Building the Bridge of Schrödinger: A Continuous Entropic Optimal Transport Benchmark
Authors: Nikita Gushchin
Alexander Kolesov
Petr Mokrov
Polina Karpikova
Andrei Spiridonov
Evgeny Burnaev
Alexander Korotin
Venue: NeurIPS 2023 Datasets and Benchmarks Poster (2023)
Keywords: entropic optimal transport
schrödinger bridge
benchmark
continuous distributions
neural methods
generative modeling

Abstract: Over the last several years, there has been significant progress in developing neural solvers for the Schrödinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is interconnected with the practically well-performing diffusion models and theoretically grounded entropic optimal transport (EOT). Still, the area lacks non-trivial tests allowing a researcher to understand how well the methods solve SB or its equivalent continuous EOT problem. We fill this gap and propose a novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. Our methodology is generic and works for a wide range of OT formulations, in particular, it covers the EOT which is equivalent to SB (the main interest of our study). This development allows us to create continuous benchmark distributions with the known EOT and SB solutions on high-dimensional spaces such as spaces of images. As an illustration, we use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution. Our code for constructing benchmark pairs under different setups is available at: https://github.com/ngushchin/EntropicOTBenchmark

Building the Bridge of Schrödinger:
A Continuous Entropic Optimal Transport Benchmark
Nikita Gushchin
Alexander Kolesov
Petr Mokrov
Skoltech
Skoltech Moscow, Russia
Skoltech
Moscow, Russia
Moscow, Russia
kolesso@sskoltech.rr
petr bettr.mokrow@sskltecchhh.
Polina Karpikova
Andrey Spiridonov
Skoltech"
Skoltech
Moscow, Russia
Moscow, Russia
polina. karpikova@skkltechh..u
andrew wwdrew..pirtei...............d.e wpt..ppprrrdooooosootoo..cccc
Evgeny Burnaev
Alexander Korotin
Skoltech" AIRII
Skoltech"
AIRI
Moscow, Russia burnaev@skoltechh.rr
Moscow, Russia korotti@sskoltechhrrr
Abstract
Over the last several years, there has been significant progress in developing neural solvers for the Schrödinger Bridge (SB) problem and applying them generative modelling. This new research field is justifiably fruitful as it is interconnected with the practically well-performing diffusion models and theoretically grounded entropic optimal transport (EOT). Still, the area lacks non non-triven trivial tests allowing researcher to understand how well the methods solve SB its equivalent continuous EOT problem. We fill this gap and propose novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. Our methodology is generic and works for a wide range OT formulations, in particular, it covers the EOT which equivalent SB (the main interest of our study). This development allows us to create continuous benchmark distributions with the known EOT and SB solutions on high-dimensiooal spaces such as spaces of images. As an illustration, we use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution. Our code for constructing benchmark pairs under different setups is available at: /tttp://githute com/ cctp///ggeccchhh/ttttoo///aaaaaa
Diffusion models are powerful tool to solve image synthesis 25 45 and image-to-image translation
tasks. Still, they suffer from the time-consuming inference which requires modeling thousands of diffusion steps. Recently, the Schrodinger Bridge (SB) has arisen as promising framework cope with this issue [15] 54). Informally. SB is special diffusion which has rather straight trajectories and finite time horizon. Thus, it may require fewer discretization steps to infer the diffusion.
50 47
Skolkovo Institute of Science and Technology 1 Artificial Intelligence Research Institute
37th Conference on Neural Information Processing Systems (NeurIPS 2023) Track on Datasets and Benchmarks.

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In addition to promising practical features, SB is known to have good and well-studied theoretical properties. Namely, it equivalent [38] the Entropic Optimal Transport problem (EOT. [3] 20
about moving the mass of one probability distribution to the other in the most efficient way. This problem has gained a genuine interest in the machine learning community thanks to its nice sample
complexity properties, convenient dual form and wide range of applications [28 44
Expectidely, recent neural EOT/SB solvers start showing promising performance in various tasks However, it remains unclear to which extent this success is actually attributed to the fact that these methods properly solve EOT/SB problem rather than to good choice of parameterization, regularization, tricks. etc. This ambiguity exists because of the lack of ways to evaluate the performance of solvers qualitatively of continuous distributions with the analytically known EOT/SB solution is narrow (Gaussians and these solutions have been obtained only recently. Hence, although papers in the field of neural EOT/SB frequently appear, we never know how well they actually solve EOT/SB.
[2 9 14
23 42
solving EOT/SB. Specifically, the class
10 41 26 7
Contributions. We develop a generic methodology for evaluating continuous EOT/SB solvers.
1. We propose a generic method to create continuous pairs of probability distributions with analyti- cally known (by construction) EOT solution between them ($3.1 $3.22
2. We use log-sum-exp of quadratic functions ($3.3)
3.4 to construct pairs of distributions ($4)
that we use as a benchmark with analytically- known EOT/SB solution for the quadratic cost.
We use these benchmark pairs evaluate (55) many popular neural EOT/SB solvers ( 2 in high-dimensional spaces, including the space of 64 x 64 celebrity faces.
In the field of neural OT, there already exist several benchmarks for the Wasserstein-2 [32 the
Wasserstein-1 [31 and the Wasserstein-2 barycenter [31] OT tasks. Their benchmark construction methodologies work only for specific OT formulations and do not generalize EOT which we study,
Background: Optimal Transport and Schrödinger Bridges Theory
We work in Euclidean space X' = R equipped with the standard Euclidean norm -| We use P(x') = P(Y) = P(R") )) to denote the sets of Borel probability distributions on X' respectively.
Classic (Kantorovich) OT formulation (27 53 48
For distributions IP P(2)), Pi E P())
and a cost function c: X' x y + R. consider the following problem (Fig. Ta
OT PP) def inf rEII(Po.P) Xxy
c(x, y)d (x,
(1)
where the optimization is performed over the set [I Pi) of transport plans, i.e., joint distributions on X' x y with marginals Po. Pr, respectively. The set [(PP,, Fi) is non-empty as always contains the trivial plan Po x Pi. With mild assumptions, a minimizer n² of (I) exists and is called an OT plan. Typical examples of are powers of Euclidean norms, i.e., c(x, = = y/1 1.
C(xin(ylx))
c(xLy1)
nylx))
Y1 Y2
X1
P
X1
P n(vlx,)
PP
X2
P X2
Ym
Xn
c(Xr.ym)
X
C(xx,n(yix,))
(a) Classic OT formulation
(b) Weak OT formulation
Figure 1: Classic (Kantorovich's) and weak OT formulations.
Weak OT formulation [22 3
Let C': : X' x P())) R U {+0)} be weak cost which takes point x E X and distribution of y E y as inputs. The weak OT cost between Po. P, is (Fig. [b]
(Po, def
inf rEI.Fo.P.))..
f C(x, (()d) = inf ren(Po.P.)).
I C(I, ((I))(()))))P(())) (2)
where (-|æ)) denotes the conditional distribution of E ) given x E X and my is the projection of i
to X' which equals P. since IIPP,, F1). Weak OT formulation (2) generalizes classic OT (I]: it
suffices to pick C(z, T-122) = Jy c(z, (y)x) to obtain I from (2). A more general case of

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weak cost is C(x, fy (((l.))d(((())) + cR (T((2=)) where e > 0 and R P()) - R is some functional (a.k.a. regularizer), e.g., variance [ 35), kernel variance (34] or entropy 3 With
mild assumptions on the weak cost function C, an OT plan π in (2) exists. We say that the family of
its conditional distributions { [eEx is the conditional OT plan.
Entropic OT formulation
13
20
It common to consider entropy-basee regularizers for
E (Po, P1)) EOT (2) de Po. P1) c, P1)
+EKL (T|PPxPP)),
(3)
def
J ((, + xEll(FoPP))
min
(4)
-
(5)
Here KL is the Kullback-Leibler divergence and H is the differential entropy, i.e., the minus KL divergence with the Lebesgue measure. Since π [I Pi). it holds that KL (T |Po
xPi)
(Po) fx H (=()[²)) dF (=())))))d((((( = - H(T) H (IP)) + (P1), i.c.. these formulations are equal up to an additive constant when Po E (X) and Pi E (a)()") and have finite entropy. Here we introduce "ac" subscript
indicate the subset of absolutely continuous distributions. With mild assumptions on c, Fu, 11, the minimizer "t exists, it is unique and called the entropic OT plan. It is
important to note that entropic OT (5) is a case of weak OT (). Indeed, for the weak cost
Ca.e (-1x)) del J. c(x,, (dr.))d(ylo)) - eH (T(+2))..
(6)
formulation immediately turns to 5). This allows us to apply the theory of weak OT to EOT.
Dual OT formulation. There exists wide range of dual formulations of OT (53
48 WOT 2
22
and EOT 20 44). We only recall the particular dual form for WOT from 3 which serves as the main theoretical ingredient for our paper. For technical reasons, from now on we consider only F1 E P,))) C P()) for some p 2 1, where subscript "p" indicates distributions with a finite p-th moment. We also assume that the weak cost C: X' x Pp())) + R RU [+-0)} } is lower bounded, convex in the second argument and jointly lower-semicontinuous in X x P)))). In this case, minimizer r of WOT (2 exists 3 Theorem 3.2] and the following dual formulation holds 3 Eq. 3.3]:
( WOT =sup { { (())))P(())) + / [w)dd
where f E C,( = y - TK continuous s.t. 3o, ER: 15(1 all 1|P + B} and jc the (
so-called weak C-transform of f which is defined by
(C(x,v)- J, f())dv()))
def inf PEP,OY)
(8)
Function f in 7 is typically called the dual variable or the Kantorovich potential.
SB problem with Wiener prior 38 Let Ω be the space of P -valued functions of time and end time = We use P(22) to denote the set of probability distributions on Ω i.e., stochastic processes.
t t (0, 1] describing trajectories R. which start at time
Consider two distributions Po (2)) and IP E (2) with finite entropy. Let F(PPo, (iloo,P)) P1) C P(22 be the subset of processes which have marginals Po and P1 at times f = 0 and t = 1, respectively. Let dW, be the differential of the standard " valued Wiener process. Let W E (()) be the Wiener process with the variance > which starts at Po at time can be represented via the following stochastic differ- ential equation (SDE): dx,- = VedW, with X, ~ Po-
n(lx)) P.
X
P.
X2
n(ylx.)
X
It
Figure 2: The bridge of Schrödinger.
The Schrödinger Bridge problem with the Wiener prior is the following:
inf TEF(Fo,P1)
KL (TI]W"'.
The inf is attained uniquely at some process T 38 Proposition 4.1]. This process turns out to be diffusion process and can be (uniquely) represented as the following SDE:
T dX, v" (X,, t)dt VedW,,
(10)
3

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where o" :R' x [0.1] R' is its drift function which we call the optimal drift. Hence, in (.. one may consider only diffusion processes C F P1) with the volatility e coinciding with the volatility of the Wiener prior W In turn, solving SB can be viewed as finding the optimal drift v".
Link between SB and EOT problem. The process T solving SB (9) is related to the solution x'' of EOT problem 5 with the quadratic cost function c(z, y) | - - yo We start with some notations. For process T E P(St), denote the joint distribution at time moments t 0, by π E P(X )). Let Ja.y be the distribution of for E (0, 1) conditioned on S' values n, at / 0,
For the solution T of SB ( it holds that x = x², where x² is the EOT plan solving 5)_ |aray i.e., informally, the "inner" part of T matches that of the prior W
Moreover, Ta.y W Wa.ag
Conditional process Waz.g is well-known the Brownian Bridge. Due to this, given æ; y, simulating le,y is rather straightforward. Thanks this aspect, SB and EOT can be treated as nearly equivalent problems. Still EOT solution x* does not directly yield the optimal drift v*.
(z.g)
the trajectories of We,,y
(z.g) of x² has the specific form (3 Theorem 2.8], namely, However, it is known that the density '(=,0)
(x-a) 7 (ar)W(ular,, e (T)p")()), where functions yo" ~0 IR - R are called the Schrödinger d(
potentials. From this equality one gets the expression for 40 (-) and the density of πT" (+|z):
dn (ylo) dy
(v) x dm (vlx)
x ((yl,,e()pp"
[N e/)
(11)
dy
up to multiplicative constants. One may recover the optimal drift u via 38 Proposition 4.1]
(x, = vV log / f NNlll, (1 ()eID)' (y)dy.
(12)
Here the normalization constant vanishes when one computes V log(-). Thus, technically, knowing the (unnormalized) density of π²,, one may recover the optimal drift o² for SB (9.
2 Background: Solving Continuous OT and SB Problems
Although OT (2). EOT (5) and SB (9) problems are well-studied in theory, solving them in practice challenging, Existing OT solvers are of two main types: discrete 44) and continuous (32). Our benchmark is designed for continuous EOT solvers; discrete OT/EOT is out of the scope of the paper.
Continuous OT assumes that distributions Po and P, are continuous and accessible only via their random samples X {Ii (Ii,....,NN,, ~ IP and Y -yy,,...,wa OT plan x² between entire Po and P, but using only X and Y. Most continuous OT solvers do this via employing neural networks implicitly learn the conditional distributions म (-1lz)) R x" --1zx). In turn, SB solvers learn the optimal drift i ss v² but it is anyway used to produce samples via solving SDE dX,, = Özz t)dt + VedW, starting from Xo- x (sampled from Fo) time
PP. The goal recover an
f(11x)
After training on available samples X and Y, continuous solvers may produce ~ 7 [reaa for previously unseen samples Fust ~ Pp. This is usually called the out-of-sample estimation. It allows applying continuous OT solver to generative modelling problems such as the image synthesis (noise- to-data) and translation (data-to-data). In both these cases, P, is data distribution, and Po is either a noise (in synthesis) or some other data distribution (in translation). Many recent OT solvers achieve tasks.
competitive performance in synthesis 12 15 46 and translation
35
Continuous OT/SB solvers are usually referred to as neural OT/SB because they employ neural
networks. There exist lot of neural OT solvers for classic OT I 46
17
56
40
30
19] see also
31 for surveys, weak OT (2) 34 35 M. entropic OT (5) 49 42 and SB ( [52 9 23 Providing a concise but still explanatory overview of them is nearly impossible as the underlying principles of many of them are rather different and non-trivial. We list only EOT/SB solvers which relevant to our benchmark and provide a brief summary of them in Table I In $ we test all these solvers on our continuous benchmark distributions which we construct in subsequent 3
Approaches to evaluate solvers. As seen from Table I each paper usually tests its solver on
restricted set of examples which rarely intersects with those from the other papers. In particular, some papers consider data->data tasks, while the others focus noise-xdata. Due to this, there is

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Underlying principle
Evaluated as BOT/SB X
lested in generation
Tested in translation
Solver
and parameterization
(noise-vdala)
(daia-ddata
Solves classic dual EOT 20 93.11 with NNs. Lears 1 more NN for the barycentric projection.
UUSPS/16616)
LSOT
49 21
MNIST (32x32)
USPS -MIIST (16x16), SVHN MNIST (3x32x32)
BOT solvers
Combines LSOT's potentials with score model for P1 to sample from st" ())) via Langevin dynamics.
SCONES
14
Gaussians
X
CelchA Upseale (3x64x664)
OT with 2 NNs (transport map and potential).
max reformulation
This generic neural solver for weak but it has not been tested with the entropic cost function.
NOT
35
dual ?
Employs energy- based modeling weak EOT 77 non-minimax;
EGNOT
Gaussians
X
Colored MNIST (3xx32x32)
Solves max-min reformulation of SB
ENOT x
with 2 NNs (potential and SDE drift).
Goussions
X
CelebA Upecale (3x64x64)) Colored MNIST 2 (3x32x332)
Alternate solving of two Half Bridge (HB) problems. HB is solved via drift estimation with GP [55]
MLE-SB 52
X
X
Single Cell data (D) 5), Mation Capture (D
salvvers
lierative Mean-Mainn Matching Proportional Fitting
DiAS 15
NNs for forward and buckward SDE drilts
Gaussians
CalebA (32x32)
-EMNIST (32x32)
Likelihood training of SB
MNIST (32x32).
FB-SDE
NNs for the V log of Schrödinger pocentials
X
GelehA (3x.2&x32).
X
CIFAR-10.3x32::
Table 1: Table of existing continuous (neural) solvers for EOT/SB.
no clear understanding of the superiority of one solver over the other. Importantly, in many cases, the quatitative evaluation is done exclusively via the metries of the downstream task. For example, 14 consider image generation or translation tasks and test the quality of generated images 24 That is, they compare generated marginal distribution गं with target IP, This allows to access the generative performance of solvers but gives no hint whether they actually learn the true EOT/SB solution. Works [14 23 42 [5]
[15
9 23
via FID
a step toward opening the veil of secrecy and test their solvers in the Gaussian case. Unfortunately, it is rather trivial and may not be representative.
3
Constructing Benchmark Pairs for OT and SB: Theory
In this section, we present our theoretical results allowing us construct pains of distributions with the EOT/SB solution known by the construction. We provide proof's in Appendix A
3.1 Generic Optimal Transport Benchmark Idea
For given distribution Py E P(X), we want construct distribution P1 Pp()) such that some OT plan π² E [I P1 for a given weak OT cost function C' between them is known by the construction. That is, T6 Po. πं PI and nt minimizes 2 In this case, (Io PP),PP) may be used as a benchmark pair with a known OT solution. Our following main theorem provides way to do so.
Theorem 3.1 (Optimal transport benchmark constructor). Let Po E P(x) be a given distribution, f E Cx)))) be given function and C' X Pp())) + R be given jointly lower semi-continuon convex in the second argument and lower bounded weak cost. Let x PPX x Y) be a distribution for which म = Fo for all X it holds that
n" (-1z) E arginf {C(I,1) J ("()))(())))) HEP,())
(13)
T" Let P, def "ं be the second marginal T² and assume that IP; E P,))). Then n" is an OT plan between Po and Pi (it minimizes (2) and I is an optimal dual potential (it maximizes 7
Thanks to our theorem, given a pair (Po. f*) E P(X) x C)) of a distribution and a potential, one may produce distribution P, for which OT plan between them is known by the construction.
This may be done by picking n²" E IT(P) whose conditionals T² (-Iz) minimize (13)
While our theorem works for rather general costs C, it may be non-trivial to compute minimizer T" *'++x) in the weak C-transform ( e.g., to sample from it or to estimate its density. Also, we note that our theorem states that x- is optimal but does not claim that it is the unique OT plan. These aspects may complicate the usage of the theorem for constructing the benchmark pairs (Po, P,) for general costs C. Fortunately, both these issues vanish when we consider EOT, see below.
3.2 Entropic Optimal Transport Benchmark Idea
For C' Ca.e 6 with > 0, the characterization of minimizers म. "+(2a) in 13 is almost explicit.
5

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Theorem 3.2 (Entropic optimal transport benchmark constructor). Let Pu E P(X) be given distribution and f E c,,l") be a given potential. Assume that c: X x + R is lower bounded and (Ic, po) + Sy cxx, v)du(y) is lower semi-continuous in X x PP)))). Furthermore, assume that there exists M E R2 such that for all x E X it holds that M2 def
((x:))) ddy S. M.. Assume that for all E X value Zr def Sy expp (L(())=(( dy is finite. Consider the joint distribution n² E P(X x )) whose first marginal distribution satisfies Th Po for all x E X it holds that dm" (ylx) dy
1 Z
(19))c(22 (f()))((zz))) -c(x
exp
(14)
c
and x- (-|x=) (-1x))PPp)) Pp)))), Then if P1 def Po, II and cost Co.e. Moreover, if fx Ca. (x, (-Ix)) (d) < xc, then n is the unique EOT plan.
= म belongs to Pp(Y), the distribution n¹ is an EOT plan for
Our result above requires some technical assumptions on and f but reader should not worry as they are easy to satisfy in popular cases such as the quadratic cost c(zz, y) 1 || y/ 3$3.). The important thing is that our result allows sampling y ~ x (-Ixr) from the conditional BOT plan by using MCMC methods [5] 111.2] since ([4) provides the unnormalized density of x (ylrr). Such sampling may be time-consuming, which is why we provide a clever approach to avoid MCMC below.
3.3 Fast Sampling With LogSumExp Quadratic Potentials.
In what follows, we propose way overcome the challenging sampling problem by considering the case c(I, y) = |اا and the special family of functions f". For brevity, for matrix A E IR DxD and b E DR we introduce Q(y b, A) der potential f to a weighted log-sum-exp (LSE)) of N quadratic functions:
exp
-
1/2 {V b) A(y b)]. Henceforth, we choose the
N
f (v) del log Σ wi Olulbnn E Ann
(15)
n=1
Here wi 2 0 and we put A, to be symmetric matrix with eigenvalues in range (-1, +oc). We say that such potentials f are appropriate. One may also check that f E C2()) as is just the LSE smoothing of quadratic functions. Importantly, for this potential f and the quadratic cost, π (-(ar is Gaussian mixture, from which one can efficiently sample without using MCMC methods. Proposition 3.3 (Entropic OT solution for LSE potentials). Let I be given uppropriate LSE
potential (15) and let IP E P Consider the plan du (x., y) = dx (ulx))dPP(a where
dx (ula)
N
Nuvumn( En) with En, del c(An + 11- pn(x)) del (An + 1) (Anb,, + +x),
dy
1
In def
Σ
def wn (2T) / (det(En)) 1 - - En).
Wn
Wns
Wn
Then it holds that P1 def
= oi belongs to P2())) and the joint distribution x² is the unique EOT plan
between Io and P1 for cost Cc.e with c(2, y) = - .1
We emphasize that although each conditional distribution x² (-(xx) is a Gaussian mixture, in general, this does not mean that n² IP; = x is Gaussian mixture, even when P. is Gaussian. This aspect does not matter for our construction, and we mention it only for the completeness of the exposition.
3.4 Schrödinger Bridge Benchmark Idea
Since there is link between EOT and SB, our approach allows us to immediately obtain solution to the Schrödinger Bridge between Fo and P2 (constructed with an LSE potential f*).
Corollary 3.4 (Solution for SB between Po and constructed P1). In the context of Theorem 3.2 let p= 2 and consider c(x, y) = 2 || - yll?. Assume that Po E (X)) C C(X') and both Po and
P1 (constructed with given f*) have finite entropy. Then it holds that 40" (v) exp (5)))) is Schrödinger potential providing the optimal drift v* for SB via formula (12
def
Although the drift is given in the closed form, its computation may be challenging, especially in high
dimensions. Fortunately, as well as for EOT, for the quadratic cost c(x, y) = ||=-vll and our LSE
2
[ potentials f*, we can derive the optimal drift explicitly.

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BOT solvers
SB solvers
Metric
LSOT
SCONES]
INOT]
EgNOT
ENOT
MILE-BB
DIISB
FB-SDE-A
FB-SDE-J
BW2 UVP
0.1
Do not wark for small c due numerical instability
cBW3-UVP
[ 85.1).
RVP
cBW3-UVP
BVP
10
Divergg with the default hyperparameters. May roquire more hyperparameter tuning.
-VV
Table 2: The summary BOT/SB solvers' quantitative performance in cBW3-UVP and BW3-UVP metrics our mixtures pairs. Detailed evaluation and coloring principles are given in Appendix B
Corollary 3.5 (SB solution for LSE potentials). Let f be given appropriate LSE potential [5 and consider a distribution IPo E (2) with finite entropy. Let IP, be the one constructed in Proposition 3.3 Then it holds that P1 has finite entropy, belongs to P2 (V) and
N
v' (x, V log
Σ
wn det(())))))
((zlbbn
1
e² 21111) 2
(16)
e(1-t)
the optimal drift for the SB between Po and Pl. Here A def (1- (1 and N del c(A,, + 1)
4 Constructing Benchmark Pairs for OT and SB: Implementation Our benchmark is implemented using PyTorch framework and is publicly available at htap:/ttuuu..cmm//mggcchchhhciii
It provides code to sample from our constructed continuous benchmark pairs (IPo F1) for various e, 44.1 for details of these pairs. In $4.2 we explain the intended usage of these pairs.
4.1 Constructed Benchmark Pairs We construct various pairs in dimensions D up 12288 and {0.1, 1,1,1,10), 103. Our mixtures pairs simulate noise->data setup and images pairs simulate data->date case.
Mixtures benchmark pairs. We consider HOT with (0.1, 1, 100 in space R.D with dimension
D E {2, 16, 64, 128). We use a centered Gaussian as Po and we use LSE function (55) with N = for constructing IP, (Proposition 3.3). In this case, the constructed distribution IP, has 5 modes (Fig.
3a 3r. Details of particular parameters (An, bn. Wn, etc.) are given in Appendix B
Images benchmark pairs. We consider EOT with E {0.1, 1, 10). As distribution Fo. we use the approximation of the distribution of 64 64 RGB images (D 12288) of CelebA faces dataset [39] Namely, we train normalizing flow with Glow architecture 29]. It is absolutely continuous by construction and allows straightforward sampling from Po. For constructing distribution P,, we also use LSE function f We fix N 100 random samples from Po for b,, and choose all A, = T. Details of wn are given in Appendix C For these parameters, samples from P1 look like noised samples from Po which are shifted to one of Ps, (Fig. g
By the construction distribution Pl. obtaining the conditional EOT plan x (ylar) between (IPo, P1) may be viewed as learning the noising model. From the practical perspective, this looks less interesting than learning the de-noising model π (x|y)) Due to this, working with images $5 we always test EOT solvers in P1 + Py direction, i.e., recovering π (xrl)) and generating clean samples form noised y. The reverse conditional OT plans π (a:ly) are not as tractable as (y/x). We overcome this issue with MCMC in the latent space of the normalizing flow (Appendix C
4.2 Intended Usage of the Benchmark Pairs
The EOT/SB solvers ( provide an approximation of the conditional OT plan 7(1xx) es T" (-lx) from which one can sample (given x ~ Pu). In particular, SB solvers recover the approximation of the optimal drift T v": it is anyway used to produce samples y ~ *(-(z)) via solving SDE d.X, (ar, L)dl + VedW, starting from X, = x at time (. = 0. Therefore, the main goal of our benchmark is to provide a way to compare such approximations म, T with the ground truth. Prior to our work, this was not possible due to the lack of non-trivial pairs (IP,, P1) with known πt" v".
7

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x
o f
Ficked x -
y'
0 y- P
2-
0-
(a) Input and target.
(b) [LEESBB].
|COOES))
(d) NOT
(e) [EgNOT]
((x0)
0
-5.0 -25
0.0
2.5
5.0
-5.0 -2.5
0.0
2.5
5.0
-5.0 -2.
0.0
5.0
-5.0
-25
0.0
5.
-5.0
-2.5
0.0
5.0
(f) True EOT plan x². (g)
[ENOT].
(h) [DiNSB].
(i) [F--SDE-]]
i LFB-SDE-]]
Figure 3: Qualitative results of EOTYSB solvers on our mixtures benchmark pair with (D, e) = (16, 1). The distributions are visualized using 2 PCA components of target distribution P1- Additional examples of performance on pairs with other E {0.1, 10} are given Appendix B
For each of the constructed pairs (Po. I. we provide the code to do main things: (a) sample
Po: (b) sample y ~ n²" (-|zz) for any given x; (c) sample pairs (xr, y) xr" from the EOT plan;
(d) sample y ~ P1: (e) compute the optimal drift v" (xx,) t). Function (c) is just a combination of
(a) and (b). For images pairs implement the extra functionality (f) to sample x ~ x= (-ly) using
MCMC. Sampling (d) is implemented via discarding x (not returning it to a user) in (x, in (c).
When training a neural EOT/SB solver, one should use random batches from (a,d). Everything coming from (b,c,e,)) should be considered as test information and used only for evaluation purposes. Also, for each of the benchmark pairs (mixtures and images), we provide hold-out test for evaluation.
5 Experiments: Testing EOT and SB Solvers on Our Benchmark Pairs
Now train various existing EOT/SB solvers from Table I on our benchmark pairs (P..P)) showcase how well they capture the ground truth EOT plan. For solvers' details, see Appendix D
MIXTURES BENCHMARK PAIRS. For quantitative analysis, we propose the following metric:
100%
cB cBw? cBW3-UVP UVP
BW (f(lxt) (11x))
(17)
Var(P.) x
For each compare conditional distributions (-11x:) and n"" (-1x)) with each other by using the Bures- Wasserstein metric [16], i.c., the Wasserstein-2 distance between Gaussian approximations of distributions. Then we average this metric w.r.t. x 1P. The final normalization 1 VVa((P))) is chosen so that the trivial baseline which maps the entire 1' to the mean of P1 provides 100% error. Metric (17) is a modification of the standard BW2 -UP 14 completeness, we also report the standard BW? UVP to check how well म matches π,.
18 for the conditional setting. For
We found that most solvers' performance significantly depends on the selected We neither have deep knowledge of many solvers nor have the resources to tune them to achieve the best performance on our benchmark pairs. Thus, we kindly invile the interested authors of solvers to improve the results for their solvers. Meanwhile, we report the results of solvers with their default configs and/or with limited tuning. Nevertheless, we present a hyperparameter study in Appendix E to show that the chosen hyperparameters are a reasonable fit for the considered tasks. Our goal here to find out and explain the issues of the methods which are due their principle rather than non-optimal hyperparameter selection.
DISCLAIMER.
The detailed results in Appendix B Here we give their concise summary (Table 2 and give qualitative example (Fig. of solvers performance on our mixtures pair with (D,c) = (16, 1).
EOT SOLVERS. [LSOT] and [SCONES solvers work only for medium/large e = 1, 10. [LSOT] learns only the barycentric projection 49 Def. 1] hence naturally experiences large errors and even collapses (Fig. 3b) [SCONES] solver works better but recovers the plan with a large error. We think
8

---

y y-W 11
[True EOT plan]
x- (+ly)
[Learned, ENOT]
X - ii(-
(a) 0.1
(b)==1
(c) 10
Figure 4: Qualitative comparison of ground truth samples x= (-ly) with samples produced by JNOTT]. With the increase of c, the diversity increases but the precision image restoration drops.
this is due to using the Langevin dynamic 14 Alg. 2] during the inference which gets stuck in modes
plus the imprecise target density approximation which we employed (Appendix) D EgNOT] solver also employs Langevin dynamic and possibly experiences the same issue (Fig. 3e Interestingly. our evaluation shows that it provides better metric in matching the target distribution Fi. [NOT] was originally not designed for EOT because it is non-trivial to estimate entropy from samples. To fix this issue, we modify the authors' code for EOT by employing conditional normalizing flow (CNF) as the generator. This allows us to estimate the entropy from samples and hence apply the solver to the EOT case. It scores good results despite the restrictiveness of the used architecture (Fig. 3d
SB SOLVERS. For [MLE-SB] ). the original authors' implementation uses Gaussian processes as para- metric approximators instead of neural nets [52]. Since other SB solvers ( ([iiffS (F--SDE-A()) and [ use neural nets, after discussion with the authors of [MLE-SBJ. we decided to use neural nets in their solver as well. All SB solvers work reasonably well, but their performance drops as the e increases. This is because it becomes more difficult to model the entire diffusion (with volatility e); these solvers may require more discretization steps. Still, the case of large is not very interesting since, in this case, the EOT is almost equal to the trivial independent plan Po x P1.
IMAGES BENCHMARK PAIRS. There only two solvers which have been tested by the authors in their papers in such a large-scale data->data setup (64 x 64 RGB images), see Table Namely, these found that [SCONES yields unstable training our benchmark pairs, probably due to too small c for it, see [4] $5.1]. Therefore, we only report the tried to run [DiffSB], J, [FB-SDE-A] solvers with
[SCONES] and [ENOT]. Unfortunately,
results of ENOT solver. For completeness, their configs from noise-ddata generative modelling setups but they diverged. We also tried [NOT] with convolutional CNF as the generator but it also did not converge. We leave adapting these solvers for high-dimensional data->data setups for future studies. Hence, here we test only [ENOT]-
In Fig. 4 we qualitatively that [ENOT] solver only for small properly learns the EOT plan x and sufficiently well restores images from the input noised inputs. As there anyway the lack of baselines in the field of neural EOT/SB, we plan to release these [ENOT] checkpoints and expect them to become a baseline for future works in the field. Meanwhile, in Appendix C we discuss possible metrics which recommend use to compare with these baselines.
6 Discussion
Potential Impact. Despite the considerable growth the field of EOT/SB, there is still no standard way to test existing neural (continuous) solvers. In our work, we fill this gap. Namely, we make step towards bringing clarity and healthy competition to this research area by proposing the first-ever theoretically grounded -ooounddd EOT/SB benchmark. We hope that our constructed benchmark pairs will

---

become the standard playground for testing continuous EOT/SB solvers as part of the ongoing effort advance computational OT/SB, in particular, in its application generative modelling.
Limitations (benchmark). We employ LSE quadratic functions (15) as optimal Kantorovich potentials to construct benchmark pairs. It is unclear whether benchmark sufficiently reflects the practical scenarios in which the EOT/SB solvers are used. Nevertheless, our methodology is generic and can be used to construct new benchmark pairs but may require MCMC to sample from them.
To show that the family of BOT plans which can be produced with LSE potentials is rich enough, we provide heuristic recipe how to construct benchmark pairs simulating given real-world datasets, see Appendix [H] As we show there, the recipe works on several non-trivien non-trivial single-cell datasets [6] g]. Thus, we conjecture that LSE potentials may be sufficient represent any complex distribution just like the well-celebrated Gaussian mixtures are capable of approximating any density 43 this inspiring theoretical question open for future studies.
We leave
For completeness, we note that our images benchmark pairs use LSE potentials and do not require MCMC for sampling from marginals Fo, II i.e., to get clean and noisy images, respectively. However, for computing the test conditional FID (Appendix(C) of EOT/SB solvers, MCMC is needed to sample clean images I x² ==ly)) conditioned on noisy inputs y. This may introduce extra sources of error.
Limitations (evaluation). We employ 1-W2-UV for the quantitative evaluation ($5) as it is popular in OT field [3 14
18, 23 42 However, it may not capture the full picture as it only compares the
1st and 2nd moments of distributions. We point to developing of novel evaluation metrics for neural OT/SB solvers as an important and helpful future research direction.
Following our disclaimer in $5 acknowledge one more time, that the syper-parameters tuning of the solvers which we test on our proposed benchmark is not absolutely comprehensive. It is possible that we might have missed something and did not manage achieve the best possible performance in each particular case. At the same time, our Appendix [D key parammeessetters and it seems that the metries reported are reasonably close to the optimal ones.
shows the extensive empirical study of the
Acknowledgements
This work was partially supported by the Skoltech NGP Program (Skoltech-MIT joint project).
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13

---

A Proofs
P1). suffices to show that π attains the optimal cost. Let CTSt(TT) be the value of weak OT functional for a plan æ,, i.e.,
Proof of Theorem 3.7 By the definition of P1, it holds that n" (I((P,,
Cost(r) xT) / C( ((z))()))))
We consider weak OT 22 between Po E P(X) and Pp((t)) and use its dual form
Cost(Po, = sup { ( (x)ddP(()) + / s(u)dP
sup (s
inf vepe (C(x, / {())de())))P((())) / ((U))B
Ix inf (C(x, v) - (()ddoe)))))) + / (')))PP()))
Now use the fact that x (+/x)) minimizes (8) for a
[ inf (C(x,v) - / (()ddv()))((((()) f (())dFi()) "EP))) =sx [C(u, / / (())dd
/ (C))dP
/ Clz, (( / / (1) (vlz) dFo(z) + ((())dPi()
x
(2)((
/ ((TT - Jr (( + / f (1))dPP
xxx
/ ((r 8" ('()) + / ('(( (())dFP(()) (v)
J ((1)) (y)d((P) म. +i)
=0 since =i=
J C(x, x= (())d((11)) = Cost(
(18)
We see that Cost(TT) is not greater than the optimal Cost(Po, P1), i.e., x² is optimal. At the same time, from the derivations above, directly follows that f is an optimal potential.
We are going to use our Theorem 3.1 First, we check that 13 holds for T² (+(x) defined by (14. Analogously to [42 Theorem 1]. for each x X, we derive
Proof of Theorem 3.2
inf (Co,(I,.V) I fr (1))d = inf wEP,()
vEP,()) { 1/ |c(x, v) - fr du(v) cH()))
(M)(M)
Minimizing Gz. one should consider only Pp.ec()) Pp()). Indeed, for Pp.ac());
holds that G2 (v²) too since c(x, y) is lower bounded and - -H(v) = +o0. We continue
H(v)
- e L log exp
(2) -(x, v)
de(y) + J. log dv(y) du(y) dy dv(v) dv(y))
inf vEPriee())
inf vEP VEP ()) {-e feef/n log (Zz-
dr (ylz)) dd(yy) + / log dy dy
dr (w(x) du(y) + / log dv(y) du dv(v)) dy
-clog 7xx
inf VEP vEPpac())) (-c log
dy
14

---

c log Zx + inf cKL (+||TT
(19)
PEPp.ue
Since x² ('(x ())) by the assumption of the current Theorem, unique minimum of (1) in (3)). Now to apply our Theorem 3.1 it remains to check that all its assumptions hold. We only have to check that Ca.e given by (6) is lower bounded, jointly lower semi-continuous and convex in the second argument.
conclude that it is the
Analogously to 19 derive
(x) = /, clz, dxz.))dul)) cH(v) -clog Mg te KL (w)1v)) -elog M,
(20)
2-clog ad
where dv=(w) def M3 exp dow)). (- This provides a lower bound on the cost Cc.-- From the first equality in (20), we see that Ca.e is jointly lower semi-continuous because the first term fy caz, (y) is jointly lower coniicontiuuous by the assumptions and the entropy term (H()) is lower coniicontinuuus in Pi()') [48 Ex 45] and hence in P₂ as well (p 2 1). The last step is to note that Cz,e(z, 1/) is convex in v thanks to the convexity of - (v).
dy
< oo, then - fx H (TT (-|z)) finite. Let U be the subset of plans x C (IPo,, Pi) where - Sx H (-((=r) is finite. It is not empty since T' E At the same time, it is a convex set and functional - - fx H H( (T (-1x=) ( is strictly convex in U thanks to the strict convexity of the (negative) entropy v + - (v)) on the set of distributions where it is finite.
Finally, if Jx Ce.e (2, (H2)
Thus, ग + fx Ca.e (1r (())2)))PP((( ()) is strictly convex U and n- is the unique minimum.
For completeness, we note that if Ixx Ca.e (ar, x (+Ix)) cost of every plan turns to be equal to too. As result, every plan is optimal.
+oo, this situation is trivial, as the
Proof of Proposition 3.3
Deriving the actual form of 7" (+12x) is an easy exercise. We substitute (15)
into 14h and use the quadratic cost c(2, y) -1s-=11?
dx (vlz) 1
(M())
c(x,
exp
dy Z.
log Dm Wn b(y (AAn) 1|2-wl 1 cxp
N
1
Zg
N
11 x
Wn Qlylbni e -Ann) exp(
Z
2e
n=1
1
N
119/ - 1/ x11))
))
Q(yibn. (An)
Zu
2e
==1
1
1
b )m An
Au (y bnn]] exp(
11y 12
Wn
exp
(y-
Z
2
2x
==1
N
An (y - bn)
119/ .0113 1)
Z
Wn
exp
2
(y
by) )T
c
2c
1
N
b 2 An (y bn) - 1
2 - -=) - =2]
1
Wn
exp
-
2 (g
Z
nml
N
1
exp - 2 (v bnn"" An (y bn) - (y x))" (y - x)) }}
(21)
Z
n=1
Next, we prove that (we write just nn, instead of pm (x) for simplicity):
(y b². An (y bm) + (y) 2) (y x)
e
15

---

(y -mn) pn)) 2 (v- mnn + (or bn) ( - (x ")1c
_
2
bn).
(22)
Indeed,
(u - bn.) ) An (u bn) + (v - x)T (y - x) =
0T An y 26 An y + b/ Am + v" -y 2x - y +
r 1
1
x -
c
/² An + I
v 2A + x)²² I
An
bn + xt I
y +
-I=
e
E
y D' y - 2(Anb + x -y + b% An bn + x
I
I
E
(An + +)
+ by An bn + xt= -I
1
5² E' y - 2 (Anbn + +x) (An + +1)
m4.
Σn²
y² "m y 2p ΣT y + b% An
1
bn + xx -I-
Un n Un - 1 -ل Un + 6% An
I
y y'''''y y 2 T 2 y +
T 1
bn + xx" -I
Un + 1% An
I
(U (m)
T Emp (y - )km) - nn
I Σm
bn
+
T
-I-
C
An bn
I
(y - pm) Ey (y - pn) + bn
T= 11 n
pn + x I
E
6
(y pm) -T (y Um)) + b/ An (Anbn + x²²²² En
Em
(Anbn + +x) + xd 1
x
x
E
c
c
An bn (Anbn + +xx²*
I
(y- (y-m)) " (2 Mn) + bi
(Anbn +x) + xT -x
e
(y mm)) ) TT (y- 1ma) + 1m A
bn
(Anba)) ,TT En
En
En
I
Anbr - 2((Abbn)
T
T
x + x -x
2
2
2
An
Em
(u - >'>'"' (y - Mm) + 1%"
(y
bn - -(Apba)
ba)
Anbr
2
En
I
bm) )T
+ x²
2
An
(y Mmm)
T TT (y Mm) + bT
bn
1/7
bn
En
I
Σn
2(An)))))
T
+ xT
(
)
An AT En An
En
1
En
(y (0-m)) pm))
TTm (y- mm)) + 17
On
2(Anb
T
x
+ xx
2
c
An
A
En
An
bn 2b An
1
Σn
-y- - "T,)" TT (y- mmm) + by
t
T
x
c
c
?
An
A
En
An
(y - )m) TT 1 (y Ma)) + 5%
bn
c
(An + 1) En
2n
1
_
2n |
2b7 n
x +
x²
(-
2
c
?
w W--ar mb) T.,' (g) -mm) + 5z An A Em An
bn 26 el
En
1
En )x
x
e
2²
6
??
16

---

b An - A Ea An bn 20% -
En
1 En
(y- (z (2) - -mn) + +0/
12
+ x²
x==
2
e
An A/ Em An
(y - Un) " (y - Mn)) + 6%h
bn
1
En }bn + (x - bn,)
1
En
-
-
(x - (mn) =
6
c
1
En
(y - pm) P' (y Un) + (x bn))
(x
bn) +
6
e²
An
1 Em
An
1
En
6
bn
b
ybn
E
2
2n x²
(U - (w-Ma)²
- En (y Un) + (x - bb,²((
(x - bnn) +
b Av A Em An
En
c
+
bbn
2n
(y1 - mn) " (y - kn) + (x - -bn)7 (-
(x - -b,) +
An(1 En An)
in
b%
+ -+
ybn
6
1
En
(0 (w-Min mmn) E (y- mnm) + (xx - -bn)²(
[xx - -bn)+ +
b%y 1n (T En (c5 - -T))
En
+ 2
ybn
En
(U - " (y - 1km) + (x - bn)²²(
(-
(x - bn))+ +
2
An
1 1+
Ln
b
6
)bn
e
En
(y) - pn) " (y- mn) + (x bbn) (--
(xx
bn) +
6²
(e - 1) En - -+ En
10n
(m- 1 (y- - pa) + (x - bn) ( [--
1 En
_ (x - bn) + b%o
En I+
En
10n
(y - pn) T Σn 1 (y - mmm) + (x - bm)) ban
Em
(xx - -b,).
2
Next, we substitute 22 into 21
N
exp 2 (v bn) r An (y - bn) + (y - (1 =
1
I
bp)
Wn
Z
e
n=1
N
1 (y- - vn) + (x - )n) ( - _ -n (xx - bw))
1
En
Wn
exp
(2
- pm) En
2
1=1
1
Wn exp(
- (y- pm) " TT, (リ- kn)) exp( expllxpppp
1
1 (1
bn) (
1
En (xx -nn)
Z,
2
n=1
N
I -_En)- =
Wry (2Tr) 2 /dtt((tn))ii Mm 2
Zy
E
1 N
I __ En Wn (2T)) 8 Q(zlbn
N(ylpn En)) =
2g
2
1=1
17

---

1
N
Σ
Cn
(yllun
2nm
Wn Myllnn 2n)
ZR
Em=1
Wn n=l
n=1
N
N
Wn
Nylluun 2nn =
Σ
Nyllun En).
R=1
7=1
Wp
n=1
which finishes the derivation of the expression for the density of x²" (-(2r). Now we prove that P1 def
"f E P2())). For each v, consider (vla)
En YnN (nnV(yl En)
dy
+ Tr Sn.). Note that Its second moment is given by Ex=1 7n (1l [x)])((()]])) (x)]1
[x)|| = (A + 1) ( (Anb + x) IlA,b,, -| + *l - (IAA, + -| b,,l, + |=]]),
(I +I-
where || applied to matrix means the operator norm. Hence, one may conclude that [ [x)] is upper bounded by some quadratic polynomial of ||z]|| i.e., there exist constants On E IR 8, R. such that I| (x)]l² S ax + B,, ||,]||² We derive
S |1a| (1) 1x1 "d (u/ar) dmy (x) =
/ X n=1
Th (Ilu [1)]1())111 Tr TrEn)dPo(x))
=dFo(w)
In (a + 9nl1=e112 Tr TrEn) (dP))
n=1
3m f ||=||//PP((x) <0
Va (an + Tr
since Po E P2(()) by the assumption of the proposition.
It remains prove that x² is the unique EOT plan. According to our Theorem 3.2 one only has to ensure that fx Cc.e (Hx)) < oo. Just for completeness, we highlight that Sx Ca x² (--xr=)) (x) is lower-bounded since Ce.e is lower bounded, see the proof of The- orem 3.2 Anyway, this indifferent for us. We recall that x is an optimal plan between Po and F1 = Ti and f is an optimal potential by our construction. Thanks to the duality, we have
/ (z. x (+1x) dPo()))) = / (2))F())))))( + / f (v)dPP(())
/ [ Zz] dlP(o)) + / (U)ddP
(23)
where in transition to (2 we used our findings of line [9). Note that fy. f (y))dIP) (y) is finite since f E C2)) is dominated by a quadratic polynomial, and we have already proved that P1 has finite second moment. It remains to upper bound the first term in (23 We note that
Zn = / exp
(w) 11 v11
) dy = (V2T))D D / exp
f (((
Neul,, el)dy
6
(V2T)) P exp (sg (((()
(f (fddy)
2 (V2TT) D exp (/,
3 + allv/l
N(yi
N(y)o,, = (24)
3 (S+all=](+e)) (25) (V2m))) exp
where in transition to line 24 used the Jesnsen's inequality and a, 3 E R some constants They exist since f E C2()).. Indeed, there exist â, β (5( s
for which (*()) 2 3 all
8 all 1| (*(() 3 all 1|2, and we a - 3 -3. In turn, line 25 uses the explicit formula for the second moment of ((yl We use (55) to upper bound the first term in
e log Zo dPo(x) & / -elog ((122T)) exp
+||l==112 + acl) } ddP(2)= } =
18

---

cD log ( - - aeD / || |I/l@dlPoo(x)
It remains to note that the last value is finite, since 1' E P2(2) by the assumption.
du (alla) dy
(((0)>1l==-p) Therefore,
Proof of Corollary
3.4
We note that
o exp
(w)
dn (v)(z)
1 |00 yll
dm (ulz), (N dy
exp
exp
(26)
dy
2c
By comparing 26) with we see that exp
(())) indeed coincides with the Schrödinger potential 12 for the optimal drift follows from
6 (y). Formula
38 Proposition 4.1
Proof of Corollary 3.5 First, we prove that constructed F1 मं actually has finite entropy. This is needed to ensure that the assumptions of [38] Proposition 4.1J. This proposition provides the formula for the optimal drift [1] via the Schrödinger potential. We write
0 S KL ( (H100 = --( /S log log.N(yl0, 1dd7
- -H(mi)+ + log(2 + / dT)
(27)
From our Proposition 3.3 it follows that P1 "ं has finite second moment, Hence, the latter constant (2 is finite. Therefore, (xr] is upper bounded. To lower bound H recall that each x (-Ix) is a mixture of N Gaussians (Proposition 3.3) with (1-independent) covariances Σ, Thus, its density def
dar (ulx) is upper bounded by { max, [2m) D/2 (det - > dy
0 which also means that
(r) dy / dr (ylz) dx)(()) (dmō(s)<
dy
We conclude that
dzi(y) di (1) (1)2 2 dy
-f |loggetri (ddr(() -log
(Ti)=->) log
(28)
i.e. (mTī) is lower-bounded as well.
Having in mind our previous Corollary, we just substitute exp (((() of LSE 15 the Schrödinger potential o (y) to 12). We derive
potential f" as
"n (x, = cV log
N(vlæ, (1 f)cl)("())dy) (y)dy
JRD
Noylo, - ()eT) expl (x) ydy =
8 v log
J
/ N(ylz, (1 exp( JRD
é log DN wn Q(vl B(yl E -1 Ann
eV Vlog log
ydy
N
cv log
Wn
(1tt)))))((l (Amn)yy
( (2 t)) exp(-(() - xTTT
I
c log
Wn Jg
2c(1 - t) (y ((-z))) c A,,ydy
"The authors of 38 consider SB with reversible Wiener prior R. i.e. the standard Brownian motion starting at the Lebesgue measure. They deal with inf infre KL (2') which matches (up to an additive constant) our formulation (9) for = 1. Indeed, using the measure disintegration theorem, one can derive KL (I) (FP) KL (T") For other > 0, the analogous equivalence holds true.
19

---

N
1
VV log
Σ
Wn
2D
exp(-(y - -z)"
2e(1 - 1) (y - x))))(((()) Ib,, - -nAA)ddy +
n=1
eV log ((2 - t)) -))=
N
1 (y x)) exp(-((x - bn) )m)² An
eV log >
exp(- (11 - -x)"
(y - bn))dy= =
Wn
R
2c - 1)
2c
A
N
2(1 1) (11 2) 1 - x) + (v - bn.) (1 t)
<V log
Wn
exp (
(y bn)
RDD
n=
Next, we use
22
but with A instead of An and Σ instead of En. Also, we denote A bn + z):
M = (A, + I)
N
(1V - x)5= - - x) + (y - bn) A) (y bn)
eV log Σ
exp (
-x)"
Wn
RRD
2(1 - 1)
n=1
N
2(1 - 8) { (2y - (1-) (2x)² -1 (wv Hit++
VV log
Un
exp
j R20
n=1
1 Σ
(x - bx) )m -
(x - -bn)] dy=
62
7((2 bnn cl - Σ2
1
VV log
Un exp
2 -t) (I -bw)
n=1
(E,)
exp
(y -
(y (h:) dy }
RDD
2
(1 t)
Wn 2(x Un: d -n
1
(EA) -1
eV log
(2 - 1)
exp
(-
2
(y) 1T)
(1 (y 11,)) dy
el - b1
/
VV log
Wh 1xx bn:
/ (2x) 8 det((1 ( 1) ) N (1 1)
22 -1) 1)
el -1 ) (2iT(1 - 1)) det( +
e log
Wn 2 bn: 22 - 1)
NVyluun (1 -j2 =
N
el 52 2/ (2x 1)) det)
VV log Σ n=1
wn 2( bn,
2 8)
el - 1tt 2²(1 1)
V log
Wn 2(x bn:
det (ett ) + e log ((2T(1 1)) 8)
=0
((z(0n. el 5/
cV log
wn / Q(z))))
2 - t)
==1
which finishes the proof.
B Mixtures Benchmark Pairs: Details and Results
Parameters for constructing benchmark pairs. In our benchmark pairs, we choose all their hyperparameters manually to make sure the constructed distributions Po, IP are visually pleasant and
20

---

distinguishable. As Fo. we always use the centered Gaussian whose covariance matrix is 0.257. We use LSE function (15) with 5 for constructing the distribution P1. each setup, all A, are the same and given in Table 3 We pick such that / 1b,, T 2z )))) We sample b, randomly from uniform distribution on sphere with the radius 5.
D 16
64
128
10
mon! 7
Table 3: Matrices A, that we use to construct our mixtures benchmark pairs.
10 random samples from F1 and """""""".. we use the 10 random samples from learned distribution Ø1. For computing expectation and covariance hold-out test set containing 1000 samples æ Po. We compute
Evaluation details. For computing BW3-UVP(M1,
matrices of x² (-|xr) analytically (Proposition 3.3) and we estimate the expectation and covariance matrix of 7(1(2) by using 10s samples. We present results of evaluation Table and Table
We present additional trivial baseline for the conditional metric eBW?- UVVP(R, m²), which is given by the independent plan Po x P1. We compare other methods with this baseline in Table
-=0.1
t=1
6= 10
D ==2 D= 16 D=
D
128
D=
D=
16
D
64
D= 128
D=
D=
16
D
64
D 128
[LSOT]
SCONES]
1.06
4.24
6.67
11.54
1.11
2.98
1.33
7.89
NOT
2.62
0.225
1,872
EgNOT
0.09
0.31
0.88
0.22
0.46
0.3
0.85
0.12
0.077
0.02
0.15
0,23
ENOT
2.9
1.8
1.4
0.22
0.4
7.8
29
1,2
18.9
28
MLE-SB
0.01
0,14
0.97
2,08
0,005
0.09
0.56
1.46
0.01
1.02
6.65
23.4
DiISB
2.88
2.81
153.22
232.67
0.87
0.99
1.12
1.56
FB-SDE-A
FB-SDE-J]
0.03
0.05
0.25
2.96
0.07
0.13
1.52
0.48
Table 4: Comparisons of BW3-UVP 1 (%) between the target IP; and learned marginal TT. Colors indicate the metric value: BW?-UVP < 0.5, BW3-UVP (0.5, 1), Bw3-UVP 1.0.
c=0.1
c=1
c 10
D=2
D=16
D=64
D=128
D=2
D= 16
D=64
D= 128
D=2
D=16
D
D 128
LSST]]
SCONES
34.88
71.34
59.12
136.44
32.9
50.84
60.44
$2.11
NOT
1.94
13.67
11.74
11.4
4.77
23.27
41.75
26.56
2,86
4.57
3.41
6.56
EgNOT
129.8
75.2
60.4
43.2
80.4
74.4
63.8
53.2
4.14
2.64
2.36
1.31
ENOT
3.64
22
13.6
12.6
1.04
9.4
21.6
48
1.4
2.4
19.6
30
MLE-SB
4.57
16.12
16.1
17.81
4.13
9.08
18.05
15,226
1.61
1.27
3.9
12.9
DiffS
73.54
59.7
1386.4
1683.6
33.76
70.86
53.42
156.46
FR-SDE-A]
53.2
1156.82
1566.44
63.48
FB-SDE-J
51.34
89.16
119.32
173.96
29.34
69.2
155.14
177.52
Independent
166.0
152.0
126.0
110.0
86.0
80.0
72.0
60.0
4.2
2.52
2.26
2.4
Table 5: Comparisons of cBW3-UVP J. (%) between the optimal plan x² and the learned plan स.
Colors indicate the ratio of the metric to the independent baseline metric: ratio < 0.2, ratio E (0.2, 0.5), ratio > 0.5.
Colors for the Table To assign color for the metric BW3-UVI and cB UVP for each in the Table 2, we use the following rule: we assign the rank if a method's metric for given dimension D has the color green, the rank 2 if method's metric B -VV has the color orange and the rank 3 a method's 's metric -VV has the color red. To get the average rank, we take the mean of 4 ranks obtained for each dimension D and round it (1.5 and 2.5 are rounded to 1 and 2 respectively).
Extra qualitative results of EOT/SB solvers. Figure 5 and Figure 6 we present the additional qualitative comparison of solvers our mixtures benchmark pairs in 1) 16 with E (0.1, 10). The figures are designed similarly to Figure 3 for (D,e) (16, 1) in the main text. Note that case 10 (Figure 6 is extremely challenging; only [EgNOT] provides more-or-less reasonable results.
21

---

Px
y fg
Ficked x Po
P--
X
(a) Input and target.
(b) [MLE-SB].
(c) [SCONES].
(d) [NOT].
(e) [EgOTT]
-2-
-4 -2
-2
-4
-2
-2
-2
(l) True EOT plan x²
(g) [NOTT]
(h) DiiISB
(i [FB-SDE-]]
i LFB-SDE-]]
Figure 5: Qualitative results of EOT/SB solvers on our mixtures benchmark pair with
(D,c) = (16, 0.1). The distributions are visualized using 2 PCA components of target distr. P1-
X- oy-
Picked x P
y
(a) Input and target.
MLLE-B]]
(c) [SCONES].
(d) [OTT]
(e) [EgNOT].
o''kn((kk)
X
X
X
5
(f) True EOT plan
(g) ENOT]
(h) DiMSB
FB-SDEE-].
( FB-SDE-J]
Figure 6: Qualitative results of EOTYSB solvers on our mixtures benchmark pair with (D,c) = (16, 10). The distributions are visualized using 2 PCA components of target distr. P1.
Computational complexity. Sampling from Po is lightspeed as it is just sampling a Normal noise. Sampling from P2 is also fast, as is the Gaussian mixture (Proposition 3.3
C Images Benchmark Pairs: Details and Results
Parameters for constructing image benchmark pairs. We fix N = 100 random samples from PP for b, and choose all A,, I. We w, such that 7n Ton V (z]bn, (+ -
GLOW details. We use the code from the repository with the default parameters:
After training, the latent variable z is sampled from N(0, N(O, That is, the image distribution Po produced by the mapping o2T, to the image space with the learned normalizing flow i.c. G=N((-o22) in our construction.
o²T) with x² = 0.49 for image generation.
22

---

MCMC in the latent space of the normalizing flow. We test BOT/SB solvers in P1 - Po direction, ie.. recovering π² (aly) and generating clean samples x from noised y. Unfortunately, the reverse conditional OT plans x (all)) are not tractable as π (ylr). However, we note that
dn (xly) dy
dm (v)zz) d
dF
(29)
dy
da
i.e., the density of n²" (=ly)) it known up to the normalizing constant. Recall that here Po is constructed
using the normalizing flow and n² (-[ar) is a Gaussian mixture (Proposition 3.3 i.e., we indeed know the values of both terms. Therefore, one may use the well-celebrated Langevin dynamics to sample from x (w]z). Unfortunately, we found that such sampling in the image space is rather slow.
To overcome this issue, we employ the Langevin sampling in the latent space of the normalizing flow. is possible since the normalizing flow is a bijection between the space of images and the latent space. We use the standard notation 2 for the latent variable and G R + Rp for the normalizing
flow, i.e., I = G(2) ~ Po for 2 ~ p(z) def [= I²1). In this case, we have
dn (=/v) dr (xll)) det JG Ja- (x)] (u(xx) dFo(z)
det Jg
dz
dæ
dy dæ
dx (ulG(z)
dP det Jo-1(x)) dx (y|G(z)) p(z),
dx dæ
dy
p(z)
and we can derive the score finction
V= log
dn (=lu) which is needed for the Langevin dynamic as da
dr dzl())
dr (v((()((2) V logp(z).
V log
(30)
dz
dy
Hence, instead of doing non-trivial Langevin in the data space with da (url)) dir one may equivalently
do the sampling in the latent space by using the score (30) and then get x G(z). We empirically found this approach works much better, presumably due to the fact that (30p is just the score of the Normal distribution which is slightly adjusted with the information coming from x' x=' (y]G(z)).
For sampling, we employ the Metropolis-adiiee 10-$,10- and 10-5 for = 10, 1 and e 0.1, respectively. It provides the theoretical guarantees that the constructed Markov chain 21, 22, ....... converges to the distribution dr (=la) For initializing the Markov chain, sample a pair (a, y) ~ x² and use = G-'(x) as the initial state for the Langevin sampling to get new samples from π (-ly). This trick allows for improving the stability of sampling and the convergence speed since it provides a good starting point. We use Langevin algorithm.
Langevin algorithm with the time steps
dz
N = 200 steps for all the setups for the Metropolis-adjustee
In Figures 7 and 8
we provide additional examples of the samples from the ground truth plan x².
0.1, 1, 10 we prepare a test set with 10a samples from Pp. We use this set to calculate the FID 24 metric between the ground truth distribution Io and the model's I's marginal distribution æ to estimate how well the model restores the target distribution. This allows to access the generative performance of solvers, i.e., the quality of generated images and matching the target distribution. However, this metric does not assess the accuracy of the recovered EOT plan.
Metric 1. For each
Metric 2. For each e = 0.1, 1, 10, we prepare a test set containing 100 "noised" samples PP, and 5K samples x >= (-ly) for each "noised" sample y, i.e., 5K x 100 images for each in consideration. We propose to compute conditional FID to evaluate the difference between the conditional plans T (-lg) and 7 (-11)). That for each y we compute FID between x² (-19) and f-lyy), and then average the result for all test y. Clearly, such an evaluation is approximately 100 x00xtimes more consuming than computing the base FID. However, it allows us fairly assess the quality of the recovered EOT solution, and we recommend this metric as the main for future EOT/SB studies.
We again emphasize that, to the best of our knowledge, there is no scalable data->data EOT/SB solver to compare against. Hence, we report the results as-is for future methods to be able compare with them the baseline.
In Tables 6
present the evaluation results for ENOT 233.
Computational complexity. Sampling x ~ Po is just applying the trained GLOW neural network to noise vectors ~ ~N[[10 I2)). Sampling y ~ Pi (or y/ takes comparable time, as it is just extra
23

---

sampling from the Gaussian mixture with z-dependent parameters (Proposition 3.3 In turn, we noted above, sampling øly requires using the Langevin dynamic and takes considerable time. To obtain 3 test sets of 5K samples y ~ x² (-(aa) per each of 100 samples ~ Po, we employed 8x A100 GPUs. This generation of test datasets took approximately week.
[True EOT plan]
X x- TI (+
(a)
(b) E
(c) 10
Figure 7: Ground truth samples x (-19) images benchmark pairs.
X 2
[True EOT plan]
yI (+Ix
(a) a 0.1
(b) e =
(c) = 10
Figure 8: Ground truth samples ~ n (-12)) on images benchmark pairs.
24

---

0.1
10
0.1
10
FID
5.99
3.21
4.9
cFID
40.5
19.8
14.47
Table 6: Test FID of [ENOT] on images benchmark pairs.
Table 7: Test conditional FID of [ENOT] on images benchmark pairs
D
Details of EOT/SB Solvers
D.1
Mixtures Benchmark Pairs
LSOT
49
We use the part of the code of [SCONES] solver from the authors' repository
corresponding to learning dual OT potentials blob/main/cpat py and the barycentric projection blob/main/bproj. -p in the Gaussian case with configuration glob/main/cconfig co..//g-fffffgpp/ py.
[SCONES] [4]. We use the aforementioned official code for training of dual OT potentials. We with 20 components approximate the score of
employ sklearn. .ixtrre GaussianMixton the target distribution. For the rest, we employ their configuration blob/main/cc/ee/// caoaiinccoiiiggi py with batch size=1024 and the learning rate for Langevin sampling 5 10-4
[NOT] This algorithm [55] Algorithm 1] is a generic algorithm for weak OT. It works for transport costs C(z,, (+,,)())))) which are straightforward to estimate
Ir (potential and transport map)
Tatepa
99
1.0
Table 8: [NOT] training parameters
by using samples of (-|1z). Entropic cost Ce.e (6) for the mixture benchmark pairs experiment.
does not fit this requirement, it is not easy to estimate entropy from samples. To do it, one has to know the density of π (-|z). Thus, the authors of [NOT] skipped EOT setting. We fill this gap and do minor modification their algorithm. As the base implementation,
https: bttps://github.
Instead of the multi-layer perceptron generator, take conditional normalizing flow with RealNVP architecture with context-dependent latent normal distribution. This enables the access to the density of π (+|2:) and allows applying [OT] algorithm EOT. Our reimplementation available at
Due to the decreased expressivity of RealNVP compared to MLP from [NOT], we do more optimiza- tion steps for the transport map before updating potential as well as larger parameter count compared to the original solver implementation for similar task. We the same set of hyperparameters across all experiments with different (c, D). The hyperparameters are summarized Table g
[EgNOT]
42
We use the official code for [EgNOT] from
htttp:://gichub..cmm/eett/t/
For our mixture benchmark pairs experiment, we adapt the author's 's setup for the Gaussian-to-Gar Gausssin experiment from their original paper [42] $5.2]. In particular, we use the same architectures of neural networks, see 42 Appendix C.2], but change the [42 Algorithm 1]. since the orig- Moassiiantt-Mixturee We hypothesize that the observed
K Kin
500
1000
0.05
1.0
1024
Table 9: [EgNOT] training parameters
for the mixture benchmark pairs experiment.
hyper-parameten inal ones do not work properly when fitting failure is due to the short-run nature of the energy-based training algorithm. We suppose that sig- nificantly increasing the number of Langevin steps K used at the training stage may leverage the problem. The specific parameerrssstrrs of [gNNT]] algorithm are the same for all (c, D) pairs and provided in Table
We initialize the learning rate as = 10-5 and decrease its value during the training. Similar to original implementation of 42 we use replay buffer but found that a high probability (p = 0.95) of samples reusage does not improve the quality and sometimes leads to unstable training. turn, we choose p 0.5. The reported numbers in Tables 4 are gathered by launching the training process
25

---

for approximately 50K iterations and reporting the best-obtained metric. We understand that such an evaluation procedure ideal and does not provide statistically significant results. However, the qualitative results reported in Table seem to show the behaviour of [EgNOT] solver on our benchmark setup and reveal the key properties of the approach.
[ENOT] 23
We use the official code from
https: petpps//giituuee
We use the same hyperparameters for this setup as the authors [23] Appendix E], except the number of discretization steps N, which we to 200 as well as for other Schrödinger Bridge based methods. We also change the learning rate of the potential to 3 10-4 for the setups with 10.
[MLE-SB]
[22]
We tested the official code from
http://www........aaaaaccaaaaa/a Giinkhhorn
Instead of Gaussian processes, we used neural network as for [NOTT. We use N 200 discretization steps as for other SB solvers, 5000 IPF iterations, and 512 samples from distributions Po and P1 in each of them. We use the Adam optimizer with Ir = 10 for optimization.
[ [15]
We utilize the official code from
with their configuration blob/main/corttnec http://aann/ooo/tttasss///////// http://aaasst/dd.ee yaml for toy problems. We increase the number of steps of dynamics to 200 and the number of steps of the IPF procedure for dimensions 16, 64 and 128 30, 40 and 60, respectively.
FB-SDE-JJ 9].
We utilize the official code from
httpp:://ggtuub..oom/gl//////-1/
with their configuration blob/main/conff biob/mmain//caffigg//aaaalll https///eeffalll.cchceeerrrrr checkerboard-to-noise toy experiment, changing the number of steps of dynamics from 100 to 200 steps. Since their hyper- parameertstsrss are developed for their 2-dimensional experiments, we increase the number of iterations for dimensions 16, 64 and 128 to 15 000.
wwwckehboard.d.....o beapkkerborr__cofiigg.oo py for the
FB-SDEE-A 9 We also take the code from the same repository as above. We base our configura- tion on the authors' one (blob/main/ htt//onffggg /onfiig///a htt//effalttttteeeeeeeeeee cesttttmo__ttoot ttt_tttssrrll_oottee corral_cooiiii ig ggy for the moon-to-spine coos-tl--ppirl experiment. As earlier, we increase the number of steps of dynamics up 200. Also, we change the number of training epochs during one IPF procedure for dimensions 16, 64 and 128 to 2,4 and correspondingly.
D.2 Images Benchmark Pairs
ENOT 23 As well as for the mixtures benchmark pairs, we use the official code from
We use the same hyperparamettrss for this setup the authors 23 Appendix F] except the batch size which we set to 16 (/blobb/main//oteeookkte.. wwwpoo//eiiiiiiieeeeeeeee www.aaeeeeeeemeettss...om ipynb).
E
Additional Study of Hyperparameters of Solvers
To show that the default solvers parameters described in Appendix are already good choice, additionally try different values of some of the most important hyperparameters. We consider each of the solvers except LSOT] because it is anyway known poorly perform due to the systematic bias For the evaluation, we consider the mixtures benchmark pair with = 64 and where most of the solvers perform reasonably well. In the tables below, use "+" to mark the hyperparameters that we use for comparisons in $4..1
in its solutions 31 32
For [ENOT] solver, we consider the number of inner and outer problem iterations during the optimization and present the results in Table TO, The obtained results show that the performance increases slowly with increasing number of iterations of both types.
26

---

Inner iters
1
10
20
Outer iters
100
131.1
130.3
74.5
129.3
1000
28.77 77
47.36
25.91
20.16
TO000
24.46
37.36
23.07
18.03
Table 10: Comparison of cBW3-UVP (%) for [ENOT] on mixtures benchmark pairs for D 64, 1 and different hyperparameters.
For IPF-based SB solvers [MLE-SB] D FB-SDE-A and [FB-SDE-]], we try different numbers IPF iterations and the number samples used in each iteration. We present the results in Tables II
12 3 14 All of the IPF-based solvers learn inversion of diffusion process at each IPF step but they differ the way how this done. The typical number IPF steps used by each algorithm is affected by this difference. The performance increases slowly with the increase of the two hyperparameters considered, at the cost of a proportional increase iterations or in the number of samples used.
Samples per iter
64
128
256
512
IPF iters
100
23.45
24.50
16.64
14.23
1000
16.95
15.35
10.71
8.74
5000
11.55
11.24
12.96
8.41*
Table 11: Comparison of cBw3--vV & (%) for [MLE-SB] on mixtures benchmark pairs for
D
64,
1 and different hyperparameters.
Samples per iter
64
256
512
1024
IPF iters
16
62.66
60.42
58.88
57.02
32
62.90
59.42
57.76*
55.08
64
62.84
59.46
57.78
55.01
Table 12: Comparison of cBW2--VP (%) for [DiASB] on mixtures benchmark pairs for D 64, 1 and different hyperparameters.
Samples per iter
64
256
512
IPF iters
15000
173.16 16
163.04
160.5*
30000
168.86
165.06
156.5
Table 13: Comparison of cBWW3-VV 1 (%) for FB-SDE-J] on mixtures benchmark pairs for
D = 64, e = 1 and different hyperparameters.
Samples per iter
64
256
512
1024
IPF iters
16
40.86
40.43
39.76
37.74
32
40.44
38.90
38.36*
35.46
40.00
38.86
38.31
35.4
Table 14: Comparison of cB 33 ABW2-UVP UVP (%) for [FB-SDE-A on mixtures benchmark pairs for D 64, = 1 and different hyperparameters.
For [SCONES] and [ggOTT] solvers, we consider the number of Langevin steps and the Langevin step size and present the results in Table 15 and Table 16 For [SCONES] the results obtained show that the performance increases slowly with increasing Langevin steps and decreasing Langevin step size. For EgNOT] the trends slightly different, since the optimal Langevin step size the interval (0.1, 0.2]. Anyway, our selected parameters are reasonable ones because specifying an enormously large number of Langevin steps for these solvers is sort of impractical.
to be
Finally, for [OT]] we consider the number of inner problem steps and the hidden size of the used neural network (conditional normalizing flow). We present results in Table 17
27

---

Langevin sleps
64
256
512
1024
Langevin step size
92.35
89.17
86.48
86.33+
93.51
90.41
88.22
87.74
Table 15: Comparison of cBW? UVP (%) for [SCONES] mixtures benchmark pairs for D = 64, = 1 and different hyperparameters.
Langevin steps
100
200
500
1000
Langevin step size
0.01
70.9
70.98
72.9
68.13
0.02
71.31
67.14
69.11
69.02
0.05
68.78
68.59
63.73*
56.84
0.1
64.52
57.45
52.35
51.9
0.2
58.22
60.08
58.93
41.31
Table 16: Comparison of cB cBW3--VP + (%) for EgNOT) on mixtures benchmark pairs for
D
64,
1 and different hyperparameters.
Hidden size
64
128
192
256
320
384
448
512
Inner steps
93.14
167.05
149.52
189.0
89.1
161.66
176.43
175.67
5
82.64
86.09
82.18
190.04
147.31
105.46
103.5
150.76
10
163.47
146.68
53.26
137.47
100.84
171.65
115.84
126.96
100
18.68
21.4
14.64
18.08
16.66
20.64*
18.71
15.15
200
61.99
52.74
58.63
53.89
52.44
55.3
55.02
54.75
Table 17: Comparison of cBW23-U -VVP J. (%) for [NOT] mixtures benchmark pairs for D = 64, = 1 and different hyperparameters.
Discussion. From the results it can be seen that for the most solvers' dependence on the considered hyperparameters is almost monotonic and the hyperparameterss chosen for the solver comparison the mixtures setup are in the region where the metric growth is almost saturated.
Qualitative Evaluation of the Drift Learned with SB methods
Our benchmark primarily aimed quantifying the recovered conditional EOT plan f(-1z). Thanks to our Proposition 3.5 our benchmark provides not only the ground truth conditional EOT plan T- (-1z), but the optimal SB drift v² (ar, well. This means that for SB solvers may additionally compare their recovered SB drift with the ground truth drift v*. Here we do this for [MLF-SB] DiffSB]] [NOTT] FB-SDE-A] [B-SSE--] solvers by using our mixtures pairs.
METRICS. Recall that T-- the Schrödinger bridge ITPp and let T3 denote the learned process:
dX, ह(x, t)dt + VedW,,
Xo ~ Pu-
Both T., and T3 diffusion processes which start at distribution Po l 0 and have fixed volatility e. Their respective drifts are v" and T. For each time E [0, 1]. consider
Li def ET,, ||00 (X 0( (X,, 4)12
(31)
c del ETg (X,,t) 7(X,, 1)
(32)
which are the expected squared differences between the ground truth v² and learned T drifts at the time l. In (3T), the expectation is w.r.t. X, coming from the SB trajectories of T.+, while in (32) w.r.t the learned trajectories from Th. Reporting this metric for all the time steps, all the mixtures pairs and solvers would be an overkill. In what follows, we use this metric for quantitative analysis.
First, for D = 16 and E (0.1, 10), we plot these metrics (as a function of time t). The results for all the solvers are shown in Figure 9 Second, provide Table 19 where for D E (2, 16, 64, 128) and E (0.1, 10) report _ metrics averaged over E [0, 1]. Namely, we report
KL (T
def
2c {'' (t and
RKL TT) def
22 ttdt.
2
(33)
2c
28

---

FBB-DE-AA
[FB-SDE-A]
DiFFSB
[DiffBB]
50
FBB-DDEJI
FBB-DD
[wa [t]
ENOOTI
40
IMLE-SB]
|MLE-BBI
30
(a) Craa 11 for (D,c) = (16, 0.1)
(b) 2.
C [4] for (D,c) (16.0.1)
100
FFB-SDE-JI
FFB-SDE-J
80
JENOTI
(ENOT]
IMLE-SB]
[MLE-SB]
60
40
H|
20
50
75
100
125
150
25
50
125
150
175
200
time
t. time
(c) C (d) for (D,c) (16, 1)
C2
[f) for (D, (16,
Figure 9: 2 metrics between the ground truth drift v" and the drift 7 learned by SB solvers.
We write "KL" and "RKL" not by an accident. Thanks to the well-celebrated Girsanov's theorem, these are indeed the forward and reverse KI divergences between processes T,, and Tg.
In all the SB solvers, consider 200 time discretization steps During testing, we evaluate L metrics (31) and (32) on the same time steps. To estimate (31) and (3)), we use 10a samples X, which are taken from random trajectories of processes T., and Tg. These trajectories are simulated via the standard Euler-Maruyama method.
200 200 ... 1} for their training.
6=0.1
t=1
D=2 2
D=
D
D= 128
D=2
D=
D
D 128
ENOT
5.49
6.59
0.86
37.53
DiMIS
6.96
12.89
12.28
>1000
>1000
>100
FB-SDE-A
6.9
11.08
10.59
>1000
>1000
>1000
FR-SDE-J
3.02
3.02
9.60
28.85
18.79
44.79
629.28
>1000
MLE-SB
0.62
2.63
4.76
7.86
0.96
1.86
9.66
31.95
Table 18: Forward KL between the ground truth SB process T. and the process T:
learned with SB solvers on our mixtures benchmark pairs.
D=2 D=
D=64
D=128
D=2
D=16
D=
D=128
ENOT
72.86
78.98
135.29
221.26
18.40
49.65
177.02
348.05
DifISB
11.85
21.16
121.43
>1000
>100
>1000
FB-SDE-
12.29
19.40
100.22
>100
>1000
>100
FB-SDE-J
8.03
12.11
17.16
49.32
64.37
123.68
>1000
>1000
MLE.SB
18.03
28.24
163.34
254.16
22.80
86.07
296.97
636.27
Table 19: Reverse KL between the ground truth SB process T.- and the process T3 learned with SB solvers on our mixtures benchmark pairs.
DISCUSSION. Interestingly, we see that the forward KL divergence shows smoother behaviour than the RKL for almost all SB solvers. According to our evaluation, the [MLE-SB] and [NNTT] solvers mostly beat every other solver the forward KI. metric. At the same time, the RKL metric of [ENOT] is surprisingly the worst. While make all these observations, we do not know how
29

---

to explain them. We hope that the question of the interpretation of the KL and RKL values will be addressed in future SB studies.
G Potential Societal Impact
Our proposed approach deals with generative models based on Entropic Optimal Transport and Schrödinger Bridge principles. Such models form and emergent subarea in the field of machine learning research and could be used for various purposes in the industry including image manipulation. artificial content rendering, graphical design, etc. Our benchmark is a step towards improving the reliability, robustness and transparency of these models. One potential negative of our work that improving generative models may lead to transforming some jobs in the industry.
H Building Benchmarks from Real Data
In this section, we present a simple heuristic recipe to build benchmark pairs similar to some given real-world data. To illustrate the recipe, we consider toy 2D data example and several single-cell datasets 36 g Code and data for the experiments in this section can be found in the
folder of our repository.
H.1 Recipe for Building Benchmark Pairs form Data.
For constructing distribution pairs similar to some given data, we consider a pair of original and target datasets obtained from the true distributions Fo and FT. respectively. We heuristically initialize the LSE potential (15) f (y) = log Ex= Wn 2, e AA, with 6,, as cluster centers obtained from the K-means clustering algorithm applied to the target data from P1. The weights W,, are chosen to be 1/N and matrices A. = Al are diagonal where is a manually-chosen parameter (shared between all A,). For any the conditional plan x (-(2:) for LSE potential f" is just Gaussian mixture and the mean of each its component is largely determined by b, (Proposition 3.3). We empirically found that the resulting constructed distribution dP (v) = dri (y) J dr (y) [ ddl from Po resembles the Gaussian mixture approximation of the target dataset if one managed to find proper value of X.
In the rest of this section, we use the described recipe to construct benchmark pairs from data to show that the LSE parameterization of the potential provides wide class of EOT/SB solutions and even allows constructing a benchmark similar to real data.
H.2 Benchmark Pairs for 2D data.
Code and data for the experiment described in this section be found in the folder of our repository.
To begin with, present the results of constructing a benchmark pair from 2D data. We consider Gaussian distribution IP as the source distribution and two moons PP as the target distribution. We aim to the previously described recipe H.. to find parameters of the LSE potential construct an EOT solution between Po and an approximation of F, denoted PP, Here we consider EOT with e = 0.05, use N = 100 for LSE potentials, and choose À = 50. The result is in Figure I0
As seen from the figure, the constructed target benchmark distribution TP, is similar to the target distribution P,. In tur, the EOT plan maps & Po to the close regions of the target distribution.
H.3 Single-cell RNA Data
Code and data for the experiment described in this section can be found in the folder wtt.o/ooorrruiiooooommlllleeeee.
of our repository.
We consider the same setup as in (3 $5.2]. We use their data from the supplementary materials The provided data displays the progression of human embryonic stem cells as they differentiate from embryoid bodies into a range of cell types, such as mesoderm, endoderm, neuroectoderm, and neural
30

---

Inous distribution
target distribution i%
targst disribution.
plana [[]]
8 00
6
0
-1
-3
Figure 10: Gaussian
Two Moons benchmark pair.
crest, throughout span of 27 days. The cell samples (approximately 2000 ones per each time period) were gathered at five distinet intervals (lg: day to 3, f1: day to 9, t2: day 12 to 15, lz: day 18 to 21, 14: day 24 27). These collected cells were evaluated via scRNAseq. subjected to quality control filtering, and then projected onto a 5-dimensional feature space utilizing principal component analysis (PCA).
To construct the benchmark pair using the LSE potential, we consider 100 and employ the train data at times to and t4. Then use the constructed benchmark plan x- (+1x) to map source data at time ty to the data at time tı and obtain benchmark target distribution samples PI. Finally, we fit TSNE [11 to the combined dataset of samples from P1 and P. and then plot their projections in Figure The resulting plots are very similar, confirming that the constructed benchmark target data resembles the considered single-cell target data.
250, = 100 and
Real larget distr distribution P:
Benchmark target distribution A
-42
Figure 11: TSNE visualization of Single-cell RNA target data and our constructed target data.
H.4 Single-cell Drugs Data
Code and data for the experiment described in this section can be found in the folder benchmark_ htt.rrrriiioooeaaeleeeeeeeeeeeee
of our repository.
Method
scGen
CAB
CellOT [8] EOT Benchmark (ours)
MMDJ
0.0241
0.0074
0.0013
0.0036
Table 20: MMDJ distances (on the test data) between the observed perturbed cells IP] and predicted responses from control cells P..
In 8), the authors consider the problem of predicting single-cell drug responses for drugs with differ- ent molecular effects, using melanoma cell lines profiled by 4i technology (single-cell technology). Utilizing blend of two melanoma tumor cell lines at a 1:1 ratio, total of 21,650 cells were imaged. Within this dataset, 11,526 cells existed the untreated control state, 2,364 received Erlotinib treatment, 2,650 underwent Imatinib treatment, 2,683 were subjected Trametinib treatment, and
31

---

2,417 were treated with a combination of Trametinib and Erlotinib. After preprocessing, each cell is described by 78 features. The train-test split with each drug is 80:20.
In this example, we consider cell data before treatment (Po) and after treatment with Erlotninib (P1).. For the construction of the benchmark pair using LSE potential, we consider N = 250, 6= and À = 20. As with the single cell RNA data H... we fit the TSNE [51] on a combined dataset of samples from P and P1 and then plot their projections in Figure 11 As seen from the visualizations,
TSNE projections the real data and the mapped data are similar.
Real target distribution
Constructed benchmark target distribution P:
-42
-20
40
20
Figure 12: TSNE visualization of Single-cell Drugs target data and our constructed target data.
In addition, we quantitatively evaluate on the test data how well the constructed target distribution P matches the true data distribution Pl. We employ the same MMD metric as the authors and present the results in Table 20 The data for the baselines scGen, cAE and the authors' method CellOT are
taken from g
As one can see, our approach is even better than two the baselines considered.
32


Reviews:
[{"review_id": "Ojx7vEjGn3", "reviewer": "Reviewer_jvUZ", "summary": "", "questions": "", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "This paper indeed provides clear guidelines for creating synthetic evaluation data which are theoretically sound. It will direct benefit the community of optimal transport study. And it will potentially benefit the research of generated model.\n\nMany technical details introduced when creating the dataset could also be baseline methods that facilitate the following research. ", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "G2fgJtF5Fi", "reviewer": "Reviewer_XL1t", "summary": "", "questions": "", "limitations": "Several limitations seem to be mentioned in the paper. Further limitations can also be found in \"Opportunities For Improvement\" above. \nThe section on societal impact could be improved, however, seems sufficient for the present paper.", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "- Novel methodology: the work presents a novel, theoretically grounded, and rather generic approach to construct benchmark pairs of probability distributions with known EOT/SB solutions.\n\n- Standardization: these benchmark problems fill a significant gap in the field and enable standardized testing of corresponding solvers.\n \n- High-dimensional evaluation: the methodology enables the evaluation of solvers in high-dimensional spaces, such as image spaces, which are relevant to many generative modeling tasks.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "clTOapDYAk", "reviewer": "Reviewer_UzdZ", "summary": "", "questions": "", "limitations": "The  most solvers’ performance significantly depends on the selected hyper-parameters.\nIt may be a serious issue in adopting this method to generate dataset.", "rating": 9, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "They has provided necessary theoretic result to pave the way for further study on the dataset for SB solvers. The dataset they obtained is of high quality.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "7mwYO6D6AG", "reviewer": "Reviewer_oGez", "summary": "", "questions": "", "limitations": "See above", "rating": 7, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "The authors explain the limitations of existing methods and the knowledge gaps in the field. The interconnection of SB and EOT with diffusion models is articulated, and the potential application of these methods in generative modelling is discussed.\n\nThe theoretical grounding of the paper is strong, with detailed information on classic and weak OT formulations, the entropic OT formulation, and the interconnection between EOT and SB. \n\nDespite some limitations (see below), the methodology provides a valuable tool for the evaluation of EOT/SB solvers and offers a detailed and carefully constructed approach to benchmarking in this complex and challenging field. The work fills a critical gap by introducing a first theoretically-grounded EOT/SB benchmark. This provides a standard way to test existing neural (continuous) solvers which previously did not exist.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "T9tNgPnaaR", "reviewer": "Reviewer_p1pQ", "summary": "", "questions": "", "limitations": "The authors have adequately addressed the limitations.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "The topic is of great importance to the research community.   \nThe idea is novel and the proofs seem to be correct.   \nThe upside is the constructed pairs of probability distribution (i.e., P0 and P1) have analytically known SB, which may be beneficial for researchers to further develop and evaluate their algorithms. \nThe writing is good. The codebase is well-organized and easy to follow.", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}]

Author rebuttal:

Meta-review: This work presents an interesting benchmark for optimal transport problems, which have become increasingly popular in the machine learning community. In particular, the authors focus on entropic optimal transport and Schrodinger bridge, and provide benchmarks based on both synthetic and real imaging data. While reviews ranged from marginal to strong support for the paper, they are all above acceptance threshold, and I believe it is generally agreed the paper would be of sufficient interest and significance for this track, and therefore I recommend acceptance.
Decision: Accept (Poster)