{s}_{Y},\phi_{Y})v^{m-(p-1)/2}\sum_{j=0}^{p-1}j^{m}$
where $h=\prod_{j=1}^{(p-1)/2}j^{b_{+}(Y)}$. But $\sum_{j=0}^{p-1}j^{m}$
equals $-1$ mod $p$ if $m>0$ and $(p-1)$ divides $m$ and is zero otherwise.
Hence
$SW^{\phi_{Y}}_{G,ind^{G}_{1}(f_{Y})}(x^{m})=\begin{cases}(-1)^{d_{Y}+1}h\,SW(Y,\mathfrak{s}_{Y},\phi_{Y})&m>0,m=0\;({\rm
mod}\;p-1),\\\ 0&\text{otherwise}\end{cases}$
Then using
$s_{G_{\mathfrak{s}},-d_{X}}(D_{X})=\sum_{l=0}^{-d_{X}}x^{l}s_{-d_{X}-l}(D_{X}),$
we obtain
$SW_{G,Z,\mathfrak{s}}^{\phi}(x^{m})=(-1)^{d_{Y}+1}h{\sum_{l}}^{\prime}e(H^{+}(X))s_{-d_{X}-l}(D_{X})SW(Y,\mathfrak{s}_{Y},\phi_{Y})v^{m+l-(p-1)/2}$
where the sum is over $l$ with $0\leq l\leq-d_{X}$, $m+l>0$ and $m+l=0\;({\rm
mod}\;p-1)$. ∎
The following corollary follows immediately from Theorem 8.4.
###### Corollary 8.5.
Let $p$ be a prime. Let $X$ be a compact, oriented, smooth $4$-manifold with
$b_{1}(X)=0$ and $b_{+}(X)>0$. Let $\mathfrak{s}$ be a spinc-structure on $X$
such that $d(X,\mathfrak{s})=0$. If $b_{+}(X)=1$ then fix a chamber. Let
$G=\mathbb{Z}_{p}$ act on the connected sum $\\#pX$ of $p$ copies of $X$ by
cyclically permuting the summands (to be precise, we take the equivariant
connected sum $S^{4}\\#pX$ where $\mathbb{Z}_{p}$ acts on $S^{4}$ by
rotation). Then
$SW_{G,\\#pX,\\#p\mathfrak{s}}(x^{m})=\begin{cases}(-1)^{d_{X}+1}hSW(X,\mathfrak{s}_{X})v^{m-(p-1)/2}&m>0,m=0\;({\rm
mod}\;p-1),\\\ 0&\text{otherwise}\end{cases}$
where $h=\prod_{j=1}^{(p-1)/2}j^{b_{+}(X)}$ for $p\neq 2$, $h=1$ for $p=2$.
## 9\. Some examples
### 9.1. Constraints on group actions
Consider the case where $G=\mathbb{Z}_{p}$ for a prime $p$. We use notation
from Section 6.6. Combining the divisibility condition Theorem 6.10 with
Theorem 6.12, we obtain the following constraint on smooth
$\mathbb{Z}_{p}$-actions:
###### Theorem 9.1.
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let
$G=\mathbb{Z}_{p}$ act smoothly on $X$ and let $\mathfrak{s}$ be a spinc-
structure preserved by $G$. If $SW(X,\mathfrak{s})\neq 0\;({\rm mod}\;p)$ and
$b_{0}\neq 1\;({\rm mod}\;2p)$ then there exists an $i$ such that $0\leq
2d_{i}-b_{0}-1\leq 2(p-2)$.
###### Proof.
If the condition $0\leq 2d_{i}-b_{0}-1\leq 2(p-2)$ is not satisfied then
Theorem 6.10 implies that $\overline{SW}^{s_{i}}_{G,X,\mathfrak{s}}=0\;({\rm
mod}\;p)$. If this holds for all $i$ then Theorem 6.12 implies that
$SW_{G,X,\mathfrak{s}}=0\;({\rm mod}\;p)$ and hence
$SW(X,\mathfrak{s})=0\;({\rm mod}\;p)$. ∎
###### Corollary 9.2.
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$. Let
$G=\mathbb{Z}_{2}$ act smoothly on $X$ and let $\mathfrak{s}$ be a spinc-
structure preserved by $G$. If $SW(X,\mathfrak{s})$ is odd and $b_{0}\neq
1\;({\rm mod}\;4)$ then there exists an $i$ such $2d_{i}=b_{0}+1$.
###### Example 9.3.
Let $K$ denote a $K3$ surface given as a degree $3$ cyclic branched cover of
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ branched over a smooth curve $\Sigma$
of bi-degree $(3,3)$. This gives an action of $G=\mathbb{Z}_{3}$ on $K$ with
fixed point set a surface of genus $4$ and self-intersection $6$. Similarly we
can realise $4(S^{2}\times S^{2})$ as the branched triple cover of an
unknotted surface in $S^{4}$ of genus $2$ [1]. This gives an action of $G$ on
$4(S^{2}\times S^{2})$ with fixed point set a surface of genus $2$ and self-
intersection zero. Now consider the equivariant connected sum
$X_{0}=4(S^{2}\times S^{2})\\#5K$ of $4(S^{2}\times S^{2})$ and five copies of
$K$ with the given $\mathbb{Z}_{3}$-action. This gives a
$\mathbb{Z}_{3}$-action on $X_{0}$ with fixed point set a single surface of
genus $22$ and self-intersection $30$. The $4$-manifold $X_{0}$ is
homeomorphic to the elliptic surface $X=E(10)$, hence the
$\mathbb{Z}_{3}$-action on $X_{0}$ also defines a continuous, locally linear
$\mathbb{Z}_{3}$-action on $X$. On the other hand we will use Theorem 9.1 to
show that there is no smooth $\mathbb{Z}_{3}$-action on $X$ with the same
fixed point data. Let $\mathfrak{s}$ denote the unique spin structure on $X$.
Then $SW(X,\mathfrak{s})=\binom{8}{4}=1\;({\rm mod}\;3)$. A smooth
$\mathbb{Z}_{3}$-action on $X$ with fixed point set a surface of genus $22$
and self-intersection $30$ will have $b_{0}=5$ by the $G$-signature theorem
and $d_{0}=0$, $d_{1}=d_{2}=5$ by the $G$-spin theorem. But this contradicts
Theorem 9.1 which requires $3\leq d_{i}\leq 4$ for some $i$. So such an action
does not exist.
### 9.2. Exotic group actions
The gluing formula gives a method of contructing group actions which are
homeomorphic but not diffeomorphic. Let $X_{1},X_{2}$ be compact oriented
smooth $4$-manifolds with $b_{1}=0$ and $b_{+}>1$. Assume that there is a
homeomorphism $\varphi:X_{1}\to X_{2}$, but that $X_{1}$, $X_{2}$ have
different mod $p$ Seiberg–Witten invariants for a prime $p$, so in particular
they are not diffeomorphic. More precisely we will assume the following. For
simplicity assume $H_{1}(X_{i};\mathbb{Z})=0$ for $i=1,2$ so that spinc-
structures can be identified with characteristics. Then we will require that
there does not exist an isometry $\psi:H^{2}(X_{1};\mathbb{Z})\to
H^{2}(X_{2};\mathbb{Z})$ for which
$SW(X_{1},\mathfrak{s}_{1})=SW(X_{2},\psi(\mathfrak{s}_{1}))\;({\rm mod}\;p)$
for every spinc-structure $\mathfrak{s}_{1}$ with
$d(X_{1},\mathfrak{s}_{1})=0$ and for some choice of orientations of
$H^{+}(X_{i})$.
Given $g>0$, let $Y_{g}$ denote the degree $p$ cyclic branched cover of an
unknotted embedded surface in $S^{4}$ of genus $g$. Then $Y_{p}$ is
diffeomorphic to $\\#^{g(p-1)}S^{2}\times S^{2}$. The branched covering
construction defines an action of $G=\mathbb{Z}_{p}$ on $Y_{g}$. Moreover
$H^{2}(Y_{g};\mathbb{Z})^{G}=0$, in particular $H^{+}(Y_{g})^{G}=0$. By [14]
there exists a $k>0$ such that $X_{1}\\#k(S^{2}\times S^{2})$ and
$X_{1}\\#k(S^{2}\times S^{2})$ are diffeomorphic. This also implies that
$pX_{1}\\#k(S^{2}\times S^{2})$ and $pX_{2}\\#k(S^{2}\times S^{2})$ are
diffeomorphic. Thus if $g(p-1)\geq k$ then we get two $\mathbb{Z}_{p}$-actions
on $X=pX_{1}\\#Y_{g}\cong pX_{2}\\#Y_{g}$. The first is obtained by taking
$Y_{g}$ with the $\mathbb{Z}_{p}$-action described above and attaching $p$
copies of $X_{1}$ which are permuted by the $\mathbb{Z}_{p}$-action. The
second action on $X$ is given by the same construction but with $X_{2}$ in
place of $X_{1}$. These two actions are equivariantly homeomorphic since we
can apply the homeomorphism $\varphi:X_{1}\to X_{2}$ to each copy of $X_{1}$.
On the other hand they are not equivarianly diffeomorphic. To see this, note
that since $H^{2}(Y_{g};\mathbb{Z})^{G}=0$, one finds that the $G$-invariant
spinc-structures on $X_{i}\\#Y_{g}$ are precisely those of the form
$ind^{G}_{1}(\mathfrak{s}_{i})\\#\mathfrak{s}$ where $\mathfrak{s}_{i}$ is a
spinc-structure on $X_{i}$ and $\mathfrak{s}$ is the unique spin structure on
$Y_{g}$. If $d(X_{i},\mathfrak{s}_{i})=0$ then Theorem 8.4 gives
$SW_{G,X,ind^{G}(\mathfrak{s}_{i})\\#\mathfrak{s}}(x^{p-1})=(-1)^{g(p-1)/2}SW(X_{i},\mathfrak{s}_{i})v^{(p-1)/2}\;({\rm
mod}\;p).$
Our assumption that $X_{1},X_{2}$ have different mod $p$ Seiberg–Witten
invariants then implies that the two different $G$-actions have different
equivariant Seiberg–Witten invariants, so they are not diffeomorphic.
### 9.3. Obstructions to enlarging group actions
Let $G$ be a finite group and $H$ a subgroup of $G$. The compatibility of the
equivariant Seiberg–Witten invariants with the restriction map from
$G$-equivariant cohomology (or $K$-theory) to $H$-equivariant implies a
condition for a smooth $H$-action to extend to $G$. This follows immediately
from Theorem 3.1 (2), but we restate it here from the perspective of extending
a group action.
###### Proposition 9.4.
Let $X$ be a compact, oriented smooth $4$-manifold with $b_{1}(X)=0$. Suppose
that a finite group $H$ acts on $X$ by orientation preserving diffeomorphisms
and suppose that $H^{+}(X)^{H}\neq 0$. Let $\mathfrak{s}$ be a $H$-invariant
spinc-structure and let $\phi\in H^{+}(X)^{H}\setminus\\{0\\}$ be a chamber.
Let $G$ be a finite group containing $H$. If the $H$-action on $X$ extends to
a smooth, orientation preserving action of $G$ which fixes $\mathfrak{s}$ and
$\phi$, then $H_{\mathfrak{s}}$ is the restriction to $H$ of an $S^{1}$
central extension $G_{\mathfrak{s}}\to G$ and for every $\theta$ in the image
of the restriction map $H^{*}_{G_{\mathfrak{s}}}(pt;A)\to
H^{*}_{H_{\mathfrak{s}}}(pt;A)$, we have that
$SW_{H,X,\mathfrak{s}}^{\phi}(\theta)$ is in the image of
$H^{*}_{G}(pt;A_{w})\to H^{*}_{H}(pt;A_{w})$.
Furthermore, if $b_{+}(X)$ is odd and a $H$-equivariant spinc-struture
$\mathfrak{o}$ on $H^{+}(X)$ is given which can be lifted to a $G$-equivariant
spinc-structure, then for every $\theta$ in the image of
$R(G_{\mathfrak{s}})\to R(H_{\mathfrak{s}})$, we have that
$SW_{H,X,\mathfrak{s}}^{\phi,K}(\theta)$ is in the image of $R(G)\to R(H)$.
###### Example 9.5.
Let us consider Proposition 9.4 in the case that
$H=\mathbb{Z}_{p}=\langle\sigma\;|\;\sigma^{p}\rangle$ is cyclic of odd prime
order and
$G=D_{p}=\langle\sigma,\tau\;|\;\sigma^{p},\tau^{2},(\tau\sigma)^{2}\rangle$
is the dihedral group of order $2p$. Both $G$ and $H$ have no non-trivial
$S^{1}$ central extensions. Let $w\in
H^{1}_{D_{p}}(pt;\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$ be the unique non-trivial
element. Recall that
$H^{2k}_{\mathbb{Z}_{p}}(pt;\mathbb{Z})\cong\mathbb{Z}_{p}$ for every $k>0$.
On the other hand a simple calculation shows that
$H^{2k}_{D_{p}}(pt;\mathbb{Z})=0$ for odd $k$ and
$H^{2k}_{D_{p}}(pt;\mathbb{Z}_{w})=0$ for even $k>0$. Also
$R(\mathbb{Z}_{p})\cong\mathbb{Z}[t]/(t^{p}-1)$ and the image of $R(D_{p})\to
R(\mathbb{Z}_{p})$ is the subring generated by $t+t^{-1}$. From this we obtain
non-trivial conditions for a smooth $\mathbb{Z}_{p}$-action on $X$ to extend
to $D_{p}$, where $X$ is a compact, oriented smooth $4$-manifold with
$b_{1}(X)=0$ and $H$ acts smoothly and orientation preservingly. Suppose
$\mathfrak{s}$ is a $H$-invariant spinc-structure on $X$ and $\phi$ is a
$H$-invariant chamber. If the $\mathbb{Z}_{p}$-action extends to a smooth
orientation preserving action of $D_{p}$ which preserves $\mathfrak{s}$ and
$\phi$. Assume $b_{+}(X)$ is odd, so $d(X,\mathfrak{s})$ is even. Then the
condition is that $SW^{\phi}_{\mathbb{Z}_{p},X,\mathfrak{s}}(x^{m})=0$
whenever $2m-d(X,\mathfrak{s})$ is positive and equals $2$ mod $4$ if $\sigma$
preserves orientation on $H^{+}$, or equals $0$ mod $4$ if $\sigma$ reverses
orientation on $H^{+}$.
Consider for instance the case that $X=\\#pY$ is the connected sum of $p$
copies of a $4$-manifold $Y$ and $\sigma$ cyclically permutes the summands.
Then by Corollary 8.5,
$SW_{\mathbb{Z}_{p},\\#pY,\\#p\mathfrak{s}_{Y}}^{\phi}(x^{p-1})$ is a non-zero
multiple of $SW(Y,\mathfrak{s}_{Y},\phi)v^{(p-1)/2}$. We obtain the following
non-existence result: if $SW(Y,\mathfrak{s}_{Y},\phi)\neq 0\;({\rm mod}\;p)$
then there does not exist an extension of the $\mathbb{Z}_{p}$-action on $X$
to a smooth action of $D_{p}$ which fixes $\\#p\mathfrak{s}_{Y}$ and $\phi$
and for which $\tau$ preserves orientation on $H^{+}(X)$ if $p=3\;({\rm
mod}\;4)$ or reverses orientation on $H^{+}(X)$ if $p=1\;({\rm mod}\;4)$.
### 9.4. Obstructions to equivariant connected sum decompositions
Let $X$ be a compact, oriented, smooth $4$-manifold with $b_{1}(X)=0$.
Consider a smooth $G$-action on $X$. If $X$ can be written as an equivariant
connected sum $X=X_{1}\\#X_{2}$ where $b_{+}(X_{1})^{G},b_{+}(X_{2})^{G}>0$,
then the equivariant Seiberg–Witten invariants vanish by Theorem 8.3. This can
be used to limit the possible ways in which $X$ can be an equivariant
connected sum.
###### Example 9.6.
We will construct two smooth $\mathbb{Z}_{3}$-actions on
$X=5\mathbb{CP}^{2}\\#23\overline{\mathbb{CP}}^{2}$ with the same fixed point
data. One of these actions will decompose as an equivariant connected sum
$X=X_{1}\\#X_{2}$ with $b_{+}(X_{1})^{G},b_{+}(X_{2})^{G}>0$, the other will
not decompose in this way. The actions on $X$ will be constructed from the
following:
* (1)
Let $K$ be the Fermat quartic
$\\{[z_{0},z_{1},z_{2},z_{3}]\in\mathbb{CP}^{3}\;|\;z_{0}^{4}+z_{1}^{4}+z_{2}^{4}+z_{3}^{4}=0\\}$
with $\mathbb{Z}_{3}$-action given by
$[z_{0},z_{1},z_{2},z_{3}]\mapsto[z_{0},z_{2},z_{3},z_{1}]$.
* (2)
Let $\mathbb{CP}^{2}_{(a)}$ denote $\mathbb{CP}^{2}$ with
$\mathbb{Z}_{3}$-action $[z_{0},z_{1},z_{2}]\mapsto[z_{0},\omega
z_{1},\omega^{a}z_{2}]$, where $\omega=e^{2\pi i/3}$.
* (3)
Let $2(S^{2}\times S^{2})$ be the branched triple cover of an unknotted torus
in $S^{4}$.
The action of $\mathbb{Z}_{3}$ on the tangent space of an isolated fixed point
will be orientation preservingly isomorphic to
$(z_{1},z_{2})\mapsto(\omega^{-1}z_{1},\omega^{a}z_{2})$, where $a$ is either
$1$ or $-1$. Let $n_{\pm}$ denote the number of isolated fixed points of type
$a=\pm 1$. Then $K$ has $(n_{+},n_{-})=(6,0)$, $\mathbb{CP}^{2}_{(1)}$ has
$(n_{+},n_{-})=(0,1)$, $\mathbb{CP}^{2}_{(2)}$ has $(n_{+},n_{-})=(3,0)$ and
$2(S^{2}\times S^{2})$ has $(n_{+},n_{-})=0$.
Let $Y$ be the equivariant sum
$Y=\overline{\mathbb{CP}}^{2}_{(2)}\\#\overline{\mathbb{CP}}^{2}_{(1)}\\#2(S^{2}\times
S^{2})$, where the first and second summands are connected along isolated
fixed points and the second and third summands are connected along non-
isolated fixed points. Then $X$ is diffeomorphic to $K\\#Y$ and so we obtain a
$\mathbb{Z}_{3}$-action on $X$ by considering $K\\#Y$ as an equivariant
connected sum. We have that $b_{+}(Y)^{G}=0$ and if we equip $2(S^{2}\times
S^{2})$ with its unique spin structure and equip each
$\overline{\mathbb{CP}}^{2}$ summand of $Y$ with a spinc-structure satisfying
$c(\mathfrak{s})^{2}=-1$, then we obtain an invariant spinc-structure
$\mathfrak{s}_{Y}$ on $Y$ for which $d_{Y}=0$. If $\mathfrak{s}_{K}$ denotes
the unique spin structure on $K$, then Theorem 8.3 gives
$SW_{\mathbb{Z}_{3},K\\#Y,\mathfrak{s}_{K}\\#\mathfrak{s}_{Y}}(1)=SW_{\mathbb{Z}_{3},K,\mathfrak{s}_{K}}(1)=SW(K,\mathfrak{s}_{K})=1.$
So $K\\#Y$ has a non-zero equivariant Seiberg–Witten invariant and can’t be
obtained as an equivariant connected sum of the form $X_{1}\\#X_{2}$ with
$b_{+}(X_{1})^{G},b_{+}(X_{2})^{G}>0$.
Next let $K^{\prime}$ be defined as follows. First take the equivariant
connected sum
$K^{\prime}_{0}=3\mathbb{CP}^{2}_{(2)}\\#\overline{\mathbb{CP}}^{2}_{(2)}$
where the three $\mathbb{CP}^{2}_{(2)}$ summands are attached to the three
isolated fixed points of $\overline{\mathbb{CP}}^{2}_{(2)}$. Then let
$K^{\prime}=K^{\prime}_{0}\\#18\overline{\mathbb{CP}}^{2}$, where the
$\mathbb{Z}_{3}$-action permutes the $18$ copies of
$\overline{\mathbb{CP}}^{2}$ in six $3$-cycles. It is easily seen that the
$\mathbb{Z}_{3}$-action on $K^{\prime}$ has the same fixed point data as $K$,
namely $(n_{+},n_{-})=(6,0)$ and no non-isolated fixed points. Hence the
$\mathbb{Z}_{3}$-actions on $K\\#Y$ and $K^{\prime}\\#Y$ have the same fixed
points as well. Moreover, $K^{\prime}\\#Y$ is diffeomorphic to $X$ so this
gives another $\mathbb{Z}_{3}$-action on $X$ with the same fixed point data.
Unlike the first action this one can be decomposed into an equivariant
connected sum $X_{1}\\#X_{2}$ with $b_{+}(X_{1})^{G},b_{+}(X_{2})^{G}>0$. This
is clear because each of the three $\mathbb{CP}^{2}_{(2)}$-summands in
$K^{\prime}_{0}$ have $b_{+}^{G}=1$.
### 9.5. Obstructions to equivariant positive scalar curvature
The $4$-manifolds of the form $a\mathbb{CP}^{2}\\#b\overline{\mathbb{CP}}^{2}$
admit metrics of positive scalar curvature. On the other hand the equivariant
Seiberg–Witten invariants can be used to find many examples of actions on such
manifolds for which there is no invariant metric of positive scalar curvature.
Let $X$ be a simply-connected $4$-manifold with $b_{+}(X)>1$ on which
$\mathbb{Z}_{p}$ acts smoothly. Assume the action has a non-isolated fixed
point and that $X$ has a non-zero mod $p$ Seiberg–Witten invariant. We assume
$X$ is not spin (if necessary we can replace $X$ by a blow up to achieve
this). Then for some $g>0$ we have that $X\\#g(p-1)(S^{2}\times S^{2})$ is
diffeomorphic to $a\mathbb{CP}^{2}\\#b\overline{\mathbb{CP}}^{2}$, where
$a=b_{+}(X)+g(p-1)$, $b=b_{-}(X)+g(p-1)$. Now recall that
$Y_{g}=g(p-1)(S^{2}\times S^{2})$ can be realised as the degree $p$ cyclic
branched cover of an unknotted surface in $S^{4}$ of genus $g$. We have
$b_{+}(Y_{g})^{\mathbb{Z}_{p}}=0$ and then $X\\#Y_{g}$ has non-zero
equivariant Seiberg–Witten invariants by Theorem 8.3. This gives a smooth
$\mathbb{Z}_{p}$-action on $a\mathbb{CP}^{2}\\#b\overline{\mathbb{CP}}^{2}$
which does not admit an invariant positive scalar curvature metric.
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# Non-local Boundary Value Problems for Brownian motions on the half line
Fausto Colantoni; Mirko D’Ovidio Department of Basic and Applied Sciences for
Engineering
Sapienza University of Rome
via A. Scarpa 10, Rome, Italy<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>
###### Abstract.
We study boundary value problems involving non-local operators for the dynamic
boundary conditions. Our analysis includes a detailed description of such
operators together with their relations with random times and random
functionals. We provide some new characterizations for the boundary behaviour
of the Brownian motion based on the interplay between non-local operators and
boundary value problems.
###### Key words and phrases:
Non-local operators, subordinators, fractional boundary condition, Brownian
motion
## 1\. Introduction
We consider dynamic boundary conditions involving non-local operators and
provide the probabilistic representation of the solutions for the associated
Cauchy problems on the real line (we provide some extension for the
$d$-dimensional case). Our analysis is based on the non-local dynamic boundary
condition
$\displaystyle\eta\,\mathfrak{D}^{\Psi}_{t}u(t,0)=-\mathbf{D}^{\Phi}_{x-}u(t,0),\quad
t>0$
for the heat equation on the positive real line. The non-local operators
appearing above are given by the Caputo-Dzherbashian (type) derivative
$\mathfrak{D}^{\Psi}_{t}$ and the Marchaud (type) derivative
$\mathbf{D}^{\Phi}_{x-}$ where $\Psi$ and $\Phi$ are Bernstein symbols
characterizing the operators. The probabilistic representation of the solution
is written in terms of a Brownian motion reflected at zero, we show that the
spatial condition controls an additive part acting on the reflecting Brownian
motion whereas the time condition introduces a time change acting on the local
time (at zero) of the reflecting Brownian motion. The additive part pushes
away from zero the process and the time change forces the process to stop for
a random amount of time. Due to the right-continuity, the stop occurs
immediately after the jump. Furthermore, the process stops for an independent
amount of time and it jumps for an independent distance from the origin.
In Section 2 we introduce some basic facts and notations about subordinators
and inverses. In Section 3 we discuss the relations between the operators we
deal with and the associated processes. In particular, the left and right
Marchaud (type) derivatives $\mathbf{D}^{\Phi}_{x-}$ and
$\mathbf{D}^{\Phi}_{x+}$ can be respectively regarded as the generator and its
adjoint for a subordinator with symbol $\Phi$. From this point of view, our
analysis completely agrees with the boundary conditions introduced by Feller
in [12]. Then, we discuss the heat equation with non-local boundary conditions
in Section 4. We provide the main result (Theorem 4.1) which generalizes the
work [10] for the time operator and the work [12] for the space operator. We
also provide a discussion on the probabilistic representation of the solution
by extending the results given in [9; 10] and the probabilistic representation
given by Itô and McKean in [16; 15]. In Section 5 we discuss on the
applications and the extension of our results. In particular, the non-local
dynamic boundary condition gives new meaning to the reflection, we provide
some characterization.
## 2\. Subordinators
We recall some basic facts just to introduce some notation. Let
$H^{\Phi}=\\{H_{t}^{\Phi},\ t\geq 0\\}$ be a subordinator (see [3] for
details). Then, $H^{\Phi}$ can be characterized by the Laplace exponent
$\Phi$, that is,
$\displaystyle\mathbf{E}_{0}[\exp(-\lambda
H_{t}^{\Phi})]=\exp(-t\Phi(\lambda)),\quad\lambda\geq 0.$ (2.1)
We denote by $\mathbf{E}_{x}$ the expected value with respect to
$\mathbf{P}_{x}$ where $x$ is the starting point. Since $\Phi(\lambda)$ is the
Laplace exponent of a subordinator, it is uniquely characterized by the pair
of non-negative real numbers $(\kappa,d)$ and by the L$\acute{e}$vy measure
$\Pi^{\Phi}$ on $(0,\infty)$ such that $\int_{0}^{\infty}(1\wedge
z)\Pi^{\Phi}(dz)<\infty$. For the symbol $\Phi$, the following L$\acute{e}$vy-
Khintchine representation holds
$\displaystyle\Phi(\lambda)=\kappa+d\lambda+\int_{0}^{\infty}(1-e^{-\lambda
z})\Pi^{\Phi}(dz),\quad\lambda>0$ (2.2)
where the killing rate $\kappa$ and the drift coefficient $d$ are given by
$\displaystyle\kappa=\lim_{\lambda\to 0}\Phi(\lambda),\quad
d=\lim_{\lambda\to\infty}\frac{\Phi(\lambda)}{\lambda}.$
The symbol $\Phi$ is a Bernstein function (non-negative, non-decreasing and
continuous, see for example [29]) uniquely associated with $H^{\Phi}$. For the
reader’s convenience we also recall that
$\displaystyle\frac{\Phi(\lambda)}{\lambda}=d+\int_{0}^{\infty}e^{-\lambda
z}\overline{\Pi}^{\Phi}(z)dz,\quad\overline{\Pi}^{\Phi}(z)=k+\Pi^{\Phi}((z,\infty))$
(2.3)
where $\overline{\Pi}^{\Phi}$ is the so called tail of the L$\acute{e}$vy
measure $\Pi^{\Phi}$. In this paper we only consider symbols for which
$\displaystyle\kappa=0,\quad d=0.$
We also define the process $L^{\Phi}=\\{L_{t}^{\Phi},\ t\geq 0\\}$, with
$L_{0}^{\Phi}=0$, as the inverse of $H^{\Phi}$, that is
$\displaystyle L_{t}^{\Phi}=\inf\\{s>0\,:\,H_{s}^{\Phi}>t\\},\quad t>0.$
We focus only on strictly increasing subordinators with infinite measures
$\Pi^{\Phi}(0,\infty)=\infty$ (see [17, Theorem 21.3]). Thus, the inverse
process $L^{\Phi}$ turns out to be a continuous process. In particular,
$H^{\Phi}$ may have jumps and the inverse has non decreasing paths. Notice
that, an inverse process can be regarded as an exit time for $H^{\Phi}$. By
definition, we also have
$\displaystyle\mathbf{P}_{0}(H_{t}^{\Phi}<s)=\mathbf{P}_{0}(L_{s}^{\Phi}>t),\quad
s,t>0.$ (2.4)
Let us introduce $h,l$ for which
$\displaystyle\mathbf{P}_{0}(H_{t}^{\Phi}\in
I)=\int_{I}h(t,x)dx,\quad\mathbf{P}_{0}(L_{t}^{\Phi}\in I)=\int_{I}l(t,x)dx,$
for a given set $I\subset(0,\infty)$. From (2.1), we have that
$\displaystyle\int_{0}^{\infty}e^{-\lambda
x}h(t,x)dx=e^{-t\Phi(\lambda)},\quad\lambda>0$
and from [23, formula (3.13)], we get
$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}l(t,x)dt=\frac{\Phi(\lambda)}{\lambda}e^{-x\Phi(\lambda)},\quad\lambda>0.$
(2.5)
By using the initial value theorem for the Laplace transform, we observe that
$\displaystyle h(t,0)=\lim_{\lambda\to\infty}\lambda e^{-t\Phi(\lambda)}$
is not always zero, but depends on $\Phi(\lambda)$ and the time variable $t$
as discussed below. See [17] for the _time dependent property_. The fact that
$h(t,x)=0$ as $x\leq 0$ seems to be quite relevant in our discussion. Indeed,
this condition has non trivial consequences. The gamma subordinator is an
example.
Further on we will also introduce the processes
$H^{\Psi}=\\{H_{t}^{\Psi},t\geq 0\\}$ and $L^{\Psi}=\\{L_{t}^{\Psi},t\geq
0\\}$ with symbol
$\displaystyle\Psi(\lambda):=\int_{0}^{\infty}(1-e^{-\lambda
z})\Pi^{\Psi}(dz),\quad\lambda>0.$
## 3\. Operators
For a continuous function $u$ on $\mathbb{R}$ extended with zero on the
negative part of the real line, that is $u(x)=0$ if $x\leq 0$, we define the
right Marchaud (type) derivative
$\displaystyle\mathbf{D}_{x+}^{\Phi}u(x)=\int_{0}^{\infty}(u(x)-u(x-y))\Pi^{\Phi}(dy)$
(3.1)
and the left Marchaud (type) derivative
$\displaystyle\mathbf{D}_{x-}^{\Phi}u(x)=\int_{0}^{\infty}(u(x)-u(x+y))\Pi^{\Phi}(dy).$
(3.2)
If $\Pi^{\Phi}$ is the L$\acute{e}$vy measure associated to a stable
subordinator, formulas (3.1) and (3.2) respectively coincide with the right
and the left Marchaud derivatives, usually denoted by
$\mathbf{D}_{x+}^{\alpha}$ and $\mathbf{D}_{x-}^{\alpha}$ respectively. The
reader can consult the famous book [28, formula (5.57) and (5.58)] or the
paper [13, section 6.1] for a nice recent discussion. The operators
$\mathbf{D}_{x+}^{\Phi}$ and $\mathbf{D}_{x-}^{\Phi}$ can be defined on
different spaces depending on $\Pi^{\Phi}$. For a general definition, given
the symbol $\Phi$, we consider $u$ bounded and locally Lipschitz continuous,
then
$\displaystyle|\mathbf{D}_{x-}^{\Phi}u(x)|$
$\displaystyle\leq\int_{0}^{1}|u(x)-u(x+y)|\Pi^{\Phi}(dy)+\int_{1}^{\infty}|u(x)-u(x+y)|\Pi^{\Phi}(dy)$
$\displaystyle\leq
K\int_{0}^{1}y\Pi^{\Phi}(dy)+2||u||_{\infty}\int_{1}^{\infty}\Pi^{\Phi}(dy)$
$\displaystyle\leq(K+2||u||_{\infty})\int_{0}^{\infty}(1\wedge
y)\Pi^{\Phi}(dy)<\infty.$ (3.3)
Indeed, for the integral in $(0,1)$ we have used the Lipschitz property for a
positive constant $K>0$ whereas for the integral in $(1,\infty)$ we have used
the boundedness of $u$. Since $\int(1\wedge z)\Pi^{\Phi}(dz)<\infty$, then the
last inequality holds. The same holds for $\mathbf{D}_{x+}^{\Phi}$. Obviously,
we may take some advantage from the explicit representation of $\Pi^{\Phi}$.
For example, if $\Phi(\lambda)=\lambda^{\alpha}$ with $\alpha\in(0,1)$, then
$\Pi^{\alpha}(dy)=\frac{\alpha}{\Gamma(1-\alpha)}\frac{dy}{y^{\alpha+1}}$. The
operators (3.1) and (3.2) are therefore defined for locally
$\gamma-$H$\ddot{o}$lder continuous functions with $\gamma>\alpha$. The case
of gamma subordinator, with $\Pi^{\Phi}(dy)=a\frac{e^{-by}}{y}dy$, may be a
bit demanding. This is due to the time dependent continuity of $h(t,x)$ at
$x=0$. In [8] we have proved that (3.1) is defined for
$\gamma-$H$\ddot{o}$lder continuous functions with $\gamma>0$.
Let us introduce the spaces
$\displaystyle W^{1,p}(0,\infty)=\\{u\in L^{p}(0,\infty):\;u^{\prime}\in
L^{p}(0,\infty)\\}$
and
$\displaystyle W^{1,p}_{0}(0,\infty)=\\{u\in W^{1,p}(0,\infty):\,u(0)=0\\}$
with $p\in[1,\infty]$. Observe that $AC(I)$ coincides with $W^{1,1}(I)$ only
if $I\subset(0,\infty)$ is bounded. We recall that $AC$ denotes the set of
absolutely continuous functions. For the interval $I$, $W^{1,\infty}(I)$
coincides with the space of Lipschitz continuous functions. If $u\in
W^{1,\infty}(0,\infty)$, then the Marchaud (type) derivatives (3.1) and (3.2)
are well defined almost everywhere. Indeed, with the first inequality of (3)
at hand, we use the fact that $u$ is essentially bounded and locally Lipschitz
almost everywhere (its derivative is bounded).
The operator $\mathbf{D}_{x+}^{\Phi}$ can be obtained by the Pillips’
representation ([25]) in the set of functions extended with zero on the
negative part of the real line. Indeed, for the shift semigroup
$\mathcal{S}_{y}u(x)=u(x-y)$ we have the representation
$\displaystyle\mathbf{D}_{x+}^{\Phi}u(x)=\int_{0}^{\infty}(u(x)-\mathcal{S}_{y}u(x))\Pi^{\Phi}(dy)$
for which
$\displaystyle\int_{0}^{\infty}e^{-\lambda
x}\mathbf{D}_{x+}^{\Phi}u(x)dx=\Phi(\lambda)\int_{0}^{\infty}e^{-\lambda
x}u(x)\,dx$ (3.4)
where we used (2.2) with $\kappa=0$ and $d=0$. We will investigate in a deep
detail the role of the Marchaud (type) derivatives
$\mathbf{D}_{x+}^{\Phi}u,\mathbf{D}_{x-}^{\Phi}u$ in the governing equations
of $H$. In order to define the Riemann-Liouville (type) derivatives on the
positive real line, we first consider a closed interval
$\bar{I}\subset(0,\infty)$ and $u\in AC(\bar{I})$. Then we extend the result
on $\mathbb{R}^{+}$ as in [18, page 79]. We now introduce the Riemann-
Liouville (type) derivatives
$\displaystyle\mathcal{D}_{(x,\infty)}^{\Phi}u(x):=-\frac{d}{dx}\int_{x}^{\infty}u(y)\overline{\Pi}^{\Phi}(y-x)dy$
(3.5)
and
$\displaystyle\mathcal{D}_{(0,x)}^{\Phi}u(x):=\frac{d}{dx}\int_{0}^{x}u(y)\overline{\Pi}^{\Phi}(x-y)dy$
(3.6)
respectively defined for function $u$ such that
$\displaystyle u(y)\overline{\Pi}^{\Phi}(y-x)\in
L^{1}(x,\infty),\quad\text{and}\quad u(y)\overline{\Pi}^{\Phi}(x-y)\in
L^{1}(0,x)\quad\forall x.$
Let us focus on (3.5). We observe that
$\displaystyle\mathcal{D}_{(x,\infty)}^{\Phi}u(x)$
$\displaystyle=-\frac{d}{dx}\int_{x}^{\infty}u(y)\overline{\Pi}^{\Phi}(y-x)dy$
$\displaystyle=\lim_{b\to\infty}-\frac{d}{dx}\int_{x}^{b}u(y)\overline{\Pi}^{\Phi}(y-x)dy$
$\displaystyle=\lim_{b\to\infty}-\frac{d}{dx}\int_{0}^{b-x}u(z+x)\overline{\Pi}^{\Phi}(z)dz$
$\displaystyle=\lim_{b\to\infty}u(b)\overline{\Pi}^{\Phi}(b-x)-\int_{0}^{b-x}u^{\prime}(z+x)\overline{\Pi}^{\Phi}(z)dz$
$\displaystyle=\lim_{b\to\infty}u(b)\overline{\Pi}^{\Phi}(b-x)-\int_{x}^{\infty}u^{\prime}(y)\overline{\Pi}^{\Phi}(y-x)dy$
where, from the final value theorem for the Laplace transform in (2.3), we
have
$\displaystyle 0=\Phi(0)=\lim_{b\to\infty}\overline{\Pi}^{\Phi}(b).$
Assuming that the growth of $u$ is asymptotically bounded, then
$\displaystyle\mathcal{D}_{(x,\infty)}^{\Phi}u(x)=-\int_{x}^{\infty}u^{\prime}(y)\overline{\Pi}^{\Phi}(y-x)dy.$
(3.7)
The right-hand side in (3.7) can be regarded as a Caputo-Dzherbashian (type)
derivative. A further relevant fact for (3.5) is the equivalence with (3.2).
If (3.7) holds, then
$\displaystyle\mathbf{D}_{x-}^{\Phi}u(x)=\mathcal{D}_{(x,\infty)}^{\Phi}u(x).$
(3.8)
Indeed, using first (3.7) and then the second formula in (2.3), we have
$\displaystyle\mathcal{D}_{(x,\infty)}^{\Phi}u(x)$
$\displaystyle=-\int_{x}^{\infty}u^{\prime}(y)\overline{\Pi}^{\Phi}(y-x)dy$
$\displaystyle=-\int_{0}^{\infty}\frac{d}{dy}u(x+y)\overline{\Pi}^{\Phi}(y)dy$
$\displaystyle=-\int_{0}^{\infty}\frac{d}{dy}u(x+y)\Pi^{\Phi}((y,\infty))dy$
$\displaystyle{=}-\int_{0}^{\infty}\int_{0}^{z}\frac{d}{dy}u(x+y)dy\Pi^{\Phi}(dz)$
$\displaystyle=-\int_{0}^{\infty}(u(x+z)-u(x))\Pi^{\Phi}(dz)$
$\displaystyle=\mathbf{D}_{x-}^{\Phi}u(x).$
Concerning (3.6), the analogous result for $\mathbf{D}_{x+}^{\Phi}$ and
$\mathcal{D}_{(0,x)}^{\Phi}$ can be proved.
For the operators (3.2) and (3.1) we introduce the following integration by
parts formula.
###### Theorem 3.1.
If $u,v\in W_{0}^{1,1}(0,\infty)$ and
$\mathbf{D}_{x-}^{\Phi}v(x),\mathbf{D}_{x+}^{\Phi}u(x)\in L^{1}(0,\infty)$,
then
$\displaystyle\int_{0}^{\infty}u(x)\left(\mathbf{D}_{x-}^{\Phi}v(x)\right)dx=\int_{0}^{\infty}\left(\mathbf{D}_{x+}^{\Phi}u(x)\right)v(x)dx.$
(3.9)
###### Proof.
First of all, from H$\ddot{o}$lder’s inequality, we observe that
$\displaystyle||u\mathbf{D}_{x-}^{\Phi}v||_{1}\leq||u||_{\infty}||\mathbf{D}_{x-}^{\Phi}v||_{1}<\infty$
because $\mathbf{D}_{x-}^{\Phi}v\in L^{1}(0,\infty)$ and
$W_{0}^{1,1}(0,\infty)$ embeds into $L^{\infty}(0,\infty)$ (see [22, section
11.2]). Similarly for $\mathbf{D}_{x+}$. By Fubini’s theorem, from the
definition (3.2),
$\displaystyle\int_{0}^{\infty}u(x)\left(\mathbf{D}_{x-}^{\Phi}v(x)\right)dx$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}u(x)[v(x)-v(x+y)]\Pi^{\Phi}(dy)dx$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}[u(x)-u(x-y)+u(x-y)][v(x)-v(x+y)]\Pi^{\Phi}(dy)dx.$
By using (3.1),
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}[u(x)-u(x-y)]v(x)\Pi^{\Phi}(dy)dx=\int_{0}^{\infty}\left(\mathbf{D}_{x+}^{\Phi}u(x)\right)v(x)dx,$
and
$\displaystyle\int_{0}^{\infty}u(x)\left(\mathbf{D}_{x-}^{\Phi}v(x)\right)dx=\int_{0}^{\infty}\left(\mathbf{D}_{x+}^{\Phi}u(x)\right)v(x)dx\
+F(u,v)$
where
$\displaystyle F(u,v)$
$\displaystyle:=\int_{0}^{\infty}\int_{0}^{\infty}\left(u(x-y)[v(x)-v(x+y)]-[u(x)-u(x-y)]v(x+y)\right)\Pi^{\Phi}(dy)dx$
$\displaystyle=\int_{0}^{\infty}\int_{y}^{\infty}u(x-y)v(x)dx\Pi^{\Phi}(dy)-\int_{0}^{\infty}\int_{0}^{\infty}u(x)v(x+y)\Pi^{\Phi}(dy)dx$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}u(x)v(x+y)dx\Pi^{\Phi}(dy)-\int_{0}^{\infty}\int_{0}^{\infty}u(x)v(x+y)\Pi^{\Phi}(dy)dx$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}u(x)v(x+y)\Pi^{\Phi}(dy)dx-\int_{0}^{\infty}\int_{0}^{\infty}u(x)v(x+y)\Pi^{\Phi}(dy)dx=0,$
hence (3.9) holds. ∎
###### Remark 3.1.
The same result for left and right Marchaud derivatives
$\mathbf{D}_{x-}^{\alpha}$ and $\mathbf{D}_{x+}^{\alpha}$ when
$x\in\mathbb{R}$ is presented in [28, (6.27)] and in [20, Exercise 1.8.2].
We are now ready to discuss the connection between operators and
subordinators.
###### Theorem 3.2.
For $0<x<y$ and $t>0$, the density $h(t,y-x)$ of $x+H_{t}^{\Phi}$ solves
$\displaystyle\partial_{t}h=-\mathbf{D}_{y+}^{\Phi}h=-\mathbf{D}_{x-}^{\Phi}h$
(3.10)
with $h(0,y-x)=\delta(y-x)$, the Dirac delta function.
###### Proof.
First we focus on $\partial_{t}h=-\mathbf{D}_{x-}^{\Phi}h$. For $\lambda>0$,
consider the Laplace transform $\widetilde{h}(t,x,\lambda)$ of the density
$h(t,x,y)=h(t,y-x)$ given by
$\displaystyle\widetilde{h}(t,x,\lambda)=\mathbf{E}_{x}[e^{-\lambda
H_{t}^{\Phi}}]=\mathbf{E}_{0}[e^{-\lambda x-\lambda H_{t}^{\Phi}}]=e^{-\lambda
x-t\Phi(\lambda)},\quad x>0,\;t>0.$
From (3.8) and (3.7), we have
$\displaystyle-\mathbf{D}_{x-}^{\Phi}\widetilde{h}(t,x,\lambda)=$
$\displaystyle-\mathcal{D}_{(x,\infty)}^{\Phi}\widetilde{h}(t,x,\lambda)$
$\displaystyle=$ $\displaystyle\int_{x}^{\infty}\left(-\lambda e^{-\lambda
y-t\Phi(\lambda)}\right)\overline{\Pi}^{\Phi}(y-x)\,dy$ $\displaystyle=$
$\displaystyle-\lambda e^{-t\Phi(\lambda)}\int_{x}^{\infty}e^{-\lambda
y}\overline{\Pi}^{\Phi}(y-x)\,dy$ $\displaystyle=$ $\displaystyle-\lambda
e^{-\lambda x-t\Phi(\lambda)}\int_{0}^{\infty}e^{-\lambda
z}\overline{\Pi}^{\Phi}(z)\,dz$ $\displaystyle=$
$\displaystyle-\Phi(\lambda)\widetilde{h}(t,x,\lambda)$
where in the last step we have used (2.3). Thus, for $\lambda>0$, the function
$\widetilde{h}$ solves the equation
$\displaystyle\frac{\partial\widetilde{h}}{\partial
t}=-\mathbf{D}_{x-}^{\Phi}\widetilde{h},\quad\widetilde{h}(0,x)=e^{-\lambda
x},\quad x>0.$ (3.11)
We now focus on $\partial_{t}h=-\mathbf{D}_{y+}^{\Phi}h$ for $y>x>0$, $t>0$
for which we have that
$\displaystyle\mathbf{D}_{y+}^{\Phi}h(t,y-x)\mathbf{1}_{(y>x)}=-\int_{0}^{\infty}\left(h(t,y-s-x)\mathbf{1}_{(x,\infty)}(y-s)-h(t,y-x)\mathbf{1}_{(x,\infty)}(y)\right)\Pi^{\Phi}(ds)$
and therefore, the Laplace transform is given by
$\displaystyle\int_{0}^{\infty}e^{-\lambda
y}\mathbf{D}_{y+}^{\Phi}h(t,y-x)dy=$
$\displaystyle-\int_{0}^{\infty}\left(e^{-\lambda(s+x)-t\Phi(\lambda)}-e^{-\lambda
x-t\Phi(\lambda)}\right)\Pi^{\Phi}(ds)$ $\displaystyle=$
$\displaystyle-e^{-\lambda x-t\Phi(\lambda)}\int_{0}^{\infty}\left(e^{-\lambda
s}-1\right)\Pi^{\Phi}(ds)$ $\displaystyle=$
$\displaystyle\Phi(\lambda)e^{-\lambda x-t\Phi(\lambda)}.$
Thus, we get that
$\displaystyle\frac{\partial\widetilde{h}}{\partial
t}=-\Phi(\lambda)\widetilde{h}=\int_{0}^{\infty}e^{-\lambda
y}\left(-\mathbf{D}_{y+}^{\Phi}h\mathbf{1}_{(y>x)}\right)dy$
which is the claimed result. ∎
Let $M>0$ and $w\geq 0$. Let $\mathcal{M}_{w}$ be the set of (piecewise)
continuous function on $[0,\infty)$ of exponential order $w$ such that
$|u(t)|\leq Me^{wt}$. Let $u\in\mathcal{M}_{w}\cap C([0,\infty)$ with
$u^{\prime}\in\mathcal{M}_{w}$. Then we define the Caputo-Dzherbashian (type)
derivative as
$\displaystyle\mathfrak{D}_{x}^{\Phi}u(x):=\int_{0}^{x}u^{\prime}(y)\overline{\Pi}^{\Phi}(x-y)dy$
(3.12)
which is a convolution type operator. Indeed, the Laplace transform writes
$\displaystyle\int_{0}^{\infty}e^{-\lambda
x}\mathfrak{D}_{x}^{\Phi}u(x)dx=\Phi(\lambda)\tilde{u}(\lambda)-\frac{\Phi(\lambda)}{\lambda}u(0),\quad\lambda>w$
(3.13)
where, as above, $\widetilde{u}$ is the Laplace transform of $u$. For explicit
representations of the operator $\mathfrak{D}^{\Phi}_{x}$ see also the recent
works [19; 30; 7].
We remark that, from the Young’s inequality, we have
$\displaystyle||\mathfrak{D}_{x}^{\Phi}u||_{p}\leq||u^{\prime}||_{p}\left(\lim_{\lambda\to
0}\frac{\Phi(\lambda)}{\lambda}\right)$ (3.14)
where $\lim_{\lambda\to 0}\Phi(\lambda)/\lambda$ is finite only in some cases
(see [5]). If $\lim_{\lambda\to 0}\Phi(\lambda)/\lambda$ is finite, then
(3.14) gives a clearly information about the cases of measurable or
essentially bounded functions. In particular, we have that for $u^{\prime}\in
L^{1}(0,\infty)$, if
$\displaystyle\lim_{\lambda\to 0}\frac{\Phi(\lambda)}{\lambda}<\infty,$
then (3.9) holds. Indeed, by considering the relation
$\displaystyle\mathfrak{D}_{x}^{\Phi}u(x)=\mathcal{D}_{(0,x)}^{\Phi}\left(u(x)-u(0)\right)$
we have equivalence between derivatives as $u(0)=0$. We can easily check that,
in this case, (3.13) coincides with (3.4).
For our purposes the operator (3.12) is used as time derivative for the
equation governing the inverse process $L^{\Phi}$, while (3.2) and (3.1) are
used as space derivatives for equations governing the process $H^{\Phi}$. Such
operators will be considered in the next section in the boundary conditions
associated with equations governing Brownian motions.
The governing equations of $H^{\Phi}$ and $L^{\Phi}$ have been studied in the
literature (see for example [19; 30]) with special attention only about the
fundamental solutions. For the reader’s convenience we conclude this section
by giving a clear statement for such equations based on the previous
discussion. Excluding densities $h$ with _time dependent property_ (as the
case of gamma subordinator), we have that
$\displaystyle C^{1,1}((0,\infty),W^{1,1}_{0}(0,\infty))\ni
h_{f}(t,x)=\int_{0}^{x}f(x-y)h(t,y)dy=\mathbf{E}_{0}[f(x-H^{\Phi}_{t})\mathbf{1}_{(t<L^{\Phi}_{t})}]$
is the unique solution to
$\begin{cases}\displaystyle\frac{\partial}{\partial
t}h_{f}(t,x)=-\mathbf{D}^{\Phi}_{x+}h_{f}(t,x),\quad(t,x)\in(0,\infty)\times(0,\infty)\\\
\displaystyle h_{f}(t,0)=0,\quad t>0\\\ \displaystyle h_{f}(0,x)=f(x)\in
W^{1,1}_{0}(0,\infty)\end{cases}$ (3.15)
and
$\displaystyle C^{1,1}(W^{1,\infty}(0,\infty),(0,\infty)\ni
l_{f}(t,x)=\int_{0}^{x}f(x-y)l(t,y)dy=\mathbf{E}_{0}[f(x-L^{\Phi}_{t})\mathbf{1}_{(t<H^{\Phi}_{t})}]$
is the unique solution to
$\begin{cases}\displaystyle\mathfrak{D}^{\Phi}_{t}l_{f}(t,x)=-\frac{\partial}{\partial
x}l_{f}(t,x),\quad(t,x)\in(0,\infty)\times(0,\infty)\\\ \displaystyle
l_{f}(t,0)=0,\quad t>0\\\ \displaystyle l_{f}(0,x)=f(x)\in
L^{p}(0,\infty)\end{cases}$ (3.16)
Notice that $l_{f}(t,\cdot)\in L^{p}(0,\infty),\forall t>0$.
Let us write
$\bar{h}_{f}(x)=\int_{0}^{\infty}h_{f}(t,x)dt\quad\textrm{and}\quad\bar{l}_{f}(x)=\int_{0}^{\infty}l_{f}(t,x)dt.$
(3.17)
We can immediately check that the Abel (type) equation
$f(x)=\mathbf{D}^{\Phi}_{x+}\bar{h}_{f}(x)$ gives the elliptic problem
associated with (3.15). On the other hand, the elliptic problem associated
with (3.16) exists only if $\lim_{\lambda\downarrow
0}\Phi(\lambda)/\lambda<\infty$, this gives a clear meaning to (3.14). Such a
result is not surprising, indeed by considering $f=\mathbf{1}$,
$\displaystyle\bar{l}_{\mathbf{1}}(x)=\int_{0}^{\infty}\mathbf{E}_{0}[\mathbf{1}_{(t<H^{\Phi}_{x})}]dt=\mathbf{E}_{0}[H^{\Phi}_{x}]=x\lim_{\lambda\downarrow
0}\frac{\Phi(\lambda)}{\lambda}$
In case the elliptic problem exists, it takes the form
$\displaystyle f(x)=\left(\lim_{\lambda\downarrow
0}\frac{\lambda}{\Phi(\lambda)}\right)\frac{\partial}{\partial
x}\bar{l}_{f}(x)$
Moving to the elliptic problems associated with (3.10) we notice that the
solution to
$\displaystyle-\mathbf{D}_{x-}^{\Phi}w_{-}(x)=-f(x)+\lambda w_{-}(x),\quad
x>0,\lambda>0$ (3.18)
is given by
$\displaystyle w_{-}(x)=\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}f(x+H_{t}^{\Phi})dt\right].$ (3.19)
Indeed, from (3.19),
$\displaystyle w_{-}(x)=$
$\displaystyle\int_{x}^{\infty}f(y)\int_{0}^{\infty}e^{-\lambda
t}h(t,y-x)\,dt\,dy$
and by using (3.10),
$\displaystyle-\mathbf{D}_{x-}^{\Phi}w_{-}(x)=$
$\displaystyle\int_{x}^{\infty}f(y)\int_{0}^{\infty}e^{-\lambda
t}(-\mathbf{D}_{x-}^{\Phi}h(t,y-x))\,dt\,dy$ $\displaystyle=$
$\displaystyle\int_{x}^{\infty}f(y)\int_{0}^{\infty}e^{-\lambda
t}(\partial_{t}h(t,y-x))\,dt\,dy$ $\displaystyle=$
$\displaystyle\int_{x}^{\infty}f(y)\left(-\delta(y-x)\right)dy+\lambda\int_{x}^{\infty}f(y)\int_{0}^{\infty}e^{-\lambda
t}h(t,y-x)\,dt\,dy$ $\displaystyle=$ $\displaystyle-f(x)+\lambda w_{-}(x).$
Thus, the solution to (3.18) is given by (3.19). Similarly, we observe that
the solution to
$\displaystyle-\mathbf{D}_{y+}^{\Phi}w_{+}(y)=-f(y)+\lambda w_{+}(y),\quad
y>0,\lambda>0$ (3.20)
is given by
$\displaystyle w_{+}(y)=\int_{0}^{y}\int_{0}^{\infty}e^{-\lambda
t}h(t,y-x)dtf(x)dx.$
Indeed, by using (3.10), we have
$\displaystyle-\mathbf{D}_{y+}^{\Phi}w_{+}(y)=$
$\displaystyle\int_{0}^{y}f(x)\int_{0}^{\infty}e^{-\lambda
t}(-\mathbf{D}_{y+}^{\Phi}h(t,y-x))\,dt\,dx$ $\displaystyle=$
$\displaystyle\int_{0}^{y}f(x)\int_{0}^{\infty}e^{-\lambda
t}(\partial_{t}h(t,y-x))\,dt\,dx$ $\displaystyle=$
$\displaystyle\int_{0}^{y}f(x)\left(-\delta(y-x)\right)dx+\lambda\int_{0}^{y}f(x)\int_{0}^{\infty}e^{-\lambda
t}h(t,y-x)\,dt\,dx$ $\displaystyle=$ $\displaystyle-f(y)+\lambda w_{+}(y).$
###### Remark 3.2.
If $\lambda\to 0$ in (3.20), then the solution can be written as
$\displaystyle w_{+}(y)=(\mathcal{I}^{\Phi}f)(y)$
where $\mathcal{I}^{\Phi}$ is a non-local integral associated to $\Phi$.
Indeed, the convolution between $f$ and $\int_{0}^{\infty}h(t,x)dt$ can be
represented as $\mathcal{I}^{\Phi}$. For example, for the $\alpha-$stable
subordinator we have that
$\displaystyle\int_{0}^{\infty}\mathbf{E}_{0}[e^{-\xi
H_{t}}]dt=\frac{1}{\xi^{\alpha}}\quad\text{where}\quad\frac{1}{\xi^{\alpha}}=\int_{0}^{\infty}e^{-\xi
x}\frac{x^{\alpha-1}}{\Gamma(\alpha)}dx.$
Then, we deduce that
$\displaystyle\int_{0}^{\infty}h(t,x)dt=\frac{x^{\alpha-1}}{\Gamma(\alpha)}$
must be considered in order to write the fractional integral.
## 4\. Non-local boundary conditions
Let us consider the heat equation on the positive half line with a Neumann
boundary condition at $x=0$, that is
$\displaystyle\begin{cases}\frac{\partial}{\partial t}u(t,x)=\Delta
u(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\ \frac{\partial}{\partial
x}u(t,x)\big{|}_{x=0}=0\quad\quad\quad\quad\quad&t>0\end{cases}.$
The probabilistic representation of the solution can be written in terms of
the reflecting Brownian motion $B^{+}=\\{B_{t}^{+},t\geq 0\\}$. We denote by
$\gamma_{t}=\gamma_{t}(B^{+})$ the local time at zero of the reflecting
Brownian motion $B^{+}$ written as
$\displaystyle\gamma_{t}=\int_{0}^{t}\mathbf{1}_{(B_{s}^{+}=0)}ds.$
Recall that $H^{\Phi}$ is the subordinator associated to $\Pi^{\Phi}$ through
(2.2) and $L^{\Phi}$ is the inverse of $H^{\Phi}$. We introduce the main
object we deal with, that is the process $B^{\bullet}=\\{B_{t}^{\bullet},t\geq
0\\}$ written as
$\displaystyle
B_{t}^{\bullet}=H^{\Phi}L^{\Phi}\gamma_{t}-\gamma_{t}+B_{t}^{+},\quad t>0.$
(4.1)
We use the following notation
$\displaystyle H^{\Phi}L^{\Phi}\gamma_{t}=H^{\Phi}\circ
L^{\Phi}\circ\gamma_{t}$
for the composition of $H^{\Phi},L^{\Phi},\gamma$. The process
$L^{\Phi}\gamma_{t}$ can be regarded as the local time of $B^{\bullet}$ at
zero [15, section 14]. We also observe that $B_{t}=B_{t}^{+}-\gamma_{t}(B)$ in
law, from the Tanaka’s formula we may consider an alternative representation
for the process (4.1) given by
$B_{t}^{\bullet}=H^{\Phi}L^{\Phi}\gamma_{t}+B_{t}$, where $B=\\{B_{t},t\geq
0\\}$ is the one-dimensional Brownian motion with local time at zero
$\gamma_{t}(B)$.
The probabilistic reading of (4.1) says that $B^{\bullet}$ is a reflecting
Brownian motion jumping away from the origin, with a length of the jump given
by the subordinator $H^{\Phi}$ running with the clock $L^{\Phi}\gamma_{t}$.
Since $\Pi^{\Phi}(0,\infty)=\infty$, then in each time interval the number of
jumps of $H^{\Phi}$ are infinite. A detailed analysis of the sample paths has
be given in [15, section 12]. We provide a discussion at the end of the paper.
We remark that $L^{\Phi}H^{\Phi}=t$ almost surely, whereas the composition
$H^{\Phi}L^{\Phi}$ is a bit demanding. In particular, for the subordinator
$H^{\Phi}$ and the inverse $L^{\Phi}$ we have that
$\mathbf{P}_{0}(H^{\Phi}{L^{\Phi}_{t}}-<t=H^{\Phi}{L^{\Phi}_{t}})=0$ ([2,
Proposition 2, section III.2]). Moreover, if $d=0$ in (2.2), then
$\mathbf{P}_{0}(H^{\Phi}{L^{\Phi}_{t}}>t)=1$ for every $t>0$ ([2, Theorem 4,
section III.2]). Thus, for a zero drift subordinator, the passage above any
given level is almost surely realized by a jump.
Now we focus on the problem with non-local dynamic boundary conditions and
provide the probabilistic representation of the solution in terms of Brownian
motions. We deal with positive functions $u$ on unbounded domain for which
$|u(t,x)|\leq Me^{cx^{2}}$, for $M,c>0$. Thus, standard arguments guarantees
uniqueness of the solution in terms of positivity (Widder’s theorem [31]) and
exponential growth ([11, section 2.3.3]) .
Let us introduce $D_{0}=D_{1}\cup D_{2}$ where
$\displaystyle D_{1}=\left\\{\varphi:\forall\,x,\;\varphi(\cdot,x)\in
W^{1,\infty}(0,\infty)\;\;\textrm{and}\;\lim_{x\downarrow
0}\mathfrak{D}^{\Psi}_{t}\varphi(t,x)\;\textrm{exists}\right\\},$
$\displaystyle D_{2}=\left\\{\varphi:\forall\,t,\;\varphi(t,\cdot)\in
W^{1,\infty}(0,\infty)\;\textrm{and}\;\;\lim_{x\downarrow
0}\mathbf{D}^{\Phi}_{x-}\varphi(t,x)\;\textrm{exists}\right\\}.$
###### Theorem 4.1.
The solution $\upsilon\in C^{1,2}((0,\infty),(0,\infty))\cap D_{0}$ to the
problem
$\displaystyle\begin{cases}\frac{\partial}{\partial
t}\upsilon(t,x)=\Delta\upsilon(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\eta\mathfrak{D}_{t}^{\Psi}\upsilon(t,x)\big{|}_{x=0}=-\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0}\quad&t>0\\\
\upsilon(0,x)=f(x)\quad&x\geq 0\end{cases}$
with $f\in C[0,\infty)\cap L^{\infty}(0,\infty)$ and $\eta\geq 0$ is unique.
Moreover, the solution $\upsilon$ has the probabilistic representation
$\displaystyle\upsilon(t,x)=\mathbf{E}_{x}\left[f(B^{\bullet}_{S_{t}^{-1}})\right],$
where $B^{\bullet}_{t}$ is defined in (4.1) and $S_{t}^{-1}$ is the right-
inverse of $S_{t}=t+H^{\Psi}\eta L^{\Phi}_{\gamma_{t}}$.
###### Proof.
The solution $\upsilon$ can be written as
$\displaystyle\upsilon(t,x)=Q_{t}^{D}f(x)+\int_{0}^{t}\frac{x}{\tau}g(\tau,x)\upsilon(t-\tau,0)d\tau$
(4.2)
where
$\displaystyle Q_{t}^{D}f(x)=\int_{0}^{\infty}(g(t,x-y)-g(t,x+y))f(y)dy$
is the Dirichlet semigroup and $g(t,z)=e^{-z^{2}/4t}/\sqrt{4\pi t}$ for which
we recall the well-known transforms
$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}g(t,x)dt=\frac{1}{2}\frac{e^{-x\sqrt{\lambda}}}{\sqrt{\lambda}},\quad\lambda>0$
(4.3)
and
$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}\frac{x}{t}g(t,x)dt=e^{-x\sqrt{\lambda}},\quad\lambda>0.$ (4.4)
Assume that
$\displaystyle\upsilon(t,x)=\mathbf{E}_{x}\left[f(B^{\bullet}\circ
S_{t}^{-1})\right],$ (4.5)
and write the $\lambda-$potential
$\displaystyle
R_{\lambda}f(x)=\mathbf{E}_{x}\left[\int_{0}^{\infty}e^{-\lambda
t}f(B^{\bullet}\circ S_{t}^{-1})dt\right],\quad\lambda>0.$ (4.6)
From the representation (4.2) we must have
$\displaystyle R_{\lambda}f(x)=R^{D}_{\lambda}f(x)+\bar{R}_{\lambda}f(x)$
where, from (4.3),
$\displaystyle R^{D}_{\lambda}f(x)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda t}Q_{t}^{D}f(x)dt$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\left(\frac{e^{-|x-y|\sqrt{\lambda}}}{\sqrt{\lambda}}-\frac{e^{-(x+y)\sqrt{\lambda}}}{\sqrt{\lambda}}\right)f(y)dy$
(4.7)
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\frac{e^{(x-y)\sqrt{\lambda}}}{\sqrt{\lambda}}f(y)dy-\frac{1}{2}\int_{0}^{\infty}\frac{e^{-(x+y)\sqrt{\lambda}}}{\sqrt{\lambda}}f(y)dy\
+$
$\displaystyle-\frac{1}{2}\int_{0}^{x}\left(\frac{e^{(x-y)\sqrt{\lambda}}}{\sqrt{\lambda}}-\frac{e^{-(x-y)\sqrt{\lambda}}}{\sqrt{\lambda}}\right)f(y)dy,\quad\lambda>0$
and, from (4.4),
$\displaystyle\bar{R}_{\lambda}f(x)=e^{-x\sqrt{\lambda}}R_{\lambda}f(0),\quad\lambda>0$
with
$\displaystyle R_{\lambda}f(x)=\int_{0}^{\infty}e^{-\lambda
t}\upsilon(t,x)dt,\quad\lambda>0$
We can easily check that
$\displaystyle\Delta R_{\lambda}f(x)=\lambda
R_{\lambda}f(x)-f(x)=\int_{0}^{\infty}e^{-\lambda t}\frac{\partial}{\partial
t}\upsilon(t,x)dt,\quad\lambda>0,x>0.$ (4.8)
We have to characterize $R_{\lambda}f(0)$ and therefore $\upsilon(t,0)$ in
order to obtain a solution of the form given in (4.2). Thus, we focus on the
boundary condition for which, first we observe that
$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}\mathfrak{D}_{t}^{\Psi}\upsilon(t,x)\big{|}_{x=0}dt=\int_{0}^{\infty}e^{-\lambda
t}\mathfrak{D}_{t}^{\Psi}\upsilon(t,x)dt\bigg{|}_{x=0},\quad\lambda>0$
and, thanks to (3.13), we have
$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}\eta\mathfrak{D}_{t}^{\Psi}\upsilon(t,x)dt\bigg{|}_{x=0}=\eta\Psi(\lambda)R_{\lambda}f(x)-\eta\frac{\Psi(\lambda)}{\lambda}f(0),\quad\lambda>0.$
(4.9)
A key point in our proof is that
$\displaystyle\mathbf{D}_{x-}^{\Phi}R_{\lambda}f(x)\big{|}_{x=0}=\big{(}\mathbf{D}_{x-}^{\Phi}R_{\lambda}^{D}f(x)+\mathbf{D}_{x-}^{\Phi}\bar{R}_{\lambda}f(x)\big{)}\big{|}_{x=0}$
can be written by considering the definition (3.2) and by taking into account
the properties of $R_{\lambda}f$. In particular, we use the fact that
$\displaystyle\lim_{x\downarrow
0}\int_{0}^{\infty}\left(R^{D}_{\lambda}f(x)-R^{D}_{\lambda}f(x+y)\right)\Pi^{\Phi}(dy)$
$\displaystyle=\int_{0}^{\infty}\left(R^{D}_{\lambda}f(0)-R^{D}_{\lambda}f(y)\right)\Pi^{\Phi}(dy)$
$\displaystyle=-\int_{0}^{\infty}R^{D}_{\lambda}f(y)\,\Pi^{\Phi}(dy)$
and
$\displaystyle\lim_{x\downarrow
0}\int_{0}^{\infty}\left(\bar{R}_{\lambda}f(x)-\bar{R}_{\lambda}f(x+y)\right)\Pi^{\Phi}(dy)$
$\displaystyle=\lim_{x\downarrow
0}\int_{0}^{\infty}\left(e^{-x\sqrt{\lambda}}-e^{-(x+y)\sqrt{\lambda}}\right)\Pi^{\Phi}(dy)\,R_{\lambda}f(0)$
$\displaystyle=\Phi(\sqrt{\lambda})\,R_{\lambda}f(0).$
Notice that, for $|f(x)|\leq K$,
$\displaystyle\int_{0}^{\infty}R^{D}_{\lambda}f(y)\,\Pi^{\Phi}(dy)\leq\frac{K}{\lambda}\int_{0}^{\infty}(1-e^{-\sqrt{\lambda}y})\,\Pi^{\Phi}(dy)=\frac{K}{\lambda}\Phi(\sqrt{\lambda}),\quad\lambda>0.$
We therefore obtain
$\displaystyle-\mathbf{D}_{x-}^{\Phi}R_{\lambda}f(x)\big{|}_{x=0}=\int_{0}^{\infty}R^{D}_{\lambda}f(y)\,\Pi^{\Phi}(dy)-\Phi(\sqrt{\lambda})\,R_{\lambda}f(0).$
This together with (4.9) leads to
$\displaystyle
R_{\lambda}f(0)=\frac{\frac{\eta}{\lambda}\Psi(\lambda)f(0)+\int_{0}^{\infty}R_{\lambda}^{D}f(l)\
\Pi^{\Phi}(dy)}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})},\quad\lambda>0.$
(4.10)
Now we observe that
$\displaystyle\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}f(B^{\bullet}\circ
S_{t}^{-1})dt\right]=\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
S_{t}}f(B^{\bullet}_{t})dS_{t}\right]=I_{1}^{\Psi}+I_{2}^{\Psi}$
where, due to the fact that $S_{t}=t+H^{\Psi}\eta L^{\Phi}_{\gamma_{t}}$,
$\displaystyle I_{1}^{\Psi}=\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}e^{-\lambda H^{\Psi}\eta
L^{\Phi}_{\gamma_{t}}}f(B^{\bullet}_{t})dt\right],\quad\lambda>0$
and
$\displaystyle
I_{2}^{\Psi}=f(0)\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}e^{-\lambda H^{\Psi}\eta L^{\Phi}_{\gamma_{t}}}dH^{\Psi}\eta
L^{\Phi}_{\gamma_{t}}\right],\quad\lambda>0$
can be obtained as follows. The computation of $I_{1}^{\Psi}$ follows from the
computation of $e_{1}$ in [15, page 215], recalling that
$\displaystyle\mathbf{E}_{0}[e^{-\lambda H^{\Psi}_{\eta t}}]=e^{-\eta
t\Psi(\lambda)},\quad\lambda>0.$
Thus, we have that
$\displaystyle I_{1}^{\Psi}=\frac{\int_{0}^{\infty}R_{\lambda}^{D}f(l)\
\Pi^{\Phi}(dl)}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})},\quad\lambda>0.$
For $I_{2}^{\Psi}$, we get that
$\displaystyle I_{2}^{\Psi}$
$\displaystyle=f(0)\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}e^{-\lambda H^{\Psi}\eta L^{\Phi}_{\gamma_{t}}}d(H^{\Psi}\eta
L^{\Phi}_{\gamma_{t}})\right]$
$\displaystyle=f(0)\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\gamma^{-1}H^{\Phi}_{t}}e^{-\lambda
H^{\Psi}_{\eta t}}dH^{\Psi}_{\eta t}\right]$
$\displaystyle=\frac{f(0)}{\lambda}\mathbf{E}_{0}\left[-1-\int_{0}^{\infty}e^{-\lambda
H^{\Psi}_{\eta t}}d\left(e^{-\gamma^{-1}H^{\Phi}_{t}}\right)\right]$
$\displaystyle=\frac{f(0)}{\lambda}\mathbf{E}_{0}\left[-1-\int_{0}^{\infty}e^{-\eta
t\Psi(\lambda)}d\left(e^{-\gamma^{-1}H^{\Phi}_{t}}\right)\right]$
$\displaystyle=\frac{f(0)}{\lambda}\mathbf{E}_{0}\left[-1-\left(-1-\eta\Psi(\lambda)\int_{0}^{\infty}e^{-\eta
t\Psi(\lambda)}e^{-\lambda\gamma^{-1}{H^{\Phi}_{t}}}dt\right)\right]$
$\displaystyle=\frac{\eta}{\lambda}\Psi(\lambda)f(0)\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\eta
t\Psi(\lambda)}e^{-\lambda\gamma^{-1}H^{\Phi}_{t}}dt\right]$
$\displaystyle=\frac{\eta}{\lambda}\Psi(\lambda)f(0)\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\eta\Psi(\lambda)L^{\Phi}_{\gamma_{t}}}e^{-\lambda
t}dL^{\Phi}_{\gamma_{t}}\right],\quad\lambda>0.$
Now set $A_{t}:=L^{\Phi}_{\gamma_{t}}$. Then
$\displaystyle\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\eta\Psi(\lambda)A_{t}}e^{-\lambda
t}dA_{t}\right]$
$\displaystyle=-\frac{1}{\eta\Psi(\lambda)}\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}de^{-\eta\Psi(\lambda)A_{t}}\right]$
$\displaystyle=-\frac{1}{\eta\Psi(\lambda)}\mathbf{E}_{0}\left[-1+\lambda\int_{0}^{\infty}e^{-\lambda
t}e^{-\eta\Psi(\lambda)A_{t}}dt\right]$
$\displaystyle=\frac{1}{\eta\Psi(\lambda)}-\frac{\lambda}{\eta\Psi(\lambda)}\mathbf{E}_{0}\left[\int_{0}^{\infty}e^{-\lambda
t}e^{-\eta\Psi(\lambda)L^{\Phi}_{\gamma_{t}}}dt\right]$
$\displaystyle=\frac{1}{\eta\Psi(\lambda)}-\frac{\lambda}{\eta\Psi(\lambda)}\mathbf{E}_{0}\left[\int_{0}^{\infty}\frac{\sqrt{\lambda}}{\lambda}e^{-\sqrt{\lambda}w}e^{-\eta\Psi(\lambda)L^{\Phi}_{w}}dw\right]$
$\displaystyle=\frac{1}{\eta\Psi(\lambda)}-\frac{\lambda}{\eta\Psi(\lambda)}\int_{0}^{\infty}\frac{\sqrt{\lambda}}{\lambda}e^{-\eta\Psi(\lambda)z}\frac{\Phi(\sqrt{\lambda})}{\sqrt{\lambda}}e^{-z\Phi(\sqrt{\lambda})}dz$
$\displaystyle=\frac{1}{\eta\Psi(\lambda)}-\frac{\Phi(\sqrt{\lambda})}{\eta\Psi(\lambda)}\frac{1}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})}$
$\displaystyle=\frac{1}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})},$
where we have used (2.5). We arrive at
$\displaystyle
I_{2}^{\Psi}=\frac{\eta}{\lambda}\Psi(\lambda)f(0)\frac{1}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})},\quad\lambda>0.$
By collecting all pieces together, we obtain
$\displaystyle I_{1}^{\Psi}+I_{2}^{\Psi}$
$\displaystyle=\frac{\int_{0}^{\infty}R_{\lambda}^{D}f(l)\
\Pi^{\Phi}(dl)}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})}+\frac{\eta}{\lambda}\Psi(\lambda)f(0)\frac{1}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})}$
$\displaystyle=\frac{\frac{\eta}{\lambda}\Psi(\lambda)f(0)+\int_{0}^{\infty}R_{\lambda}^{D}f(l)\
\Pi^{\Phi}(dl)}{\eta\Psi(\lambda)+\Phi(\sqrt{\lambda})}.$
which coincides with (4.10). Thus, we conclude that (4.6) and (4.5) hold true.
In particular, $\upsilon$ is a continuous function with Laplace transform
$R_{\lambda}f$, then $\upsilon$ is unique. ∎
###### Remark 4.1.
The lifetime of the process $B_{S^{-}1}^{\bullet}$ (which basically moves on a
given path of a reflecting Brownian motion) is infinite, that is
$R_{0}\mathbf{1}=\infty$ and $\forall\,x$, $\upsilon(\cdot,x)\notin
L^{1}(0,\infty)$ for a bounded initial datum $f$. However, there is no hope to
find $\upsilon(\cdot,x)\in L^{1}(0,\infty)$ by also asking for $f\in
L^{1}(0,\infty)$ for which we only have
$\displaystyle R_{0}^{D}f(x)=2\int_{0}^{\infty}(y\wedge x)f(y)dy<\infty$
in the potential $R_{\lambda}f=R_{\lambda}^{D}f+\bar{R}_{\lambda}f$. The last
formula can be obtained from (4). Then we ask for $f\in L^{\infty}$ in order
to obtain $\upsilon(\cdot,x)\in L^{\infty}(0,\infty)$, $\forall x$.
We now briefly discuss the composition $H^{\Psi}L^{\Phi}$ of a subordinator
with an inverse to a subordinator. The process can be seen as a non-Markovian
time-changed process, being the random time $L^{\Phi}$ non-Markovian. We use
the following notation, for a set $I\subset(0,\infty)$,
$\displaystyle\mathbf{P}_{0}(H_{t}^{\Psi}\in
I)=\int_{I}h^{\Psi}(t,x)dx\quad\mathbf{P}_{0}(L_{t}^{\Phi}\in
I)=\int_{I}l^{\Phi}(t,x)dx,$
and
$\displaystyle\mathbf{P}_{0}(H^{\Psi}L^{\Phi}_{t}\in I)=\int_{I}w(t,x)dx.$
We have that
$\displaystyle\mathfrak{D}_{t}^{\Phi}w(t,x)=-\mathbf{D}_{x+}^{\Psi}w(t,x),\quad
t>0,x>0$ (4.11)
where $\mathfrak{D}_{t}^{\Phi}$ is defined in (3.12) and
$\mathbf{D}_{x+}^{\Psi}$ is defined in (3.1). For the reader’s convenience we
give the following sketch of proof. By applying the $\lambda-$time Laplace
transform in both sides of (4.11), from (3.13), we obtain
$\displaystyle\Phi(\lambda)\widetilde{w}(\lambda,x)-\frac{\Phi(\lambda)}{\lambda}\delta(x)=-\mathbf{D}_{x+}^{\Psi}\widetilde{w}(\lambda,x).,\quad\lambda>0,t>0.$
From (3.4), by taking into account the $\xi-$space Laplace transform, we have
$\displaystyle\Phi(\lambda)\widetilde{w}(\lambda,\xi)-\frac{\Phi(\lambda)}{\lambda}=-\Psi(\xi)\widetilde{w}(\lambda,\xi),\quad\lambda>0,\xi>0$
which leads to
$\displaystyle\widetilde{w}(\lambda,\xi)=\frac{\Phi(\lambda)}{\lambda}\frac{1}{\Phi(\lambda)+\Psi(\xi)},\quad\lambda>0,\xi>0.$
The inverse of $\widetilde{w}(\lambda,\xi)$ gives the density function of
$H^{\Psi}L^{\Phi}_{t}$ which is written as
$\displaystyle w(t,x)=\int_{0}^{\infty}h^{\Psi}(s,x)l^{\Phi}(t,s)ds,\quad
t>0,x>0$
as the independence between $H^{\Psi}$ and $L^{\Phi}_{t}$ entails.
Concerning the special case of dependent processes, that is $L^{\Phi}$ the
inverse of $H^{\Phi}$, we focus on the case of stable subordinators.
###### Proposition 4.1.
Let $H^{\alpha}$ be an $\alpha-$stable subordinator and $L^{\alpha}$ its
inverse. We have:
* (i)
For $z>t>0$
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{t}}\in
dz)=\frac{\sin(\pi\alpha)}{\pi}\frac{1}{z}\left(\frac{t}{z-t}\right)^{\alpha}dz.$
* (ii)
For $0<y<t$
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{t}-}\in
dy)=\frac{\sin(\pi\alpha)}{\pi}\frac{1}{y}\left(\frac{y}{(t-y)}\right)^{\alpha}dy.$
###### Proof.
(i) From [2, Proposition 2, section III.2] we have that, for each $t\geq 0$
and $0\leq y\leq t<z$,
$\displaystyle\mathbf{P}_{0}(H^{\alpha}L^{\alpha}_{t}-\in
dy,H^{\alpha}L^{\alpha}_{t}\in dz)=U(dy)\Pi^{\alpha}(dz-y),$ (4.12)
where $U(dy)$ is the potential measure for which
$\displaystyle\int_{0}^{\infty}e^{-\lambda
y}U(dy)=\frac{1}{\lambda^{\alpha}},\quad\lambda>0.$
It turns out that $\Pi^{\alpha}$ and $U$ are written as
$\displaystyle\Pi^{\alpha}(dz)=\frac{\alpha}{\Gamma(1-\alpha)}\frac{dz}{z^{\alpha+1}}\quad\text{and}\quad
U(dy)=\frac{1}{\Gamma(\alpha)}\frac{dy}{y^{-\alpha+1}}.$
Hence, by integrating (4.12),
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{t}}\in dz)$
$\displaystyle=\int_{0}^{t}\mathbf{P}_{0}(H^{\alpha}L^{\alpha}_{t^{-}}\in
dy,H^{\alpha}L^{\alpha}_{t}\in dz)$
$\displaystyle=\frac{\alpha}{\Gamma(1-\alpha){\Gamma(\alpha)}}\int_{0}^{t}\frac{y^{\alpha-1}}{(z-y)^{\alpha+1}}dy\,dz$
$\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{1}{z}\left(\frac{t}{z-t}\right)^{\alpha}\,dz.$
(ii) From [3, Lemma 1.10], we have
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{t}-}\in
dy)=\overline{\Pi}^{\alpha}(t-y)U(dy).$
We obtain
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{t}-}\in
dy)=\frac{1}{\Gamma(1-\alpha)}\frac{1}{(t-y)^{\alpha}}\frac{y^{\alpha-1}}{\Gamma(\alpha)}dy.$
which is the claim. ∎
We observe that, for $t=1$, $H^{\alpha}{L^{\alpha}_{1}-}$ has the so-called
generalized arcsine law
$\displaystyle\mathbf{P}_{0}(H^{\alpha}{L^{\alpha}_{1}-}\in
dy)=\frac{\sin\alpha\pi}{\pi}y^{\alpha-1}(1-y)^{-\alpha}dy,$
and for $\alpha=\frac{1}{2}$ we get the well known arcsine law for the
standard Brownian motion.
### 4.1. The case $\eta=0$: Non-local space conditions
For $\eta=0$ in Theorem 4.1, since $H^{\Psi}_{0}=0$, the probabilistic
representation of the solution $\upsilon$ can be written only in terms of
$B^{\bullet}_{t}$. We observe that
$\displaystyle\int_{0}^{\infty}(\upsilon(t,l)-\upsilon(t,0))\Pi^{\Phi}(dl)$
$\displaystyle=\lim_{x\to
0}\int_{0}^{\infty}(\upsilon(t,x+l)-\upsilon(t,x))\Pi^{\Phi}(dl)$
$\displaystyle=-\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0},$ (4.13)
then our Marchaud (type) derivative coincides with the integral operator
defined in [12]. As a confirmation of this, K. Itô and P. McKean in [15,
section 12] introduced the process $B^{\bullet}$ to write the probabilistic
representation of the solution of
$\displaystyle\begin{cases}\frac{\partial}{\partial
t}\upsilon(t,x)=\Delta\upsilon(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\int_{0}^{\infty}(\upsilon(t,l)-\upsilon(t,0))\Pi^{\Phi}(dl)=0\quad&t>0.\end{cases}$
The identification of the integral operator with a Marchaud-type derivative
allows us to use some techniques from fractional calculus. Thus we write
$\displaystyle\begin{cases}\frac{\partial}{\partial
t}\upsilon(t,x)=\Delta\upsilon(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0}=0\quad&t>0.\end{cases}$
(4.14)
We now discuss the special case of Marchaud derivatives and $\alpha-$stable
subordinators $H^{\alpha}$, characterized by
Figure 1. A simulation of the $\alpha-$stable subordinator if $\alpha=0.5$,
based on [24, section 5.2].
$\displaystyle\Phi(\lambda)=\lambda^{\alpha}=\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{\infty}(1-e^{-\lambda
x})\frac{dx}{x^{\alpha+1}},\quad\alpha\in(0,1).$
Then, we write $\mathbf{D}_{x-}^{\alpha}$ in place of $\mathbf{D}_{x-}^{\Phi}$
which coincides with the well known Marchaud left derivative. We now use the
fact that (see [18, (2.2.5)] for example)
$\displaystyle\mathbf{D}_{x+}^{\alpha}u(x)$ $\displaystyle\to u(x)\quad\ \
\alpha\downarrow 0,$ (4.15) $\displaystyle\mathbf{D}_{x+}^{\alpha}u(x)$
$\displaystyle\to u^{\prime}(x)\quad\ \alpha\uparrow 1,$ (4.16)
in order to arrive at the analogue
$\displaystyle\mathbf{D}_{x-}^{\alpha}u(x)$ $\displaystyle\to u(x)\quad\ \
\alpha\downarrow 0,$ (4.17) $\displaystyle\mathbf{D}_{x-}^{\alpha}u(x)$
$\displaystyle\to-u^{\prime}(x)\quad\alpha\uparrow 1.$ (4.18)
Indeed, formula (3.9) which still holds for $\mathbf{D}_{x\pm}^{\Phi}u\in
L^{\infty}(0,\infty)$, together with (4.15) and (4.16) say that (4.17) and
(4.18) hold in some sense. Thus, roughly speaking, for $\alpha\downarrow 0$
the problem (4.14) takes the form
$\displaystyle\begin{cases}\frac{\partial}{\partial
t}\upsilon(t,x)=\Delta\upsilon(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\upsilon(t,0)=0\quad&t>0.\end{cases}$ (4.19)
and for $\alpha\uparrow 1$, (4.14) becomes
$\displaystyle\begin{cases}\frac{\partial}{\partial
t}\upsilon(t,x)=\Delta\upsilon(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\frac{\partial}{\partial x}\upsilon(t,x)\big{|}_{x=0}=0\quad&t>0.\end{cases}$
(4.20)
As reported in [3, Section 3.1.1], for $\alpha=0$ the subordinator dies
immediately, and for $\alpha=1$, it is the elementary subordinator $t$. Then,
the process (4.1) for $\alpha\downarrow 0$ becomes a reflected Brownian motion
killed at $x=0$, that is a killed Brownian motion as expected for the solution
of (4.19). For $\alpha\uparrow 1$, the subordinator becomes $H_{t}^{\alpha}=t$
and the process (4.1) is a reflected Brownian motion that does not jump at the
boundary point. It reflects and never dies. Indeed, it coincides with a
reflected Brownian motion on $[0,\infty)$ as expected for the solution of
(4.20).
Figure 2. To the left a simulation of the $\alpha-$stable subordinator if
$\alpha=0.01$, near zero. To the right a simulation of the $\alpha-$stable
subordinator if $\alpha=0.99$, near one.
### 4.2. Non-local dynamic conditions. The case $\Phi=Id$
If $\Phi=Id$, from (4.1), we have $B^{\bullet}=B^{+}$. Then, Theorem 4.1
generalizes [10, Theorem 3.1] and the solution has the following probabilistic
representation
$\displaystyle\upsilon(t,x)=\mathbf{E}_{x}\left[f(B^{+}_{\bar{V}_{t}^{-1}})\right]$
where $\bar{V}_{t}=t+H^{\Psi}\eta{\gamma_{t}}$.
### 4.3. Dynamic condition and non-local space conditions. The case $\Psi=Id$
We notice that the condition can be written as
$\Delta\upsilon(t,x)\big{|}_{x=0}=-\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0}$
by means of which we can underline the sticky nature of the motion. Indeed,
$\mathfrak{D}_{t}^{\Psi}\upsilon=\partial_{t}\upsilon$ in Theorem 4.1 and
$\upsilon$ satisfies the heat equation on $[0,\infty)$. The solution has the
probabilistic representation
$\displaystyle\upsilon(t,x)=\mathbf{E}_{x}\left[f(B^{\bullet}_{V_{t}^{-1}})\right]$
(4.21)
where $B^{\bullet}_{t}$ is defined in (4.1) and $V_{t}=t+\eta
L^{\Phi}_{\gamma_{t}}$.
### 4.4. The reflected Brownian motion in an orthant
Our results can be extended to the $d$-dimensional case by considering the
reflected Brownian motion on a positive orthant. In particular, we consider
the semimartingale reflecting Brownian motion (SRBM in short), that is a
continuous process, say $Z=\\{Z_{t}\\}_{t>0}$, with representation $Z=B+RY$
where
* -
$B=\\{B_{t}\\}_{t>0}$ is a $d$-dimensional Brownian motion;
* -
$Y=\\{Y_{t}\\}_{t>0}$ is a positive, continuous and non-decreasing
$d$-dimensional process with $Y_{0}=0$ and components
$\\{Y^{i}_{t}\\}_{i=1,\ldots,d}$ for which
$\displaystyle\int_{0}^{t}\mathbf{1}_{R^{d}_{+}\setminus
F_{k}}(Z_{s})dY^{k}_{s}=0$
with $F_{k}=\\{x\in\mathbb{R}^{d}_{+}\,:\,x_{k}=0\\}$ for
$k\in(1,2,\ldots,d)$;
* -
$R$ is a (non-negative) reflection matrix. For example, if $R$ is (positive)
diagonal, then the direction of reflection at each point of the boundary is a
positive inward normal.
Under such conditions, the process $Z$ is a weak solution to
$dZ_{t}=dB_{t}+RdY_{t}$. The process behaves like a Brownian motion inside the
orthant and it reflects instantaneously on the boundary. The process $Y$ plays
the role of local time. A discussion about SRBM can be found in [14; 27; 32].
The vector $G\circ Y$ with components $G^{i}\circ Y^{i}:=G^{i}_{Y^{i}}$ where
$G^{i}_{t}=HL_{t}-t$, $i=1,2,\ldots,d$ can be considered in order to obtain
the $d$-dimensional process $B^{\bullet}$. Indeed, for $R=diag\\{1\\}$,
$\displaystyle B^{\bullet}_{t}=B_{t}+G\circ Y_{t}+Y_{t}$ (4.22)
is a Brownian motion pushed inside the orthant by the components
$HLY^{i}_{t}-Y^{i}_{t}$, $i=1,2,\dots,d$. We can also consider
$G_{t}^{(i)}=H^{i}L^{i}_{t}-t$ where $H^{i}\perp L^{i}$ introduces the
independent vectors $\\{H^{i}_{t}\\}$ and $\\{L^{i}_{t}\\}$. The processes
$G^{i}$ and $G^{(i)}$ give rise to different problems.
Figure 3. The path of $B^{\bullet}$ in two dimensions. The starting point is
A, the process hits the boundary at B, then it jumps to the point C.
Figure 4. The path of $B^{\bullet}_{S^{-1}}$ in two dimensions. The process
can be obtained by that path of Figure 3. The new motion is trapped on the
vertical line after the jump at the point C. Then, it moves on the vertical
line as a 1-dimensional Brownian motion (see Figure 5 ) for a random amount of
time and it starts afresh from the point D as 2-dimensional process
$B^{\bullet}$
.
Figure 5. The path of
$B^{\bullet}_{S^{-1}}\big{|}({{}_{1}B^{\bullet}_{S^{-1}}=const})$ where
$B^{\bullet}_{t}$ can be written as the vector
$({{}_{1}B^{\bullet}_{t}},{{}_{2}B^{\bullet}_{t}})$.
Indeed,${{}_{1}B^{\bullet}_{t}}$ hits the zero point and jumps according with
$G^{1}_{t}$, then it stops for a random amount of time according with
$S^{-1}$. This path is represented by the vertical line of Figure 4.
### 4.5. Reflected Brownian motion in a bounded domain
Here we have no results. We only underline the main problem in dealing with
$B^{\bullet}_{S^{-1}}$ in a bounded domain. Since we have jumps as the process
approaches the boundary, then we have to control the jumping measure so that
the process can not jump outside. As far as we know, there are no results in
this direction. The authors are working on this case. Concerning $\Phi=Id$, we
have a clear picture of the problem given in [10] for a smooth domain
$\Omega\in\mathbb{R}^{d}$.
## 5\. Discussion
### 5.1. Sample paths description
In this section we recap briefly the dynamics of the processes introduced in
Section 4. The process $B^{\bullet}$, defined in (4.1), is a reflected
Brownian motion that jumps at $0$ inside $(0,\infty)$ in a random point given
by the last jump of $H^{\Phi}$. This process turns out to be right-continuous
since $H^{\Phi}L^{\Phi}$ is the composition of the right-continuous
subordinator $H^{\Phi}$ with its inverse (and continuous process) $L^{\Phi}$.
Figure 6. A possible path for $B^{\bullet}$. The case $\eta=0$.
We recall that the reflected sticky Brownian motion, say $B^{s}$, can be
represented as a time-changed reflected Brownian motion. The solution of the
problem
$\displaystyle\begin{cases}\frac{\partial}{\partial t}v_{s}(t,x)=\Delta
v_{s}(t,x)\quad&(t,x)\in(0,\infty)\times(0,\infty)\\\
\eta\frac{\partial}{\partial t}v_{s}(t,0)=\frac{\partial}{\partial
x}v_{s}(t,0)\quad&t>0\\\ v_{s}(0,x)=f(x)\quad&x\geq 0\end{cases}$
for $\eta\geq 0$ can be written as
$\displaystyle
v(t,x)=\mathbf{E}_{x}\left[f(B^{s}_{t})\right]=\mathbf{E}_{x}\left[f(B^{+}_{T_{t}^{-1}})\right],$
where $T_{t}=t+\eta\gamma_{t}$. As $B^{+}$ hits the origin, then $\gamma_{t}$
increases in a such a way that $T^{-1}$ runs slowly and slows down $B^{s}$. In
the last section we have seen that the solution of the heat equation with
boundary condition
$\eta\partial_{t}\upsilon(t,0)=-\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0}$
is given by (4.21). The process $L^{\Phi}_{\gamma_{t}}$ can be regarded as the
local time at $0$ of the process $B^{\bullet}$. Due to right-continuity of
$H^{\Phi}$, determining the jump, the time change $V^{-1}$ slows down
$B^{\bullet}$ immediately after the jump. The same arguments apply for the
boundary condition
$\eta\mathfrak{D}_{t}^{\Psi}\upsilon(t,x)\big{|}_{x=0}=-\mathbf{D}_{x-}^{\Phi}\upsilon(t,x)\big{|}_{x=0}\quad$
for which we only underline that the waiting time after a given jump is now
independent from the process. This is due to the fact that the plateaus after
the jumps are given by $H^{\Psi}$ independently from $H^{\Phi}$.
Figure 7. A possible path for $B^{\bullet}_{S^{-1}}$. The case $\eta>0$. Since
$\Psi\neq Id$ the random plateaus are given by $H^{\Psi}$ which is independent
from $B^{\bullet}$.
### 5.2. Delayed and rushed reflection
Let us consider the construction given in [10]. We respectively denote by
$\\{e_{i},\,i\in\mathbb{N}\\}$ and $\\{H^{\Psi}_{e_{i}},\,i\in\mathbb{N}\\}$
the sequences of holding times (at the boundary) for the reflecting Brownian
motion $X^{+}$ and the partially delayed reflecting Brownian motion $\bar{X}$.
The latter agrees with a delayed reflection as a boundary behavior whereas,
the process $X^{+}$ is instantaneously reflected. We say that $\bar{X}$ is
partially delayed in the sense that a delay effect only occurs on the
boundary. Thus, $B^{\bullet}_{S^{-1}_{t}}$ equals in law $\bar{X}$ in case
$\Phi=Id$. Moreover, the process $\bar{X}$ is obtained as the composition
$X^{+}_{\bar{V}^{-1}_{t}}$ written in terms of the inverse to
$\bar{V}_{t}=S_{t}:=t+H^{\Psi}_{\tau_{t}}$ with
$\tau_{t}:=\eta\,\gamma_{t}(X)$. In particular, it turns out that
$H^{\Psi}_{e_{i}}$, $i\in\mathbb{N}$ are i.i.d. random variables for which
$\displaystyle\mathbf{E}_{x}[H^{\Psi}_{e_{i}}|X^{+}]=\mathbf{E}_{x}[e_{1}|X^{+}]\mathbf{E}_{0}[H^{\Psi}_{1}]=\mathbf{E}_{x}[e_{1}|X^{+}]\,\lim_{\lambda\downarrow
0}\frac{\Psi(\lambda)}{\lambda}$ (5.1)
gives the mean holding time on the boundary for the process $\bar{X}$. In the
same spirit of the results in [6] we are able to give a classification for the
reflection near the boundary. Write
$\displaystyle c:=\lim_{\lambda\downarrow 0}\frac{\Psi(\lambda)}{\lambda}$
and assume that (5.1) holds $\forall\,x$. The process $\bar{X}$ may have (in
terms of mean holding time on the boundary):
* B1)
delayed boundary behavior if $c<1$;
* B2)
rushed boundary behavior if $c>1$;
* B3)
base (reflected) boundary behavior if $c=1$.
A special case of subordinator including either delayed or rushed effect if
the gamma subordinator for which, for $a,b>0$,
$\displaystyle\Psi(\lambda)=a\ln(1+\lambda/b)\quad\textrm{and}\quad\lim_{\lambda\downarrow
0}\frac{\Psi(\lambda)}{\lambda}=\frac{a}{b}$
and $c<1$, $c>1$ or $c=1$ depending on the ratio $a/b$. On the other hand, by
considering $\Psi=Id$, we have that $B^{\bullet}$ jumps away from the boundary
according with the last jump of $H^{\Phi}$. Due to nature of the subordinator,
the process $B^{\bullet}$ never hits the boundary, it reflects instantaneously
before hitting the boundary. In this case, the time the process spend on the
boundary can be only given (eventually) by the starting point as a point of
the boundary, case in which the process is pushed immediately away once again
for the nature of the subordinator. We recall that our analysis is based only
on strictly increasing subordinators.
Our analysis introduces the following further characterization:
* R1)
delayed/rushed reflection;
* R2)
instantaneous jumping reflection;
* R3)
reflection.
The delayed reflection can be related with a reflection on the boundary (like
an insulating boundary for example) whereas the rushed reflection seems to be
related with a reflection near the boundary (like an inflating boundary for
example). These characterizations are therefore given according with the
comparison between the total (mean) amounts of time the processes $B^{+}$ and
$\bar{X}$ would spend on the boundary. By base boundary behavior we mean the
behavior of the reflecting process $B^{+}$. We observe that instantaneous
reflection means that the time the process spends on the boundary has zero
Lebesgue measure. The reflection R1 is obtained via time change (by
considering $S^{-1}_{t}$) and depends on the symbol $\Psi$ of the boundary
condition. R1 gives rise to the behaviors B1 and B2, notice that in this case
we may consider a boundary diffusion (in the sense of Dirichlet-Neumann
generator) which can be delayed or rushed with respect to the base boundary
diffusion (according with the definition given in [6]). R2 turns out to be
associated with the symbol $\Phi$ of the boundary condition. Concerning R3 we
easily make the connection with B3, that is the case of Neumann boundary
condition.
The behaviors B1-B2-B3 can be associated with a macroscopic description of
motions on irregular domains. Indeed, irregular domains in a microscopic point
of view need a geometric characterization in order to be completely described.
By irregular domain here we mean domains with irregular boundaries, for
example the Koch domain still is regular. The definition given in [4] of trap
domain says that if a reflecting process has an infinite (mean) lifetime in a
bounded domain including a Dirichlet boundary, then that process is trapped in
the reflecting part of the boundary. That is to say, the boundary and
therefore the domain is trap for that process. In a deep detail, given a
$d$-dimensional reflecting Brownian motion $B^{+}$ on a domain
$\Omega\setminus\mathcal{B}$ where $\mathcal{B}$ is a ball with killing
boundary $\partial\mathcal{B}$ and $\Omega$ has reflecting boundary
$\partial\Omega$, then $\Omega$ is trap for the Brownian motion if
$\displaystyle\sup_{x\in\Omega\setminus\mathcal{B}}\mathbf{E}_{x}[\inf\\{t\,:\,B^{+}_{t}\notin\Omega\setminus\mathcal{B}\\}]=\infty.$
(5.2)
From the analytical point of view we ask for
$\displaystyle\sup_{x\in\Omega\setminus\mathcal{B}}\int_{\Omega\setminus\mathcal{B}}G^{+}(x,y)dy=\infty\quad\textrm{where}\quad
G^{+}(x,y)dy=\int_{0}^{\infty}\mathbf{P}_{x}(B^{+}_{t}\in dy)\,dt.$
Concerning the process $B^{\bullet}_{S^{-1}}$ on
$\Omega^{*}\subset\mathbb{R}^{d}$ with $\Phi=Id$ we provide the following
conjecture as a possible link between macroscopic and microscopic viewpoints.
We have $B^{\bullet}_{S^{-1}}\stackrel{{\scriptstyle
law}}{{=}}X^{+}_{\bar{V}^{-1}}$ as discussed above in terms of $\bar{X}$ on
$\Omega^{*}$. Our statement is the following:
The behavior of $B^{+}$ on the irregular domain $\Omega$ can be captured by
$\bar{X}$ on the smooth domain $\Omega^{*}$.
This statement is justified by the fact that the time change $\bar{V}^{-1}$
for the reflecting Brownian motion $X^{+}$ introduces the behaviors B1-B2-B3
and the (mean) total amount of time the process spend on the boundary if given
in terms of (5.1). This well accords with (5.2). We argue that different
symbols may be considered (in the macroscopic analysis) in order to describe
different boundaries (in the microscopic analysis).
We conclude this section by underlying an interesting connection with Brownian
excursions. An excursion of $B^{+}$ can be associated with no crossing trough
a boundary for $B^{+}$ and therefore, with no Poison events associated with
the crossing. For the holding time,
$\displaystyle\mathbf{P}_{x}(H^{\Psi}_{e_{i}}>t|X^{+})=\mathbf{P}_{x}(e_{i}>L^{\Psi}_{t}|X^{+})=\mathbf{E}_{0}[e^{-L^{\Psi}_{t}}]$
assuming that the exponential random variables $e_{i}$ are of parameter $1$.
The excursions of $B^{\bullet}_{S^{-1}}$ introduces the non-local Poisson
process as we can immediately see for $\Psi(\lambda)=\lambda^{\alpha}$,
$\alpha\in(0,1)$. In this case,
$\displaystyle\mathbf{P}_{x}(H^{\Psi}_{e_{i}}>t|X^{+})=E_{\alpha}(-t^{\alpha})=:\mathbf{P}_{0}(N_{L_{t}}=0)$
where $L$ is an inverse to a stable subordinator and $N$ is a Poisson process
with $\mathbf{P}_{0}(N_{t}=k)=(t^{k}/k!)e^{-t}$. The fractional Poisson
process has been introduced in [21], the literature is really vast, we refer
to [1, formula (3.19)] and [26].
## References
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* [2] Jean Bertoin. Lévy processes, volume 121. Cambridge university press Cambridge, 1996.
* [3] Jean Bertoin. Subordinators: examples and applications. In Lectures on probability theory and statistics, pages 1–91. Springer, 1999.
* [4] Krzysztof Burdzy, Zhen-Qing Chen, and Donald E. Marshall. Traps for reflected brownian motion. Mathematische Zeitschrift, 252(1):103–132, 2006.
* [5] Raffaela Capitanelli and Mirko D’Ovidio. Fractional equations via convergence of forms. Fractional Calculus and Applied Analysis, 22(4):844–870, 2019.
* [6] Raffaela Capitanelli and Mirko D’Ovidio. Delayed and rushed motions through time change. ALEA, 17:183–204, 2020.
* [7] Zhen-Qing Chen. Time fractional equations and probabilistic representation. Chaos, Solitons & Fractals, 102:168–174, 2017.
* [8] Fausto Colantoni and Mirko D’Ovidio. On the inverse gamma subordinator. Stochastic Analysis and Applications, pages 1–26, 2022.
* [9] Mirko D’Ovidio. Fractional boundary value problems. Fractional Calculus and Applied Analysis, 25(1):29–59, 2022.
* [10] Mirko D’Ovidio. Fractional boundary value problems and elastic sticky brownian motions. arXiv preprint arXiv:2205.04162, 2022.
* [11] Lawrence C. Evans. Partial differential equations, volume 19. American Mathematical Soc., 2010.
* [12] William Feller. The parabolic differential equations and the associated semi-groups of transformations. Annals of Mathematics, pages 468–519, 1952.
* [13] Fausto Ferrari. Weyl and marchaud derivatives: A forgotten history. Mathematics, 6(1):6, 2018.
* [14] Michael J. Harrison and Martin I. Reiman. Reflected brownian motion on an orthant. The Annals of Probability, 9(2):302–308, 1981.
* [15] Kiyosi Itô and Henry P. McKean, Jr. Brownian motions on a half line. Illinois journal of mathematics, 7(2):181–231, 1963.
* [16] Kiyosi Itô and Henry P. McKean, Jr. Diffusion processes and their sample paths. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. Second printing, corrected.
* [17] Sato Ken-Iti. Lévy processes and infinitely divisible distributions. Cambridge university press, 1999.
* [18] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations, volume 204 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006.
* [19] Anatoly N. Kochubei. General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory, 71(4):583–600, 2011.
* [20] Vassili N. Kolokoltsov. Markov processes, semigroups and generators, volume 38 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 2011.
* [21] Nick Laskin. Fractional poisson process. Communications in Nonlinear Science and Numerical Simulation, 8(3-4):201–213, 2003.
* [22] Giovanni Leoni. A first course in Sobolev spaces. American Mathematical Soc., 2017.
* [23] Mark M. Meerschaert and Hans-Peter Scheffler. Triangular array limits for continuous time random walks. Stochastic processes and their applications, 118(9):1606–1633, 2008\.
* [24] Mark M. Meerschaert and Alla Sikorskii. Stochastic models for fractional calculus. de Gruyter, 2019.
* [25] Ralph S. Phillips. On the generation of semigroups of linear operators. Pacific Journal of Mathematics, 2(3):343–369, 1952.
* [26] Mauro Politi, Taisei Kaizoji, and Enrico Scalas. Full characterization of the fractional poisson process. EPL (Europhysics Letters), 96(2):20004, 2011.
* [27] Martin I. Reiman and Ruth J. Williams. A boundary property of semimartingale reflecting brownian motions. Probability Theory and Related Fields, 77(1):87–97, 1988.
* [28] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev. Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon, 1993.
* [29] René L. Schilling, Renming Song, and Zoran Vondracek. Bernstein functions. de Gruyter, 2012.
* [30] Bruno Toaldo. Convolution-type derivatives, hitting-times of subordinators and time-changed $C_{0}$-semigroups. Potential Analysis, 42(1):115–140, 2015.
* [31] David V. Widder. Positive temperatures on an infinite rod. Transactions of the American Mathematical Society, 55:85–95, 1944\.
* [32] Ruth J. Williams. Semimartingale reflecting Brownian motions in the orthant. In Stochastic networks, volume 71 of IMA Vol. Math. Appl., pages 125–137. Springer, New York, 1995.
|
$E_{1}$ is simple, the first claim clearly hold for $E$, so we only need to
verify the second claim. That holds because, by the form of $E$, every node
$v$ is in $\llbracket E\rrbracket^{G^{\prime}}(v)$, but not in $\llbracket
r\rrbracket^{G^{\prime}}(v)$, as $G$ does not have any self-loops.
Finally, assume $E$ is of the form $E_{1}/E_{2}$. Note that if $E_{1}$ or
$E_{2}$ is simple, clearly claim one holds because $\llbracket
r\rrbracket^{G^{\prime}}\subseteq\llbracket E\rrbracket^{G^{\prime}}$. The
argument that follows will therefore also apply when $E_{1}$ or $E_{2}$ is
simple. We will be careful not to apply the induction hypothesis for the
second statement to $E_{1}$ and $E_{2}$.
We distinguish two cases.
* •
If $\llbracket r\rrbracket^{G^{\prime}}\subseteq\llbracket
E_{2}\rrbracket^{G^{\prime}}$, then we show that $\llbracket
r\rrbracket^{G^{\prime}}\subseteq\llbracket E\rrbracket^{G^{\prime}}$. Let
$v\in X_{i}$. We verify the following two inclusions:
* –
$\llbracket E\rrbracket^{G}(v)\supseteq X_{i}$. Let $u\in X_{i}$. If $u\neq
v$, choose a third node $w\in X_{i}$. Since $X_{i}$ is a clique,
$(v,w)\in\llbracket E_{1}\rrbracket^{G}$ because the first claim holds for
$E_{1}$. By $\llbracket r\rrbracket^{G^{\prime}}\subseteq\llbracket
E_{2}\rrbracket^{G^{\prime}}$, we also have $(w,u)\in\llbracket
E_{2}\rrbracket^{G^{\prime}}$, whence $u\in\llbracket
E\rrbracket^{G^{\prime}}(v)$ as desired. If $u=v$, we similarly have
$(v,w)\in\llbracket E_{1}\rrbracket^{G^{\prime}}$ and $(w,u)\in\llbracket
E_{2}\rrbracket^{G^{\prime}}$ as desired.
* –
$\llbracket E\rrbracket^{G}(v)\supseteq X_{\mathit{next}(i)}$. Let $u\in
X_{\mathit{next}(i)}$ and choose $w\neq v\in X_{i}$. Because the first claim
holds for $E_{1}$, we have $(v,w)\in\llbracket E_{1}\rrbracket^{G}$. By
$\llbracket r\rrbracket^{G^{\prime}}\subseteq\llbracket
E_{2}\rrbracket^{G^{\prime}}$, we also have $(w,u)\in\llbracket
E_{2}\rrbracket^{G^{\prime}}$, whence $u\in\llbracket
E\rrbracket^{G^{\prime}}(v)$ as desired.
We conclude that $\llbracket E\rrbracket^{G^{\prime}}(v)\supseteq X_{i}\cup
X_{\mathit{next}(i)}\supseteq\llbracket r\rrbracket^{G^{\prime}}$ as desired.
* •
If $\llbracket r^{-}\rrbracket^{G^{\prime}}\subseteq\llbracket
E_{2}\rrbracket^{G^{\prime}}$, then we show that $\llbracket
r^{-}\rrbracket^{G^{\prime}}\subseteq\llbracket E\rrbracket^{G^{\prime}}$.
This is analogous to the previous case, now verifying that $\llbracket
E\rrbracket^{G}(v)\supseteq X_{i}\cup X_{\mathit{prev}(i)}$.
In both cases, the second statement now follows for every node $v$. Indeed,
$v\in X_{i}\subseteq\llbracket E\rrbracket^{G^{\prime}}(v)$ but
$v\notin\llbracket r\rrbracket^{G^{\prime}}(v)$. |
# QFNN-FFD: Quantum Federated Neural Network for Financial Fraud Detection
Nouhaila Innan1, Alberto Marchisio2,3, Muhammad Shafique2,3, and Mohamed
Bennai1 1Quantum Physics and Magnetism Team, LPMC, Faculty of Sciences Ben
M’sick,
Hassan II University of Casablanca, Morocco
2eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD),
Abu Dhabi, UAE
3Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute,
NYUAD, Abu Dhabi, UAE
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
This study introduces the Quantum Federated Neural Network for Financial Fraud
Detection (QFNN-FFD), a cutting-edge framework merging Quantum Machine
Learning (QML) and quantum computing with Federated Learning (FL) for
financial fraud detection. Using quantum technologies’ computational power and
the robust data privacy protections offered by FL, QFNN-FFD emerges as a
secure and efficient method for identifying fraudulent transactions within the
financial sector. Implementing a dual-phase training model across distributed
clients enhances data integrity and enables superior performance metrics,
achieving precision rates consistently above 95%. Additionally, QFNN-FFD
demonstrates exceptional resilience by maintaining an impressive 80% accuracy,
highlighting its robustness and readiness for real-world applications. This
combination of high performance, security, and robustness against noise
positions QFNN-FFD as a transformative advancement in financial technology
solutions and establishes it as a new benchmark for privacy-focused fraud
detection systems. This framework facilitates the broader adoption of secure,
quantum-enhanced financial services and inspires future innovations that could
use QML to tackle complex challenges in other areas requiring high
confidentiality and accuracy.
###### Index Terms:
Quantum Neural Network, Quantum Federated Learning, Quantum Machine Learning,
Fraud Detection, Finance
## I Introduction
In the rapidly evolving financial technology landscape, privacy is a
fundamental pillar, crucial for upholding the trust and integrity of financial
transactions and services [1]. As digital transactions become more prevalent,
the volume of sensitive data handled by financial institutions grows
exponentially, making robust privacy measures indispensable [2]. The emergence
of Quantum Machine Learning (QML) marks a transformative era [3, 4, 5],
promising unprecedented computational capabilities by exploiting quantum
physics [6], while simultaneously raising pivotal concerns about privacy and
data security. This paper introduces the Quantum Federated Neural Network for
Financial Fraud Detection (QFNN-FFD), a pioneering framework that influences
the quantum-enhanced processing power of Quantum Computing (QC) with the
privacy-preserving attributes of Federated Learning (FL). The synergy of QML
with FL jointly improves the efficiency and accuracy of detecting fraudulent
activities, while safeguarding sensitive financial data against the ever-
looming threats of breaches and unauthorized access.
QFNN-FFD represents a significant leap forward in applying quantum
technologies to real-world economic challenges and sets a new benchmark for
privacy-centric approaches in the fintech domain. By deploying this framework,
financial institutions can potentially harness the unique advantages of
QC—such as rapid processing of large datasets—while also benefiting from the
decentralized nature of FL, which keeps sensitive data localized and reduces
the risk of central points of failure. As shown in Fig. 1, Quantum Federated
Learning (QFL) has shown superior performance in various fields [7, 8, 9],
prompting our decision to implement it in finance. Our framework has
demonstrated its capability to enhance both accuracy and privacy protection
through comparative analysis with existing models works [10, 11, 12]. This
approach meets and often surpasses current industry standards, providing a
scalable, secure framework that adapts seamlessly to diverse operational
environments while maintaining high accuracy in fraud detection under various
conditions.
(a)(b) Figure 1: Comparison of ML and FL accuracies in classical and QC
contexts across various fields and experiments. Panel (a)illustrates the
performance of different experiments within the finance sector. Panel (b)
compares QML with QFL across four domains: healthcare, IoT, computer v, and
finance. In classical computing contexts, FL generally demonstrates superior
performance compared to ML [13, 14]. In QC contexts, QFL exhibits slight
improvements over QML [15, 16, 17]. These findings highlight the potential of
QFL and provide a compelling rationale for its adoption, particularly in the
finance sector.
Our contributions significantly impact the fintech sector by providing a
scalable, secure framework that adapts to various operational environments
while maintaining high accuracy in fraud detection under different conditions.
Figure 2: The QFNN-FFD process flow. The diagram outlines the end-to-end
workflow from input through to output. Datasets are processed and fed into the
QFNN-FFD, built upon the PennyLane library. The model undergoes training and
testing for 100 iterations, incorporating a variety of noise models using
noise simulators from IBM’s Qiskit. The quantum simulator within PennyLane is
utilized to emulate a quantum environment. The output is evaluated based on
performance metrics, including accuracy, precision, recall, F1 score, and mean
squared error loss, providing a comprehensive assessment of the model’s
capability to detect fraudulent transactions.
Our novel contributions, encapsulating a comprehensive workflow that
significantly enhances fintech security measures, are listed as follows and
shown in Fig. 2:
* •
Introducing a novel QFNN-FFD that uniquely combines QML algorithms with FL
architecture to enhance both the computational capabilities and the privacy
aspects of fraud detection systems, ensuring that sensitive financial data
remains within its local environment.
* •
Demonstrating superior analytical capabilities by analyzing complex
transaction patterns more effectively than traditional models, comparative
experimental results reveal that QFNN-FFD consistently outperforms existing
fraud detection systems in terms of accuracy, thereby establishing a new
benchmark for the industry.
* •
Recognizing the challenges posed by quantum decoherence and noise by testing
our QFNN-FFD across six different quantum noise models to validate its
robustness ensures that our framework is not only theoretically but also
practically viable in real-world QC environments, maintaining high performance
under various noise conditions.
## II Background and Related Works
FL is a Machine Learning (ML) paradigm in which multiple parties [18, 19, 20],
termed clients, collaborate under the oversight of a central server to address
an ML task without exchanging their raw data.
Figure 3: Schematic representation of the FL architecture. The diagram shows
multiple users (clients), each with their local dataset, independently
training local models. These models are then transmitted as model updates to a
central server. The server aggregates these updates to improve the global
model, which is then distributed back to the users for further refinement.
This cycle ensures data privacy and security, as raw data never leaves the
local premises of each user.
As illustrated in Fig. 3, clients contribute model updates, computed from
their local datasets, to the server. Mathematically, each client $i$ computes
an update $\Delta\theta_{i}$ based on its local data $D_{i}$:
$\Delta\theta_{i}=-\eta\nabla L(\theta;D_{i}),$ (1)
where $\eta$ is the learning rate and $L$ is the loss function evaluated with
the ML model parameters $\theta$. These updates $\Delta\theta_{i}$ are then
sent to the central server, which aggregates them to update the global model
using a weighted average:
$\theta\leftarrow\theta+\sum_{i=1}^{n}\frac{|D_{i}|}{D}\Delta\theta_{i},$ (2)
where $D=\sum_{i=1}^{n}|D_{i}|$ represents the total size of data across all
clients, and $|D_{i}|$ is the size of the local dataset of client $i$. This
aggregation method effectively mitigates concerns related to privacy, data
security, and data access rights, which are particularly pertinent when
dealing with sensitive information scattered across disparate locations.
The progression of FL into the QC domain has precipitated the inception of QFL
[17, 21]. This methodology exploits quantum mechanics’ distinctive properties
to augment privacy and computational efficiency. In [22], the study delineated
the first fully operational QFL framework capable of processing exclusively
quantum data. This innovation indicated the establishment of the inaugural
quantum federated dataset, facilitating the collaborative learning of quantum
circuit parameters by quantum clients in a decentralized manner—a cornerstone
in adapting quantum technologies to federated contexts.
Subsequently, the notion of dynamic QFL frameworks was advanced in [8], which
introduced the Slimmable QFL. This framework was designed to adapt to varying
network conditions and constraints on computing resources by dynamically
modulating the training parameters of Quantum Neural Networks (QNNs).
Empirical studies demonstrated that SlimQFL sustained superior classification
accuracy under fluctuating conditions compared to conventional static QFL
methods. Moreover, integrating secure protocols, such as blind quantum
computation, into distributed learning environments enhanced the
confidentiality aspect of QFL.
The research outlined in [23] proposed a quantum protocol that leveraged the
computational capacities of remote quantum servers while safeguarding the
privacy of the underlying data. This protocol proved its robustness against
prevalent security threats, including gradient attacks, rendering it
especially beneficial in domains that demand boosted security for distributed
learning tasks.
It is essential to recognize the expansive applications of QFL across various
industries and how these applications introduce specialized implementations in
sectors requiring high data privacy and computational precision. Particularly
in the financial industry, where the confidentiality and integrity of data are
paramount, the transition from general data protection to targeted fraud
detection represents a critical evolution of QFL capabilities.
The effectiveness of QFL in securely managing and processing data within
healthcare and genomics, as explored in [24], serves as a foundation for its
application in the more complex and sensitive realm of financial transactions.
This broad applicability underscores the potential of QFL to enhance privacy
and computational efficiency in highly effective scenarios.
Advancing into financial fraud, significant research has been conducted to
apply QC and QML in detecting financial fraud. In [12], they developed quantum
protocols for anomaly detection, applying them to credit card fraud. They
compared quantum kernel methods to classical ML benchmarks, revealing the
potential for quantum approaches to achieve superior precision significantly
as the number of qubits increased. This demonstrated that quantum methods can
become increasingly advantageous with the scaling of quantum systems.
Furthermore, in [25], they explored using a Quantum Support Vector Machine
(QSVM) for real-world financial data, presenting one of the first end-to-end
applications of QML in the financial sector. Their findings highlighted the
complementary nature of QSVM to classical techniques, offering new methods for
feature exploration that enhanced fraud detection accuracy.
As the application of QML in fraud detection advances, several innovative
approaches have emerged. For instance, in [26], they explored using QML
models, including the Variational Quantum Classifier (VQC) and different QNNs.
These models showed promising results in classifying fraud and non-fraud
transactions, demonstrating QML’s potential in financial applications. Despite
their success, these models faced challenges such as needing more efficient
quantum algorithms and the ability to handle larger and more complex datasets.
In [27], the study addressed the latency in traditional fraud detection
systems by implementing a QML approach using a Support Vector Machine (SVM)
enhanced with quantum annealing solvers. This approach significantly
outperformed traditional models’ speed and accuracy, particularly on time-
series data from bank loans, which are typically highly imbalanced.
In [28], they discussed a hybrid model that combines QNNs with classical
neural networks to enhance fraud detection capabilities. This study
implemented two distinct methods: a hybrid quantum-classical neural network
and topological data analysis for noise reduction and improved classification
accuracy. Such hybrid models leverage the computational power of quantum
devices while maintaining the versatility and robustness of classical ML
frameworks.
Generative adversarial networks (GANs) have also been adapted to quantum
settings to tackle the instability and inefficiency of classical sampling
methods. [29] introduced variational quantum-classical Wasserstein GANs
(WGANs), which incorporated a hybrid quantum-classical generator with a
classical discriminator. This model was effective on a credit card fraud
dataset, providing competitive performance with classical counterparts in
terms of F1 score.
Further advancing the field, in [30], they presented an approach using data
re-uploading techniques to train single-qubit classifiers that perform
comparably to classical models under similar training conditions. This study
highlights the potential for QML to achieve significant results with minimal
quantum resources, opening new avenues for efficient quantum computations.
Moreover, in [31] and [32], they highlighted the real-time challenges in fraud
detection. They utilized quantum annealing to develop frameworks that enhance
the speed of fraud detection, addressing the critical issue of timely response
in fraudulent transaction identification.
These studies collectively demonstrate the growing capability of QML to
enhance fraud detection but often neglect the aspect of data privacy in their
computational frameworks. Most QML models focus primarily on computational
advantages without integrating robust privacy measures. Our QFNN-FFD framework
addresses this gap by combining the privacy-preserving features of FL with the
power of QC. By ensuring that data remains local and only aggregate updates
are shared, our framework enhances the security and privacy of the distributed
learning process, setting a new standard in applying quantum technologies to
sensitive financial operations.
## III QFNN-FFD Framework Design
In this section, we introduce a novel QFNN-FFD framework that integrates the
quantum computational capabilities of QML with the distributed, privacy-
preserving nature of FL, as described in Algorithm 1.
Data: QNN circuit, dataset split among $N$ clients, learning rate $\eta=0.1$,
maximum local iterations $T$.
Result: Accuracy, precision, recall, F1 score, and loss
Initialization: Parameters $\theta$ randomly initialized in $[0,1]$;
for _each client $i=1$ to $N$_ do
Initialize local model parameters $\theta_{i}\leftarrow\theta$;
for _each local iteration $t=1$ to $T$_ do
foreach _batch in local dataset_ do
Encode data into quantum states;
Apply QNN circuit with current parameters $\theta_{i}$;
Perform quantum measurements to obtain classical outputs;
Calculate loss using MSE;
Optimize $\theta_{i}$ using Adam optimizer with learning rate $\eta$;
Evaluate local model on validation set and adjust $\theta_{i}$;
If convergence criteria are met, exit loop early;
Synchronize and send optimized local parameters $\theta_{i}$ to central
server;
On central server:;
Aggregate local parameters to update global model; Broadcast updated global
parameters $\theta$ back to each client;
for _each client $i=1$ to $N$_ do
Update local model parameters $\theta_{i}\leftarrow\theta$;
Evaluate model performance on a global validation set to ensure
generalization;
Algorithm 1 QFNN-FFD Framework
### III-A QNN Circuit Design and QFL Integration
Central to this approach is a QNN circuit, shown in Fig. 5. The QNN model has
demonstrated its powerful capabilities in various applications, particularly
fraud detection. Like typical QML models, as shown in Fig. 4, it begins with
data encoding, followed by a sequence of quantum operations that form the core
of the processing circuit, and concludes with measurement to extract
actionable insights [33, 34, 35, 36].
Figure 4: General schematic of a QML model workflow. The process begins with
qubits in the zero state $(|0\rangle)$. The qubits undergo data encoding to
represent the input data in quantum form. Then, a parametrized quantum
circuit, $U(\theta)$, transforms the qubit states, where $\theta$ represents
tunable parameters. The transformed quantum states are measured, converting
quantum information into classical output. This output is evaluated using a
predefined loss function, and a classical optimization algorithm iteratively
adjusts $\theta$ to minimize the loss, thereby refining the QML model’s
performance.
The QFNN-FFD framework operates on data distributed across $N$ clients, each
possessing a subset of the overall dataset, thereby preventing the need for
central data aggregation and enhancing data privacy. Training of the QFNN-FFD
is directed in a federated manner, where local models on each client are
independently trained using their data subsets.
In the local model, the first step is to encode classical data into quantum
states through angle encoding. Each data feature $x_{i,j}$ from the vector
$\mathbf{x}_{i}$ of client $i$ is mapped onto two rotation angles,
$\theta_{i,j}$ for the $R_{y}$ rotation and $\phi_{i,j}$ for the $R_{z}$
rotation. These rotations are then applied to the qubits sequentially to
modify both their phase and orientation:
$R(\theta_{i,j},\phi_{i,j})=R_{y}(\theta_{i,j})R_{z}(\phi_{i,j}),$ (3)
where $R_{y}(\theta_{i,j})=e^{-i\theta_{i,j}Y/2}$ and
$R_{z}(\phi_{i,j})=e^{-i\phi_{i,j}Z/2}$, with $Y$ and $Z$ representing the
Pauli-Y and Pauli-Z matrices, respectively.
We apply a series of controlled operations to achieve an entangled quantum
state that captures correlations between different features. One effective
method is using a sequence of CNOT gates, which create entanglements between
successive qubits:
$U_{\text{ent}}=\prod_{k=1}^{n-1}\text{CNOT}_{k,k+1},$ (4)
where $\text{CNOT}_{k,k+1}$ applies a CNOT gate between the $k$-th and
$(k+1)$-th qubits. This sequence creates a chain of entanglements across the
qubit register, which is crucial for leveraging quantum correlations.
This setup ensures that the quantum states are intricately linked, which is
crucial for capturing complex correlations in the dataset. The full quantum
state preparation for client $i$ is thus represented by:
$\ket{\psi_{i}}=\left(\bigotimes_{j=1}^{n}R_{y}(\theta_{i,j})R_{z}(\phi_{i,j})\right)\cdot\text{CNOT}\ket{0}^{\otimes
n}.$ (5)
Figure 5: An overview of the QFNN-FFD framework. This flowchart presents the
multi-stage process, beginning with data preprocessing and distribution to
various users. Each user independently conducts a local training phase on a
QNN circuit, followed by an optimization stage. The optimized local models are
then transmitted to a central cloud server for global aggregation, culminating
in an enhanced federated model. The lower part of the figure illustrates the
quantum circuit’s structure, showcasing the intricate interplay of qubits and
quantum gates (rotations and CNOT gates) during the computation process.
### III-B Optimization and Training Process
The Adam optimizer is integral to the training process of our QFNN-FFD
framework due to its adaptive learning rate capabilities, which significantly
enhance convergence speed and efficiency. The Adam optimizer’s update rule is
particularly well-suited for the demands of quantum circuit training and is
defined as follows:
$\theta_{t+1}=\theta_{t}-\frac{\eta}{\sqrt{\hat{v}_{t}}+\epsilon}\hat{m}_{t},$
(6)
where $\eta$ represents the learning rate, $\hat{m}_{t}$ and $\hat{v}_{t}$ are
the estimates of the first and second moments of the gradients, respectively,
and $\epsilon$ is a small constant to avoid division by zero. This
configuration allows each parameter update to be adjusted dynamically based on
the individual gradients’ variability, providing a tailored approach to
parameter optimization.
In the context of our QFNN-FFD, the Adam optimizer’s role extends to
effectively minimizing the Mean Squared Error (MSE) loss function during the
training process. The MSE loss function is crucial for calibrating the model’s
predictive accuracy and is expressed as:
$L(\theta)=\frac{1}{m}\sum_{j=1}^{m}(y_{j}-\hat{y}_{j}(\theta))^{2},$ (7)
where $m$ is the batch size, $y_{j}$ are the actual labels of transactions,
and $\hat{y}_{j}(\theta)$ represents the predicted labels output by the model.
This loss function quantifies the error between the model’s predictions and
the true labels, guiding the optimizer to focus on reducing these
discrepancies. The optimization process iterates through a maximum of $T$
local iterations, refining the model’s ability to discern fraudulent
transactions accurately.
### III-C Parameter Aggregation and Model Evaluation
Following local optimization, each client’s parameters $\theta_{i}$ are
transmitted to a central server. They are aggregated through a simple
averaging process to update the global model parameters $\theta$. This cyclic
process of local optimization and global aggregation iteratively enhances the
QFNN-FFD’s performance, evaluated on a global validation set for
generalizability and efficacy. The mathematical foundation of parameter
optimization within the QFNN-FFD employs the Adam optimizer, adjusting
$\theta$ as
$\theta_{t+1}=\theta_{t}-\eta\cdot\text{Adam}(\nabla_{\theta}L(\theta_{t}))$
(8)
where $\text{Adam}(\nabla_{\theta}L(\theta_{t}))$ calculates the adjustment
based on the gradient of the loss function with respect to the parameters
$\theta$ at iteration $t$. This optimization ensures a gradual refinement of
the model’s parameters.
After the local training phases, the optimized parameters $\theta_{i}$ from
each client are securely aggregated at a central server using a federated
averaging algorithm:
$\theta_{\text{global}}=\frac{1}{N}\sum_{i=1}^{N}\theta_{i},$ (9)
This aggregation step effectively combines insights from all the distributed
models, enhancing the global model’s generalizability and robustness, steering
the QFNN-FFD towards higher accuracy in fraud detection (see Algorithm 1).
The globally updated parameters are redistributed to all clients for further
training, cycling through local optimization and global aggregation to
progressively improve the QFNN-FFD’s performance. This iterative process
enhances computational efficiency and maintains strict privacy standards.
Integrating QML with FL in our QFNN-FFD framework fosters the high-efficiency
processing of complex financial data and upholds stringent data privacy
standards. This dual advantage, coupled with the model’s mathematical rigor
and strategic parameter optimization, positions the QFNN-FFD as an effective
tool in the fight against financial fraud, marking a significant leap forward
in applying QC to real-world challenges in the financial sector.
## IV Results and Discussion
### IV-A Experimental Setup
In our study, we utilize the IEEE-CIS Fraud Detection dataset [37]. It is
divided into two primary files: identity and transaction, linked by
TransactionID. It encompasses both numerical and categorical features
essential for identifying fraudulent activities. The preprocessing steps begin
with optimizing the dataset’s memory usage by refining data types,
significantly reducing its memory footprint. This is followed by a detailed
analysis of missing values, which helps identify and quantify missing data to
inform our approach to managing these instances. Subsequently, features are
categorized and processed: categorical variables undergo one-hot encoding,
while numerical variables are standardized. To counteract the class imbalance
between fraud and non-fraud instances, an up-sampling technique is employed to
ensure equitable representation of both classes.
Our QFNN-FFD is implemented using PennyLane for model architecture and Qiskit
for simulating quantum noise [38, 39], enabling a realistic QC environment.
The framework, structured around four qubits, employs the Adam optimizer
($\eta$=0.1) across dual training phases—local and global—with up to 100
iterations for each across 15 clients. This setup is characterized by 32
initially random parameters, which are optimized through evaluations on a
training set comprising 115,386 instances (80% of the total dataset of 144,233
instances) and a validation set comprising 28,847 instances, which is 20% of
the total dataset. We focus on binary classification accuracy and MSE as key
metrics. Operational deployment occurs within an environment characterized by
a configuration consisting of 4 virtual CPUs (vCPUs), 25 gigabytes (GB) of
RAM, and a single NVIDIA Tesla V100 virtual GPU (vGPU). This setup offers a
balance of processing power, memory capacity, and advanced GPU
acceleration—crucial factors for efficiently handling the intensive
computations required by our QFNN-FFD framework.
### IV-B Accuracy and Loss Analysis
The validation accuracy and loss trends for the QFNN-FFD, as shown in Fig. 6,
provide valuable insights into the model’s performance over iterations, which
is the average outcome of 10 trials of QFNN, which ensures the reliability of
the results by accounting for variability in the model’s performance.
Initially, the model’s accuracy begins at 0.735 and demonstrates a steady
upward trend, culminating in a plateau of 0.95 at \raisebox{-0.9pt}1⃝,
consistently maintained from iteration 35 onwards. This performance plateau
signifies that the framework not only swiftly attains a high confidence level
in its fraud detection capabilities but also sustains this efficacy over time.
Alongside, the validation loss diminishes from an initial 0.275 to 0.02,
reflecting the model’s enhanced precision in identifying fraudulent
transactions.
This reduction in validation loss is significant as it suggests a substantial
enhancement in the model’s ability to differentiate between fraudulent and
legitimate transactions with minimal error, thereby reducing the likelihood of
costly false positives. The pronounced improvement in accuracy and reduction
in loss observed between iterations 10 and 20 at \raisebox{-0.9pt}2⃝ marks a
critical learning phase for the model. By iteration 35, the model achieves and
upholds a state of high accuracy and minimal loss at \raisebox{-0.9pt}3⃝,
indicative of its robust learning mechanism and stability. This phase
showcases the effective convergence of the quantum and FL components,
optimizing the model’s parameters for high-stakes decision-making
environments. The sustained model performance beyond the 35th iteration
underscores the QFNN-FFD’s ability for dependable and steady fraud prediction
within QC environments.
Moreover, the robust validation performance of QFNN-FFD highlights its
practical applicability. The high validation accuracy suggests effective
pattern recognition is crucial for fraud detection, while the low and stable
loss indicates minimized rates of false positives and negatives—essential for
the operational deployment of any fraud detection system. This balance is
particularly important in financial contexts where the cost of false negatives
can be extraordinarily high. Given the observed performance plateau,
implementing an early exit strategy in training could economize on
computational resources without compromising effectiveness, optimizing overall
efficiency. This strategy underscores the framework’s capability to deliver
high performance while efficiently managing computational demands, setting a
new standard for privacy-focused, quantum-enhanced financial services.
123Stable plateau indicatinghigh performance3Rapid improvement phase2Initial
spike inloss reduction Figure 6: Evolution of Validation Metrics as a
Function of Iteration Count. The plot illustrates the optimization trajectory
over 100 iterations, with validation accuracy demonstrating an upward trend
towards convergence, and loss exhibiting a reciprocal decrease, indicative of
the model’s improving generalization on unseen data. 321 Figure 7:
Comparative Impact of quantum noise models on QFNN-FFD Framework Accuracy. The
graph systematically evaluates the framework’s accuracy under the influence of
six quantum noise models: depolarizing, phase damping, amplitude damping,
bitflip, phaseflip, and bitphaseflip. The noise parameters are adjusted from 0
(indicating no noise) to 1 (signifying maximum noise interference), providing
insights into the relative performance stability of the QFNN-FFD framework
across a spectrum of quantum noise intensities.
### IV-C Quantum Noise Analysis
In our experiments, we expose the QFNN-FFD framework to a spectrum of quantum
noise models [40], aiming to simulate the challenging conditions of near-term
QC devices. As presented in Fig. 7, under the depolarizing noise model,
accuracy remains high at 0.97 but plummets to 0 when the noise parameter
reaches 1, indicating the model’s noise tolerance limit. In the bitflip noise,
the QFNN-FFD shows resilience, maintaining a 0.97 accuracy until the noise
parameter hits 0.3 at \raisebox{-0.9pt}1⃝, after which it significantly drops
to 0.2 at a noise level of 1, marking the model’s performance threshold. This
illustrates how bitflip errors, which flip the state of a qubit, begin to
degrade the system’s performance only at higher noise levels, demonstrating a
strong error tolerance up to a critical point. The amplitude damping noise
leads to a less severe decrease in accuracy, from 0.97 to 0.4 at
\raisebox{-0.9pt}2⃝ as noise increases, while phase damping impacts it more,
reducing accuracy to 0.09, highlighting sensitivity to phase perturbations.
These results underscore the QFNN-FFD’s varying sensitivity to different types
of quantum noise, with phase damping proving particularly detrimental. This
sensitivity is crucial for understanding which quantum error correction
techniques might be most effective in enhancing the robustness of the model.
Remarkably, against phaseflip and bitphaseflip noises, the QFNN-FFD maintains
over 0.9 accuracy up to a noise parameter of 0.7 at \raisebox{-0.9pt}3⃝, only
dropping to 0.74, demonstrating significant robustness and potential
compatibility with existing quantum technologies. This resilience against
phaseflip and bitphaseflip noises suggests that the model’s quantum circuitry
may be naturally more protected against these types of errors, possibly due to
the nature of the quantum gates used or the initial state preparation.
Such robustness implies the QFNN-FFD’s potential compatibility with current
quantum technology, where such noise is prevalent. The robust performance of
the QFNN-FFD across these diverse noise profiles strongly indicates its
applicability in quantum-enhanced fraud detection systems. The data clearly
illustrates how the QFNN-FFD could provide reliable performance, guiding
future enhancements in quantum error correction to fortify the model against
the most vulnerable types of noise. These findings are pivotal, as they
demonstrate the framework’s current efficacy and its potential for adaptation
and improvement with the maturation of quantum technologies.
### IV-D Comparison with Existing Works
TABLE I: Comparison of QML frameworks on different financial fraud datasets. Reference | Precision | Recall | F1-Score | Accuracy
---|---|---|---|---
[10] | 84 | 84.44 | 75.68 | 83.92
[11] | 96.1 | 79.5 | 86 | 94.5
[12] | 90 | – | – | –
Our QFNN-FFD | 95 | 96 | 95 | 95
Compared to the results in Table I, our QFNN-FFD outperforms other QML models
applied to similar datasets, achieving superior performance metrics. These
metrics include precision, recall, F1-score, and accuracy, where QFNN-FFD
demonstrates comprehensive superiority across all fronts. This performance is
a testament to the model’s efficacy and highlights its ability to effectively
integrate complex quantum computations within an FL framework. Unlike the
existing models [10, 11, 12], which focus solely on performance, QFNN-FFD
additionally integrates a privacy-preserving FL approach. This ensures high
detection accuracy of 95% and enhanced data privacy, establishing QFNN-FFD as
a leading solution for secure and efficient fraud detection in fintech.
### IV-E Discussion
Our results show that the framework achieves high validation accuracy,
maintains low loss across various operational conditions, and exhibits
resilience against diverse quantum noise models. Such robustness underlines
the framework’s suitability for real-world QC environments known for their
integral noise issues. In direct comparison with existing quantum and
classical models, QFNN-FFD surpasses typical performance metrics, making it a
superior choice for fraud detection. This performance is particularly notable
given the framework’s integration of privacy-preserving FL, which safeguards
sensitive financial data during detection. This dual benefit of enhanced
accuracy and increased data privacy sets QFNN-FFD apart as a leading solution
for secure and effective fraud detection in the fintech industry. Furthermore,
the framework’s ability to maintain high performance under various noise
conditions suggests its potential for broader applications beyond financial
services, including sectors where data sensitivity and security are paramount.
Integrating advanced quantum computational capabilities with robust privacy
features positions QFNN-FFD as a scalable solution for future challenges in
secure data processing and analysis.
## V Conclusion
Our research successfully demonstrates the potential of QFNN-FFD in enhancing
fraud detection within the financial sector. By integrating advanced QC
techniques with FL, we present a novel approach that significantly improves
accuracy and efficiency compared to conventional methods. Our findings reveal
that the QFNN-FFD framework, supported by a robust computational
infrastructure and optimized through sophisticated preprocessing techniques,
can effectively identify fraudulent transactions with high precision. Its
resilience against various quantum noise models is particularly noteworthy,
indicating its suitability for real-world application in the near-term QC
landscape. This resilience, coupled with the model’s ability to maintain high
performance under different noise conditions, underscores the practical value
of our approach. Furthermore, the QFNN-FFD’s adaptability to quantum noise
suggests a promising direction for future research in quantum error correction
and noise mitigation strategies. Our study contributes to the emerging field
of QC by providing an efficient framework for applying QML while ensuring
privacy to solve complex problems in finance. Expanding beyond finance, this
framework has the potential to revolutionize fields such as healthcare and
cybersecurity, where privacy and data sensitivity are paramount, thus marking
a significant milestone in the interdisciplinary application of QML. In
conclusion, the QFNN-FFD framework addresses key challenges in the fintech
sector and also sets a precedent for the deployment of quantum technologies in
privacy-critical applications, offering substantial implications for both
academic research and industry practices. It encourages further exploration
and development within the QC, QML, and FL communities, aiming to unlock new
possibilities for handling complex, large-scale data analysis tasks in an
increasingly digital and interconnected world.
## Acknowledgment
This work was supported in part by the NYUAD Center for Quantum and
Topological Systems (CQTS), funded by Tamkeen under the NYUAD Research
Institute grant CG008.
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|
# A silicon source of heralded single photons at 2 $\mu$m.
S. Signorini<EMAIL_ADDRESS>Nanoscience Laboratory, Department
of Physics, University of Trento, Via Sommarive 14, 38123, Trento, Italy M.
Sanna Nanoscience Laboratory, Department of Physics, University of Trento,
Via Sommarive 14, 38123, Trento, Italy S. Piccione Nanoscience Laboratory,
Department of Physics, University of Trento, Via Sommarive 14, 38123, Trento,
Italy M. Ghulinyan Centre for Sensors and Devices, Fondazione Bruno Kessler,
38123, Trento, Italy P. Tidemand-Lichtenberg Department of Photonics
Engineering, DTU Fotonik, Technical University of Denmark, Roskilde, 4000,
Denmark C. Pedersen Department of Photonics Engineering, DTU Fotonik,
Technical University of Denmark, Roskilde, 4000, Denmark L. Pavesi
Nanoscience Laboratory, Department of Physics, University of Trento, Via
Sommarive 14, 38123, Trento, Italy
###### Abstract
Mid infrared integrated quantum photonics is a promising platform for
applications in sensing and metrology. However, there are only few examples of
on-chip single photon sources at these wavelengths. These have limited
performances with respect to their C-band counterparts. In this work, we
demonstrate a new approach to generate heralded single photons in the mid
infrared on a silicon chip. By using a standard C-band pump, the inter-modal
spontaneous four wave mixing enables the generation of the herald idler at
1259.7 nm and the heralded signal at 2015 nm. The idler photon is easily
detected with a common infrared single photon detector while the signal photon
is upconverted to the visible before its detection. In this way, we are able
to operate a mid infrared source without the need of mid infrared detectors
and laser sources. By measuring a heralded $g^{(2)}$ of $0.23\,\pm\,0.08$ we
demonstrate the single photon behaviour of the source as well as the
feasibility of multi-photon coincidence measurements beyond 2 $\mu$m with our
setup. The source exhibits a high intrinsic heralding efficiency of
$(59\,\pm\,5)\%$, a maximum coincidence to accidental ratio of
$40.4\,\pm\,0.9$ and a generation probability of
$\left(0.72\,\pm\,0.10\right)$ W-2.
††preprint: AIP/123-QED
## I Introduction
Mid infrared (MIR) light (2 - 15 $\upmu$m) is of importance in a wide range of
technological applications. Free space telecommunication Su _et al._ (2018),
LIDAR Weibring, Edner, and Svanberg (2003), environmental monitoring Fix _et
al._ (2016), medicine and biology Evans _et al._ (2007); Bellisola and Sorio
(2012); Potter _et al._ (2001); Miller, Bourassa, and Smith (2013) are only
few of the several fields where MIR optics plays a role. In particular, gas
sensing exploits the strong absorption bands in the MIR Popa and Udrea (2019)
to enhance remarkably the sensitivity of absorption spectroscopy measurements
Petersen _et al._ (2014); Vainio and Halonen (2016); Ghorbani and Schmidt
(2017). Despite the great interest in developing MIR applications, these are
still hindered by immature optical MIR devices. Quantum optics offers new
solutions to mitigate such limitations. Sub-poissonian light can be used to
beat the shot noise limit Brida, Genovese, and Berchera (2010); Whittaker _et
al._ (2017). Entangled photons have been used to demonstrate new imaging and
spectroscopy techniques able to get rid of detection technology limitations,
namely ghost imaging Pittman _et al._ (1995); Morris _et al._ (2015) or
undetected photon measurement Lemos _et al._ (2014); Kalashnikov _et al._
(2016); Vergyris _et al._ (2020). To enable quantum enhanced MIR metrology
leveraging these quantum based measurement strategies, a source of single or
entangled photons beyond 2 $\upmu$m is required. Up to now, these techniques
have been investigated only with bulky, alignment tolerant and expensive
instrumentation, based on free space nonlinear crystals Kalashnikov _et al._
(2016); Prabhakar _et al._ (2020). To develop feasible, robust and affordable
quantum technologies, miniaturization and cost effectiveness are crucial. Such
requirements can be met by means of integrated photonics. In particular,
silicon photonics integrated circuits are characterized by mature CMOS
(complementary metal oxide semiconductor) fabrication technology, which allows
for robust, stable, low power consuming and efficient light manipulation at
the chip scale Lockwood and Pavesi (2010). On-chip MIR quantum measurements
would enable efficient and cost effective sensors, boosting the development of
MIR and quantum technologies. Recently, an on-chip silicon-on-insulator (SOI)
source of MIR pairs has been reported Rosenfeld _et al._ (2020). However, in
this work a pump in the MIR is used, and both the paired photons are beyond 2
$\upmu$m, thus requiring specific MIR technologies for both the pump and the
detection. Recently, we demonstrated that inter-modal spontaneous four wave
mixing (SFWM) can be used in silicon waveguides to generate correlated pairs
with one photon in the near infrared (NIR) and the other in the MIR by using a
standard C-band pump Signorini _et al._ (2018, 2019). However, we never
detected the MIR correlated photon. Instead we inferred its existence by
measuring the high energy photon in the pair.
In this work, we demonstrate a SOI waveguide source of heralded MIR single
photons based on inter-modal SFWM, peforming the MIR detection by means of an
upconversion system Mancinelli _et al._ (2017). The herald photon lays in the
NIR, where it can be efficiently detected with traditional InGaAs single
photon avalanche photodiodes (SPADs). Moreover, the photons are generated in
discrete bands, thus removing the need for narrow band filters to select the
operating wavelengths of signal and idler. As a result, the heralding
efficiency is increased with respect to traditional intra-modal SFWM, as
witnessed by the measured intrinsic heralding efficiency $\eta_{I}=59(5)\,\%$.
The large detuning of the generated photons is also beneficial for the pump
and Raman noise rejection, that can be easily removed with broadband filters.
The pump is a standard 1550 nm pulsed laser. Therefore, we do not require MIR
technologies to operate a source beyond 2 $\mu$m. We assessed the single
photon behaviour of the source by measuring a heralded $g^{(2)}_{h}(0)$ of
0.23(8). We monitored the idler-signal coincidences, reporting a maximum
coincidence to accidental ratio of 40.4(9), exceeding the performance of
current integrated sources of MIR heralded photons Rosenfeld _et al._ (2020).
The paper is organized in the following way: in section II we describe the
chip design and the experimental setup. In section III our approach to data
analysis is extensively described. In section IV the results relative to the
source characterization are reported. Section V concludes the paper.
Figure 1: a) Simulated intensity profiles of the TE0 and TE1 spatial modes in
the multimode waveguide. b) Experimental setup used in the experiment. For the
pump (green) we used a pulsed laser at 1550.3 nm (40 ps pulse width, 80 MHz
repetition rate), which, after passing through a band pass filter (F) and a
polarization controller (PC), is coupled to the chip via a tapered lensed
fiber. The chip schematics is shown in the bottom part. On the chip, after a
3-dB directional coupler (DC), half of the pump remains on the TE0, while the
other half is converted to the TE1 via an asymmetric directional coupler
(ADC1) (92$\%$ efficiency). In this way, the pump reaches the multimode
waveguide (MMWG) equally splitted on the TE0 and TE1 modes. In the MMWG, the
inter-modal SFWM process generates the idler (blue) and signal (red) photons
in the TE0 and TE1 modes respectively. The signal is then converted to the TE0
via another asymmetric directional coupler (ADC2). In this way, idler and
signal can be easily separated on chip. Idler and signal are then out-coupled
from the chip via two tapered lensed fibers. Pump residual and Raman noise are
rejected from the idler beam by means of a short pass filter (SP) with a cut-
off wavelength of 1335 nm. The idler is then detected via an InGaAs SPAD (ID
Quantique IDQ210), triggered by the pump, with a gate width of 1.90 ns. The
signal, after being out-coupled from the chip, is polarization rotated through
a free space half-wave plate $\left(\lambda/2\right)$ and upconverted to the
visible through an upconverter system (UC). The UC includes a long pass filter
with a cut-on wavelength of 1900 nm, which rejects the C-band pump. To be
noticed that the UC introduces noise photons collinear to the upconverted
signal and centered at the same wavelength. A bandpass filter (BP) is used to
filter away part of this noise, without filtering the upconverted signal
(purple). Then, the signal photons are analyzed by means of a Hanbury Brown
and Twiss (HBT) interferometer. The HBT interferometer is composed by a 50/50
beam splitter (BS) with two visible silicon SPADs (Excelitas SPCM-AQRH-12)
monitoring the BS reflection and transmission ports. The visible SPADs are
used in free-running mode. A time tagging unit (Swabian Time Tagger 20) is
used to monitor individual singles and coincidences between the three
detectors.
## II Chip design and experimental setup
Conventional intra-modal SFWM involves only one waveguide mode in the
conversion of two input pump photons into an idler photon and a signal photon.
On the contrary, inter-modal SFWM leverages the different chromatic
dispersions of different optical spatial modes of a photonic waveguide to
achieve phase matching Signorini _et al._ (2018). Different modal
combinations are possible, depending on the waveguide cross-section, which
also determines the generated signal and idler wavelengths. In this work, we
use the transverse electric (TE) fundamental (TE0) and first (TE1) waveguide
modes in a rib SOI waveguide. The waveguide has a width of 1.95 $\upmu$m and a
height of 0.190 $\upmu$m over a 0.3 $\upmu$m thick slab. The waveguide length
is 1.5 cm. The waveguide and the slab are in silicon, while the top and bottom
claddings are in silica. The simulated intensity profiles of the TE0 and TE1
modes are shown in Fig. 1a.
The inter-modal combination used in our work involves the pump on both the TE0
and TE1, the idler on the TE0 and the signal on the TE1. A peculiar advantage
of intermodal SFWM is the generation of the signal and idler photons on
different waveguide modes. In this way, idler and signal can be easily
separated with high efficiency through an on-chip mode converter. The
experimental setup is detailed in Fig. 1b. The upconverter (UC) is constituted
by a continuous wave (CW) laser cavity, where a Nd:YVO4 pumped intra-cavity
periodically poled lithium niobate (PPLN) allows for sum-frequency generation
(SFG) between the intra-cavity laser (1064 nm) and the input MIR photons. We
used a PPLN from HC Photonics with a length of 25 mm, tuned in temperature to
upconvert the MIR signal at 2015 nm to the visible at 696 nm. The UC used is
the same of Mancinelli et al. Mancinelli _et al._ (2017), though tuned at the
wavelengths of interest here. The transfer function of the UC is reported in
Fig. 2b, showing a full width at half maximum (FWHM) of $1.15\,\pm\,0.12$ nm.
We used a pump pulsed laser centered at $1550.30\,\pm\,0.05$ nm with 40 ps
pulse width and 80 MHz repetition rate. The generated idler spectrum is
reported in Fig. 2. We measured a discrete band centered at 1259.7 $\pm$ 0.5
nm, with a FWHM of 2.0 $\pm$ 0.3 nm. The measured FWHM of the idler is
compatible with the simulated one of 1.81 nm, as shown in Fig. 2a. According
to the energy conservation, the signal is generated at 2015.2 $\pm$ 1.5 nm.
From the measured idler bandwidth we estimated a FWHM of 5.1 $\pm$ 0.8 nm for
the signal. Therefore, the UC filters the signal photons according to the
spectrum shown in Fig. 2b.
Figure 2: a) Measured intensity spectrum of the idler beam. The fit has been
made with a gaussian function, showing a FWHM of $2.87\,\pm\,0.07$ nm. This
measurement is affected by the transfer function of the monochromator used to
perform the measurement, which enlarges the actual bandwidth of the
generation. We simulated the idler spectrum considering also the widening due
to the monochromator (orange dashed line). To evaluate the actual bandwidth of
the idler (2.0 $\pm$ 0.3 nm) we deconvolved the response function of the
monochromator. b) Measured spectral response of the upconverter. The response
has been fitted by a squared sinc function, as expected for a sum frequency
generation process. The FWHM is $1.15\,\pm\,0.12$ nm.
## III Data analysis
With SFWM, the detection probabilities per pulse for the idler ($p_{i}$),
signal ($p_{s}$), coincidences ($p_{si}$) and accidentals ($p_{acc}$) are
quadratic with the pump power $P$. In the limit of low transmission
efficiencies for the signal and idler Harada _et al._ (2009), they can be
written as
$p_{i}=\xi P^{2}\eta_{i}+d_{i},$ (1a) $p_{s}=\xi P^{2}\eta_{s}+d_{s},$ (1b)
$p_{si}=\xi P^{2}\eta_{i}\eta_{s},$ (1c) $p_{acc}=p_{i}p_{s},$ (1d)
where $\xi$ is the generation probability per pulse per squared unit power
Rosenfeld _et al._ (2020), $\eta_{i},\,\eta_{s}$ are the total transmission
efficiencies for the idler and signal channels (from generation to detection),
$d_{i},\,d_{s}$ are the dark count probabilities per pulse for the idler and
signal respectively. Eq. (1c) refers to net coincidences, thus without
accidentals. In eqs. (1), noise photons coming from the pump residual and
Raman scattering, typically linear with the pump power, have not been
considered, being negligible in our experimental setup. Singles and
coincidence rates can be calculated by multiplying the probabilities in eqs.
(1) by the repetition rate $R_{p}$ of the pump laser. Together with SFWM other
nonlinear phenomena take place in the waveguide. Two photon absorption (TPA),
cross two photon absorption (XTPA) and free carrier absorption (FCA) have to
be modelled properly in order to recover the actual generation and
transmission efficiency of the pairs. TPA, XTPA and FCA play an important role
in increasing the losses in the waveguide for both the pump and the generated
photons. As a result, the detection probabilities are no longer quadratic with
the input pump power Boyd (2019). A further effect is the nonlinearity of the
idler detector. To model the linear and nonlinear losses affecting pump,
signal and idler photons, we solved the differential equations for the pulse
propagation involving TPA, FCA and propagation losses, assuming that the pump
power is equally split on the TE0 and TE1 modes Borghi _et al._ (2017).
According to this modeling we can rewrite eqs. (1) as
$p_{si}\simeq\xi\bar{P}_{p}^{2}\bar{\eta}_{i}\bar{\eta}_{s}\eta_{ND}\equiv\bar{p}_{si},$
(2a)
$p_{i}\simeq\left(\xi\bar{P}_{p}^{2}\bar{\eta}_{i}+d_{i}\right)\eta_{ND}\equiv\bar{p}_{i},$
(2b) $p_{s}\simeq\xi\bar{P}_{p}^{2}\bar{\eta}_{s}+d_{s}\equiv\bar{p}_{s},$
(2c) $p_{acc}\simeq\bar{p}_{i}\bar{p}_{s}\equiv\bar{p}_{acc},$ (2d)
where
$\bar{P}_{p}=\sqrt{\frac{1}{L}\int_{0}^{L}P_{p}^{2}(z)dz},$ (3a)
$\bar{\eta}_{j}=\bar{\eta}_{j}^{on}\eta_{j}^{off},$ (3b)
$\bar{\eta}^{on}_{j}=\frac{1}{L}\int_{0}^{L}\eta^{on}_{j}(z)dz,$ (3c)
where $j=i,s$, $L$ is the waveguide length, $P_{p}(z)$ is the on-chip pump
power along the waveguide, $\eta^{on}_{j}(z)$ is the transmission efficiency
for a photon generated at $z$ along the waveguide accounting only for the
linear and nonlinear on-chip losses, $\eta^{off}_{j}$ is the transmission
efficiency accounting only for the losses occurring off chip (fiber-chip
coupling, filtering) and $\eta_{ND}$ models the nonlinear response of the
idler detector. Details about the derivation of eqs. (2) are reported in
Supplementary material.
## IV Results
### IV.1 Generation probability and heralding efficiency
To monitor the coincidences between signal and idler, we used a start-and-stop
detection system, using the idler as the start trigger and the signal as the
stop detection Signorini and Pavesi (2020). Coincidences are evaluated within
a coincidence window $\Delta t_{c}$. To be noticed that while for the idler
channel the detection rates (both signal and dark counts) are fixed by the
detection gate width of the idler detector (1.90 ns), for the signal the rates
depend on the coincidence window used in post processing. Therefore, given
$R_{dc,i}=620\,\textrm{cps}$ and $R_{dc,s}=2150\,\textrm{cps}$ the dark count
rates at the idler and signal detectors,
$d_{i}=R_{dc,i}/R_{p}=7.75\times 10^{-6},$ (4)
while
$d_{s}=1-\textrm{e}^{-R_{dc,s}\Delta t_{c}},$ (5)
considering a Poisson distribution for the signal noise (SPAD dark counts and
UC noise).
In order to fit the measured rates and retrieve the generation probability
$\xi$, we can reduce eqs. (2) to
$y_{i}=\frac{\bar{p}_{i}-\eta_{ND}\,d_{i}}{\bar{\eta}_{i}^{on}\eta_{ND}}=\xi\bar{P}_{p}^{2}\eta_{i}^{off}=a_{i}\bar{P}_{p}^{2},$
(6a)
$y_{s}=\frac{\bar{p}_{s}-d_{s}}{\bar{\eta}_{s}^{on}}=\xi\bar{P}_{p}^{2}\eta_{s}^{off}=a_{s}\bar{P}_{p}^{2},$
(6b)
$y_{si}=\frac{\bar{p}_{si}}{\bar{\eta}_{i}^{on}\eta_{ND}\bar{\eta}_{s}^{on}}=\xi\bar{P}_{p}^{2}\eta_{i}^{off}\eta_{s}^{off}=a_{si}\bar{P}_{p}^{2},$
(6c)
with $a_{i}=\xi\eta_{i}^{off}$, $a_{s}=\xi\eta_{s}^{off}$,
$a_{si}=\xi\eta_{i}^{off}\eta_{s}^{off}$. $y_{i}$, $y_{s}$, $y_{si}$ can be
calculated from the measured singles, coincidence and noise rates and from the
simulated $\bar{\eta}^{on}_{j}$ and the measured $\eta_{ND}$ (see
Supplementary material). Modeling exactly the nonlinear losses is a non
trivial task, being the nonlinear parameters highly variable with the
fabrication process and the geometry used. Therefore, we fit $y_{i}$, $y_{s}$,
$y_{si}$ for an input power $<$ 0.5 W (i.e. $\bar{P}_{p}<0.4$ W), where the
nonlinear losses are not the dominant ones. We use $f(x)=ax^{2}+b$ as the
fitting function, retrieving $a_{i}$, $a_{s}$ and $a_{si}$. In this way, we
can evaluate $\xi$ (in units of $W^{-2}$ of peak power) and the off-chip
transmissions, resulting in
$\xi=\frac{a_{i}\,a_{s}}{a_{si}}=\left(0.72\pm 0.10\right)W^{-2},$ (7a)
$\eta_{i}^{off}=\frac{a_{si}}{a_{s}}=\left(2.81\pm 0.17\right)\times 10^{-3},$
(7b) $\eta_{s}^{off}=\frac{a_{si}}{a_{i}}=\left(3.97\pm 0.20\right)\times
10^{-4},$ (7c)
where we used $\Delta t_{c}=1.1$ ns (3$\sigma$ bin width) and the
uncertainties are evaluated at 1 standard deviation of the fitting
coefficients. Details about the nonlinear parameters and propagation losses
used in the model are reported in Supplementary materials. From these results
we calculate the intrinsic heralding efficiency $\eta_{I}$ as Signorini and
Pavesi (2020)
$\eta_{I}=\frac{R^{net}_{si}}{\left(R_{i}-R_{dc,i}\right)\,\bar{\eta}_{s}^{off}}=59\pm
5\,\%,$ (8)
where $R^{net}_{si}$ is the measured net coincidence rate and $R_{i}$ is the
measured idler rate. By normalizing for the signal channel losses, the
$\eta_{I}$ allows to compare different sources only on the bases of their
intrinsic properties, getting rid of the setup used. Our high value comes from
the low on-chip signal losses and the moderate filtering losses to select the
signal wavelength. The heralding efficiency can be further improved by
optimizing the matching between the signal and UC bandwidths.
### IV.2 Coincidence to accidental ratio
To quantify the efficiency of coincidence detection, the coincidence-to-
accidental ratio (CAR) is used. CAR is analogous to a signal-to-noise ratio
comparing the rate of true coincidences with the accidental ones. True
coincidences come from simultaneous detection of a signal and an idler
belonging to the same pair. Coincidences between signals and idlers belonging
to different pairs or coincidences with noise photons or dark counts
contribute to the accidentals Signorini and Pavesi (2020); Harada _et al._
(2009). The measurement of CAR is carried out with the start-stop coincidence
detection described in sec. IV.1. We used the setup in Fig. 1b with a single
visible SPAD at the output of the UC after removing the beams splitter. In
fact, the CAR does not involve the intra-beam correlations. As shown in Fig.
3, the coincidences occur with a temporal delay $\delta t$ = 0 ns. The other
peaks, spaced with the laser repetition period, are due to accidentals. Please
notice that the zero-delay peak includes also accidental coincidences.
Therefore, the CAR is evaluated as
$\textrm{CAR}=\frac{\textrm{coincidence counts}}{\textrm{accidental
counts}}=\frac{N^{raw}_{si}-N_{acc}}{N_{acc}},$ (9)
with $N^{raw}_{si}$ the total coincidence counts falling in the zero delay bin
and $N_{acc}$ the accidental counts, evaluated as the average over all the
accidental peaks. The true coincidences, also called as net coincidences, are
calculated as $N_{si}^{net}=N_{si}^{raw}-N_{acc}$. Depending on the $\Delta
t_{c}$ used, the ratio between coincidence and accidentals in the individual
bin changes, changing the CAR. In Fig. 4 we report the measured CAR and the
corresponding net coincidences as a function of the on-chip peak pump power.
To be noticed that the peak power in the plot is the power at the input of the
multimode waveguide after fiber-chip coupling losses, it is not $\bar{P}_{p}$.
We report the results with a coincidence window of 1.1 ns and of 2 ns. With
the 1.1 ns window the CAR is higher, with a maximum of 40.4(9) at 115 mW. At
this power the rate of net coincidences is 0.316(3) cps. The net coincidences
are almost the same for the two windows, demonstrating that with the larger
coincidence window we are mainly introducing noise rather than signal. CAR and
net coincidences have been also simulated starting from the parameters
calculated in sec. IV.1 and sec. III. They are reported as solid lines in the
figure and are calculated as Harada _et al._ (2009)
$\textrm{CAR}=\frac{\bar{p}_{si}}{\bar{p}_{i}\,\bar{p}_{s}}=\frac{\xi\bar{P}_{p}^{2}\bar{\eta_{i}}\bar{\eta_{s}}}{\left(\xi\bar{P}_{p}^{2}\bar{\eta_{i}}+d_{i}\right)\,\left(\xi\bar{P}_{p}^{2}\bar{\eta_{s}}+d_{s}\right)},$
(10a)
$N_{si}^{net}=\xi\bar{P}_{p}^{2}\bar{\eta}_{i}\bar{\eta}_{s}\eta_{ND}R_{p}.$
(10b)
Simulated and experimental values of CAR are in agreement in the whole range
of pump power used. This agreement demonstrates that the main effects and
phenomena involved in the generation process have been properly considered and
modelled. The net coincidence rates are in agreement at low power, while at
higher power the nonlinear losses have been overestimated. A perfect agreement
would require a precise knowledge of all the nonlinear parameters of the
material.
The larger CAR here measured with respect to other works Rosenfeld _et al._
(2020) demonstrates that the overall system, considering both the generation
and detection stages, is competitive with respect to solutions already
demonstrated on the silicon platform.
Figure 3: Two-fold coincidences as a function of the delay $\delta t$ between idler (start) and signal (stop) detections. We collect the events with a coincidence window of 0.05 ns (blue). In post processing, we use a larger coincidence window, here 1.1 ns (orange), in order to take into account the majority of the coincidence events. The coincidence peak is the highest one, placed at $\delta t=0$ ns. The laser repetition period is clearly visible from the accidental peaks. In the inset, we focus on the zero-delay bin, comparing the coincidence peak shape with the post processing coincidence window. Figure 4: Measured CAR (circles) and net coincidence rates (triangles) with $\Delta t_{c}=1.1$ ns (orange) and $\Delta t_{c}=2$ ns (blue). The data are reported versus the on-chip peak pump power. The experimental points are compared with the simulated values for both the CAR (solid lines) and the net coincidence rates (dashed lines). With $\Delta t_{c}=1.1$ ns the CAR is remarkably higher with respect to the 2 ns bin, with only a limited reduction in the coincidence rate. The better performance obtained with the smaller $\Delta t_{c}$ is due to the lower noise integrated within the coincidence bin. Table 1: Comparison with state of the art MIR heralded sources. Platform | Process | Generation probability | CAR | CAR | $\mathbf{g^{(2)}_{h}(0)}$ | $\eta_{I}$ | Reference
---|---|---|---|---|---|---|---
| | (W-2) | max | @ $N^{net}_{si}\sim$ 1 Hz | | ($\%$) |
Mg:PPLN | SPDC | - | 180 $\pm$ 50 | - | - | - | Prabhakar _et al._ (2020)
SOI | intra-modal SFWM | 0.28 | 25.7 $\pm$ 1.1 | 25.7 $\pm$ 1.1 | - | 5 | Rosenfeld _et al._ (2020)
SOI | inter-modal SFWM | 0.72 $\pm$ 0.10 | 40.4 $\pm$ 0.9 | 27.9 $\pm$ 0.5 | 0.23 $\pm$ 0.08 | 59 $\pm$ 5 | This work
### IV.3 Heralded g${}^{(2)}_{h}$
To asses the single photon nature of the emission, we measured the heralded
$g^{(2)}$, that we indicate as $g^{(2)}_{h}$. Using the setup in Fig. 1b, we
tuned the delays in order to have the signal detection on one visible SPAD
coincident with the idler detection on the InGaAs SPAD. The coincidence
between these two detectors, with a coincidence window $\Delta t_{c}=$ 2 ns,
was used as the start trigger, while the detection from the remaining visibile
SPAD, that we will call "delayed signal", was used as the stop trigger. In
this way, we monitored the three-fold coincidences as a function of the delay
$\delta t$ between the start and stop events. At the same time, we measured
the two-fold coincidences between the idler and the delayed signal. We used a
coincidence window of 2 ns to monitor the three-fold coincidences. The
$g^{(2)}_{h}$ can be given as Signorini and Pavesi (2020)
$g^{(2)}_{h}(\delta t)=\frac{N_{12i}(\delta t)}{N_{1i}(0)N_{2i}(\delta
t)}N_{i},$ (11)
where $1,2,i$ label respectively the first signal detector, the second signal
detector (that is the delayed signal) and the idler detector. $N_{12i}$
corresponds to the three-fold coincidence counts, $N_{1i}$ and $N_{2i}$ are
the two-fold coincidence counts between the idler and the signal detectors,
and $N_{i}$ corresponds to the idler counts. We can normalize eq. (11) by
$N_{i}$ and $N_{1i}(0)$, such that
$g^{(2)}_{h}(\delta t)=\frac{N_{12i}(\delta t)}{\langle N_{12i}(\delta t\neq
0)\rangle}\frac{\langle N_{2i}(\delta t\neq 0)\rangle}{N_{2i}(\delta t)},$
(12)
with $\langle N_{12i}(\delta t\neq 0)\rangle$ and $\langle N_{2i}(\delta t\neq
0)\rangle$ the average of the three-folds and two-folds coincidences for
$\delta t$ different from zero. If the emission is truly at the single photon
level, $g^{(2)}_{h}(0)$ should be lower than 0.5Signorini and Pavesi (2020).
The measured $g^{(2)}_{h}(0)$ as a function of the on-chip peak pump power is
reported in Fig. 5. For an input power of 0.33 W we measured
$g^{(2)}_{h}(0)=0.23(8)$, demonstrating the single photon regime of the
source. The corresponding $g^{(2)}_{h}(\delta t)$, calculated as in eq. (12),
is reported in the inset of Fig. 5. We discarded the neighbouring bins of the
zero delay bin, affected by spurious coincidences due to photon emissions from
triggered silicon SPADs Kurtsiefer _et al._ (2001).
Figure 5: Comparison between the measured (blue points) and simulated (light
blue area) $g_{h}^{(2)}(0)$ as a function of the on-chip peak power. In the
inset is reported the measurement for the $g^{(2)}_{h}(\delta t)$ at an on-
chip peak power of 0.33 W. The bins adjacent to the zero-delayed one have been
removed due to the SPADs emitted photons.
To verify the goodness of the modeling introduced in sec. III, we used the
calculated $\xi$, $\bar{P}_{p}$, $\bar{\eta}_{i}$ and $\bar{\eta}_{s}$ in sec.
IV.1 to simulate the expected $g_{h}^{(2)}(0)$. Considering the general
formula for the heralded second order coherence, we can write
$g_{h}^{(2)}(0)=\frac{\bar{p}_{12i}\bar{p}_{i}}{\bar{p}_{1i}\bar{p}_{2i}},$
(13)
where $\bar{p}_{12i}$ is the probability per pulse of having a three-fold
coincidence. To model the experimental results, we have to consider all the
possible coincidence events that may involve signal and/or noise detections.
By considering all the possible events leading to a three-fold coincidence
(see Supplementary Material), we can rewrite $\bar{p}_{12i}$ as
$\displaystyle\bar{p}_{12i}=$
$\displaystyle\sum_{n=2}^{\infty}n^{2}(n-1)\wp(n)\,\bar{\eta}_{1}\bar{\eta}_{2}\bar{\eta}_{i}\eta_{ND}$
(14) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n^{2}\wp(n)\,(\bar{\eta}_{1}d_{2}+d_{1}\bar{\eta}_{2})\bar{\eta}_{i}\eta_{ND}$
(15) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{n=2}^{\infty}n(n-1)\wp(n)\,\bar{\eta}_{1}\bar{\eta}_{2}d_{i}\eta_{ND}$
(16) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)\,\bar{\eta}_{1}d_{2}d_{i}\eta_{ND}$
(17) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)\,d_{1}\bar{\eta}_{2}d_{i}\eta_{ND}$
(18) $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}n\wp(n)d_{1}d_{2}\bar{\eta}_{i}\eta_{ND}$
(19) $\displaystyle+$ $\displaystyle\,d_{1}d_{2}d_{i}\eta_{ND},$ (20)
with $\wp(n)$ the photon number distribution. In eq. (14), $\bar{\eta}_{i}$ is
as in eq. (3c), while $\bar{\eta}_{1}$ and $\bar{\eta}_{2}$ have to take into
account also the effect of the beam splitter, thus, according to eq. (3c),
they can be written as
$\bar{\eta}_{1}=\bar{\eta}_{s}T^{2}_{BS}\eta_{BS},$ (21a)
$\bar{\eta}_{2}=\bar{\eta}_{s}R^{2}_{BS}\eta_{BS},$ (21b)
with $T_{BS}$ and $R_{BS}$ the transmission and reflection coefficients of the
beam splitter, $T^{2}_{BS}+R^{2}_{BS}=1$, and $\eta_{BS}$ modeling the losses
of the beam splitter. In our case, $T^{2}_{BS}=R^{2}_{BS}=0.5$ and
$\eta_{BS}=1$. In eqs. (21) we are assuming the same detection efficiency for
the two visible SPADs. Considering all the events leading to a two-fold
coincidence, we can rewrite $\bar{p}_{1i}$ and $\bar{p}_{2i}$ as
$\displaystyle\bar{p}_{ki}=$
$\displaystyle\sum_{n=1}^{\infty}n^{2}\wp(n)\bar{\eta}_{k}\bar{\eta}_{i}\eta_{ND}$
(22) $\displaystyle+$
$\displaystyle\sum^{\infty}_{n=1}n\wp(n)\left(\bar{\eta}_{k}d_{i}+d_{k}\bar{\eta}_{i}\right)\eta_{ND}$
(23) $\displaystyle+$ $\displaystyle\,d_{k}d_{i}\eta_{ND},$ (24)
with $k=1,2$. To be noticed that in eq. (14) and eq. (22) we are neglecting
events with more than one photon reaching the same detector, being unlikely
with the involved transmission efficiencies (i.e. $\bar{\eta}_{i}$,
$\bar{\eta}_{1}$ and $\bar{\eta}_{2}$ are all $\ll$1). We are also neglecting
events where photon detections and dark count detections occur simultaneously
on the same detector. The photon number distribution of a squeezed source
ranges between a poissonian (infinite modes emission) and a thermal (single
mode emission) distribution Takesue and Shimizu (2010); Signorini and Pavesi
(2020). We solved eq. 14 and eq. 22 for the poissonian emission,
$\wp(n)=\frac{\mu^{n}}{n!}\textrm{e}^{-\mu},$ (25)
and for the thermal emission,
$\wp(n)=\frac{\mu^{n}}{(1+\mu)^{n+1}},$ (26)
where $\mu$ is the average number of pair per pulse. Eqs. 25 and 26 define a
lower and an upper boundary for $g^{(2)}_{h,sim}$. In computing $g^{(2)}_{h}$
we calculated $\mu$ as $\mu=\xi\bar{P}_{p}^{2}$ and we measured the noise
affecting the three channels. $d_{i}$ is the same of the CAR measurements,
$d_{1}=2.30\times 10^{-6}$ and $d_{2}=2.32\times 10^{-6}$. We simulated an
area for the expected value of the $g^{(2)}_{h}(0)$, that is upper bounded by
the thermal case and lower bounded by the poissonian case. The simulation is
reported in Fig. 5. The measured $g^{(2)}_{h}$ is compatible with the
simulated values, confirming the reliability of the modeling. We want to
stress that in this case we are not performing a fit of the measured
$g^{(2)}_{h}$ and that the experiment and the simulation are completely
independent. The experimental points in Fig. 5 are closer to the upper bound
rather than to the lower one, suggesting an emission statistics closer to the
thermal one. This is compatible with the unheralded $g^{(2)}$ of the signal
beam Signorini and Pavesi (2020), measured in Fig. 6 as a function of the pump
power. The unheralded $g^{(2)}$ results to be 1.67(2) at a power of 1.08 W,
compatible with the simulated value of 1.66 (dashed line) calculated from the
simulated joint spectral intensity (JSI) Signorini and Pavesi (2020); Borghi
(2020). The measured $g^{(2)}$ demonstrates that the source is closer to a
thermal emission, justifying the experimental $g^{(2)}_{h}$. In Fig. 6 we also
report the simulated values for a source whose statistics is between the
thermal (upper bound) and the poissonian one (lower bound). At low powers, the
dark counts dominate, and in both cases the $g^{(2)}$ goes to 1. At high
powers, the $g^{(2)}$ asymptotically increases to its actual value. In this
way, we explain the power dependent behaviour of the experimental data.
Further details about the measurement and simulation of $g^{(2)}$ are reported
in Supplementary materials.
Figure 6: The measured unheralded $g^{(2)}(0)$ (orange dots) is reported as a
function of the on-chip peak power. We report in the inset the simulated JSI,
from which we calculated the expected $g^{(2)}$ (dashed black line), that is
compatible with the experiment. The measured points fall within the simulated
values (light orange area), upper bounded by a source with thermal emission
statistics and lower bounded by a source with poissonian emission statistics
(constant $g^{(2)}=1$).
## V Conclusions
In this work, we demonstrated a heralded single photon source beyond 2 $\mu$m
based on inter-modal SFWM on a silicon chip. This source has two main
peculiarities: the discrete band generation and the large detuning between the
signal and idler photons. The discrete band generation removes the need for
tight filtering to select idler and signal wavelengths, and the generated
photons experience a higher transmission with respect to standard continuous
band sources, witnessed by the high experimental $\eta_{I}=59(5)\,\%$. The
large detuning has two advantages: on one side, it enables an easier pump and
nonlinear noise rejection; on the other side, it allows to generate the herald
photon in the NIR, benefiting of an efficient detection technology. As a last
advantage, this heralded source based on inter-modal SFWM requires a common
C-band pump laser, easier to be integrated and operated on a silicon chip. We
performed a complete characterization of the source. We demonstrated the sub-
poissonian statistics of the source by measuring $g^{(2)}_{h}(0)=0.23(8)$. We
characterized the CAR, finding a maximum value of 40.4(9), and the generation
probability per pulse, with a measured value of 0.72(10) W-2. These
performances are competitive with other reported silicon sources of MIR
photons (Table 1) demonstrating the promising perspectives of inter-modal SFWM
for bright and efficient sources of correlated photons beyond 2 $\upmu$m. The
source can be significantly improved by reducing the propagation losses and
optimizing the matching between the signal and upconverter bandwidths. With
this work we demonstrate a new approach to MIR quantum photonics, providing a
high quality source of quantum light beyond 2 $\mu$m without the need of MIR
technologies. This result paves the way towards low cost, efficient and
integrated solutions for quantum photonics beyond 2 $\upmu$m, offering new
opportunities to the developing field of MIR photonics.
## SUPPLEMENTARY MATERIAL
See supplementary material for further details about the experimental setup,
the measurements and the theoretical calculations.
###### Acknowledgements.
This work was partially supported by grants from Q@TN provided by the
Provincia Autonoma di Trento. The authors acknowledge HC Photonics, which
fabricated the PPLN crystals used for the upconversion system. S.S. wants to
thank Dr. Massimo Borghi, for fruitful discussions and precious suggestions,
and Mr. Davide Rizzotti for its careful revision of the manuscript.
## DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
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|
A Detailed View of the Broad Line Region in NGC 3783 from Velocity-Resolved Reverberation Mapping
0000-0002-2816-5398]Misty C. Bentz
Department of Physics and Astronomy,
Georgia State University,
Atlanta, GA 30303, USA
0000-0002-4645-6578]Peter R. Williams
Department of Physics and Astronomy,
University of California,
Los Angeles, CA 90095, USA
0000-0001-6279-0552]Rachel Street
LCOGT, 6740 Cortona Drive, Suite 102,
Goleta, CA 93117, USA
0000-0003-0017-349X]Christopher A. Onken
Research School of Astronomy and Astrophysics,
Australian National University,
Canberra, ACT 2611, Australia
0000-0002-6257-2341]Monica Valluri
Department of Astronomy,
University of Michigan,
Ann Arbor, MI, 48104, USA
0000-0002-8460-0390]Tommaso Treu
Packard Fellow
Department of Physics and Astronomy,
University of California,
Los Angeles, CA 90095, USA
We have modeled the full velocity-resolved reverberation response of the H$\beta$ and He2 optical broad emission lines in NGC 3783 to constrain the geometry and kinematics of the low-ionization and high-ionization broad line region. The geometry is found to be a thick disk that is nearly face on, inclined at $\sim18\degr$ to our line of sight, and exhibiting clear ionization stratification, with an extended H$\beta$-emitting region ($r_{\rm median}=10.07^{+1.10}_{-1.12}$ light days) and a more compact and centrally-located He2-emitting region ($r_{\rm median}=1.33^{+0.34}_{-0.42}$ light days). In the H$\beta$-emitting region, the kinematics are dominated by near-circular Keplerian orbits, but with $\sim 40$% of the orbits inflowing. The more compact He2-emitting region, on the other hand, appears to be dominated by outflowing orbits. The black hole mass is constrained to be $=2.82^{+1.55}_{-0.63}\times10^7$ , which is consistent with the simple reverberation constraint on the mass based on a mean time delay, line width, and scale factor of $\langle f \rangle=4.82$. The difference in kinematics between the H$\beta$- and He2-emitting regions of the BLR is intriguing given the recent history of large changes in the ionizing luminosity of NGC 3783 and evidence for possible changes in the BLR structure as a result.
§ INTRODUCTION
Black holes continue to capture our imaginations centuries after the concept was first recorded in a letter written by a country clergyman [Michell, 1784]. Today, not only are black holes a recurrent feature in science fiction, but they have become securely ensconced in science fact. We now know that supermassive ($=10^5-10^{10}$ ) black holes exist, that they inhabit the centers of most (all?) massive galaxies, and that their masses scale with several measurable properties of their host galaxies, including the bulge stellar velocity dispersion and bulge luminosity (e.g., Magorrian et al., 1998, Ferrarese & Merritt, 2000, Gebhardt et al., 2000, Gültekin et al., 2009, Kormendy & Ho, 2013).
Only a handful of methods are able to directly constrain the mass of a central, supermassive black hole through its gravitational effects on luminous matter. In the case of the Milky Way, astrometric monitoring of individual stars in the central few parsecs has resulted in a constraint on the mass of Sagittarius A* of $=(4.1\pm0.6)\times10^6$ [Ghez et al., 2000, Genzel et al., 2000, Ghez et al., 2008], while relativistic modeling of the emission from gas just outside the event horizon has constrained the mass of Pōwehi, the central black hole in M87, to $=(6.5\pm0.7)\times10^9$ [Event Horizon Telescope Collaboration et al., 2019]. Most other galaxies are not able to be studied with similar methods because we lack the necessary spatial resolution. However, many nearby galaxies ($D\lesssim100$ Mpc) may still be studied through spatially-resolved observations of the bulk nuclear gas or stellar kinematics on scales of $\sim$tens of parsecs (e.g., Gültekin et al., 2009, Kormendy & Ho, 2013). Reverberation mapping is notable among black hole mass measurement techniques because it relies on time resolution rather than angular resolution. By monitoring the spectrophotometric variability of an active galactic nucleus (AGN), the black hole mass, among other properties, may be constrained for a local Seyfert or a distant quasar (for a recent review, see Cackett et al., 2021).
Reverberation mapping makes use of the response of photoionized gas in the broad emission-line region (BLR) to variations in the continuum luminosity, a technique that was first proposed by Bahcall et al., 1972. As it is generally implemented, reverberation mapping constrains an average responsivity-weighted radius for the BLR in an AGN. Combining the radius with a measure of the line-of-sight velocity of the BLR gas via the virial theorem constrains [Peterson & Wandel, 1999, Peterson & Wandel, 2000], modulo a scale factor that accounts for the generally unknown BLR geometry and kinematics (e.g., Onken et al., 2004, Park et al., 2012, Grier et al., 2013, Batiste et al., 2017). However, high quality spectrophotometric monitoring data contain information about the gas response as a function of line-of-sight velocity, thus providing constraints on the emissivity and position of photoionized gas in a spatially-unresolved source [Blandford & McKee, 1982]. Velocity-resolved reverberation mapping, as it has come to be known, is thus able to directly constrain the BLR geometry and the black hole mass, thus avoiding the need to apply a scale factor.
The analysis of velocity-resolved reverberation mapping data can be approached as an ill-posed inverse problem, in which the goal is to recover the transfer function that describes the time delay distribution as a function of velocity across a broad emission line (e.g., Horne, 1994, Skielboe et al., 2015, Anderson et al., 2021). Or it can be approached through direct modeling, in which a framework of fully self-consistent models is built and an exploration of the available parameter space yields the family of models that best match the observational constraints (e.g., Pancoast et al., 2011). Direct modeling has the advantage that it is relatively simple to interpret the results, however its ability to match complicated data sets is limited by the phenomenology that is included and how it is parametrized. Recovery of the transfer function, on the other hand, takes advantage of the full range of details present in the observations but is nontrivial to interpret.
While the promise of velocity-resolved reverberation mapping has long been understood, it was only within the last decade or so that improvements in the quality of reverberation mapping data (e.g., Bentz et al., 2008, Bentz et al., 2009, Denney et al., 2009, Grier et al., 2012) have finally allowed the BLR structure and kinematics to be explored in detail for a handful of AGNs [Pancoast et al., 2014, Grier et al., 2017, Williams et al., 2018, Williams et al., 2020]. In general, direct modeling has found many similarities across objects, although the exact details vary: the low-ionization BLR is arranged in a thick disk-like structure at low to moderate inclination to our line of sight, and with kinematics that are dominated by near-circular Keplerian orbits but with a contribution from inflow (although Williams et al., 2018 find evidence for outflow, rather than inflow, in some of their sample). The high-ionization BLR is less well studied, and Williams et al., 2020 find several key differences in not just the kinematics but also the geometry of the low- and high-ionization BLR gas in NGC 5548. Studies that have focused on the recovery of the transfer function have generally drawn similar conclusions about the BLR structure and kinematics [Bentz et al., 2010, Horne et al., 2021]. A key finding of all these studies is that the black hole masses derived from a more simplistic reverberation analysis, involving a mean time delay and line width and an adopted scale factor of $\langle f \rangle \approx 5$, are generally in good agreement within their uncertainties with the masses derived from modeling. As expected, the largest differences are generally found for those AGNs where direct modeling derives an inclination of the BLR that is $\lesssim 15^{\circ}$ to our line of sight (cf. Figure 14 of Williams et al., 2018). Very low inclinations result in small observed line-of-sight velocities, which bias the simplistic mass estimates to low values.
We recently conducted a new reverberation mapping program focusing on the bright Southern Seyfert, NGC 3783, with the intent of improving the constraints on the black hole mass. A nearly face-on barred spiral galaxy at $z=0.0097$, NGC 3783 is one of the most well-studied AGNs in the sky. It is one of a few Seyfert 1s that may be studied in detail with VLT GRAVITY observations on spatial scales that resolve the dust torus and outer broad line region [Gravity Collaboration et al., 2021], thus it is a critical target for informing our understanding of both feeding and feedback. Furthermore, NGC 3783 is also one of a small number of Seyfert 1 galaxies that are near enough to allow a reverberation-based mass to be directly compared with masses constrained through dynamical methods. The comparison of reverberation and dynamical masses is the only independent check that we can use to investigate the reliability of the entire black hole mass scale that we currently apply across cosmic history, an important point given the different systematic biases that are inherent in each black hole mass measurement technique.
An initial assessment of the monitoring data constrained a reverberation-based black hole mass of $M_{\rm BH} =(2.3\pm0.4)\times 10^7$ M$_{\odot}$ [Bentz et al., 2021], assuming a scale factor of $\langle f \rangle = 4.82$ [Batiste et al., 2017]. However, variations in the time delay as a function of velocity across H$\beta$ and other optical emission lines were also seen in the spectra, with longer time delays observed near the line center and shorter time delays in the line wings. These initial results indicated that direct modeling would be likely to provide strong constraints on the BLR geometry and kinematics in NGC 3783, and that we might be able to probe both the low-ionization BLR through the broad H$\beta$ emission line as well as the high-ionization BLR through the He2 $\lambda4686$ broad line. Here, we present the results of that modeling and a new direct constraint on the black hole mass in NGC 3783.
§ DATA
A detailed description of the photometric and spectroscopic monitoring data are provided by Bentz et al., 2021. In summary, $V-$band photometric monitoring was carried out with the Las Cumbres Observatory global telescope (LCOGT) network of 1-m telescopes from 12 February to 30 June 2020. Notwithstanding the sudden onset of a global pandemic and the shutdown of several observatories, 209 images were acquired over this period with a median temporal sampling of 0.4 days. Spectroscopic monitoring with the robotic FLOYDS spectrograph on the 2-m Faulkes Telescope South was carried out over the same period, with 50 spectra acquired between 27 February and 26 June 2020, with a median temporal sampling of 1.7 days.
Example spectrum of NGC 3783 in black with the ULySS fit to the continuum and [O3] $\lambda\lambda 4959,5007$ doublet overplotted in red, and the continuum and O3-subtracted spectrum in blue. The vertical dotted lines mark the limits of the regions that were modeled for the H$\beta$ (4816$-$5025 Å) and He2 (4653$-$4816 Å) emission lines. With the continuum subtracted, low-level Fe2 emission is visible on the blue side of He2 and the red side of the [O3] doublet, but the analysis of Bentz et al., 2021 shows that Fe2 was not variable at a detectable level in these data.
The images and spectra were reduced in IRAF[IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.] following standard procedures. The spectra were intercalibrated using the [O3] $\lambda\lambda 4959,5007$ emission lines, which are constant in flux on timescales of a few months [Peterson et al., 2013], thus providing a correction for small wavelength shifts, differences in resolution, and offsets in flux calibration from night to night. Image subtraction methods [Alard & Lupton, 1998, Alard, 2000] were used to isolate the variable AGN point source from the constant host-galaxy emission in the images, providing a well-sampled and well-calibrated light curve of the AGN optical continuum emission. This was merged with the flux-calibrated continuum light curve measured at $5100\times(1+z)$ Å in the spectra, with data points taken within 0.25 days binned together for the final continuum light curve.
Before modeling the reverberation response, the continuum and [O3] emission lines were subtracted from each spectrum to allow the broad emission to be isolated. This was accomplished by modeling the spectral components in the high signal-to-noise mean spectrum with ULySS [Koleva et al., 2009] and then slightly adjusting that model to create a best fit for each individual spectrum before subtracting the desired model components. The continuum was fit with a powerlaw, representing the AGN continuum contribution, and a host-galaxy component parameterized by the Vazdekis models derived from the MILES library of empirical stellar spectra [Vazdekis et al., 2010]. Emission lines were fit with multiple Gaussian profiles, with 4 Gaussians needed to match each of the H$\beta$ and [O3] doublet lines and $1-4$ Gaussians needed to match other emission features in the spectrum. Once a best fit was achieved for the mean spectrum, the individual spectra were then fit one at a time, with the host-galaxy component held fixed to the best-fit template but allowed to vary in flux contribution, and with the power law and the emission-line components allowed to vary but with initial values matching their best-fit values. Once a best fit was found, the host-galaxy and power law continua and the [O3] components were then subtracted from each spectrum. Figure <ref> shows an example spectrum from a single night of observations in black, with the best-fit continuum and [O3] emission in red, and the spectrum after subtraction of those components in blue.
The H$\beta$ region was then isolated for modeling between observed wavelengths $4816-5025$ Å with the narrow emission line peak at 4910 Å, while the He2 region was isolated between $4653-4816$ Å with the narrow emission line peak observed at 4735 Å. Throughout the campaign, the rest-frame equivalent width of broad H$\beta$ relative to the starlight-corrected AGN continuum has a mean value of 139.9 Å with a median of 130.5 Å and a standard deviation of 22.4 Å. For He2, the mean rest-frame equivalent width is 15.8 Å with a median of 15.1 Å and a standard deviation of 5.4 Å.
While the blue spectra also cover the H$\gamma$ and H$\delta$ broad emission lines, and the red spectra cover the H$\alpha$ emission line, Bentz et al., 2021 described the difficulties in accurately calibrating the red spectra and the short wavelength end of the blue spectra. The integrated light curves for these emission lines clearly demonstrate significant excess noise, so we do not attempt to model them here.
§ BLR MODELS
Modeling of the BLR for H$\beta$ and for He2 was carried out with CARAMEL, a phenomenological modeling code that is described in detail by Pancoast et al., 2014. CARAMEL is capable of constraining both the geometry and kinematics of the BLR using the reverberation response across the profile of a broad emission line throughout a monitoring campaign. Here, we summarize the main components of the model.
CARAMEL represents the BLR as a large collection of massless point particles that are distributed in position and velocity space, surrounding a massive black hole whose gravity dominates the region. Each point particle processes incident continuum flux instantaneously, and the observed time delay profile of the BLR depends on the spatial distribution of point particles while the broad line wavelength profile depends on the velocity distribution of point particles.
The spatial distribution of particles is parametrized with angular and radial distributions. The radial positions of particles are drawn from a gamma distribution
\begin{equation}
p(x|\alpha,\theta) \propto x^{\alpha - 1}\exp{\left( - \frac{x}{\theta} \right)}
\end{equation}
that provides the flexibility to represent a Gaussian ($\alpha>1$), an exponential ($\alpha=1$), or a cuspier profile ($0<\alpha<1$). The gamma distribution of particles is shifted away from the location of the black hole by the Schwarzschild radius, $R_s = 2GM/c^2$, plus a minimum radius $r_{\rm min}$. To assist with interpretation of the modeling results, a change of variables is performed so that parametrization is given in terms of ($\mu$, $\beta$, $F$):
\begin{equation}
\mu = r_{\rm min} + \alpha \theta,
\end{equation}
\begin{equation}
\beta = \frac{1}{\sqrt{\alpha}},
\end{equation}
\begin{equation}
F = \frac{r_{\rm min}}{r_{\rm min} + \alpha \theta},
\end{equation}
where $\mu$ is the mean radius, $\beta$ is the shape parameter, and $F$ is $r_{\rm min}$ in units of $\mu$. The standard deviation of the shifted gamma profile is given by $\sigma_r = \mu \beta (1-F)$, and the BLR is truncated at an outer radius of $r_{\rm out} = c \Delta t_{\rm data}/2$, where $\Delta t_{\rm data}$ is the time difference between the first point in the modeled continuum light curve and the first point in the emission-line light curve. This truncation assumes that the total length of the monitoring campaign is sufficient to track reverberation signals throughout the entire BLR.
The angular distribution of the particles is then arranged in a disk with a thickness that is set by an opening angle $\theta_o$, where $\theta_o=0\degr$ is a thin disk and $\theta_o=90\degr$ is a sphere. The inclination of the disk to the observer's line of sight is set by $\theta_i$, where $\theta_i=0\degr$ is viewed face on and $\theta_i=90\degr$ is viewed edge on. The strength of line emission from different depths within the disk is parametrized by the distribution of particles as a function of depth. For a single particle, the angle of displacement from the disk midplane is given by
\begin{equation}
\theta_{d,N} = \arccos (\cos \theta_o + (1-\cos \theta_o)\times U^{\gamma})
\end{equation}
where $U$ is a random number drawn uniformly between 0 and 1. The value of $\gamma$ ranges from 1, where particles are distributed uniformly throughout the thickness of the disk, to 5, where particles are clustered at the disk face and therefore emission is preferentially from the outer skin of the BLR. An additional asymmetry parameter, $\xi$ allows for the possibility of obscuration along the midplane of the disk, where $\xi \rightarrow 0$ causes the entire back half of the disk to be obscured and $\xi = 1$ has no midplane obscuration. The final asymmetry parameter $\kappa$ is related to the weight of a particle
\begin{equation}
W(\phi) = \frac{1}{2} + \kappa \cos \phi
\end{equation}
where $W$ is the fraction of continuum flux that is reradiated back towards the observer as line flux and $\phi$ is the angle between the observer's line of sight to the source and the particle's line of sight to the source. The value of $\kappa$ ranges from $-0.5$, where particles preferentially emit back towards the ionizing source, to $0.5$, where particles preferentially radiate away from the ionizing source. In the case of $\kappa=-0.5$, an observer would see preferential emission from the far side of the disk, while preferential emission from the near side would be observed in the case of $\kappa=0.5$.
Histograms displaying the posterior distributions of the BLR model parameters for H$\beta$ (blue) and He2 (red).
Tabulated values are the median and 68% confidence intervals.
The velocity distribution of particles includes radial and tangential distributions. A fraction of the particles, $f_{\rm ellip}$, have near-circular orbits within the Keplerian potential of the central black hole with mass . The remaining particles ($1-f_{\rm ellip}$) are either inflowing ($f_{\rm flow}<0.5$) or outflowing ($f_{\rm flow}>0.5$). Whether these orbits are generally bound or unbound is determined by the parameter $\theta_e$. For a plane defined by the possible values of the radial and tangential velocities, $\theta_e$ describes the angle of the velocity components away from the escape velocity and towards the circular velocity. If $\theta_e =0$ degrees then the orbits are drawn from a Gaussian distribution centered on the escape velocity. As $\theta_e \rightarrow 90\degr$, the inflowing or outflowing orbits approach the parameter space occupied by near-circular orbits. Thus high values of $\theta_e$ indicate inflowing or outflowing orbits that are very nearly circular, $\theta_e\approx45\degr$ indicates that most of the inflowing or outflowing orbits are highly eccentric but still bound, and low values of $\theta_e$ indicate that most particles are near the escape velocity and unbound.
A contribution from macroturbulence is included in the line-of-sight component of the velocity vector for each point particle as
\begin{equation}
v_{\rm turb} = \mathcal{N} (0,\sigma_{\rm turb})|v_{\rm circ}|,
\end{equation}
where $v_{\rm circ}$ is the circular velocity and $\mathcal{N}(0,\sigma_{\rm turb})$ is a normal distribution centered on 0 and with standard deviation $\sigma_{\rm turb}$.
With the spatial and velocity distributions of the particles parametrized, the emission-line profile can then be calculated for each continuum flux measurement, assuming that the continuum flux tracks the ionizing flux from a central point source. A nonvariable narrow emission-line component is included in the modeled emission-line profiles, as is a smoothing parameter to account for the small differences in spectral resolution that arise from variable seeing conditions throughout the monitoring campaign.
To explore the full range of possible time delays arising from the BLR geometry and to properly compare the modeled emission line profiles with the measured profiles, the continuum light curve must be interpolated. CARAMEL uses Gaussian processes to both interpolate between continuum flux measurements and to extrapolate the continuum light curve beyond the beginning and end of the monitoring campaign to extend the range of time delays that may be probed. The uncertainties on the Gaussian process model parameters are included in the determination of the BLR model parameters, thus capturing the effect of the uncertainties that arise from interpolating and extrapolating the continuum data.
For each model realization, we include 2000 individual point particles to represent the BLR. The continuum light curve is interpolated and model emission-line profiles are calculated for each epoch at which an emission-line measurement was acquired. A Gaussian likelihood function compares the modeled spectra against the measured spectra and adjusts the model parameters accordingly. CARAMEL utilizes a diffusive nested sampling code, with the latest version employing DNEST4 [Brewer & Foreman-Mackey, 2018], to efficiently explore the model parameter space. DNEST4 allows for the use of a likelihood softening parameter, or statistical temperature $T$, which has the effect of increasing the measurement uncertainties. This parameter can account for underestimated measurement uncertainties or for the inability of the simplified model to capture all of the real details in the measurements. The value of $T$ is determined in the post analysis by examining the distributions of the model parameters and choosing the largest value of $T$ for which the distributions remain smooth and generally unimodal.
Finally, to check that convergence had been reached, we compared the constrained values of the model parameters from the first half of the model runs to the second half of the model runs, with the total number of model runs being 10,000. There was no discernible difference between the parameters constrained during the first half or second half of the model runs for either H$\beta$ or He2.
§ RESULTS
The top three panels display the data, one possible model, and residuals (data$-$model) for the H$\beta$ spectra, with epochs 1 and 13 and their model fits displayed immediately below to exemplify a low flux spectrum (magenta curve) and a high flux spectrum (cyan curve). The bottom two panels display the continuum and integrated H$\beta$ light curves as data points with model fits overlaid. The full ranges of the models are displayed in light turquoise with the example model corresponding to the top four panels overlaid in dark turquoise. Flux densities ($F_{\lambda}$) are in units of $10^{-15}$ erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$ while integrated flux ($F$) is in units of $10^{-13}$ erg s$^{-1}$ cm$^{-2}$. Across the six panels, it is evident that most of the gross characteristics of the data are captured by the models, although some of the finer details are not. Furthermore, the continuum model is less well constrained during time periods with multi-day gaps between the measurements. Unfortunately, these gaps resulted from the shutdown of numerous observatories in response to the global coronavirus pandemic and could not be avoided.
Modeling of the H$\beta$ emission line in NGC 3783 provides constraints on the low ionization BLR, while modeling of the He2 emission line constrains the high ionization BLR. Figure <ref> compares the posterior probability distribution functions for all the parameters of the BLR models for both H$\beta$ and He2, while the median and 68% confidence intervals for each parameter are listed in Table <ref>. We describe the resultant set of models for each emission line below.
§.§ H$\beta$
The models for H$\beta$ require a likelihood softening of $T=125$, which amounts to increasing the uncertainties on the data by a factor of $\sqrt{T} = 11.2$. Figure <ref> displays the continuum and integrated H$\beta$ emission-line light curves and the observed H$\beta$ line profiles along with model fits to all of these. In general, the emission line profiles are well-fit by the modeled profiles as are the gross flux variations of the integrated emission-line light curve, although some of the finer details of the data are not captured by the models. The small disagreements between the data and the models could be the result of uncertainties that are still underestimated for some data points, or they could signal that the models are too simplistic and do not have the full flexibility needed to match all of the real variations, or both.
The geometry of the H$\beta$-emitting BLR is found to be a relatively face-on thick disk with an opening angle of $\theta_o=34.7^{+6.2}_{-9.9}$ degrees and an inclination to our line of sight of $\theta_i=17.9^{+5.3}_{-6.1}$ degrees. The disk has an inner minimum radius of $r_{\rm min}=3.25^{+1.13}_{-1.54}$ light days with a median radius of $r_{\rm median}=10.07^{+1.10}_{-1.21}$ light days and a width of $\sigma_r=10.47^{+15.44}_{-3.82}$ light days. The disk emission is distributed radially in a near-exponential profile ($\beta=0.95^{+0.25}_{-0.25}$), and is distributed throughout the thickness of the disk with a slight preference for stronger emission near the face of the disk ($\gamma=1.84^{+1.48}_{-0.67}$) and strong but not total obscuration along the midplane ($\xi=0.23^{+0.24}_{-0.15}$). The line emission direction is rather unconstrained, with the median value centered around isotropic emission but having large uncertainties that do not discriminate between a preference for radiation towards or away from the central source ($\kappa=0.04^{+0.31}_{-0.30}$). Figure <ref> displays a representative geometric model for the H$\beta$ response in the BLR of NGC 3783, drawn from the posterior probability distribution.
Representative geometric model for the H$\beta$ response in the broad line region of NGC 3783, drawn from the posterior probability distribution. The left panel is oriented edge on, with an Earth-based observer on the +x axis, while the right panel shows the Earth-based observer's view. The transparency of each point represents the relative response of the gas to continuum fluctuations at each position, with more opaque points responsible for a stronger response. This effect is most easily viewed in the right panel, where there is less overlap between points.
Transfer function $\Psi(\lambda,\tau)$ for the example H$\beta$ model displayed in Figure <ref>. Integrating the transfer function over wavelength gives the one-dimensional lag profile $\Psi(\tau)$, which is shown on the right. Integrating the transfer function over time delay gives $\Psi(\lambda)$, or the variable emission-line profile, which is shown immediately under the transfer function. The bottom panel displays the average lag as a function of wavelength across the emission line, with the turquoise crosses showing the average time delay for 5 velocity bins across the H$\beta$ profile from Figure 6 of Bentz et al., 2021.
The associated mean and median time delays for H$\beta$ are found to be $\tau_{\rm mean}=9.05^{+0.68}_{-0.64}$ days and $\tau_{\rm median}=7.42^{+0.70}_{-0.74}$ days, which agree well with the average H$\beta$ time delay reported by Bentz et al., 2021 of $\tau_{\rm cent}=9.60^{+0.65}_{-0.72}$ days. Figure <ref> displays the transfer function, $\Psi(\lambda,\tau)$, for a representative model. Also referred to as the velocity-delay map, the transfer function displays the range of H$\beta$ responsivities as a function of time delay and velocity (or wavelength) across the broad emission line profile. The shape of the transfer function generally agrees with the cross-correlation time delays computed for different velocity bins of the H$\beta$ profile by Bentz et al., 2021, displayed here as the turquoise crosses in the bottom panel of Figure <ref>.
The black hole mass is constrained to be $\log_{10} (M_{\rm BH}/M_{\odot})=7.51^{+0.26}_{-0.13}$. Roughly 60% of the particle orbits are near circular ($f_{\rm ellip}=0.60^{+0.09}_{-0.15})$, with the other 40% strongly preferring inflow ($f_{\rm flow}<0.5$). With a low value of $\theta_e=16.1^{+18.6}_{-11.0}$ degrees, most of these are truly inflowing orbits rather than highly elliptical bound orbits. There is also a small but non-zero contribution to the kinematics from turbulence ($\sigma_{\rm turb}=0.024^{+0.050}_{-0.021}$).
§.§ He2
Same as Figure <ref>, but for He2.
The models for He2 require a likelihood softening of $T=145$, which amounts to increasing the uncertainties on the data by a factor of $\sqrt{T} = 12.0$. Figure <ref> displays the continuum and integrated He2 emission-line light curves and the observed He2 line profiles along with model fits to all of these. In general, the modeled emission line profiles fit the main features of the observations, however the lower integrated flux and larger uncertainties compared to H$\beta$ do lead to somewhat less agreement between the observations and the models. The gross flux variations of the integrated emission-line light curve also seem to be mostly captured by the models.
The geometry of the He2-emitting BLR is again found to be a relatively face-on thick disk with an opening angle of $\theta_o=23.5^{+11.8}_{-8.0}$ degrees and an inclination to our line of sight of $\theta_i=19.1^{+10.3}_{-7.0}$ degrees. The disk has an inner minimum radius of $r_{\rm min}=1.00^{+0.46}_{-0.42}$ light days with a median radius of $r_{\rm median}=1.33^{+0.34}_{-0.42}$ light days and a width of $\sigma_r=0.17^{+0.34}_{-0.13}$ light days. The disk emission is distributed radially in a Gaussian profile ($\beta=0.67^{+0.83}_{-0.45}$), although the constraints on this parameter are quite weak. The distribution of emission throughout the thickness of the disk is also not well constrained ($\gamma=2.77^{+1.55}_{-1.23}$), but there is a preference for strong obscuration along the midplane ($\xi=0.08^{+0.23}_{-0.06}$). The line emission slightly prefers radiation back towards the central source ($\kappa=-0.20^{+0.45}_{-0.24}$).
Figure <ref> displays a representative model for the He2 response in the BLR of NGC 3783, drawn from the posterior probability distribution. As expected, it is significantly more compact than H$\beta$.
Same as Figure <ref>, but for He2.
Same as Figure <ref>, but for He2.
The associated mean and median time delays for He2 are found to be $\tau_{\rm mean}=1.19^{+0.28}_{-0.30}$ days and $\tau_{\rm median}=1.16^{+0.29}_{-0.32}$ days, which are a bit more compact but agree within the uncertainties with the average He2 time delay reported by Bentz et al., 2021 of $\tau_{\rm cent}=1.95^{+1.02}_{-0.98}$ days. Figure <ref> displays the transfer function for a representative model. The shape is much more asymmetric than was found for H$\beta$, with a heavier response in the blue wing and very little response in the red wing.
The black hole mass is constrained to be $\log_{10} (M_{\rm BH}/M_{\odot})=7.13^{+0.43}_{-0.37}$. Only $1/5$ of the particle orbits are near circular ($f_{\rm ellip}=0.22^{+0.19}_{-0.16}$), while the rest of the orbits strongly preferring outflow ($f_{\rm flow}>0.5$). With a low value of $\theta_e=14.6^{+11.8}_{-10.3}$ degrees, most of these are truly outflowing orbits rather than highly elliptical bound orbits. Finally, there is again a small but non-zero contribution to the kinematics from turbulence ($\sigma_{\rm turb}=0.013^{+0.044}_{-0.011}$).
Constraints on the black hole mass in NGC 3783 from H$\beta$ (blue), He2 (red), and the joint inference using results from both emission lines (black).
Posterior distributions of the BLR model parameters for H$\beta$ before (turquoise, “unweighted”) and after (black, “weighted”) selecting only those models that agree with the joint constraint on . The unweighted distributions in turquoise are the same as the results for H$\beta$ in Figure <ref>, but are effectively smoothed with a Gaussian kernel for easier comparison with the weighted constraints. The vertical dotted lines mark the median values, while the dashed vertical lines mark the 68% confidence intervals. The parameters are generally unchanged when models that agree with the joint constraint on are preferred.
Same as Figure <ref> but for He2. The most significant changes are for the BLR radius, which shifts to larger values, and the fraction of near-circular orbits, which decreases to smaller values.
§ DISCUSSION
While both emission lines were modeled independently, they arise from the same AGN and should agree on some parameters while possibly differing for others. Comparing and contrasting the results for H$\beta$ and He2 in the context of other studies may thus shed additional light on the Seyfert nucleus of NGC 3783.
§.§ Black Hole Mass
The black hole mass of NGC 3783 is expected to be the same for both H$\beta$ and He2. And indeed, we see that there is significant overlap between the two in the top left panel of Figure <ref>. We investigated the joint inference on the black hole mass following the method described by Williams et al., 2020. We first approximated the posterior probability distribution functions of each with a Gaussian kernel density estimate and then multiplied them together. The result is shown in Figure <ref> and gives $\log_{10} (M_{\rm BH}/M_{\odot})=7.45^{+0.19}_{-0.11}$, or $=2.82^{+1.55}_{-0.63}\times10^7$ . This is consistent with the simple reverberation constraint on the mass, $M_{\rm BH} = 2.34^{+0.43}_{-0.43} \times 10^7$ , or $\log_{10} (M_{\rm BH}/M_{\odot})=7.37^{+0.07}_{-0.09}$, which is based on the mean H$\beta$ time delay and line width and an assumed scale factor of $\langle f \rangle=4.82$. We note that the uncertainties quoted for the simple mass constraint include only the measurement uncertainties on the time delay and line width, and do not include other potential uncertainties such as the object-to-object variation in the scale factor.
With a black hole mass constraint from the BLR models, we can infer a specific value of $f=6.0^{+3.5}_{-1.8}$ for NGC 3783 using the mean time delay and line width for H$\beta$. Previous investigations [Pancoast et al., 2014, Grier et al., 2017, Williams et al., 2018] have found that $f$ scales most strongly with the inclination of the system, as expected because only the line of sight velocity component is measured. NGC 3783 seems to follow the same trend that has previously been seen for other Seyferts, as the inclination angle constrained by the models together with the linear regression results of Williams et al., 2018 predict $f=6^{+16}_{-4}$, or $\log_{10} (f)=0.75^{+0.59}_{-0.61}$. Thus, the good agreement between our mass constraint and the simple reverberation constraint arises from the inclination of NGC 3783 being close to the mean inclination value for the sample of local Seyferts, and so having an individual $f$ factor that is similar to the population average.
We can also investigate any changes to the distributions of model parameters that may arise from selecting only those models that agree with the joint H$\beta$ and He2 constraint on . Figure <ref> shows the constraints on the model parameters for H$\beta$ before and after selecting only those models that agree with the joint constraint. The results are quite similar, which is unsurprising since the H$\beta$ models provided the strongest initial constraint on . Figure <ref> shows the same but for He2. In this case, we find that models that favor the joint constraint on also favor a slightly larger radius, which makes sense since the joint constraint on was at the upper end of the mass distribution for He2, and also favors an even smaller fraction ($\sim15$%) of bound near-circular orbits with the rest of the orbits outflowing. No additional changes are seen in the distributions of the model parameters if we similarly constrain the inclination angle in addition to , in which case we find a joint constraint of $\theta_i=18.2^{+3.6}_{-5.5}$ degrees.
§.§ Geometry and Kinematics
The similarities between the inclinations and opening angle constraints for H$\beta$ and He2 support the interpretation that both emission lines are probing different regions of the same thick disk of gas. While the median values of the opening angles might suggest that the H$\beta$ emitting region is more “puffed up” than the He2 emitting region, as might be expected for a bowl-shaped model of the BLR like that proposed by Goad et al., 2012, the large uncertainties on the He2 opening angle mean that the two values formally agree. As expected from the differences in their mean time delays reported by Bentz et al., 2021, the He2-emitting region is significantly more compact and close to the central ionizing source than the H$\beta$-emitting region, demonstrating clear ionization stratification within the BLR (e.g., Peterson, 1993 and references therein). There is little to no overlap between the two, with $r_{\rm min}=3.25^{+1.13}_{-1.54}$ light days for H$\beta$ compared to $r_{\rm mean}=1.40^{+0.31}_{-0.42}$ light days and $\sigma_{\rm r}=0.17^{+0.34}_{-0.13}$ light days for He2 (see Figure <ref>).
The two regions of the BLR appear, however, to be dominated by different kinematics. In the case of the H$\beta$ emitting region, the kinematics are dominated by near-circular orbits with some infall, whereas the He2 emitting region is dominated by outflow. Repeated studies of the same AGN, such as NGC 5548 [Pancoast et al., 2014, Williams et al., 2020], have found that the best-fit kinematics can change from one reverberation dataset to another, so the different kinematics may not be indicating structural differences between the inner and outer BLR in NGC 3783, but rather transient effects (“weather”). On the other hand, Korista & Goad, 2004 find that photoionization models predict He2 $\lambda 4686$ is preferentially emitted in the presence of an ionizing photon flux that is $\sim 300$ times stronger than H$\beta$. While H$\beta$-emitting gas in the BLR has been shown to be fairly stable against radiation pressure [Netzer & Marziani, 2010], He2 is preferentially emitted from lower density gas [Korista & Goad, 2004] and may be more susceptible to radiation pressure forces.
It may be that a combination of weather and photoionization physics explains the difference in kinematics between H$\beta$ and He2. NGC 3783 has demonstrated possible evidence for changes in the structure of the BLR in the recent past. Kriss et al., 2019 obtained UV spectra of NGC 3783 shortly after the discovery of a strong soft X-ray obscuring event was detected in 2016. They interpret changes in the UV broad emission lines of NGC 3783 together with the appearance of new broad absorption lines as evidence that the BLR scale height may have collapsed following a period of low ionizing luminosity that began in 2013 and continued to 2016. By late 2016, the luminosity had increased significantly and remained high through at least January 2018 [Kaastra et al., 2018], and could thus begin to drive changes in the structure of the BLR on the dynamical timescale ($\sim0.3$ years at a BLR radius of 2.0 light days, or 3 years at a radius of 10 light days). The luminosity of NGC 3783 between early 2018 and early 2020, when our observing campaign began, is unknown, but the BLR may have still been in the process of recovering from the extended low-luminosity period observed in $2013-2016$. And indeed, a rough comparison of the broad H$\beta$ profile in 2020 with the profiles observed in 2011 and 2016 (Fig. 18; Kriss et al., 2019) suggests that much of the flux deficit observed in the line core in 2016 has filled in, although the line profile has not fully returned to its 2011 state. Further multiwavelength monitoring coupled with velocity-resolved reverberation analyses could help to inform our understanding of structural changes in the BLR as a result of large changes in the ionizing luminosity.
Combined representative geometric models for the H$\beta$ response (blue) and He2 response (red) in the broad line region of NGC 3783. The left panel is oriented edge on, with an Earth-based observer on the +x axis, while the right panel shows the Earth-based observer's view. The transparency of each point represents the relative response of the gas to continuum fluctuations at each position, with more opaque points responsible for a stronger response.
Several studies of NGC 3783 have focused on attempts to model the accretion disk using the Fe K$\alpha$ emission line or the continuum emission [Brenneman et al., 2011, Patrick et al., 2011, Capellupo et al., 2017] and have found similar relatively face-on inclinations for the inner accretion disk, even when they disagree on other components of the models (such as the black hole spin). A similar inclination angle has also been found by modeling the three-dimensional structure of the spatially-resolved narrow line region on parsec scales [Fischer et al., 2013]. The consistency in inclination angles from the innermost regions of the accretion disk through the broad line region and the outermost narrow line region suggests that the spin axis of the central black hole has been stable for quite some time. With no evidence for large torques on the spin, and with the black hole spin axis apparently matching the rotation axis of this relatively face-on galaxy, the recent evolution of the supermassive black hole appears to be dominated by secular processes that are aligned with the disk of the galaxy.
The best-fit models that we find for H$\beta$ also agree well with recent interferometry results for NGC 3783 from GRAVITY [Gravity Collaboration et al., 2021], in which measurements of the broad Br$\gamma$ emission are best described by a rotating thick disk inclined at $\sim 20^{\circ}$ to our line of sight and surrounding a black hole with $\log_{10} (M_{\rm BH}/M_{\odot})=7.68^{+0.45}_{-0.43}$. Additionally, the radial extent of the Br$\gamma$-emitting region ($r_{\rm mean} = 16^{+12}_{-5}$ light days assuming $D=38.5$ Mpc) is in good agreement with the radial extent of the H$\beta$ emitting region ($r_{\rm mean}=11.4^{+1.1}_{-1.1}$ light days; see Table <ref>). A joint analysis of the GRAVITY observations with the continuum and integrated H$\beta$ light curves from Bentz et al., 2021 confirms and improves upon the results, with $M_{\rm BH}=2.54^{+0.90}_{-0.72}\times10^7$ , or $\log_{10} (M_{\rm BH}/M_{\odot})=7.40^{+0.13}_{-0.14}$, and $r_{\rm median}=16.2^{+2.8}_{-1.8}$ light days [Gravity Collaboration et al., 2021]. The black hole mass is still in excellent agreement with our findings, while the stronger constraints on the BLR radius in the joint analysis are somewhat in tension with the size of the BLR reported here ($r_{\rm median}=10.07^{+1.10}_{-1.21}$ light days). It is important to recognize that the GRAVITY results depend on the distance to NGC 3783 which is somewhat uncertain (recent studies suggest values of $35-50$ Mpc; Kourkchi et al., 2020, Robinson et al., 2021), reverberation mapping measures a responsivity-weighted radius while interferometry measures a flux-weighted radius, and photoionization effects (which are ignored in both our models and those employed in the analysis of the GRAVITY data) are known to cause different reverberation time delays for different Hydrogen recombination lines (e.g., Bentz et al., 2010). Despite these complicating factors, the good agreement between the results lends additional confidence to both.
Future work will investigate the joint constraints that may be derived from an analysis of the velocity-resolved reverberation data that we have presented here in tandem with the GRAVITY observations.
§ SUMMARY
We have modeled the full velocity-resolved response of the broad H$\beta$ and He2 emission lines in NGC 3783. The results give a black hole mass constraint that is independent of any scaling factor, and a joint analysis of the results for the two emission lines prefers $=2.82^{+1.55}_{-0.63}\times10^7$ . The geometry of the BLR is found to be a thick disk that is close to face on ($\theta_i\approx 18^{\circ}$) and exhibiting clear ionization stratification, with H$\beta$ arising from an extended region of $~\sim 3-20$ light-days, while He2 arises from a significantly more compact and centralized region of $1-2$ light days. The kinematics of the outer BLR probed by H$\beta$ are dominated by near-circular orbits with a contribution from infall, whereas the kinematics of the inner BLR probed by He2 are dominated by an unbound outflow. Given the recent history of a deficit of ionizing radiation in NGC 3783 that was observed from $2013-2016$, and the hypothesis that the BLR height collapsed as a result, it is possible that we may be seeing the BLR undergoing structural changes as it recovers.
We thank the anonymous referee for suggestions that improved the presentation of this work.
We also thank Kate Grier for helpful conversations about CARAMEL.
MCB gratefully acknowledges support from the NSF through grant AST-2009230.
TT and PRW acknowledge support by the Packard Foundation through a Packard Research Fellowship to TT and from NSF through grant NSF-AST-1907208. PRW acknowledges support from the UCLA graduate division through a Dissertation Year Fellowship.
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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).
Human-in-the-loop data curation workshop at ACM CIKM 2022, Oct 17–21, 2022,
Atlanta, GA
<EMAIL_ADDRESS>] [1] [1]
<EMAIL_ADDRESS>] [1] [1]
[1]Corresponding author. [1]KBM and RS contributed equally to the majority of
this research. Additional authors contributed to specific aspects including
initial models, data sets and project or platform development.
# From fat droplets to floating forests: cross-domain transfer learning using
a PatchGAN-based segmentation model
Kameswara Bharadwaj Mantha School of Physics & Astronomy, University of
Minnesota, Twin Cities, 116 Church St SE, Minneapolis, MN, 55455 Ramanakumar
Sankar Yuping Zheng Lucy Fortson Thomas Pengo University of Minnesota
Informatics Institute, 2231 6th St SE, Minneapolis, MN, 55455 Douglas Mashek
Medical School, University of Minnesota, Twin Cities, 420 Delaware Street SE,
Minneapolis, MN, 55455 Mark Sanders Trace Christensen Jeffrey Salisbury
Mayo Clinic, 200 First Street SW, Rochester, MN, 55905 Laura Trouille Adler
Planetarium, 1300 S DuSable Lake Shore Dr., Chicago, IL 60605 Jarrett E. K.
Byrnes Isaac Rosenthal Department of Biology, University of Massachusetts
Boston 100 Morrissey Blvd; Boston, MA, 02125 Henry Houskeeper Kyle Cavanaugh
Department of Geography, University of California Los Angeles, Los Angeles, CA
90095
(2022)
###### Abstract
Many scientific domains gather sufficient labels to train machine algorithms
through human-in-the-loop techniques provided by the Zooniverse.org citizen
science platform. As the range of projects, task types and data rates
increase, acceleration of model training is of paramount concern to focus
volunteer effort where most needed. The application of Transfer Learning (TL)
between Zooniverse projects holds promise as a solution. However,
understanding the effectiveness of TL approaches that pretrain on large-scale
generic image sets vs. images with similar characteristics possibly from
similar tasks is an open challenge. We apply a generative segmentation model
on two Zooniverse project-based data sets: (1) to identify fat droplets in
liver cells (FatChecker; FC) and (2) the identification of kelp beds in
satellite images (Floating Forests; FF) through transfer learning from the
first project. We compare and contrast its performance with a TL model based
on the COCO image set, and subsequently with baseline counterparts. We find
that both the FC and COCO TL models perform better than the baseline cases
when using $>75\%$ of the original training sample size. The COCO-based TL
model generally performs better than the FC-based one, likely due to its
generalized features. Our investigations provide important insights into usage
of TL approaches on multi-domain data hosted across different Zooniverse
projects, enabling future projects to accelerate task completion.
###### keywords:
datasets generative adversarial neural networks UNET generator patch-based
discriminator focal tversky loss transfer learning
## 1 Introduction
Citizen Science has established itself as a valuable method for distributed
data analysis enabling research teams from diverse domains to solve problems
involving large quantities of data with complexity levels requiring human
pattern recognition capabilities [1, 2]. As the largest citizen science
platform, Zooniverse.org has enabled over 2.5 million volunteers to provide
over half a billion annotations on hundreds of projects across the sciences
and humanities. Many of these projects use the resulting labels to train
machine learning algorithms typically training models from scratch e.g., [3,
4, 5, 6, 7, 8, 9, 10, 11]. To accelerate labeling efficiencies across the
platform, the Zooniverse human-machine system should take advantage of
transfer learning techniques, especially when volunteer engagement is at a
premium. When applying transfer learning, a new project would require fewer
labels from volunteers to achieve the same performance as training a model
from scratch. Volunteer labelers would thus be able to focus on tasks more
suited to humans such as anomaly detection e.g., [12].
Transfer learning (TL) is an established approach, where the feature space
from a pretrained model can be transferred to another framework and fine tuned
to perform analogous or different tasks. Feature extraction is typically
performed using Deep Convolutional Neural Networks (CNNs) such as [13, 14].
Transfer learning generally uses models trained on data that is either “out-
of-domain” (i.e., training data characteristics are different from data at
hand) or “in-domain” (data that are similar or closely relatable to the data
at hand). Quantifying the gains provided by these different TL approaches is
an active area of research, where studies find several factors to be at play
that govern its effectiveness: Accuracy and architecture choice of the
pretrained model [15], robustness of model to input adversarial noise [16],
and type of task to which the TL is being applied [17]. Recent works (e.g.,
[12, 8]) have demonstrated that transfer learning from a model pretrained on
in-domain data performs better than transfer learning from out-of-domain data.
On the other hand, some studies find that TL models based on out-of-domain
data (e.g., ImageNet or COCO datasets) perform on par with or better than the
in-domain TL models [18, 19].
In order to leverage the Zooniverse’s large library of image-label pairs
across multiple domains, there is thus a clear need to better understand the
effectiveness of cross-domain transfer learning. In particular, we are
interested in the application of transfer learning specifically to projects
that share task similarity across a wide range of domains. For example, image
segmentation tasks vary across vastly different disciplines, from cell biology
to satellite imagery. Frameworks such as the U-Net [20], Recurrent
Convolutional Networks such as Mask-RCNNs [21], and Generative Adversarial
Networks (GANs; e.g., [22, 23]) have been used to perform such object
segmentation across multiple domains and data sets. However, robust learning
of such segmentation models from scratch often requires large annotated
training samples that may not be available (e.g., medical imaging), which can
lead to poor generalizability of the learnt features to newer data, even in
related domains. While Zooniverse can provide these large annotation sets per
project, this comes at the cost of volunteer effort which we seek to optimize.
In an effort to increase project completion rates, this study investigates
potential machine performance gains through transfer learning across domains
by leveraging the shared task similarity between Zooniverse projects. We use a
PatchGAN-based [23] segmentation
model111https://github.com/ramanakumars/patchGAN/ to investigate the
effectiveness of segmenting kelp beds from satellite images. Particularly, we
test transfer learning from the COCO dataset (i.e., out-of-domain) and
microscopy imaging of lipid droplets in liver cells (pseudo-in-domain) and
compare them to their corresponding “trained from scratch” counterparts.
## 2 Methods
In this section, we detail our PatchGAN architecture [23], the training and
testing data and its preparation, and the description of the five models
analyzed in our work.
### 2.1 PatchGAN Framework
The implemented PatchGAN framework is inherited from the Pix2Pix GAN
architecture in [23], which is a conditional GAN for realizing paired image-
to-image translation. The PatchGAN architecture consists of a Generator ($G$)
and Discriminator ($D$):
The generator is composed of a U-Net [20], a U-shaped encoder-decoder neural
network, with skip connections across the bottleneck layer (Figure 1). The
encoder (decoder) comprises of $6$ downsampling (upsampling) blocks, each
consisting of $4\times 4$ convolution (transposed convolution), Leaky ReLU
activation, and a batch normalization layer. All the blocks in the inner
layers of the network also include a dropout layer which omits 50% of the
extracted features during training. The outputs of the transposed convolutions
are also concatenated with the corresponding skip connection feature map from
the encoder block.
Figure 1: U-Net Generator (top) and Discriminator (bottom) of our PatchGAN
framework.
The discriminator is a patch-wise binary classifier that takes a concatenation
of the input image and its corresponding ground truth or generated mask and
outputs a $30\times 30$ probability matrix. Each unit of this matrix
represents a 70 $\times$ 70 patch of the input image, and provides the
probability that the patch is real.
### 2.2 Data
For this study, we use three sources for our image-mask pairs: the Floating
Forests dataset, Etch-a-Cell dataset and the COCO-stuff. The former two are
Zooniverse projects focusing on image segmentation, while the latter
represents a generic image dataset that is used in computer vision,
representing an out-of-domain dataset compared to the former two. These three
data sources represent a diverse feature set on which to perform our transfer
learning experiment. Figure 2 shows an example of an image-mask pair from each
dataset.
#### 2.2.1 Floating Forests ($FF$)
Floating Forests is an ecology-based citizen science project hosted on
Zooniverse.org222https://www.zooniverse.org/projects/zooniverse/floating-
forests/ to identify kelp beds in Landsat imagery. The project presents
segments of Landsat data to Zooniverse volunteers, who draw outlines around
the kelp beds. These annotations are aggregated using a pixel-by-pixel
consensus to create masks of the kelp beds in the corresponding Landsat
segments. We use 4 channels from the Landsat data (Blue, Green, Red and near
Infrared) to train the patchGAN on the image-mask pairs. This FF data
comprises 6,967 ($350\times 350$ pix) image-mask pairs. We pre-process these
data such that each pair is cropped into four $256\times 256$ overlapping
cutouts, and augment each crop 5 times (rotation and flipping). This resulted
in $118,440$ training and $4180$ testing images.
#### 2.2.2 Etch-a-Cell: Fat Checker ($FC$)
Etch-a-Cell: Fat Checker is a cell biology project hosted on
Zooniverse.org333https://www.zooniverse.org/projects/dwright04/etch-a-cell-
fat-checker to identify lipid droplets in electron microscopy data. The
Zooniverse project presents 2D slices of the data to volunteers who annotate
the outline of the lipid droplet. The lipid mask is generated by aggregating
the annotations by multiple volunteers based on consensus. The data set
consists of $2341$ image-mask pairs and each image is $1200\times 1200\,{\rm
pix}$ in shape, with 3 channels. We split the sample into $2,106$ training and
$235$ testing sets. We transform these images and masks to work with our
PatchGAN framework by resizing them to $512\times 512$ pix and generating five
crops (four corners and one center crop). We further augment them by applying
three rotations ($90,180,270\,{\rm deg}$) per image, yielding augmented
training and testing samples of $42120$ and $4700$ images, respectively.
#### 2.2.3 COCO-Stuff
The Common Objects in COntext (COCO; [24]) is a large collection of several
real-world images with objects set in various simple to complex scenes, which
are annotated by outlines444https://github.com/nightrome/cocostuff. [25]
further processed the COCO data set to produce dense pixel-wise annotations
for them (the COCO-Stuff data set; hereafter COCO). These images and annotated
masks vary widely in their shapes, and therefore, we standardize these images
by resizing them to a $256\times 256$ pix shape. For our PatchGAN training, we
limit the training and testing data to those that host the ‘person’ class.
This amounts to $63785$ training and $2673$ testing image-mask pairs.
Figure 2: Visualization of example input image, truth mask, and patchGAN
predicted output mask.
### 2.3 Experimental Design
In this work, we investigate the potential of cross-domain transfer learning
by training $5$ models. The first $3$ models are trained from scratch –
$\Lambda_{FF}$, $\Lambda_{FC}$, and $\Lambda_{COCO}$ – using $100\%$ of their
corresponding data sets $FF$, $FC$, and $COCO$, respectively. Next, we train
the $\Lambda_{FC\rightarrow FF}$ and $\Lambda_{COCO\rightarrow FF}$ by
transferring the weights from the trained $\Lambda_{FC}$ and $\Lambda_{COCO}$
models to the $\Lambda_{FF}$. By comparing between the baseline $\Lambda_{FF}$
to the transfer learnt models $\Lambda_{FC\rightarrow FF}$ and
$\Lambda_{COCO\rightarrow FF}$, we quantify the impact of performing transfer
learning on the accelerated learning of the $\Lambda_{FF}$ model from two
distinct feature initializations. During this transfer learning exercise, we
also vary the amount of training data used from $10\%$-$100\%$.
## 3 Training & Results
In this section, we outline the training strategy and provide details of the
hyper parameters. We also present the results of our training and discuss the
outcomes of our transfer learning exercise.
### 3.1 Training Strategy
Our $\Lambda_{FF}$, $\Lambda_{FC}$, and $\Lambda_{COCO}$ models have been
trained for $50$ epochs. For the generator, we use the Focal Tversky Loss
(FTL; [26]), which is a generalized version of the Tversky Loss (TL) defined
in terms of the Tversky Index (TI) as:
$\begin{split}TI=\frac{TP}{TP+\alpha FN+\beta FP}\rightarrow
TL=(1-TI)\rightarrow FTL=(TL)^{\gamma},\end{split}$ (1)
For our training, we use $\alpha=0.7$ and $\beta=0.3$. The $\gamma$ parameter
controls the non-linearity of the TL with respect to the $TI$, enabling the
learning to focus on easier ($\gamma<1$) vs. harder ($\gamma>1$) examples. We
use $\gamma=0.75$ during our training. For the discriminator optimization, we
use the Binary Cross-Entropy (BCE) loss. Specifically, our total discriminator
loss is the average of two components: the discriminator applied on the
generated mask (i.e., against a fake label), and applied on the true mask
(i.e., the real label). For both the generator and discriminator, we use the
Adam optimizer with an initial learning rate $5\times 10^{-4}$ and $1\times
10^{-4}$ respectively, decayed exponentially by $\tau=0.95$, applied every 5
epochs.
Figure 3: Comparison of generated mask from different model runs on the
Floating Forests data, showing different performance gains from transfer
learning.
### 3.2 Transfer learning strategy
For our transfer learning based model training of $\Lambda_{FC\rightarrow FF}$
and $\Lambda_{COCO\rightarrow FF}$, we load the weights of the $\Lambda_{FC}$
and $\Lambda_{COCO}$ models into the freshly initialized $\Lambda_{FF}$ model
architecture. To account for the $3$ vs $4$ channel mismatch between the
$\Lambda_{COCO}$, $\Lambda_{FC}$ and $\Lambda_{FF}$, we load model layer
parameters excluding the input layer. For each model, we train 5 different
versions, using random subsets of $10\%,25\%,50\%,75\%$ and $100\%$ of the
full Floating Forests data, to compare TL efficiency gains from having a
smaller dataset. For these experiments, we also use only the first 6,967 un-
augmented images for re-training. We train the $\Lambda_{FC\rightarrow FF}$
and $\Lambda_{COCO\rightarrow FF}$ models with the same hyper-parameter
settings as the aforementioned “from scratch” models for $50$ epochs.
Figure 4: Comparison of mean final loss on Floating Forests validation data
across the different models.
### 3.3 Results and discussion
We find that our $\Lambda_{FF}$, $\Lambda_{FC}$ and $\Lambda_{COCO}$ generally
predict the annotation masks reasonably well (Figure 2), qualitatively
matching with the ground truths. Figures 3 and 4 show our transfer learning
results. In Figure 4, we show our average validation loss for the different
model training runs. As expected, larger training samples provide much better
performance, but we also find that the model pretrained on the COCO dataset
provides noticeably better performance on the Floating Forests data, compared
to both $\Lambda_{FC\rightarrow FF}$ and also $\Lambda_{FF}$. In fact, the
$\Lambda_{COCO\rightarrow FF}$ is able to match the performance of the
$\Lambda_{FF}$ model with between 50-75% of the training Floating Forests
dataset.
In Figure 3, we show examples highlighting the difference between the
generated masks from $\Lambda_{FF}$ and corresponding masks from
$\Lambda_{FC\rightarrow FF}$ and $\Lambda_{COCO\rightarrow FF}$. The sharpness
of the kelp beds is poorly reconstructed by the $\Lambda_{FF}$ model but is
well captured by the transfer learnt models (particularly when training
$\Lambda_{COCO\rightarrow FF}$ with more than 75% of the original data). The
transfer learnt models are also better at capturing kelp beds not identified
in the original consensus data. For example, both the ground truth and
$\Lambda_{FF}$ fail to reveal the kelp beds in the top left of the image, but
these are picked up well by the transfer learnt models.
This is likely due to the large diversity of the features in the COCO dataset,
making it a much more robust feature extraction network to transfer learn
from. Indeed, compared to $\Lambda_{FC\rightarrow FF}$, the
$\Lambda_{COCO\rightarrow FF}$ model-detected kelp beds are qualitatively
better visually (e.g., Figure 3), especially at lower training data sizes.
This is likely compounded with the lower feature diversity in both the
Floating Forest and Fat Checker data sets, given the fewer number of samples
in the training data and low variety in target classes.
#### 3.3.1 Transfer learning approaches for citizen science datasets
For the Zooniverse platform, this study provides an avenue to build quick
access for projects to use machine learning frameworks for simple tasks (e.g.,
image segmentation), by transfer learning from existing models on a small
sample of volunteer annotated data sets. However, despite the results
presented here, there are still several key questions which need to be
answered:
Domain dependency: It is unclear how much of the performance gained from COCO
was a ‘global truth’. That is, whether COCO (or similarly diverse datasets)
are immediately applicable to out-of-domain data, for all domains, or if there
are domain-specific restrictions which allow these performance gains to occur
on data such as Floating Forests. This requires more experiments with
increasingly different data sets on Zooniverse to investigate the range of
performance gains possible.
Task dependency: Previous studies on transfer learning across domains show
significant variations in performance across different task types. For
example, image classification tasks (e.g., [12, 17]) show lower gains than
image segmentation based tasks (e.g., [18]). We need to further investigate
the inherent difficulty associated with different tasks on Zooniverse
projects, and how effectively they can be transferred between domains. [12],
for example, show that significant boosts to performance is only provided by
using in-domain transfer learning.
Target data purity: For Zooniverse projects, data labels are generally
provided by volunteers and are aggregated based on volunteer consensus. In
this study, we found that transfer learning can help mitigate data purity
effects, since transfer learnt feature extraction models are generally robust
to mislabeled data. The extent to which transfer learning models are sensitive
to data purity effects needs to be further investigated.
In conclusion, we find that transfer learning can provide a significant boost
to projects that contain similar tasks on Zooniverse. However, the extent to
which this can be generalized across the full Zooniverse ecosystem is a
question of ongoing study.
## Acknowledgements
The authors would like to thank the Zooniverse volunteers without whom this
work would not have been possible. RS, KM, LF, YZ, LT would like to
acknowledge partial support from the National Science Foundation under grant
numbers IIS $2006894$ and OAC $1835530$. Partial support by RS, KM, LF, TP,
MS, TC, JS is acknowledged through Minnesota Partnership MNP IF#119.09.
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|
# Evaluating Large Language Models on the GMAT: Implications for the Future of
Business Education
Vahid Ashrafimoghari , Necdet Gürkan , Jordan W. Suchow
Stevens Institute of Technology
<EMAIL_ADDRESS>
###### Abstract
The rapid evolution of artificial intelligence (AI), especially in the domain
of Large Language Models (LLMs) and generative AI, has opened new avenues for
application across various fields, yet its role in business education remains
underexplored. This study introduces the first benchmark to assess the
performance of seven major LLMs—OpenAI’s models (GPT-3.5 Turbo, GPT-4, and
GPT-4 Turbo), Google’s models (PaLM 2, Gemini 1.0 Pro), and Anthropic’s models
(Claude 2 and Claude 2.1)— on the GMAT, which is a key exam in the admission
process for graduate business programs. Our analysis shows that most LLMs
outperform human candidates, with GPT-4 Turbo not only outperforming the other
models but also surpassing the average scores of graduate students at top
business schools. Through a case study, this research examines GPT-4 Turbo’s
ability to explain answers, evaluate responses, identify errors, tailor
instructions, and generate alternative scenarios. The latest LLM versions—
GPT-4 Turbo, Claude 2.1, and Gemini 1.0 Pro— show marked improvements in
reasoning tasks compared to their predecessors, underscoring their potential
for complex problem-solving. While AI’s promise in education, assessment, and
tutoring is clear, challenges remain. Our study not only sheds light on LLMs’
academic potential but also emphasizes the need for careful development and
application of AI in education. As AI technology advances, it is imperative to
establish frameworks and protocols for AI interaction, verify the accuracy of
AI-generated content, ensure worldwide access for diverse learners, and create
an educational environment where AI supports human expertise. This research
sets the stage for further exploration into the responsible use of AI to
enrich educational experiences and improve exam preparation and assessment
methods.
## 1 Introduction
Artificial intelligence (AI) has witnessed substantial advancements over the
past few years, which has facilitated its application across a spectrum of
fields. These applications range from augmenting personal assistant
technologies [1], to innovating healthcare practices [2], and to enriching
educational methodologies [3]. In healthcare, for instance, AI assists
professionals by organizing patient records, analyzing diagnostic images, and
identifying health issues [4]. AI applications have also been utilized in
education to enhance administrative services and academic support [3]. These
systems have been developed to simulate one-to-one personal tutoring, helping
educators in creating more effective educational settings. However, developing
these systems is challenging; it involves not only content creation and design
but also the refinement of feedback phrasing and dialogue strategies [5].
The emergence of Large Language Models (LLMs), also known as Large Generative
AI Models (LGAIMs), has been transformative for Natural Language Processing
(NLP) tasks, demonstrating considerable promise in the spheres of education
and assessment. As integral elements of AI, LLMs are adept at comprehending,
producing, and interpreting human language. With the ongoing advancement of
AI, it becomes imperative to scrutinize the proficiency and constraints of
these models within educational frameworks. Our study investigates the
efficacy of LLMs on the Graduate Management Admission Test (GMAT), an
established benchmark for entry into graduate management programs. The
objective of this evaluation is to elucidate the capabilities and limitations
of LLMs in educational environments.
The GMAT plays a crucial role in the application process for business schools
worldwide. It is designed to assess candidates’ abilities in verbal and
quantitative reasoning, analytical writing, and integrated reasoning, offering
a thorough evaluation of their preparedness for the challenging academic
environment of business school. Traditionally, preparation for the GMAT exam
has involved human tutors, who deliver their services either in classroom
settings or via online tutoring platforms. These tutors provide tailored
guidance, practice tests, and review sessions, all aimed at helping candidates
excel in the various segments of the exam.In today’s educational environment,
a wide array of companies and websites, such as Kaplan, Manhattan GMAT,
Economist, and Magoosh, offer extensive GMAT exam preparation services. These
services encompass a range of options, including self-paced online courses,
live online classes, private tutoring, and comprehensive study materials.
Their goal is to accommodate various learning styles and schedules, thereby
making GMAT preparation more accessible and adaptable for prospective business
school students.
However, the recent advancements in LLMs, such as GPT-4 Turbo, present a
unique opportunity to revolutionize the GMAT preparation process. These
sophisticated models have the potential to automate certain aspects of GMAT
preparation, offering personalized, adaptive learning experiences that can
match or even surpass traditional methods. For instance, they could provide
instant feedback on practice questions, adapt the difficulty level based on
the learner’s progress, or offer targeted practice on weak areas. Moreover,
they could be available 24/7, offering flexibility that traditional tutoring
cannot match. The exploration of this potential marks an exciting frontier in
the intersection of artificial intelligence and education, promising to make
GMAT preparation more efficient, effective, and tailored to individual needs.
Building on the possibilities introduced by LLMs, this study seeks to answer
the following research questions, each aimed at exploring the capabilities and
potential applications of these models in the context of GMAT preparation:
RQ1: How do LLMs compare to human candidates in terms of performance when
responding to the verbal and quantitative reasoning sections of the GMAT?
RQ2: What potential benefits and drawbacks are associated with the use of LLMs
in learning and education, especially for tutoring, exam preparation, and
assessment?
To address the research questions, we adopted a comprehensive approach,
initiating with an analysis of model performance on GMAT exam questions. Our
evaluation encompassed seven state-of-the-art general-purpose LLMs: GPT-3.5
Turbo, GPT-4, GPT-4 Turbo, Claude 2, Claude 2.1, PaLM 2, and Gemini 1.0 Pro,
focusing on their abilities in the quantitative and verbal reasoning sections
of the GMAT. Utilizing both the free and premium practice exams offered by the
Graduate Management Admission Council (GMAC), we aimed to discern any effects
of memorization or data leakage. The results revealed a negligible variance in
performance between the free and premium exams.Notably, GPT-4 Turbo, employing
zero-shot standard prompting, significantly outperformed the others, achieving
an average accuracy of 85.07% across three sets of official GMAT practice
exams, compared to 74.13% for GPT-4, 56.72% for GPT-3.5 Turbo, 72.14% for
Claude 2.1, 60.2% for Claude 2, 70.65% for Gemini 1.0 Pro and 50.75% for PaLM
2.
Our research goes beyond basic performance metrics to explore AI behavior in
educational settings. We provide a comparative analysis of human and AI
performance on the GMAT, discussing AI errors for targeted improvements. A
case study highlights the qualitative behavior of GPT-4 Turbo, demonstrating
its ability to articulate critical reasoning and interactively assist students
by evaluating their responses and errors, and creating counterfactual
scenarios. We conclude by reflecting on the potential impact of our findings
on business education and professional development, addressing accuracy,
fairness, and the broader implications for management practices. While
recognizing the limitations of our evaluation methods, we discuss the
necessary precautions and advancements for the real-world application of LLMs
like GPT-4 Turbo. Despite the need for careful and comprehensive evaluation,
we anticipate that AI models such as GPT-4 Turbo will have a significant and
positive impact on the business sector, becoming essential tools in
educational and professional settings.
## 2 Related Work
In recent years, several large pre-trained models have been developed,
including GPT [6], Bard [7], Claude 2 [8], BERT [9], RoBERTa [10], and the
widely used GPT-3 [11], GPT-3.5 [12], and GPT-4 [13]. These models are based
on transformer architecture [14] and have been pre-trained on massive datasets
of text to generate human-like text, answer questions, assist in translation
and summarization, and perform many NLP tasks with a single pre-training and
fine-tuning pipeline. These developments mark significant milestones in the
field of NLP and offer enormous opportunities for applications in research and
industrial contexts. We anticipate that future advancements in AI will offer
significant benefits, thus highlighting the need to explore their potential
applications in education.
LLMs have a range of applications within educational settings, enhancing
student learning. Researchers have utilized these models to develop
interactive learning tools, such as quizzes and flashcards, which in turn
improve student engagement and facilitate knowledge acquisition [15, 16]. One
of the advantages of using LLMs in education is their ability to help students
complete their practice questions more efficiently [17]. Specifically, GPT-3
has been employed to generate multiple-choice questions and answers, enhancing
reading comprehension [15]. The study indicates that automated quiz generation
not only reduces the workload for educators in creating quizzes manually but
also acts as an effective tool for students to evaluate and deepen their
knowledge throughout their study period and exam preparation. The researchers
developed a method for automatically creating prompts that encourage learners
to ask more substantive questions [18]. The findings suggest that LLMs can
significantly facilitate the promotion of curiosity-driven learning
experiences and serve as a powerful mechanism to enhance the expression of
curiosity among learners [19].
Personalized learning is a promising application area for LLMs, considering
the individual differences in learning styles and vast amount of educational
data available [20]. Researchers demonstrated the application of an advanced
GPT-3 model, specifically calibrated for chemistry, to assess answers provided
by students [21]. This technology could significantly aid educators in
appraising the quality and instructional value of student responses.
Additionally, [22] proposed a system that delivers AI-driven, personalized
feedback within a high school science task. They found that this system
effectively supported students in refining their scientific reasoning. [23]
observed that such feedback systems could enhance the ability of teacher
trainees to substantiate educational assessments. Furthermore, [24] noted that
in large-scale courses, these systems can provide feedback on written work,
potentially reducing the grading workload by 85% while maintaining high
accuracy. This enhancement also improves the perceived quality of feedback
among students. Recognizing the benefits of LLMs, Khan Academy has developed
an AI chatbot named Khanmigo, which functions as a virtual tutor and classroom
assistant. This initiative represents a step forward in incorporating LLMs
into educational platforms, aiming to enhance tutoring and coaching
experiences. The goal is to provide personalized one-on-one interactions with
students, demonstrating a practical application of LLMs in education [25].
Another notable example is the potential use of LLMs in foreign language
learning. Their features, such as an effortless, seamless, and friendly
interface, contribute to an excellent user experience. A study demonstrated
that language learners appreciated the ChatGPT prompts while performing
language tasks [26]. In another study, researchers proposed a system that
leverages LLMs to identify content that aligns with the user’s interests and
closely matches their proficiency level in a foreign language [27].
In the domain of computer science education, GPT-3 has been utilized as an
educational aid to clarify various aspects of coding, although numerous
questions related to research and instructions remain open for further
investigation [28]. In a separate endeavor, [29] have devised a method for
generating evaluative questions tailored to a data science curriculum by
refining a GPT-3 model with text-based educational material. These questions
were evaluated for their instructions merit using both an automated
categorization system employing a specifically trained GPT-3 model and expert
reviews from domain specialists. The outcomes indicated that the
GPT-3-generated questions were positively appraised by these experts, thus
supporting the adoption of large language models in the field of data science
education. In a recent study, researchers combined the capabilities of GPT-4
and GPT-3.5 to provide automated, tutor-like, high-quality programming
suggestions [30].
In digital ecosystems for education, particularly in Augmented Reality (AR)
and Virtual Reality (VR) environments [31, 32], LLMs can play a transformative
role. They can amplify key factors crucial for immersive user interaction with
digital content. For instance, LLMs can greatly enhance the natural language
processing and understanding capabilities of AR/VR systems. This enhancement
enables effective and natural communication between users and the system, such
as with a virtual teacher or virtual peers. The importance of this capability
for immersive educational technologies was identified early as a critical
aspect of usability [33] and is widely recognized as a key factor in improving
human-AI interactions [34].
In the realm of standardized testing, [35] examined ChatGPT’s performance on
the United States Medical Licensing Examination, revealing that ChatGPT’s
scores approached the passing threshold without the benefit of domain-specific
training. These findings prompted the researchers to propose that large
language models have the potential to substantially aid medical education and
even contribute to clinical decision-making processes [35]. Additionally, an
investigation into ChatGPT’s performance on four actual law school exams at
the University of Minnesota found that it averaged a C+ grade, managing to
pass in all four subjects [36]. These outcomes suggest that LLMs hold
promising implications for both medical and legal educational fields [37, 38,
39].
## 3 Methodology
In this study, we evaluated seven general-purpose LLMs on three GMAT exams
using the zero-shot standard prompting method, as described in the following.
### 3.1 Models
#### OpenAI’s GPT Family
We tested three LLMs form OpenAI’s GPT family: GPT-3.5 Turbo, GPT-4, and GPT-4
Turbo. GPT-3.5-turbo, released on March 1st, 2023, is an improved version of
GPT-3.5 and GPT-3 Davinci. GPT-3.5-turbo utilizes a transformer architecture
with a large number of parameters. It is trained on a diverse corpus of text
data, which enhances its ability to understand and generate human-like text.
This increase in scale empowers the model to tackle more complex tasks,
comprehend nuanced contexts, and ultimately achieve enhanced performance [40].
Training data sourced from diverse materials including books, articles, web
pages, and more provides GPT-3.5-turbo broad exposure to language and contexts
available across the internet. This extensive and varied training corpus
allows the model to develop a multifaceted understanding of language,
enhancing its capacity for remarkably human-like text generation. The last
update to GPT-3.5-turbo was in September 2021, and it currently powers the
freely available version of ChatGPT.
Initially launched on March 14, 2023, GPT-4 has been made accessible to the
public through the paid chatbot service, ChatGPT Plus, and also via OpenAI’s
API. As a transformer-based model, GPT-4 employs a two-step training process.
The initial pre-training phase utilizes both public data and data sourced from
third-party providers to predict the subsequent token. Following this, the
model undergoes a fine-tuning process, leveraging reinforcement learning
feedback from both humans and AI to ensure alignment with human values and
policy compliance [41].
GPT-4 Turbo is the latest version of GPT and appears to be the most advanced
AI model currently available in the market. According to the OpenAI’s website,
this enhanced version boasts increased capabilities, possesses knowledge up to
April 2023, and features an expanded context window of 128k, allowing it to
process the equivalent of 300 pages of text in one prompt. Furthermore, it
offers a cost reduction, with input tokens being 3 times more affordable and
output tokens 2 times more affordable than those of the original GPT-4 model.
Additionally, this model can generate up to 4096 output tokens [42].
While both GPT-4 and GPT-4 Turbo possess multi-modal functionalities [13, 43],
our research focuses exclusively on the text-based versions of these models.
Henceforth, any reference to GPT-4 or GPT-4 Turbo within this document will
specifically denote the text-only versions without visual processing features.
#### Anthropic’s Claude Family
Anthropic, the firm responsible for Claude, was established in 2021 by a team
of former OpenAI employees who contributed to the development of OpenAI’s
GPT-2 and GPT-3 models. In early 2023, after conducting a closed alpha with a
select group of commercial partners, Claude’s model was incorporated into
products such as Notion AI, Quora’s Poe, and DuckDuckGo’s DuckAssist. In March
2023, Claude expanded its API access to a broader range of businesses.
Subsequently, in July 2023, it launched its chatbot to the public, coinciding
with the release of the Claude 2 model. While Claude 2 may not yet match the
capabilities of GPT-4, it is rapidly advancing and consistently outperforms
most other AI models in standardized tests. Claude 2 distinguishes itself with
its capacity to manage up to 100K tokens per prompt, equivalent to
approximately 75,000 words. This is a twelve-fold increase compared to the
standard amount offered by GPT-4. Also, while GPT’s knowledge cutoff is
September 2021, Claude 2 benefits from training data extending up to early
2023[8]. Anthropic unveiled Claude 2.1 on November 23, 2023, marking the
newest update to their LLM lineup. As stated on their website, this latest
version brings enhancements crucial for enterprise applications, such as a
top-tier context window capable of handling 200K tokens, marked improvements
in reducing instances of model-generated inaccuracies, and system prompts.
Additionally, Claude 2.1 introduces a beta feature known as ’tool use.’
Alongside these advancements, Anthropic has revised their pricing structure to
increase cost-effectiveness across all their model offerings [44]. Both Clude
2.0, and Claude 2.1 are accessible through the claude.ai, Anthropic’s LLM-
based chatbot.
#### Google’s LLM Families
The Pathways Language Model (PaLM) is a LLM with 540 billion parameters,
constructed using the Transformer architecture and developed by Google AI. The
subsequent iteration, PaLM 2, was released in late 2022 and employs dynamic
pathway selection during inference to determine the most suitable pathway for
a given input [45]. Bard, Google’s conversational AI agent introduced in March
2023, was initially based on the LaMDA family of LLMs and later integrated
PaLM 2[7]. The latest generation of Google’s LLMs is the Gemini Family.
Developed from the Transformer architecture, Gemini models incorporate
architectural improvements and optimization techniques for stable, scalable
training and efficient inference on Google’s Tensor Processing Units. These
models are capable of handling a context length of 32k, employing efficient
attention mechanisms such as multi-query attention([46]). The first release,
Gemini 1.0, is available in three main sizes—Ultra, Pro, and Nano—to
accommodate a variety of applications. Currently, Gemini 1.0 Pro, a cost and
latency-optimized version of Gemini 1.0, has been integrated into the new
Bard. Google has announced plans to make Gemini 1.0 Ultra accessible through
Bard Advanced in early 2024 [47]. For this study, we analyzed data from the
legacy Bard tested between July 16 and October 2, 2023, and results obtained
from Bard after December 6, 2023, for Gemini 1.0 Pro. While Gemini models
offer multi-modal capabilities [46], our research is concentrated on the text-
only variant of Gemini 1.0 as implemented in the new Bard. Moving forward, any
mention of Gemini 1.0 Pro in this paper will refer specifically to the text-
only iteration, excluding visual processing features.
### 3.2 Graduate Management Admission Test (GMAT)
The GMAT exam serves as a differentiating factor for more than 100,000
candidates every year during the business school admissions process, with a
significant 90% of new MBA enrollments relying on a GMAT score [48]. The GMAT
exam is recognized as a reliable indicator of a student’s potential for
success. Each GMAT exam consists of 31 quantitative reasoning, 36 verbal
reasoning, 12 integrated reasoning problems, and an analytical writing essay.
The GMAT Total Score is derived from the Verbal and Quantitative sections of
the exam. This Total Score is based on the test taker’s calculated performance
before the scores for the Quantitative Reasoning and Verbal Reasoning sections
are assigned. This raw calculation is then converted into a number within the
Total Score range of 200 to 800. As only the performance in the quantitative
and verbal reasoning sections contributes to the total score, we focused
solely on the performance in these two sections, disregarding the performance
in Integrated Reasoning and Analytical Writing. The quantitative reasoning
section comprises two tasks: Problem Solving and Data Sufficiency. The verbal
reasoning section is also divided into three tasks: Reading Comprehension,
Critical Reasoning, and Sentence Correction.
In this study, we deployed seven different LLMs—GPT 3.5-turbo, GPT-4, GPT-4
Turbo, Claude2, Claude 2.1, PaLM 2, and Gemini 1.0 Pro—to generate responses
for three official GMAT practice exams. We emphasize that the materials used
in this study were officially acquired and sourced from the Graduate
Management Admission Council (GMAC), the authoritative body that oversees and
administers the GMAT examination process. Given that the GMAT exam employs an
adaptive testing format—wherein a test taker’s performance on prior questions
dictates the difficulty level of subsequent ones—we maintain that
administering the tests through the official GMAT exam website constitutes the
most authentic method for evaluating the performance of LLMs. This approach is
likely to provide the most accurate estimation of the AI’s proficiency and
competency. By leveraging the official testing platform, we can ensure that
the AI is subjected to the same dynamic adjustments in difficulty that human
test-takers face, thereby offering a fair and rigorous assessment of its
capabilities in a standardized context. Our choice of exams was strategic; to
mitigate the potential influence of publicly available free practice exams on
the results, we included premium practice exams among the three exam sets for
each model. These premium exams were not publicly accessible, thereby ensuring
a level of novelty and challenge for the LLMs.
### 3.3 Prompting
To establish a baseline for model performance and to ensure fair comparison
with human participants, we applied a simple zero-shot prompting technique for
the models being evaluated. Additionally, to reduce the potential for in-
context learning during chat-based interactions, we posed each question in a
separate chat window using LLM-based chatbots, rather than within a continuous
conversation thread. This precaution ensured that each model’s response was
independent and not influenced by previous interactions, providing a more
accurate measure of standalone performance. We standardized the prompt
template, as illustrated in Figure 1, customizing it for different problem
types, with a complete example shown in Figure 2. This approach maintained
prompt consistency across similar problems, allowing for automated model
responses with minimal human intervention, limited to entering the problem
content and prompt. In our study, we focused exclusively on the text-based
capabilities of LLMs. Consequently, for questions originally accompanied by
images, we converted them into descriptive math word problems (Figures 3 and
4). This translation process ensured that the essence of the visual
information was accurately captured and conveyed through text, allowing the
LLMs to process and respond appropriately.
Figure 1: The template employed for generating prompts for every multiple-
choice question. Elements shown in double braces are substituted with
question-specific values. Figure 2: An example of implementation of template
shown in from Figure 1. Figure 3: Original problem statement with image Figure
4: The prompt describes the problem shown in Figure 3, presented without an
accompanying image. The correct answer provided by GPT-3.5 Turbo is
highlighted in grey.
## 4 Results and Discussion
### 4.1 Performance of LLMs on the GMAT
Table 1 presents an overview of the average performance of LLMs on the GMAT
exam, both overall and across different tasks. It is worthy to note that the
overall average mentioned in the table is calculated by dividing the number of
correct answers by the total number of questions across all three exam sets
for each model.
Sections & Tasks | GPT-4 Turbo | GPT-4 | GPT-3.5 Turbo | Claude 2.1 | Claude 2 | Gemini Pro (New Bard) | PaLM 2 (Legacy Bard)
---|---|---|---|---|---|---|---
Quantitative Reasoning | 74.19 | 64.52 | 53.76 | 61.29 | 46.24 | 68.82 | 44.09
Data Sufficiency | 60.98 | 56.10 | 39.02 | 56.10 | 36.59 | 60 | 32.50
Problem Solving | 84.62 | 71.15 | 64.15 | 65.38 | 53.85 | 75.47 | 52.83
Verbal Reasoning | 94.44 | 82.41 | 61.11 | 81.48 | 72.22 | 72.22 | 56.48
Reading Comprehension | 100 | 97.44 | 70.73 | 100 | 97.44 | 87.50 | 79.49
Critical Reasoning | 96.30 | 81.48 | 55.56 | 75 | 64.29 | 66.67 | 50
Sentence Correction | 87.80 | 69.05 | 55.00 | 68.29 | 53.66 | 60.98 | 39.02
Overall Average | 85.07 | 74.13 | 57.71 | 72.14 | 60.20 | 70.65 | 50.75
Table 1: The comparison of model performance across various GMAT sections and
tasks reveals that GPT-4 Turbo outperforms the other models in all sections
and tasks.
Table 1 demonstrates that across the board, all models exhibit their highest
proficiency in the reading comprehension task. On the other hand, they
struggle the most with the data sufficiency task, which emerges as their
Achilles’ heel. For the remaining tasks—problem-solving, critical reasoning,
and sentence correction—the models’ performance falls within the average
range, indicating a balanced level of proficiency in these areas but without
the peaks observed in reading comprehension or the troughs seen in data
sufficiency.
#### Quantitative reasoning
When mathematical problems have unambiguous presentations and sequential
solutions, LLMs can apply their extensive computational power to reach correct
answers quickly and on vast scales unmatched by humans [See, 49]. However,
despite exceeding human performance on well-specified calculations, LLMs still
lack deeper comprehension of the mathematical concepts themselves. Instead,
these models depend fundamentally on pre-defined algorithms. As a result, they
fluctuate when problems require creative approaches, judgments akin to human
intuition, or the ability to integrate multiple abstract concepts. Vague or
complex multi-step problems strain LLMs even further.
#### Reading comprehension
Previous research has shown pre-trained language models are typically excel at
summarizing, classifying, and paraphrasing texts, demonstrating a keen ability
to identify the main ideas within passages [e.g., 50]. Therefore, their
consistently high performance in tackling reading comprehension problems is to
be expected. This proficiency can be attributed to their advanced language
processing capabilities, which allow them to effectively analyze and interpret
textual information. Consequently, they can accurately understand the context,
draw inferences, and provide precise responses to comprehension questions.
However, LLMs struggle with abstract or complex concepts that require a level
of reasoning or inference beyond the literal text [e.g., 51, 52].
#### Critical reasoning
LLMs are adept at pattern recognition [53] and inference-making [54], skills
that serve them well in answering critical reasoning questions focused on
identifying assumptions and assessing arguments. Yet, they can falter with
subtle or complex logical reasoning, especially when questions demand an
understanding of new evidence or hypothetical situations, as LLMs depend on
data-driven patterns rather than true understanding or practical knowledge.
#### Sentence correction
Ultimately, LLMs, pre-trained on an extensive set of grammar rules, are
proficient at identifying a broad spectrum of grammatical errors. However,
they often stall when errors require an understanding of context or language
nuances. Complex sentences, particularly those with multiple clauses or
unconventional structures, can pose challenges [55]. LLMs’ reliance on
programmed rules can lead to oversights in recognizing exceptions or
stylistically acceptable rule deviations. Additionally, LLMs sometimes cannot
grasp deeper meanings or implications of information and can struggle with
context, especially when it involves cultural, historical, or personal
references [56].
### 4.2 Performance of LLMs versus human candidates
Figure 5 illustrates the comparative performance of various AI agents,
representing different LLMs, against human participants in the GMAT exam. On
average, the LLMs attained a total score of 643.5 across all administered
exams, a score that would place them above approximately 63% of human
candidates. Notably, GPT-4 Turbo leads the pack, outshining all other AI
agents and the average human candidate, securing a spot within the top 1% of
test-takers based on the total exam score. Trailing GPT-4 Turbo, the other AI
agents—GPT-4, Claude 2.1, the new Bard (Gemini 1.0 Pro), Claude 2, GPT-3.5
Turbo, and the legacy Bard (PaLM 2)—ranked successively, surpassing 94%, 90%,
83.3%, 53%, 36.7%, and 12% of human test-takers, respectively. Echoing the
pattern observed in human candidates, all AI agents exhibited stronger results
in verbal reasoning over quantitative reasoning. The extent of this
performance gap between the two sections, however, showed variation among the
AI agents. Interestingly, the most recent versions of LLMs, such as the new
Bard (Gemini 1.0 Pro), Claude 2.1, and GPT-4 Turbo, closely emulated human
test-taker performance patterns. The discrepancy between their verbal and
quantitative scores was nearly identical to the average human candidate’s
spread. This alignment suggests that the latest LLMs are approaching human-
like balance in handling the diverse skill sets required by the GMAT. It is
important to note that the data regarding human test-takers is derived from a
sample of 282,098 individuals who sat for the GMAT between January 2020 and
December 2022, as reported in the GMAT score distribution [57].
Figure 5: This figure presents a comparative analysis of the average
performance across seven LLMs and human candidates in quantitative reasoning,
verbal reasoning, and total GMAT scores.
As depicted in Figure 6, AI agents’ performance on the GMAT exam positions
them as formidable contenders for admission into elite business schools.
According to recent admissions data from the top 50 business schools, GPT-4
Turbo stands a high chance of being accepted into all these esteemed
institutions. Remarkably, only seven business schools—Stanford Graduate School
of Business, The Wharton School of the University of Pennsylvania, Harvard
Business School, London Business School, Kellogg School of Management at
Northwestern University, HEC Paris, and Indian School of Business—boast
applicants with GMAT scores that exceed those of GPT-4 Turbo. This highlights
GPT-4 Turbo’s exceptional performance, which not only competes with but also
often exceeds that of the majority of human candidates at these prestigious
institutions. Moreover, GPT-4’s GMAT results align with or surpass the average
applicant scores at the majority of the top 50 business schools, implying a
strong likelihood of GPT-4 gaining admission to nearly all these programs. The
average GMAT scores of MBA classes at only the top 9 business schools exceed
GPT-4’s scores, underscoring its potential for acceptance. Claude 2.1 and the
new Bard, despite having nearly identical GMAT scores, perform above the
average of applicants to almost half of the top 50 business schools. This
indicates that these AI agents are competitive applicants for many top-tier
programs. Conversely, AI agents based on older LLM versions—Claude 2, GPT-3.5
Turbo, and legacy Bard—have a tiny chance of admission to the top 50 business
schools, as their average GMAT scores significantly trail behind the applicant
averages for these institutions.
Figure 6: This graph illustrates a side-by-side comparison of average GMAT
scores from seven AI models, denoted with solid lines for the latest models
and dashed lines for legacy versions, against the average, minimum, and
maximum GMAT scores recorded by students at the top 50 business schools,
according to the Financial Times 2023 rankings[58]. The data originates from
MBA class profiles publicly available on the schools’ official websites.
Inclusion criteria required schools to offer comprehensive information in
their profiles, with the compilation organized by ascending average GMAT
scores
### 4.3 Limitations of LLMs
To gain a more comprehensive understanding of the limitations of LLMs, it’s
crucial to conduct an in-depth analysis of their incorrect responses. Figures
7, 8, 9, and 10 offer a more granular perspective on the areas where LLMs
struggle. These figures specifically highlight their shortcomings in solving
problems in quantitative reasoning, reading comprehension, critical reasoning,
and sentence correction. By examining these areas of weakness, we can better
understand the challenges faced by LLMs and work towards improving their
performance in these specific domains.
A closer examination of quantitative reasoning problems answered incorrectly
by LLMs reveals 16 distinct error categories, each corresponding to specific
mathematical concepts. These categories include geometry, numbers and the
number line, sets, factors, multiples, divisibility and remainders, decimals,
fractions and percents, exponents, statistics and probability, rate, work and
mixture problems, ratio, proportions and estimations, counting methods,
inequalities, factoring and quadratic equations, algebraic expressions and
linear equations, properties of operations, sequences and series, and
functions. This classification provides a granular view of the mathematical
domains that challenge LLMs, offering insights for targeted improvement. For
example, latest versions of language model families have shown a reduction or
elimination of errors in certain categories. GPT-4 Turbo, in comparison to
earlier models, has achieved flawless performance in areas such as Ratio,
Proportions, and Estimation, and Properties of Operations. Similarly, Claude
2.1 flawlessly answers Counting Methods questions, and Gemini 1.0 Pro has a
perfect record in Sequences and Series, a category where other models still
falter. Overall, more than half of the errors made by LLMs are concentrated in
five categories: Geometry, Numbers and Number Line, Sets, Rate, Work, and
Mixture Problems, and Exponents. This pattern suggests that LLMs’ mistakes
likely arise from challenges of processing spatial, numerical, and logical
information through text. Geometry requires visualizing diagrams, which is
challenging for LLMs with text alone. Numerical concepts like absolute value
and ordering can be subtle and easily misinterpreted by LLMs. Set theory’s
logical operations, especially in complex sets, are difficult for LLMs to
process accurately. Rate and mixture problems’ multi-step ratios, as well as
the rules of exponentiation in exponent problems, often lead to errors when
LLMs fail to apply them correctly.
Reviewing the errors made by LLMs in reading comprehension, it is obvious that
they struggle with inference questions, which require the deduction of ideas
that are not explicitly stated in a passage. Additionally, they occasionally
have difficulty comprehending the main idea of passages. Similarly, incorrect
responses to critical reasoning problems, which present a challenge for LLMs,
can be categorized into eight distinct categories. These categories include
’Weaken the Argument,’ which entails identifying information that weakens the
author’s argument; ’Inference,’ which requires identifying something that must
be true based on the given information; ’Find the Assumption,’ which involves
identifying an assumption that is not explicitly stated; ’Explain a
Discrepancy,’ which requires identifying information that resolves an apparent
paradox in the argument; ’Evaluate the Argument,’ which involves identifying
information that would help determine the argument’s soundness; ’Strengthen
the Argument,’ which requires identifying information that supports the
author’s argument; ’Describe the Role,’ which involves identifying the roles
or labels of the underlined portions of the argument; and ’Find the Flaw,’
which requires identifying an illogical element in the argument. Among these
categories, questions falling under ’Weaken the Argument’ are the most
challenging for LLMs to answer, likely because it involves understanding the
author’s perspective and reasoning, as well as critically evaluating the
information presented. LLMs may find it difficult to recognize the weaknesses
in an argument or may struggle to differentiate between information that
strengthens or weakens the argument. Therefore, these types of questions pose
a particular challenge for LLMs in their critical reasoning skills.
Finally, the grammatical errors that LLMs sometimes fail to identify in
sentence correction questions can be grouped into 11 categories. These include
errors related to meaning, modifiers, subject-verb agreement, pronouns,
grammatical constructions, awkwardness or redundancies, verb forms, tenses,
comparison, parallelism, and idioms. Among these, the majority of incorrect
responses are due to the LLMs’ inability to detect errors in meaning. This
particular challenge may stem from the subtleties and variations of language
that require a deep understanding of context and intent. Errors in meaning can
often involve complex logical structures or subtle shifts in tone that LLMs
are not always equipped to handle. Modifiers, also, must be placed correctly
to avoid ambiguity, while subject-verb agreement requires careful attention to
ensure that subjects and verbs match in number. Pronouns must refer to the
appropriate noun, and grammatical constructions must be consistent and
logical. Awkwardness or redundancies in language can make sentences less clear
and concise, which LLMs might overlook. Verb forms and tenses must be used
correctly to convey the proper time frame of actions, while comparisons
require a balanced and accurate assessment of similarities or differences.
Parallelism is essential for maintaining a consistent structure in lists and
comparisons, and idioms must be used appropriately to convey the intended
meaning in a culturally specific context. In sum, the complexity of these
grammatical rules and the intricacies of their application occasionally makes
sentence correction a challenging area for LLMs.
Figure 7: A detailed categorization of LLM errors in Quantitative Reasoning
for targeted improvement Figure 8: A detailed categorization of LLM errors in
Reading Comprehension for targeted improvement Figure 9: A detailed
categorization of LLM errors in Critical Reasoning for targeted improvement
Figure 10: A detailed categorization of LLM errors in Sentence Correction for
targeted improvement
## 5 Study Limitations and Future Directions
Our paper demonstrates the remarkable capability of general purpose LLMs in
responding to quantitative and verbal reasoning questions on the GMAT exam. In
the following, we will discuss the limitations and possible expansions of our
research for the future.
### 5.1 Prompting Method
Our objective in this study is to establish a benchmark for the foundational
performance of LLMs when tasked with answering GMAT multiple-choice questions.
We aim to achieve this using a straightforward methodology, deliberately
avoiding the use of more intricate techniques such as chain-of-thought (CoT)
prompting [59], Retrieval Augmented Generation (RAG) [60] or prompt chaining
strategies [61]. Previous research has demonstrated that these advanced
methods significantly improve the capabilities of LLMs when addressing complex
queries across various fields [62, 63]. Moreover, the potential exists for the
development of new prompting methods specifically tailored to enhance the
performance of advanced language models like GPT-4 Turbo, which have not yet
been fully explored or identified. Given the complexity and adaptability of
such models, it stands to reason that a deliberate and systematic examination
of various prompting techniques, coupled with strategic fine-tuning, has the
capacity to yield significant gains in performance outcomes. This could
involve experimenting with different types of prompts, varying the complexity
and structure of the language used, or even customizing prompts to align more
closely with the model’s training data. While the pursuit of achieving the
highest possible scores on benchmarks is a valid endeavor, it is not the sole
focus of this paper. Our aim extends beyond simply pushing the limits of
benchmark performance to include a broader understanding of model capabilities
and limitations. Therefore, we acknowledge that the in-depth exploration of
these advanced prompting strategies represents a promising avenue for future
studies.
### 5.2 Scope of the study
This paper’s benchmarking parts are focused on assessing multiple-choice
questions from quantitative reasoning and verbal reasoning sections of the
GMAT, which form a significant but not complete portion of this exam. As
explained in the methodology section, the GMAT includes 12 integrated
reasoning (IR) and one essay section as well that are scored independently and
their scores will not affect the exam’s total score, so we did not include
quantitative metrics for these two sections in our benchmarks. Consequently,
the performance metrics we report may not fully reflect the LLMs capabilities
in a real GMAT exam setting. Furthermore, while a study of LLMs on the GMAT
can offer important data on certain cognitive abilities relevant to business
education, it is not sufficient to generalize its findings to the entire field
of business education without additional research that considers the full
spectrum of skills and learning experiences that business programs aim to
develop. Business education encompasses case studies, real-world problem-
solving, interpersonal skills, and ethical decision-making, which may not be
adequately captured by LLMs’ performance on standardized tests. Additionally,
success on the GMAT does not necessarily equate to success in business school
or in business practice. The ability to transfer and apply test-taking skills
to real-world scenarios is a critical aspect of business education that may
not be reflected in the study’s scope. Moreover, business schools often use a
holistic approach to evaluate candidates, considering work experience,
leadership potential, and other personal attributes alongside test scores. A
study focused on GMAT performance alone may not account for these broader
evaluative criteria.
### 5.3 Memorization
Table 2 presents the performance of various models on both free and premium
GMAT exams. It’s important to note that accessing free exams requires users to
create an account on the official GMAT exam website. Furthermore, to access
premium exams, users must purchase them. When comparing the outcomes of free
versus premium exams across all models, we observe no significant difference
in performance. This finding, coupled with the fact that our GMAT materials
come from official sources requiring login or payment for premium content,
suggests that such content likely wasn’t part of the LLMs’ training data. Even
if some overlap or contamination exists, it doesn’t appear to notably enhance
the LLMs’ GMAT performance. OpenAI’s research supports this, indicating that
despite some contamination in publicly available benchmarks, it hasn’t led to
significant performance discrepancies between contaminated and uncontaminated
samples in their assessments [13].
GMAT | GPT-4 Turbo | GPT-4 | GPT-3.5 Turbo | Claude 2.1 | Claude 2 | Gemini Pro (New Bard) | PaLM 2 (Legacy Bard)
---|---|---|---|---|---|---|---
Free Exam 1 | 88.06 | 74.63 | 61.19 | 73.13 | 62.69 | 58.21 | 43.28
Free Exam 2 | 79.10 | 71.64 | 49.25 | 73.13 | 59.70 | 77.61 | 58.21
Premium Exam | 88.06 | 76.12 | 62.69 | 70.15 | 58.21 | 76.12 | 50.75
Overall Average | 85.07 | 74.13 | 58.21 | 72.14 | 60.20 | 70.65 | 50.75
Table 2: Comparison of performance of models on the GMAT exams shows that
GPT-4 Turbo significantly surpasses the other models.
## 6 AI as Tutoring Assistant: A Case Study
LLMs can act as tutoring assistants, simplifying complex concepts for both
instructors and students. They can guide students through homework by
fostering critical thinking instead of just providing answers. LLMs can
generate practice questions, offer assignment feedback, and create tailored
study plans. They’re also adept at simulating exams, tracking progress, and
suggesting targeted practice areas, especially useful for language exams that
focus on vocabulary and grammar. Additionally, LLMs help with essay writing
and provide motivational support. As AI becomes more integrated into
education, it offers interactive and personalized learning experiences.
Engaging LLMs in interactive sessions can reveal their educational potential
and practical applications. Our study, which references the interactive
session described in [64], illustrates how a dialogue initiated by a single
critical reasoning question can showcase an AI model’s educational
capabilities. Using GPT-4 Turbo, we simulate a conversation between the model
and a student preparing for the GMAT, demonstrating the model’s ability to
correctly answer questions (Figure 11), explain the reasoning behind answers
(Figure 12), use theory-of-mind to hypothesize about errors (Figure 13),
personalize instruction (Figure 14), and engage in counterfactual reasoning by
modifying a critical reasoning problem to help the candidate consider
different outcomes (Figure 15). However, it is essential to verify the
accuracy of information generated in such interactions and in real-world
applications with expert review to ensure reliability. This investigation aims
to highlight AI’s practical applications in education and its potential to
facilitate personalized learning journeys.
Figure 11: GPT-4 Turbo selects the correct option in response to a critical
reasoning problem. Figure 12: GPT-4 Turbo evaluates a candidate’s response to
the question and explains the rationale for the correct response. Figure 13:
GPT-4 Turbo attempts to follow the candidate’s line of reasoning to identify
errors in his thought process. Figure 14: GPT-4 Turbo adjusts its tutoring
approach for personalized instruction. Figure 15: GPT-4 Turbo revises the
argument in the problem statement in order to construct a counterfactual
scenario.
## 7 Final Thoughts
The overall performance of Large Language Models (LLMs) on GMAT exam
demonstrates significant potential to revolutionize education and tutoring in
particular, while also highlighting certain challenges and limitations. We
anticipate that LLMs will become essential tools for students in their exam
preparation and for teachers in developing their pedagogical materials.
Additionally, we expect these models to significantly enhance the process of
standardized examinations. These insights suggest a rapidly evolving landscape
where AI models are not only becoming viable candidates for high-level
academic programs but are also setting new benchmarks for human applicants to
aspire to. The implications for the future of standardized testing and
admissions are profound, as AI continues to redefine the limits of performance
and potential. Nevertheless, there are some major concerns regarding the
deployment of LLMs in educational settings, specifically within the domain of
business education, that warrant careful consideration.
#### Social impacts.
One of the major concerns is that the use of LLMs in education could lead to
increased inequality [65]. Access to LLMs requires a stable internet
connection and devices capable of running such models, which may not be
available to all students. Therefore, students with access to these
technologies might gain unfair advantages over those who do not, potentially
exacerbating existing educational inequalities. Also, several studies indicate
that LLMs may perpetuate and amplify existing biases present in their training
data, potentially reinforcing stereotypes and contributing to inequality in
educational outcomes. Moreover, while research shows that face-to-face
interaction plays a crucial role in providing a sustainable learning
experience in business education [66], overreliance on LLMs for learning could
lead to a decrease in face-to-face interactions and the valuable social
aspects of learning, such as collaboration and discussion. Furthermore, the
adoption of LLMs could lead to concerns about job displacement [67], as AI
might be seen as a replacement for human teachers and tutors.
#### Risk of misinformation.
As our findings in this study showed, LLMs may generate incorrect or
misleading information, which could lead to confusion or the learning of
inaccurate material. As a result, students may inadvertently learn and
propagate these inaccuracies, leading to a spread of misinformation.
#### Impacts on personal development.
LLMs might provide answers too readily, which could discourage students from
developing their own critical thinking, problem-solving skills, and also
interpersonal skills. As a result, students might become dependent on AI,
leading to a decrease in self-efficacy and confidence in their abilities to
learn and solve problems independently.
#### Ethical considerations.
The integration of LLMs into education involves the collection and processing
of student data, raising concerns about privacy, data security, and the
potential misuse of personal information. Also, there may be concerns about
the use of copyrighted material when LLMs generate content or examples for
study purposes. Additionally, there is a risk that students might use LLMs to
complete assignments dishonestly, leading to challenges in ensuring academic
integrity and authentic assessment of student learning.
To mitigate these concerns, it’s important to use LLMs as a complement to
traditional educational methods and human interaction in business education,
rather than as a standalone solution. Additionally, ongoing monitoring,
evaluation, and adjustment of LLMs in educational settings are necessary to
ensure they are used effectively and ethically and to ensure that students are
developing their own critical thinking and problem-solving skills, rather than
becoming overly reliant on the technology. Furthermore, to regulate and
monitor the ethical use of AI in education, it is essential to establish
frameworks, guidelines, and oversight bodies.
Finally, while LLMs have demonstrated their implications in a wide range of
applications, they have several inherent limitations. Current LLMs, trained on
vast internet text sources like Reddit and Twitter, have versatile
capabilities like scriptwriting, translation, and recipe generation, but they
often stray off-topic, fabricate information, and show fluidity in
perspectives. Furthermore, while highly capable in strictly defined
calculations, current limitations persist in mathematical ambiguity and
complexity intrinsic to human cognition. Extending their precise computational
strengths while approximating human understanding and fluidity thus remains an
elusive frontier. LLMs face challenges with abstract or intricate concepts
that demand reasoning beyond the literal text. Their performance dips when
tasks require creative problem-solving, intuitive judgment, or comprehension
of context and linguistic nuances. Additionally, they may inadvertently change
the original meaning of a sentence due to their limited understanding of the
writer’s intent or the deeper connotations of the information. OpenAI tries to
mitigate some of these issues by fine-tuning LLMs with human feedback and
reinforcement learning [68], aiming for helpful, truthful, and harmless
outputs, though these goals can be subjective and challenging to define. The
statistical models of crowdsourcing, introduced by Phil Dawid and David Skene
in 1979 [69] and expanded upon in Cultural Consensus Theory [70] (CCT) by
Romney and colleagues, offer insights into assessing subjective judgments.
CCT, particularly, recognizes the lack of a singular consensus in diverse
opinions, a concept valuable for understanding and improving LLMs in handling
varied, complex information. This principle has led to further advancements,
including the fine-tuning of deep neural networks to align their outputs more
closely with consensus beliefs.[71]
## 8 Conclusion
In our study, we performed a comparative analysis of seven general-purpose
LLMs—GPT-4 Turbo, GPT-4, GPT-3.5 Turbo, Claude 2.1, Claude 2, Gemini 1.0 Pro,
and PaLM 2—focusing on their zero-shot performance in the GMAT’s quantitative
and verbal sections, which are crucial for global business school admissions.
GPT-4 Turbo stood out as the top performer, with the latest versions of the
other models also showing significant improvements over their predecessors. We
found that LLMs can achieve impressive results on the GMAT without the need
for complex prompting strategies, surpassing the average human test-taker.
This suggests that LLMs have inherent strengths that make them well-suited for
standardized testing. Our research provides a foundational understanding of
LLMs’ capabilities on the GMAT and suggests potential for further performance
optimization. Additionally, we showcased GPT-4 Turbo’s tutoring abilities,
such as explaining critical reasoning, assessing responses, using
metacognition to pinpoint student errors, and creating counterfactual
scenarios. We explored the broader implications of AI in educational settings,
suggesting that LLMs’ strong performance on the GMAT indicates their potential
value in business education and as a support tool for business professionals.
However, the potential for errors and the complexities of real-world
application call for caution. It is essential to carefully develop, evaluate,
and refine AI use, and to continue technological advancements to maximize the
benefits and minimize the risks of AI in education.
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## Appendix A Sample of Input/Output for various GMAT tasks
Figure A.1: Sample of the user’s input and the model’s response for a problem
solving task
Figure A.2: Sample of the user’s input and the model’s response for a data
sufficiency task
Figure A.3: Sample of the user’s input and the model’s response for a reading
comprehension task
Figure A.4: Sample of the user’s input and the model’s response for a critical
reasoning task Figure A.5: Sample of the user’s input and the model’s
response for a sentence correction task
|
claim (and it remains to prove) that the adiabatic limit of the eta-invariant
for the spin Dirac operator is
$\lim_{\varepsilon\to 0}\tfrac{1}{2}\eta_{\varepsilon}=\frac{1}{24}[Y]^{2},$
(5.3)
i.e. the limit of the eta-invariants is the opposite of the local boundary
term when $\beta=1$, which indeed it should be since in that case the metric
is smooth across $x=0$.
Although other derivations of the adiabatic eta invariant exist [27], we
prefer to give on here which we find intuitive and which fits nicely with
arguments above. To this end, we consider $N$, a disc bundle over a smooth
manifold $Y$, and we assume $N$ is spin. We will show below that $N$ admits a
positive scalar curvature metric. Thus, given a spin structure and metric, the
index of $\eth$ vanishes on $N$. If we furthermore note that $N$ is
diffeomorphic to $[0,1)_{x}\times X$ where $X$ is a circle bundle over $Y$,
and let $N^{\varepsilon}=[0,\varepsilon)_{x}\times X$, we may consider metrics
$g=dx^{2}+f^{2}(x)k+h,$ (5.4)
where $h$ is the pullback of a metric on $Y$, $k\in
Sym^{0,2}(N^{\varepsilon})$ $x$ and $dx$-independent and restrics to a
Riemannian metric on the fibers of $X$. We assume $f$ is smooth across $x=0$
with $f(x)=x+O(x^{2})$ which implies that $g$ is smooth on $N$. Using the
computation of the connection above, with respect to the orthonormal basis,
$X_{i},\frac{1}{f}U,\partial_{x}$ the connection one form of $g$ is
$\begin{split}\omega&=\left(\begin{array}[]{c|c|c}\widetilde{\omega}_{\widetilde{h}}-f^{2}\frac{1}{2}g^{\partial}\mathcal{R}&-fg^{\partial}\left(\widehat{\@slowromancap
ii@}+\frac{1}{2}\widehat{\mathcal{R}}\right)&0\\\ \hline\cr
fg^{\partial}\left(\widehat{\@slowromancap
ii@}+\frac{1}{2}\widehat{\mathcal{R}}\right)&0&f^{\prime}U^{\sharp}\\\
\hline\cr 0&-f^{\prime}U^{\sharp}&0\end{array}\right)\end{split}$ (5.5)
where $g^{\partial}=k+h$ is the metric on the circle bundle $X$. We will take
$f(x)=f_{\varepsilon}(x)=x\chi_{\varepsilon}(x),$
where $\chi$ is a smooth positive function that is monotone decreasing with
$\chi(x)=1$ for $x\leq 1/3$ and $\chi(x)=\beta$ for $x\geq 2/3$. Then
$f=f^{\prime}=f^{\prime\prime}=O(1/\varepsilon)$, and using
$\Omega=d\omega+\omega\wedge\omega$, we see that
$\widehat{A}_{g}=FdVol_{g}$
where $F$ is a function that is $O(1/\varepsilon)$. Since
$Vol(N_{\varepsilon})=O(\varepsilon^{2})$,
$\int_{N_{\varepsilon}}\widehat{A}_{\varepsilon}=-\frac{1}{24}\int_{N_{\varepsilon}}p_{1}\to
0\mbox{ as }\varepsilon\to 0.$
Since the index of the Dirac operator vanishes on $N^{\varepsilon}$, applying
the APS formula gives
$\begin{split}0=-\frac{1}{24}\int_{N^{\varepsilon}}p_{1}+\int_{\partial
N^{\varepsilon}}Tp_{1}(\nabla,\nabla^{\operatorname{pt}})-\tfrac{1}{2}\eta(\partial
N^{\varepsilon}),\end{split}$ (5.6)
where $\nabla^{pt}$ is as in (3), and thus the limit of the trangression forms
is exactly as computed above. Thus by taking the $\varepsilon\to 0$ limit we
obtain (5.3).
To prove part 1) it remains to prove the existence of a positive scalar
curvature metric on $N$. To this end we take the metric $g$ as in (5.4) on
$N^{\varepsilon}$ now with
$f(x)=f_{\delta}(x)=\delta\sin(x/\delta).$ (5.7)
Note that $f=O(\varepsilon),f^{\prime}=O(\varepsilon/\delta)$. Then curvature
equals
$\begin{split}\Omega&=d\omega+\omega\wedge\omega\\\
&=\left(\begin{array}[]{c|c|c}\widetilde{\Omega}_{\widetilde{h}}&0&0\\\
\hline\cr 0&0&f^{\prime\prime}dx\wedge U^{\sharp}\\\ \hline\cr
0&-f^{\prime\prime}dx\wedge
U^{\sharp}&0\end{array}\right)+O(\varepsilon)+O(\varepsilon/\delta).\end{split}$
(5.8)
Denoting our orthonormal basis by $e_{i},i=1,\dots,n$ and taking traces gives
$scal_{g}=\delta^{ik}\delta^{jl}\Omega_{ij}(e_{k},e_{l})=scal_{h}+\frac{2}{\delta^{2}}+O(\varepsilon/\delta),$
and thus taking $\varepsilon/\delta=1$ and $\delta$ small gives a positive
scalar curvature metric.
2) Since $X$ is a smooth manifold we can use Novikov additivity of the
signature to decompose the signature as
$\operatorname{sgn}(X)=\operatorname{sgn}(X\setminus
M_{\varepsilon})+\operatorname{sgn}(M_{\varepsilon}).$
Identifying $X\setminus M_{\varepsilon}$ with a disk bundle over $Y$ we have
from [25, Pg. 314] that
$\operatorname{sgn}(X\setminus
M_{\varepsilon})=\operatorname{sgn}\left(\int_{Y}e\right),$
i.e., the signature is the sign of the self-intersection number of $Y$ in $X.$
In fact this is a simple exercise using the Thom isomorphism theorem.
The Atiyah-Patodi-Singer index theorem for the signature of $M_{\varepsilon}$
yields
$\operatorname{sgn}(M_{\varepsilon})=\frac{1}{3}\int_{M_{\varepsilon}}p_{1}(\nabla)+\frac{1}{3}\int_{\partial
M_{\varepsilon}}Tp_{1}(\nabla,\nabla^{\operatorname{pt}})-\eta^{\operatorname{even}}_{\varepsilon}$
where $\eta^{\operatorname{even}}_{\varepsilon}$ is the eta-invariant of the
boundary signature operator restricted to forms of even degree. As
$\varepsilon\to 0,$ the eta invariant is undergoing adiabatic degeneration and
its limit is computed in [27, Theorem 3.2],
$\lim_{\varepsilon\to 0}=-\int_{Y}L(TY)(\coth
e-e^{-1})+\operatorname{sgn}\left(B_{e}\right)$
where $B_{e}$ is the bilinear form on $H^{0}(Y)$ given by $H^{0}(Y)\ni
c,c^{\prime}\mapsto cc^{\prime}\langle e,Y\rangle\in\mathbb{R},$ i.e., it is
again the sign of the self-intersection of $Y.$ (In comparing with [27] note
that the orientation of $\partial M_{\varepsilon}$ is the opposite of the
orientation of the spherical normal bundle of $Y$ in $X,$ and so
$\operatorname{sgn}(B_{e})=-\operatorname{sgn}(X\setminus M_{\varepsilon}).$)
The only term in $L(TY)(\coth e-e^{-1})$ of degree two is $\tfrac{1}{3}e,$ and
hence
$\operatorname{sgn}(X)=\frac{1}{3}\int_{X}p_{1}+\frac{1}{3}[Y]^{2}+\frac{1}{3}\left(-\beta^{2}[Y]^{2}\right)$
as required. (Note that we could also argue as in the Dirac case to compute
the limit of the eta invariants.)
∎
## 6\. Positive scalar curvature metrics
In this short section, we prove Theorem 3 following [23]. We recall the
statement of the theorem for the benefit of the reader:
###### Theorem 6.1.
Let $(M,g)$ be a spin space with an incomplete edge metric.
a) If the scalar curvature of $g$ is non-negative in a neighborhood of
$\partial M$ then the geometric Witt assumption (Assumption 2.1) holds.
b) If the scalar curvature of $g$ is non-negative on all of $M,$ and positive
somewhere, then $\operatorname{Ind}(\eth)=0.$
###### Proof.
a) Taking traces in (1.11), the scalar curvature $R_{g}$ satisfies
$R_{g}=R_{cone}+\mathcal{O}(1),$
where $R_{cone}$ is the scalar curvature of the cone with metric
$dx^{2}+x^{2}g_{N/Y}\rvert_{\langle\partial_{x}\rangle\oplus\tfrac{1}{x}TN/Y}$,
as in (1.7). On the other hand, by [23, Sect. 4], the scalar curvature of an
exact cone $C(Z)$ is equal to $x^{-2}(R_{Z}-\dim(Z)(\dim(Z)-1))$, where
$R_{Z}$ is the scalar curvature of $Z$. Thus $R_{g}\geq 0$ implies that
$R_{Z}\geq 0$, which by [23, Lemma 3.5] shows that Assumption 2.1 holds.
b) First off, by Theorem 1, $\eth$ is essentially self-adjoint. That is, the
graph closure of $\eth$ on $C^{\infty}_{comp}(M)$ is self-adjoint, with domain
$\mathcal{D}$ from Theorem 1, and furthermore by the Main Theorem its index
satisfies (4).
From the Lichnerowicz formula [10],
$\eth^{*}\eth=\nabla^{*}\nabla+R/4$
where $R$ is the scalar curvature. Thus, for every $\phi\in
C^{\infty}_{comp}(M)$,
$\left\|\eth\phi\right\|_{L^{2}}=\left\|\nabla\phi\right\|_{L^{2}}+\langle
R\phi,\phi\rangle_{L^{2}}.$ We conclude that for all $\phi\in
C^{\infty}_{comp}(M),$
$\left\|\eth\phi\right\|_{L^{2}}\geq\left\|\eth\phi\right\|_{L^{2}}-\langle
R\phi,\phi\rangle_{L^{2}}\geq\left\|\nabla\phi\right\|_{L^{2}}\geq 0.$ (6.1)
This implies in particular that
$\mathcal{D}_{min}(\eth)=\mathcal{D}\subset\mathcal{D}_{min}(\nabla)$, where
we recall that $\mathcal{D}_{min}(P)$ refers to the graph closure of the
operator $P$ with domain $C^{\infty}_{comp}(M)$. We claim that the index of
the operator
$\eth\colon\mathcal{D}^{+}\longrightarrow L^{2}(M;\mathcal{S}^{-}).$
vanishes, so by formula (4), Theorem 3(b) holds. In fact, the kernel of $\eth$
on $\mathcal{D}$ consists only of the zero vector, since if
$\phi\in\mathcal{D}$ has $\eth\phi=0$, then since (6.1) holds on
$\mathcal{D}$, $\nabla\phi=0$ also. By the Lichnerowicz formula again,
$R\phi=0$, but since by assumption $R$ is not identically zero, $\phi$ must
vanish somewhere and by virtue of its being parallel, $\phi\equiv 0$. ∎
## 7\. Appendix
In this appendix we prove Claim 4.8 by using standard bounds on modified
Bessel functions to prove the sup norm bound (4.9): for each $\mu$ with
$\left\lvert\mu\right\rvert>1/2$, and all $\left\lvert\eta\right\rvert$,
$\left\|\mathcal{N}_{\mu,\left\lvert\eta\right\rvert}-\mathcal{N}^{APS}_{\mu,\left\lvert\eta\right\rvert}\right\|<1-\delta,$
Among references for modified Bessel functions we recall [5, 8, 9, 49].
To begin with, using the Wronskian equation (2.27), note that
$\operatorname{Tr}\mathcal{N}_{\mu,z}=\operatorname{Tr}\mathcal{N}^{APS}_{\mu,z}=1.$
Thus the difference $\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z}$ has two
equal eigenvalues and hence its norm is the square root of the determinant. We
now assume that $\mu\geq 1/2$, since the $\mu\leq-1/2$ case is treated the
same way. Using (2.27) again, we see that
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(\mu(I_{\mu-1/2}K_{\mu+1/2}-I_{\mu+1/2}K_{\mu-1/2})\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right),\end{split}$
(7.1)
and we want to show that for some $\delta>0$ independent of $\mu\geq 1/2$ and
$z\geq 0$,
$\begin{split}-1+\delta\leq\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})\leq
1-\delta.\end{split}$ (7.2)
To begin with, we prove that
$0\leq zI_{\nu}(z)K_{\nu}(z)\leq 1/2\quad\mbox{ for }\quad\nu\geq 1/2,z\geq
0.$ (7.3)
In fact, we claim that for $\nu\geq 1/2$, $zK_{\nu}(z)I_{\nu}(z)$ is monotone.
To see that this holds, differentiate
$\begin{split}(zK_{\nu}(z)I_{\nu}(z))^{\prime}&=K_{\nu}I_{\nu}+z(K^{\prime}_{\nu}I_{\nu}+K_{\nu}I^{\prime}_{\nu})=K_{\nu}I_{\nu}(1+\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}).\end{split}$
Thus we want to show that
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)})\geq-1$.
Using [9, Eqn. 5.1], for $\nu\geq 1/2$
$\left(\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}\right)^{\prime}+\left(\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}\right)^{\prime}\leq
0,$
so the quantity
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}$
is monotone decreasing. In fact, we claim that
$\frac{zK^{\prime}_{\nu}(z)}{K_{\nu}(z)}+\frac{zI^{\prime}_{\nu}(z)}{I_{\nu}(z)}\left\\{\begin{array}[]{cc}\to
0&\mbox{ as }z\to 0\\\ \to-1&\mbox{ as }z\to\infty.\end{array}\right.$
The limit as $z\to\infty$ can be seen using the large argument asymptotic
formulas from [1, Sect. 9.7], while the limit as $z\to 0$ follows from the
recurrence relations (2.27) and the small argument asymptotics in [1, Sect.
9.6]. Thus $zK_{\nu}(z)I_{\nu}(z)$ is monotone on the region under
consideration. Using the asymptotic formulas again shows that
$zK_{\nu}(z)I_{\nu}(z)\left\\{\begin{array}[]{cc}\to 0&\mbox{ as }z\to 0\\\
\to 1/2&\mbox{ as }z\to\infty.\end{array}\right.$
so (7.3) holds.
We can now show the upper bound in (7.2). Using the Wronskian relation in
(2.27), we write
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}+\frac{1}{2}\frac{\mu}{(\mu^{2}+z^{2})^{1/2}}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(-2\mu
I_{\mu+1/2}K_{\mu-1/2}\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right)\\\
&\leq\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right)\end{split}$
(7.4)
Now, if $\mu\geq 1$, by (7.3), the right hand side in the final inequality is
bounded by $1/2$, establishing the upper bound in (7.1) in this case (with
$\delta=1/2$). If $\mu\in[1/2,1]$, we use the following inequalities of Barciz
[9, Equations 2.3, 2.4]
$\frac{zI_{\nu}^{\prime}(z)}{I_{\nu}(z)}<\sqrt{z^{2}+\nu^{2}}\qquad\mbox{ and
}\qquad\frac{zK_{\nu}^{\prime}(z)}{K_{\nu}(z)}<-\sqrt{z^{2}+\nu^{2}},$
for $\nu\geq 0,z\geq 0$. Using these inequalities and the recurrence relation
(2.27) gives
$\frac{I_{\mu-1/2}}{I_{\mu+1/2}}<\frac{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}{z},\quad\frac{K_{\mu-1/2}}{K_{\mu+1/2}}<\frac{z}{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2},$
so continuing the inequality (7.4) gives
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&\leq\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(zI_{\mu+1/2}K_{\mu+1/2}\right)\times\\\
&\qquad\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).\end{split}$
(7.5)
One checks that for $1/2\leq\mu$, the fraction in the second line is monotone
decreasing in $z$, and thus by (7.3), for $z\geq 1$ the determinant is bounded
by
$\frac{1}{4}\left(1+\frac{\sqrt{1+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{1+(\mu-1/2)^{2}}+\mu-1/2}\right)\leq\frac{1}{4}(1+(1+\sqrt{2}))\leq
1-\delta$ (7.6)
where the middle bound is obtained by checking that the fraction on the left
is monotone decreasing in $\mu$ for $\mu\geq 1/2$ and equal to $1+\sqrt{2}$ at
$\mu=1/2$. Thus, we have established the upper bound in (7.2) in the region
$z\geq 1$. For $z\leq 1$, rewrite the bound in (7.5) as
$\begin{split}\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(I_{\mu+1/2}K_{\mu+1/2}\right)\times
z\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).\end{split}$
For $\mu\geq 1/2$, by [49], the function $I_{\mu+1/2}(z)K_{\mu+1/2}(z)$ is
monotone decreasing, and by the asymptotic formulas it is goes to $1/2$ as
$z\to 0$. Thus in $0\leq z\leq 1$ the determinant is bounded about by
$\frac{1}{4}\times
z\left(1+\frac{\sqrt{z^{2}+(\mu+1/2)^{2}}+\mu+1/2}{\sqrt{z^{2}+(\mu-1/2)^{2}}+\mu-1/2}\right).$
This function is monotone increasing in $z$ for $\mu\in[1/2,1]$, so the max is
obtained at $z=1$, i.e. it is bounded by the left hand side of (7.6), in
particular by $1-\delta$ for the same $\delta$. This establishes the upper
bound in (7.2)
Finally we establish the lower bound. First, we rewrite the determinant again,
this time using the Wronskian relation in the opposite direction to obtain
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&=-\frac{1}{2}-\frac{1}{2}\frac{\mu}{(\mu^{2}+z^{2})^{1/2}}+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}\left(2\mu
I_{\mu-1/2}K_{\mu+1/2}\right.\\\
&\qquad\left.+z(I_{\mu+1/2}K_{\mu+1/2}+I_{\mu-1/2}K_{\mu-1/2})\right).\end{split}$
(7.7)
Now, recalling that $zI_{\mu+1/2}(1)K_{\mu+1/2}(1)$ is monotone increasing,
using the asymptotic formulas [1, 9.7.7, 9.7.8] we see that
$I_{\mu+1/2}(1)K_{\mu+1/2}(1)\to\frac{1}{2(\mu+1/2)}$
as $\mu\to\infty$, we use the inequality [5, Eqn. 11], namely
$I_{\mu-1/2}(z)\geq\frac{\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}}{z}I_{\mu+1/2}(z).$
On the region $z\in[0,1]$,
$\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}\geq\delta_{0}>0$. Dropping the terms with
equal order in (7.7) then gives
$\begin{split}\operatorname{det}(\mathcal{N}_{\mu,z}-\mathcal{N}^{APS}_{\mu,z})&>-1+\frac{1}{2}\frac{z}{(\mu^{2}+z^{2})^{1/2}}2\mu
I_{\mu-1/2}K_{\mu+1/2}\\\
&\geq-1+\frac{1}{2}\frac{2\mu}{(\mu^{2}+z^{2})^{1/2}}I_{\mu+1/2}K_{\mu+1/2}(\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2})\\\
&\geq-1+\delta_{0}\frac{1}{2}\frac{\mu-1/2+(z^{2}+(\mu+3/2)^{2})^{1/2}}{(\mu^{2}+z^{2})^{1/2}}\\\
&\geq-1+\delta.\end{split}$
This completes the proof of (7.2).
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monthyeardate[],
# Formal FT-based Cause-Consequence Reliability Analysis using Theorem
Proving
Mohamed Abdelghany and Sofiène Tahar
Department of Electrical and Computer Engineering,
Concordia University, Montréal, QC, Canada
<EMAIL_ADDRESS>
TECHNICAL REPORT
(January 2021)
###### Abstract
Cause-consequence Diagram (CCD) is widely used as a deductive safety analysis
technique for decision-making at the critical-system design stage. This
approach models the causes of subsystem failures in a highly-critical system
and their potential consequences using Fault Tree (FT) and Event Tree (ET)
methods, which are well-known dependability modeling techniques. Paper-and-
pencil-based approaches and simulation tools, such as the Monte-Carlo
approach, are commonly used to carry out CCD analysis, but lack the ability to
rigorously verify essential system reliability properties. In this work, we
propose to use formal techniques based on theorem proving for the formal
modeling and step-analysis of CCDs to overcome the inaccuracies of the
simulation-based analysis and the error-proneness of informal reasoning by
mathematical proofs. In particular, we use the HOL4 theorem prover, which is a
computer-based mathematical reasoning tool. To this end, we developed a
formalization of CCDs in Higher-Order Logic (HOL), based on the algebraic
approach, using HOL4. We demonstrate the practical effectiveness of the
proposed CCD formalization by performing the formal reliability analysis of
the IEEE 39-bus electrical power network. Also, we formally determine the
Forced Outage Rate ($\mathcal{FOR}$) of the power generation units and the
network reliability index, i.e., System Average Interruption Duration Index
($\mathcal{SAIDI}$). To assess the accuracy of our proposed approach, we
compare our results with those obtained with MATLAB Monte-Carlo Simulation
(MCS) as well as other state-of-the-art approaches for subsystem-level
reliability analysis.
Keywords— Cause-Consequence Diagram, Event Tree, Fault Tree, Reliability
Analysis, Safety, Formal Methods, Theorem Proving, HOL4, Monte-Carlo, FMECA,
Electrical Power Network, FOR, SAIDI.
## 1 Introduction
Nowadays, in many safety-critical systems, which are prevalent, e.g. in smart
grids [1] and automotive industry [2], a catastrophic accident may happen due
to coincidence of sudden events and/or failures of specific subsystem
components. These undesirable accidents may result in loss of profits and
sometimes severe fatalities. Therefore, the central inquiry, in many critical-
systems, where safety is of the utmost importance, is to identify the possible
consequences given that one or more components could fail at a subsystem level
on the entire system. For that purpose, the main discipline for safety design
engineers is to perform a detailed Cause-Consequence Diagram (CCD) [3]
reliability analysis for identifying the subsystem events that prevent the
entire system from functioning as desired. This approach models the causes of
component failures and their consequences on the entire system using Fault
Tree (FT) [4] and Event Tree (ET) [5] dependability modeling techniques.
FTs mainly provide a graphical model for analyzing the factors causing a
system failure upon their occurrences. FTs are generally classified into two
categories Static Fault Trees (SFT) and Dynamic Fault Trees (DFT) [6]. SFTs
and DFTs allow safety-analysts to capture the static/dynamic failure
characteristics of systems in a very effective manner using logic-gates, such
as OR, AND, NOT, Priority-AND (PAND) and SPare (SP) [4]. However, the FT
technique is incapable of identifying the possible consequences resulting from
an undesirable failure on the entire system. ETs provide risk analysis with
all possible system-level operating states that can occur in the system, i.e.,
success and failure, so that one of these possible scenarios can occur [5].
However, both of these modeling techniques are limited to analyzing either a
critical-system failure or cascading dependencies of system-level components
only, respectively.
There exist some techniques that have been developed for subsystem-level
reliability analysis of safety-critical systems. For instance, Papadopoulos et
al. in [7] have developed a software tool called HiP-HOPS (Hierarchically
Performed Hazard Origin & Propagation Studies) [8] for subsystem-level failure
analysis to overcome classical manual failure analysis of complex systems and
prevent human errors. HiP-HOPS can automatically generate the subsystem-level
FT and perform Failure Modes, Effects, and Critically Analyses (FEMCA) from a
given system model, where each system component is associated with its failure
rate or failure probability [7]. Currently, HiP-HOPS lacks the modeling of
multi-state system components and also cannot provide generic mathematical
expressions that can be used to predict the reliability of a critical-system
based on any probabilistic distribution [9]. Similarly, Jahanian in [10] has
proposed a new technique called Failure Mode Reasoning (FMR) for identifying
and quantifying the failure modes for safety-critical systems at the subsystem
level. However, according to Jahanian [11], the soundness of the FMR approach
needs to be proven mathematically.
On the other hand, CCD analysis typically uses FTs to analyze failures at the
subsystem or component level combined with an ET diagram to integrate their
cascading failure dependencies at the system level. CCDs are categorized into
two general methods for the ET linking process with the FTs [12]: (1) Small ET
diagram and large subsystem-level FT; (2) Large ET diagram and small
subsystem-level FT. The former one with small ET and large subsystem-level FT
is the most commonly used for the probabilistic safety assessment of
industrial applications (e.g., in [13]). There are four main steps involved in
the CCD analysis [14]: (1) Component failure events: identify the causes of
each component failure associated with their different modes of operations;
(2) Construction of a complete CCD: construct a CCD model using its basic
blocks, i.e., Decision box, Consequence path and Consequence box; (3)
Reduction: removal of unnecessary decision boxes based on the system
functional behavior to obtain a minimal CCD; and lastly (4) Probabilistic
analysis: evaluating the probabilities of CCD paths describing the occurrence
of a sequence of events.
Traditionally, CCD subsystem-level reliability analysis is carried out by
using paper-and-pencil-based approaches to analyze safety-critical systems,
such as high-integrity protection systems (HIPS) [14] and nuclear power plants
[15], or using computer simulation tools based on Monte-Carlo approach, as in
[16]. A major limitation in both of the above approaches is the possibility of
introducing inaccuracies in the CCD analysis either due to human infallibility
or the approximation errors due to numerical methods and pseudo-random numbers
in the simulation tools. Moreover, simulation tools do not provide the
mathematical expressions that can be used to predict the reliability of a
given system based on any probabilistic distributions and failure rates.
A more safe way is to substitute the error-prone informal reasoning of CCD
analysis by formal generic mathematical proofs as per recommendations of
safety standards, such as IEC 61850 [17], EN 50128 [18] and ISO 26262 [19]. In
this work, we propose to use formal techniques based on theorem proving for
the formal reliability CCD analysis-based of safety-critical systems, which
provides us the ability to obtain a verified subsystem-level failure/operating
consequence expression. Theorem proving is a formal verification technique
[20], which is used for conducting the proof of mathematical theorems based on
a computerized proof tool. In particular, we use HOL4 [21], which is an
interactive theorem prover with the ability of verifying a wide range of
mathematical expressions constructed in higher-order logic (HOL). For this
purpose, we endeavor to formalize the above-mentioned four steps of CCD
analysis using HOL4 proof assistant. To demonstrate the practical
effectiveness of the proposed CCD formalization, we conduct the formal CCD
analysis of an IEEE 39-bus electrical power network system. Subsequently, we
formally determine a commonly used metric, namely Forced Outage Rate
($\mathcal{FOR}$), which determines the capacity outage or unavailability of
the power generation units [22]. Also, we evaluate the System Average
Interruption Duration Index ($\mathcal{SAIDI}$), which describes the average
duration of interruptions for each customer in a power network [22].
The main contributions of the work we describe in this report can be
summarized as follows:
* $\bullet$
Formalization of the CCD basic constructors, such as Decision box, Consequence
path and Consequence box, that can be used to build an arbitrary level of CCDs
* $\bullet$
Enabling the formal reduction of CCDs that can remove unnecessary decision
boxes from a given CCD model, a feature not available in other existing
approaches
* $\bullet$
Provide reasoning support for formal probabilistic analysis of scalable CCDs
consequence paths with new proposed mathematical formulations
* $\bullet$
Application on a real-world IEEE 39-bus electrical power network system and
verification of its reliability indexes $\mathcal{FOR}$ and $\mathcal{SAIDI}$
* $\bullet$
Development of a Standard Meta Language (SML) function that can numerically
compute reliability values from the verified expressions of $\mathcal{FOR}$
and $\mathcal{SAIDI}$
* $\bullet$
Comparison between our formal CCD reliability assessment with the
corresponding results obtained from MATLAB MCS and other notorious approaches
The rest of the report is organized as follows: In Section 2, we present the
related literature review. In Section 3, we describe the preliminaries to
facilitate the understanding of the rest of the report. Section 4 presents the
proposed formalization of CCD and its formal probabilistic properties. In
Section 5, we describe the formal CCD analysis of an electrical network system
and the evaluation of its reliability indices $\mathcal{FOR}$ and
$\mathcal{SAIDI}$. Lastly, Section 6 concludes the report.
## 2 Related Work
Only a few work have previously considered using formal techniques [20] to
model and analyze CCDs. For instance, Ortmeier et al. in [23] developed a
framework for Deductive Cause-Consequence Analysis (DCCA) using the SMV model
checker [24] to verify the CCD proof obligations. However, according to the
authors [23], there is a problem of showing the completeness of DCCA due to
the exponential growth of the number of proof obligations with complex systems
that need cumbersome proof efforts. To overcome above-mentioned limitations, a
more practical way is to verify generic mathematical formulations that can
perform $\mathcal{N}$-level CCD reliability analysis for real-world systems
within a sound environment. Higher-Order-Logic (HOL) [25] is a good candidate
formalism for achieving this goal.
Prior to our work, there were two notable projects for building frameworks to
formally analyze dependability models using HOL4 theorem proving [21]. For
instance, HOL4 has been previously used by Ahmad et al. in [26] to formalize
SFTs. The SFT formalization includes a new datatype consisting of AND, OR and
NOT FT gates [4] to analyze the factors causing a static system failure.
Furthermore, Elderhalli et al. in [27] had formalized DFTs in the HOL4 theorem
prover, which can be used to conduct formal dynamic failure analysis.
Similarly, we have defined in [28] a new EVENT_TREE datatype to model and
analyze all possible system-level success and failure relationships. All these
formalizations are basically required to formally analyze either a system
static/dynamic failure or cascading dependencies of system-level components
only, respectively. On the other hand, CCDs have the superior capability to
use SFTs/DFTs for analyzing the static/dynamic failures at the subsystem level
and analyze their cascading dependencies at the system-level using ETs. For
that purpose, in this work, we provide new formulations that can model
mathematically the graphical diagrams of CCDs and perform the subsystem-level
reliability analysis of highly-critical systems. Moreover, our proposed new
mathematics provides the modeling of multi-state system components and is
based on any given probabilistic distribution and failure rates, which makes
our proposed work the first of its kind. In order to check the correctness of
the proposed equations, we verified them within the sound environment of HOL4.
## 3 Preliminaries
In this section, we briefly summarize the fundamentals of the HOL4 theorem
proving approach and existing FT and ET formalizations in HOL4 to facilitate
the reader’s understanding of the rest of the report.
### 3.1 HOL4 Theorem Proving
Theorem proving [20] is one of the formal verification techniques that use a
computerized proof system for conducting the proof of mathematical theorems.
HOL4 [21] is an interactive theorem prover, which is capable of verifying a
wide range of safety-critical systems as well as mathematical expressions
constructed in HOL. In general, given a safety-critical system to be formally
analyzed, we first model its structure mathematically, then using the HOL4
theorem prover, several properties of the system can be verified based on this
mathematical model. The main characteristic of the HOL4 theorem prover is that
its core consists only of four axioms and eight inference rules. Any further
proof or theorem should be formally verified based on these axioms and rules
or based on previously proven theorems. This ensured the soundness of the
system model analysis, i.e., no wrong proof goal can be proved. Moreover,
since the system properties are proven mathematically within HOL4, no
approximation is involved in the analysis results. These features make HOL4
suitable for carrying out the CCD-based reliability analysis of safety-
critical systems that require sound verification results. Table 1 provides the
HOL4 symbols and functions that we will use in this report.
Table 1: HOL4 Symbols and Functions HOL4 Symbol | Standard | Meaning
---|---|---
{x $|$ P(x)} | {$\lambda x$. $P(x)$} | Set of all $x$ such that $P(x)$
h :: L | $cons$ | Add an element $h$ to a list L
MAP ($\lambda$x. f(x)) X | x $\in$ X $\to$ ($\lambda$x. f) | Function that maps each element x in the list X to f(x)
$\mathrm{L}_{1}$ $++$ $\mathrm{L}_{2}$ | $append$ | Joins lists $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ together
### 3.2 Probability Theory in HOL4
Measure space is defined mathematically as ($\Omega$, $\Sigma$, and $\mu$),
where $\Omega$ represents the sample space, $\Sigma$ represents a
$\sigma$-algebra of subsets of $\Omega$, and $\mu$ represents a measure with
the domain $\Sigma$. A probability space is a measure space ($\Omega$,
$\Sigma$, and Pr), where $\Omega$ is the complete sample space, $\Sigma$ is
the corresponding event space containing all the events of interest, and Pr is
the probability measure of the sample space as 1. The HOL4 theorem prover has
a rich library of probabilities, including the functions p_space, events, and
prob. Given a probability space p, these functions return the corresponding
$\Omega$, $\Sigma$, and Pr, respectively. The Cumulative Distribution Function
(CDF) is defined as the probability of the event where a random variable X has
a value less or equal to a value t, i.e., $\mathcal{P}(X\leq t)$. This
definition can be been formalized in HOL4 as [29]:
$\vdash$ CDF p X t = distribution p X {y | y $\leq$ t}
where the function distribution takes three inputs: (i) probability space p;
(ii) random variable X; and (iii) set of real numbers, then returns the
probability of the variable X acquiring all the values of the given set in
probability space p.
### 3.3 FT Formalization
Fault Tree (FT) analysis [4] is one of the commonly used reliability
assessment techniques for critical-systems. It mainly provides a schematic
diagram for analyzing undesired top events, which can cause complete system
failure upon their occurrence. An FT model is represented by logic-gates, like
OR, AND and NOT, where an OR gate models the failure of the output if any of
the input failure events occurs alone, while an AND gate models the failure of
the output if all of the input failure events occur at the same time, and
lastly a NOT gate models the complement of the input failure event. Ahmad et
al. [26] presented the FT formalization by defining a new datatype gate, in
HOL4 as:
Hol_datatype gate = AND of (gate list) |
OR of (gate list) |
NOT of (gate) |
atomic of (event)
The FT constructors AND and OR are recursive functions on gate-typed lists,
while the FT constructor NOT operates on a gate-type variable. A semantic
function is then defined over the gate datatype that can yield an FT diagram
as:
Definition 1:
$\vdash$ FTree p (atomic X) = X $\wedge$
FTree p (OR (h::t)) = FTree p h $\cup$ FTree p (OR t) $\wedge$
FTree p (AND (h::t)) = FTree p h $\cap$ FTree p (AND t) $\wedge$
FTree p (NOT X) = p_space p DIFF FTree p X
The function FTree takes an event X, identified by a type constructor atomic,
and returns the given event X. If the function FTree takes a list of type
gate, identified by a type constructor OR, then it returns the union of all
elements after applying the function FTree on each element of the given list.
Similarly, if the function FTree takes a list of type gate, identified by a
type constructor AND, then it performs the intersection of all elements after
applying the function FTree on each element of the given list. For the NOT
type constructor, the function FTree returns the complement of the failure
event obtained from the function FTree.
Table 2: FT HOL4 Probabilistic Theorems FT Gate | Probabilistic Theorem
---|---
| prob p (FTree p (AND $\mathrm{F}_{\mathcal{N}}$)) = $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{N}}$)
| prob p (FTree p (OR $\mathrm{F}_{\mathcal{N}}$)) = 1 - $\prod$ (PROB_LIST p
(COMPL_LIST p $\mathrm{F}_{\mathcal{N}}$))
The formal verification in HOL4 for the failure probabilistic expressions of
the above-mentioned FT gates is presented in Table 2 [26]. These expressions
are verified under the following constrains: (a) $F_{\mathcal{N}}$ $\in$
events p ensures that all associated failure events in the given list
$F_{\mathcal{N}}$ are drawn from the events space p; (b) prob_space p ensures
that p is a valid probability space; and lastly (c) MUTUAL_INDEP p
$F_{\mathcal{N}}$ ensures the independence of the failure events in the given
list $F_{\mathcal{N}}$. The function $\prod$ takes a list and returns the
product of the list elements while the function PROB_LIST returns a list of
probabilities associated with the elements of the list. The function
COMPL_LIST returns the complement of the given list elements.
### 3.4 ET Formalization
Event Tree (ET) [5] analysis is a widely used technique to enumerate all
possible combinations of system-level components failure and success states in
the form of a tree structure. An ET diagram starts by an initiating event
called Node and then all possible scenarios of an event that can occur in the
system are drawn as Branches. ETs were formally modeled by using a new
recursive datatype EVENT_TREE, in HOL4 as [28]:
Hol_datatype EVENT_TREE = ATOMIC of (event) |
NODE of (EVENT_TREE list) |
BRANCH of (event) (EVENT_TREE list)
The type constructors NODE and BRANCH are recursive functions on EVENT_TREE-
typed lists. A semantic function is then defined over the EVENT_TREE datatype
that can yield a corresponding ET diagram as:
Definition 2:
$\vdash$ ETREE (ATOMIC X) = X $\wedge$
ETREE (NODE (h::L)) = ETREE h $\cup$ (ETREE (NODE L)) $\wedge$
ETREE (BRANCH X (h::L)) = X $\cap$ (ETREE h $\cup$ ETREE (BRANCH X L))
Table 3: ET HOL4 Probabilistic Theorems ET Constructor | Probabilistic Theorem
---|---
| prob p (ETREE (NODE $\mathcal{X}_{\mathcal{N}}$)) = $\sum_{\mathcal{P}}$ p
$\mathcal{X}_{\mathcal{N}}$
| prob p (ETREE (BRANCH Y $\mathcal{Z}_{\mathcal{N}}$)) = (prob p Y) $\times$
$\sum_{\mathcal{P}}$ p $\mathcal{Z}_{\mathcal{N}}$
The function ETREE takes an event X, identified by a type constructor ATOMIC
and returns the event X. If the function ETREE takes a list of type
EVENT_TREE, identified by a type constructor NODE, then it returns the union
of all elements after applying the function ETREE on each element of the list.
Similarly, if the function ETREE takes an event X and a list of type
EVENT_TREE, identified by a type constructor BRANCH, then it performs the
intersection of the event X with the union of the head of the list after
applying the function ETREE and the recursive call of the BRANCH constructor.
For the formal probabilistic assessment of each path occurrence in the ET
diagram, HOL4 probabilistic properties for NODE and BRANCH ET constructors are
presented in Table 3 [28]. These expressions are formally verified under the
same FT constrains, i.e., $\mathcal{X}_{\mathcal{N}}$ $\in$ events p,
prob_space p and MUTUAL_INDEP p $\mathcal{X}_{\mathcal{N}}$. The function
$\sum_{\mathcal{P}}$ is defined to sum the probabilities of events for a list.
## 4 Cause-Consequence Diagrams
Cause–Consequence Diagram [15] (CCD) has been developed to analyze the causes
of an undesired subsystem failure events, using FT analysis, and from these
events obtain all possible consequences on the entire system, using ET
analysis [30]. The description of the CCD basic constructors are illustrated
in Table 4 [14]. CCD analysis is mainly divided into two categories [31]: (1)
Type I that combines SFT and ET, as shown in Fig. 1 and Table 5 [12]; and (2)
Type II that combines DFT and ET without shared events in different
subsystems, as shown in Fig. 2 and Table 6 [12]. In this analysis, we focus on
the CCD-based reliability analysis at the subsystem level of Type I.
Table 4: CCD Symbols and Functions CCD Symbol | Function
---|---
| Decision Box: represents the functionality of a component. (1) NO Box:
describes the component or subsystem failure behavior. A FT of the component
is connected to this box that can be used to determine the failure probability
($\mathcal{P}_{F}$) (2) YES Box: represents the correct functioning of the
component or reliability, which can be calculated by simply taking the
complement of the failure probability determined in the NO Box, i.e., 1 -
$\mathcal{P}_{F}$
| Consequence Path: models the next possible consequence scenarios due to a
particular event
| Consequence Box: models the outcome event due to a particular sequence of
events
Figure 1: CCD Analysis Type A Table 5: SFT Symbols and Functions SFT Symbol | Function
---|---
| AND Gate: models the failure of the output if all of the input failure
events, i.e., A and B, occur at the same time (simultaneously)
| OR Gate: models the failure of the output if any of the input failure
events, i.e., C or D, occurs alone
Figure 2: CCD Analysis Type B Table 6: DFT Symbols and Functions DFT Symbol | Function
---|---
| Priority-AND (PAND) Gate: models the dynamic behavior of failing the output
when all input events occur in a sequence, i.e., A then B
| SPare (SP) Gate: models the dynamic behavior of activating the spare input
D after the failure of the main input C
Figure 3: Overview of CCD Analysis
Fig. 3 depicts the overview of the four steps of CCD analysis [3]: (1)
Components failure events: identify the causes of the undesired failure events
for each subsystem/component in the safety-critical system; (2) Construction
of a complete CCD: draw a complete system CCD model using its basic
constructors considering that the order of components should follow the
temporal action of the system; (3) CCD model reduction: remove the unnecessary
decision boxes in the system to obtain its minimal CCD representing the actual
functional behavior of the system; and (4) CCD probabilistic analysis:
evaluate the probabilities of all CCD consequence paths. The paths in a CCD
represent the likelihood of specific sequence scenarios that are possible to
occur in a system so that only one scenario can occur [30]. This implies that
all consequences in a CCD are disjoint (mutually exclusive) [14]. Assuming
that all events associated with the decision boxes in a CCD model are mutually
independent, then the CCD paths probabilities can be quantified as follows
[15]:
1. 1.
Evaluate the probabilities of each outgoing branch stemming from a decision
box, i.e., quantifying the associated FT models
2. 2.
Compute the probability of each consequence path by multiplying the individual
probabilities of all events associated with the decision boxes
3. 3.
Determine the probability of a particular consequence box by summing the
probabilities of all consequence paths ending with that consequence event
As an example, consider a Motor Control Centre (MCC) [32] consisting of three
components Relay, Timer and Fuse, as shown in Fig. 4. The MCC is designed to
control an Induction Motor (IM) and let it run for a specific period of time
then stops. The IM power circuit is energized by the closure of the Relay
Contacts (Rc), as shown in Fig. 4. Rc closes after the user press the Start
button that energizes R and at the same time energizes an ON-delay Timer (T).
The Timer opens its contacts (Tc) after a specific period of time t and
consequently the IM stops. If the IM is overloaded than its design, then the
Fuse (F) melts and protects both MCC and IM from damage. Assume that each
component in the MCC has two operational states, i.e., operating or failing.
The four steps of a CCD-based reliability analysis described by Andrews et al.
[14] are as follows [30]:
Figure 4: Schematic of an Example MCC
1. 1.
Components failure events: Assign a FT to each component in the MCC, i.e.,
$\mathrm{FT}_{Relay}$, $\mathrm{FT}_{Timer}$, $\mathrm{FT}_{Fuse}$.
2. 2.
Construction of a complete CCD: Construct a complete CCD model of the IM
control operation, as shown in Fig. 5. For instance, if the condition of the
first decision box is either satisfied or not, i.e., YES or NO, then the next
system components are considered in order, i.e., Timer and Fuse, respectively.
Each consequence in the CCD ends with either motor stops (MS) or motor runs
(MR).
3. 3.
CCD model reduction: Apply the reduction process on the obtained complete CCD
model. For instance, if the condition of the first decision box (Relay
Contacts Open) is satisfied, i.e., YES box, then the IM stops regardless of
the status of the rest of the components, as shown in Fig. 6. Similarly, if
the condition of the second decision box (Timer Contacts Open) is satisfied,
then the IM stops. So, Fig. 6 represents the minimal CCD for the IM control
operation.
Figure 5: Complete CCD Model of the MCC Figure 6: Reduced CCD Model of the MCC
4. 4.
CCD probabilistic analysis: The probabilities of the two consequence boxes MS
and MR in Fig. 6 can be expressed mathematically as:
$\begin{split}\@add@centering\centering\mathcal{P}(Consequence\\_Box_{MS})&=\mathcal{P}(Relay_{S})+\mathcal{P}(Relay_{F})\times\mathcal{P}(Timer_{S})+\\\
&\hskip
11.38109pt\mathcal{P}(Relay_{F})\times\mathcal{P}(Timer_{F})\times\mathcal{P}(Fuse_{S})\end{split}$
(1)
$\centering\mathcal{P}(Consequence\\_Box_{MR})=\mathcal{P}(Relay_{F})\times\mathcal{P}(Timer_{F})\times\mathcal{P}(Fuse_{F})\@add@centering$
(2)
where $\mathcal{P}(\mathcal{X}_{F})$ is the unreliability function or the
probability of failure for a component $\mathcal{X}$, i.e.,
$\mathrm{FT}_{\mathcal{X}}$ model, and $\mathcal{P}(\mathcal{X}_{S})$ is the
reliability function or the probability of operating, i.e., the complement of
the $\mathrm{FT}_{\mathcal{X}}$ model.
In the next section, we present, in detail, the formalization of CCDs in the
HOL4 theorem prover to analyze the failures at the subsystem level of a given
safety-critical complex system and determine all their possible cascading
dependencies of complete/partial reliability and failure events that can occur
at the system level.
### 4.1 Formal CCD Modeling
We start the formalization of CDDs by formally model its basic symbols, as
described in Table 4 in HOL4 as follows:
Definition 3:
$\vdash$ DEC_BOX p X Y = if X = 1 then FST Y else if X = 0 then SND Y else
p_space p
where Y is an ordered pair (FST Y, SND Y) representing the reliability and
unreliability functions in a decision box, respectively. The condition X = 1
represents the YES Box while X = 0 represents the NO Box. If X is neither 1
nor 0, for instance, X = 2, this represents the irrelevance of the decision
box, which returns the probability space p to be used in the reduction process
of CCDs.
Secondly, we define the CCD Consequence path by recursively applying the
BRANCH ET constructor on a given $\mathcal{N}$ list of decision boxes
($\mathrm{\small{\texttt{DEC\\_BOX}}}_{\mathcal{N}}$) using the HOL4 recursive
function FOLDL as:
Definition 4:
$\vdash$ CONSEQ_PATH p
($\mathrm{\small{\texttt{DEC\\_BOX}}}_{1}$::$\mathrm{\small{\texttt{DEC\\_BOX}}}_{\mathcal{N}}$)
=
FOLDL ($\lambda$a b. ETREE (BRANCH a b))
$\mathrm{\small{\texttt{DEC\\_BOX}}}_{1}$
$\mathrm{\small{\texttt{DEC\\_BOX}}}_{\mathcal{N}}$
Finally, we define the CCD Consequence box by mapping the function CONSEQ_PATH
on a list using the HOL4 function MAP, then applies the NODE ET constructor:
Definition 5:
$\vdash$ CONSEQ_BOX p $\mathrm{L}_{\mathcal{M}}$ = ETREE (NODE (MAP
($\lambda$a. CONSEQ_PATH p a) $\mathrm{L}_{\mathcal{M}}$))
Using the above definitions, we can construct a complete CCD model (Step 2 in
Fig. 3) for the MCC system shown in Fig. 5, in HOL4 as:
$\vdash$ MCC_COMPLETE_CCD $\mathrm{FT}_{R}$ $\mathrm{FT}_{T}$
$\mathrm{FT}_{F}$ =
CONSEQ_BOX p
[[DEC_BOX p 1 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 1 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 1 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)]]
In CCD analysis [30], Step 3 in Fig. 3 is used to model the accurate
functional behavior of systems in the sense that the irrelevant decision boxes
should be removed from a complete CCD model. Upon this, the actual CCD model
of the MCC system after reduction, as shown in Fig. 6, can be obtained by
assigning X with neither 1 nor 0, for instance, X = 2, which represents the
irrelevance of the decision box, in HOL4 as:
$\vdash$ MCC_REDUCED_CCD $\mathrm{FT}_{R}$ $\mathrm{FT}_{T}$ $\mathrm{FT}_{F}$
=
CONSEQ_BOX p
[[DEC_BOX p 1 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 2
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 2
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 2
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)]]
Also, we can formally verify the above reduced CCD model of the MCC system, in
HOL4 as:
$\vdash$ MCC_REDUCED_CCD $\mathrm{FT}_{R}$ $\mathrm{FT}_{T}$ $\mathrm{FT}_{F}$
=
CONSEQ_BOX p
[[DEC_BOX p 1 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 1
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{R}}$,$\mathrm{FT}_{R}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{T}}$,$\mathrm{FT}_{T}$);DEC_BOX p 0
($\overline{\mathrm{FT}_{F}}$,$\mathrm{FT}_{F}$)]]
where $\overline{\mathrm{FT}_{X}}$ for a component X is the complement of
$\mathrm{FT}_{X}$.
### 4.2 Formal CCD Analysis
The important step in the CCD analysis is to determine the probability of each
consequence path occurrence in the CCD [14]. For that purpose, we formally
verify the following CCD generic probabilistic properties, in HOL4 as follows:
Property 1: The probability of a consequence path for one decision box
assigned with a generic FT model, i.e., OR or AND, as shown in Fig. 7, under
the assumptions described in Table 2, respectively as follows:
Theorem 1:
$\vdash$ prob_space p $\wedge$ $F_{\mathcal{N}}$ $\in$ events p $\wedge$
MUTUAL_INDEP p $F_{\mathcal{N}}$ $\Rightarrow$
prob p
(CONSEQ_PATH p [DEC_BOX p X (FTree p (NOT (OR
$\mathrm{F}_{\mathcal{N}}$)),FTree p (OR $\mathrm{F}_{\mathcal{N}}$))]) =
if X = 1 then $\prod$ (PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{N}}$))
else if X = 0 then 1 - $\prod$ (PROB_LIST p (COMPL_LIST p
$\mathrm{F}_{\mathcal{N}}$)) else 1
Figure 7: Decision Boxes with FT Gates
For example, consider a system X consists of two components $\mathrm{C}_{1}$
and $\mathrm{C}_{2}$. Assuming the failure of either one them causes the
system failure, i.e., $\mathrm{C}_{1F}$ or $\mathrm{C}_{2F}$, We can formally
model the FT of the system ($\mathrm{FT}_{system}$), in HOL4 as:
$\vdash$ $\mathrm{FT}_{system}$ p $\mathrm{C}_{1F}$ $\mathrm{C}_{2F}$ = FTree
p (OR [$\mathrm{C}_{1F}$;$\mathrm{C}_{2F}$])
Using Theorem 1, we can obtain the probability of a decision box YES/NO
outcomes connected to the above FT model, respectively, in HOL4 as:
$\vdash$ prob p (CONSEQ_PATH p [DEC_BOX p 1
($\overline{\mathrm{FT}_{system}}$,$\mathrm{FT}_{system}$))]) =
(1 - prob p $\mathrm{C}_{1F}$) $\times$ (1 - prob p $\mathrm{C}_{2F}$)
$\vdash$ prob p (CONSEQ_PATH p [DEC_BOX p 0
($\overline{\mathrm{FT}_{system}}$,$\mathrm{FT}_{system}$))]) =
1 - (1 - prob p $\mathrm{C}_{1F}$) $\times$ (1 - prob p $\mathrm{C}_{2F}$)
Theorem 2:
$\vdash$ prob_space p $\wedge$ $F_{\mathcal{N}}$ $\in$ events p $\wedge$
MUTUAL_INDEP p $F_{\mathcal{N}}$ $\Rightarrow$
prob p
(CONSEQ_PATH p
[DEC_BOX p X (FTree p (NOT (AND $\mathrm{F}_{\mathcal{N}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{N}}$))]) =
if X = 1 then 1 - $\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$)
else if X = 0 then $\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$) else 1
For instance, in the above example, assume the failure of both components
simultaneously only causes the system failure, i.e., $\mathrm{C}_{1F}$ and
$\mathrm{C}_{2F}$. We can formally model the FT of the system, in HOL4 as:
$\vdash$ $\mathrm{FT}_{system}$ p $\mathrm{C}_{1F}$ $\mathrm{C}_{2F}$ = FTree
p (AND[$\mathrm{C}_{1F}$;$\mathrm{C}_{2F}$])
Figure 8: Two-level Decision Boxes for CCD Analysis
Using Theorem 2, we can obtain the probability of a decision box YES/NO
outcomes connected to the above FT model, respectively, in HOL4 as:
$\vdash$ prob p (CONSEQ_PATH p [DEC_BOX p 1
($\overline{\mathrm{FT}_{system}}$,$\mathrm{FT}_{system}$))]) =
1 - prob p $\mathrm{C}_{1F}$ $\times$ prob p $\mathrm{C}_{2F}$
$\vdash$ prob p (CONSEQ_PATH p [DEC_BOX p 0
($\overline{\mathrm{FT}_{system}}$,$\mathrm{FT}_{system}$))]) =
prob p $\mathrm{C}_{1F}$ $\times$ prob p $\mathrm{C}_{2F}$
Property 2: The probability of two-level decision boxes assigned to a CCD path
with all combinations of FT gates (AND-OR/OR-AND , AND-AND and OR-OR), as
shown in Fig. 8. Each combination has 4 possible operating scenarios that can
occur (0-0, 0-1, 1-0 and 1-1) and 2 other possible reduction scenarios that
may be required in Step 3 (0-2 and 1-2), which represents the removal of the
decision box Y from the path. The basic idea is to select different
combinations of decision boxes to achieve the desired system behavior and also
select the reduction combination ($>$ 1) to remove irreverent decision boxes.
This probabilistic expressions can be formally verified, in HOL4 as:
Theorem 3:
$\vdash$ prob_space p $\wedge$ ($\forall$y. y $\in$
($\mathrm{F}_{\mathcal{N}}$++$\mathrm{F}_{\mathcal{M}}$) $\Rightarrow$ y $\in$
events p) $\wedge$
MUTUAL_INDEP p ($\mathrm{F}_{\mathcal{N}}$++$\mathrm{F}_{\mathcal{M}}$)
$\Rightarrow$
prob p (CONSEQ_PATH p
[DEC_BOX p X (FTree p (NOT (AND $\mathrm{F}_{\mathcal{N}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{N}}$));
DEC_BOX p Y (FTree p (NOT (OR $\mathrm{F}_{\mathcal{M}}$)),FTree p (OR
$\mathrm{F}_{\mathcal{M}}$))]) =
if X = 0 $\wedge$ Y = 0 then
$\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$) $\times$ (1 - $\prod$
(PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$)))
else if X = 0 $\wedge$ Y = 1 then
$\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$) $\times$ $\prod$ (PROB_LIST p
(COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$))
else if X = 1 $\wedge$ Y = 0 then
(1 - $\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$)) $\times$ (1 - $\prod$
(PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$)))
else if X = 1 $\wedge$ Y = 1 then
(1 - $\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$)) $\times$ $\prod$
(PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$))
else if X = 0 $\wedge$ Y = 2 then $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{N}}$)
else if X = 1 $\wedge$ Y = 2 then (1 - $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{N}}$)) else 1
Theorem 4:
$\vdash$ prob p (CONSEQ_PATH p
[DEC_BOX p X (FTree p (NOT (AND $\mathrm{F}_{\mathcal{N}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{N}}$));
DEC_BOX p Y (FTree p (NOT (AND $\mathrm{F}_{\mathcal{M}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{M}}$))]) =
if X = 0 $\wedge$ Y = 0 then
$\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$) $\times$ $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{M}}$)
else if X = 0 $\wedge$ Y = 1 then
$\prod$ (PROB_LIST p $\mathrm{F}_{\mathcal{N}}$) $\times$ (1 - $\prod$
(PROB_LIST p $\mathrm{F}_{\mathcal{M}}$))
⋮
else if X = 1 $\wedge$ Y = 2 then (1 - $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{N}}$)) else 1
Theorem 5:
$\vdash$ prob p (CONSEQ_PATH p
[DEC_BOX p X (FTree p (NOT (OR $\mathrm{F}_{\mathcal{N}}$)),FTree p (OR
$\mathrm{F}_{\mathcal{N}}$));
DEC_BOX p Y (FTree p (NOT (OR $\mathrm{F}_{\mathcal{M}}$)),FTree p (OR
$\mathrm{F}_{\mathcal{M}}$))]) =
if X = 0 $\wedge$ Y = 0 then
(1 - $\prod$ (PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{N}}$))) $\times$
(1 - $\prod$ (PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$)))
else if X = 0 $\wedge$ Y = 1 then
(1 - $\prod$ (PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{N}}$))) $\times$
$\prod$ (PROB_LIST p (COMPL_LIST p $\mathrm{F}_{\mathcal{M}}$))
⋮
else if X = 1 $\wedge$ Y = 2 then $\prod$ (PROB_LIST p (COMPL_LIST p
$\mathrm{F}_{\mathcal{N}}$)) else 1
Property 3: A generic probabilistic property for a consequence path consisting
of complex four-level decision boxes associated with different combination of
FTs and each one consisting of $\mathcal{N}$ components (AND-OR-AND-OR/OR-AND-
OR-AND/AND-AND-OR-OR/OR-OR-AND-AND), which has 16 possible operating scenarios
that can occur and 14 other possible reduction possibilities, as shown in Fig.
9, in HOL4 as:
Theorem 6:
$\vdash$ Let
$\small{\texttt{W}}_{\mathrm{F}}$ = $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{N}}$);
$\overline{\small{\texttt{W}}}$ = 1 - $\small{\texttt{W}}_{\mathrm{F}}$;
$\small{\texttt{X}}_{\mathrm{F}}$ = 1 - $\prod$ (PROB_LIST p (COMPL_LIST p
$\mathrm{F}_{\mathcal{K}}$)); $\overline{\small{\texttt{X}}}$ = 1 -
$\small{\texttt{X}}_{\mathrm{F}}$;
$\small{\texttt{Y}}_{\mathrm{F}}$ = $\prod$ (PROB_LIST p
$\mathrm{F}_{\mathcal{M}}$);
$\overline{\small{\texttt{Y}}}$ = 1 - $\small{\texttt{Y}}_{\mathrm{F}}$;
$\small{\texttt{Z}}_{\mathrm{F}}$ = 1 - $\prod$ (PROB_LIST p (COMPL_LIST p
$\mathrm{F}_{\mathcal{J}}$)); $\overline{\small{\texttt{Z}}}$ = 1 -
$\small{\texttt{Z}}_{\mathrm{F}}$
in
prob p
(CONSEQ_PATH p
[DEC_BOX p W (FTree p (NOT (AND $\mathrm{F}_{\mathcal{N}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{N}}$));
DEC_BOX p X (FTree p (NOT (OR $\mathrm{F}_{\mathcal{K}}$)),FTree p (OR
$\mathrm{F}_{\mathcal{K}}$));
DEC_BOX p Y (FTree p (NOT (AND $\mathrm{F}_{\mathcal{M}}$)),FTree p (AND
$\mathrm{F}_{\mathcal{M}}$));
DEC_BOX p Z (FTree p (NOT (OR $\mathrm{F}_{\mathcal{J}}$)),FTree p (OR
$\mathrm{F}_{\mathcal{J}}$))]) =
if W = 0 $\wedge$ X = 0 $\wedge$ Y = 0 $\wedge$ Z = 0
then $\small{\texttt{W}}_{\mathrm{F}}$ $\times$
$\small{\texttt{X}}_{\mathrm{F}}$ $\times$ $\small{\texttt{Y}}_{\mathrm{F}}$
$\times$ $\small{\texttt{Z}}_{\mathrm{F}}$
else if W = 0 $\wedge$ X = 0 $\wedge$ Y = 0 $\wedge$ Z = 1
then $\small{\texttt{W}}_{\mathrm{F}}$ $\times$
$\small{\texttt{X}}_{\mathrm{F}}$ $\times$ $\small{\texttt{Y}}_{\mathrm{F}}$
$\times$ $\overline{\small{\texttt{Z}}}$
else if W = 0 $\wedge$ X = 0 $\wedge$ Y = 1 $\wedge$ Z = 0
then $\small{\texttt{W}}_{\mathrm{F}}$ $\times$
$\small{\texttt{X}}_{\mathrm{F}}$ $\times$ $\overline{\small{\texttt{Y}}}$
$\times$ $\small{\texttt{Z}}_{\mathrm{F}}$
⋮
else if W = 1 $\wedge$ X = 1 $\wedge$ Y = 2 $\wedge$ Z = 2
then $\overline{\small{\texttt{W}}}$ $\times$ $\overline{\small{\texttt{X}}}$
else if W = 1 $\wedge$ X = 2 $\wedge$ Y = 2 $\wedge$ Z = 2
then $\overline{\small{\texttt{W}}}$ else 1
Figure 9: Four-level Decision Boxes for CCD Analysis
For complex systems consisting of $\mathcal{N}$-level decision boxes, where
each decision box is associated with an AND/OR gate consisting of an arbitrary
list of failure events, we define three types A, B and C of possible CCD
outcomes, as shown in Fig. 10, with a new proposed mathematics as:
Figure 10: Generic $\mathcal{N}$-level CCD Analysis
Property 4 (N Decision Boxes of Type A): The probability of $n$ decision boxes
assigned to a consequence path corresponding to $n$ subsystems, where all
decision boxes are associated with FT AND models consisting of arbitrary lists
of $k$ events, can be expressed mathematically at a specific time t for three
cases as:
(A1) All outcomes of $n$ decisions boxes are NO
$\centering\mathcal{F}_{A1}(t)=\prod\limits^{n}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\@add@centering$
(3)
(A2) All outcomes of $n$ decisions boxes are YES
$\centering\mathcal{F}_{A2}(t)=\prod\limits^{n}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\@add@centering$
(4)
(A3) Some outcomes of $m$ decisions boxes are NO and the rest outcomes of $p$
decisions boxes are YES
$\centering\mathcal{F}_{A3}(t)=\Bigg{(}\prod\limits^{m}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\@add@centering$
(5)
To verify the correctness of the above-proposed new safety analysis
mathematical formulations in the HOL4 theorem prover, we define two generic
CCD functions $\mathcal{SS}^{YES}_{AND}$ and $\mathcal{SS}^{NO}_{AND}$ that
can recursively generate the outcomes YES and NO of the function FTree,
identified by gate constructors AND and NOT, for a given arbitrary list of all
subsystems failure events (SSN), respectively, in HOL4 as:
Definition 6:
$\vdash$ $\mathcal{SS}^{YES}_{AND}$ p (SS::SSN) = FTree p (NOT (AND
SS1))::$\mathcal{SS}^{YES}_{AND}$ p SSN
Definition 7:
$\vdash$ $\mathcal{SS}^{NO}_{AND}$ p (SS1::SSN) = FTree p (AND
SS1)::$\mathcal{SS}^{NO}_{AND}$ p SSN
Using above defined functions, we can verify three two-dimensional and
scalable probabilistic properties corresponding to the above-mentioned safety
equations Eq. 3, Eq. 4, and Eq. 5, respectively, in HOL4 as:
Theorem 7:
$\vdash$ prob p (CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSN)) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSN)
Theorem 8:
$\vdash$ prob p (CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSN)) =
$\prod$ (MAP ($\lambda$ b. (1 - $\prod$ (PROB_LIST p b))) SSN)
Theorem 9:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSp)]) =
$\bigg{(}\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSm)$\bigg{)}$
$\times$
$\bigg{(}\prod$ (MAP ($\lambda$ b. 1 - $\prod$ (PROB_LIST p b)) SSp)$\bigg{)}$
Property 5 (N Decision Boxes of Type B): The probabilistic assessment of $n$
decision boxes assigned to a CCD consequence path, where all decision boxes
are associated with generic FT OR models consisting of arbitrary lists of $k$
events, can be expressed mathematically for three cases:
(B1) All outcomes of $n$ decisions boxes are NO
$\centering\mathcal{F}_{B1}(t)=\prod\limits^{n}_{i=1}(1-\prod\limits^{k}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\@add@centering$
(6)
(B2) All outcomes of $n$ decisions boxes are YES
$\centering\mathcal{F}_{B2}(t)=\prod\limits^{n}_{i=1}\prod\limits^{k}_{j=2}(1-{\mathcal{F}}_{ij}(t))\@add@centering$
(7)
(B3) Some outcomes of $m$ decisions boxes are NO and some outcomes of $p$
decisions boxes are YES
$\centering\mathcal{F}_{B3}(t)=\Bigg{(}\prod\limits^{m}_{i=1}(1-\prod\limits^{k}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}\prod\limits^{k}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\@add@centering$
(8)
To verify the correctness of the above-proposed new CCD mathematical formulas
in HOL4, we define two generic functions $\mathcal{SS}^{YES}_{OR}$ and
$\mathcal{SS}^{NO}_{OR}$ to recursively generate the outcomes YES and NO of
the function FTree, identified by gate constructors OR and NOT, for a given
list of subsystems events.
Definition 8:
$\vdash$ $\mathcal{SS}^{YES}_{OR}$ p (SS::SSN) = FTree p (NOT (OR
SS1))::$\mathcal{SS}^{YES}_{OR}$ p SSN
Definition 9:
$\vdash$ $\mathcal{SS}^{NO}_{OR}$ p (SS1::SSN) = FTree p (OR
SS1)::$\mathcal{SS}^{NO}_{OR}$ p SSN
Using above defined functions, we can formally verify three scalable
probabilistic properties corresponding to Eq. 6, Eq. 7, and Eq. 8,
respectively, in HOL4 as:
Theorem 10:
$\vdash$ prob p (CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSN)) =
$\prod$
(MAP
($\lambda$ a.
(1 - $\prod$ (PROB_LIST p (compl_list p a)))) SSN)
Theorem 11:
$\vdash$ prob p (CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSN)) =
$\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSN)
Theorem 12:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSp)]) =
$\prod$
(MAP
($\lambda$ a.
(1 - $\prod$ (PROB_LIST p (compl_list p a)))) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSp)
Property 6 (N Decision Boxes of Type C): The probabilistic assessment of $n$
decision boxes assigned to a consequence path for a very complex system, where
some $m$ decision boxes are associated with generic FT AND models consisting
of $k$-events, while other $p$ decision boxes are associated with generic FT
OR models consisting of $z$-events, as shown in Fig. 10, is proposed to be
expressed mathematically for nine cases as:
(C1) All outcomes of $m$ and $p$ decisions boxes are NO.
$\centering\mathcal{F}_{C1}(t)=\Bigg{(}\prod\limits^{m}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}(1-\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\@add@centering$
(9)
Theorem 13:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
(1 - $\prod$ (PROB_LIST p (compl_list p b)))) SSp)
(C2) All outcomes of $m$ and $p$ decisions boxes are YES.
$\centering\mathcal{F}_{C2}(t)=\Bigg{(}\prod\limits^{m}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\@add@centering$
(10)
Theorem 14:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. 1 - $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSp)
(C3) All outcomes of $m$ decisions boxes are NO and all outcomes of $p$
decisions boxes are YES.
$\centering\mathcal{F}_{C3}(t)=\Bigg{(}\prod\limits^{m}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\@add@centering$
(11)
Theorem 15:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSp)
(C4) All outcomes of $m$ decisions boxes are YES and all outcomes of $p$
decisions boxes are NO.
$\centering\mathcal{F}_{C4}(t)=\Bigg{(}\prod\limits^{m}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}(1-\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\@add@centering$
(12)
Theorem 16:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. 1 - $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
(1 - $\prod$ (PROB_LIST p (compl_list p b)))) SSp)
(C5) Some outcomes of $s$ out of $m$ decisions boxes are NO, some outcomes of
$u$ out of $m$ decisions boxes are YES and all outcomes of $p$ decisions boxes
are NO.
$\centering\mathcal{F}_{C5}(t)=\Bigg{(}\prod\limits^{s}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{u}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}(1-\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\@add@centering$
(13)
Theorem 17:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSs);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSu);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSs)
$\times$ $\prod$ (MAP ($\lambda$ b. 1 - $\prod$ (PROB_LIST p b)) SSu)
$\times$ $\prod$
(MAP
($\lambda$ c.
(1 - $\prod$ (PROB_LIST p (compl_list p c)))) SSp)
(C6) Some outcomes of $s$ out of $m$ decisions boxes are NO, some outcomes of
$u$ out of $m$ decisions boxes are YES and all outcomes of $p$ decisions boxes
are YES.
$\centering\mathcal{F}_{C6}(t)=\Bigg{(}\prod\limits^{s}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{u}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{p}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\centering\@add@centering\@add@centering$
(14)
Theorem 18:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSs);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSu);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSp)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSs)
$\times$ $\prod$ (MAP ($\lambda$ b. 1 - $\prod$ (PROB_LIST p b)) SSu)
$\times$ $\prod$
(MAP
($\lambda$ c.
$\prod$ (PROB_LIST p (compl_list p c))) SSp)
(C7) Some outcomes of $s$ out of $p$ decisions boxes are NO, some outcomes of
$u$ out of $p$ decisions boxes are YES and all outcomes of $m$ decisions boxes
are NO.
$\centering\mathcal{F}_{C7}(t)=\Bigg{(}\prod\limits^{m}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{u}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{s}_{i=1}(1-\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\@add@centering$
(15)
Theorem 19:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSu);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSs)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSu)
$\times$ $\prod$
(MAP
($\lambda$ c.
(1 - $\prod$ (PROB_LIST p (compl_list p c)))) SSs)
(C8) Some outcomes of $s$ out of $p$ decisions boxes are NO, some outcomes of
$u$ out of $p$ decisions boxes are YES and all outcomes of $m$ decisions boxes
are YES.
$\centering\mathcal{F}_{C8}(t)=\hskip
11.38109pt\Bigg{(}\prod\limits^{m}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{u}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{s}_{i=1}(1-\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\@add@centering$
(16)
Theorem 20:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSm);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSu);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSs)]) =
$\prod$ (MAP ($\lambda$ a. 1 - $\prod$ (PROB_LIST p a)) SSm)
$\times$ $\prod$
(MAP
($\lambda$ b.
$\prod$ (PROB_LIST p (compl_list p b))) SSu)
$\times$ $\prod$
(MAP
($\lambda$ c.
(1 - $\prod$ (PROB_LIST p (compl_list p c)))) SSs)
(C9) Some outcomes of $s$ out of $m$ decisions boxes are NO, some outcomes of
$u$ out of $m$ decisions boxes are YES, some outcomes of $v$ out of $p$
decisions boxes are NO and some outcomes of $w$ out of $p$ decisions boxes are
YES.
$\centering\begin{split}\mathcal{F}_{C9}(t)&=\Bigg{(}\prod\limits^{s}_{i=1}\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t)\Bigg{)}\times\Bigg{(}\prod\limits^{v}_{i=1}(1-\prod\limits^{z}_{j=1}(1-{\mathcal{F}}_{ij}(t)))\Bigg{)}\\\
&\times\Bigg{(}\prod\limits^{u}_{i=1}(1-\prod\limits^{k}_{j=2}{\mathcal{F}}_{ij}(t))\Bigg{)}\times\Bigg{(}\prod\limits^{w}_{i=1}\prod\limits^{z}_{j=2}(1-{\mathcal{F}}_{ij}(t))\Bigg{)}\end{split}\@add@centering$
(17)
Theorem 21:
$\vdash$ prob p
(CONSEQ_PATH p
[CONSEQ_PATH p ($\mathcal{SS}^{NO}_{AND}$ p SSs);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{AND}$ p SSu);
CONSEQ_PATH p ($\mathcal{SS}^{NO}_{OR}$ p SSv);
CONSEQ_PATH p ($\mathcal{SS}^{YES}_{OR}$ p SSw)]) =
$\prod$ (MAP ($\lambda$ a. $\prod$ (PROB_LIST p a)) SSs)
$\times$ $\prod$ (MAP ($\lambda$ b. 1 - $\prod$ (PROB_LIST p b)) SSu)
$\times$ $\prod$
(MAP
($\lambda$ c.
(1 - $\prod$ (PROB_LIST p (compl_list p c)))) SSv)
$\times$ $\prod$
(MAP
($\lambda$ d.
$\prod$ (PROB_LIST p (compl_list p d))) SSw)
Therefore, by verifying all the above-mentioned theorems in HOL4, we showed
the completeness of our proposed formal approach and thereupon solving the
scalability problem of CCD analysis for any given large engineering complex
system at the subsystem level [33].
Property 7: A generic probabilistic expression of CONSEQ_BOX for a certain
event occurrence in the entire system as the sum of all individual
probabilities of all $\mathcal{M}$ CONSEQ_PATH ending with that event:
Theorem 22:
$\vdash$ Let
CONSEQ_PATHS $\mathrm{L}_{\mathcal{M}}$ = MAP ($\lambda$a. CONSEQ_PATH p a)
$\mathrm{L}_{\mathcal{M}}$)
in
prob_space p $\wedge$ MUTUAL_INDEP p $\mathrm{L}_{\mathcal{M}}$ $\wedge$
disjoint (CONSEQ_PATHS $\mathrm{L}_{\mathcal{M}}$) $\wedge$ ALL_DISTINCT
(CONSEQ_PATHS $\mathrm{L}_{\mathcal{M}}$) $\Rightarrow$
prob p (CONSEQ_BOX p $\mathrm{L}_{\mathcal{M}}$) = $\sum$ (PROB_LIST p
(CONSEQ_PATHS $\mathrm{L}_{\mathcal{M}}$))
where the HOL4 function disjoint ensures that each pair of elements in a given
list is mutually exclusive while the function ALL_DISTINCT ensures that each
pair is distinct. The function $\sum$ is defined to sum the events for a given
list. Remark that all above-mentioned CCD new formulations have been formally
verified in HOL4, where the proof-script amounts to about 16,000 lines of HOL4
code, which can be downloaded for use from [33]. Also, this code can be
extended, with some basic knowhow about HOL4, to perform dynamic failure
analysis of dynamic subsystems where no dependencies exist in subsystems using
DFTs, such as PAND and SP, i.e, CCD reliability analysis of Type II (see Fig.
2).
To illustrate the applicability of our proposed approach, in the next section,
we present the formal CCD step-analysis of the standard IEEE 39-bus electrical
power network and verify its reliability indexes ($\mathcal{FOR}$ and
$\mathcal{SAIDI}$), which are commonly used as reliability indicators by
electric power utilities.
## 5 Electrical Power 39-bus Network System
An electrical power network is an interconnected grid for delivering
electricity from producers to customers. The power network system consists of
three main zones [1]: (i) generating stations that produce electric power;
(ii) transmission lines that carry power from sources to loads; and (iii)
distribution lines that connect individual consumers. Due to the complex and
integrated nature of the power network, failures in any zone of the system can
cause widespread catastrophic disruption of supply [1]. Therefore a rigorous
formal cause-consequence analysis of the grid is essential in order to reduce
the risk situation of a blackout and enable back-up decisions [34]. For power
network safety assessment, reliability engineers have been dividing the power
network into three main hierarchical levels [12]: (a) generation systems; (b)
composite generation and transmission (or bulk power) systems; and (c)
distribution systems. We can use our proposed CCD formalization for the formal
modeling and analysis of any hierarchical level in the power network. In this
case study, we focus on the generation part only, i.e., hierarchical level I.
Also, we can evaluate the Force Outage Rate ($\mathcal{FOR}$) for the
generation stations, which is defined as the probability of the unit
unavailability to produce power due to unexpected equipment failure [34].
Additionally, we can determine the System Average Interruption Duration Index
($\mathcal{SAIDI}$), which is used to indicate the average duration for each
customer served to experience a sustained outage. $\mathcal{SAIDI}$ is defined
as the sum of all customer interruption durations (probability of load
failures ↯ multiplying by the mean-time-to-repair the failures and the number
of customers that are affected by these failures) over the total number of
customers served [34]:
$\centering\mathcal{SAIDI}=\frac{\sum_{{\mathcal{P}}(\mathcal{X}_{\mbox{\Lightning}})\times\mathrm{MTTR}_{\mathcal{X}}\times\mathrm{CN}_{\mathcal{X}}}}{\sum_{\mathrm{CN}_{\mathcal{X}}}}\@add@centering$
(18)
where $\mathrm{CN}_{\mathcal{X}}$ is the number of customers for a certain
location $\mathcal{X}$ while $\mathrm{MTTR}_{\mathcal{X}}$ is the mean-time-
to-repair the failure that occurred at $\mathcal{X}$. We formally define a
function $\sum\nolimits^{T}_{\mbox{\Lightning}}$ in HOL4 to sum all customer
interruption durations. Also, we formally define a generic function
$\mathcal{SAIDI}$ by dividing the output of
$\sum\nolimits^{T}_{\mbox{\Lightning}}$ over the total number of customers
served, in HOL4 as:
Definition 10:
$\vdash$ $\sum\nolimits^{T}_{\mbox{\Lightning}}$
(L::$\mathrm{\small{\texttt{L}}}_{\mathcal{M}}$)
(MTTR::$\mathrm{\small{\texttt{MTTR}}}_{\mathcal{M}}$)
(CN:$\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}$) p =
prob p (CONSEQ_BOX p $\mathrm{\small{\texttt{L}}}_{\mathcal{M}}$) $\times$
MTTR $\times$ CN + $\sum\nolimits^{T}_{\mbox{\Lightning}}$
$\mathrm{\small{\texttt{L}}}_{\mathcal{M}}$
$\mathrm{\small{\texttt{MTTR}}}_{\mathcal{M}}$
$\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}$ p
Definition 11:
$\vdash$ $\mathcal{SAIDI}$ $\mathrm{\small{\texttt{L}}}_{\mathcal{M}}$
$\mathrm{\small{\texttt{MTTR}}}_{\mathcal{M}}$
$\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}$ p =
$\dfrac{\sum\nolimits^{T}_{\mbox{\Lightning}}\mathrm{\small{\texttt{L}}}_{\mathcal{M}}\hskip
2.84526pt\mathrm{\small{\texttt{MTTR}}}_{\mathcal{M}}\hskip
2.84526pt\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}\hskip
2.84526ptp}{\sum\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}}$
where $\mathrm{\small{\texttt{L}}}_{\mathcal{M}}$ is the list of CCD paths,
$\mathrm{\small{\texttt{MTTR}}}_{\mathcal{M}}$ is the list of meantime to
repairs, and $\mathrm{\small{\texttt{CN}}}_{\mathcal{M}}$ is the list of
customer numbers. The function $\sum\nolimits^{T}_{\mbox{\Lightning}}$
(Definition 10) models the numerator of Eq. 18, which is the sum of all
customer interruption durations at different locations in the electrical power
grid. Each probability of failure is obtained by evaluating a CONSEQ_BOX
consisting of a list of $\mathcal{M}$ CONSEQ_PATH, which cause that failure.
Definition 11 represents the division of output of Definition 10 over the
total number of customers at all those locations as described in Eq. 18.
Consider a standard IEEE 39-bus electrical power network test system
consisting of 10 generators (G), 12 substations (S/S), 39 Buses (Bus), and 34
transmission lines (TL), as shown in Fig. 11 [35]. Assuming the generators
G1-G10 are of two types: (i) solar photo-voltaic (PV) power plants G1-G5; and
(ii) steam power plants G6-G10. Using the Optimal Power Flow (OPF)
optimization [36], we can determine the flow of electricity from generators to
consumers in the power network. Typically, we are only interested in
evaluating the duration of certain failure events occurrence for specific
loads in the grid. For instance, if we consider the failure of load A, which
according to the OPF is supplied from G9 and G5 only, as shown in Fig. 11,
then the failure of either one or both power plants will lead to a partial or
a complete blackout failure at that load, respectively. Assuming the failure
of two consecutive power plants causes a complete blackout of the load. Hence,
considering the disruption cases of only one supply generator, then different
partial failures for loads A, B, C and D, as shown in Fig. 11, can be obtained
by observing different failures in the power network as:
1. a.
$\begin{aligned}
\mathcal{P}(\mathrm{Load}_{A\mbox{\Lightning}})=&(1-\mathcal{FOR}_{G_{9}})\times\mathcal{FOR}_{G_{5}}+\mathcal{FOR}_{G_{9}}\times(1-\mathcal{FOR}_{G_{5}})\end{aligned}$
2. b.
$\begin{aligned}
\mathcal{P}(\mathrm{Load}_{B\mbox{\Lightning}})=&(1-\mathcal{FOR}_{G_{7}})\times\mathcal{FOR}_{G_{9}}+\mathcal{FOR}_{G_{7}}\times(1-\mathcal{FOR}_{G_{9}})\end{aligned}$
3. c.
$\begin{aligned}
\mathcal{P}(\mathrm{Load}_{C\mbox{\Lightning}})=&(1-\mathcal{FOR}_{G_{1}})\times\mathcal{FOR}_{G_{2}}+\mathcal{FOR}_{G_{1}}\times(1-\mathcal{FOR}_{G_{2}})\end{aligned}$
4. d.
$\begin{aligned}
\mathcal{P}(\mathrm{Load}_{D\mbox{\Lightning}})&=(1-\mathcal{FOR}_{G_{6}})\times(1-\mathcal{FOR}_{G_{3}})\times(1-\mathcal{FOR}_{G_{8}})\times\mathcal{FOR}_{G_{4}}\\\
&\hskip
1.99168pt+(1-\mathcal{FOR}_{G_{6}})\times(1-\mathcal{FOR}_{G_{3}})\times\mathcal{FOR}_{G_{8}}\times(1-\mathcal{FOR}_{G_{4}})\\\
&\hskip
1.99168pt+(1-\mathcal{FOR}_{G_{6}})\times\mathcal{FOR}_{G_{3}}\times(1-\mathcal{FOR}_{G_{8}})\times(1-\mathcal{FOR}_{G_{4}})\\\
&\hskip
1.99168pt+\mathcal{FOR}_{G_{6}}\times(1-\mathcal{FOR}_{G_{3}})\times(1-\mathcal{FOR}_{G_{8}})\times(1-\mathcal{FOR}_{G_{4}})\end{aligned}$
Therefore, the assessment of $\mathcal{SAIDI}$ for the Grid (G) shown in Fig.
11, including an evaluation for the $\mathcal{FOR}$ of all its power plants,
can be written mathematically as:
$\centering\mathcal{SAIDI}_{G}=\frac{\mathcal{P}(\mathrm{Load}_{A\mbox{\Lightning}})\times\mathrm{MTTR}_{\mathrm{Load}_{A}}\times\mathrm{CN}_{\mathrm{Load}_{A}}+\dots}{\mathrm{CN}_{\mathrm{Load}_{A}}+\mathrm{CN}_{\mathrm{Load}_{B}}+\mathrm{CN}_{\mathrm{Load}_{C}}+\mathrm{CN}_{\mathrm{Load}_{D}}}\@add@centering$
(19)
Figure 11: IEEE 39-bus Electrical Power Network [35]
### 5.1 Formal CCD Analysis in HOL4
We can apply our four steps of CCD formalization to verify the expression of
$\mathcal{SAIDI}$ in terms of the power plant generator components, in HOL4
as:
Step 1 (Component failure events):
The schematic FT models of a typically PV power plant consisting of 2 solar
farms [37] and a steam power plant consisting of 3 generators [34] are shown
in Fig. 13 and Fig. 13, respectively. Using the formal FT modeling, we can
formally define the FT models of both plants, in HOL4 as:
Figure 12: FT Model of a PV Power Plant
Figure 13: FT Model of a Steam Power Plant
Definition 12:
$\vdash$ $\mathrm{FT}_{PV}$ p [LF1;LF2] [DC_DC1;DC_DC2] [SA1;SA2]
[DC_AC1;DC_AC2] =
FTree p (OR [OR [LF1;DC_DC1;DC_AC1;SA1]; OR [LF2;DC_DC2;DC_AC2;SA2]])
Definition 13:
$\vdash$ $\mathrm{FT}_{STEAM}$ p [BO1;BO2;BO3] [TA1;TA2;TA3] =
FTree p (AND [AND [BO1;TA1]; AND [BO2;TA2]; AND [BO3;TA3]])
Steps 2 and 3 (Construction of a CCD and Reduction):
Construct a formal complete CCD for all loads in our case study (Fig. 11),
i.e., A, B, C, and D, then remove the irrelevant decision boxes according to
the electrical power network functional behavior. For instance, we can model
the CCD models for loads A and D, as shown in Fig. 14, respectively, in HOL4
as:
Definition 14:
$\vdash$ CCD_LOAD_A =
CONSEQ_BOX p
[[DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 1 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)];
[DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 0 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 1 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)];
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 0 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)]]
Definition 15:
$\vdash$ CCD_LOAD_D =
CONSEQ_BOX p
[[DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 1 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$);
DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX p
1 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)];
[DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 1 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$);
DEC_BOX p 1 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX p
0 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)];
⋮
[DEC_BOX p 0 ($\overline{\mathrm{FT}_{STEAM}}$,$\mathrm{FT}_{STEAM}$);DEC_BOX
p 0 ($\overline{\mathrm{FT}_{PV}}$,$\mathrm{FT}_{PV}$)]]
Figure 14: CCD Analysis of Loads A and D
Step 4 (Probabilistic analysis):
We can use our proposed formal approach to express subsystem-level
failure/reliability probabilistic expressions of electrical power grids, which
enable us to analyze the cascading dependencies with many subsystem levels,
based on any probabilistic distribution. In this work, we assumed that the
failure of each component is exponentially distributed (i.e., CDF p X t = 1
$-$ $\mathrm{e}^{(-\lambda_{X}t)}$, where $\lambda_{X}$ is the failure rate of
the variable X and t is a time index).
#### 5.1.1 $\mathcal{FOR}$ Analysis
Using Definitions 12 and 13 with the assumption that the failure states of
components are exponentially distributed, we can formally specify the
probabilistic $\mathcal{FOR}$ expression for both PV and steam power plants,
in HOL4 as:
Definition 16:
$\vdash$ $\mathcal{FOR}_{PV}$ p [LF1;LF2] [DC_DC1;DC_DC2] [SA1;SA2]
[DC_AC1;DC_AC2] =
prob p ($\mathrm{FT}_{PV}$ p ($\downarrow$ [LF1;LF2]) ($\downarrow$
[DC_DC1;DC_DC2])
($\downarrow$ [SA1;SA2]) ($\downarrow$ [DC_AC1;DC_AC2]))
Definition 17:
$\vdash$ $\mathcal{FOR}_{STEAM}$ p [BO1;BO2;BO3] [TA1;TA2;TA3] =
prob p ($\mathrm{FT}_{STEAM}$ p ($\downarrow$ [BO1;BO2;BO3]) ($\downarrow$
[TA1;TA2;TA3])
where the function $\downarrow$ takes a list of $\mathcal{N}$ components and
assigns an exponential failing event to each component in the list.
We can formally verify the above-expressions of $\mathcal{FOR}_{PV}$ and
$\mathcal{FOR}_{STEAM}$, in HOL4 as:
Theorem 23:
$\vdash$ $\mathcal{FOR}_{PV}$ p [LF1;LF2] [DC_DC1;DC_DC2] [SA1;SA2]
[DC_AC1;DC_AC2] =
$1-\mathrm{e}^{(-\lambda_{LF1}t)}$ $\times$ $\mathrm{e}^{(-\lambda_{LF2}t)}$
$\times$ $\mathrm{e}^{(-\lambda_{DC\\_DC1}t)}$ $\times$
$\mathrm{e}^{(-\lambda_{DC\\_DC2}t)}$ $\times$
$\mathrm{e}^{(-\lambda_{SA1}t)}$ $\times$ $\mathrm{e}^{(-\lambda_{SA2}t)}$
$\times$
$\mathrm{e}^{(-\lambda_{DC\\_AC1}t)}$ $\times$
$\mathrm{e}^{(-\lambda_{DC\\_AC2}t)}$
Theorem 24:
$\vdash$ $\mathcal{FOR}_{STEAM}$ p [BO1;BO2;BO3] [TA1;TA2;TA3] =
$(1-\mathrm{e}^{(-\lambda_{BO1}t)})$ $\times$
$(1-\mathrm{e}^{(-\lambda_{BO2}t)})$ $\times$
$(1-\mathrm{e}^{(-\lambda_{BO3}t)})$ $\times$
$(1-\mathrm{e}^{(-\lambda_{TA1}t)})$ $\times$
$(1-\mathrm{e}^{(-\lambda_{TA2}t)})$ $\times$
$(1-\mathrm{e}^{(-\lambda_{TA3}t)})$
#### 5.1.2 $\mathcal{SAIDI}$ Analysis
Using Theorems 1-24 with the assumption that the failure states of components
are exponentially distributed, we can formally verify $\mathcal{SAIDI}_{G}$
(Eq. 19), in HOL4 as:
Theorem 25:
$\vdash$ $\mathcal{SAIDI}$
[[CONSEQ_PATH p
[DEC_BOX p 1
(FTree p (NOT ($\mathrm{FT}_{STEAM}$ p ($\downarrow$ [BO1;BO2;BO3])
($\downarrow$ [TA1;TA2;TA3]))),
$\mathrm{FT}_{STEAM}$ p ($\downarrow$ [BO1;BO2;BO3]) ($\downarrow$
[TA1;TA2;TA3]));
DEC_BOX p 0
(FTree p (NOT ($\mathrm{FT}_{PV}$ p ($\downarrow$ [LF1;LF2]) ($\downarrow$
[DC_DC1;DC_DC2])
($\downarrow$ [SA1;SA2]) ($\downarrow$ [DC_AC1;DC_AC2]))),
$\mathrm{FT}_{PV}$ p ($\downarrow$ [LF1;LF2]) ($\downarrow$ [DC_DC1;DC_DC2])
($\downarrow$ [SA1;SA2]) ($\downarrow$ [DC_AC1;DC_AC2]))];
[DEC_BOX p 0
(FTree p (NOT ($\mathrm{FT}_{STEAM}$ p ($\downarrow$ [BO1;BO2;BO3])
($\downarrow$ [TA1;TA2;TA3]))),
$\mathrm{FT}_{STEAM}$ p ($\downarrow$ [BO1;BO2;BO3]) ($\downarrow$
[TA1;TA2;TA3]));
DEC_BOX p 1
(FTree p (NOT ($\mathrm{FT}_{PV}$ p ($\downarrow$ [LF1;LF2]) ($\downarrow$
[DC_DC1;DC_DC2])
($\downarrow$ [SA1;SA2]) ($\downarrow$ [DC_AC1;DC_AC2]))),
$\mathrm{FT}_{PV}$ p ($\downarrow$ [LF1;LF2]) ($\downarrow$ [DC_DC1;DC_DC2])
($\downarrow$ [SA1;SA2]) ($\downarrow$ [DC_AC1;DC_AC2]))]];
…]
[MTTR_LoadA;MTTR_LoadB;MTTR_LoadC;MTTR_LoadD]
[CN_LoadA; CN_LoadB; CN_LoadC; CN_LoadD] p =
$\frac{\begin{aligned}
&((1-(1-\mathrm{e}^{(-\lambda_{BO1}t)})\times(1-\mathrm{e}^{(-\lambda_{BO2}t)})\times(1-\mathrm{e}^{(-\lambda_{BO3}t)})\times\\\
&\hskip
5.406pt\quad\quad(1-\mathrm{e}^{(-\lambda_{TA1}t)})\times(1-\mathrm{e}^{(-\lambda_{TA2}t)})\times(1-\mathrm{e}^{(-\lambda_{TA3}t)}))\times\\\
&\hskip
4.2679pt(1-\mathrm{e}^{(-\lambda_{LF1}t)}\times\mathrm{e}^{(-\lambda_{LF2}t)}\times\mathrm{e}^{(-\lambda_{DC\\_DC1}t)}\times\mathrm{e}^{(-\lambda_{DC\\_DC2}t)}\times\\\
&\hskip
5.406pt\quad\quad\mathrm{e}^{(-\lambda_{DC\\_AC1}t)}\times\mathrm{e}^{(-\lambda_{DC\\_AC2}t)}\times\mathrm{e}^{(-\lambda_{SA1}t)}\times\mathrm{e}^{(-\lambda_{SA2}t)})+\\\
&\hskip
4.2679pt(1-\mathrm{e}^{(-\lambda_{BO1}t)})\times(1-\mathrm{e}^{(-\lambda_{BO2}t)})\times(1-\mathrm{e}^{(-\lambda_{BO3}t)})\times\\\
&\hskip
4.2679pt(1-\mathrm{e}^{(-\lambda_{TA1}t)})\times(1-\mathrm{e}^{(-\lambda_{TA2}t)})\times(1-\mathrm{e}^{(-\lambda_{TA3}t)})\times\\\
&\hskip
4.2679pt\mathrm{e}^{(-\lambda_{LF1}t)}\times\mathrm{e}^{(-\lambda_{LF2}t)}\times\mathrm{e}^{(-\lambda_{DC\\_DC1}t)}\times\mathrm{e}^{(-\lambda_{DC\\_DC2}t)}\times\\\
&\hskip
4.2679pt\mathrm{e}^{(-\lambda_{DC\\_AC1}t)}\times\mathrm{e}^{(-\lambda_{DC\\_AC2}t)}\times\mathrm{e}^{(-\lambda_{SA1}t)}\times\mathrm{e}^{(-\lambda_{SA2}t)})\times\\\
&\hskip
4.2679pt\mathrm{MTTR\\_LoadA}\times\mathrm{CN\\_LoadA}+\dots)\end{aligned}}{\mathrm{CN\\_LoadA}+\mathrm{CN\\_LoadB}+\mathrm{CN\\_LoadC}+\mathrm{CN\\_LoadD}}$
To further facilitate the exploitation of our proposed approach for power grid
reliability engineers, we defined a Standard Meta Language (SML) functions
[33] that can numerically evaluate the above-verified expressions of
$\mathcal{FOR}_{PV}$, $\mathcal{FOR}_{STEAM}$, and $\mathcal{SAIDI}$.
Subsequently, we compared our results with MATLAB CCD algorithm based on
Monte-Carlo Simulation (MCS) and also with other existing subsystem-level
reliability analysis techniques, such as HiP-HOPS and FMR, to ensure the
accuracy of our computations, which is presented in the next section.
### 5.2 Experimental Results and Discussion
Considering the failure rates of the power plant components
$\lambda_{\mathrm{BO}}$, $\lambda_{\mathrm{TA}}$, $\lambda_{\mathrm{LF}}$,
$\lambda_{\mathrm{DC\\_DC}}$, $\lambda_{\mathrm{DC\\_AC}}$ and
$\lambda_{\mathrm{SA}}$ are 0.91, 0.84, 0.96, 0.67, 0.22, and 0.56 per year
[38], respectively. Also, assuming that $\mathrm{MTTR}_{\mathrm{Load}_{A}}$,
$\mathrm{MTTR}_{\mathrm{Load}_{B}}$, $\mathrm{MTTR}_{\mathrm{Load}_{C}}$, and
$\mathrm{MTTR}_{\mathrm{Load}_{D}}$ are 12, 20, 15, and 10 hours/interruption
[39] and $\mathrm{CN}_{\mathrm{Load}_{A}}$, $\mathrm{CN}_{\mathrm{Load}_{B}}$,
$\mathrm{CN}_{\mathrm{Load}_{C}}$, and $\mathrm{CN}_{\mathrm{Load}_{D}}$ are
500, 1800, 900, and 2500 customers, respectively. The reliability study is
undertaken for 1 year, i.e., t = 8760 hours. Based on the given data, we can
evaluate $\mathcal{FOR}$ and $\mathcal{SAIDI}$ for the electrical power
network (Fig. 11) using following techniques:
1. 1.
Our proposed SML functions to evaluate the verified expressions of
$\mathcal{FOR}_{PV}$, $\mathcal{FOR}_{STEAM}$, and $\mathcal{SAIDI}$ in HOL4
(Theorems 23-25), as shown in Fig. 15.
Figure 15: SML Functions: $\mathcal{FOR}$ and $\mathcal{SAIDI}$ Results
2. 2.
MATLAB MCS-based toolbox that uses a random-based algorithm to obtain
$\mathcal{FOR}$ and $\mathcal{SAIDI}$ for the electrical grid. The steps
followed in this technique are as follows [40]:
* •
Read the values of failure rate $\lambda$ in f/hours and repair time r in
hours for each component
* •
Generate a random number U
* •
Calculate the predicted next Time to Fail (TTF) and Time to Repair (TTR) from
the equations
$TTF=\frac{-\ln{U}}{\lambda}\hskip 11.38109ptTTR=\frac{-\ln{U}}{r}$ (20)
* •
Repeat the above iterative process till the number of iterations exceeds 1e5
Based on the above-mentioned MCS steps, we obtain different results of
$\mathcal{FOR}$ and $\mathcal{SAIDI}$ every run of the algorithm depending on
the generated random number with a tolerance error between 4-9%. So, we
present in Table 7 the best-estimated results of $\mathcal{FOR}$ and
$\mathcal{SAIDI}$ in MATLAB based on the MCS approach with the least errors.
Subsequently, we take the mean average of all the obtained $\mathcal{FOR}$ and
$\mathcal{SAIDI}$ results for the power grid.
Table 7: MATLAB MCS: $\mathcal{FOR}$ and $\mathcal{SAIDI}$ Results Run | $\mathcal{FOR}_{PV}$ | $\mathcal{FOR}_{STEAM}$ | $\mathcal{SAIDI}$
---|---|---|---
1 | 88.55e-2 | 36.18e-3 | 5.8023
2 | 107.19e-2 | 40.03e-3 | 6.5045
3 | 93.52e-2 | 36.35e-3 | 6.0222
5 | 110.17e-2 | 43.03e-3 | 7.0495
4 | 95.24e-2 | 38.66e-3 | 6.3960
Average | 98.93e-2 | 38.85e-3 | 6.3549
3. 3.
The Failure Mode Reasoning (FMR) approach, which identifies all the failure
modes of safety-critical system inputs that can result in an undesired state
at its output. The FMR process consists of four main stages [10]:
1. (a)
Composition: Failure mode variables are defined and a set of logical
implication statements is generated that express local failure modes.
2. (b)
Substitution: Local statements will be combined to create a single global
implication statement between the critical-system inputs and outputs.
3. (c)
Simplification: The complex formula is simplified, where we trim off any
redundant statements.
4. (d)
Calculation: The probability of failure is evaluated using the component
failure rates.
Based on the above-mentioned FMR procedures, we can express the component-
level failure analysis of the PV power plant (Fig. 13) as:
$(\hat{o}=\dot{f})\Rightarrow(\hat{x_{1}}=\dot{f}\lor\hat{x_{2}}=\dot{f})$
(21)
The above equation means that if the output $o$ is False by fault then either
one of its inputs to the OR gate, i.e., $x_{1}$ or $x_{2}$, must be False by
fault. We now need to determine what can cause $\hat{x_{1}}=\dot{f}$ and
$\hat{x_{2}}=\dot{f}$. Similar to Eq. 6, we can write:
$(\hat{x_{1}}=\dot{f})\Rightarrow(\hat{x_{3}}=\dot{f}\hskip
2.84526pt\lor\hat{x_{4}}=\dot{f}\hskip 2.84526pt\lor\hat{x_{5}}=\dot{f}\hskip
2.84526pt\lor\hat{x_{6}}=\dot{f})$ (22)
$(\hat{x_{2}}=\dot{f})\Rightarrow(\hat{x_{7}}=\dot{f}\hskip
2.84526pt\lor\hat{x_{8}}=\dot{f}\hskip 2.84526pt\lor\hat{x_{9}}=\dot{f}\hskip
2.84526pt\lor\hat{x_{10}}=\dot{f})$ (23)
where $x_{3}$, $x_{4}$, $x_{5}$, $x_{6}$, $x_{7}$, $x_{8}$, $x_{9}$, $x_{10}$
are $LF_{1}$, $DC\\_DC_{1}$, $DC\\_AC_{1}$, $SA_{1}$, $LF_{2}$, $DC\\_DC_{2}$,
$DC\\_AC_{2}$, $SA_{2}$, respectively. Similarly, we can express the
component-level failure analysis of the steam power plant (Fig. 13) as:
$(\hat{o}=\dot{f})\Rightarrow(\hat{x_{11}}=\dot{f}\hskip 1.42262pt\wedge\hskip
1.42262pt\hat{x_{12}}=\dot{f}\hskip 1.42262pt\wedge\hskip
1.42262pt\hat{x_{13}}=\dot{f})$ (24)
$(\hat{x_{11}}=\dot{f})\Rightarrow(\hat{x_{14}}=\dot{f}\hskip
1.42262pt\wedge\hskip 1.42262pt\hat{x_{15}}=\dot{f})$ (25)
$(\hat{x_{12}}=\dot{f})\Rightarrow(\hat{x_{16}}=\dot{f}\hskip
1.42262pt\wedge\hskip 1.42262pt\hat{x_{17}}=\dot{f})$ (26)
$(\hat{x_{13}}=\dot{f})\Rightarrow(\hat{x_{18}}=\dot{f}\hskip
1.42262pt\wedge\hskip 1.42262pt\hat{x_{19}}=\dot{f})$ (27)
where $x_{14}$, $x_{15}$, $x_{16}$, $x_{17}$, $x_{18}$, $x_{19}$, are
$BO_{1}$, $TA_{1}$, $BO_{2}$, $TA_{2}$, $BO_{3}$, $TA_{3}$, respectively.
Table 8 shows the results of $\mathcal{FOR}_{PV}$, $\mathcal{FOR}_{STEAM}$,
and $\mathcal{SAIDI}$ based on FMR analysis using the assumed failure rates of
the power plant components.
Table 8: FMR: $\mathcal{FOR}$ and $\mathcal{SAIDI}$ Results $\mathcal{FOR}_{PV}$ | $\mathcal{FOR}_{STEAM}$ | $\mathcal{SAIDI}$
---|---|---
99.19e-2 | 38.87e-3 | 6.3728
According to Jahanian et al. [11], the soundness of the obtained FMR equations
(Eq. 21 to Eq. 27) needs to be proven mathematically.
Figure 16: HiP-HOPS: PV Plant FMECA Analysis
4. 4.
The HiP-HOPS software for failure analysis, which can perform FMECA analysis
by given architectural blocks that hierarchically describe a safety-critical
system at the subsystem level. Fig. 16 and Fig. 17 depict the FMECA analysis
of the PV and steam power plants using the HiP-HOPS software, respectively.
The probabilistic results of $\mathcal{FOR}_{PV}$, $\mathcal{FOR}_{STEAM}$,
and $\mathcal{SAIDI}$ based on HiP-HOPS analysis are equivalent to the FMR
analysis results presented in Table 8.
Figure 17: HiP-HOPS: Steam Plant FMECA Analysis
It can be observed that $\mathcal{SAIDI}$ result obtained from our formal HOL4
analysis are approximately equivalent to the corresponding ones calculated
using FMR and HiP-HOPS approaches. On the other hand, MATLAB MCS-based uses a
random-based algorithm, which estimates different results of $\mathcal{FOR}$
and $\mathcal{SAIDI}$ every generation of a random number with errors between
4-9%. This clearly demonstrates that our analysis is not only providing the
correct result but also with a formally proven reliability expressions
(Theorems 23-25) compared to simulation tools, i.e., the soundness of
subsystem-level reliability analysis. By performing the formal CCD step-
analysis of a real-world 39-bus electrical power network, we demonstrated the
practical effectiveness of the proposed CCD formalization in HOL4, which will
help design engineers to meet the desired quality requirements. Also, our
proposed formal approach can be used to analyze larger scale CCD models of
other complex electrical power system applications, such as Smartgrids [1].
## 6 Conclusions
In this work, we developed a formal approach for Cause-Consequence Diagrams
(CCD), which enables safety engineers to perform $\mathcal{N}$-level CCD
analysis of safety-critical systems within the sound environment of the HOL4
theorem prover. Our proposed approach provides new CCD mathematical
formulations, which their correctness was verified in the HOL4 theorem prover.
These formulations are capable of performing CCD analysis of multi-state
system components and based on any given probabilistic distribution and
failure rates. These features are not available in any other existing
approaches for subsystem-level reliability analysis. The proposed
formalization is limited to perform CCD-based reliability analysis at the
subsystem level that integrates static dependability analysis. However, this
formalization is generic and can be extended to perform dynamic failure
analysis of dynamic subsystems where no dependencies exist in different
subsystems. We demonstrated the practical effectiveness of the proposed CCD
formalization by performing the formal CCD step-analysis of a standard IEEE
39-bus electrical power network system and also formally verified the power
plants Force Outage Rate ($\mathcal{FOR}$) and the System Average Interruption
Duration Index ($\mathcal{SAIDI}$). Eventually, we compared the
$\mathcal{FOR}$ and $\mathcal{SAIDI}$ results obtained from our formal CCD-
based reliability analysis with the corresponding ones using MATLAB based on
Monte-Carlo Simulation (MCS), the HiP-HOPS software tool, and the Failure Mode
Reasoning (FMR) approach. As future work, we plan to integrate Reliability
Block Diagrams (RBDs) [41] as reliability functions in the CCD analysis, which
will enable us to analyze hierarchical systems with different component
success configurations, based on our CCD formalization in the HOL4 theorem
prover.
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|
acmlicensed 2024 XXXXXXXXX MAPLETRANS X Y Z 0
# On the Certification of the Kinematics of 3-DOF Spherical Parallel
Manipulators
Alexandre Lê<EMAIL_ADDRESS><EMAIL_ADDRESS>Safran
Electronics & Defense 100 avenue de Paris Massy Île-de-France France 91344
Sorbonne Université, Université de Paris Cité, Institut de Mathématiques de
Jussieu Paris Rive Gauche 4 place Jussieu Paris Île-de-France France 75252
CEDEX 05 Inria Paris 2 rue Simone Iff Paris Île-de-France France 75012 ,
Damien Chablat<EMAIL_ADDRESS>LS2N, UMR CNRS Nantes France ,
Guillaume Rance<EMAIL_ADDRESS>Safran Electronics & Defense
100 avenue de Paris Massy Île-de-France France 91344 and Fabrice Rouillier
<EMAIL_ADDRESS>Sorbonne Université, Université de Paris Cité,
CNRS, Institut de Mathématiques de Jussieu Paris Rive Gauche 4 place Jussieu
Paris Île-de-France France 75252 CEDEX 05 Inria Paris 2 rue Simone Iff Paris
Île-de-France France 75012
(2024)
###### Abstract.
Abstract. This paper aims to study a specific kind of parallel robot:
Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A
focus is made on the kinematics of such mechanisms, especially taking into
account uncertainties (e.g. on conception & fabrication parameters, measures)
and their propagations. Such considerations are crucial if we want to control
our robot correctly without any undesirable behavior in its workspace (e.g.
effects of singularities). In this paper, we will consider two different
approaches to study the kinematics and the singularities of the robot of
interest: symbolic and semi-numerical. By doing so, we can compute a
singularity-free zone in the work- and joint spaces, considering given
uncertainties on the parameters. In this zone, we can use any control law to
inertially stabilize the upper platform of the robot.
###### Key words and phrases:
parallel robots, non-linear systems, polynomial systems, singularity,
kinematics, certification, inertial stabilization
<ccs2012> <concept> <concept_id>10010520.10010553.10010554</concept_id>
<concept_desc>Computer systems organization Robotics</concept_desc>
<concept_significance>500</concept_significance> </concept> <concept>
<concept_id>10010147.10010148.10010164.10010168</concept_id>
<concept_desc>Computing methodologies Representation of
polynomials</concept_desc> <concept_significance>500</concept_significance>
</concept> <concept>
<concept_id>10010147.10010341.10010342.10010343</concept_id>
<concept_desc>Computing methodologies Modeling methodologies</concept_desc>
<concept_significance>500</concept_significance> </concept> <concept>
<concept_id>10010147.10010341.10010342.10010344</concept_id>
<concept_desc>Computing methodologies Model verification and
validation</concept_desc> <concept_significance>500</concept_significance>
</concept> <concept>
<concept_id>10010147.10010341.10010342.10010345</concept_id>
<concept_desc>Computing methodologies Uncertainty
quantification</concept_desc> <concept_significance>500</concept_significance>
</concept> </ccs2012>
[500]Computer systems organization Robotics [500]Computing methodologies
Representation of polynomials [500]Computing methodologies Modeling
methodologies [500]Computing methodologies Model verification and validation
[500]Computing methodologies Uncertainty quantification
## 1\. Context of the study
### 1.1. Introduction
In order to take a panorama picture on a moving career using high definition
cameras, a classical approach is to use a gimbal system Hilkert, (2008);
Masten, (2008). Gimbal systems can be regarded as serial robots and provide up
to 3 DOF (yaw, pitch, roll). However, those ones are limited by their
architectures: stabilizing the camera means stabilizing a mass which
downgrades the quality of inertial stabilization. If we want to improve the
inertial stablization of such devices (in terms of quality and even DOF), a
solution can found by studying other architectures: parallel robots.
### 1.2. Generalities on parallel robots
_Parallel robots_ are defined in Leinonen, (1991) as robots that control the
motion of their end-effectors by means of at least two kinematic chains going
from the end-effector towards the fixed base. In other words, parallel robots
are manipulators that are composed of two plateforms: one at the base and the
other at the top called the _moving platform_. These platforms are connected
by $n$ kinematic chains that can be regarded as robot legs. Each kinematic
chain has joints $A_{ij}$ that can be either motorized with actuactors (we
call them _active joints_) – or not (we call them _passive joints_). Finally,
two consecutive joints are linked by a _body_. A typical kinematic chain of a
parallel robot is depicted in Figure 1.
[scale=0.85]TikZ/robot_parallele_structure
Figure 1. General structure of a parallel robot
Due to their architecture, parallel robots are mechanisms presenting very good
performance in terms of dynamics, stiffness and accuracy to manipulate large
loads. Moreover, such architectures also make it possible to reduce the mass
of the movable links. Indeed, all the actuators are mainly fixed on the base
and many parts are subject to traction/compression constraints to the extent
that it possible to use less powerful actuators. Such nice properties are very
suitable for our applications. First appearance of parallel robots are
hexapods (in the middle of the 20th century) that are basically used for
flight simulations or pneumatic testing using their prismatic legs. However,
as they were less widespread than their serial counterparts, studies and
knowledge about their modeling had been limited even if they are gaining
interests in the recent years (medical, food industry, etc.).
### 1.3. Generalities on SPMs
There is no unique way Merlet, (2006); Khalil and Briot, (2015) to classify
parallel robots (some authors speak in terms of the number of DOF while others
focus on their types, e.g. revolute or prismatic joints). In this paper, we
will especially study a specific type of parallel robots: _Spherical Parallel
Manipulators_ (SPM) that are non-redundant111the number of actuators
corresponds to the DOF. SPMs only make rotational motions with their revolute
joints. In the special case of non-redundant SPMs, the mobile platform has 3
DOF that we will call _orientation_. The following figures illustrate some
remarkable (non-redundant) SPMs:
* •
Fig. 2(a) depicts the _agile eye_ by Gosselin and Hamel, (1994) ;
* •
Fig. 2(b) depicts the _agile wrist_ by Shintemirov et al., (2015) ;
* •
Fig. 2(c) depicts the _agile wrist with coaxial input shafts_ by Tursynbek et
al., (2019).
(a) Agile eye
(b) Agile wrist
(c) Coaxial agile wrist
Figure 2. Examples of non-redundant SPMs (3-RRR)
The last type of SPM is particularly suitable for our case since it allows
unlimited rolling that can be useful to obtain a panorama while stabilizing
the upper platform in the same time. However, all the non-redundant SPMs have
the same modeling that will be detailled in the next section.
## 2\. Modeling of a non-redundant 3-DOF SPM
### 2.1. Description
[scale=0.7]TikZ/poignet_agile_sc_rep_with_etas
Figure 3. Illustration of a typical SPM with conception paramaters (red $+$
green), local frames (dark blue)
Before modeling the SPM, let us describe the robot in terms of conception
parameters and frames. Figure 3 illustrates a typical SPM where these elements
are shown. According to this figure, any SPM can be described as a manipulator
that has two platforms connected with 3 legs. Each leg has 2 links (or body)
and an actuated joint at its base. The actuated joint variables (angles) will
be denoted as $\theta_{i}$ for the $i$th leg. This figure also highlights the
fact that SPMs only make pure rotations around $O$ called _center of rotation_
of the SPM. Using this property, their motions can be described with only
vectors. Those vectors must be expressed in the same frame. For convinience,
we will express all the vectors and coordinates in the reference frame
$\mathcal{F}_{0}\triangleq\left(O,\bm{x}_{0},\bm{y}_{0},\bm{z}_{0}\right)$.
There are three types of vectors: the ones describing the base denoted as
$\bm{u}_{i}$, the ones describing the moving platform denoted as $\bm{v}_{i}$
and the ones describing the intermediate joints denoted as $\bm{w}_{i}$, with
$i\in\left\llbracket 1,3\right\rrbracket$. All these vectors are concurrent in
$O$.
First $\prescript{0}{}{\bm{u}_{i}}$ (i.e. $\bm{u}_{i}$ w.r.t.
$\mathcal{F}_{0}$) can be obtained using the following transformations:
$\displaystyle\prescript{0}{}{\bm{u}_{i}}$
$\displaystyle=\bm{R}_{z}{\left(\eta_{i}\right)}\,\bm{R}_{x}{\left(\beta_{1}-\pi\right)}\,\prescript{0}{}{\bm{z}_{0}}$
(1)
$\displaystyle=\left[\;\begin{matrix}-\sin\left(\eta_{i}\right)\sin\left(\beta_{1}\right)\\\
\cos\left(\eta_{i}\right)\sin\left(\beta_{1}\right)\\\
-\cos\left(\beta_{1}\right)\end{matrix}\;\right]$
where $\bm{R}_{x}$ (resp. $\bm{R}_{y}$ and $\bm{R}_{z}$) denotes the rotation
matrix around the local $x$-axis (resp. $y$\- and $z$-axis). Then, vectors
$\bm{w}_{i}$ w.r.t. frame $\mathcal{F}_{0}$ can be expressed as:
$\displaystyle\prescript{0}{}{\bm{w}_{i}}$
$\displaystyle=\bm{R}_{z}{\left(\eta_{i}\right)}\,\bm{R}_{x}{\left(\beta_{1}-\pi\right)}\,\bm{R}_{z}{\left(\theta_{i}\right)}\,\bm{R}_{x}{\left(\alpha_{1}\right)}\,\prescript{0}{}{\bm{z}_{0}}$
(2)
$\displaystyle=\left[\;\begin{matrix}-\sin{\left(\eta_{i}\right)}\sin{\left(\beta_{1}\right)}\cos{\left(\alpha_{1}\right)}+\sin{\left(\alpha_{1}\right)}\left[\cos{\left(\eta_{i}\right)}\sin{\left(\theta_{i}\right)}-\sin{\left(\eta_{i}\right)}\cos{\left(\beta_{1}\right)}\cos{\left(\theta_{i}\right)}\right]\\\
\cos{\left(\eta_{i}\right)}\sin{\left(\beta_{1}\right)}\cos{\left(\alpha_{1}\right)}+\sin{\left(\alpha_{1}\right)}\left[\sin{\left(\eta_{i}\right)}\sin{\left(\theta_{i}\right)}+\cos{\left(\eta_{i}\right)}\cos{\left(\beta_{1}\right)}\cos{\left(\theta_{i}\right)}\right]\\\
\sin{\left(\beta_{1}\right)}\cos{\left(\theta_{i}\right)}\sin{\left(\alpha_{1}\right)}-\cos{\left(\alpha_{1}\right)}\cos{\left(\beta_{1}\right)}\end{matrix}\;\right]$
Finally, the moving platform vectors $\prescript{0}{}{\bm{v}_{i}}$ are:
$\prescript{0}{}{\bm{v}_{i}}=\bm{M}\,\bm{R}_{z}{\left(\eta_{i}\right)}\,\bm{R}_{x}{\left(-\beta_{2}\right)}\,\prescript{0}{}{\bm{z}_{0}}$
(3)
where $\bm{M}$ denotes the orientation matrix of the mobile platform and can
be expressed using several formalisms (Euler angles, Tait-Bryan angles,
quaternions, …). In our case, the $ZXY$ Tait-Bryan angles are used to describe
our orientation:
$\bm{M}\triangleq\bm{R}_{z}{\left(\chi_{3}\right)}\,\bm{R}_{x}{\left(\chi_{1}\right)}\,\bm{R}_{y}{\left(\chi_{2}\right)}$
(4)
These equations highlight the fact that a robot can be described in terms of
conception parameters and 2 types of variables: either its joint variables or
its end-effector coordinates, namely its moving platform’s orientation. The
joint variables are real values that belong to the _joint space_ $\mathcal{Q}$
and will be put into a vector
$\bm{\theta}\triangleq\left[\;\begin{matrix}\theta_{1}&\theta_{2}&\theta_{3}\end{matrix}\;\right]^{\mathsf{T}}$.
The end-effector coordinates are real values that belong to the _workspace_
$\mathcal{W}$ and will be concatenated into a vector
$\bm{\chi}\triangleq\left[\;\begin{matrix}\chi_{1}&\chi_{2}&\chi_{3}\end{matrix}\;\right]^{\mathsf{T}}$.
This is fundamental to establish any modeling of a robot. In this paper, the
_kinematics_ of SPMs are studied through their _geometric_ and _first order
kinematic models_.
### 2.2. Geometric and first order kinematic models
The _geometric model_ of a parallel manipulator is a system of equations
describing the relationships between the actuated joint variables
$\bm{\theta}$ and the coordinates (orientations) $\bm{\chi}$ of the moving
platform.
$\mathcal{W}$$\bm{\chi}$$\mathcal{Q}$$\bm{\theta}$$\bm{\theta}_{d}$Inverse
problem Forward problem Figure 4. Principle of the geometric model
An extended problem is to also consider passive intermediate joints
($\bm{\theta}_{d}$) which will not be covered in this article. Figure 4
describes the two points of view of the same problem as previously stated. As
we only focus on non-redundant SPMs, their geometric models consist in a
system $\bm{f}$ of $n_{\text{dof}}=n_{a}=3$ independent equations with
variables $\bm{\theta}$ and $\bm{\chi}$. The following system describes such a
model for SPMs:
$\bm{f}\left(\bm{\theta},\bm{\chi}\right)\triangleq\left[\;\begin{matrix}\bm{w}_{1}^{\mathsf{T}}\,\bm{v}_{1}-\cos{\left(\alpha_{2}\right)}\\\
\bm{w}_{2}^{\mathsf{T}}\,\bm{v}_{2}-\cos{\left(\alpha_{2}\right)}\\\
\bm{w}_{3}^{\mathsf{T}}\,\bm{v}_{3}-\cos{\left(\alpha_{2}\right)}\end{matrix}\;\right]=\bm{0}_{3\times
1}$ (5)
The forward geometric problem (FGM) is taking (5) with $\bm{\theta}$ being
kwown and try to solve it by finding the corresponding $\bm{\chi}$. The
solutions found are then called _assembly modes_ of the parallel robot.
Conversely, the inverse geometric problem (IGM) is taking (5) and try to solve
it by finding the corresponding $\bm{\theta}$. The solutions found are then
called _working modes_ of the parallel robot. By differentiating (5) w.r.t.
time, we get the _first order kinematic model_ which can be written as
$\bm{A}\,\dot{\bm{\chi}}+\bm{B}\,\dot{\bm{\theta}}=\bm{0}_{3\times 1}$ where
$\bm{A}\triangleq\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}\bm{f}}{\IfBeginWith{\bm{f}}{f}{\partial\mathopen{}^{0}\\!\bm{f}}{\partial\mathopen{}^{0}\bm{f}}}}{\StrSubstitute{\StrSubstitute{\bm{\chi}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{\bm{f}}}{\StrSubstitute{\StrSubstitute{\bm{\chi}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}}}}}$
denotes the _parallel Jacobian matrix_ and
$\bm{B}\triangleq\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}\bm{f}}{\IfBeginWith{\bm{f}}{f}{\partial\mathopen{}^{0}\\!\bm{f}}{\partial\mathopen{}^{0}\bm{f}}}}{\StrSubstitute{\StrSubstitute{\bm{\theta}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{\bm{f}}}{\StrSubstitute{\StrSubstitute{\bm{\theta}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}}}}}$
denotes the _serial Jacobian matrix_.
## 3\. Implementation of the geometric model
### 3.1. Requirements and strategy
The geometric model was hitherto obtained through symbolic computation (here
using Maple software 2022 version). However, in order to be implemented, we
must:
* •
define a prescribed workspace $\mathcal{W}^{\star}$: in our case we want to
ensure that our SPM can stabilize its moving platform up to $\pm 20^{\circ}$
in roll & pitch. Thus, $\mathcal{W}^{\star}$ is the set
$\mathcal{W}^{\star}\triangleq\left\\{\left(\chi_{1},\chi_{2},\chi_{3}\right)\in\mathbb{R}^{3}\text{\;s.t.\;}\left|\chi_{1}\right|\leq
20^{\circ}\text{\;and\;}\left|\chi_{2}\right|\leq 20^{\circ}\right\\}$ (6)
###### Remark 1.
We previously defined $\mathcal{W}$ as the _workspace_ of our robot which is
the set of all orientations $\bm{\chi}$ that its moving platform can reach.
However, we only focus on our _prescribed workspace_
$\mathcal{W}^{\star}\subseteq\mathcal{W}$. This distinction is useful to
upgrade/optimize the performances of our robot that are subjects to
specifications.
* •
specify conception parameters (see Tab. 1): $\beta_{1}=0$ is chosen in order
to have coaxial input shafts (see Fig. 2(c) and 3). The $\eta_{i}$s are
defined such that the platforms’ joints are regularly spaced. The other
conception parameters are chosen using the global conditioning index approach
and optimal values determined in Bai, (2010) (see Appendix A).
Parameters | $\eta_{i}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\beta_{1}$ | $\beta_{2}$
---|---|---|---|---|---
Values (rad) | $2(i-1)\pi/3$ | $\pi/4$ | $\pi/2$ | $0$ | $\pi/2$
Table 1. Exact values of conception parameters
* •
be able to certify the SPM’s FGM and IGM in order to solve them correctly.
The last point is crucial. Although the system $\bm{f}$ is non-linear (which
can make the FGM and IGM harder to solve), it can be turned into a polynomial
system through appropriate changes of variables. This allows $\bm{f}$ to
become a system $\bm{S}$ of the form
$\bm{S}\triangleq\left\\{p_{1}(\bm{U},\bm{X})=0,\dots,p_{n_{a}}(\bm{U},\bm{X})=0\right\\}$
(7)
where $\bm{U}=(U_{1},\dots,U_{d})$ is the $d$-uple of parameters,
$\bm{X}=(X_{1},\dots,X_{n})$ the $n$-uple of unknowns and $p_{i}$,
$i\in\left\llbracket 1,n_{a}\right\rrbracket$ being polynomials in the
indeterminates $\bm{U}$, $\bm{X}$ with rational coefficients. The tangent
half-angle formulas are interesting changes of variables: this substitution
has the advantage to keep the same number of equations, parameters and
variables ($n_{a}=d=n=3$).
###### Remark 2.
This would not be case for sine/cosine changes of variables that double the
number of variables and equations.
Depending on the point of view (IGM or FGM), $\bm{U}$ and $\bm{X}$ can either
be $\bm{j}\triangleq\left\\{j_{i}=\tan{(\theta_{i}/2)},i=\left\llbracket
1,3\right\rrbracket\right\\}$ or
$\bm{o}\triangleq\left\\{o_{i}=\tan{(\chi_{i}/2)},i=\left\llbracket
1,3\right\rrbracket\right\\}$. Additionally, such changes of variables are
still valid considering our joint- and workspace:
$\theta_{i},\chi_{1},\chi_{2}\neq\pm\pi\;[2\pi]$. Appendix C shows the
explicit expression of $\bm{S}$, the geometric model of our SPM in its
polynomial form. We also assume that $\bm{S}$ has a finite number of complex
solutions: for almost all $d$-uples
$\bm{u}\triangleq\left(u_{1},\dots,u_{d}\right)\in\mathbb{C}^{d}$, the system
$\left.\bm{S}\right|_{\bm{U}=\bm{u}}=\left\\{p_{1}(\bm{u},\bm{X})=0,\dots,p_{n_{a}}(\bm{u},\bm{X})=0\right\\}$
has finitly many complex solutions. Such a system is called _generically zero-
dimensionnal_ and will be solved using _Algebraic Geometry_ techniques Cox et
al., (2015) by associating $\bm{S}$ with $\mathcal{I}=\left\langle
p_{1},\dots,p_{n_{a}}\right\rangle$ being the _ideal_ of
$\mathbb{Q}[\bm{U},\bm{X}]$ generated by the polynomials
$p_{1},\dots,p_{n_{a}}$, such that
$\overline{\operatorname{proj}_{\bm{U}}\left(\mathcal{V}\left(\mathcal{I}\right)\right)}=\mathbb{C}^{d}$,
where $\operatorname{proj}_{\bm{U}}$ denotes the projection onto the parameter
space and $\overline{\mathcal{V}}$ the closure of any subset
$\mathcal{V}\subset\mathbb{C}^{d}$. Thus, the complex solutions of $\bm{S}$
define the _algebraic variety_ $\mathcal{V}(\mathcal{I})$.
However, we also want to go further by considering the robustness of our SPM:
solving (5) means assuming that the (theorical) conception parameter values
will perfectly correspond to the real case values, which is a strong
hypothesis and obviously not true. Indeed, parallel robots are unavoidably
subject to uncertainties such as fabrication parameters (assembling tolerances
of the mechanical parts) or noise in the sensors. Additionally, implementing
the kinematics requires numerical approximations (convert irrational parameter
values into rational ones).
Despite this, we want to ensure that for small deformations of parameters,
solutions found will still be close to the perfect case. In other words, we
want to _certify_ our SPM’s modeling. One of the tools dedicated to this
certification work is the notion of _discriminant variety_. This object is
closely related to the idea of _projection_ onto the space of paramaters
$\operatorname{proj}_{\bm{U}}$, as illustrated in Figure 5. The goal is to
have a set of parameters that does not meet and is far from all the numerical
unstabilities of the so called discriminant variety. First let us recall its
definition from Lazard and Rouillier, (2007); Chablat et al., (2020).
###### Definition 1 (Discriminant Variety).
The discriminant variety of $\mathcal{V}(\mathcal{I})$ w.r.t.
$\operatorname{proj}_{\bm{U}}$ denoted as $\mathcal{W}_{D}$ is the smallest
algebraic variety of $\mathbb{C}^{d}$ such that given any simply connected
subset $\mathcal{C}$ of $\mathbb{R}^{d}\,\backslash\,\mathcal{W}_{D}$, the
number of real solutions of $\bm{S}$ is constant over $\bm{U}$. In our case,
$\mathcal{W}_{D}\triangleq\mathcal{W}_{\mathrm{sd}}\cup\mathcal{W}_{c}\cup\mathcal{W}_{\infty}$
where:
* •
$\mathcal{W}_{\mathrm{sd}}$ is the closure of the projection by
$\operatorname{proj}_{\bm{U}}$ of the components of $\mathcal{V}(\mathcal{I})$
of dimension $<d$
* •
$\mathcal{W}_{c}$ is the union of the closure of the critical values of
$\operatorname{proj}_{\bm{U}}$ in restriction to $\mathcal{V}(\mathcal{I})$
and of the projection of singular values of $\mathcal{V}(\mathcal{I})$
* •
$\mathcal{W}_{\infty}$ is the set of $\bm{U}=\left(U_{1},\dots,U_{d}\right)$
such that
$\operatorname{proj}^{-1}\left(\mathcal{C}\right)\cap\mathcal{V}(\mathcal{I})$
is not compact for any compact neighborhood $\mathcal{C}$ of $\bm{U}$ in
$\operatorname{proj}_{\bm{U}}\left(\mathcal{V}(\mathcal{I})\right)$.
[]TikZ/var_dis
Figure 5. Certification by avoiding the discriminant variety $\mathcal{W}_{D}$
w.r.t. the projection onto the paramater space
In our case of non-redundant SPM, we have as many polynomials
($p_{1},\dots,p_{3}$) as unknowns ($X_{1},\dots,X_{3}$) which involves that
$\mathcal{W}_{\mathrm{sd}}=\varnothing$.
### 3.2. Uncertainty analysis
#### 3.2.1. Propagation of uncertainty on the fabrication parameters
In order to be _numerically_ implemented and from the practical point of view,
we need to ensure that the modelling is still equivalent to and valid for a
“deformed” system (e.g. approximation of irrational numbers $\sqrt{2}$,
$\sqrt{3}$, expressing the polynomial system $\bm{S}$ with only rational or
integer coefficients, uncertainties on fabrication parameters). In particular,
it is worth analyzing the impact of such uncertainties on the coefficients of
our system. This can be done by considering _interval analysis_ tools such as
_interval arithmetic_ Neumaier, (1990); Merlet, (2004) or _ball arithmetic_
van der Hoeven, (2009); Johansson, (2020) where computations are made with
intervals instead of real or float numbers. Both tools allow numerical
computations to be more rigorous by taking into account all the possible
uncertainties being purely numerical (e.g. round-off errors) or physical. In
the specific case of ball arithmetic, intervals are rather called _ball
intervals_.
###### Definition 2 (Ball interval).
A _ball interval_ $[m\pm r]$ is defined as the set of real numbers $x$ such
that $x\in\left[m-r,m+r\right]$ where $m$ denotes the _midpoint_ of the ball
interval and $r$ its _radius_.
From the computational point of view, $m$ and $r$ are binary floating-point
numbers, i.e. $m,r\in\mathbb{Z}\,2^{\mathbb{Z}}$, although all the real
numbers included in the interval are considered. This tool is implemented in
the arb C library222see https://arblib.org/ and is availiable in Maple (v.
$\geq 2022$) through the `RealBox(`$m$`,`$r$`)` function. Such a formalism is
suitable for a rigourous and reliable computation on the error bounds and will
be used in this article to analyze the propagation of fabrication parameters
uncertainties on the system of interest. By introducing a realistic
uncertainty of $r=10^{-5}\;\text{rad}$ on the fabrication parameters using the
RealBox function, the coefficients of $\bm{S}$ (depending on $o_{i}$ and
$j_{i}$, $i\in\left\llbracket 1,n_{a}\right\rrbracket$) have in the worst case
an uncertainty of $r^{\prime}_{\max}=7\times 10^{-5}$.
#### 3.2.2. On the coaxiality of the input shafts
Another important question deals with the coaxiality of the SPM’s input
shafts. Indeed, in theory, the actuators must be _perfectly_ concentric to
allow an illimited rotation around the $z$-axis (yaw). This is nevertheless
not the case in practice because of the uncertainties on the fabrication
parameters, i.e. $\beta_{1}$ in our modelling is not exactly equal to $0$ for
each leg of the SPM. Despite this unfavorable theoretical argument,
experimental prototypes Tursynbek et al., (2019) have shown that such a
mechanism is absolutely capable of moving this way. That leads to say that
among all the possible geometrical configurations induced by the uncertainties
on $\beta_{1}$, the coaxial SPM can make an illimited rotation around the yaw
axis because of the backlashes of its actuated joints. By undergoing such a
phenomenon, the robot can be associated with a virtual one having a perfect
axis of coaxiality. Thus, studying the system in its exact form makes sense.
Using the above-mentionned approach and given this context, let us certify the
IGM and FGM.
## 4\. Certifying the Inverse Geometric Model of 3-DOF SPMs
### 4.1. Workspace analysis
As previously stated, solving the IGM is equivalent to solve $\bm{S}$ with
$\bm{o}\equiv\bm{\chi}$ being (orientation) parameters related and (joint)
unknowns $\bm{j}\equiv\bm{\theta}$. The goal is to ensure that each
orientation value of our prescribed workspace $\mathcal{W}^{\star}$ has the
same number of distinct working modes. In addition, this fact must hold
despite data uncertainties such as small variations on parameters $\bm{o}$.
However, there is one special case that does not verify those properties and
that we want to avoid: numerical unstabilities of the IGM. One of them are
_Type-1 singularities_ (or _serial singularities_). These phenomena appear
when matrix $\bm{B}$ from the kinematic model degenerates: the number of
(real) distinct working modes varies and small variations on $\bm{\chi}$ in
the neighborhood require huge efforts to move $\bm{\theta}$. Therefore,
_certifiying_ implies checking if we are “far enough” from Type-1
singularities and other numerical unstabilities for all the values of
$\bm{\chi}\in\mathcal{W}^{\star}$. One way to check this singularity is to
compute the _discriminant variety_ of the IGM (in its polynomial form). This
object is convinient to compute since we deal with a parametric system in
which each polynomial equation has only 1 variable and is of degree 2 such
that
$\mathrm{IGM}\equiv\bm{S}=\left\\{p_{1}\left(j_{1},\bm{o}\right)=0,p_{2}\left(j_{2},\bm{o}\right)=0,p_{3}\left(j_{3},\bm{o}\right)=0\right\\}$
where $p_{i}=a_{i}\,j_{i}^{2}+b_{i}\,j_{i}+c_{i}$ with $i\in\left\llbracket
1,n_{a}=3\right\rrbracket$. In this particular case, studying the discriminant
variety $\mathcal{W}_{D}$ w.r.t. the projection onto the orientation space
means computing the _resultant_ of each polynomial $p_{i}$,
$\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}p_{i}}{\IfBeginWith{p_{i}}{f}{\partial\mathopen{}^{0}\\!p_{i}}{\partial\mathopen{}^{0}p_{i}}}}{\StrSubstitute{\StrSubstitute{j_{i}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{p_{i}}}{\StrSubstitute{\StrSubstitute{j_{i}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}}}}}$
w.r.t. variable $j_{i}$. Hence,
$\displaystyle\mathcal{W}_{D}(o_{1},o_{2},o_{3})$
$\displaystyle=\bigcup\limits_{i=1}^{n_{a}}\operatorname{Res}{\left(p_{i},\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}p_{i}}{\IfBeginWith{p_{i}}{f}{\partial\mathopen{}^{0}\\!p_{i}}{\partial\mathopen{}^{0}p_{i}}}}{\StrSubstitute{\StrSubstitute{j_{i}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{p_{i}}}{\StrSubstitute{\StrSubstitute{j_{i}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}},j_{i}\right)}}}}}$
(8)
$\displaystyle=\bigcup\limits_{i=1}^{n_{a}}-\operatorname{LC}{(p_{i})}\operatorname{discrim}\left(p_{i},j_{i}\right)$
where $\operatorname{discrim}\left(p_{i},j_{i}\right)=b_{i}^{2}-4a_{i}c_{i}$
is the _discriminant_ of $p_{i}$ w.r.t. variable $j_{i}$ and
$\operatorname{LC}(p_{i})$ is the _leading coefficient_ of $p_{i}(j_{i})$. As
illustrated in Figure 5, the discriminant variety $\mathcal{W}_{D}$ w.r.t.
$\operatorname{proj}_{\bm{o}}$ deals with the partition of the (orientation)
parameter space in function of the number of working modes. This amounts to
study all numerical unstabilities of the IGM according to (8) since:
* •
$\operatorname{discrim}{\left(p_{i},j_{i}\right)}=0$ describe a hypersurface
in $(o_{1},o_{2},o_{3})$ where at least two working modes are superposed which
is synonym of Type-1 singularity. In this case, there is at least one double
root and critical values $(o_{1},o_{2},o_{3})$ verifying that belong to
$\mathcal{W}_{c}$. In our case, we have
$\mathcal{W}_{c}=\left\\{\begin{aligned}
8\left(o_{3}^{2}+1\right)^{2}\left(o_{1}^{2}+2o_{1}-1\right)\left(o_{1}^{2}-2o_{1}-1\right)&=0,\\\\[8.61108pt]
-32\left(-o_{1}^{4}o_{2}^{4}+4\sqrt{3}\,o_{1}^{3}o_{2}^{3}+4o_{1}^{4}o_{2}^{2}+4\sqrt{3}\,o_{1}^{3}o_{2}-4\sqrt{3}\,o_{1}o_{2}^{3}\right.\hskip
30.00005pt\\\
\left.-o_{1}^{4}-12o_{1}^{2}o_{2}^{2}-o_{2}^{4}-4\sqrt{3}\,o_{1}o_{2}+4o_{2}^{2}-1\right)\left(o_{3}^{2}+1\right)^{2}&=0,\\\\[8.61108pt]
32\left(o_{1}^{4}o_{2}^{4}+4\sqrt{3}\,o_{1}^{3}o_{2}^{3}-4o_{1}^{4}o_{2}^{2}+4\sqrt{3}\,o_{1}^{3}o_{2}-4\sqrt{3}\,o_{1}o_{2}^{3}+o_{1}^{4}+12o_{1}^{2}o_{2}^{2}\right.\hskip
30.00005pt\\\
\left.+o_{2}^{4}-4\sqrt{3}\,o_{1}o_{2}-4o_{2}^{2}+1\right)\left(o_{3}^{2}+1\right)^{2}&=0\end{aligned}\right\\}$
(9)
* •
$\operatorname{LC}{\left(p_{i}\right)}=0$ describe a hypersurface in
$(o_{1},o_{2},o_{3})$ where at least one solution goes to infinity. Values
$(o_{1},o_{2},o_{3})$ verifying that belong to $\mathcal{W}_{\infty}$. In our
case, we have
$\mathcal{W}_{\infty}=\left\\{\begin{aligned}
-\sqrt{2}\,o_{1}^{2}o_{3}^{2}-2\sqrt{2}\,o_{1}o_{3}^{2}+\sqrt{2}\,o_{1}^{2}+\sqrt{2}\,o_{3}^{2}-2\sqrt{2}\,o_{1}-\sqrt{2}&=0,\\\\[8.61108pt]
-2\sqrt{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{3}+2\sqrt{2}\,\sqrt{3}\,o_{2}^{2}o_{3}-2\sqrt{2}\,\sqrt{3}\,o_{2}o_{3}^{2}\\\
-2\sqrt{2}\,o_{1}^{2}o_{2}^{2}o_{3}^{2}+12\sqrt{2}\,o_{3}o_{1}o_{2}+2\sqrt{2}\,o_{1}o_{2}^{2}o_{3}^{2}-\sqrt{2}\,o_{1}^{2}+\sqrt{2}\,o_{2}^{2}+2\sqrt{2}\,o_{3}^{2}+2\sqrt{2}\,o_{1}\\\
+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}o_{3}^{2}+2\sqrt{2}\,o_{1}^{2}o_{2}^{2}+\sqrt{2}\,o_{1}^{2}o_{3}^{2}-\sqrt{2}\,o_{2}^{2}o_{3}^{2}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}+2\sqrt{2}\,o_{1}o_{2}^{2}+2\sqrt{2}\,o_{1}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,o_{2}&=0,\\\\[8.61108pt]
-2\sqrt{2}+2\sqrt{2}\,o_{1}o_{2}^{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{3}-2\sqrt{2}\,\sqrt{3}\,o_{2}^{2}o_{3}+2\sqrt{2}\,\sqrt{3}\,o_{2}o_{3}^{2}\\\
+2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}-2\sqrt{2}\,o_{1}^{2}o_{2}^{2}o_{3}^{2}+12\sqrt{2}\,o_{3}o_{1}o_{2}-\sqrt{2}\,o_{1}^{2}+\sqrt{2}\,o_{2}^{2}+2\sqrt{2}\,o_{3}^{2}+2\sqrt{2}\,o_{1}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}o_{3}^{2}+2\sqrt{2}\,o_{1}o_{2}^{2}+2\sqrt{2}\,o_{1}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{2}\\\
+2\sqrt{2}\,o_{1}^{2}o_{2}^{2}+\sqrt{2}\,o_{1}^{2}o_{3}^{2}-\sqrt{2}\,o_{2}^{2}o_{3}^{2}&=0\end{aligned}\right\\}$
(10)
Both cases imply a drop in the number of working modes. Figure 6 depicts the
plot of the discriminant variety of the SPM’s IGM. We can notice that
$\mathcal{W}_{c}$ representing all the Type-1 singularities of our robot is
invariant w.r.t. $o_{3}\equiv\chi_{3}$. This makes sense since we consider the
rotation w.r.t. yaw first (see (4)) and the unlimited rolling property allows
our robot to yaw without changing its geometry. But most importantly, our
workspace $\mathcal{W}^{\star}$ does not meet the discriminant variety of the
IGM ($\mathcal{W}_{D}$ w.r.t. $\operatorname{proj}_{\bm{o}}$). More precisely,
as we deal with 3 quadratic polynomials in $j_{i}$, solving the IGM for each
orientation $\bm{\chi}\in\mathcal{W}^{\star}$ implies finding $2^{3}=8$
distinct solutions (working modes). Therefore, we can guarentee that our SPM’s
IGM will not meet any numerical unstability especially Type-1 singularities,
given our conception parameters and our prescribed workspace
$\mathcal{W}^{\star}$: we have certified the IGM of our SPM for our
application in its exact form.
(a) Critical points of the IGM (Type-1 singularities)
(b) “Infinite points” of the IGM
Figure 6. Discriminant variety of the IGM w.r.t. the projection onto the
orientation space
###### Remark 3.
From now on, we will only focus on $\mathcal{W}_{c}$, the set of the critical
points of the IGM corresponding to all Type-1 singular configurations. Indeed,
$\mathcal{W}_{\infty}$ is only a concern if we solve the IGM in its current
polynomial form whereas $\mathcal{W}_{c}$ basically depends on our SPM’s
geometry.
[]TikZ/pa_co_st1_EN_overview
Figure 7. Type-1 singularity loci of the SPM in the orientation space
Figure 7 depicts the critical values of the IGM in the $(o_{1},o_{2})$-plan by
highlighting some Type-1 singular configurations. The latter confirms the loss
of at least 1 DOF as all the configurations have a fully folded or extended
leg. A similar (but non-certified) result can be obtained using the
conditoning index approach (see Appendix B, Fig. 10(b)).
###### Remark 4.
One trivial serial singular configuration can be found by solving the first
equation of (9). We then obtain
$o_{1}=\left\\{-1\pm\sqrt{2},1\pm\sqrt{2}\right\\}$ or $\chi_{1}=\left\\{\pm
45^{\circ},\pm 135^{\circ}\right\\}$. The Type-1 singularities of interest
comes with any orientation having a roll of $\chi_{1,\mathrm{sing}}=\pm
45^{\circ}$. Starting with $\chi_{1}=0$, having
$\left|\chi_{1}\right|>45^{\circ}$ is impossible due to the first leg (the
green one in Fig. 7) being totally extended.
It might also be of interest to determine the maximum tolerance on the
fabrication parameters towards the Type-1 singularities. In the sequel, we
introduce an uncertainty on the fabrication parameters belonging to
$\bm{\varpi}$ defined as
$\bm{\varpi}\triangleq\left[\;\begin{matrix}\alpha_{1}&\alpha_{2}&\beta_{2}&\eta_{1}&\eta_{2}&\eta_{3}\end{matrix}\;\right]^{\mathsf{T}}$
(11)
Given our remarks on the coaxiality of our mechanism (see Subsection 3.2.2),
we set $\beta_{1}=0$ which also keeps the invariance of the IGM’s discriminant
variety w.r.t. $o_{3}$. Figure 8 depicts the discriminant variety of the IGM
considering such uncertainties.
Figure 8. Type-1 singularity loci of the SPM in the orientation space
considering uncertainties on the fabrication parameters
As seen on this figure, we can introduce an uncertainty up to
$10^{-1}\;\text{rad}$ on the fabrication parameters $\varpi_{i}$ before
meeting the discriminant variety of the IGM, which is in fact a comfortable
margin.
So far, we have proved that our prescribed workspace $\mathcal{W}^{\star}$ is
Type-1 singular-free. However, from the pratical point of view, it is better
to translate this information into the joint space in order to set the
actuated joint stops. These limits in $\theta_{1}$, $\theta_{2}$ and
$\theta_{3}$ play a double role since they allow our robot to move within
$\mathcal{W}^{\star}$ and avoid singular orientations at the same time. Such a
work deals with the _joint space analysis_ of our SPM and is presented in the
next subsection.
### 4.2. Joint space analysis
By taking into account the invariance in orientation w.r.t. yaw, the idea is
to set $\chi_{3}\equiv o_{3}=0$ and compute a unique working mode from the
same leaf of solution for a certain number of pair $(o_{1},o_{2})$ recovering
$\mathcal{W}^{\star}(o_{3}=0)$. Such a set is a square in the
$(o_{1},o_{2})$-plane. The uniqueness of the working mode is obtained by
choosing an initial joint vector $\bm{\theta}_{0}$ being one of the working
modes corresponding to the SPM’s equilibrium orientation, i.e. $\chi_{0,i}=0$,
$\forall\,i\in\left\llbracket 1,3\right\rrbracket$. By solving (5) being the
IGM at equilibrium, we have $\theta_{0,i}=\pm\,\pi/2$,
$\forall\,i\in\left\llbracket 1,3\right\rrbracket$. The existence of $2^{3}=8$
distinct working modes confirms the regularity of the robot at equilibrium. We
arbitrarily choose $\bm{\theta}_{0}$ with all values $\theta_{0,i}$ being
positive such that
$\theta_{i}=\frac{-b_{i}-\sqrt{\operatorname{discrim}{\left(p_{i},j_{i}\right)}}}{2\operatorname{LC}{\left(p_{i},j_{i}\right)}}$,
$\forall\,i\in\left\llbracket 1,3\right\rrbracket$. Hence, we have
$\bm{\chi}_{0}=\bm{o}_{0}=\left[\;\begin{matrix}0&0&0\end{matrix}\;\right]^{\mathsf{T}}\;\xleftrightarrow{(+++)}\;\left\\{\begin{aligned}
\bm{\theta}_{0}&=\left[\;\begin{matrix}\pi/2&\pi/2&\pi/2\end{matrix}\;\right]^{\mathsf{T}}\\\
\bm{j}_{0}&=\left[\;\begin{matrix}1&1&1\end{matrix}\;\right]^{\mathsf{T}}\end{aligned}\right.$
(12)
All the values of $o_{1}\equiv\chi_{1}$ and $o_{2}\equiv\chi_{2}$ belonging to
$\mathcal{W}^{\star}(o_{3}=0)$ are considered in the computation of the IGM
even though the square is discretized using _ball intervals_. After paving the
whole prescribed workspace with $35\times 35=1225$ ball intervals, we compute
the IGM for each ball interval by setting $\left[m\left(o_{i}\right)\pm
r\left(o_{i}\right)\right]$ such that
$-\tan{\left(\frac{\pi}{18}\right)}\simeq-\frac{177}{1000}\leq
m\left(o_{i}\right)\leq\frac{177}{1000}\simeq\tan{\left(\frac{\pi}{18}\right)}$
with $r\left(o_{i}\right)=\frac{5}{900}$, $\forall\,i\in\left\llbracket
1,2\right\rrbracket$ and $m\left(o_{3}\right)=0$ with
$r\left(o_{3}\right)=10^{-4}$. The choice of the radii $r\left(o_{1}\right)$
and $r\left(o_{2}\right)$ ensures that the ball intervalls overlap in order to
cover the whole set $\mathcal{W}^{\star}\left(o_{3}=0\right)$. The obtained
results are also expressed with the same formalism as the input data, i.e.
$\left[m\left(j_{i}\right)\pm r\left(j_{i}\right)\right]$,
$\forall\,i\in\left\llbracket 1,3\right\rrbracket$. Each joint variable
$j_{i}\equiv\theta_{i}$ has a minimum value and a maximum value such that
$\min{\left(m\left(j_{i}\right)-r\left(j_{i}\right)\right)}\leq
j_{i}\leq\max{\left(m\left(j_{i}\right)+r\left(j_{i}\right)\right)},\qquad\forall\,i\in\left\llbracket
1,3\right\rrbracket$ (13)
Those values respectively define the lower and upper bound for the joint stops
as shown in Table 2.
Joint $i$ | $\min{\left(j_{i}\right)}$ | $\max{\left(j_{i}\right)}$ | Joint stops | $\max{\left(r\left(j_{i}\right)\right)}$ | $\min{\left(\Delta{\left(p_{i},j_{i}\right)}\right)}$
---|---|---|---|---|---
$1$ | $0.6693723886$ | $1.525710784$ | $\theta_{1}\in\left[67^{\circ},114^{\circ}\right]$ | $0.05831109206017$ | $3.149730917$
$2$ | $0.6089969554$ | $2.127382005$ | $\theta_{2}\in\left[62^{\circ},130^{\circ}\right]$ | $0.18036268138497$ | $10.08465368$
$3$ | $0.4729818360$ | $1.702299683$ | $\theta_{3}\in\left[50^{\circ},120^{\circ}\right]$ | $0.15685467160577$ | $10.01625750$
Table 2. Extrema joint values obtained after the computation of the IGM of
$\mathcal{W}^{\star}\left(o_{3}=0\right)$
By considering the unlimitted rolling of our SPM, i.e.
$\chi_{3}\in\mathbb{R}$, we can define $\mathcal{Q}_{0}^{\star}$ such that
$\mathcal{Q}_{0}^{\star}\triangleq\left\\{\left(\theta_{1},\theta_{2},\theta_{3}\right)\in\mathbb{R}^{3}\;\middle|\;\begin{aligned}
67^{\circ}\leq\theta_{1}-\chi_{3}\leq 114^{\circ}\\\
62^{\circ}\leq\theta_{2}-\chi_{3}\leq 130^{\circ}\\\
50^{\circ}\leq\theta_{3}-\chi_{3}\leq
120^{\circ}\end{aligned},\quad\forall\,\chi_{3}\in\mathbb{R}\right\\}$ (14)
Given our leaf of solution, the image of $\mathcal{W}^{\star}$ through the IGM
is the set $\mathcal{Q}^{\star}$ defined as
$\mathcal{Q}^{\star}\triangleq\left\\{\bm{\theta}=\left[\;\begin{matrix}\theta_{1}&\theta_{2}&\theta_{3}\end{matrix}\;\right]^{\mathsf{T}}\in\mathbb{R}^{3}\;\middle|\;\bm{\theta}\in\mathcal{Q}_{0}^{\star}\;\text{and}\;\mathrm{FGM}\left(\bm{\theta}\right)\in\mathcal{W}^{\star}\right\\}$
(15)
###### Remark 5.
We necessarily have
$\mathcal{Q}_{0}^{\star}\supset\mathcal{Q}^{\star}\triangleq\mathrm{IGM}{\left(\mathcal{W}^{\star}\right)}$.
## 5\. Certifying the Forward Geometric Model of 3-DOF SPMs
### 5.1. Issue and adopted strategy
The goal of the FGM certification is to ensure that the number of assembly
modes stays constant for any acceptable joint reference value
$\bm{\theta}\in\mathcal{Q}^{\star}$ allowing the SPM to move within our
prescribed workspace $\mathcal{W}^{\star}$. Moreover, the previous fact must
stay true despite the above-mentioned data uncertainties. Special cases that
do not verify such conditions are numerical unstabilities including _Type-2
singularities_ (or _parallel singularities_). These phenomena appear when
matrix $\bm{A}$ from the kinematic model degenerates: the number of distinct
assembly modes changes and small variations on $\bm{\theta}$ in the
neighborhood implies huge variations on $\bm{\chi}$. The robot loses its
rigidity by gaining one (or more) uncontrollable motion: the upper platform
can move without any input joint efforts Khalil and Briot, (2015).
Consequently, such configurations should be avoided: this leads to ensure that
the set $\mathcal{Q}^{\star}$ is non-singular. Certification using the
discriminant variety done in the previous section could also theorically be
extended to the FGM. In this case, roles between parameters and variables
would be switched. However, we would obtain a parametric system in which each
equation depends of $o_{1}$, $o_{2}$ and $o_{3}$ at the same time. The
discriminant variety of the FGM is thus too substantial to compute. We need to
investigate numerical stability and robustness using another approach. One way
to ensure such properties is to prove the regularity of the FGM for any
$\bm{\theta}\in\mathcal{Q}^{\star}$ given our application, i.e. each
$\bm{\theta}\in\mathcal{Q}^{\star}$ has a _unique_ assembly mode $\bm{\chi}$
given the leaf of solution of interest. This will be done considering the
_path tracking_ problem in orientation.
### 5.2. Path tracking in orientation
A closely related problem to the FGM is the path tracking problem. In our
case, the upper platform moves with respect to the base frame. Knowing the
joint values, the calculator needs to compute the orientation (FGM) at each
step given the sampling rate. This computation can be done using a _Newton
iterative scheme_. Such a numerical method estimates the pose of the robot by
taking advantage of the fact that the unknown current orientation at time
$t+\delta t$ will be close to the orientation that was known at time $t$.
However in order to be used in a certified manner, the Newton’s method _must_
return a value that is _unique_ within its neighborhood: one way to ensure
such a condition is the use of the _Kantorovich unicity operator_ Merlet,
(2006). In the sequel, the following notation will be used for a $(n\times n)$
matrix $\bm{M}\triangleq\left[\;\begin{matrix}M_{ij}\end{matrix}\;\right]$ and
a vector $\bm{x}$ of size $n$:
* •
$\left\lVert\bm{x}\right\rVert_{\infty}\triangleq\max\limits_{i\in\left\llbracket
1,n\right\rrbracket}\left|x_{i}\right|$ denotes the maximum norm (or
$\infty$-norm) on $\mathbb{R}^{n}$.
* •
$\left\lVert\bm{M}\right\rVert_{\infty}\triangleq\max\limits_{i\in\left\llbracket
1,n\right\rrbracket}\sum_{j=1}^{n}\left|M_{ij}\right|$ denotes the row sum
norm, an induced matrix norm on $\mathbb{R}^{n\times n}$.
The Newton-Kantorovich theorem Kantorovich, (1948); Tapia, (1971); Demidovich
and Maron, (1973) states the Kantorovich test. Its aim is to investigate the
existence and uniqueness of the root of $\bm{f}(\bm{x})=\bm{0}$ in a certain
region. We will use the version formulated in Demidovich and Maron, (1973).
###### Theorem 1 (Newton-Kantorovich).
Let $\bm{f}:\mathcal{D}\subseteq\mathbb{R}^{n}\to\mathbb{R}^{n}$ a function of
class $\mathcal{C}^{2}$. Let $\bm{x}_{0}$ be a point and
$\overline{\bm{U}}\left(\bm{x}_{0}\right)$ its neighborhood defined by
$\overline{\bm{U}}\left(\bm{x}_{0}\right)\triangleq\left\\{\bm{x}\in\mathcal{D}\;\text{s.t.}\;\left\lVert\bm{x}-\bm{x}_{0}\right\rVert_{\infty}\leq
2B_{0}\right\\}$. Let
$\bm{J}_{0}\triangleq\bm{J}\left(\bm{x}_{0}\right)=\left.\partial\mathopen{}\bm{f}/\partial\mathopen{}\bm{x}\right|_{\bm{x}=\bm{x}_{0}}$
be an invertible jacobian matrix. If there exists three real constants
$A_{0}$, $B_{0}$ and $C$ such that:
1. _(i)_
$\left\lVert\bm{J}_{0}^{-1}\right\rVert_{\infty}\leq A_{0}$
2. _(ii)_
$\left\lVert\bm{J}_{0}^{-1}\,\bm{f}\left(\bm{x}_{0}\right)\right\rVert_{\infty}\leq
B_{0}$
3. _(iii)_
$\forall\,i\in\left\llbracket 1,n\right\rrbracket,\forall\,j\in\left\llbracket
1,n\right\rrbracket$ and
$\bm{x}\in\overline{\bm{U}}\left(\bm{x}_{0}\right),\;\displaystyle\sum_{k=1}^{n}\left|\IfStrEq{2}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}f_{i}(\bm{x})}{\IfBeginWith{f_{i}(\bm{x})}{f}{\partial\mathopen{}^{0}\\!f_{i}(\bm{x})}{\partial\mathopen{}^{0}f_{i}(\bm{x})}}}{\StrSubstitute{\StrSubstitute{x_{j}\partial\mathopen{}x_{k}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{2}{f_{i}(\bm{x})}}{\StrSubstitute{\StrSubstitute{x_{j}\partial\mathopen{}x_{k}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}\right|\leq
C}}}}$
4. _(iv)_
$2nA_{0}B_{0}C\leq 1$
then there is a unique solution of $\bm{f}(\bm{x})=\bm{0}$ in
$\overline{\bm{U}}\left(\bm{x}_{0}\right)$ and the (real) Newton iterative
scheme
$\bm{x}_{k+1}=\bm{x}_{k}-\bm{J}^{-1}\left(\bm{x}_{k}\right)\,\bm{f}\left(\bm{x}_{k}\right)$
with the initial estimate $\bm{x}_{0}$ quadratically converges towards this
unique solution.
###### Remark 6.
A successful Kantorovich test is a _sufficient_ condition to certify the
absence of any numerical instabilities (including singularities).
If the Kantorovich test is valid, it provides a lower bound on the radius of
the convergence domain towards the unique and guaranteed solution for Newton
schemes. Hence, its pairing with a classical Newton scheme is in the heart of
the certification of our SPM. In the case of the FGM certification, we have
$\bm{x}\equiv\bm{o}$ and
$\bm{J}\triangleq\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}\bm{f}}{\IfBeginWith{\bm{f}}{f}{\partial\mathopen{}^{0}\\!\bm{f}}{\partial\mathopen{}^{0}\bm{f}}}}{\StrSubstitute{\StrSubstitute{\bm{x}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{\bm{f}}}{\StrSubstitute{\StrSubstitute{\bm{x}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}\equiv\IfStrEq{0}{0}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}\bm{S}}{\IfBeginWith{\bm{S}}{f}{\partial\mathopen{}^{0}\\!\bm{S}}{\partial\mathopen{}^{0}\bm{S}}}}{\StrSubstitute{\StrSubstitute{\bm{o}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}{\frac{\IfStrEq{0}{0}{\partial\mathopen{}{\IfBeginWith{{f}{\partial\mathopen{}^{0}\\!{\partial\mathopen{}^{0}}{0}{\bm{S}}}{\StrSubstitute{\StrSubstitute{\bm{o}}{}\partial\mathopen{}}{\partial\mathopen{}f}{\partial\mathopen{}\\!f}}}}}}}}}}}$.
The path tracking is initialized with the SPM’s equilibrium configuration
selected in the joint space analysis (see (12)) so that the initial
orientation estimate $\bm{o}_{0}=\mathrm{FGM}{\left(\bm{j}_{0}\right)}$ is
perfectly known. Then, for a small displacement from $\bm{j}_{0}$ to
$\bm{j}_{1}\in\mathcal{Q}^{\star}$, we apply the Kantorovich test to the new
coordinates. If valid, the above-mentionned test ensures the existence of an
assembly mode $\bm{o}_{1}=\mathrm{FGM}{\left(\bm{j}_{1}\right)}$ and its
uniqueness in a certain region that includes the convergence domain of the
Newton scheme. Otherwise, the test is reapplied to coordinates that are closer
to the last valid one. Finally, the same thing goes on for any displacement
from $\bm{j}_{k}\in\mathcal{Q}^{\star}$ to
$\bm{j}_{k+1}\in\mathcal{Q}^{\star}$.
### 5.3. Implementation of the Kantorovich test
The path tracking strategy can be viewed as a semi-numerical approach to
certify the robot (more precisely its FGM in our case). It is worth recalling
that we manipulate a polynomial system $\bm{S}$ with integer coefficients.
Morever, these coefficients are expressed with classical intervals to take
into account the uncertainties.
###### Definition 3 (Interval).
An _interval_ $[x]$ is defined as the set of real numbers $x$ such that
$\underline{x}\leq x\leq\overline{x}$. This interval has a _width_
$\operatorname{width}([x])\triangleq\overline{x}-\underline{x}$ and a
_midpoint_
$\operatorname{mid}([x])\triangleq\left(\overline{x}+\underline{x}\right)/2$.
The _mignitude_ (resp. _magnitude_) of $[x]$ is given by
$\min{\left(\left|\underline{x}\right|,\left|\overline{x}\right|\right)}$
(resp.
$\max{\left(\left|\underline{x}\right|,\left|\overline{x}\right|\right)}$).
In our implementation, the intervals are defined by the binary _system
precision_ $\sigma$ such that
$\operatorname{width}{\left(\left[x\right]\right)}=1/2^{\sigma}$ and will be
denoted as $\left[x\right]_{\sigma}$ in the sequel. The constants $A_{0}$,
$B_{0}$ and $C$ are computed in a certified manner using _multiple-precision
arithmetic_ : $A_{0}$ exclusively depends on the parallel Jacobian evaluated
at the current coordinates, $B_{0}$ defines the neighborhood
$\overline{\bm{U}}$ that will be used to determine $C$. The choice of
multiple-precision (rather than double or floatting) arithmetic allows to
perform any calculation on numbers whose digits of precision are limited only
by the availiable memory of the device. Furthermore, the estimated
orientations $\bm{o}_{k+1}$ returned by the certified Newton scheme are also
computed using classical intervals. Figure 9 summarizes the way the path
tracking in orientation is implemented.
[]TikZ/utilisation_kanto_interval
Figure 9. Implementation of the path tracking in orientation
Starting with
$\bm{j}_{0}=\left[\;\begin{matrix}\left[1\right]_{9}&\left[1\right]_{9}&\left[1\right]_{9}\end{matrix}\;\right]^{\mathsf{T}}$
and
$\bm{o}_{0}=\left[\;\begin{matrix}[0,0]&[0,0]&[0,0]\end{matrix}\;\right]^{\mathsf{T}}$,
we apply the path tracking in orientation to the neighborhood of $\bm{j}_{0}$
which is a ball $\mathcal{B}_{0}\subset\mathcal{Q}^{\star}$ containing
$\bm{j}_{0}$ and whose size depends on the system precision. By taking into
account all the possible values of $\bm{j}_{1}\in\mathcal{B}_{0}$, we are
considering a family of infinite systems in $\mathcal{B}_{0}$. The success of
the Kantorovich test certifies the computation of
$\mathrm{FGM}\left(\bm{j}_{1}\right)=\bm{o}_{1}$,
$\forall\,\bm{j}_{1}\in\mathcal{B}_{0}$. The ball
$\mathcal{B}^{\prime}_{0}\subset\mathcal{W}^{\star}$ containing all the
solutions $\bm{o}_{1}$ is included in the convergence domain (provided by the
Kantorovich test) that contains the initial estimate $\bm{o}_{0}$. Applying
the Kantorovich test directly in the joint space may generate lots of
computations as we deal with three variables $j_{1}$, $j_{2}$ and $j_{3}$.
Part of the implementation strategy is also to reduce the computational cost
by considering the geometrical simplifications of our mechanism. Indeed, by
taking into account the invariance w.r.t. $o_{3}\in\mathcal{W}$, two
coordinates ($o_{1}$ and $o_{2}$) are sufficient to detect any problematic
configurations. Thus, the only values considered in the joint space are the
ones coming from the computation of
$\bm{j}=\mathrm{IGM}\left(o_{1},o_{2},o_{3}=0\right)$ A scanning of our
workspace $\mathcal{W}^{\star}\left(o_{3}=0\right)$ proves that the
Kantorovich test is valid everywhere, despite a poor binary system precision
of $\sigma=9$ bits and a displacement step of $\frac{1}{100}$ in
$\mathcal{W}^{\star}$. This certifies the FGM of our robot given our
applications.
## 6\. Conclusion and further works
In this paper, we have presented two approaches to certify the symetrical
spherical parallel manipulator with coaxial input shafts. The first approach
is the symbolic one involving the computation of the discriminant variety
w.r.t. the projection onto the parameter space for the inverse model. The
explicit expressions obtained allowed us to clearly determine a set in the
orientation space free of Type-1 singularities (and other numerical
instabilities) that includes our prescribed workspace. This strategy could not
(yet) be applied to the forward model as its coefficients and degrees are much
bigger compared to the inverse ones. Therefore, we used a semi-numerical
approach involving the Kantorovich unicity operator and a classical Newton
scheme to certify the forward model with a successful path tracking in
orientation. The numerical computations were done here considering
uncertainties on the fabrication parameters that are translated into
uncertainties on the coefficients of our system with interval and multiple
precision arithmetic. These certification tools and strategies could naturally
be extended to any other parallel robot. This work was also the opportunity to
apprehend the behavior of our mechanism in terms of motion with the
computation of the joint stops. Further works will use the basic concepts of
this article to extend the study to other spherical parallel manipulators with
different conception parameters.
We thank Jean-Pierre Merlet and Clément Gosselin for the fruitful discussions
and their feedbacks regarding the parallel mechanism of interest. We are also
grateful to people we interacted with for this article.
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## Appendix A Determination of the SPM design using the GCI approach
The _global conditioning index_ $\mathrm{GCI}$ is _numerically_ computed using
the following kinematic criteria
$\mathrm{GCI}\triangleq\dfrac{\iint_{\mathcal{W}^{\star}}\zeta{\left(\bm{J}\right)}\IfStrEq{0}{0}{\mathop{}\mathopen{}\mathrm{d}\chi_{1}}{\IfBeginWith{\chi_{1}}{f}{\mathop{}\mathopen{}\mathrm{d}^{0}\\!\chi_{1}}{\mathop{}\mathopen{}\mathrm{d}^{0}\chi_{1}}}\IfStrEq{0}{0}{\mathop{}\mathopen{}\mathrm{d}\chi_{2}}{\IfBeginWith{\chi_{2}}{f}{\mathop{}\mathopen{}\mathrm{d}^{0}\\!\chi_{2}}{\mathop{}\mathopen{}\mathrm{d}^{0}\chi_{2}}}}{\iint_{\mathcal{W}^{\star}}\IfStrEq{0}{0}{\mathop{}\mathopen{}\mathrm{d}\chi_{1}}{\IfBeginWith{\chi_{1}}{f}{\mathop{}\mathopen{}\mathrm{d}^{0}\\!\chi_{1}}{\mathop{}\mathopen{}\mathrm{d}^{0}\chi_{1}}}\IfStrEq{0}{0}{\mathop{}\mathopen{}\mathrm{d}\chi_{2}}{\IfBeginWith{\chi_{2}}{f}{\mathop{}\mathopen{}\mathrm{d}^{0}\\!\chi_{2}}{\mathop{}\mathopen{}\mathrm{d}^{0}\chi_{2}}}}\hskip
30.00005pt\text{where}\quad\zeta(\bm{J})\triangleq\dfrac{1}{\kappa{\left(\bm{J}\right)}}=\dfrac{1}{\left\lVert\bm{J}\right\rVert\,\left\lVert\bm{J}^{-1}\right\rVert}$
(16)
where $\bm{J}$ being the SPM’s _Jacobian matrix_ depending on $\bm{\chi}$,
$\bm{\theta}$ and conception parameters of Tab. 1,
$\left\lVert\bm{J}\right\rVert$ its Frobenius norm defined by
$\left\lVert\bm{J}\right\rVert\triangleq\operatorname{Tr}^{1/2}{\left(\bm{J}^{\mathsf{T}}\,\frac{1}{n_{a}}\operatorname{\mathbf{1}}_{n_{a}}\,\bm{J}\right)}$
and $0\leq\zeta{\left(\bm{J}\right)}\leq 1$ its conditioning index. With such
a method and $80\times 80$ points, we get $\mathrm{GCI}=0.93$,
$\zeta_{\min}=0.8902$ and $\zeta_{\max}=0.9487$.
## Appendix B conditioning index of the Jacobian matrices
(a) $\zeta{\left(\bm{J}\right)}$
(b) $\zeta{\left(\bm{B}\right)}<0.25$ (Type-1 singularities)
Figure 10. Conditioning index of the Jacobian matrices of the SPM
## Appendix C Geometric model of the SPM in its polynomial form
$\bm{S}\triangleq\left\\{\begin{aligned}
-\sqrt{2}\,j_{1}^{2}o_{1}^{2}o_{3}^{2}-2\sqrt{2}\,j_{1}^{2}o_{1}o_{3}^{2}+\sqrt{2}\,j_{1}^{2}o_{1}^{2}+\sqrt{2}\,j_{1}^{2}o_{3}^{2}+4\sqrt{2}\,j_{1}o_{1}^{2}o_{3}+\sqrt{2}\,o_{1}^{2}o_{3}^{2}\\\
-2\sqrt{2}\,j_{1}^{2}o_{1}-2\sqrt{2}\,o_{1}o_{3}^{2}-\sqrt{2}\,j_{1}^{2}-4\sqrt{2}\,j_{1}o_{3}-\sqrt{2}\,o_{1}^{2}-\sqrt{2}\,o_{3}^{2}-2\sqrt{2}\,o_{1}+\sqrt{2}&=0\\\\[8.61108pt]
\sqrt{2}\,o_{1}^{2}-2\sqrt{2}\,o_{3}^{2}+2\sqrt{2}\,o_{1}-2\sqrt{2}\,j_{2}^{2}-\sqrt{2}\,o_{2}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{1}^{2}o_{2}-2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{1}^{2}o_{3}\\\
+2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{2}^{2}o_{3}-2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{2}o_{3}^{2}\\\
-2\sqrt{2}\,\sqrt{3}\,j_{2}o_{1}^{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{2}o_{2}^{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{1}o_{2}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}o_{3}^{2}-2\sqrt{2}\,j_{2}^{2}o_{1}^{2}o_{2}^{2}o_{3}^{2}+12\sqrt{2}\,j_{2}^{2}o_{1}o_{2}o_{3}+8\sqrt{2}\,j_{2}o_{1}^{2}o_{2}^{2}o_{3}\\\
+12\sqrt{2}\,j_{2}o_{1}o_{2}o_{3}^{2}+2\sqrt{2}\,j_{2}^{2}o_{1}o_{2}^{2}o_{3}^{2}-\sqrt{2}\,o_{1}^{2}o_{3}^{2}+2\sqrt{2}\,o_{1}o_{3}^{2}-\sqrt{2}\,j_{2}^{2}o_{1}^{2}+\sqrt{2}\,j_{2}^{2}o_{2}^{2}\\\
+2\sqrt{2}\,j_{2}^{2}o_{3}^{2}-8\sqrt{2}\,j_{2}o_{3}+2\sqrt{2}\,j_{2}^{2}o_{1}+2\sqrt{2}\,o_{1}o_{2}^{2}-2\sqrt{2}\,\sqrt{3}\,o_{2}-2\sqrt{2}\,o_{1}^{2}o_{2}^{2}+\sqrt{2}\,o_{2}^{2}o_{3}^{2}\\\
+2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{1}^{2}o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{1}o_{2}o_{3}^{2}-8\sqrt{2}\,\sqrt{3}\,j_{2}o_{1}o_{2}o_{3}-12\sqrt{2}\,o_{3}o_{1}o_{2}\\\
+2\sqrt{2}\,j_{2}^{2}o_{1}^{2}o_{2}^{2}+\sqrt{2}\,j_{2}^{2}o_{1}^{2}o_{3}^{2}-\sqrt{2}\,j_{2}^{2}o_{2}^{2}o_{3}^{2}-4\sqrt{2}\,j_{2}o_{1}^{2}o_{3}+4\sqrt{2}\,j_{2}o_{2}^{2}o_{3}\\\
-12\sqrt{2}\,j_{2}o_{1}o_{2}+2\sqrt{2}\,j_{2}^{2}o_{1}o_{2}^{2}+2\sqrt{2}\,j_{2}^{2}o_{1}o_{3}^{2}+2\sqrt{2}\,o_{1}o_{2}^{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,j_{2}^{2}o_{2}\\\
+2\sqrt{2}\,\sqrt{3}\,j_{2}o_{1}^{2}-2\sqrt{2}\,\sqrt{3}\,j_{2}o_{2}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{3}-2\sqrt{2}\,\sqrt{3}\,o_{2}^{2}o_{3}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}+2\sqrt{2}\,o_{1}^{2}o_{2}^{2}o_{3}^{2}+2\sqrt{2}&=0\\\\[8.61108pt]
\sqrt{2}\,o_{1}^{2}-2\sqrt{2}\,o_{3}^{2}+2\sqrt{2}\,o_{1}-\sqrt{2}\,o_{2}^{2}-2\sqrt{2}\,j_{3}^{2}+8\sqrt{2}\,j_{3}o_{1}^{2}o_{2}^{2}o_{3}+12\sqrt{2}\,j_{3}o_{1}o_{2}o_{3}^{2}\\\
+2\sqrt{2}\,j_{3}^{2}o_{1}o_{2}^{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{1}^{2}o_{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{1}^{2}o_{3}-2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{2}^{2}o_{3}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}o_{1}^{2}o_{3}^{2}\\\
-2\sqrt{2}\,\sqrt{3}\,j_{3}o_{2}^{2}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{1}o_{2}-2\sqrt{2}\,j_{3}^{2}o_{1}^{2}o_{2}^{2}o_{3}^{2}+12\sqrt{2}\,j_{3}^{2}o_{1}o_{2}o_{3}\\\
-\sqrt{2}\,j_{3}^{2}o_{1}^{2}+\sqrt{2}\,j_{3}^{2}o_{2}^{2}+2\sqrt{2}\,j_{3}^{2}o_{3}^{2}-8\sqrt{2}\,j_{3}o_{3}+2\sqrt{2}\,j_{3}^{2}o_{1}-\sqrt{2}\,o_{1}^{2}o_{3}^{2}+2\sqrt{2}\,o_{1}o_{3}^{2}\\\
+2\sqrt{2}\,o_{1}o_{2}^{2}+2\sqrt{2}\,\sqrt{3}\,o_{2}-2\sqrt{2}\,o_{1}^{2}o_{2}^{2}+\sqrt{2}\,o_{2}^{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{1}^{2}o_{2}o_{3}^{2}\\\
-2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{1}o_{2}o_{3}^{2}+8\sqrt{2}\,\sqrt{3}\,j_{3}o_{1}o_{2}o_{3}-12\sqrt{2}\,o_{3}o_{1}o_{2}+2\sqrt{2}\,o_{1}o_{2}^{2}o_{3}^{2}\\\
-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{2}-2\sqrt{2}\,\sqrt{3}\,o_{1}^{2}o_{3}+2\sqrt{2}\,\sqrt{3}\,o_{2}^{2}o_{3}+2\sqrt{2}\,\sqrt{3}\,o_{2}o_{3}^{2}-2\sqrt{2}\,\sqrt{3}\,o_{1}o_{2}\\\
+2\sqrt{2}\,o_{1}^{2}o_{2}^{2}o_{3}^{2}-4\sqrt{2}\,j_{3}o_{1}^{2}o_{3}+4\sqrt{2}\,j_{3}o_{2}^{2}o_{3}-12\sqrt{2}\,j_{3}o_{1}o_{2}+2\sqrt{2}\,j_{3}^{2}o_{1}o_{2}^{2}\\\
+2\sqrt{2}\,j_{3}^{2}o_{1}o_{3}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}^{2}o_{2}-2\sqrt{2}\,\sqrt{3}\,j_{3}o_{1}^{2}+2\sqrt{2}\,\sqrt{3}\,j_{3}o_{2}^{2}+2\sqrt{2}\,j_{3}^{2}o_{1}^{2}o_{2}^{2}\\\
+\sqrt{2}\,j_{3}^{2}o_{1}^{2}o_{3}^{2}-\sqrt{2}\,j_{3}^{2}o_{2}^{2}o_{3}^{2}+2\sqrt{2}&=0\end{aligned}\right.$
(17)
|
# Results of (3,0) GHM Duals
Arpit Das Chethan N. Gowdigere and Sunil Mukhi
# New meromorphic CFTs from cosets
Arpit Das Chethan N. Gowdigere and Sunil Mukhi
###### Abstract
In recent years it has been understood that new rational CFTs can be
discovered by applying the coset construction to meromorphic CFTs. Here we
turn this approach around and show that the coset construction, together with
the classification of meromorphic CFT with $c\leq 24$, can be used to predict
the existence of new meromorphic CFTs with $c\geq 32$ whose Kac-Moody algebras
are non-simply-laced and/or at levels greater than 1. This implies they are
non-lattice theories. Using three-character coset relations, we propose 34
infinite series of meromorphic theories with arbitrarily large central charge,
as well as 46 theories at $c=32$ and $c=40$.
## 1 Introduction
A rational 2d conformal field theory (RCFT) is characterised by having a
finite set of holomorphic characters $\chi_{i}(\tau)$ and a partition function
of the form:
$\mathcal{Z}(\tau,{\bar{\tau}})=\sum_{i,j=0}^{n-1}M_{ij}\,{\overline{\chi}}_{i}({\bar{\tau}})\chi_{j}(\tau)$
(1.1)
where $n$ is the total number of linearly independent characters of some
chiral algebra.
A classification programme for rational conformal field theories in 2d was
initiated in [1, 2]. It is based on the fact that characters are vector-valued
forms that solve a modular linear differential equation (MLDE) whose order is
the number of characters. Such equations have finitely many parameters and
these can be varied to scan for solutions that satisfy the basic criteria to
be those of a conformal field theory. We will refer to such solutions as
“admissible characters”. From the study of MLDE it emerged that an important
classifier for RCFT with a given number of characters is an integer $\ell\geq
0,\ell\neq 1$ called the Wronskian index (for a detailed review, see [3]).
Admissible characters for bosonic CFTs have been constructed in [4, 5, 6, 7,
8, 9, 10, 11, 12] and for fermionic CFTs in [13, 14, 15].
Admissible characters do not, in general, correspond to any RCFT. For
meromorphic theories (those with a single character, $n=1$) this is
particularly obvious. At $c=24$ there is an infinite set of admissible
characters as we will see below, but only a finite subset corresponds to a CFT
as shown in the seminal work of Schellekens [16]. Moreover a given character
in this finite set sometimes corresponds to multiple different CFTs. That an
analogous phenomenon holds for theories with multiple characters was made
explicit in recent times [7, 8, 9] with the discovery of infinite families of
admissible characters depending on unbounded integers.
The classification of RCFTs for a fixed number of characters and Wronskian
index thus requires two steps: the classification of admissible characters,
and the sub-classification of those that describe genuine RCFT. Both steps are
easiest for small values of $n$ as well as the Wronskian index $\ell$, because
in this case the characters are completely determined by their critical
exponents [2, 5]. Given a set of admissible characters, a useful method to
find corresponding CFT comes from the coset construction of [17, 18] as
implemented for meromorphic numerator theories in [19]111See [20, 21] for
earlier discussions of meromorphic cosets.. This works as follows. We pick a
known meromorphic CFT $\cal H$ as well as a known RCFT ${\cal C}$ with $n$
characters and suitable exponents, such that the coset:
${\tilde{\cal C}}=\frac{\cal H}{\cal C}$ (1.2)
obtained by embedding the Kac-Moody algebra of ${\cal C}$ in that of ${\cal
H}$ corresponds to the given admissible characters. Whenever this can be done,
we may conclude that these characters describe the CFT ${\tilde{\cal C}}$.
This approach has been implemented in [19, 22, 23, 24] where ${\cal H}$ is a
known CFT at $c=8,16,24$ (these are completely classified) or a lattice theory
at $c=32$ (these are too many to classify but many lattice theories are known
[25, 26]). In particular, in [19] this procedure was first used to construct
the CFTs’ corresponding to admissible characters with $(n,\ell)=(2,2)$ that
had originally been found nearly three decades earlier [4] but had not been
previously identified with CFTs.
Let us briefly review some basic aspects of the meromorphic coset relation
(more details can be found in [19, 10]). The numerator theory ${\cal H}$ is
typically an extension of a non-simple Kac-Moody algebra $\oplus_{i}{\cal
G}_{r_{i},k_{i}}$ by higher-spin generators that organise a subset of Kac-
Moody characters into a single character. We denote these theories by ${\cal
E}_{1}[\oplus_{i}{\cal G}_{r_{i},k_{i}}]$ where $\cal E$ stands for
“extension” and the subscript 1 tells us that the extended theory has a single
character. There are broadly two types of meromorphic theories: those
corresponding to free bosons on an even, self-dual lattice, for which the Kac-
Moody algebra only contains simply-laced factors
($A_{r},D_{r},E_{6},E_{7},E_{8}$) at level 1, and the rest, which we call non-
lattice theories. These are characterised by the presence of non-simply-laced
factors ($B_{r},C_{r},F_{4},G_{2}$) and/or levels greater than 1. Some non-
lattice theories can be derived as orbifolds of lattice theories, while others
are more complicated to construct [27, 16, 28]. The denominator theories
${\cal C}$, at least in the original references [19, 29, 8], are taken to be
affine theories belonging to a current-algebra minimal series [30] or
occasionally the Virasoro minimal series ${\cal M}(p,q)$ [31]. The coset
theories ${\tilde{\cal C}}$ again typically have non-simple Kac-Moody algebras
that are extended by other chiral generators so that they have a smaller
number $n$ of characters than the affine theories for the same algebras.
Following the notation introduced above for meromorphic theories, we denote
these by ${\cal E}_{n}[\oplus_{i}{\cal G}_{r_{i},k_{i}}]$ where the subscript
$n$ denotes the number of characters.
If the characters of ${\cal C}$ are denoted $\chi_{i}(\tau)$, those of
${\tilde{\cal C}}$ are denoted ${\tilde{\chi}}_{i}$ and the single character
of $\cal H$ is denoted $\chi^{\cal H}$, then the coset relation is embodied in
a holomorphic bilinear relation:
$\sum_{i=0}^{n-1}d_{i}\,\chi_{i}(\tau)\,{\tilde{\chi}}_{i}(\tau)=\chi^{\cal
H}(\tau)$ (1.3)
where $d_{i}$ are positive integers and $d_{0}=1$.
When both ${\cal H}$ and $\cal C$ correspond to CFTs having a Sugawara stress
tensor in terms of Kac-Moody currents, then by embedding the currents of $\cal
C$ in those of ${\cal H}$ one defines the stress tensor of the coset theory
$\tilde{\cal C}$. At a physics level of rigour, we take this to be a proof
that ${\cal C}$ is a genuine CFT [19]. But it is also possible for Eq. (1.3)
to be satisfied when one or both of ${\cal C},{\tilde{\cal C}}$ does not have
any Kac-Moody currents. Also there are cases where none of ${\cal
C},{\tilde{\cal C}},{\cal H}$ has Kac-Moody currents (for example, ${\cal H}$
may correspond to the Monster module [32, 33]). In such cases the coset
construction of [19] is more tricky to apply, since without a Sugawara
construction we do not have an explicit expression for the stress tensor of
${\cal C}$. At the same time, the bilinear relation can certainly be verified
as easily as for the Sugawara case. So it is compelling to believe that, even
in the absence of Sugawara stress tensors, if the bilinear relation holds and
${\cal H}$ and ${\cal C}$ are both CFTs, then so is ${\tilde{\cal C}}$. One
such example [29] arises when ${\cal C}$ is the Ising model and ${\tilde{\cal
C}}$ is the Baby Monster CFT [34].
As described above, the coset relation Eq. (1.2) has been used to find
interesting theories ${\tilde{\cal C}}$ given meromorphic theories ${\cal H}$
and known denominator CFTs ${\cal C}$ to divide them by. As input, this method
has relied on known classifications of meromorphic theories at $c=8,16,24$
[27, 16] as well as special classes of lattices at $c=32$ [25, 26] to find new
$n$-character CFTs. However, if ${\cal C}$ and ${\tilde{\cal C}}$ are both
known to describe CFTs then this relation can be used in the other direction:
to argue that ${\cal H}$ also corresponds to a CFT. Thus, in principle new
meromorphic theories can be found in this way.
In the present work we propose a practical way to carry this out. The key idea
is to first generate an extension-type CFT ${\tilde{\cal C}}$ using the coset
construction, with ${\cal H}$ being a known meromorphic theory in the
Schellekens list [16] and ${\cal C}$ being a suitable affine theory embedded
in it. Next, we remove ${\cal C}$ from the story and find bilinear relations
where ${\tilde{\cal C}}$ is paired with the characters of another known CFT
${\cal C}^{\prime}$. For $n=3$, large numbers of such coset pairs were
generated in [10]. Since ${\cal C}^{\prime}$ is a different theory from ${\cal
C}$ (and not a tensor product of ${\cal C}$ with something else), the result
is a new meromorphic theory at a different central charge.
The affine algebra of the new meromorphic theory is the sum of algebras of
${\cal C}^{\prime}$ and ${\tilde{\cal C}}$, denoted
$\mathcal{E}_{1}[\mathcal{C}^{\prime}\oplus\tilde{\mathcal{C}}]$, of the
multi-character theory $\mathcal{C}^{\prime}\oplus\tilde{\mathcal{C}}$. This
extension arises from the bilinear relation satisfied by ${\cal C}^{\prime}$
and ${\tilde{\cal C}}$. In each case we explicitly find the character of
${\cal H}$. In most cases it will turn out that ${\cal
C}^{\prime}\oplus{\tilde{\cal C}}$ is non-simply-laced and/or has factors of
level $>1$, showing that it must be a non-lattice theory.
Using the above approach we predict the existence of 34 infinite families of
novel meromorphic theories with central charges $c^{\mathcal{H}}=8(m+3)$ where
$m$ is an integer $\geq 1$. Every such infinite family starts with a member at
$c^{\mathcal{H}}=24$ that can be found in Schellekens’ list [16]. We also
predict the existence of 46 novel meromorphic theories with central charges
$c^{\mathcal{H}}=32$ and $c^{\mathcal{H}}=40$. We now turn to a description of
the method and then present our results.
## 2 Background
We start by briefly summarising the admissible characters for meromorphic
theories at $c^{\mathcal{H}}=8N$ where $N$ is an integer $\geq 1$. From the
theory of modular forms (see for example [35]) we know that any polynomial
222Actually any rational function of $j(q)$ would be invariant, but we are
specialising to polynomials as we want these functions to be holomorphic in
$\mathbb{H}\setminus\\{i\infty\\}$. of the Klein $j$-invariant $j(q)$ is a
modular invariant function. The $j$-invariant has the expansion:
$j(q)=q^{-1}+744+196884q+\cdots$ (2.1)
If we are willing to accept modular invariance up to a phase then the
polynomial can be multiplied by $j(q)^{\frac{1}{3}}$ or $j(q)^{\frac{2}{3}}$.
Thus the most general modular invariant is of the form:
$\chi^{\cal
H}(\tau)=j^{\frac{N}{3}-s}(j^{s}+a_{1}\,j^{s-1}+a_{2}\,j^{s-2}+\ldots\ldots+a_{s}),$
(2.2)
where $s:=\left\lfloor\frac{N}{3}\right\rfloor$.
For $c^{\cal H}=24,32,40$, the corresponding characters are of the form:
$\displaystyle c^{\mathcal{H}}=24:\,\,\,\chi^{\cal H}(\tau)=j+{\cal N},$ (2.3)
$\displaystyle c^{\mathcal{H}}=32:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{1}{3}}(j+{\cal N})$ (2.4) $\displaystyle
c^{\mathcal{H}}=40:\,\,\,\chi^{\cal H}(\tau)=j^{\frac{2}{3}}(j+{\cal N}),$
(2.5)
where we have renamed $a_{1}$ as ${\cal N}$. In the $c^{\cal H}=24$ case we
see that all values of ${\cal N}\geq-744$ are admissible, but it is known that
only a finite subset correspond to CFTs. [16].
For $n$-character RCFT, the characters are vector-valued modular forms that
have the general form:
$\chi_{i}(q)=q^{-\frac{c}{24}}(a_{i,0}+a_{i,1}q+a_{i,2}q^{2}+\cdots),~{}~{}i=0,1,\cdots,n-1$
(2.6)
For the characters to be admissible, the coefficients $a_{i,r}$ must be non-
negative integers. Moreover we must have $a_{0,0}=1$ (non-degeneracy of the
identity character). We also define
$D_{i}=a_{i,0},m_{1}=a_{0,1},m_{2}=a_{0,2}$. These are, respectively, the
ground-state degeneracy of each of the generalised primaries, the number of
spin-1 currents and the number of spin-2 currents.
For $n=2$ there is an explicit and complete classification of all admissible
characters [8]. However for $n\geq 3$ the complete classification remains an
open problem. Nevertheless, admissible characters can be found by writing a
general Modular Linear Differential Equation of the form:
$\big{(}D^{n}+\phi_{2}(\tau)D^{n-1}+\cdots+\phi_{2n}(\tau)\big{)}\chi=0$ (2.7)
where $D$ is a covariant derivative on torus moduli space and
$\phi_{2j}(\tau)$ are meromorphic modular functions of weight $2j$. The
maximum number of poles of the $\phi_{2r}$ is called the Wronskian index
$\ell$. Consider the above equation for a fixed order $n$ and a fixed value of
$\ell$. In this case the set of $\phi_{2j}$ depends on a finite number of
parameters, and one scans the parameter space to find those values where the
coefficients $a_{i,r}$ satisfy the admissibility criteria. Of relevance in the
present work will be solutions for $n=3$ and $\ell=0$, in which case the
equation becomes:
$\big{(}D^{3}+\mu_{1}E_{4}(\tau)D+\mu_{2}E_{6}(\tau)\big{)}\chi=0$ (2.8)
While this equation was first studied long ago in [5], all its admissible
solutions with $c\leq 96$ have been classified only recently [10] and we will
make use of some of these below (all these relevant solutions are given in
table LABEL:t6 in Sec. B).
Next we summarise some results about the coset construction. In [19] the
following relation between the central charge $c$ and conformal dimensions
$h_{i}$ of the characters $\chi_{i}$, and the corresponding quantities
${\tilde{c}},{\tilde{h}}_{i}$ of the coset dual ${\tilde{\chi}}_{i}$ was
derived:
$\ell+{\tilde{\ell}}=n^{2}+\left(\frac{c+{\tilde{c}}}{4}-1\right)n-6\sum_{i=1}^{n-1}(h_{i}+{\tilde{h}}_{i})$
(2.9)
We will be interested in the case $n=3$. Also, note that $c+{\tilde{c}}$ must
be a multiple of 8, so we write it as $8N$ where $N$ is an integer $\geq 1$
333In [19] the convention was to write $c+{\tilde{c}}=24N$ where $N$ is a
multiple of $\frac{1}{3}$.. Also, since the right hand side of the bilinear
relation is a character with integer dimensions, we must have
$h_{i}+{\tilde{h}}_{i}=n_{i}$, an integer $\geq 1$, for each $i$. Thus, for
$n=3$ the relation can be written:
$\ell+{\tilde{\ell}}=6\Big{(}N+1-\sum_{i=1}^{2}n_{i}\Big{)}$ (2.10)
Let us comment on the significance of the integers $n_{i}$. These denote the
new chiral currents, of spin $n_{i}$, created by “fusing” the primaries of two
RCFTs ${\cal C},{\tilde{\cal C}}$ via a bilinear relation into a meromorphic
theory ${\cal H}$. Whenever each $n_{i}\geq 2$, we have a special situation
where the only new currents in ${\cal H}$, relative to ${\cal
C}\oplus{\tilde{\cal C}}$, have spin $\geq 2$. In such cases the Kac-Moody
algebra of ${\cal H}$ is necessarily the direct sum of that of ${\cal
C},{\tilde{\cal C}}$. On the other hand if any $n_{i}=1$, we have new Kac-
Moody currents in ${\cal H}$ that were not present in the coset pair. Viewed
from the converse perspective starting with the meromorphic theory, the first
case $n_{i}\geq 2$ arises when the denominator theory ${\cal C}$ is an affine
theory for one (or more) of the simple factors in ${\cal H}$. In that case the
coset procedure just deletes the factor(s) and the remaining ones make up the
Kac-Moody algebra of ${\tilde{\cal C}}$. But in the second case, the algebra
of ${\cal C}$ undergoes a non-trivial embedding in one of the simple factors
of ${\cal H}$. This “Higgses” that factor down to a subalgebra given by the
commutant, and some currents are lost in the process. Only the first case
$n_{i}\geq 2$ will be of relevance in the present work, while the other case
involving embeddings will be discussed in [24, 23].
From now on we focus on coset dual pairs of characters with $n=3$ and
$\ell,{\tilde{\ell}}=0,0$. From the above relation it follows that
$n_{1}+n_{2}=N+1$. While the classification of admissible $(3,0)$ characters
has not been completed, significant progress has been made in [5, 36, 37, 19,
10]. We will make use of the results in [10], which subsumes the output of
most of the previous works and provides a complete set of admissible three-
character sets with $c\leq 96$ 444In an upcoming work [24] we re-examine this
partial classification of admissible characters and attempt to identify which
of them can be shown to exist as CFTs as well as which ones can be shown not
to correspond to any CFT..
In the notation of [10], to which we refer the reader for more details, the
admissible character sets fall into five categories, labelled
$\mathbf{I},\mathbf{II},\cdots,\mathbf{V}$. Of these, all entries in
categories $\mathbf{I},\mathbf{II},\mathbf{IV}$ have already been identified
as CFTs in [10] 555with a few exceptions that correspond to generalisations of
CFTs such as Intermediate Vertex Operator Algebras (IVOA) [38].. Among these
are characters that were previously identified as CFTs in [19] and which will
play a role in the present work. We will label them GHMD where the subscript
$D$ denotes the dimension of their Kac-Moody algebra (see all GHM solutions
listed in table LABEL:t6 in Sec. B). We will also make use of characters of
type $\mathbf{III}$ and $\mathbf{V}$ which were not identified with CFTs in
[10]. These will be studied in complete detail in work to appear [24]. The
list of relevant ${\bf III}$ and ${\bf V}$ solutions can be found in table
LABEL:t6 of Appendix B.
## 3 Constructing new meromorphic CFTs
### 3.1 $(3,0)$ cosets from $c=24$ meromorphic theories
We start by using the coset construction with ${\cal H}$ being one of the
meromorphic theories in Schellekens’ list, to identify 22 sets of admissible
characters as CFTs. Table LABEL:t0 shows coset pairings of characters
$\chi_{i},{\tilde{\chi}}_{i}$ to make one of these meromorphic theories. The
purpose of this exercise is to identify the theories $\tilde{\mathcal{C}}$
that we will use later on.
The entries in the table are as follows. The first column is a serial number
labelling the 22 cases of interest. The next four columns tell us the
properties of an affine (or minimal) model ${\cal C}$ that we use as the
denominator in the coset relation. Respectively, they provide the central
charge, conformal dimensions, dimension $m_{1}$ of the Kac-Moody algebra, and
the Kac-Moody algebra itself. The next four columns provide the same
properties for a coset theory ${\tilde{\cal C}}$ that combines with ${\cal C}$
in a bilinear relation that has been verified. The last four columns tell us
the Kac-Moody algebra of the meromorphic Schellekens theory, the integers
$(d_{1},d_{2})$ appearing in the bilinear relation Eq. (1.3), the integer
${\cal N}$ that specifies the character via $\chi^{\cal H}(\tau)=j(\tau)+{\cal
N}$, and the serial number of the corresponding theory in the table of [16].
Note that the dimension of the Kac-Moody algebra of the meromorphic theory is
${\cal N}+744$.
As can be seen in the Table, rows $4-9,13-17,22$ are cases where
$\tilde{\mathcal{C}}$ is of GHM type [19]. Among the rest, rows
$1,2,10-12,18-21$ were missed in [19] for various reasons – for example that
reference did not consider coset duals where ${\cal C}$ is a tensor product of
two identical two-character theories. Note that despite this, ${\tilde{\cal
C}}$ is not a tensor product of simpler theories. Finally, row 3 is the Ising-
Baby Monster pairing implicit in [34] and discussed in the present context in
[29]. This is the only known case of a $c=24$ coset where one or both (in this
case, both) of the entries have no Kac-Moody algebra. The coset relation must
then be understood in the more general sense mentioned below Eq. (1.3), and
the corresponding meromorphic theory is the Monster CFT. We will soon see that
for $c\geq 32$, the Baby Monster theory features in coset pairings with affine
theories.
Table 1: Coset relations for $c^{\mathcal{H}}=24$ with $(n_{1},n_{2})=(2,2)$. We identify $\mathcal{M}(4,3)\cong B_{0,1}$, $A_{1,2}\cong B_{1,1}$, $C_{2,1}\cong B_{2,1}$ and BM $\equiv$ Baby Monster. We further identify $U(1)\cong D_{1,1}$, $A_{1,1}^{\oplus 2}\cong D_{2,1}$ and $A_{3,1}\cong D_{3,1}$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$ | Affine | $(d_{1},d_{2})$ | $\mathcal{N}$ | Sch.
---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | | | | | algebra | | | #
1. | $12$ | $(\frac{1}{2},\frac{3}{2})$ | $276$ | $D_{12,1}$ | $12$ | $(\frac{3}{2},\frac{1}{2})$ | $276$ | $D_{12,1}$ | $D_{12,1}^{\oplus 2}$ | $2$ | 552 | 66
2. | $12$ | $(\frac{2}{3},\frac{4}{3})$ | $156$ | $E_{6,1}^{\oplus 2}$ | $12$ | $(\frac{4}{3},\frac{2}{3})$ | $156$ | $E_{6,1}^{\oplus 2}$ | $E_{6,1}^{\oplus 4}$ | $8$ | 312 | 58
3. | $\frac{1}{2}$ | $(\frac{1}{2},\frac{1}{16})$ | $0$ | $\mathcal{M}(4,3)$ | $\frac{47}{2}$ | $(\frac{3}{2},\frac{31}{16})$ | $0$ | BM | — | $(1,1)$ | 0 | 0
4. | $\frac{3}{2}$ | $(\frac{1}{2},\frac{3}{16})$ | $3$ | $A_{1,2}$ | $\frac{45}{2}$ | $(\frac{3}{2},\frac{29}{16})$ | $45$ | $\text{GHM}_{45}$ | $A_{1,2}^{\oplus 16}$ | $(1,1024)$ | 48 | 5
| | | | | | | | | $A_{3,4}^{\oplus 3}A_{1,2}$ | | | 7
| | | | | | | | | $A_{5,6}C_{2,3}A_{1,2}$ | | | 8
| | | | | | | | | $D_{5,8}A_{1,2}$ | | | 10
5. | $\frac{5}{2}$ | $(\frac{1}{2},\frac{5}{16})$ | $10$ | $C_{2,1}$ | $\frac{43}{2}$ | $(\frac{3}{2},\frac{27}{16})$ | $86$ | $\text{GHM}_{86}$ | $D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 4}$ | $(1,512)$ | 96 | 25
| | | | | | | | | $A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}C_{2,1}$ | | | 26
| | | | | | | | | $E_{6,4}A_{2,1}C_{2,1}$ | | | 28
6. | $\frac{7}{2}$ | $(\frac{1}{2},\frac{7}{16})$ | $21$ | $B_{3,1}$ | $\frac{41}{2}$ | $(\frac{3}{2},\frac{25}{16})$ | $123$ | $\text{GHM}_{123}$ | $B_{3,1}D_{6,2}C_{4,1}B_{3,1}$ | $(1,256)$ | 144 | 39
| | | | | | | | | $B_{3,1}A_{9,2}A_{4,1}$ | | | 40
7. | $\frac{9}{2}$ | $(\frac{1}{2},\frac{9}{16})$ | $36$ | $B_{4,1}$ | $\frac{39}{2}$ | $(\frac{3}{2},\frac{23}{16})$ | $156$ | $\text{GHM}_{156}$ | $D_{8,2}B_{4,1}^{\oplus 2}$ | $(1,128)$ | 192 | 47
| | | | | | | | | $B_{4,1}C_{6,1}^{\oplus 2}$ | | | 48
8. | $\frac{11}{2}$ | $(\frac{1}{2},\frac{11}{16})$ | $55$ | $B_{5,1}$ | $\frac{37}{2}$ | $(\frac{3}{2},\frac{21}{16})$ | $185$ | $\text{GHM}_{185}$ | $B_{5,1}E_{7,2}F_{4,1}$ | $(1,1)$ | 240 | 53
9. | $\frac{13}{2}$ | $(\frac{1}{2},\frac{13}{16})$ | $78$ | $B_{6,1}$ | $\frac{35}{2}$ | $(\frac{3}{2},\frac{19}{16})$ | $210$ | $\text{GHM}_{210}$ | $B_{6,1}C_{10,1}$ | $(1,32)$ | 288 | 56
10. | $\frac{17}{2}$ | $(\frac{1}{2},\frac{17}{16})$ | $136$ | $B_{8,1}$ | $\frac{31}{2}$ | $(\frac{3}{2},\frac{15}{16})$ | $248$ | $E_{8,2}$ | $B_{8,1}E_{8,2}$ | $(1,1)$ | 384 | 62
11. | $1$ | $(\frac{1}{2},\frac{1}{8})$ | $1$ | $U(1)$ | $23$ | $(\frac{3}{2},\frac{15}{8})$ | $23$ | ${\bf III_{50}}$ | $U(1)^{\oplus 24}$ | $(8,4096)$ | 24 | 1
12. | $2$ | $(\frac{1}{2},\frac{1}{4})$ | $6$ | $A_{1,1}^{\oplus 2}$ | $22$ | $(\frac{3}{2},\frac{7}{4})$ | $66$ | ${\bf III_{45}}$ | $A_{1,1}^{\oplus 24}$ | $(64,4096)$ | 72 | 15
| | | | | | | | | $A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 4}$ | | | 16
| | | | | | | | | $A_{5,3}D_{4,3}A_{1,1}^{\oplus 3}$ | | | 17
| | | | | | | | | $A_{7,4}A_{1,1}^{\oplus 3}$ | | | 18
| | | | | | | | | $D_{5,4}C_{3,2}A_{1,1}^{\oplus 2}$ | | | 19
| | | | | | | | | $D_{6,5}A_{1,1}^{\oplus 2}$ | | | 20
13. | $3$ | $(\frac{1}{2},\frac{3}{8})$ | $15$ | $A_{3,1}$ | $21$ | $(\frac{3}{2},\frac{13}{8})$ | $105$ | $\text{GHM}_{105}$ | $A_{3,1}^{\oplus 8}$ | $(8,1024)$ | 120 | 30
| | | | | | | | | $D_{5,2}^{\oplus 2}A_{3,1}^{\oplus 2}$ | | | 31
| | | | | | | | | $A_{7,2}C_{3,1}^{\oplus 2}A_{3,1}$ | | | 33
| | | | | | | | | $D_{7,3}G_{2,1}A_{3,1}$ | | | 34
| | | | | | | | | $C_{7,2}A_{3,1}$ | | | 35
14. | $5$ | $(\frac{1}{2},\frac{5}{8})$ | $45$ | $D_{5,1}$ | $19$ | $(\frac{3}{2},\frac{11}{8})$ | $171$ | $\text{GHM}_{171}$ | $D_{5,1}^{\oplus 2}\,A_{7,1}^{\oplus 2}$ | $(8,256)$ | 216 | 49
15. | $6$ | $(\frac{1}{2},\frac{3}{4})$ | $66$ | $D_{6,1}$ | $18$ | $(\frac{3}{2},\frac{5}{4})$ | $198$ | $\text{GHM}_{198}$ | $D_{6,1}^{\oplus 4}$ | $(64,256)$ | 264 | 54
| | | | | | | | | $D_{6,1}\,A_{9,1}^{\oplus 2}$ | | | 55
16. | $7$ | $(\frac{1}{2},\frac{7}{8})$ | $91$ | $D_{7,1}$ | $17$ | $(\frac{3}{2},\frac{9}{8})$ | $221$ | $\text{GHM}_{221}$ | $D_{7,1}\,A_{11,1}\,E_{6,1}$ | $(8,64)$ | 312 | 59
17. | $9$ | $(\frac{1}{2},\frac{9}{8})$ | $153$ | $D_{9,1}$ | $15$ | $(\frac{3}{2},\frac{7}{8})$ | $255$ | $\text{GHM}_{255}$ | $D_{9,1}\,A_{15,1}$ | $(8,16)$ | 408 | 63
18. | $10$ | $(\frac{1}{2},\frac{5}{4})$ | $190$ | $D_{10,1}$ | $14$ | $(\frac{3}{2},\frac{3}{4})$ | $266$ | $E_{7,1}^{\oplus 2}$ | $D_{10,1}E_{7,1}^{\oplus 2}$ | $(1,2)$ | 456 | 64
19. | $4$ | $(\frac{1}{3},\frac{2}{3})$ | $16$ | $A_{2,1}^{\oplus 2}$ | $20$ | $(\frac{5}{3},\frac{4}{3})$ | $80$ | ${\bf V_{39}}$ | $A_{2,1}^{\oplus 12}$ | $(8748,972)$ | 96 | 24
| | | | | | | | | $A_{5,2}^{\oplus 2}C_{2,1}A_{2,1}^{\oplus 2}$ | | | 26
| | | | | | | | | $A_{8,3}A_{2,1}^{\oplus 2}$ | | | 27
20. | $\frac{28}{5}$ | $(\frac{2}{5},\frac{4}{5})$ | $28$ | $G_{2,1}^{\oplus 2}$ | $\frac{92}{5}$ | $(\frac{8}{5},\frac{6}{5})$ | $92$ | ${\bf III_{37}}$ | $E_{6,3}G_{2,1}^{\oplus 3}$ | $(50,1)$ | 120 | 32
21. | $\frac{52}{5}$ | $(\frac{3}{5},\frac{6}{5})$ | $104$ | $F_{4,1}^{\oplus 2}$ | $\frac{68}{5}$ | $(\frac{7}{5},\frac{4}{5})$ | $136$ | ${\bf III_{22}}$ | $C_{8,1}F_{4,1}^{\oplus 2}$ | $(50,1)$ | 240 | 52
22. | $4$ | $(\frac{2}{5},\frac{3}{5})$ | $24$ | $A_{4,1}$ | $20$ | $(\frac{8}{5},\frac{7}{5})$ | $120$ | $\text{GHM}_{120}$ | $A_{4,1}^{\oplus 6}$ | $(1250,1250)$ | 144 | 37
| | | | | | | | | $A_{9,2}B_{3,1}A_{4,1}$ | | | 40
Table LABEL:t0 includes a few self-dual pairs. In these cases we have:
$\chi_{0}=\tilde{\chi}_{0}$, $\chi_{1}=\tilde{\chi}_{2}$ and
$\chi_{2}=\tilde{\chi}_{1}$. Hence Eq.(1.3) (for three characters) becomes:
$\displaystyle\chi_{0}^{\mathcal{H}}=\chi_{0}^{2}+d_{1}\,\chi_{1}\chi_{2}+d_{2}\,\chi_{2}\chi_{1}=\chi_{0}^{2}+(d_{1}+d_{2})\,\chi_{1}\chi_{2}=\chi_{0}^{2}+d_{3}\,\chi_{1}\chi_{2}.$
Hence, in the $(d_{1},d_{2})$ column we just have one entry
$d_{3}=d_{1}+d_{2}$ for self-dual pairs. We follow this convention for all
tables whenever there are self-dual pairs involved in a bilinear relation.
### 3.2 New meromorphic theories at $c\geq 32$: infinite families
The next step is to combine the theories labelled ${\tilde{\cal C}}$ in Table
LABEL:t0 with suitable infinite series of affine theories to make modular
invariants with central charge $\geq 32$. One set of results is exhibited in
Table LABEL:t1, where 34 infinite families of coset pairs are described. The
theories labelled ${\tilde{\cal C}}$ are all taken from Table LABEL:t0.
However in each case the corresponding theory labelled ${\cal C}$ in Table
LABEL:t0 has been replaced by a new affine theory ${\cal C}$ labelled by an
arbitrary integer parameter $m$. The associated character is exhibited in the
last column of the table (see below for an explanation of the notation). The
$m=0$ case is the original one in Table LABEL:t0. The integers $(n_{1},n_{2})$
are, in all cases, given by $(2,m+2)$, verifying Eq. (2.10) between a pair of
three-characters with vanishing Wronskian index.
In this table, both ${\cal C}$ and ${\tilde{\cal C}}$ correspond to known
theories. Hence, from the coset pairing we conclude that the modular invariant
obtained by combining them in a bilinear relation also corresponds to a
genuine CFT. This is a meromorphic CFT that, in most cases, has to be of non-
lattice type since it either involves non-simply-laced algebras or levels
greater than 1, or both.
Table 2: Coset relations for $c^{\mathcal{H}}=8(m+3)$ with $(n_{1},n_{2})=(2,m+2)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$ | Sch. #
---|---|---|---|---|---|---|---|---|---
| | | | | | | | | ($m=0$)
1. | $\frac{16m+1}{2}$ | $\left(\frac{1}{2},\frac{16m+1}{16}\right)$ | 128$m^{2}$ \+ 8$m$ | $B_{8m,1}$ | $\frac{47}{2}$ | $\left(\frac{3}{2},\frac{31}{16}\right)$ | $0$ | BM | 0
2. | $\frac{16m+3}{2}$ | $\left(\frac{1}{2},\frac{16m+3}{16}\right)$ | 128$m^{2}$ \+ 40$m$ \+ 3 | $B_{8m+1,1}$ | $\frac{45}{2}$ | $\left(\frac{3}{2},\frac{29}{16}\right)$ | $45$ | $\mathcal{E}_{3}[A_{1,2}^{\oplus 15}]$ | 5
| | | | | | | | $\mathcal{E}_{3}[A_{3,4}^{\oplus 3}]$ | 7
| | | | | | | | $\mathcal{E}_{3}[A_{5,6}C_{2,3}]$ | 8
| | | | | | | | $\mathcal{E}_{3}[D_{5,8}]$ | 10
3. | $\frac{16m+5}{2}$ | $\left(\frac{1}{2},\frac{16m+5}{16}\right)$ | 128$m^{2}$ \+ 72$m$ \+ 10 | $B_{8m+2,1}$ | $\frac{43}{2}$ | $\left(\frac{3}{2},\frac{27}{16}\right)$ | $45$ | $\mathcal{E}_{3}[D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 3}]$ | 25
| | | | | | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}]$ | 26
| | | | | | | | $\mathcal{E}_{3}[E_{6,4}A_{2,1}]$ | 28
4. | $\frac{16m+7}{2}$ | $\left(\frac{1}{2},\frac{16m+7}{16}\right)$ | 128$m^{2}$ \+ 104$m$ \+ 21 | $B_{8m+3,1}$ | $\frac{41}{2}$ | $\left(\frac{3}{2},\frac{25}{16}\right)$ | $123$ | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$ | 39
| | | | | | | | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$ | 40
5. | $\frac{16m+9}{2}$ | $\left(\frac{1}{2},\frac{16m+9}{16}\right)$ | 128$m^{2}$ \+ 136$m$ \+ 36 | $B_{8m+4,1}$ | $\frac{39}{2}$ | $\left(\frac{3}{2},\frac{23}{16}\right)$ | $156$ | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ | 47
| | | | | | | | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ | 48
6. | $\frac{16m+11}{2}$ | $\left(\frac{1}{2},\frac{16m+11}{16}\right)$ | 128$m^{2}$ \+ 168$m$ \+ 55 | $B_{8m+5,1}$ | $\frac{37}{2}$ | $\left(\frac{3}{2},\frac{21}{16}\right)$ | $185$ | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | 53
7. | $\frac{16m+13}{2}$ | $\left(\frac{1}{2},\frac{16m+13}{16}\right)$ | 128$m^{2}$ \+ 200$m$ \+ 78 | $B_{8m+6,1}$ | $\frac{35}{2}$ | $\left(\frac{3}{2},\frac{19}{16}\right)$ | $210$ | $\mathcal{E}_{3}[C_{10,1}]$ | 56
8. | $\frac{16m+17}{2}$ | $\left(\frac{1}{2},\frac{16m+17}{16}\right)$ | 128$m^{2}$ \+ 264$m$ \+ 136 | $B_{8m+8,1}$ | $\frac{31}{2}$ | $\left(\frac{3}{2},\frac{15}{16}\right)$ | $248$ | $E_{8,2}$ | 62
9. | $8m+1$ | $\left(\frac{1}{2},\frac{8m+1}{8}\right)$ | 128$m^{2}$ \+ 24$m$ \+ 1 | $D_{8m+1,1}$ | $23$ | $\left(\frac{3}{2},\frac{15}{8}\right)$ | $23$ | $\mathcal{E}_{3}[D_{1,1}^{\oplus 23}]$ | 1
10. | $8m+2$ | $\left(\frac{1}{2},\frac{8m+2}{8}\right)$ | 128$m^{2}$ \+ 56$m$ \+ 6 | $D_{8m+2,1}$ | $22$ | $\left(\frac{3}{2},\frac{7}{4}\right)$ | $66$ | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$ | 15
| | | | | | | | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$ | 16
| | | | | | | | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$ | 17
| | | | | | | | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$ | 18
| | | | | | | | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$ | 19
| | | | | | | | $\mathcal{E}_{3}[D_{6,5}]$ | 20
11. | $8m+3$ | $\left(\frac{1}{2},\frac{8m+3}{8}\right)$ | 128$m^{2}$ \+ 88$m$ \+ 15 | $D_{8m+3,1}$ | $21$ | $\left(\frac{3}{2},\frac{13}{8}\right)$ | $105$ | $\mathcal{E}_{3}[A_{3,1}^{\oplus 7}]$ | 30
| | | | | | | | $\mathcal{E}_{3}[D_{5,2}^{\oplus 2}A_{3,1}]$ | 31
| | | | | | | | $\mathcal{E}_{3}[A_{7,2}C_{3,1}^{\oplus 2}]$ | 33
| | | | | | | | $\mathcal{E}_{3}[D_{7,3}G_{2,1}]$ | 34
| | | | | | | | $\mathcal{E}_{3}[C_{7,2}]$ | 35
12. | $8m+5$ | $\left(\frac{1}{2},\frac{8m+5}{8}\right)$ | 128$m^{2}$ \+ 152$m$ \+ 45 | $D_{8m+5,1}$ | $19$ | $\left(\frac{3}{2},\frac{11}{8}\right)$ | $171$ | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | 49
13. | $8m+6$ | $\left(\frac{1}{2},\frac{8m+6}{8}\right)$ | 128$m^{2}$ \+ 184$m$ \+ 66 | $D_{8m+6,1}$ | $18$ | $\left(\frac{3}{2},\frac{5}{4}\right)$ | $198$ | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | 54
| | | | | | | | $\mathcal{E}_{3}[A_{9,1}^{2}]$ | 55
14. | $8m+7$ | $\left(\frac{1}{2},\frac{8m+7}{8}\right)$ | 128$m^{2}$ \+ 216$m$ \+ 91 | $D_{8m+7,1}$ | $17$ | $\left(\frac{3}{2},\frac{9}{8}\right)$ | $221$ | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$ | 59
15. | $8m+9$ | $\left(\frac{1}{2},\frac{8m+9}{8}\right)$ | 128$m^{2}$ \+ 280$m$ \+ 153 | $D_{8m+9,1}$ | $15$ | $\left(\frac{3}{2},\frac{7}{8}\right)$ | $255$ | $\mathcal{E}_{3}[A_{15,1}]$ | 63
16. | $8m+10$ | $\left(\frac{1}{2},\frac{8m+10}{8}\right)$ | 128$m^{2}$ \+ 312$m$ \+ 190 | $D_{8m+10,1}$ | $14$ | $\left(\frac{3}{2},\frac{3}{4}\right)$ | $266$ | $E_{7,1}^{\oplus 2}$ | 64
17. | $8m+12$ | $\left(\frac{1}{2},\frac{8m+12}{8}\right)$ | 128$m^{2}$ \+ 376$m$ \+ 276 | $D_{8m+12,1}$ | $12$ | $\left(\frac{3}{2},\frac{1}{2}\right)$ | $276$ | $D_{12,1}$ | 66
The new meromorphic theories predicted by the above coset relations are
summarised in Table LABEL:t4. The first 15 rows have $B_{r,1}$ factors and the
next 19 rows have $D_{r,1}$ factors. We provide the Kac-Moody algebra of which
the meromorphic theory is an extension (in the first row, the extension is of
a combination of a Kac-Moody algebra with the Baby Monster module). When the
integer $m$ is equal to 0 the theory is part of the list in [16] and we
provide the serial number of that list where this entry can be found. For all
$m\geq 1$ the theory is new, to our knowledge.
Finally, for each case we provide the linear combination of character
bilinears corresponding to the character of the extension. Here we have
exhibited the way in which the 9 characters of the theory ${\cal
C}\oplus{\tilde{\cal C}}$ are combined into a meromorphic extension ${\cal
E}_{1}[{\cal C}\oplus{\tilde{\cal C}}]$ (in special cases where ${\cal
C}={\tilde{\cal C}}$ we have 6 rather than 9 characters for ${\cal
C}\oplus{\tilde{\cal C}}$, but the idea is the same). The quantities
${\hat{\chi}}_{0},{\hat{\chi}}_{2},{\hat{\chi}}_{m+2}$, labelled by their
conformal dimensions, are respectively the bilinears
$\chi_{0}{\tilde{\chi}}_{0},\chi_{1}{\tilde{\chi}}_{1},\chi_{2}{\tilde{\chi}}_{2}$
of the characters of ${\cal C},{\tilde{\cal C}}$ (these however are labelled
serially as $0,1,2$ rather than by their conformal dimensions, the latter can
be read off from the table) 666Note that the ${\tilde{\chi}}_{i}$ are not
characters of the Kac-Moody algebra, but of its three-character extension..
The ${\hat{\chi}}$’s are three of the 9 (or 6) characters in ${\cal
C}\oplus{\tilde{\cal C}}$. In general the bilinear identity involves
coefficients $d_{1},d_{2}$ (recall that $d_{0}=1$). Thus the column specifies
${\hat{\chi}}_{0}+d_{1}{\hat{\chi}}_{1}+d_{2}{\hat{\chi}}_{2}$ which defines
the new meromorphic theory ${\cal E}_{1}[{\cal C}\oplus{\tilde{\cal C}}]$.
Table 3: The $34$ infinite series of new meromorphic CFTs. In each series, $m=0$ corresponds to a Schellekens theory, whose Schellekens number S# is given in the second last column. BM denotes the Baby Monster CFT. Each theory has central charge $8m+24$. # | $\mathcal{H}$ | S# | Modular | # | $\mathcal{H}$ | S# | Modular
---|---|---|---|---|---|---|---
| | | invariant | | | | invariant
1. | $\mathcal{E}_{1}[B_{8m,1}\text{BM}]$ | $0$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,{\hat{\chi}}_{m+2}$ | 2. | $\mathcal{E}_{1}[B_{8m+1,1}A_{1,2}^{\oplus 15}]$ | $5$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$
3. | $\mathcal{E}_{1}[B_{8m+1,1}A_{3,4}^{\oplus 3}]$ | $7$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$ | 4. | $\mathcal{E}_{1}[B_{8m+1,1}A_{5,6}C_{2,3}]$ | $8$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$
5. | $\mathcal{E}_{1}[B_{8m+1,1}D_{5,8}]$ | $10$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,1024{\hat{\chi}}_{m+2}$ | 6. | $\mathcal{E}_{1}[B_{8m+2,1}C_{2,1}^{\oplus 3}D_{4,2}^{\oplus 2}]$ | $25$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$
7. | $\mathcal{E}_{1}[B_{8m+2,1}A_{2,1}^{\oplus 2}A_{5,2}^{\oplus 2}]$ | $26$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$ | 8. | $\mathcal{E}_{1}[B_{8m+2,1}A_{2,1}E_{6,4}]$ | $28$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,512{\hat{\chi}}_{m+2}$
9. | $\mathcal{E}_{1}[B_{8m+3,1}B_{3,1}C_{4,1}D_{6,2}]$ | $39$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,256{\hat{\chi}}_{m+2}$ | 10. | $\mathcal{E}_{1}[B_{8m+3,1}A_{4,1}A_{9,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,256{\hat{\chi}}_{m+2}$
11. | $\mathcal{E}_{1}[B_{8m+4,1}B_{4,1}D_{8,2}]$ | $47$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,128{\hat{\chi}}_{m+2}$ | 12. | $\mathcal{E}_{1}[B_{8m+4,1}\,C_{6,1}^{\oplus 2}]$ | $48$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+\,128{\hat{\chi}}_{m+2}$
13. | $\mathcal{E}_{1}[B_{8m+5,1}\,E_{7,2}\,F_{4,1}]$ | $53$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ | 14. | $\mathcal{E}_{1}[B_{8m+6,1}\,C_{10,1}]$ | $56$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+32\,{\hat{\chi}}_{m+2}$
15. | $\mathcal{E}_{1}[B_{8m+8,1}\,E_{8,2}]$ | $62$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ | | | |
16. | $\mathcal{E}_{1}[D_{8m+1,1}D_{1,1}^{\oplus 23}]$ | $1$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 17. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}^{\oplus 22}]$ | $15$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
18. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}]$ | $16$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 19. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}A_{5,3}D_{4,3}]$ | $17$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
20. | $\mathcal{E}_{1}[D_{8m+2,1}A_{1,1}A_{7,4}]$ | $18$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 21. | $\mathcal{E}_{1}[D_{8m+2,1}C_{3,2}D_{5,4}]$ | $19$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$
22. | $\mathcal{E}_{1}[D_{8m+2,1}D_{6,5}]$ | $20$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+4096\,{\hat{\chi}}_{m+2}$ | 23. | $\mathcal{E}_{1}[D_{8m+3,1}A_{3,1}^{\oplus 7}]$ | $30$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
24. | $\mathcal{E}_{1}[D_{8m+3,1}A_{3,1}D_{5,2}^{\oplus 2}]$ | $31$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$ | 25. | $\mathcal{E}_{1}[D_{8m+3,1}A_{7,2}C_{3,1}^{\oplus 2}]$ | $33$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
26. | $\mathcal{E}_{1}[D_{8m+3,1}\,D_{7,3}G_{2,1}]$ | $34$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$ | 27. | $\mathcal{E}_{1}[D_{8m+3,1}C_{7,2}]$ | $35$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+1024\,{\hat{\chi}}_{m+2}$
28. | $\mathcal{E}_{1}[D_{8m+5,1}A_{7,1}^{\oplus 2}D_{5,1}]$ | $49$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$ | 29. | $\mathcal{E}_{1}[D_{8m+6,1}D_{6,1}^{\oplus 3}]$ | $54$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$
30. | $\mathcal{E}_{1}[D_{8m+6,1}A_{9,1}^{\oplus 2}]$ | $55$ | ${\hat{\chi}}_{0}+64{\hat{\chi}}_{2}+256\,{\hat{\chi}}_{m+2}$ | 31. | $\mathcal{E}_{1}[D_{8m+7,1}A_{11,1}E_{6,1}]$ | $59$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+64\,{\hat{\chi}}_{m+2}$
32. | $\mathcal{E}_{1}[D_{8m+9,1}A_{15,1}]$ | $63$ | ${\hat{\chi}}_{0}+8{\hat{\chi}}_{2}+16\,{\hat{\chi}}_{m+2}$ | 33. | $\mathcal{E}_{1}[D_{8m+10,1}E_{7,1}^{\oplus 2}]$ | $64$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+2\,{\hat{\chi}}_{m+2}$
34. | $\mathcal{E}_{1}[D_{8m+12,1}D_{12,1}]$ | $66$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$ | | | |
We now work out an example in detail. This is a typical case in this table and
should clarify the procedure that has been used for all cases. We pick Row 6
of Table LABEL:t1, involving the coset pairing of $B_{8m+5,1}$ with ${\cal
E}_{3}[E_{7,2}\oplus F_{4,1}]$. The resulting meromorphic theory is in row 13
of Table LABEL:t4. Notice that $B_{r,1}$ is a three-character affine theory
for all $r$ (the same is true for the $D_{r,1}$ series, see Appendix A for
details). We now claim that the two factors pair to a meromorphic character
with $c^{\mathcal{H}}=8(m+3)$ and $(n_{1},n_{2})=(2,m+2)$ and that the
bilinear identity is:
$\displaystyle\chi_{0}^{\mathcal{H}}$
$\displaystyle=\chi_{0}\tilde{\chi}_{0}+\chi_{\frac{1}{2}}\tilde{\chi}_{\frac{3}{2}}+\chi_{\frac{16m+11}{16}}\tilde{\chi}_{\frac{21}{16}}={\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}$
(3.1)
The notation has been explained above. It should be kept in mind that for
$m=0$ the last two terms in the last expression remain distinct although both
would be denoted by ${\hat{\chi}}_{2}$, one comes from the bilinear in
$h=\frac{1}{2},{\tilde{h}}=\frac{3}{2}$ while the other comes from the
bilinear in $h=\frac{11}{16},{\tilde{h}}=\frac{21}{16}$.777However for self-
dual pairs the two ${\hat{\chi}}_{2}$’s are indeed same at $m=0$. However for
$m\geq 1$, the case of interest here, there is no ambiguity in the notation.
Now we explain how the bilinear identity is proved for all $m$, with the
$m$-independent coefficients $(d_{1},d_{2})=(1,1)$ in this family. To start
with, we have explicitly verified the bilinear relation, to order $q^{2000}$
in the $q$-series, for $32\leq c^{\cal H}\leq 72$, which corresponds to $1\leq
m\leq 6$ by comparing the series expansion on both sides. We find
$(d_{1},d_{2})=(1,1)$ in all these cases, and the modular invariant on the RHS
of the bilinear relation to be:
$\begin{split}&c^{\mathcal{H}}=24:\,\,\,\chi^{\cal H}(\tau)=j-504,\\\
&c^{\mathcal{H}}=32:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{4}{3}}-456\,j^{\frac{1}{3}},\\\
&c^{\mathcal{H}}=40:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{5}{3}}-152\,j^{\frac{2}{3}},\\\
&c^{\mathcal{H}}=48:\,\,\,\chi^{\cal H}(\tau)=j^{2}+408\,j-129024,\\\
&c^{\mathcal{H}}=56:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{7}{3}}+1224\,j^{\frac{4}{3}}-374784\,j^{\frac{1}{3}},\\\
&c^{\mathcal{H}}=64:\,\,\,\chi^{\cal
H}(\tau)=j^{\frac{8}{3}}+2296\,j^{\frac{5}{3}}-659456\,j^{\frac{2}{3}},\\\
&c^{\mathcal{H}}=72:\,\,\,\chi^{\cal
H}(\tau)=j^{3}+3624\,j^{2}-839680\,j-33030144.\end{split}$ (3.2)
This led us to conjecture that $(d_{1},d_{2})=(1,1)$ independent of $m$, and
then immediately to a proof of the conjecture.
For the proof we switch to the notation of Eq.(2.2) and find a formula for the
coefficients in this example for all $m$. For this we first write:
$\displaystyle{\hat{\chi}}_{0}+d_{1}\,{\hat{\chi}}_{2}+d_{2}\,{\hat{\chi}}_{m+2}=q^{-\frac{c^{\mathcal{H}}}{24}}\left(\text{an
integral power series of q}\right)$ (3.3)
After cancelling the fractional power of $q$ – if any – from both sides of the
above equation, we see that:
$\displaystyle a_{r}(m)=$ $\displaystyle\,\text{coefficient of}\ q^{r}\
\text{in LHS of Eq.(\ref{b8m5_ghm185_9char00})}$ $\displaystyle-$
$\displaystyle\ \text{coefficient of}\ q^{r}\ \text{in}\
\left(j^{\frac{m+3}{3}}+a_{1}\,j^{\frac{m}{3}}+\ldots+a_{r-1}j^{\frac{m+3}{3}-(r-1)}\right).$
(3.4)
For $r=1$ we can write more explicitly:
$\displaystyle a_{1}(m)=$ $\displaystyle\ \text{coefficient of}\ q\ \text{in
LHS of Eq.(\ref{b8m5_ghm185_9char00})}\ -\ \text{coefficient of}\ q\
\text{in}\ j^{\frac{m+3}{3}}$ $\displaystyle=$ $\displaystyle\
128\,m^{2}+168\,m+240-248(m+3).$ (3.5)
Note that though $a_{1}(m)$ has been derived from Eq.(3.3), it only comes from
comparing $\mathcal{O}(q)$ coefficient on both sides of Eq.(3.3) and hence the
formula for $a_{1}(m)$ is independent of $d_{1}$ and $d_{2}$. This is because
on the LHS of Eq.(3.3) $d_{1}$ appears at $\mathcal{O}(q^{2})$ and $d_{2}$
appears at $\mathcal{O}(q^{m+2})$.
The characters $\chi_{i}$ of the $B$-series affine theory at level 1 are known
to be given by Jacobi $\theta$-constants. Using this representation one can
show that $m_{2}(m)$ is quartic in $m$ (recall that $m_{2}$ is the second-
level degeneracy for the identity character). This in turn would imply that
the coefficient of $q^{2}$ in $\chi^{\cal H}$ would be quartic in $m$ and
hence $a_{2}$ would be quartic in $m$. Using this, we now prove that
$d_{1}(m)$ is independent of $m$.
We first employ Mathematica to solve (for $32\leq c^{\cal H}\leq 72$ as in
Eq.(3.2) above):
$\displaystyle(m,\text{coefficient of}\ q^{2}\ \text{in}\ \chi^{\cal H})=$
$\displaystyle[(1,272432),(2,560268),(3,1121832),(4,2159876),$
$\displaystyle(5,3942688),(6,6804092)].$ (3.6)
which returns:
$\displaystyle\text{coefficient of}\ q^{2}\ \text{in }\chi^{\cal
H}=121108+\frac{337268}{3}\,m+\frac{89056}{3}\,m^{2}+\frac{19456}{3}\,m^{3}+\frac{8192}{3}\,m^{4}.$
(3.7)
From the above we can get a general formula for $a_{2}$ using Eq.(2.2),
$\displaystyle a_{2}(m)=$ $\displaystyle
121108+\frac{337268}{3}\,m+\frac{89056}{3}\,m^{2}+\frac{19456}{3}\,m^{3}+\frac{8192}{3}\,m^{4}$
$\displaystyle-4124(m+3)-30752(m+2)(m+3)-248\,m\,a_{1}.$ (3.8)
Comparing the $\mathcal{O}(q^{2})$ term on both sides of Eq.(1.3)), we find:
$\displaystyle\tilde{m}_{2}+\tilde{m}_{1}m_{1}(m)+m_{2}(m)+d_{1}(m)\,D_{1}(m)\tilde{D}_{1}$
$\displaystyle=\,a_{2}(m)+248\,m\,a_{1}(m)+4124(m+3)+30752(m+2)(m+3).$ (3.9)
Now from row 6 of Table LABEL:t1 we read off:
$\begin{split}&{\tilde{m}}_{1}=185\\\ &m_{1}(m)=128m^{2}+168m+55,\end{split}$
(3.10)
while from the $\theta$-constant representation of the characters of $B_{r,1}$
we get:
$\begin{split}D_{1}(r)&=2r+1=16m+11\\\
m_{2}(r)&=1+\frac{25}{6}r+\frac{23}{6}r^{2}-\frac{2}{3}r^{3}+\frac{2}{3}r^{4},\end{split}$
(3.11)
where $r=8m+5$ for the present example.
Finally, solving the MLDE for the ($m$-independent) $\tilde{\cal C}$ theory in
row 6 of Table LABEL:t1 gives:
$\begin{split}{\tilde{m}_{2}}&=56351,\qquad{\tilde{D}}_{1}=4921.\end{split}$
(3.12)
Inserting Eqs.(3.10), (3.11), (3.12) in Eq. (3.9), we get:
$\displaystyle 56351+$ $\displaystyle
185(128\,m^{2}+168\,m+55)+\left(1+\frac{25}{6}(8m+5)+\frac{23}{6}(8m+5)^{2}\right.$
$\displaystyle\left.-\frac{2}{3}(8m+5)^{3}+\frac{2}{3}(8m+5)^{4}\right)+d_{1}(m)4921(16m+11)$
$\displaystyle=$
$\displaystyle\,a_{2}(m)+248\,m\,a_{1}(m)+4124(m+3)+30752(m+2)(m+3)$ (3.13)
from which, using Eq. (3.8) and Eq. (3.5), we find that $d_{1}(m)=1$ for all
$m$. Similarly one can argue that $d_{2}=1$ $\forall\,m\geq 1$.
The corresponding results for the remaining infinite families can be explained
in the same way. In all cases the coefficients $d_{1},d_{2}$ in the bilinear
relation for all $m$ are the same as those obtained for $m=0$, i.e. the
$c^{\cal H}=24$ case. Thus we have verified (still to order $q^{2000}$ in the
$q$-series), that this new theory satisfies a bilinear relation with
${\tilde{\cal C}}$ forming a modular invariant with $c^{\cal H}=8(m+3)$ for
all $m$.
Now let us give below, explicitly, the $3$-character extension of the twelve
character theory $E_{7,2}F_{4,1}$,
$\displaystyle\tilde{\chi}_{0}=\chi^{E}_{0}\chi^{F}_{0}+\chi^{E}_{\frac{7}{5}}\chi^{F}_{\frac{3}{5}}$
$\displaystyle\tilde{\chi}_{\frac{3}{2}}=\chi^{E}_{\frac{3}{2}}\chi^{F}_{0}+\chi^{E}_{\frac{9}{10}}\chi^{F}_{\frac{3}{5}}$
$\displaystyle\tilde{\chi}_{\frac{21}{16}}=\chi^{E}_{\frac{21}{16}}\chi^{F}_{0}+\chi^{E}_{\frac{57}{80}}\chi^{F}_{\frac{3}{5}}$
(3.14)
where $\chi^{E}_{i}$’s represent the six characters of $E_{7,2}$ and
$\chi^{F}_{i}$s represent the two characters of $F_{4,1}$. The above
expressions are obtained by comparing the characters of the extension (found
from the MLDE approach) with the leading behaviour of the characters
$\chi^{E},\chi^{F}$ which is given by the dimensions of integrable
representations.
Using the above result, now we can explicitly write the $1$-character
extension of the $36$-character theory $B_{8m+5,1}E_{7,2}F_{4,1}$:
$\displaystyle\chi^{\mathcal{H}}$
$\displaystyle={\hat{\chi}}_{0}+{\hat{\chi}}_{2}+{\hat{\chi}}_{m+2}=\chi_{0}\tilde{\chi}_{0}+\chi_{\frac{1}{2}}\tilde{\chi}_{\frac{3}{2}}+\chi_{\frac{16m+11}{16}}\tilde{\chi}_{\frac{21}{16}}$
$\displaystyle=j^{1/3}\left(j+128\,m^{2}+168\,m+240-248(m+3)\right),$
$\displaystyle=\chi_{0}\chi^{E}_{0}\chi^{F}_{0}+\chi_{0}\chi^{E}_{\frac{7}{5}}\chi^{F}_{\frac{3}{5}}+\chi_{\frac{1}{2}}\chi^{E}_{\frac{3}{2}}\chi^{F}_{0}+\chi_{\frac{1}{2}}\chi^{E}_{\frac{9}{10}}\chi^{F}_{\frac{3}{5}}+\chi_{\frac{16m+11}{16}}\chi^{E}_{\frac{21}{16}}\chi^{F}_{0}+\chi_{\frac{16m+11}{16}}\chi^{E}_{\frac{57}{80}}\chi^{F}_{\frac{3}{5}},$
(3.15)
where ${\hat{\chi}}_{i}$s represent the nine characters of
$B_{8m+5,1}\oplus\mathcal{E}_{3}[E_{7,2}F_{4,1}]$, $\chi_{i}$’s represent the
three characters of $B_{8m+5,1}$, $\tilde{\chi}_{i}$’s represent the three
characters of $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$.
### 3.3 New meromorphic theories at $c=32,40$ : finite families
In this section we exhibit a finite set of novel meromorphic theories with
central charges $c^{\mathcal{H}}=32$ and $c^{\mathcal{H}}=40$ only. As in the
previous cases, the bilinear relations for these examples have also been
verified to order $q^{2000}$. Tables LABEL:t2 and LABEL:t3 exhibit the coset
pairings which correspond respectively to $c^{\mathcal{H}}=32$ with
$(n_{1},n_{2})=(2,3)$, and $c^{\mathcal{H}}=40$ with $(n_{1},n_{2})=(3,3)$.
There exists no new family for $c^{\mathcal{H}}=40$ with
$(n_{1},n_{2})=(2,4)$. At $c^{\mathcal{H}}=32$ we find that there are 7
additional meromorphic theories. Out of these, 3 are novel non-lattice
meromorphic theories (second and third line of row 1, and row 2). The
remaining 4 are lattice meromorphic theories whose lattices can be found in
[39]. Similarly, at $c^{\mathcal{H}}=40$, there are 39 additional meromorphic
theories, of which 32 are novel non-lattice theories and the remaining 7 are
lattice theories.
Table 4: Coset relations for $c^{\mathcal{H}}=32$ with $(n_{1},n_{2})=(2,3)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$
---|---|---|---|---|---|---|---|---
1. | $12$ | $\left(\frac{2}{3},\frac{4}{3}\right)$ | 156 | $E_{6,1}^{\oplus 2}$ | $20$ | $\left(\frac{4}{3},\frac{5}{3}\right)$ | $80$ | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$
| | | | | | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
| | | | | | | | $\mathcal{E}_{3}[A_{8,3}]$
2. | $\frac{68}{5}$ | $\left(\frac{4}{5},\frac{7}{5}\right)$ | 136 | $\mathcal{E}_{3}[C_{8,1}]$ | $\frac{92}{5}$ | $\left(\frac{6}{5},\frac{8}{5}\right)$ | $92$ | $\mathcal{E}_{3}[E_{6,3}G_{2,1}]$
3. | $14$ | $\left(\frac{3}{4},\frac{3}{2}\right)$ | 266 | $E_{7,1}^{\oplus 2}$ | $18$ | $\left(\frac{5}{4},\frac{3}{2}\right)$ | $198$ | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$
| | | | | | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$
4. | $15$ | $\left(\frac{7}{8},\frac{3}{2}\right)$ | 255 | $\mathcal{E}_{3}[A_{15,1}]$ | $17$ | $\left(\frac{9}{8},\frac{3}{2}\right)$ | $221$ | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$
Table 5: Coset relations for $c^{\mathcal{H}}=40$ with $(n_{1},n_{2})=(3,3)$. # | $c$ | $(h_{1},h_{2})$ | $m_{1}$ | $\mathcal{C}$ | $\tilde{c}$ | $(\tilde{h}_{1},\tilde{h}_{2})$ | $\tilde{m}_{1}$ | $\tilde{\mathcal{C}}$
---|---|---|---|---|---|---|---|---
| | | | | | | |
1. | $20$ | $\left(\frac{1}{2},\frac{5}{2}\right)$ | 780 | $D_{20,1}$ | $20$ | $\left(\frac{5}{2},\frac{1}{2}\right)$ | $780$ | $D_{20,1}$
2. | $20$ | $\left(\frac{4}{3},\frac{5}{3}\right)$ | 80 | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ | $20$ | $\left(\frac{5}{3},\frac{4}{3}\right)$ | $80$ | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$
| | | | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ | | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
| | | | $\mathcal{E}_{3}[A_{2,1}^{\oplus 10}]$ | | | | $\mathcal{E}_{3}[A_{8,3}]$
| | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$ | | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$
| | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}C_{2,1}]$ | | | | $\mathcal{E}_{3}[A_{8,3}]$
| | | | $\mathcal{E}_{3}[A_{8,3}]$ | | | | $\mathcal{E}_{3}[A_{8,3}]$
3. | $20$ | $\left(\frac{7}{5},\frac{8}{5}\right)$ | 120 | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$ | $20$ | $\left(\frac{8}{5},\frac{7}{5}\right)$ | $120$ | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$
| | | | $\mathcal{E}_{3}[A_{4,1}^{\oplus 5}]$ | | | | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$
| | | | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$ | | | | $\mathcal{E}_{3}[A_{9,2}B_{3,1}]$
4. | $17$ | $\left(\frac{3}{2},\frac{9}{8}\right)$ | 221 | $\mathcal{E}_{3}[A_{11,1}E_{6,1}]$ | $23$ | $\left(\frac{3}{2},\frac{15}{8}\right)$ | $23$ | $\mathcal{E}_{3}[D_{1,1}^{\oplus 23}]$
5. | $\frac{35}{2}$ | $\left(\frac{3}{2},\frac{19}{16}\right)$ | 210 | $\mathcal{E}_{3}[C_{10,1}]$ | $\frac{45}{2}$ | $\left(\frac{3}{2},\frac{29}{16}\right)$ | $45$ | $\mathcal{E}_{3}[A_{1,2}^{\oplus 15}]$
| | | | $\mathcal{E}_{3}[C_{10,1}]$ | | | | $\mathcal{E}_{3}[A_{3,4}^{\oplus 3}]$
| | | | $\mathcal{E}_{3}[C_{10,1}]$ | | | | $\mathcal{E}_{3}[A_{5,6}C_{2,3}]$
| | | | $\mathcal{E}_{3}[C_{10,1}]$ | | | | $\mathcal{E}_{3}[D_{5,8}]$
6. | $18$ | $\left(\frac{5}{4},\frac{3}{2}\right)$ | 198 | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | $22$ | $\left(\frac{7}{4},\frac{3}{2}\right)$ | $66$ | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$
| | | | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | | | | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$
| | | | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | | | | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$
| | | | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | | | | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$
| | | | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | | | | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$
| | | | $\mathcal{E}_{3}[D_{6,1}^{\oplus 3}]$ | | | | $\mathcal{E}_{3}[D_{6,5}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{1,1}^{\oplus 22}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{3,2}^{\oplus 4}A_{1,1}^{\oplus 2}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{5,3}D_{4,3}A_{1,1}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{7,4}A_{1,1}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[D_{5,4}C_{3,2}]$
| | | | $\mathcal{E}_{3}[A_{9,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[D_{6,5}]$
7. | $\frac{37}{2}$ | $\left(\frac{3}{2},\frac{21}{16}\right)$ | 185 | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | $\frac{43}{2}$ | $\left(\frac{3}{2},\frac{27}{16}\right)$ | $86$ | $\mathcal{E}_{3}[D_{4,2}^{\oplus 2}C_{2,1}^{\oplus 3}]$
| | | | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | | | | $\mathcal{E}_{3}[A_{5,2}^{\oplus 2}A_{2,1}^{\oplus 2}]$
| | | | $\mathcal{E}_{3}[E_{7,2}F_{4,1}]$ | | | | $\mathcal{E}_{3}[E_{6,4}A_{2,1}]$
8. | $19$ | $\left(\frac{3}{2},\frac{11}{8}\right)$ | 171 | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | $21$ | $\left(\frac{3}{2},\frac{13}{8}\right)$ | $105$ | $\mathcal{E}_{3}[A_{3,1}^{\oplus 7}]$
| | | | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[D_{5,2}^{\oplus 2}A_{3,1}]$
| | | | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{7,2}C_{3,1}^{\oplus 2}]$
| | | | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[D_{7,3}G_{2,1}]$
| | | | $\mathcal{E}_{3}[D_{5,1}A_{7,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[C_{7,2}]$
9. | $\frac{39}{2}$ | $\left(\frac{3}{2},\frac{23}{16}\right)$ | 156 | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ | $\frac{41}{2}$ | $\left(\frac{3}{2},\frac{25}{16}\right)$ | $123$ | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$
| | | | $\mathcal{E}_{3}[D_{8,2}B_{4,1}]$ | | | | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$
| | | | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[D_{6,2}C_{4,1}B_{3,1}]$
| | | | $\mathcal{E}_{3}[C_{6,1}^{\oplus 2}]$ | | | | $\mathcal{E}_{3}[A_{9,2}A_{4,1}]$
The new meromorphic theories predicted by the above coset relations are
summarised in Table LABEL:t5. The format for the columns of this table is
similar to that of Table LABEL:t4. The first 7 entries are theories at $c=32$
and the next 39 are theories at $c=40$.
Table 6: The finite set of 46 new meromorphic CFTs at central charges $32$ and $40$. # | $\mathcal{H}$ | $c^{\mathcal{H}}$ | $\chi^{\mathcal{H}}$ | # | $\mathcal{H}$ | $c^{\mathcal{H}}$ | $\chi^{\mathcal{H}}$
---|---|---|---|---|---|---|---
1. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$ | 2. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 2}C_{2,1}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$
3. | $\mathcal{E}_{1}[A_{8,3}E_{6,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+972{\hat{\chi}}_{2}+2^{2}\cdot 3^{7}{\hat{\chi}}_{3}$ | 4. | $\mathcal{E}_{1}[C_{8,1}E_{6,3}G_{2,1}]$ | $32$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{2}+1250{\hat{\chi}}_{3}$
5. | $\mathcal{E}_{1}[D_{6,1}^{\oplus 3}E_{7,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+256{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$ | 6. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}E_{7,1}^{\oplus 2}]$ | $32$ | ${\hat{\chi}}_{0}+256{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$
7. | $\mathcal{E}_{1}[A_{11,1}A_{15,1}E_{6,1}]$ | $32$ | ${\hat{\chi}}_{0}+512{\hat{\chi}}_{2}+64{\hat{\chi}}_{3}$ | | | |
8. | $\mathcal{E}_{1}[D_{20,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2{\hat{\chi}}_{3}$ | 9. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 20}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
10. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}A_{5,2}^{\oplus 2}C_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 11. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 10}A_{8,3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
12. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 4}C_{2,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 13. | $\mathcal{E}_{1}[A_{5,2}^{\oplus 2}A_{8,3}C_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$
14. | $\mathcal{E}_{1}[A_{8,3}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{3}\cdot 3^{12}{\hat{\chi}}_{3}$ | 15. | $\mathcal{E}_{1}[A_{4,1}^{\oplus 10}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$
16. | $\mathcal{E}_{1}[A_{9,2}^{\oplus 2}B_{3,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$ | 17. | $\mathcal{E}_{1}[A_{4,1}^{\oplus 5}A_{9,2}\,B_{3,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{2}\cdot 5^{8}{\hat{\chi}}_{3}$
18. | $\mathcal{E}_{1}[A_{11,1}D_{1,1}^{\oplus 23}E_{6,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 19. | $\mathcal{E}_{1}[A_{1,2}^{\oplus 15}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
20. | $\mathcal{E}_{1}[A_{3,4}^{\oplus 3}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 21. | $\mathcal{E}_{1}[A_{5,6}C_{2,3}C_{10,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
22. | $\mathcal{E}_{1}[C_{10,1}D_{5,8}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 23. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 22}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
24. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 25. | $\mathcal{E}_{1}[A_{1,1}A_{5,3}D_{4,3}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
26. | $\mathcal{E}_{1}[A_{1,1}A_{7,4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 27. | $\mathcal{E}_{1}[C_{3,2}D_{5,4}D_{6,1}^{\oplus 3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
28. | $\mathcal{E}_{1}[D_{6,1}^{\oplus 3}D_{6,5}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 29. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 22}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
30. | $\mathcal{E}_{1}[A_{1,1}^{\oplus 2}A_{3,2}^{\oplus 4}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 31. | $\mathcal{E}_{1}[A_{1,1}A_{5,3}A_{9,1}^{\oplus 2}D_{4,3}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
32. | $\mathcal{E}_{1}[A_{1,1}A_{7,4}A_{9,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 33. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}C_{3,2}D_{5,4}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$
34. | $\mathcal{E}_{1}[A_{9,1}^{\oplus 2}D_{6,5}]$ | $40$ | ${\hat{\chi}}_{0}+2^{19}{\hat{\chi}}_{3}+2^{12}{\hat{\chi}}_{3}$ | 35. | $\mathcal{E}_{1}[C_{2,1}^{\oplus 3}D_{4,2}^{\oplus 2}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
36. | $\mathcal{E}_{1}[A_{2,1}^{\oplus 2}A_{5,2}^{\oplus 2}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 37. | $\mathcal{E}_{1}[A_{2,1}E_{6,4}E_{7,2}F_{4,1}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
38. | $\mathcal{E}_{1}[A_{3,1}^{\oplus 7}A_{7,1}^{\oplus 2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 39. | $\mathcal{E}_{1}[A_{3,1}A_{7,1}^{\oplus 2}D_{5,1}D_{5,2}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$
40. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}A_{7,2}C_{3,1}^{\oplus 2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 41. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}D_{5,1}D_{7,3}G_{2,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$
42. | $\mathcal{E}_{1}[A_{7,1}^{\oplus 2}C_{7,2}D_{5,1}]$ | $40$ | ${\hat{\chi}}_{0}+2^{6}{\hat{\chi}}_{3}+2^{17}{\hat{\chi}}_{3}$ | 43. | $\mathcal{E}_{1}[B_{3,1}B_{4,1}C_{4,1}D_{6,2}D_{8,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
44. | $\mathcal{E}_{1}[A_{4,1}A_{9,2}B_{4,1}D_{8,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | 45. | $\mathcal{E}_{1}[B_{3,1}C_{4,1}C_{6,1}^{\oplus 2}D_{6,2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$
46. | $\mathcal{E}_{1}[A_{4,1}A_{9,2}C_{6,1}^{\oplus 2}]$ | $40$ | ${\hat{\chi}}_{0}+{\hat{\chi}}_{3}+2^{15}{\hat{\chi}}_{3}$ | | | |
## 4 Conclusions
The meromorphic theories we have identified in the present work have been
summarised in Tables LABEL:t4 and LABEL:t5. As mentioned in the Introduction,
the proposals in this work are made at a physics level of rigour and
considerable evidence provided. It should be possible in future to convert
these to a set of mathematically rigorous statements and proofs. We have
obtained infinitely many theories that can be thought of as generalisations of
34 non-lattice theories in Ref. [16] to arbitrary central charge $c=8N$. The
fact that infinitely many generalisations exist is ultimately due to
properties of the $B_{r,1}$ and $D_{r,1}$ affine theories, which have three
characters for all $r$ and whose modular transformation matrix is periodic in
$r$.
It is interesting to see the Baby Monster module make an appearance in this
discussion, in row 1 of Table LABEL:t4. This appears to illustrate a general
phenomenon: modules with and without Kac-Moody algebras appear on a similar
footing in general meromorphic theories (and presumably in more general
RCFTs). This is in contrast to the $c=24$ case where there is one entry (the
Monster CFT) that is the extension of modules without a Kac-Moody algebra,
namely ${\cal E}_{1}[{\cal M}(4,3)\,BM]$, while the rest are extensions of
only Kac-Moody algebras. Such a perspective may be helpful to derive at least
partial classifications for meromorphic theories at $c=32$, even though a
complete classification is understood to be virtually impossible due to the
enormous number (of order $10^{9}$) of possible theories.
The present work makes no claim to being complete. The goal has only been to
provide several examples of an interesting phenomenon: the prediction of
meromorphic theories at higher $c$ starting from coset pairings for
meromorphic theories at lower $c$. The present results already suggest many
new possibilities, for example we may conjecture that starting at every
$c^{\cal H}=8(2p+3)$ (where $p\in\mathbb{N}\cup\\{0\\}$) there is an infinite
series labelled by another parameter $m$ with $(n_{1},n_{2})=(p+2,p+m+2)$.
This series is the extension $\mathcal{E}_{1}[D_{8p+12,1}D_{8p+8m+12,1}]$. The
special case for $p=0$ and arbitrary $m$ corresponds to row 17 of table
LABEL:t1 which describes a series of meromorphic theories starting at $c^{\cal
H}=24$ given by $\mathcal{E}_{1}[D_{12,1}D_{8m+12,1}]$. Also the special case
$p=1$ and $m=0$ can be found in row 1 of table LABEL:t3 which describes a
meromorphic theory at $c^{\cal H}=40$ given by,
$\mathcal{E}_{1}[D_{20,1}D_{20,1}]$. Our conjecture subsumes these examples
and suggests infinite generalisations thereof. More generally, we have not yet
considered cases with multiple $B_{r,1}$ or $D_{r,1}$ factors. We hope to
return to all these and many more cases in the future [24].
The recursive nature of the process described here is a potentially very
useful feature. If a meromorphic theory is first discovered by any method at
any central charge $c=8N$, one can attempt to construct its generalisations at
higher values of $c$ using our approach.
Our results provide more evidence for the close relationship between general
RCFTs and meromorphic CFT. It is not true, as was implicitly thought earlier
(and is still occasionally claimed in the literature), that meromorphic
theories constitute some sort of “exotic” outliers in the space of all RCFTs.
Rather, the most general RCFTs are the ones with extensions of the usual
Virasoro and Kac-Moody algebras. The most familiar RCFTs, such as minimal
Virasoro and affine theories, are merely unextended special cases of the RCFT
landscape which is mostly populated by extensions. Meromorphic theories are
such extensions, and are intimately linked to $n$-character RCFT for $n>1$
through coset relations.
## Acknowledgements
AD and CNG would like to thank Jagannath Santara for collaboration on the
papers [3] and [10] and for very helpful discussions. AD would like to thank
Daniele Dorigoni, Nabil Iqbal and Madalena Lemos for useful discussions on Lie
algebras. He would also like to thank Sigma Samhita for her immense help in
the type setting of the tables required for this work. He would also like to
express his gratitude to Gabriel Arenas-Henriquez and Jose Cerqueira-sa for
helpful discussions on Mathematica. SM would like to thank Brandon Rayhaun for
very helpful discussions, and Keshav Dasgupta and Alex Maloney for hospitality
at McGill University, Montreal where part of this work was done. He is also
grateful for support from a grant by Precision Wires India Ltd. for String
Theory and Quantum Gravity research at IISER Pune.
## Appendix A Some general features of $B_{r,1}$ and $D_{r,1}$ affine
theories
* •
For $B_{r,1}$ we have the three characters as follows,
* –
$\chi_{0}$: Identity (adj. irrep): $[0,\cdots,0]$ (which contains information
about the dimension of Lie algebra (= coefficient of $\mathcal{O}(q)$ =
$2r^{2}+r$) and hence is the adjoint irrep of $SO(2r+1)$).
* –
$\chi_{\frac{1}{2}}$: Fundamental irrep: $\sigma_{1}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2r+1$) matches with that of the
fundamental irrep of $SO(2r+1)$).
* –
$\chi_{\frac{2r+1}{16}}$: Spinor irrep: $\sigma_{r}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2^{r}$) matches with that of the
spinor irrep of $SO(2r+1)$)888Dimension of spinorial representation of
$SO(n\,\text{odd})=2^{\frac{n-1}{2}}$.
* •
For $D_{r,1}$ we have the three characters as follows,
* –
$\chi_{0}$: Identity (adj. irrep): $[0,\cdots,0]$ (which contains information
about the dimension of Lie algebra (= coefficient of $\mathcal{O}(q)$ =
$2r^{2}-r$) and hence is the adjoint irrep of $SO(2r)$)
* –
$\chi_{\frac{1}{2}}$: Fundamental irrep: $\sigma_{1}$ (whose dimension (=
coefficient of $\mathcal{O}(q^{0})$ term = $2r$) matches with that of the
fundamental irrep of $SO(2r)$)
* –
$\chi_{\frac{r}{8}}$: Spinor irrep: $\sigma_{r}\equiv\sigma_{r-1}$ (whose
dimension (= coefficient of $\mathcal{O}(q^{0})$ term = $2^{r-1}$) matches
with that of the spinor irrep of $SO(2r)$)999Dimension of spinorial
representation of $SO(n\,\text{even})=2^{\frac{n}{2}-1}$. In this case (even
dimension) we have chiral spinor irreps ($\sigma_{r}\equiv\sigma_{r-1}$) but
at the level of characters they will be treated on equal footing.
Now note that we have an infinite family whenever we have a factor of
$B_{r_{1},1}$ or $D_{r_{1},1}$ in a theory and the idea is to generate an
infinite set by increasing $r_{1}\to r_{1}+8k$ (with $k\in\mathbb{Z}$). Now,
we can explicitly compute and check the $(d_{1},d_{2})$ values for upto
$c^{\mathcal{H}}=40$ and we observe that we get the same $(d_{1},d_{2})$ for a
given coset pair forming a particular infinite family at every stage of
$c^{\cal H}$. This is because the only thing that differs in the bilinear
relation of one stage to another for a given infinite family is the rank (in
steps of 8) of $B_{r_{1},1}$ or $D_{r_{1},1}$ factors. However, increasing
$r_{1}\to r_{1}+8$ doesn’t change the number of irreps of $B_{r_{1},1}$ or
$D_{r_{1},1}$. Hence, we observe again that the values of $(d_{1},d_{2})$ at
one value of $c^{\cal H}$ will be the same for all other values of $c^{\cal
H}$.
The above discussion shows a very crucial point about 3-characters. As we
discussed above we were able to generate infinite families of meromorphic
theories because each of these theories had a $B_{r_{1},1}$ or $D_{r_{1},1}$
as their factors and as we know from [3], every $B_{r_{1},1}$ or $D_{r_{1},1}$
has 3 characters (except $D_{4,1}$ which has 2 due to triality). Thus, we
could generalise the Schellenkens coset pairs which always resulted in a
9-character theory and from Schellekens we know that the non-$B_{r_{1},1}$ or
non-$D_{r_{1},1}$ factors always admit a unique $3$-character extension.
## Appendix B Relevant admissible character solutions
In this section we present (for completeness) the relevant admissible
character solutions to the $(3,0)$ MLDE and GHM solutions from [10] and [19]
respectively.
Table 7: Some admissible character solutions to the $(3,0)$ MLDE and GHM solutions. # | $c$ | $h_{1}$ | $h_{2}$ | $m_{1}$ | $D_{1}$ | $D_{2}$ | # | $c$ | $h_{1}$ | $h_{2}$ | $m_{1}$ | $D_{1}$ | $D_{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
${\bf III_{22}}$ | $\frac{68}{5}$ | $\frac{4}{5}$ | $\frac{7}{5}$ | $136$ | $119$ | $68$ | ${\bf III_{37}}$ | $\frac{92}{5}$ | $\frac{6}{5}$ | $\frac{8}{5}$ | $92$ | $1196$ | $299$
${\bf V_{39}}$ | $20$ | $\frac{4}{3}$ | $\frac{5}{3}$ | $80$ | $5$ | $4$ | ${\bf III_{45}}$ | $22$ | $\frac{3}{2}$ | $\frac{7}{4}$ | $66$ | $77$ | $11$
${\bf III_{50}}$ | $23$ | $\frac{3}{2}$ | $\frac{15}{8}$ | $23$ | $575$ | $23$ | | | | | | |
$\text{GHM}_{45}$ | $\frac{45}{2}$ | $\frac{3}{2}$ | $\frac{29}{16}$ | $45$ | $4785$ | $45$ | $\text{GHM}_{86}$ | $\frac{43}{2}$ | $\frac{3}{2}$ | $\frac{27}{16}$ | $86$ | $5031$ | $43$
$\text{GHM}_{105}$ | $21$ | $\frac{3}{2}$ | $\frac{13}{8}$ | $105$ | $637$ | $21$ | $\text{GHM}_{120}$ | $20$ | $\frac{7}{5}$ | $\frac{8}{5}$ | $120$ | $4$ | $13$
$\text{GHM}_{123}$ | $\frac{41}{2}$ | $\frac{3}{2}$ | $\frac{25}{16}$ | $123$ | $5125$ | $41$ | $\text{GHM}_{156}$ | $\frac{39}{2}$ | $\frac{3}{2}$ | $\frac{23}{16}$ | $156$ | $5083$ | $39$
$\text{GHM}_{171}$ | $19$ | $\frac{3}{2}$ | $\frac{11}{8}$ | $171$ | $627$ | $19$ | $\text{GHM}_{185}$ | $\frac{37}{2}$ | $\frac{3}{2}$ | $\frac{21}{16}$ | $185$ | $4921$ | $2368$
$\text{GHM}_{198}$ | $18$ | $\frac{3}{2}$ | $\frac{5}{4}$ | $198$ | $75$ | $9$ | $\text{GHM}_{210}$ | $\frac{35}{2}$ | $\frac{3}{2}$ | $\frac{19}{16}$ | $210$ | $4655$ | $35$
$\text{GHM}_{221}$ | $17$ | $\frac{3}{2}$ | $\frac{9}{8}$ | $221$ | $561$ | $17$ | $\text{GHM}_{255}$ | $15$ | $\frac{3}{2}$ | $\frac{7}{8}$ | $255$ | $455$ | $15$
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|
# Some comments on piecewise-projective groups
of the line
Nicolas Monod École Polytechnique Fédérale de Lausanne (EPFL)
CH–1015 Lausanne, Switzerland
###### Abstract.
We consider groups of piecewise-projective homeomorphisms of the line which
are known to be non-amenable using notably the Carrière–Ghys theorem on
ergodic equivalence relations. Replacing that theorem by an explicit fixed-
point argument, we can strengthen the conclusion and exhibit uncountably many
“amenability gaps” between various piecewise-projective groups.
This note is dedicated with admiration to Rostislav Grigorchuk at the occasion
of his 70th birthday. Slava taught me about Thompson’s group when I was a
student — and many times since then.
###### Contents
1. 1 Introduction
2. 2 Notation and preliminaries
3. 3 Amenability and fixed points
4. 4 Refining and strengthening non-amenability
5. 5 Additional comments
## 1\. Introduction
The purpose of this note is to revisit and to strengthen the non-amenability
of the group $H(\mathbf{R})$ of piecewise-projective homeomorphisms of the
real line and of many of its subgroups.
The motivation to revisit the proof given in [Mon13] is that the method it
introduced to establish non-amenability relied on the theory of equivalence
relations, specifically on a remarkable theorem of Carrière–Ghys [CG85]
adressing a conjecture of Connes and Sullivan. We shall show that the non-
amenability can be established from first principles and without the
Carrière–Ghys theorem. This possibility was alluded to in Remark 11 of
[Mon13]; this time, we give an elementary group-theoretical proof of non-
amenability. Namely, in Section 3.C we exhibit concrete convex compact spaces
on which the groups act without fixed point. This applies to $H(\mathbf{R})$
and to subgroups defined over arithmetic rings such at $\mathbf{Q}$,
$\mathbf{Z}[\sqrt{2}]$, $\mathbf{Z}[1/p]$ and their uncountably many variants.
The two motivations to strengthen non-amenability are, first, that a number of
_amenability-like_ properties were established in [Mon13], most prominently
the fact that $H(\mathbf{R})$ does not contain non-abelian free subgroups.
Secondly, the best-known piecewise-projective group is Thompson’s group $F$,
for which amenability remains a notorious open question. In fact, our
criterion fails precisely for $F$ and we have repeatedly been asked: does the
non-amenability of the various other groups imply or suggest anything for
Thompson’s group?
Towards this question, we shall prove that there are infinitely many “layers”
of non-amenability stacked atop one another within the subgroups of
$H(\mathbf{R})$. To any set $S$ of prime numbers, we associate the subgroup
$\Gamma_{S}$ of $H(\mathbf{R})$ obtained by restricting the coefficients of
the projective transformations to the ring $\mathbf{Z}[1/S]$ of $S$-integers.
This defines a poset of $2^{\aleph_{0}}$ subgroups and $F$ lies at the bottom
in the group $\Gamma_{\varnothing}$ with integral coefficient.
###### Theorem I.
Let $S,S^{\prime}$ be any sets of prime numbers with $S\subsetneqq
S^{\prime}$.
Then $\Gamma_{S}$ is not co-amenable in $\Gamma_{S^{\prime}}$.
The definition of co-amenability is recalled below (Section 4.A); informally,
this means that $\Gamma_{S^{\prime}}$ is “even less amenable” than
$\Gamma_{S}$.
In particular, we obtain a mathematically well-defined version of this
heuristic answer to the above question: no, the non-amenability of our various
subgroups of $H(\mathbf{R})$ does not give any hint for Thompson’s group.
Informally: for $S$ non-empty, $\Gamma_{S}$ is non-amenable _regardless_ of
the amenability status of $F$. Had $F$ been co-amenable in $\Gamma_{S}$, then
our non-amenability results would have implied the non-amenability of $F$.
Contrariwise, if $F$ is non-amenable, then $\Gamma_{S}$ is still “even less
amenable” than $F$.
The reader might regret that one can nest only countably many subgroups
$\Gamma_{S}$ into a chain of pairwise not co-amenable subgroups, whereas the
poset of all $\Gamma_{S}$ consists of continuum many subgroups of the
countable group $H(\mathbf{Q})$.
Not to worry: I follows from a more general statement that indeed allows us to
distinguish _any two_ $\Gamma_{S}$ from the perspective of “mutual non-
amenability”. A concrete way to state this is as follows:
###### Theorem II.
Let $S,S^{\prime}$ be any sets of prime numbers with $S\neq S^{\prime}$.
Then there exists a convex compact (explicit) $H(\mathbf{Q})$-space in which
one and only one of the two subgroups $\Gamma_{S}$ and $\Gamma_{S^{\prime}}$
admits a fixed point.
As it turns out, the latter statement is formally stronger than I even when
$S\subseteq S^{\prime}$. This will be explained in Section 4.A once the notion
of relative co-amenability will have been recalled. This “amenability gap”
between subgroups can also be reflected in the relative Furstenberg boundary
$\partial(G,G_{1})$ introduced in [BK21, Mon21]; II implies:
###### Corollary III.
The relative Furstenberg boundaries $\partial(H(\mathbf{Q}),\Gamma_{S})$,
where $S$ ranges over all subsets of prime numbers, are pairwise non-
isomorphic compact $H(\mathbf{Q})$-spaces.
The next result goes back to completely general rings and states that the non-
amenability of $H(A)$ can be strengthened to an amenability gap with respect
to the integral group $\Gamma_{\varnothing}$. In contrast to all other results
stated in this introduction, it will be proved using the relational method
introduced in [Mon13].
###### Theorem IV.
If $A<\mathbf{R}$ is any ring other than $\mathbf{Z}$, then
$\Gamma_{\varnothing}$ and a fortiori the Thompson group are not co-amenable
in $H(A)$.
Finally, we mention that the methods used in this note can easily be adapted
to some variants of the above groups. As an example, we show in 5.1 below that
the Thompson group is also not co-amenable in a larger group of
$C^{1}$-diffeomorphisms.
The considerations of relative co-amenability between various subgroups of a
given group $G$ can be formulated using some sort of “spectrum” $\dot{G}$
which is a poset functorially attached to $G$. Loosely speaking, the points of
$\dot{G}$ are defined by subgroups of $G$, where two subgroups are identified
whenever they concurrently do or do not admit a fixed point in any given
convex compact $G$-space, see Section 5.B.
Then II can be reformulated as stating that the poset of all sets of prime
numbers embeds fully faithfully into $\dot{G}$ for $G=H(\mathbf{Q})$. More
generally, each $\dot{\Gamma}_{S}$ contains a copy of the poset of all subsets
of $S$, see 5.4.
## 2\. Notation and preliminaries
The group $H(\mathbf{R})$ consists of all homeomorphisms $h$ of the real line
for which the line can be decomposed into finitely many intervals so that on
each interval, $h$ is given by $h(x)=g(x)=\dfrac{ax+b}{cx+d}$ for some
$g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ in $\mathbf{SL}_{2}(\mathbf{R})$.
This $g$ depends on the interval; since we consider homeomorphisms of the
line, the singularity $x=-d/c$ cannot lie in the corresponding interval.
The element $g$ is locally uniquely defined in $\mathbf{PSL}_{2}(\mathbf{R})$
but we use matrix representatives in $\mathbf{SL}_{2}(\mathbf{R})$ whenever no
confusion is to be feared.
We write the projective line $\mathbf{P}^{1}_{\mathbf{R}}$ as
$\mathbf{R}\cup\\{\infty\\}$ using projective coordinates $[x:1]$ for
$x\in\mathbf{R}$ and $\infty=[1:0]$, but we also use the signed symbols
$\pm\infty$ to clarify the relative position of points in $\mathbf{R}$ and how
they might converge to $\infty$ in $\mathbf{P}^{1}_{\mathbf{R}}$.
By restricting the coefficients of the matrix $g$ and the breakpoints in
$\mathbf{R}$, we obtain a wide variety of subgroups of $H(\mathbf{R})$:
Given a (unital) ring $A<\mathbf{R}$ and a set $B\subseteq\mathbf{R}$ such
that $B\cup\\{\infty\\}\subseteq\mathbf{P}^{1}_{\mathbf{R}}$ is
$\mathbf{SL}_{2}(A)$-invariant, we denote by $H_{B}(A)$ the subgroup where
matrices are chosen in $\mathbf{SL}_{2}(A)$ and breakpoints in $B$. A first
example is $H_{\mathbf{Q}}(\mathbf{Z})$, which is isomorphic to Thompson’s
group $F$, as observed by Thurston (this fact follows by identifying the
Stern–Brocot subdivision tree of $\mathbf{Q}\cup\\{\infty\\}$ with the dyadic
subdivision tree of $[0,1]$ used in the more common PL definition of $F$
[Imb97]; the proof given in [CFP96, §7] is a variant of this argument, with
piecewise-projective transformations of $[0,1]$ itself).
The easiest way to construct a piecewise-projective transformation of
$\mathbf{R}$ is to cut $\mathbf{P}^{1}_{\mathbf{R}}$ in two. Any hyperbolic
$g\in\mathbf{SL}_{2}(A)$ has exactly two fixed points in $\mathbf{P}^{1}$
which thus divide $\mathbf{P}^{1}_{\mathbf{R}}$ into two intervals. Then we
can define an element of $H(\mathbf{R})$ by the identity on the component
containing $\infty$ and by $g$ on the other. For that reason, we worked in
[Mon13] with the set $B$ of all such fixed points, and denoted the resulting
group simply by $H(A)$.
It is perhaps even simpler to restrict only the matrix coefficients and work
with the group of all piecewise-$\mathbf{SL}_{2}(A)$ homeomorphisms of the
line. In the above notation, this group is $H_{\mathbf{R}}(A)$. In the main
setting of this article, namely $S$-integers $A=\mathbf{Z}[1/S]$, these two
conventions coincide anyway:
###### Lemma 2.1.
Let $S$ be any non-empty set of prime numbers. Then all breakpoints of any
element of $H(\mathbf{R})$ with matrix coefficients in $\mathbf{Z}[1/S]$ are
fixed points of hyperbolic elements in $\mathbf{SL}_{2}(\mathbf{Z}[1/S])$.
Thus the group $\Gamma_{S}$ defined in the introduction coincides with the
countable group $H(\mathbf{Z}[1/S])$.
###### Proof.
Write $A=\mathbf{Z}[1/S]$. We first note that since $S$ contains at least some
prime $p$, the points $\infty$ and $0$ are fixed points of the hyperbolic
element $h=\left(\begin{smallmatrix}p&0\\\ 0&1/p\end{smallmatrix}\right)$ of
$\mathbf{SL}_{2}(A)$.
Let now $x\in\mathbf{R}$ be any breakpoint of an element $g\in\Gamma_{S}$.
Considering the matrix representatives $g_{-},g_{+}\in\mathbf{SL}_{2}(A)$
describing $g$ on the intervals left and right of $x$, we have $g_{-}x=g_{+}x$
by continuity. Therefore $g_{-}^{-1}g_{+}$ is an element of
$\mathbf{SL}_{2}(A)$ fixing $x$. This element is non-trivial since $x$ is a
breakpoint; it is therefore hyperbolic or parabolic. In the first case we are
done, and in the second case $x$ is rational because the characteristic
polynomial of $g_{-}^{-1}g_{+}$ has only one solution and has its coefficients
in $A<\mathbf{Q}$. Since $\mathbf{SL}_{2}(\mathbf{Z})$ acts transitively on
$\mathbf{Q}\cup\\{\infty\\}$, there is $k\in\mathbf{SL}_{2}(\mathbf{Z})$ with
$k.0=x$. Now the conjugate $khk^{-1}$ is a hyperbolic element of
$\mathbf{SL}_{2}(A)$ fixing $x$. ∎
Next, we recall the Frankensteinian cut-and-paste from [Mon13]. Since we aim
for elementary proofs of non-amenability, we give both the basic dynamical
explanation and the explicit algebraic computation from [Mon13, Prop. 9]
(especially since the latter has some “$\neq$” instead of “$=$” in the journal
version).
###### Proposition 2.2.
For every $g\in\mathbf{SL}_{2}(\mathbf{R})$ and $x\in\mathbf{R}$ with
$gx\neq\infty$, there is a piecewise-projective homeomorphism $h$ of
$\mathbf{R}$ which coincides with $g$ on a neighbourhood of $x$.
Moreover, one can take $h\in H(A)$ whenever $A<\mathbf{R}$ is a ring
containing the coefficients of $g$.
###### Dynamical explanation.
The homeomorphism $h$ of $\mathbf{R}$ is obtained as follows. Keep $g$ on a
small interval around $x$ and continue everywhere else with some translation
$y\mapsto y+n$. Thus we just need our interval boundaries to be two points
left and right of $x$ where $g$ coincides with the translation by $n$. The
basic dynamics of projective transformations such as $g$ imply that for _any_
large enough $n\in\mathbf{R}$, there are exactly two such points where this
holds for $n$ or possibly $-n$. The only exception is if $g$ was already
affine, in which case the proposition is trivial. ∎
We now give a completely explicit algebraic proof to keep track of the
coefficients involved, following [Mon13]. The above translation by $n$ will
now correspond to a column operation on the matrix for $g$ and therefore we
take $n\in\mathbf{Z}$ to remain in $A$.
###### Algebraic proof.
We can assume $g\infty\neq\infty$. Claim: there is
$q_{0}\in\mathbf{SL}_{2}(A)$ with $q_{0}\infty=g\infty$ and with two fixed
points $\xi_{-},\xi_{+}\in\mathbf{R}\subseteq\mathbf{P}^{1}_{\mathbf{R}}$ such
that the open interval of $\mathbf{R}$ determined by $\xi_{-},\xi_{+}$
contains $gx$ but not $g\infty$.
To deduce the proposition from this claim, consider the
piecewise-$\mathbf{SL}_{2}(A)$ homeomorphism $q$ of
$\mathbf{P}^{1}_{\mathbf{R}}$ which is the identity on the above interval and
is given by $q_{0}$ on its complement. This is indeed a homeomorphism since
the interval is defined by fixed points of $q_{0}$. Now $h=q^{-1}g$ fixes
$\infty$ and is the desired element $h\in H(A)$. (Notice that the breakpoints
of $h$ are $g^{-1}\xi_{\pm}$, which are the fixed points of the hyperbolic
matrix $g^{-1}q_{0}g$.)
To prove the claim, represent $g$ as $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$
and define $q_{0}$ by $\begin{pmatrix}a&b+na\\\ c&d+nc\end{pmatrix}$, where
$n\in\mathbf{Z}$ is an integer yet to be chosen. Then
$q_{0}\infty=[a:c]=g\infty$ holds. Moreover, $gx\neq\infty$ forces $c\neq 0$
and we can hence assume $c>0$. Therefore, the trace
$\tau=a+d+nc\in A$
satisfies $\tau^{2}>4$ as soon as $|n|$ is large enough. Then we have two real
eigenvalues $\lambda_{\pm}=\frac{1}{2}(\tau\pm\sqrt{\tau^{2}-4})$ and the
corresponding eigenvectors $\begin{pmatrix}x_{\pm}\\\ c\end{pmatrix}$, where
$x_{\pm}=\lambda_{\pm}-d-nc$, represent the points $\xi_{\pm}=[x_{\pm}:c]$.
Now we compute $\lim_{n\to+\infty}\xi_{-}=-\infty$ and
$\lim_{n\to+\infty}\xi_{+}=[a:c]=g\infty$, with the latter limit approaching
from the left because $x_{+}<a$. Conclusion: in case $gx<g\infty$, any
sufficiently large $n$ will ensure
$\xi_{-}\ <\ gx\ <\ \xi_{+}\ <\ g\infty,$
which yields the claim for that case. In the other case, when $gx>g\infty$,
the result is obtained in the exact same way but with $n\to-\infty$. ∎
## 3\. Amenability and fixed points
### 3.A. Convex compact spaces
If our goal is to give a simple and transparent proof of non-amenability, we
should choose which one of the many equivalent definitions of amenability to
use:
###### Definition 3.1.
A group $G$ is amenable if every convex compact $G$-space $K\neq\varnothing$
admits a fixed point.
It is implicit in this definition that $G$ acts on $K$ by affine
homeomorphisms and that the convex compact structure of $K$ is induced by some
ambient locally convex Hausdorff topological vector space containing $K$.
However, our starting point is a very concrete example of $K$ without fixed
point, maybe the simplest and best-known such example:
###### Example 3.2 (Measures on the projective line).
Let $K=\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{R}})$ be the space of
probability measures on $\mathbf{P}^{1}_{\mathbf{R}}$, endowed with the weak-*
topology. This is the pointwise convergence topology when measures are seen as
functionals on the space $C$ of continuous functions on
$\mathbf{P}^{1}_{\mathbf{R}}$. Then $K$ is a convex compact $G$-space for
$G=\mathbf{SL}_{2}(\mathbf{R})$ and it admits no $G$-fixed point. In fact, if
$g\in G$ is a hyperbolic matrix, then any probability measure fixed by $g$ is
supported on the two points $\xi_{\pm}\in\mathbf{P}^{1}_{\mathbf{R}}$ fixed by
$g$, namely the two points defined by the eigenvectors of $g$. More precisely,
given any $x\neq\xi_{\pm}$, $g^{n}x$ converges to the eigenvector with largest
eigenvalue as $n\to\infty$ (this is particularly clear after diagonalising
$g$). Since this forces the same convergence for any compact interval in
$\mathbf{P}^{1}_{\mathbf{R}}\smallsetminus\\{\xi_{\pm}\\}$, it implies the
statement for measures.
In conclusion, no point of $K$ can be fixed simultaneously by two hyperbolic
matrices without common eigenvector. This classical “ping-pong” argument shows
much more: suitable powers of two such matrices freely generate a free group.
###### Remark 3.3.
The definition of amenability is generalised to topological groups by
requiring the action in 3.1 to be jointly continuous, i.e. the map $G\times
K\to K$ must be continuous. This is therefore a weaker condition than the
amenability of the underlying abstract group $G$. The action in 3.2 is jointly
continuous for the usual topology on $G$. It therefore witnesses that $G$ is
non-amenable also when viewed as a topological group.
###### Example 3.4 (Measures on the $p$-adic projective line).
The exact same arguments from 3.2 hold over $\mathbf{Q}_{p}$ for any prime
$p$. This shows that $K=\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$
is a convex compact $G$-space without fixed point for
$G=\mathbf{SL}_{2}(\mathbf{Q}_{p})$. Note however that the subgroup
$\mathbf{SL}_{2}(\mathbf{Z}_{p})$ admits a fixed point in $K$ because it is a
compact group in the $p$-adic topology for which the action is continuous.
Next we introduce our only other source of convex compact spaces: a
construction that takes a convex compact space $K_{0}$ and yields a new space
$K$ by considering $K_{0}$-valued maps. We will only apply this construction
to the case where $K_{0}$ is given by 3.2 or 3.4.
###### Example 3.5 (Spaces of functions).
Start with a convex compact $G$-space $K_{0}$ and define $K$ to be the space
of all measurable maps $f\colon\mathbf{P}^{1}_{\mathbf{R}}\to K_{0}$, where
two maps are identified when they agree outside a null-set.
To define the compact topology on $K$, we assume for simplicity that $K_{0}$
is metrisable and realised in the dual $C^{\prime}$ of some Banach space $C$
with the weak-* topology. This is the case of 3.2 and 3.4, namely
$C=C(\mathbf{P}^{1})$ is the space of continuous functions on the projective
line and $C^{\prime}$ the space of measures. Now $K$ is endowed with the
weak-* topology obtained by viewing it in the dual of
$L^{1}(\mathbf{P}^{1}_{\mathbf{R}},C)$. The compactness then follows from
Alaoglu’s theorem.
(Even beyond the simple needs of the present article, the above assumptions
would not be restrictive: metrisability is not needed, and the realisation in
some $C^{\prime}$ can always be obtained by taking $C$ to be the space of
continuous affine functions on $K$.)
### 3.B. Piecewise action on maps
We now endow the convex compact spaces of maps from 3.5 with group actions. If
$K_{0}$ is a convex compact $G$-space and moreover $G$ acts on
$\mathbf{P}^{1}_{\mathbf{R}}$, then $G$ acts on
$f\colon\mathbf{P}^{1}_{\mathbf{R}}\to K_{0}$ by $(g.f)(x)=g(f(g^{-1}x))$,
where $x\in\mathbf{P}^{1}_{\mathbf{R}}$. Thus $f\in K$ is a $G$-fixed point if
and only if $f$ is $G$-equivariant as a map. It is understood here that the
$G$-action on $\mathbf{P}^{1}_{\mathbf{R}}$ is supposed _non-singular_ , that
is, preserves null-sets. This ensures that the $G$-action is both well-defined
and by homeomorphisms.
If for instance $G=\mathbf{SL}_{2}(\mathbf{R})$ and
$K_{0}=\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{R}})$, then this $G$-action
on $K$ admits a fixed point, namely the map sending $x$ to the Dirac mass at
$x$.
The crucial interest of 3.5 is that the action defined above makes sense more
generally for _piecewise_ groups:
Let $H$ be a group of piecewise-$\mathbf{SL}_{2}(A)$ transformations of
$\mathbf{P}^{1}_{\mathbf{R}}$, where $A<\mathbf{R}$ is any subring. At this
point it is not even important that $h\in H$ should be a homeomorphism; we
only need to know that the interior points of the “pieces” cover
$\mathbf{P}^{1}_{\mathbf{R}}$ up to a null-set, which is notably the case if
we have finitely many intervals as pieces. It is then well-defined to consider
the projective part $h|_{x}\in\mathbf{PSL}_{2}(A)$ of $h$ at
$x\in\mathbf{P}^{1}_{\mathbf{R}}$, except for a null-set in
$\mathbf{P}^{1}_{\mathbf{R}}$ that we shall ignore. Notice that for
$h,h^{\prime}\in H$ we have the chain rule
$(hh^{\prime})|_{x}=(h|_{h^{\prime}x})(h^{\prime}|_{x})$.
Given now a convex compact $\mathbf{PSL}_{2}(A)$-space $K_{0}$, we define the
$H$-action on the space $K$ of measurable maps
$f\colon\mathbf{P}^{1}_{\mathbf{R}}\to K_{0}$ by
$(h.f)(x)=h|_{h^{-1}x}(f(h^{-1}x)),\kern
14.22636ptx\in\mathbf{P}^{1}_{\mathbf{R}}.$
This $H$-action on $K$ is perfectly suited to the cut-and-paste method
recalled in Section 2. Indeed, noting that the chain rule gives
$h|_{h^{-1}x}=(h^{-1}|_{x})^{-1}$, we see that $f$ is $H$-fixed if and only if
$f(hx)=h|_{x}(f(x))\kern 14.22636pt\text{for all $h\in H$ and a.e.
$x\in\mathbf{P}^{1}_{\mathbf{R}}$.}$
The key point is that this equation only involves the _local_ behaviour of $h$
at $x$. Therefore, it immediately implies the following.
###### Proposition 3.6.
Suppose that for every $g\in\mathbf{PSL}_{2}(A)$ and almost every
$x\in\mathbf{P}^{1}_{\mathbf{R}}$ there is $h\in H$ with $h|_{x}=g$.
Then $f\in K$ is $H$-fixed if and only if it is $\mathbf{PSL}_{2}(A)$-fixed.
###### Proof.
Suppose that $f$ is $H$-fixed. We want to show, given
$g\in\mathbf{PSL}_{2}(A)$ and $x\in\mathbf{P}^{1}_{\mathbf{R}}$, that
$f(gx)=gf(x)$ holds. The element $h$ of the assumption satisfies
$f(hx)=h|_{x}f(x)$, which is exactly what is claimed since $h|_{x}=g$. The
converse is tautological. ∎
### 3.C. The fundamental non-amenability argument
We already have all the elements to deduce that many piecewise-projective
groups $H(A)$ are non-amenable. We begin with the cases of $A=\mathbf{Z}[1/p]$
and more generally of $S$-integral coefficients.
###### Theorem 3.7.
Let $S$ be any non-empty set of prime numbers and choose $p\in S$.
Then the group $\Gamma_{S}=H(\mathbf{Z}[1/S])$ has no fixed point in the
convex compact space $K$ of measurable maps
$\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$.
In particular, this group is non-amenable.
###### Proof.
It suffices to consider $S=\\{p\\}$. The cut-and-paste principle of 2.2 shows
that we are in the setting of 3.6. Thus it suffices to show that
$\mathbf{SL}_{2}(\mathbf{Z}[1/p])$ has no fixed point in the space $K$ of maps
$f\colon\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$.
We write $\Lambda=\mathbf{SL}_{2}(\mathbf{Z}[1/p])$,
$G_{\mathbf{R}}=\mathbf{SL}_{2}(\mathbf{R})$,
$G_{\mathbf{Q}_{p}}=\mathbf{SL}_{2}(\mathbf{Q}_{p})$ and
$G=G_{\mathbf{R}}\times G_{\mathbf{Q}_{p}}$. By elementary reduction theory
(recalled in 3.8 below), the diagonal image of $\Lambda$ in $G$ is discrete
and of finite covolume.
We extend the $\Lambda$-action on $K$ to a $G$-action by
$((g_{1},g_{2}).f)(x)=g_{2}(f(g_{1}^{-1}x)),\kern 14.22636ptg_{1}\in
G_{\mathbf{R}},g_{2}\in G_{\mathbf{Q}_{p}}.$
This is a continuous action without fixed points: a map fixed by
$G_{\mathbf{R}}$ would be constant and its constant value would be
$G_{\mathbf{Q}_{p}}$-fixed, but $G_{\mathbf{Q}_{p}}$ has no fixed points in
$\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$, see 3.4. However, any
point fixed by $\Lambda$ would yield another point fixed by $G$ after
integration over the quotient $G/\Lambda$. Explicitly, if $f\in K$ is
$\Lambda$-fixed, then the orbit map $g\mapsto g.f$ descends to $G/\Lambda$ and
hence $\int_{G/\Lambda}g.f\,d(g\Lambda)\in K$ is $G$-fixed. ∎
###### Remark 3.8 (Reduction theory).
Since the proof of 3.7 used that $\Lambda$ is discrete and of finite covolume
in $G_{\mathbf{R}}\times G_{\mathbf{Q}_{p}}$, it is worth recalling that this
is elementary:
Discreteness holds because $\mathbf{Z}[1/p]$ is discrete in
$\mathbf{R}\times\mathbf{Q}_{p}$ by definition of the $p$-adic topology. As
for a fundamental domain, we can take $D\times\mathbf{SL}_{2}(\mathbf{Z}_{p})$
whenever $D$ is a fundamental domain for the modular group
$\mathbf{SL}_{2}(\mathbf{Z})$ in $\mathbf{SL}_{2}(\mathbf{R})$. Indeed,
$\mathbf{SL}_{2}(\mathbf{Z}_{p})$ is a compact-open subgroup of
$G_{\mathbf{Q}_{p}}$ (again because the corresponding fact holds for
$\mathbf{Z}_{p}$ in $\mathbf{Q}_{p}$ by definition of the topology) and
${\Lambda\cap\mathbf{SL}_{2}(\mathbf{Z}_{p})}$ is
$\mathbf{SL}_{2}(\mathbf{Z})$. Finally, a very explicit domain $D$ of finite
volume is familiar since Gauss, namely the pre-image in
$\mathbf{SL}_{2}(\mathbf{R})$ of the strip $|{\mathrm{Re}(z)|\leq 1/2}$,
${|z|\geq 1}$ in the upper half-plane.
The case of other number fields, as in 3.9 below, is handled by restriction of
scalars, see e.g. Siegel, Satz 12 p. 233 in [Sie39]. (The generalisation to
arbitrary reductive groups, which soars far beyond our needs, is the
Borel–Harish-Chandra theorem [BHC62, Thm. 12.3], [Bor63, §8].)
The proof of 3.7 is based on the $p$-adic completion of the number field
$\mathbf{Q}$. It holds exactly the same way for the two other types of places,
namely real and complex completions:
###### Example 3.9 (Other places).
The argument given above for the ring $A=\mathbf{Z}[1/p]$ can be applied to
any other ring $A$ of $S$-integers (or integers) in any other real number
field $F<\mathbf{R}$, except $A=\mathbf{Z}$. Indeed, when we used
$\mathbf{Q}_{p}$ in the proof of 3.7, we only needed some completion of
$F=\mathbf{Q}$ _in addition_ to the given completion $\mathbf{R}$ used to
define the action on the variable $x\in\mathbf{P}^{1}_{\mathbf{R}}$.
In particular, this also works if the second completion happens to be
$\mathbf{R}$ as well. For instance, consider $A=\mathbf{Z}[\sqrt{2}]$, which
is the ring of integers of $\mathbf{Q}(\sqrt{2})$ [Sam70, §2.5]. We let
$\Lambda=\mathbf{SL}_{2}(A)$ act on
$\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{R}})$ via the _second_ embedding
$\Lambda\to G_{\mathbf{R}}$ given by the negative root of two. Reasoning
exactly as above, the action of $\Lambda$ on the space $K$ of maps
$\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{R}})$
has no fixed points because $\Lambda$ is a lattice in $G_{\mathbf{R}}\times
G_{\mathbf{R}}$. Therefore, 3.6 shows that $H(A)$ has no fixed points in $K$
and in particular it is non-amenable.
Likewise, if a real number field $F<\mathbf{R}$ is not totally real, it can
happen that the only other Archimedean completion is complex. This is the case
for instance of pure cubic fields, e.g. $F=\mathbf{Q}(\sqrt[3]{2})$. Denoting
its ring of integers by $\mathscr{O}_{F}$, we can thus use that
$H(\mathscr{O}_{F})$ has no fixed points in the convex compact space of maps
$\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{C}})$.
###### Remark 3.10 (Produced spaces).
It is well-known that if a (discrete) group $\Gamma$ contains a non-amenable
group $\Lambda<\Gamma$, then $\Gamma$ is also non-amenable. With
$\Gamma=H(\mathbf{R})$ in mind, the reader might ask: does this fact also
admit an elementary proof in terms of fixed points in convex compact spaces?
The question is legitimate since this fact has no analogue for general
topological groups, whereas the fixed-point condition is completely general.
There is indeed such a direct argument. Let $K_{0}\neq\varnothing$ be a convex
compact $\Lambda$-space without fixed points. Define the produced space $K$ by
$K=\big{\\{}f\colon\Gamma\to K_{0}:f(hg)=hf(g)\
\forall\,h\in\Lambda,g\in\Gamma\big{\\}}$
which is convex and compact in the product topology, i.e. the topology of
pointwise convergence for maps $\Gamma\to K_{0}$. This is a $\Gamma$-space for
the right translation action $(\gamma f)(g)=f(g\gamma)$. Now any
$\Gamma$-fixed point $f$ in $K$ would be constant, and by construction the
constant value $f(g)$ would be a $\Lambda$-fixed point in $K_{0}$.
Our only scruple could be to verify that $K$ is non-empty; we note that any
transversal $S\subseteq\Gamma$ for $\Lambda$ gives by restriction an
isomorphism $K\cong K_{0}^{S}$.
## 4\. Refining and strengthening non-amenability
### 4.A. Relativisation
A key concept here is the notion of co-amenability for a subgroup $G_{1}<G$ of
a group $G$, due to Eymard [Eym72]. Informally, it means that $G$ is “as
amenable as $G_{1}$”, or “amenable conditioned on $G_{1}$ being so”. Neither
$G_{1}$ nor $G$ need be amenable when the question of co-amenability is
raised. If for instance $G_{1}$ is a normal subgroup of $G$, then co-
amenability amounts simply to the amenability of the quotient group $G/G_{1}$.
To motivate the general definition, recall that a group $G$ is amenable iff
every non-empty convex compact $G$-set admits a $G$-fixed point.
###### Definition 4.1.
A subgroup $G_{1}<G$ of a group $G$ is co-amenable in $G$ if every convex
compact $G$-set with a $G_{1}$-fixed point admits a $G$-fixed point.
This notion, which makes sense in the generality of topological groups and
their subgroups, has been extensively studied by Eymard [Eym72] in the context
of locally compact groups.
There is a further generalisation ([Por13, §2.3], [CM14, §7.C]) that compares
two subgroups $G_{1},G_{2}<G$ that are not necessarily nested:
###### Definition 4.2.
The subgroup $G_{1}$ is co-amenable to $G_{2}$ relative to $G$ if every convex
compact $G$-space which has a $G_{1}$-fixed point also has a $G_{2}$-fixed
point.
Again, this definition extends to arbitrary topological groups simply by
requiring that all actions on convex compact spaces be jointly continuous.
Back to the discrete case, we record the following standard equivalences.
###### Lemma 4.3.
Given two subgroups $G_{1},G_{2}$ of a group $G$, the following are
equivalent:
1. (i)
$G_{1}$ is co-amenable to $G_{2}$ relative to $G$;
2. (ii)
the $G_{2}$-action on $G/G_{1}$ admits an invariant mean;
3. (iii)
the restriction to $G_{2}$ of the quasi-regular $G$-representation
$\lambda_{G/G_{1}}$ weakly contains the trivial representation.
The reformulation (iii) suggests to use II as an input for C*-algebra
arguments; this is pursued in [GM23].
###### Proof of 4.3.
The equivalence of (ii) and (iii) is the usual Hulanicki–Reiter condition for
any action on any set, here the $G_{2}$-action on $G/G_{1}$. The implication
(i)$\Rightarrow$(ii) follows by considering the convex compact $G$-space of
all means on $G/G_{1}$, noting that it indeed has a $G_{1}$-fixed point,
namely the Dirac mass at the trivial coset.
Finally, for (ii)$\Rightarrow$(i), consider an arbitrary convex compact
$G$-space $K$ which contains some $G_{1}$-fixed point $x\in K$. The orbital
map $g\mapsto gx$ induces a $G$-equivariant map $G/G_{1}\to K$ and hence by
push-forward we obtain a $G_{2}$-invariant mean on $K$. The barycentre of this
mean is a $G_{2}$-fixed point. (We recall here that the usual notion of
barycentre for probability measures applies to means. Indeed, probability
measures are states on continuous functions, while means are states on bounded
functions; but any continuous function on $K$ is bounded.) ∎
With the statements of I and II in mind, the importance of relative co-
amenability is as follows. By exhibiting a large poset of subgroups $G_{i}<G$
that are pairwise _not_ co-amenable to one another relative to $G$, we
strengthen the non-amenability of $G$. Every chain of inclusions in such a
poset can be thought of as a gradient of increasing non-amenability.
We underline that even when the subgroups are nested $G_{1}<G_{2}<G$, this
relative non-amenability is stronger than simply stating that $G_{1}$ is not
co-amenable in $G_{2}$:
###### Example 4.4.
Consider a situation with $G_{1}<G_{2}<G$ where $G_{1}$ is co-amenable in $G$
but not in $G_{2}$. Examples are given in [MP03] and [Pes03], for instance
$G=(\bigoplus_{\mathbf{Z}}F_{2})\rtimes\mathbf{Z}$,
$G_{2}=\bigoplus_{\mathbf{Z}}F_{2}$ and $G_{1}=\bigoplus_{\mathbf{N}}F_{2}$.
Then $G_{1}$ is still co-amenable to $G_{2}$ _relative to $G$_.
We can now complete the proof of both I and II from the introduction. Recall
that for any set $S$ of prime numbers, $\Gamma_{S}$ denotes the group of
piecewise-$\mathbf{SL}_{2}(\mathbf{Z}[1/S])$ homeomorphisms of the line and
that when $S$ is non-empty, $\Gamma_{S}$ coincides with $H(\mathbf{Z}[1/S])$.
The more explicit statement is as follows:
###### Theorem 4.5.
Let $S,S^{\prime}$ be any sets of primes. If $p$ is in
$S^{\prime}\smallsetminus S$, then the convex compact $H(\mathbf{Q})$-space of
measurable maps
$\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$
admits a $\Gamma_{S}$-fixed point but no $\Gamma_{S^{\prime}}$-fixed point.
In particular, $\Gamma_{S}$ is not co-amenable to $\Gamma_{S^{\prime}}$
relative to $H(\mathbf{Q})$.
A fortiori, if $S\subsetneqq S^{\prime}$ then the subgroup
$\Gamma_{S}<\Gamma_{S^{\prime}}$ is not co-amenable in $\Gamma_{S^{\prime}}$.
###### Proof.
We proved in 3.7 that $\Gamma_{S^{\prime}}$ has no fixed point. By contrast,
$\Gamma_{S}$ fixes the constant map whose value is the
$\mathbf{SL}_{2}(\mathbf{Z}_{p})$-invariant measure on
$\mathbf{P}^{1}_{\mathbf{Q}_{p}}$ (see 3.4) since $\mathbf{Z}[1/S]$ is
contained in $\mathbf{Z}_{p}$. ∎
The fact that III follows from 4.5 is established in Corollary 4.2 of [Mon21].
As discussed in the introduction, the above results also illustrate that the
open problem of the (non-)amenability of Thompson’s group
$H_{\mathbf{Q}}(\mathbf{Z})$ is completely decoupled from the non-amenability
of the various groups $H(A)$ with $A$ non-discrete. In the above poset,
$H_{\mathbf{Q}}(\mathbf{Z})$ lies in $\Gamma_{\varnothing}$ and is therefore
not co-amenable in any of the other $\Gamma_{S}$, nor even co-amenable to
$\Gamma_{S}$ relative to $H(\mathbf{Q})$.
### 4.B. Other rings
The goal of this note until this point was to establish variations and
stronger forms of the non-amenability theorem from [Mon13], but without the
theory of measured equivalence relations. Instead, we argued from first
principles using simple cut-and-paste and leveraging the very basic fact that
$\mathbf{SL}_{2}(\mathbf{Z})$ has finite covolume in
$\mathbf{SL}_{2}(\mathbf{R})$.
This was enough to be able to deal with the rings $A=\mathbf{Z}[1/S]$ and
applies also to the ring of integers in any other real number fields, such as
$A=\mathbf{Z}[\sqrt{2}]$, or their $S$-arithmetic generalisations.
In order to justify IV from the introduction even when $A$ is not some
$S$-arithmetic ring, we prove the following, which does rely on the
Carrière–Ghys theorem.
###### Theorem 4.6.
Let $\Lambda<H(\mathbf{R})$ be any subgroup containing $\Gamma_{\varnothing}$.
Suppose that $\Lambda$ has the same same orbits in
$\mathbf{P}^{1}_{\mathbf{R}}$ as a countable dense subgroup
$\Delta<\mathbf{SL}_{2}(\mathbf{R})$ (after discarding a null-set in
$\mathbf{P}^{1}_{\mathbf{R}}$).
Then $\Gamma_{\varnothing}$ is not co-amenable in $\Lambda$.
The same statement holds with $\Gamma_{\varnothing}$ replaced throughout by
Thompson’s group.
This can be applied for instance when $\Lambda$ is the group studied by
Lodha–Moore [LM16].
To prove 4.6, we shall need the following general fact about relative
amenability in relation to relations; this is a relative of the fact related
in Proposition 9 of [Mon22].
###### Proposition 4.7.
Let $\Lambda$ be a countable group with a non-singular action on a standard
probability space $X$. Let $\Gamma<\Lambda$ be a co-amenable subgroup of
$\Lambda$.
If the orbit equivalence relation produced by $\Gamma$ on $X$ is amenable,
then so is the relation produced by $\Lambda$.
###### Proof of 4.7.
Let $L\neq\varnothing$ be a compact metrisable space and let
$\alpha\colon\Lambda\times X\to\operatorname{Homeo}(L)$ be a relation cocycle
to the group of homeomorphisms of $L$. Being a _cocycle_ means that
$\alpha(\lambda,\cdot)$ is a measurable map for each $\lambda\in\Lambda$ and
that for $\lambda,\eta\in\Lambda$ the chain rule
$\alpha(\lambda\eta,x)=\alpha(\lambda,\eta x)\alpha(\eta,x)$ holds for a.e.
$x\in X$. Being a _relation_ cocycle means that $\alpha(\lambda,x)$ depends
only on the pair $(\lambda x,x)$ of points.
Hjorth proved (Theorem 1.1 in [Hjo06]) that the relation produced by $\Lambda$
on $X$ is amenable if and only if for any such $L$ and $\alpha$, there exists
a measurable map $f\colon X\to\operatorname{Prob}(L)$ such that for every
$\lambda\in\Lambda$ and a.e. $x\in X$
$f(\lambda x)=\alpha(\lambda,x)f(x).$
This is equivalent to $f$ being $\Lambda$-fixed for the action
$(\lambda.f)(x)=\alpha(\lambda,\lambda^{-1}x)(f(\lambda^{-1}x)).$
Exactly as in Section 3.B, we have now a convex compact $\Lambda$-space $K$ of
maps. Hjorth’s criterion tells us that $\Gamma$ has a fixed point in $K$, and
therefore the co-amenability of $\Gamma$ in $\Lambda$ yields the conclusion. ∎
###### Proof of 4.6.
The Carrière–Ghys theorem states that the orbit equivalence relation produced
by $\Delta$ on $\mathbf{SL}_{2}(\mathbf{R})$ is not an amenable equivalence
relation. As recalled in [Mon13], this implies that the relation produced on
$\mathbf{P}^{1}_{\mathbf{R}}$ is not amenable either. (This follows from the
fact that $\mathbf{P}^{1}_{\mathbf{R}}$ can be realised as the quotient of
$\mathbf{SL}_{2}(\mathbf{R})$ by an _amenable_ subgroup, namely the upper
triangular matrices.) Thus the relation produced by $\Lambda$ is not amenable.
On the other hand, the relation produced by $\Gamma_{\varnothing}$ is amenable
since it coincides with the relation produced by
$\mathbf{SL}_{2}(\mathbf{Z})$, which is amenable because
$\mathbf{SL}_{2}(\mathbf{Z})$ is discrete in $\mathbf{SL}_{2}(\mathbf{R})$,
see [Zim84, 4.3.2].
At this point, the statement follows from 4.7. ∎
Note that 4.6 implies in particular IV for any countable ring
$\mathbf{Z}\lneqq A<\mathbf{R}$.
There remains one issue, namely that the ring is not supposed countable in IV.
The remedy should be to consider $\Gamma_{\varnothing}<H(A^{\prime})<H(A)$ for
some countable ring $\mathbf{Z}\lneqq A^{\prime}<A$. The problem with this
approach is that given nested subgroups $\Gamma<\Lambda_{1}<G$, there is
generally no implication between the co-amenability of $\Gamma$ in
$\Lambda_{1}$ or in $G$. Indeed, examples to this end can be found in [Pes03]
and [MP03], as already mentioned in 4.4.
We can overcome this difficulty if we observe that (co-)amenability, being an
analytic property, is countably determined:
###### Proposition 4.8.
Let $\Gamma<G$ be a countable co-amenable subgroup of a group $G$.
For every countable $\Lambda_{1}<G$ containing $\Gamma$, there is a countable
subgroup $\Lambda<G$ containing $\Lambda_{1}$ such that $\Gamma$ is co-
amenable in $\Lambda$.
If we are moreover given any cofinal family $\mathscr{H}$ of countable
subgroups $H<G$ which is closed under countable increasing unions, then we can
choose $\Lambda\in\mathscr{H}$.
(_Cofinal_ means that every countable subgroup of $G$ is contained in some
$H\in\mathscr{H}$. In our case, $\mathscr{H}$ will consist of the groups of
the form $H(A)$.)
We note that 4.8 has a trivial converse:
Let $\mathscr{H}$ be a family as above and let $\Gamma<G$ be a countable
subgroup. Suppose that for every countable $\Lambda_{1}<G$ containing $\Gamma$
there is $\Lambda\in\mathscr{H}$ containing $\Lambda_{1}$ such that $\Gamma$
is co-amenable in $\Lambda$. Then $\Gamma$ is co-amenable in $G$.
This directly follows from the definition of co-amenability (4.1) by a
compactness argument.
###### Proof of 4.8.
We inductively construct an increasing sequence of countable subgroups
$\Lambda_{n}<G$, each enumerated as $\\{\lambda_{n,j}:j\in\mathbf{N}\\}$, and
a sequence of functions $v_{n}\in\ell^{1}(G/\Gamma)$ such that
$\big{\|}\lambda_{i,j}v_{n}-v_{n}\big{\|}<\frac{1}{n}\|v_{n}\|\kern
14.22636pt\forall\,i,j\leq n$
holds for the quasi-regular (translation) $G$-representation on
$\ell^{1}(G/\Gamma)$. Supposing $\Lambda_{n}$ already given, the existence of
$v_{n}$ follows from the co-amenability of $\Gamma$ in $G$ (see e.g. [Eym72,
p. 44]). We then define $\Lambda_{n+1}$ to be the subgroup generated by the
union of $\Lambda_{n}$ and the pre-image in $G$ of the support of $v_{n}$,
noting that the latter is a countable set. If some $\mathscr{H}$ is given, we
replace $\Lambda_{n+1}$ by an element of $\mathscr{H}$ containing it.
Define now $\Lambda$ to be the union of all $\Lambda_{n}$. Since the support
of every $v_{n}$ is contained in $\Lambda/\Gamma\subseteq G/\Gamma$, any
accumulation point of the sequence $v_{n}$ in the space of means on
$\ell^{\infty}(G/\Gamma)$ defines in fact a mean on
$\ell^{\infty}(\Lambda/\Gamma)$. This mean is $\Lambda$-invariant, thus
verifying another equivalent characterisation [Eym72] of the co-amenability of
$\Gamma$ in $\Lambda$. ∎
###### Proof of IV.
Suppose for a contradiction that $\Gamma_{\varnothing}$ is co-amenable in
$G=H(A)$ for a ring $\mathbf{Z}\lneqq A<\mathbf{R}$. Let $\mathscr{H}$ be the
family of all $H(A^{\prime})$, where $A^{\prime}$ ranges over all countable
rings with $\mathbf{Z}\lneqq A^{\prime}<A$. Let $\Lambda_{1}=H(A^{\prime})$ be
one such group and let $\Lambda=H(A^{\prime\prime})$ be the countable group
given by 4.8. Thus $A^{\prime\prime}$ is a countable ring in $A$ and
$\mathbf{SL}_{2}(A^{\prime\prime})$ is dense in $\mathbf{SL}_{2}(\mathbf{R})$
because $A^{\prime}<A^{\prime\prime}$. Therefore, the co-amenability of
$\Gamma_{\varnothing}$ in $\Lambda$ contradicts 4.6. ∎
## 5\. Additional comments
### 5.A. Breaking up smoothly and rationally
The breakpoint convention chosen in [Mon13] to define $H(A)$ has an aesthetic
drawback in the case of $A=\mathbf{Z}$. Namely, the fixed points of hyperbolic
elements are surds (solutions of quadratic equations) while the fixed points
of parabolic elements are rational. In particular, the analogue of 2.1 does
not hold in that case.
Thus the Thompson group $H_{\mathbf{Q}}(\mathbf{Z})$ and the group
$H(\mathbf{Z})$ are two different subgroups of $\Gamma_{\varnothing}$ (though
$H(\mathbf{Z})$ contains isomorphic copies of Thompson’s group, see [Sta21]).
In addition to having rational breakpoints, the Thompson group
$H_{\mathbf{Q}}(\mathbf{Z})$ exhibits another pleasant quality: since
$\mathbf{Z}$ has no non-trivial positive unit, these piecewise-projective
elements are actually $C^{1}$-smooth. This is the maximal smoothness for
breakpoints because projective transformations are entirely determined by
their $C^{2}$-jet at any point. We could therefore also ask to strengthen our
non-amenability results for groups of $C^{1}$-diffeomorphisms.
The method of proof employed above can indeed be adapted to this setting; here
is an example.
###### Theorem 5.1.
The Thompson group $H_{\mathbf{Q}}(\mathbf{Z})$ is not co-amenable in the
group $H_{\mathbf{Q}}(\mathbf{Q})$ of piecewise-$\mathbf{SL}_{2}(\mathbf{Q})$
homeomorphisms of the line with rational breakpoints.
It is also not co-amenable in the smaller subgroup
$H_{\mathbf{Q}}^{C^{1}}(\mathbf{Q})$ of $C^{1}$-diffeomorphisms.
All that is needed is to revisit the cut-and-paste of 2.2 and perform a more
cosmetic type of surgery:
###### Proposition 5.2.
For every $g\in\mathbf{SL}_{2}(\mathbf{Q})$ and $x\in\mathbf{R}$ with
$gx\neq\infty$, there is a piecewise-$\mathbf{SL}_{2}(\mathbf{Q})$
homeomorphism $h$ of $\mathbf{R}$ with breakpoints in $\mathbf{Q}$ and which
coincides with $g$ on a neighbourhood of $x$.
Furthermore, we can choose $h$ to be a $C^{1}$-diffeomorphism of $\mathbf{R}$.
###### Proof of 5.2.
We first justify why we can take breakpoints in $\mathbf{Q}$. Thus, in the
proof of 2.2, we want the fixed points $\xi_{\pm}$ of $q_{0}$ to be rational.
Equivalently, the root $\sqrt{\tau^{2}-4}$ must be rational, recalling that
$\tau$ is the trace $a+d+nc$.
The point is that we were completely free to choose $n$ as long as we can take
$|n|$ arbitrarily large and with a sign prescribed by the relative position of
$gx$ and $g\infty$. Since we work now in $\mathbf{SL}_{2}(\mathbf{Q})$, we
just need to show that there are such $n\in\mathbf{Q}$ with moreover
$\sqrt{\tau^{2}-4}\in\mathbf{Q}$.
The solution to this Diophantine problem was already know to Euclid (see Book
X, Proposition 29 in the _Elements_), as follows. Given any integer $N$, let
define $n\in\mathbf{Q}$ by $n=(N+1/N-a-d)/c$. Then $\tau=N+1/N$ and thus
$\tau^{2}-4=(N-1/N)^{2}$ is indeed a square. Moreover, $n$ can indeed be
chosen arbitrarily large of either sign simply by letting $N\to\pm\infty$ in
$\mathbf{Z}$.
We now turn to the $C^{1}$ condition, which will require additional
dissections to assemble a polytomous spline.
The only singularities introduced in the proof of 2.2 arise from the two
points $\xi_{\pm}$, where $q$ has breakpoints, transitioning from $q_{0}$ to
the identity and back. The strategy is to smoothen $q$ near one breakpoint at
the time, which is sufficient provided the modification can be done in a small
enough neighbourhood of the breakpoint. Since $\xi_{\pm}$ are rational and
$\mathbf{SL}_{2}(\mathbf{Q})$ acts doubly transitively on the rational points
of the projective line, it suffices to prove the following claim:
For any $\epsilon>0$ and for any $p_{0}\in\mathbf{SL}_{2}(\mathbf{Q})$ fixing
$0$ and $\infty$, there exists a $C^{1}$-smooth
piecewise-$\mathbf{SL}_{2}(\mathbf{Q})$ homeomorphism $p_{1}$ of $\mathbf{R}$
with breakpoints in $\mathbf{Q}$ and which coincides with the identity on
$(-\infty,-\epsilon]$ and with $p_{0}$ on $[\epsilon,+\infty)$.
The assumptions on $p_{0}$ imply that it is given by a diagonal matrix, or
equivalently that there is $a\in\mathbf{Q}$ with $p_{0}(x)=a^{2}x$ for all
$x$; without loss of generality, $a>0$. Let $\epsilon_{1}>0$ be rational with
$\epsilon_{1}\leq\epsilon,\epsilon/a$ and define
$u\in\mathbf{SL}_{2}(\mathbf{Q})$ by
$u=\frac{1}{a+1}\begin{pmatrix}2a&\epsilon_{1}a(a-1)\\\
(1-a)/(\epsilon_{1}a)&2\end{pmatrix}$
(The conceptual explanation for this choice is that $u$ is a unipotent, which
allows us to match derivatives.) We now define $p_{1}$ as follows for
$x\in\mathbf{R}$:
$p_{1}(x)=\begin{cases}x&\text{ if }x\in(-\infty,-\epsilon_{1}a],\\\
u(x)=\dfrac{2ax+\epsilon_{1}a(a-1)}{(1-a)x/(\epsilon_{1}a)+2}&\text{ if
}x\in(-\epsilon_{1}a,\epsilon_{1}],\\\ p_{0}(x)=a^{2}x&\text{ if
}x\in(\epsilon_{1},+\infty).\end{cases}$
Thus we only have to check that $p_{1}$ is indeed a $C^{1}$-smooth
homeomorphism. Note first that the denominator in $u(x)$ vanishes only at
$x={2\epsilon_{1}a/{(a-1)}}$, which is outside the interval
$(-\epsilon_{1}a,\epsilon_{1}]$ where $u$ is applied. Now we turn to the
breakpoints, where $p_{1}$ is continuous since
$u(-\epsilon_{1}a)=-\epsilon_{1}a$ and $u(\epsilon_{1})=a^{2}\epsilon_{1}$.
Furthermore, computing
$u^{\prime}(x)=\left(\frac{(1+a)a\epsilon_{1}}{(1-a)x+2a\epsilon_{1}}\right)^{2}$
we find that $u^{\prime}(x)=1$ at $x=-\epsilon_{1}a$ and $u^{\prime}(x)=a^{2}$
at $x=\epsilon_{1}$. This verifies that $p_{1}$ is a $C^{1}$-smooth
homeomorphism. ∎
###### Remark 5.3.
If we consider $h$ as a transformation of $\mathbf{P}^{1}_{\mathbf{R}}$ fixing
$\infty$ rather than as a transformation of $\mathbf{R}$, then it remains true
that $h$ is $C^{1}$. Indeed, recall from the proof of 2.2 that $h$ is just a
translation in some neighbourhood of $\infty$; this fact has not been altered
by the smoothing operation of the above proof. Thus $h$ is projective (in
particular smooth) in a neighbourhood of $\infty$ in
$\mathbf{P}^{1}_{\mathbf{R}}$.
###### Proof of 5.1.
It suffices to exhibit a convex compact $H_{\mathbf{Q}}(\mathbf{Q})$-space
admitting a $H_{\mathbf{Z}}(\mathbf{Q})$-fixed point but no point fixed by the
smooth group $H_{\mathbf{Q}}^{C^{1}}(\mathbf{Q})$. Choose any prime $p$. Then
the space of measurable maps
$\mathbf{P}^{1}_{\mathbf{R}}\to\operatorname{Prob}(\mathbf{P}^{1}_{\mathbf{Q}_{p}})$
will do. Indeed, 5.2 allows us to apply 3.6. Thus it suffices to show that
$\mathbf{SL}_{2}(\mathbf{Q})$ has no fixed point in $K$. This follows from the
case of its subgroup $\mathbf{SL}_{2}(\mathbf{Z}[1/p])$ established in the
proof of 3.7. ∎
### 5.B. Organising the layers of non-amenability
This last section is purely descriptive and is placed in the context of
completely general topological groups $G$ and arbitrary subgroups of $G$.
We propose to consider some sort of “spectrum” $\dot{G}$ recording the layers
of non-amenability to be found between subgroups $G$; in particular, $\dot{G}$
will be reduced to a point if and only if $G$ itself is amenable.
Recall that given any two subgroups $L,H<G$, we say that $L$ is co-amenable to
$H$ relative to $G$ if any (jointly continuous) convex compact $G$-space with
an $L$-fixed point has an $H$-fixed point.
We write $L\succeq_{G}H$, of simply $L\succeq H$ when the ambient group does
not vary. This is a pre-order relation.
We denote by $\dot{G}$ the quotient of the set of all subgroups of $G$ under
the equivalence relation associated to the pre-order $\succeq_{G}$. Thus
$\dot{G}$ is a poset and we still denote its order relation by $\succeq_{G}$
or $\succeq$. We denote the equivalence class of a subgroup $H<G$ by
$[H]_{G}\in\dot{G}$, or simply $[H]$.
It is sufficient to consider closed subgroups of $G$ because the closure
$\overline{H}$ of $H$ in $G$ satisfies $[\overline{H}]=[H]$. Furthermore,
$[H]$ only depends on the conjugacy class of $H$. For two subgroups $H,L<G$ we
trivially have
$H<L\ \Longrightarrow\ [H]\preceq_{G}[L].$
In complete generality, $\dot{G}$ admits an upper bound, $[G]$, and a lower
bound, $[1]$. The former coincides with the set of all co-amenable subgroups
of $G$, while the latter coincides with the set of all relatively amenable
subgroups of $G$. (Relative amenability, defined in [CM14], boils down to
amenability in the case of discrete groups.)
In particular, $\dot{G}$ is reduced to a point if and only if $[G]=[1]$, which
happens if and only if $G$ is amenable. It can happen that $\dot{G}$ consists
precisely of two points, e.g. when $G$ is a non-amenable Tarski monster.
Regarding functoriality, we note that any morphism $\varphi\colon G\to H$ of
topological groups induces a morphism of posets $\dot{G}\to\dot{H}$, where for
$L<G$ the point $[L]_{G}$ is mapped to $[\varphi(L)]_{H}$. This follows
immediately from pulling back convex compact $H$-spaces to $G$-spaces via
$\varphi$.
This map $\dot{G}\to\dot{H}$ is onto if $G\to H$ is onto, but monomorphisms
might induce non-injections, for instance in the setting of 4.4. Indeed, if
$L<G<H$ is such that $L$ is co-amenable in $H$ but not in $G$, then the
inclusion morphism $G\to H$ maps the point $[L]_{G}\in\dot{G}$ to
$[L]_{H}\in\dot{H}$, but we have $[L]_{G}\neq[G]_{G}$ and
$[L]_{H}=[H]_{H}=[G]_{H}$.
It would be interesting to find examples where $\dot{G}$ can be completely
described (without being trivial). We expect that for most familiar discrete
groups this object is too large to be described, even though the case of
Tarski monsters shows that there are exceptions.
II can be reformulated as exhibiting a huge part of $\dot{G}$ for various
piecewise-projective groups $G$, as follows. The poset of all sets of prime
numbers is isomorphic to the poset of subgroup $\Gamma_{S}<H(\mathbf{Q})$.
Then II states that this uncountable poset is fully faithfully represented (as
a poset) into $\dot{H}(\mathbf{Q})$.
More generally, fixing $S$, we can consider the poset of all subsets
$T\subseteq S$, which again gives us a poset of subgroups which is uncountable
as soon as $S$ is infinite. The non-co-amenability of various $\Gamma_{T}$
relative to $H(\mathbf{Q})$ a fortiori implies non-co-amenability relative to
$\Gamma_{S}$. Thus II implies:
###### Corollary 5.4.
The canonical map $T\to[\Gamma_{T}]$ is a fully faithful embedding of the
poset of all subsets $T\subseteq S$ into $\dot{\Gamma}_{S}$.∎
On the other hand we expect that non-discrete groups will provide more
tractable examples. For instance, what is $\dot{G}$ for
$G=\mathbf{SL}_{n}(\mathbf{R})$ ?
## References
* [BHC62] Armand Borel and Harish-Chandra, _Arithmetic subgroups of algebraic groups_ , Ann. of Math. (2) 75 (1962), 485–535.
* [BK21] Alex Bearden and Mehrdad Kalantar, _Topological boundaries of unitary representations_ , Int. Math. Res. Not. IMRN (2021), no. 12, 9425–9457.
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* [CM14] Pierre-Emmanuel Caprace and Nicolas Monod, _Relative amenability_ , Groups Geom. Dyn. 8 (2014), no. 3, 747–774.
* [Eym72] Pierre Eymard, _Moyennes invariantes et représentations unitaires_ , Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 300.
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* [Hjo06] Greg Hjorth, _The Furstenberg lemma characterizes amenability_ , Proc. Amer. Math. Soc. 134 (2006), no. 10, 3061–3069.
* [Imb97] Michel Imbert, _Sur l’isomorphisme du groupe de Richard Thompson avec le groupe de Ptolémée_ , Geometric Galois actions, 2, London Math. Soc. Lecture Note Ser., vol. 243, Cambridge Univ. Press, Cambridge, 1997, pp. 313–324.
* [LM16] Yash Lodha and Justin Tatch Moore, _A nonamenable finitely presented group of piecewise projective homeomorphisms_ , Groups Geom. Dyn. 10 (2016), no. 1, 177–200.
* [Mon13] Nicolas Monod, _Groups of piecewise projective homeomorphisms_ , Proc. Natl. Acad. Sci. USA 110 (2013), no. 12, 4524–4527.
* [Mon21] by same author, _Furstenberg boundaries for pairs of groups_ , Ergodic Theory Dyn. Syst. 41 (2021), no. 5, 1514–1529.
* [Mon22] by same author, _Lamplighters and the bounded cohomology of Thompson’s group_ , Geom. Funct. Anal. 32 (2022), no. 3, 662–675.
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* [Pes03] Vladimir Germanovich Pestov, _On some questions of Eymard and Bekka concerning amenability of homogeneous spaces and induced representations_ , C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 76–81.
* [Por13] Jürg Portmann, _Counting integral points on affine homogeneous varieties and Patterson–Sullivan theory_ , Ph.D. thesis, ETH Zürich, 2013\.
* [Sam70] Pierre Samuel, _Algebraic theory of numbers_ , Houghton Mifflin Co., Boston, Mass., 1970, Translated from the French by Allan J. Silberger.
* [Sie39] Carl Ludwig Siegel, _Einheiten quadratischer Formen_ , Abh. Math. Sem. Univ. Hamburg 13 (1939), no. 1, 209–239.
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|
# Skipper: Improving the Reach and Fidelity of Quantum Annealers by Skipping
Long Chains
Ramin Ayanzadeh Georgia Institute of TechnologyAtlantaUSA and Moinuddin
Qureshi Georgia Institute of TechnologyAtlantaUSA
###### Abstract.
Quantum Annealers (QAs) operate as single-instruction machines, lacking a SWAP
operation to overcome limited qubit connectivity. Consequently, multiple
physical qubits are _chained_ to form a program qubit with higher
connectivity, resulting in a drastically diminished effective QA capacity by
up to 33x. We observe that in QAs: (a) chain lengths exhibit a power-law
distribution, a few _dominant chains_ holding substantially more qubits than
others; and (b) about 25% of physical qubits remain unused, getting isolated
between these chains. We propose _Skipper_ , a software technique that
enhances the capacity and fidelity of QAs by skipping dominant chains and
substituting their program qubit with two readout results. Using a 5761-qubit
QA, we demonstrate that Skipper can tackle up to 59% (Avg. 28%) larger
problems when eleven chains are skipped. Additionally, Skipper can improve QA
fidelity by up to 44% (Avg. 33%) when cutting five chains (32 runs). Users can
specify up to eleven chain cuts in Skipper, necessitating about 2,000 distinct
quantum executable runs. To mitigate this, we introduce _Skipper-G_ , a greedy
scheme that skips sub-problems less likely to hold the global optimum,
executing a maximum of 23 quantum executables with eleven chain trims.
Skipper-G can boost QA fidelity by up to 41% (Avg. 29%) when cutting five
chains (11 runs).
## 1\. Introduction
Quantum computers (QCs) have the potential to solve certain problems beyond
the capabilities of classical computers (arute2019quantum, ;
villalonga2020establishing, ; preskillNISQ, ; king2021scaling, ; wu2021strong,
). Two main types of QCs exist: digital machines, exemplified by industry
leaders like IBM (IBMQ, ), Google (GoogleAI, ), IonQ (IonQ, ), and Quantinuum
(quantinuum, ), and analog devices such as superconducting _Quantum Annealers_
(_QAs_) developed by D-Wave (D-Wave, ), as well as neutral atom platforms by
QuEra (QuEra, ) and PASQAL (PASQAL, ). Both digital and analog QCs have
polynomial equivalent computing power (aharonov2008adiabatic, ;
albash2018adiabatic, ). For instance, QAs have demonstrated their potential in
tackling real-world applications such as finance (elsokkary2017financial, ),
drug discovery (mulligan2020designing, ), cryptography (peng2019factoring, ;
hu2020quantum, ), Boolean Satisfiability (SAT) (su2016quantum, ;
ayanzadeh2020reinforcement, ; ayanzadeh2018solving, ; ayanzadeh2019sat, ),
planning and scheduling (inoue2021traffic, ; rieffel2015case, ;
venturelli2015quantum, ; tran2016hybrid, ), linear algebra (o2018nonnegative,
), and signal processing (ayanzadeh2019quantum, ; ayanzadeh2020ensemble, ),
extending beyond application-specific acceleration.
While both QC types are accessed via the cloud (AmazonBraKet, ;
MicrosoftAzure, ; D-Wave, ), their operation models and design trade-offs
differ significantly (ayanzadeh2022equal, ). In digital QCs (namely the gate-
based or circuit model quantum computing), as shown in Fig. 1(a), qubits
undergo a scheduled sequence of quantum operations defined by the quantum
algorithm to directly manipulate their states (nielsen2010quantum, ).
Conversely, as shown in Fig. 1(b), analog QCs operate as single-instruction
systems, where the qubit environment is incrementally modified based on the
evolution of a physical system, called _Hamiltonian_ , thereby allowing
natural qubit evolution and indirect state alteration (ayanzadeh2022equal, ;
albash2018adiabatic, ; mcgeoch2020theory, ).
Figure 1. Operation Model of QCs: (a) Digital QCs execute compiled quantum
circuits. (b) Analog QAs execute the embedded problem Hamiltonian. _Operating
as single-instruction machines, analog QAs do not incorporate quantum
circuits._
Full connectivity of qubits at scale is infeasible. In digital QCs, compilers
introduce SWAP operations to make physical qubits adjacent
(zulehner2018efficient, ; murali2019noise, ; tannu2019not, ). Conversely,
analog QCs cannot apply operations to qubits, thus preventing the use of SWAPs
for qubit routing. Instead, QAs employ _embedding_ (zbinden2020embedding, ;
pelofske20234, ; pelofske2019solving, ; pelofske2022solving, ;
barbosa2021optimizing, ) where multiple physical qubits are _chained_ (or
entangled) to represent a program qubit with higher connectivity, as shown in
Fig. 2(a). Compiling quantum circuits in digital QCs preserves qubit
utilization (1-to-1 mapping between program and physical qubits), however,
embedding in QAs can substantially increase physical qubit utilization
(ayanzadeh2022equal, ). For instance, the 5761-qubit QA can accommodate up to
177 program qubits with all-to-all connectivity, highlighting nearly 33x
reduced _logical capacity_.
(a) (b)
Figure 2. (a) Embedding seven program qubits $(Q_{i})$ onto a $5\times 7$
grid of physical qubits. (b) Max embeddable Barabasi–Albert (BA) graphs on a
5761-qubit QA device for different preferential attachment factors (${m}$),
ranging from sparse BA-1 ($m=1$) to highly dense BA-6 ($m=6$) structures.
Given that the hardware graph remains fixed after fabrication, QAs’ logical
capacity is primarily determined by the topology of the problem graph. Real-
world applications typically involve irregular “Power-Law” graphs
(agler2016microbial, ; clauset2016colorado, ; gamermann2019comprehensive, ;
goh2002classification, ; house2015testing, ; mislove2007measurement, ;
pastor2015epidemic, ), and Barabasi–Albert (BA) graphs are widely considered
representative of such real-world graphs (albert2005scale, ;
barabasi1999emergence, ; barabasi2000scale, ; gray2018super, ;
kim2022sparsity, ; lusseau2003emergent, ; wang2019complex, ;
zadorozhnyi2012structural, ; zbinden2020embedding, ). Figure 2(b) illustrates
the largest embeddable BA graphs on a 5761-qubit QA, ranging from sparse (with
attachment factor $m=1$, BA-1) to nearly fully connected (with $m=6$, BA-6)
structures. As $m$ increases linearly, the logical capacity experiences a
superpolynomial reduction, converging to the 177-node fully connected graph.
We observe that chain lengths in QAs follow a “Power-Law” distribution, where
a few _dominant chains_ are significantly longer than most other chains (see
section 4.1 for more information). Moreover, we observe that a significant
portion of physical qubits, nearly 25%, remain unused as they become trapped
in long chains. Furthermore, we observe that long chains can reduce the
fidelity of QAs too. The qubits within a chain might take different values
post-measurement, called _broken chains_. Broken chains can negatively impact
QAs’ reliability, and longer chains are more likely to break.
In this study, we aim to improve the capacity and fidelity of QAs through
eliminating dominant chains, as they account for a substantial portion of
qubit utilization and are the main reason for isolating physical qubits. We
propose _Skipper_ , which _prunes_ these chains by removing their
corresponding program qubits and replacing them with two possible measurement
outcomes: -1 and +1. By eliminating a dominant chain, Skipper accomplishes two
objectives: (a) releasing physical qubits previously used within pruned
chains, and (b) releasing all qubits previously trapped with the pruned chain.
This can enable us to use all released physical qubits to accommodate more
program qubits.
However, identifying and pruning dominant chains is nontrivial. Chains are
formed post-embedding. First, when a (long) chain is eliminated, the remaining
embedding is likely not to be optimum, necessitating re-embedding the new
problem to maximize the reliability of QAs. Embedding itself is nontrivial, as
it can take several hours for problems at scale. Moreover, embedding
techniques are heuristic, and they may fail to find an embedding successfully
for a problem, requiring multiple attempts. Second, pruning the longest chain
can change the position of the second-longest chain when re-embedding the
problem, necessitating an embedding for every pruned chain. To this end,
Skipper adopts a greedy approach to prune $c$ chains by sorting program qubits
based on their degree and removing the top $c$ qubits simultaneously. We
observe that this greedy approach exhibits desirable, near-optimal behavior
for $c\geq 5$ chain cuts.
Importantly, the number of chain cuts in Skipper is user-defined; the system
allows for a maximum of eleven chains to be cut, and this does not scale with
the problem size, offering flexibility within the user’s budget constraints.
Each chain cut bifurcates the search space of the initial problem; therefore,
trimming eleven chains can lead to up to 2048 sub-problems, and Skipper
examines all of them to ensure guaranteed recovery. Our experiments with a
5761-qubit QA by D-Wave demonstrate that Skipper can address up to 59% (Avg.
28.3%) larger problems when up to eleven chains are trimmed. Additionally,
Skipper can significantly enhance QA fidelity by up to 44.4% (Avg. 33.1%),
when trimming up to five chains and running 32 quantum executables.
Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ). Skipper
enhances both the capacity and fidelity of analog QAs. However, FrozenQubits
has a negligible impact on the capacity of digital QCs, where one program
qubit is represented with one physical qubit. Furthermore, FrozenQubits’
performance diminishes as graph density increases, whereas Skipper effectively
handles graphs ranging from sparse to dense structures.
The quantum cost of Skipper can present affordability challenges for certain
users. We introduce _Skipper-G_ , a greedy approach that bypasses sub-spaces
less likely to include the global optimum. Consequently, it runs at most 23
quantum executables, compared to the 2048 required by Skipper for trimming up
to eleven chains. It is worth noting that Skipper-G is proposed to improve QA
fidelity, with its effect on increasing capacity being negligible. Our
experiments demonstrate that Skipper-G can boost QA fidelity by up to 40.8%
(Avg. 29.2%), with five chain cuts and 11 runs.
Overall, this paper makes the following contributions:
1. (1)
We show that in QAs, the chain length exhibits a “Power-Law” distribution,
with a few dominant chains having significantly more qubits. Moreover, we
demonstrate that approximately 25% of physical qubits remain unused as they
become trapped within long chains.
2. (2)
We introduce _Skipper_ that enhances the capacity and reliability of QAs by
cutting dominant chains, thereby addressing up to 59% (Avg. 28.3%) larger
problems and improving QA fidelity by up to 44.4% (Avg. 33.1%), when up to
eleven and five chains are cut, respectively. To our knowledge, Skipper is the
first proposal to simultaneously enhance both the capacity and fidelity of
QAs.
3. (3)
We demonstrate that the quantum cost of Skipper in enhancing QA fidelity can
be substantially reduced (to only 23 runs, compared to over 2000 runs in
Skipper) by bypassing sub-spaces unlikely to contain optimal solutions.
4. (4)
We propose _Skipper-G_ , a greedy scheme that enhances QA fidelity by up to
40.8% (Avg. 29.2%), with five chain cuts and only 11 runs (compared to 32 runs
in Skipper).
## 2\. Background and Motivation
### 2.1. Quantum Computers: Digital vs. Analog
QCs fall into two categories: digital and analog. Digital QCs, like those from
IBM and Google, apply precise quantum operations—defined by the quantum
algorithm—to qubits in order to directly manipulate their state
(nielsen2010quantum, ). Conversely, analog QCs, like those from D-Wave and
QuEra, do not directly manipulate the state of qubits. Instead, they apply
precise changes—defined by the quantum program—to the environment in which the
qubits reside, allowing the qubits to evolve and change their states naturally
(albash2018adiabatic, ; ayanzadeh2022equal, ).
### 2.2. Quantum Annealers: Analog Quantum Accelerators
Quantum annealing is a meta-heuristic for tackling optimization problems that
runs on classical computers. _Quantum Annealers_ (_QAs_) are a form of analog
QCs that can sample from the ground state (the configuration with the lowest
energy value) of a physical system, called Hamiltonian (albash2018adiabatic, ;
ayanzadeh2021multi, ; ayanzadeh2022equal, ). QAs by D-Wave are single-
instruction optimization accelerators that can only sample from the ground
state of the following problem Hamiltonian (or Ising model):
(1) $\mathcal{H}_{p}:=\sum_{i}{\mathbf{h}_{i}\mathbf{z}_{i}}+\sum_{I\neq
j}{J_{ij}\mathbf{z}_{i}\mathbf{z}_{j}}$
acting on spin variables $\mathbf{z}_{i}\in{-1,+1}$, where
$\mathbf{h}_{i}\in\mathbb{R}$ and $J_{ij}\in\mathbb{R}$ are linear and
quadratic coefficients, respectively (ayanzadeh2022equal, ).
### 2.3. Operation Model of Single-Instruction QAs
QAs operate as single-instruction computers, and during each execution trial,
they only draw a single sample to approximate the global minimum of (1).
Therefore, we _cast_ real-world problems into Hamiltonians, where $\mathbf{h}$
and ${J}$ are defined in such a way that its global minimum represents the
optimal solution to the problem at hand (albash2018adiabatic, ;
ayanzadeh2022equal, ). The abstract problem Hamiltonian is then _embedded_
into the connectivity map of the QA hardware to generate an executable Quantum
Machine Instruction (QMI) (minorminerGithub, ; cai2014practical, ). Casting
and embedding in QAs are akin to designing and compiling quantum circuits in
digital QCs, respectively (Fig. 1). The QMI is executed for several trials,
and the outcome with the lowest objective value is deemed as the ultimate
result (ayanzadeh2022equal, ).
### 2.4. Anneal Time: Current Technological Barriers
As the energy gap between the global minimum and the adjacent higher state
diminishes linearly, the required annealing time for successful adiabaticity
grows exponentially (albash2018adiabatic, ; das2008colloquium, ), surpassing
the limits of contemporary QAs (yan2022analytical, ). Nonetheless, QAs, akin
to other QCs, are advancing; subsequent generations are expected to bypass
present technological constraints. Specifically, incorporating $XX$ terms into
the time-dependent Hamiltonian can ebb the annealing time scaling from
exponential to linear (nishimori2017exponential, ).
Figure 3. Embedding example.
### 2.5. Embedding for QAs
The connectivity of QA qubits is sparse, thereby limiting users to only
specify $J_{ij}$ for those qubits that are physically connected. Thus, the
abstract problem Hamiltonian is _embedded_ into QA hardware where a program
qubit ($Q_{i}$) with higher connectivity is represented by multiple physical
qubits ($q_{i}$) called _chain_ (Fig. 3). Satisfying the following conditions
is sufficient to guarantee that both the abstract Hamiltonian and the embedded
Hamiltonian executed on the QA hardware have identical ground states:
1. (1)
All chains representing program qubits must be a connected component
graph—i.e., there must be a path between any two qubits within a chain.
2. (2)
There must be at least one connection between chains whose corresponding
program qubits are connected.
3. (3)
The quadratic coefficient $J_{ij}$ is distributed equally among the couplers
connecting $Q_{i}$ and $Q_{j}$.
4. (4)
The linear coefficient $\mathbf{h}_{i}$ is distributed equally among all
physical qubits of the corresponding chain.
5. (5)
Inter-chain quadratic coefficients must be large enough to guarantee that all
qubits within a chain take an identical value—i.e., a very high penalty for
broken chains.
### 2.6. Prior Work Limitations
#### 2.6.1. Circuit Cutting in Digital QCs
Circuit cutting techniques, namely CutQC (tang2021cutqc, ), partition quantum
circuits into smaller sub-circuits, enabling larger quantum circuits to be run
on smaller QCs. However, a similar approach is infeasible in the analog
quantum realm because: (a) analog QAs do not incorporate quantum circuits to
cut its wires; and (b) partitioning graphs by edge/node removal is nontrivial
(e.g., highly dense graphs are non-partitionable).
#### 2.6.2. Solving Larger Problems on Smalle QAs
Previous methods for solving larger problems on smaller QAs
(pelofske2022solving, ; okada2019improving, ) employ iterative or alternating
approaches involving approximations, leading to reduced reliability as problem
size increases. Additionally, convergence—even to a local optimum—is not
guaranteed with these techniques. Conversely, Skipper explores the entire
search space comprehensively without resorting to approximations, and since it
is not iterative, it does not face convergence issues.
#### 2.6.3. Application-Specific Policies
Recent studies have proposed methods for tackling larger instances in various
domains, such as Boolean Satisfiability (SAT) (tan2023hyqsat, ), Max-Clique
(pelofske2023solving, ; pelofske2019solving, ; pelofske2022parallel, ;
pelofske2021decomposition, ), and compressive sensing with matrix uncertainty
(ayanzadeh2019quantum, ; mousavi2019survey, ). However, these techniques are
tailored to their specific applications and cannot be easily adapted to other
domains. In contrast, Skipper is versatile and can be applied to any problem
Hamiltonian. Moreover, reduction to SAT and Max-Clique often leads to a
polynomial increase in program qubits, expanding the problem size.
#### 2.6.4. FrozenQubits
Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ), with both
methods aiming to eliminate high-degree program qubits. While the impact of
FrozenQubits on addressing larger problems in digital QCs is minimal due to
the one-to-one correspondence between program and physical qubits, Skipper, on
the other hand, is capable of solving larger problems on QAs and enhancing QA
fidelity. Moreover, unlike FrozenQubits, whose performance declines with
increasing graph density, Skipper maintains effectiveness across a spectrum of
graph densities, from sparse to dense structures.
### 2.7. Goal of This Paper
Figure 4(a) shows the maximum and average chain lengths for different graph
topologies embedded on a 5761-qubit QA. A few dominant chains contain over
7.9x as many qubits as the average chain lengths. Furthermore, Fig. 4(b)
displays the number of unused qubits when embedding the largest possible
graphs on a 5761-qubit QA for different graph topologies, indicating that more
than 25% of physical qubits remain unutilized, primarily due to dominant
chains.
The severe underutilization of QA qubits, along with utilizing several
physical qubits to represent a single program qubit, severely diminishes the
capacity of QAs by up to 33x. For instance, while current D-Wave QAs boast
over 5,700 qubits, they can accommodate at most 177 program qubits with full
connectivity. The aim of this paper is to enable QAs to tackle larger problems
by pruning dominant chains, while also enhancing the fidelity of the QAs.
(a) (b)
Figure 4. Maximum embeddable BA graphs on 5761-qubit QA: (a) Avg and Max
chain lengths, and (b) Number of unutilized qubits.
## 3\. Methodology
### 3.1. Hardware Platform
For our evaluations, we utilize the D-Wave Advantage System (version 6.2),
which features over 5,760 qubits and more than 40,000 couplers, accessed via
the D-Wave Leap cloud service (D-Wave, ). We employ the default annealing time
of 20 microseconds and adhere to the anneal schedule recommended for this
device. Each problem is run for 4,000 trials to comply with the two seconds
maximum job duration limit.
### 3.2. Software Platform
We utilize the _minorminer_ tool (minorminerGithub, ; cai2014practical, ) to
find embeddings for arbitrary problem Hamiltonians on QA hardware. In our
experiments, we set a timeout of 1,000 seconds, a maximum of 20 failed
attempts for improvement, and conduct 20 trials. To program the D-Wave QAs, we
employ the Ocean SDK (dwave_ocean_github, ).
### 3.3. Benchmarking
Although current QAs feature over 5,700 qubits, their single-instruction
operation model limits them to a few hundred program qubits with higher
degrees, which is far below the number of variables required for real-world
applications. Consequently, in this study, we employ synthetic benchmarks
instead of real-world problems. In many real-world applications, graphs often
exhibit a “Power-Law” distribution (agler2016microbial, ; clauset2016colorado,
; gamermann2019comprehensive, ; goh2002classification, ; house2015testing, ;
mislove2007measurement, ; pastor2015epidemic, ), and the _Barabasi–Albert_
(BA) algorithm (albert2005scale, ; barabasi1999emergence, ) is considered
representative of these real-world graph structures
(ayanzadeh2023frozenqubits, ; barabasi2000scale, ; gray2018super, ;
kim2022sparsity, ; lusseau2003emergent, ; wang2019complex, ;
zadorozhnyi2012structural, ; zbinden2020embedding, ). The BA graphs are
generated with a preferential attachment factor $m$, enabling us to vary the
density of the graphs by adjusting $m$—with higher values of $m$ yielding
denser graphs. We generate BA graphs with $m$ values ranging from $m=1$ (BA-1)
to $m=6$ (BA-6) to capture a broad spectrum of topologies, from sparse to
nearly fully connected networks, thus effectively representing the dynamics of
various real-world systems (clauset2016colorado, ). Edge weights are assigned
randomly following a standard normal distribution, which is a common approach
in QA benchmarking (das2008colloquium, ; ayanzadeh2022equal, ;
ayanzadeh2021multi, ).
### 3.4. Figure of merit
In our evaluations, we use the _Energy Residual_ (_ER_) to assess the fidelity
of QA as
(2) $\downarrow\textrm{Energy Residual (ER)}=\left|E_{min}-E_{global}\right|,$
where $E_{global}$ represents the global minimum of the benchmark problem, and
$E_{min}$ corresponds to the best solution obtained by the QA. Ideally, an ER
value closer to zero is desirable as it indicates a solution that closely
aligns with the ground state of the problem Hamiltonian. We conducted
intensive classical computations using state-of-the-art tools
(ayanzadeh_ramin_2021_5142230, ) to determine the global minima of the
benchmarks.
## 4\. Skipper: Skipping Dominant Chains
We introduce _Skipper_ , a software technique to enhance the capacity and
fidelity of QAs through strategically skipping dominant qubit chains.
### 4.1. Key Insights
#### 4.1.1. Not All Program Qubits are Equal
In digital QCs, the individual fabrication of physical qubits, such as
superconducting ones, results in inevitable performance variations
(tannu2019not, ). Compilers, therefore, aim to prioritizing high-quality ones
and limit the reliance on those of lower quality (tannu2019not, ;
li2018tackling, ; noiseadaptive, ). However, in analog QCs, our observations
reveal a significant variability at the level of program qubits.
Figure 5(a) shows the histogram of chain lengths (in log-scale) for the BA-3
graph type after embedding onto a 5761-qubit QA device, revealing a _Power-
Law_ distribution with some notably longer _dominant chains_ and a majority of
considerably shorter chains. Figure 5(b) presents the maximum and average
chain lengths as the number of nodes in BA-3 graphs increases, notably
magnifying the variability in chain lengths with the increase in problem size.
These intriguing observations extends beyond the BA-3 graph type, and we
observe it in all benchmark graphs, including BA-1 to BA-6, spanning from
sparse to nearly fully connected graphs. Additionally, we observe the
nonuniformity of chain lengths in regular and fully connected graphs.
(a) (b)
Figure 5. (a) Histogram of chain lengths for a 600-node BA-3 graph (log-
scale), indicating a “Power-Law“ distribution of chain lengths. (b) Max and
Avg chain lengths of BA-3 graphs, embedded on a 5761-qubit QA.
#### 4.1.2. QA Qubits are significantly Underutilized
We observe that, on average, 25% of physical qubits remain unused as they get
trapped by chains. Additionally, we observe that the dominant chains
significantly contribute to this qubit isolation. This underutilization of QA
qubits, along with utilizing several physical qubits to represent a single
program qubit, severely diminishes the capacity of QAs by up to 33x. For
instance, the 2048-qubit and 5760-qubit QAs by D-Wave can accommodate a
maximum of 64 and 177 fully connected program qubits, respectively.
#### 4.1.3. Diminishing Returns with Increased QA Trials
QAs are noisy and prone to errors, leading to a systematic bias during the
execution of quantum programs. This bias causes deviations from the global
optimum, reducing the reliability of QAs (albash2018adiabatic, ;
mcgeoch2020theory, ; ayanzadeh2022equal, ; ayanzadeh2021multi, ). The bias
arises from repeating the same quantum program across multiple iterations,
exposing all trials to a similar noise profile (ayanzadeh2022equal, ).
Figure 6 shows that when the number of trials in QA is increased, the output
distribution reaches saturation. This indicates that the gap between the ideal
solution and the QA does not reduce despite drawing more samples. Moreover,
due to the operation model of QAs as single-instruction computers, strategies
commonly used in gate-based QCs (tannu2019ensemble, ; patel2020veritas, ) to
address this bias are not applicable.
Figure 6. The Energy Residual (ER) in QAs tends to plateau with an increasing
number of trials, and the global minimum often remains unreachable by QAs.
Figure 7. Overview of Skipper.
### 4.2. Overview of Skipper
Figure 7 shows the overview of Skipper. Skipper leverages insights into the
distribution of chain lengths and the severe underutilization of qubits in
QAs, employing a strategic approach to prune dominant chains and replace the
corresponding program qubit with two potential measurement outcomes (+1 and
-1). This process involves eliminating each chain, which partitions the search
space into two sub-spaces. Skipper explores all sub-spaces, guaranteeing an
exact recovery of the optimum solutions.
Eliminating a dominant chain accomplishes two significant objectives: firstly,
it frees up physical qubits previously used within pruned chains, and
secondly, it eliminates the isolation of solitary qubits resulting from
dominant chains. As a result, Skipper enables the handling of larger problems
by accommodating a significantly higher number of program qubits.
Additionally, Skipper significantly enhances QA fidelity by substantially
mitigating the impact of dominant chains, a primary factor in compromising QA
reliability. While Skipper utilizes more quantum resources due to the need to
execute $2^{c}$ unique quantum programs for the removal of $c$ chains, it
doesn’t correspondingly enhance QAs (baseline) performance, as demonstrated in
Fig. 6.
### 4.3. Chain Skips: How, Where, and When to Skip?
Figure 8 illustrates the elimination of two chains from a problem with five
variables, creating two and four _independent_ sub-problems, respectively. To
skip a chain, the program qubit is replaced with +1 and -1 (two measurement
outcomes), removing the node and its connected edges from the problem graph.
Unlike digital QCs, where removing one program qubit results in reducing the
physical qubit utilization by one, in QAs, removing one program qubit
liberates all the physical qubits involved in its corresponding chain.
Figure 8. By replacing $Q_{0}$ with +1 and -1 among five spin variables
(baseline), two sub-problems each with four spin variables are obtained
($c=1$). Fixing $Q_{1}$ in these two sub-problems with +1 and -1 results in
four sub-problems with three spin variables ($c=2$). The same embedding is
utilized for all sub-problems at each level of the binary tree.
Identifying dominant chains to trim in Skipper is nontrivial. In digital QCs,
high-degree program qubits necessitate more CNOT gates, enabling direct
identification prior to circuit compilation. However, in QAs, it is not
feasible to directly recognize program qubits linked to longer chains, thus
requiring embedding techniques to identify them. Furthermore, it is not always
optimal to prune the dominant chain. In Fig. 9(a), the dominant chain is
$Q_{0}$ and consists of ten physical qubits. Pruning $Q_{0}$ (Fig. 9(b)),
liberates all ten physical qubits, leaving the other chains intact. However,
as shown in Fig. 9(c), removing $Q_{2}$ and re-embedding the problem not only
releases the five physical qubits associated with $Q_{2}$ but also effectively
reduces $Q_{0}$ to a singleton chain, totaling fourteen qubits released.
Skipper adopts a greedy approach to prune $c$ chains by sorting program qubits
based on their degree and removing the top $c$ qubits simultaneously. The
removal of a single program qubit can have a substantial impact on other
chains, as shown in Fig. 9(c). In the context of irregular graphs that often
follow the Power-Law distribution in real-world applications, this greedy
approach exhibits a desirable, near-optimal behavior for $c\geq 5$ chain cuts.
Figure 9. (a) Embedding of five program qubits on a grid. (b) Freeing ten
qubits by pruning the dominant chain $Q_{0}$. (c) Fourteen qubits freed by
pruning $Q_{2}$.
### 4.4. Skip Count: A Cost-Performance Tradeoff
Skipper permits users to trim up to eleven chains. Each chain skipped
bifurcates the search space; therefore, trimming up to eleven chains can lead
to a maximum of 2048 sub-problems. Skipper executes all corresponding QMIs to
ensure exact solution recovery. However, the nontrivial embedding process and
the need to execute up to 2048 embeddings can create a bottleneck for Skipper.
Fortunately, the identical structure of all sub-problems at the $c$-th level
in the binary tree enables sharing the same embedding across them (Fig. 8).
### 4.5. Unembedding: Remediating Broken Chains
The sub-problem resulting from cutting chains consists of $n-c$ program
qubits. However, after embedding, the problem executed on the QA hardware
encompasses $N$ physical qubits, where $c\ll n\ll N$. As a result, the QA
produces outcomes as $N$-bit strings, with each program qubit collectively
represented by multiple bits in a chain. Therefore, Skipper _unembeds_ these
outcome samples, converting them back into the space of program qubits.
Ideally, all physical qubits within a chain should have identical values in a
given QA sample. The value of the associated program qubit is then determined
by observing any one of the physical qubits within it (e.g., program qubit
$Q_{0}$ in Fig. 10). However, QAs are inherently open systems, as interactions
with the environment are unavoidable in QCs, and the annealing process tends
to be diabatic since truly adiabatic processes are often unfeasible
(ayanzadeh2021multi, ). As a result, qubits within a chain can take different
values, an issue known as _broken chains_ (grant2022benchmarking, ;
pelofske2020advanced, ; king2014algorithm, ; barbosa2021optimizing, ).
To remediate broken chains, Skipper employ the _majority voting_ approach. For
instance, in Fig. 10, although $Q_{1}$ exhibits a broken chain with varying
qubit values, the unembedding process assigns a value of -1, reflecting the
majority of -1 values within the chain (4 versus 1). However, not all chains
have an odd length, and forcing the embedding to produce odd chain lengths is
nontrivial. Unembedding even length chains with mostly identical qubit values
(e.g., $Q_{2}$ in Fig. 10) is not challenging, as majority voting can
effectively determine the value of the program qubit. However, as demonstrated
by $Q_{3}$ in Fig. 10, a chain of even length can contain an equal number of
-1 and +1 values, referred to as _balanced chains_ , a condition where
majority voting fails. Skipper manages balanced chains by counting them and
implementing distinct strategies based on their quantity. For problems with
fewer than ten balanced chains, Skipper discards their qubit values and uses a
brute-force approach (with up to 1024 possible configurations), selecting the
configuration that yields the lowest energy value. If the number of balanced
chains exceeds ten, Skipper randomly assigns values to the corresponding
program qubits. When a broken chain occurs, Skipper can optionally apply
Single-Qubit Correction (SQC) (ayanzadeh2021multi, ; ayanzadeh2022equal, )
postprocessing to maintain a feasible solution for the original problem.
Figure 10. Unembedding examples
### 4.6. Decoding Sub-Problem Results
After unembedding, each sample will encompass $n-c$ bits, while the original
problem includes $n$ variables. The decoding process reintroduces the values
of the $c$ pruned program qubits, which were fixed during the sub-problem
formulation by assigning fixed values to these variables.
### 4.7. Postprocessing
Theoretically, QAs sample from a Boltzmann distribution, exponentially
favoring lower energy values, and thus should locate the global optimum in few
attempts. However, like other QCs, QAs are vulnerable to noise and various
error sources that degrade their fidelity. To enhance the reliability of QAs,
we can optionally apply postprocessing heuristics to the resulting samples
(ayanzadeh2021multi, ).
### 4.8. Deriving the Final Output
In Skipper, all sub-problems are executed independently, each one
corresponding to a separate sub-space of the primary problem. Consequently, in
Skipper, the sample with the lowest energy or objective value is deemed as the
ultimate output, with the originating sub-space of this global optimum being
of no consequence.
### 4.9. Overhead of Skipper
Let ${c}$ represent the number of skipped chains, $e$ denote the edges in the
problem graph, $r$ symbolize the number of trials on the QA, while $n$ and $N$
correspond to the number of program and physical qubits, respectively.
Quantum overhead: Skipper allows for up to eleven chain cuts, necessitating
the execution of at most 2048 distinct quantum executables, each running
independently.
Classical overhead: We separate the embedding overhead of Skipper from all
other classical pre/post-processing modules, as the embedding is the primary
factor influencing the end-to-end runtime of the proposed techniques in this
paper (refer to section 8). Given the fact that $c\ll n\ll N\ll r$, Skipper
demonstrates a classical time complexity of
$O\left(2^{c}\left(rN+c\right)\right)$. This is representative of the
unembedding and decoding processes for outcome samples of sub-problems.
Embedding overhead: In Skipper, all sub-problems of the binary tree at the
$c$-th level share a single embedding, leading to $O(1)$ embedding complexity.
Note that we assume all sub-problems are executed on the same QA hardware or
that all devices have the same working graph topology, allowing them to share
the embedding.
Memory utilization: The memory utilization in Skipper scales according to
$O(rN2^{c})$.
## 5\. Skipper Evaluation Results
We evaluate Skipper using Barabasi–Albert (BA) graphs (barabasi1999emergence,
) with different preferential attachment factor values: $m=1$ (BA-1) to $m=6$
(BA-6).
### 5.1. Solving Larger Problems
#### 5.1.1. Impact on Chain Length
Figure 11(a) illustrates that increasing the number of chain cuts ($c$) in
Skipper leads to a reduction in the average chain length of the embeddings.
Figure 11(b) demonstrates that Skipper decreases the mean chain length by up
to 1.32x (with an average of 1.22x) when cutting up to eleven chains.
Figure 12(a) shows that the maximum chain length of the embeddings decreases
as $c$ in Skipper increases. Figure 12(b) shows that cutting up to $c=11$
chains in Skipper reduces the maximum chain length by up to 9.14x (average
1.86x). Our observations indicate that long chains are the primary
contributing factor to the underutilization of physical qubits.
(a) (b)
Figure 11. Relative Avg. chain length in Skipper compared to baseline (lower
is better). (a) Relative Avg. chain length for different graphs as cut size
($c$) increase. (b) Overall relative mean chain lengths for up to 11 chain
cuts.
(a) (b)
Figure 12. Relative Max chain length in Skipper compared to the baseline
(lower is better). (a) Relative Max chain length for different graphs as $c$
increases. (b) Overall relative max chain lengths for up to 11 chain cuts.
#### 5.1.2. Impact on Qubit Utilization
Figure 13(a) displays the average and maximum number of physical qubits when
up to eleven chains are pruned. In Fig. 13(b), Skipper reduces
underutilization of QA qubits by up to 57% (average 22.14%) with up to eleven
trimmed chains.
(a) (b)
Figure 13. (a) Utilization of Physical Qubits in Skipper across Different
Graph Types. (b) Relative Number of Unused Physical Qubits in Skipper for up
to 11 Chain Cuts, Compared to the Baseline. Lower is better.
#### 5.1.3. Impact on Capacity of QAs
The QA capacity to accommodate specific graph types, from BA-1 to BA-6, is
determined by the largest number of program qubits of each type that can be
embedded on the QA. Figure 14(a) shows that QA capacity in Skipper improves
with increasing $c$ across various graph topologies.
Figure 14(b) demonstrates that Skipper enables the embedding of larger
problems onto current QAs, with an increase of up to 59.61% (average 28.26%).
It is important to note that this growth in the number of program qubits
necessitates a substantial increase in the number of physical qubits, as one
program qubit is represented by multiple physical qubits.
Skipper’s performance remains consistent regardless of the increasing density
of problem graphs (from BA2 to BA6).
(a) (b)
Figure 14. Relative QA capacity in Skipper compared to baseline. (a) Relative
capacity for different graphs as cuts increase. (b) Overall relative capacity
for up to 11 chain cuts. Higher is better.
### 5.2. Boosting QA Reliability
In addition to enhancing QA capacity, Skipper can be employed to improve the
reliability of QAs.
#### 5.2.1. Impact on Embedding Quality
QAs do not incorporate circuits, thus precluding the use of the Probability of
Successful Trials metric commonly employed to assess compilation quality in
digital QCs (ayanzadeh2023frozenqubits, ; alam2020circuit, ; nishio, ;
tannu2022hammer, ). Prior studies suggest that embeddings with similar chain
lengths can produce better solutions (boothby2016fast, ;
venturelli2015quantum, ; rieffel2015case, ; choi2008minor, ). Figure 15(a)
demonstrates that trimming up to eleven chains in Skipper reduces the average
variance in chain lengths by 2.93x (up to 70.19x).
(a) (b)
Figure 15. (a) Relative variance of chain lengths and (b) relative embedding
time in Skipper compared to the baseline when trimming up to eleven chains.
Lower is better.
#### 5.2.2. Impact on Embedding Time
Figure 15(b) demonstrates that pruning up to eleven chains in Skipper leads to
a significant reduction in embedding time, with a maximum improvement of
17.13x (average improvement of 7.12x).
#### 5.2.3. Impact on Fidelity
Figure 16(a) shows that as Skipper skips more chains, the Energy Residual (ER)
decreases, indicating a progressive approach towards the global optimum.
Additionally, Fig. 16(b) demonstrates a significant reduction in ER by up to
44.4% (average 33.08%), when up to five chains are cut using Skipper, compared
to the baseline.
(a) (b)
Figure 16. Relative Energy Residual (ER) in Skipper compared to baseline
(lower is better). (a) Relative ER as $c$ increase. (b) Overall relative ER
for up to five chain cuts.
## 6\. Skipper-G: A Marcovian Approach
We propose _Skipper-G_ , a greedy scheme that reduces the quantum cost of
Skipper by skipping the examination of sub-problems unlikely to contain the
global optimum. However, this strategy entails a trade-off: Skipper-G achieves
marginally lower fidelity gains compared to Skipper and is ineffective for
enhancing QA capacity.
### 6.1. Insight: Not ALL Sub-Spaces Include Global Optimum
Skipper employs a Breadth-First Search (BFS) strategy to examine sub-problems,
as depicted in Fig. 17(a). Trimming each chain bifurcates the search space,
with skipping $c$ chains resulting in a binary tree of depth $c$. To ensure
successful recovery, Skipper evaluates all leaf nodes, running a separate QMI
for each sub-space at the tree’s last level. Notably, Skipper does not examine
intermediate nodes (or sub-spaces) since all chains are trimmed
simultaneously.
Users define the number of chain cuts in Skipper, with the option to skip up
to eleven chains based on their budgetary constraints. For instance, if a user
opts for the maximum allowable eleven cuts, Skipper must run 1024 QMIs when
all linear coefficients are zero (ayanzadeh2023frozenqubits, ), and up to 2048
QMIs otherwise. Nonetheless, these sub-problems are independent, allowing for
parallel execution by Skipper. Notably, Skipper’s overall runtime remains
comparable to the baseline, attributed to the significantly reduced embedding
time, as detailed in Section 5.2. However, the quantum costs incurred on QCs
are substantially higher than those on classical platforms, which may present
affordability issues for some users.
Not every sub-space contains the global optimum. Leveraging this insight, we
introduce _Skipper-G_ (_greedy Skipper_), which reduces the quantum cost of
Skipper by adopting a Depth-First Search (DFS) strategy (Fig. 17(b)), to
bypass sub-spaces unlikely to include the global optimum. When pruning the
maximum of eleven chains, Skipper-G executes 23 QMIs, in contrast to Skipper’s
potential 2048 QMIs.
Figure 17. (a) Skipper utilizes a Breadth-First Search (BFS) strategy,
examining all leaf nodes (intermediate nodes are not examined). (b) Skipper-G
adopts a Depth-First Search (DFS) strategy, examining only two nodes at each
level of the binary tree (including intermediate nodes). Example: With $c=11$
chain cuts, Skipper and Skipper-G execute at most 2048 and 23 QMIs,
respectively.
### 6.2. How Skipper-G Work?
Figure 18 illustrates the overview of the Skipper-G scheme. In Skipper-G,
similar to Skipper, users can determine the number of chain cuts, with the
possibility of skipping up to eleven chains, depending on their budget
constraints. However, unlike Skipper where all chains are cut simultaneously,
Skipper-G employs an iterative approach, cutting one chain in each iteration.
As illustrated in Fig. 17(b), Skipper-G initiates by setting the root node
(i.e., the baseline with no chains cut) as the current node and executing the
corresponding quantum program. For each chain cut, Skipper-G performs the
following steps:
1. (1)
In the current node (problem), the dominant chain is trimmed by setting its
corresponding program qubit to either +1 or -1, resulting in two child nodes.
If the current node at level $c$ has the index $x$, then its left and right
children at level $c+1$ will have indices $2x$ and $2x+1$, respectively (e.g.,
node $x=1$ at the third level leads to nodes 2 and 3 in Fig. 17(b)).
2. (2)
The quantum programs corresponding to the children are executed on the QA.
3. (3)
The best offspring is set as the current node.
### 6.3. Branch and Bound Criteria
When evaluating a node in Skipper-G, a quantum program is executed on a QA
device for multiple trials. Each trial produces an outcome with an associated
objective value. The assessment of node quality in Skipper-G is based on the
following feature (lower is better):
(3) $\downarrow
f(Z)=\left|\frac{1}{\text{E}_{\text{min}}\times\text{EV}}\right|,$
where $Z$ denotes the set of obtained samples, and $\text{E}_{\text{min}}$ and
EV represent the minimum and the expected value of the energy values in $Z$,
respectively. The lower the value of $f$, the greater the likelihood that a
child includes the global optimum in its corresponding subspace during the
traversal of the associated binary tree. This feature balances the best sample
with the overall quality of all samples, reducing the likelihood of getting
trapped in local optima.
### 6.4. Overhead of Skipper-G
Skipper-G is capable of trimming up to eleven chains, which necessitates a
maximum of 23 distinct QMI executions. Although Skipper-G examines two nodes
at each level of the binary tree, these nodes, due to their identical
structures, can utilize a single embedding. Consequently, Skipper-G requires
$c$ embeddings for $c$ chain cuts. Nonetheless, since these embeddings can be
executed in parallel, the embedding time for Skipper-G remains similar to that
of the baseline, as the root node’s embedding is expected to be more time-
consuming than the embeddings of the smaller intermediate nodes.
Figure 18. Overview of Skipper-G.
### 6.5. Evaluation Results
Figure 19(a) illustrates the ER for various chain cut counts in Skipper-G,
showing that progressively trimming more dominant chains leads to a decrease
in the ER, approaching the global minimum. Additionally, Fig 19(b) reveals
that pruning up to five chains in Skipper-G can reduce the gap between the
global optimum and the best QA sample by as much as 40.75% (Avg. 29.19%)
compared to the baseline.
Skipper marginally outperforms Skipper-G, achieving a 3.89% greater reduction
in ER, albeit at the expense of significantly higher quantum resource
utilization. Skipper-G includes the baseline at the root of the binary tree,
ensuring it performs no worse than the baseline.
(a) (b)
Figure 19. Relative ER for five chain cuts compared to the baseline (lower is
better). (a) Skipper-G: Relative ER with increasing $C$. (b) Overall Relative
ER (lower is better) for Skipper vs. Skipper-G.
## 7\. Skipper and Skipper-G in Classical Realm
Unfortunately, neither Skipper nor Skipper-G can be utilized to enhance the
fidelity of optimization techniques used in classical realm. In the classical
domain, the hardness of optimization problems depends on the number of
variables and the graph topology. For instance, while planar graphs are
tractable (dei2006exact, ; hadlock1975finding, ) in classical realm, neither
regular nor power-law graphs become planar simply by eliminating a few nodes.
Additionally, eliminating ten nodes from a 1000-node graph results in sub-
graphs with 990 nodes, which typically remain intractable in the classical
realm.
Similarly, Skipper is not suitable for tackling larger problems in the
classical realm. Its primary goal is to address the sparse connectivity of
qubits, a key factor limiting the capacity of QAs. However, the full
connectivity of classical bits does not present a similar limitation.
## 8\. Workflow Analysis
The runtime of quantum programs is mainly determined by queuing delays,
execution modes through cloud services (which vary across providers), and
embedding time, rather than the execution time on quantum hardware
(microseconds to milliseconds). To offer a holistic examination of the runtime
between the proposed techniques and the baseline, we employ the following
analytical model:
(4)
$T=T_{\text{emb}}+N_{\text{QMI}}\left({T_{\text{queue}}+T_{\text{QMI}}+T_{\text{net}}}\right)+T_{\text{classical}},$
where $T_{\text{emb}}$ is the embedding time, $N_{\text{QMI}}$ is the number
of quantum executables, $T_{\text{queue}}$ is the job wait time,
$T_{\text{QMI}}$ is the QMI execution time, $T_{\text{net}}$ is the network
delay, and $T_{\text{classical}}$ is the classical pre/post-processing time.
For $r$ trials, $T_{QMI}=t_{p}+\Delta+r\times t_{s}$, where $t_{p}$ is the raw
signal preparation time, $\Delta$ is the 10ms QA initialization time, and
$t_{s}$ is the single annealing/readout time. Given that D-Wave limits
$T_{QMI}$ to two seconds, we assume $T_{QMI}=2$ in all cases. We assume one
second for $T_{\text{net}}$ for each job.
We assume a baseline embedding time of 30 minutes, decreasing proportionally
with skipped chains (as discussed in section 5.2). For example, pruning ten
chains reduces the embedding time to three minutes. All embeddings can be
computed in parallel, making $T_{\text{emb}}$ in Skipper-G the maximum time
for individual embeddings. Additionally, we allocate one second each for pre-
and post-processing.
We examine two access scenarios: _shared_ and _dedicated_ , with one and zero-
second queuing times, respectively. Figure 20 compares the end-to-end runtime
of the baseline and our proposed techniques with $c=11$, resulting in up to
1024 QA runs. Skipper shows significantly greater advantages over others in
the dedicated access mode.
Figure 20. Overall Runtime comparison.
## 9\. Related Work
Prior studies can be broadly classified into two categories: (a) techniques
for solving larger problems on QAs, which are relevant primarily to Skipper;
and (b) approaches for enhancing QA fidelity, which are considered related
work to both Skipper and Skipper-G.
Prior research on solving larger problems with smaller QAs
(pelofske2022solving, ; okada2019improving, ) are iterative schemes, which
tend to lose reliability as problem size increases due to reliance on
approximations. Conversely, Skipper explores the entire search space without
resorting to approximations. Recent studies have introduced schemes for
addressing larger instances of Boolean Satisfiability (SAT) (tan2023hyqsat, ),
Max-Clique (pelofske2023solving, ; pelofske2019solving, ;
pelofske2022parallel, ; pelofske2021decomposition, ), and compressive sensing
with matrix uncertainty (ayanzadeh2019quantum, ) problems. However, these
methods are specific to their respective applications and are not transferable
to other domains, whereas Skipper is versatile and applicable to any
application.
Policies for improving the fidelity of QAs can be classified as: (a)
Preprocessing (pelofske2019optimizing, ; ayanzadeh2022equal, ;
ayanzadeh2020reinforcement, ), modifying QMIs before submission; (b)
Postprocessing (ayanzadeh2021multi, ), enhancing outcomes using heuristics;
(c) Hybrid strategies (ayanzadeh2022quantum, ; ayanzadeh2020leveraging, ),
combining heuristics and QAs for reliability; (d) Logical analog qubits
(jordan2006error, ; sarovar2013error, ; young2013error, ; pudenz2014error, ;
vinci2015quantum, ; venturelli2015quantum, ; vinci2016nested, ;
matsuura2016mean, ; mishra2016performance, ; matsuura2017quantum, ;
vinci2018scalable, ; pearson2019analog, ; matsuura2019nested, ;
mohseni2021error, ), spreading qubit information over multiple physical
qubits; and ensembling policies (ayanzadeh2022equal, ;
ayanzadeh2020reinforcement, ; ayanzadeh2020ensemble, ), subjecting the quantum
program to different noise profiles to suppress the bias. These proposals are
orthogonal to Skipper and Skipper-G and can effectively boost the reliability
of our proposed techniques.
Skipper is inspired by FrozenQubits (ayanzadeh2023frozenqubits, ) in digital
QCs. While FrozenQubits enhances fidelity of optimization applications in
digital QCs, Skipper excels in addressing larger problems and enhancing QA
fidelity. More importantly, while FrozenQubits’ performance diminishes with
increased problem graph density, Skipper and Skipper-G maintain their
performance, demonstrating the effectiveness of our proposal in handling
sparse to dense graphs.
## 10\. Conclusion
We propose _Skipper_ , a software scheme designed to enhance the capacity and
fidelity of QAs. Observing that chain lengths in QAs follow a “Power-Law”
distribution, with a few _dominant chains_ containing significantly more
qubits than others, Skipper prunes these chains. This approach replaces their
corresponding program qubits with two possible measurement outcomes, freeing
all qubits in the dominant chains and an additional 25% of isolated qubits
previously entrapped in chains. Our experiments on a 5761-qubit QA by D-Wave
show that Skipper allows QAs to solve problems up to 59% larger (Avg. 28.3%)
when up to eleven chains are skipped. Additionally, by removing five chains,
Skipper substantially improves QA fidelity by up to 44.4% (Avg. 33.1%).
The number of chain cuts in Skipper is user-defined; users can trim up to
eleven chains, which necessitates running an average of 1024 (and up to 2048)
distinct quantum executables. However, this may lead to affordability concerns
for some users. To mitigate this, we introduce _Skipper-G_ , a greedy scheme
that prioritizes examining sub-spaces more likely to contain the global
optimum. When up to eleven chains are pruned, Skipper-G runs a maximum of 23
quantum executables. Our experiments show that Skipper-G enhances QA fidelity
by up to 40.8% (Avg. 29.2%), requiring only 11 quantum executable runs for up
to five chain cuts, compared to Skipper’s 32 runs.
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|
#
Ben-Gurion University of the Negev
The Faculty of Natural Sciences
Department of Mathematics
Poles of degenerate Eisenstein series and Siegel-Weil identities for
exceptional split groups
Thesis Submitted in Partial Fulfillment of the Requirements for the Master of
Sciences Degree
By: Hezi Halawi
Under the Supervision of: Dr Nadya Gurevich
Beer Sheva, August 2016
Ben-Gurion University of the Negev
The Faculty of Natural Sciences
Department of Mathematics
Poles of degenerate Eisenstein series and Siegel-Weil identities for
exceptional split groups
Thesis Submitted in Partial Fulfillment of the Requirements for the Master of
Sciences Degree
By: Hezi Halawi
Under the Supervision of: Dr Nadya Gurevich
Signature of Student: Date:
Signature of Supervisor: Date:
Signature of Chairperson of the Committee for Graduate Studies: Date:
Beer Sheva, August 2016
## Abstract
Let $G$ be a linear split algebraic group. The degenerate Eisenstein series
associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the
spherical section $f^{0}$ is studied in the first part of the thesis. In this
part, we study the poles of $E_{P}(f^{0},s,g)$ in the region
$\operatorname{Re}s>0$. We determine when the leading term in the Laurent
expansion of $E_{P}(f^{0},s,g)$ around $s=s_{0}$ is square integrable. The
second part is devoted to finding identities between the leading terms of
various Eisenstein series at different points. We present an algorithm to find
this data and implement it on SAGE. While both parts can be applied to a
general algebraic group, we restrict ourself to the case where $G$ is split
exceptional group of type $F_{4},E_{6},E_{7}$, and obtain new results.
###### Contents
1. Abstract
|
About split quaternion algebras over quadratic fields and symbol algebras of
degree $n$
Diana SAVIN
Abstract. In this paper we determine sufficient conditions for a quaternion
algebra to split over a quadratic field. In the last section of the paper, we
find a class of non-split symbol algebras of degree $n$ (where $n$ is a
positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic
field.
Key Words: quaternion algebras, symbol algebras; quadratic fields, cyclotomic
fields; Kummer fields; $p-$ adic fields
2010 AMS Subject Classification: 11R18, 11R37, 11A41, 11R04, 11R52, 11S15,
11F85.
1\. Introduction
Let $K$ be a field with char$K\neq 2.$ Let $K^{\ast}=K\backslash\\{0\\},$
$a,b$ $\in K^{\ast}.$ The quaternion algebra $H_{K}\left(a,b\right)$ is the
$K$-algebra with $K$-basis $\left\\{1;e_{1};e_{2};e_{3}\right\\}$ satisfying
the relations: $e^{2}_{1}=a,$ $e^{2}_{2}=b,$ $e_{3}=e_{1}\cdot
e_{2}=-e_{2}\cdot e_{1.}$
Let $n$ be an arbitrary positive integer, $n\geq 3$ and let $\xi$ be a
primitive $n$-th root of unity. Let $K$ be a field with char$K\neq 2,$ char$K$
does not divide $n$ and $\xi$$\in$ $K.$ Let $a,b$ $\in K^{\ast}$ and let $A$
be the algebra over $K$ generated by elements $x$ and $y$ where
$x^{n}=a,y^{n}=b,yx=\xi xy.$
This algebra is called a symbol algebra and it is denoted by
$\left(\frac{a,~{}b}{K,\xi}\right).$ For $n=2,$ we obtain the quaternion
algebra. Quaternion algebras and symbol algebras are central simple algebras
of dimension $n^{2}$ over $K$, non-commutative, but associative algebras (see
[Mil; 08]).
In this article we find sufficient conditions for a quaternion algebra to
split over a quadratic field. In the paper [Sa; 16] we found a class of
division quaternion algebra over the quadratic field
$\mathbb{Q}\left(i\right)$ ($i^{2}=-1$), respectively a class of division
symbol algebra over the cyclotomic field $\mathbb{Q}\left(\xi\right),$ where
$\xi$ is a primitive root of order $q$ (prime) of unity. In the last section
of this article we generalize these results for symbol algebras of degree
$n\geq 3,$ not necessarily prime.
2\. Preliminaries
We recall some results of the theory of cyclotomic fields, Kummer fields and
$p-$ adic fields, associative algebras, which will be used in our paper.
Let $n$ be an integer, $n\geq 3$ and let $K$ be a field of characteristic
prime to $n$ in which $x^{n}-1$ splits; and let $\xi$ be a primitive $n$ th
root of unity. The following lemma (which can be found in [Ca, Fr; 67]) gives
information about certain extension of $K.$
Lemma 2.1. If $a$ is a non-zero element of $K,$ there is a well-defined normal
extension $K\left(\sqrt[n]{a}\right),$ the splitting field of $x^{n}-a.$ If
$\alpha$ is a root of $x^{n}=a,$ there is a map of the Galois group
$G\left(K\left(\sqrt[n]{a}\right)/K\right)$ into $K^{*}$ given by
$\sigma\longmapsto\sigma\left(\alpha\right)/\alpha;$ in particular, if $a$ is
of order $n$ in $K^{*}/\left(K^{*}\right)^{n},$ the Galois group is cyclic and
can be generated by $\sigma$ with $\sigma\left(\alpha\right)=\xi\alpha.$
Moreover, the discriminant of $K\left(\sqrt[n]{a}\right)$ over $K$ divides
$n^{n}\cdot a^{n-1};$ $p$ is unramified if $p$ $\nmid$ $na.$
Let $A\neq 0$ be a central simple algebra over a field $K.$ We recall that if
$A$ is a finite-dimensional algebra, then $A$ is a division algebra if and
only if $A$ is without zero divisors ($x\neq 0,y\neq 0\Rightarrow xy\neq 0$).
$A$ is called split by $K$ if $A$ is isomorphic with a matrix algebra over
$K.$ If $K\subset L$ is a fields extension, we recall that $A$ is called split
by $L$ if $A\otimes_{K}L$ is a matrix algebra over $L$. The Brauer group
(Br($K$), $\cdot$) of $K$ is Br($K$)$=\left\\{\left[A\right]|A\ is\ a\
central\ simple\ K-\ algebra\right\\},$ where, two classes of central simple
algebras are equals $\left[A\right]=\left[B\right]$ if and only if there are
two positive integers $r$ and $s$ such that
$A\otimes_{K}M_{r}\left(K\right)\cong B\otimes_{K}M_{s}\left(K\right).$ The
group operation in Br($K$) is : ”$\cdot$”:
Br($K$)$\times$Br($K$)$\longrightarrow$Br($K$),
$\left[A\right]\cdot\left[B\right]=\left[A\otimes_{K}B\right],$ for
$\left(\forall\right)$ $\left[A\right],\left[B\right]$$\in$Br($K$) (see [Mil;
08], [Ko; 00]). A result due Albert-Brauer-Hasse-Noether says that for any
number field $K,$ the following sequence is exact:
$0\longrightarrow
Br\left(K\right)\longrightarrow\oplus_{v}Br\left(K_{v}\right)\longrightarrow\mathbb{Q}/\mathbb{Z}\longrightarrow
0$
Remark 2.1. ([Led; 05]). Let $n$ be a positive integer, $n\geq 3$ and let
$\xi$ be a primitive $n$-th root of unity. Let $K$ be a field such that
$\xi$$\in$$K,$ $a,b\in$$K^{*}$. If $n$ is prime, then the symbol algebra
$\left(\frac{a,~{}b}{K,\xi}\right)$ is either split either a division algebra.
Theorem 2.1. ([Lin; 12]) (Albert-Brauer-Hasse-Noether). Let $H_{F}$ be a
quaternion algebra over a number field $F$ and let $K$ be a quadratic field
extension of $F.$ Then there is an embedding of $K$ into $H_{F}$ if and only
if no prime of $F$ which ramifies in $H_{F}$ splits in $K.$
Proposition 2.1. ([Ki, Vo; 10]). Let $F$ be a number field and let $K$ be a
field containing $F.$ Let $H_{F}$ be a quaternion algebra over $F.$ Let
$H_{K}=H_{F}\otimes_{F}K$ be a quaternion algebra over $K.$ If $[K:F]=2,$
then $K$ splits $H_{F}$ if and only if there exists an $F$-embedding
$K\hookrightarrow H_{F}.$
3\. Quaternion algebras which split over quadratic fields
Let $p,q$ be two odd prime integers, $p\neq q.$ If a quaternion algebra
$H\left(p,q\right)$ splits over $\mathbb{Q},$ of course it splits over each
algebraic number fields. It is known that if $K$ is an algebraic number field
such that $[K:\mathbb{Q}]$ is odd and $\alpha,\beta$$\in$$\mathbb{Q}^{*},$
then the quaternion algebra $H_{K}\left(\alpha,\beta\right)$ splits if and
only if the quaternion algebra $H_{\mathbb{Q}}\left(\alpha,\beta\right)$
splits (see [Lam; 04]). But, when $[K:\mathbb{Q}]$ is even there are
quaternion algebras $H\left(\alpha,\beta\right)$ which does not split over
$\mathbb{Q},$ but its split over $K.$ For example, the quaternion algebra
$H\left(11,47\right)$ does not split over $\mathbb{Q},$ but it splits over the
quadratic field $\mathbb{Q}\left(i\right)$ (where $i^{2}=-1$).
We want to determine sufficient conditions for a quaternion algebra
$H\left(p,q\right)$ to split over a quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right).$ Let $\mathcal{O}_{K}$ be the ring of
integers of $K.$ Since $p$ and $q$ lie in $\mathbb{Q},$ the problem whether
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ is split reduces to
whether $H_{\mathbb{Q}}\left(p,q\right)$ splits under scalar extension to
$\mathbb{Q}\left(\sqrt{d}\right).$
It is known that, for each prime positive integer
$Br\left(\mathbb{Q}_{p}\right)\cong\mathbb{Q}/\mathbb{Z}$ (the isomorphism is
$inv_{p}:Br\left(\mathbb{Q}_{p}\right)\rightarrow\mathbb{Q}/\mathbb{Z}$) and
for $p=\infty,$ $Br\left(R\right)\cong\mathbb{Z}/2\mathbb{Z}.$
We obtain sufficient conditions for a quaternion algebra $H\left(p,q\right)$
to split over a quadratic field.
Theorem 3.1. Let $d\neq 0,1$ be a free squares integer, $d\not\equiv 1$ (mod
$8$) and let $p,q$ be two prime integers, $q\geq 3,$ $p\neq q.$ Let
$\mathcal{O}_{K}$ be the ring of integers of the quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right)$ and $\Delta_{K}$ be the discriminant of
$K.$ Then, we have:
i) if $p\geq 3$ and the Legendre symbols
$\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ splits;
ii) if $p=2$ and the Legendre symbol $\left(\frac{\Delta_{K}}{q}\right)\neq
1,$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right)$ splits.
Proof. i) Applying Albert-Brauer-Hasse-Noether theorem, we obtain the
following description of the Brauer group of $\mathbb{Q}$ and of the Brauer
group of the quadratic field $\mathbb{Q}\left(\sqrt{d}\right).$
${0}$${Br\left(\mathbb{Q}\right)}$${\oplus_{p}Br\left(\mathbb{Q}_{p}\right)\cong\left(\oplus_{p}\mathbb{Q}/\mathbb{Z}\right)\oplus\mathbb{Z}/2\mathbb{Z}}$${\mathbb{Q}/\mathbb{Z}}$${0}$${0}$${Br\left(\mathbb{Q}\left(\sqrt{d}\right)\right)}$${\oplus_{P}Br\left(\mathbb{Q}\left(\sqrt{d}\right)_{P}\right)\cong\left(\oplus_{P}\mathbb{Q}/\mathbb{Z}\right)}$${\mathbb{Q}/\mathbb{Z}}$${0}$$\scriptstyle{\oplus_{p}\varphi_{p}\oplus
0}$
where $\varphi_{p}$ is the multiplication by $2$ when there is single
$P$$\in$Spec$\left(\mathcal{O}_{K}\right)$ above the ideal $p\mathbb{Z}$ i.e.
$p\mathbb{Z}$ is inert in $\mathcal{O}_{K}$ or $p\mathbb{Z}$ is ramified in
$\mathcal{O}_{K},$ respectively $\varphi_{p}$ is the diagonal map
$\mathbb{Q}/\mathbb{Z}\rightarrow\mathbb{Q}/\mathbb{Z}\oplus\mathbb{Q}/\mathbb{Z}$
if there are two primes $P,$ $P^{{}^{\prime}}$ of $\mathcal{O}_{K}$ above
$p\mathbb{Z}$ i.e. $p\mathbb{Z}$ is totally split in $\mathcal{O}_{K}.$ Using
this description we determine sufficient conditions for a quaternion algebra
$H\left(p,q\right)$ to split over a quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right).$
It is known that $\Delta_{K}=d$ (if $d\equiv 1$ (mod $4$)) or $\Delta_{K}=4d$
(if $d\equiv 2,3$ (mod $4$)). Since $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ it results
$\left(\frac{d}{p}\right)=-1$ or $\left(\frac{d}{p}\right)=0,$ respectively
$\left(\frac{d}{q}\right)=-1$ or $\left(\frac{d}{q}\right)=0.$ Applying the
theorem of decomposition of a prime integer $p$ in the ring of integers of a
quadratic field (see for example [Ire, Ros; 90], p. 190), it results that $p$
is ramified in $\mathcal{O}_{K}$ or $p$ is inert in $\mathcal{O}_{K},$
respectively $q$ is ramified in $\mathcal{O}_{K}$ or $q$ is inert in
$\mathcal{O}_{K}.$ So, $p$ and $q$ do not split in $K.$
Let $\Delta$ denote the discriminant of the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right).$
It is known that a prime positive integer $p^{{}^{\prime}}$ ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ if
$p^{{}^{\prime}}|2\Delta$ ([Ko], [Ko; 00]). This implies
$p^{{}^{\prime}}|2pq.$
Since $d\not\equiv 1$ (mod $8$) and the decomposition of $2$ in
$\mathcal{O}_{K}$ (see [Ire, Ros; 90], p. 190), it results that $2$ does not
split in $K.$
From the previously proved and applying Theorem 2.1 and Proposition 2.1, it
results that the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ splits.
ii) Let $p^{{}^{\prime}}$ be a prime positive integer which ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right).$ In this case the
condition $p^{{}^{\prime}}|2\Delta$ implies $p^{{}^{\prime}}|2q.$ With similar
reasoning as i) we get that the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(2,q\right)$ splits.
Remark 3.1. The conditions $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ from Theorem 3.1 are not necessary
for the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(q,p\right)$ splits. For example, if
$d=-1,$ the conditions $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1$ are equivalent to
$p$$\equiv$$q$$\equiv$$3$(mod $4$). We consider the quaternion algebra
$H_{\mathbb{Q}\left(i\right)}\left(5,29\right),$ so $p=5$$\equiv$$1$(mod $4$)
and $q=29$$\equiv$$1$(mod $4$). Doing some calculations in software MAGMA, we
obtain that the algebra $H_{\mathbb{Q}\left(i\right)}\left(5,29\right)$
splits. Analogous, for $p=5$$\equiv$$1$(mod $4$) and $q=19$$\equiv$$3$(mod
$4$), we obtain that the algebra
$H_{\mathbb{Q}\left(i\right)}\left(5,19\right)$ splits. Another example: if
$d=3,$ $p=7,q=47.$ We have $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$ but
$\left(\frac{\Delta_{K}}{q}\right)=1.$ However the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{3}\right)}\left(7,47\right)$ splits. Another remark
is that the quaternion algebra $H_{\mathbb{Q}}\left(7,47\right)$ does not
split.
We wonder what happens with a quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(p,q\right)$ from Theorem 3.1 when
instead of $p$ or $q$ we consider an arbitrary integer $\alpha.$ Immediately
we obtain the following result:
Corollary 3.1. Let $d\neq 0,1$ be a free squares integer, $d\not\equiv 1$ (mod
$8$) and let $\alpha$ be an integer and $p$ be an odd prime integer. Let
$\mathcal{O}_{K}$ be the ring of integers of the quadratic field
$K=\mathbb{Q}\left(\sqrt{d}\right)$ and $\Delta_{K}$ be the discriminant of
$K.$ If the Legendre symbols $\left(\frac{\Delta_{K}}{p}\right)\neq 1,$
$\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ for each odd prime divisor $q$ of
$\alpha$ then, the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right)$ splits.
Proof. We want to determine the primes $p^{{}^{\prime}}$ which ramifies in
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right),$ i.e the primes
$p^{{}^{\prime}}$ with the property $p^{{}^{\prime}}|2\Delta.$ This implies
$p^{{}^{\prime}}|2\alpha\cdot p.$ Since $\left(\frac{\Delta_{K}}{p}\right)\neq
1,$ $\left(\frac{\Delta_{K}}{q}\right)\neq 1,$ for each odd prime divisor $q$
of $\alpha,$ using a reasoning similar with that of Theorem 3.1, we get that
such primes does not exist, so the quaternion algebra
$H_{\mathbb{Q}\left(\sqrt{d}\right)}\left(\alpha,p\right)$ splits.
4\. Symbol algebras of degree $n$
In the paper [Sa; 16] we found a class of division quaternion algebras over
the quadratic field $\mathbb{Q}\left(i\right)$ ([Sa; 16], Th. 3.1) and a class
of division symbol algebras of degree $q$ (where $q$ is an odd prime positive
integer) over a $p$\- adic field or over a cyclotomic field ([Sa; 16], Th.
3.2). Here we generalize theorem 3.2 from [Sa; 16], when $A$ is a symbol
algebra over the $n$-th cyclotomic field, where $n$ is a positive integer,
$n\geq 3.$
Theorem 4.1. Let $n$ be a positive integer, $n\geq 3,$ $p$ be a prime positive
integer such that $p\equiv 1$ (mod $n$), $\xi$ be a primitive root of order
$n$ of unity and let $K=\mathbb{Q}\left(\xi\right)$ be the $n$ th cyclotomic
field. Then there is an integer $\alpha$ not divisible by $p,$ $\alpha$ is not
a $l$ power residue modulo $p,$ for $\left(\forall\right)$
$l$$\in$$\mathbb{N},$ $l|n$ and for every such an $\alpha$, we have:
i) if $A$ is the symbol algebra $A=\left(\frac{\alpha,p}{K,\xi}\right),$ then
$A\otimes_{K}\mathbb{Q}_{p}$ is a non-split algebra over $\mathbb{Q}_{p};$
ii) the symbol algebra $A$ is a non-split algebra over $K.\vskip 3.0pt plus
1.0pt minus 1.0pt$
Proof. i) Let be the homomorphism
$f:\mathbb{F}_{p}^{\ast}\rightarrow\mathbb{F}_{p}^{\ast},$
$f\left(x\right)=x^{n}.$ Since $p\equiv 1$ (mod $n$), it results
$Ker\left(f\right)=\left\\{x\in{F}_{p}^{\ast}|x^{n}\equiv 1\ (mod\
p)\right\\}$ is non -trivial, so $f$ is not injective. So, $f$ is not
surjective. It results that there exists $\overline{\alpha}$ (in
$\mathbb{F}_{p}^{\ast},$) which does not belongs to
$\left(\mathbb{F}_{p}^{\ast}\right)^{n}.$ Let $\beta$ be an $n$ th root of
$\alpha$ (modulo $p$). Since $\alpha$ is not a $l$ power residue modulo $p,$
for $\left(\forall\right)$ $l$$\in$$\mathbb{N},$ $l|n,$ it results that the
extension of fields
$\mathbb{F}_{p}\left(\overline{\beta}\right)/\mathbb{F}_{p}$ is a cyclic
extension of degree $n.$ Applying a consequence of Hensel’s lemma (see for
example [Al,Go;99]) and the fact that $p\equiv 1$ (mod $n$), it results that
$\mathbb{Q}_{p}$ contains the $n$-th roots of the unity, therefore
$\mathbb{Q}\left(\xi\right)\subset\mathbb{Q}_{p}.$ Let the symbol algebra
$A\otimes_{K}\mathbb{Q}_{p}=\left(\frac{\alpha,p}{\mathbb{Q}_{p},\xi}\right).$
Applying Lemma 2.1, it results that the extension
$\mathbb{Q}_{p}\left(\sqrt[n]{\alpha}\right)/\mathbb{Q}_{p}$ is a cyclic
unramified extension of degree $n,$ therefore a norm of an element from this
extension can be a positive power of $p,$ but cannot be $p.$ According to a
criterion for splitting of the symbol algebras (see Corollary 4.7.7, p. 112
from [Gi, Sz; 06]), it results that
$\left(\frac{\alpha,p}{\mathbb{Q}_{p},\xi}\right)$ is a non-split algebra.
ii) Applying i) and the fact that $K\subset\mathbb{Q}_{p},$ it results that
$A$ is a non-split algebra.
Remark 4.1. Although Theorem 4.1 is the generalization of Theorem 3.2 from
[Sa; 16] for symbol algebras of degree $n,$ there are some differences between
these two theorems, namely:
\- One of the conditions of the hypothesis of Theorem 3.2 from [Sa; 16] is:
$\alpha$ is not a $q$ power residue modulo $p.$ With a similar condition in
the hypothesis of Theorem 4.1, namely: $\alpha$ is not a $n$ power residue
modulo $p,$ Theorem 4.1 does not work. We give an example in this regard: let
$p=7,$ $n=6,$ $\alpha=2.$ $2$ is not a $6$ power residue modulo $7,$ but $2$
is a quadratic residue modulo $7.$ Let $\beta$ be an $6$ th root of $\alpha$
(modulo $7$). We obtain that the polynomial $Y^{6}-\overline{2}$ is not
irreducible in $\mathbb{F}_{7}\left[Y\right].$ We have
$Y^{6}-\overline{2}=\left(Y^{3}-\overline{3}\right)\cdot\left(Y^{3}+\overline{3}\right)$
(in $\mathbb{F}_{7}\left[Y\right]$). So, the extension of fields
$\mathbb{F}_{7}\subset\mathbb{F}_{7}\left(\overline{\beta}\right)$ has not the
degree $n=6.$ For this reason, in the hypothesis of Theorem 4.1 we put the
condition: $\alpha$ is not a $l$ power residue modulo $p,$ for
$\left(\forall\right)$ $l$$\in$$\mathbb{N},$ $l|n;$
\- In Theorem 3.2 from [Sa; 16] we proved that $A\otimes_{K}\mathbb{Q}_{p}$ is
a non-split symbol algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a non-
split symbol algebra over $K$) and applying Remark 2.1. this is equivalent to
$A$ is a division symbol algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a
division symbol algebra over $K$). But, Remark 2.1 holds iff $n$ is prime. For
this reason, the conclusion of Theorem 4.1 is: $A$ is a non-split symbol
algebra over $\mathbb{Q}_{p}$ (respectively $A$ is a non-split symbol algebra
over $K$).
Conclusions. In the last section of the paper, we found a class of non-split
symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$)
over a $p-$ adic field, respectively over a cyclotomic field. In a further
research we intend to improve Theorem 4.1 from this paper, for to find a class
of division symbol algebras of degree $n$ (where $n$$\in$$\mathbb{N}^{*}$,
$n\geq 3$) over a cyclotomic field.
References
[Al, Go; 99] V. Alexandru, N.M. Gosoniu, Elements of Number Theory (in
Romanian), Ed. Bucharest University, 1999.
[Ca, Fr; 67] J. W. S. Cassels, A. Fr$\ddot{o}$hlich (editors), Algebraic
Number Theory (Proceedings of an instructional conference organized by the
London Mathematical Society), Academic Press, 1967.
[Gi, Sz; 06] P. Gille, T. Szamuely, Central Simple Algebras and Galois
Cohomology, Cambridge University Press, 2006.
[Ire, Ros; 90] K. Ireland, M. Rosen, A classical introduction to modern number
theory, Springer-Verlag, 1990.
[Ki, Vo; 10] M. Kirschmer, J. Voight, Algorithmic enumeration of ideal classes
for quaternion orders, SIAM J. Comput. (SICOMP) 39 (2010), no. 5, 1714-1747.
[Ko] D. Kohel, Quaternion algebras, echidna.maths.usyd.edu.au/kohel/alg/doc/
AlgQuat.pdf
[Ko; 00] D. Kohel, Hecke module structure of quaternions, Proceedings of Class
Field Theory - Centenary and Prospect (Tokyo, 1998), K. Miyake, ed., Advanced
Studies in Pure Mathematics, 30, 177-196, 2000.
[Lam; 04] T. Y. Lam, Introduction to Quadratic Forms over Fields, American
Mathematical Society, 2004.
[Led; 05] A. Ledet, Brauer Type Embedding Problems , American Mathematical
Society, 2005.
[Lin; 12] B. Linowitz, Selectivity in quaternion algebras, Journal of Number
Theory 132 (2012), pp. 1425-1437.
[Mil; 08] J.S. Milne, Class Field Theory, http://www.math.lsa.umich.edu/
jmilne.
[Sa; 16] D. Savin, About division quaternion algebras and division symbol
algebras, Carpathian Journal of Mathematics, 32(2) (2016), p. 233-240.
[Vo; 10] J. Voight, The Arithmetic of Quaternion Algebras. Available on the
author’s website: http://www.math.dartmouth.edu/ jvoight/ crmquat/book/quat-
modforms-041310.pdf, 2010.
Diana SAVIN,
Faculty of Mathematics and Computer Science, Ovidius University,
Constanta 900527, Bd. Mamaia no.124, România
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
|
# Leveraging Large Language Model and Story-Based Gamification in Intelligent
Tutoring System to Scaffold Introductory Programming Courses: A Design-Based
Research Study
Chen Cao<EMAIL_ADDRESS>0000-0003-4368-0336 University of
SheffieldUnited KingdomS10 2TN
(2023)
###### Abstract.
Programming skills are rapidly becoming essential for many educational paths
and career opportunities. Yet, for many international students, the
traditional approach to teaching introductory programming courses can be a
significant challenge due to the complexities of the language, the lack of
prior programming knowledge, and the language and cultural barriers. This
study explores how large language models and gamification can scaffold coding
learning and increase Chinese students’ sense of belonging in introductory
programming courses. In this project, a gamification intelligent tutoring
system was developed to adapt to Chinese international students’ learning
needs and provides scaffolding to support their success in introductory
computer programming courses.
My research includes three studies: a formative study, a user study of an
initial prototype, and a computer simulation study with a user study in
progress. Both qualitative and quantitative data were collected through
surveys, observations, focus group discussions and computer simulation. The
preliminary findings suggest that GPT-3-enhanced gamification has great
potential in scaffolding introductory programming learning by providing
adaptive and personalised feedback, increasing students’ sense of belonging,
and reducing their anxiety about learning programming.
††conference: IUI’23: 28th International Conference on Intelligent User
Interfaces; March 27 – 31, 2023; Sydney, Australia††booktitle: IUI’23: 28th
International Conference on Intelligent User Interfaces, March 27 – 31, 2023,
Sydney, Australia††journalyear: 2023††copyright: rightsretained††conference:
28th International Conference on Intelligent User Interfaces; March 27–31,
2023; Sydney, NSW, Australia††booktitle: 28th International Conference on
Intelligent User Interfaces (IUI ’23 Companion), March 27–31, 2023, Sydney,
NSW, Australia††doi: 10.1145/3581754.3584111††isbn: 979-8-4007-0107-8/23/03
## 1\. Problem Statement
Computing and technology are increasingly ubiquitous and have become a
necessary part of many educational paths, professional opportunities, and
industries (Pedro et al., 2019). Consequently, the importance of programming
knowledge and coding skills has grown significantly in recent years.
Introductory programming courses, often referred to as computer science 1
(CS1) courses are designed to introduce students to the fundamentals of
programming and coding (Becker and Quille, 2019). However, these courses can
be challenging for many international students, who may lack prior programming
expertise and are confronted with a language and cultural barriers(Khanal and
Gaulee, 2019).
Recent years have also seen an increased number of Chinese international
students enrolling in universities in the UK(de Wit, 2020). Despite the clear
benefits of international student enrollment, Chinese international students
can often feel isolated in their new environment and struggle to integrate
into university life(Chen and Zhou, 2019). In the context of programming
education, Chinese international students often face hardship due to a lack of
prior exposure to coding concepts and language (Alaofi and Russell, 2022). As
a result, they are more likely to experience anxiety and a feeling of low
belonging. However, previous research showed bias and stereotypes when
describing Chinese students’ learning behaviours in global higher education
(Heng, 2018). Chinese international students are often perceived as passive
and reluctant learners when adapting to British educational systems (Zhu and
O’Sullivan, 2022). This stereotype, however, fails to take into account the
cultural and educational factors driving these behaviours.
At the same time, gamification has seen a surge in popularity in the
educational sector. It has been used as a tool to engage students and
encourage learning in a range of contexts (Welbers et al., 2019). This
presents an opportunity to investigate the potential of gamification to
scaffold coding learning and increase the sense of belonging among Chinese
international students in introductory programming courses. Additionally, the
use of educational technologies, such as intelligent tutoring systems in
educational settings has been growing steadily in recent years, as an
increasing number of educators recognize the potential of this approach to
engage and motivate learners (Szymkowiak et al., 2021). An intelligent
tutoring system is particularly attractive due to its ability to personalise
the learning experience, offering tailored activities to meet the needs of
diverse learners (Shemshack et al., 2021). In addition, an intelligent
tutoring system can provide feedback in a timely and effective manner,
offering learners the opportunity to improve their skills in a supportive and
engaging environment (AlShaikh and Hewahi, 2021).
The research background of this thesis is rooted in the idea that AI-enhanced
gamification can be used to scaffold learning and increase belonging among
Chinese international students in introductory programming courses. AI-
enhanced gamification combines the use of gamification techniques with AI-
driven intelligent tutoring systems to create a personalised learning
experience. This approach encourages learners to engage in the learning
process and offers them the opportunity to practise and improve their coding
skills in a supportive and engaging environment.
This research is motivated by the need to bridge the gaps in programming
knowledge between Chinese international students and their domestic peers. It
is also driven by the urgent need to create an inclusive learning environment
that is accessible and welcoming to all students. The research findings could
have a considerable impact on the teaching and learning of programming for
Chinese international students. It has the potential to provide insight into
how AI-enhanced gamification can be used to make the learning process easier
and more effective for these students, as well as to foster a sense of
belonging. The findings could inform instructors, administrators, and
policymakers of the most effective strategies for teaching and learning to
program for this population. This study can also be expanded to support other
underrepresented student groups (such as female STEM students and other
international students from different countries and cohorts) who also need a
sense of belonging to their peers, faculty, and subject-related careers.
## 2\. Research aim and questions
The main focus of this research project is to explore the potential of AI-
enhanced gamification to scaffold coding learning and increase the sense of
belonging among Chinese international students in introductory programming
courses. Specifically, the focus will be on designing, evaluating, and
refining the use of AI-enhanced gamification to improve learning outcomes and
increase motivation among Chinese international students in introductory
programming courses.
The research project is framed within the context of design-based research,
which emphasises the importance of designing, implementing, and evaluating a
learning environment, with the goal of optimising the learning experience. The
research project is guided by two research questions:
* •
RQ1: what are the challenges that Chinese international students are facing
regarding developing a sense of belonging and code learning in introductory
programming courses?
* •
RQ2: how can AI and gamification be used to scaffold coding learning and
increase the sense of belonging among Chinese international students in
introductory programming courses?
## 3\. Related work
This research is situated within the larger context of educational technology
and game-based learning. The use of technology in education has become
increasingly popular, with a growing number of educators recognising the
potential of this approach to engage and motivate learners (Szymkowiak et al.,
2021). This study specifically focuses on the application of AI-enhanced
gamification to scaffold coding learning among Chinese international students.
There has been a considerable amount of research examining the challenges and
barriers associated with teaching introductory programming courses, e.g.
(Alam, 2022). These studies have highlighted the need for more effective
pedagogical approaches to engage learners in CS1 courses. Several researchers
have proposed the use of game-based learning techniques such as serious games
and simulations to increase engagement and motivation among students
(Papadakis and Kalogiannakis, 2019) . Gamification combines elements of gaming
(e.g., points, rewards, leaderboards) with traditional educational activities
to create an engaging learning experience.
The use of intelligent tutoring systems (ITS) in programming education has
also been studied extensively. Research has shown that ITS can improve student
performance and reduce cognitive load in programming courses. For example,
Grenander et al. (Grenander et al., 2021) proposed an AI-enhanced educational
system that was designed to provide personalised feedback based on individual
learners’ needs and evaluated its effectiveness using deep discourse analysis.
Similarly, Eguchi (Eguchi et al., 2021) investigated the use of AI-enhanced
games to support STEM education for children with visual impairments. They
found that the AI-enhanced game had a positive impact on engagement and
motivation among participants. Furthermore, ITS can provide personalised
instruction and targeted remediation, which can be particularly beneficial for
those who lack prior experience in programming. In particular, GPT-3
(Generative Pre-trained Transformer 3), an advanced language model developed
by OpenAI, has been used to improve the performance of ITSs by providing
natural language understanding capabilities(Tack and Piech, 2022).
This research project is also informed by recent studies on belonging in
higher education. It has been argued that belonging is an important factor
when it comes to understanding and supporting the learning experiences of
international students (Cureton and Gravestock, 2019). Studies have also shown
that international students often face difficulties in developing a sense of
belonging due to language and cultural barriers (Cena et al., 2021).
## 4\. Research method
This research project applies design-based research (DBR) as its
methodological approach. DBR is an iterative process that emphasises the
importance of designing, implementing, and evaluating educational technology
to optimise the learning experience. This approach has been widely used in
educational settings to investigate the effectiveness of new technologies and
approaches in engaging and motivating learners. The use of DBR provides a
unique opportunity to combine theoretical insights with practical
implementation to create meaningful interventions that are responsive to the
needs of diverse learners.
The research is divided into three phases: 1) a survey to identify Chinese
students’ needs and the challenges they met regarding a sense of belonging and
learning experience in introductory programming courses; 2) a design probe
with a story-based gamification prototype to increase Chinese students’ sense
of belonging and improve their learning experience in CS1 courses with user
evaluation; 3) the development and systematic evaluations of an intelligent
tutoring system leveraging large language model and gamification features for
CS1 courses.
The study used a mixed-methods approach, incorporating both quantitative and
qualitative data. The quantitative data were collected through a survey
measuring students’ sense of belonging, academic performance, and academic
emotions, and a computer simulation study evaluating the performances of the
ITS. The qualitative data were collected through participatory observations
and focus group interviews to explore students’ experiences and perceptions in
more depth. The data were analyzed using descriptive statistics, thematic
analysis, and computational analysis.
## 5\. Preliminary results
### 5.1. Study 1: Formative study
The initial survey was conducted as a formative study in the first semester of
the 21/22 academic year. Based on the questionnaire with 57 Chinese
international students and in-depth interviews with nine of them, the study
found a number of unique challenges faced by the participants in the CS1
courses. Chinese students generally found it difficult to cope with the
demands of the programming course. A number of factors were identified as
contributing to this difficulty, including the challenges to adapt to new
teaching methods emphasizing independent thinking, critical thinking, and
innovative learning; less interaction with teachers and classmates;
disconnection between the knowledge and real-life cases and not receiving
enough academic support with timely help and real-time feedback. As a result,
they have low engagement in classes, low retention and negative academic
emotions. One of the key factors that have been identified as contributing to
these unsmooth academic transition experiences is a lack of intrinsic
motivation, especially a lack of sense of belonging.
### 5.2. Study 2: A prototype of story-based ITS
The second study designed, deployed and evaluated a prototype of a story-based
gamification design on a learning management system Blackboard. Based on the
initial findings from the survey, the prototype adopted a set of gamification
features, including animated trailers, story-telling, role-play, teamwork,
points, leaderboards and feedback. The study was conducted over a period of
two weeks with a total of 34 Chinese students enrolled in the introductory
programming course INF6032 Big Data Analytics, which is one of the core
modules of the MSc in Data Science programme at the University of Sheffield.
The practical session in week 6 was set in the context of airline companies’
survival during the pandemic. In week 7, the story was about solving a
criminal case. The narrative storytelling of each session with real-life cases
was designed to make students feel more connected to the programme.
The findings from questionnaires (n=32), focus groups (n=32) and participatory
observation revealed that Chinese students generally had positive perceptions
of the effect of story-based gamification design on their sense of belonging
in the introductory programming course. The majority of students reported
feeling more motivated to learn, more engaged in the course, and more
connected to their classmates as a result of the story-based gamification
design. In addition, the findings suggested that story-based gamification
design had a positive effect on Chinese students’ sense of belonging in the
introductory programming course. Most students reported feeling a greater
sense of belonging in the course after the story-based gamification design was
implemented. It was also found that Chinese students have some unique cultural
needs which should be considered in the story-based gamification design. These
findings suggest that story-based gamification design can be an effective
means of improving Chinese students’ sense of belonging in the introductory
programming course.
### 5.3. Study 3: A model of the story-based ITS empowered by GPT-3
The reflection upon the findings from the previous study encouraged the
development of the current model of the story-based intelligent tutoring
system (ITS). The ITS empowered by GPT-3 was designed to better fit the needs
of individual learners, and increase their sense of belonging to the class,
institution and career-related subject, in order to further foster inclusion
in higher education.
The design of the user interface is inspired by the English TV series Sherlock
with abundant modern English elements, which is ideal to engage international
students coming to the UK for higher education. The user interface of the ITS
was built on a popular open-source template111Bootstrap 4 UI kit,
https://demos.creative-tim.com/now-ui-kit-react/#/index?ref=nukr-github-readme
in React, a mainstream JavaScript library for developing front-ends. The
overall design principles of the web app are inspired by the famous detective,
Sherlock Holmes. The colour scheme, typography and navigation are all based on
the idea of the Mind Palace, with a focus on the dark blue and white colours
to convey a feeling of sophistication and intelligence.
The development of the learning system consists of four main components: the
instructional content, the gamification mechanism design, the user interface,
and the generative language model. The learning content consists of the
introduction of programming knowledge, demo explanations and exercises to
provide the learning materials to the users. The game engine was designed to
provide a fun and engaging learning experience. It consisted of a set of
gamification elements, including alternative reality, points, badges,
personalized feedback and encouragements, levels and challenges, an avatar, a
progress bar and an exploratory word cloud. The user interface is responsible
for displaying the learning content and gamification mechanisms to the users
and accepting user input. The generative language model is used for providing
AI capabilities such as Q&A and explaining code to the system.
The system is designed to be a web-based application. It consists of three
main components, which are the front end, the back end and the GPT-3 platform.
The front end used React library to develop the user interface and enable
interactions. The back end is responsible for data storage, processing and
retrieval. In this system, Firebase is applied to connect with the interface
and synchronise data. All the states (user behaviours and inputs) in the front
end were centralised with Redux and synchronised to Firebase, where all the
user data, such as the user’s avatar, learning progress and chat history with
the AI conversational agents were stored for further analysis.
GPT-3 was used in this system to train the AI model for the intelligent
agents. The agents were designed to interact with the users, and provide
guidance and support throughout the learning process. In addition, the agents
were also responsible for marking the user’s progress and giving feedback.
There were four chatbots providing support to students with different
functions, including the intelligent tutoring bot Sherlock, the peer and
critical thinking bot Watson, the career engagement bot Inspector Lestrade and
the emotion support bot Mrs Hudson. When the student asked a question,
Sherlock answered it first. Then Watson, Inspector Lestrade and Mrs Hudson
generated follow-up questions to continue the conversation, in order to foster
inquiry-based learning and improve students’ computational thinking and
understanding of job market needs.
In the interface of marking and feedback, students can submit the codes for
quizzes and get real-time and tailored feedback about their answers. The model
first extracts the code snippet from the user input and then generates an
explanation for the code snippet by calling the GPT-3 API with the code
snippet and a prompt as input. The prompt is a natural language question that
can be used to generate an explanation for the code snippet. The code-
explaining module uses a set of predefined prompts to generate explanations
for the code snippets. The prompts are selected to cover a wide range of
programming concepts.
A computer simulation study and a pilot user study were conducted to evaluate
the ITS. By using prompt programming, a dataset containing 360 rounds of
simulated questions that students may ask in the class and answers were
simulated based on the learning objectives and instructors’ reflections. The
findings from semantic similarity analysis, topic modelling and sentiment
analysis indicated that GPT-3 empowered agents performed well in providing
feedback and having insightful conversations, but the quality of answers
depend on the form and level of the questions. The ITS was also piloted with
different stakeholders, including potential users, instructors, practitioners
and researchers to assess the usability, acceptability and effectiveness of
the system. Findings from the pilot study indicated that the system performs
well in terms of usability and acceptability. Most participants reported
feeling positive about their experience with the intelligent tutoring system,
with many saying they were impressed and felt supported by the AI agents. They
also commented positively on the gamification elements in the system, such as
the intriguing storyline of Sherlock.
## 6\. Implications and future work
The current studies suggested that GPT-3 as a large language model trained on
a vast amount of text data is particularly well-suited to answering questions
at the remembering and understanding levels of Bloom’s taxonomy. These levels
involve recalling and comprehending information, which is a strength of large
language models like GPT-3. IT also performed well in answering questions at
the higher levels of the taxonomy, such as analysis, synthesis, and evaluation
if the questions are well-formed. Generally, GPT-3’s ability to answer
questions in the educational scenario of CS1 courses is promising but limited
by its lack of domain-specific knowledge and expertise.
Despite the promising results of this study, there are still a number of areas
for further research and development. Firstly, more detailed studies should be
conducted to investigate the long-term impact of AI-enhanced gamification on
students’ learning outcomes and sense of belonging in introductory programming
courses. Secondly, contextual factors such as culture, language and prior
knowledge should be taken into account when designing AI-enhanced gamification
systems to better support Chinese international students. Thirdly, more
sophisticated machine learning models can be explored to improve the
performance of AI agents. Finally, more user studies need to be conducted to
explore how AI-enhanced gamification can be used to foster collaboration among
team members, reduce anxiety and facilitate learning transfer.
In order to fully realize the potential of the GPT-3 enhanced ITS to increase
student sense of belonging in higher education, more research must be
conducted in real educational scenarios. More user studies are needed to
assess how GPT-3 affects student academic performance, motivation, and sense
of belonging. In addition, researchers must continue to refine GPT-3’s
capabilities to improve the ITS’s ability to pick up on subtle social cues and
to develop a better understanding of student emotions and motivations.
## References
* (1)
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* Alaofi and Russell (2022) Suad Alaofi and Seán Russell. 2022. The Influence of Foreign Language Classroom Anxiety on Academic Performance in English-based CS1 Courses. _ACM International Conference Proceeding Series_. https://doi.org/10.1145/3555009.3555020
* AlShaikh and Hewahi (2021) Fatema AlShaikh and Nabil Hewahi. 2021. Ai and machine learning techniques in the development of Intelligent Tutoring System: A review. In _2021 International Conference on innovation and Intelligence for informatics, computing, and technologies (3ICT)_. IEEE, 403–410.
* Becker and Quille (2019) Brett A Becker and Keith Quille. 2019. 50 years of cs1 at sigcse: A review of the evolution of introductory programming education research. In _Proceedings of the 50th acm technical symposium on computer science education_. 338–344.
* Cena et al. (2021) Elida Cena, Stephanie Burns, and Paul Wilson. 2021\. Sense of Belonging, Intercultural and Academic Experiences among International Students at a University in Northern Ireland. _Journal of International Students_ (2021).
* Chen and Zhou (2019) Jia Chen and George Zhou. 2019. Chinese international students’ sense of belonging in North American postsecondary institutions: A critical literature review. _Brock Education Journal_ 28, 2 (2019), 48–63.
* Cureton and Gravestock (2019) Debra Cureton and Phil Gravestock. 2019. ‘We Belong’: differential sense of belonging and its meaning for different ethnicity groups in higher education. _Compass: Journal of Learning and Teaching_ (2019).
* de Wit (2020) Hans de Wit. 2020\. Internationalization of Higher Education: The Need for a More Ethical and Qualitative Approach. _Journal of International Students_ 10 (2020).
* Eguchi et al. (2021) Amy Eguchi, Hiroyuki Okada, and Yumiko Muto. 2021\. Contextualizing AI education for K-12 students to enhance their learning of AI literacy through culturally responsive approaches. _KI-Künstliche Intelligenz_ 35, 2 (2021), 153–161.
* Grenander et al. (2021) Matt Grenander, Robert Belfer, Ekaterina Kochmar, Iulian V Serban, François St-Hilaire, and Jackie C K Cheung. 2021. Deep Discourse Analysis for Generating Personalized Feedback in Intelligent Tutor Systems. https://github.com/UKPLab/sentence-transformers
* Heng (2018) Tang T. Heng. 2018\. Different is not deficient: contradicting stereotypes of Chinese international students in US higher education. _Studies in Higher Education_ 43 (1 2018), 22–36. Issue 1. https://doi.org/10.1080/03075079.2016.1152466
* Khanal and Gaulee (2019) Jeevan Khanal and Uttam Gaulee. 2019. Challenges of international students from pre-departure to post-study: A literature review. _Journal of International Students_ 9, 2 (2019), 560–581.
* Papadakis and Kalogiannakis (2019) Stamatios Papadakis and Michail Kalogiannakis. 2019. Evaluating the effectiveness of a game-based learning approach in modifying students’ behavioural outcomes and competence, in an introductory programming course. A case study in Greece. _International Journal of Teaching and Case Studies_ 10, 3 (2019), 235–250.
* Pedro et al. (2019) Francesc Pedro, Miguel Subosa, Axel Rivas, and Paula Valverde. 2019. Artificial intelligence in education: Challenges and opportunities for sustainable development. (2019).
* Shemshack et al. (2021) Atikah Shemshack, Jonathan Michael Spector, et al. 2021\. A comprehensive analysis of personalized learning components. _Journal of Computers in Education_ 8, 4 (2021), 485–503.
* Szymkowiak et al. (2021) Andrzej Szymkowiak, Boban Melović, Marina Dabić, Kishokanth Jeganathan, and Gagandeep Singh Kundi. 2021\. Information technology and Gen Z: The role of teachers, the internet, and technology in the education of young people. _Technology in Society_ 65 (2021), 101565.
* Tack and Piech (2022) Anaïs Tack and Chris Piech. 2022. The AI Teacher Test: Measuring the Pedagogical Ability of Blender and GPT-3 in Educational Dialogues. _ArXiv_ abs/2205.07540 (2022).
* Welbers et al. (2019) Kasper Welbers, Elly A Konijn, Christian Burgers, Anna Bij De Vaate, Allison Eden, and Britta C Brugman. 2019. Gamification as a tool for engaging student learning: A field experiment with a gamified app. _E-Learning and Digital Media_ 16, 2 (2019), 92–109.
* Zhu and O’Sullivan (2022) Haiping Zhu and Helen O’Sullivan. 2022. Shhhh! Chinese students are studying quietly in the UK. _Innovations in Education and Teaching International_ 59, 3 (2022), 275–284.
|
tcb@breakable
# Early Classification of Time Series:
Taxonomy of Methods and Extensive Benchmark
Aurélien Renault<EMAIL_ADDRESS>
Orange Innovation, Châtillon, France
AgroParisTech, Palaiseau, France
Alexis Bondu<EMAIL_ADDRESS>
Orange Innovation, Châtillon, France
Antoine Cornuéjols<EMAIL_ADDRESS>
AgroParisTech, Palaiseau, France
Vincent Lemaire<EMAIL_ADDRESS>
Orange Innovation, Lannion, France
(September 2023)
###### Abstract
In many situations, the measurements of a studied phenomenon are provided
sequentially, and the prediction of its class needs to be made as early as
possible so as not to incur too high a time penalty, but not too early and
risk paying the cost of misclassification. This problem has been particularly
studied in the case of time series, and is known as Early Classification of
Time Series (ECTS). Although it has been the subject of a growing body of
literature, there is still a lack of a systematic, shared evaluation protocol
to compare the relative merits of the various existing methods. This document
begins by situating these methods within a principle-based taxonomy. It
defines dimensions for organizing their evaluation, and then reports the
results of a very extensive set of experiments along these dimensions
involving nine state-of-the art ECTS algorithms. In addition, these and other
experiments can be carried out using an open-source library in which most of
the existing ECTS algorithms have been implemented (see https://github.com/ML-
EDM/ml_edm).
## 1 Introduction
In hospital emergency rooms (?), in the control rooms of national or
international power grids (?), in government councils assessing critical
situations, there is a time pressure to make early decisions. On the one hand,
the longer a decision is delayed, the lower the risk of making the wrong
decision, as knowledge of the problem increases with time. On the other hand,
late decisions are generally more costly, if only because early decisions
allow one to be better prepared. For example, a cyber-attack that is not
detected quickly enough gives hackers time to exploit the security flaw found.
A number of applications involve making decisions that optimizes a trade-off
between accuracy of the prediction and its earliness. The problem is that
favoring one usually works against the other. Greater accuracy comes at the
price of waiting for more data.
Such a compromise between the Earliness and the Accuracy of decisions has been
particularly studied in the field of Early Classification of Time Series
(ECTS) (?), and introduced by ? (?). ECTS consists in finding the optimal time
to trigger the class prediction of an input time series observed over time.
Successive measurements provide more and more information about the incoming
time series, and ECTS algorithms aim to optimize online the trade-off between
two conflicting objectives, namely, the earliness and accuracy of class
predictions. More formally, we have the following problem.
##### Problem statement:
In the ECTS problem, measurements of an input time series are observed over
time. At time $t$, the incomplete time series
${\mathbf{x}}_{t}\,=\,\langle{x_{1}},\ldots,{x_{t}}\rangle$ is available where
${x_{i}}_{(1\leq i\leq t)}$ denotes the time indexed measurements. These
measurements can be single or multi-valued. The input time series belongs to
an unknown class $y\in{\cal Y}$. The task is to make a prediction
$\hat{y}\in{\cal Y}$ about the class of the incoming time series, at a time
$\hat{t}\in[1,T]$ before the deadline $T$ which corresponds to the length of
the full time series. A misclassification cost is incurred when a prediction
is made, denoted by $\mathrm{C}_{m}(\hat{y}|y)$. Furthermore, there exists a
delay cost $\mathrm{C}_{d}(\hat{t})$ that expresses the time pressure and
encourages early decisions (defined in Section 2.2). The choice of the best
trigger time should optimize a compromise between these costs that move in
opposite direction. The choice of the best triggering time must optimize a
compromise between the two costs, which are moving in opposite directions.
To the best of our knowledge, ? (?) are the earliest explicitly mentioning
“classification when only part of the series are presented to the classifier”.
Since then, several researchers have continued their efforts in this direction
and have published a large number of research articles. A recent and extensive
review of the ETCS approaches can be found in the paper written by ? (?),
including the applications that motivated the researchers to work in this
area, and covering about fifty relevant papers selected from the $213$ papers
found by search engines at the time of this writing.
As pointed out by ? (?), the ECTS problem is a special case of optimal
stopping (?, ?), where the decision to be made concerns both: (i) when to stop
receiving new measurements in order to (ii) predict the class of the incoming
time series.
In the same paper, the ECTS problem has been extended into a more general one,
which consists in optimizing the decision times of Machine Learning (ML)
models in a wide range of settings where data is collected over time. The
authors proposed a set of open research questions to the community, in order
to widen the range of applications that are amenable to the ECTS framework
(i.e. dealing with other learning tasks, other types of data, other
application contexts etc.).
However, despite the growing interest in ECTS over the last twenty years,
there still remains a need for a shared taxonomy of approaches and an agreed
well-grounded evaluation methodology. Here, in particular, we list limits that
hamper a fair comparison of ECTS methods and algorithms:
1. 1.
Costs taken into account for evaluating the performance of the proposed method
are not always clearly stated. It seems natural to distinguish between the
misclassification costs $\mathrm{C}_{m}(\hat{y}|y)$, and the delay cost
$\mathrm{C}_{d}(t)$, and to add them in order to define the cost of making a
decision at time $t$. More generally, the delay cost may depend on the true
class $y$ and the predicted one $\hat{y}$, and a single cost function
$\mathrm{C}_{m}(\hat{y}|y,t)$ integrating misclassification and delay costs
should then be used. For the sake of clarity, we keep the simple notation
which distinguishes both cost functions in the rest of this paper. But in all
cases, it is essential to state the framework used and the associated
evaluation metric.
2. 2.
The performance of the proposed methods should be evaluated against a range of
possible types of cost functions. It is usual to evaluate “by default” the
methods using a $\ell_{0-1}$ loss function that penalizes a wrong
classification by a unity cost, and to consider a linear delay cost function:
$\mathrm{C}_{d}(t)=\lambda\,t$, for a value $\lambda>0$. However, lots of
applications rather involve unbalanced miss-classification costs, and possibly
also non linear delay costs. This is the case, for instance, in maintenance
applications where wrongly not recognizing a critical situation is much more
costly than wrongly predicting a problem and taking steps to fix it, and where
delay cost may rise as a an exponential function of time:
$\mathrm{C}_{d}(t)=\lambda\,e^{t}$ . It is therefore quite important to assess
the adaptability of the methods to various representative problem settings.
3. 3.
The contributions of the various components of a ECTS algorithm should be
clearly delineated. The predominant approach to ECTS is to have a decision
component which is in charge of evaluating the best moment to make the
prediction about the class of the incoming times series, and a classifier one
which makes the prediction itself. In order to fairly compare the triggering
methods, which are at the heart of ECTS, the classifier used should be the
same. We call these methods “separable methods”.
An alternative approach relies on having a system that classifies the incoming
time series $\mathbf{x}_{t}$ at each time with a prediction $\hat{y}$ in the
set {‘postpone decision’, $y_{1},\ldots,y_{N}$} where $N$ is the number of
classes. Therefore, within this approach, a single system decides either to
wait at least one more time step, or to predict a class and stops the process.
In this case, no distinction can be made between a decision component and a
classifier one, and the whole system is evaluated as such. In the spirit of
deep neural networks, we call this type of methods “end-to-end” to underline
the fact that a single learning system is in charge of all operations, here
decision and classification. Of course, this precludes a comparison involving
only the choice of the decision component with other methods.
4. 4.
As with other supervised learning tasks, performance should be compared with
that of “baseline” algorithms. In the case of ECTS tasks, two naive baselines
are: (1) make a prediction as soon as it is allowed, and (2) make a prediction
at $T$, after the entire time series has been observed. In our experiments
reported in Section 4, we have added a third baseline, less simple than the
two afore-mentioned ones, but still too obvious so that, to our knowledge, it
has never been published as an original method. This is a confidence-based
method where a decision is triggered as soon as the confidence for the
likeliest class given $\mathbf{x}_{t}$ is greater than a threshold. (Formally,
let $\hat{y}=\operatorname*{arg\,max}_{y\in\mathcal{Y}}p(y|\mathbf{x}_{t})$,
then a prediction is made (i.e. $\hat{y}$) as soon as
$p(\hat{y}|\mathbf{x}_{t})\geq\varepsilon$, for some threshold
$\varepsilon\in[0,1]$.)
5. 5.
Precautions should be taken when using datasets of training time series to
ensure that no bias enters unwillingly into the training and evaluation
process. A case in point, concerns the normalization often used in time series
datasets. ? (?) have reported that 71% of the reference time series
classification datasets used to evaluate ECTS methods are made up of
z-normalized time series, i.e. with measurements independently modified on
each complete series to obtain a mean of zero and a standard deviation equal
to 1. Clearly, this setting is not applicable in practice, as z-normalization
would require knowledge of the entire incoming time series. In a research
context, previous work has used such training sets to test the proposed
algorithms. As ? (?, ?) note, this preprocessing is irreversible and can
generate a problem for ECTS by introducing a temporal leakage of information.
In order to assess its impact, we report in Section B.5 of Appendix B a
comparison of results for z-normalized and non-normalized time series.
Up until now, it has been difficult to conduct fair comparisons between
competing methods. Often, published performances are based on choices
concerning data sets, the precise performance measure used, hyperparameter
values and evaluation protocols (e.g., the split between training and test
sets) that are not entirely explicit or, in any case, are difficult to
reproduce. This is why we have recoded all the methods, specified a shared
evaluation protocol with variants that can be employed by everyone, and
searched for a collection of data sets that can be widely used to test and
compare new as well as existing methods. We hope this will be a useful
resource for the scientific community working in this field.
As this community shows a growing interest in the ECTS problem, granted by the
increasing number of applications that fall in its range, it is timely (1) to
propose a framework into which to cast the various approaches and thus
indicate avenues for future research, and (2) a well-grounded evaluation
methodology. Specifically, this paper makes the following contributions:
* •
A taxonomy is proposed in Section 2, classifying approaches in the literature
according to their design choices.
* •
Extensive experiments have been performed, meeting the above-mentioned
shortcomings. (1) The experimental protocol in Section 4.1 explicitly defines
the costs used during training and evaluation, and varies the balance between
misclassification and delay costs by using a large range of cost values. (2)
Experiments are performed repeatedly for several types of cost function, i.e.
balanced or unbalanced misclassification cost, and linear or exponential delay
cost (see Sections 4.2 and 4.3) and many intermediate results are available in
the supplementary materials. (3) Ablation and substitution studies are
conducted in Section 4.4 with the aim of evaluating the impact of
methodological choices, such as the choice of classifier, its calibration, or
even z-normalization of training time series. (4) The experiments include
three baseline approaches, rarely considered in the literature, which often
give surprisingly good results. (5) In addition to the reference data used in
the ECTS field, a collection of some thirty non-z-normalized datasets is
proposed and provided to the community.
* •
An open source library is being made available111https://github.com/ML-
EDM/ml_edm in order to enable reproducible experiments, as well as facilitate
the scientific community’s development of future approaches. Particular care
has been taken to ensure the quality of the code, so that this library may be
used to develop real-life applications.
The rest of this paper is organised as follows : The section 2 proposes and
describes a new ECTS taxonomy, the different choices to be made in a well-
founded way when designing an ECTS method and a set of four questions that
need to be answered to make these choices.
Section 3 presents a comprehensive view of the ECTS field, along the suggested
taxonomy and the four questions raised in Section 2. In Section 4 we present
the pipeline developed in order to realize extensive experimentation and we
report the main results obtained for different cost setting. This benchmark is
supported by a library released as Open Source for dissemination and used in
the ECTS research community. Finally Section 5 concludes this paper. Appendix
A lists the datasets used for the experiments, and complementary results are
provided in Appendix B.
## 2 Organizing the ECTS approaches: a taxonomy
The aim of this section is to outline in a principled way the various choices
that need to be made when designing one ECTS method.
#### General form of an ECTS model
An ECTS approach aims at optimizing a trade-off between accuracy and earliness
of the prediction, and thus must be evaluated on this ground. The correctness
of the prediction is measured by the misclassification cost
$\mathrm{C}_{m}(\hat{y}|y)$ where $\hat{y}$ is the prediction and $y$ is the
true class. The time pressure is sanctioned by a delay cost
$\mathrm{C}_{d}(t)$ that is assumed to be positive and, in most applications,
an increasing function of time. We thus consider:
* •
$\mathrm{C}_{m}(\hat{y}|y):{\cal Y}\times{\cal Y}\rightarrow\mathbb{R}$, that
corresponds to the misclassification cost of predicting $\hat{y}$ when the
true class is $y$.
* •
$\mathrm{C}_{d}(t):\mathbb{R}^{+}\rightarrow\mathbb{R}$, the delay cost that,
usually, is a non-decreasing function over time.
An ECTS function involves a predictor $\hat{y}(\mathbf{x}_{t})$, which
predicts the class of an input time series $\mathbf{x}_{t}$ for any
$t\in[1,T]$. The cost incurred when a prediction has been triggered at time
$t$ is given by a loss function
$\mathcal{L}(\hat{y}({\mathbf{x}}_{t}),y,t)=\mathrm{C}_{m}(\hat{y}({\mathbf{x}}_{t})|y)+\mathrm{C}_{d}(t)$.
The best decision time $t^{*}$ is given by:
$t^{\star}=\operatorname*{arg\,min}_{t\in[1,T]}\mathcal{L}(\hat{y}({\mathbf{x}}_{t}),t,y).$
(1)
Let $s^{\star}\in\mathcal{S}$ an optimal ECTS function belonging to a class of
functions $\mathcal{S}$, whose output at time $t$ when receiving
$\mathbf{x}_{t}$ is:
$s^{\star}({\mathbf{x}}_{t})=\left\\{\begin{array}[]{ll}\emptyset&\mbox{if
extra measures are queried;}\\\ y^{\star}=\hat{y}(x_{t^{\star}})&\mbox{when
prediction is triggered at $t=t^{\star}$;}\end{array}\right.$ (2)
ECTS is however an online optimization problem, where at each time step $t$ a
function $s({\mathbf{x}}_{t})$ must decide to make a prediction or not.
Equation 1 is thus no longer operational since it requires a complete
knowledge of the time series. In practice, the function $s({\mathbf{x}}_{t})$
triggers a decision at $\hat{t}$, based on a partial description
${\mathbf{x}}_{\hat{t}}$ of the incoming time series ${\mathbf{x}}_{T}$ (with
$t\leq T$). The goal of an ECTS system is to chose a triggering time $\hat{t}$
as close as possible to the optimal one $t^{*}$, at least in terms of cost,
minimizing
$\mathcal{L}(\hat{y}({\mathbf{x}}_{\hat{t}}),\hat{t},y)-\mathcal{L}(\hat{y}({\mathbf{x}}_{t^{\star}}),t^{\star},y)$
as much as possible.
From a machine learning point of view, the goal is to find a function
$s\in{\cal S}$ that best optimizes the loss function $\mathcal{L}$, minimizing
the true risk over all time series distributed according to the
distribution222 Notice that the notation $\mathcal{X}$ is an abuse that we use
use to simplify our purpose. In all mathematical rigor, the measurements
observed successively constitute a family of time-indexed random variables
$\mathbf{x}=(\mathbf{x}_{t})_{t\in[1,T]}$. This stochastic process
$\mathbf{x}$ is not generated as commonly by a distribution, but by a
filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\in[1,T]}$ which is defined as a
collection of nested $\sigma$-algebras (?) allowing to consider time
dependencies. Therefore, the distribution $\mathcal{X}$ should also be re-
written as a filtration. $\mathbb{P}_{\mathcal{X}}$ that governs the time
series in the application:
$\small\normalsize\operatorname*{arg\,min}_{s\in\mathcal{S}}\operatornamewithlimits{\mathbb{E}_{\>\mathbf{x}\sim\mathbb{P}_{\mathcal{X}}}^{\>}}\left[\mathcal{L}(\hat{y}({\mathbf{x}}_{\hat{t}}),\hat{t},y)\right]$
(3)
The questions then are:
1. 1.
Which form can take the function $s(\cdot)$? We will distinguish end-to-end
architecture from separable ones.
2. 2.
How the criterion to be optimized accounts for the trade-off between accuracy
and earliness? We will see that the costs implied, about misclassification and
delay, are variously explicit in the existing methods.
3. 3.
How the when question of the stopping problem can be approached? This will
lead us to distinguish between cost-informed and cost-uninformed ones on one
hand, and between anticipation-based methods versus myopic ones, on the other
hand.
4. 4.
How the prediction problem itself can be solved given that $\mathbf{x}_{t}$
belongs to a different input spaces at each time step $t$?
This set of questions and possible choices for their solution are illustrated
in Figure 1. We turn successively to each one in the following.
Figure 1: Proposed ECTS taxonomy
### 2.1 The different forms of the function $s(\cdot)$
An ECTS function must solve both the question of (i) when to stop receiving
new measurements and decide to make a prediction and (ii) how to make the
prediction about the class of the incoming time series $\mathbf{x}_{t}$.
In the separable approach, these questions are solved using two separate
components. The classification one deals with making a prediction:
$\mathbf{x}_{t}\mapsto\hat{y}$, while the trigger function decides when to
predict. Within this perspective, the classification component is learned
independently of the trigger one, while the latter uses the results of the
classification component in order to trigger a decision. A simple triggering
strategy is to decide it is time to make a prediction as soon as the
classification component is sufficiently sure of its prediction. We formalize
separable approaches by: $s(\mathbf{x}_{t})=(g\circ h)(\mathbf{x}_{t})$ where
$g$ is the decision or trigger function, and $h$ is the prediction function.
In the end-to-end approaches, a single component decides when to make a
prediction and what that prediction is. Thus, the function $s$, defined in
Equation 2, is responsible both for choosing the time $\hat{t}$ for making the
prediction, and for the prediction itself $\hat{y}$.
The question that naturally arises is which type of architecture (i.e. end-to-
end or separable) performs best. On the one hand, in separable approaches, the
classification component is trained independently of the triggering one, which
can be detrimental. On the other hand, separating the ECTS problem into two
inherently simpler sub-problems could be an advantage. In this paper, we do
not delve any further into this question, which we leave for future work.
### 2.2 Choice of the optimizing criterion during training
This section covers optimization criteria existing in the literature and used
to train ECTS functions. In the following, these criteria are listed by level
of cost awareness, and we differentiate between the following situations:
1. 1.
The first one, we call cost-informed at training time.
$\mathbb{P}_{\mathcal{X}}$ being unknown, instead of using Equation 3
describing the true risk, one tries to minimize the empirical risk, also
called average cost in the ECTS literature, for a training set of $M$ time
series:
$\displaystyle
AvgCost\;=\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}(\hat{y}_{i},y_{i},\hat{t})\;=\;\frac{1}{M}\sum_{i=1}^{M}\mathrm{C}_{m}(\hat{y}_{i}|y_{i})+\mathrm{C}_{d}(\hat{t}_{i})$
(4)
AvgCost is the most appropriate criterion to both train and evaluate ECTS
approches. The following presents proxy measures of AvgCost.
2. 2.
The second situation is a sub-case of cost-informed, and can be qualified as
cost-proxy at training time. There, while the accuracy and earliness of
prediction are taken into account, the optimization criterion combines them in
a proxy which is not the AvgCost. Classical proxys include:
* •
The Harmonic Mean (see ?):
$HM=\frac{2\times Accuracy\times(1-Earliness)}{Accuracy+(1-Earliness)}$ (5)
with:
$\displaystyle Accuracy$
$\displaystyle=\frac{1}{M}\sum_{i=1}^{M}\mathbb{1}(\hat{y}_{i}=y_{i})$ (6)
$\displaystyle Earliness$ $\displaystyle=\frac{1}{M\times
T}\sum_{i=1}^{M}\hat{t}_{i}$ (7)
where $\hat{t}_{i}$ is the time that the ECTS function decides to make
prediction for the time series $\mathbf{x}^{i}$.
* •
The CF (i.e. Cost Function) criterion with $0\leq\alpha\leq 1$ (see ?, ?, ?):
$CF=\alpha\times(1-Accuracy)+(1-\alpha)\times Earliness$ (8)
When the cost of misclassification
$\mathrm{C}_{m}(\hat{y},y)=\mathbb{1}(\hat{y}=y)$, and the delay cost is
$\mathrm{C}_{d}(t)=\hat{t}/T$ and $\alpha=0.5$, CF becomes a particular case
of AvgCost.
The choice of using costs or approximating them by a proxy is a technical
issue, which may, for example, be relevant to making the loss function
differentiable by approximation. For this reason, in the remainder of this
paper, cost-informed/cost-proxys at training time situations are grouped
together under the term cost-informed at training time, this difference not
being essential.
3. 3.
The third situation is qualified as cost-uninformed-train. This is the case of
methods that only set the threshold on the confidence of the prediction in
order to trigger the prediction, regardless of the costs incurred on the
training set. Another example is methods that use rigid rules, such as “decide
as early as the first measurement is available”.
It is a question whether any of these approaches fare better than the others.
To measure this, we must use the Average Cost (see Equation 4) measured on a
test set, which represents the ground truth of the ECTS problem. It should be
noted that many of the proposed methods have been evaluated on the basis of
other criteria in the literature, with the result of not allowing a rigorous
comparison between them. We will come to this problem in Section 4.
The rest of this section is specific to separable ECTS approaches, which
represent a large part of the literature.
### 2.3 Information used by the trigger function during inference
The trigger function can draw on different types of information. In the
simplest case, it can decide irrespectively of the incoming time series
$\mathbf{x}_{t}$. This is the case, for example, of the rule that would say:
“wait until half of the measurements are available, then make a prediction”.
The corresponding trigger function can be said to be blind.
Apart from this extreme case, it is interesting to distinguish two dimensions.
First, the trigger function may or may not take the misclassification and
delay costs into account. Second, it can also make its decision at each time
step solely on the basis of past observations, or it can anticipate possible
futures to help its decision.
For instance, confidence-based methods (seen Section 2.3.1) do not explicitly
take costs into account in the triggering decision.
#### 2.3.1 Confidence-based approaches
Confidence-based approaches are widely used in the literature. The simplest
trigger model of this kind consists in monitoring a quantity related to the
confidence of the prediction over time and triggering class prediction as soon
as a threshold value is exceeded. The confidence metric monitored can take
different forms. For example, a baseline approach, referred to as Proba
Threshold333Baseline implemented in the aeon (?) library :
https://urlz.fr/qmWl in the remainder of this paper, involves monitoring
$\max_{y\in{\cal Y}}p(y|{\mathbf{x}}_{t})$ the highest conditional probability
estimated by the classifier. This baseline example is qualified as instant-
based method, since it takes as input only the last confidence score available
at time $t$. Another type of approaches, qualified as sequence-based, monitors
the entire sequence of past confidence scores, and triggering a prediction is
made conditionally on a particular property of this sequence. Accordingly,
trigger functions can either take as input a scalar value, e.g.
$g(\max_{y\in{\cal Y}}p(y|{\mathbf{x}}_{t}))$, in the case of instant based
approaches, or a sequence of scalar values, e.g. $g(\\{\max_{y\in{\cal
Y}}p(y|{\mathbf{x}}_{\tau})\\}_{1\leq\tau\leq t})$, in the case of sequence-
based approaches (see Section 3.1).
#### 2.3.2 cost-informed at testing time
Given that an ECTS approach will ultimately be evaluated on the average cost
of using it (see Equation 4), it seems natural to exploit the cost values at
testing time, in order to trigger predictions at optimal moments. Methods such
as Economy (?) and 2step/NoCluster (?) do that. They can thus be qualified as
“cost-informed at testing time”.
Other approaches use instead trigger functions that, once learned, do not take
into account the cost at testing time, but rely on other measures such as, for
instance, the confidence of the prediction. This is the case of the SR
approach (?). Therefore, these approaches can be qualified as “cost-uninformed
at testing time”.
Notice that some approaches are cost-informed during training but not during
inference. This is the case with the SR approach, which is cost-informed at
training time since it uses costs to optimize its parameters, and cost-
uninformed at testing time since the resulting trigger function does not use
cost values during inference. Table 1 shows these two different properties for
each approach in the literature.
#### 2.3.3 Blind, Myopic and Anticipation-based decisions
Some separable approaches consider the output of the classifier $h$ at time
$t$ to decide whether this is the right time to make a prediction. For
instance, stopping the process when the confidence in the classification
$h_{t}(\mathbf{x}_{t})$ is above some threshold, or when the difference of
confidence between the best two predictions exceeds some value. These methods
can be described as myopic since they only look at the current time step $t$,
without trying to guess the future.
But there is a another possibility. As was first noted by ? (?), the ECTS
problem can be cast as a LUPI (Learning Using Privileged Information) problem
(?).
In this scenario, the learner can benefit at the training time of privileged
information that will not be available at test time. Formally, the training
set can be expressed as
$\mathcal{T}=\\{(\mathbf{x}_{i},\mathbf{x}_{i}^{\star},y_{i})\\}$, where
$\mathbf{x}_{i}$ is what is observable and $\mathbf{x}_{i}^{\star}$ is some
additional information not available when the prediction must be made. This is
exactly what happens in the ECTS problem. Whereas at test time, only
$\mathbf{x}_{t}$ is available, during training the complete time series are
known. This brings the possibility to learn what are the likely future of an
incoming time series $\mathbf{x}_{t}$ provided it comes from the same
distribution. Hence, it becomes also possible to guess the cost to be
optimized for all future time steps, and therefore to wait until the moment
seems the best. This type of approach can be said anticipation-based (also
called non-myopic in the literature). Because more information from the
training set is exploited, it can be expected that these methods outperform
myopic and blind ones.
Is this confirmed by experience? Are there situations where the advantage is
significant? Our experiments in Section 4 provide answers to these questions.
### 2.4 Choice of the classification component
One source of difficulty when devising an ECTS method in the separable setting
is that inputs differ from one time step to another. The number of
measurements, and hence the input dimension, varies. Two approaches have been
used to deal with the problem.
1. 1.
A set of classifiers $\\{h_{t}\\}_{t\in[1,T]}$ is learned, each dedicated to a
given time step $t$, and thus a given input dimension. In practice, authors
often chose a limited subset of timestamps, usually a set of twenty (one
measurement every 5% of the length of the time series), to restrict the number
of classifiers to learn and therefore the associated computational cost.
2. 2.
A single classifier $h$ is used for all possible incoming time series
$\mathbf{x}_{t}$. One way of doing this is to “project” an input
$\mathbf{x}_{t}$ of dimension $t\times d$, if $d$ is the dimension of an
observation at time $t$ (i.e. multi-valued time series), into a fixed
dimensional vector whatever $t$ and $d$. This may simply be the mean value and
standard deviation of the available measurements (multiplied by the dimension
$d$) or the result of a more sophisticated feature engineering as tested by ?
(?). Deep learning architectures can also be used to learn an encoding of the
time series in an intermediate layer. For instance, ? (?) use a CNN
architecture, and ? (?) a FCN one. ? (?) show that using deep neural
architectures often perform well for time series classification.
Both approaches have their own limitations. On the one hand, using a set of
classifiers, each independently dedicated to a time step, does not exploit
information sharing. On the other hand, using a single classifier seems to be
a more difficult task, as the representation of $\mathbf{x}_{t}$ can be
different at times $t$ and $t+1$ and all further time steps which can lead to
additional difficulty for the classifier while moreover requiring a more
demanding feature engineering step. Therefore, here also, it is interesting to
measure experimentally whether one dominates the other. This will be the
subject of future work.
## 3 State of the art
In this section, we position various existing proposed methods along the
dimensions of the taxonomy presented in Section 2 (see Table 1). Then, we
examine the methods and how they exemplify solutions to the general questions
raised in the taxonomy. Thus, while each method combines solutions to all
questions, in the following, we underline how each one brings an original
solution to one specific problem. For instance, the Economy approach is
separable, anticipation-based, and cost-informed at testing time. However,
we’re focusing here on the anticipation-based dimension, since this is one of
the first methods to have emphasized it and introduced an original solution
for this aspect.
References | Classifier(s) (collection ✓) | End2end | Confidence | Anticipation | Cost informed
---|---|---|---|---|---
EDSC (?) | Shapelet | ✓ | ✓ | | ✗
ECTS’ (?) | 1NN | | ✓ | ✓ | ✗
Reject (?) | SVM | | ✓ | | ✗
RelClass (?) | QDA, Linear SVM | | ✓ | ✓ | ✗
iHMM (?) | HMM | | ✓ | | ✗
2step/NoCluster (?) | Linear SVM (✓) | | | ✓ | train & test
ECDIRE (?) | Gausian Process (✓) | | ✓ | | ✗
Stopping Rule (?) | Gaussian Process (✓) | | ✓ | | train
EARLIEST (?) | LSTM | | | | train
ECEC (?) | WEASEL (✓) | | ✓ | | train
DDQN (?) | MLP | ✓ | | ✓ | train
TEASER (?) | WEASEL (✓) | | ✓ | | train
ECONOMY-$\gamma$-max (?) | XGBoost $+$ tsfel (✓) | | | ✓ | train & test
DETSCNet (?) | TCN | ✓ | | | train
CALIMERA (?) | MiniROCKET (✓) | | | ✓ | train
ELECTS (?) | LSTM | ✓ | | | train
SOCN (?) | FCN | | ✓ | | train
EarlyStop-RL (?) | MLP | ✓ | | ✓ | train
Table 1: Table of published methods for the ECTS problem with their properties
along dimensions underlined in the taxonomy.
### 3.1 Confidence-based approaches
Most ECTS methods to date are separable, confidence-based, cost-informed at
training time, and are not anticipation-based. They implement separately the
prediction and the triggering components, they learn them using the costs,
hence they are cost-informed at training time, but they decide to trigger a
decision based on the information available at the current time step $t$
without trying to anticipate the likely future, and they base their decision
upon the confidence of the predictions made by the classifier.
There exist two families of confidence-based approaches. In the first one,
only the last time step is considered, a score based on confidence estimations
is monitored at each time step and a class prediction is triggered as soon as
a threshold on this score is exceeded. By contrast, in the second, a sequence
of estimated scores is monitored, and the condition to trigger a decision
depends upon some property of this sequence.
#### 3.1.1 Instant-based decision criterion
$\bullet$ One basic method is to monitor $\max_{y\in{\cal
Y}}p(y|{\mathbf{x}}_{t})$, the highest conditional probability estimated by
the classifier, which is a simple measure of classifier confidence over time.
As soon as it exceeds a value, which is an hyper parameter of the method, a
prediction is made. We call this method Proba Threshold and use it as baseline
for comparison later in our experiments.
$\bullet$ The Reject method (?) uses ensemble consensus as a confidence
measure. For each time step, first (i), a pool of classifiers is trained by
varying their hyperparameters (i.e. SVMs); then (ii), the most accurate of
these are selected; and (iii) the pair of classifiers minimizing their
agreement in predictions is chosen to form the ensemble. Finally, the
prediction is triggered as soon as both classifiers in the ensemble predict
the same class value. In this case, the monitored confidence measure is binary
(agreement or disagreement), there is no trigger threshold and thus this
trigger model is free of hyper-parameters444The Reject approach involves
choosing the number of classifiers trained in step (i) and selected in step
(ii) that could be considered as hyper-parameters of the monitored confidence
measure..
$\bullet$ Hidden Markov Models (HMMs) are naturally suited to the
classification of online sequences. An HMM is learned for each class, and at
each time step $t$, the class to be preferred is the one with the highest a
posteriori probability given $\mathbf{x}_{t}$. However, the decision to make a
prediction now or to postpone it must then involve a threshold so that the
prediction is only made if the a posteriori probability of the best HMM is
sufficiently high or is greater than that of the second-best. In reaction to
this, ? (?) propose to replace the standard HMM with imprecise HMMs based on
the concept of credal classifier. This eliminates the need to choose a
threshold, since a decision is made when one classification “dominates”
(according to a criterion based on probability intervals) all the others.
$\bullet$ Rather than considering only the largest value predicted by the
classifier, it is appealing to consider also the difference with the second
largest value, since a large difference points to the fact that there is no
tie between predictions to expect.
This is one dimension used in the Stopping Rule (SR) approach (?).
Specifically, the output of the system is defined as:
$g(h({\mathbf{x}}_{t}))=\begin{cases}\emptyset&\mbox{if extra measures are
queried;}\\\ \hat{y}=\operatorname*{arg\,max}_{y\in{\cal
Y}}p(y|{\mathbf{x}}_{t})&\mbox{when
}\gamma_{1}\,p_{1}+\gamma_{2}\,p_{2}+\gamma_{3}\,\frac{t}{T}>0\end{cases}$ (9)
where $p_{1}$ is the largest posterior probability $p(y|{\mathbf{x}}_{t})$
estimated by the classifier $h$, $p_{2}$ is the difference between the two
largest posterior probabilities, and $\frac{t}{T}$ represents the proportion
of the incoming time series at time $t$. The parameters $\gamma_{1}$,
$\gamma_{2}$ and $\gamma_{3}$ are learned from the training set.
$\bullet$ Using the same notations as SR, the Early Classification framework
based on class DIscriminativeness and RELiability (Ecdire) (?) finds the
earliest timestamp for which a threshold applied on $p_{1}$ is reached
(defined as in Equation 9). Then, the quantity $p_{2}$ is monitored, and a
second threshold is applied to trigger the prediction.
$\bullet$ A different class of methods relies on searching telltale
representations of subsequences, such that if the incoming time sequence
$\mathbf{x}_{t}$ matches one or more of these representations, then its class
can be predicted. Typically, these representations take the form of shapelets
that discriminate well one class from the others (?). For instance, the Early
Distinctive Shapelet Classification (Edsc) method learns a distance threshold
for each shapelet, based on the computation of the Euclidean distance between
the considered subsequence and all others valid subsequences in the training
set (?). It selects a subset of them, based on a utility measure that combines
precision and recall, weighted by the earliness. A prediction is made as soon
as $\mathbf{x}_{t}$ matches one of these shapelets well enough. Because this
family of methods is computationally expensive, extensions have been developed
to reduce the computational load (?, ?). Other extensions aimed at improving
the reliability of the predictions (?, ?), and tackling multivariate time
series (?, ?, ?, ?).
#### 3.1.2 Sequence-based decision criterion
Other approaches propose sequence-based confidence measures specifically
designed for the ECTS problem.
$\bullet$ The Effective Confidence-based Early Classification (Ecec) (?)
proposes a confidence measure based on the sequence of predicted class values,
from the first one observed to the current timestamp. At each time step, this
approach exploits the precision of the classifier to estimate the probability
for each possible class value $y\in{\cal Y}$ of being correct if predicted.
Then, assuming that successive class predictions are independent, the proposed
confidence measure represents the probability that the last class prediction
is correct given the sequence of predicted class values. The proposed
confidence measure is monitored over time, and prediction is triggered if this
measure exceeds a certain threshold $\gamma$ tuned as the single hyper-
parameter.
$\bullet$ The Teaser (Two-tier Early and Accurate Series classifiER) (?)
approach considers the problem of whether or not a prediction should be
triggered as a classification task, the aim of which is to discriminate
between correct and bad class predictions. As the authors point out, the
balance of this classification task varies according to the time step
considered $t\in[1,T]$. Indeed, assuming there is an information gain over
time, there are fewer and fewer bad decisions as new measurements are received
(or even no bad decisions after a while, i.e.
$\forall\;t>t^{\prime}\;(0<t^{\prime}\leq T)$ for some datasets). To exploit
this idea, a collection of one-class SVMs is used, learning hyper-spheres
around the correct predictions for each time step. A prediction is triggered
when it falls within these hyper-spheres for $\nu$ consecutive time steps
($\nu$ being a parameter of the method).
$\bullet$ The Second-Order Confidence Network approach (Socn) (?) considers,
as does Teaser, the same classification task aiming to discriminate between
correct and bad predictions. To learn this task, a transformer (?) is used,
taking as input the complete sequence of conditional probabilities estimated
by the classifier $h$, from the first time step, up to the current time step.
A confidence threshold $\nu$ is learned by minimizing the same cost function
as (?), above which the prediction is considered reliable and therefore
triggered.
### 3.2 Anticipation-based methods
One way of designing approaches that anticipate future measurements is to
achieve classification of an incomplete time series while guaranteeing a
minimum probability threshold according to which the same decision would be
made on the complete series. This is the case of the Reliability
Classification (RelClass) approach (?). Assuming that the measurements are
i.i.d. and generated by a Gaussian process, this approach estimates
$p({\mathbf{x}}_{T}|{\mathbf{x}}_{t})$ the conditional probability of the
entire time series ${\mathbf{x}}_{T}$ given an incomplete realization
${\mathbf{x}}_{t}$ and thus derives guarantees of the form:
$p\bigl{(}h_{T}({\mathbf{x}}_{T})=y|{\mathbf{x}}_{t}\bigr{)}\,=\,\int_{{\mathbf{x}}_{T}\text{
s.t.
}h_{T}({\mathbf{x}}_{T})=y}\,p({\mathbf{x}}_{T}|{\mathbf{x}}_{t})\,d{\mathbf{x}}_{T}\,\geq\,\gamma$
where ${\mathbf{x}}_{T}$ is a random variable associated with the complete
times series, $\gamma$ is a confidence threshold, and $h_{T}$ is the
classifier learned over complete times series. At each time step $t$,
$p(h_{T}({\mathbf{x}}_{T})=y|{\mathbf{x}}_{t})$ is evaluated and a prediction
is triggered if this term becomes greater than the threshold $\gamma$, which
is the only hyper-parameter to be tuned.
Another way of implementing anticipation-based approaches is to exploit the
continuations of training time series, which are full-length. One of the first
method for ECTS (?) has been derived into such an anticipation-based approach
(?). The first, called Early Classification on Time Series (Ects) (?),
exploits the concept of Minimum Prediction Length (MPL), defined as the
earliest time step for which the predicted label should not change for the
incoming time series $\mathbf{x}_{t}$ from $t$ to $T$. This is estimated by
looking for the 1NN of $\mathbf{x}_{t}$ in the training set, and check whether
from $t$ onward, its predicted label did not change. To be more robust, the
MPL is defined based on clusters computed on full-length training time series
to estimate the best decision time. The approach has been extended later on to
speed up the learning stage (?). This method looks in its own way at the
likely future of $\mathbf{x}_{t}$ \- i.e. an incomplete time series belongs to
a cluster whose continuations are known - and thus can be considered as an
anticipation-based method.
? (?) present a method that claims explicitly to be “non myopic” in that a
decision is taken at time $t$ only insofar as it seems that no better time for
prediction is to be expected in the future. In order to do this, the family of
Economy methods estimates the future cost expectation based on the incoming
time series $\mathbf{x}_{t}$. This can be done since the training data
consists of full-length time series and therefore a Learning Using Privileged
Information (LUPI) (?) is possible.
More formaly, the objective is to trigger a decision when
$\mathbb{E}_{y,\hat{y}}[\mathcal{L}(\hat{y},y,t)|\mathbf{x}_{t}]$ is minimal,
with:
$\displaystyle\mathbb{E}_{y,\hat{y}}[\mathcal{L}(\hat{y},y,t)|\mathbf{x}_{t}]=\sum_{y\in{\cal
Y}}P(y|\mathbf{x}_{t})\,\sum_{\hat{y}\in{\cal
Y}}P(\hat{y}|y,\mathbf{x}_{t})\,C_{m}(\hat{y}|y)\;+\;C_{d}(t)$ (10)
A tractable version of Equation 10 has been proposed by introducing an
additional random variable which is the membership of $\mathbf{x}_{t}$ to the
groups of a partition ${\cal G}$:
$\mathbb{E}_{y,\hat{y}}[\mathcal{L}(\hat{y},y,t)|\mathbf{x}_{t}]=\sum_{g_{k}\in{\cal
G}}P(g_{k}|\mathbf{x}_{t})\sum_{y\in{\cal Y}}P(y|g_{k})\sum_{\hat{y}\in{\cal
Y}}P(\hat{y}|y,g_{k})C_{m}(\hat{y}|y)+C_{d}(t)$ (11)
In technical terms, training approaches from the Economy framework involve
estimating the three probability terms of Equation 11, for the current time
step $t$, as well as for future time steps $t+\tau\in[t+1,T]$, with:
* •
$P(g_{k}|\mathbf{x}_{t})$ the probability of $\mathbf{x}_{t}$ belonging to the
groups $g_{k}\in{\cal G}$,
* •
$P(y|g_{k})$ the prior probability of classes in each group,
* •
$P(\hat{y}|y,g_{k})$ the probability of predicting $\hat{y}$ when the true
class is $y$ within the group $g_{k}$.
A key challenge in this framework is to design approaches achieving the most
useful partition for predicting decision costs expectation. In the first
article which presents this framework (?), a method, called Economy-$K$, is
designed as follows. (i) A partition of training examples is first performed
by a K-means algorithm ; (ii) then a simple model uses the Euclidean distance
as a proxy of the probability that $\mathbf{x}_{t}$ belongs to each group;
(iii) the continuation of training time series within each group is exploited
to predict the cost expectation for future time steps.
In order to avoid the clustering step with the associated choice of hyper
parameters, (?) presented a variant called NoCluster which uses the 1 nearest
neighbor in the training set in order to guess the likely future of
$\mathbf{x}_{t}$.
Then, Economy-$\gamma$ was introduced in (?) which relies on a supervised
method to define a confidence-based partition of training time series. The
algorithm, dedicated to binary classification problems, is designed as
follows: (i) a collection of partitions is constructed by discretizing the
output of each classifier $\\{h_{i}\\}_{i\in[1,T]}$ into equal-frequency
intervals, the groups thus formed correspond to confidence levels for each
time step; (ii) at the current time $t$, the incoming time series
$\mathbf{x}_{t}$ belongs to only one group, since the output of the classifier
$h_{t}$ falls within a particular confidence level; (iii) then, a Markov chain
model is trained to estimate the probabilities of the future time step
confidence levels. Economy-$\gamma$-max (?) generalizes this approach to
multi-class problems, aggregating the multiple conditional probabilities in
the classifiers’ output by using only the most probable class value.
Finally, Calimera (?) uses anticipation about the future from another
perspective. Instead of trying to guess the likely continuation of
$\mathbf{x}_{t}$ which allows one to compute expected future costs, and
therefore to wait until there seems no better time to make a prediction, their
method is based on predicting directly the difference in cost between
predicting the class now or wait at least one more time step. If this
difference is positive, then it is better to postpone the prediction. They
advocate furthermore, that a calibration step should intervene above the
regression in order to make a decision.
### 3.3 Reinforcement Learning based methods
To learn an ECTS function, it is possible to use an agent that learns what to
do by exploring the outcomes associated with different decision strategies as
it interacts with the world. The ECTS problem can therefore be recast as a
reinforcement learning (RL) problem.
In RL, an agent must learn to associate an action $a$ with each observable
state $s$ so that the expected gain be maximized. Let us suppose that the
agent-environment interactions break naturally into subsequences which we call
episodes
At each time step $t$, the agent perceives the environment’s state $s_{t}$
(i.e. $\mathbf{x}_{t}$), choses an action $a_{t}$ (i.e. either make a
prediction now and measure the gain
$G_{t}=-\mathrm{C}_{m}(\hat{y}|y)-\mathrm{C}_{d}(\hat{t})$, or postpone the
decision and receive the next measurement $x_{t+1}$) according to its current
policy $\pi$, where $\pi(a|s)=p(a_{t}=a|s_{t}=s)$ is the probability that
action $a$ is taken given the state $s$.
The goal is for the agent to learn an optimal policy $\pi^{\star}$ from
sequences of interactions with its environment $\langle
s_{t},a_{t},s_{t+1}\rangle$. This can be done by computing the utility
function $Q(s,a)$ defined for all (state, action) pairs. By definition:
$Q_{\pi}(s,a)\;=\;\operatornamewithlimits{\mathbb{E}_{\>\pi}^{\>}}\left[G_{t}|S_{t}=s,A_{t}=a\right]$
(12)
and the optimal policy can be derived from:
$Q^{\star}(s,a)\;=\;\operatornamewithlimits{max}_{\pi}Q_{\pi}(s,a)$ (13)
It suffices at each observed state $s$ to chose action $a^{\star}$ such as:
$a^{\star}\;=\;\operatorname*{arg\,max}_{a}Q^{\star}(s,a)$ (14)
One way to learn the function $Q^{\star}$ is using the Q-learning algorithm
(?, ?)555Note that other approaches are possible in the reinforcement learning
scenario, like learning the utility function $V(s)$ and using TD-learning, or
even learning the policy $\pi$ directly. The Q-learning approach is however
widely used. and its variants for continuous definition of environment’s
states, such as Deep Q-learning (?).
In End-to-end approaches, only one function is responsible for both stopping
and making a prediction about the class of $\mathbf{x}_{t}$ whereas, in
separable approaches involves two combined functions, respectively dedicated
to triggering the prediction and to the classification itself. In the case of
Reinforcement Learning based approaches, this distinction takes the following
form:
1. 1.
The separable approach. RL can be used to learn the Trigger function only,
once the classifiers $h_{t}$ have been learnt. In that case, the set of
actions $\mathcal{A}_{t}$ at each time step $t$ is restricted to two elements:
{‘decide now’, ‘postpone decision’}. The $Q$ function evaluates for each state
the expected gain for each of the two possibilities, allowing one to decide
what to do at each time step.
2. 2.
The End-to-End approach. RL can also be used to learn at once both when to
trigger a prediction and what prediction to make. In principle, it suffices to
extend the set of actions to $\mathcal{A}_{t}=\\{`\textit{postpone
decision'},c_{1},\ldots,c_{N}\\}$, where there are $N$ classes. Either, the
agent postpones the decision, or it predicts a class for the incoming time
series $\mathbf{x}_{t}$.
In the literature, Reinforcement Learning-based ECTS approaches frequently
include Deep Learning modules:
##### Separable RL approaches:
The Earliest (Early and Adaptive Recurrent Label ESTimator) uses a RNN
architecture (?) to make the prediction and a Reinforcement Learning agent
trained jointly using policy gradient, to trigger prediction or not. If a
prediction is triggered, the hidden representation given by the RNN is sent to
a Discriminator, whose role is to predict a class, given this representation.
The model has been adapted to deal with irregularly sampled time series (?). ?
(?) extend the ECTS framework to channel filtering, using here also
Reinforcement Learning.
##### End-to-end RL approaches:
? (?, ?) use a Deep Q-Network (?), alongside a specifically designed reward
signal, encouraging the agent to find a good trade-off between earliness and
accuracy. Those types of approaches also naturally extend to online settings
where time series are not of fixed length. (?) introduce EarlytStop-RL, in
which model-free RL is used to address the problem of early diagnosis of lung
cancer.
### 3.4 Deep Learning based methods
Alongside the Reinforcement Learning based approaches, there exist deep
learning methods that do not use RL.
The Decouple ETSC Network (Detscnet) (?) architecture leverages a gradient
projection technique in order to jointly learn two sub-modules: one for
variable-length series classification, the other for the early exiting task.
The End-to-end Learned Early Classification of Time Series method (Elects)
leverages a LSTM architecture, adding a stopping prediction head to the
network and adapting the loss function to promote good early predictions (?).
## 4 Experiments & Results
This section presents the extensive set of experiments carried out in order to
provide a consistent and fair evaluation of a wide range of the existing
literature’s methods. We first describe the experimental protocol used. We
then turn to the experiments and their results. Figure 4 provides a synthetic
view of the organization of these experiments.
* •
Section 4.1 introduces the experimental protocol as well as the global
evaluation methodologies.
* •
In Section 4.2, the main state-of-the-art and the three baseline methods are
evaluated using a widely used cost setting, i.e. with a a binary balanced
misclassification cost and a linear delay cost.
* •
In Section 4.3, methods are tested in an anomaly detection scenario where the
misclassification cost matrix is severely imbalanced, with false negatives
being much more costly than false positives, and where the delay cost is no
longer linear with time but increases exponentially with time.
* •
Finally, Section 4.4 briefly describes a set of other experimental setups,
derived from either standard setting or the anomaly detection one, including
for instance testing the impact of z-normalization. Complementary results can
be found in Appendix B.
4.2 Standard cost setting$\displaystyle C_{d}$ linear - $\displaystyle C_{m}$
balanced4.3 Anomaly detection cost setting$\displaystyle C_{d}$ exponential -
$\displaystyle C_{m}$ imbalanced4.4 Conplementary resultsAblation/substitution
studiesB.3 Removing calibrationDerived from 1.B.4 Impact of base classifierB.5
Impact of $\displaystyle z$-normalizationB.6 Exponential delay cost
onlyDerived from 2.B.6 Imbalanced cost onlyB.6 Standard cost settingUsed
datasets:Classically used datasets collectionProposed, non $\displaystyle z$
normalized, datasets collection$\displaystyle Z$ normalized version of
theproposed datasets collectionImbalanced version of datasets collection
Figure 2: Experiments diagram: each line corresponds to one (or two) full
benchmark runs, representing in total twelve full benchmarks. While this
Section mainly discusses results about the cost settings in Subsections 4.2
and 4.3, many other alternative experiments are briefly analyzed in Subsection
4.4 and are more detailed in the Appendix B.
The experiments presented here aim to evaluate the effects of design choices
on method performance and thus to provide answers to the questions:
* •
Do anticipation-based methods perform better than blind or myopic ones?
* •
Do methods that are cost-informed for their decision (i.e. explicitly
estimating costs) perform better than methods that are cost-uninformed (e.g.
confidence-based) (see Section 2.3.2).
* •
How the various methods fare when modifying the form of the delay cost and/or
the misclassification cost matrix?
### 4.1 Experimental Protocol
This section covers the shared part of the experimental protocol for all
experiments irrespective of the choice of the cost functions (see Sections 4.2
and 4.3 for this).
#### 4.1.1 Evaluation of the performance
The ECTS problem explicitly balances the pressure to make a correct prediction
and a time pressure. The correctness of the prediction is measured by the
misclassification cost $\mathrm{C}_{m}(\hat{y}|y)$ where $\hat{y}$ is the
prediction and $y$ is the true class. The time pressure is sanctioned by a
delay cost $\mathrm{C}_{d}(t)$ that is assumed to be positive and, in most
applications, an increasing function of time. Sometimes, the two can be
combined when the misclassification cost is function of the time :
$\mathrm{C}_{md}(\hat{y}|y,t)$.
Therefore, for each test times series $\mathbf{x}^{i}$, an ECTS method incurs
a cost assumed to be of the following additive form:
$\mathrm{C}_{m}(\hat{y}_{i}|y_{i})\,+\,\mathrm{C}_{d}(\hat{t})$, where
$\hat{t}$ is the time when the system decided to make a prediction, this
prediction being $\hat{y}_{i}$.
For a test set of $M$ time series, the average cost of a method is:
$AvgCost_{\text{test}}\;=\;\frac{1}{M}\sum_{i=1}^{M}C_{m}(\hat{y}_{i}|y_{i})+C_{d}(\hat{t}_{i})$
(15)
This is accordingly the criterion with which we evaluated the methods in the
experiments reported in this paper.
In addition, in order to assess how the methods adapt to various balances
between the misclassification and the delay costs, we vary the settings of
these costs by weighting them during training and testing. The performance of
the methods is therefore evaluated using the weighted average cost, as defined
in Equation 16, for different values of the costs balance $\alpha$, ranging
from $0$ to $1$, with a $0.1$ step:
$\displaystyle AvgCost_{\alpha}=\frac{1}{N}\sum_{i=0}^{N}\alpha\times
C_{m}(\hat{y}_{i}|y_{i})+(1-\alpha)\times C_{d}(\hat{t}_{i})$ (16)
Small values of $\alpha$ correspond to a high delay cost and a small
misclassification cost ; inversely, large values of $\alpha$ give more weight
to the misclassification cost with a lower delay cost.
#### 4.1.2 Optimization of the parameters of the methods
We have optimized the parameters of all tested methods using AvgCost during
training as the optimization criterion. Most of the methods rely on some
simple validation grid-search, for which the bounds have been set according to
the original published papers. When possible, granularity of the grid has been
adapted to keep similar computation times between competitors. As well,
default hyper-parameters have been set according to the values reported in the
original papers, with one exception. Because the original version of Teaser
uses the Harmonic Mean (see Section 2.2), we have kept this setting (the
resulting method being Teaser${}_{\textit{HM}}$), and we have added a variant
called Teaser${}_{\textit{Avg}}$ optimized using AvgCost.
#### 4.1.3 Comparing the trigger methods
In order to carry out a fair comparison between the tested methods, we
isolated as far as possible the triggering function, responsible for deciding
when to stop receiving measurements, from the prediction one, responsible for
predicting a label to the incoming time series. As advocated by ? (?), we have
chosen the MiniROCKET algorithm (?) to be the base classifier for all methods.
It is indeed recognized as among the best performing classifier in the Time
Series classification literature as well as one of the fastest one. Of course,
distinguishing the decision component from the prediction one is only possible
for the “separable” methods. In our experiments, we chose to not evaluate
“end-to-end” methods, leaving that for future work.
Trigger models: Nine trigger models were selected from the literature based on
their usage and their performances666The EDSC algorithm (?), even though
available in the provided library, is not included in the following
experiments, due to high space and time complexity (which hinders fair
comparisons).
* •
Economy-$\gamma$-Max (?): triggers a decision if the predicted cost
expectation is the lowest at time $t$ when compared with the expected cost for
all future time steps (cf. Section 3.2, Anticipation-based).
* •
Calimera (?): triggers a decision when a regressor model which predicts the
difference between the current observed cost and the minimum cost in the
future is negative (cf. Section 3.2, Anticipation-based).
* •
Stopping Rule (?): uses a trigger function based on a linear combination of
confidence estimates and a delay measure linear on time (cf. Section 3.1,
Confidence-based).
* •
Teaser${}_{\textit{HM}}$ (?): employs a trigger module consisting of a
collection of $T$ One Class SVM learned over the training set in order to
isolate good predictions from bad ones. A prediction is triggered once $\nu$
consecutive predictions have been classified as ‘good’ by these OneClass SVM
($\nu$ being tuned to maximize the harmonic mean between Earliness and
Accuracy) (cf. Section 3.1, Confidence-based).
* •
Teaser${}_{\textit{Avg}}$ (?): same algorithm as above. $\nu$ is now tuned
maximizing the $AvgCost$ criterion, in order to allow the method to adapt to
different cost settings.
* •
Ecec (?): defines a confidence measure, based on the aggregated confidence of
the predictions up to time $t$, and triggers a prediction if it exceeds a
threshold, tuned by explicit grid-search (cf. Section 3.1, Confidence-based).
* •
Ecdire (?): determines “safe” timestamps, based on classifier performance,
from which predictions about possible classes can be made. Predictions cannot
be triggered if those timestamps have not been reached. In addition, the
difference between the two highest predicted probabilities must also exceed a
certain threshold. (cf. Section 3.1, Confidence-based).
* •
Ects (?): computes the first time $t$ for which nearest neighbors of the
incoming time series $\mathbf{x}_{t}$ in the training set were given a label
that did not change by the classifier (cf. Section 3.2, Anticipation-based).
All these methods have been re-implemented using Python, reproducing results
close to the published ones. Except the code for the Ects implementation,
which has been taken from ? (?). Hyper-parameters are the ones chosen in the
original published methods. Code to reproduce the experiments is available
publicly at https://github.com/ML-
EDM/papers_experiments/tree/main/ects_reloading.
Baselines: Furthermore, in order to evaluate the benefits, if any, of the
various methods, it is telltale to compare them with simple ones. We chose
three such baselines:
* •
Asap (As Soon As Possible) always triggers a prediction at the first possible
timestep.
* •
Alap (As Late As Possible) always waits the complete series to trigger the
prediction.
* •
Proba Threshold is a natural, confidence-based, cost-informed at training
time, baseline: it triggers a prediction if the estimated probability of the
likeliest prediction exceeds some threshold, found by grid search (cf. Section
3.1, Confidence-based).
#### 4.1.4 Calibration of the classifications
Like ? (?), we add a calibration step when learning the classifiers, i.e.
Platt’s scaling (?). Indeed, as we are dealing with collections of
independently trained classifiers, the prediction scores may not remain
consistent with one another over the time dimension. However, the trigger
methods usually have their parameters set with the same values for all time
steps. This is the case, for example with the Proba Threshold approach. In
addition, some approaches such as Calimera and Economy-$\gamma$-Max exploit
the estimated posterior probabilities
$\\{p(y|\textbf{x}_{t})\\}_{y\in\mathcal{Y}}$ to estimate the future cost
expectation. It is therefore highly desirable for all classifiers, at all
times, to have their output calibrated.
#### 4.1.5 Datasets and training protocol
Datasets777All original datasets of the paper can be downloaded, already
prepared and splitted, from https://urlz.fr/qRqu: In order to be able to
directly compare our results to past experiments, we first use the usual TSC
datasets from the UCR Archive (?) with the default split. In total, we have
used 77 datasets from the UCR Archive, i.e. the ones with enough training
samples to satisfy our experimental protocol end-to-end (blue cylinder in
Figure 4). In this way, most of the datasets used by either ? (?) and ? (?) or
by ? (?) are contained in our experiments.
A second collection of non z-normalized data sets is also provided. In this
way, the associated potential information leakage is avoided (see Section 1).
Any difference with the performance obtained on the z-normalized data sets can
thus signal the danger of z-normalization with firm evidence. Considering the
limited amount of non z-normalized datasets within the UCR archive (?), we
have decided to look for complementary new datasets so as to provide another
collection of datasets. To this end, the Monash archive for extrinsic
regression (?), provided 20 new time series datasets, for which we have
discretized the the numerical target variable into binary classes based on a
threshold value. For instance, if this threshold is equal to the median value
of the regression target, the resulting classification datasets will be
balanced in term of classes (as in Section 4.2). Note that this threshold can
be chosen differently to get imbalanced datasets (as in Section 4.3.2),
several thresholds could also be used to increase number of classes. As a
result, we get a new set of classification tasks, as has recently been done by
? (?). Finally, 35 datasets have been gathered: 15 from the original archive
and 20 from the Monash extrinsic regression archive. (orange cylinder in
Figure 4).
##### Splitting strategy:
When not using predefined splits, the train sets are split into two distinct
sets in a stratified fashion: a first one to train the different classifiers,
corresponding to 60% of the training set and another one to train the trigger
model, trained over the 40% left. The set used to train the classifiers, is
itself split into two different sets in order to train calibrators, using 30%
of the given data. Because of this procedure, we have been led to exclude some
of the datasets, due to their limited training set size.
All the experiments have been performed using a linux operating machine, with
an Intel Xeon E5-2650 2.20GHz (24-cores) and 252GB of RAM. Proceeding all
datasets (including both blue and orange cylinders) over all competing
approaches takes between 9-10 days, using MiniROCKET classifier, which is the
most efficient tested.
### 4.2 Experiments with balanced misclassification and linear delay costs
This first setting is the one most widely used in the literature to date.
#### 4.2.1 Cost definition
The misclassification cost is symmetrical and balanced, the delay cost is
linear. They can be defined as follows:
$\displaystyle C_{m}(\hat{y}|y)$ $\displaystyle=\mathbb{1}(\hat{y}\neq y)$
$\displaystyle C_{d}(t)$ $\displaystyle=\frac{t}{T}$
Thus, for each dataset, the AvgCost is bounded between 0 and 1.
#### 4.2.2 Results and analysis
For comparability reasons, this first set of experiments is analyzed over the
classical ECTS benchmark used in the literature so far (blue cylinder in
Figure 4). Results over the new, non z-normalized, datasets can be found in
Appendix B.
(a) Evolution of the mean ranks, for every $\alpha$, based on the AvgCost
metric. Shaded areas correspond to 90% confidence intervals.
(b) Alpha is now fixed to $\alpha=0.5$. Wilcoxon signed-rank test labeled with
mean AvgCost.
Figure 3: The ranking plot (a) shows that, across all values of $\alpha$, a
top group of four approaches distinguishes itself. The significance of this
result is supported by statistical tests. Specifically, we report this for
$\alpha=0.5$ as shown in (b).
Figure 3(b) provides a global views about the relative performances of the
tested methods. The Wilcoxon-Holm Ranked test provides an overall statistical
analysis. It examines the critical difference among all techniques to plot the
method’s average rank in a horizontal bar. Lower ranks denote better
performance, and the methods connected by a horizontal bar are similar in
terms of statistical significance. When evaluated by their average rank on all
data sets with respect to the average cost (Equation 16), here for
$\alpha=0.5$, four methods significantly outperform the others:
Methods | Confidence | Anticipation | Cost informed
---|---|---|---
Stopping Rule | ✓ | | train
Proba Threshold | ✓ | | train
Economy-$\gamma$-max | | ✓ | train & test
Calimera | | ✓ | train
Figure 3(a) allows a closer look, this time varying the relative costs of
misclassification and delaying prediction using Equation 16, where a small
value of $\alpha$ means that delay cost is paramount. $90\%$ level confidence
intervals have been computed using bootstrap888Resample with replacement has
been done a large number of times (10.000 $\times$) and are reported as shaded
colors on the figure. The statistic of interest is studied, here the mean, by
examining the bootstrap distribution at the desired confidence level.. Again,
the same four methods top the others for almost every values of $\alpha$. Not
surprisingly, the baseline Asap (predict as soon as possible) is very good
when the delay cost is very high, while Alap (predict at time $T$) is very
good when there is no cost associated with delaying decision.
It is remarkable that, in this cost setting, the simple Proba Threshold method
exhibits a strong performance for almost all values of $\alpha$. It is
therefore worth including in the evaluation of new methods. However, while
Figures 3(a), 3(b) are useful for general analysis, they do not provide
insights about how the Accuracy vs Earliness trade-off is optimized for each
of the competitor. Figure 4 provides some explanation for this.
Figure 4: Pareto front, displaying for each $\alpha$ the Accuracy on the $y$
axis and $Earliness$ on the $x$ axis. Best approaches are located on the top
left corner. In zoomed boxes, on the right of the Figure, points corresponding
to a single $\alpha$ are highlighted, while other points are smaller and gray.
Each of the trigger model is optimizing the trade-off in its own way,
resulting in many different approaches having points in the Pareto dominant
set.
In this figure, the two evaluation measures: ‘Accuracy’ and ‘Earliness’, are
considered as dimensions in conflict. The Pareto front is the set of points
for which no other point dominates with respect to both Accuracy and
Earliness. It is drawn here when varying their relative importance using
$\alpha$ (in the set $\\{0,0.1,0.2,\ldots,1.0\\}$).
One must note first that, as Ects and Ecdire are cost-uninformed, their
performance do not vary with $\alpha$. Whatever the relative weight between
accuracy and earliness, they make their prediction approximately after having
observed half of the times series and they reach an average accuracy
respectively near 0.64 and 0.77. They are clearly dominated by the other
methods. This is also the case for Teaser${}_{\textit{HM}}$, which, while
being cost-informed at training time, also only appears once in the figure.
Indeed, no weighting mechanism is provided in the original version of the
algorithm, where the harmonic mean is used as an optimization criterion (see
Equation 5).
Each of the leading methods Stopping Rule, Proba Threshold, Economy and
Calimera have at least one point on the Pareto front and generally exhibits a
combined performance very close to it. A closer look reveals how each approach
optimizes the earliness vs. accuracy trade-off differently for a fixed cost.
If we consider $\alpha=0.8$, for example, it appears that Economy takes its
decision earlier than Proba Threshold, itself being more precocious than Ecec.
Because this is also an area of problems where the delay cost is low, by doing
so, Economy prevents itself from benefiting from waiting for more measurements
and increasing its performance. Hence its slight downward slope on Figure 3(a)
for high values of $\alpha$.
It is worth noting that the two naive baselines Asap and Alap perform better
than the majority of approaches on seven $\alpha$ values out of ten. This is
specially the case when the delay cost is large, i.e. for
$\alpha\in[0.1,0.3]$, for which the Asap baseline is as competitive as top
performers. Globally, the performance of Proba Threshold is remarkable in this
cost setting. Even though, it is simply based on a single threshold on the
confidence in the current prediction, its performance makes it one of the best
methods.
The results computed over the proposed datasets ensemble (i.e. orange
cylinder) are displayed in Figure 8 of Appendix B. No significant changes can
be observed in the ranking of competing approaches.
One question is how such simple method, like Proba Threshold, can adapt to
scenarios where the misclassification and delay costs, not being symmetrical
for the misclassification cost, and not linear for the delay one, reflect
other application settings.
### 4.3 Experiments with unbalanced misclassification and non-linear delay
costs
While the previous section has provided a first assessment of how the various
methods adapt to different respective weights for the misclassification and
the delay costs, it nonetheless assumed that the misclassification costs were
balanced (e.g. 0 if correctly classified and 1 otherwise) and that the delay
cost was a linear function of time.
There are however applications where these assumptions do not hold, for
instance predictive maintenance or hospital emergency services, are
characterized by (i) imbalanced misclassification costs (e.g. it is more
costly to have to repair a machine than to carry out a maintenance operation
that turns out not being necessary) and by (ii) non linear delay cost (e.g.
usually, the later the surgical operation is decided, the costlier it is to
organize it and the larger the risk for the patient). In the following, we
call all applications presenting these characteristics “anomaly detection”
applications.
The question arises as to how the various ECTS algorithms behave in this case,
depending on their level of cost awareness and whether or not they are
anticipation-based. This is what is investigated in the series of experiments
reported in this section.
(a) Exponential delay cost:
$C_{d}(t)=(1-\alpha)\exp(\frac{t}{T}*\log(100))$
(b) Misclassification cost matrix
(three class problem)
Figure 5: Representative delay cost (a) and misclassification ones (b) for an
anomaly detection scenario. In our experiments, $\alpha\in[0,1]$.
#### 4.3.1 Cost definition for anomaly detection
In order to study the behavior of the various algorithms on scenarios
corresponding to anomaly detection, we set the unbalanced misclassification
cost matrix such that a false negative (i.e. missing an anomaly) was 100 times
costlier than a false positive (i.e. wrongly predicting an anomaly) (see
Figure 5(b)). For this last situation, the cost was arbitrarily set to 1. The
delay cost is defined as an exponential function of time. In order to have a
cost commensurable with the misclassification one, we decided that waiting for
the entire time series to be seen, at $T$, would cost 100 (see Figure 5(a)),
starting at 1 for $t=0$ and reaching 100 when $t=T$.
#### 4.3.2 Results and analysis
In this part, as a new cost setting is explored, there is no need to produce
comparable results from previous works. Thus, we choose to use the new non
z-normalized datasets collection (orange cylinder in Figure 4). In order for
the imbalanced misclassification cost to make sense, those datasets have been
altered so that the minority class represents 20% of all labels. As explained
in Section 4.1, some extrinsic regression datasets are turn into
classification ones. In these cases, the threshold value has been set to the
second decile of the regression target. For the original classification
datasets, the minority class has been sub-sampled when necessary.
figurec
(a) Evolution of the mean ranks, for every $\alpha$, based on the AvgCost
metric. Shaded areas correspond to 90% confidence intervals.
(b) Alpha is now fixed to $\alpha=0.5$. Wilcoxon signed-rank test labeled with
mean AvgCost.
Figure 6: The ranking plot (a) shows that, across all $\alpha$, a top group
composed by three approaches distinguish. This result is significant as
supported by statistical tests. Specifically, for $\alpha=0.5$ as shown in
(b).
Results from the Wilcoxon-Holm Ranked test (both regarding the average rank
and the value for AvgCost) (see Figure 6(b)) and from the AvgCost plot (see
Figure 6(a)) with varying values of $\alpha$ (in Equation 16) show that now
the best method overall is Economy which is both cost-informed at training and
testing time, beside being anticipating-based. However, Stopping-Rule is a
very strong contender while being cost-informed at training time but not at
testing time and confidence-based. There is a reason for it. When Stopping
Rule equals or overpasses Economy, this is for high values of $\alpha$ when
the delay cost loses its importance, therefore leaving the misclassification
cost to reign and confidence-based methods to be good.
It may come as a surprise that Calimera lags behind Economy for
$\alpha\in[0,0.4]$, despite being similarly based on the estimation of future
cost expectation. One reason for this is that the cost expectation is achieved
by considering only the predicted class. This poor estimate of the cost
expectancy becomes critical when the delay cost is important.
Similarly, Proba Threshold is surprisingly good in this scenario, even if it
is no longer in the top tier. Looking solely at prediction confidence, we
might expect it to be blind to the rapid increase in delay cost in the anomaly
detection scenario. However, it is noticeable that the cost of delay only
increases sharply after around 60% of the complete time series has been
observed, which is generally sufficient to exceed the confidence threshold.
Hence, Proba Threshold does not suffer from high delay costs that are to come,
and exhibits a good performance here.
Figure 7 plots the Pareto front considering two axes based on decision costs.
The horizontal axis corresponds to the average delay cost incurred for each
example, normalized by the worst delay cost paid at $t=T$. It is better to be
on the left of the $x$-axis. The vertical axis corresponds to 1 minus the
misclassification cost incurred for each example, normalized by the worst
prediction cost. It is better to be high on the $y$-axis.
We observe that the Pareto front is composed almost exclusively of points
corresponding to the Economy method. This is consistent with the evaluation
based on the AvgCost metric. This figure highlights the fact that the design
of approaches capable of handling arbitrarily parameterized decision costs
requires a cost-informed application framework.
Figure 7: Pareto front, displaying for each $\alpha$, the normalized version
of the AvgCost, decomposed over delay and misclassification cost on $x$-axis
and $y$-axis respectively. Best approaches are located on the top left corner.
Due to the exponential shape of the delay cost, the $x$-axis is on log scale.
### 4.4 Other experiments: ablation and substitution studies
In this section, complementary experiments, namely ablation studies as well as
sanity checks are briefly discussed. For the sake of brevity, the figures
supporting the analysis are reported in Appendix B.
#### Impact of removing calibration
? (?) assert that calibration of the classifiers is paramount for the
performance of ECTS algorithms. In order to test this claim, we have repeated
the experiments removing the calibration step. The examples used for
calibration have been removed as well during training, so that all else
remains the same as before.
The results of Figure 11 show that indeed Calimera suffers greatly if no
calibration is done. Indeed, this approach relies on estimating the
expectation of future costs via a regression problem, and a miscalibration may
have a negative impact on the built targets. For its part, Proba Threshold
suffers somewhat mildly. This is no surprise as they rely on a single
threshold on the confidence of the prediction for all time steps.
#### Impact of the choice of base classifier
All methods have been compared using the same classifier: MiniROCKET so that
only the decision components differ. However, the choice of the base
classifiers could induce a bias favoring or hampering some methods. In order
to clarify this, we have repeated the experiments replacing MiniROCKET with
two base classifiers: WEASEL 2.0 (?), and the XGBoost classifier (?) using
features produced by tsfresh (?). Both of these classifiers have already been
tested within the ECTS literature by ? (?, ?) and ? (?) respectively. Figure
12 and 13 in Appendix B report the results respectively with these two
classification methods. One can observe that the results are not significantly
altered with the same overall ordering of the methods when varying the value
of $\alpha$. Furthermore, our results on AvgCost show that performances tends
to be better for all methods using MiniROCKET. It is thus to be preferred
given its simplicity and good performances.
#### Impact of z-normalization
Considering the newly proposed ensemble of datasets, we were not able to
identify any problems of information leakage over time. This inconclusive
result simply indicates that the variance of the time series measurements is
not informative for theses datasets, which still could be the case considering
past published results. For further details, please refer to Section B.5 of
Appendix B.
## 5 Conclusion
The first contribution of this research work is the coding of all the methods
tested and making the codes available in a repository open to everyone. In
this way, the experiments reported can be duplicated and further studies
carried out. Furthermore, the deposited datasets and the experimental
framework provide a ground for fair comparisons bewteen competing methods. We
claim that the AvgCost is the appropriate measure by which to evaluate the
performance of the methods. This is indeed what will be “paid” at the end of
the day by a practitioner using a method. We have accordingly characterized a
number of methods from the literature.
Our extensive experiments have shown that:
* •
It is worthwhile to resort to dedicated ECTS methods, and more so in scenarios
like anomaly detection with asymmetrical misclassification cost and
exponential delay cost.
* •
However, it is noticeable that Proba Threshold, a baseline method, is
surprisingly good overall in the standard setting with symmetrical
misclassification cost and linear delay cost, exhibiting comparable
performance to confidence-based myopic methods such as Stopping-Rule and
anticipation-based cost-informed ones such as Calimera and Economy.
* •
Calibration of the classifiers has a large impact on some methods such as
Calimera in particular, less so on other methods like Proba Threshold and
Ecdire.
In this paper, we have proposed a reading guide to highlight the main
characteristics of ECTS methods, namely (i) the importance of the two
components: decision and prediction which are distinct in the “separable”
architecture and not in the “end-to-end” one, (ii) the distinction between
anticipation-based and myopic methods, and (iii) between cost-informed and
cost-uninformed techniques. On the basis of these dimensions, it becomes easy
to imagine new methods that combine them in original ways, which could lead to
new properties and better performance for solving the problem of early
classification of time series, which, present in many applications has
potentially great impacts.
To go a step further, future work could be carried out to study the
literature’s approaches applied in as yet unexplored cost settings. For
example, in many applications, the delay cost depends on the true class and
the predicted one, and thus a single cost function integrating
misclassification and delay costs should then be used. The use of this general
cost form requires the adaptation of some state-of-the-art methods and has not
yet been studied.
In addition, in real ECTS applications, it is up to the business expert to
define the costs, which is not an easy task in practice. Future work could
study the impact of: (i) noisy costs, (ii) and cost drift between training and
testing stages. This would make it possible to identify the most resilient
literature approaches.
Finally, in the case of existing separable approaches, the misclassification
cost is not exploited for training the classification function. Future work
could investigate the interest of using cost-sensitive classifiers in the case
of ECTS.
## A Data description
### A.1 UCR Time Series Classification datasets
Table 2: UCR TSC datasets : 77 datasets from the UCR archive have been retained to run the experiments over the 128 contained in the full archive. Those are the ones with fixed length, without missing values and with enough training samples to execute our experiments pipeline end-to-end. Italic datasets are not included in experiments using default split for this reason. | Train | Test | Length | Class | Type
---|---|---|---|---|---
Data | | | | |
ACSF1 | 100 | 100 | 1460 | 10 | Device
Adiac | 390 | 391 | 176 | 37 | Image
Beef | 30 | 30 | 470 | 5 | Spectro
BeetleFly | 20 | 20 | 512 | 2 | Image
BME | 30 | 150 | 128 | 3 | Simulated
Car | 60 | 60 | 577 | 4 | Sensor
CBF | 30 | 900 | 128 | 3 | Simulated
Chinatown | 20 | 345 | 24 | 2 | Traffic
ChlorineConcentration | 467 | 3840 | 166 | 3 | Sensor
CinCECGTorso | 40 | 1380 | 1639 | 4 | Sensor
Coffee | 28 | 28 | 286 | 2 | Spectro
Computers | 250 | 250 | 720 | 2 | Device
CricketX | 390 | 390 | 300 | 12 | Motion
CricketY | 390 | 390 | 300 | 12 | Motion
CricketZ | 390 | 390 | 300 | 12 | Motion
Crop | 7200 | 16800 | 46 | 24 | Image
DiatomSizeReduction | 16 | 306 | 345 | 4 | Image
DistalPhalanxOutlineCorrect | 600 | 276 | 80 | 2 | Image
Earthquakes | 322 | 139 | 512 | 2 | Sensor
ECG200 | 100 | 100 | 96 | 2 | ECG
ECG5000 | 500 | 4500 | 140 | 5 | ECG
ECGFiveDays | 23 | 861 | 136 | 2 | ECG
ElectricDevices | 8926 | 7711 | 96 | 7 | Device
EOGVerticalSignal | 362 | 362 | 1250 | 12 | EOG
EthanolLevel | 504 | 500 | 1751 | 4 | Spectro
FaceAll | 560 | 1690 | 131 | 14 | Image
FaceFour | 24 | 88 | 350 | 4 | Image
FacesUCR | 200 | 2050 | 131 | 14 | Image
FiftyWords | 450 | 455 | 270 | 50 | Image
Fish | 175 | 175 | 463 | 7 | Image
FordA | 3601 | 1320 | 500 | 2 | Sensor
FreezerRegularTrain | 150 | 2850 | 301 | 2 | Sensor
GunPoint | 50 | 150 | 150 | 2 | Motion
Ham | 109 | 105 | 431 | 2 | Spectro
HandOutlines | 1000 | 370 | 2709 | 2 | Image
Haptics | 155 | 308 | 1092 | 5 | Motion
Herring | 64 | 64 | 512 | 2 | Image
HouseTwenty | 34 | 101 | 3000 | 2 | Device
InlineSkate | 100 | 550 | 1882 | 7 | Motion
InsectEPGRegularTrain | 62 | 249 | 601 | 3 | EPG
InsectWingbeatSound | 220 | 1980 | 256 | 11 | Sensor
ItalyPowerDemand | 67 | 1029 | 24 | 2 | Sensor
LargeKitchenAppliances | 375 | 375 | 720 | 3 | Device
Lightning2 | 60 | 61 | 637 | 2 | Sensor
Lightning7 | 70 | 73 | 319 | 7 | Sensor
Mallat | 55 | 2345 | 1024 | 8 | Simulated
Meat | 60 | 60 | 448 | 3 | Spectro
MedicalImages | 381 | 760 | 99 | 10 | Image
MelbournePedestrian | 1200 | 2450 | 24 | 10 | Traffic
MixedShapesRegularTrain | 500 | 2425 | 1024 | 5 | Image
MoteStrain | 20 | 1252 | 84 | 2 | Sensor
NonInvasiveFetalECGThorax1 | 1800 | 1965 | 750 | 42 | ECG
NonInvasiveFetalECGThorax2 | 1800 | 1965 | 750 | 42 | ECG
OSULeaf | 200 | 242 | 427 | 6 | Image
OliveOil | 30 | 30 | 570 | 4 | Spectro
PhalangesOutlinesCorrect | 1800 | 858 | 80 | 2 | Image
Plane | 105 | 105 | 144 | 7 | Sensor
PowerCons | 180 | 180 | 144 | 2 | Power
ProximalPhalanxOutlineCorrect | 600 | 291 | 80 | 2 | Image
RefrigerationDevices | 375 | 375 | 720 | 3 | Device
Rock | 20 | 50 | 2844 | 4 | Spectrum
ScreenType | 375 | 375 | 720 | 3 | Device
SemgHandGenderCh2 | 300 | 600 | 1500 | 2 | Spectrum
ShapesAll | 600 | 600 | 512 | 60 | Image
SmoothSubspace | 150 | 150 | 15 | 3 | Simulated
SonyAIBORobotSurface1 | 20 | 601 | 70 | 2 | Sensor
SonyAIBORobotSurface2 | 27 | 953 | 65 | 2 | Sensor
StarLightCurves | 1000 | 8236 | 1024 | 3 | Sensor
Strawberry | 613 | 370 | 235 | 2 | Spectro
SwedishLeaf | 500 | 625 | 128 | 15 | Image
Symbols | 25 | 995 | 398 | 6 | Image
SyntheticControl | 300 | 300 | 60 | 6 | Simulated
ToeSegmentation1 | 40 | 228 | 277 | 2 | Motion
Trace | 100 | 100 | 275 | 4 | Sensor
TwoLeadECG | 23 | 1139 | 82 | 2 | ECG
TwoPatterns | 1000 | 4000 | 128 | 4 | Simulated
UMD | 36 | 144 | 150 | 3 | Simulated
UWaveGestureLibraryX | 896 | 3582 | 315 | 8 | Motion
UWaveGestureLibraryY | 896 | 3582 | 315 | 8 | Motion
UWaveGestureLibraryZ | 896 | 3582 | 315 | 8 | Motion
Wafer | 1000 | 6164 | 152 | 2 | Sensor
Wine | 57 | 54 | 234 | 2 | Spectro
WordSynonyms | 267 | 638 | 270 | 25 | Image
Worms | 181 | 77 | 900 | 5 | Motion
Yoga | 300 | 3000 | 426 | 2 | Image
### A.2 Proposed, non z-normalized, datasets
| Train | Test | Length | Class | Type
---|---|---|---|---|---
Data | | | | |
BME | 30 | 150 | 128 | 3 | Simulated
Chinatown | 20 | 345 | 24 | 2 | Traffic
Crop | 7200 | 16800 | 46 | 24 | Image
DodgerLoopDay | 78 | 80 | 288 | 7 | Sensor
EOGVerticalSignal | 362 | 362 | 1250 | 12 | EOG
GestureMidAirD1 | 208 | 130 | 360 | 26 | Trajectory
GunPointAgeSpan | 135 | 316 | 150 | 2 | Motion
HouseTwenty | 34 | 101 | 3000 | 2 | Device
InsectEPGRegularTrain | 62 | 249 | 601 | 3 | EPG
MelbournePedestrian | 1200 | 2450 | 24 | 10 | Traffic
PLAID | 537 | 537 | Vary | 11 | Device
Rock | 20 | 50 | 2844 | 4 | Spectrum
SemgHandGenderCh2 | 300 | 600 | 1500 | 2 | Spectrum
SmoothSubspace | 150 | 150 | 15 | 3 | Simulated
UMD | 36 | 144 | 150 | 3 | Simulated
AcousticContaminationMadrid_nmv | 166 | 72 | 365 | 2 | Environnement
AluminiumConcentration | 440 | 189 | 2542 | 2 | Environnement
BitcoinSentiment | 232 | 100 | 24 | 2 | Sentiment
ChilledWaterPredictor | 321 | 138 | 168 | 2 | Energie
CopperConcentration | 440 | 189 | 2542 | 2 | Environnement
Covid19Andalusia | 142 | 62 | 91 | 2 | Santé
DailyOilGasPrices | 133 | 58 | 30 | 2 | Économie
DhakaHourlyAirQuality | 1447 | 621 | 24 | 2 | Environnement
ElectricityPredictor | 567 | 243 | 168 | 2 | Energie
FloodModeling3 | 429 | 184 | 266 | 2 | Environnement
HouseholdPowerConsumption1 | 1001 | 430 | 1440 | 2 | Energie
HotwaterPredictor | 245 | 106 | 168 | 2 | Energie
MadridPM10Quality_nmv | 4845 | 2077 | 168 | 2 | Environnement
ParkingBirmingham_eq | 1391 | 597 | 14 | 2 | Environnement
PrecipitationAndalusia_nmv | 470 | 202 | 365 | 2 | Environnement
SierraNevadaMountainsSnow | 350 | 150 | 30 | 2 | Environnement
SolarRadiationAndalusia_nmv | 470 | 202 | 365 | 2 | Energie
SteamPredictor | 210 | 90 | 168 | 2 | Energie
TetuanEnergyConsumption | 254 | 110 | 144 | 2 | Energie
WindTurbinePower | 596 | 256 | 144 | 2 | Energie
Table 3: New datasets collection : 35 datasets from both the UCR archive
(dashed line) and the Monash UEA extrinsic regression archive. When missing
values and/or varying lengths, replace missing values with 0 and pad series to
maximum length with 0. All of the datasets are not $z$-normalized and thus
don’t suffer from potential data leakage. Italic datasets are not included
when classes are imbalanced as problems become too difficult for the chosen
classifiers.
## B Supplementary results
### B.1 Additional figures : Standard cost setting
Figure 8: Standard cost setting, non $z$-normalized proposed datasets (orange
cylinder). Compared to Figure 3(a), the global ranking is not altered much.
One can observe that for $\alpha\in[0.5,0.7]$ the top group is now more
populated, gathering the first six approaches, probably due to the limited
amount of datasets available in this case.
### B.2 Additional figures : Anomaly detection cost setting
Figure 9: Anomaly detection cost setting, original UCR datasets (blue
cylinder). Compared to Figure 6(a), one can see that Calimera is now clearly
dominating all other methods for all $\alpha$. The global ranking remains
globally stable otherwise.
### B.3 Removing calibration
Figure 10: Standard cost setting, original UCR datasets (blue cylinder). The
calibration step is now removed, i.e. the outputs from the decision function
is now simply passed through a softmax function. Both Calimera and Proba
Threshold suffer heavily from using uncalibrated scores. Figure 11: Pairwise
comparison (?), calibration ($calib$) vs no calibration ($no\>\>calib$),
$\alpha=0.8$. Square colors are indexed on the mean AvgCost difference. For
example, Calimera has a lower mean AvgCost when trained over calibrated
scores: it appears in dark blue. The Wilcoxon p-value is equal to 0.0288,
which is lower than significance level equal to 0.05. Thus, Calimera
statistically under-performs when using uncalibrated scores. This is also the
case for the Proba Threshold method.
### B.4 Changing the base classifier
Figure 12: Standard cost setting, original UCR datasets (blue cylinder). The
base classifier is now Weasel 2.0 (?). Results are very close to those exposed
in Figure 3(a). Figure 13: Standard cost setting, original UCR datasets (blue
cylinder). The base classifier is now a pipeline including features extraction
with tsfresh (?) and classification using XGBoost (?). Results are a bit
noisier than those exposed in Figure 3(a).
### B.5 Impact of z-normalization
Clearly, using z-normalized datasets is not applicable in practice, as it
would require knowledge of the entire incoming time series. In a research
context, previous work has used such training sets to test the proposed
algorithms. Our goal here, is to assess whether this could have a large impact
on the performances. For example, when a normalized time series has a low
variance at the beginning, we can expect a high variance in the rest of the
series since the mean variance is 1. There is therefore an information leakage
that can be exploited by a ECTS algorithm, while this is not representative of
what happens in real applications. A proposal such as the one presented in
(?), where the z-normalization of available time series is repeated at each
time step, has its own problems. In particular, it means that if a single
classifier is used for all time steps, the representation of $\mathbf{x}_{t}$
can be different at times $t$ and $t+1$ and all further time steps which can
induce confusion for the classifier.
On the one hand, z-normalization induces an information leakage that could
help methods to unduly exploit knowledge about the future of incoming time
series. On the other hand, any normalization rescale the signal and therefore,
potentially, hinder the recognition of telltale features. So, does
z-normalization affect the performance of ECTS methods? And if yes, in which
way?
In order to answer this question, we took the new datasets collection
described in Section 4.1. They are indeed not z-normalized originally. We
duplicate and z-normalized them to get a second collection. As explained in
Section 4.1, some extrinsic regression datasets has been converted into
classification one. Here, the threshold value chosen to discretize the output
into binary classes has been set the median of the regression target. In this
way, classes within those datasets are equally populated.
In these experiments, the delay cost is linear as in Section 4.2 and as in
most of the literature. Figure 14 reports pairwise comparisons done on the 35
datasets. We look at $\alpha=0.8$, as this is the only value for which
significant differences are observed. One can see that most of the trigger
models do not actually benefit from the z-normalization. Quite the opposite:
out of nine trigger models, only one, i.e. Ecdire, actually has a better mean
AvgCost when being trained on z-normalized data. Regarding the remaining
methods, both Stopping Rule and TeaserAvg performs significantly worse when
operating on z-normalized data. Those trends are quite similar for other
$\alpha$ values, without any significance on the statistical tests though.
Thus, while z-normalization has some impact, since privileged information from
the future can be leaked, our experiments, for the proposed datasets
collection at least, show that this does not alter the overall results
reported in the literature, and are globally in accordance with the results
presented in Section 4.
Figure 14: Pairwise comparison (?), $z$-normalization ($Z$) vs no
$z$-normalization ($\bar{Z}$), $\alpha=0.8$. Square colors are indexed on the
mean AvgCost difference. For example, Calimera has a lower mean AvgCost when
trained over non $z$-normalized datasets by 1.3e-2 and appears in light blue.
It beats the $z$-normalized version over 25 datasets, loses over 9 and are
tied on 1. The Wilcoxon p-value is equal to 0.2224, which is higher than
significance level equal to 0.05. Thus, no statistical difference can be
observed for the considered approach. Figure 15: Multi-comparison-matrices
(?). The upper triangle, with dark blue contours, displays the comparison of
the competitive methods trained over non z-normalized dataset (orange
cylinder). The values within this triangle has to be read by lines, i.e. for a
considered line, red shades indicate better performances, blue shades weaker
performances. The lower triangle, with dark red contours, is the comparison of
the methods trained over the same datasets, z-normalized (chocolate cylinder).
The values within this triangle has to be read by columns, i.e. for a
considered column, red shades indicate better performances, blue shades weaker
performances. The complete figure being symmetrical indicates that
z-normalization does not impact much relative ranking between methods.
### B.6 Anomaly detection cost setting : an ablation study
#### Exponential delay cost only
Figure 16: Exponential delay cost, symmetric binary misclassification cost,
non z-normalized proposed imbalanced datasets (orange cylinder with a whole).
#### Imbalanced misclassification cost only
Figure 17: Linear delay cost, non symmetric imbalanced misclassification cost,
non z-normalized proposed imbalanced datasets (orange cylinder with a whole).
#### Standard cost setting, imbalanced datasets
Figure 18: Linear delay cost, symmetric binary misclassification cost, non
z-normalized proposed imbalanced datasets (orange cylinder with a whole).
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|
# Evaluating Prompting Strategies for Grammatical Error Correction Based on
Language Proficiency
Min Zeng1∗ Jiexin Kuang1 Mengyang Qiu2 Jayoung Song3 Jungyeul Park1
1Department of Linguistics, The University of British Columbia, Canada
2Department of Psychology, Trent University, Canada
3Department of Asian Studies, Pennsylvania State University, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
*Equally contributed authors.
(
Preprint version. To appear in LREC-COLING 2024, short paper.
)
###### Abstract
This paper proposes an analysis of prompting strategies for grammatical error
correction (GEC) with selected large language models (LLM) based on language
proficiency. GEC using generative LLMs has been known for overcorrection where
results obtain higher recall measures than precision measures. The writing
examples of English language learners may be different from those of native
speakers. Given that there is a significant differences in second language
(L2) learners’ error types by their proficiency levels, this paper attempts to
reduce overcorrection by examining the interaction between LLM’s performance
and L2 language proficiency. Our method focuses on zero-shot and few-shot
prompting and fine-tuning models for GEC for learners of English as a foreign
language based on the different proficiency. We investigate GEC results and
find that overcorrection happens primarily in advanced language learners’
writing (proficiency C) rather than proficiency A (a beginner level) and
proficiency B (an intermediate level). Fine-tuned LLMs, and even few-shot
prompting with writing examples of English learners, actually tend to exhibit
decreased recall measures. To make our claim concrete, we conduct a
comprehensive examination of GEC outcomes and their evaluation results based
on language proficiency.
## 1 Introduction
Large language models (LLMs) like Generative Pre-trained Transformers (GPT)
have emerged as a transformative force in natural language processing (NLP)
and artificial intelligence. These models, boasting billions of parameters,
have been trained on an extensive corpus of internet text, making them highly
effective across a wide spectrum of language tasks, such as translation,
summarization, and question answering, often achieving state-of-the-art
results (Brown et al., 2020).
One such application of LLMs is Grammatical Error Correction (GEC). GEC is a
challenging task in NLP that involves detecting and correcting grammatical
mistakes in written text. LLMs like GPT have shown promising results in this
domain, with their ability to generate fluent, grammatically correct text
(e.g., Coyne et al., 2023; Loem et al., 2023). However, despite their
impressive performance, these models are not without limitations. For example,
LLMs have a tendency to overcorrect, leading to higher recall but lower
precision measures (Fang et al., 2023).
Grammatical Error Correction has been a pivotal task in NLP, with numerous
methodologies and systems being developed over the years to improve its
performance. Prior to the advent of LLMs, the most effective GEC systems have
predominantly adopted one of two paradigms: sequence-to-sequence Neural
Machine Translation (NMT)-based approaches and sequence tagging edit-based
approaches. The unique characteristic of GEC, notably the high overlap between
the source and target sentences, has led to the development of edit-based
approaches. These models employ a transformer-based architecture, akin to
their NMT-based counterparts. However, instead of predicting entire sentences,
they are trained to anticipate a sequence of editing operations, such as
delete, append, and replace, significantly enhancing the speed of inference
while preserving high performance (Omelianchuk et al., 2020).
The advent of LLMs has ushered in a new era for GEC. A notable example of is
the work by Rothe et al. (2021), where they leveraged the power of LLMs,
specifically the mT5 model with up to 11 billion parameters. Their work
establishes a new set of baselines for GEC and simplifies the typical GEC
training pipelines composed of multiple fine-tuning stages.
In addition to this fine-tuning approach, recent studies have begun to explore
the potential of the prompt-based approach in the application of LLMs for GEC,
which focuses more on the design of effective prompts that guide the model’s
generation of corrected sentences. For example, Loem et al. (2023)
investigated the impact of task instructions and the number of examples on the
performance of GPT-3 in GEC tasks. They found that instructive instructions
significantly improved GPT-3’s performance in both zero-shot and few-shot
settings, and the performance became more consistent as it received more
examples.
Another area which should be taken into account is L2 learners’ language
proficiency levels. Considering that there is a significant relationship
between learners’ language proficiency levels and types of errors they make
(Yuksel et al., 2017), having language proficiency as one of the variables in
the model might enhance the performance of the model. To be specific,
exploring the relationships between GEC using LLMs, especially, GPT, and
language proficiency levels could reduce the notable limitation of LLMs, that
it its tendency to overcorrection, leading to higher recall but lower
precision measures (Fang et al., 2023).
Building upon these observations, this paper intends to explore the
performance of LLMs in GEC by examining the interaction between LLMs’
performance and the language proficiency levels of the learners. We focus our
exploration on how prompting strategies and fine-tuning impact GEC
performance, with particular attention given to zero-shot and few-shot
prompting. Our goal is to provide a comprehensive understanding of the
strengths and limitations of LLMs in GEC, aiming to illuminate ways in which
their performance can be optimized for language learners of different
proficiency levels, which has hardly been explored thoroughly.
## 2 Language Proficiency
For prompting GEC using GPTs, we use the Cambridge English Write & Improve
(W&I) corpus, which is manually annotated with CEFR proficiency levels,
consisting of beginner level A, intermediate level B, and advanced level C
(Yannakoudakis et al., 2018). It was introduced at the Building Educational
Applications 2019 Shared Task: Grammatical Error Correction (BEA2019) (Bryant
et al., 2019). The text data was from writings of L2 English learners. It has
a propensity that sentences from data of higher proficiency are longer than
lower proficiency: average tokens per sentence in training data sets A, B, and
C are 17.538, 18.304, and 19.212, respectively. For a characteristic example
of proficiency A, the case of in in the ungrammatical sentence (2) is
corrected with In in its counterpart correction (2). It showcases a typical
replacement orthography error, to be more specific, a capitalization error. We
can also observe that the sentence contains an error with, which is corrected
with that (R:PREP). Although it is grammatically accurate to use agree with as
a transitive phrasal verb, an object clause of the verb in the example
sentence is not grammatical. In this case, the error annotation scheme
maintains the structure of the clause while replacing the preposition instead.
*in addition more and more scientists agree with alien really exist *In addition, more and more scientists agree that aliens really exist.
We analyze the error distribution in training data of different language
proficiency levels, in which the distribution of errors in the data sets of
proficiency levels B and C is similar: missing punctuation marks (M:PUNCT),
replacement prepositions (R:PREP), and missing determinants (M:DET) are the
most apparent types of errors. Additionally, proficiency A includes an extra
error type, replacement orthography (R:ORTH), which is defined for case or
whitespace errors. Table 1 shows the ratios of the most frequent error types
of training data in W&I, which we investigate thoroughly in $\S$4.
Proficiency A | Proficiency B | Proficiency C
---|---|---
M:PUNCT | 0.0933 | M:PUNCT | 0.1134 | M:PUNCT | 0.1183
R:ORTH | 0.0602 | R:PREP | 0.0589 | R:PREP | 0.0517
R:PREP | 0.0506 | M:DET | 0.0442 | M:DET | 0.0345
R:VERB:TENSE | 0.0455 | R:VERB | 0.0414 | R:VERB | 0.0323
R:VERB | 0.0419 | R:VERB:TENSE | 0.0393 | R:VERB:TENSE | 0.0273
Table 1: Most frequent errors and their ratio in W&I
## 3 Experimental Results
For experiments, we use the development data set of W&I from BEA2019, which
distinguishes language proficiency levels into A, B and C. We follow the
experimental setting described in Suzgun et al. (2022) for GPT-2 (gpt2-xl)
inferences, and we also adapt it to GPT-3.5 (text-davinci-003). Instead of
using the test data set for the BEA2019 (Bryant et al., 2019), we use the
development data set for evaluation to control proficiency levels. To evaluate
the performance of language proficiency levels A, B, and C, we report ERRANT
results (Bryant et al., 2017) as metrics that include true positive, false
positive, false negative, precision, recall, and more importantly, F0.5 scores
which emphasize precision than recall. Table 3 summarizes the prompting GEC
results for different language models, including GPT-2, GPT-3.5, and fine-
tuned GPT-2. We used default setting in Suzgun et al. (2022) for inference
parameters:
model | gpt2-xl
---|---
tokenizer | gpt2-xl
num_examplars | 0-4 shots
max_model_token_length | 256 if num_examplars is 0
| else 512
delimiter left and right | { }
1-shot | ungrammatical | This is important thing.
---|---|---
A | grammatical | This is an important thing.
2-shot | ungrammatical | Water is needed for alive.
A | grammatical | Water is necessary to live.
3-shot | ungrammatical | And young people spend time more ther lifestile.
A | grammatical | And young people spend more time on their lifestyles.
4-shot | ungrammatical | Both of these men have dealed with situations in an unconventional manner and the results are with everyone to see.
A | grammatical | Both of these men have dealt with situations in an unconventional manner and the results are plain to see.
Table 2: Prompt examples
We used the prompts described in Table 2, and the following setting for fine-
tuning parameters:
epochs | 5
---|---
using masked language modeling | False
block size (train) | 128
per_device_train_batch_size | 4
save_steps | 10000
save_total_limit | 2
| | A | B | C | all
---|---|---|---|---|---
| | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5
GPT-2 | zero-shot | 70 | 3944 | 2878 | 0.0174 | 0.0237 | 0.0184 | 45 | 5204 | 2453 | 0.0086 | 0.018 | 0.0096 | 28 | 4860 | 1058 | 0.0057 | 0.0258 | 0.0068 | 143 | 14008 | 6389 | 0.0101 | 0.0219 | 0.0113
| 1-shot | 86 | 3447 | 2862 | 0.0243 | 0.0292 | 0.0252 | 58 | 4240 | 2440 | 0.0135 | 0.0232 | 0.0147 | 28 | 3730 | 1058 | 0.0075 | 0.0258 | 0.0087 | 172 | 11417 | 6360 | 0.0148 | 0.0263 | 0.0163
| 2-shot | 103 | 4175 | 2845 | 0.0241 | 0.0349 | 0.0257 | 69 | 5442 | 2429 | 0.0125 | 0.0276 | 0.0141 | 30 | 4905 | 1056 | 0.0061 | 0.0276 | 0.0072 | 202 | 14522 | 6330 | 0.0137 | 0.0309 | 0.0154
| 3-shot | 140 | 4445 | 2808 | 0.0305 | 0.0475 | 0.0329 | 95 | 5710 | 2403 | 0.0164 | 0.038 | 0.0185 | 38 | 4979 | 1048 | 0.0076 | 0.035 | 0.009 | 273 | 15134 | 6259 | 0.0177 | 0.0418 | 0.02
| 4-shot | 133 | 4347 | 2815 | 0.0297 | 0.0451 | 0.0319 | 84 | 5422 | 2414 | 0.0153 | 0.0336 | 0.0171 | 31 | 4790 | 1055 | 0.0064 | 0.0285 | 0.0076 | 248 | 14559 | 6284 | 0.0167 | 0.038 | 0.0189
GPT-3.5 | zero-shot | 1203 | 3770 | 1740 | 0.2419 | 0.4088 | 0.2634 | 940 | 4693 | 1556 | 0.1669 | 0.3766 | 0.1878 | 407 | 4183 | 677 | 0.0887 | 0.3755 | 0.1047 | 2550 | 12646 | 3973 | 0.1678 | 0.3909 | 0.1894
| 1-shot | 1300 | 3086 | 1643 | 0.2964 | 0.4417 | 0.3173 | 1068 | 3562 | 1428 | 0.2307 | 0.4279 | 0.2541 | 472 | 3086 | 612 | 0.1327 | 0.4354 | 0.1541 | 2840 | 9734 | 3683 | 0.2259 | 0.4354 | 0.2499
| 2-shot | 1443 | 2983 | 1500 | 0.326 | 0.4903 | 0.3494 | 1116 | 3157 | 1380 | 0.2612 | 0.4471 | 0.2849 | 486 | 2592 | 598 | 0.1579 | 0.4483 | 0.1814 | 3045 | 8732 | 3478 | 0.2586 | 0.4668 | 0.2839
| 3-shot | 1477 | 2646 | 1466 | 0.3582 | 0.5019 | 0.38 | 1114 | 3164 | 1382 | 0.2604 | 0.4463 | 0.2841 | 479 | 2416 | 605 | 0.1655 | 0.4419 | 0.1891 | 3070 | 8226 | 3453 | 0.2718 | 0.4706 | 0.2969
| 4-shot | 1330 | 2328 | 1613 | 0.3636 | 0.4519 | 0.3784 | 1089 | 2424 | 1407 | 0.31 | 0.4363 | 0.329 | 457 | 1870 | 627 | 0.1964 | 0.4216 | 0.2199 | 2876 | 6622 | 3647 | 0.3028 | 0.4409 | 0.323
FT GPT-2 | zero-shot | 1118 | 1479 | 1830 | 0.4305 | 0.3792 | 0.4192 | 928 | 1203 | 1570 | 0.4355 | 0.3715 | 0.421 | 383 | 792 | 703 | 0.326 | 0.3527 | 0.331 | 2429 | 3474 | 4103 | 0.4115 | 0.3719 | 0.4029
| 1-shot | 1127 | 1668 | 1821 | 0.4032 | 0.3823 | 0.3989 | 925 | 1325 | 1573 | 0.4111 | 0.3703 | 0.4022 | 382 | 913 | 704 | 0.295 | 0.3517 | 0.3048 | 2434 | 3906 | 4098 | 0.3839 | 0.3726 | 0.3816
| 2-shot | 1107 | 1700 | 1841 | 0.3944 | 0.3755 | 0.3904 | 937 | 1359 | 1561 | 0.4081 | 0.3751 | 0.401 | 383 | 919 | 703 | 0.2942 | 0.3527 | 0.3043 | 2427 | 3978 | 4105 | 0.3789 | 0.3716 | 0.3774
| 3-shot | 1073 | 1860 | 1875 | 0.3658 | 0.364 | 0.3655 | 874 | 1596 | 1624 | 0.3538 | 0.3499 | 0.353 | 381 | 1168 | 705 | 0.246 | 0.3508 | 0.2616 | 2328 | 4624 | 4204 | 0.3349 | 0.3564 | 0.339
| 4-shot | 1032 | 1911 | 1916 | 0.3507 | 0.3501 | 0.3505 | 818 | 1815 | 1680 | 0.3107 | 0.3275 | 0.3139 | 359 | 1310 | 727 | 0.2151 | 0.3306 | 0.2313 | 2209 | 5036 | 4323 | 0.3049 | 0.3382 | 0.311
SOTA | gector | 1046 | 632 | 2054 | 0.6234 | 0.3374 | 0.533 | 785 | 458 | 1836 | 0.6315 | 0.2995 | 0.5169 | 315 | 208 | 845 | 0.6023 | 0.2716 | 0.4843 | 2146 | 1298 | 4735 | 0.6231 | 0.3119 | 0.5194
| t5 | 1338 | 741 | 1762 | 0.6436 | 0.4316 | 0.586 | 1018 | 620 | 1603 | 0.6215 | 0.3884 | 0.5549 | 377 | 351 | 783 | 0.5179 | 0.325 | 0.4629 | 2733 | 1712 | 4148 | 0.6148 | 0.3972 | 0.5541
Table 3: Prompting results using GPT-2 (gpt2-xl and ft = fine-tuned), GPT-3.5
(text-davinci-003) and SOTA results by models of gector (Omelianchuk et al.,
2020) and t5 (Rothe et al., 2021).
When evaluating the efficacy of few-shot strategies on GPT-2 and GPT-3.5, it
is evident that the few-shot prompting method exhibits better performance
compared to the zero-shot prompting method. For instance, in the all data set
which combines corpus of three language proficiency levels, we observe that
the 4-shot F0.5 scores for GPT-2 and GPT-3.5 are 0.0495 and 0.323
respectively, which are higher than the zero-shot F0.5 scores for GPT-2 and
GPT-3.5. It is also noticeable that the 4-shot approach consistently yields
higher F0.5 scores in comparison to the zero-shot approach. However, this
trend is not observed for the fine-tuned GPT-2 model on different language
proficiency levels. For example, in the all data set, the F0.5 score for the
4-shot approach is lower than the F0.5 score for the zero-shot approach.
Therefore, based on our experimental findings, it is feasible to conclude that
few-shot techniques may not have a significant impact on fine-tuned GPT-2
models.
In addition, GPT-2 exhibits a large decreasing rate of recall as the language
proficiency levels increase from A to C. Specifically, there is a notable
increase in the dropping rate of precision from 50.57% (0.0174 in A versus
0.0086 in B) to 33.72% (0.0086 in B versus 0.0057 in C). However, the fine-
tuned GPT-2 shows a better trend for the precision rate. From proficiency
level A to proficiency level C, the precision score increases from 0.4305 in A
to 0.4355 in B (+1.16%) and then drops to 0.326 (-25.14%) in C. It indicates
the fine-tuned model is more robust for different proficiency level data sets.
## 4 Analysis and Discussion
Unless specified otherwise, our analysis and discussion are based on results
of the fine-tuned gpt2-xl using zero-shot which we achieve the best results.
### Label-by-label evaluation approach
We implement a label-by-label evaluation method. As Bryant et al. (2017)
suggested, we provide edit operation-based and POS-based errors as well as
detailed breakdown composed errors (m—r—u with POS) to investigate further the
relationship between GEC and different proficiency levels. For example, Table
4 shows different types of error evaluation results. When comparing correcting
missing operation errors with all errors, it has higher F0.5 scores where it
suggests that GEC using GPT performs better in the specific missing error
regardless of language proficiency. M:PUNCT (missing punctuation marks) is the
most frequent error among all error types in three language proficiency, which
outperforms the entire results for all proficiency levels. This reflects the
general characteristics of the performance of GEC using GPT. R:VERB (replacing
verbs) consistently performs poorly compared to the entire results, and this
has the same tendency for all r edit errors where the proficiency C achieves
especially lower results. We observed that GEC using GPT contradicts to the
problem of over-correction for lower proficiency levels because of the much
higher numbers of FN in A and B.
| | TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---|---
M:PUNCT | A | 189 | 171 | 134 | 0.525 | 0.5851 | 0.536
| B | 203 | 132 | 133 | 0.606 | 0.6042 | 0.6056
| C | 95 | 96 | 80 | 0.4974 | 0.5429 | 0.5059
R:VERB | A | 21 | 60 | 113 | 0.2593 | 0.1567 | 0.2293
| B | 17 | 55 | 113 | 0.2361 | 0.1308 | 0.2033
| C | 6 | 43 | 51 | 0.1224 | 0.1053 | 0.1186
m | A | 318 | 436 | 372 | 0.3703 | 0.3571 | 0.1691
| B | 336 | 347 | 344 | 0.4919 | 0.4941 | 0.2458
| C | 157 | 222 | 168 | 0.4142 | 0.4830 | 0.2180
Table 4: Detailed breakdown evaluation results for the most frequent errors,
and missing operation errors (FT GPT2, zero-shot).
### Is recall higher than precision in prompting GPT for the GEC task?
Consistent higher recall compared to precision showcases a tendency of over-
correction in prompting GPT for the GEC task. We have observed that
proficiency levels A and B, however, do not exhibit such a propensity. It
holds true even for GPT-3.5, where recall consistently surpasses precision.
Nevertheless, the difference between precision and recall measurements in
levels A and B is considerably smaller compared to level C.
### Results using various F-scores
Table 5 shows results of FT GPT-2 and GPT-3.5 obtained with different
F-scores, where $\beta=$ 0.5, 1, and 2. The result implies that FT GPT-2 is
less prone to over-correction in comparison to GPT-3.5 because the F2 scores
are mostly higher in GPT-3.5. In traditional approaches in GEC, such as SOTA
results in Table 3, where the total numbers of TP and FP are relatively small,
F0.5 would be relevant to measure GEC results. Since recent approaches by
prompting GPT in the GEC task illustrate much higher numbers, especially FP,
it appears that the F1-score proves to be a more effective indicator in GEC
results.
| FT GPT-2 | GPT-3.5
---|---|---
| F0.5 | F1 | F2 | F0.5 | F1 | F2
A | 0.4192 | 0.4032 | 0.3885 | 0.3784 | 0.4030 | 0.4310
B | 0.4210 | 0.4010 | 0.3827 | 0.3291 | 0.3625 | 0.4034
C | 0.3310 | 0.3388 | 0.3470 | 0.2199 | 0.2680 | 0.3430
all | 0.3907 | 0.4029 | 0.3792 | 0.3590 | 0.3230 | 0.4040
Table 5: Different F-scores with F0.5, F1 and F2. FT GPT-2 results are based
on 0-shot, while GPT-3.5 (text-davinci-003) results are based on 4-shot.
### Comparison between prompting GPT and SOTA
State-of-the-art (SOTA) results continue to demonstrate superior performance
compared to prompting GPT in the GEC task in all aspects of results including
precision and recall measures regardless of proficiency levels. Our assumption
is primarily based on the fact that SOTA models are usually subjected to
extensive fine-tuning processes.
### Discussion
In this section, we present the evaluation outcomes using our own
implementation to count the numbers of TP, FP, and FN, which are different
from the ERRANT scores. We leave it as future work to further investigate and
explore potential improvements.
Additionally, while we examine a correlation between proficiency level C and
native in prompting GPT in GEC as shown in Table 6, we are unable to identify
any comparable behavior in prompting GPT in GEC for native-like proficiency C
and native proficiency. Hawkins and Buttery (2010) analyze that some error
types are more notable in B1 and B2 levels than C1 and C2 levels, such as
missing preposition and form of determiner. For example, there are more errors
like missing preposition (M:PREP) or replacement of determiners (R:DET) in B
than in C, which confirm what the previous work proposes. Table 7 shows a
behavior of prompting GPT in the GEC task proficiency specific errors, in
which finding their correlation could be excessively challenging because of
the performance of GEC for proficiency level C. We consider results of the
proficiency level C as unnatural behavior, which deviates significantly from
what is considered typical prompting GPT in GEC. We also leave it as future
work.
| TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---
C | 383 | 792 | 703 | 0.326 | 0.3527 | 0.331
N | 2429 | 3474 | 4103 | 0.4115 | 0.3719 | 0.4029
Table 6: Results between proficiency level C and native | | TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---|---
M:PREP | B | 24 | 29 | 31 | 0.4528 | 0.4364 | 0.4494
| C | 9 | 23 | 17 | 0.2812 | 0.3462 | 0.2922
R:DET | B | 15 | 30 | 41 | 0.3333 | 0.2679 | 0.3178
| C | 7 | 12 | 23 | 0.3684 | 0.2333 | 0.3302
Table 7: Detailed breakdown evaluation results for M:PREP and R:DET
## 5 Conclusion
In this paper, we investigated the strengths and limitations of prompting GPT
for the GEC task based on different language proficiency levels. We used our
own implementations to calculate relevant metrics for label-by-label analysis,
which are different from the current standard ERRANT scores by using m2 files.
We observed a tendency of over-correction in prompting GPT, and it is more
obvious in the recent version of GPTs, where recall consistently surpasses
precision. Additionally, since prompting GPT generates much higher false
positive numbers in results, the F1-score, rather than the F0.5-score, would
be a more effective measure in GEC results.
## 6 Ethics Statement
To confirm Behavioural Research Ethics at the University of British
Columbia,111https://ethics.research.ubc.ca/behavioural-research-ethics authors
have obtained a certificate of the Tri-Council Policy Statement: Ethical
Conduct for Research Involving Humans (TCPS 2): Course on Research Ethics
(CORE-2022).222https://tcps2core.ca
## Acknowledgement
This work was supported in part by Oracle Cloud credits and related resources
provided by Oracle for Research.
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# Transverse MHD waves as signatures of braiding-induced magnetic reconnection
in coronal loops
A. Ramada C. Sukarmadji Department of Mathematics, Physics and Electrical
Engineering, Northumbria University
Newcastle upon Tyne, NE1 8ST, UK Patrick Antolin Department of Mathematics,
Physics and Electrical Engineering, Northumbria University
Newcastle upon Tyne, NE1 8ST, UK
###### Abstract
A major coronal heating theory based on magnetic reconnection relies on the
existence of braided magnetic field structures in the corona. In this small-
angle reconnection scenario, numerical simulations indicate that the
reconnected magnetic field lines are driven sideways by magnetic tension and
can overshoot from their new rest position, thereby leading to low-amplitude
transverse MHD waves. This provides an efficient mechanism for transverse MHD
wave generation, and the direct causality also constitutes substantial
evidence of reconnection from braiding. However, this wave-generation
mechanism has never been directly observed. Recently, the telltale signature
of small-angle reconnection in a sheared coronal structure has been identified
through nanojets, which are small, short-lived, and fast jet-like bursts in
the nanoflare range transverse to the guide-field. We present for the first
time IRIS and SDO observations of transverse MHD waves in a coronal loop that
directly result from braiding-induced reconnection. The reconnection is
identified by the presence of nanojets at the loop apex which release
nanoflare-range energy. We find that the oscillations have an energy flux on
the order of $10^{6}-10^{8}$ erg cm-2 s-1, which is within the budget to power
active region loops. The estimated kinetic and thermal energy from the
nanojets is also sufficient to power the transverse waves and sustain the
observed heating at the loop apex. This discovery provides major support to
(a) existing theories that transverse MHD waves can be a signature of
reconnection, (b) the existence of braiding in coronal structures and (c) the
coronal reconnection scenario identified by nanojets.
The Sun (1693) — Solar corona (1483) — Solar magnetic fields (1503) — Solar
coronal waves (1995) — Magnetohydrodynamics (1964)
## 1 Introduction
Magnetohydrodynamic (MHD) waves and magnetic reconnection are the two leading
theories for the coronal heating problem, which has been a subject of
investigation for decades. MHD waves can carry large amounts of energy (Uchida
& Kaburaki, 1974), making them a suitable candidate for heating through their
dissipation (Wentzel 1979; Klimchuk 2006; Van Doorsselaere et al. 2020). This
was supported by the discovery of transverse oscillations in loop-like
structures (Aschwanden et al. 1999; Nakariakov et al. 1999), and followed up
by many other works reporting similar oscillations (Aschwanden et al. 2002;
Verwichte et al. 2004; Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Van
Doorsselaere et al. 2009; Okamoto et al. 2015; Anfinogentov et al. 2015; Li &
Long 2023). However, direct observational evidence of wave-based heating
remains scarce (Van Doorsselaere et al., 2020).
Among transverse oscillations, the kink oscillations (Nakariakov et al., 2021)
are the most prominently observed and often found to be decayless (Tian et al.
2012; Wang et al. 2012; Nisticò et al. 2013; Anfinogentov et al. 2013). Many
external driving mechanisms have been proposed for these decayless
oscillations, including footpoint driving (Nisticò et al. 2013), quasi-steady
flows (Nakariakov et al. 2016), or by Alfvénic vortex shedding (Karampelas &
Van Doorsselaere 2021). Proposed internal mechanisms include the combination
of resonant absorption and the Kelvin-Helmholtz instability (Antolin et al.
2016; Antolin & Van Doorsselaere 2019) or coronal rain (Kohutova & Verwichte
2017), but there is still a lack of observational evidence.
Another major coronal heating candidate is the Parker nanoflare theory
(Parker, 1988), which conjectures the existence of myriad energy bursts at the
order of $10^{24}$ erg generated by small-scale magnetic reconnection events
driven by braiding. The braided state of a loop is thought to be the result of
slow footpoint motions at photospheric level (Pontin & Priest, 2022). Energy
releases within the nanoflare range have been previously reported by e.g.
Testa et al. (2013) and Testa et al. (2014), and Chitta et al. (2018) have
also suggested that chromospheric reconnection from flux cancellation in loop
footpoints may facilitate nanoflare-sized energy release in loops. However, a
direct link to coronal reconnection could not be established in these heating
events.
The discovery of nanojets by Antolin et al. (2021) provided direct evidence of
nanoflare-based heating driven by small-scale component reconnection. Nanojets
are small-scale and short-lived bursts, around 500 km in width and 1000 km in
length on average, that last no longer than 15 s on average. They are a result
of very fast transverse motion of reconnected field lines driven by magnetic
tension, combined with localised heating (the nanoflare). In Antolin et al.
(2021), they were observed in a loop-like structure and driven by the loss of
stability of a nearby prominence. This was then followed by observations of
nanojets in loop-like structures with coronal rain Sukarmadji et al. (2022),
with the Kelvin-Helmholtz instability (KHI) and Rayleigh-Taylor instability
(RTI) as the likely underlying drivers. The different observations of nanojets
in a variety of environments with different drivers further suggests that they
may be common, and could contribute significantly to the heating of the solar
corona.
It has been long known that magnetic reconnection can produce all kinds of MHD
waves (e.g. Petschek 1964, Parker 1991, Kigure et al. 2010), however, it is
unclear which waves would be predominantly observed in the Parker nanoflare
theory. The kink instability as a trigger of reconnection and driver of
coronal heating has been extensively studied through numerical simulations of
twisted magnetic fields (Browning et al. 2008; Hood et al. 2009; Bareford et
al. 2013; Hood et al. 2016; Reid et al. 2018; Reid et al. 2020). In
particular, Hood et al. (2009), Hood et al. (2016), Reid et al. (2018), and
Reid et al. (2020) proposed the existence of twisted coronal braids or
strands, some of which would become unstable thereby setting a cascade of
nanoflare-sized reconnection events affecting neighboring stable strands.
Although not investigated in detail, these works show the generation of
transverse MHD waves during the reconnection process. Observationally,
Kohutova et al. (2020) have detected torsional Alfvén waves produced from a
reconnection event, although the configuration leading to the reconnection
corresponds to the presence of 2 separate coronal structures and not a single
braided structure. To date, there are no direct observations of small-angle
reconnection events leading to kink waves. Yet, as mentioned previously, kink
waves are ubiquitous in the solar corona and their origin is highly debated.
We present in this paper first observations of transverse oscillations driven
by small-angle reconnection events in a coronal loop, where the reconnections
are identified by the presence of nanojets. We will look into the properties
of the waves produced, and discuss the heating contributed from the observed
event. In Section 2, we present the observation and the present structures.
Section 3 discusses the reconnection nanojets, followed with a discussion of
the transverse waves produced in Section 4. We will discuss the energy budget
in Section 5 and provide conclusions in Section 6.
## 2 Observations
An observation of AR 12192 was taken by the Interface Region Imaging
Spectrograph (IRIS; De Pontieu et al. 2014) on the 29th of October 2014
between 08:37:04-13:43:35 UT, observing in the SJI 1330 filtergram with
spatial resolution, temporal cadence, and exposure time of $0.16\arcsec$, 9.6
s, and 8 s respectively. This is a large coarse 8-step raster observation
centered at (x,y) = ($956\arcsec$,$-262\arcsec$) with a field-of-view (FOV) of
$119\arcsec\times 119\arcsec$. We use the level 2 data for our analysis, along
with coaligned observations from the Atmospheric Imaging Assembly (AIA; Lemen
et al. 2012). During the observing period, the area produced a series of C to
M-class flare events with a number of surges and quiescent and flaring coronal
rain.
Our focus will be on a hot loop shown in Figure 1. We initially observe a
diffuse bright loop-like region at 11:49:37 UT only visible in AIA 94 and 131
and faintly in the SJI 1330 (the Fe XXI emission), which disappears after
13:18:01 UT. Following this disappearance, the loop is seen to form at
13:23:37 UT in SJI 1330, AIA 304, 171, 193, 211, and faintly in 131 and 335,
suggesting catastrophic cooling and the appearance of coronal rain. The time
range of interest starts at 13:30:19 UT, when the cool loop exhibits a
secondary heating event and is seen in all changes but only faintly in AIA 94,
335 and 1600. The structure remains visible until the end of the observing
period of IRIS, and until 14:19:07 UT in AIA. Unfortunately, the IRIS slit
does not cross our loop of interest, and therefore we can only obtain plane-
of-sky (POS) values for all of our measurements.
The loop appears first at a measured height at the apex of $29\pm 5$ Mm as
measured in the POS from the solar surface, and a length of $120\pm 20$ Mm in
the POS. Within the loop, we observe coronal rain flowing with POS velocities
of 20-32 km s-1 at the apex and 100-120 km s-1 along the legs of the loop. The
rain strands have widths of $600\pm 45$ km, and the apparent width of the loop
at the apex seen in the upper transition region or coronal AIA passbands is
$4059\pm 489$ km.
We used the Basis Pursuit Method, based on the Differential Emission Measure
(DEM) Analysis (Cheung et al. 2015) to estimate the emission distribution with
respect to temperature. The DEM weighted electron number density of the loop
$\langle n_{e}\rangle_{DEM}$ depends on the emission $DEM(T)$ in the LOS $l$
for a given temperature bin
$DEM(T)=n_{e}n_{H}\frac{dl}{dT},$ (1)
where $n_{e}$ and $n_{H}$ are the electron and hydrogen number densities,
respectively. $\langle n_{e}\rangle_{DEM}$ follows
$\langle n_{e}\rangle_{DEM}=\sqrt{\frac{\int_{T}DEM(T)_{loop}dT}{1.2l}},$ (2)
where we assume a fully ionised plasma with $10\%$ Helium abundance and $l$ is
the length of the emitting plasma in the loop along the LOS.
$DEM(T)_{loop}=DEM(T)_{LOS}-DEM(T)_{env}$ with $DEM(T)_{LOS}$ is the DEM at a
given temperature bin for a LOS crossing the loop, and $DEM(T)_{env}$ is the
DEM for a LOS neighboring the loop in the same temperature bin. We thus assume
that the foreground and background along this neighboring LOS is the same as
that crossing the loop so that the subtraction gives the DEM of the loop. We
averaged the EM values from the DEM calculation that has converged, for pixels
contained in the loop for each temperature bin. The temperature bins used are
$log(T)=5.5-6.4$ as they show emission from the loop.
We assume that $l$ is similar to the POS width of the loop to obtain an
electron number density of $4.7\pm 0.5\times 10^{9}$ cm-3, which corresponds
to the optically thin hot plasma surrounding the coronal rain strands. The
number density of the cool rain strands is estimated through pressure balance
and taking the peak temperature response for SJI 1330 ($10^{4.3}$ K), to find
$3.8\pm 0.8\times 10^{11}$ cm-3. This matches previous measurements of coronal
rain densities in observations (Antolin et al. 2015; Froment et al. 2020) and
numerical simulations (Li et al. 2022; Antolin et al. 2022a) .
## 3 Reconnection Nanojets
Figure 1: First two rows shows the snapshots of the loop at the time when the
nanojets are most visible in SJI 1330, AIA 131, 171, 193, 211, and 304. The
bottom row shows snapshots for selected emission bins from Log(T) = 5.8, 6.2
and 6.4. Three of the clearest nanojets (left to right, N1, N2, and N3) are
marked with the white arrows. An animation of this figure is available online
showing the time evolution of the loop. Figure 2: The two panels of the left
column show the same snapshot of the apex of the loop with nanojets. In the
top panel, the three most visible nanojets are marked, and in the bottom
panel, slices along the trajectory of the nanojets A, B, and C are taken to
produce the time-distance diagrams on the first three panels of the right
column. The nanojets in the time-distance diagrams are marked with N1-N6. The
white vertical lines in the time-distance diagrams mark the time of the
snapshot from the left column. The region contained by the white contour line
in the bottom panel of the left row shows the region used to produce the
light-curve plot shown in the bottom panel of the right column. The light
curves are constructed by summing over the intensity values within the contour
at a given timestamp and then normalised. In the bottom we show a schematic
for our interpretation of how a nanojet forms from reconnection due to
misalignments between $\vec{B}_{1}$ and $\vec{B}_{2}$. The resulting
configuration is likely to be affected by the tension from the reconnection,
which overshoots the resting configuration and therefore produces
oscillations, as seen at $t=t_{3}$. An animation of this figure without the
schematic is available online, showing the nanojet formation in the SJI, and
the white vertical line in the time distance diagrams and light curves
following the timestamp of the SJI image. The 7 other observed nanojets are
also marked in the animation with white arrows (nanojets pointing upwards) and
cyan arrows (nanojets pointing downwards).
The reconnection is identified by the presence of nanojets at the apex of the
loop, shown in Figure 1 and the top left panel of Figure 2. After 13:35:35 UT,
we begin to observe around 10 small jet-like structures characteristic of
nanojets forming at the loop apex with 7 of them oriented upwards (away from
the solar surface) and the remaining oriented downwards (towards the solar
surface). Following Antolin et al. (2021) and Sukarmadji et al. (2022), we
identify these structures as nanojets if they have the small, transverse (to
the loop) jet-like features in the image and the running difference image
(current image subtracted with image from previous timestamp) with widths and
lengths of around 500 km and 1000 km, respectively, and are accompanied with a
transverse motion fairly perpendicular to the loop and with short timescales
of less than $25$ s. We investigated 3 of the clearest ones (N1, N2, and N3)
to have a measure of their properties. The mean of their POS lengths and
widths are 1500 km and 341 km, with values ranging from $838-2781$ km and
$320-405$ km respectively. The lifetimes of all three nanojets are $19\pm 5$
s. We also measured the DEM weighted electron number densities and
temperatures of the nanojets. The electron number density is measured through
Equation 2, using the emission from the pixel containing the nanojet and
assuming that $l$ is similar to the POS width of the nanojet. The nanojet’s
temperature is measured through
$T=T_{t_{0}}+\Delta T,$ (3)
averaged over all the nanojet pixels, where $T_{t_{0}}$ is the DEM weighted
temperature of the region before the nanojet forms, following
$T_{t_{0}}=\frac{\int DEM(T)_{t_{0}}TdT}{\int DEM(T)_{t_{0}}dT},$ (4)
and $\Delta T$ is the temperature change at the nanojet timestamp, measured by
the variation of the DEM:
$\Delta T=\frac{\int\Delta DEM(T)TdT}{\int\Delta DEM(T)dT},$ (5)
where $\Delta DEM(T)=DEM(T)_{nanojet}-DEM(T)_{t_{0}}$. $DEM(T)_{nanojet}$ is
the DEM at the nanojet timestamp and $DEM(T)_{t0}$ is the DEM at a timestamp
before the nanojet forms. We find a mean number density and temperature of
$1.2\times 10^{10}$ cm-3 and 2.3 MK, with values ranging from $0.9-1.4\times
10^{10}$ cm-3 and $2.2-2.4$ MK, respectively. The nanojets have an average POS
speed of 156 km s-1, ranging from $91-290$ km s-1.
To obtain a measure of the kinetic and thermal energy, the kinetic energy
($E_{K}$) and thermal energy ($E_{T}$) is calculated through
$E_{K}=\frac{1}{2}1.27V\langle n_{e}\rangle_{DEM}m_{p}v^{2}$ and
$E_{T}=\frac{3}{2}2.09V\langle n_{e}\rangle_{DEM}kT$, where $V$ is the nanojet
volume, $m_{p}$ is the proton mass, $k$ is the Boltzmann constant, and $T$ is
the average nanojet temperature (following Equation 3). The factors 1.27 and
2.09 comes from assuming a 10% Helium abundance and a highly ionized plasma
111Although coronal rain corresponds to partially ionised plasma, numerical
simulations show that its ionisation fraction is relatively high (Antolin et
al. 2022b).. We assume that the nanojet has a cylindrical structure with
radius and length set by the observed mean values, and $v$ is set equal to the
mean POS velocity. This gives us energy releases within the nanoflare range,
with a kinetic and thermal energy release average of $7.8\times 10^{24}$ erg
and $1.4\times 10^{25}$ erg per nanojet, with values of $0.8-21.7\times
10^{24}$ erg and $0.9-2.3\times 10^{25}$ erg, respectively. The average total
energy per nanojet is therefore $2.2\times 10^{25}$ erg, and as we observe
around 10 nanojets, the total estimated energy released from all nanojets is
$2.2\times 10^{26}$ erg. From these measured cases, the nanojets have similar
morphologies, dynamics, and energy release as in Antolin et al. (2021) and
Sukarmadji et al. (2022), although slightly smaller in size. We also observe
local brightenings in the hot AIA channels in the regions where we observe
nanojets, as shown in in Figure 1, supporting the presence of localised
heating in the loop-like structure. The nanojets are particularly seen in AIA
131, 171, 193, 211, 304, and the DEM bin Log(T) = 5.8 and 6.4.
These signatures are clearer in a light curve plot integrating over the loop
region containing the nanojets, shown in Figure 2 (see figure caption for
methods), where the occurrence of nanojets coincides with peaks of the average
loop apex intensity in various passbands (all except 335). The light curve
intensity for the SJI 1330, AIA channels 211, 193, 171, 131 ramps up to two
intensity peaks as the nanojets form, with a minor first peak when N5 and N6
form, and a major peak when N1-N4 form (shown in the time-distance diagrams).
The intensity in all channels starts to decrease at 13:41:58 UT (or $t=383$ s
in the light-curve) after the nanojets have disappeared. This is with the
exception for AIA 304, where there is continuous increase afterwards likely
due to the hot Si XI emission (1.5 MK) within the 304 passband, as suggested
by the temperature variation.
## 4 Transverse Oscillations
Figure 3: The IRIS observation of our data taken in the SJI 1330 filtergram.
The two panels in the left column show the loop-like structure of interest
(top), and the zoomed-in portion from the top figure’s box with slices 1-5
across the loop (bottom). The 5 slices produce the time-distance diagrams
shown in the right column, where the solid white vertical lines indicate the
time of the left column’s snapshots. Time distances 1 and 2 show the slopes
indicating the transverse velocities of the strands. After $t=150$ s, we can
see oscillating structures in the time-distance diagrams. An animation of this
figure is available online, showing the evolution of the loop-like structure
where the white vertical line in the time distance diagrams follows the
timestamp of the panels showing the SJI images. The end of the time distance
diagram is also the end of the IRIS observing period.
After 13:37:59 UT (around $t=150$ s in the time-distance diagrams), the
nanojets that form are followed by the upward motion of several nearby rain
strands, initiating a transverse motion as seen in the time-distance diagrams
of Figure 2 and Figure 3 between t = $150-300$ s. The initial upward motion of
the strands originating from the nanojets are identified by the diagonal
bright slopes across the loop, where the slopes indicate velocities of $40-48$
km s-1 (two examples are marked in Figure 3). The moving strands surpass the
upper edges of the loop at $t=290$ s. After $t=290$ s, it can be seen that the
strands start an oscillatory motion while continuing to move upward at
gradually slower speeds until the end of the IRIS observation. A schematic of
these dynamics is shown in the bottom panel of Figure 2, where the nanojets’
transverse motion overshoots the resting configuration resulting in an
oscillatory field motion. These oscillations have a measured period of $97\pm
4$ s.
Figure 4: Top row shows the SJI 1330 and AIA 304 image with the slice (white
line) for the time distance diagram plots in the second and third row. The
white vertical line in the time distance diagrams marks the time of the SJI
and AIA image. The time-distance diagrams have the white dots marking the
position of the oscillations which are selected based on an intensity
threshold that separates the loop from its surrounding. The third row shows
the combined oscillation data from the SJI 1330 (first 232 seconds, in green)
and AIA 304 (remaining 232 seconds, in blue) and the calculated upward moving
trend (red dots). Each point in the upward moving trend is calculated by
averaging a given point’s value with 8 of its neighbouring points. The IRIS
and AIA points are subtracted with the global trend points to recover the
detrended oscillations, shown in the bottom plot. We observe 2 clear periods
before it damps.
The signatures of multiple strands oscillating can also be seen from the
presence of multiple waves in Figure 3’s time-distance diagrams, and an
example is shown in panel 5 where two waves are labeled as A and B. Note that
these 2 waves are out-of-phase with each other, suggesting that each wave in
the time-distance diagram comes from individual strands. Despite not being in
phase with one another, all of the waves have similar periods with only 2-5
seconds differences, suggesting similar conditions across the strands. In
total, we observe around $7\pm 1$ strands that oscillates from the SJI images
and time-distance diagrams. The oscillations are also most visible in the AIA
304 and 171, and faintly in 131, 193, 211, and 1600.
We show time-distance diagrams of the oscillations in selected channels in
Figures 4 and 5. Note that the sinusoidal shape of the oscillations in the AIA
is not as clear as it is in the SJI, due to the small displacement and low
spatial resolution of AIA. This means that it may be a challenge to identify
them as oscillations had there been no accompanying observations from IRIS.
In Figures 4 and 5, there appears to be a damping of the oscillation at the
extended observing time of AIA after IRIS’s observing time has ended. From
Figure 4, we can estimate the wave properties by combining the observed
oscillations in IRIS and AIA to obtain the de-trended oscillations (see
text/caption for further details). From the de-trended oscillations, two
periods are seen with a measured maximum POS displacement ($x_{max}$) $321\pm
30$ km. We fitted these values in a sinusoidal function of
$x(t)=x_{max}\sin(\frac{2\pi t}{T})$ using the period $T$ to obtain a velocity
profile of $v(t)=w\cos(2\pi t/T)$, with an amplitude $w=\frac{2\pi
x_{max}}{T}$ of $21\pm 2$ km s-1. The loop appears to oscillate for two
periods before it damps, but it must be noted that the oscillations that we
managed to recover well are only from the IRIS data thanks to its higher
spatial resolution.
## 5 Energy Budget of the Reconnection Event and Waves
As the oscillations occur at the apex of the loop, the fundamental mode is the
most likely to be excited. The wave can either be a standing mode or a
propagating wave. Following the nanojets, we initially observe strands that
oscillate out-of-phase with each other, but eventually appear to oscillate
collectively in the upper part of the loop. For this case, we can then assume
that all strands oscillate with a global kink mode. However, we can also
consider a multiple kink wave scenario, in which individual strands oscillate
with their own kink mode (e.g. in Figure 3). The SJI time-distance diagrams in
Figures 2 and 3 also show 7 strands that oscillate individually. We therefore
have four possible cases: A global kink mode in which the strands oscillate as
a whole, multiple kink modes guided by individual strands, and for each of
these two cases we have either a standing (fundamental mode) or a propagating
wave.
In the case of a fundamental kink mode, the period $P$ of the fundamental mode
is
$P=\frac{2L}{c_{k}},$ (6)
where L is the length of the loop and the phase speed $c_{k}$ is
$c_{k}=\sqrt{\frac{\rho_{i}v_{A_{i}}^{2}+\rho_{e}v_{A_{e}}^{2}}{\rho_{i}+\rho_{e}}}.$
(7)
$c_{k}$ depends on the number density inside $\rho_{i}$ and outside $\rho_{e}$
the waveguide (the loop or the strand in the global kink mode or individual
strands cases, respectively), and the corresponding Alfvén speeds
$v_{A}=\frac{B}{\sqrt{\mu\rho}}$ inside and outside are written as $v_{A_{i}}$
and $v_{A_{e}}$, where the magnetic field strength B is expected to vary
little under coronal conditions. The energy flux of kink modes in a bundle of
loop $E_{flux}$ can be calculated from Equation 8 (Van Doorsselaere et al.,
2014), using the transverse velocity amplitude $w$ measured from the
oscillations and the filling factor $f$, following
$E_{flux}=\frac{1}{2}f(\rho_{i}+\rho_{e})w^{2}c_{k}.$ (8)
The total energy can also be estimated following Van Doorsselaere et al.
(2014) through
$E_{total}=\pi
R^{2}L\left(\frac{1}{2}(\rho_{i}+\rho_{e})w^{2}-f\frac{1}{4}\rho_{e}\frac{c^{2}_{k}+v^{2}_{Ae}}{c^{2}_{k}}w^{2}\right).$
(9)
R is the radius of the loop (we have used half of the apparent width of the
loop apex from Section 2), and $L$ is the length of the loop portion that is
oscillating.
For the global kink mode and propagating wave case, the filling factor can be
estimated from the area occupied by the observed number of strands in IRIS
within the oscillating loop’s cross-section observed in AIA (Van Doorsselaere
et al., 2014). Observationally, we observe 6 strands inside the loop portion
(we observe 7 oscillating strands, but 1 has dampen by the time the
oscillating portion forms), but it must be noted that this is a lower bound
since there may be other strands that overlap one another. The oscillating
loop width that contains all the oscillating strands is $3272\pm 386$ km,
whereas an individual strand has a measured width of $600\pm 45$ km. Assuming
a circular geometry, if we fill the 6 strands in the loop we will have a
filling factor of $0.20\pm 0.05$. We will also assume that the external number
density (outside the loop, $\rho_{e}$) is $10^{8}$ cm-3, and use the internal
number density of the loop (surrounding the strands observed in the SJI,
$\rho_{i}$) from the DEM analysis of $4.7\pm 0.5\times 10^{9}$ cm-3 for this
case.
Using the values for $\rho_{i}$ and $\rho_{e}$ obtained above, the measured
loop length of $120\pm 20$ Mm in the POS for $L$, the wave period of $97\pm 4$
s, we have $c_{k}=2474\pm 425$ km s-1 for the global standing kink mode.
Assuming that the magnetic field inside the loop and outside the loop are
similar, the estimated magnetic field is $B=66\pm 11$ G to match the observed
period for the fundamental mode. The $E_{flux}$ and $E_{total}$ for this case
are $1.2\pm 0.5$ $\times 10^{6}$ erg cm-2 s-1 and $2.3\pm 1.0\times 10^{25}$
erg, respectively. Whereas for the global propagating mode case, we have used
the measured minimum phase speed $v_{ph}$ for $c_{k}$ and the length of the
loop portion that appears to oscillate of $16800\pm 720$ km for $L$, to obtain
$B=32\pm 17$ G. The $E_{flux}$ and $E_{total}$ for this case is $0.6\pm 0.4$
$\times 10^{6}$ erg cm-2 s-1 and $3.3\pm 1.5\times 10^{24}$ erg, respectively.
For the multiple kink mode case, the filling factor is 1 since we are
resolving the strands with IRIS, and we will use the coronal rain number
density of $3.4\pm 0.7\times 10^{11}$ cm-3 for $\rho_{i}$ and the DEM weighted
number density for $\rho_{e}$. For the standing waves case we find that
$B=555\pm 96$ G, and $E_{flux}$ and $E_{total}$ for a single strand are
$4.3\pm 1.3\times 10^{8}$ erg cm-2 s-1 and $4.4\pm 1.3\times 10^{25}$ erg,
respectively. We have $7\pm 1$ strands oscillating, so the total energy
released is $3.1\pm 1.7\times 10^{26}$ erg. For the multiple propagating kink
modes case (using $v_{k}=c_{k}$), we obtain $B=274\pm 145$ G, and $E_{flux}$
and $E_{total}$ of $2.1\pm 1.2\times 10^{8}$ erg cm-2 s-1 and $6.1\pm
1.5\times 10^{24}$ erg, respectively. For 7 strands, $E_{total}$ is $4.3\pm
2.2\times 10^{25}$ erg.
The above are lower bound estimates since all measurements are only in the
POS. Assuming that the Doppler velocity component is of the same order as the
POS component, then v increases by $\sqrt{2}$. Taking account these
considerations, this leads to energy flux ranges of $1.2\pm 0.4$ $\times
10^{6}$ erg cm-2 s-1 to $8.6\pm 1.3$ $\times 10^{8}$ erg cm-2 s-1 from all
four cases. Whereas the estimated wave energy will range between $6.5\pm
1.5\times 10^{24}$ erg to $6.1\pm 2.5\times 10^{26}$ erg.
We can also calculate the total thermal energy released from the reconnection
events between a given time $t_{0}$ (right before any nanojet occurrence) and
$t$. This value $\Delta TE(t)$ can be calculated using the DEM values, for a
portion of the loop that contains nanojets and is oscillating using
$\Delta TE(t)=\frac{3}{2}2.09V\langle n_{e}\rangle_{DEM}k_{B}\langle\Delta
T(t)\rangle_{DEM}$ (10)
with the assumption of 10% Helium abundance and a highly ionized plasma.
$\langle{n}_{e}\rangle_{DEM}$ is the DEM weighted electron number density of
the loop (from Section 2), $V=\pi R^{2}L_{osc}$ is the loop portion’s volume
assuming a cylindrical structure and the length of the oscillating loop
$L_{osc}$ of $16800\pm 720$ km. $\langle\Delta T(t)\rangle_{DEM}$ is the
average temperature variation of the loop following Equation 5. This isolates
the temperature change from the reconnection events associated with the
nanojets that contributes to the thermal energy. We calculate the DEM for the
time period starting from 13:35:11 UT - 13:41:59 UT (meaning that $t_{0}$ =
13:35:11 UT). Figure 5 plots the $\Delta TE(t)$ for the loop portion, and we
find that there is a continuous increase in the thermal energy to a maximum of
$4.0\times 10^{25}$ erg.
Figure 5: Top row shows snapshots of the loop apex in IRIS SJI 1330, AIA 304,
and AIA 171. The white line in each panel is the slice taken for the
respective time-distance diagrams shown in the next three rows, where t = 0 s
corresponds to 13:35:11 UT. The time-distance diagrams have the white dots
marking the position of the oscillations which are selected based on an
intensity threshold that separates the loop from its surrounding. The vertical
white line in the time-distance diagram shows the time of the snapshots in the
respective panels above. In the time-distance diagrams, the green vertical
lines marks the period where we observe clear oscillations in IRIS. The plot
in the bottom row shows the thermal energy release change from $t=t_{0}$, for
the region of the loop that appears to be oscillating, estimated from the DEM
for the region bounded by the white contour in the AIA images.
## 6 Discussions and Conclusions
Our observations suggest that the transverse waves are produced by small-angle
reconnection events within the structure, where the reconnection signatures
can be identified by the nanojets. Prior to the nanojets, the loop did not
show any oscillations as seen in the time-distance diagrams of Figures 3 and
5. It is only when the nanojets form that we observe the separation of
individual strands and their subsequent oscillation, which is then followed by
multiple strands that eventually appear to oscillate collectively after all of
the nanojets have formed. This indicates that the nanojets and the transverse
MHD waves share the same generation mechanism, i.e. magnetic reconnection, and
that nanojets reflect the energy available to power the oscillations.
In the small-angle reconnection scenario, the reconnected magnetic field lines
are driven sideways by magnetic tension but overshoot from their new rest
position, thereby leading to transverse waves. This scenario suggests an
efficient mechanism for transverse MHD wave generation. If common (as
conjectured by Parker), it therefore provides an alternative explanation to
the observed ubiquity of small-amplitude transverse MHD waves in the corona.
The period of our observed oscillation is $97\pm 4$ s with a maximum
displacement and amplitude of $321\pm 30$ km and $21\pm 2$ km s-s
respectively. We have considered four cases, depending on whether the observed
kink mode is a global kink mode (case in which the strands oscillate in phase
on average) or a multiple kink mode (case in which each strand oscillates
independently). Furthermore, we consider 2 cases for the modes: standing or
propagating modes. The estimated magnetic field strengths are $32-66$ G for
the global cases, and $274-555$ G for the individual strands cases. These
values are considerably high but still expected from an active region
producing a series of C to M-class flares (e.g. Asai et al. 2001; Landi et al.
2021; Wei et al. 2021).
The produced oscillations also have similar periods suggesting similar
conditions across the strands, and we observe that they occur for 2 periods
before they damp. The fact that the kink waves strongly damp in a loop that is
visible in the hot AIA channels throughout the event strongly suggest wave
dissipation and heating. Based on the measured density, filling factor and
wave properties, we estimate that the energy flux from the waves is on the
order of $10^{6}-10^{8}$ erg cm-2 s-1 for all cases, which is sufficient to
balance the energy losses for active regions (Withbroe & Noyes 1977). These
values are lower bound estimates since we only measure projected velocities,
but they indicate that braiding-induced reconnection has enough energy to
power active region coronal loops.
If the dynamics at the origin of the nanojets are what triggers the kink mode,
then we may expect that the kinetic energy from the waves should match the
kinetic energy of the total number of nanojets. We divide the wave’s total
energy released ($E_{total}$) with the average kinetic energy released by a
nanojet of $7.8\times 10^{24}$ erg to obtain an estimate of how many nanojets
are required to match the wave energy for each case: For the global standing
kink mode, we only require 2-3 nanojets, which is less than the 10 clearest
nanojets observed by eye. For the global propagating wave case, we need less
than 1 nanojet’s worth of kinetic energy.
For the multiple standing kink mode, we require around 39-40 nanojets, which
is substantially more than the observed number of nanojets. We did observe
around 10-12 other nanojet-like features that were too small or faint,
suggesting that such numbers are indeed possible. Whereas for the multiple
propagating waves, we only require around 5-6 nanojets.
An expected feature from nanojets is the strand separation that accompanies
the small-angle reconnection (Antolin et al. 2021; Sukarmadji et al. 2022),
which would overshoot the resting field configuration. However, this strand
separation may not always lead to transverse oscillations. For example, if two
internal misalignments trigger nanojets that have opposite directions, the
resulting oscillation may be a sausage mode rather than a kink mode. Also, if
the separation is accompanied by a displacement of the footpoints then minimal
or no overshoot is produced. This suggests that the reconnection events may
need to have very specific conditions to produce sufficient overshoot to
trigger transverse waves.
The thermal energy increase from the DEM values at the apex also shows an
increase on the order of $10^{25}$ erg, with a maximum value of $4.0\times
10^{25}$ erg just after the nanojets have stopped forming. Part of the kinetic
energy of the nanojets is also likely converted into heat, and the thermal
energy increase is on the same order of magnitude than the nanojet’s average
total energy release of $2.2\times 10^{25}$ erg. If we assume that the thermal
energy increase comes from the nanojet’s total (kinetic and thermal) energy,
this would mean that we only need around 2 nanojets in total. This means that
only a few nanojets is required to sustain the heating seen at the apex of the
loop.
We observe strands appearingly misaligned to one another, similar to the loops
observed in Sukarmadji et al. (2022). Furthermore, the rain flows along the
legs of the loop also appear to be misaligned, suggesting a braided structure.
The entire event starts with a few nanojets, which produce transverse motion
and likely create more misalignments triggering the following nanojet clusters
that occur over the next five minutes. This is similar to an MHD avalanche,
which is expected from previous MHD simulations of braided structures, that
produce bursty nanoflare-sized heating (Hood et al. 2009; Hood et al. 2016;
Reid et al. 2018; Reid et al. 2020).
The event from this work is evidence that kink waves can be a signature of
braiding-induced magnetic reconnection, and that the generated kink waves can
be used as a diagnostic of the energy released through reconnection. It is
likely that a large proportion of heating is still undetected through AIA: The
fact that the oscillations are barely resolved in the AIA channels may wrongly
suggest that there is very little wave energy in the corona. The oscillations
and nanojets are only clear in IRIS, and were also only clearly detected
thanks to the presence of coronal rain in the strands.
A major open question is how often the small-angle reconnection leads to kink
waves, and whether a constant generation of nanojets could support the
decayless kink oscillations commonly observed. If this is indeed the case,
then braided field lines should be expected in oscillating loops as we require
numerous misalignments to consistently produce nanojets that would sustain a
decayless oscillation. However, the kink waves observed in this event damp
very quickly, leading to a question of whether unresolved reconnection
processes power decayless oscillations.
P. A. acknowledges funding from his STFC Ernest Rutherford Fellowship (No.
ST/R004285/2). IRIS is a NASA small explorer mission developed and operated by
LMSAL with mission operations executed at NASA Ames Research Center and major
contributions to downlink communications funded by ESA and the Norwegian Space
Centre. SDO is part of NASA’s Living With a Star Program. All data used in
this work are publicly available through the websites of the respective solar
missions.
## 7 Appendix 1
Figure 6: Left column, top to bottom: The SJI snapshot with the nanojets
marked, the running difference image (current snapshot subtracted with the
previous snapshot), and the SJI image showing the slices A, B, and C used for
the time-distance diagrams from the SJI images and the running difference
images on the right column. The nanojets in the time-distance diagrams are
marked with N1-N6. The white vertical lines in the time-distance diagrams mark
the time of the snapshots from the left column. The causality between the
nanojets and the oscillations is seen from how the strands only move upwards
following a nanojet, which is then followed by an oscillation. An example of
this trajectory is marked by the white dots in the time-distance diagram B,
where it starts with nanojet N6, and is followed by a transverse upward motion
that surpasses the upper part of the loop which then initiates the oscillatory
motion at t = 300 s of the diagram. An animation of this figure is available
online, showing the nanojet formation in the SJI, and the white vertical line
in the time distance diagrams following the timestamp of the SJI images.
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|
# How to Use the IEEEtran LaTeX Templates
IEEE Publication Technology Department Manuscript created October, 2020; This
work was developed by the IEEE Publication Technology Department. This work is
distributed under the LaTeX Project Public License (LPPL) ( http://www.latex-
project.org/ ) version 1.3. A copy of the LPPL, version 1.3, is included in
the base LaTeX documentation of all distributions of LaTeX released 2003/12/01
or later. The opinions expressed here are entirely that of the author. No
warranty is expressed or implied. User assumes all risk.
###### Abstract
This document describes the most common article elements and how to use the
IEEEtran class with LaTeX to produce files that are suitable for submission to
the Institute of Electrical and Electronics Engineers (IEEE). IEEEtran can
produce conference, journal and technical note (correspondence) papers with a
suitable choice of class options.
###### Index Terms:
Class, IEEEtran, LaTeX, paper, style, template, typesetting.
## I Introduction
Welcome to the updated and simplified documentation to using the IEEEtran
LaTeX class file. The IEEE has examined hundreds of author submissions using
this package to help formulate this easy to follow guide. We will cover the
most commonly used elements of a journal article. For less common elements we
will refer back to the “IEEEtran_HOWTO.pdf”.
This document applies to version 1.8b of IEEEtran.
The IEEEtran template package contains the following example files:
* bare_jrnl.tex
* bare_conf.tex
* bare_jrnl_compsoc.tex
* bare_conf_compsoc.tex
* bare_jrnl_comsoc.tex
These are “bare bones” templates to quickly understand the document structure.
It is assumed that the reader has a basic working knowledge of LaTeX. Those
who are new to LaTeX are encouraged to read Tobias Oetiker’s “The Not So Short
Introduction to LaTeX”, available at:
http://tug.ctan.org/info/lshort/english/lshort.pdf which provides an overview
of working with LaTeX.
## II The Design, Intent and
Limitations of the Templates
The templates are intended to approximate the final look and page length of
the articles/papers. Therefore, they are NOT intended to be the final produced
work that is displayed in print or on IEEEXplore®. They will help to give the
authors an approximation of the number of pages that will be in the final
version. The structure of the LaTeXfiles, as designed, enable easy conversion
to XML for the composition systems used by the IEEE’s outsource vendors. The
XML files are used to produce the final print/IEEEXplore® pdf and then
converted to HTML for IEEEXplore®. Have you looked at your article/paper in
the HTML version?
## III LaTeX Distributions: Where to Get Them
IEEE recommends using the distribution from the TeXUser Group at
http://www.tug.org. You can join TUG and obtain a DVD distribution or download
for free from the links provided on their website:
http://www.tug.org/texlive/. The DVD includes distributions for Windows, Mac
OS X and Linux operating systems.
## IV Where to get the IEEEtran Templates
The IEEE Template Selector will always have the most up-to-date versions of
the LaTeX and MSWord templates. Please see: https://template-
selector.ieee.org/ and follow the steps to find the correct template for your
intended publication. Many publications use the IEEETran LaTeX templates,
however, some publications have their own special templates. Many of these are
based on IEEEtran, but may have special instructions that vary slightly from
those in this document.
## V Where to get LaTeX help - user groups
The following on-line groups are very helpful to beginning and experienced
LaTeX users. A search through their archives can provide many answers to
common questions.
* http://www.latex-community.org/
* https://tex.stackexchange.com/
## VI Document Class Options in IEEEtran
At the beginning of your LaTeX file you will need to establish what type of
publication style you intend to use. The following list shows appropriate
documentclass options for each of the types covered by IEEEtran.
* Regular Journal Article
* $\backslash$documentclass[journal]IEEEtran
* Conference Paper
* $\backslash$documentclass[conference]IEEEtran
* Computer Society Journal Article
* $\backslash$documentclass[10pt,journal,compsoc]IEEEtran
* Computer Society Conference Paper
* $\backslash$documentclass[conference,compsoc]IEEEtran
* Communications Society Journal Article
* $\backslash$documentclass[journal,comsoc]IEEEtran
* Brief, Correspondence or Technote
* $\backslash$documentclass[9pt,technote]IEEEtran
There are other options available for each of these when submitting for peer
review or other special requirements. IEEE recommends to compose your article
in the base 2-column format to make sure all your equations, tables and
graphics will fit the final 2-column format. Please refer to the document
“IEEEtran_HOWTO.pdf” for more information on settings for peer review
submission if required by your EIC.
## VII How to Create Common Front Matter
The following sections describe general coding for these common elements.
Computer Society publications and Conferences may have their own special
variations and will be noted below.
### VII-A Paper Title
The title of your paper is coded as:
\title{The Title of Your Paper}
Please try to avoid the use of math or chemical formulas in your title if
possible.
### VII-B Author Names and Affiliations
The author section should be coded as follows:
\author{Masahito Hayashi
\IEEEmembership{Fellow, IEEE}, Masaki Owari
\thanks{M. Hayashi is with Graduate School
of Mathematics, Nagoya University, Nagoya,
Japan}
\thanks{M. Owari is with the Faculty of
Informatics, Shizuoka University,
Hamamatsu, Shizuoka, Japan.}
}
Be sure to use the $\backslash$IEEEmembership command to identify IEEE
membership status. Please see the “IEEEtran_HOWTO.pdf” for specific
information on coding authors for Conferences and Computer Society
publications. Note that the closing curly brace for the author group comes at
the end of the thanks group. This will prevent you from creating a blank first
page.
### VII-C Running Heads
The running heads are declared by using the $\backslash$markboth command.
There are two arguments to this command: the first contains the journal name
information and the second contains the author names and paper title.
\markboth{Journal of Quantum Electronics,
Vol. 1, No. 1, January 2021}
{Author1, Author2,
\MakeLowercase{\textit{(et al.)}:
Paper Title}
### VII-D Copyright Line
For Transactions and Journals papers, this is not necessary to use at the
submission stage of your paper. The IEEE production process will add the
appropriate copyright line. If you are writing a conference paper, please see
the “IEEEtran_HOWTO.pdf” for specific information on how to code ”Publication
ID Marks”.
### VII-E Abstracts
The abstract is the first element of a paper after the $\backslash$maketitle
macro is invoked. The coding is simply:
\begin{abstract}
Text of your abstract.
\end{abstract}
Please try to avoid mathematical and chemical formulas in the abstract.
### VII-F Index Terms
The index terms are used to help other researchers discover your paper. Each
society may have it’s own keyword set. Contact the EIC of your intended
publication for this list.
\begin{IEEEkeywords}
Broad band networks, quality of service
\end{IEEEkeywords}
## VIII How to Create Common Body Elements
The following sections describe common body text elements and how to code
them.
### VIII-A Initial Drop Cap Letter
The first text paragraph uses a “drop cap” followed by the first word in ALL
CAPS. This is accomplished by using the $\backslash$IEEEPARstart command as
follows:
\IEEEPARstart{T}{his} is the first paragraph
of your paper. . .
### VIII-B Sections and Subsections
Section headings use standard LaTeX commands: $\backslash$section,
$\backslash$subsection and $\backslash$subsubsection. Numbering is handled
automatically for you and varies according to type of publication. It is
common to not indent the first paragraph following a section head by using
$\backslash$noindent as follows:
\section{Section Head}
\noindent The text of your paragraph . . .
### VIII-C Citations to the Bibliography
The coding for the citations are made with the LaTeX $\backslash$cite command.
This will produce individual bracketed reference numbers in the IEEE style. At
the top of your LaTeX file you should include:
\usepackage{cite}
For a single citation code as follows:
see \cite{ams}
This will display as: see [1]
For multiple citations code as follows:
\cite{ams,oxford,lacomp}
This will display as [1, 2, 3]
### VIII-D Figures
Figures are coded with the standard LaTeX commands as follows:
\begin{figure}[!t]
\centering
\includegraphics[width=2.5in]{fig1}
\caption{This is the caption for one fig.}
\label{fig1}
\end{figure}
The [!t] argument enables floats to the top of the page to follow IEEE style.
Make sure you include:
\usepackage{graphicx}
at the top of your LaTeXfile with the other package declarations.
To cross-reference your figures in the text use the following code example:
See figure \ref{fig1} ...
This will produce:
See figure 1 . . .
Figure 1: This is the caption for one fig.
### VIII-E Tables
Tables should be coded with the standard LaTeX coding. The following example
shows a simple table.
\begin{table}
\begin{center}
\caption{Filter design equations ...}
\label{tab1}
\begin{tabular}{| c | c | c |}
\hline
Order & Arbitrary coefficients &
coefficients\\
of filter & $e_m$ & $b_{ij}$ \\
\hline
1& $b_{ij}=\hat{e}.\hat{\beta_{ij}}$,
& $b_{00}=0$\\
\hline
2&$\beta_{22}=(~1,-1,-1,~~1,~~1,~~1)$ &\\
\hline
3& $b_{ij}=\hat{e}.\hat{\beta_{ij}}$,
& $b_{00}=0$,\\
\hline
\end{tabular}
\end{center}
\end{table}
To reference the table in the text, code as follows:
Table~\ref{tab1} lists the closed-form...
to produce:
Table I lists the closed-form . . .
TABLE I: A Simple Table Example. Order | Arbitrary coefficients | coefficients
---|---|---
of filter | $e_{m}$ | $b_{ij}$
1 | $b_{ij}=\hat{e}.\hat{\beta_{ij}}$, | $b_{00}=0$
2 | $\beta_{22}=(~{}1,-1,-1,~{}~{}1,~{}~{}1,~{}~{}1)$ |
3 | $b_{ij}=\hat{e}.\hat{\beta_{ij}}$, | $b_{00}=0$,
### VIII-F Lists
In this section, we will consider three types of lists: simple unnumbered,
numbered and bulleted. There have been numerous options added to IEEEtran to
enhance the creation of lists. If your lists are more complex than those shown
below, please refer to the “IEEEtran_HOWTO.pdf” for additional options.
A plain unnumbered list
* bare_jrnl.tex
* bare_conf.tex
* bare_jrnl_compsoc.tex
* bare_conf_compsoc.tex
* bare_jrnl_comsoc.tex
coded as:
\begin{list}{}{}
\item{bare\_jrnl.tex}
\item{bare\_conf.tex}
\item{bare\_jrnl\_compsoc.tex}
\item{bare\_conf\_compsoc.tex}
\item{bare\_jrnl\_comsoc.tex}
\end{list}
A simple numbered list
1. 1.
bare_jrnl.tex
2. 2.
bare_conf.tex
3. 3.
bare_jrnl_compsoc.tex
4. 4.
bare_conf_compsoc.tex
5. 5.
bare_jrnl_comsoc.tex
coded as:
\begin{enumerate}
\item{bare\_jrnl.tex}
\item{bare\_conf.tex}
\item{bare\_jrnl\_compsoc.tex}
\item{bare\_conf\_compsoc.tex}
\item{bare\_jrnl\_comsoc.tex}
\end{enumerate}
A simple bulleted list
* •
bare_jrnl.tex
* •
bare_conf.tex
* •
bare_jrnl_compsoc.tex
* •
bare_conf_compsoc.tex
* •
bare_jrnl_comsoc.tex
coded as:
\begin{itemize}
\item{bare\_jrnl.tex}
\item{bare\_conf.tex}
\item{bare\_jrnl\_compsoc.tex}
\item{bare\_conf\_compsoc.tex}
\item{bare\_jrnl\_comsoc.tex}
\end{itemize}
### VIII-G Other Elements
For other less common elements such as Algorithms, Theorems and Proofs, and
Floating Structures such as page-wide tables, figures or equations, please
refer to the “IEEEtran_HOWTO.pdf” section on “Double Column Floats.”
## IX How to Create Common Back Matter Elements
The following sections demonstrate common back matter elements such as
Acknowledgments, Bibliographies, Appendicies and Author Biographies.
### IX-A Acknowledgments
This should be a simple paragraph before the bibliography to thank those
individuals and institutions who have supported your work on this article.
\section{Acknowledgments}
\noindent Text describing those who
supported your paper.
### IX-B Bibliographies
References Simplified: A simple way of composing references is to use the
$\backslash$bibitem macro to define the beginning of a reference as in the
following examples:
[6] H. Sira-Ramirez. “On the sliding mode control of nonlinear systems,”
Systems & Control Letters, vol. 19, pp. 303–312, 1992.
coded as:
\bibitem{Sira3}
H. Sira-Ramirez. ‘‘On the sliding mode
control of nonlinear systems,’’
\textit{Systems \& Control Letters},
vol. 19, pp. 303--312, 1992.
[7] A. Levant.“Exact differentiation of signals with unbounded higher
derivatives,” in Proceedings of the 45th IEEE Conference on Decision and
Control, San Diego, California, USA, pp. 5585–5590, 2006.
coded as:
\bibitem{Levant}
A. Levant. ‘‘Exact differentiation of
signals with unbounded higher
derivatives,’’ in \textit{Proceedings
of the 45th IEEE Conference on
Decision and Control}, San Diego,
California, USA, pp. 5585--5590, 2006.
[8] M. Fliess, C. Join, and H. Sira-Ramirez. “Non-linear estimation is easy,”
International Journal of Modelling, Identification and Control, vol. 4, no. 1,
pp. 12–27, 2008.
coded as:
\bibitem{Cedric}
M. Fliess, C. Join, and H. Sira-Ramirez.
‘‘Non-linear estimation is easy,’’
\textit{International Journal of Modelling,
Identification and Control}, vol. 4,
no. 1, pp. 12--27, 2008.
[9] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez. “Stabilization of
food-chain systems using a port-controlled Hamiltonian description,” in
Proceedings of the American Control Conference, Chicago, Illinois, USA, pp.
2245–2249, 2000.
coded as:
\bibitem{Ortega}
R. Ortega, A. Astolfi, G. Bastin, and H.
Rodriguez. ‘‘Stabilization of food-chain
systems using a port-controlled Hamiltonian
description,’’ in \textit{Proceedings of the
American Control Conference}, Chicago,
Illinois, USA, pp. 2245--2249, 2000.
### IX-C Accented Characters in References
When using accented characters in references, please use the standard LaTeX
coding for accents. Do not use math coding for character accents. For example:
\’e, \"o, \‘a, \~e
will produce: é, ö, à, ẽ
### IX-D Use of BibTeX
If you wish to use BibTeX, please see the documentation that accompanies the
IEEEtran Bibliography package.
### IX-E Biographies and Author Photos
Authors may have options to include their photo or not. Photos should be a
bit-map graphic (.tif or .jpg) and sized to fit in the space allowed. Please
see the coding samples below:
\begin{IEEEbiographynophoto}{Jane Doe}
Biography text here without a photo.
\end{IEEEbiographynophoto}
or a biography with a photo
\begin{IEEEbiography}[{\includegraphics
[width=1in,height=1.25in,clip,
keepaspectratio]{fig1.png}}]
{IEEE Publications Technology Team}
In this paragraph you can place
your educational, professional background
and research and other interests.
\end{IEEEbiography}
Please see the end of this document to see the output of these coding
examples.
## X Mathematical Typography
and Why It Matters
Typographical conventions for mathematical formulas have been developed to
provide uniformity and clarity of presentation across mathematical texts. This
enables the readers of those texts to both understand the author’s ideas and
to grasp new concepts quickly. While software such as LaTeX and MathType® can
produce aesthetically pleasing math when used properly, it is also very easy
to misuse the software, potentially resulting in incorrect math display.
IEEE aims to provide authors with the proper guidance on mathematical
typesetting style and assist them in writing the best possible article.
As such, IEEE has assembled a set of examples of good and bad mathematical
typesetting. You will see how various issues are dealt with. The following
publications have been referenced in preparing this material:
* _Mathematics into Type_ , published by the American Mathematical Society
* _The Printing of Mathematics_ , published by Oxford University Press
* _The LaTeXCompanion_, by F. Mittelbach and M. Goossens
* _More Math into LaTeX_ , by G. Grätzer
* AMS-StyleGuide-online.pdf, published by the American Mathematical Society
Further examples can be seen at http://journals.ieeeauthorcenter.ieee.org/wp-
content/uploads/sites/7/IEEE-Math-Typesetting-Guide.pdf
### X-A Display Equations
A simple display equation example shown below uses the “equation” environment.
To number the equations, use the $\backslash$label macro to create an
identifier for the equation. LaTeX will automatically number the equation for
you.
$x=\sum_{i=0}^{n}2{i}Q.$ (1)
is coded as follows:
\begin{equation}
\label{deqn_ex1}
x = \sum_{i=0}^{n} 2{i} Q.
\end{equation}
To reference this equation in the text use the $\backslash$ref macro. Please
see (1)
is coded as follows:
Please see (\ref{deqn_ex1})
### X-B Equation Numbering
Consecutive Numbering: Equations within an article are numbered consecutively
from the beginning of the article to the end, i.e., (1), (2), (3), (4), (5),
etc. Do not use roman numerals or section numbers for equation numbering.
Appendix Equations: The continuation of consecutively numbered equations is
best in the Appendix, but numbering as (A1), (A2), etc., is permissible.
Hyphens and Periods: Hyphens and periods should not be used in equation
numbers, i.e., use (1a) rather than (1-a) and (2a) rather than (2.a) for sub-
equations. This should be consistent throughout the article.
### X-C Multi-line equations and alignment
Here we show several examples of multi-line equations and proper alignments.
A single equation that must break over multiple lines due to length with no
specific alignment.
$\text{The first line of this example}\\\ \text{The second line of this
example}\\\ \text{The third line of this example}$ (2)
is coded as:
\begin{multline}
\text{The first line of this example}\\
\text{The second line of this example}\\
\text{The third line of this example}
\end{multline}
A single equation with multiple lines aligned at the = signs
$\displaystyle a$ $\displaystyle=c+d$ (3) $\displaystyle b$
$\displaystyle=e+f$ (4)
is coded as:
\begin{align}
a &= c+d \\
b &= e+f
\end{align}
The align environment can align on multiple points as shown in the following
example:
$\displaystyle x$ $\displaystyle=y$ $\displaystyle X$ $\displaystyle=Y$
$\displaystyle a$ $\displaystyle=bc$ (5) $\displaystyle x^{\prime}$
$\displaystyle=y^{\prime}$ $\displaystyle X^{\prime}$
$\displaystyle=Y^{\prime}$ $\displaystyle a^{\prime}$ $\displaystyle=bz$ (6)
is coded as:
\begin{align}
x &= y & X & =Y & a &=bc\\
x’ &= y’ & X’ &=Y’ &a’ &=bz
\end{align}
### X-D Subnumbering
The amsmath package provides a subequations environment to facilitate
subnumbering. An example:
$\displaystyle f$ $\displaystyle=g$ (7a) $\displaystyle f^{\prime}$
$\displaystyle=g^{\prime}$ (7b) $\displaystyle\mathcal{L}f$
$\displaystyle=\mathcal{L}g$ (7c)
is coded as:
\begin{subequations}\label{eq:2}
\begin{align}
f&=g \label{eq:2A}\\
f’ &=g’ \label{eq:2B}\\
\mathcal{L}f &= \mathcal{L}g \label{eq:2c}
\end{align}
\end{subequations}
### X-E Matrices
There are several useful matrix environments that can save you some
keystrokes. See the example coding below and the output.
A simple matrix:
$\begin{matrix}0&1\\\ 1&0\end{matrix}$ (8)
is coded as:
\begin{equation}
\begin{matrix} 0 & 1 \\
1 & 0 \end{matrix}
\end{equation}
A matrix with parenthesis
$\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}$ (9)
is coded as:
\begin{equation}
\begin{pmatrix} 0 & -i \\
i & 0 \end{pmatrix}
\end{equation}
A matrix with square brackets
$\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ (10)
is coded as:
\begin{equation}
\begin{bmatrix} 0 & -1 \\
1 & 0 \end{bmatrix}
\end{equation}
A matrix with curly braces
$\begin{Bmatrix}1&0\\\ 0&-1\end{Bmatrix}$ (11)
is coded as:
\begin{equation}
\begin{Bmatrix} 1 & 0 \\
0 & -1 \end{Bmatrix}
\end{equation}
A matrix with single verticals
$\begin{vmatrix}a&b\\\ c&d\end{vmatrix}$ (12)
is coded as:
\begin{equation}
\begin{vmatrix} a & b \\
c & d \end{vmatrix}
\end{equation}
A matrix with double verticals
$\begin{Vmatrix}i&0\\\ 0&-i\end{Vmatrix}$ (13)
is coded as:
\begin{equation}
\begin{Vmatrix} i & 0 \\
0 & -i \end{Vmatrix}
\end{equation}
### X-F Arrays
The array environment allows you some options for matrix-like equations. You
will have to manually key the fences, but you’ll have options for alignment of
the columns and for setting horizontal and vertical rules. The argument to
array controls alignment and placement of vertical rules.
A simple array
$\left(\begin{array}[]{cccc}a+b+c&uv&x-y&27\\\
a+b&u+v&z&134\end{array}\right)$ (14)
is coded as:
\begin{equation}
\left(
\begin{array}{cccc}
a+b+c & uv & x-y & 27\\
a+b & u+v & z & 134
\end{array} \right)
\end{equation}
A slight variation on this to better align the numbers in the last column
$\left(\begin{array}[]{cccr}a+b+c&uv&x-y&27\\\
a+b&u+v&z&134\end{array}\right)$ (15)
is coded as:
\begin{equation}
\left(
\begin{array}{cccr}
a+b+c & uv & x-y & 27\\
a+b & u+v & z & 134
\end{array} \right)
\end{equation}
An array with vertical and horizontal rules
$\left(\begin{array}[]{c|c|c|r}a+b+c&uv&x-y&27\\\ \hline\cr
a+b&u+v&z&134\end{array}\right)$ (16)
is coded as:
\begin{equation}
\left(
\begin{array}{c|c|c|r}
a+b+c & uv & x-y & 27\\
a+b & u+v & z & 134
\end{array} \right)
\end{equation}
Note the argument now has the pipe ”$|$” included to indicate the placement of
the vertical rules.
### X-G Cases Structures
Many times we find cases coded using the wrong environment, i.e., array. Using
the cases environment will save keystrokes (from not having to type the
$\backslash$left$\backslash$lbrace) and automatically provide the correct
column alignment.
${z_{m}(t)}=\begin{cases}1,&{\text{if}}\ {\beta}_{m}(t)\\\
{0,}&{\text{otherwise.}}\end{cases}$
is coded as follows:
\begin{equation*}
{z_m(t)} =
\begin{cases}
1,&{\text{if}}\ {\beta }_m(t),\\
{0,}&{\text{otherwise.}}
\end{cases}
\end{equation*}
Note that the “&” is used to mark the tabular alignment. This is important to
get proper column alignment. Do not use $\backslash$quad or other fixed spaces
to try and align the columns. Also, note the use of the $\backslash$text macro
for text elements such as “if” and “otherwise”.
### X-H Function Formatting in Equations
In many cases there is an easy way to properly format most common functions.
Use of the $\backslash$ in front of the function name will in most cases,
provide the correct formatting. When this does not work, the following example
provides a solution using the $\backslash$text macro.
$d_{R}^{KM}=\underset{d_{l}^{KM}}{\text{arg
min}}\\{d_{1}^{KM},\ldots,d_{6}^{KM}\\}.$
is coded as follows:
\begin{equation*}
d_{R}^{KM} = \underset {d_{l}^{KM}}
{\text{arg min}} \{ d_{1}^{KM},
\ldots,d_{6}^{KM}\}.
\end{equation*}
### X-I Text Acronyms inside equations
This example shows where the acronym “MSE” is coded using $\backslash$text{}
to match how it appears in the text.
$\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\hat{Y_{i}})^{2}$
\begin{equation*}
\text{MSE} = \frac {1}{n}\sum _{i=1}^{n}
(Y_{i} - \hat {Y_{i}})^{2}
\end{equation*}
### X-J Obsolete Coding
Avoid the use of outdated environments, such as eqnarray and $$ math
delimiters, for display equations. The $$ display math delimiters are left
over from PlainTeX and should not be used in LaTeX, ever. Poor vertical
spacing will result.
### X-K Use Appropriate Delimiters for Display Equations
Some improper mathematical coding advice has been given in various YouTubeTM
videos on how to write scholarly articles, so please follow these good
examples:
For single-line unnumbered display equations, please use the following
delimiters:
\[ . . . \] or
\begin{equation*} . . . \end{equation*}
Note that the * in the environment name turns off equation numbering.
For multiline unnumbered display equations that have alignment requirements,
please use the following delimiters:
\begin{align*} . . . \end{align*}
For single-line numbered display equations, please use the following
delimiters:
\begin{equation} . . . \end{equation}
For multiline numbered display equations, please use the following delimiters:
\begin{align} . . . \end{align}
## XI LaTeX Package Suggestions
Immediately after your documenttype declaration at the top of your LaTeX file
is the place where you should declare any packages that are being used. The
following packages were used in the production of this document.
\usepackage{amsmath,amsfonts}
\usepackage{algorithmic}
\usepackage{array}
\usepackage[caption=false,font=normalsize,
labelfont=sf,textfont=sf]{subfig}
\u00sepackage{textcomp}
\usepackage{stfloats}
\usepackage{url}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{balance}
## XII Additional Advice
Please use “soft” (e.g., `\eqref{Eq}`) or `(\ref{Eq})` cross references
instead of “hard” references (e.g., `(1)`). That will make it possible to
combine sections, add equations, or change the order of figures or citations
without having to go through the file line by line.
Please note that the `{subequations}` environment in LaTeX will increment the
main equation counter even when there are no equation numbers displayed. If
you forget that, you might write an article in which the equation numbers skip
from (17) to (20), causing the copy editors to wonder if you’ve discovered a
new method of counting.
BibTEX does not work by magic. It doesn’t get the bibliographic data from thin
air but from .bib files. If you use BibTEX to produce a bibliography you must
send the .bib files.
LaTeX can’t read your mind. If you assign the same label to a subsubsection
and a table, you might find that Table I has been cross referenced as Table
IV-B3.
LaTeX does not have precognitive abilities. If you put a `\label` command
before the command that updates the counter it’s supposed to be using, the
label will pick up the last counter to be cross referenced instead. In
particular, a `\label` command should not go before the caption of a figure or
a table.
Please do not use `\nonumber` or `\notag` inside the `{array}` environment. It
will not stop equation numbers inside `{array}` (there won’t be any anyway)
and it might stop a wanted equation number in the surrounding equation.
## XIII A Final Checklist
1. 1.
Make sure that your equations are numbered sequentially and there are no
equation numbers missing or duplicated. Avoid hyphens and periods in your
equation numbering. Stay with IEEE style, i.e., (1), (2), (3) or for sub-
equations (1a), (1b). For equations in the appendix (A1), (A2), etc..
2. 2.
Are your equations properly formatted? Text, functions, alignment points in
cases and arrays, etc.
3. 3.
Make sure all graphics are included.
4. 4.
Make sure your references are included either in your main LaTeX file or a
separate .bib file if calling the external file.
## References
* [1] Mathematics into Type, American Mathematical Society. Online available:
* [2] T.W. Chaundy, P.R. Barrett and C. Batey, The Printing of Mathematics, Oxford University Press. London, 1954.
* [3] The LaTeXCompanion, by F. Mittelbach and M. Goossens
* [4] More Math into LaTeX, by G. Grätzer
* [5] AMS-StyleGuide-online.pdf, published by the American Mathematical Society
* [6] H. Sira-Ramirez. “On the sliding mode control of nonlinear systems,” Systems & Control Letters, vol. 19, pp. 303–312, 1992.
* [7] A. Levant. “Exact differentiation of signals with unbounded higher derivatives,” in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, California, USA, pp. 5585–5590, 2006.
* [8] M. Fliess, C. Join, and H. Sira-Ramirez. “Non-linear estimation is easy,” International Journal of Modelling, Identification and Control, vol. 4, no. 1, pp. 12–27, 2008.
* [9] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez. “Stabilization of food-chain systems using a port-controlled Hamiltonian description,” in Proceedings of the American Control Conference, Chicago, Illinois, USA, pp. 2245–2249, 2000.
Jane Doe Biography text here without a photo.
---
| IEEE Publications Technology Team In this paragraph you can place your
educational, professional background and research and other interests.
---|---
|
spelled out, appropriately relaxed, correspond to deformations of a symmetric
product orbifold CFT by a “single-trace” $T\bar{T}$ and $J\bar{T}$
deformation, i.e. an operator of the form
$\sum_{i=1}^{N}T_{i}\bar{T}_{i}\;\;\;\;\;\;\;\mbox{or}\;\;\;\;\;\;\;\sum_{i=1}^{N}J_{i}\bar{T}_{i}$
(4.8)
where the index $i$ runs over the $N$ copies of the CFT. For specific choices
of the seed CFT, such theories have been proposed to be holographically dual
to the background obtained from the NS5 decoupling limit of the NS5-F1 system
[32] and, respectively, to a decoupling limit that yields a warped AdS3
spacetime [23, 16].
We concentrate on the single-trace $T\bar{T}$ deformation (the $J\bar{T}$ case
is entirely analogous). We will moreover focus on the untwisted sector only,
where the effect of the single-trace $T\bar{T}$ deformation on the spectrum is
clearly understood, as it acts independently in each copy. In this case, one
can build a Virasoro algebra $L_{m}^{\mu,i}$ associated to each of the copies,
and expect that the corresponding $L_{0}^{\mu,i}$ is related via (4.4) to the
Hamiltonian $H_{i}$ associated to that copy. The total Hamiltonian can then be
written as
$H=\sum_{i=1}^{N}\frac{\sqrt{1+4\mu\left(L_{0}^{\mu,i}+\bar{L}_{0}^{\mu,i}\right)+4\mu^{2}\left(L_{0}^{\mu,i}-\bar{L}_{0}^{\mu,i}\right)^{2}}-1}{2\mu}$
(4.9)
The total Virasoro generators can be built as
$L_{m}^{\mu}=\sum_{i}L_{m}^{\mu,i}$ and are manifestly symmetric under the
interchange of the copies. Since the various copies commute with each other,
we find
$[L_{m}^{\mu},H]=\sum_{i}\alpha_{m}(H^{i},P^{i})L_{m}^{\mu,i}$ (4.10)
Given this commutator, it is straightforward to construct a generator whose
eigenvalue is conserved
$L^{\mu}_{m,S}(t)=\sum_{i}e^{i\alpha_{m}(H^{i},P_{i})t}L_{m,S}^{\mu,i}=\sum_{i}L^{\mu,i}_{m,S}(t)$
(4.11)
Since the states in the symmetric product orbifold are symmetrized, the
expectation values of $H_{i}$ across the copies should be equal, thus
producing an overall time-dependent phase factor, which would be interesting
to reproduce from a gravitational calculation. For such a symmetric state, in
which $\langle H_{i}\rangle=\langle H\rangle/N$, it is interesting to note
that the departure of $\alpha_{m}$ from its usual CFT value is
$\langle\alpha_{m}\rangle=m\hbar-\frac{2m\mu\hbar}{N}\frac{\langle
H\rangle-\langle P\rangle}{1+2\mu m\hbar}+\mathcal{O}(1/N^{2})$ (4.12)
suggesting that even if the non-locality scale of the theory is set by $\mu$,
the departure of the non-local Virasoro algebra from a standard one, as
measured by the commutator $\alpha_{m}$, is suppressed by powers of $1/N$, and
thus possibly hard to detect in the dual gravity picture. If the corrections
associated to the inclusion of twisted sectors do not significantly modify
this picture, this ‘large N’ supression mechanism of the non-local commutator
could provide an explanation for the ubiquity of Virasoro asymptotic symmetry
groups in three-dimensional spacetimes that are not asymptotically AdS3.
_More general deformations_
As noted in the introduction, there exist many examples of string backgrounds
that are obtained through non-trivial decoupling limits, and which could be
dual to non-local, UV complete two-dimensional QFTs, which appear to fulfill
at least the first condition on having a Virasoro algebra.
To understand whether a direct relation between $L_{0}^{\lambda}$ and $H$ can
be established, as required by our second condition, one can try to analyse
the spectrum of the deformed theory (representing the $H$ eigenvalues) as a
function of the undeformed one (representing the eigenvalues of
$L_{0}^{\lambda}$). In $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs, the two are
related in a universal manner that does not depend at all on the details (or
particular spectrum) of the seed CFT. For this reason, the universal relation
between the deformed and undeformed spectra in these theories can be lifted to
a general relation between the operators $H$ and $L_{0}^{\lambda}$.
For the more general deformations we are interested in, the situation is more
nuanced242424We thank Gregory Korchemsky for discussions of this point.. On
the one hand, reducing the existence of the Virasoro symmetry to a question
about the spectrum of the theory is an enormous simplification, because the
spectrum is an observable that can be computed not just in the field theory -
where tracking the irrelevant deformation is hard, if not impossible - but
also in the dual supergravity or string theory picture, where the description
of the deformation may significantly simplify252525For example, in the case of
TsT transformations, their field-theoretical description is usually
intractable, whereas their effect in the dual string theory description is to
simply change the boundary conditions of the worldsheet fields [33].. On the
other hand, it is harder to argue that knowledge of the deformed spectrum will
immediately imply a relation between $H$ and $L^{\lambda}_{0}$, because: i)
the deformations are less universal, and therefore they will only exist in
specific seed CFTs, whose spectra may not be generic and ii) even if one
succeeds in computing the spectrum, this will usually be for a subsector of
the theory (e.g., described by supergravity excitations, or string solutions),
and thus one will not in general have access to the relation between
$L_{0}^{\lambda}$ and $H$ for all possible eigenvalues that they can take.
This being said, it does appear to be true that in many string-theoretical
examples of continuous deformations, either exactly marginal [34], or of the
more general kind that change the asymptotics [35], the deformed dimensions
are (especially at strong coupling) smooth, universal functions of the
undeformed ones for entire subsectors of the theory. One may therefore be able
to write at least an approximate relation between $H$ and $L_{0}^{\lambda}$ in
these subsectors, which may allow one to show, via an argument that parallels
the one given for the $J\bar{T}$ and $T\bar{T}$ deformations, that the
respective subsector is acted upon by a Virasoro symmetry. Such an aproximate
relation could be sufficient to explain why various asymptotic symmetry group
analyses in the dual gravity picture were uncovering Virasoro symmetries. Note
the latter may even act locally, perhaps through a ‘large N’ supression
mechanism of the non-local part of the commutator with $H$, similar to the one
we hinted at in the case of single-trace $T\bar{T}$ deformations. Whether the
full theory has Virasoro symmetry would however depend on the exact expression
for $L_{0}^{\lambda}$ in terms of $H$ and other operators in the theory, which
would presumably need to be rather special, and likely difficult to establish.
## 5 Discussion
In this article, we have analysed the symmetries of $J\bar{T}$ \- deformed
CFTs, at both the classical and the quantum level and showed that in a certain
basis, they consist of two commuting copies of the Virasoro-Kac-Moody algebra,
with the same central extension as that of the undeformed CFT. While the
possibility of having Virasoro symmetries in a non-local theory may seem
surprising, we provided a concrete mechanism for reconciling the non-locality
of the model with these symmetries: the zero mode of the Virasoro algebra is
not identified with the Hamiltonian, as in usual CFTs; rather, it is a non-
linear function of it. We showed such a condition was still compatible with
the existence of an infinite set of Virasoro conserved charges and discussed
possible extensions of this mechanism to more general non-local, UV complete
QFTs.
There are many interesting future directions to explore. First, our
construction of the symmetry generators is somewhat abstract, even in the
classical case, where we did provide entirely explicit formulae for the
conserved charges. It would be worth having a better physical picture of the
action of these non-local symmetries on the fields, as well as a better
understanding of the physical implications of an operator-dependent time
dependence in the generators. One may be able to address these questions by
focussing on a simple example, such as the $J\bar{T}$ \- deformed free boson.
Still on the technical side, it would be interesting to develop an algorithmic
procedure for explicitly computing quantum corrections to the $J\bar{T}$
operator, to all orders in the coupling. As explained in the main text, this
would involve reconstructing the right-moving translation current $T_{\alpha
V}$ from knowledge of the right-moving conserved charges. While this exercise
is clearly possible in principle, it is less clear how to do it in practice,
due to the non-linear nature of the charges. Perhaps reconstructing the right-
moving spectral-flow-invariant current, whose relation to the associated
conserved charges is simpler (2.49), may be a good place to start.
Another interesting direction would be to understand the Virasoro symmetries
discussed in this article from a dual gravitational point of view. The
holographic description of $J\bar{T}$ \- deformed CFTs is given by AdS3
gravity coupled to a Chern-Simons gauge field, with mixed boundary conditions
connecting the two [36]. While the field-dependent Virasoro symmetries of
$J\bar{T}$ \- deformed CFTs were first uncovered in the asymptotic symmetry
group analysis of these boundary conditions, that analysis did not take into
account the subtleties associated with finite size which, as we have recently
learned, play an essential role in rendering the charges well-defined in
compact space. It would thus be very interesting to understand how the fully
non-local piece of the transformation is implemented in the gravitational
description, and in particular whether the current formalisms for computing
asymptotic conserved charges already incorporate such structures, or not.
Since field-dependent asymptotic symmetries are not uncommon in space-times
with non-trivial asymptotics, this analysis could be rather useful in future
applications of asymptotic symmetry group methods to non-AdS holography.
So far, we have only described the action of the non-local Virasoro generators
on the spectrum of the theory. An obvious next step is to study the
consequences of these symmetries on other observables, such as correlation
functions. For this, one should find a basis of operators that transform
nicely under them, analogously to primary operators in usual CFTs. Should one
be able to fix the form of correlation functions using symmetry arguments
alone, one may attempt to give an axiomatic definition of non-local CFTs that
does not specifically refer to their construction in terms of an irrelevant
deformation, which does not appear very natural from a renormalisation group
point of view.
All the questions that we listed above also apply to the $T\bar{T}$
deformation. In this case, it would be interesting to first achieve an
explicit classical realisation of the symmetry generators, as we already have
for $J\bar{T}$ \- deformed CFTs. One can also ask whether an analogue of the
unflowed generators exists in these theories, given that the quantum argument
we provided only predicts the existence of the analogues of the flowed ones.
After finding the classical symmetries, it would be interesting to understand
how they match to the asymptotic symmetry group analysis in the dual AdS3 with
mixed boundary conditions [37].
Finally, a rather interesting avenue to explore is to find more general
examples of non-local CFTs, along the lines of our discussion in section 4.
Possibly the simplest such generalizations are the single-trace analogues of
the $T\bar{T}$ and $J\bar{T}$ deformations that we have already mentioned,
whose advantage is to have a rather concrete definition. In this setting, it
appears worthwhile to better understand the effects of these single trace
deformations on the twisted sectors. By doing so, one would not only be able
to test the holographic duality proposed in [32] at a more refined level, but
also make new field-theoretical predictions that should be reproduced by the
gravitational side of the correspondence. One can also consider deformations
of this duality and understand what is the most general structure of the
deformed CFT that still allows for the existence of a Virasoro symmetry, as
well as whether, in the holographic duals to the string backgrounds discussed
in the introduction, this symmetry is exact or, rather, emergent in the large
$N$, large gap limit.
#### Acknowledgements
The author would like to thank Zohar Komargodsky, Gregory Korchemsky, Mark
Mezei, Nikita Nekrasov, Anton Pribytok, Sylvain Ribault, Samson Shatashvili
and especially Blagoje Oblak for interesting discussions. This research was
supported in part by the ERC Starting Grant 679278 Emergent-BH.
## Appendix A Properties of $\alpha_{m}^{l,r}$
The quantities $\alpha_{m}^{l,r}$ are operator-valued functions, defined via
the commutation relations of the flowed generators with the right-moving
Hamiltonian
$[\widetilde{\bar{L}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{L}}{}^{\lambda}_{m}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{L}}{}^{\lambda}_{m}$
(A.1)
The $\alpha_{m}^{l,r}$ need not commute with
$\widetilde{\bar{L}}{}^{\lambda}_{n}$, reason for which they can in principle
be different. They do commute with $H_{R}$, being a function of it. Their
explicit form to all orders in $\hbar$ can be found using (A.1) and the
following relations
$[\widetilde{\bar{L}}{}{}_{m}^{\lambda},\widetilde{\bar{L}}{}_{0}^{\lambda}]=m\hbar\widetilde{\bar{L}}{}^{\lambda}_{m}\;,\;\;\;\;\;\;\widetilde{\bar{L}}{}^{\lambda}_{0}=R_{v}H_{R}-\lambda
H_{R}\bar{J}_{0}-\frac{k\lambda^{2}}{4}H_{R}^{2}$ (A.2)
both of which are assumed to be exact in $\hbar$. Commuting the second
equation with $\widetilde{\bar{L}}{}_{m}^{\lambda}$, we find
$\frac{k\lambda^{2}}{4}(\alpha^{r}_{m})^{2}+(R-\lambda
Q_{K})\alpha^{r}_{m}-m\hbar=0$ (A.3)
$\frac{k\lambda^{2}}{4}(\alpha^{l}_{m})^{2}-(R-\lambda
Q_{K})\alpha^{l}_{m}+m\hbar=0$ (A.4)
where, as before, $Q_{K}=J_{0}+\frac{\lambda k}{2}H_{R}$. The solutions are
given by
$\alpha_{m}^{r}=-2\frac{R-\lambda Q_{K}-\sqrt{(R-\lambda Q_{K})^{2}+\hbar
km\lambda^{2}}}{k\lambda^{2}}\approx\frac{m\hbar}{R-\lambda
Q_{K}}-\frac{km^{2}\lambda^{2}\hbar^{2}}{4(R-\lambda
Q_{K})^{3}}+\mathcal{O}(\hbar^{3})$ (A.5)
$\alpha^{l}_{m}(E_{R})=2\frac{R-\lambda Q_{K}-\sqrt{(R-\lambda
Q_{K})^{2}-\hbar km\lambda^{2}}}{k\lambda^{2}}\approx\frac{m\hbar}{R-\lambda
Q_{K}}+\frac{km^{2}\lambda^{2}\hbar^{2}}{4(R-\lambda
Q_{K})^{3}}+\mathcal{O}(\hbar^{3})$ (A.6)
This reproduces the classical result (2.27) to leading order in $\hbar$. It
may be sometimes useful to rewrite this using
$\widetilde{\bar{L}}{}^{\lambda}_{0}$ instead of $Q_{K}$, using the identity
$(R-\lambda Q_{K})^{2}=(R-\lambda
J_{0})^{2}-\lambda^{2}k\widetilde{\bar{L}}{}^{\lambda}_{0}$ (A.7)
Note that, to the extent that $Q_{K}$ has real eigenvalues, $\alpha_{m}^{r}$
is well defined for arbitrarily large positive $m$, while $\alpha_{m}^{l}$ is
well defined for large negative $m$. One can consequently choose the best way
to write (A.1), dpending on the value of $m$.
Using the fact that the commutator
$[\widetilde{\bar{K}}{}_{m}^{\lambda},\widetilde{\bar{L}}{}_{0}^{\lambda}]=m\hbar\widetilde{\bar{K}}{}_{m}^{\lambda}$,
one can also show that
$[\widetilde{\bar{K}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{K}}{}^{\lambda}_{m}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{K}}{}^{\lambda}_{m}$
(A.8)
for the same values of $\alpha_{m}^{l,r}$.
It is interesting to work out the commutator of
$\widetilde{\bar{L}}{}^{\lambda}_{m}$ with an arbitrary function of $H_{R}$.
We find
$[\widetilde{\bar{L}}_{m},f(H_{R})]=\widetilde{\bar{L}}_{m}(f(H_{R})-f(H_{R}-\alpha_{m}^{r}))=(f(H_{R}+\alpha^{l}_{m})-f(H_{R}))\widetilde{\bar{L}}_{m}$
(A.9)
and similarly for $\widetilde{\bar{K}}{}_{m}^{\lambda}$. One can use this to
show in particular that the difference between $\alpha_{m}^{l}$ and
$\alpha_{m}^{r}$ is consistent with (A.1), because
$\alpha_{n}^{r}(H_{R})=\alpha_{n}^{l}(H_{R}-\alpha_{n}^{r})\;,\;\;\;\;\;\alpha_{n}^{l}(H_{R})=\alpha_{n}^{r}(H_{R}+\alpha_{n}^{l})$
(A.10)
Other useful relations are
$\alpha_{n}^{r}(H_{R}-\alpha^{r}_{m})=\alpha^{r}_{m+n}(H_{R})-\alpha^{r}_{m}(H_{R})\;,\;\;\;\;\;\;\alpha_{n}^{l}(H_{R}+\alpha_{m}^{l})=\alpha_{m+n}^{l}-\alpha_{m}^{l}$
(A.11)
which can be used to show that
$[\widetilde{\bar{L}}{}^{\lambda}_{m},\alpha_{n}^{r}]=\widetilde{\bar{L}}{}^{\lambda}_{m}(\alpha_{m}^{r}+\alpha_{n}^{r}-\alpha^{r}_{m+n})\;,\;\;\;\;\;\;[\widetilde{\bar{L}}{}^{\lambda}_{m},\alpha_{n}^{l}]=(\alpha^{l}_{m+n}-\alpha^{l}_{m}-\alpha^{l}_{n})\widetilde{\bar{L}}{}^{\lambda}_{m}$
(A.12)
and similarly for $\widetilde{\bar{K}}{}^{\lambda}_{m}$. These relations,
together with $\alpha^{l,r}_{0}=0$, can be used to show that any combination
of the flowed right-moving generators and $H_{R}$ satisfies the the Jacobi
identities.
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# FastTrack: Fast and Accurate Fact Tracing for LLMs
Si Chen
Virginia Tech
<EMAIL_ADDRESS>
&Feiyang Kang
Virginia Tech
<EMAIL_ADDRESS>
&Ning Yu
Netflix Eyeline Studios
<EMAIL_ADDRESS>
&Ruoxi Jia
Virginia Tech
<EMAIL_ADDRESS>
###### Abstract
Fact tracing seeks to identify specific training examples that serve as the
knowledge source for a given query. Existing approaches to fact tracing rely
on assessing the similarity between each training sample and the query along a
certain dimension, such as lexical similarity, gradient, or embedding space.
However, these methods fall short of effectively distinguishing between
samples that are merely relevant and those that actually provide supportive
evidence for the information sought by the query. This limitation often
results in suboptimal effectiveness. Moreover, these approaches necessitate
the examination of the similarity of individual training points for each
query, imposing significant computational demands and creating a substantial
barrier for practical applications. This paper introduces FastTrack, a novel
approach that harnesses the capabilities of Large Language Models (LLMs) to
validate supportive evidence for queries and at the same time clusters the
training database towards a reduced extent for LLMs to trace facts. Our
experiments show that FastTrack substantially outperforms existing methods in
both accuracy and efficiency, achieving more than 100% improvement in F1 score
over the state-of-the-art methods while being $\times 33$ faster than TracIn.
## 1 Introduction
Recent years have witnessed large language models (LLMs) demonstrating
remarkable abilities in absorbing vast knowledge from extensive text corpora,
yielding impressive advancements in NLP tasks such as question answering (QA).
However, these models often produce seemingly coherent yet unfounded outputs,
known as ‘hallucinations’ (Agrawal et al., 2023), posing risks in high-stake
scenarios such as healthcare and finance, where reliability is of paramount
importance (Master of Code, 2023). This critical challenge has motivated
research on _fact tracing_ (Akyürek et al., 2022), which aims to identify the
training data that serves as the knowledge source for LLMs’ generation.
Striving to provide a pathway to understanding and mitigating the issue of
hallucination, Akyürek et al. (2022) proposed a benchmark for fact tracing,
formulating it as a challenging task that involves searching for training data
that has fact-support correspondence (i.e., supportiveness) with given
queries. Current methods, however, tend to miss the mark and overly rely on
similarity measures between individual training samples and the target query,
such as gradient similarity (Pruthi et al., 2020; Koh and Liang, 2017),
embedding similarity (Rajani et al., 2020), or lexical similarity (Robertson
et al., 1995; Lv and Zhai, 2011). As a natural result, these approaches may
fail to differentiate between samples that merely look similar and those that
actually contain the supporting information sought by the query–even in
considerably simple cases. This prominent issue limits their effectiveness in
identifying supportive training examples, preventing them from being effective
in broader use cases (Akyürek et al., 2022). Besides, some of these methods,
such as Pruthi et al. (2020); Koh and Liang (2017), carry a significant
computational overhead in analyzing a given query. Providing intellectual
inspiration for research exploration, nonetheless, its computational demand
can be unaffordable for most practical scenarios. Despite soaring interest in
this emerging problem, current research still falls short of the critical need
by a large margin.
We summarize the desiderata for fact-tracing methods as the following:
$\diamond$ D-i. Effective and Accurate. For a target query, fact-tracing
methods need to identify all supporting facts in the training corpus and
achieve both high precision and recall simultaneously.
$\diamond$ D-ii. Computationally Tractable. Fact-tracing methods need to be
scalable with both the number of queries and the number of training samples to
be examined. $\diamond$ D-iii. Practically Robust. Fact-tracing prioritizes
general-purposed, principled methods that are plausible for deployment and
transferable between use cases.
Current methods all miss one or more of these principles. Specifically,
gradient-similarity-based methods (Pruthi et al., 2020; Koh and Liang, 2017)
are notoriously computationally demanding (D-ii). Also, gradients are
considerably susceptible to noises, rendering their performance rather
unstable even with extensive hyper-parameter tuning (Akyürek et al., 2022;
Park et al., 2023) (D-i, D-iii). Lexical-similarity-based methods (Robertson
et al., 1995; Lv and Zhai, 2011) are typically faster, but relying on queries
and samples with supporting facts being similarly phrased. This assumption is
not necessarily true in realistic use cases (D-iii). Table 4 shows that the
performance for such methods may drop a large margin under slight rephrasing
of the text (D-i). Therefore, these methods are neither practical nor reliable
(as illustrated in Sec. 5.2).
Figure 1: _FastTrack achieves the best tradeoffs between fact tracing efficacy
and efficiency._ The x-axis the the computational time of evaluating 100
queries using a 10k corpus, and the y-axis is the tracing performance when
using top-k thresholds (if applicable). TDA methods yield consistently low
performance across top-k thresholds, making them look like dots in the plot.
Determining whether a training example supports a factual statement in a query
demands reasoning abilities beyond sample similarities where support for a
factual assertion often arises through the inference of connections among
related pieces of information. The dilemma with these approaches is that no
single representation works in all cases and the similarity in these pre-
defined spaces may easily fail to capture the nuance of supportiveness
effectively. Inspired by the recent advancement in LLM’s abilities in natural
language understanding (NLU), a natural idea is to directly evaluate the
supportiveness between each training sample and the target query using an LLM.
Unprecedented in-context learning (ICL) capabilities make these models notably
versatile and easily adaptable to novel cases with minimal customization,
effectively bridging the realistic gap between fact-tracing methods and real-
world scenarios. Admittedly, our preliminary investigation shows that this
idea indeed enhances the efficacy in the identification of supportive training
samples to an impressive extent. Nevertheless, this idea faces immediate
challenges when applied to a practical-sized training corpora: traversal
evaluation for all training sample-query pairs requires a massive number of
queries to the LLM, unaffordable in both computation time and costs, hindering
it from being practically useful.
To address this dilemma, we propose FastTrack, which is a two-stage scheme
decomposed into offline and online components. In the first stage, we build
semantic indexes for the training corpus through hierachical clustering. Such
process is completely offline and only need to be run once. During online
stage, these pre-built semantic indexes facilitate the retrieval of relevant
clusters for any given query, significantly reducing the search range.
FastTrack then runs a fine-grained examination by employing a LLM to evaluate
the supportiveness of training data in the retrieved clusters. While prior
work (Akyürek et al., 2022) requires careful selection of small candidate set
of size around 500 for practical evaluation, FastTrack enables a balance
between computational feasibility and fine-grained analysis. This enables it
to accommodate large corpus of size 10k or even 100k, while ensuring both
satisfactory efficiency and efficacy (high precision and recall).
Our contributions are summarized as follows:
* •
We propose a novel two-stage pipeline FastTrack and show it is easily
adaptable without needing to train a model. ( meets D-iii)
* •
We evaluate FastTrack’s performance on various datasets with baseline methods.
FastTrack achieves notable F1 scores of 0.72 on FTRACE-TREx and 0.91 on
VITAMINC, more than doubling the performance of the best existing methods.
(meets D-i)
* •
We show FastTrack to offer a substantial edge in efficiency, being
$\mathbf{33\times}$ faster than the TDA method TracIn for a corpus of 10k
samples, and readily applicable to larger datasets with more than 100k
samples. (meets D-ii)
Figure 2: Illustration of FastTrack workflow. Stage 1, which is completely
offline, reorganizes the training corpus into a semantic tree for easier
navigation; Stage 2 retrieves relevant clusters using fuzzy keyword matching,
then employs LLMs to assess candidate samples, retrieving those with a score
of 1.
## 2 Related Work
#### Training Data Attribution (TDA).
TDA aims to trace model predictions back to the training examples that
responsible for these predictions, which shares a similar goal with fact
tracing. Prior work (Akyürek et al., 2022) proposes to use two main types of
TDA methods as baselines: gradient-based and embedding-based attributions.
Gradient-based methods, such as TracIn Pruthi et al. (2020), estimate the
attribution score of training data on predictions by calculating the cosine
similarity between the gradients of the training data and the query.
Embedding-based methods employs the model’s internal representations to
determine the relevance of training examples to a given test prediction
(Rajani et al., 2019). The attribution score is defined as a cosine product of
hidden representations.
To retrieve supporting training data for a given query $z_{\text{query}}$, one
need to score every training data and rank them by their influence score. As
it could be computationally infeasible for gradient-based TDA scoring all
training data in large datasets, Akyürek et al. (2022) only evaluates on
carelly selected small subsets (i.e., around 500) for each query. This
limitation motivates us to design a framework that is both more
computationally efficient and more effective.
#### Information Retrieval (IR).
IR focuses on retrieving relevant documents in a large collection given
specific queries (Izacard et al., 2021). Though not originally designed for
fact tracing task, prior work (Akyürek et al., 2022) found it effective and
outperforms principled TDA methods by a large margin. IR splits into two
categories: term-frequency-based methods like BM25(Thakur et al., 2021; Zhou
et al., 2022), which score each training data base on the token overlap with
the given query, inversely weighted with the frequency of such tokens, and
neural network-based methods (Izacard et al., 2021; Ni et al., 2021), which,
despite their advanced capabilities, often require extensive manual
annotations, making them less suited for fact tracing due to the absence of
necessary annotations. Recent attempts to adapt neural methods through zero-
shot learning have not matched BM25’s performance (Thakur et al., 2021; Zhou
et al., 2022). Therefore, following prior work, we select BM25 as the baseline
for fact tracing due to its superior retrieval quality without the need for
annotated data.
All of the methods above focus on _relevance_ while neglecting the
_supportiveness_ of the connection between training data and the query. In
this paper, we introduce FastTrack, the first supportiveness-aware approach
for fact tracing, offering substantial benefits in real scenarios where
training data may contain conflicting information.
## 3 Methodology
Fact tracing aims to identify knowledge source of a particular query. While
similar to TDA, it focuses more on the fact-support correspondance between
training data and query. This distinction is crucial: existing methods often
retrieve relevant examples but fail to provide factual support, misaligning
with the objective. The strong capability of LLMs such as ChatGPT makes it a
perfect solution to provide justification based on ‘supportiveness’. However,
directly doing pair-level comparison could be very time-consuming: Given a
corpus of size $N$ and $m$ queries, the computation complexity is
$\mathcal{O}(mN)$.
In this section, we introduce an original two-stage framework FastTrack, as
illustrated in Figure 2. In the first stage, FastTrack leverages a recursive
clustering scheme to mine the semantic structure in the training corpus, which
enables a coarse matching for a given query. This significantly refines the
search range, making it feasible to perform a fine-grained examination of each
candidate training examples in the second stage.
### 3.1 Semantic Clustering
The goal of the first stage is to create semantically meaningful indexes in an
offline setting. This one-time process allows for the efficient utilization of
these indexes in subsequent online stages, eliminating the need for re-
computation. In this paper, we propose to employ a simple hierarchical
clustering process over training data embeddings to recover underlying tree
structures of the data. This process reorganize the entire training corpus
into a more structure format, laying the groundwork for more effective data
navigation and retrieval. We first apply k-means clustering on the sample
embeddings to mine its semantic structure. The clustering is conducted
recursively where larger clusters will be further clustered until the size of
all the clusters is within a certain threshold.
The key of our method lies in transcending the limitations of conventional
clustering algorithms, which typically do not assign semantically meaningful
labels to each cluster. By harnessing the power of Large Language Models
(LLMs), FastTrack assigns a carefully selected set of keywords to each
cluster, serving as its semantic label. This strategic integration not only
renders the clustering outcomes interpretable but also significantly
simplifies the process of navigating through the corpus in response to
specific queries. We note that such semantic clustering only need to be
applied once offline, effectively allowing us to leverage the massive amount
of compute in pre-training for free.
### 3.2 LLM as a Sample-Level Tracer
With the structured and semantically meaningful clusters, we can now online
process each query for fact tracing efficiently. The first step is to retrieve
relevant clusters for a given query. A simple example for such cluster-level
retrieval is to apply fuzzy match 111https://github.com/seatgeek/thefuzz to
identify those clusters that shared similar keywords as the query.
Furthermore, the efficacy of clustering can be enhanced through ensemble of
different clustering outcomes, as detailed in Table 2.
Now, with the retrieved clusters, the second step is to identify the
groundtruth supporting data from this narrowed pool. We frame this stage as a
binary verification problem: given a specific query, we classify each
candidate training example into two categories based on its ‘supportiveness’.
An example is considered ’grounding’ if it supports the query. A direct way to
perform such classification is to instruct the LLM to evaluate a single
training example against a query for supportiveness, assigning a score of 1
for supportiveness and 0 otherwise. Although effective, this one-at-a-time
scoring method can still be computationally and financially costly. To futher
enhance efficiency and speed up the process, we devised the prompting strategy
to evaluate a batch of training data in a single inference run. This batch
processing approach significantly cuts down the time required for evaluations,
reducing the number of necessary inferences by a factor of $b$, where $b$ is
the number of candidate examples in a batch. The example prompt used in our
experiments can be found in Appendix F. Following the LLM’s evaluation,
examples that are assigned a score of 1, indicating supportiveness, are
systematically retrieved. The detailed workflow of FastTrack is presented in
Algorithm 1.
## 4 Experimental Setup
### 4.1 Datasets
#### FTRACE-TREx.
The FTRACE-TRex dataset is proposed by (Akyürek et al., 2022), with 27k
queries created using LAMA (Petroni et al., 2019) and 1M masked training
examples extracted from TREx (Elsahar et al., 2018) as the attribution set.
Each training example is a cloze-style sentence with either the subject or
object masked. The groundtruth training example for each query is defined as
the examples that express the same fact, regardless of the masking position.
To address the computational overhead, Akyürek et al. (2022) proposes to
construct a small, separate candidate set for each query (around 500). We
follow a similar setup, but use a larger, fixed candidate pool to better
reflect real-world scenarios: we randomly sample 100 queries from the entire
query set for evaluation, and build the candidate pool by including all the
corresponding groundtruth, supplementing with random samples to form a corpus
of 10k.
#### VITAMINC.
We incorporate the VITAMINC dataset (Schuster et al., 2021) as a means to
evaluate fact tracing methods’ ability to mirror real scenarios where training
corpus of LMs containing contradictions or misinformation. The VITAMINC
dataset is built based on factual revisions to Wikipedia: each single factual
revision yields a contrastive pair of contexts, where one context refutes the
given claim and the other supports it. The original VITAMINC dataset presented
each entry in the format of _claim_ , _evidence_ , and _label_ , where the
label indicates if the evidence ’SUPPORTS’, ’REFUTES’, or provide ’NOT ENOUGH
INFO’ to the evidence. To use it for fact tracing purposes, we build the
attribution set by collecting 10k unique pieces of evidence (acting as
training data). Then the query set is built by collecting corresponding claims
that can be supported by the evidence. 222Due to the labeling format of the
original dataset, some claims may have more than one supporting evidence but
we do not know. To address such an issue, we manually inspect 100 queries for
their groundtruth data and use these queries for evaluation. We provide the
data we manually inspect along with this submission.
### 4.2 Baselines
Following Akyürek et al. (2022), we compare our method FastTrack with TDA
methods (i.e., TracIn, Embed) and the most representative IR method (i.e.,
BM25).
#### TracIn.
TracIn (Pruthi et al., 2020) is a recent gradient-based TDA method that has
demonstrated strong empirical results and tractability. Following the setup of
Akyürek et al. (2022), we use an optimized version of TracIn by rescaling
gradients with Adafactors accumulators, applying unit-normalization to the
gradients, and selecting the best-performing layer. Data in FTRACE-TREx are
cloze-style examples, hence we finetune an MT5 model (Xue et al., 2021)
following Akyürek et al. (2022) to predict the masked tokens. We note that
gradient similarity is only meaningful when query and training data have the
same question-answer construction, and it is difficult to construct the
VITAMINC dataset in this way. Hence, we omit the evaluation of TracIn on
VITAMINC dataset.
#### Embed.
Embedding-based similarity is another popular branch for fact tracing tasks.
Here we refer to Equation 2 as baseline Embed. For FTRACE-TREx dataset, we use
the same fine-tuned MT5 model as for TracIn, selecting the best-performing
layer as the final result. For the VITAMINC dataset, we finetune a BERT model
(Kenton and Toutanova, 2019) on our constructed attribution set.
#### BM25.
We use a publicly available implementation of BM25 (Lv and Zhai, 2011) as our
baselines 333https://pypi.org/project/rank-bm25/. We tokenize queries and
training examples by space, removing any masked tokens. We proceed with the
default settings for all hyperparameters, ensuring a standardized approach for
our baseline comparisons.
### 4.3 Tracing Performance Evaluation
TDA methods and BM25 score a given test query against every training example
and then sort all examples based on their scores. This results in a top-k
precision and recall performance measurement, where the k is the threshold of
taking the top k ranked examples as the retrieved supporting training data
(Akyürek et al., 2022). In contrast, our method directly retrieves the
supporting training data without ranking. To enable a unified comparison, we
use F1 score as the main metric. We report the best-performing F1 score and
the corresponding precision and recall for each method.
## 5 Empirical Results
### 5.1 Overall Performance
We first evaluate the overall performance of different methods on FTRACE-TREx
and VITAMINC datasets in Table 1. Hyperparameters for all methods are
presented in Appendix C.
Table 1: Comparison of fact tracing performance. We present the best F1 scores
among top-k for each method; precisions and recalls are chosen at the
threshold lead to optimal F1 score. Among all methods, FastTrack performs the
best. *The last row gives the upper bound performance achievable in the first
cluster-level retrieval stage.
| FTRACE-TREx | VITAMINC
---|---|---
| F1 | Precision | Recall | F1 | Precision | Recall
TracIn | 0.02 | 0.19 | 0.01 | - | - | -
Embed | 0.01 | 0.08 | 0.01 | 0.48 | 0.54 | 0.46
BM25 | 0.40 | 0.49 | 0.52 | 0.55 | 0.59 | 0.53
Ours | 0.72 | 0.81 | 0.69 | 0.91 | 0.88 | 0.98
Ours* | 0.86 | 0.92 | 0.83 | 1.00 | 1.00 | 1.00
Fact tracing is a challenging task. Previous work (Akyürek et al., 2022)
proposes several techniques to optimize TDA methods but found that even BM25
with no tuning outperforms TDA, and all these methods are far from perfect. In
Table 1 we show similar findings, where TracIn and Embed resulted in F1 score
lower than $0.1$ on FTRACE-TREx dataset. We also observe that TracIn’s
performance is highly dependent on the chosen model checkpoint. Specifically,
the performance noted in our main results table was achieved using the final
80k-step checkpoint, with earlier checkpoints yielding even weaker outcomes
(as shown in Appendix E).
Takeaway: FastTrack delivers impressive tracing performance, yielding both
high precision and recall, improving the F1 score by >80% compared to the
best-performing baseline BM25.
All baseline methods retrieve training examples based on their ‘relevance’ to
the given query, which could violate the goal of fact tracing. This
discrepancy becomes evident in real-world scenarios, where datasets, unlike
the scientifically accurate and consistent ones often evaluated in prior
research, contain conflicting information. Our evaluation on VITAMINC dataset
reveals that such methods yield low precision due to their relevance-focused
logic. Notably, FastTrack significantly outperforms all baselines, achieving
an F1-score of 0.91, demonstrating its effectiveness in accurately identifying
grounding training data for queries.
Takeaway: FastTrack not only excels in fact-tracing performance but also
offers the optimal balance between computational speed and effectiveness. It
outperforms competitors significantly, running 33 times faster than TracIn in
evaluating 100 queries (Figure 1).
### 5.2 Failure Analysis
In this section, we qualitatively examine some failure examples of different
tracing methods to shed light on the future direction of fact tracing.
#### When does BM25 fail?
BM25 operates based on token overlap, and retrieves examples with high lexical
similarity to the query, regardless of their factual consistency. As shown in
the example below, while the first retrieved example is correct, the second
contradicts the query, and the third is entirely unrelated.
Query: Alloy Digital’s network has a monthly reach of more than 100 million
unique visitors. BM25 Retrieved:
Rank-1: Defy Media: According to comScore, Alloy Digital’s network reaches
over 221 million unique visitors each month, including more than half of the
aged 12-34 internet users. Rank-2: According to comScore, Alloy media
platforms reach over 95 million unique visitors each month, including over
half of the age 12-34 internet users. Rank-3: The franchise has sold more than
26 million units worldwide with the release of 2018 ’s installment.
BM25’s performance can be poor even when there are no such data conflicts. We
further conduct experiment on FTRACE-TREx dataset where we paraphrase each
query using an open-sourced paraphraser
444https://huggingface.co/humarin/chatgpt_paraphraser_on_T5_base. The
performance of BM25 before and after paraphrasing is shown in Table 4, where
both precision and recall drop by a wide margin.
#### When do TDA methods fail?
TracIn conducts a first-order approximation and uses the dot product of the
model’s gradients between each train-test sample pair to measure this
contribution However, we find its actual performance is fragile and can be
affected by a number of factors.
_1) TracIn’s performance is highly dependent on having the exact same
construct of question-answer pairs._ LMs for QA tasks typically use an
encoder-decoder architecture, such as T5/MT5. The gradient is calculated with
respect to the loss of the word/token being predicted. However, gradient
similarity between a train-test sample pair is only meaningful when these are
the same QA questions with identical question-answer pairs. In other words,
even for sample pairs where the texts are the same, if the construction of
question-answer is different, the loss and gradient may be unrelated. This
aligns with our evaluation results: we find that TracIn cannot identify those
groundtruth training examples with supporting facts but having different QA
construction. This results in arbitrarily poor performance on some queries, as
the cosine similarity between gradients - which are high-dimensional vectors -
can be dominated by unrelated factors and fail to capture the actual
correlation between samples.
_2) TracIn tends to retrieve sentence with the same masked token._ Such
finding has also been observed in (Akyürek et al., 2022). This likely occurs
because the same masked token produces similar training gradients.
Query: Comptroller of Maryland is a legal term in ____. (Maryland) TracIn
Retrieved:
Rank-1: The ____ Comptroller election of 2010, was held on November 2, 2010.
(Maryland) Rank-2: It is found in Alabama, Florida, Louisiana, ____,
Mississippi, North Carolina and Virginia. (Maryland)
As illustrated in the example above, the top-ranked retrieved example is
correct, where the training example and query share the same masked target
token. However, the second retrieved example does not provide any relevant
fact, only the masked token to predict is the same.
The other TDA method evaluated in this paper, Embed, relies on hidden space
similarity search. The dilemma for this approach is that no single
representation works for all tasks (Vaze et al., 2023), which is more
pronounced in these QA problems. The similarity of text pairs could be
measured from different perspectives and the one that is best captured does
not necessarily focus on the "supporting fact". Another major issue with this
approach is that similar texts always receive similar scores, rendering the
results end up in clumps. If the front-running clump is wrong, all samples in
the clump are wrong, yields zero top-k accuracy. For example, for the same
query "_Comptroller of Maryland is a legal term in <MASK>_", the top 3
retrieved examples of Embed are:
Rank-1: the Mayor of ____. (Moscow) Rank-2: Embassy in Cyprus is located in
____. (Nicosia) Rank-3: He served on the ____ of Edmonton. (town council)
These retrieved examples, to varying degrees, relate to the query by involving
1) public offices and elected officials, 2) political or geographical
entities, and 3) individuals with governmental roles. In fact, The groundtruth
example belongs to a similar category. Yet, embedding similarity cannot detect
fact-support correspondence between samples and cannot distinguish different
levels of sample similarities.
## 6 Ablation Study and Analysis
#### In-depth Analysis of FastTrack.
The first stage of FastTrack\- cluster level retrieval - decides the
performance upper bound of our methods. If relevant clusters are not
identified during this phase, it becomes impossible to recover them in the
later stage. We report the upper bound performance achievable in the last row
of Table 1, to reveal the limitation origins from the first stage.
Specifically, this upper bound assumes perfect accuracy in the second stage,
meaning if the correct cluster is identified, we achieve 100% precision and
recall on this cluster. As shown in Table 1, the upper bound of FastTrack has
a short gap to perfect. The precision is 0.92 while the recall is only 0.83.
Such failure origins from the first stage could come from two sides:
_1) clustering algorithm._ The clustering algorithms group data with similar
embedding together. Although in general, we observe that groundtruth training
data for a specific query usually falls within 4 clusters on average, which
means the clustering algorithm successfully groups relevant training data into
the same cluster, there still exists the case that for some clusters the
groundtruth training data is the minority. In such case, groundtruth data
could be ignored when assigning the cluster semantically meaningful keywords,
making this cluster hard to retrieve. In practice, this can be improved by
using an ensemble - we observe that an ensemble of three yields a performance
upper bound of precision 0.92, recall 0.83; while single clustering yields an
upper bound of precision 0.81, recall 0.65.
Table 2: Upperbound performance of FastTrack when using single and ensemble
embeddings on FTRACE-TREx.
| Single | Two-Ensemble | Three-Ensemble
---|---|---|---
Precision | 0.81 | 0.89 | 0.92
Recall | 0.65 | 0.78 | 0.83
_2) cluster retrieval method._ We currently employ simple fuzzy matches to
capture clusters that share similar keywords as the query. However, the
training data may present the query on a different surface. Future studies
could leverage more advanced tools to enhance the process.
Table 1 shows that there exists a gap between performance upper bound and
final performance. This gap comes from ChatGPT’s limitation, where it
misclassified a few examples. We show two interesting types of
misclassification here:
Query: President of the Executive Yuan is a legal term in _____. (Taiwan)
False negative examples (mask removed): 1\. He has interviewed financial
services regulators including Sean Chen (politician), the Premier of Taiwan,
when he was the Chairman of the Financial Supervisory Commission (Republic of
China) of Taiwan and negotiated the financial Memorandum of Understanding with
China. 2\. Hsich Tung-min was the ninth Governor of Taiwan Province
(1972-1978) and the sixth and first local Taiwanese Vice President of the
Republic of China (1978-1984) under President Chiang Ching-Kuo. GPT-4
analysis:
The term "President of the Executive Yuan" is not mentioned in any of the
texts. The texts mention various political positions in Taiwan, such as the
Premier of the Republic of China and the President of Taiwan, but none of them
refer to the President of the Executive Yuan. Therefore, it cannot be inferred
from the texts that "President of the Executive Yuan" is a legal term in
Taiwan.
In the above example, GPT-4 did not recognize that the ’Executive Yuan’s
leader is the ‘Premier of Taiwan’, indicating a gap in connecting related
concepts. The second failure example appears to be a labeling error. Another
example is that GPT-4 struggles with complex logical reasoning involving
dates; for instance, it incorrectly equates the information from different
dates, focusing merely on numerical comparisons (see Appendix E). Failure
cases at this stage mainly stem from LLM’s own bottleneck. These challenges
represent a significant area of ongoing research and are beyond the scope of
our current study. We acknowledge these limitations and suggest them as
critical avenues for future investigation to enhance the capabilities and
applications of LLMs.
#### Embeddings Schemes.
We use Sentence-Transformer 555https://www.sbert.net as the embedding model to
perform clustering in our main evaluation. To test the sensitivity of
FastTrack on different choices of embeddings, we also test some state-of-the-
art embedding models such as Cohere Embed
v3666https://txt.cohere.com/introducing-embed-v3/ and Mistral-
Embed777https://docs.mistral.ai/api/. As shown in Table 6, FastTrack
consistently achieves comparable top-performance upperbounds across various
embedding models, underscoring its adaptability to different embedding
choices.
#### Corpus Size.
Moving forward, we aim to tackle a more challenging scenario: we use the same
query set of VITAMINC, but augment the attribution set with additional non-
relevant examples until the total reaches 100k. This setting is designed to
evaluate our method’s robustness in scenarios that better resemble real-world
applications. As shown in Table 7, both methods exhibit a slight decline in
performance, yet FastTrack consistently outperforms BM25 by a significant
margin. BM25’s performance drop is ascribed to the inclusion of new examples
that exhibit high lexical overlap with the queries, while our method, mainly
stems from the clustering stage, where the clustering logic has been impacted
after a more diverse sample are included. We leave a detailed analysis in
Appendix E.
Table 3: Performance of BM25 and FastTrack when dealing with different corpus
size. Both of the methods encounter a slight performance drop, while FastTrack
is still $1.66\times$ better than BM25.
| VITAMINC-10k | VITAMINC-100k
---|---|---
| F1 | Precision | Recall | F1 | Precision | Recall
BM25 | 0.55 | 0.59 | 0.53 | 0.53 | 0.56 | 0.50
Ours | 0.91 | 0.88 | 0.98 | 0.88 | 0.85 | 0.92
Ours* | 1.00 | 1.00 | 1.00 | 0.95 | 0.95 | 0.95
## 7 Conclusion
In this paper, we introduced FastTrack, a pioneering two-stage framework
designed to address the shortcomings in current fact tracing methodologies,
particularly their neglect of the ’supportiveness’ of evidence. FastTrack
substantially improves the tracing performance by more than 100% in F1 score,
and offers computational efficiency, capable of handling large datasets up to
100k in size. We also provide a thorough analysis of each tracing method to
shed light on future direction in fact tracing. It’s important to note that
the current performance bottleneck primarily stems from the limitations of GPT
models. Therefore, future efforts could focus on fine-tuning a model
specifically for tracing purposes.
## Limitations
While our proposed method, FastTrack, has shown considerable success, it’s
important to acknowledge that its performance is ultimately constrained by the
capabilities of GPT models. Thus, future work could explore techniques to
fine-tune a LLM specifically targeting tracing purpose. Another limitation of
FastTrack is its capacity to process only a limited number of training
examples in each batch. This presents an opportunity for future improvements
by incorporating techniques that can handle longer contexts. By doing so, it
may be possible to decrease the necessity for multiple inferences, thereby
optimizing the process.
## Ethics Statement
Our research advances the accuracy and efficiency of fact tracing techniques,
elucidating the connections between the training data of LLMs and their
generated assertions. Our method ensures data privacy integrity, as it solely
utilizes publicly accessible data samples, and is not designed for the
inadvertent generation of unintended content by LLMs. While mindful of the
potential for redistributing web data to inadvertently disseminate
misinformation, we foresee no other ethical concerns with our methods.
Committed to the ethos of open science, our study champions reproducibility,
transparency, and the facilitation of further inquiry. To this end, we will
grant unrestricted access to all materials related to our research. This
includes a comprehensive, meticulously documented repository encompassing all
scripts, models, and codes necessary for preprocessing and evaluation,
allowing for the full replication of our experiments. Beyond making these
resources available, we are dedicated to their ongoing maintenance and to
providing timely support for any inquiries or clarifications. This commitment
underlines our dedication to fostering a collaborative, inclusive research
community.
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## Appendix A Algorithm of FastTrack
Input: Query set $Q$, training corpus $D$, instruction prompt for keyword
assignment $Inst_{\text{key}}$, instruction prompt for supportiveness
evaluation $Inst_{\text{eval}}$
Output: Retrieved Samples $D_{\text{sel}}$
1
/* Stage 1: Semantic Clustering (Offline) */
2 $D_{emb}\leftarrow SentenceTransformer(D)$ Leaf Clusters
$C=\\{c_{0},c_{1},\ldots,c_{n-1}\\}\leftarrow$ Hierarchical clustering on
$D_{emb}$ using k-Means (k=10)
3 Semantic Labels
$J=\\{j_{0},j_{1},\ldots,j_{n-1}\\}\leftarrow\text{LLM}(\\{c_{0},c_{1},\ldots,c_{n-1}\\},Inst_{\text{key}})$
4
/* Stage 2: Tracing (Online) */
5 for _each query $q\in Q$_ do
6 $D_{q}\leftarrow\\{\\}$
7 $C_{\text{sel}}\leftarrow\text{fuzzymatch}(q,J,C)$
8 $Batches\leftarrow$ partition $C_{sel}$ into batches of size $b$
9 for _each batch $B\in Batches$_ do
10 $S_{B}\leftarrow\text{LLM}(q,B,Inst_{\text{eval}})$
11 $D_{q}\leftarrow D_{q}\cup\\{z\mid z\in B,s_{i}=1\\}$
12
13 end for
14 $D_{\text{sel}}\leftarrow D_{\text{sel}}\cup D_{q}$
15 end for
Algorithm 1 FastTrack Workflow
## Appendix B Extended Related Work
#### Training Data Attribution (TDA).
TDA aims to trace model predictions back to the training examples that
responsible for these predictions. As it shares a similar goal with fact
tracing, prior work (Akyürek et al., 2022) proposes to use two popular
families of TDA methods as baselines. Gradient-based attribution, for
instance, focuses on quantifying the direct influence a particular training
example $z$ has on the loss at a test example $z_{\text{query}}$, when using a
model parameterized by $\theta$. A notable technique within this category is
TracIn Pruthi et al. (2020). It employs a first-order Taylor approximation to
estimate the loss change on $z_{\text{query}}$ when taking a gradient step on
training example $z$ at time $t$. The resulting attribution score is simply a
dot product of gradients at a particular step t:
$\mathcal{I}_{\mathrm{t}}\left(z,z_{\text{query
}}\right)=\nabla_{\theta}L\left(z_{\text{query
}},\theta_{\mathrm{t}}\right)^{\top}\nabla_{\theta}L\left(z,\theta_{\mathrm{t}}\right)$
(1)
Embedding-based attribution employs the model’s internal representations to
determine the relevance of training examples to a given test prediction
(Rajani et al., 2019). The attribution score is defined as a cosine product of
hidden representations:
$\mathcal{I}\left(z,z_{\text{query }}\right)=\frac{LM_{\text{inter.
}}(z)^{\top}LM_{\text{inter. }}\left(z_{\text{query
}}\right)}{\left\|LM_{\text{inter. }}(z)^{\top}\right\|\left\|LM_{\text{inter.
}}\left(z_{\text{query }}\right)\right\|}$ (2)
To retrieve supporting training data for a given query $z_{\text{query}}$, one
need to score every training data and rank them by their influence score.
However, TDA methods fail to justify groundeness and often perform worse than
simple IR baseline (i.e., BM25) (Akyürek et al., 2022). Moreover, it could be
computationally infeasible for gradient-based TDA scoring all training data in
large datasets, and relies on evaluation on carelly selected smallsubset
(i.e., around 500) for each query. This limitation motivates us to design a
framework that is both more computationally efficient and more effective.
#### Information Retrieval (IR).
IR focuses on retrieving relevant documents in a large collection given
specific queries (Izacard et al., 2021). Though not theoretically justified
for fact tracing task, prior work (Akyürek et al., 2022) found it could serve
as a possible solution and outperforms principled TDA methods by a large
marigin. There are two branches of IR methods: term-frequency based (Mikolov
et al., 2013; Lv and Zhai, 2011; Robertson et al., 1995) and neural network
based (Karpukhin et al., 2020; Xiong et al., 2020; Izacard et al., 2021; Ni et
al., 2021). A classic example of the former one is BM25 (Robertson et al.,
1995; Lv and Zhai, 2011), which represents the best performing variant of
lexical similarity-based IR methods. When using BM25 for fact tracing, one can
treat examples as a bag of words, and score each training data base on the
token overlap with the given query, inversely weighted with the frequency of
such tokens. On the other hand, neural network-based methods often require
labor-intensive annotations on query-document pairs (Karpukhin et al., 2020;
Xiong et al., 2020). This making them impractical in the fact tracing scenario
where such annotations are not available. While some recent works (Izacard et
al., 2021; Ni et al., 2021) propose to overcome the limitation using zero-shot
learning, they usually results in an inferior retrieval quality, even worse
than a non-parametric BM25 (Thakur et al., 2021; Zhou et al., 2022). Thus, we
follow Akyürek et al. (2022) and choose BM25 as IR baseline for fact tracing.
Similar to TDA methods, IR methods also focus on _relevance_ while neglecting
the _supportiveness_ of the connection between training data and the query. In
this paper, we introduce FastTrack, the first supportiveness-aware approach
for fact tracing, offering substantial benefits in scenarios where training
data contains conflicting information, such as time-sensitive facts in news
reports.
## Appendix C Baseline Implementation Details
#### TracIn
We follow the setup of Akyürek et al. (2022), optimize TracIn by recaling
gradient with Adafactor accumulators, applying unit-normalization to the
gradients and selecting the best performing layer, which is first encoder
layer. In practice, it is found that aggregating over multiple checkpoints
often does not lead to improved performance but raises the computational
burden. Thus, it is preferred to just use a single checkpoint that gives the
best results (Just et al., 2023). For FTRACE-TREx dataset, we use a MT5 model
finetuned on it for 80k steps.
#### Embed
We use the same MT5 checkpoint as for TracIn on FTRACE-Trex dataset. For
VITAMINC dataset, we finetune a Bert model by randomly masking some tokens of
the training data. We observe that best performing layer is the last encoder
layer of Bert and use results of this layer as the final results.
## Appendix D FastTrack Implementation Details
A detailed algorithm of FastTrack is given in 1. When perform hierachical
clustering, we employ k-means (Na et al., 2010). The clustering process is
applied recursively: clusters with more than $c$ samples will be clustered
again until it contains less then $c$ samples. We use $c=100,k=10$ for all
experiments.
For keyword assignment, we set temperature to be 0 and output length to be 256
when calling GPT-4 api. For the batch process at Stage 2, we use a batch size
$b=15$. We set temperture to be 0 and output length to be 1024. Prompts used
can be found in Section F.
## Appendix E More Results
#### BM25’s performance before and after paraphrasing queries
BM25 operates based on token overlap, and retrieves examples with high lexical
similarity to the query, regardless of their factual consistency. Its
performance can be poor even when there are no such data conflicts. We further
conduct experiment on FTRACE-TREx dataset where we paraphrase each query using
an open-sourced paraphraser
888https://huggingface.co/humarin/chatgpt_paraphraser_on_T5_base. The
performance of BM25 before and after paraphrasing is shown in Table 4, where
both precision and recall drop by a wide margin.
Table 4: BM25’s performance before and after paraphrasing queries in FTRACE-
TREx dataset. Notably, BM25 exhibits a 21 percentage point drop in precision,
while FastTrack maintains consistent performance, achieving a precision of
0.81 and a recall of 0.69.
| Top-1 | Top-10 | Top-25
---|---|---|---
| Precision | Recall | Precision | Recall | Precision | Recall
Before | 0.83 | 0.06 | 0.66 | 0.36 | 0.49 | 0.52
After | 0.62 | 0.05 | 0.48 | 0.28 | 0.38 | 0.42
#### TracIn’s performance when using different model checkpoints
Table 5: TracIn’s performance using checkpoints at gradient steps 30k and 80k
| Top-1 | Top-10 | Top-25
---|---|---|---
| Precision | Recall | Precision | Recall | Precision | Recall
30k | 0.10 | 0.003 | 0.02 | 0.01 | 0.01 | 0.01
80k | 0.19 | 0.01 | 0.05 | 0.02 | 0.03 | 0.03
Table 6: Upperbound performance of FastTrack using different clustering
algorithms on FTRACE-TREx. Different embedding models do not bring much
effects on the performance upperbound, demonstrating the robustness of
FastTrack on the choice of embeddings. *We list GPU time if it is an open-
sourced model deployed on our server and costs if it is accessed through
queries to the API.
Embedding Scheme | Precision | Recall | Time/Costs*
---|---|---|---
Sentence-Transformer | 0.81 | 0.65 | 0.16 min
Cohere Clustering | 0.80 | 0.63 | $ 0.04
Cohere Classification | 0.75 | 0.57 | $ 0.04
Cohere Search-Query | 0.79 | 0.60 | $ 0.04
Cohere Search-Document | 0.82 | 0.56 | $ 0.04
Mistral-Embed | 0.69 | 0.53 | $0.05
#### FastTrack’s performance with larger corpus size.
Akyürek et al. (2022) benchmark tracing methods only on a curated small
candidate set with about 500 examples for each query. In contrast, our
benchmark has assessed the efficacy and efficiency of our method on a
significantly larger corpus, containing 10k instances. Moving forward, we aim
to tackle a more challenging scenario: we use the same query set of VITAMINC,
but augment the attribution set with additional non-relevant examples until
the total reaches 100k. This setting is designed to evaluate our method’s
robustness in scenarios that better resemble real-world applications. As shown
in Table 7, both methods exhibit a slight decline in performance, yet
FastTrack consistently outperforms BM25 by a significant margin.
BM25’s performance drop is ascribed to the inclusion of new examples that
exhibit high lexical overlap with the queries, thereby impairing BM25’s
effectiveness. For FastTrack, the performance drop is primarily observed in
the initial stage, where some clusters are not successfully retrieved. This
results from the clustering logic changes after more diverse sample are
included. Specifically, if a cluster contains only a few groundtruth examples,
these may be overlooked during the semantic labeling process, leading to false
negative retrieval. Despite this minor reduction in performance, FastTrack
consistently outperforms BM25 by a significant margin.
Table 7: Performance of BM25 and FastTrack when dealing with different corpus
size. Both of the methods encounter a slightly performance drop, while
FastTrack still $1.66\times$ better than BM25.
| VITAMINC-10k | VITAMINC-100k
---|---|---
| F1 | Precision | Recall | F1 | Precision | Recall
BM25 | 0.55 | 0.59 | 0.53 | 0.53 | 0.56 | 0.50
Ours | 0.91 | 0.88 | 0.98 | 0.88 | 0.85 | 0.92
Ours* | 1.00 | 1.00 | 1.00 | 0.95 | 0.95 | 0.95
#### Failure example of FastTrack.
Example below shows that GPT-4 struggles with complex logical reasoning
involving dates; for instance, it incorrectly equates the information from
different dates, focusing merely on numerical comparisons.
Query: As of 14 March , there have been more than 20 confirmed COVID-19 cases
in Kerala . False positive examples: 1: 2020 coronavirus pandemic in Kerala:
As of 28th March 2020 , there are 182 confirmed cases of the virus and more
than 1,15,000 people are under surveillance in the state . 2: 2020 coronavirus
pandemic in Kerala: As of 29th March 2020 , there are 202 confirmed cases of
the virus and more than 1,41,000 people are under surveillance in the state .
GPT-4 analysis:
Text 1: States that "As of 28th March 2020, there are 182 confirmed cases of
the virus in Kerala." This text supports the claim because if there were 182
cases by the 28th of March, there must have been more than 20 cases by the
14th of March. Text 2: States that "As of 29th March 2020, there are 202
confirmed cases of the virus in Kerala." This text also supports the claim
because if there were 202 cases by the 29th of March, there must have been
more than 20 cases by the 14th of March.
## Appendix F Prompting Templates
Figure 3: Prompt template for keyword assignment. Prompt: Analyze the
following group of sentences and identify 5 to 10 of phrases that capture the
main topics, focusing on the key entities. Group of sentences: [cluster texts]
Output the keywords in the following format: #keywords: your keywords here.
[note: Extract 5 to 10 keywords only. The keywords should capture the main
topic and seperated by comma.] Figure 4: Prompt template for supportiveness
evaluation on FTRACE-TREx dataset. Prompt: I will give you a claim and
multiple texts. Carefully evaluate each text, check if the text supports the
claim. For example, Fact: Member of the Scottish Parliament is a legal term in
Scotland. Group of Texts: Text 1: Dennis Robertson is a Scottish politician,
and has been an Member of the Scottish Parliament (MSP) for Aberdeenshire West
since 2011, after defeating the Liberal Democrat incumbent, Mike Rumbles, by a
majority of 4,112 votes. Text 2: The West Lothian question, also known as the
English question, refers to whether MPs from Northern Ireland, Scotland and
Wales, sitting in the House of Commons of the United Kingdom, should be able
to vote on matters that affect only England, while MPs from England are unable
to vote on matters that have been devolved to the Northern Ireland Assembly,
the Scottish Parliament and the Welsh Assembly. #analysis: A "legal term"
refers to a term or expression that is associated with or used in the formal
context of a particular country. Text1 mentions that "Member of the Scottish
Parliament (MSP)" is in Scotland; because we can infer that member of the
scottish parliament is used in the formal political context of Scotland, it
implicitly establishes that Member of the Scottish Parliament is a legal term
in Scotland. Text2 mentions the Scottish Parliament but does not state that
"Member of the Scottish Parliament" is a legal term used within the context of
the Scotland. #scores: 1, 0 Fact: [query] Group of Texts: [indexed candidate
training data] Now evaluate each text carefully in the group and output in the
following format: #analysis: your analysis here. #scores: your scores here.
(score each text 0 or 1 according to the analysis.) Figure 5: Prompt template
for supportiveness evaluation on VITAMINC dataset. Prompt: I will give you a
claim and multiple texts. Carefully evaluate each text, check if the text
supports the claim. For example, Claim: Black Mass earned less than $ 43.6
million in North America as of 22 April 2020. Group of Texts: Text 1: Black
Mass has grossed less than $ 42.6 million in North America as of 22 April
2020. Text 2: Black Mass has grossed less than $ 42.6 million in North America
as of 12 April 2020. Text 3: Black Mass has grossed more than $ 42.6 million
in North America as of 22 April 2020. Text 4: Black Mass has grossed more than
$ 43.6 million in North America as of 22 April 2020. #analysis: Text 1: States
that "Black Mass has grossed less than $42.6 million in North America as of 22
April 2020." This text supports the claim because if it grossed less than
$42.6 million, it also grossed less than $43.6 million. Text 2: States that
"Black Mass has grossed less than $42.6 million in North America as of 12
April 2020." This text does not directly suppport or refute the claim because
it provides information as of 12 April. Without specific information on the
movie’s earnings trends or events that might have affected its box office
performance between 12 April and 22 April 2020, it is impossible to determine
whether the gross was less than $43.6 million as of 22 April. Text 3: States
that "Black Mass has grossed more than $42.6 million in North America as of 22
April 2020." This text does not directly support or refute the claim because
it does not provide enough information to determine whether the gross was less
than $43.6 million. It only indicates that the gross was more than $42.6
million, which could still be less than $43.6 million. Text 4: States that
"Black Mass has grossed more than $43.6 million in North America as of 22
April 2020." This text does not support the claim because it directly
contradicts it, indicating that the gross was more than $43.6 million.
#scores: 1, 0, 0 Fact: [query] Group of Texts: [indexed candidate training
data] Now evaluate each text carefully in the group and output in the
following format: #analysis: your analysis here. #scores: your scores here.
(score each text 0 or 1 according to the analysis.)
|
# Mixed vine copula flows for flexible modelling of neural dependencies
††thanks: Corresponding author: Lazaros Mitskopoulos,
<EMAIL_ADDRESS>
Lazaros Mitskopoulos
School of Informatics
University of Edinburgh
Edinburgh
<EMAIL_ADDRESS>
Theoklitos Amvrosiadis
Centre for Discovery Brain Sciences
University of Edinburgh
Edinburgh
<EMAIL_ADDRESS>
Arno Onken
School of Informatics
University of Edinburgh
Edinburgh
<EMAIL_ADDRESS>
###### Abstract
Recordings of complex neural population responses provide a unique opportunity
for advancing our understanding of neural information processing at multiple
scales and improving performance of brain computer interfaces. However, most
existing analytical techniques fall short of capturing the complexity of
interactions within the concerted population activity. Vine copula-based
approaches have shown to be successful at addressing complex high-order
dependencies within the population, disentangled from the single-neuron
statistics. However, most applications have focused on parametric copulas
which bear the risk of misspecifying dependence structures. In order to avoid
this risk, we adopted a fully non-parametric approach for the single-neuron
margins and copulas by using Neural Spline Flows (NSF). We validated the NSF
framework on simulated data of continuous and discrete type with various forms
of dependency structures and with different dimensionality. Overall, NSFs
performed similarly to existing non-parametric estimators, while allowing for
considerably faster and more flexible sampling which also enables faster Monte
Carlo estimation of copula entropy. Moreover, our framework was able to
capture low and higher order heavy tail dependencies in neuronal responses
recorded in the mouse primary visual cortex during a visual learning task
while the animal was navigating a virtual reality environment. These findings
highlight an often ignored aspect of complexity in coordinated neuronal
activity which can be important for understanding and deciphering collective
neural dynamics for neurotechnological applications.
_K_ eywords neural dependencies $\cdot$ higher-order dependencies $\cdot$
heavy tail dependencies $\cdot$ vine copula flows $\cdot$ neural spline flows
$\cdot$ mixed variables
## 1 Introduction
Coordinated information processing by neuronal circuits in the brain is the
basis of perception and action. Neuronal ensembles encode sensory and
behavior-related features in sequences of spiking activity which can exhibit
rich dynamics at various temporal scales [1]. Acquiring an understanding of
how multivariate interactions in neural populations shape and affect
information transmission is not only important for neural coding theory but
will also inform methodological frameworks for clinically translatable
technologies such as Brain Computer Interfaces (BCIs). Both of these research
programs have enjoyed a surge of activity as a result of recent advances in
imaging technologies [2] and high-yield electrophysiology both for human [3]
and animal studies [4]. BCIs can mediate neural signal transduction for moving
prosthetic limbs or external robotic devices in paralyzed patients or they can
aid communication with patients suffering from locked-in syndrome [5].
Therefore, successful clinical use relies on accurate reading and relaying of
information content transmitted via population spiking responses. Doing so can
be quite challenging from a data analytic perspective as moderate to high-
dimensional brain activity can be considerably complex, exhibiting non trivial
multivariate neural dependencies [6]. Moreover, the resulting behavioral
output variables (e.g. limb movement) might display vastly different
statistics to neural variables like spike trains or event-related potentials
(ERPs). These challenges highlight the importance of developing novel
analytical tools that can handle the complexity within neural population
activity and its relation to behavior. Such tools should also have broad
applicability over different types of data (e.g. spike counts, local field
potentials, EPRs).
The present need for novel methods stems from the fact that the majority of
past work on neural dependencies has focused on pairwise shared response
variability between neurons, also known as noise correlations [7, 8, 9, 10].
Neural responses are known to exhibit considerable variability even when
repeatedly presented with the same stimulus, but this might be part of
collective dynamical patterns of activity in a neural population. The typical
assumption in this line of research is that the noise in neural responses is
Gaussian, and thus, firing rates are modelled with multivariate normal
distributions where a certain covariance matrix specifies all pairwise linear
correlations [11, 12]. While this approach may provide a reasonable
approximation for correlations in coarse time-scales, its validity can be
disputed for spike-counts in finer time-scales. First of all, real spike
counts are characterized by discrete distributions and they exhibit a positive
skew instead of a symmetric shape [13]. Also, spike data do not usually
display elliptical dependence as in the normal distribution but tend to be
heavy tailed [14], which geometrically translates to having probability mass
concentrated on one of the corners. Finally, although the multivariate normal
distribution is characterized by only second-order correlations, recent
studies have indicated that higher order correlations are substantially
present in local cortical populations and have a significant effect on the
informational content of encoding [15, 16, 17, 18, 19] as well as on the
performance of decoding models [20]. Therefore, dissecting the structure of
multivariate neural interactions is important to the study of neural coding
and clinical applications such as BCIs that rely on accurate deciphering of
how neural activity translates to action. This calls for an alternative
approach that goes beyond pairwise correlations.
A statistical tool which is suited for the study of multivariate dependencies
is that of copulas. Copula-based approaches have enjoyed wide usage in
economics for modelling risk in investment portfolios [21], but have received
limited attention in neuroscience. Intuitively, copulas describe the
dependence structures between random variables, and in conjunction with models
of the individual variables, they can form a joint statistical model for
multivariate observations. When observations come from continuous variables
their associated copulas are unique, independent from marginal distributions
and invariant to strictly monotone transformations [22]. However, data in
neuroscience research are often discrete (e.g. spike counts) or they may
contain interactions between discrete and continuous variables with vastly
different statistics (e.g. spikes with local field potentials or behavioral
measurements such as running speed or pupil dilation). Despite the
indeterminacy of copulas in these cases, they are still valid constructions
for discrete objects and mixed interactions and one can apply additional
probabilistic tools to obtain consistent discrete copulas [23]. Previous work
has successfully applied copula-based methods to discrete or mixed settings
[24, 25, 26, 27, 28] using copulas from parametric families that assume a
certain type of interaction. Although parametric models are a powerful tool
for inference, their application can bear a risk of misspecification by
imposing rather rigid and limiting assumptions on the types of dependencies to
be encountered within a dataset with heterogeneous variables or multiscale
dynamical processes. This risk is especially amplified for dependencies
between more than two variables as available multivariate copulas are quite
limited in number and assume a particular type of dependence structure for all
variables which can ignore potentially rich interaction patterns. As the set
of commonly used bivariate parametric copulas is much larger, a common
alternative is to decompose multivariate dependencies into a cascade of
bivariate copulas organized into hierarchical tree structures called vines or
pair copula constructions [29]. Nodes of the vines correspond to conditional
distributions and edges correspond to copulas that describe their interaction.
This formulation allows for a flexible incorporation of various dependence
structures in a joint model. Previous studies that employed vine copulas in
mixed settings used parametric models [24, 25, 26, 27, 28] For the present
study, given the aforementioned intricacies of neuronal spiking statistics, we
aim to explore the potential of non-parametric methods as a more flexible
alternative for estimating discrete and continuous vine copulas. Existing non-
parametric approaches have focused on kernel-based methods [30, 31] or
jittering and continuous convolutions with specified noise models to obtain
pseudo-continuous data [32, 33].
Another model-free method that shows promise is that of normalizing flows, a
class of generative models for density estimation that allow for flexible
sampling [34, 35]. Recently, some authors have attempted to employ normalizing
flows for non-parametric modeling of copulas using simulated and standard
benchmark datasets [36, 37]. An application of these models to recordings from
neural populations is still missing and has the potential to shed light on the
structure of coordinated neural activity, thereby potentially improving BCIs
that take this structure into account.
In this study, we aimed to conduct a thorough investigation into flow-based
estimation of vine copulas with continuous and discrete artificial data in
various settings with different but known dependence structures and number of
variables. Furthermore, we sought to demonstrate the potential of this
framework to elucidate interaction patterns within neural recordings that
contain heavy tails and extend beyond bivariate dependencies. For this reason,
we chose to investigate neural responses in the mouse V1 while the animal is
navigating in a virtual reality environment. Studying neural interfaces in
rodents has been important for pre-clinical testing of BCIs to probe potential
limitations that can inform applications in humans [38, 39]. The test case we
chose serves as a proof-of-concept study but it can also provide meaningful
insights on how spatial navigation related cues and/or behavioral variables
modulate visual activity, which can inform future clinical research on BCIs.
## 2 Materials and Methods
### 2.1 Copulas
Multivariate neuronal interactions can be described probabilistically by means
of copulas, which embody how spiking statistics of individual neurons, i.e.
the marginal distributions, are entangled in intricate ways to produce the
observed joint population activity. The central theoretical foundation for
copula-based approaches is Sklar’s theorem [40]. It states that every
multivariate cumulative distribution function (CDF) $F_{x}$ can be decomposed
into its margins, in this case the single-neuron distributions
$F_{1},...F_{d}$, and a copula $C$ (Figure 1A) such that:
$F_{x}(x_{1},\dots,x_{d})=C(F_{1}(x_{1}),\dots,F_{d}(x_{d}))$ (1)
Figure 1: Mixed vine copula flows. A. Samples from mixed variables of any
joint probability density function (pdf) can be decomposed into their margins
and a copula. Copulas are extracted by transforming continuous and discrete
samples to uniform through the probability integral transform. B. Graphical
illustration of a C-vine for 4 variables. Nodes and edges of the first tree
denote the variables and bivariate dependencies respectively. Edges of
subsequent trees denote dependencies that condition on one or more variables.
C. Illustration of forward and inverse normalizing flow transformation f of
base distribution Z and target distribution Y.
Copulas are also multivariate CDFs with support on the unit hypercube and
uniform margins and their shape describes the dependence structure between
random variables in a general sense which goes beyond linear or rank
correlations [22]. Following Sklar’s theorem, it is possible to obtain copulas
from joint distributions using:
$C(u_{1},\dots,u_{d})=F_{x}(F^{-1}_{1}(u_{1}),\dots,F^{-1}_{d}(u_{d})),$ (2)
Conversely, it is also possible to construct proper joint distributions by
entangling margins with copulas. These operations rely on the probability
transform $F$ and its generalized inverse, the quantile transform $F^{-1}$.
The probability transform maps samples to the unit interval:
$:F(X)\rightarrow{U\sim{U_{[0,1]}}}$, where $U_{[0,1]}$ denotes the uniform
distribution on the unit interval. Since copulas for discrete data depend on
margins and are not unique [23], additional tools are required to obtain
consistent mapping to copula space. We employed the distributional transform:
$G(X,V)=F_{x-}(x)+V(F_{x}(x)-F_{x-}(x))$ (3)
where $F_{x-}(x)=Pr(X<x)$ as opposed to the regular expression for the CDF,
$F_{x}(x)=Pr(X<=x)$, and $V$ is a random variable uniformly distributed on
[0,1] independent of $X$. This extension to the probability transformation
effectively converts a discrete copula to a pseudo-continuous variable by
adding uniform jitter in between discontinuous intervals in the support of
discrete variables and makes it possible to use non-parametric estimators
designed for continuous data. An example with continuous and discrete
observations is illustrated in Figure 1A. This case of a mixed interaction is
described by a Clayton copula which displays an asymmetric heavy tail. The
empirical copula can be discovered by subjecting the variables to the
probability transform with an added distributional transform for the discrete
one. This operation dissects the dependence information that is embedded in
the joint probabilities of these two variables.
### 2.2 Pair copula constructions
The curse of dimensionality encountered in large datasets can pose
considerable challenges for copula modeling. Pair copula constructions [41]
offer a flexible way of scaling copula models by factorising the multivariate
distribution into a cascade of bivariate conditional distributions that can be
described by bivariate copulas. The latter can be modelled parametrically, as
previous studies have already done [28, 27] or with non-parametric tools,
which is the present study’s approach.
The space of possible factorizations is prohibitively large so vine copula
structures, a special type of pair copula constructions can be employed to
facilitate inference and sampling [29]. They can be represented as
hierarchical sets of trees which account for a specific graph of multivariate
interactions among elements of the distributions and assume conditional
independence for the rest. In the present study, we focused on the canonical
vine or C-Vine (Figure 1B) in which each tree in the hierarchy has a node that
serves as a connection hub to all other nodes. The C-Vine decomposes the joint
distribution $f$ into a product of its margins and conditional copulas $c$.
$f_{X}(x_{1},...,x_{d})=\prod_{k=1}^{d}f(x_{k})\prod_{j=1}^{d-1}\prod_{i=1}^{d-j}c_{j,i+j|1,...,j-1}(F(x_{j}|x_{1},...,x_{j-1}),F(x_{i+j}|x_{1},...,x_{j-1}))$
(4)
where $c_{i,j|A}$ denotes the pair copula between elements $i$ and $j$ given
the elements in the set $A$", which is empty in the surface tree but it
increases in number of elements with deeper trees.
### 2.3 Copula Flows
We modeled the margin and copula densities non-parametrically using a specific
class of normalizing flows, that is called Rational-Quadratic Neural Spline
Flows (NSF) [42]. In general, normalizing flows are a class of generative
models that construct arbitrary probability densities using smooth and
invertible transformations to and from simple probability distributions [34]
(Figure 1C). In essence, this is an application of the change of variables
formula:
$p_{x}(x)=p_{u}(T^{-1}(x))\det\left|\frac{\partial T^{-1}(x)}{\partial
x}\right|,$ (5)
where $p_{x}(x)$ is the density of the observations and $p_{u}$ is the base
density of random variable $U=T^{-1}(X)$, which is a known and convenient
distribution such as the normal or the uniform distribution. The
transformation $T$ is usually a composition of invertible transformations that
can be perceived and implemented as an artificial neural network with a
certain number of layers and hidden units. Its parameters have to be learned
through training in order to achieve an invertible mapping between the two
distributions, while scaling appropriately by the determinant of the Jacobian
matrix which keeps track of volume changes induced by $T$.
Since our main goal was to apply normalizing flows on a copula-based framework
for neural dependencies, it was natural to choose the uniform distribution on
[0,1] as a base distribution, so that backward and forward flow
transformations for the margins approximate the probability transform and its
inverse, the quantile transform respectively, so as to map observations to
copula space and back. Furthermore, a uniform base distribution for copula
flows can be leveraged to generate new (simulated) observations via inverse
sampling. Different types of normalizing flows exist in the literature which
involve simple affine or more flexible non-affine transformations albeit with
the cost of sacrificing invertibility in some cases [35]. Our choice of
employing NSF [42] in this study for modelling both margin and copula
densities was in virtue of the fact that they combine the flexibility of non-
affine flows while maintaining easy invertibility by approximating a quasi-
inverse with piecewise spline-based transformations around knot points of the
mapping.
### 2.4 Sequential Estimation and Model Selection
To fit the C-vine model with NSF to data, we applied the Inference for Margins
procedure, which first fits the margins and then fits the copulas deriving
from pairs of margins. For the first step, we fit each margin with NSF and
then proceeded with the copulas of a particular canonical vine formulation
[29]. Fitting NSF to bivariate copulas in the vine was conducted sequentially
starting with the copulas of the surface layer of the tree. Subsequently,
conditional marginals for the tree in the layer next in depth were estimated
using the copulas of the previous layer via h-functions [43]. Then,
conditional copulas were constructed by transforming the conditional marginals
in the same layer according to (2). This procedure was followed until all
copulas in the vine decomposition were estimated. For each copula, we used a
random search procedure [44] to determine the best performing NSF
hyperparameter configuration on a validation set containing 10% of the data.
The hyperparameters that were tuned during training were the number of hidden
layers and hidden units as well as the number of knots for the spline
transforms. This sequential estimation and optimization scheme for NSF-based
vine copulas was followed in both our analyses with artificial data as well as
data from neuronal recordings.
### 2.5 Other non-parametric estimators
The other non-parametric estimators used for comparisons against NSF included
four versions of Kernel Density Estimators (KDE), namely one with log-linear
local likelihood estimation (tll1), one with log-quadratic local likelihood
estimation (tll2), one with log-linear local likelihood estimation and nearest
neighbour bandwidths (tll1nn) and one with log-quadratic local likelihood
estimation and nearest neighbour bandwidths (tll2nn) [33]. Lastly, an
estimator based on Bernstein polynomials [45] was also used for the
comparisons. The implementations for all these five non-parametric estimators
are from kdecopula package [33].
### 2.6 Artificial data
The NSF framework for vine copulas was validated on artificial data with known
dependency structures and was compared against other non-parametric
estimators. We constructed several test cases where data was continuous or
discrete, consisting of 4 (low dimensional) or 8 (higher dimensional)
variables exhibiting weak or strong dependencies that were characterized by
different copula families, namely either Clayton, Frank or Gaussian copulas.
These three types of parametric copulas display different behavior in the tail
regions [46]. Clayton copulas have an asymmetric tail dependence whereby
probability mass is concentrated in one corner of the copula space indicating
a single heavy tail region (see example copula in Figure 1A). On the contrary,
Frank copulas do not have heavy tails and probability mass is allocated
uniformly and symmetrically along the correlation path. Gaussian copulas are
also symmetric and without particularly heavy tails, but probability mass
concentration in the tail regions is larger compared to Frank copulas. The
strength of dependencies was determined by varying the $\theta$ parameter for
Clayton ($\theta=2$ for weak and $\theta=5$ for strong dependence) and Frank
copulas ($\theta=3$ for weak and $\theta=7$ for strong dependence) and the
off-diagonal entries of the 2x2 correlation matrix for Gaussian copulas (0.4
for weak dependence and 0.8 for strong dependence). We constructed and drew
simulated samples from all vines with the aforementioned specifications using
the mixed vines package developed by Onken and Panzeri (2016). Training for
all the estimators was conducted with 5000 simulated samples for each variable
in the artificial data. The training procedure was repeated 10 times.
Performance was measured with Kullback-Leibler (KL) divergences of 8000 copula
samples generated by each of the estimators to an equal number of ground truth
copula samples from the mixed vines package [28]. To estimate the KL
divergences from sample data, a k-nearest neighbour algorithm was used [47],
which was implemented in [48]. The resulting KL divergences from the 10
repetitions and the different copulas in each vine were aggregated to measure
the overall performance of the framework in each test case. We statistically
compared performances by all estimators via Kruskal-Wallis significance tests
at level of significance equal to 0.05. Bonferonni correction was used for
multiple comparisons. Moreover, we calculated copula entropies from the NSF
copula densities via classical Monte Carlo estimation:
$h(c(u_{1},u_{2}))=\mathbb{E}_{c}[-\log_{2}c(u_{1},u_{2})]\approx-\frac{1}{k}\sum_{k=1}^{K}(\log_{2}c(u^{k}_{1},u^{k}_{2})),$
(6)
where $h$ denotes the entropy and $\mathbb{E}_{c}$ denotes the expectation
with respect to the copula $c$. The expectation is approximated by summing
over a $K$ number of samples which needs to be sufficiently large ($K=8000$ in
our study). Negative copula entropy provides an accurate estimate of mutual
information between variables which does not depend on the marginal statistics
of each variable [40, 49] and is a measure of the degree to which knowing one
variable can reduce the uncertainty of the other one. All analyses including
sampling and entropy calculations were conducted on an ASUS laptop with
Intel(R) 4 Cores, i5-8300 CPU, 2.30 GHz.
### 2.7 Experimental data
In order to assess our framework’s applicability to neuronal activity we used
2-photon calcium imaging data of neuronal activity in the mouse primary visual
cortex, that were was collected at the Rochefort lab (see [50] for more
details). Briefly, V1 layer 2/3 neurons labeled with the calcium indicator
GCamP6s were imaged while the animal was headfixed, freely running on a
cylindrical treadmill and navigating through a virtual reality environment
(160 cm). Mice were trained to lick at a specific location along the virtual
corridor in order to receive a reward. In addition to neuronal activity,
behavioral variables such as licking and running speed were monitored
simultaneously. Over the course of 5 days, mice learned to lick within the
reward zone to receive the water reward. This visual detection task was used
to investigate V1 neuronal activity before, during and after learning [50].
The goal of the experiments was to elucidate how repeated exposure to a
stimulus modulates neural population responses, particularly in the presence
of a stimulus-associated reward. Our analysis was based on deconvolved spike
trains instead of the calcium transients. Spiking activity had been
reconstructed using the MLspike algorithm [51] (see [50] for more details).
The data we used in this study were limited to one mouse on day 4 of the
experiment when the animal was an expert at the task. Moreover, in order to
provide a proof-of-concept example and illustrate the complete vine copula
decomposition as well as the performance of the NSF framework, we selected a
subset of 5 neurons out of the 102 V1 neurons that were monitored in total for
that particular mouse. Each of the 5 selected neurons showed non-negligible
positive or negative rank correlation with every other neuron in that subset
(Kendall’s $\tau>0.3$ or Kendall’s $\tau<-0.3$), suggesting that they might be
part of a module warranting a more detailed investigation of the dependence
structures within.
## 3 Results
### 3.1 Validation on artificial data
In order to demonstrate the potential of the NSF vine copula framework to
capture multivariate dependencies of arbitrary shape and data type (continuous
vs discrete) we conducted a simulation study. The set of generated samples
that were used for training the NSF (n=5000) included cases with 3 different
types of copulas (Clayton, Frank and Gaussian) with each dictating a different
set of dependence structures for weak and stronger dependencies, continuous
and discrete data as well as 4 and 8 dimensions. This ensemble provided a wide
range of settings from simpler symmetric dependencies to skewed and heavy
tailed interactions that can be encountered in neural population spiking
responses.
Overall, performance of NSF as assessed by the KL divergence of the NSF-
generated copula samples with those from the ground truth copulas was broadly
within comparable levels to that of the KDE estimators while Bernstein
estimators often performed the worst (Figure 2 and Figure 3). However, it is
worth noting that relative performances varied slightly on a case-by-case
level. For example, with weaker dependencies in 4 dimensional data with all
copulas, NSF performed slightly but significantly worse than the other
estimators (Kruskal-Wallis tests, $p<0.05$) except Bernstein estimators
(Figure 2). The latter even outperformed NSF in one exceptional case with
weakly dependent 4D discrete data with Frank copulas. This trend of slightly
worse NSF performance relative to all else except Bernstein estimators was
also observed in 4D continuous and discrete data for all copulas with stronger
dependencies (Figure 3). However, NSF closed that gap and performed similarly
with the group of KDE estimators in cases where data was 8D, either continuous
or discrete and was entangled with Clayton copulas with weak dependencies or
Frank copulas with both weak and stronger dependencies (Kruskal-Wallis tests,
$p>0.05$). Notably, in the case of Clayton copulas with stronger dependencies
for 8D data, NSF outperformed all other estimators (Kruskal-Wallis tests,
$p<0.05$). These findings might suggest that the flexibility of the NSF
framework can be more beneficial with higher data dimensionality and
dependence structures that are characterized by heavier tails.
Figure 2: NSFs perform comparably to existing non-parametric estimators.
Boxplots of performance of NSFs on all bivariate copulas from artificial data
compared to Kernel Density Estimators with either log-linear local likelihood
estimation (tll1), log-quadratic local likelihood estimation (tll2), log-
linear local likelihood estimation and nearest neighbour bandwidths (tll1nn),
log-quadratic local likelihood estimation and nearest neighbour bandwidths
(tll2nn) and Bernstein estimator. Simulations shown in this figure had weak
dependencies described by Clayton ($\theta=2$), Gaussian ($0.4$ in the off-
diagonal) and Frank copulas ($\theta=3$) for 4 and 8 dimensional vines with
continuous and discrete variables. Figure 3: Same conventions as Figure 2 but
for simulations with strong dependencies described by Clayton ($\theta=5$),
Gaussian ($0.8$ in the off-diagonal) and Frank copulas ($\theta=7$) .
As copulas offer a detailed and invariant to marginal statistics view into
multivariate dependencies, their negative entropy provides an accurate
estimate of mutual information between variables. Thus, calculating copula
entropies can be useful not only in undestanding coordinated information
processing but also in BCI settings where dependencies might be an important
feature needing to be accounted for. Despite the largely similar performance
to KDEs, NSFs showed a remarkable advantage regarding drawing samples from the
trained models and estimating copula entropies. Inverting the kernel
transformation in KDE estimators and rejection sampling in Bernstein
estimators are considerably more computationally expensive compared to a
flexible and substantially faster sampling scheme in NSFs which directly
approximate the CDF and inverse CDF of margins and copulas (Figure 4).
Plotting the copula entropies from all the estimators against the KL
divergence for every particular iteration of fitting in every bivariate copula
from the vine revealed an inverse relationship between the two quantities.
Namely, better performance appeared to relate to higher copula entropy and
thus mutual information for all distinct copulas, vine dimensions and data
types (Figure 5). This could mean that bivariate copulas with higher KL
divergence were overfit to the point of diminishing the informational content
of the interaction captured by the copula. It is noteworthy that Clayton
copulas from 8D vines that were well fit by NSFs exhibited significantly
higher copula entropy compared to the other estimators (Kruskal-Wallis,
$p<0.01$). This could indicate an potential advantage of NSFs over the other
estimators with heavy-tailed data, which might warrant further future
investigation.
Figure 4: NSFs vastly outperform other non-parametric estimators on sampling.
A. Boxplots of time (mins) required to sample from trained NSF compared to the
other non-parametric estimators. The vertical axis is plotted on logarithmic
scale. B. Boxplots of time (mins) required to estimate copula entropy from the
trained NSF compared to the other non-parametric estimators. Figure 5: Scatter
plots of performance against copula entropy for NSFs versus other non-
parametric estimators on artificial data. All other conventions are the same
as Figures 2 and 3. Simulations shown in this figure had strong dependencies.
### 3.2 NSF elucidate dependence structures in rodent V1 spiking activity and
behavior
Having validated the vine flow copula framework with artificial data, we
subsequently focused on spiking activity from a subset of 5 V1 layer 2/3
neurons while the animal was navigating a virtual reality corridor (Figure 6A)
[50]. A parametric vine copula framework with Gaussian processes [14] was
recently applied to calcium transients from the same V1 layer 2/3 neurons
[50]. In the present study, we focused instead on modeling the deconvolved
spiking activity with flow-based vine copulas. Our use of a non-parametric
tool aimed at escaping potentially restricting assumptions that parametric
copula families might place on the dependence structures among neurons. Our
analysis aimed to detect such dependencies both as a function of time and
position since previous findings indicate that V1 neuronal activity can be
modulated by behaviorally relevant variables such as a reward being given at a
particular location.
Raster plots across all trials on day 4 for the subset of the selected 5
neurons in Figure 6A illustrate how some of the neurons were more active for a
short span of the virtual corridor within and after the reward zone (120-140
cm) while the activity of others was more spread out across the corridor and
fell off at the onset of the reward zone. The strength of rank correlation
between pairs of these neurons can be assessed by measuring the Kendall’s
$\tau$ from their spiking activities. However, this approach reduces the study
of dependencies into single numbers that only provide a limited description of
the interactions. In contrast, a copula-based approach can provide a detailed
account of the actual shapes of neuronal dependencies which are usually
characterized by a concentration of probability mass in the tail regions.
In similar fashion to the analysis on artificial data, we fit a C-vine whereby
NSFs were fit to the spiking distributions of the neurons, i.e. the margins as
a first step and then NSFs were sequentially fit to the empirical copulas
(blue scatter plots in Figure 6B) from the surface tree level to the deeper
one. These empirical copulas were obtained by transforming the data with the
distributional transform. The 5 neurons were ordered according to the degree
to which they exhibited statistical dependence with the other neurons as
measured by the sum of Kendall’s $\tau$ for each neuron. The copula densities
(Figure 6C) and samples (red scatter plots in Figure 6D) from the trained NSFs
were able to accurately capture the dependence structures between the neurons.
All interactions were characterized by the presence of heavy tails in
different corners of the uniform cube. Heavy tail dependencies at the top
right part of the cube signified neurons that were more co-dependent when both
neurons were active compared to other activity regimes (e.g. Figure 6C top row
$1,2$ and $1,5$). For example, neurons like neuron 1 and neuron 2 displayed
weak or no significant interaction until their activity was modulated by
experimental or behavioral variables that are associated with the reward zone.
Conversely, heavy tails at the bottom right (Figure 6C $2,3|1$) or top left
(Figure 6C $4,5|1,2,3$) signified inverse relations of spiking activity
between these neurons, i.e. being active at different locations in the
corridor. Furthermore, it is worth noting that the probability mass
concentration in the tail regions was different among neurons, with some pairs
displaying lighter tails than others (e.g. Figure 6C $3,4|1,2$).
Figure 6: Tail dependencies in position dependent V1 activity are captured by
NSF vine copulas. A. Illustration of mouse navigating a virtual environment
with grating stimuli until it reaches the reward zone (120-140 cm) where it is
required to lick in order to receive water reward. Raster plots for 5 neurons
in V1 rodent data across trials and position (bin size of 2.5cm) in the
virtual corridor. Grey region denotes the reward zone B. Scatter plots of
empirical copula samples (blue dots) in a 5D vine extracted from the spike
data. Axis labels denote which unconditional or conditional neuron margins
have been transformed to copula space. Margin indices after vertical slash
denote those in the conditioning set C. Copula densities of the 5-D vine from
the trained NSF. D. Simulated samples (red dots) from the trained NSF copula
densities of the 5-D vine.
As neuronal activity in V1 is modulated by introducing a reward in a certain
location of the virtual corridor, it is also interesting to investigate neural
responses and dependencies in time around that reward. Therefore, we also
analyzed neural interactions as revealed by the spike counts of the same 5
neurons 3.5 seconds before and after the reward (bin size was 300 ms). Only
successful trials where the mouse licked within the reward zone were included
in this analysis. Raster plots in Figure 7A show a variety of spiking patterns
relative to the timing of the reward across trials. The copula densities
(Figure 7C) and samples (red scatter plots in Figure 7D) from the trained NSFs
closely captured the dependence structures as before. Moreover, as before,
these dependence structures were characterized by heavy tailed regions either
in the top right (e.g. $1,2$ in Figure 7C) or top left corners (e.g. $2,5|1$
in Figure 7C) indicating stronger neuronal interactions for some regimes of
spiking activity and not otfhers. The more apparent block structure in this
case compared to the previous one was a result of the fewer states of spike
counts with the bin size we chose. For example, probability mass in the copula
space for neurons that would fire from 0 to 5 spikes in a given time bin,
would have to be assigned to blocks that correspond to the relative
proportions of jointly observed spike counts.
Figure 7: Tail dependencies in time dependent V1 activity are captured by NSF
vine copulas. A. Raster plots for same 5 neurons in V1 rodent data across
trials and time with respect to reward (3.5 seconds before until 3.5 seconds
after reward) in the virtual corridor. Grey vertical line denotes reward
acquisition. B. Scatter plots of empirical copula samples (blue dots) from the
5-D vine. All other conventions are the same with Figure 6 C. Copula densities
of the 5-D vine from the trained NSF. D. Simulated samples (red dots) from the
trained NSF copula densities of the 5-D vine.
A copula-based approach can also offer a tool for studying dependencies of
neuronal activity with behavioral variables which might exhibit vastly
different statistics, such as running speed and number of licks. In Figure 8
we showcase how copulas modelled with NSFs can elucidate the shape of
interaction of running speed and licks with the spiking activity of an example
neuron (neuron 16 from the dataset, which is not included in the 5 neurons in
the previous analysis) binned with respect to position (bin size was 2.5 cm)
in the virtual corridor. This neuron increased its activity considerably
within the reward zone (Figure 8B) which coincided with greatly reduced
running speed (Figure 8A) as the mouse stopped to lick (Figure 8C) and receive
the reward. Licking was thus positively co-dependent with spiking activity,
$\tau=0.25,p<0.001$ for number of licks and running speed was negatively co-
dependent with spiking activity $\tau=-0.21,p<0.001$. However, transforming
the variables to copula space revealed mostly uniform copulas with a
relatively heavier tail in the bottom right for running speed (copulas in
Figure 8A) and in the top right for licking (copulas in Figure 8C). This
finding indicated that these behavioral variables had a rather weak or no
interaction for most of the span of the virtual corridor except for the part
when the neuron was mostly active, which was the reward zone.
Figure 8: Dependencies of spiking activity with behavioral variables. A. Left:
Color plot of running speed (cm/sec) across the virtual corridor for all
trials. White vertical lines denote beginning and end of reward zone. Center:
Empirical copula of neuron 16 spiking activity (discrete) with running speed
(continuous). Right: Copula density from the trained NSF. Red circles indicate
probability concentration in the tail. B. Raster plot of neuron 16 spiking
activity across trials and position in the virtual corridor. C. Left:
Grayscale-coded plot of number of licks across the virtual corridor for all
trials. Red vertical lines denote beginning and end of reward zone. Center:
Empirical copula of neuron 16 spiking activity (discrete) with number of licks
(discrete). Right: Copula density from the trained NSF
Considering all the aforementioned, the flow-based vine copula framework shows
remarkable promise for flexibly capturing complicated interaction patterns
within neural populations. The rich picture of these interactions and
potential insights into neural circuit function would have remained undetected
by only measuring pairwise rank correlations or pairwise Pearson correlations
which would not have indicated any heavy tailed interactions.
## 4 Discussion
In this study, we proposed a fully non-parametric approach for estimating vine
copula densities for both continuous and discrete data using NSF, a subtype of
normalizing flows. Overall, the framework performed comparably to existing
non-parametric approaches on artificial data while benefiting from more
flexible and faster sampling. Furthermore, we demonstrated how flow-based vine
copulas can shed light on the structure of dependencies in neural spiking
responses.
The intricate shapes in bivariate copulas of spikes involving skewness and
heavy tails would have been assumed as non-existent in a conventional approach
based on pairwise linear correlations. Additionally, the arrangement of the
discovered copulas into block structures (Figure 6B) would have been harder to
capture by commonly applied parametric copula models (e.g. Clayton, Frank or
Gaussian copulas) and thus lead to misleading conclusions. Therefore, we
showed that non-parametric modeling with normalizing flows can be a valuable
tool especially in the case of copula-based methods for discrete data and
mixed data.
The development of such tools is crucial for understanding how coordinated
neural activity transmits information about external stimuli or internal
signals that guide planning and action. Copula-based methods have the capacity
to provide a description of neural activity that includes the intricacies of
dependencies and allows for higher-order interactions to occur within a
certain set of conditions specified by the vine structure. Such descriptions
can potentially offer significant insights that will aid and inform the
development of neural coding theory. At the same time, decoding models that
take into account the aforementioned features elucidated by copula methods,
could potentially be valuable for the development of reliable Brain-Computer
Interfaces. Some lines of evidence suggest that higher-order interaction
patterns might have a significant presence and role in shaping collective
neural dynamics and conveying information [16, 17, 18] but further research
needs to be conducted to gain a deeper understanding of the processes
involved.
A limitation of our study is that the type of normalizing flows we used was
designed to model variables with continuous data. We therefore included an
additional rounding operation for discrete margins that were generated by
trained NSF. This issue with discrete margins could be addressed in future
work that can improve upon the framework by incorporating normalizing flow
models that are more naturally suited to handle discrete or mixed data and do
not need the ad hoc modifications that were employed in the present project.
SurVAE flows that were developed recently [52] as an attempt to unify
variational autoencoders and normalizing flows might potentially be better
suited for discrete data as the kind of surjective transformations may better
account for discontinuities in discrete cumulative distribution functions.
Future directions include the analysis of entire neural populations as opposed
to small subsets of a population. A possible extension to modelling population
dependencies with vine copula methods can also involve leveraging the power of
dimensionality reduction methods. A well established trait of neural
population activity is that latent collective dynamics that drive responses
occupy a subspace that is much lower in dimensionality than the number of
neurons recorded [53]. Thus, dimensionality reduction methods could be applied
prior to copula modeling so that the latter can be employed on a set of low
dimensional latent factors that capture the most prominent latent processes.
Combining such methods can potentially provide even less computationally
demanding frameworks that can be useful for clinical translation.
## Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.
## Author Contributions
LM and AO conceptualized the study and designed the methodology. LM conducted
data analysis, visualization of findings under the supervision of AO, and
wrote the first draft version. TA collected and curated the neuronal
recordings data. LM, TA and AO reviewed and edited the article. All authors
approved the submitted version.
## Funding
This work was supported by the Engineering and Physical Sciences Research
Council (grant [EP/S005692/1], to AO) and the Precision Medicine Doctoral
Training Programme (Medical Research Council grant number [MR/N013166/1], to
TA). The funders had no role in study design, data collection and analysis,
decision to publish, or preparation of the manuscript.
## Acknowledgments
We would like to thank Nathalie Rochefort and Tom Flossman for constructive
feedback on the analysis and manuscript.
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|
# FIESTA: Fast IdEntification of State-of-The-Art models using adaptive bandit
algorithms
Henry B. Moss
STOR-i Centre for
Doctoral Training,
Lancaster University
&Andrew Moore
School of Computing
and Communications,
Lancaster University
<EMAIL_ADDRESS>
&David S. Leslie
Dept. of Mathematics
and Statistics,
Lancaster University
&Paul Rayson
School of Computing
and Communications,
Lancaster University
###### Abstract
We present FIESTA, a model selection approach that significantly reduces the
computational resources required to reliably identify state-of-the-art
performance from large collections of candidate models. Despite being known to
produce unreliable comparisons, it is still common practice to compare model
evaluations based on single choices of random seeds. We show that reliable
model selection also requires evaluations based on multiple train-test splits
(contrary to common practice in many shared tasks). Using bandit theory from
the statistics literature, we are able to adaptively determine appropriate
numbers of data splits and random seeds used to evaluate each model, focusing
computational resources on the evaluation of promising models whilst avoiding
wasting evaluations on models with lower performance. Furthermore, our user-
friendly Python implementation produces confidence guarantees of correctly
selecting the optimal model. We evaluate our algorithms by selecting between
$8$ target-dependent sentiment analysis methods using dramatically fewer model
evaluations than current model selection approaches.
## 1 Introduction and Background
Natural Language Processing (NLP) is a field driven by empirical evaluations.
Authors are under pressure to demonstrate that their models or methods achieve
state-of-the-art performance on a particular task or dataset, which by
definition requires reliable model comparison. As models become more numerous,
require larger computational resources to train, and the performance of
competing models gets closer, the task of reliable model selection has not
only become more important, but also increasingly difficult. Without full
disclosure of model settings and data splits, it is impossible to accurately
compare methods and models.
To be able to perform meaningful model comparisons, we need to be able to
reliably evaluate models. Unfortunately, evaluating a model is a non-trivial
task and the best we can do is to produce noisy estimates of model performance
with the following two distinct sources of stochasticity:
1. 1.
We only have access to a finite training dataset, however, evaluating a model
on its training data leads to severe over-estimates of performance. To
evaluate models without over-fitting, practitioners typically randomly
partitioning data into independent training and testing sets, producing
estimates that are random quantities with often high variability for NLP
problems Moss et al. (2018). Although methods like bootstrapping Efron and
Tibshirani (1994) and leave-one-out cross validation Kohavi (1995) can provide
deterministic estimates of performance, they require the fitting of a large
number of models and so are not computationally feasible for the complex
models and large data prevalent in NLP. Standard NLP model evaluation
strategies range from using a simple (and computationally cheap) single train-
test split, to the more sophisticated $K$-fold cross validation, CV Kohavi
(1995).
2. 2.
The vast majority of recent NLP models are non-deterministic and so their
performance has another source of stochasticity, controlled by the choice of
random seed during training. Common sources of model instability in modern NLP
include weight initialisation, data sub-sampling for stochastic gradient
calculation, negative sampling used to train word embeddings Mikolov et al.
(2013) and feature sub-sampling for ensemble methods. In particular, the often
state-of-the-art LSTMs (and its many variants) have been shown to exhibit high
sensitivity to random seeds Reimers and Gurevych (2017).
For reliable model selection, it is crucial to take into account both sources
of variability when estimating model performance. Observing a higher score for
one model could be a consequence of a particularly non-representative train-
test split and/or random seed used to evaluate the model rather than a genuine
model improvement. This subtlety is ignored by large scale NLP competitions
such as SemEval with evaluations based on a pre-determined train-test split.
Although more precise model evaluations can be obtained with higher
computation, calculating overly precise model evaluations is a huge waste of
computational resource. On the other hand, our evaluations need to provide
reliable conclusions (with only a small probability of selecting a sub-optimal
model). It is poorly understood how to choose an appropriate evaluation
strategy for a given model selection problem. These are task specific,
depending on model stability, the closeness in performance of competing models
and subtle properties of the data such as the representativeness of train-test
splits.
In contrast to common practice, we consider model selection as a sequential
process. Rather than using a fixed evaluation strategy for each model (which
we refer to as a non-adaptive approach), we start with a cheap evaluation of
each model on just a single train-test split, and then cleverly choose where
to allocate further computational resources based on the observed evaluations.
If we decide to further test a promising model, we calculate an additional
evaluation based on another data split and seed, observing both sources of
evaluation variability and allowing reliable assessments of performance.
To perform sequential model fitting, we borrow methods from the multi-armed-
bandit (MAB) statistical literature Lai and Robbins (1985). This field covers
problems motivated by designing optimal strategies for pulling the arms of a
bandit (also known as a slot machine) in casinos. Each arm produces rewards
from different random distributions which the user must learn by pulling arms.
In particular, model selection is equivalent to the problem of best-arm-
identification; identifying the arm with the highest mean. Although appearing
simple at a first glance, this problem is deceptively complex and has provided
motivation for efficient algorithms in a wide range of domains, including
clinical trials Villar et al. (2015) and recommendation systems Li et al.
(2010).
Although we believe that we are the first to use bandits to reduce the cost
and improve the reliability of model selection, we are not the first to use
them in NLP. Recent work in machine translation makes use of another major
part of the MAB literature, seeking to optimise the long-term performance of
translation algorithms Nguyen et al. (2017); Sokolov et al. (2016); Lawrence
et al. (2017). Within NLP, our work is most similar to Haffari et al. (2017),
who use bandits to minimise the number of data queries required to calculate
the F-scores of models. However, this work does not consider the stochasticity
of the resulting estimates or easily extend to other evaluation metrics.
The main contribution of this paper is the application of three intuitive
algorithms to model selection in NLP, alongside a user-friendly Python
implementation: FIESTA (Fast IdEntification of State-of-The-
Art)111https://github.com/apmoore1/fiesta. We can automatically identify an
optimal model from large collections of candidate models to a user-chosen
confidence level in a small number of model evaluations. We focus on three
distinct scenarios that are of interest to the NLP community. Firstly, we
consider the fixed budget (FB) model selection problem (Section 4.1), a
situation common in industry, where a fixed quota of computational resources
(or time constraints for real-time decisions) must be appropriately allocated
to identify an optimal model with the highest possible confidence. In
contrast, we also consider the fixed confidence (FC) problem (Section 4.2),
which we expect to be of more use for researchers. Here, we wish to claim with
a specified confidence level that our selected model is state-of-the-art
against a collection of competing models using the minimal amount of
computation. Finally, we also consider an extension to the FC scenario, where
a practitioner has the computational capacity to fit multiple models in
parallel. We demonstrate the effectiveness of our procedures over current
model selection approaches when identifying an optimal target-dependent
sentiment analysis model from a set of eight competing candidate models
(Section 5).
## 2 Motivating example
We now provide evidence for the need to vary both data splits and random seeds
for reliable model selection. We extend the motivating example used in the
work of Reimers and Gurevych (2017), comparing two LSTM-based Named Entity
Recognition (NER) models by Ma and Hovy (2016) and Lample et al. (2016),
differing only in character representation (via a CNN and a LSTM
respectively). We base model training on Ma and Hovy (2016), however,
following the settings of Yang et al. (2018) we use a batch size of 64, a
weight decay of $10e^{-9}$ and removed momentum. We ran each of the NER models
five times with a different random seed on 150 different train, validation,
and test splits222The original CoNLL data was split with respect to time
rather than random sub-sampling, explaining the discrepancy with previous
scores on this dataset using the same models.. Reimers and Gurevych (2017)
showed the effect of model instability between these two models, where
changing the model’s random seeds can lead to drawing different conclusions
about which model performed best. We extend this argument by showing that
different conclusions can also be drawn if we instead vary the train-test
split used for the model evaluation (Figure 1). We see that while data splits
0 and 2 correctly suggest that the LSTM is optimal, using data split 1
suggests the opposite. Therefore, it is clear that we must vary both the
random seeds and train-test splits used to evaluate our models if we want
reliable model selection.
Figure 1: The left plot shows the distribution of results when varying the
data splits and random seeds, with the dashed lines representing the quartile
values. The three right plots each represent a different single data split
over five runs on different random seeds. The lines represent a single run
result.
## 3 Problem Statement
Extending notation from Arcuri and Briand (2014), we can precisely state the
task of selecting between a collection of $N$ candidate models
$S=\\{m_{1},m_{2},..m_{N}\\}$ as finding
$m^{*}=\operatorname*{argmax}_{m\in S}\mathcal{M}(m).$ (1)
$m^{*}$ is the best model according to some chosen evaluation metric
$\mathcal{M}$ that measures the performance of that model, e.g accuracy,
F-score or AUC (for an summary of model evaluation metrics see Friedman et al.
(2001)).
As already argued, Equation (1) paints an overly simplistic picture of model
selection. In reality we only have access to noisy realisations of the true
model score $\mathcal{M}(m)$ and direct comparisons of single realisations of
random variables are unreliable. Therefore, we follow the arguments of Reimers
and Gurevych (2018) and consider a meaningful way of comparing noisy model
evaluations: namely, finding the model with largest expected performance
estimate across different train-test splits and random seeds. Defining the
mean performance of model $m$ as $\mu_{m}$, we see that the task of model
selection is equivalent to the accurate learning and comparison of these $N$
unknown means:
$m^{*}=\operatorname*{argmax}_{m\in S}\mu_{m}.$
We can now set up the sequential framework of our model selection procedure
and precisely state what we mean by reliable model selection. At each step in
our algorithm we choose a model to evaluate and sample a performance estimate
by randomly generating a data split and random seed. After collecting
evaluations, we can calculate sample means for each model, which we denote as
$\hat{\mu}_{m}$. After running our algorithm for $T$ steps, reliable model
selection corresponds to knowing how confident we should be that our chosen
model $\hat{m}_{T}=\operatorname*{argmax}\hat{\mu}_{m}$ is in fact the true
optimal model $m^{*}$, i.e. we wish to make a precise statement of the form;
$\displaystyle\mathds{P}\left(\hat{m}_{T}=m^{*}\right)\geq 1-\delta,$ (2)
where $1-\delta$ represents this confidence.
In Section 1 we motivated two distinct goals of a sequential model selection
routine, which we can now state as:
1. 1.
Fixed budget model selection (FB): We wish to find the best model using only a
fixed budget of $T$ model evaluations. The aim is to collect the $T$
evaluations that allow us to claim (2) with the largest possible confidence
level $1-\delta$.
2. 2.
Fixed confidence model selection (FC): We wish to find the best model to a
pre-specified confidence level. The aim is to collect the minimal number of
model evaluations that allow us to claim (2).
Although an algorithm designed to do well in one of these scenarios will
likely also do well in the other, we will see that to achieve the best
performance at either FB or FC model selection, we require subtly different
algorithms.
## 4 Algorithms
We now examine model selection from a bandit viewpoint, summarising three
bandit algorithms and relating their use to three distinct model selection
scenarios. Although the underpinning theoretical arguments for these
algorithms are beyond the scope of this work, we do highlight one point that
is relevant for model selection; that scenarios enjoying the largest
efficiency gains from moving to adaptive algorithms are those where only a
subset of arms have performance close to optimal Jamieson et al. (2013). Model
selection in NLP is often in this scenario, with only a small number of
considered models being close to state-of-the-art, and so (as we demonstrate
in Section 5) NLP has a lot to gain from using our adaptive model selection
algorithms.
### 4.1 Fixed Budget by Sequential Halving
Algorithm 1 Sequential Halving for Fixed Budget Model Selection
0: Computational Budget $T$, Set of $N$ candidate models $S$
while $|S|\neq 1$ do
Evaluate each model $m$ in $S$
$\Big{\lfloor}\frac{T}{|S|\lceil\log_{2}N\rceil}\Big{\rfloor}$ times
Update the empirical means $\hat{\mu}_{m}$
Remove $\big{\lfloor}\frac{|S|}{2}\big{\rfloor}$ models with worst
$\hat{\mu}_{m}$ from $S$
end while
return Chosen model $S$
FB best-arm identification algorithms are typically based on successively
eliminating arms until just a single (ideally) optimal arm remains Jamieson et
al. (2013); Jamieson and Nowak (2014); Audibert and Bubeck (2010). We focus on
the sequential halving (SH) algorithm of Karnin et al. (2013) (Algorithm 1).
Here we break our model selection routine into a series of
$\big{\lfloor}log_{2}N\big{\rfloor}$ rounds, each discarding the least
promising half of our candidate model set, eventually resulting in a single
remaining model. Our computational budget $T$ is split equally among the
rounds to be equally budgeted among the models remaining in that round. This
allocation strategy ensures an efficient use of resources, for example the
surviving final two models are evaluated
$2^{\big{\lfloor}log_{2}N\big{\rfloor}}-1$ times as often as the models
eliminated in the first round. An example run of the algorithm is summarised
in Table 1.
Round | Candidate Models | # Evaluations
---|---|---
1 | $S=\\{m_{1},m_{2},m_{3},m_{4}\\}$ | 2
2 | $S=\\{m_{2},m_{4}\\}$ | 4
output: | $S=\\{m_{2}\\}$ |
Table 1: An example of sequential elimination selecting between four models
with a budget of $T=16$. After two evaluations of each model, two models are
eliminated. The remaining budget is then used to reliably decide between the
remaining pair. Standard practice would evaluate each model an equal four
times, wasting computational resources on sub-optimal models.
In the bandit literature Karnin et al. (2013), this algorithm is shown to have
strong theoretical guarantees of reliably choosing the optimal arm, as long as
the reward-distributions for each arm are bounded (limited to some finite
range). This is not a restrictive assumption for NLP, as the majority of
common performance metrics are bounded, for example accuracy, recall,
precision and F-score are all constrained to lie in $\left[0,1\right]$. We
will demonstrate the effectiveness of sequential halving for model selection
in Section 5.
### 4.2 Fixed Confidence by TTTS
For fixed confidence model selection, where we wish to guarantee the selection
of an optimal arm at a given confidence level, we cannot just discard arms
that are likely to be sub-optimal without accurately estimating this
likelihood of sub-optimality. Although approaches that sequentially eliminate
arms (like our sequential halving algorithm) do exist for FC best-arm
identification Jamieson et al. (2014); Karnin et al. (2013); Audibert and
Bubeck (2010); Even-Dar et al. (2002), the best theoretical guarantees for the
FC problem come from algorithms that maintain the ability to sample any arm at
any point in the selection procedure Garivier and Kaufmann (2016); Jamieson
and Nowak (2014). Rather than seeking to eliminate half the considered models
at regular intervals of computation, a model is only evaluated until we can be
sufficiently confident that it is sub-optimal. Unfortunately, the performance
guarantees for these methods are asymptotic results (in the number of arms and
the number of arm pulls) and have little practical relevance to the (at most)
tens of arms in a model selection problem.
Our practical recommendation for FC model selection is a variant of the well-
known Bayesian sampling algorithm, Thompson sampling, known as top-two
Thompson sampling (TTTS) Russo (2016). We will see that this algorithm can
efficiently allocate computational resources to quickly find optimal models.
Furthermore, this approach provides full uncertainty estimation over the final
choice of model, providing the confidence guarantees required for FC model
selection.
Our implementation makes the assumption that the evaluations of each model
roughly follow a Gaussian distribution, with different means and variances.
Although such assumptions are common in the model evaluation literature
Reimers and Gurevych (2018) and for statistical testing in NLP Dror et al.
(2018), they could be problematic for the bounded metrics common in NLP.
Therefore we also experimented with modelling the logit transformation of our
evaluations, mapping our evaluation metric to the whole real line. However,
for our examples of Section 5 we found that this mapping provided a negligible
improvement in reliability and so was not worth including in our experimental
results. This may not be the case for other tasks or less well-behaved
evaluation metrics and so we include this functionality in the FIESTA package.
Algorithm 2 Top-Two Thompson Sampling
0: Desired Confidence $1-\delta$, Set of $N$ candidate models $S$
Initialise a uniform belief $\pi$
Evaluate each model in $S$ three times 333We enforce a minimum of three
evaluations to ensure that the t distribution in our posterior remains well-
defined
Update belief $\pi$
while $\max_{m\in S}\pi_{m}\leq 1-\delta$ do
Sample distinct $m_{1}$ and $m_{2}$ according to $\pi$
Randomly choose between $m_{1}$ and $m_{2}$
Evaluate chosen model
Update belief $\pi$
end while
return Chosen model $\operatorname*{argmax}_{m\in S}\pi_{m}$
To provide efficient model selection, we use our current believed probability
that a given model is optimal $\pi_{m}=\mathds{P}\left(m^{*}=m\right)$
(producing a distribution over the models $\pi=\\{\pi_{1},..,\pi_{N}\\}$) to
drive the allocation of computational resources. Standard Thompson sampling is
a stochastic algorithm that generates a choice of model by sampling from our
current belief $\pi$, i.e. choosing to evaluate a model with the same
probability that we believe is optimal (see Russo et al. (2018) for a concise
introduction). Although this strategy allows us to focus computation on
promising arms, it actually does so too aggressively. Once we believe that an
arm is optimal with reasonably high confidence, computation will be heavily
focused on evaluating this arm even though we need to become more confident
about the sub-optimality of competing models to improve our confidence level.
This criticism motivates our chosen algorithm TTTS (summarised in Algorithm
2), where instead of sampling a single model according to $\pi$, we sample two
distinct models. We then uniformly choose between these two models for the
next evaluation, allowing a greater exploration of the arms and much improved
rates of convergence to the desired confidence level Russo (2016). We use this
new evaluation to update our belief and continue making evaluations until we
believe that a model is optimal with a higher probability than $1-\delta$ and
terminate the algorithm. An example run of TTTS is demonstrated on a synthetic
example in Figure 2, where we simulate from $5$ Gaussian distributions with
means $\\{0.65,0.69,0.69,0.70,0.71\\}$ and standard deviation $0.01$ to mimic
accuracy measurements for a model selection problem.
Figure 2: TTTS seeking the optimal model with confidence $0.99$ from $5$
synthetic models. The background represents our evolving belief $\pi$ in the
optimal model and the lines represent the proportion of the total evaluations
made on each model. We start evaluating the models uniformly but our adaptive
algorithm quickly focuses resources on the best models.
We now explain how we calculate $\pi$ (our belief in the location of the
optimal model) using well-known results from Bayesian decision theory (see
Berger (2013) for a comprehensive coverage). As justified earlier, we assume
that the evaluations of model $m$ are independently distributed with a
Gaussian distribution $\mathcal{N}(\mu_{m},\sigma_{m}^{2})$ for some unknown
mean $\mu_{m}$ and variance $\sigma_{m}^{2}$. Although we are primarily
interested in learning $\mu_{m}$, we must also learn $\sigma_{m}^{2}$ in order
to make confidence guarantees about the optimality of our selected model.
Therefore, as well as keeping track of the sample means for the evaluations of
each model $\hat{\mu}_{m}$, we also keep track of the sample variances
$\hat{S}_{m}$ and counters $T_{m}$ of the number of times each model has been
evaluated. To facilitate inference, we choose a uniform prior for the unknown
$\mu_{m}$ and $\sigma_{m}$. Not only is this a conjugate prior for Gaussian
likelihoods, but it is also shown to encourage beneficial exploratory
behaviour when using Thompson sampling on Gaussian bandit problems Honda and
Takemura (2014) and so allows fast identification of optimal arms (or models).
After observing $T_{m}$ evaluations of each model and producing estimates
$\hat{\mu}_{m}$ and $\hat{S}_{m}$, our posterior belief for each deviation
between the true and observed model means $\mu_{m}-\hat{\mu}_{m}$ satisfies
(as derived in Honda and Takemura (2014));
$\sqrt{\frac{T_{m}(T_{m}-2)}{\hat{S}_{m}}}\left(\mu_{m}-\hat{\mu}_{m}\right)|\,\hat{\mu}_{m},\hat{S}_{m}\sim
t_{T_{m}-2},$
where $t_{d}$ is a Student’s t-distribution with $d$ degrees of freedom.
$\pi$ is then defined as the probability vector, such that $\pi_{m}$ is the
relative probability that $\mu_{m}$ is the largest according to this posterior
belief. Unfortunately, there is no closed form expression for the maximum of
$N$ t-distributions and so FIESTA uses a simple Monte-Carlo approximation
based on the sample maxima of repeated draws from our posteriors for
$\mu_{m}$. In practice this is very accurate and did not slow down our
experiments, especially in comparison to the time saved by reducing the number
of model evaluations.
### 4.3 Batch Fixed Confidence by BTS
NLP practitioners often have the computational capacity to fit models in
parallel across multiple workers, evaluating multiple models or the same model
across multiple seeds at once. Their model selection routines must therefore
provide batches of models to evaluate. Our proposed solution to FB model
selection naturally provides such batches, with each successive round of SH
producing a collection of model evaluations that can be calculated in
parallel. Unfortunately, TTTS for FC model selection successively chooses and
then waits for the evaluation of single models and so is not naturally suited
to parallelism.
Extending TTTS to batch decision making is an open problem in the MAB
literature. Therefore, we instead consider batch Thompson sampling (BTS), an
extension of standard Thompson sampling (as described in Section 4.2) to batch
sampling from the related field of Bayesian optimisation Kandasamy et al.
(2018). At each step in our selection process we take $B$ model draws
according to our current belief $\pi$ that the model is optimal, where $B$
represents our computational capacity. This is in contrast to the single draw
in standard Thompson sampling and the drawn pair in TTTS. In addition, this
approach extends to the asynchronous setting, where rather than waiting for
the whole batch of $B$ models to be evaluated before choosing the next batch,
each worker can draw a new model to evaluate according to the updated $\pi$.
This flexibility provides an additional efficiency gain for problems where the
different models have a wide range of run times.
## 5 Experiments
We now test our three algorithms on a challenging model selection task typical
of NLP, selecting between eight Target Dependent Sentiment Analysis (TDSA)
models based on their macro F1 score. We consider two variants of four re-
implementations of well-known TDSA models: ATAE Wang et al. (2016), IAN Ma et
al. (2017), TDLSTM Tang et al. (2016) (without target words in the left and
right LSTM), and a non-target-aware LSTM method used as the baseline in Tang
et al. (2016).
These methods represent state-of-the-art within TDSA, with only small
differences in performance between TDLSTM, IAN, and ATAE (see figure 3). All
the models are re-implemented in PyTorch Paszke et al. (2017) using AllenNLP
Gardner et al. (2018). To ensure the only difference between the models is
their network architecture the models use the same optimiser settings and the
same regularisation. All words are lower cased and we use the same Glove
common crawl 840B token 300 dimension word embedding Pennington et al. (2014).
We use variational Gal and Ghahramani (2016) and regular Hinton et al. (2012)
dropout for regularisation and an ADAM Kingma and Ba (2014) optimiser with
standard settings, a batch size of $32$ and use at most $100$ epochs (with
early stopping on a validation set). Many of these settings are not the same
as originally implemented, however, having the same training setup is required
for fair comparison (this explains the differences between our results and the
original implementations). To increase the difficulty of our model selection
problem, we additionally create four extra models by reducing the dimensions
of the Glove vectors to 50 and removing dropout. Although these models are
clearly not state-of-the-art, they increase the size of our candidate model
set and so provide a more complicated model selection problem (an intuition
discussed in Appendix A).
All of the TDSA experiments are conducted on the well-studied SemEval 2014
task 4 Restaurant dataset Pontiki et al. (2014) and we force train-val-test
splits to follow the same ratios as this dataset’s official train-test split.
Each individual model evaluation is then made on a randomly generated train-
test split and random seed to access both sources of evaluation variability.
Figure 3: F1 scores for our candidate TDSA models. After $500$ evaluations of
each model on different data splits and model seeds we see that the TDLSTM is
the state-of-the-art model.
### 5.1 Fixed Budget Model Selection
We use the TDSA model selection problem to test fixed budget model selection.
To thoroughly test our algorithm, we consider an additional four models based
on 200 dimensional Glove vectors, bringing the total number of models to 12.
We compare our approach of sequential halving to the standard non-adaptive
approach of splitting the available computational budget equally between the
12 candidate models. For example, we would allocate a budget of $24$ model
evaluations as evaluating each model two times and selecting the model with
the highest sample mean.
Figure 4 compares the proportion of $10,000$ runs of sequential halving that
correctly identify the optimal model with the proportion identified by the
non-adaptive approach with the same computational budget. Sequential halving
identifies the optimal model more reliably ($\approx 15\%$ more often) than
the current approach to FB model selection in NLP. Using sequential halving
with $204$ evaluations almost always ($99\%$ of runs) selects the optimal
model, whereas the non-adaptive approach is only correct $85\%$ of the time.
Figure 4: Proportion of the runs correctly selecting the optimal TDSA model using sequential halving against the standard non-adaptive approach. Sequential halving consistently identifies the optimal model at a significantly higher rate across a wide range of budgets. | # | evaluations | with | Non-Adaptive | # | evaluations | with | TTTS
---|---|---|---|---|---|---|---|---
$\delta$ | min | mean | max | | $\%$ correctly
---
selected
min | mean | max | | $\%$ correctly
---
selected
0.05 | 48 | 281 | 1552 | 100 | 27 | 130 | 518 | 100
0.1 | 40 | 206 | 1192 | 99 | 24 | 96 | 460 | 99
0.2 | 32 | 128 | 608 | 96 | 24 | 65 | 274 | 97
Table 2: Number of evaluations required to select a TDSA model at a range of
confidence levels across $500$ runs of TTTS and a standard non-adaptive
approach.
### 5.2 Fixed Confidence Model Selection
We perform fixed confidence model selection on the eight TDSA candidate models
(the full models and those based on 50 dimensional vectors). We compare TTTS
to a non-adaptive approach where all models are evaluated at each step,
irrespective of the results of earlier evaluations (the standard approach for
model selection in NLP). We run this non-adaptive approach until we reach the
required confidence level calculated using the same Bayesian framework as in
TTTS.
We run each approach $500$ times and note the number evaluations required to
get to a range of confidence levels (Table 2) alongside the proportion that
correctly identify the optimal model. TTTS requires substantially less model
evaluations (in terms of the minimum, mean and max across our runs) to reach a
given confidence level than the non-adaptive approach, achieving the same
reliability at half the cost (on average). TTTS is able to quickly identify
sub-optimal models and so can avoid wasting resources repeatedly evaluating
the whole candidate set.
| # | evaluations | with | BTS-4 | # | evaluations | with | BTS-8
---|---|---|---|---|---|---|---|---
$\delta$ | min | mean | max | | $\%$ correctly
---
selected
min | mean | max | | $\%$ correctly
---
selected
0.05 | 28 | 282 | 1392 | 100 | 88 | 315 | 1128 | 100
0.1 | 24 | 144 | 520 | 100 | 56 | 178 | 784 | 100
0.2 | 24 | 76 | 280 | 98 | 32 | 106 | 352 | 99
Table 3: Number of evaluations of required to select a TDSA model at a range
of confidence levels across $500$ runs of BTS selecting batches of 4 and 8
models.
Finally, we test our proposed approach to batch FC model selection by running
exactly the same experiment but using BTS to choose collections of four and
eight models at a time (Table 3). As expected, performance degrades as we
increase batch size, with batches of four allowing more fine grained control
over model evaluations than using batches of eight. In particular, due to the
exploitative nature of Thompson sampling, we see that selecting models to a
very high confidence (95%) requires more computation with BTS than the
standard non-adaptive approach. However, BTS does reach the other confidence
levels faster and correctly identifies the optimal model more often. However,
as TTTS performs significantly better across all confidence levels, we
emphasise the need for a less-exploitative version of BTS with adjustments
similar to those used in TTTS.
## 6 Conclusions
The aim of this paper has been to propose three algorithms for model selection
in NLP, providing efficient and reliable selection for two distinct realistic
scenarios: fixed confidence and fixed budget model selection. Crucially, our
research further calls into question the current practice in NLP evaluation as
used in the literature and international competitions such as SemEval. Our
algorithms adaptively allocate resources to evaluate promising models, basing
evaluations across multiple random seeds and train-test splits. We demonstrate
that this allows significant computational savings and improves reliability
over current model selection approaches.
Although we have demonstrated that our algorithms perform well on a complex
model selection problem typical of NLP, there is still work to be done to
create algorithms more suited to these problems. Future research directions
include making selection routines more robust to evaluation outliers, relaxing
our Gaussian assumptions and developing more effective batch strategies.
## 7 Acknowledgements
The authors are grateful to reviewers, whose comments and advice have greatly
improved this paper. The research was supported by an EPSRC Doctoral Training
Grant and the STOR-i Centre for Doctoral Training. We thank Dr Chris Jewell at
the Centre for Health Informatics, Computing, and Statistics, Lancaster
University for the loan of a NVIDIA GP100-equipped workstation for this study.
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## 8 Appendix
## Appendix A Characterising the Difficulty of a Model Selection Problem
We briefly summarise a result from the best-arm identification literature,
providing intuition for our experiment section through a mechanism to
characterise the difficulty of a model selection problem. Intuitively, model
selection difficulty increases with the size of the set of candidate models
$N$ and as the performance of sub-optimal models approaches that of the
optimal model (and becomes harder to distinguish), i.e. as
$\mu_{m^{*}}-\mu_{m}$ gets small for some sub-optimal arm $m$. In fact, it is
well known in the MAB literature that it is exactly these two properties that
characterise the complexity of a best-arm-identification problem, confirming
our intuition for model selection. Mannor and Tsitsiklis (2004) show that the
number of arm pulls required for the identification of a best arm at a
confidence level $1-\delta$ has at least a computational complexity of
$O(H\log(1/\delta))$, where
$H=\sum_{m^{\prime}\in
S\setminus\\{m*\\}}\frac{1}{\left(\mu_{m^{*}}-\mu_{m}\right)^{2}}.$
|
# Cosmological Dark Matter from a Bulk Black Hole
Sylvain Fichet<EMAIL_ADDRESS>ICTP South American Institute for
Fundamental Research & IFT-UNESP, R. Dr. Bento Teobaldo Ferraz 271, São Paulo,
Brazil Centro de Ciencias Naturais e Humanas (CCNH), Universidade Federal do
ABC, Santo Andre, 09210-580 SP, Brazil Eugenio Megías<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de
Física Teórica y Computacional,
Universidad de Granada, Avenida de Fuente Nueva s/n, E-18071 Granada, Spain.
Mariano Quirós<EMAIL_ADDRESS>Institut de Física d’Altes Energies (IFAE) and
The Barcelona Institute of Science and Technology (BIST), Campus UAB, 08193
Bellaterra (Barcelona) Spain
###### Abstract
We study the cosmology of a three-brane in a specific five-dimensional scalar-
gravity (i.e. soft-wall) background, known as the linear dilaton background.
We discover that the Friedmann equation of the brane-world automatically
contains a term mimicking pressureless matter. We propose to identify this
term as dark matter. This dark matter arises as a projection of the bulk black
hole on the brane, which contributes to the brane Friedmann equation via both
the Weyl tensor and the scalar stress tensor. The nontrivial matter-like
behavior is due to an exact cancellation between the Weyl and scalar
pressures. We show that the Newtonian potential only receives a mild short-
distance correction going as inverse distance squared, ensuring compatibility
of the linear dilaton brane-world with observed 4D gravity. Our setup can be
viewed as a consistent cosmological description of the holographic theories
arising in the linear dilaton background. We also present more general scalar-
gravity models where the brane cosmology features an effective energy density
whose behavior smoothly interpolates between dark radiation, dark matter and
dark energy depending on a model parameter.
## I Introduction
Five-dimensional (5D) gravity coupled to a scalar field has proven to be a
fecund playground, leading to a host of theoretical results and models of the
real world (see e.g. Karch _et al._ (2006); Gursoy and Kiritsis (2008);
Gursoy _et al._ (2008); Gubser and Nellore (2008); Falkowski and Perez-
Victoria (2008); Batell and Gherghetta (2008); Batell _et al._ (2008); Cabrer
_et al._ (2010); von Gersdorff (2010); Cabrer _et al._ (2011); Megías and
Quirós (2019, 2021a, 2021b)). Our focus in this letter is a specific 5D
scalar-gravity background (i.e. a soft-wall background) which is sometimes
referred to as the “linear dilaton background”. This model is known to have
peculiar thermodynamic Gursoy and Kiritsis (2008) and field theoretical
properties Cabrer _et al._ (2010); Megías and Quirós (2019, 2021a, 2021b).
For example, all quantum fields living on the linear dilaton background have a
spectral distribution that features a gapped continuum. This feature has been
recently used in extensions of the Standard Model Csáki _et al._ (2022a, b).
In this work we put the linear dilaton background at finite temperature and
posit a flat 3-brane moving over the background, in the spirit of brane-world
models (see e.g. Brax and van de Bruck (2003)). We discover a surprising
property: from the viewpoint of a brane observer, the local Friedmann equation
automatically contains an effective energy term that may be identified as dark
matter. This dark matter emerges as a nontrivial effect from the bulk physics
projected on the brane. It originates from a combination of the 5D Weyl tensor
and of the bulk scalar vev, as we will demonstrate further below.
To bring this result into context, we remind that there is a notorious analog
in pure Anti-de Sitter (AdS) background, that has been gradually uncovered and
studied in Shiromizu _et al._ (2000); Binetruy _et al._ (2000); Hebecker and
March-Russell (2001); Langlois _et al._ (2002); Langlois and Sorbo (2003). In
pure AdS the net effect of the bulk physics projected on the brane gives rise
to radiation, which is identified as cosmological dark radiation in the
context of a brane-world. This remarkable fact is in direct connection with
the fact that the bulk black hole in AdS-Schwarzschild background corresponds
to the thermal state in the holographic CFT, perhaps one of the most
fascinating entries of the AdS/CFT correspondence Aharony _et al._ (2000). In
our case, by performing the analogous calculation with the linear dilaton
background, we discover that the bulk black hole gives rise to dark matter.
Decades of astronomical observations point to the existence of dark matter.
Determining its nature is a pressing question in fundamental physics. While a
common hypothesis is that dark matter may be a new particle (that remains so
far elusive), our study leads to a fundamentally different viewpoint. Our
setup provides, in a sense, an origin to cosmological dark matter via a
modification of gravity. See e.g. Clifton _et al._ (2012) for a few other
attempts to explain dark matter via modified gravity.
In this paper we present the derivation of our central result, the effective
Friedmann equation of the linear dilaton brane-world, that is shown to contain
dark matter. We present nontrivial consistency checks of this result. We also
compute the deviation to the Newtonian potential. We then outline more general
models featuring a variety of equations of state depending on a model
parameter, and discuss some conceptual points and prospects. Extra
developments and technical details are laid out in Fichet _et al._ (2022),
which can be considered as a companion to this letter.
## II The 5D scalar-gravity system
We consider the general scalar-gravity action in the presence of a brane,
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{g}\left(\frac{M_{5}^{3}}{2}{}^{(5)}R-\frac{1}{2}(\partial_{M}\phi)^{2}-V(\phi)\right)$
(1) $\displaystyle-$
$\displaystyle\int_{\textrm{brane}}d^{4}x\sqrt{\bar{g}}\,(V_{b}(\phi)+\Lambda_{b})+\ldots\,$
${}^{(5)}R$ is the 5D Ricci scalar, $\phi$ is the scalar field, $M_{5}$ is the
fundamental 5D Planck scale, $\bar{g}_{\mu\nu}$ is the induced metric on the
brane, $g\equiv|\det g_{MN}|$ and $\bar{g}\equiv|\det\bar{g}_{\mu\nu}|$ are
the metrics determinants, $\Lambda_{b}$ is the brane tension, $V$ and $V_{b}$
are the bulk and brane-localized potentials for $\phi$. We assume that the
brane potential sets the scalar field vacuum expectation value (vev) to a
nonzero value $\langle\phi\rangle=v_{b}$, with $V_{b}(v_{b})=0$. The bulk
potential is explicitly given further below. The ellipses encode the Gibbons-
Hawking-York term York (1972); Gibbons and Hawking (1977) and the action of
quantum fields living on the 5D background.
The 5D metric is written in a frame suitable for brane cosmology as
$\displaystyle ds^{2}$
$\displaystyle=g_{MN}dx^{M}dx^{N}\equiv-n^{2}(r)d\tau^{2}+\frac{r^{2}}{\ell^{2}}d\mathbf{x}^{2}+b^{2}(r)dr^{2}\,.$
(2)
We allow the existence of a black hole horizon encoded in the $n(r)$ and
$b(r)$ factors, the position of the horizon being given by
$n(r_{h})=0=1/b(r_{h})$. Latin indices $(M,N,\cdots)$ refer to 5D coordinates,
Greek indices $(\mu,\nu,\cdots)$ refer to 4D coordinates.
The 3-brane is localized at the position $r=r_{b}$. Our frame (2) is
appropriate to describe cosmology as seen from the brane standpoint. The
induced metric on the brane is
$\displaystyle ds^{2}$ $\displaystyle=\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}\equiv-
dt^{2}+\frac{r_{b}^{2}}{\ell^{2}}d\mathbf{x}^{2}\,,$ (3)
where we have introduced the brane cosmic time $dt=n(r_{b})d\tau$. According
to this metric, if the brane moves along $r$ in the 5D background, the
observer perceives expansion of the four-dimensional (4D) universe with Hubble
parameter $H=\dot{r_{b}}/r_{b}$, where $\dot{r_{b}}\equiv\partial_{t}r_{b}$.
We choose that $r_{b}$ equals $\ell$ at present times, such that
$r_{b}=a(t)\ell$ where $a(t)$ is the standard scale factor. An overview of the
brane-world is shown in Fig. 1.
Figure 1: Overview of the scalar-gravity system. The brane and the horizon are
located respectively at $r=r_{b}$ and $r=r_{h}$. The scalar vacuum expectation
value is fixed by a brane potential to a value $v_{b}$ which remains constant
when $r_{b}$ varies. The scalar VEV (plain) evolves in the bulk, the
blackening factor of the metric (dotted) diverges at the horizon.
The 5D equations of motion of the system are
$\displaystyle{}^{(5)}G_{MN}=\frac{1}{M_{5}^{3}}T^{\phi}_{MN}\,,\quad\frac{1}{\sqrt{g}}\partial_{M}\left(\sqrt{g}g^{MN}\partial_{N}\phi\right)=\frac{\partial
V}{\partial\phi}\,,$ (4)
with ${}^{(5)}G_{MN}={}^{(5)}R_{MN}-\frac{1}{2}g_{MN}{}^{(5)}R$ and
$T^{\phi}_{MN}=\partial_{M}\phi\partial_{N}\phi-
g_{MN}\left[\frac{1}{2}(\partial_{A}\phi)^{2}+V(\phi)\right]$. More
explicitly, the equations of motion for the 5D background in the cosmological
frame are Megías _et al._ (2018)
$\displaystyle\frac{n^{\prime\prime}(r)}{n(r)}-\left(\frac{n^{\prime}(r)}{n(r)}-\frac{1}{r}\right)\left(\frac{b^{\prime}(r)}{b(r)}-\frac{2}{r}\right)=0\,,$
(5)
$\displaystyle\frac{n^{\prime}(r)}{n(r)}+\frac{b^{\prime}(r)}{b(r)}-r\bar{\phi}^{\prime}(r)^{2}=0\,,$
$\displaystyle\frac{n^{\prime}(r)}{n(r)}+\frac{1}{r}+r\,b^{2}(r)\bar{V}(\bar{\phi})-\frac{r}{2}\bar{\phi}^{\prime}(r)^{2}=0\,,$
$\displaystyle\bar{\phi}^{\prime\prime}(r)+\left(\frac{n^{\prime}(r)}{n(r)}-\frac{b^{\prime}(r)}{b(r)}+\frac{3}{r}\right)\bar{\phi}^{\prime}(r)-b^{2}(r)\frac{\partial\bar{V}}{\partial\bar{\phi}}=0\,,$
with the dimensionless field $\bar{\phi}\equiv\phi/(3M_{5}^{3})^{1/2}$ and
$\bar{V}\equiv V/(3M_{5}^{3})$. Importantly, even though one of these
differential equations seems redundant, it cannot be ignored because it still
implies a nontrivial algebraic relation between the integration constants.
Also notice that the integration constants can depend on $r_{b}$ through the
boundary conditions; the brane location thus influences the 5D background.
We turn to gravity from the brane viewpoint. The effective 4D Einstein
equation seen by a brane observer is computed from the 5D Einstein equation by
projecting on the brane via the Gauss equation together with the Israel
junction condition Shiromizu _et al._ (2000). Introducing the unit vector
$n_{M}$ normal to the brane that satisfies $n_{M}n^{M}=1$ and
$\bar{g}_{MN}=g_{MN}-n_{M}n_{N}$, the 4D Einstein equation on the brane is
$\displaystyle{}^{(4)}G_{\mu\nu}=\frac{1}{M^{2}_{\rm
Pl}}\left(T_{\mu\nu}^{b}+T^{\rm
eff}_{\mu\nu}\right)+O\left(\frac{T_{b}^{2}}{M^{6}_{5}}\right)$ (6)
with
${}^{(4)}G_{\mu\nu}={}^{(4)}R_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}{}^{(4)}R$,
and $T^{b}_{\mu\nu}$ the stress tensor of brane-localized matter. The
“holographic” effective stress tensor $T^{\rm
eff}_{\mu\nu}=\tau^{W}_{\mu\nu}+\tau^{\phi}_{\mu\nu}+\tau^{\Lambda}_{\mu\nu}$
contains:
i) The projection of the 5D Weyl tensor ${}^{(5)}C^{M}{}_{NPQ}$ on the brane
$\displaystyle\frac{1}{M^{2}_{\rm
Pl}}\tau^{W}_{\mu\nu}=-{}^{(5)}C^{M}{}_{NPQ}n_{M}n^{P}\bar{g}_{\mu}{}^{N}\bar{g}_{\nu}{}^{Q}\,,$
(7)
leading to corresponding values of the energy density $\rho^{W}$ and pressure
$P^{W}$ given by
$\displaystyle\rho^{W}$
$\displaystyle=\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}-\frac{1}{r_{b}}\right)\,,$
$\displaystyle P^{W}$
$\displaystyle=\frac{1}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}-\frac{1}{r_{b}}\right)\,,$
(8)
where we have made use of Eqs. (5).
ii) The projection of the bulk stress tensor
$\displaystyle\frac{1}{M^{2}_{\rm Pl}}\tau^{\phi}_{\mu\nu}$
$\displaystyle=\frac{2}{3M_{5}^{3}}\left[T^{\phi}_{MN}\bar{g}_{\mu}{}^{M}\bar{g}_{\nu}{}^{N}+\big{(}T^{\phi}_{MN}n^{M}n^{N}-\frac{1}{4}T^{\phi,M}_{M}\big{)}\bar{g}_{\mu\nu}\right]$
$\displaystyle=\frac{3}{2}\left(\frac{\bar{\phi}^{\prime}(r_{b})^{2}}{2b^{2}(r_{b})}-\bar{V}\right)\bar{g}_{\mu\nu}\,,$
(9)
leading to the values of $\rho^{\phi}$ and $P^{\phi}$, after using the EoM
(5),
$\displaystyle\rho^{\phi}$
$\displaystyle=-\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)\,,$
$\displaystyle P^{\phi}$
$\displaystyle=\frac{3}{2}\frac{M^{2}_{Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)\,.$
(10)
iii) The contribution from the brane tension
$\displaystyle\frac{1}{M^{2}_{\rm
Pl}}\tau^{\Lambda}_{\mu\nu}=-\frac{\Lambda_{b}^{2}}{12M_{5}^{6}}\bar{g}_{\mu\nu}\,,$
(11)
which yields the values of $\rho^{\Lambda}$ and $P^{\Lambda}$ as
$\rho^{\Lambda}=-P^{\Lambda}=\frac{M^{2}_{\rm
Pl}\Lambda_{b}^{2}}{12M_{5}^{6}}\,.$ (12)
The brane tension is ultimately tuned to set the effective 4D cosmological
constant to zero.
We work in the low-energy regime
$|T_{\mu\nu}^{b}|\ll\frac{M_{5}^{6}}{M_{\rm Pl}^{2}}$ (13)
which justifies neglecting the higher order terms in (6). This restriction
implies further simplifications below.
## III Dark matter from the linear dilaton black hole
The linear dilaton background is defined by the bulk (super)potential
$\bar{W}(\bar{\phi})=\frac{2}{\ell}e^{\bar{\phi}},\quad\bar{V}(\bar{\phi})=-\frac{3}{2\ell^{2}}e^{2\bar{\phi}}\,.$
(14)
Solving the equations of motion (5) with the potential (14) we find for the 5D
background
$\displaystyle n(r)$ $\displaystyle=$
$\displaystyle\frac{r}{\ell}\sqrt{1-\frac{r_{h}^{3}}{r^{3}}}\,,$ (15)
$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{\ell}{r_{b}}\frac{e^{-\bar{v}_{b}}}{\sqrt{1-\frac{r_{h}^{3}}{r^{3}}}}\,,$
(16) $\displaystyle\bar{\phi}(r)$ $\displaystyle=$
$\displaystyle\bar{v}_{b}-\log\left(\frac{r}{r_{b}}\right)\,,$ (17)
where $r_{h}$ (an integration constant) is the location of the black hole
horizon in the brane cosmology frame. The domain of the variable $r$ is the
interval $[0,\ell]$, where $r=0$ is the metric singularity and $r=\ell$ is the
value of the brane location today, while $0\leq r_{h}\leq r_{b}\leq\ell$.
Importantly, we can notice that a power of $3$ appears in the Schwarzschild
factors, in contrast with pure AdS5 where, instead, there would be a power of
$4$.
We then evaluate the brane effective Einstein equation by plugging the bulk
solutions into Eq. (6), and deduce the Friedmann equation. The mass scale that
naturally appears in the physical quantities is
$\eta=\frac{M_{5}^{3}}{M^{2}_{\rm Pl}}=\frac{e^{\bar{v}_{b}}}{\ell}\,.$ (18)
Using the low-energy assumption (13) which here becomes
$\rho_{b}\ll\eta^{2}M^{2}_{\rm Pl}$ (19)
we obtain the first Friedmann equation on the brane,
$3M^{2}_{\rm Pl}\left(\frac{\dot{r}_{b}}{r_{b}}\right)^{2}=\rho_{b}+\rho_{\rm
eff}+O\left(\frac{\rho_{b}^{2}}{\eta^{2}M^{2}_{\rm Pl}}\right)\,$ (20)
with
$\rho_{\rm eff}=3\eta^{2}M_{\rm Pl}^{2}\frac{r^{3}_{h}}{r^{3}_{b}}\,.$ (21)
The $\rho_{\rm eff}$ energy density term is the critical result. It is a
nontrivial effect from the bulk physics: a combination of the Weyl tensor and
of the scalar stress tensor contributions. This holographically-induced
$\rho_{\rm eff}$ scales as $r_{b}^{-3}$, therefore it behaves as a
nonrelativistic matter term in the 4D Friedmann equation. (The analogous
calculation in AdS would instead give a $r^{-4}_{b}$ scaling, i.e. radiation).
In the brane-world paradigm, we identify the Standard Model fields as brane-
localized modes that give rise to the brane energy density $\rho_{b}$. The
effective energy density $\rho_{\rm eff}$ in Eq. (20) is then naturally
identified as the dark matter energy density. In other words, the linear
dilaton brane-world automatically features dark matter.
From the expression of $\rho_{\rm eff}$ in Eq. (21), the fraction of dark
matter energy in the Universe $\Omega_{\rm DM}=\rho_{\rm DM}/\rho_{\rm crit}$
(with $\rho_{\rm crit}=3H^{2}M^{2}_{\rm Pl}$) induced by the linear dilaton
background is then
$\Omega_{\rm
DM}=\left(\frac{\eta}{H}\right)^{2}\left(\frac{r_{h}}{r_{b}}\right)^{3}\,.$
(22)
At present times we have $r_{b}=\ell$, $\Omega_{{\rm DM},0}=0.26$ and
$H_{0}=1.47\times 10^{-42}\,\textrm{GeV}$. This provides a constraint between
the model parameters given by
$r_{h}\simeq 0.64\ell\left(\frac{H_{0}}{\eta}\right)^{2/3}\,.$ (23)
As in the Standard Cosmology, this dark matter dominates the universe for
temperatures $T\lesssim 0.7\,\textrm{eV}$ and is subdominant with respect to
radiation for higher temperatures.
The origin of the $r^{-3}_{b}$ scaling is better understood as follows. The
effective energy density and pressure, which appear in the Friedmann and
continuity equations, are defined as
$\rho_{\rm eff}=\rho^{W}+\rho^{\phi}+\rho^{\Lambda},\quad P_{\rm
eff}=P^{W}+P^{\phi}+P^{\Lambda}\,,$ (24)
where $\rho^{W}$ and $P^{W}$ are given by Eq. (8), $\rho^{\phi}$ and
$P^{\phi}$ by Eq. (10), and $\rho^{\Lambda}$ and $P^{\Lambda}$, after imposing
the condition for cancellation of the cosmological constant,
$\Lambda_{b}=6\eta^{2}M^{2}_{\rm Pl}$, are given by
$\rho^{\Lambda}=-P^{\Lambda}=3\eta^{2}M^{2}_{\rm Pl}\,.$ (25)
A straightforward application of the BH solutions (15) and (16) yields
$\rho^{W}+\rho^{\phi}=-3\eta^{2}M^{2}_{\rm
Pl}\left(1-\frac{r_{h}^{3}}{r_{b}^{3}}\right)\,,$ (26)
which combined with $\rho^{\Lambda}$ from (25), yields the result which
appears in Eq. (21).
On the other hand, for the effective pressure $P_{\rm eff}$ using again Eqs.
(15) and (16) we get
$P^{W}+P^{\phi}=\frac{2M^{2}_{\rm
Pl}}{b^{2}(r_{b})r_{b}}\left(\frac{n^{\prime}(r_{b})}{n(r_{b})}+\frac{1}{r_{b}}\right)=3\eta^{2}M^{2}_{\rm
Pl}\,,$ (27)
which combined with Eq. (25) yields $P_{\rm eff}=0$, leading to the equation
of state $w_{\rm eff}=P_{\rm eff}/\rho_{\rm eff}=0$
This explains the $r^{-3}_{b}$ scaling and ensures that the 4D conservation
equation i.e. that the 4D Bianchi identity $D^{\mu}{}^{(4)}G_{\mu\nu}=0$ is
satisfied. The cancellation we report here is nontrivial, as it is unclear if
there exists a symmetry that enforces it.
Another nontrivial consistency check is at the level of the 5D conservation
equation projected on the brane, which takes the general form Tanaka and
Himemoto (2003); Langlois and Sorbo (2003)
$\dot{\rho}_{\rm eff}+4H\rho_{\rm eff}+HT_{\,\,\mu}^{{\rm
eff}\,\mu}=-2T_{MN}u^{M}n^{N}\,.$ (28)
Notice the $4H$ factor arising due to 5D spacetime. On the rhs, $n^{N}$ is the
unit vector normal to the brane and outward-pointing, and $u^{M}$ is the brane
velocity vector satisfying $u_{M}u^{M}=-1$ Tanaka and Himemoto (2003);
Langlois and Sorbo (2003). In the low-energy regime we have
$u^{M}\approx\left(\frac{1}{n},{\bm{0}},Hr_{b}\right)\,,\,\quad
n^{M}\approx\left(Hr_{b}\frac{b}{n},{\bm{0}},\frac{1}{b}\right)\,$ (29)
up to $O\left(H^{2}/\eta^{2}\right)$. Using the explicit expression of
$T_{MN}$ obtained from our scalar-gravity solutions, Eqs. (15)–(17), it turns
out that $T_{MN}u^{M}n^{N}=0$ in the low-energy regime. The calculation
involves again beautiful cancellations, and it is detailed in Fichet _et al._
(2022). One can then easily verify that the 5D conservation equation is
satisfied by the effective energy density (21), ensuring that the framework is
fully consistent.
The low-energy regime Eq. (19) implies $H\ll\eta$ since the total energy
density is $\rho\sim H^{2}M^{2}_{\rm Pl}$; it is the only assumption made
throughout the calculations. We worked at first order in $H/\eta$. The
cancellations observed in $T_{\mu\nu}^{\rm eff}$ and in the 5D conservation
equation occur up to small $O(H^{2}/\eta^{2})$ factors.
## IV The Newtonian potential
The Newtonian potential for the LD model at present times can be deduced from
the graviton brane-to-brane propagator $G_{\bm{2}}$ using the optical theorem
Fichet _et al._ (2022). We find the discontinuity of this propagator to be
$\text{Disc}_{s}[G_{\bm{2}}(\sqrt{s})]=2\pi\delta(s)+\frac{\sqrt{\frac{s}{\sigma^{2}}-1}}{s}\theta\Big{(}s\geq\sigma^{2}\Big{)}\,,$
(30)
where $\sigma=3\eta/2$ is the mass gap. The $\delta$ term corresponds to the
4D graviton. The second term, which encodes the rest of the 5D graviton
fluctuations, forms a gapped continuum characteristic of the linear dilaton
background Megías and Quirós (2021a). From this discontinuity we deduce that
the Newtonian potential of the linear dilaton brane-world is
$V_{N}(R)=-\frac{m_{1}m_{2}}{M^{2}_{\rm Pl}\,R}\bigg{(}1+\Delta(R)\bigg{)}\,,$
(31)
with
$\Delta(R)\approx\begin{cases}\frac{4}{3\pi\sigma R}~{}{\rm
if}~{}R\ll\frac{1}{\sigma}\\\ O\Big{(}e^{-\sigma R}\Big{)}~{}{\rm
if}~{}R\gg\frac{1}{\sigma}\end{cases}\,.$ (32)
We see that the deviation from the Newtonian potential appears essentially
below the distance scale $1/\sigma$ corresponding to the inverse mass gap. The
deviation to the potential goes as $\propto 1/R^{2}$, unlike the AdS case,
where it goes as $1/R^{3}$. Micron-scale fifth force experiments such as
Smullin _et al._ (2005) mildly constrain the $\sigma$ scale as $\sigma\gtrsim
10$ meV. This constraint, along with Eq. (23), translates into an upper bound
on the location of the bulk black hole horizon, $r_{h}\lesssim 2.3\times
10^{-21}\ell$.
## V Extensions and uniqueness of the linear dilaton brane-world
In the previous sections we have seen that the bulk black hole from the LD
braneworld model characterized by the exponential potential Eq. (14) leads to
a pressureless matter term on the brane. We may wonder whether such a behavior
of $\rho_{\rm eff}$ is specific to the LD model or if it appears in other 5D
scalar-gravity solutions. In the next subsections we provide hints of
uniqueness by extending the model in two different directions. We consider a
model with an exponential superpotential (like that of the LD model) but with
a different exponent, and a model where a constant is added to the exponential
superpotential. In both cases the scalar-gravity solutions will depend on a
parameter which reproduces the LD model for particular values, but generalizes
it. These more general scalar-gravity solutions are interesting per se. We
leave an extended investigation for future work. Our focus here is mostly on
illustrating the uniqueness of the behavior of $\rho_{\rm eff}$ in the LD
model.
### V.1 A generalized exponential potential
In this section we generalize the (super)potential of the LD model given by
Eq. (14) to
$\bar{W}(\bar{\phi})=\frac{2}{\ell}e^{\nu\bar{\phi}},\quad\bar{V}(\bar{\phi})=-\frac{4-\nu^{2}}{2\ell^{2}}e^{2\nu\bar{\phi}}\,,$
(33)
where the LD model is reproduced for the value $\nu=1$, while the AdS model is
reproduced for the value $\nu=0$.
The solution to the 5D equations of motion (5) is given by
$\displaystyle n(r)$
$\displaystyle=\frac{r}{\ell}\sqrt{1-\left(\frac{r_{h}}{r}\right)^{4-\nu^{2}}}\,,$
(34) $\displaystyle b(r)$
$\displaystyle=\left(\frac{r}{\ell}\right)^{\nu^{2}-1}\left(\frac{\ell}{r_{b}}\right)^{\nu^{2}}\frac{e^{-\nu\bar{v}_{b}}}{\sqrt{1-\left(\frac{r_{h}}{r}\right)^{4-\nu^{2}}}}\,,$
(35) $\displaystyle\bar{\phi}(r)$
$\displaystyle=\bar{v}_{b}-\nu\log\left(\frac{r}{r_{b}}\right)\,,$ (36)
and the relation between the 5D and 4D Planck scales is given by
$M_{5}^{3}=\frac{1}{2}\bar{W}_{b}M_{\rm Pl}^{2}=\eta M_{\rm
Pl}^{2},\quad\eta\equiv\frac{1}{\ell}e^{\nu\bar{v}_{b}}\,.$ (37)
After using the relation for vanishing of the cosmological constant
$\Lambda_{b}=3M_{5}^{3}\bar{W}_{b}=6\eta M_{5}^{3}$, one readily gets the
brane vacuum energy and pressure as
$\rho^{\Lambda}=-P^{\Lambda}=3\eta^{2}M_{\rm Pl}^{2}\,.$ (38)
Using the solution (34)-(35) one easily gets
$\rho_{\rm eff}=3\eta^{2}M_{\rm
Pl}^{2}\frac{r_{h}^{4-\nu^{2}}}{r_{b}^{4-\nu^{2}}},\quad P_{\rm
eff}=\eta^{2}M_{\rm
Pl}^{2}(1-\nu^{2})\frac{r_{h}^{4-\nu^{2}}}{r_{b}^{4-\nu^{2}}}\,,$ (39)
which yields an equation of state
$w_{\rm eff}=\frac{1-\nu^{2}}{3}\,.$ (40)
We can see that the dark matter behavior ($w_{\rm eff}=0$) appears only for
$\nu=1$.
Interestingly, the “holographic” effective energy density in this model
interpolates from dark radiation behavior ($w_{\rm eff}=1/3$) for $\nu=0$ to
dark energy behavior ($w_{\rm eff}=-1$) for $\nu=2$. For $0\leq\nu\leq 2$ the
solution satisfies the continuity equation automatically, cf. Eq. (28).
Finally, let us point out that the singularity at $r=0$ is a good one for
$\nu\leq 2$ Cabrer _et al._ (2010). 111For $\nu=2$ the solution to the 5D
equations of motion (5) is $n(r)=r/\ell$,
$b(r)=c_{b}(\ell/r_{b})(r/r_{b})^{3}$ and
$\bar{\phi}(r)=\bar{v}_{b}-2\log\left(r/r_{b}\right)$, where $c_{b}$ is an
arbitrary constant. This corresponds to a solution with no black hole for
which $\rho_{\rm eff}=-P_{\rm eff}=3\eta^{2}M_{\rm
Pl}^{2}\left(1-1/(c_{b}\eta\ell)^{2}\right)+\Lambda_{4}M_{\rm Pl}^{2}$, where
we have not assumed cancellation of the cosmological constant. If one fixes
$c_{b}\eta\ell=1$, then $\rho_{\rm eff}=-P_{\rm eff}=\Lambda_{4}M_{\rm
Pl}^{2}$ consistently with the 4D Einstein equations. Detailed investigation
is left for a future work.
### V.2 Asymptotically AdS linear dilaton model
We can also define a slightly different model interpolating between AdS and
the linear dilaton background. The model is defined by the bulk potential
$\bar{V}(\bar{\phi})=\frac{1}{8}\bar{W}^{\prime}(\bar{\phi})^{2}-\frac{1}{2}\bar{W}(\bar{\phi})^{2}\,,$
(41)
where $\bar{W}(\bar{\phi})=2(1+e^{\bar{\phi}})/\ell$ Megías and Quirós (2021a,
b). In the brane cosmology frame, the behavior of the effective energy term
depends on the parameter
$c=\exp(-\bar{v}_{b}+e^{-\bar{v}_{b}})\equiv(\eta\ell)^{-1}$. We find that
$\rho_{\rm eff}$ behaves as in AdS in the limit $c\to\infty$, and as in the
linear dilaton background in the limit $c\to 0$, with
$\rho_{\rm eff}\simeq\begin{cases}3\eta^{2}M_{\rm
Pl}^{2}\frac{r_{h}^{3}}{r^{3}_{b}}\,\;\quad{\rm if}~{}~{}c\ll 1\\\
\frac{3}{\ell^{2}}M_{\rm Pl}^{2}\,\frac{r_{h}^{4}}{r_{b}^{4}}\quad\quad{\rm
if}~{}~{}c\gg 1\end{cases}\,.$ (42)
We can recognize the dark radiation behavior for $c\gg 1$ and the dark matter
behavior, Eq. (21), for $c\ll 1$.
We confirm all these results via numerical solving of the 5D conservation
equation (28). More details are given in Ref. Fichet _et al._ (2022), where
we also discuss the transition region. We find that for arbitrary values of
$c$, the equation of state smoothly interpolates between matter and radiation
behavior, $\rho_{\rm eff}\propto a^{-3[1+w_{\rm eff}(c)]}$ with $P_{\rm
eff}/\rho_{\rm eff}=w_{\rm eff}(c)$. The numerical value of the equation-of-
state parameter $w_{\rm eff}(c)$ is exhibited in Fig. 2 where a continuous
transition appears between $w_{\rm eff}=0$, for dark matter, and $w_{\rm
eff}=1/3$ for dark radiation.
Figure 2: Plot of the equation-of-state parameter $w_{\textrm{eff}}\equiv
P_{\rm eff}/\rho_{\rm eff}$ as a function of $c$, within the asymptotically
AdS linear dilaton model.
In summary, we find that the asymptotic AdS/linear dilaton background, created
by the potential in Eq. (41), gives rise to a cosmological brane-world in
which the behavior of the “holographic” effective energy density can range
from dark radiation to dark matter, as controlled by the $c$ parameter.
## VI Discussion
We now discuss a few conceptual points and relations to the literature.
#### Birth of the bulk black hole.
In a typical cosmological scenario, analogously to the AdS brane-world, the
bulk horizon is created by energy leaked from the brane into the continuum of
bulk gravitons and other bulk fields. See Hebecker and March-Russell (2001);
Langlois _et al._ (2002); Langlois and Sorbo (2003) for a consistent analysis
in AdS, and Fichet (2022) for the rate in arbitrary background. The radiation
feeds the bulk black hole, which typically grows with time. This feeding
mechanism is efficient at early times while at late times, the radiation is
negligible hence the horizon does not evolve anymore. This corresponds to the
low-energy regime in our analysis. The process of dumping energy into the
bulk, known since Gubser (2001), is either similar or truely equivalent (via
AdS/CFT) to the process of heating up a CFT sector (see e.g. Gubser (2001);
Hebecker and March-Russell (2001); von Harling and McDonald (2012, 2012); Brax
_et al._ (2019); Hong _et al._ (2020)).
#### What is the dark matter made of?
The dark matter arising in our linear dilaton brane-world is purely made of
the curvature of spacetime. However this curvature is the result of populating
the bulk with gravitons. Deep in the bulk these gravitons are strongly
interacting, and their net effect is the presence of the bulk horizon, which
is seen by the brane observer. Since the continuum of gravitons is involved,
our result shares, in a sense, some similarity with the proposal of “continuum
dark matter” made in Csáki _et al._ (2022a, b); Csaki _et al._ (2022). It is
plausible that our analysis provides the consistent framework needed to
understand cosmology in such models.
## VII Prospects
Overall, the results reported in this letter hint at an alternative view of
dark matter which certainly deserves further investigation. We thus end with a
discussion of future directions.
#### Cosmological perturbations.
The key calculation presented in this letter shows that the linear dilaton
background could explain dark matter in the homogeneous universe. Computing
perturbations and structure formation is a task beyond the present work,
however the roadmap is clear: the study of cosmological perturbations in our
model belongs to the realm of the fluid/gravity correspondence Bhattacharyya
_et al._ (2008); Hubeny _et al._ (2012). The dark matter of our brane-world
model amounts to a (non-conformal) “holographic fluid”, whose properties such
as viscosities need to be carefully computed and compared to observations.
#### Dark matter at galactic scales.
Our brane-world model may explain dark matter at cosmological scale, however
nothing is said about galactic scales. To understand how the dark matter
emerging in our model behaves at galactic scales we would have to compute less
symmetric solutions of the 5D scalar-gravity system, as needed to describe
e.g. halos. One should thus investigate $SO(3)$-symmetric solutions, possibly
assisted by matter sources on the brane. This is left for future
investigation.
#### Dark matter decay.
In analogy with AdS, the bulk black hole may in principle be able to decay via
Hawking radiation into the brane, see Rocha (2008, 2009) for an analysis in
AdS. Since the bulk black hole is the origin of dark matter, Hawking decay
amounts in our model to “dark matter decay”. It would be very interesting to
study this mechanism and its observational consequences, as well as its
implications for holography. We leave it as an open question to investigate.
#### Continuum signatures.
In our model the graviton is accompanied by a gapped continuum that can be
experimentally tested, as exemplified by the correction to the Newtonian
potential Eq. (32). Standard Model fields can be included in the model by
introducing 5D bulk fields and identifying the brane-localized modes as the
Standard Model fields. Analogously to the graviton, each Standard Model field
is accompanied with a gapped continuum which has generally mild coupling to
the brane. Such a setup looks typically like a dark sector Brax _et al._
(2019). The phenomenology of continuum sectors is an active topic of
investigation, see e.g. Katz _et al._ (2016); Csáki _et al._ (2019); Lee
(2018); Gao _et al._ (2020); Fichet (2020); Costantino _et al._ (2020);
Chaffey _et al._ (2021); Csáki _et al._ (2022a, b); Csaki _et al._ (2022).
The present study reinforces the motivation for such models and, in a sense,
starts to explore their cosmology.
###### Acknowledgements.
We thank Philippe Brax, Csaba Csaki and Philip Tanedo for useful discussions.
The work of SF has been supported by the São Paulo Research Foundation
(FAPESP) under grants #2011/11973, #2014/21477-2 and #2018/11721-4 and by
CAPES under grant #88887.194785. EM would like to thank the ICTP South
American Institute for Fundamental Research (SAIFR), São Paulo, Brazil, for
hospitality and partial finantial support of FAPESP Grant 2016/01343-7 from
Aug-Sep 2022 where part of this work was done. The work of EM is supported by
the project PID2020-114767GB-I00 funded by MCIN/AEI/10.13039/501100011033, by
the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020
Operational Programme under Grant A-FQM-178-UGR18, and by Junta de Andalucía
under Grant FQM-225. The research of EM is also supported by the Ramón y Cajal
Program of the Spanish MICIN under Grant RYC-2016-20678. The work of MQ is
partly supported by Spanish MICIN under Grant PID2020-115845GB-I00, and by the
Catalan Government under Grant 2021SGR00649. IFAE is partially funded by the
CERCA program of the Generalitat de Catalunya.
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|
2022 Prospects for low-frequency radio astronomy in S. A.
# Science research from the Instituto Argentino de Radioastronomía
P. Benaglia11affiliation: Instituto Argentino de Radioastronomía (CONICET –
CIC – UNLP), C.C.5, (1984) Villa Elisa, Buenos Aires, Argentina (paula@iar-
conicet.gov.ar).
###### Abstract
In this talk, I will present some figures and milestones of the written
production of the Instituto Argentino de Radioastronomía (IAR), as well as a
personal review of the scientific achievements carried out in recent years by
the researchers working at the IAR. I will also briefly describe the
scientific objectives of the IAR’s flagship project, the Multipurpose
Interferometric Array (MIA), in the context of the instrumental projects that
have lately been or are being installed on Argentine soil.
En esta charla, presentaré algunos números e hitos de producción escrita desde
el Instituto Argentino de Radioastronomía (IAR), y una revisión personal de
logros científicos llevados a cabo en los últimos años por investigadores que
trabajan en el IAR. También, y en el marco de los proyectos instrumentales
recientemente instalados – o en vías de serlo – en suelo argentino, describiré
sucintamente los objetivos científicos del proyecto insignia del IAR, el
Multipurpose Interferometric Array o MIA (por sus siglas en inglés).
Publications, bibliography history and philosophy of astronomy miscellaneous
## 0.1 Introduction
The Instituto Argentino de Radioastronomía (IAR) was founded in October 1962,
originally under the name Instituto Nacional de Radioastronomia; for details
see Romero (2023), this volume. Less than three years later, in April 1965,
its 30-m single dish radiometer detected for the first time the neutral
hydrogen (HI) line at 1.4 GHz. Observations of this transition were the
strongest driver for building an observatory at our latitudes, since HI was
discovered as a tracer of Galactic structure tracer, and an important part of
the southern sky was inaccessible from most of the radio telescope sites
operating at the time (in the northern hemisphere).
The first paper published with IAR affiliation was Varsavsky (1966). And the
first publication using data obtained from observations with the IAR’s first
radio telescope was that of Mészáros (1968).
## 0.2 Some numbers and milestones
In the 60 yr of its existence, about 1500 works have been published by authors
affiliated to the IAR. The number includes articles in periodic journals,
proceedings of professional meetings, theses at all levels, technical reports,
books (Fig. 1).
Figure 1: Some publications including books, dissertations, proceedings, cover
page articles in peer-reviewed journals, etc.
The production of theses includes doctoral theses and ‘licenciatura’ thesis
(equivalent to a Bachelor’s and Master’s degree in Argentine National
Universities), among other formats. The works correspond mainly to
presentations given at the National University of La Plata and the University
of Buenos Aires. In terms of awards, the doctoral theses developed at the IAR
received three times the prize for the best theses of the biennium, awarded by
the Asociación Argentina de Astronomía (www.astronomiaargentina.org), and one
of the equivalent (Giambiagi prize) of the Argentine Physics Association. In
2020, a doctoral thesis conducted both at the IAR and the Karlsruhe Institute
of Technology became the first one in the institution to be completed as a
double doctorate (University of Karlsruhe and National University of San
Martín), and the volume was selected by Springer as one of the world’s best
theses of the year and published in book form in Springer’s “Great Theses”
series.
Several articles first authored by IAR researchers have appeared on the front
pages of major journals, others have received honors from the Gravity Research
Foundation, and once even the Top Scientific Contribution Award, from the
American Institute of Physics as one of the most cited papers of the year in
The Astrophysical Journal.
The IAR’s first 30-m dish radiometer - christened Carlos M. Varsavsky, as its
first Director - was the second instrument of its kind to operate
systematically in the southern hemisphere, along with the Parkes radio
telescope in Australia. This led to many discoveries in the southern sky.
Together with the second radio telescope to be built, now called Esteban
Bajaja and completed around 1980, both continuum and line observations could
be carried out, mainly at 1.4 GHz (atomic hydrogen), and 1.6 GHz (CO molecular
transition).
Until the nineties, a great deal of time was devoted to mapping the
distribution and velocity of the HI. Colomb et al. (1980) (CPH) presented
images and profiles covering positions with declination ${\delta\leq
30^{\arcdeg}}$, each $1^{\arcdeg}$. The CPH survey complemented that of Heiles
& Habing (1974), which was carried out using the Hat Creek radio telescope,
with 28-m dish. The sensitivity of the data was between 0.1 and 1 K.
After a change of receivers, among other things, the southern sky was again
surveyed at the 1.4 GHz center frequency, with higher sensitivity: the system
temperature of $\sim$25 K allowed an rms noise below 0.01 K. Bajaja et al.
(2005), and Kalberla et al. (2005) present the HI distribution and kinematic
information, complementing the work of Hartmann & Burton (1997). The continuum
emission with full polarization information has also been surveyed, but with
the latest radiometer, achieving a sensitivity of 15 mK in the Stokes
parameters Q and U (Testori et al., 2008), for the coordinates not covered by
Reich & Reich (1986).
Results on certain, in some respects special, sources were achieved. For
example, with the radio telescopes of the IAR the passage of the tail of the
comet Halley was observed (Bajaja et al., 1987) 4 h a day for 3 months in 1986
to measure OH lines in absorption. The data allowed to derive the abundance of
the molecule and the OH production rate. HI observations led to the discovery
of the supernova remnant Vela Jr (Combi et al., 1999). The extreme variability
of Active Galactic Nuclei of blazar type was reported for the first time,
after being measured with one of the IAR dishes (Romero et al., 1994). In
2010, the first stellar bow shock with evidence of non-thermal emission was
discovered using the Very Large Array, with a follow-up study of the
polarization of the emission (Benaglia et al., 2021). The Long Baseline
Australian Array was used to map the fifth - at that time - colliding wind
region of a massive binary stellar system (HD 93120AaAb) and to estimate
important stellar/wind parameters (Benaglia et al., 2015). Fernández López et
al. (2014) published results of long-term mosaic observations of the Serpens
South molecular cloud with the Combined Array for Research in Millimeter-wave
Astronomy, revealing new features in the gas dynamics.
## 0.3 Main lines of research
The research done at the IAR, let’s say in the last decade or so, covers
several fields, in astrophysics, physics, computer science, mathematics, etc.
There is a lot of activity in the area of extragalactic sources, such as AGN
and their (super)massive black holes, extended to gravitational waves, cosmic
rays, sources studied by means of electromagnetic cascades, neutrino
astrophysics, field theory and relativity.
Compact objects such as neutron stars or black holes, X-ray binary studies,
have a common ground with those of HE (gamma-ray) sources or other candidates
to produce HE radiation (i.e. colliding wind binaries, stellar bow shocks).
Stellar objects are widely studied, including their parent molecular clouds,
star formation scenarios both of massive and low-mass types, interaction with
the interstellar medium (ISM) and the ISM itself, the supernova stage and
remnants, and planetary science.
The studies include the construction of mathematical models, add numerical
simulations, and signal processing algorithms.
In the following, examples of research at the IAR are presented as divided in
three groups: those carried out with the IAR radio telescopes, those where
theoretical developments dominate, and those where observations with
instruments around the world play a key role.
## 0.4 Research with IAR radiometers
After about fifteen years out of use, the radio telescopes at the IAR have
been refurbished (Gancio et al., 2020), especially challenged by a major
project to study transients, meaning by that sources that experiment changes
in radiation with duration of the order of a second of time or much less. The
observations resumed in 2018, and the data collection as a regular operation
started in 2019. The mentioned project is organized in the framework of a
scientific and technological cooperation between IAR and the Rochester
Institute of Technology (USA). The group involved at IAR is called Pulsar
Monitoring in Argentina (PuMA) (see Lousto, 2023, this volume).
The updated radio telescopes have digital back-ends (CASPER ROACH cards with
4x400-MHz of bandwidth). These are capable of recording two polarizations at
1420 MHz. They can be remotely controlled, and have access to an atomic clock,
GPS and GNSS for timing purposes. Sources with declinations less than
$-7{\arcdeg}$ can be tracked for 4 h per day.
The new architecture is optimal for studies involving pulsar timing arrays,
targeted searches for continuous gravitational wave sources, monitoring of
magnetars and glitching pulsars, and short time scale interstellar
scintillation. The timing precision is better than 1$\mu$s (e.g. Zubieta et
al., 2023, this volume).
Another advantage is the geographical location of the IAR: sources that are
invisible from Australia at 12 h daily interval and from South Africa at 5 h
daily interval can be accessed with the IAR radio telescopes. In this way,
alerts of detections between these three places and their instruments allow to
follow a transient phenomenon along its entire occurrence.
## 0.5 Theory, simulations, models
With the creation of the Group of Relativistic Astrophysics and Radio
Astronomy (GARRA, by its Spanish acronym), the scientific work at IAR was
given a strong capacity for the development of theoretical studies. Articles
such as Romero et al. (2007), chosen as the cover of the journal Astronomy and
Astrophysics, are good first examples of such research. The authors focused on
a challenging gamma-ray binary (then usually referred to as a microquasar), LS
I $+$61 303, a Be-star with a compact companion, to elucidate the nature of
the compact object, between pulsar-neutron star or black hole. The light
curve, the spectrum of the observed TeV gamma-ray emission, and the required
energy budget were analyzed. In particular, they modeled the interaction
between the components of the system for both hypotheses (Fig. 2). For this,
they obtained time on the powerful supercomputer HITACHI SR11000 at Hokkaido
University. The results were consistent with the second case.
Detailed models of the spectral energy distribution of massive binary systems
(colliding-wind binaries) and the interaction with the interstellar medium
were also developed; see del Valle & Romero (2012), del Palacio et al. (2018),
del Palacio et al. (2022).
Figure 2: Wind collision interface geometry for the pulsar case in the orbital
plane (top) and in a perpendicular plane (bottom), from Romero et al. (2007).
In Vieyro et al. (2019) the authors studied the case of a core-collapse
supernova inside the relativistic jet of an active galactic nucleus. After the
analyzing the dynamical evolution of the supernova ejecta impacted by the jet
and the observed gamma-ray light curve, they computed the spectral energy
distribution for two different galaxy hosts, a radio galaxy and a blazar. They
concluded that the first scenario appears to be much more common and easier to
detect than the other.
Collaborations on planet formation are presented in San Sebastián et al.
(2019). The article deals with modelling the fragmentation of planetesimal in
the formation of giant planets, taking into account the relative velocities
and compositions of the planetesimals and the accretion produced by their
fragmentation.
An example of progress in cosmology is presented in Pérez & Romero (2022) and
its references, related to a universe with contractions and bounces. The
survival of certain structures along them, in this case black holes, is
studied using a generalized McVittie’s metric.
García et al. (2022) studied physical and geometrical properties of the corona
of the microquasar GRS 1915$+$105\. They applied a variable comptonization
model vKompth, – developed by IAR’s Ph.D. student C. Bellavita (Bellavita et
al., 2022) –, supported by archival X-ray observations, and found consistent
trends in the evolution of the corona size, temperature, and feedback (see
also Méndez et al., 2022).
As can be seen, Ph.D. theses are of particular interest at the IAR. For
example, L. Combi studied binary black holes - in principle of equal mass,
spinning, and approaching merger. Three-dimensional general relativistic
magnetohydrodynamical simulations were performed on a system geometry that
included a circumbinary disk and mini-disks around each black hole. These
allowed to study the gas dynamics and system evolution, the morphology and
variability of the electromagnetic flux densities, and to analyze the
accretion (see Fig. 3). The results on realistic synthetic light curves and
spectra are very valuable for future observations with instruments like the
Laser Interferometer Space Antenna (LISA) (Combi et al., 2022).
Figure 3: Surface density snapshot of a binary black hole system for two
epochs (Combi et al., 2022).
In the same subject and with a similar treatment, E. M. Gutiérrez Ph.D.
project consisted more in the study of the radiation from supermassive binary
black holes (Gutiérrez et al., 2022). Simulations including blackbody
radiation from an optically thick accretion disk, and hard X-rays from
optically thin corona allowed to obtain spectra, images and light curves (Fig.
4).
The Ph.D. Thesis by F. Fogantini was focused on the phenomenology of accreting
high-mass X-ray binaries (HMXBs). In this Thesis, it was demonstrated how
geometrical or line-of-sight effects have a strong impact on the observed
spectral and variability properties of archetypical HMXBs sources like SS 433
(Fogantini et al., 2023).
Figure 4: Stokes I instantaneous radiation distribution of a binary
supermassive black hole approaching periastron (Gutiérrez et al., 2022).
## 0.6 Observing with interferometric arrays
The regular acquisition of observing time with instruments in foreign soil
started around 2000, and is growing steadily mainly pursued by members of the
fringe research group (Formation in Radio Interferometry - arGEntina).
In recent years, the field of high-mass star formation has been shaken by the
discovery of explosive outflows, in addition to the well-known bipolar
outflows; (see Zapata et al., 2009, and references therein). Only a few such
regions have been described. One is G5.98–0.39, an ultra-compact HII region
with formation of massive stars. Zapata et al. (2020) and Fernández López et
al. (2021) spotted it with the Atacama Large Millimeter/submillimeter Array
(ALMA), and found dozens of CO filaments, and expanding warmer SiO gas at the
origin. The energy released was inferred to be $\sim 10^{46}-10^{49}$ erg.
There is a north-south filamentary structure, a compact HII region, and a
possibly expanding dusty belt, which harbours an O5V star. Polarized emission
could be measured in the filaments, $\sim$4.4%, coming from magnetically
aligned dust grains. As Fig. 5 shows, the magnetic field lines in the central
belt of dust are radially aligned.
Figure 5: Electrical vector position angle rotated 90$\arcdeg$, superpimposed
on Stokes I continuum emission towards G5.89–0.39 (full details in Fernández
López et al., 2021).
In Guzmán Ccolque et al. (2022), part of first author’s Ph.D. thesis is
presented, on the object IRAS 16076–5134, a high-mass star-forming region
studied with ALMA band-7 CO archive data. Fourteen cores were detected. The
imaged morphology and kinematics suggest a dispersal, explosive outflow, with
filament-like CO ejections from a central position (see Fig. 6), quasi
isotropic; several filaments show a linear velocity gradient.
Figure 6: Red-shifted and blue-shifted condensations in IRAS 16076–513 (Guzmán
Ccolque et al., 2022).
Another example of recent Ph.D. project was the one focused on galaxy groups
and certain types of galaxies (dwarf-, low surface brightness-, super thin-,
local galaxies). The groups are a very common environment, easy to study
because of the low relative velocities involved. The sources are studied by
means of HI-line data. Questions such as which is the role of the environment
in galaxy evolution, or how galaxy mergers, ram-pressure stripping,
gravitational interactions and intragroup medium affect star formation and
morphology are investigated. One of the contributed papers (Saponara et al.,
2021), aimed at the superthin galaxy Fourcade-Figueroa, consisted in modeling
the HI distribution, deriving the rotation curve, to finally obtain, also
through modeling, characteristics of the dark matter halo (see Fig. 7).
Figure 7: HI rotation curve model of the Fourcade-Figueroa galaxy. The orange
line shows the rotation curve due to the stellar disk, the blue line shows the
contribution due to the gas disk, the green line shows that due to the dark
matter halo and the red colour shows the best-fitting model rotation curve.
See details in Saponara et al. (2021).
Since radio interferometric observations provide a very high angular
resolution, the products they deliver can be complemented by others at shorter
wavelengths. For example, processes that take place in the interstellar medium
are studied by combining radio and infrared data. This is the case of the
bright HII region RCW 49 and its very rich ionizing cluster Westerlund 2.
Figure 8 shows the distribution of gas, dust, and stars along the field, at
arcsec angular scales. The gas is probed by data from the Australia Telescope
Compact Array, represented by a 40-pointing mosaic observation; the dust and
stars by Spitzer images (Benaglia et al., 2013). There are HE sources in the
field, imaged by means of H.E.S.S. and Fermi LAT data. The work discusses
possible radio counterparts.
Figure 8: RCW 49 field as seen in the radio continuum (9 GHz, in red) and
Spitzer-GLIMPSE band 1 (3.6 $\mu$m) in blue and band 4 (8 $\mu$) in green
(Benaglia et al., 2013).
Nearly 200 h of observing time were devoted to the survey of the Cygnus OB2
association and its surroundings with the Giant Metrewave Radio Telescope (see
Benaglia et al., 2020, and related papers). The observations were made at two
MHz bands. About 4000 sources were detected and catalogued (1000 in the two
bands), some of them for the first time. The database allowed further studies
of the massive early-type stars, protoplanetary disks, young stellar objects,
double-lobed objects, and counterparts to unidentified high-energy sources.
Observations with instruments that record emission outside the radio window
are also carried out for projects leaded by IAR researchers, for instance,
using XMM-Newton, Chandra, NuSTAR. An example is described in the work by
Saavedra et al. (2022) on the binary source OAO 1657–415 – an accreting X-ray
pulsar with a high-mass companion –. The authors identified pulsations in
NuSTAR data and explained their origin and characteristics, estimated the
value of the dipolar magnetic field at the pulsar surface and a obtained a
lower limit on the distance of the source.
## 0.7 Scientific research with MIA
The Multipurpose Interferometric Array, in its full configuration, is expected
to be formed at least by 32 elements/antennas with 5-m diameter dishes; an
expansion to 64 antennas is also planned (see full description in Gancio et
al., 2023, this volume). The largest 55 km baseline will provide an angular
resolution close to 1 arcsec in L-band. The final coverage is expected to be
between 100 MHz and 2.3 GHz.
With the above parameters, MIA observations can contribute to advancing
studies on four major topics: transient sources and timing measurements,
sources of non-thermal radiation, neutral hydrogen, from rest to redshifted
velocities, and astrophysical plasmas.
The high-precision timing settings will allow the detection of transient
counterparts of gamma-ray bursts, the study of fast radio bursts, pulsars and
gravitational waves, and the observation of flares from magnetars.
The short frequencies at which the MIA receivers will operate are ideal to
probe sources where non-thermal radiation is important. This, combined with
MIA’s high temporal resolution capabilities, will make the instrument optimal
for studying the counterparts of unidentified gamma-ray sources, performing
multifrequency studies of AGN variability, spectro-temporal studies of X-ray
and gamma-ray binaries, studies of the morphology and spectral distribution of
supernova remnants, the mapping of continuum non-thermal extended sources, and
a long list of other objects.
MIA’s high bandwidth, from GHz to a few MHz is well suited to observe the HI
line of atoms at rest, but also at large distances, i.e. high redshift
sources, with a resolution down to 1 arcsec. This will allow studies of nearby
and/or extended galaxies, of the interstellar medium with high angular
resolution, and, under certain conditions, of HI at cosmological distances to
characterize the Epoch of Reionization.
Emission from astrophysical plasmas is relatively strong even at short
centimeter wavelengths. MIA will be very useful for the physical and
kinematical characterization of HII regions, for the study of the OH maser
variability in star-forming regions and evolved massive stars, and star
forming regions in general.
## References
* Bajaja et al. (1987) Bajaja, E., Morras,R., Poeppel, W. G. L. et al, 1987, ApJ 322, 549
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# Negligible effect of brain MRI data preprocessing for tumor segmentation.
Ekaterina Kondrateva Polina Druzhinina Alexandra Dalechina Svetlana
Zolotova Andrey Golanov Boris Shirokikh Mikhail Belyaev Anvar Kurmukov
Artificial Intelligence Research Institute (AIRI), Moscow, Russia Skolkovo
Institute of Science and Technology, Moscow, Russia National Medical Research
Center for Neurosurgery, Moscow Gamma Knife Center, Moscow, Russia
###### Abstract
Magnetic resonance imaging (MRI) data is heterogeneous due to differences in
device manufacturers, scanning protocols, and inter-subject variability. A
conventional way to mitigate MR image heterogeneity is to apply preprocessing
transformations such as anatomy alignment, voxel resampling, signal intensity
equalization, image denoising, and localization of regions of interest.
Although a preprocessing pipeline standardizes image appearance, its influence
on the quality of image segmentation and on other downstream tasks in deep
neural networks has never been rigorously studied.
Experiments on three publicly available datasets evaluate the effect of
different preprocessing steps in intra- and inter-dataset training scenarios.
Results demonstrate that most popular standardization steps add no value to
network performance; moreover, preprocessing can hamper performance. Our
results suggest that image intensity normalization approaches do not
contribute to model accuracy because of the reduction of signal variance with
image standardization. Additionally, the contribution of skull-stripping in
data preprocessing is almost negligible if measured in terms of estimated
tumor volume.
The only essential transformation for accurate deep learning analysis is the
unification of voxel spacing across the dataset. In contrast, inter-subjects
anatomy alignment in the form of non-rigid atlas registration is not necessary
and intensity equalization steps (denoising, bias-field correction and
histogram matching) do not improve performance.
The study code is accessible online111https://github.com/MedImAIR/brain-mri-
processing-pipeline.
###### keywords:
Brain MRI , segmentation , preprocessing , nnU-net , UNETR , SAM
## 1 Introduction
In recent years, modern deep neural networks (DNN) have steadily improved the
quality of automatic segmentation pipelines in medical imaging. Specifically,
for the task of brain tumor segmentation, the performance of DNNs has achieved
human-level efficiency Chang et al. [2019]. This advancement can be explained
by the improvement of DNN architectures and the growth of the training
datasets. For example, the size of the BraTS competition Bakas et al. [2018]
training dataset increased from $100$ subjects in 2013 to $2000$ subjects in
2021. Simultaneously, top-performing algorithms have progressed from random
forest and gradient boosting trees on radiomics features: first to fully
convolutional networks, then to u-shaped UNet and UNet-like networks, and
finally to Vision transformers.
In contrast, the preprocessing steps used to prepare data for analysis seem to
have undergone considerably fewer changes. For instance, the set of
preprocessing steps for brain MRI images has remained relatively stable and
has been reproduced across the majority of papers on the topic from the early
2010s until now. Here we challenge the conventional pipelines for MRI image
processing and question their necessity for accurate prediction with regard to
new advanced deep learning machinery.
The traditional brain MRI preparation steps could be divided into four
distinct categories:
* 1.
The first category is subject-wise image alignment, typically in the form of
rigid registration of one MRI sequence to another, (e.g. T2-FLAIR onto T1 with
contrast enhancement). This step is mandatory if one uses multiple MR
modalities to predict a single segmentation map to ensure correct alignment
between ground truth annotation and corresponding image’s voxels.
* 2.
The second category is voxels resampling to some standard. The most common
methods are voxel resampling to homogeneous spacing (often $1\times 1\times
1\text{ mm}^{3}$) and non-rigid registration to some anatomical atlas.
* 3.
The third category includes steps that affect voxels’ intensity distribution,
such as bias-field correction (such as N4 correction Tustison et al. [2010]),
intensity normalization (typically in a form of image-wise z-scoring), image
denoising methods (e.g. SUSAN Smith and Brady [1997]), and histogram
equalization Nyúl et al. [2000].
* 4.
Finally, the last step that is preserved in almost all the papers is skull
stripping as a method to localize regions of interest (the brain tissue),
implement feature selection to ease localization, and reduce the amount of
False Positives Chang et al. [2019].
While the motivation behind applying these steps is clearly to standardize
image appearance and remove different sources of domain shift Kondrateva et
al. [2021], these steps are computationally costly and their utility for deep-
learning segmentation lacks investigation. Specifically, it is widely known
that increasing variability of the data by data augmentation (image resizing,
non linear intensity transformations, applying noise, etc.) leads to improved
DNN performance Wightman et al. [2021]. However, data preprocessing works
quite in the opposite way by reducing data variance.
In this study we analyze the most popular preprocessing steps for brain MRIs
and measure their influence on deep-learning based tumor segmentation tasks.
We analyze different preprocessing strategies and recommend the minimal
pipeline required for accurate segmentation with the benefits of lower
computational costs and improved reproducibility.
## 2 Related works
Image preprocessing is a de-facto standard first step in almost all deep
learning pipelines for medical image analysis Nixon and Aguado [2019]. In this
domain, data preparation is convenient due to two major causes:
* 1.
diversity in scanning protocols and therefore diverse spatial resolutions and
image intensity profiles Kurmukov et al. [2021],
* 2.
large image sizes and small sample sizes, thus leading to high-dimension
learning compounding negative effects on model generalisability Berisha et al.
[2021].
For example, a typical multi-institutional brain MRI dataset consists of
images with varying resolutions and acquisition parameters (depending on
scanning protocol). Therefore, a majority of studies utilize data
preprocessing pipelines. We select several recent publications on brain MRI
segmentation to identify the most common preprocessing steps (see Table 1).
Table 1: Common preprocessing steps for multi-modal brain MRI image analysis.
Checkmarks represent the step mentioned in the study and x-marks are placed if
the step is missing or unclear.
Preprocessing step | Resample to image size | Resample to spacing | Atlas registration | Bias-field correction | Denoising | Histogram matching | Skull stripping
---|---|---|---|---|---|---|---
Győrfi et al. [2021] | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✓
Ranjbarzadeh et al. [2021] | ✓ | ✓ | ✓ | ✗ | ✗ | ✗ | ✓
Pei et al. [2020] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Menze et al. [2021] | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓
Ermiş et al. [2020] | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓
Eijgelaar et al. [2020] | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
Rathore et al. [2017] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Wang et al. [2019] | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
Bakas et al. [2018] | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
Kickingereder et al. [2019] | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓
Bakas et al. [2022] | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
The overall brain MRI data preprocessing pipeline can be divided into four
groups of methods.
### 2.1 Intra-subject alignment (rigid registration)
During this step different MR sequences from a single patient are reoriented
in a similar way and rigidly registered. This step is applied in all studies
that analyze multi-sequence MRI Győrfi et al. [2021], Ranjbarzadeh et al.
[2021], Pei et al. [2020], Menze et al. [2021], Ermiş et al. [2020], Eijgelaar
et al. [2020].
### 2.2 Inter-subject alignment
This step standardizes the size of images across the dataset. Most of the
observed papers use voxel resampling to an isotropic voxel (e.g. $1\times
1\times 1\text{ mm}^{3}$), or to the same image resolution (in voxels, e.g.
$256\times 256\times 256$), or both, by means of non-rigid atlas registration
Győrfi et al. [2021], Ranjbarzadeh et al. [2021], Pei et al. [2020], Menze et
al. [2021] and Ermiş et al. [2020], Eijgelaar et al. [2020], Rathore et al.
[2017], Bakas et al. [2018]. The only exceptions are two studies that analyze
the data acquired with a unified scanning protocol (isotropic voxel)
Kickingereder et al. [2019], Wang et al. [2019].
### 2.3 Non-linear intensity correction and enhancement
Multi-institutional studies typically use some intensity or noise correction
approaches, with the most popular being histogram matching Győrfi et al.
[2021], Pei et al. [2020], Rathore et al. [2017], bias-field correction Győrfi
et al. [2021], Pei et al. [2020], Eijgelaar et al. [2020], Rathore et al.
[2017], and denoising Pei et al. [2020], Rathore et al. [2017].
Histogram equalization (harmonization or matching) methods standardize images
by aligning their intensity distributions Nyúl et al. [2000]. Denoising
algorithms filter the image whilst preserving the underlying structure Smith
and Brady [1997]. Finally, bias-field correction methods mitigate the effects
of magnetic field variation Tustison et al. [2010], see Figure 2.
Most of the observed studies apply z-scoring Ranjbarzadeh et al. [2021], Pei
et al. [2020], Menze et al. [2021], Ermiş et al. [2020], Rathore et al.
[2017], Wang et al. [2019], Kickingereder et al. [2019] prior to analysis.
This is a common data normalization approach for all computer vision
algorithms Patro and Sahu [2015] and is not specific to medical imaging.
### 2.4 Skull stripping
Finally, all but one of the observed papers Wang et al. [2019] use skull
stripping, arguing that non-brain tissue is a significant source of error for
downstream tumor segmentation Chang et al. [2019]. Authors point out that
skull stripping reduces the number of False Positives and improves
segmentation quality.
### 2.5 Novelty of the proposed study
In general, researchers experimenting on single-institutional data or data
collected under unified acquisition protocol tend to use fewer preprocessing
steps. On the contrary, the analysis of heterogeneous multi-center data
typically includes a data preparation pipeline. For example, the data
preparation pipeline for the most-known benchmark dataset for multi-
institutional brain MRI
segmentation222https://cbica.github.io/CaPTk/preprocessing_brats.html includes
image reorientation, altas registration Rohlfing et al. [2010], bias-field
correction, and skull stripping Thakur et al. [2020].
These MRI preprocessing methods have been exploited by scientists for several
decades. Yet to date there is no consensus regarding which of the methods
should be applied for deep-learning-based analysis. In the present work we
demonstrate that the effect of most standardization steps is negligible even
for relatively small data collection.
To the best of our knowledge, there have been two similar attempts to analyze
the influence of preprocessing steps for medical imaging tasks. Authors of
Moradmand et al. [2020] test how preprocessing influences radiomics features
calculation (for brain MRI), and de Raad et al. [2021] investigate the
influence of data augmentation for three medical imaging tasks (brain and knee
MRI and liver CT segmentation).
### 2.6 Contributions
In our study we focus on preprocessing steps used primarily for brain MRI. Our
contribution is three-fold. First, we numerically estimate the effect of 7
common preprocessing steps: resizing images to equal size,inter-subject atlas
registration,resampling voxels to equal size, bias-field correction, image
denoising, histogram matching, skull stripping; with state-of-the-art deep
learning architectures that are both convolutional and attention-based. This
effect is assessed in two scenarios: training from scratch and fine-tuning
from a larger dataset.
Second, we propose an explanation for the observed low (or even negative)
influence of image intensity correction techniques on model accuracy. For that
we compare segmented tissue with the rest of the brain before and after
preprocessing.
Third, we suggest a minimal preprocessing pipeline for multi-sequence, multi-
protocol brain MRI segmentation studies, consisting of two steps333We assume
intra-subject sequence alignment and z-scoring to be mandatory first and last
preprocessing steps, respectively.:
1. 1.
voxel resampling (to roughly align images across the dataset),
2. 2.
skull stripping.
The last step results in a marginal but statistically significant improvement
in terms of segmentation quality, especially for smaller ($<100$ subjects)
datasets, though it could be omitted for larger datasets with little drop in
accuracy.
## 3 Methods
Figure 1: Study experimental pipeline. Steps in arrows are mandatory, steps in
blocks are optional. Steps 4.a-d are performed after step 4. The minimal
preprocessing pipeline consists of steps 1 and 5. Thus, we have 8 different
preprocessing pipelines in total.
In the current study we test how different preprocessing pipelines affect the
quality of a downstream deep learning segmentation model. We compare eight
preprocessing pipelines using two segmentation models (convolutional and
attention based), Figure 1. Additionally, we test if the optimal preprocessing
pipeline changes in a transfer learning scenario (vs training from scratch) by
fine-tuning a model pretrained on a different dataset (with the same
preprocessing).
In the following sections we provide information on experimental design and a
detailed description of the preprocessing steps, neural network setups,
quality metrics, and datasets.
### 3.1 Experimental design
We compare preprocessing pipelines (Figure 1) in two scenarios:
1. 1.
Training a segmentation model from scratch.
2. 2.
Fine-tuning a segmentation model from a related or a similar task.
Thus, we check if different preprocessing pipelines result in better in-domain
training, and an improved fine-tuning/knowledge transfer in a cross-dataset
regime. In the second scenario we only consider whole model fine-tuning as it
is the most common approach to knowledge transfer Wang et al. [2021], Dai et
al. [2019].
### 3.2 Preprocessing steps
In all experiemnts we use multi-sequence datasets consisting of 4 MRI
sequences: T1 weighted (T1), T1 weighted with contrast enhancement (CT1), T2
weighted (T2), and T2-FLAIR (FLAIR). We start each preprocessing pipeline with
rigid registration (rotation, shift and linear interpolation of voxel size) of
every MR sequence to CT1 or FLAIR (depending on the dataset) independently for
each subject. This step aligns different MR sequences (from the same subject)
with each other and the existing tumour annotation. After that, we proceed
with preprocessing steps as described in Figure 1. We end each preprocessing
pipeline with image-wise Z-scoring:
$X_{s}=\frac{X-\text{mean}(X)}{\text{std}(X)}.$
#### 3.2.1 Inter-subject alignment
After the intra-subject rigid registration, we apply one of three methods to
align images across the dataset (inter-subject):
* 1.
resizing images to the same size $240\times 240\times 155$ voxels;
* 2.
resampling to an isotropic voxels’ size $1\times 1\times 1\ \text{mm}^{3}$;
* 3.
a non-rigid atlas registration to SRI24 atlas Rohlfing et al. [2010].
The latter results in both of the same images of size $240\times 240\times
155$, and an isotropic voxel $1\times 1\times 1\ \text{mm}^{3}$. All
transformations were performed by ANTs utilities Avants et al. [2009].
#### 3.2.2 Image enhancement
The next step is applying one of three algorithms of image intensity
correction:
* 1.
bias-field correction,
* 2.
denoising,
* 3.
histogram standardization.
Figure 2: Visualization of three methods of image intensity correction used in
the ablations study. The top raw (b-d) showing the image segment with tumor
after preprocessing of (a) an example CT1 MR image from GBM data after
resampling to [1,1,1] voxel size; (b) N4 correction of the image; (c) SUSAN
denoising of the image; (d) Histogram standartization to GBM sample; The
bottom row (e-g) showing image residuals as voxelwise difference between the
original resampled image and the preprocessed one.
Bias-field is a smooth, non-heterogeneous and low-frequency signal which
corrupts MRI images. It is generally believed that algorithms which
postprocess MRI images such as segmentation or classification do not produce
satisfactory results without correcting for the bias field signal Juntu et al.
[2005]. To test the impact of such a correction we use a popular N4 algorithm
Tustison et al. [2010], implemented in Simple-ITK Beare et al. [2018] as a
containerized solution from the CaPTk toolbox.
SUSAN is a method of nonlinear image smoothing and denoising Smith and Brady
[1997] that, while old, is still used nowadays for noise reduction. We use an
FSL implementation of SUSAN. These two steps were applied for each MR image
individually.
Finally, to homogenize grayvalue distribution across images in the dataset, we
apply a histogram equalization Nyúl et al. [2000], as implemented in TorchIO
library Pérez-García et al. [2021]. To estimate the parameters of histogram
matching we use training folds, and the same transformation is applied to the
data from test folds. Histogram matching is applied sequence-wise, meaning
that we equalize voxels’ histograms for each MR sequence separately Li et al.
[2021].
These steps are applied after resampling images to an isotropic voxel (step 4,
Figure 1). As we show in the results, there is no statistical differences
between steps 2-4. We thus use voxel resampling as the most intuitive and
simple one.
#### 3.2.3 Skull stripping
Skull stripping is a method to localize the region of interest and filter out
potential False Positives. To test how skull stripping affects model
performance, we use HD-BET Isensee et al. [2019], which is the most accurate
publicly available brain extraction tool. We extract brain mask on CT1 and
apply it to all other MR modalities after image alignment (this results in no
loss of tumor mask).
### 3.3 Models architecture and training
We test the effect of preprocessing on two deep-learning segmentation
architectures: 3D nn-UNet Isensee et al. [2021] and vision transformer-based
3D UNETR Hatamizadeh et al. [2022].
In all experiments we trained networks for 300 epochs or until convergence
(using 20 epochs patience as a stopping criterion) without data augmentations,
as data augmentation can interfere with measurements of preprocessing effects.
All experiments were performed on a 3-fold cross-validation (subject-wise)
with the average time for one experiment being 20 hours on 32 GB Tesla V100.
### 3.4 Perfomance metrics
The accuracy of brain MRI semantic segmentation is conventionally evaluated
using the Dice Similarity coefficient (referred to as Dice) and the 95th
percentile of the Hausdorff distance between the predicted labels and the
provided ground truth, these metrics are used for SOTA models evaluation on
benchmark datasets. Bakas et al. [2022].
In the current work we focus on overlap-based metric Dice coefficient as the
standard for assessing the quality of segmentation and provide clear and
meaningful interpretations:
$\text{Dice}\ (A,B)=\frac{2\cdot|A\cap B|}{|A|+|B|},$ (1)
where $A$ and $B$ are 3D binary arrays. In addition, to measure the error in
terms of tumor volume estimation, which is the simplest clinically relevant
feature Chang et al. [2019], we use Mean Absolute Error (MAE):
$\text{MAE}(V_{\text{true}},V_{\text{estimated}})=\frac{1}{n}\sum_{i=1}^{n}|V^{i}_{\text{true}}-V^{i}_{\text{estimated}}|.$
(2)
Mean and standard deviation for Dice score and absolute errors were obtained
out of fold. We also asses absolute volumetric error of estimation (for
evaluation of result significance from the clinical perspective).
To measure the differences in voxel intensity between healthy and tumored
tissues for different image enhancement experiments, we use Kullback-Leibler
divergence (KL) between corresponding intensity histograms:
$\text{KL}(H_{\text{healthy}},H_{\text{tumor}})=\sum_{i=1}^{\text{\\#bins}}H^{i}_{\text{healthy}}\cdot\log{\frac{H^{i}_{\text{healthy}}}{H^{i}_{\text{tumor}}}}$
(3)
We report KL values for histograms with fixed bin size (100 bins), and we have
tested the Freedman-Diaconis heuristic Freedman and Diaconis [1981] (500-700
bins depending on subject) and Sturge’s rule Sturges [1926] (20-60 bins). All
approaches result in different absolute KL values but preserve a relative
trend, without affecting the experiment’s conclusion.
To assess the results’ significance and compare experiments with different
data preprocessing, we test the Null hypothesis of the Means equality in two
samples with Welch’s t-test Bonferroni corrected for multiple comparisons. If
the $p_{\text{value}}$ for the test has not exceeded $0.05$, we declare that
there is no evidence in the data for rejecting the Null hypothesis. For
simplicity, from now on we will refer to it as statistically significant.
### 3.5 Data description
We explore the effect of data preprocessing for tumor segmentation on multi-
modal brain MRI data (a task similar to an extensively studied BraTS Menze et
al. [2014]). We selected three largest publicly available multi-domain with
original DICOM data from TCIA Clark et al. [2013a]: Glioblastoma Multiform
(GBM) and Lower Grade Glioma (LGG) Beers et al. [2018b], Bakas et al. [2017b],
Beers et al. [2018a], and an in-house dataset of 180 patients with
glioblastoma (BGPD)Zolotova et al. [2023].
All three datasets contain 4 MR sequences (T1, T2, CT1, FLAIR) available in a
raw DICOM format, without any prior preprocessing. All three datasets comprise
of multi-protocol studies (Table 3). A summary of all datasets is presented in
Table 2. A more detailed description is given below.
Table 2: Description of open-source multi-institutional datasets on brain
tumor segmentation used in the study. All datasets have multimodal MR set per
patient, including: CT1, T1, T2 and FLAIR modalities.
Dataset | Dataset name | Size | Diagnosis | Preprocessing | Segmentation classes | Annotation source
---|---|---|---|---|---|---
Beers et al. [2018a] | LGG | 38 | pre-operative low-grade glioma | ✗ | WT, ET | semi-automatic
Beers et al. [2018a] | GBM | 102 | pre-operative glioblastoma | ✗ | WT, ET, TC | semi-automatic
Zolotova et al. [2023] | BGPD | 180 | pre-radiotherapy glioblastoma | ✗ | GTV | manual
#### 3.5.1 Glioblastoma Multiforme (GBM) Dataset
GBM is an open-source data collection Bakas et al. [2017a, c], Clark et al.
[2013b], Beers et al. [2018b] which originally included 262 patients with
preprocessed images. We selected 102 patients with accessible segmentation
labels for data without
preprocessing444https://wiki.cancerimagingarchive.net/pages/viewpage.action?pageId=41517733.
The annotations are semi-automatic, obtained using a GLISTRBoost Bakas et al.
[2015] tool with manual correction, and include segmentation for the whole
tumor (WT), tumor core (TC) and enhancing tumor (ET). The data subset is
comprised of data from $3$ manufacturers and at least $28$ different study
protocols.
#### 3.5.2 Low Grade Glioma (LGG) Dataset
LGG is an open-source data collection Pedano et al. [2016] that originally
included 199 patients. We selected a subset of 38 unique patients with
accessible segmentation labels for data without preprocessing. The annotations
are semi-automatic, obtained using a GLISTRBoost Bakas et al. [2015] tool with
manual correction, and include segmentation for WT and ET ($38$ out of the
selected $39$ images do not contain TC segmentation map and are thus excluded
from the experiments for LGG). The ata subset is comprised of data from $3$
manufacturers and at least $11$ different study protocols.
#### 3.5.3 Burdenko Glioblastomas Progression Dataset (BGPD)
BGPD Zolotova et al. [2023] is an MRI collection of patients who underwent
radiotherapy in Burdenko Neurosurgery Institute for Radiotherapy in Moscow,
Russia. It contains the data of 180 unique patients. Segmentation maps were
imported from a radiotherapy planning system and correspond to Gross Tumor
Volume (GTV). The collection is highly heterogeneous and comprised of data
from $4$ manufacturers and at least $51$ different study protocols.
Table 3: Variability of study protocols for T1 and T2 MRI sequences for GBM,
BGPD and LGG data collections. All datasets contain images from 3 major MRI
scanner manufacturers: GE, Siemens, and Toshiba.
Aquisition parameter | GBM | BGPD | LGG
---|---|---|---
T1 | T2 | T1 | T2 | T1 | T2
Echo Time, ms | min | $2.1$ | $20$ | $1.8$ | $18.4$ | $3.7$ | $16.1$
max | $19$ | $120$ | $23$ | $120$ | $15$ | $120$
$\\#\text{unique}$ | $28$ | $38$ | $51$ | $67$ | $11$ | $17$
Repetition Time, ms | min | $5$ | $2020$ | $7.4$ | $567$ | $8$ | $897$
max | $3379.6$ | $6650$ | $3119.2$ | $8200$ | $3232$ | $10000$
$\\#\text{unique}$ | $56$ | $36$ | $50$ | $57$ | $38$ | $18$
Voxel volume, $\text{mm}^{3}$ | min | $0.5$ | $0.2$ | $0.1$ | $0.1$ | $0.6$ | $0.5$
max | $5.2$ | $5.2$ | $5.3$ | $4.8$ | $13.2$ | $35.2$
$\\#\text{unique}$ | $32$ | $32$ | $53$ | $60$ | $17$ | $19$
## 4 Results
Table 4: nnU-net and UNETR segmentation performance for the three datasets:
GBM and LGG (WT label), BGPD (GTV label). Segmentation accuracy presented in
Dice scores from three fold cross-validation as $\text{Mean}\ (\text{STD})$
multiplied by 100, the higher — the better. Models trained for 300 epochs.
Arrows denote the statistically significant difference
($p_{\text{value}}<0.05$) compared to step 4. Resampling to spacing,
$\uparrow$ (increase), $\downarrow$ (decrease).
| nnU-net | UNETR
---|---|---
Data preprocessing | GBM | BGPD | LGG | GBM | BGPD | LGG
1\. Inter-modality registration | $44\ (28)\downarrow$ | $36\ (29)\downarrow$ | $67\ (27)$ | $39\ (26)\downarrow$ | $35\ (30)\downarrow$ | $66\ (23)$
2\. Resampling to image size | $85\ (11)$ | $73\ (19)$ | $72\ (24)$ | $82\ (12)$ | $67\ (20)$ | $66\ (26)$
3\. Atlas registration | $85\ (11)$ | $75\ (16)$ | $71\ (25)$ | $82\ (13)$ | $68\ (21)$ | $67\ (25)$
4\. Resampling to spacing | $85\ (12)$ | $74\ (18)$ | $70\ (25)$ | $83\ (14)$ | $67\ (21)$ | $67\ (23)$
4.a Bias field correction | $82\ (13)\downarrow$ | $75\ (17)$ | $67\ (25)$ | $80\ (13)$ | 72$\ (19)\uparrow$ | $62\ (22)$
4.b Denoising | $84\ (12)$ | $74\ (17)$ | $70\ (26)$ | $83\ (13)$ | $69\ (21)\uparrow$ | $65\ (25)$
4.c Histogram matching | $83\ (16)$ | $75\ (16)$ | $68\ (26)$ | $81\ (16)$ | $68\ (18)$ | $63\ (26)$
4.d Skull stripping | $87\ (11)$ | $76\ (14)\uparrow$ | $77\ (21)\uparrow$ | $85\ (11)$ | $72\ (18)\uparrow$ | $75\ (19)\uparrow$
Our results are four-fold. First, we analyze the effect of image resampling
(Steps 2-4 Figure 1). Second, we analyze image enhancement methods (Steps
4.a-4.c Figure 1). Next, we discuss the utility of skull-stripping in terms of
segmentation metrics and volume estimates. Finally, we analyze if there is
optimal preprocessing in a transfer learning scenario. We end the results
section with our recommendations on MRI images preprocessing for deep
learning-based tumor segmentation.
### 4.1 Inter-subject image alignment
Table 4 shows validation results of nnU-net and UNETR architectures for the
three datasets.
First, for two larger datasets (GBM and BGPD) we observe that introducing some
resampling strategy to homogenise voxel volume across the dataset is always
beneficial, Table 4 step 1 versus steps 2-4. Recall that without any voxel
resampling, the differences between voxels’ volumes are as large as 10 times
for GBM ($0.5\ \text{mm}^{3}$ versus $5.2\ \text{mm}^{3}$), and 53 times for
BGPD ($0.1\ \text{mm}^{3}$ versus $5.3\ \text{mm}^{3}$), see Table 3. Thus,
for a 3D convolutional neural network, the receptive field of a convolution
filter will differ by a factor of 53. Interestingly, while for the LGG dataset
we also have a difference between voxels’ spacing by a factor of 22, there are
no significant differences in performance between steps 1-4. The latter might
be the consequence of a relatively small sample size.
Second, we observe that applying non-rigid Atlas registration (step 3) lead to
the same results compared to a faster Resampling to the same image size (step
2) or Resampling all images to the same voxel spacing (step 4). For both NN
architectures in all datasets, it is not possible to reject the Null
hypothesis about the equality of means ($p_{\text{values}}>0.1$, accounting
multiple comparisons correction). Recall that images on step 3 are both the
same image size and have a voxel of the same volume555Resampling to same image
size results in almost equal voxel volumes $1.07\ (0.34)\ \text{mm}^{3}$, mean
(std) for BGPD dataset..
### 4.2 Image enhancement
We compare three different intensity normalization steps commonly used in
brain MRI analysis pipelines: Bias field correction (step 4.a), Denoising
(step 4.b) and Histogram matching (step 4.c). As there were no significant
differences found between resampling approaches, steps 4.a-c were performed
after resampling images to the same voxel volume (step 4).
First, for a convolutional nnU-net, intensity correction transformations could
be completely omitted. As show in Table 4, there is no statistically
significant improvement of either of steps 4.a-c compared to step 4
($p_{\text{values}}>0.1$) for all three datasets. In most of the cases, the
average segmentation performance is actually worse (compared to no intensity
normalization, step 2) by an absolute value, though the only statistically
significant drop in performance is Bias field correction on GBM dataset (82
mean Dice score (step 4.a) compared to 85 mean Dice score (step 4),
$p_{\text{value}}=0.014$).
Second, for an attention-based model, the general trend stays the same, except
for the BGPD dataset and steps 4.a and b. Here, we observe a small but
statistically significant increase in performance. We do not have a reasonable
explanation for the effect (see A), though we acknowledge that for all
datasets, UNETR architecture results in worse performance compared to nnU-net.
This might be the effect of a relatively small sample size, as transformer-
based architectures require more training data.
### 4.3 Skull stripping
A brain mask application before training results in a moderate but
statistically significant Dice score improvement for BGPD and LGG datasets
(both nnU-net and UNETR) over the experiment without skull stripping, see
Table 4 4 and 4.d. For the GBM dataset, the average segmentation quality is
larger and the standard deviation is smaller in the experiment with skull
stripping, though after a multiple comparisons correction these differences
are not statistically significant ($p_{\text{value}}=0.09$).
### 4.4 Volumetric errors
In addition to the Dice scores, we provide the errors in tumour volume
estimates for different preprocessing steps.
We report the results for nnU-net in Table 5, as it performs better in our
experiments. In terms of volume estimates, errors follow the same trend as
with Dice scores: inter-subject alignment is always beneficial, image
enhancement does not result in any improvements, and skull stripping
systematically improves the quality.
In volume estimates, skull stripping improves tumor segmentation quality for
GBM and LGG datasets (step 4.d., Table 5) and does not for the BGPD dataset
(for which there is a statistically significant improvement in terms of Dice
scores). This result itself is aligned with the results in Dice scores, yet it
is an argument for using clinically relevant metrics in addition to
segmentation metrics.
Volumetric errors of model predictions on BGPD data with and without skull
stripping are not statistically significant (MAE 26 (42) mL for skull
stripping and MAE 28 (49) mL without skull stripping,
$p_{\text{value}}=0.308$), thus from a clinical perspective this additional
step is not completely justified. Complete volumetric measurements for all
experiments are provided in the A. Additionally, we check if error in volume
estimate depends on tumor total size (Figure 3), and do not observe any
dependence.
Table 5: Estimated MAE of model prediction from ground truth label for the three datasets: GBM and LGG (WT label), BGPD (GTV label). Results are represented in mL, values are Mean (STD), the lower — the better. Volumes estimates are based on nnU-net. Arrows denote statistically significant difference ($p_{\text{value}}<0.05$) compared to step 4. Resampling to spacing $\uparrow$ (increase), $\downarrow$ (decrease). Data preprocessing | GBM | BGPD | LGG
---|---|---|---
1.Inter-modality registration | $63\ (47)\downarrow$ | $55\ (62)\downarrow$ | $34\ (39)$
2.Resampling to image size | $15\ (14)$ | $28\ (43)$ | $23\ (24)$
3.Atlas registration | $13\ (11)$ | $27\ (45)$ | $31\ (33)$
4.Resampling to spacing | $14\ (13)$ | $28\ (49)$ | $32\ (31)$
4.a Bias field correction | $15\ (15)$ | $27\ (49)$ | $34\ (37)$
4.b Denoising | $13\ (13)$ | $27\ (47)$ | $32\ (34)$
4.c Histogram matching | $14\ (12)$ | $27\ (47)$ | $42\ (61)$
4.d Skull stripping | $10\ (9)\uparrow$ | $26\ (42)$ | $19\ (17)\uparrow$
Figure 3: The relation between tumor volume and its’ estimated volume from
predicted segmentation mask for two nnU-net experiments on BGPD: 4. Resample
to spacing and 4.d Resample to spacing with skull stripping.
(a) GBM dataset, WT label
(b) BGPD dataset, GTV label
Figure 4: nnU-net performance on GBM and BGPD datasets. Horizontal:
segmentation accuracy presented in Dice scores from three fold cross-
validation as $\text{Mean}\ (\text{STD})$ multiplied by 100. Vertical:
preprocessing experiments 1-4.d for models trained with random weights
initialization for 100 epochs (blue), 300 epochs (green) and fine-tuning for
100 epochs from pretrained weighs on another dataset (red).
### 4.5 Transfer learning.
We test if MRI data preprocessing can facilitate model transfer from another
dataset. In particular, we explore if model fine-tuning after training on
preprocessed data is better than model fine-tuning on non-preprocessed data.
We repeat the main ablation study with nnU-net models pretrained on the GBM
dataset for 300 epochs and fine-tune them on BGPD with the same data
preprocessing for 100 epochs (and pretraining on BGPD with fine-tuning of GBM
for corresponding experiments). We compare the Dice scores on three fold
cross-validation for a model trained for 100 and 300 epochs from random weight
initialization, and a model trained on 100 epochs from pretrained weights. The
results are presented in Figure 4.
First, we observe a definitive improvement in nnU-net performance on the GBM
dataset with weight transfer and fine tuning for 100 epochs (red bars),
compared to a training from scratch for 300 epochs (green bars), Figure 4(a).
For the BGPD dataset, pretraining on the GBM sample results in better
segmentation performance compared to training for 100 epochs (blue bars and
red bars), but worse compared to training from scratch for 300 epochs (green
bars), Figure 4(b). This effect could be a consequence of the different sample
sizes, as BGPD is almost two times larger than GBM — thus pretraining on the
BGPD improves GBM segmentation, but not vice versa.
Second, we do not observe any differences in segmentation performance for
either of the preprocessing steps on both datsets. For example, for the GBM
dataset and preprocessing step 4.b Denoising, a model trained for 100 epochs
results in 84 (12) Dice score (STD), the same if trained for 300 epochs, and
86 (10) Dice score if fine-tuned from BGPD. Yet, with the same data
preprocessing on BGPD, we see a decrease in the mean Dice score with weight
transfer from the GBM dataset to BGPD. Similarly, for step 4.d, the
improvement of the segmentation quality on the GBM dataset is not observable
in the BGPD sample. From these experiments we conclude that no preprocessing
step among those studied improves model performance with weight transfer for
both datasets. Numerical results depicted in Figure 4 are accessible in C.
Lastly, we perform an experiment with model fine-tuning from two large
datasets BraTS2021, consisting of 2000 subjects Baid et al. [2021] and EGD
dataset with 774 subjects.van der Voort et al. [2021], with multiclass labels,
similar to GBM and LGG ones. According to our results, model fine-tuning from
a larger sample — the most exploited method of transfer learning Ardalan and
Subbian [2022] — can be reached by longer training, irrespective of the size
of the dataset for weight transfer, see B Table C.
### 4.6 Our recommendations for brain MRI preprocessing for deep learning
segmentation.
The overall results suggest the following recommendations:
* 1.
It’s essential to align multi-modal MRI data between subjects for analysis,
and even fast methods like image or voxel resizing yield comparable to atlas
registration segmentation accuracy.
* 2.
Bias-field correction, denoising, and histogram matching are unnecessary in
MRI segmentation pipelines based on UNet-like or UNETR architectures.
* 3.
Although skull stripping can improve segmentation performance, its impact on
clinical measurements, such as differences in lesion volume estimates, is
relatively small. Therefore, depending on the clinical task and the need for
fast processing times, this step may not be necessary.
* 4.
Preprocessing MRI data does not help with transfer learning while fine-tuning
models on other datasets. Moreover, there’s almost no significant difference
between fine-tuning models from other data and just doing longer training on
the original sample.
## 5 Conclusion
We perform a rigorous ablation study of the most conventional preprocessing
steps used in the analysis of brain MRI images, including atlas registration,
voxel resampling and image resizing, histogram matching, bias-field
correction, denoising and skull stripping.
Although the image reprocessing steps might be useful for annotators and make
distinct properties of the image more recognizable for the human eye, we show
that only image alignment and voxel resampling are essential for accurate
prediction with DNN models. We conclude that predictions after atlas
registration do not significantly differ from ones with equal voxel
resampling. We observe that bias-field correction, denoising, and histogram
matching reduce data variance and do not affect DNN performance positively. We
point out that skull stripping can lead to a measurable increase in accuracy
and facilitate model convergence. On the other hand, brain extraction is very
computationally expensive, and its incorporation into a pipeline does not
affect clinically relevant volumetric measurements.
Thus we believe that skipping all steps excluding image alignment and voxel
resampling from the brain MRI deep learning pipeline may reduce computational
costs and improve reproducibility across studies.
These recommendations will be especially relevant for MRI data preprocessing
for semi-automated labeling with Segment Anything Model (SAM) and
modifications Kirillov et al. [2023]. SAM is a vision-transformer based
architecture, shown to be extremely useful for data annotation, yet still not
surpassing the SOTA solutions for brain MRI data segmentation Wu et al.
[2023]. In the current work we define necessary preprocessing steps needed for
MRI data annotation and further training, that will ensure reproducible across
the studies and best segmentation accuracy.
### 5.1 Work limitations
Our findings on data preprocessing strategies suggest that overall research
reproducibility will benefit if one discards custom preprocessing steps,
including different skull stripping, various implementations of bias field
correction, denoising, etc. Yet we observed that the results of transfer
learning and model training from scratch are strictly related to datasets’
homogeneity and size. These effects could be different on datests of thousands
of images de Raad et al. [2021].
In the current study we focused on conventional brain MRI data preparation
methods. The newly-developed methods of MRI harmonization, as multi site image
harmonization by cumulative distribution function (CDF) alignment (MICA Wrobel
et al. [2020]) or robust intensity distribution alignment (RIDA) Sederevicius
et al. [2022] could be outperforming the most conventional algorithms for
histogram matching Nyúl et al. [2000]. Advanced image intensity enhancement
methods can be compared with the explored ones, i.e. with orthogonal moments
da Silva et al. [2022] for MR image enhancement. These analyses were outside
the scope of the original study.
### 5.2 Authors contributions
Conceptualization and methodology, A.K., E.K. and P.D.; data pipeline
organization and preprocessing pipeline E.K.; model training and analysis
P.D.; interpretation of results, A.K. and A.D., S.Z., A.G.; writing original
draft A.K., E.K. and P.D.; writing—review and editing, M.B., B.S. and A.D. All
authors have read and agreed to the published version of the manuscript.
### 5.3 Declaration of Competing Interest
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
### 5.4 Acknowledgments
This work was conducted in the Artificial Intelligence Research Institute in
Moscow, Russia in collaboration with the National Medical Research Center for
Neurosurgery and the Moscow Gamma Knife Center. The work of A.D., S.Z., A.G.
and A.K. was supported by the Russian Foundation for Basic Research grant
18-29-01054. The work of P.D. and E.K. was supported by the Russian Science
Foundation grant 21-71-10136.
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## Appendix A The effect of image enhancement
Figure 5: Computing the distance between healthy tissue and tumor tissue
regions.
Why do image enhancement methods not facilitate segmentation?
We attempt to explain why popular intensity normalization steps have
questionable effect on segmentation performance. Our hypothesis is that while
these steps equalize modalities appearance across the data, they also reduce
the differences between voxels’ intensities within each individual image. We
compare intensity distribution for healthy brain voxels and voxels inside a
tumor mask using Kullback-Leibler divergence (Table 6). We expect that if
preprocessing steps increases KL divergence it should result in increased
segmentation quality, and vice versa. In most of the cases this supposition
holds. Two exceptions are Denoising for Tumor core segmentation and Histogram
matching for Enhancing tumor segmentation.
We report KL values for histograms with fixed bin size (30 bins), and we test
the Freedman-Diaconis heuristic (500-700 bins depending on subject) and
Sturge’s rule (20-60 bins). Both approaches result in different absolute KL
values but preserve a relative trend, without affecting the experiment
conclusion.
In most cases, if the KL divergence decreases (differences between healthy and
tumor tissue decrease), model performance decreases, too, and vice versa
(steps 4.a-d in comparison to step 4, Table 6). In almost all cases, lower KL
values corresponds to lower performance, e.g. for bias-field correction, the
KL between healthy brain and WT tissue is equal to 0.47, compared to 0.61 of
atlas registration, which coincides with a segmentation quality drop (from
86.4 for atlas registration to 84.9 for bias-field corrected data). On the
contrary, for denoised data, the KL are either the same or slightly larger
compared to atlas data: 0.63 vs 0.61 for WT; 4.16 vs 4.01 for TC and 7.11 vs
6.70 for ET, which completely coincides with segmentation performance. The
only comparison that does not follow this explanation is bf-correction for TC
(it has a lower KL compared to atlas data, but slightly better segmentation
quality).
Table 6: KL divergence values (KL) and JS distance (JS) between intensity
histograms for masked brain w/o tumor and tumor region. Lower values
correspond to smaller differences in voxel intensities between healthy (brain
mask without Whole Tumor(WT) for GBM and RT for BGPD sample) and tumor tissue.
Stars represent statistically significant difference to 4. Resampling to
spacing distances.
Data preprocessing | GBM, KL | GBM, JS | BGPD, KL | BGPD, JS
---|---|---|---|---
4\. Resampling to spacing | $0.830(0.450)$ | $0.303(0.095)$ | $1.640(1.015)$ | $0.477(0.129)$
4.a Bias-field correction | $0.660(0.300)$ | $0.259(0.087)$ | $1.145(0.723)$ | $0.465(0.120)$
4.b Denoising | $0.870(0.470)$ | $0.308(0.097)$ | $1.702(1.045)$ | $0.484(0.129)$
4.c Histogram matching | $0.770(0.470)$ | $0.304(0.095)$ | $1.650(1.029)$ | $0.478(0.130)$
## Appendix B Model architectures and training
Table 7: Hyperparameters for nnU-net and UNETR models. Parameter name | nnU-net | UNETR
---|---|---
learning rate | $0.0003$ | $0.0001$
weight decay | $0$ | $0.00001$
momentum | $0.99$ | $0.99$
patch size | $[128,128,128]$ | $[128,128,128]$
batch size | $2$ | $2$
nnU-net. We use NVIDIA’s nn-Unet implementation for the BraTS2021
challenge666github.com/NVIDIA/DeepLearningExamples/tree/
ddbcd54056e8d1bc1c4d5a8ab34cb570ebea1947/PyTorch/Segmentation/nnUNet. The
following changes were applied by the authors on top of the default nnU-net
architecture Isensee et al. [2021]: increasing the encoder depth to $7$,
modifying the number of convolution channels to $[64,96,128,192,256,384,512]$,
and using deep supervision — two additional output heads at the decoder
levels. The model has multi-modal input of four modalities [T1, CT1, T2,
FLAIR], plus a channel with one-hot encoding for foreground voxels, generated
by image thresholding. We train nn-Unet with a patch size of [128,128,128]
voxels and a batch size of two, a learning rate of $0.0003$ and a Adam
optimizer with momentum of $0.99$.
UNETR. We use vision a transformer-based model UNETR Hatamizadeh et al. [2022]
with an embedding size of $512$ for a 1D sequence of a 3D input of the same
$5$ channels with patches of [128,128,128] voxels and the resolution of each
patch equals 16. The model has $10$ heads and is trained with learning rate
$0.0001$, with a weight decay of $0.00001$, and an Adam optimizer with
momentum of $0.99$.
Model optimization. For the BGPD dataset we train the model to predict one
class label. For the GBM dataset we train the model on three classes,
according to the BraTS data labelling: WT, ET, TC. The model is then trained
with the complex loss function. Each label class is optimized separately with
the weighted sum of binary Cross-Entropy and the Dice loss (the trade-off
weight value to 1 for both losses). The final complex loss function is
optimized for a combination of class labels: the whole tumor (WT) (describes
the union of the tumor core (ET), the non-enhancing part of the tumor core
(NET) and the peritumoral edema ED), the tumor core (TC) (the union of the ET
and NET), and the ET.
Figure 6: Visualization of the two models’ predictions with respect to the
method of image intensity correction. The top raw (b-d) showing the image
segmentation on 3 classes with nnUnet model (a) an example CT1 MR image from
GBM data after resampling to [1,1,1] voxel size; (b) N4 correction of the
image; (c) SUSAN denoising of the image; (d) Histogram standartization; The
bottom row showing tumor segmentation with UNETR model. Colors: ET - red, TC -
white, WT - yellow.
## Appendix C Additional illustration of the results
In this section we show additional illustrations of the results not
represented in the test’s main body. In Tables 8 and 9 we show the results of
other segmentation classes for multiclass tumor segmentation mask (WT, ET and
TC) for the GBM dataset and (WT and ET) for the LGG dataset.
In Table 10 we show the numerical representation of graphical results from
Figure 4.
Table 8: nnU-net and UNETR segmentation performance for three-class (Whole
tumor, Enhancing tumor, Tumor core) segmentation on GBM dataset. Numbers are
Dice scores mean (std) multiplied by 100. Trained for 300 epoch, columns 1 and
4 are duplicated from Table 4.
| nnU-net | UNETR
---|---|---
Data preprocessing | WT | ET | TC | WT | ET | TC
1\. Inter-modality registration | $44\ (28)\downarrow$ | $37\ (30)\downarrow$ | $30\ (27)\downarrow$ | $39\ (26)\downarrow$ | $32\ (26)\downarrow$ | $28\ (24)\downarrow$
2\. Resampling to image size | $85\ (11)$ | $80\ (18)$ | $74\ (19)$ | $82\ (12)$ | $75\ (18)$ | $72\ (19)$
3\. Atlas registration | $85\ (11)$ | $80\ (17)$ | $72\ (20)$ | $82\ (13)$ | $74\ (18)$ | $69\ (21)$
4\. Resampling to spacing | $85\ (12)$ | $80\ (16)$ | $73\ (19)$ | $83\ (14)$ | $75\ (20)$ | $71\ (22)$
4.a Bias field correction | $82\ (13)\downarrow$ | $80\ (17)$ | $72\ (19)$ | $80\ (13)$ | $75\ (20)$ | $71\ (21)$
4.b Denoising | $84\ (12)$ | $80\ (17)$ | $73\ (20)$ | $83\ (13)$ | $75\ (20)$ | $70\ (22)$
4.c Histogram matching | $83\ (16)$ | $78\ (20)$ | $72\ (21)$ | $81\ (16)$ | $72\ (24)$ | $68\ (24)$
4.d Skull stripping | $87\ (11)$ | $82\ (16)$ | $76\ (19)$ | $85\ (11)$ | $79\ (16)$ | $75\ (19)$
Table 9: nnU-net and UNETR segmentation performance for two-classes (Whole
tumor, Enhancing tumor) segmentation on LGG dataset. Numbers are Dice scores
$\text{mean}(\text{std})$ multiplied by 100, columns 1 and 3 are duplicated
from Table 4.
| nnU-net | UNETR
---|---|---
Data preprocessing | WT | ET | WT | ET
1\. Inter-modality registration | $67\ (27)\downarrow$ | $47\ (29)$ | $66\ (23)$ | $49\ (26)$
2\. Resampling to image size | $72\ (24)$ | $52\ (27)$ | $66\ (26)$ | $49\ (27)$
3\. Atlas registration | $71\ (25)$ | $52\ (29)$ | $67\ (25)$ | $49\ (27)$
4\. Resampling to spacing | $70\ (25)$ | $48\ (28)$ | $67\ (23)$ | $50\ (25)$
4.a Bias field correction | $67\ (25)$ | $45\ (29)$ | $62\ (22)$ | $45\ (25)$
4.b Denoising | $70\ (26)$ | $51\ (30)$ | $65\ (25)$ | $48\ (27)$
4.c Histogram matching | $68\ (26)$ | $49\ (29)$ | $63\ (26)$ | $45\ (28)$
4.d Skull stripping | $77\ (21)\uparrow$ | $54\ (25)\uparrow$ | $75\ (19)\uparrow$ | $54\ (25)\uparrow$
Table 10: nnU-net performance on GBM and BGPD datasets, numerical
representation of Figure 4. Segmentation accuracy presented in Dice scores
from three fold cross-validation as $\text{Mean}\ (\text{STD})$ multiplied by
100. Models trained with random weights initialization for 100 epochs, 300
epochs and fine-tuning for 100 epochs from pretrained weighs on other dataset
(BGPD-GBM, GBM-BGPD finetune). Columns 2 and 5 are duplicated from Table 4.
| GBM | BGPD
---|---|---
Data preprocessing | 100 epochs | 300 epochs | BGPD-GBM | 100 epochs | 300 epochs | GBM-BGPD
1\. Inter-modality registration | $42\ (27)$ | $44\ (28)$ | $43\ (27)$ | $34\ (28)$ | $36\ (29)$ | $36\ (29)$
2\. Resampling to image size | $83\ (13)$ | $85\ (10)$ | $86\ (10)$ | $71\ (18)$ | $73\ (19)$ | $74\ (19)$
3\. Atlas registration | $84\ (12)$ | $85\ (11)$ | $86\ (10)\uparrow$ | $72\ (19)$ | $75\ (16)$ | $72\ (17)$
4\. Resampling to spacing | $84\ (13)$ | $85\ (12)$ | $86\ (11)$ | $72\ (18)$ | $74\ (18)$ | $72\ (19)$
4.a Bias field correction | $83\ (13)$ | $82\ (13)$ | $85\ (11)\uparrow$ | $72\ (17)$ | $75\ (17)$ | $73\ (18)$
4.b Denoising | $84\ (12)$ | $84\ (12)$ | $86\ (10)\uparrow$ | $71\ (19)$ | $74\ (17)$ | $73\ (19)$
4.c Histogram matching | $82\ (16)$ | $83\ (16)$ | $84\ (26)$ | $72\ (16)$ | $75\ (16)$ | $74\ (16)$
4.d Skull stripping | $86\ (11)$ | $87\ (11)$ | $87\ (11)$ | $76\ (15)$ | $76\ (14)$ | $75\ (16)$
Table 11: nnU-net performance with and w/o model transfer from large datasets
(BraTS2021 Baid et al. [2021] and EGD van der Voort et al. [2021].
Segmentation accuracy presented in Dice scores from three fold cross-
validation as $\text{Mean}\ (\text{STD})$ multiplied by 100. Models were
trained for 300 epochs and fine-tuned for 100 epochs from pretrained weights
on the larger dataset (BraTS-BGPD, EGD-GBM, BraTS-EGD finetune). Columns 1 and
3 are duplicated from Table 4.
Data preprocessing | BGPD, | | | | |
---|---|---|---|---|---|---
| BraTS-BGPD, | | | | |
| GBM, | | | | |
| EGD-GBM, | | | | |
| EGD, | | | | |
| BraTS-EGD, | | | | |
| Finetune | | | | |
3\. Atlas registration | $75\ (17)$ | $74\ (20)$ | $85\ (12)$ | $84\ (11)$ | $-$ | $-$
4.d Skull stripping | $76\ (14)$ | $75\ (17)$ | $87\ (11)$ | $86\ (10)$ | $83\ (12)$ | $83\ (12)$
|
11institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
11email<EMAIL_ADDRESS>
# Modeling the secular evolution of embedded protoplanetary discs
J. Mauxion G. Lesur S. Maret
(Received October 27, 2023; accepted March 25, 2024)
###### Abstract
Context. Protoplanetary discs are known to form around nascent stars from
their parent molecular cloud as a result of angular momentum conservation. As
they progressively evolve and dissipate, they also form planets. While a lot
of modeling efforts have been dedicated to their formation, the question of
their secular evolution, from the so-called class 0 embedded phase to the
class II phase where discs are believed to be isolated, remains poorly
understood.
Aims. We aim to explore the evolution between the embedded stages and the
class II stage. We focus on the magnetic field evolution and the long-term
interaction between the disc and the envelope.
Methods. We use the GPU-accelerated code Idefix to perform a 3D, barotropic,
non-ideal magnetohydrodynamic (MHD) secular core collapse simulation that
covers the system evolution from the collapse of the pre-stellar core until
$100\,\mathrm{kyr}$ after the first hydrostatic core formation and the disc
settling while ensuring sufficient vertical and azimuthal resolutions (down to
$10^{-2}$ au) to properly resolve the disc internal dynamics and non-
axisymmetric perturbations.
Results. The disc evolution leads to a power-law gas surface density in
Keplerian rotation that extends up to a few $10\,\mathrm{au}$. The magnetic
flux trapped in the disc during the initial collapse decreases from
$100\,\mathrm{mG}$ at disc formation down to $1\,\mathrm{mG}$ by the end of
the simulation. After the formation of the first hydrostatic core, the system
evolves in three phases. A first phase with a small ($\sim 10\,\mathrm{au}$),
unstable, strongly accreting ($\sim 10^{-5}\,\mathrm{M_{\odot}\,yr^{-1}}$)
disc that loses magnetic flux over the first $15\,\mathrm{kyr}$, a second
phase where the magnetic flux is advected with a smooth, expanding disc fed by
the angular momentum of the infalling material, and a final phase with a
gravitationally-regulated $\sim 60\,\mathrm{au}$ disc accreting at at few
$10^{-7}\,\mathrm{M_{\odot}\,yr^{-1}}$. The initial isotropic envelope
eventually feeds large-scale vertically-extended accretion streamers, with
accretion rates similar to that onto the protostar ($\sim
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$). Some of the streamer material collides
with the disc’s outer edge and produces accretion shocks, but a significant
fraction of the material lands on the disc surface without producing any
noticeable discontinuity.
Conclusions. While the initial disc size and magnetisation are set by magnetic
braking, self-gravity eventually drives accretion, so that the disc ends up in
a gravitationally-regulated state. This evolution from magnetic braking to
self-gravity is due to the weak coupling between the gas and the magnetic
field once the disc has settled. The weak magnetic field at the end of the
class I phase ($B_{z}\sim 1\,\mathrm{mG}$) is a result of the magnetic flux
dilution in the disc as it expands from its initial relatively small size.
This expansion should not be interpreted as a viscous expansion, as it is
driven by newly accreted material from large-scale streamers with large
specific angular momentum.
###### Key Words.:
Stars: formation – Protoplanetary discs – Magnetohydrodynamics – Methods:
numerical
## 1 Introduction
Protoplanetary discs are ubiquitous in star-forming systems. Once they have
formed, they are believed to be the main reservoir of mass that feeds the
protostar and forms planets. In the early stages of their evolution, they are
still embedded in a massive infalling envelope. As the system evolves, the
envelope is progressively accreted onto the disc, which acts as a buffer
between the envelope and the protostar. It is, therefore, crucial to
understand the long-term evolution of such discs. They likely result from a
complex interplay between mass input from the envelope and mass removal
through accretion onto the protostar, outflowing material, and planet
formation.
In essence, protoplanetary discs are the result of angular momentum
conservation in a process that gathers mass from a $10^{3}\,\mathrm{au}$ scale
down to a few tens of $\mathrm{au}$. The initial configuration and later
evolution of the disc are set by the amount of angular momentum stored in the
gas by the time of its formation, and the mechanisms that are able to modify
this amount.
On the one hand, it is now clear from core-collapse models of growing physical
complexity (see Tsukamoto et al. 2022, for a review) that we must both account
for the initial magnetisation of the pre-stellar core and its complex
chemistry to consistently reproduce the range of sizes and masses inferred in
young discs from observational surveys (Maury et al. 2019; Maret et al. 2020;
Tobin et al. 2020; Sheehan et al. 2022). Yet, robust conclusions about the
relative importance of these ingredients and the influence of the initial
conditions are still lacking.
On the other hand, results from numerical models emphasize the importance of
magnetisation and self-gravity in the disc formation and early evolution.
Masson et al. (2016) find that ambipolar diffusion is crucial during the
collapse and for disc formation, as it decouples the magnetic field and the
gas sufficiently to prevent the magnetic braking catastrophe, that suppresses
the disc in ideal MHD simulations (Price & Bate 2007). Hennebelle et al.
(2016) support this result and show that the size of the newborn disc is set
through a balance between magnetic braking (driven by toroidal magnetic field
amplification) and ambipolar diffusion.
This magnetically-regulated phase is often followed by a gravitationally-
regulated one. In their work, Tomida et al. (2017); Xu & Kunz (2021a, b) find
a disc that becomes so massive and resistive that it is mainly controlled by
angular momentum transport induced by self-gravity. In such a situation, they
argue that the disc is stuck in a feedback loop where mass influx from the
envelope promotes the generation of self-gravitating density waves that heat
the gas, thus stabilizing the disc. Hence, while we have a good understanding
of the relative importance of magnetisation and self-gravity during the early
disc evolution, their influence on a secular scale remains to be explored.
As it falls onto the disc, the envelope can provide additional angular
momentum and promote disc growth. The classic picture of an isotropic infall
through a flattened, pressure-supported envelope (also known as pseudo-disc)
is questioned by recent core collapse simulations of sufficiently massive
molecular cloud accounting for turbulent, non-ideal magnetohydrodynamics (MHD,
Kuffmeier et al. 2017; Kuznetsova et al. 2019; Lebreuilly et al. 2021). In
these simulations, the infall from the envelope is anisotropic and takes the
form of filamentary or sheet-like structures. Such structures are reminiscent
of the large-scale accretion streamers observed these recent years in embedded
systems (Yen et al. 2019; Pineda et al. 2020; Murillo et al. 2022; Valdivia-
Mena et al. 2022)
Hence, to understand the secular evolution of protoplanetary discs, one must
understand the accretion mechanisms at stake in the disc. Magnetic braking,
controlled by diffusion effects and the magnetic field intensity, is the best
candidate. Yet, in cases where it becomes inefficient, self-gravity takes
over. Thus, understanding the secular evolution of the magnetic field is key
to understanding the regulation processes of the disc. The field may also play
a role in the formation of anisotropies in the envelope.
Modeling the formation and evolution of protoplanetary discs is challenging
because of the complex physics, the implied ranges of size and time, and the
tri-dimensional nature of some key processes. For this reason, most collapse
models either under-resolve the disc vertical structure or limit the
computation time to the early class I stage. Yet, Lubow et al. (1994) showed
that the magnetic field radial diffusion efficiency depends on the disc
thickness. It is also important to correctly capture phenomena such as
magnetic field line twisting or spiral density wave generation. Thus, it is
crucial to properly resolve the disc vertical extent while integrating for a
significant time after the disc formation to study the disc and field
evolution on a secular scale.
This paper is organized as follows: in Sect. 2 we present our method and
numerical setup, as well as the initial conditions of our model. Section 3
follows the overall evolution of the setup, starting from the isothermal
collapse and browsing the secular evolution of the disc. In Sect. 4 we draw
the accretion history of the disc as a complex interplay between magnetic
braking, gravitational instability, and angular momentum influx from the
envelope while Sect. 5 probes the disc-envelope interaction that arises in the
form of a large-scale accretion streamer. Finally, Sect. 6 confronts our
results with observational and numerical constraints. We conclude in Sect. 7.
## 2 Method
We aim at performing a long timescale core collapse simulation using the
finite volume code Idefix (Lesur et al. 2023). This section presents the code
and setup general properties.
### 2.1 Governing equations and integration scheme
The framework of our simulation lies in the context of non-relativistic, non-
viscous, and locally isothermal non-ideal magnetohydrodynamics (MHD). The code
solves for the classic mass, momentum, and Maxwell’s equations:
$\displaystyle\partial_{t}\rho+\mathbf{\nabla}\cdot(\rho\mathbf{u})=0,$ (1)
$\displaystyle\partial_{t}(\rho\mathbf{u})+\mathbf{\nabla}\cdot(\rho\mathbf{u}\otimes\mathbf{u})=-\mathbf{\nabla}P-\rho\mathbf{\nabla}\phi_{g}+\frac{\mathbf{J}\times\mathbf{B}}{c},$
(2)
$\displaystyle\partial_{t}\mathbf{B}=-\mathbf{\nabla}\times\mathbf{\mathcal{E}},$
(3) $\displaystyle\mathbf{\nabla}\cdot\mathbf{B}=0,$ (4)
$\displaystyle\mathbf{J}=\frac{c}{4\pi}\mathbf{\nabla}\times\mathbf{B},$ (5)
where $\rho$, $P$, $\mathbf{u}$, $\mathbf{B}$ and $\mathbf{J}$ are
respectively the density, the thermal pressure, the gas velocity, the magnetic
field threading the medium and the electrical current. $c$ is the speed of
light and $\phi_{g}$ is the gravitational potential. It is the sum of a point
mass contribution from central mass $\phi_{pm}$ (see Appendix. B) and a self-
gravitational contribution $\phi_{sg}$ which is connected to the density
distribution via the Poisson equation:
$\Delta\phi_{sg}=4\pi G\rho$ (6)
with $G$ the gravitational constant. We assume the point mass to be fixed at
the center and therefore neglect any reaction to the accretion of non-zero
linear momentum material. The electromotive field $\mathbf{\mathcal{E}}$ is
derived from the non-ideal Ohm’s law in the case of ambipolar and Ohmic
diffusions and reads:
$\mathbf{\mathcal{E}}=-\mathbf{u}\times\mathbf{B}+\frac{4\pi}{c}\eta_{O}\mathbf{J}-\frac{4\pi}{c}\eta_{A}\mathbf{J}\times\mathbf{b}\times\mathbf{b}$
(7)
where $\eta_{O}$ and $\eta_{A}$ are the Ohmic and ambipolar diffusion
coefficients, and $\mathbf{b}$ is a unitary vector aligned with the magnetic
field.
Idefix solves the above equations using a conservative Godunov method (Toro &
Toro 2009) with a Constrained Transport (CT) scheme to evolve the magnetic
field (Evans & Hawley 1988). The parabolic terms associated with non-ideal
effects are computed separately using a super-timestepping Runge-Kutta-
Legendre scheme (Meyer et al. 2014; Vaidya et al. 2017). To prevent the
accumulation of round-off errors on $\mathbf{\nabla}\cdot\mathbf{B}$ induced
by the super-timestepping, we use a modified CT scheme in which the primitive
variable evolved by the code is the vector potential $A$ on cell edges in
place of the magnetic field $B$ on cell faces, as recommended by Lesur et al.
(2023). We implemented a biconjugate gradient stabilized (BICGSTAB) method
with preconditioning that iteratively solves the Poisson equation (see
Appendix A and Appendix B for the method and its application to our problem).
Finally, a grid coarsening procedure (Zhang et al. 2019) is applied in the
azimuthal direction close to the axis to increase the integration timestep
without loss of resolution in the equatorial region.
### 2.2 Grid and geometry
The simulation is performed in a spherical coordinate system
$(r,\theta,\varphi)$, but we also introduce the cylindrical coordinate system
$(R,Z,\varphi)$ that is useful for the analysis.
The radius ranges from $r_{in}=1$ to $r_{out}=10^{5}$ $\mathrm{au}$. The
radial axis is discretized over 576 cells. A first patch of 512 cells follows
a logarithmic progression from 1 to $10^{4}$ $\mathrm{au}$. The remaining
cells are distributed from $10^{4}$ to $10^{5}$ $\mathrm{au}$ with a stretched
spacing. The $\theta$ angle is mapped between $0$ and $\pi$ over $128$ cells.
Near the poles, $32$ cells (for each side) are spread on a stretched grid,
with increasing resolution towards the midplane. An additional 64 cells are
used from $\theta=1.27$ to $\theta=1.87$ with uniform spacing to ensure a
satisfying resolution in the equatorial region. The $\varphi$ coordinate
covers the full $2\pi$ with $64$ cells evenly distributed. The total size of
the computational domain is $576\times 128\times 64$.
The configuration reaches a maximum resolution of $\leavevmode\nobreak\
10^{-2}$ $\mathrm{au}$ in the $r$ and $\theta$ directions and
$\leavevmode\nobreak\ 10^{-1}$ $\mathrm{au}$ in the $\varphi$ direction at
$R=1\,\mathrm{au}$, that scale linearly with the radius around the midplane.
Overall, the Jeans length $\lambda_{J}$ is resolved by more than $20$ cells in
the radial and polar direction and at least $4$ cells in the azimuthal
direction.
The disc vertical extent is properly sampled with at least 10 cells per scale
height $H$ at the inner boundary, where $H=\epsilon R$ is the disc geometrical
scale height and assuming a canonical aspect ratio $\epsilon=0.1$. We checked
that the fiducial azimuthal resolution of $64$ cells is sufficient to
accurately capture non-axisymmetric perturbations by running a more resolved
test, for a shorter time, with $256$ azimuthal cells. We found no qualitative
difference between the two.
### 2.3 Equation of state
In our setup, we do not solve the energy equation. Instead, we prescribe a
barotropic equation of state (EOS) following Marchand et al. (2016). As our
spatial resolution is too coarse to capture the second hydrostatic core
formation, this EOS reduces to:
$T=T_{0}\sqrt{1+\left(\frac{n}{n_{1}}\right)^{2(\gamma-1)}}$ (8)
where $n$ is the gas particle density, $T_{0}$ is the initial gas temperature,
$\gamma=7/5$ is the adiabatic index and $n_{1}=10^{11}\,\mathrm{cm^{-3}}$ is
the critical gas particle density.
Consequently, our effective thermal behavior could be summarized in two
stages: an isothermal phase while $n<n_{1}$ followed by an adiabatic one. We
define the formation of the first hydrostatic core as the moment where the
central density reaches $n_{1}$. It corresponds to $t=0$ in our simulation.
### 2.4 Non-ideal diffusivities
The simulation takes into account Ohmic and ambipolar diffusions. To compute
the associated diffusivity coefficients, we compute the steady-state
abundances of the main charge carriers. For this, we use the chemical network
described in Appendix C. The network is solved using the code Astrochem (Maret
& Bergin 2015) for a range of the gas densities $\rho$ and the magnetic field
intensities $B$ (when relevant). The resulting diffusivities are stored in a
table and, for every timestep, we read the table and perform an interpolation
on-the-fly in each cell depending on the $\rho$ and $B$ value.
### 2.5 Boundary conditions, internal boundaries, and restart
The inner and outer boundary conditions are similar to a classic outflow
condition, in the sense that the material can only leave the domain in the
radial direction. The azimuthal magnetic field $B_{\varphi}$ is set to zero to
prevent the angular momentum from being artificially conveyed out of the
numerical domain via magnetic braking. The remaining quantities are just
copied in the ghost cells from the last active one.
In the $\theta$ direction, we use an ”axis” boundary condition. It is
specially designed to prevent the loss of magnetic field in the polar region
(see appendix of Zhu & Stone 2018). For the azimuthal direction, we set a
classic periodic boundary condition.
For the self-gravity solver, the boundary conditions are the same as for the
dynamical solver in the $\theta$ and $\varphi$ directions. In the radial
direction, the gravitational potential is set to zero at the outer boundary.
We define a specific ”origin” inner boundary condition that expands the grid
down to the center (see Appendix A.2).
We implemented three internal numerical boundaries, mainly to prevent the
timestep from dropping, while ensuring physical accuracy. These features
include an Alfvén speed limiter, diffusivity caps (following Xu & Kunz 2021a,
b), and an advection timestep limiter. A detailed discussion is provided in
Appendix D.
The full integration is performed following two steps. We first integrate the
problem assuming a 2D axisymmetric geometry (with a single azimuthal cell)
until just before the first core formation (this takes about one free-fall
time). The axisymmetric assumption allows us to save computation time during
the initial collapse in which the flow is quasi-axisymmetric. We then continue
the integration in full 3D geometry before the first core formation, for a
$100\,\mathrm{kyr}$ integration.
Because the first step is 2D, the initial conditions for the second step are
axisymmetric, which may prevent the emergence of non-axisymmetric
perturbations. To alleviate this problem, we add a white noise of amplitude
$\pm 0.1\,u_{\varphi}$ to the azimuthal velocity when starting the 3D
simulation. We checked that the angular momentum is conserved when adding this
white noise.
### 2.6 Initial conditions
The initial conditions mostly follow Masson et al. (2016). We consider a
$M_{0}=1\,\mathrm{M_{\odot}}$ spherical cloud of initial radius
$r_{0}=2500\,\mathrm{au}$ and uniform particle density $n_{0}\simeq 2\times
10^{6}\,\mathrm{cm^{-3}}$. It is embedded in a $100$ times more diluted halo
of radius $10^{5}\,\mathrm{au}$. The associated free-fall time is
$t_{ff}=\sqrt{3\pi/32G\rho_{0}}\approx 22.1\,\mathrm{kyr}$, with $\rho_{0}$
the initial uniform gas mass density111$\rho_{0}\approx 9\times
10^{-18}\,\mathrm{g\,cm^{-3}}$ ..
The thermal over gravitational energy ratio is
$\alpha=(5r_{0}c_{s0}^{2})/(2M_{0}G)=0.25$, corresponding to an initial
isothermal sound speed $c_{s0}\simeq 0.188\,\mathrm{km\,s^{-1}}$. The initial
temperature222We assume a mean mass per neutral particle $m_{n}=2.33m_{p}$,
corresponding to the composition of the solar nebula. is, therefore,
$T_{0}=c_{s0}^{2}m_{n}/k_{B}\simeq 10\,\mathrm{K}$.
The core is subject to solid body rotation with a ratio of rotational over
gravitational energy $\beta=(\Omega_{0}^{2}r_{0}^{3})/(3M_{0}G)=0.02$
corresponding to a rotation rate $\Omega_{0}\simeq 3.9\times
10^{-13}\,\mathrm{rad\,s^{-1}}$. One difference with Masson et al. (2016) is
that the background is also rotating, with a profile
$\Omega(r)=\Omega_{0}(r/r_{0})^{-2}$ for $r>r_{0}$ which corresponds to
constant specific angular momentum along (spherical) radial lines (following
Xu & Kunz 2021a).
Another difference is that the _whole_ domain is initially threaded by a
uniform vertical magnetic field $B_{0}$ (and not only the central core). We
set a mass-to-flux ratio333$\mu=\frac{M_{0}/(B_{0}\pi
r_{0}^{2})}{(M/\phi)_{cr}}$. The core is therefore supercritical ($\mu>1$).
$\mu=2$ in unit of the critical value for collapse
$(M/\phi)_{cr}=(3/2)(63G)^{-1/2}$ (Mouschovias & Spitzer Jr 1976) which
corresponds to $B_{0}\simeq 4\times 10^{-4}\,\mathrm{G}$.
## 3 Overall evolution
This section focuses on the qualitative properties of the run. First, we
present the behavior of the gas and attached magnetic field during the first
isothermal collapse phase and subsequent disc formation. Second, we examine
the disc secular evolution properties. Third, we look at the evolution of the
disc in terms of dynamics, size, and mass repartition.
### 3.1 From pre-stellar collapse to disc formation
Figure 1: Snapshots of the azimuthally averaged particle density (color) with
attached poloidal magnetic field lines (white contours) and poloidal velocity
stream (grey arrows). From left to right: the first snapshot is a large-scale
view focusing on the cloud morphology significantly before the first core
formation, while the three last snapshots zoom into the first core during and
after its formation.
We show in Fig. 1 a few snapshots of the azimuthally averaged gas particle
density with attached field lines and poloidal velocity stream slightly before
and just after the first hydrostatic core formation.
The first snapshot is a large-scale view of the collapsing core. It
illustrates how the magnetic field acts upon the infalling material:
vertically, the motion is aligned with the field and the gas is free-falling.
Radially, the misalignment generates a magnetic tension that slows the
collapse. The result is a flattening of the core, as well as a pinching of the
magnetic field lines that are dragged along the midplane by the gas. The clear
interplay between gas motions and field line deformation is a result of the
low diffusivities involved at this stage of the collapse. They are inefficient
at decoupling the gas and the magnetic field, which remains in a near-ideal
MHD regime.
The three last snapshots focus on what happens at a small scale, just after
the first hydrostatic core formation. As the core particle density increases,
it reaches the critical value $n_{1}$ (cf. Eq. (8)). The gas becomes
adiabatic, which provides thermal support to the core that stops collapsing
vertically (second snapshot). In the meantime, the radial collapse catches up
and drags the magnetic field which acquires an hourglass shape. A torus-like,
pressure-supported structure arises: it’s the first hydrostatic core formation
(third snapshot).
The large densities also increase the ambipolar and Ohmic diffusions. Inside
the first core, the gas and the magnetic field are decoupled, and angular
momentum accumulates. As a consequence, a small, rotationally-supported disc
settles (last snapshot). The newborn disc is fed by the remnant, vertically
pressure-supported gas that could never reach the radial hydrostatic
equilibrium. We refer to this midplane, pressure-supported gas as the ”pseudo-
disc” in the following. The ”envelope” denomination is more generic and
corresponds to any material that is not belonging to the disc or the seed.
### 3.2 Disc secular evolution properties
Figure 2: Azimuthally averaged surface density (top left), rotation rate (top
right), poloidal magnetic field intensity (bottom left), and plasma parameter
$\beta_{P}$ (bottom right) versus the radius, computed in the midplane. Each
color corresponds to one snapshot, the darker being the disc formation epoch,
while the lighter is associated with the later times of the simulation. In the
top right panel, the black dashed line is the theoretical Keplerian rotation
rate with $M_{\star}\approx 0.7M_{\odot}$.
In Fig. 2 we present the azimuthally-averaged midplane surface density
$\Sigma$, rotation rate $\Omega$, poloidal magnetic field intensity $B_{Z}$,
and plasma parameter $\beta_{P}$ as a function of the radius, starting from
the first core formation until the later times of the simulation.
We compute the surface density as $\Sigma\equiv\int_{0}^{\pi}\rho r\sin\theta
d\theta$, where the density is integrated along the polar angle. The density
barely contributes out of the disc, which itself covers a small $\theta$ range
around the midplane. Therefore, this polar integration in spherical
coordinates is a convenient approximation of the vertical integration in
cylindrical coordinates.
At $5\,\mathrm{kyr}$ for $R\lesssim 10\,\mathrm{au}$, the gas follows a steep
power-law starting from a flat, high $\Sigma$. Further out, it follows a
shallow power-law with low $\Sigma$. As time goes on, the steep power law
becomes shallower while the transition radius increases up to
$200\,\mathrm{au}$.
The rotation rate is compared with the Keplerian prediction
$\Omega_{K}=\sqrt{GM_{\star}}R^{-3/2}$ with $M_{\star}$ the seed mass and $R$
the cylindrical radius444$\Omega_{K}$ is derived taking $M_{\star}$ at
$50\,\mathrm{kyr}$, because $M_{\star}$ is roughly constant afterwards.. At
$5\,\mathrm{kyr}$ for $R\lesssim 10\,\mathrm{au}$, the gas is in Keplerian
rotation. Further out, it is sub-Keplerian. As time goes on, the Keplerian
transition radius increases up to $200\,\mathrm{au}$. Thus, the steep, inner
$\Sigma$ region is associated with rotation-supported material while the outer
shallow $\Sigma$ region is sub-Keplerian.
At $5\,\mathrm{kyr}$ for $R\lesssim 10\,\mathrm{au}$, the poloidal magnetic
field shows a plateau. Further out, it follows a power-law. The intensity of
the plateau decreases with time down to a few $\mathrm{mG}$ and the plateau
transition radius increases up to a few $10\,\mathrm{au}$. Initially, the
plateau is associated with the rotation-supported, steep $\Sigma$ region while
the power-law is associated with the pressure-supported, shallow $\Sigma$
region.
The plateau is characteristic of the non-ideal MHD regime, responsible for
decoupling the gas and $B_{Z}$, while the power-law indicates a near-ideal MHD
regime due to the gas being less dense. A slight bump is observed at the
transition radius and can be explained as follows: in the pseudo-disc region,
$B_{Z}$ is dragged to inner radii by infalling material. Reaching the non-
ideal region, most of this field cannot be conveyed any further and piles up.
Finally, the plasma parameter is defined as $\beta_{P}=P_{th}/P_{mag}$, where
$P$ is the thermal pressure and $P_{mag}=B^{2}/8\pi$ is the magnetic one.
Thus, $\beta_{P}\gg 1$ indicates a thermally-dominated gas, while
$\beta_{P}\ll 1$ indicates a magnetically-dominated one.
At $5\,\mathrm{kyr}$ for $R\lesssim 10\,\mathrm{au}$, it follows a steep
power-law starting from a high $\beta_{P}\approx 10^{5}$. Further out, the
profile is slowly increasing from $\beta_{P}\approx 1$ and stays close to this
limit value between the two regimes. Hence, there is a correlation between the
magnetized, high surface density, rotationally-supported gas, and the
thermally-dominated region.
As time goes on, the steep power law becomes shallower while the transition
radius increases up to $200\,\mathrm{au}$. The innermost region is even more
thermally-dominated, reaching $\beta_{P}\approx 10^{9}$. The outer region
becomes magnetically-dominated, with $\beta_{P}\approx 10^{-1}$. We note that
for any snapshot, the limit value $\beta_{P}\approx 1$ is located near the
transition radius in the three other profiles.
### 3.3 Dynamics, size and mass repartition
Figure 3: Gas surface density in the equatorial plane. From left to right:
three characteristic snapshots illustrating the successive behaviors of the
disc at $5$, $30$, and $70\,\mathrm{kyr}$ respectively. Figure 4: Top panel:
radius of the disc over time. The dash-dotted black lines emphasize the radii
where the mass accretion rate is inspected in the next panel. Middle panel:
mass accretion rate over time near the protostar ($R=5\,\mathrm{au}$, red) and
at the maximum stable outer radius ($R=60\,\mathrm{au}$, green). Dashed lines
correspond to expanding material. Bottom panel: protostar mass (orange), disc
mass (blue), the total mass of the disc-protostar system (grey), and envelope
mass (brown) over time. Data for the accretion rate are convolved in time
using a $10\,\mathrm{kyr}$ window (equivalent to $8$ orbits at
$100\,\mathrm{au}$).
The disc morphology is investigated in Fig. 3, which shows the gas surface
density in the equatorial midplane at $5$, $30$, and $70\,\mathrm{kyr}$. It
covers the different dynamical states experienced by the disc during the
secular integration: first, the disc is small and subject to spiral density
waves. Second, it smoothes out and builds up, apart from one large, streamer-
like spiral arm. Finally, the outer disc triggers new spirals that propagate
to the inner radii.
In the middle panel, the disc exhibits a slightly eccentric morphology. We
caution that this is probably a consequence of the fixed point mass
assumption. In principle, while accreting mass with linear momentum, the point
mass should move. Because this motion is not taken into account, there is a
non-zero indirect term from gravity which makes the disc eccentric. That being
said, we think it does not significantly affect the results, and the
eccentricity disappears afterwards (see right panel).
Figure 4 gives an overview of the evolution of the disc radius
($R_{\mathrm{d}}$), the accretion rate ($\dot{M}$) and the mass repartition
between the seed ($M_{\star}$), the disc ($M_{\mathrm{d}}$) and the envelope
($M_{\mathrm{env}}$) over the $100\,\mathrm{kyr}$ of the simulation.
The top panel follows the evolution of the disc radius $R_{\mathrm{d}}$. To
compute it, we assume that cells satisfying $u_{\varphi}\geq 0.9V_{K}$,
$u_{\varphi}>u_{R}$ and $u_{\varphi}>c_{s}$ are part of the disc ($V_{K}$ and
$c_{s}$ are the local Keplerian and sound velocities). The disc radius is then
defined as the azimuthal median of the outermost radii that satisfy this
criterion in the midplane.
As the disc forms after a few $\mathrm{kyr}$, its radius reaches
$10\,\mathrm{au}$ and immediately starts decreasing until $15\,\mathrm{kyr}$.
Then, it increases smoothly up to $60\,\mathrm{au}$ after $30\,\mathrm{kyr}$
where it remains constant until $45\,\mathrm{kyr}$. After this point, the
radius is subject to chaotic fluctuations around $60\,\mathrm{au}$ and remains
so until the end of the simulation.
The middle panel follows the evolution of $\dot{M}$ near the protostar
($R=5\,\mathrm{au}$) and at the maximum stable outer radius
($R=60\,\mathrm{au}$). We perform an azimuthal integration over $2\pi$ and a
vertical integration between $\pm H$. In the following, H always refers to the
disc geometrical scale height. Data are convolved in time using a
$10\,\mathrm{kyr}$ window (equivalent to $8$ orbits at $100\,\mathrm{au}$). We
caution that, in doing so, we smooth out short-timescale events occurring in
the disc to focus on secular events. For instance, the fact that the accretion
rate oscillates on the spiral dynamical timescale (Tomida et al. 2017) is
verified, but hidden due to the large smoothing window.
We first focus on $\dot{M}(R=5\,\mathrm{au})$, which gives a good proxy for
the accretion onto the protostar. As the disc forms, the protostar accretes at
a strong rate of a few $10^{-5}\,\mathrm{M_{\odot}\,yr^{-1}}$ that immediately
starts decreasing until $20\,\mathrm{kyr}$. There, it reverses with a negative
rate around $10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$, which means that gas is
expanding consistently with $R_{\mathrm{d}}$ and the protostar is no more
accreting. After $35\,\mathrm{kyr}$, the expansion episode stops and standard
protostar accretion is back. It is first small, with strong variations between
$10^{-8}$ and a few $10^{-7}\,\mathrm{M_{\odot}\,yr^{-1}}$. After
$45\,\mathrm{kyr}$ the range increases and stabilizes between a few $10^{-7}$
and $10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$.
For $\dot{M}(R=60\,\mathrm{au})$, the material is not part of the disc until
$30\,\mathrm{kyr}$ and only marginally belongs to the disc afterwards due to
radius variability, such that we essentially probe the pseudo-disc accretion.
As the disc forms, the pseudo-disc accretes at a strong rate of a few
$10^{-5}\,\mathrm{M_{\odot}\,yr^{-1}}$ that immediately start decreasing until
$30\,\mathrm{kyr}$. There, it reverses with a negative rate of a few
$10^{-7}\,\mathrm{M_{\odot}\,yr^{-1}}$. It is shortly and slightly expanding
because $R_{\mathrm{d}}$ stops growing there. After $35\,\mathrm{kyr}$, the
expansion episode stops and standard pseudo-disc accretion is back. Note that
the $\dot{M}(R=60\,\mathrm{au})$ is about one order of magnitude larger than
$\dot{M}(R=5\,\mathrm{au})$, indicating that the disc density structure is
still evolving and that proper steady-state has not yet been reached.
Finally, the bottom panel shows the evolution of the mass repartition between
the seed, the disc, and the envelope. $M_{\star}$ accounts for any mass
falling below $R_{in}$. $M_{\mathrm{d}}$ is computed by summing $\rho dV$ over
any cell matching the disc criterion. The envelope mass is what is left of the
initial $1\,\mathrm{M_{\odot}}$ cloud.
As the disc forms, $M_{\star}$ grows to $0.7\,\mathrm{M_{\odot}}$ until
$15\,\mathrm{kyr}$ and stagnates afterwards. In the meantime, $M_{\mathrm{d}}$
reaches a maximum $0.15\,\mathrm{M_{\odot}}$ and immediately starts decreasing
until $15\,\mathrm{kyr}$. Then, it increases smoothly to
$0.15\,\mathrm{M_{\odot}}$ after $30\,\mathrm{kyr}$ where it remains constant
until $45\,\mathrm{kyr}$. After this point, the disc mass keeps increasing
while oscillating and remains so until the end of the simulation. The final
disc mass is $0.25\,\mathrm{M_{\odot}}$.
$M_{\mathrm{env}}$ is rapidly decreasing until $10\,\mathrm{kyr}$. Most of the
lost mass ends up in the seed, the rest becomes part of the disc. After
$5\,\mathrm{kyr}$, the envelope mass becomes negligible compared to
$M_{\star}$, and the decaying slope is shallower. The envelope is mainly
accreted onto the disc. After $80\,\mathrm{kyr}$, $M_{\mathrm{env}}$ becomes
negligible compared to $M_{\mathrm{d}}$.
## 4 Accretion history
In this section, we study the accretion history of the disc based on the
evolution of key physical quantities (surface density, magnetic field…). We
isolate the main driving mechanisms for accretion, address their relevance
throughout the disc evolution, and derive a comprehensive scenario over three
accretion phases.
### 4.1 Driving accretion mechanisms
Figure 5: From top to bottom we focus on $t=5$, $30$ and $70\,\mathrm{kyr}$
respectively. Each panel presents $G_{R\varphi}$ (green), $M_{Z\varphi}$
(blue) and $H_{Z\varphi}$ (red) versus the radius. Solid and dashed lines are
associated with positive and negative stresses respectively. The black dotted
line is the corresponding disc radius. Data are convolved in time using a
$10\,\mathrm{kyr}$ window (equivalent to $8$ orbits at $100\,\mathrm{au}$).
Figure 6: Mass accretion rate versus the radius. Each color corresponds to one
snaphsot at $5$, $30$ and $70\,\mathrm{kyr}$ respectively. The lighter, the
later. Dashed lines represent the disc radius associated with each epoch. Data
are convolved in time using a $10\,\mathrm{kyr}$ window (equivalent to $8$
orbits at $100\,\mathrm{au}$).
Protoplanetary discs are rotationally supported structures. In this context,
accretion is only possible if there are one or several mechanisms capable of
extracting angular momentum from the gas.
To properly account for each of these mechanisms, we derive the conservation
of angular momentum in the cylindrical coordinates system $(R,Z,\varphi)$ for
the case of a self-gravitating, magnetized rotating disc (adapted from Lesur
2021a):
$\overline{\rho u_{R}}\partial_{R}\left(\Omega
R^{2}\right)+\frac{1}{R}\partial_{R}\left(R^{2}\underbrace{\overline{\Pi_{R\varphi}}}_{\text{radial
stress}}\right)+R\underbrace{\left[\langle\Pi_{Z\varphi}\rangle\right]_{-H}^{H}}_{\text{surface
stress}}=0$ (9)
with
$\displaystyle\Pi_{R\varphi}=\rho
u_{R}V_{\varphi}+\frac{g_{R}g_{\varphi}}{4\pi
G}-\frac{B_{R}B_{\varphi}}{4\pi}$ (10) $\displaystyle\Pi_{Z\varphi}=\rho
u_{Z}V_{\varphi}+\frac{g_{Z}g_{\varphi}}{4\pi
G}-\frac{B_{Z}B_{\varphi}}{4\pi}$ (11)
where $V_{\varphi}=u_{\varphi}-\frac{1}{2H}\overline{u_{\varphi}}$, g is the
gravitational field and
$\begin{array}[]{ccc}\langle
X\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}Xd\varphi&\text{ and
}&\overline{X}=\int_{-H}^{H}\langle X\rangle dZ\end{array}$ (12)
for any quantity $X$ and $[X]_{-H}^{H}=X(Z=H)-X(Z=-H)$.
There are therefore six mechanisms acting upon the angular momentum transport
in our case: hydrodynamical transport (first term in Eqs. (10)-(11)),
gravitational transport (second term) and magnetic transport (third term),
each of them generating both a radial stress (Eq. (10)) and a surface stress
(Eq. (11)).
Among these quantities, we want to focus on the three main levers identified
in previous works (Xu & Kunz 2021a, b) as the preponderant mechanisms at stake
for such massive, embedded and magnetized discs: the radial gravitational
stress $G_{R\varphi}$, the magnetic braking $M_{Z\varphi}$ (corresponding to
the surface magnetic stress), and the surface hydrodynamical stress
$H_{Z\varphi}$. Each of them is labeled as follows:
$\displaystyle
G_{R\varphi}\equiv\frac{1}{R}\partial_{R}\left[R^{2}\frac{\overline{g_{R}g_{\varphi}}}{4\pi
G}\right]$ (13) $\displaystyle M_{Z\varphi}\equiv-R\left[\frac{\langle
B_{Z}B_{\varphi}\rangle}{4\pi}\right]_{-H}^{H}$ (14) $\displaystyle
H_{Z\varphi}\equiv R\left[\langle\rho u_{Z}V_{\varphi}\rangle\right]_{-H}^{H}$
(15)
Figure 5 presents a comparison of each stress for three snapshots (with time
increasing from top to bottom). A positive torque is associated with the
extraction of angular momentum from the disc while a negative torque brings
angular momentum to the disc.
At $5\,\mathrm{kyr}$, the gravitational stress is positive and dominant in the
disc. It transports angular momentum from the inside out. In the meantime,
magnetic braking is positive and dominant in the pseudo-disc. It extracts
angular momentum from the gas. The hydrodynamical stress can be significant
but is never dominant in the innermost $200\,\mathrm{au}$.
At $30\,\mathrm{kyr}$, the hydrodynamical stress is negative and dominant in
the disc and inner pseudo-disc. It brings angular momentum from the envelope.
In the meantime, magnetic braking is positive and dominant in the outer
pseudo-disc. Yet, its intensity has significantly decreased. The gravitational
stress can be significant but is never dominant in the innermost
$200\,\mathrm{au}$.
At $70\,\mathrm{kyr}$, the hydrodynamical stress is negative and dominant in
the inner disc. In the meantime, the gravitational stress is positive and
dominant in the outer disc and the pseudo-disc. The magnetic braking is
essentially positive but never significant in the innermost
$200\,\mathrm{au}$.
The relative importance of each stress can be connected with Fig. 6, which
shows the accretion rate versus the radius for the same snapshots. $\dot{M}$
is computed as in the middle panel of Fig. 4. A positive rate corresponds to
accretion while a negative one is associated with expansion.
At $5\,\mathrm{kyr}$, both the disc and the pseudo-disc efficiently accrete at
a few $10^{-5}\,\mathrm{M_{\odot}\,yr^{-1}}$. At $30\,\mathrm{kyr}$, the disc
expands with $\dot{M}$ ranging between $-10^{-7}$ and
$-10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$ and the pseudo-disc accretes at
$10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$. The disc is therefore growing because
of inside-out expansion and accumulation at the edge. At $70\,\mathrm{kyr}$,
the inner disc has no clear trend. It switches between accretion and
expansion. In the meantime, the outer disc and the pseudo-disc accrete at
$\dot{M}\sim 10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$.
Accretion therefore results from the relative importance of each stress in the
disc and the pseudo-disc. Understanding the secular evolution of these
stresses is the key to understanding the different accretion behaviors.
### 4.2 Secular accretion scenario
Figure 7: Spacetime diagrams of the surface density (top panel), poloidal
magnetic flux (middle panel), and Toomre parameter (bottom panel). Colors and
radii are in log scale. The dashed white line corresponds to the disc radius,
and dash-dotted white lines delimit the three accretion phases. Data were
convolved in time using a $10\,\mathrm{kyr}$ window (equivalent to $8$ orbits
at $100\,\mathrm{au}$). Figure 8: From left to right we focus on $t=5$, $30$,
and $70\,\mathrm{kyr}$ respectively, corresponding to each accretion phase.
Each panel presents the specific angular momentum $j$ in colors, with the
attached poloidal velocity flow $\@vec{V_{p}}$ as white quivers.
In the following, we derive the disc accretion scenario in three phases by
looking into the physical quantities underlying each stress.
First, spiral density waves are known to be an efficient angular momentum
carrier. In our simulation, they are observed simultaneously with high
accretion rates in the disc, making them a good candidate to explain
accretion. Typically, Gravitational Instabilities (GI) are triggered when
$M_{d}\gtrsim\epsilon M_{\star}$ (Armitage 2011, Eq. 12). The aspect ratio of
the disc is $\epsilon\lesssim 0.1$ during the simulation, while
$M_{\mathrm{d}}>0.1M_{\star}$. Thus, our disc lies in the adequate regime,
indicating that our spiral density waves are probably triggered by GI.
In order to characterize the stability of the disc with respect to its own
gravity, we use a simplified version of the Toomre parameter $Q$ (Toomre 1964;
Goldreich & Lynden-Bell 1965a; Goldreich & Lynden-Bell 1965b). Assuming that
the gas is in Keplerian rotation, the simplified Toomre parameter $Q_{K}$
writes as (Xu & Kunz 2021a, b):
$Q_{K}=\left(\frac{c_{s}\Omega_{K}}{\pi G\Sigma}\right)_{Z=0}$ (16)
Many studies discuss the critical value for stability, which depends on the
assumptions on the disc thickness or the perturbations linearity (Toomre 1964;
Goldreich & Lynden-Bell 1965a; Goldreich & Lynden-Bell 1965b; Gammie 2001;
Wang et al. 2010). Based on these works, we expect the disc to be unstable
when $Q_{K}\sim 1$ while keeping in mind that the lower $Q_{K}$, the more
likely the GI.
Second, the magnetic braking is a function of the poloidal magnetic field.
Hence, the braking efficiency is controlled by the amount of poloidal magnetic
flux stored in the gas. It is computed from the magnetic vector potential
$\mathbf{A}$:
$\phi_{B}=R\langle A_{\varphi}(Z=0)\rangle$ (17)
To complete our diagnostics, the hydrodynamical vertical stress is a function
of the specific angular momentum $j=Ru_{\varphi}$ transported by the poloidal
velocity flow $\@vec{V_{p}}=u_{R}\@vec{e_{R}}+u_{Z}\@vec{e_{Z}}$.
We present in Fig. 7 a series of spacetime diagrams connecting the surface
density of the gas (top panel), the poloidal magnetic flux (middle panel), and
the Toomre parameter (bottom panel). Fig 8 shows the specific angular momentum
with the attached poloidal flow. We discuss below the different phases that we
find from these figures.
#### 4.2.1 Phase I: a small, GI-driven disc
From the top panel of Fig. 7, we see that until $15\,\mathrm{kyr}$, a high
surface density region concentrates to the innermost $10\,\mathrm{au}$
corresponding to the disc. At the transition, there is a sharp drop in
density, and the pseudo-disc is left with a low surface density. In parallel,
the middle panel shows that a lot of magnetic flux is stored in the gas,
especially within the pseudo-disc region, while the bottom panel presents a
Toomre parameter that is of the order of unity in the disc region, which is,
therefore, GI unstable.
In addition, the left panel of Fig. 8 shows that the pseudo-disc provides
material to the disc that has low specific angular momentum. At intermediate
altitudes, the envelope provides material with higher $j$ but does not reach
the inner regions corresponding to the disc. At high altitudes, it provides
material to the disc through its surfaces, but with low specific angular
momentum.
Thus, in the first phase, the disc feeds the protostar thanks to GI-triggered
spiral density waves that efficiently remove angular momentum at low radii.
The instability is itself sustained by the mass influx from the pseudo-disc,
driven by a powerful magnetic braking. There is a specific angular momentum
contribution from the pseudo-disc/envelope to the disc, but it is not
significant.
#### 4.2.2 Phase II: disc expansion fed by the envelope
From the top panel of Fig. 7, we see that between $15$ and $45\,\mathrm{kyr}$,
the surface density increases at $R\gtrsim 10\,\mathrm{au}$, while the disc
grows from inside out. This is synchronous with a continuous outwards
advection of the magnetic flux in the middle panel. We emphasize that at
$10\,\mathrm{au}$, $\phi_{B}$ decreases by roughly $2$ orders of magnitude in
the $20$ first $\mathrm{kyr}$. In the meantime, from the bottom panel, the
Toomre parameter exhibits quite complex behavior. For $R\lesssim
10\,\mathrm{au}$, $Q_{K}$ is close to unity. This trend, along with the
persistence of two low $Q_{K}$ rings until the end of the simulation, predicts
that the inner disc should trigger spiral density waves which we do not
observe for these times and for these radii. Instead, two persistent ring-like
features are observed in the surface density. Such rings are ubiquitous in
self-gravitating disc simulations (Durisen et al. 2005; Boley et al. 2006;
Michael et al. 2012; Steiman-Cameron et al. 2013, 2023) and are believed to be
a common product of GI. Between $10$ and $60\,\mathrm{au}$, a significant
decrease down to $Q_{K}\sim 5$ is observed, as a response to the surface
density increase, but this is not enough to enter a GI regime. The disc
therefore smoothes out.
In addition, the middle panel of Fig. 8 shows that the pseudo-disc has become
more efficient at providing angular momentum to the disc. At intermediate and
high altitudes, the envelope now provides material with a large specific
angular momentum, among which a significant part reaches the disc.
Thus, in the second phase, the disc stabilizes and smoothes out. It can’t
accrete, since no angular momentum transport mechanism is efficient enough in
this phase. On the contrary, it gains angular momentum from the envelope. The
net result is a radial expansion of the disc, and the surface density power
law becomes shallower. In the process, the accumulated magnetic flux is
advected outwards in a flux-freezing manner, hence a decrease in the
magnetisation. This is discussed in Sect. 4.3.
#### 4.2.3 Phase III: GI-driven outer disc
From the top panel of Fig. 7, we see that the disc radius expansion is halted
around $45\,\mathrm{kyr}$ and stabilizes at $60\,\mathrm{au}$ until the end of
the simulation. In the meantime, the surface density front stops around
$100\,\mathrm{au}$, with a slight tendency to decrease by the end of the
simulation. The flux outwards advection in the middle panel stops as well, and
there is even some late inwards advection in the disc. In the bottom panel, at
roughly $55\,\mathrm{kyr}$, $Q_{K}$ becomes of the order of unity in the
outermost part of the disc. The low value propagates to inner radii
afterwards, until the end of the simulation.
In addition, the right panel of Fig. 8 shows that the pseudo-disc starts to
accrete low $j$ material again. At intermediate altitudes, the specific
angular momentum is the strongest, near the outer disc surfaces. From high
altitudes, the envelope still provides material with significant $j$ to the
inner disc.
Thus, in the third phase, the disc does not receive enough angular momentum to
keep expanding. The pseudo-disc is still braked and accretes onto the disc.
The surface density profile steepens at the disc edge and triggers a new GI
producing new spirals that propagate to inner radii. Hence the outer disc can
accrete. This final state holds for the remaining $55\,\mathrm{kyr}$
suggesting that the disc ends up in a GI-regulated state. We explore this idea
in the following.
### 4.3 Flux-freezing during disc expansion
Figure 9: Ideal magnetic flux velocity transport versus the radius. Each color
corresponds to one snapshot at $5$, $30$, and $70\,\mathrm{kyr}$ respectively.
The lighter, the later. Dashed lines represent the disc radius associated with
each epoch. Data are convolved in time using a $10\,\mathrm{kyr}$ window
(equivalent to $8$ orbits at $100\,\mathrm{au}$).
In Sect. 4.2.2, we find that while the disc is expanding, the magnetic flux is
advected along with the gas consistently with a flux-freezing behavior. Here
we discuss whether advection is indeed the dominant contribution to the
magnetic field transport during the disc secular evolution. We define the
midplane, azimuthally-averaged, ideal, poloidal magnetic flux velocity
transport $V_{B}^{id}$ as (adapted from Lesur 2021b):
$V_{B}^{id}=\frac{\langle\mathcal{E}^{id}_{\varphi}(Z=0)\rangle}{\langle
B_{Z}(Z=0)\rangle}$ (18)
where $\mathcal{E}_{\varphi}^{id}=u_{R}B_{Z}$ comes from the first term of Eq.
(7). Physically, it corresponds to the magnetic flux variation associated to
advection.
Figure 9 presents $V_{B}^{id}$ versus the radius for each phase, along with
the location of the disc radius. It is normalized by the Keplerian velocity
$V_{K}$. A positive velocity is associated with outwards transport while a
negative velocity is associated with inwards transport. In the following, we
compare the prediction for the flux transport from advection only with the
actual evolution from Fig. 7, middle panel. If they match, we conclude that
the advection is mainly responsible for the flux transport, i.e the gas
exhibits a flux-freezing behavior.
At $5\,\mathrm{kyr}$, the ideal transport predicts strong ($\approx-0.1$)
inwards advection for the flux inside the disc, while we see it starts
spreading in Fig. 7 (middle panel, phase I). Thus, diffusive transport
dominates over advection. At $30\,\mathrm{kyr}$, the ideal transport predicts
significant ($\approx 0.01$) outwards advection for the flux inside the disc,
which is consistent with the flux recession in Fig. 7 (middle panel, phase
II). Thus, the flux is advected by the spreading disc. At $70\,\mathrm{kyr}$,
the ideal transport predicts low ($<0.001$) outwards advection in the inner
smooth disc and a slightly stronger ($>0.001$) outwards advection in the outer
spiral-driven disc, while the flux seems slightly advected back inwards on the
long term in Fig. 7 (middle panel, phase III). This discrepancy suggests that
accretion is not the main driver of flux transport in this phase.
Thus, we conclude that diffusion is responsible for flux leaking essentially
during phase I and III. Conversely, during phase II, advection overcomes
diffusion and the field is just diluted in the expanding disc.
### 4.4 A GI-regulated final secular evolution
Figure 10: Disc surface density versus radius. Each color focuses on one
accretion phase at $5$, $30$, and $70\,\mathrm{kyr}$ respectively. The
lighter, the later. The black dashed line is the predicted critical surface
density from Eq. (19).
In Sect. 4.2, we conclude that the disc angular momentum process is dominated
by GI-driven spiral density waves when accreting. Self-gravitating discs are
prone to enter a marginally unstable, self-regulated gravito-turbulent state
where the Toomre parameter is maintained around the critical value $Q\sim 1$
(Gammie 2001).
From Eq. (16), $Q_{K}$ is controlled by two main levers : surface density
$\Sigma$ and temperature $T$ (through $c_{s}=\sqrt{k_{B}T/m_{n}}$). Yet, we
use a barotropic EOS, such that $T$ is set by the density. In this specific
case, Xu & Kunz (2021a) argues that the disc stability is controlled by the
surface density alone. It can enter GI through $\Sigma$ increase, as a result
of mass influx from the environment. Once unstable, such a disc can be brought
back into stability only by lowering $\Sigma$, if authorized to spread or to
significantly accrete.
To probe this phenomenon, we derive a critical surface density $\Sigma_{cr}$
as a function of the radius, above which the disc should become unstable :
$\Sigma_{cr}(R)=\Sigma_{0}R^{(\gamma+2)/(\gamma-3)}=\Sigma_{0}\left(\frac{R}{1\,\mathrm{au}}\right)^{-2.125}$
(19)
where
$\Sigma_{0}=\left[\frac{c_{s,0}}{\pi}\sqrt{\frac{M_{\star}}{G}}(\epsilon\rho_{1})^{(1-\gamma)/2}\right]^{2/(3-\gamma)}\approx
2\times 10^{5}\,\mathrm{g\,cm^{-2}}$ (20)
with $\rho_{1}=n_{1}\cdot m_{n}$ the critical mass density for which the gas
becomes adiabatic.
$\Sigma_{cr}$ is calculated from Eq. (16) with the following assumptions :
* •
$Q_{K}=1$.
* •
$c_{s}=c_{s,0}\left(\frac{\rho}{\rho_{1}}\right)^{(\gamma-1)/2}$ (adiabatic
regime).
* •
$\rho=\frac{\Sigma}{H}=\frac{\Sigma}{\epsilon R}$.
Most of the data used for the calculation are detailed in Sects. 2.3 and 2.6,
and we take $\gamma=1.4$, $\epsilon=0.1$ and $M_{\star}=0.7\,M_{\odot}$.
The critical surface density is reported as a dashed black line in Fig. 10,
along with the measured disc surface density for three snapshots spanning over
each phase.
At $5\,\mathrm{kyr}$, the steep power-law at $R\lesssim 10\,\mathrm{au}$
matches the critical value between $2$ and $4\,\mathrm{au}$, and stands below
further out. This is consistent with the spiral-driven disc in phase I. At
$30\,\mathrm{kyr}$, the power-law spreads, while staying below $\Sigma_{cr}$.
This is consistent with the smooth discs in phase II. At $70\,\mathrm{kyr}$,
the inner disc stands below the critical surface density while the outer disc
saturates at $\Sigma_{cr}$, around which it oscillates. This is consistent
with the spiral-driven outer disc in phase III.
Hence, any deviation from the stability regime enforced by self-gravity leads
to negative feedback that promotes accretion. In this sense, the disc is
shaped by self-gravity. In the case where a sustained influx of material
increases locally the surface density, the disc enters a self-regulated state
where $R_{\mathrm{d}}$ stabilizes around $60\,\mathrm{au}$. This emphasizes
the role of the environment interacting with the disc.
## 5 Interaction with the parent molecular cloud : a large-scale accretion
streamer
The most striking evidence of an interaction between the molecular cloud and
the protostar-disc system is the appearance of a large-scale, streamer-like
spiral arm driving asymmetric infall between the remnant cloud and the central
protostar-disc system. Here, we discuss the accretion streamer morphology and
kinematics and investigate on how it connects to the central system. We also
compute some observational properties.
### 5.1 Streamer spatial and velocity distribution : a sheet-like morphology
Figure 11: Top row: large-scale equatorial slice of the gas surface density
with associated equatorial velocity flow (white quivers). A dashed white line
indicates the azimuth of the main streamer, for which poloidal slices are
performed. Middle row: large-scale poloidal slice of the gas particle density
with associated poloidal velocity flow (white quivers). Bottom row: large-
scale poloidal slice of the gas poloidal velocity, normalized by the free-fall
velocity, with associated poloidal velocity flow (white quivers). Each column
corresponds to a late time snapshot, respectively at $60$ (left column) and
$70\,\mathrm{kyr}$ (right column). In each plot, a line-integrated convolution
treatment is applied to emphasize the gas streamlines. The innermost
$100\,\mathrm{au}$ are masked by a grey circle to allow for adequate contrast.
Figure 12: Large-scale equatorial slice at $t=60\,\mathrm{kyr}$. Colors are
the plasma parameter $\beta_{P}$.
Figure 11 presents a large-scale equatorial slice of the gas surface density
(top row) along with poloidal slices of the gas particle density (middle row)
and poloidal velocity (bottom row) performed at the azimuth of the streamer
for two late time snapshots. The poloidal velocity is normalized by the free-
fall velocity $V_{ff}=\sqrt{2}V_{K}$.
Focusing on the top row, we see that the environment of the protostar-disc
system is divided between a low surface density ”bubble” and an extended
region of growing surface density that culminates in an azimuthally localized
channel of gas at lower radii. This is the main accretion streamer, towards
which the equatorial velocity flow is converging. The streamer structure
extends up to approximately $1000\,\mathrm{au}$ and connects to the disc. A
second converging flow is observed in the low-density ”bubble”, corresponding
to an additional fainter streamer. The streamers rotate between the two
snapshots.
The middle row shows that the gas particle density is vertically stratified,
with densities ranging between $10^{6}\,\mathrm{cm^{-3}}$ close to the polar
axis and $10^{8}\,cm^{-3}$ in the midplane. Overall, the streamer converges in
a sheet-like morphology that is even better identified when represented in
three dimensions 555A 3D model of the streamer at $70\,\mathrm{kyr}$ based on
particle density contours is available on
https://sketchfab.com/3d-models/streamer-1407-fid-66107da9c4854078aadac140de9f4e73..
Finally, the bottom row illustrates two different dynamical behaviors. Far
from the midplane, the flow is near free-fall. The gas is channeled towards
lower radii at constant velocity and falls directly above (and below) the
disc. Near the midplane, the large-scale velocity is smaller. A gradient is
observed towards lower radii as the gas catches up with more elevated
material, reaching near free-fall velocity. In this case, the gas falls
directly onto the disc edge. The associated velocity variation suggests that
material is shocking in this region.
In Fig. 12, we present a large-scale equatorial slice at $t=60\,\mathrm{kyr}$
showing the plasma parameter $\beta_{P}$. Interestingly, the streamer and the
central protostar-disc system are characterized by $\beta_{P}>1$, indicating
that the gas in these regions is thermally-dominated. On the contrary, the
surrounding low-density ”bubble” is characterized by $\beta_{P}<1$, indicating
that the gas is magnetically-dominated. This configuration points towards a
magnetic origin for the streamer formation, such as the interchange
instability proposed by Mignon-Risse et al. (2021) in the context of massive
star formation or the magnetic process discussed in Tu et al. (2023) for the
collapse of a turbulent, low-mass protostellar core.
### 5.2 Connecting to the central system : looking for shock signatures
Figure 13: Three-dimensional representation of the disc and streamer with two
attached streamlines at time $70\,\mathrm{kyr}$. Red surfaces correspond to
the disc, while green surfaces represent the streamer. Both streamlines start
at $R=350\,\mathrm{au}$ and $\varphi=\varphi_{streamer}$ with respectively
$Z=0$ (orange solid line) and $Z=100\,\mathrm{au}$ (blue solid line). Black
dots indicate shocks (or rarefactions) along the streamlines while the grey
dot corresponds to a density transition (see Fig. 14 for their
identification). For an animated version of the figure with a large-scale
visualisation of the streamer, see https://cloud.univ-grenoble-
alpes.fr/s/ZNwHrbWg8A24Trb. Figure 14: Top : particle density along the
streamlines starting at $Z_{0}=0\,\mathrm{au}$ and $Z_{0}=100\,\mathrm{au}$
respectively. Bottom: gas velocity projected along each streamline. Colors and
dots are the same as in Fig. 13.
Figure 13 provides a three-dimensional representation of the disc and streamer
with two attached streamlines at time $70\,\mathrm{kyr}$. The disc and
streamer are displayed for representation purposes. In this plot, cells
associated with the disc must have gas particle density over
$10^{9}\,\mathrm{cm^{-3}}$. The streamer corresponds to cells where $n\geq
10^{6}\,\mathrm{cm^{-3}}$ and $u_{r}<0$ to ensure we focus on infalling
material belonging to the parent molecular cloud. We additionally require
$r>200\,\mathrm{au}$ to exclude the central system. Streamlines are integrated
with starting points of cylindrical radius $R_{0}=350\,\mathrm{au}$ and
azimuth $\varphi_{0}=\varphi_{streamer}$, with respectively $Z_{0}=0$ and
$Z_{0}=100\,\mathrm{au}$ to probe the gas in the midplane and at elevated
location.
The midplane streamline remains in the streamer and the midplane until it
reaches the large-scale spiral arm where it is abruptly deflected (see Fig.
14) and entrained in the spiral motion. In contrast, the elevated streamline
is channeled directly onto the innermost part of the disc, if not directly
falling onto the seed (see the animated version of the plot for a better
understanding).
The question of the shock is addressed by Fig. 14. The top panel shows the gas
particle density in the streamline as a function of the position on the
streamline while the bottom panel is the normalized gas velocity parallel to
the streamline $V_{\parallel}/V_{ff}$.
For the midplane streamline, a first shock signature is observed at
$100\,\mathrm{au}$ with both discontinuities in density and velocity
consistent with the encounter between the streamer and the spiral arm. The
density jumps from roughly $10^{8}$ to almost $10^{10}\,\mathrm{cm^{-3}}$ and
the gas loses more than half of its velocity. A fainter secondary rarefaction
signature is observed around $170\,\mathrm{au}$ where density and velocity
drop again. We caution that the gas is not in a steady state, hence the
streamline may not be representative of the gas kinematics after the first
deflection. For the elevated streamline, we observe a sharp increase in
density corresponding to the moment where the gas connects with the disc.
However, the velocity along the streamline remains near free-fall at any
position, and only a faint, smooth velocity gradient is observed. This
corresponds to a smooth density transition, rather than a shock signature.
### 5.3 Streamer mass and infall rate : impact on protostar-disc accretion
To close our analysis of the streamer, we compute its mass and infall rate. We
stay as close as possible to the computation method provided by Valdivia-Mena
et al. (2022).
We stick to the definition in Sect. 5.2 to flag cells belonging to the
streamer. The mass is then computed by summing $\rho dV$ over each cell. We
obtain $M_{streamer}\approx 0.02\,\mathrm{M_{\odot}}$ corresponding to
$3\mathrm{\%}$ of the protostar’s mass and $10\mathrm{\%}$ of the disc mass at
$70\,\mathrm{kyr}$.
The infall rate $\dot{M}_{in}$ is the mass rate at which the streamer is
infalling onto the protostar-disc system. It is not to be confused with the
accretion rate $\dot{M}_{\star}$ directly onto the protostar. Valdivia-Mena et
al. (2022) compute it through $\dot{M}_{in}=M_{streamer}/t_{ff,streamer}$
where $t_{ff,streamer}$ is computed using a streamline model. On each point of
the streamline, they interpolate the velocity $V_{\parallel}$ and length
variation $dl$. They can therefore compute the time needed to reach the disc.
We do the same using a streamline starting at
$(R,Z,\varphi)=(R_{streamer},0,\varphi_{streamer})$, with $R_{streamer}\approx
1400\,\mathrm{au}$ the outermost radius of the streamer. We then sum
$dl/V_{\parallel}$ along the streamline to get $t_{ff,streamer}\approx
13\,\mathrm{kyr}$ giving $\dot{M}_{in}\approx 1.5\times
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$. Compared to the accretion rate
$\dot{M}_{\star}\equiv\dot{M}(R=5\,au)\approx 5\times
10^{-7}\,M_{\odot}\,yr^{-1}$ at the same time, we get a ratio of infall to
accretion of $\dot{M}_{in}/\dot{M}_{\star}\approx 3$.
## 6 Discussion
In this section, we confront our disc secular evolution and final GI-regulated
state with observations. We also discuss the compatibility of our accretion
streamer with what is observed by comparing its properties with the
literature.
### 6.1 Disc secular evolution and GI self-regulation
In the first $15\,\mathrm{kyr}$ of its life, the newborn disc is small,
compact, and magnetized. It lies in the non-ideal regime and accretes through
GI-driven spiral density waves. This is because efficient magnetic braking
promotes accretion in the well-coupled MHD pseudo-disc, which in return
increases the disc mass and makes it unstable. During the second phase,
between $15$ and $45\,\mathrm{kyr}$, the accretion in the pseudo-disc becomes
less efficient and the disc can stabilize. In the meantime, the envelope
provides high angular momentum material to the disc that can therefore expand.
As a result, the accumulated magnetic flux is advected outwards and
magnetisation decreases. In the third phase, lasting for the remaining
$55\,\mathrm{kyr}$, expansion is halted in the disc and mass accumulates at
the edge from the pseudo-disc. At some point, the outer disc is massive enough
to be unstable and the disc’s final state is GI-regulated.
As a complement, we would like to emphasize an interesting result regarding
the plasma parameter $\beta_{P}$. In subsection 4.2, we find that during phase
II the disc is expanding, and its magnetic flux is advected along with the
gas. In such a case, assuming the disc mass $M_{d}$ and poloidal magnetic flux
$\phi_{d}$ to be constant, one can show that $\beta_{P}(R_{d})\propto
R_{d}^{2}$. This is verified in Fig. 2, bottom right panel. At
$5\,\mathrm{kyr}$, $\beta_{P}(R_{d})\approx 10$. Later on, once the disc
radius has increased by roughly one order of magnitude, the associated plasma
parameter is $\beta_{P}(R_{d})\approx 10^{3}$. The verified dependency is a
confirmation that once a given amount of flux is stored in the disc, it
follows the gas evolution, and magnetisation evolves accordingly.
Numerically speaking, our collapse and disc’s early evolution are in line with
other works. Magnetic decoupling occurs by the time of the first core
formation leading to a plateau with $B_{z}\simeq 100\,\mathrm{mG}$ threading
the disc, as expected in core-collapse simulations including ambipolar
diffusion (Masson et al. 2016; Vaytet et al. 2018; Xu & Kunz 2021b). The
initial disc size of roughly $10\,\mathrm{au}$ is consistent with a magnetic
regulation (Hennebelle et al. 2016). A decreased magnetisation is then
observed for simulations that properly resolve the disc vertical extent (Xu &
Kunz 2021b). The disc becomes massive and resistive enough to be
gravitationally-regulated (Tomida et al. 2017; Xu & Kunz 2021a, b). The piling
up of magnetic field at the transition between the diffusive and ideal MHD
regimes is reminiscent of the magnetic wall proposed by Li & McKee (1996) and
observed also in Xu & Kunz (2021b), though with a fainter accumulation that
could be explained by the differences in the diffusivity tables.
On the observational side, resolution is often missing to infer the surface
density profile in class 0/I systems and we lack robust tracers to unveil the
magnetisation. The few studies available for the surface density find a power-
law index between $-1.7$ and $-1.5$ (Yen et al. 2014; Aso et al. 2015, 2017).
This is shallower than what we constrain to justify our GI-regulation
mechanism ($\approx-2$). On the other hand, Fiorellino et al. (2023) find that
many class I have a disc-to-star mass ratio above 0.1, which they claim is
typical for a GI-regulated disc. That being said, spirals are observed in only
a few class 0/I protostars (Tobin et al. 2018; Lee et al. 2020), while most of
these objects do not show structures (Ohashi et al. 2023). Yet, it is worth
mentioning that these discs can be optically thick, in which case the spirals
can be hidden.
The Class II stage is slightly better constrained. The review from Andrews
(2020) summarizes class II disc properties inferred from observations. The
constrained surface density profile has an index of $\approx-1$, again
shallower than our finding, making GI regulation unlikely. Conversely, Zeeman
measurements of the poloidal magnetic field give an upper limit of $\sim
1\,\mathrm{mG}$ at a few $10\,\mathrm{au}$ (Vlemmings et al. 2019) consistent
with our final disc magnetisation.
Thus, most of the properties of our evolved disc are consistent with current
observations, with the exception of the GI-regulated state, characterized by a
steep surface density profile and prominent spiral density waves, which seem
to be uncommon, even in young discs. The inclusion of internal heating, which
is missing in the current model, could help stabilize the disc. Yet, it
wouldn’t change the final picture. Indeed, in our simulation, the triggering
of GI is inevitable when magnetic braking is neutralized by an inefficient
magnetic coupling because no other mechanism is able to evacuate the disc
angular momentum as mass piles up.
This however ignores the role of MHD disc winds that could be launched from
the ionised surface layers of the disc. In our simulation, a large-scale
outflowing structure arises, but it is launched far from the disc surface. The
importance of such outflows with elevated launching points is discussed in
many core-collapse simulations (Machida & Hosokawa 2013; Tsukamoto et al.
2015; Masson et al. 2016; Marchand et al. 2020) and they are proposed as a
means to redistribute angular momentum. However, in our simulation, we find
that the outflow is launched from a low density region (a ”vacuum”) which
density is set by the numerical limiters (Alfvén and density floor).
Therefore, in our view, its origin remains numerical.
Hence, we conclude that no proper MHD disc wind is found in our simulation,
while many ”disc only” simulations including surface ionisation exhibit MHD
disc winds (Béthune et al. 2017; Bai & Stone 2017; Suriano et al. 2019; Lesur
2021b). These surface ionisation processes are due to stellar far UV and X-ray
photons that increase the ionisation fraction by orders of magnitude. These
effects are absent in our simulation, in which we only consider cosmic ray
ionisation in our chemical network. Hence it is tempting to associate the GI-
regulated disc we obtain to the lack of stellar irradiation at the disc
surface. This possibility should be investigated in the future.
### 6.2 Large-scale streamer driving accretion in the envelope
The long-term interaction of the envelope with the protostar-disc system leads
to the formation of an accretion streamer. It is composed of near free-fall
gas organized in a sheet-like configuration. It connects to the protostar-disc
system either by shocking onto the disc edge in the midplane or by smoothly
accreting onto the disc surfaces from higher altitudes. Quantitatively, the
streamer maximum radius is $\approx 1400\,\mathrm{au}$ with mass and infall
rates of $0.02\,\mathrm{M_{\odot}}$ and $1.5\times
10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$, corresponding to $3\mathrm{\%}$ of the
seed mass and $3$ times the protostar accretion respectively.
Asymmetric large-scale structures are ubiquitous in numerical core-collapse
models with sufficiently massive molecular clouds, accounting for turbulence
(Kuffmeier et al. 2017; Lam et al. 2019; Kuznetsova et al. 2019; Wurster et
al. 2019; Lebreuilly et al. 2021) or not (Mignon-Risse et al. 2021). The
resulting envelope is often messy, exhibiting streamers with filamentary or
sheet-like morphologies. Tu et al. (2023) propose a magnetic origin of the
streamer formation, which is also supported by Mignon-Risse et al. (2021) and
consistent with our work.
Accretion streamers have been observed both in deeply-embedded class 0 (Pineda
et al. 2020; Murillo et al. 2022) and in class I sources (Yen et al. 2019;
Valdivia-Mena et al. 2022; Hsieh et al. 2023). They are rotating, free-
falling, and connect to the disc (Pineda et al. 2020; Valdivia-Mena et al.
2022). The meeting point is either associated with a smooth transition between
the infalling streamer velocity and the Keplerian disc velocity (Yen et al.
2019; Pineda et al. 2020) or it can present a sharp velocity drop consistent
with shock tracing signatures (Valdivia-Mena et al. 2022). It is remarkable
that both these conclusions are in agreement with the kinematics in our
streamer.
A large variety of streamer sizes have been observed
($10^{3}-10^{4}\,\mathrm{au}$). The streamer mass ranges between $10^{-3}$ and
$10^{-1}\,\mathrm{M_{\odot}}$, and can be a significant fraction of the
protostellar mass ($0.1-10\mathrm{\%}$ of $M_{\star}$, Pineda et al. 2020;
Valdivia-Mena et al. 2022; Hsieh et al. 2023). The infall rate of the streamer
is $\dot{M}_{in}=10^{-6}\,\mathrm{M_{\odot}\,yr^{-1}}$, and can be of the same
order as the protostellar accretion rate $\dot{M}_{\star}$, if not higher.
Pineda et al. (2020); Valdivia-Mena et al. (2022); Hsieh et al. (2023) find
$\dot{M}_{in}/\dot{M}_{\star}\approx 1$, $5-10$ and $\geq 0.05$ respectively.
Our computations are all lying in the observed range.
## 7 Conclusion
We performed a 3D large timescale, non-ideal core collapse simulation in order
to probe the secular evolution of embedded protoplanetary discs while paying
specific attention to the magnetic field evolution and the disc’s long-term
interaction with the surrounding infalling envelope.
We follow the cloud collapse until the first hydrostatic core formation
leading to disc settling and integrate for an additional $100\,\mathrm{kyr}$.
The simulation lasts for about $20\,\mathrm{\%}$ of the class I stage (Evans
et al. 2009). Yet in the meantime, $90\,\mathrm{\%}$ of the envelope is
accreted by the seed or onto the disc, pointing towards an ending class I.
This faster evolution of the numerical model with respect to the observations
is probably a consequence of the dynamically unstable initial condition.
We achieve a resolution of $10$ cells per scale height (assuming
$\epsilon=0.1$) in order to properly capture magnetic field diffusion, field
line twisting, and GI-induced spirals. Our main results are:
1. 1.
The disc experiences three accretion phases. In particular, once the disc has
settled, magnetic braking mainly controls accretion in the pseudo-disc.
Conversely, self-gravity controls angular momentum transport in the disc
through spiral density waves triggered via Toomre instability. When gas is
expanding, it is thanks to the envelope providing high angular momentum
material through the disc surfaces.
2. 2.
During phase II, the disc expands while keeping its mass and flux roughly
constant. As a result, the plasma parameter at the disc edge follows
$\beta_{P}(R_{d})\propto R_{d}^{2}$. This dependency is evidence that the
initial amount of magnetic flux is conserved throughout the disc evolution,
and magnetisation evolves accordingly to the gas motion.
3. 3.
The disc ends up in a GI-regulated state maintaining $R_{\mathrm{d}}$ around
$60\,\mathrm{au}$. Its surface density profile is shaped by a critical surface
density radial profile of index $\approx-2$.
4. 4.
The early evolution of the disc reproduces well the results from core-collapse
models such as disc compacity, magnetic field decoupling, magnetic and later
GI regulation. However, no MHD disc wind is found in our simulation. This is a
natural outcome of efficient decoupling, yet it contrasts with disc-only
models that systematically find MHD disc winds. We conjecture that this could
be due to a lack of stellar ionisation processes.
5. 5.
The expanding Keplerian gas and the decrease of the magnetic field
qualitatively match class II observations. Yet, the observed power-law surface
density is too steep to trigger gravitational instabilities, and the presence
of Toomre-driven spirals is not supported by observations.
6. 6.
After $30\,\mathrm{kyr}$, the envelope is organized in a large-scale, sheet-
like accretion streamer that feeds the disc. It smoothly connects to its
surfaces from elevated locations and shocks onto the outer rim in the
midplane. It is a significant reservoir of mass whose infall rate is
comparable to the accretion rate of the protostar.
In conclusion, this secular run shows that the neutralisation of magnetic
braking due to an efficient decoupling leads the disc to a nearly pure
hydrodynamical state where GI is the only means to accrete. Consequently, the
disc is stuck in a GI-regulated state shaping its surface density profile and
final radius.
Yet, this scenario is difficult to support, due to the lack of observed
spirals in embedded systems. This suggests that current models overestimate
the importance of diffusion after disc formation. It would be interesting to
explore additional ionisation processes susceptible to recovering a
magnetically dominated accretion. The choice of initial conditions and the
impact of grain size may also act upon the diffusions. This will be the topic
of a forthcoming paper.
###### Acknowledgements.
We wish to thank our anonymous referee for valuable comments and suggestions
that improved the present paper. We are thankful to Matthew Kunz, Benoit
Commerçon and Patrick Hennebelle for fruitful discussions about the physics of
collapsing cores. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (Grant Agreement No. 815559 (MHDiscs)). This work was supported by the
”Programme National de Physique Stellaire” (PNPS), ”Programme National Soleil-
Terre” (PNST), ”Programme National de Hautes Energies” (PNHE), and ”Programme
National de Planétologie” (PNP) of CNRS/INSU co-funded by CEA and CNES. This
work was granted access to the HPC resources of IDRIS under the allocation
2023-A0140402231. Some of the computations presented in this paper were
performed using the GRICAD infrastructure (https://gricad.univ-grenoble-
alpes.fr), which is supported by Grenoble research communities. Data analysis
and visualisation in the paper were conducted using the scientific Python
ecosystem, including numpy (Harris et al. 2020), scipy (Virtanen et al. 2020)
and matplotlib (Hunter 2007).
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## Appendix A Self-gravity solver
### A.1 Implementation in Idefix
The Idefix code is upgraded with a self-gravity module. By resolving Eq. (6),
it infers the self-gravitational potential from the gas density distribution.
The Laplacian operator is discretized using second-order finite difference
with self-consistent boundary conditions. The resulting matricial system is
solved iteratively by a biconjugate gradient stabilized (BICGSTAB) algorithm.
It uses the Kokkos routines encapsulated in Idefix to handle parallelisation
(Trott et al. 2022; Lesur et al. 2023).
A Jacobian preconditioner $P$ can be used to fasten convergence. It is
designed as the diagonal part of the discretized Laplacian. It proved to be
efficient at easing convergence when the grid is irregular.
While the BICGSTAB algorithm is the one used in the present work, we
implemented two additional methods: a conjugate gradient (CG) and a minimum
residual (MINRES). There is a loss of generality when switching from one to
another: CG requires the operator to be symmetric positive definite, while
MINRES only assumes symmetry and BICGSTAB has no constraint. On the other
hand, improving generality increases computational cost and/or slows down
convergence. Hence, the implementation of several solvers provides an optimum
solution depending on the problem.
### A.2 ”origin” boundary condition
To circumvent the problem of the singularity at the center of the grid, we
implement a specific ”origin” inner boundary condition for the self-gravity
solver. It expands the grid radially inwards with a constant spacing so as to
entirely fill the inner ”hole” of the grid. We assume that the gas density is
zero in this extension of the domain.
The Poisson equation for the gravitational potential is then solved by the
self-gravity solver on this extended domain. Because the domain includes the
origin, there is no need to prescribe any inner boundary condition in this
approach.
### A.3 Validation tests
Figure 15: Static validation test: radial profile of the gravitational field
along $(\theta,\varphi)=(\pi/2,0)$. The radius is in log scale. The grid
configuration and boundary conditions are the same as our fiducial run, but we
halved the resolution on each axis, uniformely for each patch. The density
distribution is a uniform off-centered sphere of radius $1000$, located at
$(r,\theta,\varphi)=(3000,\pi/2,0)$. We set $\rho_{0}=\frac{3}{4\pi}$ and
$G\equiv 1$ and the quantities are displayed in code units. The blue line is
the theoretical profile, red dots are the computed data. Figure 16:
Convergence rate of the gravitational potential (blue dotted line) as a
function of the grid spatial resolution using the BICGSTAB method. It is based
on the off-centered sphere test. It exhibits second order spatial convergence.
Figure 17: Dynamic validation test: amplitude of density fluctuations in log
scale as a function of time following Jean’s instability for $\lambda_{J}=1/3$
where Poisson equation is solved at every timestep. All quantities are
dimensionless. The blue line is the theoretical prediction for the most
unstable mode ($\lambda=10\,\mathrm{u.c.}$), the red line is the computed
result.
We validate our implementation of self-gravity with two tests. The first one
is a static test and confirms that the gravitational potential retrieved from
the solver is accurate compared to the predicted one. The second is a dynamic
test, where the dynamical solver is coupled with the self-gravity solver. It
is based on Jean’s instability and makes sure we properly capture mode growth.
Figure 15 shows the radial profile along $(\theta,\varphi)=(\pi/2,0)$ of the
gravitational field in code units, inferred from an off-centered spherical,
uniform density distribution. We compute the gravitational field rather than
the potential to get rid of the integration constant and to make the
comparison easier.
We took the same grid configuration and boundary conditions as our fiducial
run. We halved the resolution on each axis, uniformely for each patch, in
order to reduce the computation time while keeping the grid anisotropy. The
density distribution is uniform inside an off-centered sphere of radius
$1000\,\mathrm{au}$. The center is located at
$(r,\theta,\varphi)=(3000,\pi/2,0)$. We emphasize that only the self-gravity
solver is tested here. Thus, as the physics is unimportant, we set
$\rho_{0}=\frac{3}{4\pi}$ and $G\equiv 1$, and the quantities are displayed in
code units. We set the convergence threshold to $10^{-5}$.
The theoretical and computed solutions are well matching thanks to the high
resolution and low convergence threshold. The convergence rate for this test
is about $600$ iterations, starting from a zero initial guess potential. After
this first ”burn-in” computation, the solver requires between 1 and $\sim
10^{2}$ iterations to converge, depending on the dynamics of the gas (it is
$10$ on average for our fiducial run). We checked that the scheme is second-
order accurate for the gravitational potential (see Fig. 16).
Figure 17 shows the amplitude of density fluctuations in log scale as a
function of time following Jean’s instability with the Poisson equation solved
every timestep. Both quantities are adimensionned by background density
$\rho_{0}$ and $c_{s0}/L$ respectively, $L=10\,\mathrm{u.c.}$ being the domain
size.
The setup is $1$D cartesian with periodic boundary conditions. The $x$
coordinate is meshed with $1000$ uniform cells and ranges between $0$ and $L$.
The density distribution is initialized with a Gaussian perturbation of
amplitude $10^{-4}$. Setup is adiabatic with $\gamma=5/3$, background density
and pressure are $\rho_{0}=3$, $P_{0}=1/\gamma$ in code units which gives
$\lambda_{J}=1/3$ ($G\equiv\pi$).
For a given wavelength $\lambda$, the expected growth rate is given by
$s=2\pi(c_{s}/\lambda)\sqrt{|1-(\lambda/\lambda_{J})^{2}|}$ with
$\lambda>\lambda_{J}$. The mode of the largest wavelength is therefore the
most unstable. Then, for $\lambda=L$, the theoretical growth rate is
$s_{th}=188.4\,c_{s}/L$, and the associated perturbation should dominate the
evolution of density perturbation.
This is confirmed by red dots, associated with the computed evolution of
density perturbation, which matches the theoretical linear prediction for the
most unstable mode (blue solid line) where $c_{s0}t/L$ is in the range
$0.1-0.4$. A linear regression in this portion of the slope gives
$s_{cpt}=186.6$ corresponding to a relative error of $0.9\mathrm{\%}$. Hence,
the dynamic is properly captured by our self-gravity solver.
## Appendix B Gravity step
The gravity calculation is performed just before the dynamical step. It
triggers the gravitational potential computation from various sources. In our
case, that includes self-gravity (see Appendix A) and point mass contribution.
In order to properly account for the _whole_ gravitational feedback, the
missing inner seed is assimilated to a point mass with:
$\phi_{pm}=-\frac{GM_{pm}}{r}$ (21)
where $M_{pm}$ and $\phi_{pm}$ are respectively the mass and associated
potential of the point mass.
The initial mass is the one enclosed in a uniform sphere of radius $r_{in}$
and density $\rho_{0}$. We sum up mass fluxes over the inner shell during the
integration to update the central mass according to mass transits. The net
gravitational potential used for the dynamical integration is then
$\phi_{g}=\phi_{pm}+\phi_{sg}$.
One can specify the frequency of the gravity step. Béthune & Rafikov (2019)
showed that updating the gravitational potential every $4$ dynamical timestep
does not substantially impact the system evolution (see their test on Jeans’
instability).
We conducted our own test and obtained a relative error of $8\mathrm{\%}$ on
the growth rate of the most unstable mode when computing gravity every $4$
dynamical timestep. We consider this variation acceptable and choose to
compute gravity every $4$ timestep in our simulation to speed up the
integration.
## Appendix C Chemical network
The magnetic diffusivities depend on the abundances of the charge carriers. To
compute these abundances, we consider a simple chemical network based on
Umebayashi & Nakano (1990) and Kunz & Mouschovias (2009). The network includes
the following reactions:
$\displaystyle\mathrm{H_{2}}\xrightarrow{\mathrm{CR}}\mathrm{H_{2}^{+}}+\mathrm{e^{-}}$
(22)
$\displaystyle\mathrm{H_{2}^{+}}+\mathrm{H_{2}}\rightarrow\mathrm{H_{3}^{+}}+\mathrm{H}$
(23)
$\displaystyle\mathrm{H_{3}^{+}}+\mathrm{CO}\rightarrow\mathrm{H_{2}}+\mathrm{HCO^{+}}$
(24)
$\displaystyle\mathrm{Fe}+\mathrm{HCO^{+}}\rightarrow\mathrm{Fe^{+}}+\mathrm{H}+\mathrm{CO}$
(25)
$\displaystyle\mathrm{HCO^{+}}+\mathrm{e^{-}}\rightarrow\mathrm{H}+\mathrm{CO}$
(26)
$\displaystyle\mathrm{Fe^{+}}+\mathrm{e^{-}}\rightarrow\mathrm{Fe}+\mathrm{photon}$
(27) $\displaystyle\mathrm{e^{-}}+\mathrm{grain}\rightarrow\mathrm{grain^{-}}$
(28) $\displaystyle\mathrm{e^{-}}+\mathrm{grain^{+}}\rightarrow\mathrm{grain}$
(29)
$\displaystyle\mathrm{Fe^{+}}+\mathrm{grain}\rightarrow\mathrm{Fe}+\mathrm{grain^{+}}$
(30)
$\displaystyle\mathrm{Fe^{+}}+\mathrm{grain^{-}}\rightarrow\mathrm{Fe}+\mathrm{grain}$
(31)
$\displaystyle\mathrm{HCO^{+}}+\mathrm{grain}\rightarrow\mathrm{H}+\mathrm{CO}+\mathrm{grain^{+}}$
(32)
$\displaystyle\mathrm{HCO^{+}}+\mathrm{grain^{-}}\rightarrow\mathrm{H}+\mathrm{CO}+\mathrm{grain}$
(33)
$\displaystyle\mathrm{H}+\mathrm{H}\xrightarrow{\mathrm{grain}}\mathrm{H_{2}}$
(34)
The ionization of $\mathrm{H_{2}}$ (Eq. 22) is solely due to cosmic rays. We
neglect shielding and focussing effects of cosmic rays, so the ionization rate
$\zeta=\mathrm{1.3\times 10^{-17}\,\mathrm{s^{-1}}}$ is assumed to be
constant. The reaction rates for the ion-neutral (Eqs. 23, 24 and 25), the
dissociative recombination (Eq. 26), and the radiative recombination (Eq. 27)
reactions are given by:
$k=\alpha\,{\left(\frac{T}{300\,\mathrm{K}}\right)}^{\beta}$ (35)
where $T$ is the temperature and $\alpha$ and $\beta$ are the prefactor and
temperature exponent, respectively. The values of $\alpha$ and $\beta$ for
each reaction are given in Table 1.
Table 1: Rate coefficients for the ion-neutral, dissociative recombination and radiative recombination reactions. Reaction | $\alpha$ | $\beta$
---|---|---
| $(\mathrm{cm^{3}\,s^{-_{1}}})$ |
Eq. (23) | $\mathrm{2.1\times 10^{-9}}$ | 0
Eq. (24) | $\mathrm{1.7\times 10^{-9}}$ | 0
Eq. (25) | $\mathrm{2.5\times 10^{-9}}$ | 0
Eq. (26) | $\mathrm{2.0\times 10^{-7}}$ | -0.75
Eq. (27) | $\mathrm{2.8\times 10^{-12}}$ | -0.86
The reaction rates for electron attachment (Eq. 28) and charge exchange
reactions on neutral grains (Eqs. 30 and 32) are given by:
$k=\pi a^{2}{\left(\frac{8k_{B}T}{\pi m}\right)}^{1/2}\left[1+{\left(\frac{\pi
e^{2}}{2ak_{B}T}\right)}^{1/2}\right]S$ (36)
where $a$ is the grain radius, $k_{B}$ is the Boltzmann constant, $m$ is the
mass of the electron or the ion, $e$ is the electron charge, and $S$ is a
sticking coefficient, assumed to be 0.6 for electrons and 1 for ions,
respectively. For electron attachement (Eq. 29) and charge exchange reactions
on charged grains (Eqs. 31 and 33), the reaction rates become:
$k=\pi a^{2}{\left(\frac{8k_{B}T}{\pi
m}\right)}^{1/2}\left(1+\frac{e^{2}}{ak_{B}T}\right)\left[1+{\left(\frac{2}{2+\left(ak_{B}T/e^{2}\right)}\right)}^{1/2}\right]S$
(37)
We assume that grains are spherical with a radius $a=0.1\,\mathrm{\mu m}$. The
gas-to-dust mass ratio is assumed to be equal to 100\. Assuming that the
grains have a mass density of $3\,\mathrm{g\,cm^{-3}}$, this gives a grain
abundance with respect to H nuclei of $\mathrm{1.3\times 10^{-12}}$.
Finally, we assume the following reaction rate for the $\mathrm{H_{2}}$ on
grains (Eq. 34):
$k=\alpha{\left(\frac{T}{300\,\mathrm{K}}\right)}^{1/2}$ (38)
with $\alpha=\mathrm{4.95\times 10^{-17}}\,\mathrm{s^{-1}}$.
Table 2 gives the initial abundances. We assume that all hydrogen is in
molecular form and that all carbon is in the form of CO. Iron is assumed to be
ionized, so it has the same initial abundance as free electrons. Grains are
assumed to be initially neutral. The abundances of all species in the chemical
network are computed as a function of time using Astrochem (Maret & Bergin
2015), until the steady-state equilibrium is reached.
Table 2: Initial abundances with respect to H nuclei of the species considered in chemical network. Species | Abundance
---|---
$\mathrm{H_{2}}$ | 0.5
$\mathrm{CO}$ | $\mathrm{8.4\times 10^{-5}}$
$\mathrm{Fe^{+}}$ | $\mathrm{2.05\times 10^{-6}}$
$\mathrm{e^{-}}$ | $\mathrm{2.05\times 10^{-6}}$
Fig. 18 shows the abundances of the main charge carriers at the steady-state
and the ionization fraction (i.e. the total abundance of positively or
negatively charged species with respect to H nuclei), as function of the H
number density, for a grain radius of $a=0.1\,\mathrm{\mu m}$ and a gas
temperature given by Eq. (8). The abundances of the main charge carriers are
in agreement with Umebayashi & Nakano (1990, see their Fig. 2, which
corresponds to the same grain size and the similar initial abundances that
those adopted here). The ionization fraction decreases with the density for
$n_{\mathrm{H}}<10^{11}\,\mathrm{cm^{-3}}$ and remains constant at higher
densities. For densities lower than $10^{11}\,\mathrm{cm^{-3}}$, the main
charge carriers are free electrons and $\mathrm{Fe^{+}}$ ions. At higher
densities, the main charge carriers are positively and negatively charged
grains. The transition between the two regimes occurs when the ion fraction
becomes comparable to the number density of grains.
Figure 18: Abundances of the charge carriers at the steady-state as a function
of the H number density for $a=0.1\,\mathrm{\mu m}$ and
$\zeta=\mathrm{1.3\times 10^{-17}\,s^{-1}}$. The black dotted line shows the
ionization fraction.
Fig. 19 shows the corresponding magnetic diffusivities (see Eqs. 3.4, 3.5 and
3.6 in Lesur 2021a) as a function of the H number density, for a magnetic
field intensity
$B=0.1\mathbf{\left(n_{\mathrm{H}}/\mathrm{cm^{-3}}\right)^{0.5}}\,\mathrm{\mu
G}$. The diffusivities are in agreement with Xu & Kunz (2021a, see their Fig.
C2).
Figure 19: Magnetic diffusivities as a function of the H number density for
$a=0.1\,\mathrm{\mu m}$, $\zeta=\mathrm{1.3\times 10^{-17}\,s^{-1}}$, and
$B=0.1\mathbf{\left(n_{\mathrm{H}}/\mathrm{cm^{-3}}\right)^{0.5}}\,\mathrm{\mu
G}$. The dashed line corresponds to a negative diffusivities.
## Appendix D Internal boundaries
Figure 20: Top: Alvén speed in code units versus the radius, where
$\varphi=0$. Solid lines are the midplane values ($\theta\approx\pi/2$) while
dashed lines are closer to the pole ($\theta\approx\pi/3$). The lighter the
color, the later the snapshot. The black dotted line is the Alvén cap
$V_{A}=V_{A,max}r$ where $V_{A,max}=1\,\mathrm{u.c.}$. Bottom: ambipolar
(solid) and Ohmic (dashed) diffusions versus the radius. Color coding and
black dotted line are the same as for the top panel, with the diffusivity cap
$\eta=\eta_{0}r^{2}$ where $\eta_{0}\approx 7.1\times
10^{18}\,\mathrm{cm^{2}\,s^{-1}}$.
We use three internal boundaries in order to prevent a dramatic drop of the
timestep without significant loss of accuracy: an Alvén speed limiter,
diffusivity caps, and an advection timestep limiter.
In code units, Alvén speed is defined by
$V_{A}=\frac{B_{\textsc{Idefix}}}{\sqrt{\rho}}$ (39)
where $B_{\textsc{Idefix}}\equiv B/\sqrt{4\pi}$.
Consequently, in strongly magnetized, low-density regions it can become very
high and require very low timesteps, incompatible with the large timescale
integration.
To alleviate this problem we introduce an Alvén speed limiter: for any cell of
radius $r$ if $V_{A}>V_{A,max}r$ the density is replaced with
$\rho_{new}=B_{\textsc{Idefix}}^{2}/(V_{A,max}r)^{2}$. The associated
velocities are updated following $u_{i,new}=u_{i}\cdot\rho/\rho_{new}$ to
satisfy as much as possible momentum conservation. Only $u_{\varphi}$ is left
untouched. In this simulation, we set $V_{A,max}$ to $1$ in code units.
Figure 20, the top panel presents the Alvén speed profile at $\varphi=0$ in
the midplane (solid lines) and near the pole (dashed lines) for snapshots of
increasing time. In the midplane, except for the very beginning, the Alvén
speed limiter is never triggered. This is not the case near the pole. Because
of cavity carving, the areas above and below the seed are strongly magnetized
with low density and we need to limit the Alvén speed up to
$100\,\mathrm{au}$. That being said, the cavity region is barely discussed in
this work because it is poorly described in the actual framework.
We also use diffusivity caps following Xu & Kunz (2021a, b). The timestep
associated with diffusion processes is proportional to $\Delta l^{2}/\eta$,
where $\Delta l$ is the typical cell size and $\eta$ is the diffusivity
coefficient associated with the diffusion process. Particularly strong values
of $\eta_{A}$ and $\eta_{0}$ are therefore susceptible to dramatically slow
down the integration.
We solve the problem by introducing a diffusivity cap such that for any cell
of radius $r$, if $\eta_{A,O}>\eta_{0}r^{2}$, the diffusivity coefficient is
replaced with $\eta_{A,O}=\eta_{0}r^{2}$. Here, $\eta_{0}\,\approx 7.1\times
10^{18}\,\mathrm{cm^{2}\,s^{-1}}$ is the conversion factor from code units to
physical units.
Figure 20, bottom panel shows the ambipolar (solid lines) and Ohmic (dashed
lines) diffusivity profiles at $\varphi=0$ for the same snapshots as the top
panel. The cap is triggered for ambipolar diffusion only in the very
beginning. Ohmic diffusion, however, is limited for radii below
$5\,\mathrm{au}$ as soon as the disc forms and remains so until the end. This
is expected as Ohmic diffusion increases with gas density.
The last internal boundary, the advection timestep limiter, is a consequence
of the first one. In the innermost regions, gas affected by the Alvén limiter
can’t be launched in the outflow. Conversely, it is falling back onto the
seed, reaching high velocities that limit the timestep. To solve this problem,
we use a timestep limiter such that where $dt_{adv}<dt_{min}$, the velocity
components are updated following $u_{i,new}=u_{i}\cdot dt_{min}/dt_{adv}$. We
set $dt_{min}=1\,\mathrm{u.c.}$
We monitored the total mass and angular momentum in the system during the
integration to ensure that none of these routines was significantly affecting
their balance.
|
# Combinatorial exploration of quantum spin liquid candidates in the
herbertsmithite material family
Alex Hallett Materials Department, University of California, Santa Barbara,
California 93106, USA Catalina Avarvarei Materials Department, University of
California, Santa Barbara, California 93106, USA John W. Harter
<EMAIL_ADDRESS>Materials Department, University of California, Santa Barbara,
California 93106, USA
###### Abstract
Geometric frustration of magnetic ions can lead to a quantum spin liquid
ground state where long range magnetic order is avoided despite strong
exchange interactions. The physical realization of quantum spin liquids
comprises a major unresolved area of contemporary materials science. One
prominent magnetically-frustrated structure is the kagome lattice. The
naturally occurring minerals herbertsmithite [ZnCu3(OH)6Cl2] and Zn-
substituted barlowite [ZnCu3(OH)6BrF] both feature perfect kagome layers of
spin-$1/2$ copper ions and display experimental signatures consistent with a
quantum spin liquid state at low temperatures. To investigate other possible
candidates within this material family, we perform a systematic first-
principles combinatorial exploration of structurally related compounds
[$A$Cu3(OH)${}_{6}B_{2}$ and $A$Cu3(OH)${}_{6}BC$] by substituting non-
magnetic divalent cations ($A$) and halide anions ($B$, $C$). After optimizing
such structures using density functional theory, we compare various structural
and thermodynamic parameters to determine which compounds are most likely to
favor a quantum spin liquid state. Convex hull calculations using binary
compounds are performed to determine feasibility of synthesis. We also
estimate the likelihood of interlayer substitutional disorder and spontaneous
distortions of the kagome layers. After considering all of these factors as a
whole, we select several promising candidate materials that we believe deserve
further attention.
## I Introduction
In a quantum spin liquid (QSL), frustrated antiferromagnetic exchange
interactions prevent localized spins from ordering at low temperatures,
instead forming a fluid-like phase. The large degeneracy of this state can
give rise to novel phenomena such as fractionalized quasiparticles, emergent
gauge fields, and long-range entanglement [1, 2, 3, 4]. The kagome lattice of
corner-sharing triangles is known to have high geometric frustration and is
capable of hosting such a phase. A leading QSL material candidate possessing
this structure is herbertsmithite [ZnCu3(OH)6Cl2], which contains perfect
kagome layers of spin-$1/2$ copper cations separated by non-magnetic Zn and Cl
ions [5, 6], as shown in Fig. 1(a,c). Indeed, although herbertsmithite has
strong antiferromagnetic exchange interactions, no magnetic phase transition
is observed down to sub-kelvin temperatures [7, 8, 9, 10, 11], and an array of
experimental and theoretical work favors a possible QSL scenario [12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 19, 24].
Figure 1: Crystal structures of herbertsmithite and Zn-barlowite. (a)
Herbertsmithite viewed along the $c$-axis, showing the kagome arrangement of
Cu ions. (b) Zn-barlowite viewed along the $c$-axis. (c) Herbertsmithite
viewed along the [110] direction, showing the shifted stacking arrangement of
the kagome layers. (d) Zn-barlowite viewed along [110], showing the stacking
of the kagome layers and the inequivalence of the Br and F sites.
Despite its many promising features, herbertsmithite is prone to cation
substitutional disorder, where Cu may occupy interlayer sites and Zn may
occupy intralayer kagome sites [7, 14, 25]. The precise amount of this
disorder is debated. Several studies suggest that while there is minimal
substitution of Zn on the kagome layers, the interlayer sites can be occupied
by up to 15% Cu [26, 27, 12, 28], resulting in a decidedly off-stoichiometric
compound. These interlayer “orphan” spin-$1/2$ Cu2+ defects are highly
problematic for the QSL state, causing weak ferromagnetic interactions between
kagome layers and distorting the surrounding matrix of magnetic ions [13]. Zn-
substituted barlowite (Zn-barlowite), a structurally related compound and
another potential QSL candidate [29, 30], is thought to have a much lower
interlayer disorder concentration, largely due to the greater chemical
distinction between the interlayer and intralayer sites, as shown in Fig.
1(b,d) [31, 32]. Experiments indicate that in Zn-barlowite, off-center
interlayer $C_{2v}$ sites can contain up to 5% Cu defects. Like
herbertsmithite, however, Zn-barlowite does not order magnetically, even with
these large concentrations of magnetic defects [33, 34, 35]. While progress on
this class of materials is encouraging, it is nevertheless desirable to
further minimize orphan Cu spins to realize a clean QSL ground state.
Synthesizing compounds structurally similar to herbertsmithite and Zn-
barlowite is a promising route to discover new QSL candidates. For example,
Mg-substituted herbertsmithite, MgxCu4-x(OH)6Cl2 (tondiite), has been
successfully synthesized and shows no magnetic phase transition down to 1.8 K
[36, 37, 38], and a Cd analog [CdCu3(OH)6Cl2] shows no magnetic ordering down
to 2 K, although it exhibits significant distortions of the kagome planes
[39]. Synthesis of the bromide analog of herbertsmithite [ZnCu3(OH)6Br2] was
attempted but unsuccessful [40]. A Zn-barlowite related structure, Zn-
claringbullite [ZnCu3(OH)6ClF], shows no obvious magnetic transition down to 2
K, but a perfectly stoichiometric compound was not achieved [41]. While the Mg
analog of barlowite cannot be synthesized due to the insolubility of MgF2 in
water, the bromide analog was attempted [MgCu3(OH)6Br2], but did not have the
Zn-barlowite structure and ordered antiferromagnetically at 5.4 K [42].
Clearly, more work is needed to search for and identify viable candidates in
this material family. Only a few computational studies exist exploring cation
substitution in barlowite [31, 32], and a complete exploration of the
structural families of herbertsmithite and Zn-barlowite using computational
methods has not been performed. In this paper, we use ab initio calculations
to systematically explore compounds within the herbertsmithite and Zn-
barlowite families. We compare the thermodynamic stability, structural
properties, and tendency towards disorder. After considering all these
criteria together, we select promising QSL candidates that merit further
experimental and theoretical examination.
## II Computational Procedure
We carry out a systematic exploration of the structural relatives of
herbertsmithite [$A$Cu3(OH)${}_{6}B_{2}$] and Zn-barlowite
[$A$Cu3(OH)${}_{6}BC$] by substituting closed-shell (spinless) $2+$ cations
($A$ = Ba, Be, Ca, Cd, Ge, Hg, Mg, Pb, Sn, Sr, Zn) and halide anions ($B,C$ =
Br, Cl, F, I). We investigate all 44 possible herbertsmithite relatives. While
there are 176 possible Zn-barlowite relatives, we eliminate compounds where
$B=C$ because the herbertsmithite structure always has lower energy in these
cases. We also do not consider compounds in which the less electronegative
anion occupies the $C$ site [the site occupied by F in Fig. 1(b,d)]. All
hydrogen bonds are oriented towards the $C$ site, so the more electronegative
ion will always occupy this position to minimize energy. Thus, a total of 66
relatives in the Zn-barlowite family were selected for consideration.
We perform high-throughput calculations where the structural optimization of
each candidate is followed by a static calculation to extract the ground-state
energy and to compute phonon frequencies at the $\Gamma$ point to confirm
structural stability. In addition to confirming the stability of the relaxed
structures, we perform convex hull calculations to determine if synthesis of
the candidate compounds is thermodynamically feasible. For the most promising
materials, we also calculate defect formation energies and full phonon
dispersions throughout the first Brillouin zone to verify stability at
$k$-points away from the zone center.
All structures were calculated by allowing the lattice parameters, cell
volume, and atomic positions to fully relax using density functional theory
(DFT) as implemented in the Vienna ab initio simulation package (vasp) [43,
44, 45]. We used the supplied projector augmented wave potentials [46] within
the generalized gradient approximation and Perdew-Burke-Ernzerhof scheme [47].
Electronic wave functions were expanded in a plane wave basis set with an
energy cutoff of 800 eV, and reciprocal space was sampled using an $8\times
8\times 8$ $k$-point mesh for herbertsmithite-related structures and an
$8\times 8\times 5$ $k$-point mesh for Zn-barlowite-related structures. A
$\Gamma$-centered mesh is necessary due to the hexagonal symmetry of Zn-
barlowite. The spacing between $k$-points was $\sim$0.15 Å-1 for both
structural families, and this spacing was also used for calculating the
energies of binary compounds used in the convex hull analysis. All structures
were relaxed until forces on the atoms were less than 1 meV/Å. Calculations
were non-spin-polarized. Input files for all calculations can be found in the
Supplemental Material [48].
Figure 2: Structural stability and thermodynamics of candidate compounds. (a)
Lowest optical phonon frequency for herbertsmithite-related candidates. (b)
Lowest optical phonon frequency for Zn-barlowite-related candidates. (c)
Convex hull energies for herbertsmithite-related candidates. (d) Convex hull
energies for Zn-barlowite-related candidates. Structurally unstable compounds
(identified by $f_{0}<0$) are denoted with an ‘X’. Cations are shown on the
vertical axis and anions on the horizontal axis, in order of increasing ionic
radius from bottom to top and left to right, respectively. The reference
compound (either herbertsmithite or Zn-barlowite) is shown in white and marked
with an asterisk. Compounds with parameter values more favorable than the
reference compounds are shown with warm colors, and values less favorable are
shown with cool colors. Figure 3: Structural properties of candidate
compounds. (a) Cu-O-Cu bond angle for herbertsmithite-related candidates. (b)
Cu-O-Cu bond angle for Zn-barlowite-related candidates. (c) Interplane kagome
distance for herbertsmithite-related candidates. (d) Interplane kagome
distance for Zn-barlowite-related candidates. Structurally unstable compounds
are denoted with an ‘X’. Cations are shown on the vertical axis and anions on
the horizontal axis, in order of increasing ionic radius from bottom to top
and left to right, respectively. The reference compound (either
herbertsmithite or Zn-barlowite) is shown in white and marked with an
asterisk. Compounds with parameter values more favorable than the reference
compounds are shown with warm colors, and values less favorable are shown with
cool colors. Figure 4: Dependence of structural properties on ion size. (a)
Cu-O-Cu bond angle versus anion radius. For Zn-barlowite, the radius plotted
is that of the most electronegative anion. Blue (red) traces correspond to Zn-
barlowite (herbertsmithite) relatives. Different cations are plotted as
separate traces where darker (lighter) traces correspond to smaller (larger)
ion sizes. (b) Interplane kagome distance versus cation radius for
herbertsmithite (red) and Zn-barlowite (blue) relatives. Separate traces are
plotted for each anion, where small (large) anions are plotted in dark (light)
shades. (c) Cu-O-Cu bond angle versus the anion $B$ to anion $C$ ratio for
stable compounds. Separate traces are plotted for different cations. (d)
$c$-axis length versus cation size (left, dashed line) and $a$-axis length
versus anion size (right, solid lines). (e) Frequency of the lowest optical
phonon mode versus $c$-axis length for Zn-barlowite (blue) and herbertsmithite
(red) relatives. Stable (unstable) compounds are shown with filled (empty)
markers. The group IV elements (Ge, Sn, Pb) are plotted with darker colors
because they are almost always unstable, regardless of their $c$-axis length.
(f) Cu-O-Cu bond angle versus $a$-axis length for Zn-barlowite (blue) and
herbertsmithite (red) relatives. Stable (unstable) compounds are shown with
filled (empty) markers. Compounds containing group IV cations (shown in darker
colors) tend to be unstable and have much smaller bond angles.
## III Results and Discussion
### III.1 Phonon Calculations
Phonon calculations at the $\Gamma$ point for the fully-relaxed structures
were performed in vasp within the finite differences approximation to confirm
structural stability. As expected, many structures have unstable phonon modes.
Fig. 2(a,b) shows the frequency of the lowest energy optical phonon mode,
$f_{0}$, for all compounds. In all subsequent plots, the unstable compounds
(with $f_{0}<0$) are marked with an ‘X’ to distinguish them from structurally
stable and potentially viable candidates. Cations are shown on the vertical
axis and anions on the horizontal axis, in order of increasing ionic radius
from bottom to top and left to right, respectively. The reference compound,
either herbertsmithite or Zn-barlowite, is shown in white and marked with an
asterisk. Compounds with parameter values more favorable than the reference
compound are shown with warm colors, and values less favorable are shown with
cool colors. For example, a higher frequency of the lowest energy optical mode
indicates higher dynamical stability, so higher frequencies are shown with
warm colors. Most compounds containing group IV elements (Ge, Sn, Pb) tend to
be unstable, with the exception of GeCu3(OH)6F2 and PbCu3(OH)6F2. Compounds
containing larger cations are generally unstable, as well as Zn-barlowite
relatives containing Be.
### III.2 Convex Hull Calculations
The convex hull of a compound is useful for determining if synthesis is
thermodynamically feasible, usually through a comparison of the compound’s
formation energy to the sum of the energies of all other possible combinations
of crystal structures that could be created from the same set of elements in
the same ratios. Due to the prohibitive size of the phase space for our
candidate materials, we perform a simplified procedure. Instead of considering
all possible crystal structures, we consider only simple binary ionic
compounds [e.g. $A$(OH)2, $AB_{2}$], which are most likely to yield the lowest
convex hull energies (see Supplemental Material [48]). Starting structures for
these binary compounds were obtained from the Materials Project [49] and then
re-relaxed with our settings.
Insulators with energies less than $\sim$50 meV above the convex hull tend to
be stable [50]. We therefore use an energy cutoff of 50 meV/atom as our
criteria for thermodynamic stability when identifying candidate materials. The
calculated energy above the hull for each compound is shown in Fig. 2(c,d).
Energies higher than the reference compound are considered unfavorable and are
represented with cool colors, while energies lower than the reference compound
are favorable and represented with warm colors. Again, the reference compounds
are shown in white and marked with an asterisk, and compounds with structural
instabilities (as determined by phonon calculations) are marked with an ‘X’.
There does not appear to be a clear connection between convex hull energy and
structural stability or ion size.
### III.3 Comparing Structural Parameters
In addition to structural and thermodynamic stability, we use Cu-O-Cu bond
angles and spacings between kagome layers as additional metrics to rank the
candidate compounds. A Cu-O-Cu bond angle approaching 180∘ leads to a large
antiferromagnetic superexchange interaction while minimizing undesirable
Dzyaloshinskii–Moriya interactions. Larger bond angles are therefore highly
desirable. Additionally, a greater separation between the kagome layers
isolates the two-dimensional magnetic subsystems and suppresses unwanted
coupling between planes. In Fig. 3, these two structural properties are
displayed for all candidate compounds. Squares corresponding to specific
compounds are colored and marked according to the same system described for
Fig. 2, where bond angles and interplane distances larger (smaller) than the
reference compounds are favorable (unfavorable) and represented with warm
(cool) colors, and structurally unstable compounds continue to be marked with
an ‘X’. Compounds with larger cation and anion radii generally lead to larger
bond angles and interplane distances, but also tend to be structurally
unstable. Compounds containing group IV elements are unstable and tend to have
smaller bond angles.
In Fig. 4, we investigate the effects of ion size on the physical properties
of the candidate compounds in more detail. In Fig. 4(a), the Cu-O-Cu bond
angle is plotted versus anion radius for the structurally stable materials.
The anion size plotted on the horizontal axis for Zn-barlowite relatives
refers to the $C$-site anion that occupies the same position as F in the
reference compound [ZnCu3(OH)6BrF] because it has the largest influence on
bond angle. For all materials, bond angle increases with increasing anion
size, and for a given anion, the bond angle also increases with increasing
cation size. Figure 4(b) shows the kagome plane spacing versus cation radius
for stable compounds, with separate traces for each anion. As expected, a
larger cation radius leads to greater distance between the kagome layers. For
a given cation, interplane distance also increases with increasing anion size.
In Fig. 4(c), we find that while the $C$-site anion has the greatest effect on
the Cu-O-Cu bond angle, larger bond angles are obtained when the $B$-site
anion is similar in size to the $C$-site anion.
We examine the effect of ion size on the lattice parameters of stable
compounds in Fig. 4(d). The $c$-axis length primarily increases with cation
size while the $a$-axis length primarily increases with anion size, although
anion size has a much weaker affect on the $a$-axis than cation size does on
the $c$-axis. The frequency of the lowest optical phonon mode ($f_{0}$) is
plotted against $c$-axis length in Fig. 4(e) for both stable (filled markers)
and unstable (empty markers) structures. Of all the structural parameters, the
$c$-axis length has the highest correlation with $f_{0}$. For herbertsmithite
relatives, as the $c$-axis increases, $f_{0}$ decreases, meaning compounds
tend to be less dynamically stable. Compounds containing group IV ions (Ge,
Sn, Pb) are plotted in darker shades for both structural families because
nearly all compounds containing these elements are unstable. Of the compounds
not containing group IV ions, $c$-axis lengths that are very small or very
large lead to structural instabilities. Compounds containing cations from
groups IIA and IIB which are close in size to Zn tend to be most stable. Fig.
4(f) shows Cu-O-Cu bond angle versus $a$-axis length. We find that a larger
$a$-axis leads to a larger bond angle, which agrees with the results in Fig.
4(a), where bond angle is positively correlated with anion radius, and Fig.
4(d), which shows the positive correlation between anion size and the length
of the $a$-axis. It should be noted that many unstable compounds containing
group IV elements have much smaller bond angles than most other candidates.
We also explored correlations between Cu-O-Cu bond angle, interplane distance,
and in-plane Cu-Cu bond length. These plots can be found in the Supplemental
Material [48]. The Cu-O-Cu bond angle has a weak positive correlation with
interplane distance. There is also a positive correlation between in-plane Cu-
Cu distance and Cu-O-Cu bond angle, as both are influenced by the length of
the $a$-axis, which increases with increasing anion size. There is no obvious
correlation between the interplane kagome distance and the in-plane Cu-Cu bond
length, as the interplane distance depends mostly on cation size, and in-plane
bond length depends on anion size. Overall, for both structural families,
compounds with cations of intermediate size (Mg, Zn, Cd, and Hg) are most
stable. Compounds containing group IV elements (Ge, Sn, Pb) are mostly
unstable. Larger anions and cations lead to favorable structural properties,
such as larger bond angles and interplane distances, but may also lead to
distortions of the kagome layers or other structural instabilities.
Table 1: Properties of the most promising QSL candidate materials as compared to the reference materials. The references (herbertsmithite and Zn-barlowite) are highlighted in gray, and the final candidates (with no instabilities throughout the Brillouin zone) are marked with asterisks. Compound | $f_{0}$ (THz) | $E_{\mathrm{hull}}$ (meV/atom) | $E_{d}^{f}$ (eV) | $\theta$ (deg) | $d_{\mathrm{inter}}$ (Å) | $d_{\mathrm{in}}$ (Å)
---|---|---|---|---|---|---
BaCu3(OH)6I2 | 0.41 | 42.6 | 2.42 | 128.0 | 6.09 | 3.53
CaCu3(OH)6Br2 | 0.50 | 30.7 | 0.87 | 125.7 | 5.19 | 3.53
CaCu3(OH)6Cl2 | 0.70 | 44.8 | 0.57 | 125.8 | 5.06 | 3.51
MgCu3(OH)6Br2 * | 2.23 | 36.0 | 0.36 | 125.2 | 4.65 | 3.57
ZnCu3(OH)6Cl2 | 2.63 | 41.2 | 0.13 | 125.0 | 4.58 | 3.53
CaCu3(OH)6IBr | 0.77 | 31.6 | 0.74 | 127.7 | 5.20 | 3.58
CaCu3(OH)6ICl * | 0.94 | 19.2 | 0.72 | 125.4 | 5.17 | 3.54
MgCu3(OH)6ClF * | 1.09 | 39.6 | 0.39 | 118.1 | 4.60 | 3.38
MgCu3(OH)6BrCl | 0.35 | 26.9 | 0.30 | 126.1 | 4.61 | 3.56
ZnCu3(OH)6BrF | 1.41 | 38.6 | 0.10 | 118.0 | 4.69 | 3.39
ZnCu3(OH)6ClF | 0.89 | 43.1 | 0.07 | 118.5 | 4.64 | 3.38
### III.4 Defect Formation Energy
Herbertsmithite and Zn-barlowite are both susceptible to cation disorder. In
herbertsmithite, the Jahn-Teller active $d^{9}$ Cu2+ ion occupies the
tetragonally elongated site in the center of the CuO4Cl2 octahedra. The
$d^{10}$ Zn2+ ions are not Jahn-Teller active, and occupy the higher-symmetry
trigonally compressed octahedral sites between the kagome layers. Due to the
electronic configurations of the ions and distinct coordination environments,
it is not favorable for Zn to occupy the in-plane sites within the kagome
layer. However, herbertsmithite is the $x=1$ end member of the Zn-paratacamite
family [ZnxCu4-x(OH)6Cl2], and there is a preference for some Cu to exist on
the interlayer site instead of full occupation with Zn alone [7]. The
equilibrium occupation of the interlayer site by Cu has been estimated to be
as large as 15% in herbertsmithite [26, 27].
In Zn-barlowite, the interlayer site has a trigonal prismatic geometry, making
it even less favorable for the Jahn-Teller active Cu2+ ion. As a result, the
interlayer Cu occupation is only $\sim$5% in Zn-barlowite [33], confirming
early computational predictions [31, 32]. Site-specific x-ray diffraction
measurements have shown that there are two distinct interlayer sites in Zn-
barlowite: an off-center $C_{2v}$ site and a central $D_{3h}$ site. The
interlayer Cu defects occupy the $C_{2v}$ sites. It should be noted that even
for large concentrations of magnetic impurities on the interlayer site, Zn-
barlowite does not show signs of magnetic ordering, indicating that the
possible QSL phase is somewhat robust against interlayer magnetic impurities
[33].
An ideal QSL candidate will have only non-magnetic ions on the interlayer
sites, and therefore must have a high energy cost for interlayer Cu
substitution. We calculated the formation energy of such defects in a select
number of our most promising candidates (those structurally stable, with
$E_{\mathrm{hull}}<50$ meV/atom, and with bond angles and interplane distances
larger than the reference compounds). Since nearly all experimental and
computational studies indicate that there is negligible substitution of non-
magnetic ions within the kagome layers, we consider only interlayer defects.
The general expression for the formation energy of a charge-neutral
substitutional defect is
${E_{d}^{f}=E[\mathrm{defect}]-E[\mathrm{bulk}]+(\mu_{A}-\mu_{\mathrm{Cu}})=\Delta
E_{s}+\Delta\mu,}$
where $\Delta E_{s}$ is the difference in energy between a structure with a
single defect and the pristine bulk structure and $\Delta\mu$ is the chemical
potential difference of $A$ and Cu. To calculate $E[\mathrm{defect}]$, we
construct defect structures from $2\times 2\times 2$ supercells of
herbertsmithite relatives and $2\times 2\times 1$ supercells of Zn-barlowite
relatives, with a single Cu substitution. A depiction of our defect
configuration can be found in the Supplemental Material [48]. We relax the
atomic positions of the defect structures and subtract the energy of the
original defect-free structure to obtain $\Delta E_{s}$.
Figure 5: Phonon dispersions of final candidates. (a) The phonon dispersion
for MgCu3(OH)6Br2 (blue) overlaid with the reference dispersion for
herbertsmithite (gray). (b) The phonon dispersion for CaCu3(OH)6ICl (blue)
overlaid with the reference dispersion for Zn-barlowite (gray). (c) The phonon
dispersion for MgCu3(OH)6ClF (blue) overlaid with the reference dispersion for
Zn-barlowite (gray). The absence of imaginary phonon frequencies in all three
cases confirms the structural stability of these candidate compounds.
The chemical formulas for the defect-containing and defect-free configurations
are not equivalent, so the chemical potential difference
$\Delta\mu=\mu_{A}-\mu_{\mathrm{Cu}}$ must be considered. Interlayer defects
are primarily created during the initial growth of the material. During
synthesis of $A$Cu3(OH)${}_{6}B_{2}$, the chemical potentials of the
constituent elements must satisfy the inequality
${\mu_{A}+3\mu_{\mathrm{Cu}}+6\mu_{\mathrm{OH}}+2\mu_{B}>E[A\mathrm{Cu}_{3}(\mathrm{OH})_{6}B_{2}].}$
Individual chemical potentials must all be less than zero ($\mu_{A}<0$,
$\mu_{B}<0$, $\mu_{\mathrm{OH}}<0$, and $\mu_{\mathrm{Cu}}<0$). Additionally,
the formation of unwanted side products must be avoided, imposing the
additional inequalities
${\mu_{A}+2\mu_{B}<E[AB_{2}],}$
${\mu_{\mathrm{Cu}}+2\mu_{B}<E[\mathrm{Cu}B_{2}],}$
${\mu_{A}+2\mu_{\mathrm{OH}}<E[A(\mathrm{OH})_{2}].}$
Similar inequality constraints exist for $A$Cu3(OH)${}_{6}BC$. A higher defect
formation energy is preferable to minimize disorder. To maximize $E_{d}^{f}$,
we must maximize the chemical potential difference $\Delta\mu$ subject to the
above inequality constraints. The defect formation energies calculated with
these optimal values of $\Delta\mu$ are given in Table 1. All candidate
compounds investigated had a higher energy cost for interlayer defects than
herbertsmithite and Zn-barlowite except ZnCu3(OH)6ClF (Zn-substituted
claringbullite).
Two previous computational studies investigated doping selectivity in
barlowite [31, 32]. In both cases, the authors investigated the likelihood of
substituting various non-magnetic ions into the interlayer and intralayer
sites of barlowite, in contrast to the present work where we examine the
energy cost of a Cu defect on an interlayer site in fully-substituted
$A$-barlowite ($A$ = Zn, Mg, Ca). Despite differences in the methodology used
to construct defect structures and calculate the chemical potential
differences, our findings are generally consistent with those studies, which
suggested Zn and Mg to be the most favorable ions for synthesizing barlowite-
related compounds. More details on our defect formation energy calculations
can be found in the Supplemental Material [48].
### III.5 Selecting Promising Candidates
After eliminating all compounds with structural instabilities at the $\Gamma$
point, formation energies greater than 50 meV/atom above the convex hull, and
Cu-O-Cu bond angles smaller than the reference compounds, 9 candidate
materials remained. For these candidates, we calculated the defect formation
energy $E_{d}^{f}$. To determine a final ranking, we used the following
criteria:
1. 1.
Structural stability ($f_{0}>0$)
2. 2.
Convex hull energy (E${}_{\mathrm{hull}}<50$ meV/atom)
3. 3.
Defect energy cost ($E_{d}^{f}[\mathrm{candidate}]>E_{d}^{f}[\mathrm{ref}]$)
4. 4.
Cu-O-Cu bond angle ($\theta>\theta^{\mathrm{ref}}$)
All compounds satisfying these criteria are listed with their associated
properties in Table 1. Complete data sets for all 44 herbertsmithite relatives
and 66 Zn-barlowite relatives can be found in the Supplemental Material [48].
We also verified structural stability by calculating the full phonon
dispersion throughout the entire Brillouin zone using the finite displacement
method within the phonopy code [51]. Such calculations can identify structural
instabilities associated with an enlargement of the unit cell. Dispersion
curves were calculated for all candidates in Table 1. However, only one
compound in the herbertsmithite family and two compounds in the Zn-barlowite
family were found to be stable throughout the entire Brillouin zone. The
dispersion curves of these compounds are shown in Fig. 5, while dispersions
for all compounds in Table 1 can be found in the Supplemental Material [48].
Surprisingly, while Zn-claringbullite [ZnCu3(OH)6ClF] is known to have perfect
kagome layers at room temperature [41], our ground state dispersion shows
instabilities at the $M$ and $K$ points (see Supplemental Material [48]). The
instabilities we observe in DFT may be avoided by thermal fluctuations at room
temperature, which could explain the discrepancy between our calculations and
the experimental results. Two other Zn-barlowite-related candidate compounds
listed in Table 1, CaCu3(OH)6IBr and MgCu3(OH)6BrF, showed similar
instabilities, and therefore may also be stable at room temperature (see
Supplemental Material [48]).
Our calculations identify MgCu3(OH)6Br2 as a potential candidate within the
herbertsmithite family, as well as CaCu3(OH)6ICl and MgCu3(OH)6ClF in the Zn-
barlowite family. However, some practical considerations related to synthesis
may require further investigation. For instance, the Mg analog of Zn-barlowite
[MgCu3(OH)6BrF] has not been synthesized due to the insolubility of MgF2 in
water. While synthesis of Zn-barlowite using NH4F yields a structurally
equivalent compound, crystals obtained using this method show a similar
magnetic transition to barlowite, suggesting possible differences in defect
structures between the two synthesis methods [52]. The insolubility of MgF2
may therefore present difficulty in synthesizing our candidate MgCu3(OH)6ClF
[41]. Synthesis of MgCu3(OH)6Br2 has been attempted, but the desired product
was a Zn-barlowite analog [42]. The synthesis method, which followed the
typical hydrothermal procedure, resulted in a compound with $P\bar{3}m1$
symmetry, which may mean that the herbertsmithite $R\bar{3}m$ structure is not
favored in this reaction. It is possible that other synthesis methods could
yield different results. To our knowledge, no experimental studies have been
performed on the Ca analog of either herbertsmithite or Zn-barlowite, nor any
related compounds containing I.
## IV Conclusion
In summary, we performed a systematic combinatorial exploration of
herbertsmithite and Zn-barlowite material relatives and identified those with
properties that may enhance the likelihood of an ideal QSL ground state. We
found several promising candidates—MgCu3(OH)6Br2, CaCu3(OH)6ICl, and
MgCu3(OH)6ClF—that are structurally stable, thermodynamically feasible to
synthesize, have high energy costs for interlayer defects, and whose
structural properties may result in antiferromagnetic superexchange
interactions stronger than herbertsmithite or Zn-barlowite. These compounds,
if they can be synthesized, may prove to be better QSL candidates than their
well-studied counterparts.
## Acknowledgments
We would like to thank Siavash Karbasizadeh for helpful discussions. This work
was supported by the Air Force Office of Scientific Research under AFOSR award
no. FA9550-21-1-0337. C.A. acknowledges support from the UCSB Quantum Foundry
Internship Program, which is funded by the National Science Foundation (NSF)
through Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for
Quantum Materials Science, Engineering, and Information (Q-AMASE-i): Quantum
Foundry at UC Santa Barbara (DMR-1906325). Use was made of computational
facilities purchased with funds from the NSF (CNS-1725797) and administered by
the Center for Scientific Computing (CSC). The CSC is supported by the
California NanoSystems Institute and the Research Science and Engineering
Center (MRSEC; NSF DMR-1720256) at UC Santa Barbara.
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|
# Evidence-centered Assessment for Writing with Generative AI
Yixin Cheng
Monash University
<EMAIL_ADDRESS>
Kayley Lyons
University of Melbourne
<EMAIL_ADDRESS>
Guanliang Chen
Monash University
<EMAIL_ADDRESS>
Dragan Gašević
Monash University
<EMAIL_ADDRESS>
Zachari Swiecki
Monash University
<EMAIL_ADDRESS>
###### Abstract
We propose a learning analytics-based methodology for assessing the
collaborative writing of humans and generative artificial intelligence. Framed
by the evidence-centered design, we used elements of knowledge-telling,
knowledge transformation, and cognitive presence to identify assessment
claims; we used data collected from the CoAuthor writing tool as potential
evidence for these claims; and we used epistemic network analysis to make
inferences from the data about the claims. Our findings revealed significant
differences in the writing processes of different groups of CoAuthor users,
suggesting that our method is a plausible approach to assessing human-AI
collaborative writing.
_K_ eywords Generative Artificial Intelligence $\cdot$ Assessment $\cdot$
Evidence-centered Design $\cdot$ Epistemic Network Analysis
## 1 Introduction
Effectively communicating ideas via writing is a critical human skill. Every
day, many of us send text messages, draft emails, and make notes; in many
specialised domains, such as research, writing is a core form of discourse.
The process of writing has, of course, changed over time; writing tools have
transformed from mere storage receptacles to tools that help us craft more
effective writing, such as word processors, spellcheckers, and grammar
checkers [1]. Recently, however, there has been a step-change in tools for
writing. Whereas prior tools helped us to save and process our own writing,
tools-based on _generative artificial intelligence_ (GAI) can now compose
writing for us.
This technological advance has already had far reaching implications for
society. Arguably, these implications have been—and will continue to be—the
most profound for education. Education relies, in part, on assessment. Broadly
speaking, we want students to learn skills that will be valuable to their
well-being and the well-being of society. But to know and communicate whether
learning has occurred, we need to be able to _evidence_ that learning.
Assessments are the structures that make such inferences about learning
possible [2]. A key feature of assessments is that they assign credit to both
pragmatic actions—actions that advance toward a goal state—and epistemic
actions—actions that reduce cognitive effort [3]. However, when students use
tools like ChatGPT [4] to help them write, these actions are mixed between the
human and the tool [5], making the assignment of credit difficult. How, for
example, should a teacher assess writing crafted partly by a human and partly
by AI?
In the last year, educational institutions across the world have scrambled to
put policies in place that seek to address questions like the one above. For
example, the Tertiary Education Quality and Standards Agency (TEQSA),
responsible for regulating and the quality of all providers of higher
education in Australia, recently convened a panel of experts to produce a
whitepaper on “assessment reform in the age of artificial intelligence” that
outlines broad changes institutes of higher education will be expected to make
in response to AI [6]. Rather than cut students off from AI, its authors argue
that assessments should prepare students for a world where collaboration with
AI is commonplace by redesigning assessments to emphasize: (a) appropriate and
authentic engagement with AI; (b) the process of learning; and (c)
opportunities for students to work with each other and AI.
We agree with these recommendations. In turn, we argue that there is a need to
re-develop assessments to account for interactions between humans and AI in
ways that afford reasoned arguments about learning claims from verifiable and
persistent evidence. In this paper, we propose and test such an assessment
informed by (a) the evidence-centered design (ECD) assessment framework (b)
extant theories of the cognitive processes involved in writing (c) recent
interfaces to GAI, and (d) learning analytic process models. Our results
suggest that this method can find expected differences between the processes
people use when writing with GAI, such as differences between genres. This
work is a proof of concept that contributes to a deeper understanding of
human-AI collaboration and suggests a path forward for innovative assessments
in this new age of AI.
## 2 Background and Theory
Educational assessments are a means for determining whether and how students
have learned. While several frameworks for assessment design have been
proposed, the ECD framework has proven to be particularly successful and
adaptable to a variety of domains [2]. ECD defines a Conceptual Assessment
Framework that incorporates three high-level models: a _student model_ , _task
model_ , and _evidence model_. Together, these components suggest a coherent
and logical approach for designing assessments that are aligned with the
constructs they intend to measure.
### 2.1 ECD Student Model
The student model identifies and describes the claims about learning that the
assessment aims to measure. Our study aimed to assess the processes students
enact while writing with GAI as opposed to the quality of the final writing
product. There is a long history of assessing writing ability—as well as
learning more generally—in terms of products like essays, posts, and articles.
For example, automated essay scoring [7] and writing analytics [8] have
developed into active sub-fields of psychometrics and learning analytics,
respectively. Such research traditionally uses natural language processing
(NLP) techniques to assess writing quality and rhetorical structure. More
recently, work has transitioned from machine learning and traditional NLP
models to analyses based on generative language models [9, 10].
While assessing products is undoubtedly important, assessing process has been
recognised as valuable in the learning sciences and learning analytics
communities because it offers a more complete view of student abilities,
facilitates effective feedback, and aids in creating adaptive scaffolds for
personalized needs [11, 12]. We argue that a process-oriented view of
assessment will help to resolve the many of the challenges current educators
face when incorporating or accounting for GAI in their assessment design. Much
of the worry surrounding assessment and GAI is centered on the only evidence
of learning being a product with unknown providence [13]. Altering assessment
to instead focus on writing processes will make it clearer who (the student or
the AI) produced which actions and how these actions might positively or
negatively impact student learning.
While few studies of how people co-write with GAI have been reported to date,
the interfaces of various GAI tools suggest particular kinds of interaction
patterns. ChatGPT [4], for example, includes a text area for prompting the AI
and the ability to copy text, regenerate text, and like or dislike a response.
Similarly, Google’s _Bard_ [14] lets users modify responses within the
interface and export them to emails or word documents directly from the tool.
These affordances suggest that when using GAI writers may engage in varied
processes of prompting, as well as exploring, using, and modifying AI
generated responses.
#### 2.1.1 Knowledge Telling and Knowledge Transforming in Writing
Bereiter and Scardamalia’s work on knowledge building offers a lens with which
to view these kinds of interactive writing behaviors [15]. They distinguish
between _knowledge telling_ —a linear process of writing where individuals
focus on transcribing information without deep engagement or critical
restructuring—and knowledge transformation, a more critical engagement that
restructures arguments thoughtfully, synthesizes different viewpoints, and
creates narratives that are both cohesive and compelling. Raković [16]
extended this work to argue that knowledge transformation is a set of actions
whereby writers comprehend, evaluate, and select information sources to
fostering novel connections among previously disconnected fragments of
knowledge to facilitate learning.
In the context of writing with GAI, we operationalize knowledge telling and
knowledge transformation in terms of how learners select, modify, and apply
AI-generated text. More specifically, knowledge telling might appear as
learners accepting suggestions from the AI without alteration, following the
path laid out by the AI at any given moment. Knowledge transformation, on the
other hand, can be seen when users take AI suggestions as a basis for revision
and modify them to fit their needs. Learners may also engage in actions such
as declining initial GAI suggestions and seeking alternatives several times;
or they might deliberate between different sets of suggestions. Such example
defy a simple categorization under knowledge telling or knowledge
transformation. However, we argue below that the cognitive presence framework
can help us to understand these interactions.
#### 2.1.2 Cognitive Presence
Garrison situated [17] cognitive presence within the Community of Inquiry
(CoI) framework—a theoretical framework that promotes educational experience
through computer-mediated communication. Cognitive presence refers to the
extent to which learners are able to construct and confirm meaning through
communication in a specific sociocultural context. It is manifested through a
four-phase cycle that includes a _triggering event_ —the initial encounter
with a problem or question that ignites curiosity; _exploration_ —where
learners actively seek information to deepen their understanding to the
question; _integration_ —where gathered information is synthesized to
formulate coherent responses; and finally, _resolution_ —where synthesized
knowledge is used to propose a solution [17]. Prior work has largely focused
on the context of online discussion—for example, automatically identifying
cognitive presence in messages [18] and associating cognitive presence with
particular speech acts [19]. However, cognitive presence has yet to be
investigated in the context of collaborative writing between humans and AI.
In this study, we operationalize cognitive presence in terms of interactions
that take place while co-writing with GAI. Triggering events may occur when
learners approach GAI with an initial question or problem they encounter while
they are writing. Exploration may occur as learners actively engage with the
GAI to brainstorm or search through various suggestions. Integration may occur
as learners synthesize the ideas and suggestions received from the AI, working
towards constructing a cohesive piece of writing. In this way, integration is
similar to knowledge transformation as described above. Finally, resolution
may occur when learners fine-tune their piece with the aid of the AI, seeking
advice on polishing the language and structure to reach a satisfactory final
product.
The combination of knowledge telling, knowledge transformation, and cognitive
presence suggests a set of claims to include in the student model for
assessing the processes involved in co-writing with GAI: using generative AI
responses as is (knowledge telling); modifying generative AI responses to fit
your own goals (knowledge transformation/integration); requesting information
from GAI (triggering); comparing different GAI outputs (exploration); and
arriving at a final product that aligns with the human writer’s intent
(resolution).
### 2.2 ECD Task model
The task model specifies the tasks and task environment students will interact
with to evidence their learning. As mentioned before, several commercial
interfaces to GAI exist. These tools allow individuals to prompt GAI and
integrate the outputs into their own writing. While powerful, they fall short
of requirements of an efficient task model because the data they capture is
not easily accessible for assessment purposes. This lack of the accessibility
leaves educators with few options: either attempt to ban students from using
GAI—which is likely to fail because it is notoriously difficult to invigilate
students’ online activity [20]—or try to document students’ use of GAI—which
is difficult and time-consuming. It is hard to imagine a scenario where a
teacher reviews hours of videos or analyzes numerous screenshots of AI
interactions for each of their students. A more viable solution would be
platforms designed to automatically capture and preserve evidence related to
writing with GAI.
One such platform is CoAuthor [21]—an online tool designed to (a) afford
interactions with GAI and (b) passively capture those interactions for
analysis. CoAuthor was designed as a test-bed for investigating human-AI
collaborative writing. It provides users with writing prompts either for
argumentative writing (e.g. essays) or creative writing (e.g. stories) and
asks users to compose a response. As they write, they can seek suggestions
from GAI that continue their writing. Users are free to explore, accept, or
dismiss suggestions and modify them if accepted. While users are constrained
in their interactions with GAI compared to commercial tools—for example, they
cannot prompt the AI with their own questions—CoAuthor provides a suitable
task environment for assessment purposes because it automatically logs and
makes accessible fine-grained interactions with the system. All keystrokes,
cursor movements, and interactions with GAI are logged in both a human and
machine readable format. Here, we use existing data collected as part of the
CoAuthor project to demonstrate what a valid assessment for writing with GAI
might look like.
### 2.3 ECD Evidence Model
The ECD evidence model describes how students’ responses to the task will be
related to the claims in the student model. Recently, the sub-field of writing
analytics has developed a variety of methods for evidencing claims from
writing data [22]. To understand writing processes, trace data (e.g. keystroke
logging, mouse movement, eye tracking, etc.) [13] are typically analyzed using
process models. For example, recent research [23, 24, 25] has explored
students’ cognitive and metacognitive processes during essay writing via trace
data by applying techniques like semi-Markov processes [26].
To better understand writing processes and process models, researchers often
use process graphs, which show the relationships or connections between
different writing behaviors. For example, Leijten and Waes [27] used the
process graph Inputlog111https://www.inputlog.net to visualize writing
progression over time. They further adapted work into a network visualization
with Pajek222http://mrvar.fdv.uni-lj.si/pajek/ to highlight the key
information sources used during writing. Other researchers have used
representations called revision maps to investigate writing processes. For
example, Southavilay et al. [28] developed a revision map to visually track
changes in paragraphs of a collaborative document over time using color-coded
rectangles to signify the nature and extent of edits. Likewise, Shibani and
colleagues [29] created a revision graph to analyze the editing process in
essay drafts using nodes to represent sentences and edges to depict the
organizational changes across drafts.
Building on revision maps and process graphs, Shibani et al. [30] introduced
CoAuthorViz. This more advanced process graph visualizes co-writing behaviors
with GPT-3 suggestions, drawing upon trace data from _CoAuthor_. Different
from traditional process graphs, it emphasizes three key constructs:
"Suggestion Selection", "Suggestion Modification", "Empty GPT-3 Call", and
"User Text Addition" detailing the sequence of writing actions at the sentence
level. These graphical representations offer insights into sentence-level
differences in compositions and revisions.
The process models and graph approaches described above are commendable for
their focus on the relationships between features that characterise writing;
however, they have important limitations as components of an evidence model.
First, traditional process graphs and revision maps are useful for
understanding individual or aggregated writing processes; however they do a
poor job of effectively comparing processes between individuals or groups.
This is primarily due to their lack of a coordinated semantic space for
network comparisons. Second, while Markov-models, traditional process graphs,
and revision maps excel at showing detailed relationships, they lack
corresponding summary statistics that can be used to quantify the network
interpretation. Such summary statistics are useful because they afford process
comparisons at scale using standard statistical techniques.
The process graph introduced in CoAuthorViz distinguishes itself by analyzing
writing across two groups of users and incorporating summary statistics.
Despite these advancements, the tool has several notable limitations. First,
the constructs included in the model are relatively broad and may therefore
miss the fine-grained behaviors essential for analyzing writing in a nuanced
and theoretically driven way. For example, "Suggestion Selection" consists of
three distinct behaviors—suggestion seeking, exploration, and acquirement, but
these are aggregated together and analyzed as a single construct that is not
explicitly related to theories that describe the cognitive aspects of writing.
Second, their comparison of group of users was based on the final written
product, not writing processes.
To address these limitations, we used epistemic network analysis (ENA) as the
major component of our evidence model. ENA is a widely used learning analytic
technique for modeling actions via network models [31]. ENA coordinates
multiple networks in the same embedding space to facilitate visual and
statistical comparisons of learning processes at scale. While ENA and related
techniques have been used to model metacognitive processes during writing
tasks [32], no work has explored ENA as method for evidencing the cognitive
behaviors inherent to co-writing with GAI.
### 2.4 Research Questions
Our study aimed to demonstrate an assessment method for human-AI collaborative
writing. Framed by ECD, we used elements of knowledge-telling, knowledge
transformation, and cognitive presence to identify claims for our student
model (see hypotheses below); we used data collected from the CoAuthor writing
tool as a proxy for our task model; and we used ENA to evidence claims for the
student model. Ideally, this assessment approach would distinguish high-
quality human-AI collaborative writing processes from low-quality ones.
Unfortunately, the CoAuthor dataset does not include writing quality scores
that could be correlated with writing process measures for this purpose.
However, the dataset does contain a number of conditions that should be
distinguishable with a valid assessment method. In particular, written
products and trace data were divided by how much ownership—i.e., the
proportion of content generated by the human author versus the GAI—the human
had over the final product; whether the author was responding to a creative
writing prompt or an argumentative writing prompt; and whether the GAI had a
high or low temperature setting—that is, whether its outputs had high or low
textual variability. To provide a proof of concept for our assessment method,
we compared these conditions according to the following research questions:
RQ1
Did authors with higher ownership over the final product write differently
than those with lower ownership over the final product?
RQ2
Did authors who responded to creative writing prompts write differently than
those who responded to argumentative prompts?
RQ3
Did authors who use GAI with a higher temperature setting—that is, more
variable output—write differently than those who used a lower temperature
setting?
Regarding RQ1, we hypothesized that: (1) authors who had lower ownership over
the final product would have writing processes more characterised by knowledge
telling and triggering events because they sought and relied more on ready-
made suggestions provided by the GAI; (2) conversely, authors with higher
ownership over the final product would have processes more characterised by
knowledge transformation, exploration, integration, and resolution because
they tended to adapt GAI suggestions to better fit their writing.
To address RQ2, we built upon genre theory [33], which suggests major
differences between fiction (e.g. creative) and non-fiction (e.g.
argumentative) writing. Namely, fiction writing is inherently less bound to a
"reality status" or tight relationship to the real world. Therefore, authors
in this genre are more free to explore the boundaries of imagination and
creativity without the limits of factual accuracy. On the other hand, non-
fiction writing is grounded in reality, requiring a structured approach to
presenting logical arguments based on facts and evidence. Thus, for RQ2 we
hypothesized that: (3) authors responding to creative writing prompts would
have writing processes more characterised by knowledge telling, triggering
events, and exploration as they would likely need to adapt GAI responses less
to maintain a reality status; and (4) authors responding to argumentative
writing prompts would have writing processes more characterised by knowledge
transformation, integration, and resolution as they may have needed to align
GAI outputs (which are known to include falsehoods [34]) with reality.
High temperature settings mean that GAI will tend to produce varied outputs to
the same request, while low temperature settings mean that GAI will tend to
produce repetitive outputs. Regarding RQ3, we hypothesized that: (5) authors
interacting with lower temperature GAI would have processes more characterized
by knowledge transformation, integration, and resolution as they would need to
actively think and rework the available suggestions; (6) while those
interacting with higher temperature GAI would have processes more
characterized by exploration and knowledge telling as they might find it
easier to select suggestions that align closely with their intended message
without significantly transforming the information.
## 3 Method
As shown in Figure 1, our methodological steps included (a) pre-processing the
keystroke-level data from CoAuthor; (b) qualitatively analyzing these data to
derive codes related to knowledge telling, transformation, cognitive presence,
and general writing behaviors for subsequent analysis; (c) the development,
validation, and application of classifiers for automatic assignment of the
codes to the data; (d) applying ENA to the data to create networks that
described the writing processes of authors in different experimental
conditions; and (e) using summary statistics from ENA to statistically compare
the experimental conditions via mixed-effects regression analysis.
Figure 1: Methodology overview (the leftmost image sourced from [21])
### 3.1 Data
We used data collected from the CoAuthor project as a proxy for the task model
of our proposed assessment methodology. These data include digital records of
1,445 writing sessions from 60 authors recruited from Amazon Mechanical Turk.
Authors were asked to respond to one or more prompts. Prompts could be from
the creative writing condition, which were sampled from the r/WritingPrompts
subreddit, or from the argumentative writing condition, which were sampled
from a New York Times prompt library. During the writing sessions, authors
interacted with a user interface that included a standard text editor as well
as a tab that allowed them to request up to five suggestions from GPT-3 for
continuing the currently written text.
Overall, elements of the dataset we used included two distinct types of data:
* •
Trace Data: Sequential actions and events throughout the writing session.
These included user actions such as text insertion, deletion, and cursor
movement, as well as system interactions like opening, getting, and closing
suggestions. Furthermore, it traced the progression of the written content on
an event-by-event basis.
* •
Metadata: Details such as the session time, participant ID, session ID, prompt
condition, prompt ID, GPT temperature, as well as metrics that summarise the
interactions between the user and GPT such as ownership—that is the percentage
of the final written text produced by the human author vs. GPT-3.
### 3.2 Pre-processing
The original data included 830 creative writing sessions and 615 argumentative
writing sessions which averaged 1,862 events per session. We removed eight
sessions from the analysis due to participants removing the original prompt
and commencing a different writing piece, and one session due to missing
metadata about identification of the author. In addition, we identified and
manually corrected or excluded events with formatting errors. Our final
dataset consisted of 822 creative writing sessions and 614 argumentative
writing sessions that included more than 2.6M events. For sessions using
argumentative writing prompts, low/high temperature values were 0.2 and 0.9;
for creative writing prompts they were 0.3 and 0.75. Author and GAI ownership
was defined using a median split on the percentage of characters authored by a
human (mdn = 76%).
### 3.3 Qualitative Analysis
ENA requires a coded dataset for analysis—that is, data where the relevant
actions have been labelled. To derive codes, we combined a top-down approach
informed by theory with a bottom-up approach that sought to identify relevant
themes in the data without explicit reference to theoretical frameworks[35].
We qualitatively investigated 40 randomly sampled sessions for code
identification. Two authors discussed and refined these codes to arrive at the
final list below in Table 1.
Table 1: Qualitative codes, definitions, and identifiers.
Code | Definition | Identifiers
---|---|---
compose | Writing new content at the end of the existing text | event name is "text-insert" in the end of text (space removed where applicable)
relocate | Rearranging sentences | The index of same sentence changes in last-current document
reflect | Revising the content after or near completing a draft | Revise content after finishing 90% content
seek suggestion | Obtaining suggestions from GPT | Event name is ’suggestion-get’
dismiss suggestion | Dismissing suggestions from GPT | Event name is ’suggestion-close’ and event source is ’user’
accept suggestion | Accepting a suggestion from GPT | Event name is ’suggestion-select’
hover suggestion | Hovering over the suggestions | Event name is ’suggestion-hover’
cursor forward | Moving the position of cursor forward | Event name is "cursor-forward"
cursor backward | Moving the position of cursor backward | Event name is "cursor-backward"
cursor select | Selecting text | Event name is "cursor-select"
revise user | Revising content they wrote | The inserts or deletes in-text content and ownership of revising sentence is ’user’
revise suggestion | Revising a GPT suggestion | The user inserts or deletes in-text content and ownership of revising sentence is ’api’
low modification | Making minor adjustments to sentences without altering their core meaning | Sentence semantic similarity > 0.8
high modification | Making significant changes to sentence meaning | Sentence semantic similarity < 0.8
In our analysis, codes were related to the student model in terms of the
presence or absence of co-occurrences or _connections_ between them (see 3.5).
We interpreted _knowledge telling_ in terms of connections among seek
suggestion, accept suggestion, revise suggestion, and low modification;
_knowledge transformation_ and _integration_ in terms of connections among
seek suggestion, accept suggestion, revise suggestion, high modification, and
relocate; _triggering events_ in terms of connections to seek suggestion and
accept suggestion; _exploration_ in terms of connections among seek
suggestion, accept suggestion, hover suggestion, and dismiss suggestion; and
_resolution_ in terms of connections among seek suggestion, accept suggestion,
and reflect. In addition to codes that we mapped to knowledge
telling/transformation/cognitive presence, our qualitative analysis suggested
that including codes related to composing, revising one’s own text, and cursor
movements would be useful for more completely understanding writing processes.
### 3.4 Automated Classifier Development
Given the scale of the data, we developed automated classifiers to label it
for our codes. This was simple to implement for the majority of codes as all
that was required was a string match with the corresponding event name in the
data. However, to code for revision behaviors, a more complex algorithm was
required (see below). The details of this algorithm, along with paper-related
data and analysis can be found in our Github
repository333https://github.com/yixin-cheng/coAuthor.
Sentence-level segmentation and analysis allowed us to identify and code the
high and low modifications behaviors. By using NLTK sentence tokenizer and
identifying ending markers, we collected all sentences in their initial form
including their cursor range and ownership (i.e., prompt, user, or api), into
a list. Following this, the list was used to track these sentences under the
following conditions: Sentence Removal, Sentence Merger, and Sentence
Revision. In the case of sentence removal, the corresponding sentence is
marked as ’None’ in our sentence list. For sentence revisions, the sentence
list is updated after every revision. We excluded sentence pairs with a single
word difference identified as a misspelling by the spellchecker Python package
from being counted as a revision. For sentence mergers, we used TF-IDF to
generate word embeddings and calculated the cosine similarity between the
updated sentence and the original sentence. The most similar sentence was
replaced with the merged version, with others marked as ’None’. The final
sentence list’s accuracy was corroborated by its exact match with the final
story or essay versions from the data, validating our proposed algorithm.
Next, we identified the cursor location of revisions between initial and final
sentence lists. The codes revise user and revise suggestion, were identified
based on the combination of sentence ownership and updates to the sentence
list. To code low modification and high modification, we used Sentence-BERT
[36] to compute sentence cosine similarities between initial and final
sentence pairs. We used a threshold value of 0.8 to distinguish high and low
modification [37].
To test the reliability of the high/low modification classifiers and the
validity of the codes, we randomly sampled 80 events from the data. Two human
raters then manually coded these data for the presence/absence of high/low
modifications. Next, for both codes we did pairwise comparisons between each
set of human coding and the automated classifications. The codes were
considered valid and reliable if all pairs of ratings achieved a Cohen’s kappa
greater than 0.65 and Shaffer’s rho less than 0.05. All kappa and rho values
for these two codes exceeded these thresholds. The end result of the automated
coding process was a dataset with binary values that indicated the presence or
absence of a given code for a given event.
### 3.5 Epistemic Network Analysis
To analyze the coded dataset, we used the ENA implementation for R [38]. ENA
creates a separate network representation for each unit of analysis where the
nodes of the networks are codes and edges between nodes indicate the relative
frequency of co-occurrence between those codes in a given unit’s data. To
construct these networks, we used the following ENA specifications:
* •
Units of analysis: A separate network was created for each combination of
session ID and user ID.
* •
Codes: All codes listed in Table 1.
* •
Conversations: The data was grouped by session ID, user ID, and sentence for
co-occurrence accumulation. That is, codes in all events associated with a
given session, user, and sentence could co-occur.
* •
Window Size: An "infinite" stanza window was used to identify the co-
occurrence between codes within the data for each unit of analysis. This
window begins at the first event in a given conversation and expands to
include all events within that conversation. In this way, all coded events
within the conversation co-occur, but the result is a more fine-grained
accumulation of the co-occurrence structure in the data.
ENA projects the networks into a low dimensional embedding space using
singular value decomposition to produce ENA scores for each network. This
space distinguishes networks in terms of the linear combination of co-
occurrence variables that explain the most variance in the data. To help
interpret the dimensions of this space, ENA co-registers the network graphs in
the embedding space such that the position of the nodes and the connections
they define align with the most influential co-occurrences on that dimension.
Thus, researchers can visually interpret the dimensions according to the
connections at the extremes. For more details, see [39].
### 3.6 Regression Analysis
To directly address our research questions, we conducted a regression analysis
of the ENA results that compared the ownership, genre, and temperature
conditions. Our models regressed the ENA scores for each dimension on
categorical predictors for ownership (user vs. GAI), prompt type
(argumentative vs. creative), and temperature (high vs. low). For each writing
session, these values were derived from the associated metadata in the
CoAuthor dataset.
Authors may have participated in multiple writing sessions. As a result, the
ENA scores were nested within participants. In addition, each participant may
have written multiple responses to prompts of the same kind (e.g., userID
A118B participated in four argumentative writing session and three creative
writing sessions), meaning that participants could also be nested into prompt
kinds. To accommodate this nested structure, we used _mixed-effects_
regression models [40]. To test for the effect of nesting, we calculated the
intraclass correlation coefficient (ICC) for the ENA scores grouped within
participant and prompt using the ICC package for R. Confidence intervals
around the ICC scores suggested that nesting was significant for the
participant variable, but not the prompt variable. In turn, we included the
participant identifier as a random effect (intercept) in all regression
models.
The outputs of interest were the regression coefficients of the ownership
prompt type, and temperature variables, which represent the difference between
the mean outcome scores of the two levels of each variable. We conducted the
regression modelling using the lmer and lmerTest packages for R [41, 42].
lmerTest computes significance tests for the regression coefficients using
Satterthwaite’s method [43]. Confidence intervals around the regression
coefficients were calculated via bootstrapping using the percentile method
[44]. Finally, we calculated the effect size of any significant regression
coefficients of interest using Cohen’s $d$ [45].
### 3.7 Network Comparisons
To interpret the dimensions of the ENA embedding spaces and better visualise
differences between subgroups, we examined the corresponding mean ENA
networks. That is, for given subgroup in the data, we averaged the edge
weights of their associated networks and plotted them in the embedding space.
To compare any two subgroups, we computed their network difference graphs by
subtracting their corresponding edge weights to show which co-occurrences were
more frequent in one group relative to the other.
## 4 Results
### 4.1 ENA Embedding Space
Figure 2: ENA embedding space
The resultant ENA embedding space is shown in Figure 2. The first dimension
primarily distinguishes between authors seeking suggestions from GAI
(seekSugg) on the left and revising their own writing (reviseUser) on the
right. A shift from left to right reflects the degree to which authors tended
to rely on suggestions versus composing and editing their own writing. The
second dimension primarily distinguishes between composing (compose) at the
top and behaviors related to suggestions and modifications (seekSugg, high
modification, and low modification) at the bottom. A higher position along
this dimension indicates a greater emphasis on composition, whereas a lower
position indicates a greater emphasis on modifications and suggestion use. The
dimensions of the space account for 22.4% and 11.3% of the total variation in
the data, respectively.
### 4.2 Regression Results
| X | Y
---|---|---
Intercept | $-0.03$ | $0.05$
| $(0.04)$ | $(0.03)$
prompt types_creative | $0.01$ | $\mathbf{-0.06}^{***}$
| $(0.02)$ | $(0.01)$
ownership_high | $\mathbf{0.09}^{***}$ | $-0.00$
| $(0.02)$ | $(0.02)$
temperature_high | $0.02$ | $-0.02$
| $(0.02)$ | $(0.01)$
AIC | $569.90$ | $33.03$
BIC | $601.52$ | $64.64$
Log Likelihood | $-278.95$ | $-10.51$
Num. obs. | $1435$ | $1435$
Num. groups: author_id | $60$ | $60$
Var: worker_id (Intercept) | $0.07$ | $0.02$
Var: Residual | $0.08$ | $0.05$
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table 2: Regression models with ENA scores on either dimension as the
dependent variable. Standard errors in parentheses.
The results of the regression analysis are shown in Table 2. We report two
models that include author_id as a random effect, prompt types, ownership, and
temperature as fixed effects, and ENA scores on either the first or second
dimension of the ENA space as the outcome. Testing for interactions between
the fixed effects did not yield significant results, so we report only the
main effects here.
On the first dimension (X), only the ownership variable was significant.
Authors with more ownership over their final written product (user ownership)
were significantly higher on the dimension than those with less ownership (GAI
ownership): $t=4.35$, $p<0.001$, $d=0.24$, 95% CI $[0.05,0.13]$. Given the
interpretation of the first dimension above, this result suggests that user
ownership authors tended to focus on composing and revising their own writing
significantly more than GAI ownership authors.
On the second dimension (Y), only the prompt condition variable was
significant. When the authors responded to creative writing prompts, they were
significantly lower on the dimension compared to when they responded to
argumentative prompts: $t=-4.71$, $p<0.001$, $d=-0.22$, 95% CI
$[-0.09,-0.04]$. This result suggests that when the authors responded to
creative writing prompts, they tended to focus on editing their writing and
using GAI suggestions significantly more than when they responded to
argumentative writing prompts.
The temperature variable was not statistically significant on either dimension
suggesting that we have insufficient empirical evidence to falsify the null
hypothesis of no difference between the high and low temperature conditions.
### 4.3 User vs. GAI ownership
Figure 3 shows the network comparison for the _ownership_ variable; user
ownership on the left (a) and GAI ownership in the middle (b). Both networks
include a large number of connections to the compose node suggesting that
authors tended to link their composition process to a variety of other
behaviors. For example, both networks show relatively strong connections
between compose and seekSugg, acceptSugg, reviseUser, and cursor movements.
The network differences are shown in plot (c). Here, red edges indicate more
frequent co-occurrences in the GAI ownership group; blue edges indicate more
frequent co-occurrences in the user ownership group. The graph indicates that
the GAI ownership authors made stronger connections between compose and
seekSugg, compose and acceptSugg, and reviseSugg and lowModification. User
ownership authors, on the other hand, made stronger connections between
compose and cursor movements and compose and reviseUser. The centroids of the
networks—that is average ENA scores (blue and red squares)—corroborate the
regression analysis, with the GAI ownership authors being further to the left
on the first dimension than the user ownership authors, on average.
Figure 3: Ownership
### 4.4 Creative vs. Argumentative
Figure 4 shows the network comparisons for the prompt types variable; the mean
network for responses to creative writing prompts is on the left (a), and the
mean network for responses to argumentative writing prompts is in the middle
(b). As before, in both networks, connections to the Compose node feature
prominently. The subtracted network is shown in subplot (c). Here, blue edges
represent connections that occurred more frequently in the creative condition
and red edges represent connections that occurred more frequently in the
argumentative condition. The graph indicates that when responding to creative
writing prompts, authors tended to explore AI generated suggestions more,
having stronger connections among seekSugg, hoverSugg, and compose. When
responding to argumentative prompts, authors tended to focus more on composing
their own writing and revising AI-generated suggestions—as indicated by
stronger connections between compose and cursorSelect, compose and acceptSugg,
and reviseSugg and lowModification. The centroids of the two networks
corroborate the regression analysis, with the argumentative condition being
higher on the second dimension than the creative condition, on average.
Figure 4: Prompt type
### 4.5 High vs. Low Temperature
Figure 5 contrasts for the networks for the high (a) and low (b) temperature
conditions. The network subtraction (c) indicates that the authors in the high
temperature condition made stronger connections between compose and seekSugg,
compose and reviseUser, and compose and cursor movements. The authors in the
high temperature condition made stronger connections between compose and
acceptSugg. The centroids of the two networks overlap on both dimensions,
corroborating the regression results of no significant difference between the
two conditions.
Figure 5: Temperature
## 5 Discussion and Conclusions
In this study, we sought to demonstrate an assessment method for human-AI
collaborative writing. Framed by ECD, we used elements of knowledge-telling,
knowledge transformation, and cognitive presence to identify claims for our
student model; we used data collected from the CoAuthor writing tool as a
proxy for our task model; and we used ENA to evidence claims about the student
model using data from the task model. More specifically, we compared the co-
writing behaviors of users across three conditions: high vs. low ownership;
creative vs. argumentative prompt types; and high vs. low temperature. We
found statistically significant differences between the process of authors in
the high/low ownership conditions and the creative/argumentative conditions.
Specifically, authors with GAI ownership over their final product tended to
rely more on GAI suggestions, while those with user ownership tended to focus
more on composing and revising their own writing. When responding to creative
writing prompts, authors tended to explore GAI suggestions more, while those
responding to argumentative prompts tended focus on composing their own
writing and making small revisions to GAI writing.
In terms of more specific assessment claims, the results support Hypothesis
1—that frequent users of GAI outputs would tend more towards knowledge telling
and triggering events, as evidenced by stronger connections between compose
and seekSugg, as well as compose and acceptSugg. Hypothesis 2—that less
frequent users of GAI would tend toward knowledge transformation, exploration,
integration, and resolution was not supported. Authors with more ownership
over their written product sought and accepted some GAI suggestions but did
not tend to revise them. Instead their revisions tended to be made on their
own text with only slight modifications as evidenced by stronger connections
among compose, reviseUser, and lowModification.
The results were mixed for Hypothesis 3—that authors responding to creative
writing prompts would tend toward knowledge telling, triggering events, and
exploration. On the one hand, they did focus more on triggering events and
exploration, as evidenced by stronger connections between seekSugg and
hoverSugg, as well as compose and hoverSugg. On the other hand, evidence of a
greater focus on knowledge telling is less clear. Authors that responded to
argumentative prompts had stronger connections between acceptSugg and compose.
But they also had stronger connections between reviseSugg and lowModification,
indicating at least some level of transformation that was less prevalent for
those responding to creative writing prompts. The results were similarly mixed
for Hypothesis 4—that authors responding to argumentative writing prompts
would be characterized by knowledge transformation, integration, and
resolution. There is some evidence for this given stronger connections between
reviseSugg and lowModification, but the lack of connections to
highModificaiton and reflect limit this interpretation.
Hypothesis 5 and Hypothesis 6—that authors interacting with lower temperature
GAI would tend more toward knowledge transformation, integration, and
resolution, while authors interacting with higher temperature GAI would tend
more toward knowledge telling and exploration—were not supported. While
authors in the high temperature condition did have stronger connections to
acceptSugg and authors in low temperature condition had stronger connections
to seekSugg, their overall co-occurrence patterns were highly similar.
Our study has several limitations. First, as with any study, our results are
limited by the data at hand. In this case, our data comes from self-selected
participants who engaged with the CoAuthor platform. This potentially
introduces a selection bias, as these individuals represent a subset with
specific characteristics or preferences, which might not be fully
representative of the wider population.
Second, the tasks and environment we used are not necessarily representative
of the broader selection of GAI tools for writing. CoAuthor constrains user
interactions with GAI by only allowing them to seek suggestions that continue
the current text. Widely used tools, however, afford less restricted
interactions where users can phrase questions to the GAI any way they want, as
well as include "steering" instructions that tell the GAI how to respond in
general. Moreover, the available CoAuthor dataset contains interactions with
the now outdated GPT-3. It is possible that interactions with more
contemporary versions of GAI might yield different results.
Third, our coding scheme, and thus our proposed student model, only focused on
a narrow subset of cognitive aspects of writing related to observable
behaviors in the data. Our analysis did not consider other potentially
important features such as the semantic content of the writing and the
metacognitive strategies being used. Future studies will explore these as
possible assessment targets.
Finally, our proposed assessment method is plausible but its inherent
complexity could restrict its scalability and accessibility. We aim to address
these limitations by developing an adaptable and flexible API that includes
customizable parameters, such as event names, to meet diverse task
requirements such as post-event or real-time analysis in human-AI writing
environments. This solution seeks to bridge the gap between the current proof
of concept and its practical, large-scale application.
Despite these limitations, our work provides a proof of concept for the
evidence-centered assessment of writing composed with the help of GAI. Our
methodology posits specific features of human-AI collaborative writing to
target; adopts an existing task model to produce assessment data; and
leverages process models to relate these data to differences between the
writing processes of participants. We hope that this work will continue to
expand such that we have assessments not just for writing, but a variety of
meaningful interactions between learners and AI. Such assessments will be
crucial to preparing learners for an age in which effective human-AI
collaboration is an essential skill.
## Acknowledgments
This research was in part supported by the Australian Research Council
(DP220101209, DP240100069) and Jacobs Foundation (CELLA 2 CERES).
## Citation
Yixin Cheng, Kayley Lyons, Guanliang Chen, Dragan Gašević, and Zachari
Swiecki. 2024. Evidence-centered Assessment for Writing with Generative AI. In
The 14th Learning Analytics and Knowledge Conference (LAK ’24), March 18–22,
2024, Kyoto, Japan. ACM, New York, NY, USA, 16 pages.
https://doi.org/10.1145/3636555.3636866
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|
# Quantum enhanced non-interferometric quantitative phase imaging
Giuseppe Ortolano Quantum metrology and nano technologies division, INRiM,
Strada delle Cacce 91, 10153 Torino, Italy DISAT, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy Alberto Paniate Quantum
metrology and nano technologies division, INRiM, Strada delle Cacce 91, 10153
Torino, Italy DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24,
10129 Torino, Italy Pauline Boucher Carmine Napoli Quantum metrology and
nano technologies division, INRiM, Strada delle Cacce 91, 10153 Torino, Italy
Sarika Soman Silvania F. Pereira Imaging Physics Dept. Optics Research
Group, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg
1, 2628CJ Delft, The Netherlands Ivano Ruo Berchera Quantum metrology and
nano technologies division, INRiM, Strada delle Cacce 91, 10153 Torino, Italy
Marco Genovese Quantum metrology and nano technologies division, INRiM,
Strada delle Cacce 91, 10153 Torino, Italy
###### Abstract
Quantum entanglement and squeezing have significantly improved phase
estimation and imaging in interferometric settings beyond the classical
limits. However, for a wide class of non- interferometric phase
imaging/retrieval methods vastly used in the classical domain e.g.,
ptychography and diffractive imaging, a demonstration of quantum advantage is
still missing. Here, we fill this gap by exploiting entanglement to enhance
imaging of a pure phase object in a non-interferometric setting, only
measuring the phase effect on the free-propagating field. This method, based
on the so-called “transport of intensity equation”, is quantitative since it
provides the absolute value of the phase without prior knowledge of the object
and operates in wide-field mode, so it does not need time-consuming raster
scanning. Moreover, it does not require spatial and temporal coherence of the
incident light. Besides a general improvement of the image quality at a fixed
number of photons irradiated through the object, resulting in better
discrimination of small details, we demonstrate a clear reduction of the
uncertainty in the quantitative phase estimation.
Although we provide an experimental demonstration of a specific scheme in the
visible spectrum, this research also paves the way for applications at
different wavelengths, e.g., X-ray imaging, where reducing the photon dose is
of utmost importance.
## I Introduction
Quantum imaging Berchera and Degiovanni (2019); Moreau et al. (2019); Genovese
(2016) and sensing Degen et al. (2017); Pirandola et al. (2018); Petrini et
al. (2020) have provided genuine and valuable advantages in many measurement
applications ranging from fundamental physics Aasi et al. (2013); Pradyumna et
al. (2020) to biology Taylor and Bowen (2016); Casacio et al. (2021); Petrini
et al. (2022) from microscopy Schwartz and Oron (2012); Gatto Monticone et al.
(2014); Samantaray et al. (2017); Tenne et al. (2019)to optical sensors Lawrie
et al. (2019); Lee et al. (2021).
In particular, given the importance of optical phase measurement, appearing in
all the science fields, a considerable effort has been made to exploit quantum
entanglement or squeezing for this task. Quantum phase estimation through
first-order interference involving the mixing of two optical modes in a linear
Polino et al. (2020); Demkowicz-Dobrzanski et al. (2015) or non-linear
Chekhova and Ou (2016); Kalashnikov et al. (2016); Töpfer et al. (2022)
interaction is well understood. The ultimate uncertainty bound with quantum
optical states is known to scale with the number of probing particles $N$ as
$N^{-1}$, the so-called ’Heisenberg scaling’. In contrast, for the classical
probing state, it is limited to $N^{-\frac{1}{2}}$, referred to as the
standard quantum limit (SQL) or shot noise limit. Although the quantum
advantage would, in principle, be disruptive for $N\gg 1$ in a realistic
scenario, the gain over the SQL is rather, in the form of a constant depending
on the optical losses Demkowicz-Dobrzanski et al. (2015). Proofs of principle
of quantum-enhanced linear interferometry with the so-called entangled NOON
states have been achieved, for example, in phase contrast Ono et al. (2013)
and polarization scanning microscopy Israel et al. (2014), usually limited to
the case of N=2. However, the generation and preservation of NOON states
involving a higher number of particles are extremely demanding, so their
usability in a real-world application is questionable. More practical is the
use of squeezed states Caves (1981); Xiao et al. (1987); Aasi et al. (2013);
Schnabel (2017); Gatto et al. (2022). Non-linear interferometry involving a
parametric amplifier instead of a beam splitter for mixing the light mode is
promising for some applications, especially because the detection can be done
at a wavelength different from the probing one Kalashnikov et al. (2016);
Töpfer et al. (2022) and the quantum advantage is independent of the detection
noise Manceau et al. (2017). Moreover, with some remarkable exceptions
Camphausen et al. (2021); Frascella et al. (2019), these interferometric
schemes do not provide spatial resolution or require raster scanning for
extended samples.
Other phase imaging methods born in the quantum domain exploit second-order
intensity correlation (or two-photon coincidence) among signal and idler beams
of SPDC to retrieve the phase information. In contrast, the first-order
intensity measurement of either the signal or the idler arm does not show
interference Gatti et al. (2004). These techniques include ghost imaging and
diffraction Strekalov et al. (1995); Valencia et al. (2005); Zhang et al.
(2005); Shapiro and Boyd (2012); Losero et al. (2019) quantum holography Lemos
et al. (2014); Vinu et al. (2020); Devaux et al. (2019); Defienne et al.
(2021), quantum Fourier ptychography Aidukas et al. (2019) and phase
reconfigurable contrast microscopy Hodgson et al. (2023). In general, the
signal-to-noise ratio (SNR) is smaller compared to direct first-order
measurement if considering the same number of impinging photons on the object.
However, some advantages can be found in some cases at few photon illumination
levels, for example, the rejection of independent external noise Meda et al.
(2017); Erkmen and Shapiro (2009); Brida et al. (2011); Morris et al. (2015),
robustness through turbulence and scattering Dixon et al. (2011); Bina et al.
(2013).
Here we present a quantitative non-interferometric quantum-enhanced phase
imaging (NIQPI) scheme exploiting quantum correlations that do not belong to
any of the techniques mentioned above since it does not involve either
interference or second-order intensity correlations. In fact, only first-order
intensity in both branches is measured, so the full-field phase retrieval is
obtained in real-time by single-shot measurement. We will demonstrate, both
theoretically and experimentally that the method can provide a clear advantage
compared to the corresponding classical direct imaging thanks to the quantum
correlation.
The NIQPI protocol exploits the scheme depicted in Fig. 1. We consider two
quantum correlated beams produced by the spontaneous down-conversion process
(SPDC), usually dubbed as signal beam ($s$) and idler ($i$) beam, with
intensity patterns that are perfectly identical in the far-field, point-by-
point. Even the shot noise fluctuation is, in principle, perfectly reproduced
in the two beams, which is impossible in the classical domain. The far field
of SPDC is imaged at the sensors of a highly efficient and low-noise CCD
camera. Only the signal beam probes the object, while the idler one is used as
the reference for the noise. When the object is placed close to the far field
but not exactly there, it produces an intensity perturbation on the signal
photons propagation that is registered at the CCD camera. In particular, by
measuring the signal intensity pattern $I(\bm{x},\pm dz)$ at the detection
plane for two different ’defocused’ object positions along the $z$-axis,
namely, $+dz$ and $-dz$, it is possible to reconstruct the phase profile
$\phi(\bm{x},z=0)$, by solving the so-called transport of intensity equation
(TIE) Teague (1983):
$-k\frac{\partial}{\partial
z}I(\bm{x},z)=\nabla_{\bm{x}}\cdot\left[I(\bm{x},0)\nabla\phi(\bm{x},0)\right]$
(1)
where the derivative is approximated by the finite difference of two
measurements out of focus, $\frac{\partial}{\partial
z}I(\bm{x},z)\approx[I(\bm{x},dz)-I(\bm{x},-dz)]/(2dz)$ and $I(\bm{x},z=0)$ is
the far field intensity of the source. TIE is experimentally easy and
computationally efficient as compared to the conventional phase retrieval
techniques and, under suitable hypotheses, the method leads to a unique and
quantitative wide-field image of the phase profile Teague (1983); Paganin and
Nugent (1998); Zuo et al. (2020). However, the reconstruction obtained in the
signal arm can be strongly affected by the detection noise and by the shot
noise if low illumination is used (see $M\&M$ for a detailed discussion). On
the one hand, a faithful reconstruction through Eq. (1) requires a small
defocus distance $|dz|$ in order to well approximate the derivative on its
right-hand side. But, on the other hand, if $|dz|$ is too small, the effect of
the phase gradient on the measured intensity becomes negligible and can be
covered entirely by the shot noise. Here, we show that the noise pattern
acquired on the idler beam can be used to reduce the effect of the shot noise
in the signal beam, enhancing the overall phase image reconstruction and
reducing the uncertainty on the quantitative spatially resolved phase
retrieval Ortolano et al. (2019). NIQPI can work with partially coherent light
and has some advantages compared to interferometric schemes: it can be
directly applied to wide-field transmission microscopy settings and it is
intrinsically more stable than an interferometric setup Zuo et al. (2020).
Moreover, since NIQPI is based on free propagation effect, it can be realized
without using lenses and optical components, thus being particularly suitable
in extreme UV or X-Ray imaging, where optical components are not efficient but
where SPDC sources are available and quantum enhanced detection has been
already demonstrated Borodin et al. (2016); Sofer et al. (2019).
Figure 1: Scheme of the NIQPR. Two correlated beams labeled signal ($s$) and
idler ($i$) are generated by the spontaneous parametric down conversion (SPDC)
pumped by a CW laser @405nm and propagate through an imaging system composed
of two lenses ($L_{1}$ is the far field lens with focal length $F=1$ cm and
$L_{2}$ is the imaging lens with focal length of 3 cm) and a test object. An
interference filter (IF) is used to select a bandwidth of 40 nm around the
degenerate wavelength (@810nm) and to block the pump. $L_{2}$ images the far
field plane on the camera chip around the focal plane with a magnification
factor of about 8. The object is placed near to the far field of the source,
and only the probe beam interacts with it. Phase information can be retrieved
from intensity measurements taken at some out of focus ($\pm dz$) planes.
Figure 2: Sample. Pure phase objects used in the experiment are sketched.
Figure 3: Experimental Reconstruction of the $``\pi$” sample as a function of
the defocusing distance. First row presents the phase reconstruction when 100
intensity patterns are used. Second and third rows show the single frame
reconstructions for the classical and the quantum case, respectively. The size
of each image is $80\times 80$ $\text{pix}^{2}$
## II Results
In our experiment, the number of photons per pixel per frame is about
$n\approx 10^{3}$, so that for the purpose of this work we can substitute the
continuous quantity $I(\bm{x})$ appearing in Eq. (1) with the number of
photons detected by the pixel at the coordinate $\bm{x}$. Actually, before the
TIE algorithm, we apply an averaging filter of size $d=4$ to the intensity
image, that consists in replacing the count in each pixel by the average count
of a neighborhood of size $4\times 4\;\text{pix}^{2}$ around it, so that the
final image conserves the same number of pixels. However, the averaging filter
does not have any influence on the classical reconstructions, neither positive
nor negative, while it improves the quantum reconstruction (see discussion in
M$\&$M and related Fig. 7). From now on we will refer to $I(\bm{x})$ with that
meaning, namely after the application of such averaging filter.
It is essential to point out that the SPDC source operates in the regime of
very low photon number per spatio-temporal mode. In this limit, the photon
statistics follows a Poisson distribution (see M$\&$M Sec.IV for details). So,
aside from the negligible contribution of electronic readout noise, the
measurement on the single beam is shot noise limited.
We image pure phase objects reported in Fig. 2 with 66 $\pm$ 3 nm thickness
estimated by profilometer (DektakXT, Bruker). It corresponds to a phase shift
of 0.230 $\pm$ 0.012 rad @ 810 nm, the central degeneracy frequency of the
SPDC photons. The samples have been realized by etching structures on a UV-
fused Silica glass window using buffered oxide etch.
Fig. 3 shows the experimental reconstructions of the $``\pi$”-shaped phase
sample of Fig. 2 as a function of the defocussing distance $dz$. Each pixel of
the phase image corresponds to a transverse resolution of about $5\mu$m in the
object plane. As a reference, the first row of Fig. 3 shows the phase
retrieved averaging 100 shots, so the shot noise effect is estimated to be
negligible compared to the other sources of disturbance. However, even in this
case, the reconstruction at small $dz$ is not perfect because of the well-
known sampling error due to the discretization of the image, while at large
defocussing the finite approximation of the derivative in $z$ fails,
essentially producing blurred images. These two opposite trends determine a
defocussing distance for which the reconstruction is optimal. The second row
of Fig. 3 shows the reconstructions obtained by single frame intensities
$I_{s}(\bm{x_{s}},\pm dz)$ measured at the CCD camera in the signal arm. In
this case, the shot noise dominates and yields a drop in the reconstruction
quality for all values of $dz$. How the noise on the intensity propagates to
the phase reconstruction through the TIE is discussed in M&M. In particular,
the region of smaller $dz$ is the most affected since the intensity variation
produced by the phase gradient is still small and is almost completely hidden
in the shot noise.
In order to take advantage of the quantum correlations here, we propose to
replace into the Eq. (1) the single beam intensity with the following one
Moreau et al. (2017); Losero et al. (2018); Ruo-Berchera et al. (2020):
$I_{s-i}(\bm{x},z)=I_{s}(\bm{x_{s}},z)-k_{opt}\delta I_{i}(\bm{x_{i}},0).$ (2)
where $\delta X\equiv\langle X\rangle-X$ represents the quantum fluctuation of
the operator $X$, and $\langle\cdot\rangle$ is the quantum expectation value.
In fact, the second term in Eq. (2) is meant to compensate for the quantum
fluctuation of the signal pattern exploiting the local correlation between
probe and reference beams. The factor $k_{opt}$ is a parameter chosen to
minimize the residual fluctuation $\langle\delta^{2}I_{s-i}\rangle$ and can be
evaluated experimentally by calibration of the system since it is related to
the detection efficiency. A phenomenological model describing noise reduction
is discussed in M$\&$M. It turns out that the fluctuation of the quantity in
Eq. (2) is reduced with respect to the shot noise according to the following
expression:
$\displaystyle\langle\delta^{2}I_{s-i}\rangle$ $\displaystyle=$
$\displaystyle\left[1-\left(1-\alpha\right)^{2}\eta^{2}\right]\langle
I(0)\rangle,$ (3)
where $0<\eta<1$ is the heralding efficiency, namely the probability of
detecting an idler photon in the pixel in $\bm{x}_{i}$ conditioned to the
detection of the correlated signal photon in the pixel in $\bm{x}_{s}$ (see
M$\&$M section). The parameter $\alpha$ is the average fraction of photons
that deviate from the original path due to the phase object and depends on the
average phase gradient. It can be experimentally evaluated as the spatial
average of the quantity
$\alpha(\bm{x})\equiv\langle|I(\bm{x}_{s},0)-I(\bm{x}_{s},dz)|\rangle/\langle
I(\bm{x}_{s},0)\rangle$. Eq. (3) states that the intensity fluctuation is
reduced below the shot noise by a factor that depends on the efficiency in
detecting quantum correlation and that it is effective if the object is weakly
affecting the intensity distribution, namely when $\alpha\ll 1$. In our
experiment, following the absolute calibration method reported in Meda et al.
(2015); Avella et al. (2016); Samantaray et al. (2017), we estimate
$\eta=0.57$ for the particular case of averaging filter size $d=4$. The value
of $\alpha$ for the faint object considered is very small, for example we
estimated $\alpha=7\cdot 10^{-3}$ for $dz=0.1$ mm.
The third row of Fig. 3 reports the reconstructions when the shot noise has
been reduced using quantum correlations between probe and reference, according
to Eq. (3). A general improvement of the reconstruction can be appreciated. As
expected, the noise reduction is more evident at smaller $dz$ leading to an
improvement in the reconstruction of higher spatial frequency.
Figure 4: Pearson correlation between reconstructed and reference images. The
light-blue and yellow curves are the result of a Fourier optics based
simulation. The line-width is the confidence interval of one standard
deviation after an average over 100 reconstructions for each $dz$. The
experimental points are represented as purple and yellow dots with uncertainty
bar also corresponding to one standard deviation. The red curve corresponds to
the reconstruction obtained by summing of 100 intensity patterns, where the
shot noise becomes negligible (in this case, quantum and classical correlation
overlaps).
A quantitative analysis of the quality of the reconstructions and of the
quantum advantage can be performed by evaluating the Pearson correlation
coefficient between the reference phase image and the reconstructed one. The
Pearson coefficient is defined as,
$\mathcal{C}=\frac{\sum_{\bm{x}}(\phi_{r}(\bm{x})-\bar{\phi_{r}})(\phi(\bm{x})-\bar{\phi})}{\sqrt{\text{Var}[\phi_{r}]\text{Var}[\phi]}}$
(4)
where $\bar{\phi}$ and Var$[\phi]$ denote the spatial mean and variance of the
phase image $\phi$, and provides a simple and commonly used figure of merit to
quantify the similarity between the two images. Fig. 4, shows the Pearson
coefficient as a function of the defocusing. Each curve has a correspondence
with each image strip in Fig. 3. The red curve corresponds to the
reconstruction using 100 frames, where shot noise is negligible (corresponding
to the first strip in Fig. 3). The lower curves present the performance of
single frame experimental reconstructions, both quantum and classical,
obtained from a simulation. Experimental points are well in agreement with
these simulations. As expected, according to this figure of merit, an optimal
reconstruction is reached for the intermediate value of defocusing. The
quantum advantage is confirmed in terms of correlation with the reference
image.
Figure 5: Phase estimation. A The estimated value of the phase step (average
of the rectangular selected region) is plotted at different defocusing
distances. Experimental points for the classical (yellow dot) and the quantum
(purple dot) phase retrieval are compared with the simulations (reported for
one standard deviation confidence band). For comparison, we also report the
nominal value, estimated by the profilometer in reflection, of the phase step
difference between the etched/non-etched areas. B The uncertainty in the phase
estimation for quantum and classical cases demonstrate the quantum advantage.
Besides the correct reconstruction of the complex phase profile assessed by
the correlation coefficient, in many cases, it is of utmost importance to
achieve a quantitative estimation of the phase. Fig. 5A reports the phase
value estimated as a function of $dz$, where, for the analysis, we have
selected the region indicated in the red rectangle in the insets. The results
indicate that the phase step is reconstructed without bias compared to the
nominal value (red horizontal line) up to $dz=100\ \mu$m for both the
classical and the quantum case. The experimental points and their error bars
agree with the confidence bandwidths provided by the simulations. However, the
uncertainty on the estimated value is smaller for the quantum case. The
quantum advantage, reported in Fig. 5B, is relatively constant in the range
considered up to a 40$\%$. The estimated phase value drops down for higher
defocusing distances because of the blurring of the image evident from the
first row in Fig. 3. However, it is clear that in this region, the method does
not provide useful reconstructions simply because the approximation of the
derivative in Eq. (1) is no longer valid.
We have also tested a different object, the pattern of regular squares
represented in Fig. 2. In Fig. 6A we report two examples of reconstructions,
at $dz=50\ \mu$m and $dz=100\ \mu$m, respectively. In Fig. 6B, the Pearson
coefficient is reported alongside the simulations. The quantum advantage is
comparable to the one obtained for the $``\pi"$, showing its robustness and
independence from the particular spatial shape of the sample. Although the
quantitative analysis of the Pearson coefficient confirms a similar quantum
advantage as the one reported in Fig. 4, by looking at the images, it appears
that the quantum advantage in the localization of dots could be even larger,
indicating the possibility of significant advantages for specific tasks
related to the recognition of finer spatial details.
In summary, these results demonstrate, for the first time, a significant
advantage of quantum phase imaging, that can be further extended in the future
with various potentially significant applications.
Figure 6: Single frame reconstruction of the squares pattern. A Examples of
classical and quantum reconstructions of the sample with squares in Fig.2
(right-hand side) for two different defocussing distances. B Pearson
correlation coefficient between the reconstructed phase image by a single
intensity frame and the reference image as a function of the defocussing.
## III Conclusions
Here, we have demonstrated a genuine quantum enhancement in non-
interferometric quantitative phase imaging, showing that the spatially multi-
mode quantum correlations can be used to reduce the detrimental effect of
quantum noise in phase reconstruction. The present NIQPI scheme exports the
classical methods known as the transport of intensity equation to the quantum
regime, which provides real-time wide-field phase imaging and the quantitative
local estimation of the phase. The last aspect is fundamental for many
applications, providing reliable information on the object’s internal
parameters related to the phase.
We point out that, compared to the imaging of an amplitude object Brida et al.
(2010); Samantaray et al. (2017); Ruo-Berchera et al. (2020); Losero et al.
(2018), the propagation of the shot noise of the intensity measurement to the
retrieved phase in the NIQPI is not as trivial. On the one side, the noise
reduction allows reaching smaller defocussing distances for a better
approximation of the derivative in the TIE, thus providing a more faithful
reconstruction of the phase details. On the other side, artifacts due to the
noise appear at low spatial frequencies (see discussion in M$\&$M and Fig. 3)
and are known to affect mainly the reconstruction of slow phase curvature,
which produces weaker signal intensity signals Paganin et al. (2004). In this
work, in order to obtain a quantitative validation of the protocol, we studied
binary phase objects with sharp borders. However, it is expected that for an
object with smoother phase changes, e.g., biological samples, the quantum
advantage can be even more significant.
## IV Materials and Methods
### IV.1 Phase retrieval by TIE
A non-interferometric method Teague (1983) to retrieve the phase of an object
consists of probing the object with a beam and measuring the intensity
$I(\bm{x},z=0)$ at the object plane of coordinate $\bm{x}$ and its derivative
along the propagation axis $z$. The derivative is computed by a finite
difference of two measurements out-of-focus of a distance $dz$,
$\frac{\partial}{\partial z}I(\bm{x},z)\approx\Delta I(\bm{x},dz)/(2dz)$ with
$\Delta I(\bm{x},dz)=I(\bm{x},dz)-I(\bm{x},-dz)$. Under paraxial
approximation, the phase is retrieved using the TIE reported in Eq. (1).
Using energy conservation considerations, this equation has been proven valid
even with partially coherent sources Paganin and Nugent (1998). This feature
makes the TIE approach perfectly suited for being used with light from SPDC,
where transverse and longitudinal coherence lengths can be much smaller than
the object size and the whole illuminating beam. This is not a secondary
aspect since it is exactly due to the multimode nature of the emission that
correlation shows a local character and shot noise can be removed pixel-by-
pixel in the image. The solution of the Eq. (1) is unique provided that the
on-focus intensity $I(\bm{x},0)$ and the intensity derivative along $z$ are
known and the phase is continuous.
Following the analysis in Paganin et al. (2004), we assume that the intensity
is varying sufficiently slowly that the effects of phase curvature dominate
the intensity derivative, so that the right side of Eq. (1) can be safely
approximated as $I_{0}\nabla^{2}\phi(\bm{x},0)$. Then, we consider for a
moment that the only contribution to the finite difference $\Delta
I(\bm{x},\delta z)$ is the noise fluctuation on the intensity measurement,
$\sigma(\bm{x})$ . In this case, substituting the latter in Eq. (1), one has
that the phase artifacts in the reconstruction due to the noise are:
$-k\frac{\sigma(\bm{x})}{\sqrt{2}I_{0}\delta
z}=\nabla_{\bm{x}}^{2}\phi_{noise}(\bm{x}).$ (5)
The noise is assumed independent in the two planes $+\delta z$ and $-\delta
z$, so it has been combined in quadrature. The Eq. (5) can be solved by taking
the Fourier transform on both sides, leading to
$k\frac{\tilde{\sigma}(\bm{q})}{4\pi^{2}\sqrt{2}I_{0}\delta
z|\bm{q}|^{2}}=\tilde{\phi}_{noise}(\bm{q})$ (6)
where the tilde indicate the Fourier transform and $\bm{q}$ is the spatial
frequency. The damping factor $|\bm{q}|^{2}$ of the higher frequencies at the
denominator of Eq. (6) and the fact that the quantum noise (shot noise) has a
flat white spectrum $\sigma_{SN}(\bm{q})=\sigma_{SN}$, indicate that the
effect of shot noise is to generate artifacts especially at lower frequencies,
which are not intrinsically suppressed by the phase retrieval algorithm. This
noise at low-frequencies is evident in the single frame images reported in
Fig. 3. Moreover, in the direct propagation problem, higher frequencies of the
phase object generate a stronger effect on the intensity. Thus, based on these
remarks, the regions with rapid changes in the phase (higher frequency) are
better reconstructed than the ones characterized by slow curvature.
### Experimental details: Source, Sample, Detection
Source: In the experiment, we use SPDC in the low gain regime in which a
photon of the pump beam (p) (CW laser @405nm), thanks to the interaction with
a bulk beta-barium borate non-linear crystal as long as 15 mm, have a small
probability of converting in a couple of photons, usually called signal (s)
and idler (i), subject to conservation of energy,
$\omega_{p}=\omega_{s}+\omega_{i}$, and of momentum,
$\textbf{k}_{p}=\textbf{k}_{s}+\textbf{k}_{i}$. Thus, under the plane wave
pump approximation, signal and idler photons are perfectly correlated in
frequency and direction $\bm{q}_{s}=-\bm{q}_{i}$ (assuming $\bm{q}_{p}=0$),
although their individual spectrum is broadband both in space and frequency.
In the far field, obtained at the focal plane of a thin lens in a $f-f$
configuration, a transverse mode $\bm{q}$ is mapped in a single transverse
position $\bm{x}$ according to the transformation
$(2cf/\omega)\bm{q}\rightarrow\bm{x}$, so that momentum correlation translate
in a position correlation, $\bm{x}_{s}=-\bm{x}_{i}$ (for degenerate frequency
$\omega_{s}\approx\omega_{i}$). Signal and idler photons generate two
symmetrical intensity noise patterns, and pairs of symmetric pixels of a
camera will detect the same number of photons in the ideal lossless scenario
in the same time window. Thus, quantum fluctuation affecting the object plane
in the signal beam can be measured independently on the idler beam. The
coherence time of the SPDC sources is typically of hundreds of fs and the
spatial coherence in the far field is proportional to the inverse of the pump
transverse size. The number of photons per spatial-temporal mode is very low,
$\sim 10^{-8}$, in general, the time bandwidth of the detector is orders of
magnitude smaller than the inverse of the coherence time. Although the single
SPDC mode is thermal, in the limit above, the detected multi-thermal photon
statistics are indistinguishable from a Poisson distribution Meda et al.
(2017).
For a Gaussian distributed pump with angular full-width-half-maximum (FWHM) of
$\Delta q$ the spatial cross-correlation is also Gaussian with FWHM of $\Delta
x=2\sqrt{2\log 2}\sigma=(2cf/\omega_{p})\Delta q$: if a signal photon is
detected in the position $\bm{x}_{s}$ the twin idler photon will be detected
according to that Gaussian probability centered in $\bm{x}_{i}=-\bm{x}_{s}$.
In the experiment we have estimated $\Delta x\approx 5\mu$m
Test sample: The structures are etched on to a fused Silica glass window
(WG41010-A, Thorlabs) with an anti-reflection coating on one side. The window
is coated with positive PMMA resist and the design is exposed using electron
beams. The exposed structures are developed using a MIBK-IPA solution. After
development, the window is submerged in a buffered oxide etch for 30 seconds
to etch the structures into the window. The etch depth is determined by the
submergence time. The unexposed resist is then removed using acetone solution.
Detection: We measure the SPDC emission and the effect of the phase object by
imaging the far field of the source at the sensor of a CCD camera operated in
the conventional linear mode. Each pixel delivers a count proportional to the
number of incident photons. The proportionality coefficient is the product of
the (electronic gain) that has been carefully calibrated and the quantum
efficiency of the camera, nominally above 95% @810 nm. The electronic readout
noise is 4$e^{-}/(\text{pix}\cdot\text{frame})$. The number of photons
detected per pixel per frame is $10^{3}$, where the integration time of the
camera is set to $100$ ms, meaning that the shot noise dominates compared to
the electronic noise.
Because of the finite cross-correlation area defined in the previous section
of the M$\&$M, in order to collect most of the correlated photons, two
symmetrically placed detectors (or pixels) must have areas larger than the
cross-coherence area. Pixel size is 13 $\mu$m and a binning of $3\times 3$ is
performed to set the resolution to 5 $\mu$m at the object plane, which matches
the measured cross-coherence area. Actually, the heralding efficiency $\eta$,
i.e. the probability of detecting an idler photon conditioned to the prior
detection of the twin signal photon, depends on the pixel size $L$ and
possible misalignment $\bm{\Delta}$ of the two pixels compared to the optimal
positions, according to this expression:
$\eta(L,\bm{\Delta})=\eta_{0}L^{-2}\int_{L\times L}d\bm{x}_{s}\int_{L\times
L}d\bm{x}_{i}\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(\bm{x}_{i}+\bm{x}_{s}+\bm{\Delta})^{2}}{2\sigma^{2}}}$
(7)
where, $\eta_{0}$ is the single photon detection efficiency. As the the pixel
size $L$ increases with respect to the coherence area $\Delta x$, we have that
$\eta\longmapsto\eta_{0}$.
As a consequence of that, in Eq. (3), the noise reduction depends on the pixel
size used for the measurement. This trade-off between the quantum advantage
and the spatial resolution of the intensity measurement has been reported and
analyzed in the context of sub-shot-noise imaging of amplitude objects
Samantaray et al. (2017); Ruo-Berchera et al. (2020). However, in the present
imaging of pure phase objects, the resolution issue has less impact. In fact,
as it is described in the first section of this M$\&$M, the solution of the
TIE tends by itself to suppress the higher frequency component of the
intensity perturbation. Thus, to some extent, a reduction of resolution in the
intensity measurement does not affect the phase reconstruction. In the
experiment, in order to increase the heralding efficiency, and thus the
quantum enhancement, we use an averaging filter to the intensity image that
substitutes the count in each pixel by the average of a neighborhood of size
$d\times d\;\text{pix}^{2}$ around it. The quantum correlations are then
enhanced because the effective integration area is larger, while the number of
pixels in the final image is unvaried. In Fig. 7, we report the quality of the
phase reconstruction, evaluated in terms of the Pearson correlation with the
reference image in Fig. 2, as a function of the averaging size. On the one
hand, the quantum reconstruction is enhanced as expected when the effective
resolution in the intensity measurement decreases ($d$ increases). On the
other hand, the classical reconstruction is unaffected, confirming that
classically we do not have any negative issue related to the poorer resolution
in the intensity pattern. In summary, moderate use of the averaging filter to
enhance the quantum effects is perfectly legitimate in this context.
Figure 7: Pearson correlation as a function of averaging filter size. The
purple (yellow) dots represent the values corresponding to the quantum
(classical) experimental reconstructions. The quantum (classical) confidence
bands at one standard deviation are also shown in turquoise (yellow).
### Model for the noise reduction
According to the scheme in Fig. 1, the signal beam of SPDC probes the object,
while the idler beam is used as a reference for the noise. When the object is
inserted with a defocusing distance $dz$, the photons in the signal beam are
deflected, creating local depletion or accumulation of photons at the
detection plane, and the perturbed intensity can be written as:
$I_{s}(\bm{x},z)=I_{s}(\bm{x},0)-\Delta I_{-}(\bm{x})+\Delta I_{+}(\bm{x}),$
(8)
where $I_{s}(\bm{x},0)$ is the unperturbed pattern and $\bm{x}$ indicates the
position of a pixel. The quantity $\Delta I_{-}(\bm{x})$ $(\Delta
I_{+}(\bm{x})$) represents the photons that are deflected out from (into) the
position $\bm{x}$. From now on, to simplify the notation, the spatial average
of the quantities is simply indicated by dropping the spatial dependence on
$\bm{x}$. Since the total number of photons is conserved, the spatial average
of the number of photons per pixel is unchanged, i.e. $I_{s}(z)=I_{s}(0)$ and
thus $\Delta I_{-}=\Delta I_{+}$. The loss of photons can be described as the
action of a beam splitter of transmittance $1-\alpha$ (average value) so that,
the quantum expectation value for the $\Delta I_{-}$ is simply $\langle\Delta
I_{-}\rangle=\alpha\;\langle I_{s}(0)\rangle=\langle\Delta I_{+}\rangle$ Meda
et al. (2017). In this work, we are interested in small perturbations that can
be hidden or strongly affected by the quantum noise, so we will assume
$\alpha\ll 1$.
In order to reduce spatial intensity fluctuation we replace in the TIE the
quantity in Eq. (8) with the one in Eq. (2) involving the idler measurement.
The optimal factor $k_{opt}$ appearing there is chosen to minimize the
residual fluctuation, by imposing $\frac{\partial}{\partial
k}\langle\delta^{2}I_{s-i}(\bm{x},z)\rangle=0$. We obtain,
$\displaystyle k_{opt}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{\langle\delta I_{s}(\bm{x}_{s},z)\delta
I_{i}(\bm{x}_{i},0)\rangle}{\langle\delta^{2}I_{i}(\bm{x}_{i},0)\rangle},$ (9)
$\displaystyle\langle\delta^{2}I_{s-i}(\bm{x},z)\rangle$ $\displaystyle=$
$\displaystyle\langle\delta^{2}I_{s}(\bm{x}_{s},z)\rangle-\frac{\langle\delta
I_{s}(\bm{x}_{s},z)\delta
I_{i}(\bm{x}_{i},0)\rangle^{2}}{\langle\delta^{2}I_{i}(\bm{x}_{i},0)\rangle}.$
According to the Poisson distribution of the detected photon, we can replace
the variance of the intensities appearing in Eq. (9) with the respective
quantum mean values. In particular, by performing the spatial averaging, one
gets $\langle\delta^{2}I_{i}(0)\rangle=\langle
I_{i}(0)\rangle=\langle\delta^{2}I_{s}(z)\rangle=\langle I_{i}(z)\rangle$. For
the calculation of the covariance in Eq. (9), note that $I_{s}(\bm{x}_{s},z)$
and $I_{i}(\bm{x}_{i},0)$ are correlated only for the fraction of photons that
are not lost, namely not deviated from the path due to phase effect on the
propagation along $z$. Thus, after spatial averaging Meda et al. (2017):
$\displaystyle\langle\delta I_{s}(z)\delta I_{i}(0)\rangle$ $\displaystyle=$
$\displaystyle(1-\alpha)\langle\delta I_{s}(0)\delta I_{i}(0)\rangle$ (10)
$\displaystyle=$ $\displaystyle\eta(1-\alpha)\langle I_{s}(0)\rangle.$ (11)
The last equality is justified again using the Poisson hypothesis, and
introducing the heralding efficiency $\eta$ that spoils the otherwise perfect
signal-idler correlation. By using Eq. (11), and the Poisson hypothesis above,
we can rewrite the Eq.s (9) as,
$\displaystyle k_{opt}$ $\displaystyle=$ $\displaystyle(1-\alpha)\,\eta$ (12)
$\displaystyle\langle\delta^{2}I_{s-i}\rangle$ $\displaystyle=$
$\displaystyle\left[1-\left(1-\alpha\right)^{2}\eta^{2}\right]\langle
I_{s}(0)\rangle$ (13)
## V Acknowledgments
This project 20FUN02 POLight has received funding from the EMPIR programme co-
financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme. S.S. would like to acknowledge Dr.
Iman E. Zadeh for his supervision for the sample fabrication.
### Author contributions
GO and IRB devised the scheme of NIQPI. PB participated in the realization of
the setup and preliminary measurement with GO, while AP and CN performed the
final experimental acquisitions, which were realised in the laboratories of
the research sector directed by MG. The samples have been prepared and
characterized by SS and SP. Simulations and data analysis have been carried
out by GO and AP with the help of IRB. IRB and MG supervised the whole
project. IRB wrote the paper with the contribution of all authors.
### Competing Interests
The authors declare no competing interest.
### Data availability
All data needed to evaluate the conclusions are reported in the paper. Further
data, for reproducibility of the results, will be available in a public
repository linked to the published paper.
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# Waveguide-integrated and portable optomechanical magnetometer
Fernando Gotardo1,2 Benjamin J. Carey1,2 Hamish Greenall1,2 Glen I.
Harris1,2 Erick Romero1,2 Douglas Bulla4 Elizabeth M. Bridge5 James S.
Bennett3 Scott Foster4 and Warwick P. Bowen1,2 1School of Mathematics and
Physics, The University of Queensland, St Lucia, Queensland 4067, Australia.
2ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland
4067, Australia.
3Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4072,
Australia.
4Australian Government Department of Defence Science and Technology,
Edinburgh,
South Australia 5111, Australia.
5Quantum Brilliance, Acton, Australian Capital Territory 2601, Australia
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article
Optomechanical magnetometers enable highly sensitive magnetic field sensing.
However, all such magnetometers to date have been optically excited and read-
out either via free space or a tapered optical fiber. This limits their
scalability and integrability, and ultimately their range of applications.
Here, we present an optomechanical magnetometer that is excited and read-out
via a suspended optical waveguide fabricated on the same silicon chip as the
magnetometer. Moreover, we demonstrate that thermomechanical noise limited
sensitivity is possible using portable electronics and laser. The magnetometer
employs a silica microdisk resonator selectively sputtered with a
magnetostrictive film of galfenol (FeGa) which induces a resonant frequency
shift in response to an external magnetic field. Experimental results reveal
the retention of high quality-factor optical whispering gallery mode
resonances whilst also demonstrating high sensitivity and dynamic range in
ambient conditions. The use of off-the-shelf portable electronics without
compromising sensor performance demonstrates promise for applications.
## 1 Introduction
In recent years, optomechanical sensors have emerged as a powerful new type of
sensor for stimuli ranging from temperature [1] to pressure [2], forces [3, 4]
and acceleration [5]. Such sensors leverage both optical and mechanical
resonances to enable high sensitivity and high spatial resolution.
Optomechanical magnetometers are one example [6, 7, 8, 9], that are attractive
due to the crucial role that high-sensitivity magnetometers play in
applications ranging from fundamental research to medical diagnostics [10,
11], mineral exploration and surveying [12, 13], magnetic anomaly detection
[14, 15], and navigation [16, 17, 18]. Owing to their photonic nature, they
are light-weight, small-sized, low power [16, 7, 19], and can be exceedingly
resilient to detriments such as electrical interference and radiation.
At their current state of development, optomechanical magnetometers have
achieved tens-of-micron spatial resolution with sensitivities ranging from
several n$\mathrm{T}/\sqrt{\mathrm{Hz}}$ down to tens of
p$\mathrm{T}/\sqrt{\mathrm{Hz}}$ [20, 7, 16, 8]. The demonstrated sensitivity
is competitive with SQUID and diamond magnetometers of similar size but
without the need for cryogenics or high-powered optical and RF components [16,
21, 22], with theoretical models suggesting that sensitivities in the low, or
even sub-, femtotesla may be possible in future [23].
To-date, optomechanical magnetometers have used free-space or tapered-fiber
coupling for optical excitation and readout [22, 24, 9]. This prevents them
from being fully integrated on a silicon chip. Furthermore, those
demonstrations used the magnetostrictive material Terfenol-D to convert
magnetic fields into a measurable strain [6, 7]. Which is difficult to
reproducibly deposit and sensitive to corrosion and oxidation [25]. Works with
this material have also relied on high performance laser and electronic
systems [19]. Together, this introduces significant challenges for
applications outside of laboratory environments. The work reported here seeks
to address these challenges.
We develop an optomechanical magnetometer that is efficiently coupled to an
on-chip suspended waveguide, by employing galfenol (Fe82Ga18) for the first
time to convert magnetic fields to a measurable mechanical signal. This
provides low-cost sputter-coated thin-films with improved resilience to
corrosion and oxidation [25] and good magnetostriction ($\sim$400 ppm) at
lower saturation fields[26, 25]. We also use portable electronic and laser
systems to control and read-out the magnetometer, showing that they allow
performance that is limited by fundamental thermomechanical noise rather than
laser or electronic noise. Together, this represents progress towards robust,
portable and high performance mangetometers that could be employed in diverse
research and industrial settings.
## 2 Design and Simulation
### 2.1 Device Design and Functionality
The device design concept is depicted in Figure 1. It is based around a 100
$\upmu$m-diameter silica microdisk cavity on a silicon chip. This microdisk is
capable of supporting optical whispering galley modes (WGMs) throughout the
visible and near-infrared spectrum as well as megahertz frequency mechanical
resonances. A 1.5 $\upmu$m wide silica waveguide is fabricated from the same
layer of silica as the microdisk. Both microdisk and waveguide are undercut so
that the optical modes are confined to the silica, rather than leaking into
the higher-refractive index silicon substrate. The microdisk is suspended from
a central silicon pedestal. The waveguide is suspended using thin silica
tethers that are patterned along its length to reduce warping of the waveguide
that can be caused by the intrinsic stress present in the as-fabricated SiO2
films. Buckling of the waveguide could lead to severe bending losses, and out-
of-plane buckling can lead to inconsistent coupling between the waveguide and
optical cavity. The tethers are sub-wavelength (240 nm width) in order to
minimise optical scattering of light propagating within the waveguide. As
silica is lower refractive index than many other waveguide material (e.g.,
silicon) the guided wavelength is longer allowing for minimal scattering. The
waveguide is broadened (inverse-tapered) at input and output to efficiently
mode-match light into and out-of tapered optical fibers.
Figure 1: Design of the integrated SiO2 magnetometer. Here, laser light is
coupled to the trumpet waveguide via a tapered fiber. The waveguide is narrow
at the centre to optimize evanescent field coupling to the disk cavity. The
galfenol is sputtered on top of the cavity. The tethers support the waveguide,
preventing buckling.
The microdisk is coated with galfenol in a disk of diameter sufficiently
smaller than the disk diameter so as not to introduce optical absorption.
Galfenol is chosen because of its high magnetostriction, low fragility, and
low volatility [25]. When a magnetic field is applied, the expansion of the
galfenol induces strain in the microdisk, changing the optical path length and
hence the optical resonance frequency. The magnetostrictive response is
amplified at frequencies close to mechanical resonances of the microdisk,
leading to enhanced modulation of the intracavity optical field.
The light is coupled into the disk evanescently from the suspended waveguides
which are designed to support single mode propagation around 1550 nm. The
coupling of light from an optical fiber into the on-chip waveguide despite the
geometric mismatch (SMF-28 optical fiber has core and cladding diameters of 10
$\upmu$m compared to the 300 nm thickness of the waveguide) is facilitated by
mode-matched taper-drawn fibers (similar to those presented in [27]) and
trumpeted waveguide (4 $\upmu$m down to 1.5 $\upmu$m over a length of 30
$\upmu$m). This allows adiabatic coupling of light to and from the waveguides.
Further, the 4 $\upmu$m wide flared section acts as a semi-rigid anchor point
for the fiber, and its size reduces the requirement for extremely precise
positioning. This allows greater tolerance to imperfections in important
integration processes, such as bonding the fiber tip in place.
#### 2.1.1 Finite-element Simulations
Finite Element Method (FEM) simulations performed with Ansys-Lumerical
software for the optical performance of the device are presented in Figure 2
a). Here the coupling from fiber to waveguide is studied by monitoring the
optical mode cross-section ($yz$) along the propagation direction ($x$). The
cross-sections shown in (i-iii) correspond to cross-sections of only fiber
(i), fiber & waveguide (ii), and only waveguide (iii). For simplicity, both
the fiber taper and the waveguide were considered uniform at the coupling
region. The taper was chosen to have a 1 $\upmu$m diameter with the optimum
waveguide width of 4 $\upmu$m then found using recursive simulations. From the
simulated cross-sections, we see that the optical mode migrates from the fiber
into the waveguide. We obtain a fiber-to-waveguide coupling efficiency of 60%
by taking the ratio of the optical power contained within the fiber at point
(i) and within the waveguide at point (iii).
Within the same simulation the waveguide-to-disk coupling and disk resonances
were also studied. Here the optical excitation frequency was swept and the
optical intensity across the geometry was recorded at each frequency. The
transmission efficiency across the device could then be calculated by
comparing the integrated intensity over the cross-sections of the input and
output of the waveguide (i.e., $T=I_{\text{out}}/I_{\text{in}}$, as measured
at points (iii) & (vi)). Fig. 2(a)(vi) shows the expected periodic
transmission dips when the frequency of the light matches WGMs of the
microdisk. The transmission is predicted to drop by as much as 70%, indicating
that efficient coupling of light into WGMs should be possible. The
correspondence of the dips with WGMs is confirmed in Fig. 2 a)(v), which shows
confinement of the light to the periphery of the disk when driven on resonance
(at a wavelength of 1551 nm in this case). To further corroborate the
confinement of the WGMs we performed an axisymmetric eigenmode-solver FEM
simulation in COMSOL Multiphysics (Figure 2 a) (iv)). This confirmed that the
WGM is contained within the outer 5 $\upmu$m of the disk, as expected.
The Free Spectral Range (FSR) of the optical resonances and corresponding
coupling were calculated from the simulated transmission in Fig. 2(a)(vi). We
find a simulated FSR of approximately 7 nm. This compares well to the expected
FSR given the circumference of the microdisk of:
$\Delta\lambda\approx\frac{\lambda^{2}}{n(\lambda)L}\approx 7.6\;\mathrm{nm},$
(1)
where $n(\lambda)$ is the effective refractive index of the cavity mode (taken
from Lumerical simulations to be 1.01) and $L$ is the length of the cavity
i.e., $L=$ 100$\pi\;\mathrm{\upmu m}$.
Figure 2: FEM simulations of the integrated optomechanical magnetometer.
a) The optical properties of the device. The cross-sectional optical mode
profiles (i-iii) at their corresponding green rectangles demonstrate the
evolution of the optical modes from fiber to SiO2 trumpet waveguide and (iv)
the cross-sectional optical mode inside the microdisk. (v) Depicts the optical
mode at the planar cross-section (yellow rectangle) of the device, and (vi)
shows the optical transmission spectrum of the system. (b) Mechanical
simulation revealing the resonance frequencies and their flexual mode-shapes
As evidenced by the results in Figure 2 a) (iv & v), the optical field of the
Whispering Gallery Mode (WGM) extends negligibly into the centre of the SiO2
disk. Hence, the addition of the optically absorbing magnetostrictive layer to
the disk’s centre should not significantly affect the quality of the optical
modes contained therein [20].
Using COMSOL Multiphysics, we performed further simulations to assess the
mechanical properties of the microdisk. As shown in Figure 2 b), we found the
mechanical eigenmodes by using fully three-dimensional geometry of the
released devices. This was necessitated because of the inclusion of stress-
release slots (discussed in 3) that break the axial symmetry of the mechanical
modes. The physical properties of the galfenol were taken from the datasheet
supplied by TdVib LLC. The lowest frequency mechanical flexural mode and two
lowest frequency crown modes are shown in Figure 2 b), with mechanical
frequencies of 3.12, 3.21, and 3.26 MHz, respectively.
## 3 Device Fabrication
The fabrication process used to produce the devices is outlined in Fig. 3 a)
(i). SiO2 (300 nm) on Si substrate wafers (500 $\upmu$m, 4") were diced into
square 15$\times$15 mm chips, large enough to fit more than 100 devices per
chip. Electron beam lithography was used to define patterns for galfenol
deposition and markers for subsequent lithography steps in the following way.
Two layers of PMMA resist were spin-coated (PMMA 950k and 495k at 2500 RPM)
onto of the SiO2/Si substrate, then patterned with an Electron-beam Pattern
Generator (EBPG) (Raith EBPG 5150) with 100 kV accelerating voltage and a 1200
$\upmu$C/cm2 dosage. Post exposure, the chips were developed in methyl
isobutyl ketone (MIBK) and rinsed with Isopropyl Alcohol (IPA).
To produce the markers, 5 nm of Ti and 50 nm of Au were e-beam evaporator
deposited (Temescal FC-2000) follow by a lift-process via submersion into
acetone and IPA. The galfenol films were then sputtered by magnetron DC
sputtering in an argon atmosphere (150 W, 2 mTorr) with a (3 inches dia.)
galfenol target. A seeding layer (5 nm Ta, plus a 16 nm Cu) and capping layer
(5 nm Ta) were used to promote adhesion and inhibit corrosion respectively.
Afterwards, the lift-off process was repeated, resulting in a 300 nm thick, 60
$\upmu$m diameter circular thin-film of galfenol on top of the SiO2 layer.
Figure 3: Representation of the fabrication process. a) PMMA e-beam resist
deposition. b) EBPG exposure and development. c). Galfenol sputter deposition.
d) lift-off. e) ARP e-beam resist deposition. f) Second EBPG exposure and
development. g) RIE of SiO2. h) Resist removal. i) Released devices after
undercutting by XeF2 etching of the Si layer.
SEM images of the final devices depicting j) trumpet wave guide and k) the
optomechanical cavity with galfenol layer in its centre and coupling
waveguide.
With markers produced and galfenol deposited, the waveguide and the disk
cavity structures were then defined. For this, 20 nm thick Ti prime adhesion
helper was spin-coated (4000 rpm) and baked (150°C, 15 min) follow by a layer
of ARP 6200.09 (CSAR-62, All Resist) 350 nm thick (1250 rpm spin-coat, 180°C
for 5 min bake). The chip was then patterned with the Raith EBPG 5150 (100 kV,
260 $\upmu$C/cm2). Proximity effect correction was performed using GenISys
Beamer software to ensure precision and reproducibility in the EBPG process.
Post exposure, the patterns were developed with All Resist AR600-546 and
rinsed with o-xylene and IPA.
RIE was used remove the unwanted SiO2 using an Oxford Instruments PlasmaPro
80. Here 25 sccm CHF3 and 45 sccm Ar at 200 W RF power for 12 min
anisotropically etched all the way through the SiO2 layer exposing the silicon
substrate. 50 sccm O2 at 50 W was then used to remove any residual resist.
Finally, the SiO2 structures were undercut by etching of the supporting
silicon with Xenon Difluoride (XeF2) gas (SPTS XACTIX e2 Series XeF2 etcher).
Here, 10 pulses of XeF2 gas at a pressure of 2 torr provides an isotropic etch
rate of about 1.4 $\upmu$m per pulse with a selectivity of >1000:1. This
removed the Si beneath the silica waveguide and WGMs of the microdisk whilst
leaving both the silica layer and galfenol unmarred.
SEM (Jeol 7800) imaging of the devices was performed in order to assess their
structural integrity. Fig. 3 j) & k) shows SEMs of the trumpet-shaped
waveguide ends for fiber coupling and the waveguide near the resonator,
supported by tethers to the main body of the wafer. It is apparent that the
waveguide shows no signs of buckling or collapse after the release process. It
can also be observed that the undercut beneath the silica layer is
approximately 18 $\upmu$m. This undercut extends under the disk, leaving
behind a silicon (Si) pedestal which is obscured by the galfenol coating
above. Measurements on a device with no galfenol revealed a pedestal width of
15 $\upmu$m (measured with a Zeta 300 3D optical profiler).
Stress release slots in the resonator were found to be necessary to prevent
buckling of the disk due to the inherent strain within both the SiO2 layer and
the galfenol film. However, as discussed in section 2, these slots are
expected to have negligible effect on the optical modes because they are
outside of the region containing appreciable intensity. The mechanical
simulation of Fig. 2 b) fully accounts for the effect of the slots on the
mechanical eigen-frequencies.
A critical parameter for consideration during fabrication of the device is the
distance between the disk and the waveguide at the coupling region ($d$). As
the light is coupled evanescently the coupling efficiency ($\kappa$) follows
the relation $\kappa\propto e^{-d}$ [28]. Devices with a range of waveguide-
microdisk coupling distances were fabricated in order to produce resonators
with optimum coupling strengths. The devices with near-critical coupling were
further investigated.
## 4 Device Performance
The experimental setup used to assess the performance of the integrated
magnetometers is depicted in Fig. 4a). Here, a continuously tuneable laser
(EXFO T100S-HP) supplied light to the resonator via tapered optical fibers
with a house-built test rig featuring two 3-axis translation stages for
precise positioning of the fibers. The transmitted light was then guided to a
Thorlabs (PDA100-CS) photodector (PD) and the photocurrent was analyzed with a
spectrum analyzer (Siglent SSA 3021x). A function generator (Rigol DG-1022)
was used to directly drive a home-wound coil (8 turns, 10 mm dia.) held
approximately 1 mm above the chip, producing a field of $\sim$50 $\upmu$TPP at
the surface of the chip (drive voltage of 10 VPP).
Figure 4: Experimental performance of the magnetometers. a) Schematic
demonstrating the experimental setup with corresponding photographs of the
laser system b), and devices on test-rig c) used for the measurements. d)
Optical transmission spectra of the devices with accompanying high-resolution
spectra around one of the WGM resonances (inset of e) which was used for the
sensing investigations. e) Power spectrum of the transmitted light with and
without externally applied magnetic field.
The emission wavelength of the laser was swept and the voltage output from the
PD (and hence power via the known responsivity and transimpedance gain of the
PD) was recorded to characterise the optical mode spectrum of fabricated
devices. The optical transmission spectrum of a typical device is presented in
Fig. 4 d), showing many dips in the transmission dips each associated to one
WGM. The observed FSR of $\approx$ 7 nm is in good agreement with the FSR as
determined from the FEM seen in Section 2 and Fig. 2a)(vi). On this device
(with a designed waveguide-microdisk separation of 550 nm) we find that the
WGM at a wavelength of 1551 nm (enclosed by the dashed box in Fig. 4 d)) is
close to critically coupled, with a transmission dip of $\sim 95\%$.
Because the 1551 nm WMG mode is close to critically coupled, we select it to
perform magnetic field measurements. A high-resolution sweep across the mode
is shown in the inset of Fig. 4e). From this the optical $Q$ of the cavity is
estimated to be:
$Q_{opt}\approx\frac{\lambda_{0}}{\Delta\lambda_{FWHM}}\approx 10^{5}.$ (2)
For many applications, it is desirable to use a low cost, low power, and
compact laser source, together with compact electronic systems, rather than
the high performance EXFO fiber-laser and associated electronics used in this
work to date. Here, we test whether it is possible to do this without
sacrificing performance. A commercially available Distributed Feedback (DFB)
laser (Eblana EP1550) with a portable laser driver (Koheron CTL101) was used
to couple light onto and off the chip (Figure 4 b)) were used for all
subsequent measurements.
Tuning the DFB laser to the side of the 1551 nm WGM allows shifts in the
resonance frequency to be directly observed as changes in the optical
intensity. This allows optical detection of mechanical vibrations, and hence
magnetic field, without the need for interferometric detection [6]. Analysing
the resulting photocurrent on a spectrum analyser reveals the mechanical mode
spectrum shown in Fig. 4b). Three mechanical modes are observed at frequencies
of at 3.55, 3.58, and 3.64 MHz. We attribute the discrepancy between the
measured and simulated mechanical frequencies to the inherent stress of the
galfenol film ($\sigma$=500 MPa) adding a stiffening effect to the mechanical
resonances.
The noise-floor of the measurement consists of two components. At frequencies
far away from the mechanical resonance frequencies it is dominated by laser
noise. This is evidenced by an increase in noise when the laser tuned to the
side of the WGM compared to when it is at a frequency far away from the mode.
At frequencies close to the mechanical resonance frequencies, it is dominated
by the thermomechanical noise. We can therefore conclude that the compact
electronic systems used introduce no degradation in performance and, close to
the mechanical resonances, neither does the optical noise of the DFB laser.
To determine the magnetic field sensitivity of the device, we apply a magnetic
field at the frequency of the most prevalent mechanical mode (3.55 MHz). This
induces a sharp peak in the Power Spectrum Density (PSD) (Figure 4e) orange-
trace), evidencing that magnetic fields can be detected. With this particular
applied field (BAC = 50 $\upmu$T) we measure a Signal-to-Noise Ratio (SNR) of
17.5 dB. The magnetic sensitivity of the device at 3.55 MHz could then be
calculated using:
$S=B_{AC}\left({10^{\frac{SNR}{10}}\times\mathrm{RBW}}\right)^{-1/2}$ (3)
where RBW is the resolution bandwidth of the spectrum analyser [6]. This
yielded a sensitivity of 2 $\upmu$$\mathrm{T}/\sqrt{\mathrm{Hz}}$, which is
comparatively less sensitive than previously demonstrated optomechanical
magnetometers that present sub n$\mathrm{T}/\sqrt{\mathrm{Hz}}$ sensitivities
[16].
This reduced sensitivity can be attributed to geometric design of the device.
With these devices the galfenol lies in part, above the pedestal, where the
silicon greatly suppresses both mechanical motion and imbued strain. Further,
the mechanical eigenmodes have very little motion where the galfenol resides,
thus do not experience the maximum possible driving force from the
magnetostriction. These effects provide a reduction of the force exerted onto
the optical eigenmodes from magnetostrictive stress. Thus, the sensitivity
could be considerably improved through the use of device geometry optimized
for deformation of the optical path from the magnetostrictive strain of the
galfenol layer.
Despite the modest sensitivity this work achieves thermomechanically limited
sensing with suspended waveguide coupling and a galfenol thin-film atop the
optomechanical resonator whilst utilising portable electronics and DFB laser.
## 5 Conclusion
Optomechanical magnetometers promise to enable a range of research and
industrial applications. Many of these will require fully integrated
magnetometers operating with compact lasers and electronics. In this work we
make progress towards this goal, demonstrating an optomechanical magnetometer
that is integrated on a silicon chip with a suspended optical waveguide,
utilises galfenol as a magnetostrictive material to provide improved
resilience to corrosion and oxidation, and achieves thermomechanical noise-
limited performance using a DFB laser and compact electronic systems. Funding
The Commonwealth of Australia (represented by the Defence Science and
Technology Group) supports this research through a Defence Science
Partnerships agreement. This work was financially supported by the Australian
Research Council (ARC) Centre of excellence for Engineered Quantum systems
(EQUS): Grant No. CE170100009, and by Orica Australia Pty Ltd.
Acknowledgments The Authors acknowledge the highly valuable advice and support
provided by Rodney Appleby. The authors also acknowledge the University of
Queensland’s Centre for Microscopy and Micro-analysis (CMM) and the Queensland
node of the Australian National Fabrication Facility (ANFF). The equipment and
staff expertise of the CMM and ANFF enabled the fabrication of the devices.
Disclosures The authors declare no conflicts of interest.
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*[FEM]: Finite Element Method
*[FSR]: Free Spectral Range
*[WGM]: Whispering Gallery Mode
*[EBPG]: Electron-beam Pattern Generator
*[IPA]: Isopropyl Alcohol
*[RIE]: Reactive-ion Etching
*[SEM]: Scanning Electron Microscope
*[Si]: silicon
*[PD]: photodector
*[DFB]: Distributed Feedback
*[PSD]: Power Spectrum Density
*[SNR]: Signal-to-Noise Ratio
|
Extremal correlation coefficient for functional data
Mihyun Kim$^1$ and Piotr Kokoszka$^2$
${}^1$Department of Statistics, West Virginia University, Morgantown, WV, USA
${}^2$Department of Statistics, Colorado State University,
Fort Collins, CO, USA
Address for correspondence: Mihyun Kim,
Department of Statistics,
West Virginia University, Morgantown, WV 26506, USA.
Email<EMAIL_ADDRESS>
We propose a coefficient that measures dependence in
paired samples of functions. It has properties similar to the
Pearson correlation, but differs in significant ways:
1) it is designed to measure dependence between curves, 2)
it focuses only on extreme curves. The new coefficient is derived
within the framework of regular variation in Banach
spaces. A consistent estimator is proposed and justified
by an asymptotic analysis and a simulation study.
The usefulness of the new coefficient is illustrated on
financial and and climate functional data.
Keywords: correlation, extremes, functional data.
§ INTRODUCTION
Due to the growing impact of extreme events related, for example,
to financial downturns or unusual weather, there has been increasing
interest in developing statistical tools to study patterns of extreme curves.
This is to a large extent due to the increasing availability of high resolution
data; time series of asset prices can be constructed at practically
any temporal resolution, and modern weather databases contain
measurements at hourly or even higher frequencies. Such data
can be interpreted as curves, e.g., one curve per day,
and provide more complete information than a single number
per day, like the total return or the maximum temperature.
We propose a coefficient that quantifies the tendency of paired extreme curves to exhibit similar patterns simultaneously. Two examples of the type of questions that the tool deals with are the following: 1) During a stock market crisis, such as the market decline due to the COVID-19 pandemic, do
returns of different sectors of the economy exhibit similar extreme daily trajectories? 2) How likely is location A to experience a similar
daily pattern of temperature as location B (on the same day) during a heat
wave? The coefficient we propose focuses not just on extreme total
return or extreme maximum daily temperature,
but also on the shapes of extreme curves.
There has been some research focusing on probabilistic and
statistical methods for extreme curves.
Extreme value theory in the space of continuous functions is
studied in Chapters 9 and 10 of [6].
Principal component analysis of extreme curves has been studied by [20], [18], and [2]. Extremal properties of scores of functional data were studied by [21] and Kim and Kokoszka ([16], [17]).
Additional, more closely related papers are introduced as we
develop our approach.
We propose a method for quantifying extremal dependence of paired functional samples, for which there are currently no appropriate tools.
We note that there has been considerable research aimed at quantifying extremal dependence for heavy-tailed random vectors. Ledford and Tawn ([23], [24],
[25]) introduced the coefficient of tail dependence, which was later generalized to the extremogram by
[5]. The extremal dependence measure based on the angular measure of a regularly varying random vector was introduced by [29] and further investigated by
[22]. [15] recently introduced a unified approach for representing tail dependence using random exceedence sets. Those measures for extremes are designed for random vectors in a Euclidean space. Thus applying any such measures to functional data requires some sort of dimension reduction, e.g., principal component analysis, or data compression like converting daily temperature curves to daily average or maximum values. The reduced data are then analyzed using those tools for multivariate extremes, see, e.g., [27], [7], and [17]. This approach is convenient, but it does not fully utilize all relevant information that functional data contain.
We develop a new measure, the extremal correlation coefficient, that captures the extremal dependence of paired functional samples utilizing the information in the sizes and shapes of the curves. The measure is constructed by computing an inner product of pairs of extreme curves. This approach is closely related to the concept of cosine similarity that is often used, for example, to quantify document similarity in text analysis. While the cosine similarity computes the inner product between two vectors to see whether they are pointing in the same direction using all available pairs, our measure calculates the inner product between paired extreme curves to see whether they look alike in extremes. Similar ideas have been applied in non-extreme contexts of functional data analysis. [9] introduced a measure, called dynamical correlation, that computes the inner product of all pairs of standardized curves. The concept was further studied by
[31] where an autocorrelation measure, termed spherical autocorrelation, for functional time series was proposed. These measures are however computed based on the total body of functional data, and so are not suitable for describing extremal dependence.
The coefficient we develop quantifies extremal dependence in pairs of heavy-tailed functional observations. It is conceptually appealing, as it shares desirable features with the classic correlation coefficient: 1) its values range from -1 to 1, 2) it measures the strength and direction of linear relationship between two extreme curves, 3) if the extremal behavior of two curves is independent, the coefficient is zero. Moreover, it can be used in practice with a relatively simple numerical implementation. We thus hope that such interpretable and tractable tool makes a useful contribution. No measures of extremal dependence for pairs of curves are currently available.
Turning to mathematical challenges, the concept of vague convergence, see e.g., Chapters 2 and 6 of [30], cannot be readily used. The vague convergence, which now provides a standard mathematical framework for extremes in Euclidean spaces, can be defined only on locally compact spaces. Since every locally compact Banach space has finite dimension, a different framework must be used for functional data in Hilbert spaces. We use the theory of regularly varying measures developed by [14] who introduced the notion of $M_0$ convergence, which works for regularly varying measures on complete separable metric spaces. The $M_0$ convergence is further studied by [27], where it is applied to regularly varying time series in a separable Banach space. Within this framework,
to establish the the consistency of the estimator we propose.
To do so, we proceed through a
a number of $M_0$ convergence results that allow us to apply an abstract Bernstein-type inequality. A method of computing the extremal
correlation coefficients analytically (in relatively simple cases)
is also developed.
The remainder of the paper is organized as follows. In Section <ref>, we review regularly varying random elements in Banach spaces. In Section <ref>, we extend the concept of regular variation to bivariate random elements in a product Banach space. The extremal correlation coefficient is introduced in Section <ref>, where its asymptotic properties are also studied. Section <ref> contains a simulation study, and Section <ref> illustrates applications to intraday return curves and daily temperature curves.
Theoretical justification of our approach requires more detailed background and some technical derivations, which are placed in the
online Supplementary Material. Preliminary results for the proof of Theorem <ref> are presented in Section <ref>, followed by its proof in Section <ref>. In Section <ref>, the proof of Lemma <ref> is presented.
§ REGULAR VARIATION IN BANACH SPACES
This section presents background needed to understand the development in Sections <ref> and <ref>. In Functional Data Analysis, observations are typically treated as elements of $L^2 := L^2(\cT)$, where the measure space $\cT$ is such that $L^2(\cT)$ is a separable Hilbert
space, equipped with the usual inner product $\lip x,y\rip = \int_{\cT} x(t)y(t) dt$. The $L^2$-norm is then $\| x\| = \lip x, x\rip^{1/2}$ = $ \lp \int_{\cT} x(t)^2 dt \rp^{1/2}$. An introduction to Functional Data Analysis is presented in [19], while a detailed mathematical treatment is given in [13]. While we refer to the elements of
$L^2$ as curves, due to the examples we consider, the set
$\cT$ can be a fairly abstract space (a metric Polish space),
for example a spatial domain.
An extreme curve in $L^2$ is defined as a functional object whose size, measured by the $L^2$-norm, is large. The norm can be large for various reasons as long as the area under the squared curve on $\cT$ is large. For example, curves that are far away from the sample mean or that fluctuate a lot around the sample mean will be extreme according to this definition. Extreme functional observations are thus very different from extreme scalar or multivariate observations because there is a multitude of ways in which a curve can be extreme. We informally call functional data heavy-tailed if the probability that an extreme curve occurs is relatively large.
We now briefly review the $M_0$ convergence in a separable Banach space $\mbB$. In what follows, $\bzero$ is the zero element. Fix a norm $\| \cdot \|_{\mbB}$ and let $B_{\eg} := \{z \in \mbB: \| z\|_{\mbB} <\eg\}$ be the open ball of radius $\eg>0$ centered at the origin. A Borel measure $\mu$ defined on $\mbB_0:=\mbB \setminus \{\bzero\}$ is said to be boundedly finite if $\mu(A)<\infty$, for all Borel sets that are bounded away from $\bzero$, i.e., $A \cap B_{\eg} = \emptyset$, for some $\eg>0$. Let $M_0 (\mbB)$ be the collection of all such measures on $\mbB_0$. For $\mu_n, \mu \in M_0(\mbB)$, the sequence of $\mu_n$ converges to $\mu$ in the $M_0$ topology ($\mu_n \stackrel{M_0}{\longrightarrow} \mu$),
if $\mu_n(A) \to \mu(A)$, for all bounded away from $\bzero$, $\mu$–continuity Borel sets $A$, i.e., those with $\mu(\partial A)=0$, where $\partial A$ is the boundary of $A$. Equivalently, $\mu_n \stackrel{M_0}{\longrightarrow} \mu$, if $\int_{\mbB} f(x) \mu_n(dx) \to \int_{\mbB} f(x) \mu(dx)$ for all $f \in \cC_0(\mbB)$, where $\cC_0 (\mbB)$ is the class of bounded and continuous functions $f:\mbB_0 \to \mbR$ that vanish on a neighborhood of $\bzero$.
We now define regular variation for random elements in $\mbB$, see Theorem 3.1 of [14] and Chapter 2 of [27]. This concept formalizes the idea of heavy-tailed observations in infinite dimensional spaces.
A random element $X$ in $\mbB$ is regularly varying with index $-\ag$, $\ag> 0$, if there exist a sequence $b(n) \to \infty$ and a measure $\mu$ in $M_0(\mbB)$ such that
\begin{equation} \label{e:def1-H}
nP \lp \frac{X}{b(n)} \in \cdot \rp
\stackrel{M_0}{\longrightarrow} \mu, \ \ \ \ n \to \infty,
\end{equation}
where the exponent measure $\mu$ satisfies $\mu(tA) = t^{-\ag}\mu(A)$ for Borel sets $A \subset \mbB_0$.
A possible choice for $b(n)$ is the quantile function, defined by $P(\| X \|_{\mbB} >b(n)) = n^{-1}$. Roughly speaking, the tail probability of $X$ decays like a power function, $P(\| X \|_{\mbB} > t ) \approx C t^{-\ag}$, as $t\to \infty$.
The following lemma, see [14],
states an equivalent definition of a regularly varying element in $\mbB$.
A random element $X$ in $\mbB$ is regularly varying with index $-\ag$, $\ag> 0$, if and only if there exist a sequence $b^{\prime}(n) \to \infty$ and a probability measure $\Gg$ on $\mbS:=\{x \in \mbB : \|x \|_{\mbB}=1\}$ (called the angular measure) such that for any $y>0$,
\begin{equation} \label{e:def2-H}
nP \lp \| X \|_{\mbB} >b^{\prime}(n)y, X/\| X \|_{\mbB} \in \cdot \rp
\stackrel{w}{\longrightarrow} c y^{-\ag}\Gg, \ \ \ \ n \to \infty,
\end{equation}
for some $c>0$.
The first three orthonormal basis elements in $L^2[0,1]$ defined in basis (left-most); simulated data when $\Gg$ concentrates on $\phi_1$(second from the left); on $\phi_2$ (third left); on $\phi_3$ (fourth left).
If Definition <ref> (or condition def2-H) holds,
we write $X \in RV(-\ag, \Gg)$. The polar representation def2-H provides an intuitive interpretation of regular variation in $\mbB$. It characterizes regular variation of $X$ in $\mbB$ using two components, the tail index $\ag$ and the angular probability measure $\Gg$. The tail index $\ag$ quantifies how heavy the tail distribution of $\|X\|_{\mbB}$ is, e.g., the probability of extreme curves occurring gets higher as $\ag$ gets smaller. While the tail index $\ag$ determines the frequency of occurrence of extreme curves, the angular measure $\Gg$, defined on the unit sphere $\mbS$, fully characterizes the distribution of the shape of the scaled extreme curves, $X/\|X\|_{\mbB}$. To illustrate this, consider a set of orthonormal functions in $L^2([0,1])$ of the form
\begin{equation} \label{e:basis}
\phi_j(t) = \sqrt{2} \sin \lp \lp j-\frac{1}{2}\rp \pi t\rp, \ \ \ \ j=1,2,\ldots, \ \ t \in [0,1].
\end{equation}
The first three functions are shown in the left-most plot of Figure <ref>. We consider a finite-dimensional subspace of $L^2([0,1])$, spanned by the first 9 $\phi_j$'s, for the purpose of simulations. The data generating process is
$X(t) = \sum_{j=1}^{9} Z_{j} \phi_j(t)$,
where $\bZ=[Z_1, \ldots, Z_9]$ is a 9-dimensional random vector with independent components. Suppose that $Z$ is a random variable following a Pareto distribution with tail index $\ag=3$ and $N$ is a normal random variable with mean 0 and variance .5. We consider the following three cases for $\bZ$:
1. $\bZ = [Z, N, N, N, \ldots, N]^{\top}$; the angular measure $\Gg$ concentrates on $\phi_1$.
2. $\bZ = [N, Z, N, N \ldots, N]^{\top}$; the angular measure $\Gg$ concentrates on $\phi_2$.
3. $\bZ = [N, N, Z, N \ldots, N]^{\top}$; the angular measure $\Gg$ concentrates on $\phi_3$.
Figure <ref> displays simulated data with sample size of 100 for each of the three cases. The plots of simulated data clearly show that the angular measure $\Gg$ represents the distribution of the shapes of extreme curves in that they are dominated by the shape of the functional axis $\phi_j$ on which $\Gg$ concentrates.
§ BIVARIATE REGULAR VARIATION IN BANACH SPACES
In order to describe the extremal dependence of two regularly varying random elements $X$ and $Y$ in $L^2$, we need to identify their joint probabilistic behavior. We again study it in the more general space $\mbB^2$. We propose the following definition.
A bivariate random element $[X,Y]^{\top
}$ in $\mbB^2$ is said to be jointly regularly varying with index $-\ag$, $\ag> 0$, if there exist a sequence $b(n) \to \infty$ and a measure $\mu$ in $M_0(\mbB^2)$ such that
\begin{equation} \label{e:def-RVXY}
nP \lp \frac{(X,Y)}{b(n)} \in \cdot \rp
\stackrel{M_0}{\longrightarrow} \mu, \ \ \ \ n \to \infty,
\end{equation}
where the joint exponent measure $\mu$ satisfies $\mu(tA) = t^{-\ag}\mu(A)$ for Borel sets $A \subset \mbB_0^2$.
We assume that one-dimensional marginal distributions of $\mu$ are non-degenerate, i.e., $\mu_X:=\mu(\cdot \times \mbB)$ and $\mu_Y:=\mu(\mbB \times \cdot)$ are measures in $M_0(\mbB)$ satisfying analogs of def1-H. Since $X$ and $Y$ are normalized by the same function $b(n)$, the marginal distributions are tail equivalent. A possible choice for $b(n)$ is the quantile function, defined by
\[
P(\|(X,Y)\|_{\mbB^2} >b(n)) = n^{-1}.
\]
With this choice, we have that
\[
nP \lp \frac{(X,Y)}{b(n)} \in \cA_1 \rp = \frac{P \lp (X,Y) \in b(n)\cA_1 \rp }{P(\|(X,Y)\|_{\mbB^2} >b(n)) } =\frac{P \lp (X,Y) \in \cA_{b(n)} \rp }{P \lp (X,Y) \in \cA_{b(n)} \rp}=1, \]
where $\cA_{r}$ is defined by
\begin{equation} \label{e:A}
\mathcal A_r =\{(x,y)\in \mbB^2: \|(X,Y)\|_{\mbB^2} \ge r\}, \ \ \ \ r>0.
\end{equation}
Thus, it follows from the $M_0$ convergence in def-RVXY and Lemma <ref> that
\begin{equation} \label{e:mu1}
\mu(\cA_1 )=\mu\{(x,y) \in \mbB^2: \|(X,Y)\|_{\mbB^2} >1\}=1,
\end{equation}
which implies that $\mu$ is a probability measure on $\cA_1$. Throughout the paper, we set
\[
\|(x,y)\|_{\mbB^2} := \|x\|_{\mbB} \vee \|y\|_{\mbB}.
\]
This choice of norm works well with the extremal correlation coefficient defined in Section <ref>.
In order to derive the joint angular probability measure of $X$ and $Y$, we consider the polar coordinate transformation $T:\mbB_0^2 \to \lp [0,\infty)^2 \setminus \{\bzero\} \rp \times \mbS^2$, defined by
\begin{equation}\label{e:T}
T(x,y) = \lp \|x\|_{\mbB}, \|y\|_{\mbB}, \frac{x}{\|x\|_{\mbB}}, \frac{y}{\|y\|_{\mbB}} \rp=: (r_X, r_Y, \thg_X, \thg_Y), \ \ \ \ (x,y) \in \mbB_0^2.
\end{equation}
Using $T$, we obtain an equivalent formulation for a regularly varying random element in $\mbB^2$.
A bivariate random element $[X,Y]^{\top}$ in $\mbB^2$ is regularly varying with index $-\ag$, $\ag>0$, if and only if there exists an exponent measure $\nu$ in $M_0([0,\infty)^2\setminus\{\bzero\})$ and an angular probability measure $\Gg$ in $M_0(\mbS^2)$ such that
\begin{equation} \label{e:pol-RVXY}
nP \lp \frac{(\|X\|_{\mbB}, \|Y\|_{\mbB})}{b(n)} \in \cdot, \lp X/\|X\|_{\mbB}, Y/\|Y\|_{\mbB} \rp\in \cdot \rp
\stackrel{M_0}{\longrightarrow} \nu \times \Gg, \ \ n \to \infty,
\end{equation}
where $b(n)$ is the increasing sequence in def-RVXY. (We note that $\mu$ in def-RVXY is a measure on $\mbB_0^2$, while $\nu$ in pol-RVXY is a measure on $[0,\infty)^2 \setminus \{\bzero\}$.)
We only show that def-RVXY implies pol-RVXY since showing the converse is similar. Take any $f \in \cC_0([0,\infty)^2 \times \mbS^2)$. It then follows from the change of variables that
\begin{align*}
& \int_{[0,\infty)^2} \int_{\mbS^2} f(r_X, r_Y, \thg_X, \thg_Y) \ nP \lp \frac{(\|X\|_{\mbB}, \|Y\|_{\mbB})}{b(n)} \in (dr_X, dr_Y), \lp \frac{X}{\|X\|_{\mbB}}, \frac{Y}{\|Y\|_{\mbB}} \rp \in (d\thg_X, d\thg_Y)\rp \\
% & = \int_{[0,\infty)^2 \times \mbS^2} f(r_X, r_Y, \thg_X, \thg_Y) \ \frac{1}{V(u)}P \lp \lp u^{-1}X, u^{-1}Y \rp \rp \circ T^{-1} (dr_X, dr_Y, d\thg_X, d\thg_Y) \\
&= \int_{\mbB^2} f(T(x,y)) \ nP \lp \frac{X}{b(n)} \in dx, \frac{Y}{b(n)} \in dy\rp.
\end{align*}
Since $f \in \cC_0([0,\infty)^2 \times \mbS^2)$, there exists a set $A$, bounded away from $\bzero$ in $[0,\infty)^2 \times \mbS^2$, such that $f(T(x,y)) =0$ if $T(x,y) \notin A$. Then we have that $f(T(x,y)) =0$ if $(x,y) \notin T^{-1}(A)$. Since $T^{-1}(A)$ is bounded away from $\bzero$ in $\mbB^2$, we have that $f \circ T \in \cC_0(\mbB^2)$. Then by the $M_0$ convergence in Definition <ref>, we have that
\begin{align*}
&\int_{\mbB^2} (f \circ T)(x,y) \ nP \lp \frac{X}{b(n)} \in dx, \frac{Y}{b(n)} \in dy\rp \\
&\to \int_{\mbB^2} (f \circ T)(x,y) \mu(dx,dy) = \int_{T(\mbB^2)} f(r_X, r_Y, \thg_X, \thg_Y) \mu\circ T^{-1}(dr_X, dr_Y, d\thg_X, d\thg_Y).
% &=c\int_{\mbS^2} \int_{[0,\infty)^2} f(r_X, r_Y, \thg_X, \thg_Y) \nu(dr_X, dr_Y) \Gg (d\thg_X, d\thg_Y)
\end{align*}
To investigate the form of $\mu\circ T^{-1}$, take any $t>0$ and Borel set $S \subset \mbS^2$. It then follows from the homogeneity property of $\mu$ that
\begin{align*}
&\mu\circ T^{-1} ( [0,\infty)^2\setminus[0,t]^2 \times S) \\
&= \mu \lbr (x,y) \in \mbB^2_0: \|x\|_{\mbB} \vee \|y\|_{\mbB} >t, (x/\|x\|_{\mbB}, y/\|y\|_{\mbB}) \in S \rbr \\
&= \mu \lbr (x,y) \in \mbB^2_0: \|x\|_{\mbB} \vee \|y\|_{\mbB} >t \rbr \times \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{ t^{-\ag}\mu \lbr (x,y) \in \mbB^2_0: \|x\|_{\mbB} \vee \|y\|_{\mbB} >1, (x/\|x\|_{\mbB}, y/\|y\|_{\mbB}) \in S \rbr}{ t^{-\ag}\mu \lbr (x,y) \in \mbB^2_0: \|x\|_{\mbB} \vee \|y\|_{\mbB} >1 \rbr}.
\end{align*}
It then follows from mu1 that
\[
\mu\circ T^{-1} ( [0,\infty)^2\setminus[0,t]^2 \times S) = \nu([0,\infty)^2\setminus[0,t]^2) \Gg(S),
\]
\begin{align*}
&\nu(A) := \mu \lbr (x,y) \in \mbB^2_0: (\|x\|_{\mbB}, \|y\|_{\mbB}) \in A \rbr,\ \ \ \ A \subset [0,\infty)^2 \setminus \{\bzero\};\\
&\Gg(S) := \mu \lbr (x,y) \in \mbB^2_0: \|x\|_{\mbB} \vee \|y\|_{\mbB} > 1, (x/\|x\|_{\mbB}, y/\|y\|_{\mbB}) \in S \rbr, \ \ \ \ S\subset \mbS^2.
\end{align*}
Thus, $\mu \circ T^{-1}$ has the product form such that on $([0,\infty)^2\setminus \{\bzero\}) \times \mbS^2$
\begin{equation} \label{e:nuGg}
\mu \circ T^{-1} = \nu\times \Gg,
\end{equation}
which completes the proof.
Convergence pol-RVXY can be understood as a polar representation of the bivariate regular variation of $[X,Y]^{\top}$ in $\mbB^2$.
The difference between $\nu$ in pol-RVXY and $\mu$ in def-RVXY is that $\mu$ describes the joint behavior of extreme curves $X$ and $Y$ in $\mbB_0^2$, but $\nu$ describes extremal dependence between the sizes $\|X\|_{\mbB}$ and $\|Y\|_{\mbB}$ in $[0,\infty)^2\setminus \bzero$.
The joint behavior of $X$ and $Y$ in extremes is thus characterized by two measures, $\nu$ on $[0,\infty)^2 \setminus \{\bzero\}$ and $\Gg$ on $\mbS^2$. The measure $\nu$ describes the joint behavior of $\|X\|_{\mbB}$ and $\|Y\|_{\mbB}$ in extremes. If $\nu$ has its mass only on the axes, then $\|X\|_{\mbB}$ and $\|Y\|_{\mbB}$ are asymptotically independent, i.e., if one curve shows an extreme behavior, there is negligible probability of the other curve also showing an extreme behavior. If $\nu$ has mass only on the line $\{t(1, 1), t > 0\}$, then $\|X\|_{\mbB}$ and $\|Y\|_{\mbB}$ show asymptotic full dependence, i.e., extreme curves occur simultaneously in $X$ and $Y$.
We remark that the measure $\nu$ on $[0,\infty)^2 \setminus \bzero$ is homogeneous because for any $A \subset [0,\infty)^2 \setminus \{\bzero\}$ and $t>0$,
\begin{align*}
\nu(tA) &= \mu \lbr (x,y) \in \mbB^2_0: (\|x\|_{\mbB}, \|y\|_{\mbB}) \in tA \rbr\\
&= \mu \lbr t(x^{\prime},y^{\prime}) \in \mbB^2_0: (\|x^{\prime}\|_{\mbB}, \|y^{\prime}\|_{\mbB}) \in A \rbr \\
&= t^{-\ag}\mu \lbr (x^{\prime},y^{\prime}) \in \mbB^2_0: (\|x^{\prime}\|_{\mbB}, \|y^{\prime}\|_{\mbB}) \in A \rbr\\
\end{align*}
The joint angular probability measure $\Gg$ characterizes how the shapes of scaled $X$ and $Y$ are related in extremes. If the extreme curves are exactly proportional, i.e., $X=\la Y$, $\la \neq 0$, the scaled curves share the same extreme functional elements. This means that $\Gg$ concentrates on the “line" $\{(\phi_j,\phi_j),\ j\in \cJ \subset \mbN \}\subset\mbS^2$, where $\{\phi_j,\ j \ge 1\}$ is a set of orthonormal elements in $\mbS$. If the shapes of two curves do not match in extremes, then $\Gg$ concentrates on $\{(\phi_j,\phi_k)\}\subset\mbS^2$, where $ j\in \cJ \subset \mbN$, $k \in \cK\subset \mbN$, $\cJ \cap \cK = \emptyset$. This situation corresponds to vanishing extremal covariance defined in Section <ref>.
The marginal extreme behavior of $X$ can be obtained by integrating all possible values of $Y$ in def-RVXY, or $\|Y\|_{\mbB}, Y/\|Y\|_{\mbB}$ in pol-RVXY. Then $X$ has its marginal measure $\mu_X$, and equivalently $X \in RV(-\ag, \Gg_X)$, where $\Gg_X$ is the marginal angular measure of $X$. Similarly, $Y$ has its marginal $\mu_Y$, and equivalently $Y \in RV(-\ag, \Gg_Y)$, where $\Gg_Y$ is the marginal angular measure of $Y$.
§ EXTREMAL CORRELATION COEFFICIENT FOR FUNCTIONAL DATA
The range of $\sg_{XY}$, extremal covariance between $X$ and $Y$, depending on the extremal dependence between $\|X\|$ and $\|Y\|$ and the level of similarity between $X/\|X\|$ and $Y/\|Y\|$ in extremes.
$\|X\|$ and $\|Y\|$ 2c$\|X\|$ and $\|Y\|$
Asymptotic Independence 2cAsymptotic Dependence
$X/\|X\|$ and $Y/\|Y\|$
$\sg_{XY}= 0$ look similar $\sg_{XY} >0$
look orthogonal $\sg_{XY}\approx 0$
look opposite $\sg_{XY}<0$
In this section, we introduce the extremal correlation coefficient for functional data. It focuses on the extreme part of the joint distribution of regularly varying random elements $X$ and $Y$ in $L^2$. It measures the tendency of paired curves to exhibit similar extreme patterns by computing a suitable inner product between $X$ and $Y$ conditional on large $[X, Y]^{\top}$.
Given a regularly varying bivariate random element $[X,Y]^{\top}$ in $L^2\times L^2$ with joint exponent measure $\mu$, we define the extremal covariance between $X$ and $Y$ by
\begin{equation} \label{e:cov}
\sg_{XY} = \int_{\|x\| \vee \|y\|>1} \lip x, y \rip \mu(dx, dy).
\end{equation}
Recall that by mu1, $\mu I_{\{\|x\| \vee \|y\|>1\}}$ is a probability measure. The extremal covariance is thus an extreme analog of the classic covariance in that $\sg_{XY}$ measures how much two random curves vary together in extremes. To see this more closely, we recall the transformation $T$ defined in T and the relation $\mu\circ T^{-1} = \nu \times \Gg$ in nuGg. It then follows from the change of variables that
\begin{equation} \label{e:cov_pol}
\sg_{XY} = \int_{r_X \vee r_Y >1} r_Xr_Y \ \nu (dr_X,dr_Y) \int_{\mbS^2} \lip \thg_X, \thg_Y \rip \Gg (d\thg_X,d\thg_Y).
\end{equation}
The extremal covariance of $X$ and $Y$ can be thus factorized into the extremal dependence between $\|X\|$ and $\|Y\|$ and the level of similarity of the shapes between $X/\|X\|$ and $Y/\|Y\|$. If $\|X\|$ and $\|Y\|$ are asymptotically independent, i.e., extreme curves do not occur simultaneously, then $\nu$ concentrates on the coordinate axes. This implies that $\int_{r_X \vee r_Y >1} r_Xr_Y \ \nu (dr_X,dr_Y)=0$, so $\sg_{XY}=0$ regardless of how $X$ and $Y$ look like in extremes. If $\|X\|$ and $\|Y\|$ are asymptotically dependent, i.e., extreme norms tend to occur simultaneously, then $\int_{r_X \vee r_Y >1} r_Xr_Y \ \nu (dr_X,dr_Y)> 0$ and so there are three possible ranges for $\sg_{XY}$ depending on the relative shape of extreme $X/\|X\|$ and $Y/\|Y\|$: 1) $\sg_{XY}>0$ if the shapes look similar, 2) $\sg_{XY} \approx 0$ if the shapes do not match, i.e., the curves are orthogonal, or 3) $\sg_{XY}<0$ if they look opposite. These properties are summarized in Table <ref>.
We further define the extremal correlation coefficient by
\begin{equation} \label{e:cor}
\rho_{XY} = \frac{\sg_{XY}}{\sg_{X}\sg_{Y}},
\end{equation}
\begin{align*}
\sg_{X} = \lbr \int_{\|x\| \vee \|y\|>1} \|x\|^2 \mu(dx,dy) \rbr^{1/2}, \ \ \ \
\sg_{Y} = \lbr \int_{\|x\| \vee \|y\|>1} \|y\|^2 \mu(dx,dy) \rbr^{1/2}.
\end{align*}
The coefficient $\rho_{XY}$ has properties analogous to the classic correlation coefficient: 1) $-1\le \rho_{XY} \le 1$, 2) $\rho_{XY}$ measures the strength and direction of linear relationships between $X$ and $Y$ in extremes, 3) If $X$ and $Y$ are independent, then $\rho_{XY}=0$ since independence implies asymptotic independence between $\|X\|$ and $\|Y\|$.
To motivate our estimation approach, we first show that $\sg_{XY}$ is a limit of the expected inner product of $X$ and $Y$ conditional on large values of $[X,Y]^{\top}$.
Let $[X,Y]^{\top}$ be a regularly varying random element in $L^2 \times L^2$. Then,
\[
\sg_{XY} = \lim_{n\to \infty}E\lb \lip \frac{X}{b(n)}, \frac{Y}{b(n)} \rip \Bigg|\|X\| \vee \|Y\| >b(n)\rb.
\]
Considering $f: L^2 \times L^2 \to \mbR$ defined by $(x, y) \to \lip x, y \rip I_{\|x\| \vee \|y\|>1}$, we have that
\begin{align*}
&E\lb \lip {b(n)}^{-1}X, {b(n)}^{-1}Y \rip | \|X\| \vee \|Y\| >{b(n)}\rb \\
&=\frac{1}{P(\|X\|\vee\|Y\|>{b(n)})} E\lb \lip {b(n)}^{-1}X, {b(n)}^{-1}Y \rip I_{\|X\| \vee \|Y\| >{b(n)}}\rb\\
&=\int_{L^2 \times L^2} f(x,y) \frac{P({b(n)}^{-1}X \in dx, {b(n)}^{-1}Y \in dy)}{P(\|X\|\vee\|Y\|>{b(n)})}.
\end{align*}
Note that $f$ is bounded and vanishes on a neighborhood of $\bzero$ in $L^2 \times L^2$. Also, the discontinuity set of $f$ is the boundary of $\mathcal A_1 = \{(x,y) \in L^2 \times L^2: \|x\| \vee \|y\| \ge 1\}$, and it follows from Lemma <ref> that $\mu(\partial \mathcal A_1)=0$. Therefore, by def-RVXY and Lemma A.1 of [27], we get the claim.
Based on Proposition <ref>, we propose an estimator for $\sg_{XY}$ defined by
\begin{equation} \label{e:ec}
\hat{\sg}_{n,k} =\frac{1}{k}\sum_{i=1}^n \lip \frac{X_i}{R_{(k)}}, \frac{Y_i}{R_{(k)}} \rip I_{R_i \ge R_{(k)} },
\end{equation}
where $[X_i, Y_i]^{\top}, 1 \le i \le n$, are i.i.d. copies of $[X,Y]^{\top}$, $R_i:=\|X_i\| \vee \|Y_i\|$ and $R_{(k)}$ is the $k$th largest order statistic with the convention $R_{(1)} = \max \{ R_1, \ldots, R_n \}$. An estimator for $\rho_{XY}$ is then defined by
\begin{equation} \label{e:ecc}
\hat{\rho}_{n,k} = \frac{\sum_{i=1}^n \lip X_i, Y_i \rip}{\sqrt{\sum_{i=1}^n \|X_i\|^2} \sqrt{\sum_{i=1}^n \|Y_i\|^2}} \ I_{R_i \ge R_{(k)} }.
\end{equation}
These estimators take only the $k$ largest pairs of $[X_i, Y_i]^{\top}$, $1\le i\le n$, as inputs. This approach falls into so-called peaks-over-threshold framework in that it relies only on $k$ largest observations whose magnitude exceeds a certain threshold. Asymptotic properties in this framework are typically derived as $k$ goes to infinity with $n$, in such a way that $k/n \to 0$. We assume throughout the paper that this condition holds.
We will work under the following assumption.
The bivariate random element $[X,Y]^{\top}$ in $L^2 \times L^2$ has mean zero and is regularly varying with index $-\ag$, $\ag >2$. The observations $[X_1,Y_1]^{\top}, [X_2,Y_2]^{\top}, \ldots$ are independent copies of $[X,Y]^{\top}$.
We state in the following theorem that the estimator $\hat{\sg}_{n,k}$ is consistent for the extremal covariance. All proofs of the theoretical results introduced in this section are presented in Sections <ref> and <ref> of the Appendix, as they require a number of preliminary results and technical arguments.
Under Assumption <ref>,
\[
\hat{\sg}_{n,k} \convP \sg_{XY},
\]
where $\hat{\sg}_{n,k}$ and $\sg_{XY}$ are defined in ec and cov, respectively.
From Theorem <ref>, the consistency of $\hat{\rho}_{n,k}$ for $\rho_{XY}$ follows from Slutsky's theorem.
Under Assumption <ref>,
\[
\hat{\rho}_{n,k} \convP \rho_{XY},
\]
where $\hat{\rho}_{n,k}$ and $\rho_{XY}$ are defined in ecc and cor, respectively.
We end this section with a discussion on the condition $\ag>2$ in Assumption <ref>. The condition is needed because the definition of extremal correlation coefficient presumes the existence of the second moment of the underlying processes. It can be lifted for the following alternative measure
\begin{equation} \label{e:ga}
\ga_{XY} := \int_{\mbS^2} \lip \thg_X, \thg_Y \rip \Gg(d\thg_X, d\thg_Y),
\end{equation}
which is the angular density factor in cov_pol. This measure itself could be considered a measure for extremal correlation since $-1 \le \ga_{XY} \le 1$.
Just as Proposition <ref>, it can be shown that
\[
\ga_{XY} = \lim_{n\to \infty} E \lb \lip \frac{X}{\|X\|}, \frac{Y}{\|Y\|} \rip \Bigg| \|X\| \vee \|Y\| >b(n)\rb.
\]
From this relation, an estimator for $\ga_{XY}$ can be defined by
\[
\hat{\ga}_{n,k} = \frac{1}{k} \sum_{i=1}^N \lip \frac{X_i}{\|X_i\|}, \frac{Y_i}{\|Y_i\|} \rip I_{R_i \ge R_{(k)}},
\]
and its consistency can be proven in almost the same manner as the proof of Corollary 4.2 of [2], where $\ag>0$ is assumed.
Although the assumption $\ag>2$ could be relaxed by employing $\ga_{XY}$ as an extremal correlation measure, it should be noted that $\ga_{XY}$ does not account for whether extreme curves $X$ and $Y$ occur simultaneously, as $\rho_{XY}$ in cor does. To see this, suppose that $X=Z_1\phi$ and $Y=Z_2\phi$, where $Z_1$, $Z_2$ are independent random variables satisfying $P(Z_{1}>z) = z^{-\ag}$, $z>0$, and $\phi$ is any unit norm
element of $L^2$. Then, $\ga_{XY} = 1$, but $\rho_{XY}=0$.
One can argue that the measure $\rho_{XY}=0$ is more appropriate
because the extremes of $X$ and $Y$ occur independently,
and the objective of our coefficient is to measure similarity
of shapes in paired extreme curves. The identity $\rho_{XY}=0$
holds because the measure $\nu$ in cov_pol concentrates on the coordinate axes, so
\[
\sg_{XY} = \int_{r_X \vee r_Y >1} r_Xr_Y \ \nu (dr_X,dr_Y) \int_{\mbS^2} \lip \thg_X, \thg_Y \rip \Gg (d\thg_X,d\thg_Y) = 0\times 1=0,
\]
and the condition $\ag >2$ is needed for the first integral to exist.
§ A SIMULATION STUDY
We perform a simulation study to demonstrate that the proposed estimator, $\hat{\rho}_{n,k}$, consistently estimates the extremal correlation. We generate functional observations in such a way that the theoretical value of $\rho_{XY}$ can be computed analytically, so that we can see how close $\hat{\rho}_{n,k}$ is to the true value.
The design of our study is as follows. Suppose that $Z_1$, $Z_2$ are i.i.d. random variables in $\mbR$ satisfying $P(|Z_{1}|>z) = z^{-\ag}$ with equal chance of $Z_1$ being either negative or positive. Also, let $N_1$, $N_2$, $N_3$ be i.i.d. normal random variables in $\mbR$ with mean 0 and variance 1. Consider $\{\phi_j, j \ge 1 \}$ defined by basis and recall that it is an orthonormal basis in $L^2([0,1])$. These functions are simulated on a grid of 100 equally–spaced points on the unit interval $[0,1]$. We consider the following data generating processes, for $-1 \le \rho \le 1$,
\begin{align} \label{e:XY}
X(t) &= Z_{1}\phi_1(t) + N_1\phi_2(t) + N_2\phi_3(t); \\ \nonumber
Y(t) &= \rho Z_{1}\phi_1(t) + (1-\rho^2)^{1/2} Z_{2}\phi_2(t) + N_3\phi_3(t).
\end{align}
This generates extreme curves dominated by the shape of the functional axis $\phi_1$ for $X$ and by either $\phi_1$ or $\phi_2$ for $Y$. The following lemma gives an analytic formula for $\rho_{XY}$. Its proof is provided in Section <ref> of the Appendix.
Let $\bZ = [Z_1, Z_2]^{\top}$ be a random vector in $\mbR^2$ consisting of iid components $Z_j$ whose magnitude is regularly varying with tail index $\ag>2$, i.e., for some $c_{+}$, $c_{-}\ge 0$,
\[
P(Z_1 >z) \sim c_{+}z^{-\ag}, \ \ \ \ P(Z_1 <-z) \sim c_{-}z^{-\ag},
\]
where $f(z) \sim g(z)$ if and only if $\lim_{z\to \infty}f(z)/g(z)=1$.
Also, let $\{\phi_j,\ j \ge 1\}$ be a set of orthonormal elements in $\mbS$. Then, for $X$ and $Y$ in XY,
\[
\rho_{XY} = \frac{\rho}{\{\rho^2 + (1-\rho^2)^{\ag/2}\}^{1/2}}.
\]
Empirical biases (standard errors) of $\hat{\rho}_{n,k}$ in ecc when $\ag=2.1$. The number of upper order statistics is $k = 2\lfloor n^{1/2} \rfloor$.
$\rho_{XY}$ $N=100$ $N=500$ $\rho_{XY}$ $N=100$ $N=500$
0 0.000 (0.051) 0.000 (0.023)
0.1 0.005 (0.086) 0.015 (0.096) -0.1 -0.008 (0.097) -0.009 (0.070)
0.2 0.012 (0.138) 0.014 (0.107) -0.2 -0.009 (0.133) -0.016 (0.116)
0.3 -0.012 (0.159) 0.012 (0.137) -0.3 0.002 (0.166) -0.013 (0.142)
0.4 -0.021 (0.176) 0.007 (0.149) -0.4 0.010 (0.180) -0.011 (0.156)
0.5 -0.041 (0.178) 0.002 (0.164) -0.5 0.030 (0.179) 0.008 (0.161)
0.6 -0.066 (0.180) -0.011 (0.168) -0.6 0.056 (0.180) 0.012 (0.162)
0.7 -0.071 (0.179) -0.029 (0.153) -0.7 0.076 (0.174) 0.032 (0.160)
0.8 -0.093 (0.155) -0.040 (0.142) -0.8 0.097 (0.158) 0.032 (0.138)
0.9 -0.112 (0.127) -0.043 (0.108) -0.9 0.111 (0.127) 0.041 (0.102)
1.0 -0.116 (0.066) -0.040 (0.021) -1.0 0.118 (0.069) 0.041 (0.021)
We consider $\rho_{XY} \in \{0, \pm.1, \pm.2, \ldots, \pm .9, \pm 1 \}$ and $\ag \in\{2.1, 3,4,5\}$, from which values of $\rho$ can be obtained by Lemma <ref>. For each $\rho$, we generate $[X_i, Y_i]^{\top}$, $1 \le i \le N$, that are i.i.d. copies of $[X,Y]^{\top}$, with sample sizes $N \in \{100, 500\}$. In each case, 1000 replications are generated.
Empirical biases (standard errors) of $\hat{\rho}_{n,k}$ when $\ag=2.1$. The KS method is used to choose optimal $k$s. On average, it selects $k=8$ ($N=100$) and $k=25 \sim 32$ ($N=500$).
$\rho_{XY}$ $N=100$ $N=500$ $\rho_{XY}$ $N=100$ $N=500$
0 -0.001 (0.083) 0.000 (0.041)
0.1 0.026 (0.139) 0.032 (0.127) -0.1 -0.033 (0.153) -0.026 (0.117)
0.2 0.042 (0.204) 0.033 (0.156) -0.2 -0.043 (0.198) -0.040 (0.179)
0.3 0.022 (0.225) 0.035 (0.190) -0.3 -0.033 (0.228) -0.032 (0.195)
0.4 0.013 (0.229) 0.023 (0.193) -0.4 -0.023 (0.234) -0.028 (0.204)
0.5 -0.006 (0.232) 0.019 (0.204) -0.5 -0.006 (0.231) -0.002 (0.215)
0.6 -0.032 (0.235) 0.006 (0.211) -0.6 0.021 (0.233) -0.002 (0.207)
0.7 -0.034 (0.221) -0.019 (0.188) -0.7 0.039 (0.217) 0.025 (0.198)
0.8 -0.049 (0.195) -0.033 (0.173) -0.8 0.054 (0.192) 0.024 (0.170)
0.9 -0.067 (0.153) -0.032 (0.122) -0.9 0.066 (0.146) 0.034 (0.124)
1.0 -0.064 (0.056) -0.029 (0.026) -1.0 0.065 (0.059) 0.030 (0.027)
In order to compute $\hat{\rho}_{n,k}$ in ecc, we must select $k$ largest pairs of curves. We first consider $k=2\lfloor N^{1/2} \rfloor$, where $\lfloor x \rfloor$ is the integer part of $x$, to demonstrate the performance of the estimator with a deterministic form of $k$. Additionally, we provide a method for determining the optimal value of $k$ as a guiding tool in practical applications.
Our approach to identifying the optimal value of $k$ is motivated by the theoretical property of the size of pairs, $R_i=\|X_i\| \vee \|Y_i\|$. By Lemma <ref> (i), if $(X, Y)$ are regularly varying in $L^2 \times L^2$ with tail index $\ag$, then $R = \|X\| \vee \|Y\|$ is also regularly varying in $\mbR_+$ with the same tail index $\ag$. Therefore, we choose $k$ that results in successful tail estimation for $R$ in finite samples. In the literature on tail estimation, various methods for selecting upper-order statistics have been introduced, e.g., [11], [10], [8], and [4], just to name a few. We adopt one of the methods proposed by [3]. It chooses $k$ that minimizes the“Kolmogorov-Smirnov"(KS) distance between the empirical tail and the theoretical tail of a Pareto distribution, which is shown in their paper to exhibit relatively good/stable performance in finite samples. This method is implemented by the function mindist of the R package tea.
We now report empirical biases (average minus theoretical value) and standard errors computed as sample standard deviations. For $\ag=2.1$, the results with $k=2\lfloor N^{1/2} \rfloor$ are presented in Table <ref>, and the results with the optimal $k$s based on the KS method in Table <ref>. We put the results with the KS method when $\ag \in \{3,4,5\}$ in Section <ref> of Appendix since they show similar results as when $\ag=2.1$. The conclusions are summarized as follows.
* The estimator is consistent as both the bias and the standard errors decrease with increasing sample sizes, across almost all values of $\rho_{XY}$. The data-driven method for selecting $k$ yields slightly higher standard errors than the deterministic method, but it produces relatively smaller biases when $|\rho_{XY}|$ is close to 1.
* The bias tends to increase in magnitude as $|\rho_{XY}|$ approaches 1. This could be due to the effect of the boundary, $\rho_{XY} \in \{-1, 1\}$. These barriers make the estimator underestimate the true value.
* The standard errors are observed to be non-uniform across $\rho_{XY}$, they roughly behave like a quadratic function of $\rho_{XY}$ with its peak at $\pm.5$.
The last finding suggests that the asymptotic variance of $\hat{\rho}_{n,k}$ could be proportional to $|\rho_{XY}|(1-|\rho_{XY}|)$, just like for the classic correlation coefficient. The derivation of the asymptotic distribution of
$\hat \rho_{XY}$ is postponed to future work.
§ APPLICATIONS TO FINANCIAL AND CLIMATE DATA
In this section, we compute the extremal correlation coefficient
for a number of paired functional data samples that fall into
two categories: intraday returns and daily temperatures.
Our objective is to show that the coefficient provides meaningful and useful information.
§.§ Extremal dependence of intraday returns
on sector ETFs
In this section, we study pairwise extremal dependence of cumulative intraday return curves (CIDRs) of Exchange Traded Funds (ETFs) reflecting
performance of key sectors of the U.S. economy. We work with nine
Standard & Poor's Depositary Receipt ETFs listed in Table <ref>.
Our objective is to
measure the tendency of paired CIDRs to exhibit similar extreme daily trajectories during the market decline caused by the COVID-19 pandemic.
The CIDRs are defined as follows. Denote by $P_i(t)$ the price of an asset
on trading day $i$ at time $t$. For the assets in our example, $t$ is time in minutes between 9:30 and 16:00 EST (NYSE opening times) rescaled to the unit interval $(0,1)$. The CIDR on day $i$ is the curve
\[
R_i(t) = \ln P_i(t) - \ln P_i(0), \ \ \ t\in [0,1],
\]
where $P_i(0)$ is the opening price on day $i$. The curves $R_i$ show how the return accumulates over the trading day, see Figure <ref>.
We consider all full trading days between Jan 02, 2020 and July 31, 2020 ($N=147$).
The nine sector ETFs and the estimates $\hat{\ag}$ computed by applying the Hill estimator to $\| R_i\|$ with the function mindist of the R package tea.
Ticker Sector $\hat{\ag}$
XLY Consumer Discretionary 3.8
XLP Consumer Staples 2.6
XLE Energy 4.2
XLF Financials 4.0
XLV Health Care 3.9
XLI Industrials 3.7
XLB Materials 3.4
XLK Technology 4.7
XLU Utilities 3.8
The CIDR of four ETFs on the four most extreme days during the Covid-19 market decline. Curves of matching color and type
represent curves on the same day; XLF is paired with XLK and
XLY with XLB.
Estimates of the pairwise extremal correlation coefficients of CIDRs across the nine sectors.
Recall that the mathematical framework def-RVXY from
which $\rho_{XY}$ is derived assumes that the marginal distributions of $X$ and $Y$ are tail equivalent.
Using the estimates $\hat\ag$ in Table <ref>
and a power transformation, we standardize all tails to $\ag=2.5$.
For completeness, we recall this method. Given $X \in RV(-\ag_X, \Gg_X)$ and $Y \in RV(-\ag_Y, \Gg_Y)$, consider the transformation
\[
g_X(x) = \frac{x}{\|x\|^{1-{\ag_X}/\ag}}, \ \ \ \ g_Y(y) = \frac{y}{\|y\|^{1-{\ag_Y}/\ag}},\ \ \ \ x,y \in L^2,
\]
where $\ag$ is a desired tail index. Applying $g_X$ and $g_Y$ to $X$ and $Y$, respectively, makes them tail equivalent because $P(\|g_X(X)\|>\cdot)$ and $P(\|g_Y(Y)\|>\cdot)$ are regularly varying with $-\ag$. Since this method adjusts only the scale of curves, the transformed curves still retain their original shapes.
After applying the above transform, we select an optimal $k$ for each pair using the KS method described in Section <ref> to compute $\hat{\rho}_{n,k}$.
Figure <ref> shows estimates of the pairwise extremal correlation coefficient across the nine ETF sectors. All pairs exhibit positive extremal correlations ($\hat{\rho}_{n,k} = 0.39 \sim 0.96$), and 56% of the pairs have strong extremal correlations above 0.7. We see that the CIDRs overall exhibit matching patterns of cumulative
intraday returns on extreme market volatility days during the Covid-19 market turbulence. To the first approximation, on such days, almost all sectors drop together or increase together. However, our coefficient reveals more subtle
information as well. For example, extreme return curves of
XLF (Financials) are exceptionally strongly correlated with extreme curves for
XLV, XLB and XLK (Health Care, Materials, Technology), but relatively weakly
correlated with XLU (Utilities). We use this example for illustration and
do not aim at an analysis of the stock market or the economy, but we
note that some findings are interesting. One might expect that the
financial sector (mostly banks) will be strongly affected by the
technology sector (mostly large IT companies like Google or Microsoft)
because such mega corporations dominate the U.S. stock market.
The similarity of extreme return curves for XLF and XLK is illustrated in the two left panels of Figure <ref>. One could
also expect that the stocks of banks will be less affected by the
performance of utility companies whose revenues are to a large extent
fixed. But it is less obvious that banks are strongly correlated
with Health Care and Materials sectors. As another comparison, consider
XLY (Consumer Discretionary) and XLB (Materials) that show a weak extremal correlation, $\hat{\rho}_{n,k}=0.39$. Their extreme curves exhibit dissimilar patterns, see the right two panels in Figure <ref>.
§.§ Extremal correlation between daily temperature curves
The three locations in the United States: Fort Collins, CO; Colorado Springs, CO; Austin, TX. The pairwise extremal correlation of daily temperature curves between the three locations is evaluated.
In this section, we evaluate the tendency of paired daily temperature curves to exhibit similar extreme patterns across three locations in the United States. The three locations are marked in Figure <ref>. We focus on the pairwise extremal dependence of those curves during the 2021 heat wave. Although this example focuses on temperature curves, our tool can be used
for analyzing other curves during extreme weather events; for example,
daily precipitation patterns or river flows during floods. A correlation of extreme data during past events may help with planning a resilient
infrastructure that can better withstand the next extreme weather event.
We use hourly temperature measurements provided by the European Centre for Medium-Range Weather Forecasts (ECMWF). The data are part of their ERA5 (Fifth Generation of ECMWF atmospheric reanalyses) dataset, and represent the temperatures of air at 2 meters above the surface of land, sea or inland waters. We refer to [12] for more details on the ERA5 data. We partition the hourly data into daily curves, with each day's curve beginning at UTC$+0$, to produce concurrent daily temperature curves across locations in different time zones. We denote the temperature (in Celsius) on day $i$ at hour $t$ and at location $s \in \{ {\rm Fort \ Collins},\ {\rm Colorado \ Springs}, \ {\rm Austin}\}$ by
$X_i(s, t)$, $i=1, \ldots, N$.
Figure <ref> depicts examples of daily temperature curves at the three locations. The data are taken from May 17, 2021 to Aug 31, 2021 ($N$ = 107).
Extreme daily temperature curves (in Celsius) during the 2021 heat wave (local time on the x-axis). Curves of matching color represent the same days when both Fort Collins and Colorado Springs experienced extreme patterns simultaneously.
Prior to computing $\hat{\rho}_{n,k}$ for each pair of the three locations, daily curves are centered by the mean function, $\bar{X}_N (s,t) = \frac{1}{N}\sum_{i=1}^N X_i(s, t)$, for each location $s$.
We then compute the tail index estimate $\hat{\ag}$ of the
norms $\| X_i(s,t) -\bar{X}_N (s,t)\|$, for each location $s$, shown in Table <ref>. The results suggest that the marginal distributions of those curves across the three locations are not tail-equivalent. We apply the power transformation method, described in Section <ref>, to get the tail index $\ag=2.5$ at all locations. We then apply the KS method, described in Section <ref>, to the centered curves to select an optimal $k$ for each pair.
Tail index estimates $\hat{\ag}$ and estimates of the pairwise extremal correlation coefficients, $\hat{\rho}_{n,k}$, of daily temperature curves across Fort Collins, CO, Colorado Springs, CO, and Austin, TX.
Location $\hat{\ag}$ Fort Colorado Austin
Collins Springs
Fort Collins 5.5 1 0.94 0.79
Colorado Springs 3.6 0.94 1 0.74
Austin 4.4 0.79 0.74 1
Table <ref> reports estimates of the pairwise extremal correlation coefficient across the three locations. There are positive and strong extremal correlations among all pairs ($\hat{\rho}_{n,k} = 0.74 \sim 0.94$), suggesting a high degree of association between the daily temperature extreme patterns across the three locations, even between different
climatic regions like the Front Range foothills and the southern edge of the Great Plains.
We see that the proximity in geographical locations corresponds to greater similarity in extreme patterns, showing that $\hat{\rho}_{n,k}$ is
a meaningful and useful dependence measure.
We Thank Professor Hong Miao of the Department of Finance and Real
Estate at Colorado State University for preprocessing the financial data used in Section <ref>. We thank Professor Joshua French of
the Department of Mathematical and Statistical Sciences at the University of Colorado Denver
for preprocessing the temperature data used in Section <ref>.
Conflicts of interest: None declared.
P.K. was partially supported by the United States National Science Foundation
grant DMS–2123761.
Data availability
The raw high frequency financial data used in Section <ref> can be acquired from tickdata.com, as well as from many other providers.
The reanalysis temperature data used in Section <ref>
can be downloaded from ecmwf.int. The R code used in this paper can be found at github.com/veritasmih/ecc.
Supplementary material
Supplementary material is available at Journal of the Royal Statistical Society: Series B online
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Supplementary Material
§ PRELIMINARY RESULTS
In this section, we put together preliminary results needed to prove Theorem <ref>. These results allow us to streamline the exposition of proofs of the main result. Recall that, c.f., A, $\mathcal A_r =\{(x,y)\in \mbB_0^2: \|(x,y)\|_{\mbB^2} \ge r\}$, $r>0$, where $\|(x,y)\|_{\mbB^2} = \|x\|_{\mbB} \vee \|y\|_{\mbB}$.
Suppose $\mu$ is a measure in $M_0(\mbB^2)$ satisfying $\mu(t\cdot)=t^{-\ag}\mu(\cdot)$, $t>0$. Then, $\mathcal A_r$ is a $\mu$–continuity set, i.e., $\mu(\partial \mathcal A_r)=0$.
We assume $\mu(\partial{\mathcal A_r})>0$ and get a contradiction. Since ${\mathcal A_r}\supset \bigcup_{n\ge 1} \partial(n^{1/\ag}{\mathcal A_r})$, it follows from the homogeneity property of $\mu$ that
\[
\mu({\mathcal A_r}) \ge \sum_{n=1}^{\infty} \mu(\partial(n^{1/\ag}{\mathcal A_r})) = \sum_{n=1}^{\infty} \mu(n^{1/\ag}\partial{\mathcal A_r}) =\sum_{n=1}^{\infty} n^{-1}\mu(\partial{\mathcal A_r})=\infty.
\]
It contradicts to the fact that $\mu$ is boundedly finite.
Recall that $R= \|(X, Y)\|$, $R_i = \|(X_i, Y_i)\|$, and $R_{(k)}$ is the $k$th largest order statistic with the convention $R_{(1)} = \max \{ R_1, \ldots, R_n \}$. Let $b(n)$ be the quantile function such that $P(R>b(n))=n^{-1}$. We claim in the following lemma that regular variation of $[X,Y]^{\top}$ in $L^2\times L^2$ implies regular variation of $R$ in $[0,\infty)$.
Let $M_{+}(0,\infty]$ be the space of Radon measures on $(0,\infty]$, and $\nu_{\ag}(r, \infty] = r^{-\ag}$. If $[X,Y]^{\top}$ is regularly varying in $L^2\times L^2$ according to Definition <ref>, then
(i) $R$ is a nonnegative random variable whose distribution has a regularly varying tail with index $-\ag$,
(ii) $\frac{1}{k} \sum_{i=1}^n I_{R_i / b(n/k)} \convP \nu_{\ag}$, in $M_{+}(0,\infty]$,
(iii) $R_{(k)}/b(n/k) \convP 1$, in $[0,\infty)$,
(iv) $\frac{1}{k} \sum_{i=1}^n I_{R_i / R_{(k)}} \convP \nu_{\ag}$ in $M_{+}(0,\infty]$.
For statement $(i)$, observe that for $r>0$,
\[
\frac{n}{k}P \lp R \ge rb(n/k) \rp = \frac{n}{k}P \lp \frac{(X,Y)}{b(n/k)} \in \cA_{r} \rp,
\]
where $\cA_r$ is defined in A. It then follows from def-RVXY and mu1 that
\[
\frac{n}{k}P \lp R \ge rb(n/k) \rp \to \mu(\cA_r)= r^{-\ag}\mu(\cA_1)=r^{-\ag}.
\]
Therefore, by Theorem 3.6 of [30], $(i)$ holds. By Theorem 4.1 of [30], it can be shown that $(i)$ implies $(ii)$. Also, it follows from Step 1 in the proof of Theorem 4.2 of [30] that $(ii)$ implies $(iii)$ and from Step 2 in that proof that $(ii)$ and $(iii)$ imply $(iv)$.
The following lemma is used to prove Lemmas <ref> and <ref>.
Suppose $\ga_n$ converges vaguely to $\nu_\ag$ in $M_+(0,\infty]$. Then for any compact interval $K\subset (0,\infty]$,
\[
\int_K r^2\ga_n(dr) \to \int_K r^2 \nu_\ag(dr).
\]
Since the function $r\mapsto r^2 I_{K}$ is not continuous,
we use an approximation argument. Set $K= [a, b]$, for $0<a<b \le \infty$. Construct
compact intervals $K_j \searrow K$ and nonnegative continuous functions
$f_j$ such that $I_K \le f_j \le I_{K_j}$. By the triangle inequality,
\begin{align*}
\left | \int_K r^2 \ga_n(dr) - \int_K r^2 \nu_\ag(dr)\right |
&\le \left | \int r^2 I_K(r) \ga_n(dr) - \int r^2 f_j(r) \ga_n (dr)\right |\\
& \ \ +
\left | \int r^2 f_j(r) \ga_n (dr) - \int r^2 f_j(r) \nu_\ag (dr)\right |\\
& \ \ +
\left | \int r^2 f_j(r) \nu_\ag (dr) - \int r^2 I_K(r) \nu_\ag(dr)\right |\\
&=: A_{n, j}^{(1)} + A_{n, j}^{(2)} + A_{j}^{(3)}.
\end{align*}
Fix $\ep > 0$. There is $j^\star$ such that for $j \ge j^\star$,
\[
A_{j}^{(3)} \le c\int \lb f_j(r) - I_K(r) \rb \nu_\ag(dr)
\le c \nu_\ag( K_j \setminus K^\circ) < \ep/2,
\]
where $c=b^2I_{b\neq \infty}+a^2I_{b=\infty}$.
Similarly $ A_{n, j}^{(1)} \le c \ga_n ( K_j \setminus K^\circ)$,
so for every fixed $j$,
\[
\limsup_{n\to \infty} A_{n, j}^{(1)} \le M^2
\limsup_{n\to \infty} \ga_n( K_j \setminus K^\circ)
\le M^2 \nu_\ag( K_j \setminus K^\circ)
\]
because $K_j \setminus K^\circ$ is compact, cf. Proposition 3.12 in
[28]. Thus,
\[
\limsup_{n\to\infty}
\left | \int_K r^2 \ga_n(dr) - \int_K r^2 \nu_\ag(dr)\right |
\le \ep + \limsup_{n\to\infty} A_{n, j^\star}^{(2)} = \ep.
\]
Since $\ep$ is arbitrary, we get the claim.
The following two lemmas are used to prove Lemma <ref> and Proposition <ref>.
Under Assumption <ref>, for any $M>0$,
\[
\frac{n}{k} E \lb \lp \frac{R}{b(n/k)} \rp^2 I_{R \ge Mb(n/k) } \rb \to \frac{\ag}{\ag-2}M^{2-\ag}.
\]
Observe that
\[
\frac{n}{k} E \lb \lp \frac{R}{b(n/k)} \rp^2 I_{R \ge Mb(n/k) } \rb
=\int_M^{\infty} r^2 \frac{n}{k}P\lp \frac{R}{b(n/k)} \in dr\rp,
\]
\[
\frac{\ag}{\ag-2}M^{2-\ag} = \int_{M}^{\infty} r^2 \nu_{\ag}(dr).
\]
By Lemma <ref> (i), we have that in $M_+(0,\infty]$
\[
\frac{n}{k}P\lp \frac{R}{b(n/k)} \in \cdot \rp \convv \nu_{\ag}.
\]
Therefore, we get the claim by Lemma <ref> with $K= [M, \infty]$.
The function $h$ on $M_+(0,\infty]$ defined by $h(\ga) = \int_1^M r^2 \ga(dr)$ is continuous at $\nu_\ag$.
Suppose $\ga_n$ converges vaguely to $\nu_\ag$. Then, by Lemma <ref> with $K= [1, M]$, it can be shown that
\[
\lim_{n\to\infty} \int_1^M r^2 \ga_n(dr) = \int_1^M r^2 \nu_\ag(dr).
\]
The following lemma is the key argument to prove Proposition <ref>.
Under Assumption <ref>, the following statements hold:
\begin{align}
\frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{R_{(k)}} \rp^{2} I_{R_i \ge R_{(k)} } &\convP \frac{\ag}{\ag-2}; \label{e:2k}\\
\frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} I_{R_i \ge b(n/k) } &\convP \frac{\ag}{\ag-2}. \label{e:2b}
\end{align}
The proofs for 2k and 2b are almost the same, so we only prove 2k to save space. Let $\hat{\ga}_{n,k}=\frac{1}{k} \sum_{i=1}^n I_{R_i / R_{(k)}}$, and recall that $\hat{\ga}_{n,k} \convP \nu_{\ag}$ (see Lemma <ref> (iv)). Since
\[
\frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{R_{(k)}} \rp^{2} I_{R_i \ge R_{(k)} }= \int_1^\infty r^2 \hat{\ga}_{n,k}(dr),
\]
we need to show that
\[
\int_1^\infty r^2 \hat{\ga}_{n,k}(dr)
\convP \int_1^\infty r^{2} \nu_\ag(dr)=\frac{\ag}{\ag-2}.
\]
To prove this convergence, we use the second converging together theorem, Theorem 3.5
in [30], (also stated as Theorem 3.2 of [1]). Let
\begin{align*}
V_{n,k} = \int_1^\infty r^2 \hat{\ga}_{n,k}(dr), \ \ \
& V= \int_1^\infty r^2 \nu_\ag(dr);\\
V_{n,k}^{(M)} = \int_1^M r^2 \hat{\ga}_{n,k}(dr), \ \ \
& V^{(M)}= \int_1^M r^2 \nu_\ag(dr).
\end{align*}
To show the desired convergence $V_{n,k} \convP V$ (equivalently, $V_{n,k} \convd V$), we must verify that
\begin{equation} \label{e:VnM}
\forall \ M > 1, \ \ \ V_{n,k}^{(M)} \convd V^{(M)}, \ \ \ \ {\rm as} \ n \to\infty;
\end{equation}
\begin{equation} \label{e:VM}
V^{(M)} \convd V, \ \ \ {\rm as} \ M\to\infty;
\end{equation}
\begin{equation} \label{e:bill}
\forall \ \eg > 0, \ \ \ \lim_{M\to\infty} \limsup_{n\to\infty}
P\lp | V_{n,k}^{(M)}- V_{n,k} | > \eg \rp = 0.
\end{equation}
Convergence VnM follows from Lemma <ref> (iv)
and Lemma <ref>. Convergence VM holds since for $\ag>2$
\[
\int_M^\infty r^2 \nu_\ag(dr)=\int_M^\infty r^2 \ag r^{-\ag-1} dr =\frac{\ag}{\ag-2} M^{2-\ag} \to 0, \ \ \ \ {\rm as}\ M \to \infty.
\]
It remains to show that $\forall \eg>0$,
\[
\lim_{M\to\infty} \limsup_{n\to\infty}
P\lp | V_{n,k}^{(M)}- V_{n,k} | > \eg \rp = \lim_{M\to\infty} \limsup_{n\to\infty}
P\lp \int_M^\infty r^2 \hat\ga_{n,k}(dr) > \eg \rp = 0.
\]
Fix $\eg> 0$ and $\eta> 0$. Observe that
\[
P\lp \int_M^\infty r^2 \hat\ga_{n,k}(dr) > \eg \rp
\le Q_1(n) + Q_2(n),
\]
\[
Q_1(n) = P\lp \int_M^\infty r^2 \hat\ga_{n,k}(dr) > \eg, \
\left | \frac{R_{(k)}}{b(n/k)} - 1 \right | < \eta \rp,\ \
Q_2(n) = P \lp \left | \frac{R_{(k)}}{b(n/k)} - 1 \right | \ge \eta\rp.
\]
By Lemma <ref> (iii), $ \limsup_{n\to\infty} Q_2(n) = 0$. For $Q_1(n)$, we start with the bound
\begin{align*}
&\le P\lp \int_M^\infty r^2 \hat\ga_{n,k}(dr) > \eg, \
\frac{R_{(k)}}{b(n/k)} > 1- \eta \rp\\
&= P \lp \int_M^\infty r^2 \frac{1}{k} \sum_{i=1}^n
I_{R_i/ R_{(k)} \in dr} > \eg, \
\frac{R_{(k)}}{b(n/k)} > 1- \eta \rp.
\end{align*}
Conditions $R_i/ R_{(k)} > M$ and ${R_{(k)}}/{b(n/k)} > 1- \eta$
imply ${R_{i}}/{b(n/k)} > M(1- \eta)$, so
\begin{align*}
&\le P \lp \int_{M(1-\eta)}^\infty r^2 \frac{1}{k} \sum_{i=1}^n
I_{R_i/b(n/k) \in dr} > \eg \rp\\
&= P \lp \frac{1}{k} \sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^2
I_{R_i \ge M(1-\eta)b(n/k) } >\eg\rp.
\end{align*}
Then, it follows from Markov's inequality and Lemma <ref> that
\[
Q_1(n) \le \frac{1}{\eg} \frac{n}{k} E \lb \lp \frac{R_1}{b(n/k)} \rp^2
I_{R_1 \ge M(1-\eta)b(n/k) } \rb \to \frac{1}{\eg} \frac{\ag}{\ag-2} \{M(1-\eta)\}^{2-\ag}, \ \ \ {\rm as} \ n \to \infty.
\]
This bound goes to 0 as $M \to \infty$ since $\ag>2$.
The following lemma follows from Theorem 3.8 of [26]. It states a Bernstein type inequality, which is the key technique to prove Proposition <ref>.
Let $\bZ_n=(Z_1, \ldots, Z_n)$ with the $Z_i$ taking values in a
Lebesgue measurable subset $\cZ$ of an Euclidean space. Let $f$ be a real-valued function defined on $\cZ^n$. For $(z_1, \ldots, z_i) \in \cZ^{i}$, $1 \le i \le n$, put
\begin{equation} \label{e:gi}
g_i(z_1, \ldots, z_i):=E\lb f(\bZ_n)| Z_j=z_j, 1 \le j \le i\rb -E\lb f(\bZ_n)| Z_j=z_j, 1 \le j \le i-1\rb.
\end{equation}
Define the maximum deviation by
\begin{equation} \label{e:b}
b := \max_{1\le i\le n}\sup_{(z_1, \ldots, z_i) \in \cZ^i} g_i(z_1, \ldots, z_i),
\end{equation}
and define the supremum sum of variances by
\begin{equation} \label{e:v}
\hat{v} := \sup_{(z_1, \ldots, z_n) \in \cZ^n} \sum_{i=1}^n \var\lb g_i(z_1, \ldots, z_{i-1}, Z_i^{\prime})\rb,
\end{equation}
where $Z_i^{\prime}$ is an independent copy of $Z_i$ conditional on $Z_j=z_j$, $1 \le j \le i-1$. If $b$ and $\hat{v}$ are finite, then for any $\eg \ge 0$,
\[
P \lp f(\bZ_n) - E[f(\bZ_n)] \ge t \rp \le \exp \lp \frac{-\eg^2}{2(\hat{v}+b\eg/3)} \rp.
\]
§ PROOF OF THEOREM <REF> IN SECTION <REF>
Recall ec, i.e., the definition:
\[
\hat{\sg}_{n,k} =\frac{1}{k}\sum_{i=1}^n \lip \frac{X_i}{R_{(k)}}, \frac{Y_i}{R_{(k)}} \rip I_{R_i \ge R_{(k)} }.
\]
To prove the consistency of $\hat{\sg}_{n,k}$ for the extremal covariance $\sg_{XY}$, we consider the following sequence of random variables
\begin{align} \label{e:sg}
\sg_{n,k} &:=\frac{1}{k}\sum_{i=1}^n \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip I_{R_i \ge b(n/k) }.
\end{align}
Note that $\sg_{n,k}$ is not observable since $b(\cdot)$ is unknown. However, $b(n/k)$ can be estimated by its consistent estimator $R_{(k)}$, and it can be shown that replacing $b(n/k)$ by $R_{(k)}$ ensures that the difference between $\sg_{n,k}$ and $\hat{\sg}_{n,k}$ is asymptotically negligible, which will be shown in Proposition <ref>. Thus, the key argument for establishing the consistency is to show that $\sg_{n,k}$ converges in probability to $\sg_{XY}$, which is proven in the following proposition.
Under Assumption <ref>,
\[
\sg_{n,k} \convP \sg_{XY}.
\]
\begin{equation} \label{e:sgbar}
\bar{\sg}_{n,k} := E\lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip \Bigg|\|X_1\| \vee \|Y_1\| >b(n/k)\rb.
\end{equation}
Then, by Proposition <ref>, $\bar{\sg}_{n,k} \to \sg_{XY}$, so it remains to show that $|\sg_{n,k} - \bar{\sg}_{n,k}| \convP 0$.
Let $\bZ_n = (Z_1, \ldots, Z_n)$, where $Z_i = (X_i, Y_i)$, and $\bz_n = (z_1, \ldots, z_n)$, where $z_i = (x_i, y_i)$, for $1 \le i \le n$. Consider a map $f: (L^2 \times L^2)^n \to \mbR$ defined by
\[
f(\bz_n) := \lmo \frac{1}{k}\sum_{i=1}^n \lip \frac{x_i}{b(n/k)}, \frac{y_i}{b(n/k)} \rip I_{r_i \ge b(n/k) }- \frac{n}{k}E\lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip I_{R_1 >b(n/k)}\rb \rmo.
\]
Then, we have that
\[
|\sg_{n,k} - \bar{\sg}_{n,k}| = f(\bZ_n) - E[f(\bZ_n)] + E[f(\bZ_n)].
\]
We aim to show that $f(\bZ_n) - E[f(\bZ_n)] \convP 0$ and $E[f(\bZ_n)] \to 0$.
To establish the convergence, $f(\bZ_n) - E[f(\bZ_n)] \convP 0$, we use the Bernstein type concentration inequality in Lemma <ref>. Since the $(X_i,Y_i)$ are independent, the deviation function in gi has the following form
\[
g_i(z_1, \ldots, z_i)= E \lb f(z_1, \ldots, z_{i-1},z_i, Z_{i+1}, \ldots, Z_n)-f(z_1, \ldots, z_{i-1},Z_i, Z_{i+1}, \ldots, Z_n) \rb.
\]
Then, using the fact that $\lmo |x|- |y| \rmo \le |x-y|$, we have that
\begin{align*}
g_i(z_1, \ldots, z_i)
&\le \frac{1}{k} E\lb \lmo \lip \frac{x_i}{b(n/k)}, \frac{y_i}{b(n/k)} \rip I_{r_i \ge b(n/k) } - \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip I_{R_i \ge b(n/k) } \rmo \rb \\
&\le \frac{1}{k} \lbr \frac{|\lip x_i, y_i\rip|}{b(n/k)^2} + \frac{k}{n} \frac{n}{k} E \lb \lp \frac{R_i}{b(n/k)} \rp^2 I_{R_i \ge b(n/k) } \rb \rbr\\
&\le \frac{1}{k} \lbr \frac{\|x_i\| \| y_i\|}{b(n/k)^2} + \frac{n}{k} E \lb \lp \frac{R_i}{b(n/k)} \rp^2 I_{R_i \ge b(n/k) } \rb \rbr.
\end{align*}
Since $(x_i,y_i) \in L^2\times L^2$ and $\frac{n}{k} E \lb \lp {R_i}/{b(n/k)} \rp^2 I_{R_i \ge b(n/k) } \rb \to \ag/(\ag-2)$ by Lemma <ref>, we have that $g_{i}(z_1, \ldots, z_i) \le {c_1}/{k}$, for some constant $c_1>0$. Therefore, the maximum deviation $b$ in b is bounded by $c_1/k$.
Next we investigate the upper bound for the sum of variances $\hat{v}$ in v. Since $E[g_i(z_1, \ldots, z_{i-1}, Z_i^{\prime})]=0$ by the law of total probability, we have that
\begin{align*}
&\var\lb g_i(z_1, \ldots, z_{i-1}, Z_i^{\prime})\rb \\
&= E [g_i^2(z_1, \ldots, z_{i-1}, Z_i^{\prime})]\\
&= E \lb \lbr f(z_1, \ldots, z_{i-1},Z_i^{\prime}, Z_{i+1}, \ldots, Z_n)-f(z_1, \ldots, z_{i-1},Z_i, Z_{i+1}, \ldots, Z_n) \rbr^2 \rb \\
&\le \frac{1}{k^2} E\lb \lbr \lip \frac{X_i^{\prime}}{b(n/k)}, \frac{Y_i^{\prime}}{b(n/k)} \rip I_{R_i^{\prime} \ge b(n/k) } - \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip I_{R_i \ge b(n/k) } \rbr^2 \rb \\
&\le \frac{2}{k^2} E \lb \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip^2 I_{R_i \ge b(n/k) } \rb \\
&\le \frac{2}{k^2} \lbr \frac{k}{n} \frac{n}{k} E \lb \lp \frac{R_i}{b(n/k)} \rp^2 I_{R_i \ge b(n/k) } \rb \rbr.
\end{align*}
It then again follows from Lemma <ref> that $\var\lb g_i(z_1, \ldots, z_{i-1}, Z_i^{\prime})\rb \le c_2/(nk)$ for some $c_2>0$. Then the supremum sum of variances $\hat{v}$ is bounded above by $c_2/k$. Therefore by Lemma <ref>, for any $\eg>0$
\[
P \lp f(\bZ_n) - E[f(\bZ_n)] \ge \eg \rp \le \exp \lp \frac{-k\eg^2}{c_1+c_2\eg/3} \rp.
\]
If we apply this inequality to $-f(\bZ_n)$, then we obtain the following `two-sided' inequality
\[
P \lp |f(\bZ_n) - E[f(\bZ_n)]| \ge \eg \rp \le 2\exp \lp \frac{-k\eg^2}{c_1+c_2\eg/3} \rp.
\]
From this, we obtain that $f(\bZ_n) - E[f(\bZ_n)] \convP 0$.
Next, to show $E[f(\bZ_n)] \to 0$, we set, for $1 \le i \le n$
\[
\Delta_i = \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip I_{R_i \ge b(n/k) }- E\lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip I_{R_1 >b(n/k)}\rb.
\]
Then, we have that
\begin{align*}
E \lb f(\bZ_n) \rb = \frac{n}{k} E \lb \lmo \frac{1}{n} \sum_{i=1}^n \Delta_i \rmo \rb &\le \frac{n}{k} \lbr E \lb \lp \frac{1}{n} \sum_{i=1}^n \Delta_i \rp^2 \rb \rbr^{1/2}\\
&= \frac{n}{k} \lbr E \lb \frac{1}{n^2} \sum_{i=1}^n \Delta_i^2 + \frac{1}{n^2} \sum_{i \neq j} \Delta_i\Delta_j \rb \rbr^{1/2}.
\end{align*}
Since the $\Delta_i$ are independent, $E[\Delta_i\Delta_j]=0$, for $i \neq j$. Therefore,
\begin{align*}
&E \lb f(\bZ_n) \rb \\
&\le \frac{\sqrt{n}}{k} \lbr E \lb \Delta_1^2 \rb \rbr^{1/2} \\
&= \frac{\sqrt{n}}{k} \lbr E \lb \lp \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip I_{R_1 \ge b(n/k) } - E\lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip I_{R_1 >b(n/k)}\rb \rp^2\rb \rbr^{1/2}\\
& = \frac{\sqrt{n}}{k} \lbr \var \lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip I_{R_1 \ge b(n/k) } \rb \rbr^{1/2} \\
& \le \frac{\sqrt{n}}{k} \lbr E \lb \lip \frac{X_1}{b(n/k)}, \frac{Y_1}{b(n/k)} \rip^2 I_{R_1 \ge b(n/k) } \rb \rbr^{1/2}\\
& \le \frac{\sqrt{n}}{k} \lbr E \lb \lp \frac{R_1}{b(n/k)} \rp^2 I_{R_1 \ge b(n/k) } \rb \rbr^{1/2}.
\end{align*}
Therefore, by Lemma <ref> we have that
\[
E \lb f(\bZ_n) \rb \le \frac{\sqrt{n}}{k} \lbr \frac{k}{n} \frac{n}{k} E \lb \lp \frac{R_1}{b(n/k)} \rp^2 I_{R_1 \ge b(n/k) } \rb \rbr^{1/2} \le \frac{c_3}{\sqrt{k}},
\]
for some $c_3>0$, which completes the proof.
Under Assumption <ref>,
\[
|\hat{\sg}_{n,k} - \sg_{n,k} | \convP 0.
\]
Consider the following decomposition
\[
|\hat{\sg}_{n,k} - \sg_{n,k} | \le P_1(n) + P_2(n),
\]
\begin{align*}
&P_1(n) :=\lmo \frac{1}{k}\sum_{i=1}^n \lip \frac{X_i}{R_{(k)}}, \frac{Y_i}{R_{(k)}} \rip \lbr I_{R_i \ge R_{(k)} } - I_{R_i \ge b(n/k) }\rbr \rmo,\\
&P_2(n) :=\lmo \frac{1}{k}\sum_{i=1}^n \lbr \lip \frac{X_i}{R_{(k)}}, \frac{Y_i}{R_{(k)}} \rip - \lip \frac{X_i}{b(n/k)}, \frac{Y_i}{b(n/k)} \rip \rbr
I_{R_i \ge b(n/k) } \rmo.
\end{align*}
We will show that each of the two parts goes to 0. We first focus on $P_1(n)$. Observe that
\begin{align*}
P_1(n) &\le \lp\frac{b(n/k)}{R_{(k)}} \rp^2 \frac{1}{k}\sum_{i=1}^n \lmo \lip \frac{X_i}{R_i}, \frac{Y_i}{R_i} \rip \rmo \lp \frac{R_i}{b(n/k)} \rp^{2} \lmo I_{R_i \ge R_{(k)} } - I_{R_i \ge b(n/k) } \rmo \\
&\le \lp\frac{b(n/k)}{R_{(k)}} \rp^2 \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} \lmo I_{R_i \ge R_{(k)} } - I_{R_i \ge b(n/k) } \rmo \\
& = \lp\frac{b(n/k)}{R_{(k)}} \rp^2 \lmo \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} I_{R_i \ge R_{(k)} } - \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} I_{R_i \ge b(n/k) } \rmo \\
& = \lmo \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{R_{(k)}} \rp^{2} I_{R_i \ge R_{(k)} } - \lp\frac{b(n/k)}{R_{(k)}} \rp^2 \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} I_{R_i \ge b(n/k) } \rmo
\end{align*}
Then, by Lemma <ref> (iii), we have that $\lp {b(n/k)}/{R_{(k)}} \rp^2 \convP 1$. By Lemma <ref> that $\frac{1}{k}\sum_{i=1}^n \lp{R_i}/{R_{(k)}} \rp^{2} I_{R_i \ge R_{(k)} } \convP \ag/(\ag-2)$ and $\frac{1}{k}\sum_{i=1}^n \lp {R_i}/{b(n/k)} \rp^{2} I_{R_i \ge b(n/k) }\convP \ag/(\ag-2)$. Therefore, we have that $P_1(n) \convP 0$.
Now we work on $P_2(n)$. Observe that
\begin{align*}
P_2(n) & = \lmo \frac{1}{k}\sum_{i=1}^n \lip \frac{X_i}{R_i}, \frac{Y_i}{R_i} \rip R_i^2 \lp \frac{1}{R_{(k)}^2} - \frac{1}{b(n/k)^2}\rp
I_{R_i \ge b(n/k) } \rmo \\
& \le \lmo \frac{b(n/k)^2}{R_{(k)}^2} - 1 \rmo \frac{1}{k}\sum_{i=1}^n \lmo \lip \frac{X_i}{R_i}, \frac{Y_i}{R_i} \rip \rmo \lp \frac{R_i}{b(n/k)} \rp^{2}
I_{R_i \ge b(n/k)} \\
& \le \lmo \frac{b(n/k)^2}{R_{(k)}^2} - 1 \rmo \frac{1}{k}\sum_{i=1}^n \lp \frac{R_i}{b(n/k)} \rp^{2} I_{R_i \ge b(n/k)}.
\end{align*}
By Lemma <ref>, we have that $\frac{1}{k}\sum_{i=1}^n \lp {R_i}/{b(n/k)} \rp^{2} I_{R_i \ge b(n/k)} = O_P(1)$, and by Lemma <ref> (iii), we have that $b(n/k) / R_{(k)} \convP 1$. Thus, $P_2(n) \convP 0$.
Proof of Theorem <ref>. It follows from Propositions <ref> and <ref>.
§ PROOF OF LEMMA <REF> IN SECTION <REF>
We begin by noting that since $Z_1$ and $Z_2$ are independent, there exists $\nu$ in $M_+(\mbR_+^2)$ such that
\begin{equation} \label{e:dRV2}
nP \lp \frac{(|Z_1|, |Z_2|)}{b(n)} \in \cdot \rp \convv \nu,
\end{equation}
and for $\bx = [x_1, x_2]^{\top}$
\[
\nu([0, \bx]^c) = c\{(x_1)^{-\ag} +(x_2)^{-\ag}\}.
\]
With the choice of $b(n)$ defined by
\begin{align} \label{e:bn}
n^{-1} &= P(\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n))\\ \nonumber
&= P(|Z_1| \vee (\rho^2 Z_1^2 + (1-\rho^2) Z_2^2)^{1/2} > b(n)),
\end{align}
we set $c= 1/(1+ (1-\rho^2)^{\ag/2})$ to ensure that $\nu$ is a probability measure on $\{(z_1,z_2): |z_1| \vee ( \rho^2 z_1^2 + (1-\rho^2) z_2^2 )^{1/2} >1 \}$.
We claim that
\begin{align} \label{e:sxy}
& \sg_{XY}=\rho \frac{c\ag}{\ag-2};\\ \label{e:sx}
& \sg_{X}^2 =\frac{c\ag}{\ag-2};\\ \label{e:sy}
& \sg_{Y}^2 =\lbr \rho^2 + (1-\rho^2)^{\ag/2} \rbr \frac{c\ag}{\ag-2}.
\end{align}
We first work on sxy. Since the terms with the $N_j$ do not affect the extremal behavior of $X$ and $Y$, we have that by Proposition <ref>
\begin{align*}
&\sg_{XY} \\
&= \lim_{n \to \infty} E \lb \lip \frac{Z_1 \phi_1}{b(n)}, \frac{\rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2}{b(n)}\rip \Bigg| \| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n)\rb \\
&= \lim_{n \to \infty} \frac{1}{P(\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n))} \times \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E \lb \lip \frac{Z_1 \phi_1}{b(n)}, \frac{\rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2}{b(n)}\rip I_{\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n)}\rb \\
&= \lim_{n \to \infty} \frac{1}{P(\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n))} E \lb \rho \frac{Z_1^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb
\end{align*}
It then follows from dRV2 and bn that
\begin{align*}
\sg_{XY} &= \lim_{n \to \infty} n E \lb \rho \frac{Z_1^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb \\
&= \lim_{n \to \infty} \int_{\mbR^2_+} \rho z_1^2 I_{ |z_1| \vee ( \rho^2 z_1^2 + (1-\rho^2) z_2^2 )^{1/2} >1} nP\lp \frac{|Z_1|}{b(n)} \in dz_1, \frac{|Z_2|}{b(n)}\in dz_2 \rp \\
&= \int_{\mbR_+^2} \rho z_1^2 I_{ |z_1| \vee ( \rho^2 z_1^2 + (1-\rho^2) z_2^2 )^{1/2} >1} \ \nu(dz_1, dz_2) \\
& = \int_{\mbR_+} \rho z_1^2 I_{ \{(z_1, 0): z_1>1\} } \ c\nu_{\ag}(dz_1) + \int_{\mbR_+} \rho z_1^2 I_{ \{(0, z_2): z_2>1/(1-\rho^2)^{1/2} \}} \ c\nu_{\ag}(dz_2) \\
&= \int_{1}^{\infty} \rho z_1^2 \ c\nu_{\alpha}(dz_1) + 0 = \rho \frac{c\ag}{\ag-2}.
\end{align*}
Analogously, for sx we can show that
\begin{align*}
&\sg_{X}^2 \\
&= \lim_{n \to \infty} E \lb \lip \frac{Z_1 \phi_1}{b(n)}, \frac{Z_1 \phi_1}{b(n)}\rip \Bigg| \| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n)\rb \\
&= \lim_{n \to \infty} \frac{1}{P(\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n))} E \lb \frac{Z_1^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb \\
&= \lim_{n \to \infty} nE \lb \frac{Z_1^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb \\
\end{align*}
Next, we work on sy. Observe that
\begin{align*}
&\sg_{Y}^2 \\
&= \lim_{n \to \infty} E \lb \frac{\|\rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \|^2}{b(n)^2} \Bigg| \| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n)\rb \\
&= \lim_{n \to \infty} \frac{1}{P(\| Z_1 \phi_1 \| \vee \| \rho Z_1 \phi_1+\sqrt{1-\rho^2} Z_2 \phi_2 \| > b(n))} \times \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E \lb \frac{\rho^2 Z_1^2 +(1-\rho^2) Z_2^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb.
\end{align*}
Then, again it follows from dRV2 and bn that
\begin{align*}
\sg_{Y}^2 &= \lim_{n \to \infty} nE \lb \frac{\rho^2 Z_1^2 +(1-\rho^2) Z_2^2 }{b(n)^2} I_{|Z_1| \vee (\rho^2 Z_1^2 +(1-\rho^2)Z_2^2 )^{1/2} > b(n)} \rb \\
&= \lim_{n \to \infty} \int_{\mbR^2_+} \lbr \rho^2 z_1^2 + (1-\rho^2)z_2^2 \rbr I_{ |z_1| \vee ( \rho^2 z_1^2 + (1-\rho^2) z_2^2 )^{1/2} >1} nP\lp \frac{|Z_1|}{b(n)} \in dz_1, \frac{|Z_2|}{b(n)}\in dz_2 \rp \\
&= \int_{\mbR_+^2} \lbr \rho^2 z_1^2 + (1-\rho^2)z_2^2 \rbr I_{ |z_1| \vee ( \rho^2 z_1^2 + (1-\rho^2) z_2^2 )^{1/2} >1} \ \nu(dz_1, dz_2) \\
& = \int_{\mbR_+} \rho^2 z_1^2 I_{ \{(z_1, 0): z_1>1\} } \ c\nu_{\ag}(dz_1) + \int_{\mbR_+} (1-\rho^2)z_2^2 I_{ \{(0, z_2): z_2>1/(1-\rho^2)^{1/2} \}} \ c\nu_{\ag}(dz_2) \\
&= \int_{1}^{\infty} \rho^2 z_1^2 \ c\nu_{\alpha}(dz_1) + \int_{1/(1-\rho^2)^{1/2}}^{\infty} (1-\rho^2)z_2^2 \ c\nu_{\alpha}(dz_1) \\
&= \lbr \rho^2 + (1-\rho^2)^{\ag/2} \rbr \frac{c\ag}{\ag-2}.
\end{align*}
§ CONSISTENCY OF $\HAT{\RHO}_{N,K}$ FOR $\AG \IN \{3,4,5\}$
Empirical biases (standard errors) of $\hat{\rho}_{n,k}$ when $\ag=3$. The KS method is used to choose optimal $k$s. On average, it selects $k=8 \sim 9$ ($N=100$) and $k=29 \sim 33$ ($N=500$).
$\rho_{XY}$ $N=100$ $N=500$ $\rho_{XY}$ $N=100$ $N=500$
0 -0.002 (0.117) 0.002 (0.062)
0.1 -0.011 (0.133) 0.000 (0.081) -0.1 0.002 (0.130) 0.003 (0.076)
0.2 -0.009 (0.167) -0.012 (0.109) -0.2 0.014 (0.156) 0.008 (0.116)
0.3 -0.045 (0.185) -0.020 (0.130) -0.3 0.035 (0.187) 0.022 (0.124)
0.4 -0.061 (0.193) -0.041 (0.134) -0.4 0.055 (0.194) 0.036 (0.145)
0.5 -0.097 (0.202) -0.052 (0.150) -0.5 0.084 (0.190) 0.064 (0.149)
0.6 -0.127 (0.201) -0.078 (0.159) -0.6 0.126 (0.211) 0.080 (0.152)
0.7 -0.156 (0.197) -0.106 (0.148) -0.7 0.161 (0.196) 0.108 (0.150)
0.8 -0.183 (0.184) -0.130 (0.135) -0.8 0.194 (0.195) 0.118 (0.141)
0.9 -0.219 (0.165) -0.142 (0.122) -0.9 0.222 (0.168) 0.142 (0.121)
1.0 -0.222 (0.125) -0.139 (0.079) -1.0 0.230 (0.132) 0.141 (0.082)
Empirical biases (standard errors) of $\hat{\rho}_{n,k}$ when $\ag=4$. The KS method is used to choose optimal $k$s. On average, it selects $k=8$ ($N=100$) and $k=25 \sim 29$ ($N=500$).
$\rho_{XY}$ $N=100$ $N=500$ $\rho_{XY}$ $N=100$ $N=500$
0 0.000 (0.172) 0.003 (0.094)
0.1 -0.030 (0.167) -0.019 (0.093) -0.1 0.027 (0.163) 0.019 (0.097)
0.2 -0.053 (0.188) -0.052 (0.114) -0.2 0.064 (0.184) 0.051 (0.116)
0.3 -0.123 (0.198) -0.085 (0.127) -0.3 0.102 (0.180) 0.083 (0.130)
0.4 -0.157 (0.192) -0.118 (0.137) -0.4 0.147 (0.202) 0.121 (0.136)
0.5 -0.211 (0.206) -0.155 (0.150) -0.5 0.197 (0.196) 0.163 (0.140)
0.6 -0.253 (0.202) -0.204 (0.146) -0.6 0.256 (0.199) 0.209 (0.154)
0.7 -0.306 (0.202) -0.249 (0.156) -0.7 0.315 (0.205) 0.255 (0.154)
0.8 -0.366 (0.207) -0.291 (0.150) -0.8 0.366 (0.206) 0.291 (0.150)
0.9 -0.415 (0.193) -0.340 (0.141) -0.9 0.415 (0.200) 0.329 (0.142)
1.0 -0.410 (0.170) -0.326 (0.124) -1.0 0.428 (0.171) 0.325 (0.126)
Empirical biases (standard errors) of $\hat{\rho}_{n,k}$ when $\ag=5$. The KS method is used to choose optimal $k$s. On average, it selects $k=7 \sim 8$ ($N=100$) and $k=18 \sim 20$ ($N=500$).
$\rho_{XY}$ $N=100$ $N=500$ $\rho_{XY}$ $N=100$ $N=500$
0 -0.007 (0.199) -0.004 (0.159)
0.1 -0.046 (0.208) -0.043 (0.144) -0.1 0.053 (0.195) 0.041 (0.145)
0.2 -0.087 (0.204) -0.095 (0.145) -0.2 0.097 (0.202) 0.096 (0.158)
0.3 -0.173 (0.210) -0.147 (0.149) -0.3 0.155 (0.201) 0.142 (0.161)
0.4 -0.229 (0.198) -0.203 (0.151) -0.4 0.212 (0.209) 0.196 (0.161)
0.5 -0.289 (0.202) -0.263 (0.164) -0.5 0.279 (0.206) 0.264 (0.156)
0.6 -0.351 (0.217) -0.324 (0.172) -0.6 0.356 (0.207) 0.330 (0.184)
0.7 -0.411 (0.206) -0.388 (0.171) -0.7 0.424 (0.210) 0.397 (0.157)
0.8 -0.491 (0.209) -0.458 (0.170) -0.8 0.491 (0.207) 0.446 (0.162)
0.9 -0.556 (0.207) -0.511 (0.156) -0.9 0.554 (0.207) 0.505 (0.163)
1.0 -0.548 (0.171) -0.502 (0.143) -1.0 0.564 (0.178) 0.509 (0.149)
|
## I Title
All-optical polarization and amplitude modulation of second harmonic
generation in atomically thin semiconductors
## II Author list
Sebastian Klimmer,1 Omid Ghaebi,1 Ziyang Gan,2 Antony George,2,3 Andrey
Turchanin,2,3 Giulio Cerullo,4 Giancarlo Soavi1,3
## III Affiliations
1Institute of Solid State Physics, Friedrich Schiller University Jena, 07743
Jena, Germany
2Institute of Physical Chemistry, Friedrich Schiller University Jena, 07743
Jena, Germany
3Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena,
Germany
4Dipartimento di Fisica, Politecnico di Milano, 20133 Milan, Italy
## IV Abstract
###### Abstract
Second harmonic generation is of paramount importance in several fields of
science and technology, including frequency conversion, self-referencing of
frequency combs, nonlinear spectroscopy and pulse characterization. Advanced
functionalities are enabled by the modulation of the harmonic generation
efficiency, which can be achieved with electrical or all-optical triggers.
Electrical control of the harmonic generation efficiency offers large
modulation depth at the cost of low switching speed, in contrast to all-
optical nonlinear devices which provide high speed and low modulation depth.
Here we demonstrate all-optical modulation of second harmonic generation in
MoS2 with close to 100 % modulation depth and speed limited only by the
fundamental pulse duration. This result arises from the combination of the D3h
crystal symmetry and the deep sub-wavelength thickness of the sample and it
can therefore be extended to the whole family of transition metal
dichalcogenides, thus providing a large flexibility in the design of advanced
nonlinear optical devices such as high-speed integrated frequency converters,
broadband autocorrelators for ultra-short pulse characterization and tunable
nanoscale holograms.
## V Main text
Stemming from the first demonstration of optical harmonic generation [1],
nonlinear optics has been in the spotlight of science and technology for more
than half-century. In particular, second harmonic generation (SHG) is a
second-order nonlinear process widely used for frequency conversion, self
referencing of frequency combs [2], crystal symmetry and Rashba effect studies
[3, 4], sensing [5], interface spectroscopy [6] and ultra-short pulse
characterization [7]. Besides free-space applications, there is an increasing
interest towards the realization of micro-scale integrated nonlinear devices.
Here, a major challenge comes from the centro-symmetric nature of silicon (Si)
and silicon nitride (Si3N4), which forbids second-order nonlinearities. Large
efforts have been devoted to the integration of nonlinear crystals such as
lithium niobate [8, 9] or to symmetry breaking in Si and Si3N4, for instance
via strain [10], electric fields [11] or the photogalvanic effect [12].
Two-dimensional (2D) materials, such as graphene and transition metal
dichalcogenides (TMDs), hold great promise for nonlinear optical applications.
They have strong and broadband optical response [13, 14], combined with the
possibility of harmonic generation enhancement at excitonic resonances in TMDs
[15] and at multi-photon resonances in the graphene’s Dirac cone [16, 17]. In
addition, thanks to their flexibility and mechanical strength [18], they are
easy to integrate on photonic platforms. Various functionalised devices for
sensing and frequency conversion have been demonstrated on fibers [19],
waveguides [20] and microrings [21], while direct patterning of TMDs has been
used to realize atomically thin meta-lenses [22, 23] and nonlinear holograms
[24, 25]. In addition, harmonic generation in 2D materials can be efficiently
tuned by external electrical [26, 27, 16, 28] or all-optical excitation [29,
30], offering an additional degree of freedom for the design of advanced
nanoscale devices.
However, all the electrical and all-optical schemes that have been proposed to
date for SHG modulation in 2D materials have significant downsides. On one
hand, electrical modulation has been demonstrated in tungsten diselenide
(WSe2) monolayers [26] by tuning the oscillator strength of neutral and
charged exciton resonances through electrostatic doping, and in molybdenum
disulfide (MoS2) homo-bilayers [27], by breaking the naturally occurring
inversion symmetry through electrical gating, in the latter case with large
modulation depth up to a factor of 60. However, electronics is intrinsically
slower compared to optics and photonics. On the other hand, all-optical SHG
modulation has been achieved by quenching of the exciton oscillator strength
following ultrafast optical excitation in MoS2 [29, 30]. This approach offers
high modulation speed, limited in principle only by the excited state/exciton
lifetime ($\sim$ tens of ps). However, the largest depth in all-optical SHG
modulation reported to date in TMDs is 55 % [29], with a strong dependence on
the excitation wavelength and fluence. In addition, this scheme for all-
optical SHG modulation is only effective for excitation and frequency
conversion above-gap or at excitonic resonances, while it is not applicable
for below-gap excitation, leading to a naturally limited spectral bandwidth.
Here we demonstrate a novel approach for the all-optical control of the second
harmonic (SH) polarization in MoS2 and show that this can be used for all-
optical modulation of the SH efficiency with modulation depth close to 100 %
and speed limited only by the fundamental frequency (FF) pulse duration. Our
method relies solely on symmetry considerations in combination with the deep
sub-wavelength thickness of the sample and thus does not require resonant
enhancement or above-gap excitation for its implementation. Moreover, the same
approach can be extended to any 2D material belonging to the D3h symmetry
group, thus for instance to any material of the TMDs’ family. Our findings
provide a new strategy for the tuning of the fundamental properties of light
(polarization and amplitude) in the nonlinear regime and in the 2D thickness
limit, and thus pave the way to the design of novel advanced functionalities
in high-speed frequency converters, nonlinear all-optical modulators and
transistors [31, 32], interferometric autocorrelators for ultra-short pulse
characterization and tunable atomically thin holograms [24].
### Nonlinear optical characterization
Figure 1: MoS2 symmetry properties, optical selection rules and all-optical
SHG modulation scheme. (a) Top view of a MoS2 cystal. The arrows inside the
hexagon highlight the D3h three-fold rotational symmetry. (b) Schematic of the
resulting SH polarization for different combinations of FF along AC
(horizontal arrow) and ZZ directions (vertical arrow). (c-d) Sketch of the
all-optical SH polarization modulation. The control pulse is polarized along
the ZZ direction while the probe pulse is polarized along the AC direction of
the MoS2 sample. (c) When the delay between the control and probe pulses is
larger than the FF pulse duration, both will generate a SH signal polarized
along the AC direction. (d) For zero-delay between the probe and control
pulses the SH signal will be emitted along the ZZ direction. Figure 2: All-
optical modulation set-up and SHG characterization. (a) Sketch of the setup
used for the experiments. For the FF we used an OPO tunable between $\sim$1.3
and 2.0 µm. The FF beams are separated into two perpendicular replicas inside
a Mach-Zehnder interferometer and subsequently focused onto the MoS2 sample
with a 40x microscope objective. The back-scattered SH signal is spectrally
filtered and detected with a single-photon avalanche diode. (b) Power
dependence of the SHG signal for all wavelengths used in our experiments. The
grey dashed line is a guide to the eye to indicate a slope of 2, typical of
the SHG process. (c) Polar plot of the normalized SH intensity as a function
of the excitation polarization angle $\theta$ with SH polarization detection
always parallel to the FF. Blue circles show experimental data and the solid
blue line indicates the $\text{cos}^{2}[3(\theta-\phi_{0})]$ fit.
For the experiments, we used high quality monolayer MoS2 flakes, fabricated
with a modified chemical vapour deposition method [33, 34] on thermally
oxidized Si/SiO2 substrates. In contrast to thin films, where surface SHG is
allowed due to an out-of-plane component of the second-order susceptibility
[35, 6], TMDs belong to the D3h symmetry group and thus have only one non-
vanishing in-plane component of the nonlinear optical susceptibility (see
Methods) [35, 36, 37, 38, 39, 40].
$\displaystyle\chi^{(2)}\equiv\chi_{yyy}^{(2)}=-\chi_{yxx}^{(2)}=-\chi_{xyx}^{(2)}=-\chi_{xxy}^{(2)},$
(1)
where x and y refer to the in-plane Cartesian coordinates of the SH
polarization and of the two FFs. A sketch of the hexagonal lattice for MoS2 is
shown in Fig. 1(a), where the Cartesian coordinates are defined with respect
to the two main lattice orientations: the armchair (AC) and zigzag (ZZ)
directions. In this framework, the SH intensity I2ω as a function of the FF
for any TMD along the AC and ZZ directions can be written as
$\displaystyle I_{AC}^{2\omega}$ $\displaystyle\sim$ $\displaystyle\lvert
E_{AC}^{2}-E_{ZZ}^{2}\rvert^{2},$ (2) $\displaystyle I_{ZZ}^{2\omega}$
$\displaystyle\sim$ $\displaystyle\lvert 2E_{AC}E_{ZZ}\rvert^{2},$ (3)
where EAC and EZZ correspond to the FF fields with polarization along the AC
and ZZ directions respectively [41, 42, 43, 39]. Thus, the SHG from two
electric fields with the same polarization (either along AC or ZZ) will always
result in an emitted SH intensity with polarization along the AC direction, as
depicted in Fig. 1(b). This is indeed the case of all the SHG experiments
performed to date on 2D materials [15, 36, 43, 39]. On the other hand, two
ultrashort FFs with perpendicular polarization (along the AC and ZZ
directions) and with the same amplitude will generate a SH signal along the AC
direction if they do not overlap in time (Fig. 1(c)), while they will generate
a SH signal along the ZZ direction at zero-delay (Fig. 1(d)), thus leading to
an ultrafast $90^{\circ}$ polarization switch within the FFs pulse duration.
Finally, for circularly polarized FFs, the emitted SH has opposite circular
polarization due to valley-dependent selection rules (see analysis of equation
(4)) [44, 45, 46].
Figure 3: Ultrafast polarization and amplitude all-optical switching. (a) SH
intensity for 1480 nm FF wavelength (black line) measured along the zigzag
direction as a function of the delay between the two perpendicularly polarized
FFs (illustrated in the inset). The blue line corresponds to a Gaussian fit of
the autocorrelation curve. The grey shaded area represent the noise level at
$\sim$3.5 fW of our experiments. (b) Normalized SH intensity and noise level
for all FF wavelengths used in our experiments. (c) Double logarithmic plot of
the emitted SH power as a function of the incident FF power along the AC
direction. The grey dashed line is a guide to the eye representing a slope of
1.
The SHG measurements were performed with the set-up shown in Fig. 2(a) and
described in the Methods. In order to realize all-optical polarization
switching and SH modulation it is crucial to first characterize the relative
orientation between the FFs and the MoS2 sample. To do so, we first performed
SHG experiments using a tunable near-IR optical parametric oscillator (OPO) as
a FF. The emitted SHG power for the FF wavelengths used in our experiments
(between 1360 nm and 1560 nm) is shown in Fig. 2(b). The slope of 2 in the
double logarithmic plot is a further proof of a genuine SHG process. The
crystal orientation of the MoS2 sample was determined for each FF wavelength
by SH polarization dependent experiments with an additional polarizer in front
of the detector to measure the SH parallel to the excitation polarization [47,
36, 43, 39]: the SH intensity is proportional to
$\text{cos}^{2}[3(\theta-\phi_{0})]$, where $\theta$ is the FF polarization
angle and $\phi_{0}$ is the rotation of the AC direction relative to the
p-polarization in the laboratory frame. Fig. 2(c) is an example of the SH
polar plot for a FF wavelength of 1400 nm and shows that the AC direction is
tilted by $\phi_{0}=13.6^{\circ}+n\cdot 60^{\circ}$ ($n$ integer) with respect
to p-polarization in the lab coordinates. In addition, Fig. 1(c) confirms the
absence of any detectable strain in our sample, since an uniaxial strain would
result in a symmetric attenuation along its direction of action [38, 39]. The
small asymmetry of the lobes along the two AC directions ($\sim$ 15∘/70∘ and
$\sim$ 190∘/250∘) is attributed to polarization scrabbling effects due to the
use of a dichroic mirror in reflection geometry [48]. Finally, based on the
results in Fig. 2(b), we determine the modulus of the complex second-order
nonlinear susceptibility at the FF wavelengths used in our experiments. To do
so, we estimate the optical losses of the setup from the SH emission at the
sample position to the detection on the Single Photon Avalanche Detector
(SPAD) and calculate the SH tensor element $\chi^{(2)}$ of MoS2, as described
in the Methods. We thus obtained effective second-order susceptibility values
for our FF wavelengths 1360 nm, 1400 nm, 1480 nm and 1560 nm of $\sim$ 282.1
pm/V, $\sim$ 153.7 pm/V, $\sim$ 44.7 pm/V, $\sim$ 24.2 pm/V respectively. The
highest value obtained at 1360 nm FF wavelength is due to exciton resonant SH
enhancement [50, 15]. All values are in good agreement with those previously
reported by experiments performed at similar FF wavelengths [42, 36, 51, 50,
52] and predicted by theory [53, 54]. It is worth noting that for single layer
TMDs, where interlayer interference [49] is absent, SHG is insensitive to the
phase of the nonlinear optical response.
Figure 4: Phase-locked all-optical SH modulation along armchair and zigzag
directions. (a) SH power for 1480 nm FF wavelength along AC (blue) and ZZ
(red) directions as a function of the relative delay between the two
perpendicularly polarized FFs. The inset shows the Poincaré-sphere with
polarization directions on its orthodromes. The red line indicates the change
in polarization when tuning the delay between the two FFs with a birefringent
delay line. (b) Schematic image of the modified common-path delay generator.
The two perpendicular components of an incident pulse, polarized at 45∘ with
respect to the AC/ZZ directions, are delayed by two birefringent alpha barium
borate crystals with perpendicular optical axis, whereby the variable
thickness of the second one allows to control the delay with interferometric
precision. (c) Zoom-in of (a), showing the detected SH power for 1480 nm FF
along the AC (blue) and ZZ directions (red) close to zero delay.
### Nonlinear all-optical modulation
Having defined the AC and ZZ directions of our sample, we now demonstrate all-
optical SH polarization and amplitude modulation. We separate the FF beam into
two perpendicular replicas, align them along the sample’s AC and ZZ directions
using a half-waveplate and control their relative delay with a motorized
mechanical stage (see Methods for details). For large delays (i.e., longer
than the FF pulse duration) between the two perpendicular FFs, SH will be
emitted by each individual FF along the AC direction following equation (2).
Instead, at zero-delay, when the two FFs overlap perfectly in time, the SH
intensity along the AC direction will go to zero and the SH signal will be
emitted only along the ZZ direction. Fig. 3(a) shows the SH average power
emitted along the ZZ direction as a function of the delay between the two
perpendicularly polarized FFs and for a FF wavelength of 1480 nm. The Gaussian
fit, blue curve in Fig. 3(a), has a FWHM of $\sim$ 250 fs, corresponding to
the autocorrelation function of our OPO pulse with duration of $\sim$ 175 fs.
Moreover, the SH signal along the ZZ direction is now ideally background free,
demonstrating the potential of ultrafast SH polarization switching and close
to 100 % amplitude modulation of our approach.
We further note that this result is solely based on symmetry considerations
and thus provides ultra-broadband response, not limited to above-gap or
resonant exciton pumping. Indeed, we obtained the same result for all the FF
wavelengths used in our experiments, as shown in Fig. 3(b). The possibility to
emit SH along perpendicular directions (AC and ZZ) with the same efficiency is
a unique feature that arises from the combination of symmetry and deep sub-
wavelength thickness of TMDs, which relaxes the phase-matching constraints
typical of harmonic generation. This result could have an immediate
application in background free ultra-broadband and ultra-short pulse
characterization. For instance, in the most advanced commercial systems [55]
for ultra-short pulse characterization one has to switch between collinear and
non-collinear geometries in order to collect either the interferometric or the
background-free intensity autocorrelation signals respectively. In contrast,
in our approach both signals are accessible using the same geometry and by
simply switching the SH detection from AC to ZZ. Further, following equation
(3), the power scaling of the emitted SH along the ZZ direction is linear with
respect to each of the FF intensities. This is confirmed by the power-
dependent SHG measurements reported in Fig. 3(c), where we show the emitted SH
power along the ZZ direction at zero delay between the two FFs and as a
function of the AC polarized FF power.
To gain more insight into the temporal evolution of the emitted SH
polarization and amplitude, we finally scan the delay between the two
perpendicularly polarized FFs with interferometric precision and measure the
emitted SH along both AC and ZZ directions, as shown in Fig. 4(a). In order to
control the delay between two perpendicular pulses with the desired sub-
optical cycle precision we used the common-path delay generator sketched in
Fig. 4(b) and described in the Methods. As expected, for delays longer than
our pulse duration the SH power is emitted only along the AC direction, blue
curve in Fig. 4(a), and no signal is detected along the ZZ direction, red
curve in Fig. 4(a). Instead, for delays close to zero we observe a strong
ultrafast modulation of the SH power emitted along the AC direction. This can
be better appreciated looking at Fig. 4(c), which shows the emitted SH power
along AC and ZZ directions at 1480 nm for delays between -10 fs and +10 fs.
It is useful to note that for delays much shorter compared to the pulse
duration our interferometric measurement is the analogue of tuning the
polarization of one single FF pulse along the orthodrome of the Poincaré-
sphere, inset in Fig. 4(a). This corresponds to a rotation of the FF
polarization from $-45^{\circ}$ with respect to the AC-ZZ directions (at zero
delay) to left/right circular polarization (at delay $\tau=\pm\frac{T}{4}$,
where $T$ is the FF optical cycle) to $+45^{\circ}$ with respect to the AC-ZZ
directions at delay $\tau=\pm\frac{T}{2}$. This result is consistent with the
theoretical SH polarization $\textbf{P}^{(2)}$ generated by an arbitrary
elliptically polarized FF [46] after a simple basis transformation to account
for the rotation by $-45^{\circ}$ with respect to AC/ZZ directions:
$\displaystyle\textbf{P}^{(2)}=\epsilon_{0}\chi^{(2)}\lvert\textbf{E}\rvert^{2}\begin{pmatrix}\widehat{ZZ}\pm
i\,\text{sin}(2\vartheta)\widehat{AC}\end{pmatrix},$ (4)
Here $\vartheta=0^{\circ}$ denotes a linearly polarized FF at $45^{\circ}$
with respect to the AC/ZZ direction and $\vartheta=45^{\circ}$ corresponds to
a circularly polarized FF. This clearly shows that the SH component emitted
along the AC direction oscillates with a period of $\frac{T}{2}$ as a function
of the FF polarization, in contrast to the SH emitted along the ZZ direction.
This underpins the interferometric precision required in order to fully
capture the modulation along the AC direction. The experimental results show a
weak modulation also for the SH emitted along the ZZ direction, although this
is not expected from theory. This could arise from weak strain (i.e., below
the limit detectable by our SHG polarization measurements [56, 39]), small
deviations in the alignment of the detection polarizer with respect to the
AC/ZZ directions or from additional terms in the $\chi^{(2)}$ arising from the
valley degree of freedom [57, 48]. Looking at Fig. 4(c) one can indeed
appreciate that at zero delay (FF at $-45^{\circ}$ with respect to AC/ZZ
directions) the SH is emitted only along the ZZ direction, while at
$\frac{T}{4}$ delay the emitted SH components along AC and ZZ are identical,
as expected for circular polarization.
### Discussion
In conclusion, we have demonstrated all-optical polarization switching and
amplitude modulation of SHG in MoS2. Our approach surpasses all previously
reported electrical and all-optical attempts of SH tuning in terms of
modulation depth and speed, providing $90^{\circ}$ polarization switch, close
to 100 % modulation depth and speed limited only by the FF pulse duration.
Moreover, our method is intrinsically broadband since it only relies on the
crystal symmetry of TMDs. We thus foresee a direct impact of our results on a
variety of photonic devices, such as high-speed frequency converters,
nonlinear all-optical modulators and transistors [31, 32], autocorrelators for
ultra-short pulse characterization and atomically thin optically tunable
nonlinear holograms [24].
## VI Methods
Polarization dependent SH intensity. The vectorial components of the second-
order polarization $P^{(2)}(2\omega)$ for a material with D3h symmetry (such
as TMDs) are given by
$\displaystyle\begin{pmatrix}P^{(2)}_{x}(2\omega)\\\ P^{(2)}_{y}(2\omega)\\\
P^{(2)}_{z}(2\omega)\end{pmatrix}=\epsilon_{0}\begin{pmatrix}0&0&0&0&0&\chi^{(2)}_{xxy}\\\
\chi^{(2)}_{yxx}&\chi^{(2)}_{yyy}&0&0&0&0\\\
0&0&0&0&0&0\end{pmatrix}\begin{pmatrix}E^{2}_{x}(\omega)\\\
E^{2}_{y}(\omega)\\\ E^{2}_{z}(\omega)\\\ 2E_{y}(\omega)E_{z}(\omega)\\\
2E_{x}(\omega)E_{z}(\omega)\\\ 2E_{x}(\omega)E_{y}(\omega)\end{pmatrix}$
where $\chi^{(2)}_{yyy}=-\chi^{(2)}_{xxy}=-\chi^{(2)}_{yxx}=\chi^{(2)}$. If we
now consider a TMD oriented in such way that the ZZ and AC directions lie
along the x and y Cartesian coordinates respectively and we neglect the z
(out-of-plane) direction, we obtain the following expression:
$\displaystyle\begin{pmatrix}P^{(2)}_{ZZ}(2\omega)\\\
P^{(2)}_{AC}(2\omega)\end{pmatrix}=\epsilon_{0}\chi^{(2)}(2\omega,\omega,\omega)\begin{pmatrix}2E_{ZZ}(\omega)E_{AC}(\omega)\\\
E^{2}_{ZZ}(\omega)-E^{2}_{AC}(\omega)\end{pmatrix}$
Finally, since the SH intensity is proportional to the absolute square value
of the second-order polarization, we retrieve equations (2) and (3) of the
main text:
$\displaystyle I_{AC}^{2\omega}=\lvert P^{(2)}_{AC}\rvert^{2}\sim\lvert
E^{2}_{AC}-E^{2}_{ZZ}\rvert^{2}$ $\displaystyle I_{ZZ}^{2\omega}=\lvert
P^{(2)}_{ZZ}\rvert^{2}\sim\lvert 2E_{AC}E_{ZZ}\rvert^{2}$
SHG setup. For the FF we used an OPO (Levante IR from APE) pumped by an
ytterbium-doped mode locked laser (FLINT12, Light Conversion) with a
repetition rate of 76 MHz and 100 fs pulse duration and generating pulses
tunable between $\sim$1.3 µm and 2.0 µm. The FF is then separated into two
perpendicular replicas whose relative delay is tuned with two different
approaches: a computer controlled motorized stage (M-414.2PD, PI) in a Mach-
Zehnder interferometer configuration and a commercial common-path birefringent
interferometer (GEMINI, NIREOS) [58]. Compared to standard home-built
interferometers, the GEMINI provides sub-wavelength interferometric stability
with precise control on the delay between the two replicas with attosecond
precision. The polarization of the FFs was tuned using a half-waveplate
(AHWP05M-1600, Thorlabs) and the power on the sample was controlled by two
polarizers (LPNIR050 and WP12L-UB, both Thorlabs). Finally, the two collinear
and perpendicularly polarized FFs were focused on the sample using a custom
built microscope equipped with a 40x reflective objective (LMM-40X-P01,
Thorlabs). The back-scattered SH signal is spectrally separated using a
dichroic mirror (DMSP950, Thorlabs), further spectrally purified by filters
(FELH0650, FESH850, FESH0950, all Thorlabs) and detected with a single-photon
avalanche diode (C11202-050, Hamamatsu). The SH polarization was measured
using a wire grid polarizer (WP12L-UB, Thorlabs).
Estimate of the optical losses of the setup. In order to quantify the SH
signal generated directly at the sample position, optical losses of the
different components of the setup must be considered. While the transmission
coefficients for the investigated SH wavelengths of the filters and the
dichroic mirror (all > 96 %) were taken from the manufacturer’s website, the
values for polarizers and the microscope objective were determined
experimentally. A transmission of $\sim$ 79 % was determined for the wire grid
polarizer while for the reflective objective we determined a transmission of
50 %. Last, the responsivity of the single-photon avalanche diode was taken
into account, which ranges depending on the investigated SH wavelength between
$\sim$ 17 % and $\sim$ 31 %. In total, we estimated our optical losses from
the SH emission to the detector to be $\sim$ 86-92 % depending on the
wavelength.
Calculation of the second-order nonlinear susceptibility. The sheet-SH tensor
element $\chi_{S}^{(2)}$ can be calculated from the FF and SH average powers
using the equation [42]:
$\displaystyle\chi_{S}^{(2)}=\sqrt{\frac{c^{3}\epsilon_{0}f\pi
r^{2}t_{FWHM}(1+n_{2})^{6}P_{SHG}(2\omega)}{16\sqrt{2}S\omega^{2}P_{FF}^{2}(\omega)}},$
(5)
where $c$ is the speed of light, $\epsilon_{0}$ is the permittivity of free-
space, $f$ = 76 MHz is the pump laser repetition rate, $r\sim$ 1.85 µm is the
focal spot radius, $t_{FWHM}\sim$ 200 fs is the pulse full width at half
maximum, $n_{2}\sim$ 1.45 is the substrate refractive index, $S=0.94$ is a
shape factor for Gaussian pulses, $\omega$ is the FF angular frequency, and
$P_{SHG}(2\omega)$ \- $P_{FF}(\omega)$ are the SH and FF average powers. The
effective bulk-like second-order susceptibility $\chi_{eff}^{(2)}$ can be
subsequently calculated from equation (5) as
$\chi_{eff}^{(2)}=\frac{\chi_{S}^{(2)}}{d_{MoS_{2}}}$, where the thickness
$d_{MoS_{2}}$ of MoS2 is 0.65 nm [18, 46].
Pulse duration of the FFs. In order to prove that our method is solely limited
by the pulse duration of the FFs, we performed a standard characterization of
temporal profile of our OPO source at different wavelengths (Extended Data
Fig. 1). For the measurements we used a home built autocorrelator based on a
Michelson interferometer equipped with a motorized and computer controlled
translation stage (HPS60-20X-M5, Optosigma). Two identical and temporally
delayed replicas of the OPO pulse were then focused onto a 1 mm thick
$\beta$-barium borate crystal (BBO-652H, Eksma Optics) in non-collinear
geometry and the background free SHG autocorrelation intensity was detected on
a Si photodetector (DET10A2, Thorlabs). From this we obtained values for the
autocorrelation in the range of 217 fs to 310 fs with gaussian fits,
corresponding to pulse durations between 150 fs and 220 fs.
## VII References
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## VIII Acknowledgments
The authors greatly acknowledge Habib Rostami and Fabrizio Preda for helpful
discussions. This work was supported by the European Union’s Horizon 2020
Research and Innovation program under Grant Agreement GrapheneCore3 881603
(G.S.). This publication is part of the METAFAST project that received funding
from the European Union’s Horizon 2020 Research and Innovation program under
Grant Agreement No. 899673 (G.S. and G.C.). The authors acknowledge the German
Research Foundation DFG (CRC 1375 NOA project numbers B2 (A.T.) and B5 (G.S.))
and the Daimler und Benz foundation for financial support (G.S.).
## IX Author Contributions Statement
S.K. and G.S. conceived the experiments. S.K. and O.G. performed the all-
optical modulation measurements. Z.G., A.G. and A.T. fabricated and provided
the high quality MoS2 sample. S.K., G.C and G.S. wrote the manuscript with
contribution from all the authors. All the authors participated to the
discussion and commented on the manuscript.
## X Competing Interests Statement
The authors declare no competing interest.
## XI Figure Legends/Captions (for main text figures)
MoS2 symmetry properties, optical selection rules and all-optical SHG
modulation scheme. (a) Top view of a MoS2 cystal. The arrows inside the
hexagon highlight the D3h three-fold rotational symmetry. (b) Schematic of the
resulting SH polarization for different combinations of FF along AC
(horizontal arrow) and ZZ directions (vertical arrow). (c-d) Sketch of the
all-optical SH polarization modulation. The control pulse is polarized along
the ZZ direction while the probe pulse is polarized along the AC direction of
the MoS2 sample. (c) When the delay between the control and probe pulses is
larger than the FF pulse duration, both will generate a SH signal polarized
along the AC direction. (d) For zero-delay between the probe and control
pulses the SH signal will be emitted along the ZZ direction.
All-optical modulation set-up and SHG characterization. (a) Sketch of the
setup used for the experiments. For the FF we used an OPO tunable between
$\sim$1.3 and 2.0 µm. The FF beams are separated into two perpendicular
replicas inside a Mach-Zehnder interferometer and subsequently focused onto
the MoS2 sample with a 40x microscope objective. The back-scattered SH signal
is spectrally filtered and detected with a single-photon avalanche diode. (b)
Power dependence of the SHG signal for all wavelengths used in our
experiments. The grey dashed line is a guide to the eye to indicate a slope of
2, typical of the SHG process. (c) Polar plot of the normalized SH intensity
as a function of the excitation polarization angle $\theta$ with SH
polarization detection always parallel to the FF. Blue circles show
experimental data and the solid blue line indicates the
$\text{cos}^{2}[3(\theta-\phi_{0})]$ fit.
Ultrafast polarization and amplitude all-optical switching. (a) SH intensity
for 1480 nm FF wavelength (black line) measured along the zigzag direction as
a function of the delay between the two perpendicularly polarized FFs
(illustrated in the inset). The blue line corresponds to a Gaussian fit of the
autocorrelation curve. The grey shaded area represent the noise level at
$\sim$3.5 fW of our experiments. (b) Normalized SH intensity and noise level
for all FF wavelengths used in our experiments. (c) Double logarithmic plot of
the emitted SH power as a function of the incident FF power along the AC
direction. The grey dashed line is a guide to the eye representing a slope of
1.
Phase-locked all-optical SH modulation along armchair and zigzag directions.
(a) SH power for 1480 nm FF wavelength along AC (blue) and ZZ (red) directions
as a function of the relative delay between the two perpendicularly polarized
FFs. The inset shows the Poincaré-sphere with polarization directions on its
orthodromes. The red line indicates the change in polarization when tuning the
delay between the two FFs with a birefringent delay line. (b) Schematic image
of the modified common-path delay generator. The two perpendicular components
of an incident pulse, polarized at 45∘ with respect to the AC/ZZ directions,
are delayed by two birefringent alpha barium borate crystals with
perpendicular optical axis, whereby the variable thickness of the second one
allows to control the delay with interferometric precision. (c) Zoom-in of
(a), showing the detected SH power for 1480 nm FF along the AC (blue) and ZZ
directions (red) close to zero delay.
## XII Data Availability
The data that support the plots within this paper and other findings of this
study are available from the corresponding author on reasonable request.
Source data are provided with this paper.
|
††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work
# The effect of fast noise on the fidelity of trapped-ions quantum gates
Haim Nakav Physics of Complex Systems, Weizmann Institute of Science and
AMOS, Rehovot 7610001, Israel Ran Finkelstein Physics of Complex Systems,
Weizmann Institute of Science and AMOS, Rehovot 7610001, Israel Current
address : Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA Lee Peleg Physics of Complex Systems,
Weizmann Institute of Science and AMOS, Rehovot 7610001, Israel Nitzan
Akerman Physics of Complex Systems, Weizmann Institute of Science and AMOS,
Rehovot 7610001, Israel Roee Ozeri Physics of Complex Systems, Weizmann
Institute of Science and AMOS, Rehovot 7610001, Israel
###### Abstract
High fidelity single and multi-qubit operations compose the backbone of
quantum information processing. This fidelity is based on the ability to
couple single- or two-qubit levels in an extremely coherent and precise
manner. A necessary condition for coherent quantum evolution is a highly
stable local oscillator driving these transitions. Here we study the effect of
fast noise, that is noise at frequencies much higher than the local oscillator
linewidth, on the fidelity of one- and two-qubit gates in a trapped-ion
system. We analyze and measure the effect of fast noise on single qubit
operations including resonant $\pi$ rotations and off-resonant sideband
transitions . We further analyze the effect of fast phase noise on the Mølmer-
Sørensen two-qubit gate. We find a unified and simple way to estimate the
performance of all of these operations through a single parameter given by the
noise power spectral density at the qubit response frequency. While our
analysis focuses on phase noise and on trapped-ion systems, it is relevant for
other sources of fast noise as well as for other qubit systems in which spin-
like qubits are coupled by a common bosonic field. Our analysis can help in
guiding the deign of quantum hardware platforms and gates, improving their
fidelity towards fault-tolerant quantum computing.
## I Introduction
Quantum coherence is the fundamental resource in any quantum information
processing task. Much of the remarkable progress achieved in the field over
the past two decades was achieved by careful examination and reduction of
dephasing mechanisms through improvements in control technology, and
mitigation of environmental noise [1, 2]. Yet, single and two-qubit gates are
still often limited by the finite coherence between the qubit and the
classical local oscillator (LO) used to control and measure the quantum
system.
In quantum information processing, phase noise typically limits the coherence
time of quantum registers and compromises the fidelity of quantum gates. Many
studies investigated this effect, and techniques to mitigate it were proposed
and implemented. These include dynamical decoupling [3, 4, 5, 6, 7, 8, 9], and
decoherence-free subspaces [10, 11, 12, 13, 14]. These methods were designed
to mitigate the effects of slow phase noise ($\sigma_{z}$ errors), which
refers to noise within the linewidth of the oscillator as it acts on the qubit
transition. However, noise that is faster than this linewidth affects qubit
performance differently. In some cases, which will be discussed in this paper,
such noise acts as a bit flip error, rather than a $\sigma_{z}$ error,
rendering the above-mentioned methods ineffective.
Fast LO noise can result from various mechanisms. One example of such a source
is a side effect of reducing the magnitude of slow phase noise. In order to
narrow the linewidth of an oscillator often an external servo loop is added to
suppress the slow phase noise of the LO with respect to a stable reference.
While subduing the slow phase noise, the servo loop generates excess noise at
the higher frequencies close to its unity gain response [15]. This noise
feature, which is prevalent mainly in narrow linewidth lasers, is sometimes
referred to as servo bump and results in fast phase noise.
Traditionally, noise at these high frequencies was thought to average out over
the much longer time scale of typical quantum evolution. However, it was
recently realized that such noise limits various quantum operations [16, 17,
18, 19, 20, 21, 22] if it overlaps with a frequency at which the quantum
system resonantly responds.
As an example, we consider a single qubit rotation on the Bloch sphere shown
in Fig. 1(a). Phase or frequency instabilities lead to fast fluctuations of
the Rabi vector in the Bloch sphere equatorial plane, resulting in randomly
modified trajectories [purple trajectories on the Bloch sphere in Fig. 1(a)].
Specifically, the fast noise frequency components that will not average out
and lead to significant rotation errors are those that overlap the Rabi
frequency. Fig 1(b) showcases an example of such a modified spectrum. In fact,
the effect of fast noise goes beyond impacting single qubit rotations and can
have deleterious effects on the fidelity of two-qubit gates. For example,
Starting from $\mid\uparrow\uparrow\rangle$, the Mølmer-Sørensen (MS) gate
shown in Fig. 1(c,d) under fully coherent evolution forms a perfect Bell-
state. However, the presence of fast noise that overlaps the carrier
transition leads to incoherent errors and reduced fidelity.
Here, we simulate and measure the effect of fast noise on the fidelity of
single- and two-qubit gates in trapped ion qubits. Single-qubit rotations are
performed via resonant pulses and their fidelity was directly measured for
different phase-noise spectra. Two-qubit entanglement is performed using the
MS gate that uses electromagnetic fields that are close to resonance with the
sideband of a common ion-phonon mode. We find that the magnitude of the error
induced by fast phase noise in quantum gates predominantly scales with the
overlap of the noise power spectral density (PSD) with the relevant response
frequency. That is, the Rabi frequency in the case of single qubit gates and
the detuning from carrier transition in the case of sideband transitions and
MS gates. While this study focuses on the effect of phase noise on trapped-ion
qubit gates, our results are broadly relevant for any fast noise sources, such
as amplitude noise as well. Our results are also relevant for other quantum
computing technologies in which two-qubit gates are realized through coupling
to a common bosonic mode, such as in superconducting qubits coupled through a
microwave resonator.
Figure 1: The effect of fast noise on the fidelity of trapped-ions quantum
gates. (a) Illustration of the Bloch sphere of a qubit where the Rabi vector
orientation fluctuates due to fast phase noise. The orange dashed line is the
noise-free trajectory for this qubit operation, whereas the purple lines are
trajectories of the qubit in the presence of fast noise. (b) Typical laser
phase power spectral densities (PSD). The red curve represents a typical PSD
for a free-running laser which is characterized by a white frequency noise or,
equivalently, $1/f^{2}$ phase noise. The purple curve represents the PSD of
the same noise process after applying an external feedback loop that
suppresses the slow noise while increasing the noise at the edge of its
bandwidth. The equivalent time traces are shown in the inset. (c) Mølmer-
Sørensen (MS) gate in the presence of fast noise can result in incoherent
coupling to other levels resulting in leakage of population to other motional
states, decreasing the gate fidelity. (d) The MS interaction dynamics,
visualized through the two populations without phase noise (dashed lines) and
with noise (solid lines). At gate time (green area), we expect to obtain a
Bell state, due to phase noise the qubits reach a slightly erroneous state
which includes an incoherent mixture of Bell states.
## II Experimental and numerical setup
We perform stochastic master equation simulations and controlled experiments
to evaluate the effect of fast noise on the fidelity of quantum operations.
Specifically, here we focus on the PSD typical of oscillators where fast noise
is amplified by a frequency-stabilization feedback. For spectral frequencies
far from the carrier typical oscillators exhibit a white frequency noise
spectrum or equivalently, a brown phase noise spectrum
$S(\omega)=S_{0}/\omega^{2}$ [23]. To obtain a random vector with such a PSD
and a target root-mean square (RMS) we use a vector of Gaussian distributed
independent samples as a seed for a generalized Gauss-Markov stochastic model
[24, 25]. This noise is then convoluted with the impulse response of a phase-
locked loop system. The gain and positions of the system’s poles can be tuned
to shape the spectral response and the resulting output PSD. This results in a
spectral region of amplified noise or servo-bump, around the unity gain
frequency of the simulated system as shown in Fig. 1(b). The random time-
series generated through this process are used for both the experiments and
the simulations presented below. For the simulations, we numerically solve the
master equation with a time-dependent phase in the $\sigma_{x}$ drive term. In
each iteration we sample a different random phase vector, while all vectors
are drawn from the same PSD. The numerical simulations are performed using the
QuTiP[26] python package. For the different configurations described below, we
use different expectation values obtained from the time-evolution of the full
density matrix. These values are averaged over the different noise
realizations to obtain the results presented here.
For the experiment, we use an external cavity diode laser stabilized to a
high-finesse cavity. We further utilize the cavity as an extremely narrow
filter and take light transmitted through it to perform injection locking on
an additional diode, yielding the desired power with low noise. We synthesize
phase noise using an arbitrary waveform generator and inject it to the
experiment through modulation with an acousto-optical modulator (AOM). The
laser is then used to drive the optical qubit transition
$|S_{1/2};m_{j}{=}1/2\rangle-|D_{5/2};m_{j}{=}3/2\rangle$ in a single
${}^{88}\mathrm{Sr}^{+}$ ion trapped in a linear Paul trap.
## III Single qubit gates
We begin by studying the most simple case of single-qubit rotations in the
presence of fast phase noise. In Fig. 2(a) we show the results of continuously
driving the qubit transition with a Rabi frequency $\Omega=2\pi\times 100$ kHz
and an additional synthesized phase noise characterized by a peak in the PSD
around 200 KHz. The experimental measured excited state population (circles)
exhibits Rabi oscillations with a decoherence rate which agrees well with the
stochastic numerical simulation (solid line) accounting for the the same noise
PSD [27]. We further study numerically the $\pi$-pulse error for a wide range
of noise spectra and find that the gate error is linearly dependent on the
noise PSD at the Rabi frequency. The relevant PSD in this case is the PSD of
the electric field (rather than the phase) driving the transition,
$\mathrm{E}=\mathrm{E_{0}}\cos[\omega t+\phi(t)]$, where $\phi(t)$ is the
time-dependent phase which includes the noise term. To evaluate the field PSD
in units of Rabi frequency we normalize the entire spectrum such that the area
under the carrier peak is $\Omega^{2}$. We term the new spectral density as
the Rabi PSD (RSPD). The dependence of a $\pi$-pulse error vs. the RPSD is
shown in Fig. 2(c) where we see a linear dependence. For pulses that are long
as compared with the inverse of relevant noise features, this dependence makes
sense, as the RPSD at the Rabi frequency generates an effective coupling
between the Rabi-dresses states. This effect was widely studied in the context
of noise spectroscopy [28, 17, 29].
For short pulses however, such as $\pi$ rotations, we find a slightly
different picture. In Fig. 2(b) we plot the noise PSD and the calculated $\pi$
pulse infidelity $1-F_{\pi}$ for different Rabi frequencies. We find that the
largest infidelity does not occur when the Rabi frequency exactly overlaps the
peak in the PSD, but rather at lower Rabi frequencies. Considering the $\pi$
pulse length $t_{\pi}=\pi/\Omega$ we find that the effective bandwidth of such
a short pulse is on the order of the Rabi frequency and thus samples, in
practice, the entire relevant region of the noise PSD equally. However, the
pulse length is still inversely proportional to the Rabi frequency such that
longer pulses sample the noise for a longer time. This combined integrated
response leads to a shift in the maximal infidelity towards lower Rabi
frequencies [30]. The sensitivity of short gate fidelity to noise integrated
over a wide spectral window, renders the proportionality factor between the
gate error and the RPSD at the Rabi frequency, seen in Fig. 2(c) to be on the
order of $10^{-2}$, dependent on the details of the noise spectrum.
Figure 2: Single qubit rotation infidelity due to fast phase noise. (a)
Simulation (solid line) and measurements (filled circles) of Rabi oscillations
for an ion driven by a noisy oscillator. Experimental error bars are smaller
than the marker size. (b) The infidelity of a single qubit $\pi$ rotation
calculated at different Rabi frequencies (orange markers), overlaid on the
phase noise PSD characterizing the simulated driving laser. (c) The infidelity
of a single qubit $\pi$ rotation at a fixed Rabi frequency as a function of
the Rabi PSD at the Rabi frequency multiplied by the gate time.
## IV Two qubit gates
_Sideband transitions.–_ Here we consider the Mølmer-Sørensen gate [31, 32]
which is a composite operation in which the spin and motion of trapped ions
are entangled during the gate through driving of motional sideband
transitions. We thus start by disassembling this operation into its primitive
constituents. We first study numerically and experimentally the interplay of
fast noise and coherent driving of sideband transitions in a single trapped
ion. Beyond their role in the MS gate, off-resonant coupling fields are widely
used in experiments with atoms and molecules. Such fields are used in
generating dressed states, trapping, cooling and local addressing of atomic
qubits. It is thus of general relevance to understand the effect of fast noise
on sideband transitions.
We begin by studying the coupling of noise to the spin degree of freedom. We
effectively remove any coupling to motional degrees of freedom by detuning the
laser central frequency by a few hundreds of KHz from both the carrier
transition as well as from any sideband transition. The synthesized noise PSD
here has a broad peak around typical trap frequencies $\nu\approx 700$ KHz.
Following a drive of variable time, we image the ion to determine its spin
state. In Fig. 3 we plot the excited spin state population as a function of
drive time for different levels of synthesized noise, each averaged over 31
realizations. We observe incoherent spin pumping due to fast noise overlapping
the carrier transition. We find good agreement with the numerical simulation,
which for a sufficient number of realizations converges well to a simple
pumping rate model $\mathrm{P_{e}}=0.5[1-\mathrm{exp}(-\Gamma t)]$. In Fig.
3(b) we plot the pumping rate $\Gamma$ as a function of the RPSD at the
detuning frequency from the carrier transition $\Delta$. We find that the
pumping rate is proportional to the RPSD at the carrier transition frequency,
_i.e._ $\Gamma\simeq 2\cdot\mathrm{PSD}(f=\Delta)$.
In a second experiment, we effectively trace out the spin degree of freedom
and measure the occupation of the motional modes. We drive the blue sideband
transition for an integer number of spin cycles, and then perform thermometry
by driving the blue sideband transition again with a noise-free laser and
infer the occupation of motional states from sideband Rabi thermometry [33,
34]. In Fig 4, we plot the measured average number of phonons $\bar{n}$
following an integer number of blue sideband cycles. We find excess heating in
the presence of fast noise, which grows as a function of the drive time. As a
reference, we repeat the measurement in the absence of synthesized noise and
find negligible amount of residual heating, which may be attributed to
inherent noise in our laser.
This heating effect is another outcome of the incoherent carrier coupling in
the presence of fast phase noise and driving sideband transition. As seen in
the inset of Fig. 4, fast noise drives population incoherently on the carrier
transition which results in transfer of the ion to higher phonon states. Our
numerical simulation in this regime (solid line in Fig. 4) reveals the
interplay between coherent sideband driving and incoherent population of
vibrational modes. As a result, the contrast of coherent Rabi oscillations is
reduced along with a constant increase in the average number of phonons. When
this number reaches $\bar{n}=1$, the distribution becomes thermal, and we can
assign an effective temperature to the ion. Our analysis shows that further
driving of the ion would lead to a linear increase in temperature, which we
expect to saturate when the motional spread of the ion exceeds the limit of
the Lamb-Dicke regime such that carrier and sideband excitations are
suppressed.
Figure 3: Incoherent pumping due to carrier coupling. (a) Simulation (solid
line) and measurements (filled circles) of fraction of atoms in the excited
state as a function of off-resonant drive time for drives with different power
spectral densities. (b) Incoherent pumping rate as a function of the Rabi
frequency power spectral density at the detuning from the carrier transition.
Figure 4: Heating due to fast noise in driven sideband transitions. In the
main panel we plot the average number of occupied motional states $\bar{n}$
inferred from thermometry following an integer number of blue sideband driving
cycles. The red markers show measurements following evolution with added
synthesized noise while the blue markers show measurements with no added noise
(dashed line is a guide to the eye). The inset shows the relevant level scheme
where unwanted carrier transitions (orange dashed arrows) are induced due to
fast noise. Figure 5: Simulated Mølmer-Sørensen gate error under noisy drive.
We calculate and plot the gate infidelity as a function of the RPSD at the
trap frequency $\nu$ (filled circles). We find that infidelity depends
linearly in the RPSD at this specific frequency with a proportionality factor
close to unity. This is shown by the linear fit to the numerical results
(dashed line). The standard error is smaller than the marker size.
_Full numerical simulation of two qubit gates.–_ We now combine our
observation on both incoherent spin and motional dynamics to study the effect
of fast phase noise on the MS gate. We use a similar numerical noise PSD where
the noise overlaps with the trap frequency and a Hamiltonian with Rabi
frequency $\Omega=20$ kHz, $\nu=200$ kHz, and the Lamb-Dicke parameter
$\eta=0.15$. We solve the stochastic master equation for two qubits and their
common motional state modeled as a quantum harmonic oscillator with 30 modes
and find the gate fidelity as the overlap between the resulting density matrix
and the target Bell-state. This is averaged over 1000 realizations of the same
phase noise PSD. We repeat this for different noise amplitudes and plot the
gate fidelity vs. the RPSD at the carrier multiplies with the gate time in Fig
5. Once again, we find that this single parameter quantifies the performance
of this two-qubit gate. We fit this trend with a linear model and find that
the fidelity is proportional to the RPSD at the trap frequency with a
proportionality factor near unity $1-F\simeq T\cdot RPSD(f=\nu)$. The nearly
unity proportionality factor results from the fact that MS gates are long as
compared with the inverse spectral width of typical noise features.
Similarly to our findings in a single ion, the mechanism behind this
infidelity can be either due to incoherent spin pumping or heating of the ion.
Since the MS gate is insensitive to the ion’s temperature in the Lamb-Dicke
regime, the leading source of infidelity is the incoherent spin pumping due to
noise overlapping the carrier transition. This mechanism results in bit-flip
errors during the gate that can further propagate. However, repeated operation
of the MS gate as needed in a quantum circuit could eventually lead to
considerable heating and larger gate errors.
## V Discussion
In this paper, we study the effect of fast noise on the fidelity of quantum
operations. For a broad class of single and two qubit operations including
single qubit rotations with resonant transitions, off-resonant driving of
multi-level single qubits, and two-qubits entangling gates, we identify a
single parameter which quantifies the rate of errors or decoherence during the
drive. This parameter is the noise Rabi PSD at the relevant frequencies, that
is the Rabi frequency for resonant rotations and the detuning from the carrier
transition for off-resonant drives and the MS gate.
We find that the infidelity of single qubit rotations via resonant drive is
set by the spectral overlap of the Rabi frequency and the noise PSD. For off-
resonant drives or operations on sideband transitions, we identify two main
channels for errors. The first one is due to a incoherent spin-pumping on the
carrier transition and the second is coupling to higher excited motional
modes, which takes the system out of the Hilbert space of exchanging a single
motional quanta and results in effective heating. The latter becomes more
significant with more drive cycles as the motional excitation accumulates. We
show that the heating mechanism has a negligible contribution to errors in a
single operation of a two-qubit gate, whereas the incoherent spin pumping
plays a pivotal role.
Assuming a trapped-ions quantum register with a carrier Rabi frequency of
$\Omega=2\pi\times$100 kHz and a motional mode with a Lamb-Dicke parameter
$\eta=0.05$, a MS gate using this mode will last 100 $\mu$s. Our findings
indicate that in order to achieve an error below $10^{-4}$ we require the
phase noise overlapping the carrier, in terms of RPSD to be under $\thicksim
1[\frac{Hz^{2}}{Hz}]$. In terms of dBc this represents a requirement of -100
dBc/Hz on the RPSD in this frequency range. For weak noise the RPSD is
proportional to the phase PSD and we have the -100 dBc/Hz requirement on phase
PSD as well. This is not necessarily an easy goal to achieve. As an example,
using a narrow linewidth laser operating on an optical qubit, with 100 kHz
wide servo bump overlapping with the trap frequency, no more than $\simeq
10^{-5}$ of the laser intensity can be contained within this servo bump.
The fact that the gate error is well approximated by the RPSD at a single
frequency results from the fact that typically the gate time is longer than
the correlation time of the noise at these frequencies. In our case, the servo
bump is spectrally wider than the Fourier width of the gate. This is not
always true in trapped ion systems, and in particular we can find short single
qubit $\pi$-times that deviate from this assumption. In these cases a proper
overlap integral is necessary to estimate the gate error [30].
Our analysis indicates that for full quantum control of operations at the high
fidelity frontier it is important to characterise and study such fast noise
mechanisms in any system. We note that our analysis is valid for any noise at
the relevant spectral window which may overlap characteristic Hamiltonian
energy scale and that the intricate dynamics coupling spin and motion under a
noisy drive in the MS gate are relevant to any system where a bosonic mode
mediates interaction between two spin-like qubits, such as gates between
superconducting qubits mediated by a resonator. Our findings, analysis and
detailed numerical calculations may thus guide the design and further
improvement of quantum hardware and tailored quantum gates.
## VI acknowledgements
This work was supported by the Israeli Science Foundation and the Goldring
Family Foundation. We thank Yotam Shapira and Tom Manovitz for fruitful
discussions.
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|
11institutetext: University of Twente, Netherlands 22institutetext: Philipps-
University Marburg, Germany 33institutetext: Hospital Group Twente (ZGT), The
Netherlands 44institutetext: University of Mannheim, Germany
44email<EMAIL_ADDRESS>44email:
<EMAIL_ADDRESS>44email:
<EMAIL_ADDRESS>
# Prototype-based Interpretable Breast Cancer Prediction Models: Analysis and
Challenges
Shreyasi Pathak 11 0000-0002-6984-8208 Jörg Schlötterer 2244
0000-0002-3678-0390 Jeroen Veltman 33 0000-0002-6824-3987 Jeroen Geerdink 33
0000-0001-6718-6653 Maurice van Keulen 11 0000-0003-2436-1372 Christin Seifert
22 0000-0002-6776-3868
###### Abstract
Deep learning models have achieved high performance in medical applications,
however, their adoption in clinical practice is hindered due to their black-
box nature. Using explainable AI (XAI) in high-stake medical decisions could
increase their usability in clinical settings. Self-explainable models, like
prototype-based models, can be especially beneficial as they are interpretable
by design. However, if the learnt prototypes are of low quality then the
prototype-based models are as good as black-box. Having high quality
prototypes is a pre-requisite for a truly interpretable model. In this work,
we propose a prototype evaluation framework for coherence (PEF-C) for
quantitatively evaluating the quality of the prototypes based on domain
knowledge. We show the use of PEF-C in the context of breast cancer prediction
using mammography. Existing works on prototype-based models on breast cancer
prediction using mammography have focused on improving the classification
performance of prototype-based models compared to black-box models and have
evaluated prototype quality through anecdotal evidence. We are the first to go
beyond anecdotal evidence and evaluate the quality of the mammography
prototypes systematically using our PEF-C. Specifically, we apply three state-
of-the-art prototype-based models, ProtoPNet, BRAIxProtoPNet++ and PIP-Net on
mammography images for breast cancer prediction and evaluate these models
w.r.t. i) classification performance, and ii) quality of the prototypes, on
three public datasets. Our results show that prototype-based models are
competitive with black-box models in terms of classification performance, and
achieve a higher score in detecting ROIs. However, the quality of the
prototypes are not yet sufficient and can be improved in aspects of relevance,
purity and learning a variety of prototypes. We call the XAI community to
systematically evaluate the quality of the prototypes to check their true
usability in high stake decisions and improve such models further.
###### Keywords:
Explainable AI Prototype-based models Breast cancer prediction Mammography.
## 1 Introduction
Deep learning has achieved high performance on various medical applications,
e.g. breast cancer prediction [17, 16]. However, its adoption to clinical
practice is hindered due to its black-box nature. Explainable AI (XAI) aims to
address this gap by explaining the reasoning of black-box models in a post-hoc
manner (post-hoc explainable methods) or developing models which are
interpretable by design (self-explainable methods).
Post-hoc explainable methods explain a trained deep neural network with a
separate explainer model, e.g. Grad-CAM can highlight the important image
regions to explain the prediction from a neural network [11]. However, these
explanations are not always reliable or trustworthy [12], as they come from a
second explainer model. Self-explainable models were developed to address this
issue – one of the most prominent ones being prototype-based models, e.g.
ProtoPNet [2].
Breast cancer prediction using mammography is a challenging medical task as
the regions-of-interest (ROIs) are small compared to the whole mammogram and
presence of dense breast tissue may make it difficult to read the mammogram.
Depending on the finding it can be difficult for a radiologist to decide
whether to perform a biopsy, wait for follow up or diagnose the lesion as
certainly benign. Learning relevant malignant and benign prototypes from the
training data and showing the reasoning behind a model’s prediction might
assist radiologists in taking data based decision. Existing works [21, 20]
have extended prototype-based model, ProtoPNet [2] on breast cancer prediction
using whole mammography images. [1] extended ProtoPNet for breast cancer
prediction on ROIs from mammography images. However, most works show anecdotal
evidence for the explanations. The quality of the learnt prototypes from
mammography images have not been extensively evaluated yet.
A recent survey [8] found that most XAI methods are usually evaluated based on
anecdotal evidence by selecting a few good examples. However, anectodal
evidence is not sufficient to judge the quality of the explanations. Nauta et
al. [8] proposed a framework of twelve properties for evaluating the quality
of explanations. One of the properties, Coherence, evaluates whether the
explanations are correct with respect to domain knowledge. In the medical
domain, it is highly important to evaluate the explanation quality in terms of
Coherence, before the model can be adopted in a clinical setting.
In this work, we go beyond anecdotal evidence and propose a prototype
evaluation framework for coherence (PEF-C) to quantitatively evaluate the
quality of prototypes and to provide a broader evaluation of prototype-based
models based on the domain knowledge. Specifically, we apply state-of-the-art
prototype based models on breast cancer prediction using mammography images
and evaluate the quality of the prototypes both quantitatively and
qualitatively. Our contributions are as follows:
1. 1.
We propose PEF-C, a Prototype Evaluation Framework for Coherence, for
evaluating the quality of prototypes in prototype-based models. Our framework
can be used as a comprehensible and generalizable evaluation framework for
prototype-based models of medical applications.
2. 2.
We reproduced a state-of-the-art breast cancer prediction model,
BRAIxProtoPNet++, which had no public source code. We release the source code
of the evaluation framework, the implementations of all black-
box111https://github.com/ShreyasiPathak/multiinstance-learning-mammography and
interpretable models222https://github.com/ShreyasiPathak/prototype-based-
breast-cancer-prediction for further research.
3. 3.
We systematically compare prototype-based models and blackbox models on three
standard benchmark datasets for breast cancer detection, and show that
interpretable models achieve comparative performance. Our analysis of
prototype quality identifies differences between prototype-based models and
points towards promising research directions to further improve model’s
coherence.
## 2 Related Work
Prototype-based models. Prototype-based models were first introduced with
ProtoPNet [2], an intrinsically interpretable model with similar performance
to black-box models. ProtoTree [9] reduces the number of prototypes in
ProtoPNet and uses a decision tree structure instead of a linear combination
of prototypes as decision layer. ProtoPShare [14] and ProtoPool [13] also
optimize towards reducing the number of prototypes used in classification.
TesNet [22] disentangles the latent space of ProtoPNet by learning prototypes
on a Grassman manifold. PIP-Net [6] addresses the semantic gap between latent
space and pixel space for the prototypes, adds sparsity to the classification
layer and can handle out-of-distribution data. XProtoNet [4] is a prototype-
based model developed for chest radiography and achieved state-of-the-art
classification performance on a chest X-ray dataset.
Interpretability in Breast Cancer Prediction. [23] developed a expert-in-a-
loop interpretation method for interpreting and labelling the internal
representation of the convolutional neural network for mammography
classification. BRAIxProtoPNet++ [21] extended ProtoPNet for breast cancer
prediction achieving a higher performance compared to ProtoPNet and
outperforming black-box models in classification performance. InterNRL [20]
extended on BRAIxProtoPNet++ using a student-teacher reciprocal learning
approach. IAIA-BL [1] extended ProtoPNet with some supervised training on
fine-grained annotations and applied on ROIs instead of whole mammogram
images.
Evaluating XAI. Most XAI methods are usually evaluated using anecdotal
evidence [8]. However, the field is moving towards standardizing evaluation of
explanations [8, 7] similar to the standardized approach for evaluating
classification performance. In this work, we develop a prototype evaluation
framework for coherence (PEF-C) and use it in the context of breast cancer
prediction. We use ProtoPNet, BRAIxProtoPNet++ and PIP-Net on breast cancer
prediction using mammography images and evaluate on 3 public datasets - CBIS-
DDSM [15], VinDr [10] and CMMD [3] for classification performance. We further
use PEF-C to evaluate the learnt prototypes of the 3 models on CBIS-DDSM
dataset.
## 3 Preliminaries
In this section, we provide a summary of the state-of-the-art models used in
this paper to make this paper self-contained.
### Problem Statement
We pose breast cancer prediction using mammography as a classification
problem. We define it as a binary classification problem for the datasets
CBIS-DDSM and CMMD containing 2 classes, benign and malignant as labels and a
multiclass classification problem for the dataset, VinDr, containing the 5 BI-
RADS [18] categories as labels (cf. Sec. 5.1). Suppose,
$\\{(x_{1},y_{1}),$…$,(x_{n},y_{n}),$…$,(x_{N},y_{N})\\}\in\mathcal{X}\times\mathcal{Y}$
is a set of N training images with each image having a class label
$y_{n}\in\\{0,1\\}$ ($0$: benign, $1$: malignant) for binary classification
and a class label $y_{n}\in\\{0,1,2,3,4\\}$ ($0$: BI-RADS 1, $1$: BI-RADS 2,
$2$: BI-RADS 3, $3$: BI-RADS 4, $4$: BI-RADS 5) for multiclass classification.
Our objective is to learn the prediction function $f$ in
$\hat{y}_{n}=f(x_{n})$ where $\hat{y}_{n}$ is the predicted label.
### Black-box Models
We used pretrained ConvNext [5] and EfficientNet [19] models and changed the
classification layer to 2 classes and 5 classes with a softmax activation
function. GMIC [17] is a state-of-the-art breast cancer prediction model
containing a global module with feature extractor for learning the features
from a whole image, a local module with feature extractor for learning the
features from the patches and a fusion module to concatenate the global and
local features. The global module generates a saliency map, which is used to
retrieve the top-k ROI candidates. Local module takes these ROI candidates
(patches) as input and aggregates the features using a multi-instance learning
method. The global, local and fusion features are separately used for
prediction and passed to the loss function for training GMIC. During
inference, only fusion features are used for prediction. We used their model
for binary classification with 1 neuron in the classification layer with a
sigmoid activation function. For multiclass classification, we used their
setup of multilabel classification and replaced sigmoid with a softmax
activation function.
### Prototye-based Models
ProtoPNet [2] consists of a convolutional feature extractor backbone, a layer
with prototype vectors and a classification layer. The feature extractor
backbone generates a feature map of $H\times W\times D$ where H, W, and D are
height, width and depth of the feature map. A prototype vector of size
$1\times 1\times D$ is passed over $H\times W$ spatial locations to find the
patch in the feature map that is most similar to the prototype vector. The
prototype layer contains equal number of prototype vectors for each class. In
the classification layer, the prototype vectors are connected to their own
class with a weight of 1 and to other classes with a weight of -0.5. After the
most similar patch for each prototype is saved, the patch is visualized in the
pixel space using upsampling.
BRAIxProtoPNet++ [21] extends on ProtoPNet to improve the classification
performance by including a global network and distilling the knowledge from
the global network to the ProtoPNet. It also increases the prototype diversity
by taking all prototypes from different training images.
PIP-Net [6] does not use a separate prototypical layer like ProtoPNet and
BRAIxProtoPNet++, but uses regularization to make the features in the last
layer of the convolutional backbone interpretable. For a feature map of
dimension $H\times W\times D$, the model can have a maximum of $D$ prototypes.
Each patch of dimension $1\times 1\times D$ in the feature map is softmaxed
over $D$ such that one patch tries to match to maximally one prototype. The
maximum similarity score is taken from each of the $D$ channels, which is then
used in the classification layer. Further, the classification layer is
optimized for sparsity to keep only the positive connections towards a
decision.
Table 1: Overview of prototype evaluation measures. Global prototypes (GP) is the number of all prototypes, relevant prototypes (RP) contain ROIs. Categories refer to fine-grained part annotations from a lexicon (abnormality type and BIRADS descriptors in our application case). Total number of categories (TC) are all possible categories from the lexicon, unique categories (UC) is the size of the set of categories found by the model. Level refers to whether the measure is calculated at the local (L), i.e., at the image-level or global (G) model level. Property | Question | Level | Measure
---|---|---|---
Compactness | What is the explanation size? | G | $GP$
| | L | $LP$
Relevance | How many of the learnt prototypes are relevant for diagnosis? | G | $|RP|/GP$
Specialization | Are the prototypes meaningful enough to be assigned names of abnormality categories? | G | Purity${}_{RP}/|RP|$
Uniqueness | Does the model activate more than one prototype for one abnormality category? | G | $UC_{RP}/|RP|$
Coverage | How many abnormality categories were the model able to learn? | G | $UC_{RP}/TC$
Localization | To what extent is the model able to find correct ROIs? | L | IoU, DSC
Class-specific | Is the model associating the prototypes with the correct class? | G | $\frac{Align(W_{RP},CD_{C_{RP}})}{|RP|}$
## 4 Prototype Evaluation Framework
We developed a Prototype Evaluation Framework for Coherence (PEF-C) to i)
quantitatively evaluate the quality of the learnt prototypes and ii) provide a
broader assessment of the prototype-based models, based on domain knowledge.
We use our PEF-C to assess the status quo of the prototype-based models on
breast cancer prediction using mammography. We evaluate the following 7
properties, summarized in Table 1:
1. 1.
Compactness measures the size of the local and global explanation of the
model. The lower the explanation size, the easier it is to comprehend for
users. Note that while this property is associated with the presentation
aspect of the explanations and not with coherence [8], we decided to include
it here as the measurement of further properties is based on the explanation
size. We calculate this as follows:
Global prototypes (GP): Total number of prototypes with non-zero weights to
the classification layer.
Local prototypes (LP): Average number of prototypes that are active per
instance, i.e., whose contribution to the classification decision (prototype
presence score $p$ times weight $w$ to the classification layer) is non-zero.
2. 2.
Relevance measures the number of prototypes associated with the ROIs among the
whole set of prototypes. The larger the number of relevant prototypes, the
better the capability of the model in finding the relevant regions. We measure
this property by calculating the ratio between the number of prototypes
activated on at least one ROI (relevant prototypes (RP)) and the number of
global prototypes:
$Rel.=\frac{|RP|}{GP}$
Specifically, to determine $RP$, we take the top-k instances with highest
prototype presence scores for each prototype (i.e., images, where the
prototype is most activated). If the corresponding patch for the prototype in
any of these top-k instances overlaps with the center of a ROI in this image,
we consider it relevant. We set $k=10$ to focus on the most representative
instances, but still allow a bit of variation (outliers).
3. 3.
Specialization measures the purity of prototypes by matching the extent of its
alignment to lexical categories. A truly interpretable prototype represents a
single semantic concept that is comprehensible to humans, i.e., can be named
or given a meaning by users. For example in our case, a prototype is not fully
interpretable if it gets activated both on mass or calcification, as we don’t
know whether to call this prototype a mass prototype or a calcification
prototype. The higher the purity, the more interpretable the prototypes.
$Spec.=\frac{\sum_{p\in RP}Purity_{p}}{|RP|}$
where $RP$ is the set of relevant prototypes as defined above and $Purity_{p}$
is the share of the majority category in top-k patches (relevant prototypes
overlap with an ROI center and each ROI is labelled with a category). If all
top-k patches belong to the same category, the prototype is pure. Again,
$k=10$ from before. For a detailed evaluation by category, we assign the
majority category and report purity scores per category. If the category
lexicon constitutes a hierarchy like in our case, we can determine purity also
on more fine-grained category levels. In our application case, the top-level
categories are abnormality types (mass and calcification) and sub-categories
constitute fine-grained BIRADS descriptors, e.g., shape or margin of a mass
and morphology or distribution of a calcification.
4. 4.
Uniqueness measures the number of unique categories of prototypes among all
relevant prototypes.
$Uniq.=\frac{UC_{RP}}{|RP|}$
where $UC_{RP}$ stands for unique categories (UC) of all relevant prototypes.
Each prototype has a single (majority) category assigned from the purity
calculation of the Specialization measure before. Ideally, we want each
prototype to represent a different, unique category and not multiple
prototypes that represent the same. The uniqueness measure should always be
evaluated in combination with relevance and coverage (next measure), as
trivially, a single relevant prototype would yield the maximum uniqueness
score of 1.
5. 5.
Coverage measures the fraction of all categories in the dataset that are
represented by prototypes.
$Cov.=\frac{UC_{RP}}{TC}$
where $TC$ is the total number of categories.
Ideally, a model has at least one prototype associated with each category,
such that it is able to incorporate all variations in its reasoning process.
6. 6.
Localization measures the capability of the model in finding the correct ROIs
through the activated prototypes.
$Loc.=\frac{\sum_{t=1}^{T}IoU(ROI_{t},Patch_{t})}{T}$
where IoU is the intersection over union (IoU) over groundtruth ROI and the
patch activated by the prototype, $t$ stands for the $t^{th}$ test instance of
the total $T$ test instances. IoU can also be replaced with dice similarity
coefficient (DSC). We calculate and report both in our experiments.
This property is a local measure, meaning that it’s calculated per test
instance and averaged over the whole test set. The higher the localization,
the better the model is at finding the correct ROIs.
7. 7.
Class-specific measures the extent to which prototypes are associated with the
correct class. For example, a prototype that has an assigned category (cf.
Purity calculation in Specialization measure) that is more often found to be
benign in our dataset, should have a higher weight to the benign class in the
classification layer. A high class-specific score means the model is good at
associating categories with the correct class. We measure this property by the
following:
$Cl\text{-}spec.=\frac{\sum_{p\in RP}Align(W_{p},CD_{C_{p}})}{|RP|}$
where $Align$ stands for the alignment between the weights of a relevant
prototype towards the classes in the classification layer, $W_{RP}$, and the
class distribution (CD) of the category (C) assigned to this prototype,
$CD_{C_{RP}}$. $Align$ is a binary assessment: If a prototype has the highest
weight to the majority class of its category, the score is 1 and 0 otherwise.
The majority class is obtained from ROI category annotations in the dataset
(category class distribution $CD_{C_{RP}}$).
## 5 Experimental Setup
In this section, we describe the datasets, training details, setup of our
prototype evaluation framework and the visualization method of the prototypes.
### 5.1 Datasets
We performed our experiments on three commonly used mammography datasets:
CBIS-DDSM [15] is a public dataset containing 3,103 mammograms from 1,566
patients including mediolateral oblique (MLO) and craniocaudal (CC) views. The
dataset contains 1,375 malignant and 1,728 benign images. For our experiments,
we used the official train (2458 images) and test split (645 images) provided
by the dataset.
CMMD [3] is a Chinese public dataset collected between years 2012 and 2016,
containing 5,202 mammograms of CC and MLO views from 1,775 patients. It
contains 2,531 malignant and 2,570 benign images. We split the dataset in a
patient-wise manner into training set (3,199 images, D1-0001 to D2-0247) and
test set (2,002 images, D2-0248 to D2-0749) following [20] for fair comparison
to their results.
VinDr-Mammo [10] is a public dataset from two hospitals in Vietnam, containing
5,000 mammography cases (20,000 mammogram images) including CC and MLO views.
The images are annotated with BIRADS category 1 to 5 with 1 being normal
mammogram and 5 being most malignant. We used the official training (16,000
images) and test (4,000 images) split of the dataset for our experiments.
### 5.2 Training Details
We split the training set patient-wise into 90% train and 10% validation for
all datasets. We performed hyperparameter tuning on CMMD (and when needed also
on CBIS-DDSM) dataset and the hyperparameters with the highest validation AUC
were selected as the hyperparameter for our experiments on all datasets. The
models were trained till a certain epoch (details in the following paragraph)
and he model of the epoch with the highest validation AUC was selected for
test set inference.
EfficientNet-B0 [19] was trained with a fixed learning rate (lr) of 0.001 and
weight decay (wd) of 1e-5 for 60 epochs. ConvNext [5] was trained with a fixed
lr=1e-5 and wd=1e-5 for 50 epochs. GMIC [17] was trained with a fixed learning
rate $\approx$ 6e-5 and weight decay $\approx$ 3e-4 (exact values can be found
in our repository1) for 50 epochs. ProtoPNet [2] was trained in 3 stages - i)
warm optimizer stage where only the add on layer and the prototype vectors are
trained with lr=1e-4 for 5 epochs, ii) fine tuning stage where the whole
network except the last layer is trained for 55 epochs with lr=1e-5 for the
backbone, lr=1e-4 for the add on layers and the prototype vector, iii) the
last layer is trained every 11th epoch for 20 iterations with lr=1e-2.
BRAIxProtoPNet++ [21] was trained in 3 stages - i) GlobalNet training where
the GlobalNet and its classification layer was trained with lr=1e-5 for 3
epochs, ii) ProtoPNet training where the add on layer, the prototype vector
and the ProtoPNet classification layer was trained with lr=1e-5 for 3 epochs,
and iii) whole model fine-tuning stage, where all the layers were trained with
lr=1e-5 for the backbone, lr=1e-4 for the add on layer and the prototype
vector and lr=1e-2 for the classification layers for 54 epochs. This schedule
of BRAIxProtoPNet++ was followed for CBIS-DDSM dataset, however, for CMMD and
VinDr, the (iii) stage used a fixed lr=1e-5 for all layers. In PIP-Net [6],
the feature extractor backbone was trained with a fixed lr=0.00001 and the
classifier layer was trained with a lr=0.05 in a cosine annealing with warm
restart schedule. PIP-Net was pretrained in a self-supervised manner for 10
epochs, followed by full finetuning for 60 epochs following [6].
All images were resized to 1536x768 following [21, 20]. Data augmentation of
random horizontal flipping, shearing, translation and rotation was applied
(details can be found in our repository).
We report the average performance and standard deviation over three random
weight initializations with different seeds.
### 5.3 Prototype Evaluation Framework Setup
We calculated the seven properties of PEF-C for the three prototype-based
models, ProtoPNet, BRAIxProtoPNet++ and PIP-Net. All analysis were perfomed on
CBIS-DDSM, because it is the only dataset with fine-grained annotations of
abnormality types – mass and calcification, mass BIRADS descriptors - shape
and margin, and calcification BIRADS descriptors - morphology and
distribution. We took the top-$k$, where $k=10$ similar patches from the
training set for each prototype to calculate the relevance, specialized,
uniqueness and coverage.
For purity calculation in specialization, we define categories following the
hierarchical level of BIRADS lexicon – at the top level, abnormality type is
the category, i.e. mass or calcification. At a sub-level, mass abnormalities
(lesions) can be described with or categorized by shape and margin
descriptors. At the same level, calcification abnormalities can be described
with or categorized by a morphology and distribution. For uniqueness, we
define categories by combining the descriptors at all levels. A category for
mass is defined with a combination of abnormality type - shape - margin and
for calcification, with a combination of abnormality type - morphology -
distribution. For example, an abnormality category of a mass can be mass-oval-
circumscribed, i.e. a mass with oval shape and circumscribed margin, and an
abnormality category for a calcification can be calcification-pleomorphic-
segmental, i.e. a calcification with pleomorphic morphology and segmental
distribution. Following this rule, we found that the total abnormality
categories in the CBIS-DDSM is 132 (70 mass categories, and 62 calcification
categories) and we used this value for calculation of coverage property.
For calculating the class-specific measure, we selected the abnormality
categories that have instances for both malignant and benign classes in the
dataset.
Localization: We calculate the intersection over union (IoU) and Dice
Similarity Coefficient (DSC) for the top-1 (IoU1, DSC1), top-10 (IoU10, DSC10)
and all (IoUAll, DSCAll) activated prototypes with the annotated ROI. For
selecting the top-1 and top-10 prototypes, we score the prototypes using the
absolute magnitude of the similarity score $\times$ weight of the prototype
with the groundtruth class.
Table 2: Performance comparison of SOTA breast cancer prediction models. Showing mean and standard deviation; values normalized to percentages. | CBIS | VinDr | CMMD
---|---|---|---
| F1 | AUC | F1wM | AUCM | F1 | AUC
Black box modles
EfficientNet [19] | $58\pm 6$ | $76\pm 1$ | $65\pm 4$ | $77\pm 1$ | $73\pm 3$ | $83\pm 1$
ConvNext [5] | $67\pm 3$ | $\textbf{80}\pm 2$ | $\textbf{72}\pm 1$ | $\textbf{83}\pm 0$ | $\textbf{74}\pm 2$ | $84\pm 1$
GMIC [17] | $\textbf{68}\pm 3$ | $\textbf{80}\pm 2$ | $64\pm 3$ | $80\pm 1$ | $72\pm 5$ | $83\pm 1$
Part-prototpe models with ConvNext backbone
ProtoPNet [2] | $65\pm 1$ | $77\pm 1$ | $58\pm 13$ | $79\pm 1$ | $70\pm 0$ | $83\pm 1$
BRAIxProtoPNet++ [21] | $64\pm 4$ | $79\pm 1$ | $63\pm 11$ | $81\pm 0$ | $72\pm 3$ | $\textbf{86}\pm 2$
PIP-Net [6] | $63\pm 3$ | $75\pm 2$ | $63\pm 3$ | $78\pm 1$ | $70\pm 1$ | $81\pm 1$
### 5.4 Visualization of prototypes
We visualize the local and global explanations of ProtoPNet, BRAIxProtoPNet++
and PIP-Net on CBIS-DDSM and CMMD dataset for qualitative evaluation. We
randomly select a test instance for the local explanation and show the top-3
activated prototypes. For global explanation, we show the top-10 image patches
for each of eight reasonable looking prototypes. ProtoPNet and PIP-Net have
different approaches for visualizing the prototypes. ProtoPNet does not have a
fixed patch size. It takes the $x$ percentile from the upsampled similarity
map to visualize the patch corresponding to the prototype. PIP-Net has a fixed
patch size for prototype visualization. It takes the highest similarity score
from the feature map, upsamples it to the patch size in the image space and
uses that patch to visualize the prototype. For local visualization, we used
the visualization method specific to each model, where we set percentile as
99.9 for ProtoPNet and BraixProtoPNet++ to select the relevant region and for
PIP-Net, we set the patch size as $130\times 130$. However, we used a fixed
patch size of $130\times 130$ for visualization of global explanation for all
models. We also used the same fixed patch size for calculation of purity
across all models for a fair comparison of the scores.
## 6 Results and Discussion
In this section, we report the classification performance of black-box vs
prototype-based models, the qualitative evaluation of the prototypes through
local and global explanation and the quantitative evaluation of the prototypes
using our prototype evaluation framework for coherence.
### 6.1 Performance Comparison of Black-box vs Prototype-based Models
We compared the classification performance of three intrinsically
interpretable prototype-based models (ProtoPNet [2], BRAIxProtoPNet++ [21] and
PIP-Net [6]), and three black-box model (EfficientNet [19], ConvNext [5] and
GMIC [17]) (cf. Table 2). ConvNext achieves the highest performance (F1 score)
for VinDr (F1: 0.72) and CMMD (F1: 0.74) dataset and GMIC achieves the highest
performance (F1: 0.68) for CBIS, with ConvNext having competitive performance
(F1: 0.67). However, prototype-based models have competitive performance with
black-box models, with BRAIxProtoPNet++ achieving highest AUC (0.86) on CMMD.
BRAIxProtoPNet++ outperforms ProtoPNet and PIP-Net in terms of F1 score on all
datasets. Note that though PIP-Net has a lower AUC score compared to ProtoPNet
and BRAIxProtoPNet++, the F1 score is competitive with the other models even
with a much lower number of activated prototypes. BRAIxProtoPNet++ did not
have a public code. We implemented their model and reach a nearby performance
to the ones reported in [20] on CMMD (BRAIxProtoPNet++
AUC${}^{\text{Reported}}$ 88% vs AUCOurs 86%; ProtoPNet
AUC${}^{\text{Reported}}$ 85% vs AUCOurs 84%).
### 6.2 Local and Global Visualization of Prototypes
For global explanation, we visualized the top-10 image patches (based on the
highest similarity score) activated by each prototype for the 3 prototype-
based models for CBIS-DDSM (cf. Fig. 1) and CMMD dataset (cf. Fig 2). We can
see prototypes representing the mass abnormality in the visualization from
CBIS-DDSM. In ProtoPNet global explanation (cf. Fig 1(a)), the prototype in
the second row contains some patches with mass ROI of irregular shape and
spiculated margin and the fourth row contains some mass ROIs of oval shape and
ill-defined margin, among other different looking patches. This shows that all
patches activated by a prototype in ProtoPNet may not be visually similar.
Further, separate prototypes may learn the same abnormality category as shown
in Fig 1(b), where prototypes in 5th, 6th and 7th row all contain mass ROIs of
irregular shape and spiculated margin. Further, the same prototype can get
activated for different abnormality categories. For example, in Fig. 1(c),
prototype in 4th row contains most ROIs of category mass with irregular shape
and spiculated margin. However, that prototype also contains ROIs of category
calcification with pleomorphic and amorphous morphology and segmental
distribution. Further, there are prototypes representing normal tissue and
black background in all prototype-based models. This is expected, however,
such prototypes should have lower weights to the classes as they usually do
not have much contribution to benign and malignant prediction. In general, it
can be observed that i) duplicate prototypes might be learnt by the models
(quite easily visually observed in ProtoPNet (cf. Fig 2(a))), ii) all top-10
patches activated by one prototype may not look visually the same or may not
belong to the same abnormality category, iii) not all learnt prototypes belong
to the ROIs, but can belong to edges and background with reasonable weights in
the classification layer.
(a) ProtoPNet
(b) BRAIxProtoPNet++
(c) PIP-Net
Figure 1: Global visualization of 3 prototype-based models trained on CBIS-
DDSM dataset. Each row represents one prototype visualized with the top-10
activated image patches.
(a) ProtoPNet
(b) BRAIxProtoPNet++
(c) PIP-Net
Figure 2: Global visualization of 3 prototype-based models trained on CMMD
dataset. Each row represents one prototype visualized with the top-10
activated image patches.
We visualized local explanation for one test instance from CMMD dataset for
ProtoPNet (cf. Fig. 3), BRAIxProtoPNet++ (cf. Fig. 4) and PIP-Net (cf. Fig. 5)
as an anecdotal evidence. We show the top-3 activated prototypes for each
model. For ProtoPNet (cf. Fig. 3), we observed that i) one of the activated
prototypes contained the correct ROI, ii) however, the top activated
prototypes also represented some irrelevant black region and edges, iii) the
test image patch does not always look similar to the prototype from the
training set, and iv) similar looking prototype may get activated for both
benign and malignant classes. For BRAIxProtoPNet++ (cf. Fig. 4), we observed
that i) the region surrounding the actual ROI gets activated rather than the
actual ROI, ii) most of the activated prototypes point to the same region (we
have only shown for 3, but we found other prototypes to also point to the same
region as the top 3), and iii) the activated prototypes of the benign class
(not predicted class) represent the background. For PIP-Net, we observed that
i) the top activated prototypes contain the ROIs, ii) the test image patch has
some similarity to the prototype from the training set, and iii) the benign
prototypes don’t belong to the ROI, but to the other regions of the breast.
Overall, the local explanation of PIP-Net for this test image looked more
reasonable compared to the other models.
(a) Prototypes activated for malignant class
(b) Prototypes activated for benign class
Figure 3: Local explanation from ProtoPNet showing the top-3 activated
prototypes for the malignant class and the benign class. Example image: CMMD,
malignant test case D2-0249, view RCC, predicted class malignant
(a) Prototypes activated for malignant class
(b) Prototypes activated for benign class
Figure 4: Local explanation from BRAIxProtoPNet++ showing the top-3 activated
prototypes for the malignant class and the benign class. Example image: CMMD,
malignant test case D2-0249, view RCC, predicted class malignant
(a) Prototypes activated for malignant class
(b) Prototypes activated for benign class
Figure 5: Local explanation from PIP-Net showing the top-3 activated
prototypes for the malignant class and the benign class. Example image: CMMD,
malignant test case D2-0249, view RCC, predicted class malignant
### 6.3 Automatic Quantitative Evaluation of Prototypes
We report the results of our prototype evaluation framework in Table 3 for
ProtoPNet, BRAIxProtoPNet++ and PIP-Net on CBIS-DDSM dataset. We observe that
PIP-Net has much lower number of global and local prototypes increasing the
sparsity of the model (compactness). ProtoPNet has higher relevance score
compared to the others suggesting more prototypes get activated on ROIs,
however, the standard deviation is quite high reducing the reliability of the
score. Specialization scores show that ProtoPNet has the highest purity at the
first granularity level of abnormality type, however, PIP-Net has the highest
purity in the second granularity level. This shows that though ProtoPNet is
better at learning prototypes representing only mass or calcification, PIP-Net
is better at learning prototypes representing a particular type of mass or
calcification, which is in line with the ‘semantic correspondence’
characteristic of the model. Further, PIP-Net has the highest uniqueness score
suggesting that most relevant prototypes belong to a unique category. However,
the high uniqueness score is also an effect of the low global prototypes. For
example, in one run of PIP-Net, 16 out of 48 global prototypes were relevant
and of these 16, 10 were unique, resulting in a high uniqueness score of
0.625. Whereas in one run of BRAIxProtoPNet++, 135 out of 400 global
prototypes were relevant, of which 33 were unique resulting in a lower
uniqueness score of 0.24. Therefore, it is better to look at the uniqueness
measure together with the coverage score, which shows BRAIxProtoPNet++ to have
a higher coverage than PIP-Net due to higher unique categories. We also
measured class-specific score, i.e. how much do the class weights of the
relevant prototypes (associated with a category) align with the class
distribution of that category from the dataset. PIP-Net has the highest class-
specific score suggesting that more relevant prototypes (representing the
ROIs) are associated the correct class compared to the other models. However,
PIP-Net also has a low number of relevant prototypes, compared to the others,
which can result in a higher class-specific score. Lastly, the localization
score shows that PIP-Net is better in finding the ROI with the top-1 activated
prototype. At top-10 activated prototypes, ProtoPNet has similar IoU to PIP-
Net and with all activated prototypes, BRAIxProtoPNet++ has the highest IoU
due to its high number of global prototypes (400) compared to PIP-Net (90).
The IoU6 score of the black-box model, GMIC, is similar to the IoU1 of the
prototype based models and with only 10 activated prototypes, the prototype
based models outperform GMIC is finding the correct ROI. This shows that
prototype-based models can be a good candidate for unsupervised ROI
extraction.
Table 3: Comparison of prototype-based models using our prototype evaluation
framework on CBIS-DDSM dataset. Mean and standard deviation reported across
model runs with different seeds.
Property | ProtoPNet | BRAIxProtoPNet++ | PIP-Net
---|---|---|---
Compactness | | |
Global | $400\pm 0.0$ | $400\pm 0.0$ | $90\pm 83$
Local | | |
Positive | $314\pm 20$ | $200\pm 0$ | $27\pm 31$
Negative | $84\pm 20$ | $200\pm 0$ | -
Sparsity | 0% | 0% | 70%
Relevance | $\textbf{0.36}\pm 0.23$ | $0.30\pm 0.07$ | $0.28\pm 0.05$
Specialization | | |
Abnorm. Type | $\textbf{0.37}\pm 0.02$ | $0.22\pm 0.09$ | $0.31\pm 0.13$
Mass Shape | $0.29\pm 0.04$ | $0.29\pm 0.07$ | $\textbf{0.35}\pm 0.08$
Mass Margin | $0.34\pm 0.03$ | $0.33\pm 0.10$ | $\textbf{0.38}\pm 0.04$
Calc. Morph. | $0.30\pm 0.22$ | $0.24\pm 0.06$ | $\textbf{0.32}\pm 0.09$
Calc. Distribution | $0.24\pm 0.11$ | $0.24\pm 0.03$ | $\textbf{0.27}\pm 0.09$
Uniqueness | $0.13\pm 0.07$ | $0.31\pm 0.08$ | $\textbf{0.53}\pm 0.12$
Coverage | $0.12\pm 0.05$ | $\textbf{0.25}\pm 0.02$ | $0.10\pm 0.08$
Class-specific | $0.52\pm 0.05$ | $0.61\pm 0.03$ | $\textbf{0.65}\pm 0.08$
Localization | | |
IoU1 | $0.04\pm 0.01$ | $0.04\pm 0.02$ | $\textbf{0.07}\pm 0.00$
IoU10 | $\textbf{0.13}\pm 0.02$ | $0.10\pm 0.01$ | $\textbf{0.13}\pm 0.01$
IoUAll | $0.23\pm 0.03$ | $\textbf{0.28}\pm 0.01$ | $0.18\pm 0.06$
DSC1 | $0.06\pm 0.02$ | $0.06\pm 0.03$ | $\textbf{0.10}\pm 0.0$
DSC10 | $\textbf{0.20}\pm 0.03$ | $0.15\pm 0.02$ | $0.19\pm 0.01$
DSCAll | $0.34\pm 0.04$ | $\textbf{0.39}\pm 0.02$ | $0.26\pm 0.09$
* Note: The black-box model, GMIC has an IoU of $0.05\pm 0.0$ and DSC of $0.09\pm 0.0$ on CBIS-DDSM over 6 ROI candidates that are extracted from the GMIC model.
## 7 Conclusion and Future Work
We extensively compare state-of-the-art black-box models to state-of-the-art
prototype-based models in the context of breast cancer prediction using
mammography. We found that though black-box model, ConvNext has higher
classification performance compared to prototype-based models, prototype-based
models are competitive in performance. To go beyond anecdotal evidence in
evaluating the quality of the prototypes, we propose a prototype evaluation
framework for coherence (PEF-C) to quantitatively evaluate the prototype
quality based on domain knowledge. Our framework needs regions-of-interest
(ROIs) annotation and some fine-grained labels for automatic evaluation of
prototype quality. We used such annotations from a public dataset, CBIS-DDSM
and evaluate our framework on 3 prototype-based models. We found that around
30% prototypes were relevant, prototypes were around 37% pure at the first
granularity level of mass and calcification and 35% to 27% pure at the second
granularity level of a specific type of mass or calcification. We also found
that learnt prototypes covered around 25% of the total abnormality categories
from the dataset (categories from the BI-RADS lexicon) and around 60-65% of
the relevant prototypes are associated with the correct class. Further,
prototype-based model outperformed the black-box model, GMIC, in localizing
the ROIs, suggesting that prototype-based models can be a good candidate for
unsupervised region of interest extraction. However, the intersection over
union scores for top-10 activated prototypes are quite low (around 0.13). This
suggests that we still need to improve on learning relevant prototypes which
can get activated on all ROIs of the same category. We found slightly higher
scores for PIP-Net w.r.t some properties, but PIP-Net also had a much lower
number of learnt prototypes resulting in low coverage score. Having lower
number of prototypes (compactness) makes the explanation easier for users to
comprehend, but we need to increase the number of unique and relevant
prototypes learnt by the model to improve the model’s detection of various
ROIs.
In conclusion, our analysis shows that prototype-based models still need to
improve on i) purity of the prototypes, ii) reducing irrelevant prototypes,
iii) learning more unique prototypes such that the prototypes can cover more
abnormality categories and iv) finding the ROIs more accurately. In the
future, it would be also interesting to perform user evaluation of the learnt
prototypes to assess their quality with domain-experts. Our vision is to
integrate the interpretable model in the clinical workflow for the purpose of
assisting the clinicians with diagnosis. We envision a two-phase approach: a
bootstrap-phase where clinicians study the prototypes of the global
explanation and discuss their meaning giving them a name and a description;
and a use-and-tune phase where clinicians during their everyday diagnosis see
and amend the local explanation along with the prototype names and
descriptions. In this way, the documentation of the meaning of the prototypes
and thereby the quality of the model can gradually improve in a natural
manner.
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