SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 3.2 (Entropic optimal transport benchmark constructor). Let  $\mathbb{P}_0\in \mathcal{P}(\mathcal{X})$  be a given distribution and  $f^{*}\in \mathcal{C}_{p}(\mathcal{Y})$  be a given potential. Assume that  $c:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$  is lower bounded and  $(x,\mu)\mapsto \int_{\mathcal{V}}c(x,y)d\mu (y)$  is lower semi-continuous in  $\mathcal{X}\times \mathcal{P}_p(\mathcal{Y})$ . Furthermore, assume that there exists  $M\in \mathbb{R}_{+}$  such that for all  $x\in \mathcal{X}$  it holds that  $M_x\stackrel {\text{def}}{=}\int_{\mathcal{V}}\exp \left(-\frac{c(x,y)}{\epsilon}\right)dy\leq M$ . Assume that for all  $x\in \mathcal{X}$  value  $Z_{x}\stackrel {\text{def}}{=}\int_{\mathcal{V}}\exp \left(\frac{f^{*}(y) - c(x,y)}{\epsilon}\right)dy$  is finite. Consider the joint distribution  $\pi^{*}\in \mathcal{P}(\mathcal{X}\times \mathcal{Y})$  whose first marginal distribution satisfies  $\pi_0^* = \mathbb{P}_0$  and for all  $x\in \mathcal{X}$  it holds that
$$
\frac {d \pi^ {*} (y | x)}{d y} = \frac {1}{Z _ {x}} \exp \left(\frac {f ^ {*} (y) - c (x , y)}{\epsilon}\right) \tag {14}
$$
and  $\pi^{*}(\cdot |x)\in \mathcal{P}_{p}(\mathcal{Y})$ . Then if  $\mathbb{P}_1\stackrel {\text{def}}{=}\pi_1^*$  belongs to  $\mathcal{P}_p(\mathcal{V})$ , the distribution  $\pi^{*}$  is an EOT plan for  $\mathbb{P}_0,\mathbb{P}_1$  and cost  $C_{c,\epsilon}$ . Moreover, if  $\int_{\mathcal{X}}C_{c,\epsilon}\big(x,\pi^{*}(\cdot |x)\big)d\mathbb{P}_{0}(x) < \infty$ , then  $\pi^{*}$  is the unique EOT plan.
Our result above requires some technical assumptions on  $c$  and  $f^*$  but a reader should not worry as they are easy to satisfy in popular cases such as the quadratic cost  $c(x, y) = \frac{1}{2} \| x - y \|^2$  (§3.3). The important thing is that our result allows sampling  $y \sim \pi^*(\cdot|x)$  from the conditional EOT plan by using MCMC methods [5, §11.2] since (14) provides the unnormalized density of  $\pi^*(y|x)$ . 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 a way to overcome the challenging sampling problem by considering the case  $c(x,y) = \frac{||x - y||^2}{2}$  and the special family of functions  $f^{*}$ . For brevity, for a matrix  $A\in \mathbb{R}^{D\times D}$  and  $b\in \mathbb{R}^D$ , we introduce  $\mathcal{Q}(y|b,A)\stackrel {\mathrm{def}}{=}\exp \left[-\frac{1}{2} (y - b)^TA(y - b)\right]$ . Henceforth, we choose the potential  $f^{*}$  to be a weighted log-sum-exp (LSE) of  $N$  quadratic functions:
$$
f ^ {*} (y) \stackrel {\text {d e f}} {=} \epsilon \log \sum_ {n = 1} ^ {N} w _ {n} \mathcal {Q} \left(y \mid b _ {n}, \epsilon^ {- 1} A _ {n}\right) \tag {15}
$$
Here  $w_{n} \geq 0$  and we put  $A_{n}$  to be a symmetric matrix with eigenvalues in range  $(-1, +\infty)$ . We say that such potentials  $f^{*}$  are appropriate. One may also check that  $f^{*} \in \mathcal{C}_{2}(\mathcal{V})$  as it is just the LSE smoothing of quadratic functions. Importantly, for this potential  $f$  and the quadratic cost,  $\pi^{*}(\cdot |x)$  is a Gaussian mixture, from which one can efficiently sample without using MCMC methods.
Proposition 3.3 (Entropic OT solution for LSE potentials). Let  $f^{*}$  be a given appropriate LSE potential (15) and let  $\mathbb{P}_0 \in \mathcal{P}_2(\mathcal{X}) \subset \mathcal{P}(\mathcal{X})$ . Consider the plan  $d\pi^{*}(x,y) = d\pi^{*}(y|x)d\mathbb{P}_{0}(x)$ , where
$$
\begin{array}{l} \frac {d \pi^ {*} (y | x)}{d y} = \sum_ {n = 1} ^ {N} \gamma_ {n} \mathcal {N} (y | \mu_ {n} (x), \Sigma_ {n}) w i t h \Sigma_ {n} \stackrel {{d e f}} {=} \epsilon (A _ {n} + I) ^ {- 1}, \mu_ {n} (x) \stackrel {{d e f}} {=} (A _ {n} + I) ^ {- 1} (A _ {n} b _ {n} + x), \\ \gamma_ {n} \stackrel {{d e f}} {{=}} \widetilde {w} _ {n} / \sum_ {n = 1} ^ {N} \widetilde {w} _ {n}, \qquad \widetilde {w} _ {n} \stackrel {{d e f}} {{=}} w _ {n} (2 \pi) ^ {\frac {D}{2}} \sqrt {\det (\Sigma_ {n})} \mathcal {Q} (x | b _ {n}, \frac {1}{\epsilon} I - \frac {1}{\epsilon^ {2}} \Sigma_ {n}). \\ \end{array}
$$
Then it holds that  $\mathbb{P}_1\stackrel {\text{def}}{=}\pi_1^*$  belongs to  $\mathcal{P}_2(\mathcal{Y})$  and the joint distribution  $\pi^{*}$  is the unique EOT plan between  $\mathbb{P}_0$  and  $\mathbb{P}_1$  for cost  $C_{c,\epsilon}$  with  $c(x,y) = \frac{1}{2}\| x - y\|^2$ .
We emphasize that although each conditional distribution  $\pi^{*}(\cdot |x)$  is a Gaussian mixture, in general, this does not mean that  $\pi^{*}$  or  $\mathbb{P}_1 = \pi_1^*$  is a Gaussian mixture, even when  $\mathbb{P}_0$  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 a link between EOT and SB, our approach allows us to immediately obtain a solution to the Schrödinger Bridge between  $\mathbb{P}_0$  and  $\mathbb{P}_1$  (constructed with an LSE potential  $f^*$ ).
Corollary 3.4 (Solution for SB between  $\mathbb{P}_0$  and constructed  $\mathbb{P}_1$ ). In the context of Theorem 3.2, let  $p = 2$  and consider  $c(x,y) = \frac{1}{2}\| x - y\|^2$ . Assume that  $\mathbb{P}_0 \in \mathcal{P}_{2,ac}(\mathcal{X}) \subset \mathcal{P}(\mathcal{X})$  and both  $\mathbb{P}_0$  and  $\mathbb{P}_1$  (constructed with a given  $f^*$ ) have finite entropy. Then it holds that  $\varphi^*(y) \stackrel{\text{def}}{=} \exp\left(\frac{f^*(y)}{\epsilon}\right)$  is a Schrödinger potential providing the optimal drift  $v^*$  for SB via formula (12).
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  $c(x,y) = \frac{|x - y|^2}{2}$  and our LSE (15) potentials  $f^*$ , we can derive the optimal drift explicitly.
EOT solvers	SB solvers
ε	Metric	[LSOT]	[SCONES]	[NOT]	[ EgNOT]	[ENOT]	[MLE-SB]	[DiffSB]	[FB-SDE-A]	[FB-SDE-J]
0.1	BW2-UVP	xDo not work for small εdue to numerical instability[14, §5.1].
cBW2-UVP
1	BW2-UVP
cBW2-UVP
10	BW2-UVP	Diverge with the default hyperparameters.May require more hyperparameter tuning.
cBW2-UVP
Table 2: The summary of EOT/SB solvers' quantitative performance in cB $\mathbb{W}_2^2$ -UVP and B $\mathbb{W}_2^2$ -UVP metrics on our mixtures pairs. Detailed evaluation and coloring principles are given in Appendix B.
Corollary 3.5 (SB solution for LSE potentials). Let  $f^{*}$  be a given appropriate LSE potential (15) and consider a distribution  $\mathbb{P}_0 \in \mathcal{P}_{2,ac}(\mathcal{X})$  with finite entropy. Let  $\mathbb{P}_1$  be the one constructed in Proposition 3.3. Then it holds that  $\mathbb{P}_1$  has finite entropy, belongs to  $\mathcal{P}_{2,ac}(\mathcal{Y})$  and
$$
v ^ {*} (x, t) = \nabla_ {x} \log \sum_ {n = 1} ^ {N} w _ {n} \sqrt {\det (\Sigma_ {n} ^ {t})} \mathcal {Q} (x | b _ {n}, \frac {1}{\epsilon (1 - t)} I - \frac {1}{\epsilon^ {2} (1 - t)} \Sigma_ {n} ^ {t}) \tag {16}
$$
is the optimal drift for the SB between  $\mathbb{P}_0$  and  $\mathbb{P}_1$ . Here  $A_{n}^{t}\stackrel {\text{def}}{=}(1 - t)A_{n}$  and  $\Sigma_n^t\stackrel {\text{def}}{=} \epsilon (A_n^t +I)^{-1}$ .
4 Constructing Benchmark Pairs for OT and SB: Implementation
Our benchmark is implemented using PyTorch framework and is publicly available at
It provides code to sample from our constructed continuous benchmark pairs  $(\mathbb{P}_0,\mathbb{P}_1)$  for various  $\epsilon$  see §4.1 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 to 12288 and  $\epsilon \in \{0.1, 1, 10\}$ . Our mixtures pairs simulate noise→data setup and images pairs simulate data→data case.
Mixtures benchmark pairs. We consider EOT with  $\epsilon \in \{0.1, 1, 10\}$  in space  $\mathbb{R}^D$  with dimension  $D \in \{2, 16, 64, 128\}$ . We use a centered Gaussian as  $\mathbb{P}_0$  and we use LSE function (15) with  $N = 5$  for constructing  $\mathbb{P}_1$  (Proposition 3.3). In this case, the constructed distribution  $\mathbb{P}_1$  has 5 modes (Fig. 3a, 3f). Details of particular parameters  $(A_n, b_n, w_n, \text{etc.})$  are given in Appendix B.
Images benchmark pairs. We consider EOT with  $\epsilon \in \{0.1, 1, 10\}$ . As distribution  $\mathbb{P}_0$ , we use the approximation of the distribution of  $64 \times 64$  RGB images ( $D = 12288$ ) of CelebA faces dataset [39]. Namely, we train a normalizing flow with Glow architecture [29]. It is absolutely continuous by the construction and allows straightforward sampling from  $\mathbb{P}_0$ . For constructing distribution  $\mathbb{P}_1$ , we also use LSE function  $f^*$ . We fix  $N = 100$  random samples from  $\mathbb{P}_0$  for  $b_n$  and choose all  $A_n \equiv I$ . Details of  $w_n$  are given in Appendix C. For these parameters, samples from  $\mathbb{P}_1$  look like noised samples from  $\mathbb{P}_0$  which are shifted to one of  $\mu_n$  (Fig. 4).
By the construction of distribution  $\mathbb{P}_1$ , obtaining the conditional EOT plan  $\pi^{*}(y|x)$  between  $(\mathbb{P}_0,\mathbb{P}_1)$  may be viewed as learning the noising model. From the practical perspective, this looks less interesting than learning the de-noising model  $\pi^{*}(x|y)$ . Due to this, working with images in §5, we always test EOT solvers in  $\mathbb{P}_1\to \mathbb{P}_0$  direction, i.e., recovering  $\pi^{*}(x|y)$  and generating clean samples  $x$  form noised  $y$ . The reverse conditional OT plans  $\pi^{*}(x|y)$  are not as tractable as  $\pi^{*}(y|x)$ . 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 (§2) provide an approximation of the conditional OT plan  $\widehat{\pi}(\cdot|x) \approx \pi^*(\cdot|x)$  from which one can sample (given  $x \sim \mathbb{P}_0$ ). In particular, SB solvers recover the approximation of the optimal drift  $\widehat{v} \approx v^*$ ; it is anyway used to produce samples  $y \sim \widehat{\pi}(\cdot|x)$  via solving SDE  $dX_t = \widehat{v}(x,t)dt + \sqrt{\epsilon} dW_t$  starting from  $X_0 = x$  at time  $t = 0$ . Therefore, the main goal of our benchmark is to provide a way to compare such approximations  $\widehat{\pi}, \widehat{v}$  with the ground truth. Prior to our work, this was not possible due to the lack of non-trivial pairs  $(\mathbb{P}_0, \mathbb{P}_1)$  with known  $\pi^*, v^*$ .
(a) Input and target.
(b) [MLE-SB].
(c) [SCONES].
(d) [NOT].
(e) [EgNOT].
(f) True EOT plan  $\pi^{*}$ .
(g) [ENOT].
(h) [DiffSB].
(i) [FB-SDE-A].
Figure 3: Qualitative results of EOT/SB solvers on our mixtures benchmark pair with
(j) [FB-SDE-J].
For each of the constructed pairs  $(\mathbb{P}_0,\mathbb{P}_1)$ , we provide the code to do 5 main things: (a) sample  $x\sim \mathbb{P}_0$ ; (b) sample  $y\sim \pi^{*}(\cdot |x)$  for any given  $x$ ; (c) sample pairs  $(x,y)\sim \pi^{*}$  from the EOT plan; (d) sample  $y\sim \mathbb{P}_1$ ; (e) compute the optimal drift  $v^{*}(x,t)$ . Function (c) is just a combination of (a) and (b). For images pairs we implement the extra functionality (f) to sample  $x\sim \pi^{*}(\cdot |y)$  using MCMC. Sampling (d) is implemented via discarding  $x$  (not returning it to a user) in  $(x,y)$  in (c).
When training a neural EOT/SB solver, one should use random batches from  $(\mathbf{a},\mathbf{d})$ . Everything coming from  $(\overline{\mathbf{b}},\mathbf{c},\mathbf{e},\mathbf{f})$  should be considered as test information and used only for evaluation purposes. Also, for each of the benchmark pairs (mixtures and images), we provide a hold-out test for evaluation.
5 Experiments: Testing EOT and SB Solvers on Our Benchmark Pairs
Now we train various existing EOT/SB solvers from Table 1 on our benchmark pairs  $(\mathbb{P}_0,\mathbb{P}_1)$  to 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:
$$
\mathrm {c B W} _ {2} ^ {2} - \operatorname {U V P} \left(\widehat {\pi}, \pi^ {*}\right) \stackrel {\text {def}} {=} \frac {100 \%}{\frac {1}{2} \operatorname {V a r} \left(\mathbb {P} _ {1}\right)} \int_ {\chi} \mathrm {B W} _ {2} ^ {2} \left(\widehat {\pi} (\cdot | x), \pi^ {*} (\cdot | x)\right) d \mathbb {P} _ {0} (x). \tag{17}
$$
For each  $x$  we compare conditional distributions  $\widehat{\pi}(\cdot|x)$  and  $\pi^*(\cdot|x)$  with each other by using the Bures-Wasserstein metric [16], i.e., the Wasserstein-2 distance between Gaussian approximations of distributions. Then we average this metric w.r.t.  $x \sim \mathbb{P}_0$ . The final normalization  $\frac{1}{2}\mathrm{Var}(\mathbb{P}_1)$  is chosen so that the trivial baseline which maps the entire  $\mathbb{P}_0$  to the mean of  $\mathbb{P}_1$  provides 100% error. Metric (17) is a modification of the standard  $\mathbb{BW}_2^2$ -UVP [14, 23, 42, 33, 18] for the conditional setting. For completeness, we also report the standard  $\mathbb{BW}_2^2$ -UVP to check how well  $\widehat{\pi}_1$  matches  $\pi_1^*$ .
DISCLAIMER. We found that most solvers' performance significantly depends on the selected hyper-parameters. 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 invite the interested authors of solvers to improve the results for their solvers. Meanwhile, we report the results of solvers with their default config 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 is to find out and explain the issues of the methods which are due to their principle rather than non-optimal hyperparameter selection.
The detailed results are in Appendix B. Here we give their concise summary (Table 2) and give a qualitative example (Fig. 3) of solvers' performance on our mixtures pair with  $(D, \epsilon) = (16, 1)$ .
EOT SOLVERS. [LSOT] and [SCONES] solvers work only for medium/large  $\epsilon = 1, 10$ . [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
Figure 4: Qualitative comparison of ground truth samples  $x \sim \pi^{*}(\cdot |y)$  with samples produced by [ENOT]. With the increase of  $\epsilon$ , the diversity increases but the precision of 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).  $\lceil \mathbf{EgNOT}\rceil$  solver also employs Langevin dynamic and possibly experiences the same issue (Fig. 3e). Interestingly, our evaluation shows that it provides a better metric in matching the target distribution  $\mathbb{P}_1$ .  $\lceil \mathbf{NOT}\rceil$  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  $\lceil \mathbf{MLE - SB}\rceil$ , the original authors' implementation uses Gaussian processes as parametric approximators instead of neural nets [52]. Since other SB solvers ( $\lceil \mathbf{DiffSB}\rceil$ ,  $\lceil \mathbf{FB - SDE - A}(\mathbf{J})\rceil$  and  $\lceil \mathbf{ENOT}\rceil$ ) use neural nets, after discussion with the authors of  $\lceil \mathbf{MLE - SB}\rceil$ , we decided to use neural nets in their solver as well. All SB solvers work reasonably well, but their performance drops as the  $\epsilon$  increases. This is because it becomes more difficult to model the entire diffusion (with volatility  $\epsilon$ ); these solvers may require more discretization steps. Still, the case of large  $\epsilon$  is not very interesting since, in this case, the EOT is almost equal to the trivial independent plan  $\mathbb{P}_0\times \mathbb{P}_1$ .
IMAGES BENCHMARK PAIRS. There are only two solvers which have been tested by the authors in their papers in such a large-scale data  $\rightarrow$  data setup (64 × 64 RGB images), see Table 1. Namely, these are [SCONES] and [ENOT]. Unfortunately, we found that [SCONES] yields unstable training on our benchmark pairs, probably due to too small  $\epsilon$  for it, see [14, §5.1]. Therefore, we only report the results of [ENOT] solver. For completeness, we tried to run [DiffSB], [FB-SDE-A] solvers with their config from noise  $\rightarrow$  data 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  $\rightarrow$  data setups for future studies. Hence, here we test only [ENOT].
In Fig. 4, we qualitatively see that  $\lceil\mathsf{E}\mathsf{N}\mathsf{O}\mathsf{T}\rceil$  solver only for small  $\epsilon$  properly learns the EOT plan  $\pi^{*}$  and sufficiently well restores images from the input noised inputs. As there is anyway the lack of baselines in the field of neural EOT/SB, we plan to release these  $\lceil\mathsf{E}\mathsf{N}\mathsf{O}\mathsf{T}\rceil$  checkpoints and expect them to become a baseline for future works in the field. Meanwhile, in Appendix C, we discuss possible metrics which we recommend to use to compare with these baselines.
Potential Impact. Despite the considerable growth of 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 a step towards bringing clarity and healthy competition to this research area by proposing the first-ever theoretically-grounded 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 to advance computational OT/SB, in particular, in its application to generative modelling.
Limitations (benchmark). We employ LSE quadratic functions (15) as optimal Kantorovich potentials to construct benchmark pairs. It is unclear whether our 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 EOT plans which can be produced with LSE potentials is rich enough, we provide a heuristic recipe on how to construct benchmark pairs simulating given real-world datasets, see Appendix H. As we show there, the recipe works on several non-trivial single-cell datasets [36, 8]. Thus, we conjecture that LSE potentials may be sufficient to represent any complex distribution just like the well-celebrated Gaussian mixtures are capable of approximating any density [43]. We leave this inspiring theoretical question open for future studies.
For completeness, we note that our images benchmark pairs use LSE potentials and do not require MCMC for sampling from marginals  $\mathbb{P}_0,\mathbb{P}_1$ , 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  $x\sim \pi^{*}(\cdot |y)$  conditioned on noisy inputs  $y$ . This may introduce extra sources of error.
Limitations (evaluation). We employ  $\mathbb{BW}_2^2$ -UVP for the quantitative evaluation (§5) as it is popular in OT field [33, 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, we acknowledge one more time, that the hyper-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 to achieve the best possible performance in each particular case. At the same time, our Appendix E shows the extensive empirical study of the key hyper-parameters and it seems that the metrics reported are reasonably close to the optimal ones.
This work was partially supported by the Skoltech NGP Program (Skoltech-MIT joint project).
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