Entering the Era of Discrete Diffusion Models: A Benchmark for Schrödinger Bridges and Entropic Optimal Transport
Authors: Xavier Aramayo Carrasco
Grigoriy Ksenofontov
Aleksei Leonov
Iaroslav Sergeevich Koshelev
Alexander Korotin
Venue: ICLR 2026 Poster (2026)
Keywords: Benchmark
Schrödinger Bridge
Entropic Optimal Transport
Optimal Transport
Unpaired Learning
Discrete Spaces
Discrete Diffusion Models
Generative Modeling
Primary area: generative models
TL;DR: We present a benchmark enabling systematic evaluation of Entropic Optimal Transport and Schrödinger Bridge methods on discrete spaces, and introduce new methods tailored to this setting.

Abstract: The Entropic Optimal Transport (EOT) problem and its dynamic counterpart, the Schrödinger bridge (SB) problem, play an important role in modern machine learning, linking generative modeling with optimal transport theory. While recent advances in discrete diffusion and flow models have sparked growing interest in applying SB methods to discrete domains, there remains no reliable way to assess how well these methods actually solve the underlying problem. We address this challenge by introducing a benchmark for SB on discrete spaces. Our construction yields pairs of probability distributions with analytically known SB solutions, enabling rigorous evaluation. As a byproduct of building this benchmark, we obtain two new SB algorithms, DLightSB and DLightSB-M, and additionally extend prior related work to construct the $\alpha$-CSBM algorithm. We demonstrate the utility of our benchmark by evaluating both existing and new solvers in high-dimensional discrete settings. This work provides the first step toward proper evaluation of SB methods on discrete spaces, paving the way for more reproducible future studies. The code for the benchmark and all associated experiments is available at [this repository](https://github.com/gregkseno/catsbench).

Published as a conference paper at ICLR 2026
ENTERING THE ERA OF DISCRETE DIFFUSION MODELS: A BENCHMARK FOR SCHRÖDINGER BRIDGES AND ENTROPIC OPTIMAL TRANSPORT
Xavier Aramayo Carrasco* i
Grigoriy Ksenofontov*
Aleksei Leonov
Applied AI Institute, Moscow, Russia
Applied AI Institute, MIRAI. Moscow, Russia
AI Foundation and Algorithm Lab MIRAI
Moscow, Russia
laroslav Koshelev
Alexander Korotin
AI Foundation and Algorithm Lab Moscow, Russia
Applied AI Institute, AXXX, Moscow, Russia
ABSTRACT
The Entropic Optimal Transport (EOT) problem and its dynamic counterpart, the Schrödinger bridge (SB) problem, play an important role in modern machine learning, linking generative modeling with optimal transport theory. While recent advances in discrete diffusion and flow models have sparked growing interest in applying SB methods to discrete domains, there remains no reliable way to assess how well these methods actually solve the underlying problem. We address this challenge by introducing a benchmark for SB on discrete spaces. Our construc- tion yields pairs of probability distributions with analytically known SB solutions, enabling rigorous evaluation. As a byproduct of building this benchmark, we ob- tain two new SB algorithms. DLightSB and DLightSB-M, and additionally extend prior related work to construct the o- B-CSMM algorithm. We demonstrate the utility of our benchmark by evaluating both existing and new solvers in nigh-dimmnsional discrete settings. This work provides the first step toward proper evaluation of SB methods on discrete spaces, paving the way for more reproducible future studies. The code for the benchmark and all associated experiments is available at https: http:/gittttt com/ hwtom//greeeeee//cacceee cateeeeee/caattaeeeeee sbench
INTRODUCTION
The Entropic Optimal Transport (Cuturi 2013
EOT) problem and its dynamic counterpart, the Schrödinger bridge (Schrödinger 1931 SB), have recently attracted significant attention the ma-
chine learning community due to their relevance for generative modeling and unpaired learning. A variety of methods have been developed to solve these problems in continuous data spaces such as
(Daniels al. 2021 Gushchin al. 2023a 2024b
2024 Vargas al. 202 1 Chen 2024 Korotin et al. 2024
Mokrov et al.)
et al. 2021 Shi et al. 2023 De Bortoli et al.
Gushchin et al. 2024a).
At the same time, much real world data are discrete by nature, including text (Austin al. 2021 Gat ct al. 2024), molecular graphs (Vignac et al.) 2022; Qin et al. 2024 Luo et al. 2024). and protein sequences (Campbell al. 2024). Others are discrete by constraction, such vector-quantized
representations of images and audio (Van Den Oord et al. 2017 Esser et al., 2021).
Given the prevalence of such discrete data and the rapid progress in discrete diffusion//low models (Hoogeboom et al. 2021] Austin et al. [202T] Campbell et al.) 2022 Lou et al. 2023) Sahoo et al.)
2024 Campbell et al. 2024) Gat et al.] 2024), research on SBs has attracted growing attention
in recent years. For instance, several works have already taken first steps in this direction (Kim) ct al. 2024 DDSBMiKsenofotto & Korotin 2025 CSBM), adapting diffusion methodologies from VVignac et al. 2022, DiGress Austin et al.] 2021 D3PM), respectively. Yet, beyond these initial
studies, practical, broadly applicable solvers for discrete-space EOT/SB remain largely absent.
"Equal contribution
1 'Conesponnencee to: xavier. aramayo20gmaa. gmail. coccom com MMscoo Independent Research Institute of Artificial Intelligence

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Published as a conference paper at ICLR 2026
To make progress this front, it is also important to have reliable ways to evaluate SB solvers. In practice, EOT/SB methods are often assessed using proxy metrics such as FID (Heusel et al. 2017) or mean-ssuaard-errro between input and output. While being useful, these metrics only indurectly reflect whether a method truly solves the EOT/SB problem, since they can be strongly influenced by parameterization, regularization, and other implementation details. Evaluation benchmarks address this limitation by providing a controlled setting in which solvers can be compared against known SB solutions. This allows performance differences be attributed directly the underlying algorithms. Consistent with this goal, several optimal transport (OT) benchmarks have been proposed (Korotin al. 2021 2022), and similar efforts have recently appeared for continuous- state SB (Gushchin et al. 2023b). However, no analogous benchmark currently exists for discrete data.
From the discussion above, two key limitations in discrete-spant space EOT/SB research emerge: (1) lack of benchmark to assess the performance of the solvers and (2) the limited availability of discrete- space EOT/SB solvers. this work, we address both issues; our contributions are detailed below:
Methodology. We present a general methodology to create pairs of discrete probability distri- butions with known SB solutions (33.1) To overcome tractability issues of the methodology in discrete spaces, we introduce CP-based parameterization ($3.2). This parameterization yields a closed-form SB and enables practically feasible benchmark construction.
Algorithms. We present various adaptations of continuous-state SB algorithms to discrete settings and evaluate them on our benchmark. Specifically, we develop the following algorithms:
DLighttB (34.3) and DLightSB-M ($4.4) solvvrs obtained as a byproduct of the bench- mark, mirroring LightSB and LightSB-N -M (Korotin et al. 2024 Gushchin et al. (024a);
a-CSBM ($4.2) a solver that combines CSBM (Ksenofontov & Korotin 2025 with the online update strategy of a-DSBM (De Bortoli et al. 2024).
Notation. We consider a discrete state space X' = SD, where S = 10. S 1} is the set of S categories and D the dimensionality. Each x E is D-dimensional vector x = (a ..). Time is discretized as [t= 18 with 0=ty <...<..., <tN+1= =1. This gives 2 time points and defines the path space X N with the tuple Jin (att...... Ttx N collecting the intermediate states. The set P())+2) comprises all discrete time stochas- tic processes on the path space, with +2)
P(X denoting the subset of Markov processes. Any E M(2 admits forward and backward representations: q( Tin 21) q(xo) II I (xt,,) g(21) II N-1 q(32,_,, 1x,,) Finally, q(+l-) is used to denote conditional (ro + 11) and transition (It,-. + +t,,) distributions.
BACKGROUND
This section recalls EOT and SB, which form the core framework for our benchmark and methods. We begin with the dynamic SB formulation and its connection to the static SB problem ($2.I) We then recall different reference processes examples ($2.2) that induce practical cost functions and thereby connect SB to the EOT framework ($2.3) Finally, we introduce the discrete-space generative EOT/SB task and specify the benchmark evaluation criteria used in this work ($2.4)
2.1 DYNAMIC AND STATIC SCHRÖDINGER BRIDGES ON DISCRETE SPACES
Dynamic Schrödinger Bridge. The original SB problem (Shhödinneer 1931 1932 Léonard 2013 seeks find a process q² E +22 interpolating between an initial distribution Po tu and a final distribution at + 1. This distribution is found by minimizing the Kullback-eeible (KL) divergence with respect given Markov reference process qed N+2 subject the
marginal constraints Po(0o) q(zr) and pr (21) = g(z1). One finds the following optimal process:
q² = arg min KL (g(20, Fi. 21 ( Rmn x1)). (2oPP)
(1)
where II (Po. P1) C P(x denotes the subset of X-valued stochastic processes which have Po and p1 marginals times to 0 and f 1. respectively. In other words, the dynamic SB problem seeks the stochastic process q that minimally deviates from a reference process gre while respecting the boundary distributions Po and p1.
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Static Schrödinger Bridge. The previously introduced dynamic SB problem also allows static formulation. Linking them begins with observing that T admits the following decomposition:
KI. (1(20,, xn "ll (1a, (1a, 21)) + (y( KL. (q(3in ( xi )10 ( (ro, mn))]
min y
(2)
We further note that the conditional KL term in 2 vanishes when q( To. x1) = q (=m x1). Thus, we restrict q to the set of processes that satisfy this condition. This set is known as the reciprocal class of qed and optimization reduces to the first KL term alone, leading directly to the static SB problem:
denoted by R. (xx+2) C P()x++2). Under this restriction, the
q (2u; 21) arg min KL (g(20. vn (2o,
gell(po.Pt)
where Hp P1) C P(X2) denotes the set of joint distributions with marginals Po and pr, and (Io, is the optimal joint distribution.
Notably, the static SB formulation is closely related to EOT problem. This connection is estab- lished since the reference process induces corresponding cost function. To make this link explicit, the next sections, we first introduce commonly used reference processes on discrete spaces and then derive the corresponding connection between static SB and EOT.
2.2 EXAMPLES OF SUITABLE REFERENCE PROCESSES
The key ingredient in both SB formulations is the Markov reference process qae In discrete space, is typically modeled a discrete-time Markov chain with strictly positive transitions, i... (I,,,, (Tt,,)) >0 for all (tt, As in standard discrete diffusson models (Austin et al, 2021. we focus on factorizable reference processes g (xt,, [x,,,,)) TT (x=%, (=?,-)) and thus present the construction in the one-dimensional case. We further assume time-homogeneity (1""((1f _=),,) Q E [0, 1]588 for all n), so that the cumulative transition distributions for
n-steps defined by the matrix power Q [Q""
Remark. The reference process qee can also be defined in continuous time, where transitions are specified by transition rates (see, e.g., (Campbell al. 2022)). In this setting, controlling the dy- namics is often less direct; moreover, discrete-time Markov chains form strictly larger class, since not every chain admits continuous-ting analogue (the embeddability problem (Kingman) 1962). For the convenience of benchmark construction, we therefore focus on the discrete- setting.
We now introduce two popular diffusion like transitions: uniform (Hoogeboom al. 2021 Camp-
bell et al. 2022 and Gaussian-like (Austin et al. 2021).
Uniform Reference Process "qii For unordered data, where no relation exists between cate- gories, natural choice is the so-called uniform transition matrix. In this case, for each dimension d, the elements of the transition matrix Quef are defined by
1-7. 7 if xd xta-1' if ay + ot
Qudcl
8-11
where 7 10 is stochasticity parameter. This reference process assigns equal probability to transitioning into any different category, thereby ignoring any inherent ordering among categories. In Appendix we provide a closed-form expression for (=(((=)) Tot in the uniform case.
Gaussian Reference Process Sauss), For ordered data, where categories are expected exhibit meaningful relations, Gaussian-like transition matrix is more appropriate. With the stochasticity parameter 7 > 0 and the maximum category distance A S 1, the transition probabilities are
exp
cQ
D #
I
(5)
482 [T2)²
exp
The diagonal entries take the remaining probability so that each row sums 1.
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2.3 ENTROPIC OPTIMAL TRANSPORT ON DISCRETE SPACES
Following the construction of the Markov reference process, the static SB problem ($3) takes
a form equivalent to the EOT problem (Cuturi 2013). Concretely, expressing ged (To I1)
( (2o) and setting (xo0) (o(z)) lets us rewrite the minimization in as
min
KL x1)lll (x0, x1))
qe gell(po.pL)
g xT)
min (EI)(po.P1)
Σ q(20, xi) log grot (moo) (20)
20.21
min gell(po.P1)
-4(4)
Σ
((ra, 01 (11
Σ 20.
(zo, 11) log (1o)
(6)
H(po)=ccont
min (E))
E Eg( log go (x) [mn)]] 11 (1) const
min I ((ro_a))ng(c(x00: r1)] (()) const, (EIT(po.P1)
where H (q) is the entropy q, while H(p)) remains constant when minimizing over q. Thus, the static SB formulation becomes equivalent to the entropy-regularie optimal transport problem with cost c( z1)
log (z) (z)). This establishes a direct correspondence between both SB formulations and EOT, which collectively referred as the EOT/SB problem, enabling the construction a unified benchmark for all three problems.
2.4 DISCRETE ATIVE PROBLEM SETUP AND EVALUATION PROTOCOL.
Building on this background. we formalize the generative discrete-space EOT/SB task. This is a well-established problem in the literature (Kim al., 2024] Ksenofontov & Korotin 2025). In short, the goal learn an optimal conditional distribution that transports a probability distribution on discrete spaces using available empirical samples. Formally, consider the following setup:
We assume the learner is given empirical datasets {a0 hi and (ar jer,, a at E X, consisting of i.i.d. samples from the unknown distributions Po: P1 ((2)) where X' is discrete state space. Then, the task is to use these samples to find solution q" to the EOT/SB problem 3 6 between Po and p1 for a given reference qee Morcover, the solution should support out-of -smple generation so that for any new (I:(0)) one can generate ==oow ~ model (x1 (=:)")).
Despite recent progress in developing discrete-space SB methods for this task, there is still no standard evaluation protocol. The main obstacle is the lack discrete datasets with known ground-truth EOT/SB solutions, i.e., pairs of marginals (Po.Pı) for which the optimal conditional ('(r (n)) is available. Access to such ground truth enables direct comparison with learned model moodel (z1 (ro), allowing evaluation of how accurately method solves the underlying EOT/SB prob- lem rather than relying on proxy metries. Inspired by (Gushchin et al.) 2023b), we therefore propose pipeline that generates ground-truth benchmark instances, applicable discrete-space solvers
Remark. Our paper is not related to the discrete EOT, which includes solvers such as the Sinkhorn algorithm (Cuturi 2013) gradient-based methods (Dvurechensky al.] 2018). These approaches
are designed for a non-generative problem setting, see (Ksenofontov & Korotin 2025 $2.3). They
treat samples as empirical distributions po (20) [To] Lien 0_(0) P1(31) TT. Zjen 5 60. The
resulting joint distribution is then bi-stochastic Jal x |/|] matrix, which does not support out- of-sample generation. While some extensions attempt to provide inference for unseen data (Hütter
& Rigollet 2021 Pooladian & Niles- Weed 2021 Manole et al., 2024 Deb et al.) 2021), they
designed for continuous spaces (X=R") rather than the discrete (2' considered in our work.
3 BENCHMARK
In this section, we address the absence of evaluation benchmarks for discrete-space BOT/SB solvers by proposing novel benchmark construction. In $3.1 we present the theoretical foundations of the construction. Next, we make it tractable via a CP parameterization in $3.2) Finally, we use this parameterization construct high-dimensional Gaussian mixture benchmark in $3.3 We provide detailed proofs for all stated theoretical results in Appendix

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3.1 MAIN THEOREM FOR BENCHMARK CONSTRUCTION
For an initial distribution Po E P()), we aim to construct target distribution p1 E ((()) such that the optimal joint distribution 9" (æo, 21) between them is known by our construction. The resulting pair (po: P1) together with g" can then be used as benchmark data for evaluating SB methods. Our following theorem plays the key role in the construction of benchmark pairs.
Theorem 3.1 (Benchmark Pair Construction for Discrete-Space Space EOT/SB). Let Po E P(X) be given initial distribution on discrete space X' and v- X R be given scalar- valued function. Consider a joint distribution q" E P((2)) such that q" (²(1)) = pu Po(2o) and (xi (2r) v² (ar (ffd (In (r9) define pı (x1) == q" (x1) as its second marginal. Then q" together with the reference process qe defines the discrete-space EOT/SB 636 between po and p1.
Similar results in continuous spaces appear in Gushchin et al. 2023b In Theorem 3.1 we provide a discrete-space analog which shows that any pair (Po, v²) induces corresponding pair (Pos P1) with closed-fone form q" (z1 (ro) on discrete space. We refer to the latter pair as benchmark pair. However, this construction specifies q" (x1 (r)) only up to proportionality, necessitating the normalized form:
q' (1) (mo) (= u"
where ca(x)) : Ez 8- (x1)g (11 xo) is the normalization constant. Although this provides closed-form expression for q", implementing it in high-dimensional spaces (|X') S)) remains computationally challenging. In particular, evaluating the normalizing constant and sampling from q² are non-trivial tasks. To address these challenges, we introduce a CP-parameeeriiztiion that allows efficient computation and sampling, as detailed in the next section.
3.2 PRACTICAL PARAMETERIZATION
We parameterize the scalar-valued function 8" using a rank-1 Canonical Polyadic (CP) decomposi- tion, which captures interactions across dimensions and provides compact yet expressive represen- tation. Such decompositions act as universal approximators, capable of modeling complex functions when the rank is sufficiently large (Cohen et al. 2016 Basharin al. 2025). Thus, V* is written as
K
D
0n (x1) =
Σ
3x
II Mal
(8)
Expression ( defines mixture of K factorizable distributions, each with weight 8g 2 O. For each mixture component and dimension d, probabilitics are defined by non-negative vectors ry E R. referred to as CP cores, where r [[ denotes the probability of state aÇ-
Proposition 3.1 (Tractable Parameterization of Conditional Distributions). Let qre be a fac- torizable Markov reference process on a discrete space X. Using the CP decomposi- tion of the scalar-valued function v² in 88. the optimal conditional distribution satisfies:
a' (xi(z0)=
c'(xo)
Σ BR II (1(
(9)
Σ
Bk
II d=1
Σ
1 (2412))
(10)
(1O)
where c² (xx) is the normalization constant. This formulation expresses of (xn (mr) as mixture of K factorizable distributions, each weighted by a scalar coefficient By-
key consequence of Proposition 3.1 that c* (2r)) and q (xn (zo) are computationally tractable, since both expressions factorize into products of mic-dimensional sums. In particular, instead of summing over the full joint space of size 5D, computing c (2o) requires only K products of D scalar sums over S states, reducing the complexity from O(S)) O(KDS). The evaluation of q" (I1 (mo) is similar. In practice, computations are performed using log-sum-exp operations for nu- merical stability. To sample from q" (x) (Za),
ancestral sampling: first choose mixture component with probability proportional 3r I2 Eng r [nd] get (xq ag)), and then, condi- tioned on k sample each coordinate off independently from q" (arf =g)) x r((a ned zg))
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Previously. described the benchmark construction in the static SB (equivalently, EOT) setting, with focus on q (21 (o).. Nevertheless, a similar result can be obtained for the dynamic SB and its forward Markov representation defined by transition distributions q (It,, (zt,_.). More precisely, these transitions can be obtained by reweighting the reference process transitions (Ie (=)
with time-dependent scalar-valued functions 07 (Georgiou & Pavon 2015 Thm. 2):
Di,, (IFt.)
9º (2 (xt,,)) = =0d (It, (et..)
1i (It...) = LE (eilx)) (21)). (1I)
oi (1)
The next proposition provides tractable form of the corresponding transition distributions.
Proposition 3.2 (Tractable Parameterization of Conditional SB Transition Distributions). Let qra be a factorizable Markov reference process on a discrete space X. Using the CP decomposition of the scalar-valued function v" in (8) together with the definition of the time lime-dependent functions of,, 17y, the optimal transition distributions satisfy:
K
(3( Ire, x geth (Ft,, Itt
Σ
3x
II
(12)
d=1
where us. [+,,] Esr ( (of 20 )r (11) Sampling is done via ancestral sampling.
By the same argument as for (z1 (z))xr) the transition distributions g" (xt, (zt,_.) are tractable. In practice, normalization of (12) is carried out the log-domain by subtracting the log-sum-exp over all states to obtain numerically stable transition probabilities.
3.3 HIGH-DIMENSIONAL GAUSSIAN MIXTURES BENCHMARK CONSTRUCTION
We now instantiate the proposed construction. We Po as discretized Gaussian De (2, 16, 64) dimensions with S = 50 categories. For v", we use = components with uniformly initialized weights 3 E IR K The CP cores are initialized using discretized Gaussian probabilities, with means uniformly sampled on a sphere of radius 5 and a standard deviation fixed to [1.5, 1.5.255) for the lowest to highest dimensions, respectively. Given Po and v", we then construct p1. This initialization produces a target resembling a discretized Gaussian mixture with clear visual structure. We construct pairs under different reference processes: gumus with 7 E {0.02, 0.05] and q quiil with {0.005, 0.01}. using N + 1 128 for both. Figure 1b shows the resulting benchmark pairs.
4 SOLVERS FOR EVALUATION
In this section, we recall available discrete EOT/SB solvers well as addressing the limited availability such approaches. We begin by recalling Categorical Schrödinger Bridge Match- ing (CSBM) (Ksenofontov & Korotin 2025), which is an existing SB method tailored to categorical distributions. We then introduce C-CSBM, which incorporates the online update strategy of (De Bor- toli et al. 2024) into the CSBM framework. Next, propose Discrete Light Schrödinger Bridge (DLighTBB) obtained as a byproduct of our benchmark construction (33) and extending (Korotin al. 2024) to the discrete setting. Finally, we present Discrete Light Schrödinger Bridge Matching (DLighrSB-M) dynamic extension of DLightSB following (Gushchin et al., (024a).
4.1 CATEGORICAL SCHRÖDINGER BRIDGE MATCHING (CSBM)
In (Ksenofontov & Korotin 2025
Theorem 3.1), the discrete-space dynamic SB problem is ad- dressed by the discrete-time Iterative Markovian Fitting (D-IMF) procedure, whose convergence is established by extending the discrete-time existence theorem of (Gushchin et al., 2024b Theorem 3.6) the discrete space and time setting. The exponential convergence rates of DMF can be found in (Sokolov & Korotin| 2025). This constructive method uses the fact that the dynamic SB g is both reciprocal (q' E R +2)+ and Markov (q" E N+ )). The D-IMF algorithm alternates be- tween projections onto these two sets, starting from an initial process q (Ir, a1 ( (m x1), where (20, 21) E II(po, Pı). e.g., (z()))p1(z1). and converges to the SB q" in KL. Namely,
projи
21(2))
g2+1
1= 1,0,1,..
proj
where
(projRe((g))(() Finy x1) 21)
q( xn ( (oil
Ay E P(XN++2).
(13)

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Tin Uma x1) (
arg min KL (g(20, Tin z1) #1)l| Ein x1)), Aq E R (xN++). (14)
mEM(X2112)
Loss. Fortunately, the reciprocal part 13 is straightforward via ancestral sampling. In turn, to make Markovian step (14) tractable, the authors parameterize the transitions of m, writing m qo and minimize the following objective, defined up to an additive constant independent of 0:
(0) (10.00))
Σ B Be (os (mo,))) (1I.(1"((1r (st,,, [rt, 1 21) |l go (3 l3t,, .)]]
-Eai ( [log go (2 (2
(15)
In practice, the D-IMF procedure implemented in a bidirectional manner (see (Ksenofontov & Korotin, 2025, $3.2.5)). That is, the forward and backward models are trained alternately at each Markovian step. Notably, the KI. loss can be replaced by any divergence from the Bregman family (e.g., the mean squared error (MSE)), introducing additional hyperparameter in experimental setup. For details on this equivalence, see [Ksenofontov Korotin 2025 App. C.I).
IMF was introduced in the Discrete Diffusion Schrödinger Bridge Matching (Kim et al.] 2024, DDSBM) paper, which performs the Markovian projection (14) by matching the generator matrices of continuous-time Markov chains. As it reduces to the same loss and inference process due to the necessity to discretize time, we report results only for CSBM.
Remark. A continuous-
4.2 a-CATEGORICAL SCHRÖDINGER BRIDGE MATCHING (a-CSBM)
Running the IMF procedure bidirectionally is often beneficial, see (Kholkin et al. 2026), but doubles the computational burden, since two neural networks must be trained to model the forward and backward representations. To slightly mitigate this cost, recent work has proposed an online alternative to IMF, called a-IMF (De Bortoli et al. 2024 Peluchetti 2025).
In this approach, the exact projections in (3) and are replaced by partial updates (De Bortoli al. 2024 Eq. 9). Concretely, rather than running each projection to full convergence, one performs a single optimization step at each iteration i. Although each update is incomplete, the alternating steps still steer the learned distribution toward the double projection Inspired by these advances in the continuous setting, we extend the same ideas to CSBM ($4.1) viewing the discrete version of a-IMF as heuristic analogue of the original procedure.
(projjn( (-)), in IMF,
Loss. Since the method does not require each projection to fully converge, we can take a single optimization step for both representation directions simultaneously. This allows extend the CSBM bidirectional setup 4.1 by jointly updating models using objective (15) in both directions:
(()) = KL (Fag Tm. z1) 21 (2 Zin, 21)) + KL (fag (Zo. Tin, x1) x1) 11@6 (Zo. Fin Fmn. :))) (16)
where - and + denote the direction of the Markov representation (typically implemented by con- ditioning neural network a direction variable), and r'g denotes (q)))=((())) with top-graddeet.
Limitation. To avoid evaluating the full space of size SP, transition probabilities go (z,,, (zt,.,) are factorized across dimensions, reducing it to x S. However, this parametrization constitutes a key limitation of (SB), introduces approximation error.
4.3 DISCRETE LIGHT SCHRÖDINGER BRIDGE (DLIGHTSB)
Now we take different direction and propose new solvers for discrete-space EOT/SB problems. Specifically,, we focus on methods that arise directly as byproduct of our benchmark construction. We begin by introducing new static SB solver, which entitle DLightSB. In what follows, adopt the same CP benchmark parameterization ($3.2) for the conditional distribution q( (æt).. In particular, treat the weights 3g and the CP cores r learnable parameters and collect them in 0 [Bk, r.), yielding the model (21 (20) with valur- function Ug (02)).
Loss. To optimize the parameters f, we follow the approach of Korotin et al. (2024). Concretely, we derive a discrete objective that is equivalent to the direct KL objective KL (g ( up to additive constant. Importantly, this reformulation removes the dependence on the unknown optimal joint distribution 9² and enables a feasible optimization procedure, as shown in the following proposition.

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Proposition 4.1 (Feasible Discrete Reformulation of the Direct KL Objective). Under the paranetrization (9) of go (11 2o). the objective KL (q" (190) admits the following reformulation:
KL (q" [00)) [()) L", where
((0) - (o() log co (zo) Epr (mi)) log (0(21)
(17)
and L E R. is constant value not depending on 0, therefore, it can be omitted.
Notably, the expectations in Proposition 4.1 can be efficiently estimated via Monte Carlo sampling, and the resulting objective can be optimized using stochastic gradient descent with respect to A.
4.4 DISCRETE LIGHT SCHRÖDINGER BRIDGE MATCHING (DLIGHTSB-M)
Finally, introduce DLightSB-Na dynamic variant of DLightSB, which also construction. Inspired by Guuhhhhn et al. 2024a), we introduce a discrete-space optimal projection that recovers the SB through a single projection step. In particular, rather than projecting reciprocal process r R (2: onto the Markov set ((+ as in the D-IMF procedure (14). we
benchmark
directly project onto the of all SBs, defined as follows:
S(X = { SB E P2) such that 3 40 .4 E P(X),
ST
arg min @ (
KL (alla"*))
(18)
To justify this projection in discrete settings, we show that (Gushchin et al. 2024a Theorem 3.1) extends to an arbitrary Markov reference process yedt
Proposition 4.2 DDscrete-Space Optimal Projection with an Arbitrary Reference Process), Let r (x*++)) be reciprocal process defined with reference process gre E MMX and joint distribution r r(wo; #1) E [T((o.P)) P1). Then, the optimal projection of onto the set of all SBs N(x)+2) is the SB q" between the desired marginals po and pı, i.e.,
arg min (SEES(N442)
KL (r(lqsh).
(19)
Loss. To complete the construction, we need to parameterize g. so that the minimization is re- stricted to N(2)+22). To address this we recall that g E (M (84.1 and it is fully deter- mined by its forward transitions g( (xt.-)). We therefore adopt the closed-form transition param- eterization 12) with parameters A = {Be, r in DLightSB, yielding the transitions qe (Z',,, (r,,,)) that define an SB. Finally, under this parameterization, the discrete-space optimal projection (19 can be optimized using the Markovian projection objective (15) via stochastic gradient descent.
Importantly, since the reference process ge factorizes across dimensions, the KL and cross-entropy over dimensions. Concretely, each
(CE) terms in (15) admit equivalent reformulation term reduces to sum over dimensions of KL or CE between nad (x² | = and the model's
d-th transition. The latter is obtained by summing out all dimensions except the d-th one (see (34)
in the proof of Proposition 3.2), which is tractable under our parameterization.
Limitation. In high dimensions, the CP parameterization requires large number of components K capture complex structure. This makes DLightSB(-M) methods computationally demanding.
5
EVALUATION
In this section, we first introduce the evaluation metrics and describe the baselines (5 evaluate the solvers from $4 on our benchmark setup ($3.3) and report the results in 5.2
We then
5.1
METRICS AND BASELINES
Evaluating generative models on discrete data is challenging because widely used metrics, such as generative perplexity for text or FID for images (Heusel et al. 2017), domain-specific. Following dabular-data evaluation work (Zhang et al. 2024 Shi et al. 2025). y we adopt the Shape Score and
Trend Score metrics. Further, introduce an E0T/SB- metric given by the Trajectory KI..
8

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Published as a conference paper at ICLR 2026
malt 0.005
(a) Input/Target (b) Benehmark
(c) CSBM
(d) C-CSBM
(c) DLightSB (f) DLightSB-M
Figure 1; Samples from all considered methods on our high-dimensional Gaussian mixture benchmark. Top row: g (y = 0.005). Bottom row: Gaussian benchmark (7 = 0.02). CSBM, a-CSBM. and DLightSB-M
trained with KL loss (N+ 64).
Shape and Trend Score. These metrics quantify how well solver matches the benchmark at the level of dimension-wise and pair-wise marginals. For each coordinate d (Shape) and each pair (dm dn) (Trend), we compute an adjusted total variation score and then average over dimensions:
SSM
TSM om
1 (x*) 20(x")1
Σ
1-2
Σ
2
zmm zonn
where the tilde denotes empirical distribution, i.e., ((x)= E 856)
Conditional Metrics. In our evaluation, we primarily report conditional variants of these metrics. These are computed by averaging SSM and TSM over sampled zu Po and multiple generated sn for each zi- This approach provides a direct measure the fidelity of the learned conditional distribution qo (3r1 (aro) and quantifies how well the EOT/SB solver solves the underlying problem.
Trajectory KL divergence. Additionally, we utilize the dynamic SB and its Markov property compute the forward and reverse KL divergences between processes, namely KL (g" Tin, x1) 21)][lg((0o Tin, æ) and KL (ga(xo, Lin, x1 l14" (Fo, Tin, æ1)), respectively, i.e.
N+1 D
KL (2o, Vin: 11) 11 (x Fin, 21))
Σ
Σ
KL (a" ( )ago (xtm-.))
d=1
An analogous decomposition holds for the reverse KL divergence. This metric relies on factor- ization across dimensions specifically for methods that model only marginals (e.g., CSBM, 34.1 In contrast, for methods that do not impose such restriction (e.g., DLightSB, $4.3). the required transitions are obtained by marginalizing out all coordinates except the d-th dimension, see (34).
Baselines. To provide simple point of reference for our benchmark, we consider three simple baselines. The first, Independent, ignores the joint distribution between xn and æ and samples æ directly from P1- The second, Reference, takes x be sample from the reference process ge The third, Feature-wise SB, solves an SB problem separately for each dimension using the analytical D- IMF procedure (Ksenofontov & Korotin 2025 Section 4.1) with empirically estimated marginals Po and p1. This baseline assumes factorization across dimensions and samples each 2 independently.
5.2 RESULTS
Benchmark Usage Protocol. To simulate the generative discrete-space space BOT/SB task discussed in $2.4 we use our benchmark pair constructor differently for training and testing. For training, we sample z ~ Po and -r ~ P1 independently. We further remove dataset-size limitations that could affect direct comparison of methods by infinitely sampling the training data, as enabled by our benchmark. For testing, unconditional metrics and the Trajectory KL are computed on fixed set of 20 000 precomputed benchmark pairs (Zo, .1) q" (1). Conditional metries evaluated on 000 x; samples for each 157 unique Zo. where the Tn are drawn from this 20 000 set and the corresponding ii sampled from both the method and the benchmark. We also compare

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Published as a conference paper at ICLR 2026
15:15
1
unioom
uniiers
unifee
umioom
graedan
unifim
unifoem
Method
Loss
0.02
0.00G
0,01
0.02
0.06
0,005
0.01
0.02
0.0G
0.00G
0.01
0.05
0.000
0.01
0.00
0.005
0.0L
0.02
0.00
0,00G
0.01
Andepesdeer
10.31
0.161
11:70
01.30
08.77
0.15
10.06
0.00
11.50
08.35
0.57
14.25
0.488
11.31
Reformcee
0:27
0.445
0.30
0.42
0.23
0.20
0.30
0.53
0.44
0.87
0.96
0.36
0.39
00.25
0.22
0.27
0.11
0.06
0.00
0.16
6.20
0.12
01.14
0.11
Frotureieell SW
0.:7
0.88
0.88
01.80
01.81
1.70
0..61
0.67
0.103
0.46
73
0.71
0.31
0..55
(10
0.74
0.00
0.55
0.45
0:33
0.64
0. 8.40
0:53
0.46
D6.gh15B
10.05
0..93
0.05
0.98
0.85
0.98
0.992
0.03
0.944
0.91
0.085
0.92
0..91
0.25
0.887
0.87
0.26
0.84
0.84
0.84
00.85
0.75
0.82
0.77
CSBM
MSE
KL
64
64
16
10:73
0:35
0.56
0.29
01.22
0.87
0.73
0.77
0.77
0.87
6.500
0.52
0.27
0.92
0.75
0.82
0.77
0.71
0.81
(182
0.664
0.22
0.67
0.39
B065
0.71
0.50
0164
0.77
0,06
0,50
1,51
8.85
D002
0.62
0.50
0.06
0.88
0.62
0.60
0.54
0.44
0.26
0.75
0153
0.79
0..0
0.551
0.00
0.81
0.35
0.75
0.67
0.77
0.995
0..90
0.754
0.72
0.57
0..1
0.30
0.57
0.48
0.61
0.05
0.50
0.041
6.11
03.0
0.50
0.71
0.70
01465
01:00
01.65
0.54
0.46
0.51
0.48
00:00
11.22
0.51
0.500
0.66
0.64
0.64
0.68
0.51
MSE
64
00.58
0.555
0.71
08:86
0.54
6.85
0.67
01.5
0.700
0.84
0177
0,.00
0.57
01633
0.05
0.31
0.60
0.86
0:77
0.61
0.00
0.5%
0.60
0.54
0.:11
0.75
0.70
0.777
0.79
0.26
0.29
0.70
0.653
0.49
0.30
0.55
00449
01750
0.60
0.66
0:47
0:45
0.46
0.40
11:00
11:31
0.001
0.81
11:11
0..92
0.83
0.76
0.778
0.58
0.85
11.00
11.05
70
0.53
11.22
0.701
0,80
0:47
6:30
DUghISB
0.65
0..02
0.02
0187
0.02
0.52
0.5%
0.84
0.50
0.86
0.81
0.55
0.82
0.71
6.67
MSE
0.70
0.55
0.74
(a) Conditional Shape Score (1)
(b) Conditional Trend Score (t)
Table I: Conditional metrics on our high -iseeninnnii Gaussian mixture benchmark. The best-performing
method is highlighted in hold, and the second is underlined. Color code: vermillion for < 0.5, orange for
[0.5, 0.75), yellow for [0.75, 0.85), and bluish-green for 2 0.85.
different training setups by varying N across CSBM, a-CSBM, and DLightSB-M. For the same set of methods, we experiment with two loss functions: KL and MSE.
In the main text, we report only the conditional metries, as they more accurately reflect the per- formance of the EOT/SB solvers. In Appendix D.2 we provide experiments to validate conditional metries against the unconditional ones. Further experimental details are provided in Appendix C
High-Dimensional Gaussian Mixtures. Here, we report results on the high nigh-dimensional Gaussian mixture benchmark constructed as in $3.3 using the methods from 34 Visual results are shown in Figure 1 for guams (y=00 0.02) and g quiif (y= 0.005). while additional plots provided in Appendix D.2
The qualitative results are presented in Tables la lb
and 6
First, across all benchmarks, each baseline underperforms for reasons specific to its construction. In particular, Reference consistently degrades, indicating that the scalar value function 8' in ! strongly reweights the reference process and yields a challenging q" to approximate. The Indepen- dent baseline deteriorates when the stochasticity parameter is small, while Feature-wise SB degrades as the dimension increases. Together, these failures highlight the target properties that our bench- marks are designed to test in EOT/SB solvers.
Second, DLightSB consistently achieves the strongest performance across all setups. We attribute this to the DLightSB solver being constructed according to the same principle underlying our bench- mark. DLightSB-M, which incorporates the same inductive bias, attains comparable results with a slight drop in metrics, likely due to the additional variance introduced by the KL minimization loss. While this alignment could be seen as a limitation of the benchmark, we take the opposite view: DLightSB(-M) can be treated as oracle-like method in this setting, since its inductive bias makes it less informative as a measure of general solver performance. For an analysis of the reverse benchmark designed to probe this bias, see Appendix [D.]]
Finally, CSBM and a-CSBM perform noticeably worse than DLightSB(-M). Notably, a-CSBM attains comparable quality to CSBM while halving the computational cost, making it more efficient alternative. Increasing N generally improves the metrics. As for the loss function, KL consistently outperforms MSE, likely because MSE minimizes pointwise squared error and yields smotthed solutions that blur modes. This effect can be observed in Figure 2
DISCUSSION
Our work addresses key gaps in disceeete-space EOT/SB research by introducing the first standard- ized benehmark for these methods, along with new approaches. This contribution provides ground- truth data and consistent evaluation metrics for rigorous assessment of underlying solvers. The benchmark reveals fundamental limitations of both existing and proposed approaches: DLightSB(- M) faces severe memory constraints in high dimensions, while CSBM and o-CSBM remain sensitive to parameter choices and require long training times. These findings underscore the need for more scalable architectures and stable training procedures, and guide future research in this area.
Reprodueibility. We provide the experimental details in Appendix C and the code required to re- produce the conducted experiments is available in this repository (see RE ADME. md for details).
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LLM Usage. Large Language Models (LLMs) were used only to assist with rephrasing sentences and improving the clarity of the text. All scientific content, results, and interpretations this paper were developed solely by the authors.
ACKNOWLEDGMENTS
The work supported by the grant for research centers in the field of Al provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C113925P4F0002 and the agreement N339-10-20255-033.
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A PROOFS
Proof of Theorem 3.1 We start from the expression the static SB problem in
min (Ell(po.P1)
KL
21
min géll(po.p))
- ((
Σ
g 21) log प (21 (zo)
const
30.21
min (ETT(Po-P1)
Σ mo.01
( 2l0g
Σ m0.001
a(=i, x1 ( const
(20)
Noting that the joint distribution can be written as (( 21) = (()))g and enforcing the marginal constraints En, (2oo))g(e (ar)) pi (x1) and Ex, q(2) (20) (equivalently ((xo) (zoo)) the corresponding Lagrangian can be formulated as
(x))
C(a) Σ po (2 (20) log (po(2o)g((11
Σ po(20)g((11120) log (xn))))))))))))))))))))))
20:21
m0.31
+ Σ M(x1)
Σ (0(01110)00000) - pı p((
Σ T(xo)
Σ
q((22100) po(zo)
+
F1
150
x1
Σ Po (100)((211 log)o(zoo) +
Σ
po(200g((11 (2o) log ((011)00)=
20.21
20.21
(21)
Exg po( log (1to)
Σ po(20)a(21120) log (21(20) + Ex(r)) 20.21
M(x1) (0(e)1o))00 pı (x11)
Σ ((ro)
where A(x1) and r(ro) denote the Lagrange multipliers associated with the marginal constraints on I1 and No, respectively. Taking the pointwise partial derivative C(q) with respect to g( (xo)
ac po(zo) (log (211(())) 1 (00-Po(ro)) log ( (Z1))
(22)
0z(z1| (zo)
Therefore, the process q can be written as
r(xO)
(<21|=0) exp(-A(z)) 1)g (21 exp
(23)
po po(
We can then define
0(01) == exp(- X(x1), ((x) exp (-
T(30))
-po(ro))
Then q(x) (r0) (z and the normalization constraint Ex, q(x1 (2o) = forces
Σ oa( (m
c(x0))
Thus the optimal conditional distribution q (an (mo) is uniquely defined by
(x (Zo)"" (x1) " (x)
with c² (=0)) Σ (x) (24)
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Published as a conference paper at ICLR 2026
By construction qº (2o) po(zo), and P1 (21) := (x = Ex Pu(za)q" (21 is its second marginal, q" P1). Because the KL divergence is strictly convex and the feasible set [(po.P)) P1) is convex, the first-order conditions are sufficient for optimality. Hence q" is the unique minimizer of min qEll(po.Pi)
KL (glla")). Therefore q² in Equation (4 together with the reference
process qed defines the discrete-space EOT/SB between and P1.
Assuming the CP parameterization introduced in ( and further assum- ing that the reference process factorizes across dimensions as (x1 (mo) = II g0d (x91=g)) the normalized conditional distribution q" (x1 can be rewritten as
Proof of Proposition 3.7
q (x11
1
Σ
3
r Mall
(
(=(0))
(25)
K
BK II d=1
(1910)))
c (2o)
where the reference factors can be merged with the rank-1 components because they are indepen- dent of the mixture index it and factorize over dimensions. From here, it is possible to obtain the normalizing constant (xoo by summing over all possible values of æ1 SP, where of E $ S 1}. The normalizing constant can then be rewritten as
((x0)) = Σ
Be
II
o
21
k=1 d=1
D
8
Σ
II
(26)
A=1
EEP d=1
8-1 Σ
Mitl af (2)1))))
A=
The exchange between the product and the sum is valid here because the summation is separable across dimensions, i.c., each factor depends only on its corresponding coordinate x
where Ex = -
-
Proof of Proposition
4./ We start from the standard KL minimization problem from the LightSB paper (Korotin al. 2024 and define in discrete space.
KL (g) = Σ q" xz) log
(2((00
Σ q" log q" (#o, 22)
Σ q" log g( 21)
20121
--((-")- Σ q" (Ir, an log ((50, - H(q²)
Σ q"" log (1(z0)g(21(20))
m0.31
--H(gg) Σ q" (xo, x1 log ((x))
Σ q" (2 I1) log (egl(()11))
(po(
Z0,21
-H((g)) Σ log pa (wo)
Σ 9" (20,
Σ (=0, log (())
=g (20)
Now using
7
( (z0) we can get
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KL (g-1l9) - H(a))
Σ log logPo(zo)Pu(zo)) - Σ q² (*v, 01) log
g0r (011200))
(xo)
=-((a")
Σ log (og)o(=
Σ
q² (5o, 11 log (11
20.31
U0 (x1)
E (2o, 11)
(x())
20.31
Σ 2o
q (1) log (x0)
E (20, x1) log =
20.21
Σ Po (x)) log c² (xa) - Ex (x)) log (x1) L
Zg
21
= (20)) (2a) logro(1ro) (e)) [ L³
That concludes the proof.
Proof of Proposition 4.2
KL (r(2o, Fin x. x1) (1ro Fin F.1)
KL (r(zo, (x)) + KL (Iinker 1) Ill l (Iin /200 x1)
(27)
Σ (1)
Σ 20.21
r(zo, 21 log 9 ( x2)
= -H(((x0,01))
SB (x)) (x1
-H(r(1o,21))- - Σ r(zo, x1) log
(28)
SB (Io)
--((((20,21))
Σ r(00, x1) log u (x1)=
Σ
r(wo, 21) log (x: (200
Σ
r(an, 21 log (8
20.21
(H((000021))
Σ log (x1)
Σ r(zol=1)
(29)
= (x))
=I=Enn 8r (5olar)
Σ r(zo, x1 log (x (20) + Σ log SB (Io)
((x))
Σ r(01/20)
(30)
=p(xa)=pp (aro)
=1= (011222)
SB (x11.
-H(((Z0,21)) Σ
r( 11 log " (mo) - Σ (20, log
eSB (Io)
30.21
=0,
y SB (m))
= = - Σ (20, x1) log
SB (No)
20.21
Σ
q" ("o, (o) log (x (20) + Σ 9" (20, x1) log " (21(2r)
(31)
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Published as a conference paper at ICLR 2026
1 (2) (are (21 ("o, log
=-
Σ 20,30
+ C1 + Σ (xo, (x))))))
(21l
cSB (xr)
(ro)
=0²
= C
Σ
q' (xp. 11 Di) log (x)
Zo 21
Σ 20.21
q (2o, x1) log (20,
Σ
o (20, (1) (20 (32)
20,21
(1) ( x2)
Σ (1) log 10.21
+ C2
Σ
(2o, (ro, v1)
20.3g
= KL (x0, t1 11lg (zo, x1)) + C33
In (27 J, we use the disintegration of the KL. divergence to transition from the dynamic the static formulation. In (28). we apply our parameterization from (7). Next, in (29) and (30) we use the properties the reciprocal process r, which has the true marginals at add zero term to introduce (x1 (20) with the expectation taken over the optimal joint distribution
and 1. In (31), we
9º (mo, (1). Finally, in (32). we obtain the entropy term, completing the expression for the desired KI. divergence. This establishes that optimizing KI. (r(rr, 21 )ly (Irn 21)) with respect to y equivalent to optimizing KL (q" (zo, 21 (xo,
up to an additive constant.
Proof of Proposition 3.2 We first derive the transitional distributions of the SB by recalling its well-
known characterization (Georgiou Pavon 2015 Thm. 2):
D7, (ZI..)
9' (It,, Ire, 1/af (Ite, (ee, 1) Pi. (Ft,
o7, (2 = Bigrot (xg (o
Using the CP parametrization of o" from 8 and exploiting the factorization of qre the scalar-valued functions dr., can written as:
K
D
S-1
01 (IL..
Σ Be
E
E Eyrle
3x
II
Σ
(
rfl191.
=
where k.e. satisfy the following recursive relation:
Ug [
ref (2t (2gn)) uhete (rmull
u = Th.
Thus, obtain the following transition distributions:
K
q" (Itz ((t_.))
( /1t,.)
3x
(33)
j=1
To obtain the d-th marginal transition distribution, marginalize over xt."" = tol, Fise as follows:
qº a2 lve, .) x Σ
I 1=1
d set (or)
Σ
3x
I
oee k.
j=1
Fen
q (ar 1x 1) Σ Baud R.
III
Σ
rell (s)) lo .) 1o
3d
(by recursion)
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Finally, obtain the desired expression up to normalization:
qº (etni)) x 12 by 1
Br uk
(34)
j=1
3
B CLOSED-FORM OF THE CONDITIONAL DISTRIBUTION FOR THE UNIFORM REFERENCE PROCESS
It is also useful to derive closed-form expression for (x (=))) Since this conditional distribution given by the (N+ 11)-st transition of the reference chain, can be obtained from the (N 1)-th power of the transition matrix Q In the uniform case, this yields:
N+1
QN+1 (1
S
N+1
1 1-
1-11-781
8-1
(1-75-1)
5-1
I+
11
(35)
where 1-
1-
T T ERS is vector of ones and I is an Sx S identity matrix.
Proof of Equation 35
Let Q be the transition matrix in rewritten as
Q = - (1-7)1 +
11 -))
8-1
$ -11- S-
-11
7 11
1+ S-1
where / is the identity matrix and 11 is the all-ones matrix. Let
S a= 1 1
Y S-1
so that Q T and note that a +Sbb 1. We compute Q +1 using the binomial expansion.
Since and 11 commute:
Q" = (al + +b11 y
Σ 11
n bb (11T)*
Using (I1T)* 11-111 for l > and separating the k: 0 term:
Q" a / +
k 6 g '11
k=1
a 1
Σ
a (6S) 11
The binomial expansion gives:
(a + bS)" = Σ a nー (bS)*
11
= a" + Σ
n1
a - ((S)/*
k
k
k=
k=
Since a + + 1, we have (a + 1S)" = 1, so Σn=
(
a" -
(bS)* = a", Thus,
1
=² = a"
11
S
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Published as a conference paper at ICLR 2026
Substituting N + and 75- yields
(1
N-1
psd = 1 5-1 1+
8
N+1
1
(1-9
(1-93-1
1-1
3-1
11
S
From here can be seen that @ converges to (1/S)11T when (N+1) + co, that is uniform distribution over the number of categories S. In the case of the Gaussian reference process, the closed-form expression can also be obtained, but it much more complex.
C EXPERIMENT DETAILS
This section provides detailed descriptions all methods and their configurations.
Shared Aspects. Across all experiments, we use the AdamW optimizer with fixed beta values of 0.95 and 0.99. For the high-dimensional Gaussian benchmark ($5.2). Notably, for diffusion- methods, we fully sample the Markov chain, contrast Austin al. (2021), which applies an argmax operation at the final timestep.
CSBM and a- a-CSBM. For CSBM and a-CSBM, we use the official implementation from Kseno-
fontov & Korotin 22025:
To stabilize training and improve final performance, we apply Exponential Moving Average (EMA) parameter updates with a decay rate of 0.999, tuned consistently across all experiments. Unlike omit the Liimple loss during training. We employ simple MLP with three
Austin et al. 2021). hidden layers of size [128, 128, 128) and ReLU activations. Time conditioning is implemented via an embedding layer of the same size dimensions, D. Both methods
trained for D-IMF iter- ations, using 120 000 gradient updates in the first iteration and 40 000 in each subsequent iteration. For a-CSBM, following De Bortoli et al. 20244. For CSBM, we use a learning rate of 10-1, The D-IMF procedure for both solvers initialized using an independent joint distribution (q (zo, x1)
learning rate of 10-3 and halve the batch size for training single model,
po (o(z))p (x1)).
DLightSB and DLightSB-M. For all benchmark experiments, both methods use K 1000 components initialized from data samples and are trained for 100 000 gradient updates. The learning rate is set to 10-2 for both, with DLightSB-M using independent joint distribution (q" (zo, x1) = Po (2))pt (I1)).
Computational Resources and Training Time. All high-dimensional Gaussian mixture bench- mark experiments were conducted on 1 A100 GPU unless otherwise specified, with training times reported inclusive of evaluation. For D = 2, training is relatively short: CSBM and a-CSBM each complete within about 5 hours, DLightSB-M within 4 hours, and DLightSB in roughly 20 minutes. For D = 64, CSBM completes in under 14 hours, a-CSBM in under 9 hours, DLightSB-M in just under 2 days (on 2 A100 GPUs), and DLightSB in under 7 hours.
D ADDITIONAL EXPERIMENTS
D.1 REVERSE BENCHMARK
In this section, try to overcome the inherited inductive bias of DLig solvers. By con- struction, the forward conditional distribution (x1]ap) admits a CP decomposition, while the re- verse distribution qº (ze (21) does not. As result, when the benchmark is used in the reverse direc- tion with the same marginals Po and pI, DLightSB(-M methods can no longer rely on the inductive bias that benefits them in the forward setup.
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Unfortunately, in this setup, the true conditional distributions not available, pute conditional metrics. To overcome this restriction, we decided to compute the Classifier Two Sample Test (Lopez-Paz & Oquab 2017 C2ST) metric, ROC AUC of classifier between pairs (Io, x1 ) ~ (r )q" (2o and (on, ri) ~ pi (31 (0 (z1) As the classifier, we used two layer MLP with ReLU activations that takes input the concatenation of one-hot vectors of To and æ1- We present C2ST scores in the following table.
cannot com-
D 16
D
gaussian
uniform
gaussian
uniform
gaussian
uniform
Method
Loss
+
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
DLightSB
0.93
1.00
1.00
0.99
0.96
0.97
0.99
1.00
0.97
0.99
0.99
0.99
16
0.99
0.99
1.00
1.00
0.98
0.99
1.00
0.99
0.99
0.99
0.99
1.00
KL
64
1.00
1.00
0.99
1.00
0.99
0.98
0.99
0.98
1.00
1.00
0.99
1.00
CSBM
MSB
0.95
1.00
0.99
1.00
0.98
1.00
0.99
0.99
1.00
0.98
0.99
0.99
0.99
1.00
0.99
1.00
1.00
1.00
Table 2: C2ST metric (1) high-dimensional Gaussian mixture benchmark. Color code: vermillion for 0.5, orange for [0.5, 0.75). yellow |0.75, 0.85), and bluish-green for 0.85.
As be seen from Table 2 computed metric values are not informative. Across all methods, the metric values are nearly Identical, indicating that the classifier not capable of distinguishing generated samples from real ones. As result, decided to discard this metric.
D.2 ADDITIONAL METRICS AND PLOTS
iepiriy 1051
Taree
Teapt 7itnd Nardlry
Teryd (9teet
atriet
Maychhey
Imput Target
(c)
(d)
(e) CSBM
in in pon)
Tapettry one
(g) a C-CBM
(i) 3BC-CBBM
① CLB.CSBMMN
(0) DLighSB
() DLighTBBMM KL/16
(m) DLightSB-M (n) DLighhSB-M (o) DLightSBMM
Figure Samples from all methods on high-dimensional Gaussian mixture benchmark using Gaussian reference process q with 7 = 0.02.
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D
16
guussian
uniform
gaussian
uniform
guussian
uniform
Method
Loss
N +
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
Reference
0.5
0.6
0.33
0.5
01.5
0.7
0.3
0.5
0.5
0.6
0.3
0.5
Feuture-wise
0.0
0.0
0.0
0.1
0.1
0.0
0.0
0.0
0.0
DLighSSB
0.3
0.1
KL
8.0
32.1
4.7
18.4
180.1
70.8
104.9
16.8
22.0
15.2
14.1
CSBM
64
1.5
7.4
1.0
1.5
2.8
21.0
7.9
8.8
22.8
2.3
4.0
3.6
MSB
14.0
23.8
7.9
28.7
182.9
108.0
156.8
47.5
24.9
36.8
25.8
9.5
10.3
3.2
3.1
45.0
25.2
32.1
59.3
2.1
14.8
10.9
KI.
16
6.44
23.7
4.2
6.4
13.3
103.4
61.1
81.5
26.1
3.5
4.9
5.2
a-CSEM
1.3
4.9
0.7
0.7
3.55
1.1
2.1
2.2
24.5
2.1
4.5
4.6
17.5
52.8
72.3
46.2
17.9
MSE
64
1.0
3.1
0.9
0.8
9.1
4.7
5.6
4.4
58.8
2.0
15.5
11.0
16
01.1
0.0
0.0
0.0
3.2
0.1
0.2
0.2
6.2
1.3
26.4
22.3
KL
0.2
0.0
0.0
1.2
0.4
0.3
6.6
0.8
22.9
7.3
DLighhSB-M
MSE
0.8
0.0
0.2
22.0
0.5
0.3
26.3
1.4
8.8
48.3
64
1.0
0.0
0.2
0.1
24.8
0.3
0.6
0.4
58.8
2.3
41.0
17.9
Table 3: Trajectory KL divergence (L) on the high-dimensione Gaussian mixture benehmark. The best performing method is highlighted in bold, and the second is underlined. Color code: bluish-green for < 0.5, yellow for [0.5, 2), orange for [2, 10), and vermillion for 2 10.
gaussian
gaussian
uniform
gaussian
uniform
Method
Loss
N
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
Reference
2.1
3.0
0.8
1.3
2.2
4.6
1.4
1.5
1.4
3.5
1.0
1.2
Fenture wise SB
0.2
0.0
0.0
0.2
0.5
0.3
0.3
0.0
01.4
0.3
0.4
DLightSB
0.0
0.0
0.3
16
1.5
3.7
1.0
1.1
7.7
22.1
50.9
71.4
15.8
13.0
16.6
20.2
KL
CSBM
64
0.4
1.8
0.4
0.4
2.7
7.1
10.3
10.8
22.3
2.5
11.3
10.8
MSE
3.3
4.3
88.3
122.3
35.4
12.2
41.8
31.7
64
10.2
1.8
2.1
2.2
4.4
9.1
22.2
24.4
75.6
2.6
38.5
39.0
KL
16
1.7
3.3
0.8
0.9
8.1
17.5
41.8
53.88
23..6
5.0
20..5
27.3
a CSBM
0.7
0.4
0.3
4.4
1.0
7.3
10.7
22.7
3.0
17.1
16.2
16
2.7
1.7
1.1
18.5
52..5
75.4
54.1
5.6
83.6
89.5
MSE
64
0.5
1.1
1.4
1.3
11.4
2.8
16.9
20.0
75.9
3.3
46.7
45.3
KL
16
0.5
0.0
0.1
0.2
21.7
1.0
7.4
5.9
6.4
2.3
322.2
329.7
DLightSB
64
0.4
12.3
6.8
1.9
292.6
110.7
16
1.1
0.3
0.3
0.2
182.3
4.5
6.5
6.6
25.4
15.8
70.3
406.0
MSE
1.4
0.4
0.2
156.4
8.2
8.4
32.4
27.6
313.8
104.1
Table 4: Trajectory reverse KL divergence (L) the high-dimensional Gaussian mixture benchmark. The best-performing method is highlighted bold, and the second is underlined. Color code: bluish-green for < 0.5, yellow for [0.5, 2), orange for [2, 10), and vermillion for 2 10.
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64
gaussian
uniform
gaussion
uniform
gaussian
uniform
Method
Loss
N+1
0.002
0.05
0.0005
0.01
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
Independent
0.98
0.99
0.98
0.98
0.99
0.99
0.98
0.98
0.99
0.98
0.98
0.98
Reference
0.32
0.48
0.41
0.43
0.41
0.31
0.41
0.36
0.49
0.48
0.47
Feature-wise SB
0.92
0.99
0.98
0.98
0.92
0.98
0.98
0.98
0.98
0.98
0.98
0.98
DLightSB
0.97
0.97
0.98
0.98
0.89
0.97
0.96
0.98
0.97
0.95
0.97
0.97
KL
0.77
0.72
0.92
0.91
0.82
0.78
0.85
0.79
0.91
0.90
0.84
0.89
CSBM
64
0.91
0.89
0.96
0.97
0.91
0.85
0.92
0.93
0.94
0.94
0.79
0.88
MSE
0.52
0.72
0.84
0.56
0.82
0.74
0.80
0.74
0.81
0.89
0.81
0.81
0.84
0.84
0.81
0.74
0.96
0.83
0.82
16
0.74
0.75
0.93
0.91
0.79
0.81
0.85
0.78
0.93
0.89
0.81
0.87
e-CSBM
64
0.82
0.89
0.96
0.97
0.92
0.93
0.93
0.95
0.95
0.93
0.80
0. 0.96
MSE
0.73
0.89
0.90
0.79
0.85
0.82
0.93
0.83
0.82
64
0.92
0.88
0.89
0.93
0.80
0.90
0.87
0.90
0.74
0.93
0.82
0.85
KL
16
0.90
0.96
0.97
0.97
0.86
0.93
0.96
0.98
0.89
0.80
0.83
0.86
DLightSB M
0.88
0.95
0.95
0.96
0.90
0.95
0.95
0.96
0.89
0:90
0.80
0.93
MSF
16
0.75
0.95
0.85
0.93
0.69
0.96
0.93
0.92
0.72
0.88
0.89
0.79
0.73
0.95
0.81
0.92
0.65
0.62
0.68
Table 5: Shape Score metric (f) high-dimensional Gaussian mixture benchmark. The best-perforning method is highlighted bold, and the second is underlined. Color code: vermillion for < 0.5, orange for [0.5, 0. 0,75), yellow for [0.75, 0.85), and bluish-green for 2 0.85.
gaussian
uniform
gaussian
uniform
gaussian
uniform
Method
Loss
N+1
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
0.02
0.05
0.005
0.01
Independent
0.97
0.97
0.96
0.96
0.96
0.96
0.94
0.95
0.95
0.92
0.92
0.92
Reference
0.17
0.30
0.25
0.29
0.19
0.12
0.14
0.14
0.37
0.23
0.23
0.20
Featur-wise SR
0.71
0.70
0.78
0.82
0.81
0.66
0.55
0.51
0.93
0.67
0.73
0.68
DLightSB
16
0.96
0.96
0.96
0.96
0.84
0.95
0.93
0.94
0.94
0.90
0.91
0.91
KL
0.70
0.64
0.87
0.87
0.73
0.75
0.67
0.85
0.83
0.78
0.53
CSBM
64
0.89
0.85
0.93
0.93
0.87
0.81
0.87
0.88
0.90
0.89
0.73
0.83
MSE
16
0.50
0.65
0.77
0.79
0.73
0.62
0.69
0.62
0.72
0.81
0.74
0.76
0.32
0.78
0.76
0.74
0.81
0.77
0.78
0.83
0.63
0.90
0.77
0.77
16
0.66
0.66
0.89
0.87
0.71
0.71
0.78
0.70
0.88
0.83
0.74
0.81
Q-CSBM
0.81
0.85
0.93
0.93
0.89
0.90
0.88
0.91
0.91
0.88
0.74
0.81
MSE
16
0.63
0.64
0.83
0.53
0.71
0.69
0.76
0.73
0.87
0.75
0.75
64
0.89
0.84
0.83
0.86
0.72
0.87
0.81
0.84
0.63
0.88
0.75
0.79
KI.
16
0.87
0.95
0.94
0.95
0.78
0.91
0.91
0.92
0.83
0.74
0.64
0.67
DLighiSB-M
0.85
0.93
0.93
0.94
0.84
0.92
0.91
0.92
0.83
0.85
0.64
0.83
MSE
16
0.67
0.93
0.80
0.91
0.54
0.92
0.88
0.88
0.60
0.83
0.79
0.51
0.92
0.76
0.89
0.49
0.46
Table 6: Trend Score (1) high- nigh-dimensional Gaussian mixture benchmark. The pest-pern method is highlighted in bold, and the second underlined. Color code: vermillion for 0.5, orange for [0.5, 0.75). yellow for [0.75, 0.85). and bluish-grecn for 2 0.85.
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Published as a conference paper at ICLR 2026
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()) DLightSBMM KL/16
(m) DLightSB-M KL/64 (n) DLightSB-M MSE/16 (o) DLightSB-M MSE/64
Figure 3: Samples from all methods on the high- sigh-dimensional Gaussian mixture benchmark using the Gaussian reference process gaasss with y = 0.05.
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Figure Samples from all methods on the nigh-dimensional Gaussian mixture benchmark using the uniform reference process n with = 0.005.
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Published as a conference paper at ICLR 2026
Te
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(m) DLightSB-M KL/64 (n) DLightSB-M MSE/16 (o) DLighiSB-M MSE/64
Figure Samples from all methods on the high-dimensional Gaussian mixture benchmark using the uniform reference process g mii with 0.01.
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Reviews:
[{"review_id": "TrfOQW8QBc", "reviewer": "Reviewer_MD5S", "summary": "The paper identifies a valid and significant gap in the machine learning literature: the absence of a standardized, ground-truth benchmark for evaluating Schrödinger Bridge (SB) solvers on high-dimensional discrete state spaces. The authors' stated contributions are:\n1.  A method for constructing benchmark pairs of probability distributions $(p_0, p_1)$ on discrete spaces ($\\mathcal{X} = \\mathbb{S}^D$) for which the analytical SB solution $q^*$ is known by design.\n2.  The use of a Canonical Polyadic (CP) decomposition to parameterize a key scalar function ($v^*$), rendering the benchmark construction and sampling computationally tractable in high dimensions.\n3.  The introduction of three new or adapted solvers (DLightSB, DLightSB-M, and $\\alpha$-CSBM) as \"byproducts\" of this framework.\n4.  An empirical evaluation of these new solvers against the existing CSBM method, using the proposed benchmark, which concludes that the DLightSB and DLightSB-M methods demonstrate superior performance.\n\nDespite the commendable goal, the paper's execution suffers from several fundamental weaknesses in its methodology, evaluation, and framing that undermine its primary conclusions and limit its utility as a general benchmark for the community.", "questions": "#### **On the Benchmark Design and Evaluation Bias**\n\n1.  The paper's primary \"winning\" solver, DLightSB, is based on a Canonical Polyadic (CP) decomposition. The benchmark's ground truth is *also* constructed using a CP decomposition (Prop 3.1).  How can we be sure that the superior results of DLightSB are not an artifact of it being given the correct inductive bias, a bias that competing methods like CSBM (using an MLP) do not have?\n\n2.  The paper's results show that DLightSB \"wins\" on a benchmark built on \"the same principle used by the DLightSB solver.\" Does this not confirm a circular evaluation? How would DLightSB perform on a benchmark constructed with a *different* non-factorizable structure, such as one based on a deep neural potential?\n\n3.  The ground-truth benchmark is generated with a low rank ($K=4$), while the DLightSB solver is given a capacity of $K=1000$. Does this 250-fold capacity advantage not make the task trivial for a solver that already shares the problem's functional form? What insights can be gained from a benchmark that is this simple and architecturally-matched to the solver?\n\n#### **On the Choice of Methods and Comparisons**\n\n4.  The remark in Section 2.4 dismisses \"tabular EOT\" solvers as non-generative. This seems to overlook the entire class of modern EOT solvers that parameterize the dual potentials ($\\phi, \\psi$) with neural networks, which are both generative and generalizable. Why were these potential-fitting neural EOT methods—which would provide a true \"apples-to-apples\" comparison against DLightSB—not included as a baseline?\n\n5.  The evaluation compares DLightSB (a *static, dual-potential solver*) against CSBM (a *dynamic, primal-path solver*). Is this not a \"apples-to-oranges\" comparison? DLightSB solves a simpler, static problem, while CSBM solves a much harder, dynamic one. How can we conclude DLightSB is a better \"Schrödinger Bridge solver\" from this comparison?\n\n#### **On the Choice of Evaluation Metrics**\n\n6.  The entire purpose of the benchmark is that the ground-truth potential $v^*$ is known. Why, then, were the evaluation metrics (Shape Score, Trend Score) based on comparing *output samples*? Why did the authors not use the most direct and rigorous metric: the error between the learned potential $v_\\theta$ and the true potential $v*$?\n\n#### **On the Theoretical Contributions and Framing**\n\n7.  Theorem 3.1 appears to be a restatement of the well-known dual solution to the EOT problem, where one potential is set to 1 by construction. Can the authors clarify the theoretical novelty of this theorem beyond this restatement?\n\n8.  The paper's main innovation appears to be the *engineering* choice of using a CP decomposition to make the benchmark tractable. However, this choice seems to *require* the ground-truth problem to be low-rank ($K=4$). Does this not create a fundamental trade-off where the benchmark can be either (a) computable but trivial, or (b) complex but not computable?\n\n10. The paper's \"winning\" solvers (DLightSB, DLightSB-M) are admitted to have \"severe memory constraints\" and become \"prohibitive for high-dimensional data.\" This seems to create a contradictory message. What is the value of a solver that only \"wins\" on a simple, biased benchmark and is not scalable to the very high-dimensional problems it claims to address?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "1. Given a source marginal, and a scalar valued function (what is known in the literature as potentials) it is easy to obtain a corresponding target marginal and the solution SB distribution. The construction is simple and practical and serves as a good synthetic test bed.\n\n2. The parametrization of the scalar valued potential using rank-1 Canonical Polyadic decompositions seems to be versatile (in theory) given that they can act as universal approximators and possess nice theoretical properties.", "weaknesses": "### 1. Stated Theoretical Contribution: Benchmark Construction\n\nThe paper's central theoretical claim is the introduction of a method for constructing ground-truth benchmark pairs for the Schrödinger Bridge (SB) problem on high-dimensional discrete spaces. This method yields pairs of probability distributions $(p_0, p_1)$ for which the analytical SB solution $q^*$ is known by design, enabling rigorous evaluation.\n\nThis construction is formally presented in **Theorem 3.1**, which states:\n\n1.  One begins with a known source (initial) distribution $p_0 \\in \\mathcal{P}(\\mathcal{X})$.\n\n2.  One also defines a scalar-valued function $v^*: \\mathcal{X} \\rightarrow \\mathbb{R}$.\n\n3.  The \"ground-truth\" joint distribution $q*$  is then constructed such that its $x_0$ marginal is $p_0$ and its conditional distribution $q ^* (x| y)$ is proportional to the product of this new function $v*$ and the reference transition kernel $q^{ref}(x_1|x_0)$:\n\n    $$q ^* (x_1|x_0) \\propto v^*(x_1) q^{ref}(x_1|x_0)$$,\n\nor \n\n   $$q ^* (x_1|x_0) =v^*(x_1) q^{ref}(x_1|x_0) / c(x_0).$$\n\n4.  The target distribution $p_1$ is then *defined* as the second marginal of this $q ^* (x_0, x_1) = p_0(x_0) q ^* (x_1|x_0)$.\n5.  The theorem concludes that the $q^*(x_0, x_1)$ constructed this way is the true static SB solution between the resulting $p_0$ and $p_1$.\n\n\n### 2. Marginal Theoretical Novelty\n\nThe theoretical novelty of this construction (Theorem 3.1) is only marginal. It is a \"reverse-engineered\" benchmark based on a restatement of the well-established dual solution to the Entropic Optimal Transport (EOT) problem, which the paper itself shows is equivalent to the static SB problem (Section 2.3).\n\n* **Standard EOT Duality:** The solution $\\pi ^*$ to the EOT problem $\\min_{\\pi \\in \\Pi(p_0, p_1)} \\text{KL}(\\pi || R)$ is uniquely characterized by two potentials (Lagrange multipliers) $\\phi$ and $\\psi$ such that the optimal coupling has the form:\n\n    $$\\pi^*(x_0, x_1) \\propto \\phi(x_0) \\psi(x_1) R(x_0, x_1)$$\n\n* **Deriving Theorem 3.1 from first principles:** The paper's construction can be easily derived from the known result above.\n\n    * The paper's *reference measure* $R(x_0, x_1)$ is $q^{ref}(x_0, x_1)$, which is assumed to have the $p_0$ marginal: $R(x_0, x_1) = p_0(x_0) q^{ref}(x_1|x_0)$.\n    * Suppose we construct the optimal coupling as $q ^* (x_0, x_1) = \\phi(x_0) v ^* (x_1) q^{ref}(x_0, x_1)$, and we want its first marginal to be $p_0$. Integrating out $x_1$ gives $p_0(x_0) = \\phi(x_0) p_0(x_0) \\sum_{x_1} v ^* (x_1) q^{ref}(x_1 | x_0)$, so canceling out $p_0$ terms gives $\\phi(x_0) = 1/c(x_0) = 1/ \\sum_{x_1} v ^* (x_1) q^{ref}(x_1 | x_0)$. This is exactly the paper's construction. \n\nSince one can easily deduce the construction in Theorem 3.1 from a very well-known first principle in SB/EOT literature, it doesn't appear as a significant contribution in this work. \n\n### 3. The \"Single-Potential Problem\" Simplification\n\nThe paper's methodology *chooses* to construct a benchmark for a simplified theoretical problem.\n\n* The general, two-potential SB problem requires solving for *both* potentials $\\phi(x_0)$ and $\\psi(x_1)$ to satisfy the two marginal constraints $\\pi_0 = p_0$ and $\\pi_1 = p_1$.\n* The paper's benchmark, the only unknown potential is for the output domain; the input domain's potential is then determined. (Note that the potential $\\phi$ on the input space is determined by the potential $v ^*$ on the output domain and the reference transition kernel $q^{ref}(x_1 | x_0)$.)\n* This theoretical simplification is crucial because the paper's \"winning\" solver, **DLightSB**, is *explicitly* a static, dual-problem solver designed to solve this very particular problem. \n\n\n\n### 4. Theoretical Limitation of the Chosen Method\n\nThe paper's actual innovation is not Theorem 3.1, but the methodology to make it computationally tractable. The theoretical constructions above are trivial in low dimensions, but in the paper's high-dimensional setting ($\\mathcal{X} = \\mathbb{S}^D$), the potential $v ^*$ and the normalization constants are intractable, requiring sums over $S^D$ states. \n\nThe paper's *actual* contribution is the application of a Canonical Polyadic (CP) decomposition to parameterize $v^*$. As shown in **Proposition 3.1**, this factorization cleverly turns the intractable normalization sum into a product of sums, making the computation feasible. This is a practical, **engineering contribution**, not a new theory for constructing benchmarks.\n\nThis reliance on a CP parameterization for tractability imposes a severe *theoretical limitation* on the benchmark itself: it guarantees the ground-truth problem is trivially simple.\n\n* The computational tractability of the CP method is only achieved if the rank $K$ is small.\n* The authors, in their experiments, use **$K=4$** to generate the ground-truth data.\n* This means the \"challenging\" benchmark problem is, by construction, an *extremely simple, low-rank function*.\n* This defeats the entire purpose of a benchmark, which is to test a method's ability to solve a complex problem. The paper's theoretical framework *requires* the problem to be simple for the ground truth to be computable.\n\n\n### 5. Fundamental Methodological Bias and \"Apples-to-Oranges\" Evaluation\nThe paper's central weakness is a circular and biased evaluation methodology. The primary conclusion—that DLightSB and DLightSB-M are the superior solvers—is an artifact of the benchmark's design rather than a demonstration of general algorithmic strength.\n\n* **Architectural Circularity:** The tractability of the benchmark (Prop. 3.1) relies entirely on parameterizing the scalar function $v^*$ using a **CP decomposition**. The authors' \"new\" solvers, DLightSB and DLightSB-M, are *also* explicitly parameterized using the **exact same CP decomposition**. This creates an evaluation that is not a fair comparison of general SB solvers. Instead, it tests how well a CP-based model can fit a function already known to have a CP structure. The authors explicitly acknowledge this circularity: \"We attribute this to the benchmark pairs being built on the same principle used by the DLightSB solver\".\n\n* **\"Apples-to-Oranges\" Comparison:** The benchmark problem itself represents a *simplified* version of the full SB problem. The authors' \"generative\" construction (Theorem 3.1) fixes $p_0$ and the reference dynamics, which effectively reduces the problem to solving for a *single* static potential, $v^*$. The paper then compares two fundamentally different classes of algorithms:\n    * **DLightSB (The \"Apple\"):** A **static, dual-problem solver** whose entire purpose is to fit this single potential $v_\\theta$.\n    * **CSBM (The \"Orange\"):** A **dynamic, primal-problem solver** based on Iterative Markovian Fitting (D-IMF), which is designed to learn the *full time-dependent transition probabilities* $m(x_{t_n} | x_{t_{n-1}})$.\n\nThis evaluation is inherently unfair. A specialized static solver (DLightSB) that is custom-built for this simplified problem (and given the correct architecture) is predictably going to outperform a general-purpose dynamic solver (CSBM) that is designed for a much harder task.\n\n### 6. Triviality of the Benchmark Task\nThe reliance on the CP parameterization for tractability forces the ground-truth benchmark to be exceptionally simple, defeating the purpose of a benchmark, which should test a challenging problem.\n\n* The ground-truth benchmark is generated using a CP decomposition with a rank of **$K=4$**.\n* The authors' \"winning\" solvers, DLightSB and DLightSB-M, are then initialized with a capacity of **$K=1000$** components.\n\nThe task is thus to fit a simple $K=4$ mixture model using a massively over-parameterized ($K=1000$) model that shares the *identical* functional form. The 250-fold capacity advantage, combined with the correct inductive bias, makes the problem trivial for the DLightSB solvers. A benchmark that is this simple and architecturally-biased provides little insight into how these solvers would perform on complex, real-world problems.\n\n### 7. Flawed Evaluation Metrics and Exclusion of Relevant Competitors\nThe paper's evaluation protocol is indirect and avoids the most rigorous tests of its own contributions by excluding the most relevant class of competing algorithms.\n\n* **Exclusion of Neural EOT Solvers:** The paper's \"Remark\" dismisses all \"tabular EOT\" solvers as \"non-generative.\" This is a strawman argument that ignores the modern, active field of research that *does* learn generalizable, generative couplings by parameterizing the dual potentials ($\\phi, \\psi$) with neural networks. A fair and informative benchmark would have *required* a comparison between **DLightSB (CP-potential)** and a **Neural EOT Solver (MLP-potential)**. This \"apples-to-apples\" comparison would have directly tested the paper's only true innovation: the claim that a CP decomposition is a good inductive bias for this problem.\n\n* **Inadequate \"Black-Box\" Metrics:** By excluding other potential-fitting methods, the paper is forced into the \"apples-to-oranges\" (DLightSB vs. CSBM) comparison. Because these models learn different objects (potentials vs. dynamics), a direct comparison is impossible. The authors are thus forced to use weak, indirect metrics: **Shape Score** and **Trend Score**. These metrics are inadequate for two reasons:\n    1.  **They Don't Measure the *Real* Ground Truth:** The *entire point* of the benchmark is that the true potential $v*$ is known. The most rigorous metric would be a direct comparison of the learned potential $v_\\theta$ to $v*$. The paper avoids this.\n\n    2.  **They Don't Guarantee Optimality:** These metrics only measure low-order statistics of the *output samples*. They confirm that the model can approximate the *final marginal* $p_1$, but they provide no guarantee that the model has learned the *true, optimal transport coupling* $q ^*$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RZryaC8Jo6", "reviewer": "Reviewer_dFbH", "summary": "This paper presents a benchmark for SB methods on discrete spaces. The contributions of this work seem to be two-fold: (1) a theoretical investigation for constructing a benchmark for discrete SB problems and (2) an actual benchmark along with initial results, including three new implementations of SB algorithms. The authors constructed a practical benchmark based on Gaussian mixtures. The implemented DLightSB algorithms (discrete variants) achieved the best results.", "questions": "* Is there a table that comprehensively summarizes the basic statistics of the GMM benchmarks? (D={2, 16, 64})", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "Based on my reading, the paper has the following strengths. \n* Solid theoretical investigation and a well-grounded evaluation strategy for discrete SB problems. \n* An actual benchmark that will benefit the SB community. \n* Comprehensive empirical experiments and new discrete SB algorithms", "weaknesses": "I believe this work in its current format lacks novelty in many regards.\n\n* The theoretical arguments presented in this paper can be seen as either well-known or a reformulation of continuous SB work. I generally view expanding discrete theoretical work to continuous spaces as more challenging and practical, not the other way around.\n* The proposed benchmark setup relies heavily on Gaussian mixtures, which were originally intended for representing distributions in continuous spaces, and this tarnishes one of the central contributions of the paper (focusing on the new discrete problem), as the statistics will share many characteristics with continuous problems.\n* It is widely known in the SB community that LightSB algorithms excel in Gaussian mixture EOT benchmarks, since LightSB actually has a parametric prior of Gaussian mixture models (see Korotin et al. and Gushchin et al.). Therefore, I am somewhat certain that the authors were highly knowledgeable of this information before the experimental design and engineered DLightSB and DLightSB-M to show similarly good performance on their platform. In other words, these results contain very similar implications to the two LightSB papers, which makes the novelty of this paper weak, since not much new information is given beyond the references.\n* The sheer number of benchmarks is lacking, and four important algorithms (DLightSB, DLightSB-M, and α-CSBM) can be generally considered as reformulations from the work of a single research group. The authors are encouraged to put some more effort into implementing other discrete formulations of competitive SB algorithms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NoZjladN4Z", "reviewer": "Reviewer_ECS7", "summary": "In this paper, an analytic benchmark for discrete optimal transport is developed. In order to assess capabilities of current algorithms for solving the schrodinger bridge problem on discrete state spaces, there is a need for a benchmark for which the ground truth solution is known. The authors construct this analytic solution by writing the optimal transition distribution as a scalar function multiplying the reference transition distribution. Then a specific mixture of factorized forms is proposed for the scalar function such that the true schrodinger bridge transition distribution becomes analytically calculable. The authors propose two new light schrodinger bridge algorithms for discrete state spaces and compare them with existing methods on their benchmark.", "questions": "Why do CSBM algorithms have better performance on the benchmark with higher dimensionality? I would expect that performance would degrade with higher dimensions given a fixed model capacity but perhaps because of the low dimensional structure in the benchmark this is not the case?\n\nWouldn't it be possible to create a metric that doesn't just take into account single marginals or pairs of marginals by looking at distribution metrics? For example you could take a given starting point x0, sample p(x1 | x0) from the learned SB and the true SB then compute a distributional metric between those two sample distributions. Then you could average these values over different draws of x0.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The use of the mixture of factorized functions for the scalar tilting function looks novel to me and is an interesting way to construct a non-trivial problem for which the schrodinger bridge is known. The same factorization technique is then also used to construct the Light Schrodinger algorithms which show very good performance compared to prior bridge matching approaches. It is important to have these Light Schrodinger bridge baseline algorithms for further comparison during algorithm development as they serve as useful reference points.", "weaknesses": "The benchmark seems to already be too easy because the DLightSB method is already almost saturating the performance metrics. This means the benchmark is already losing its discriminative power between the best methods. Ideally, the benchmark would contain sub-tasks for which all methods struggle. Could the task be made more difficult by increasing the number of mixture components in the scalar function?\n\nIn the introduction in line 47 you state that one motivation for introducing this benchmark is to check whether methods actually solve the SB problem separating out 'true algorithmic performance from artifacts of specific parameterizations'. However, then on L431 you hypothesize that DLightSB performs the best on this benchmark because it is built on the same factorization principle as the benchmark itself. This would then seem to go against the stated goal of the benchmark to try and not be influenced by specific parameterizations of the method. Perhaps the benchmark needs to be made more complex as to not be 'defeated' by the DLightSB algorithm.\n\nThe differences between your two proposed methods DLightSB and DLightSB-M could have been investigated more. You hypothesize that DLightSB-M works worse due to error accummulation. This could have been verified or falsified with a further experiment, for example, changing the number of discretization time steps in the diffusion process to see if the discrepancy between the two methods gets smaller.\n\nEquations (19) and (20) are quite hard to parse due to the circular definition of the set \\mathcal{S}. The set itself is used within its own definition in the argmax and makes the whole equation hard to understand.\n\nOverall, I think the contribution of the rank-1 Canonical Polyadic decomposition for constructing both ground truth SBs and for learning LightSBs is a worthy contribution to the conference notwithstanding the limitations in evaluations of the proposed methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XiZcRNLk50", "reviewer": "Reviewer_FM4X", "summary": "The authors proposed a mixture-like benchmark for SB on discrete spaces to evaluate the effectiveness of different methods. The authors also proposed a light version of DSB and a smoothing version of the CSBM algorithm, α-CSBM,  to enhance the training stability and efficiency. Interesting comparisons are made on toy examples to show the strength and weakness.", "questions": "1. Could you provide an additional section (like section 4.4 in CSBM) to verify the scalability?\n\n2. i didn't get line 352: \"be generalized to an arbitrary reference process qref, thereby enabling the application of the optimal projection in discrete space settings.\" If some algorithm accuracy is compromised (like low rank), does it still guarantee optimal projection? it is a bit confusing. \n\n3. I haven't fully checked CSBM yet. Is the algorithm simulation-free or not? To me, low-rank may be interesting but the simulation-free property is more important for the scalability.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Providing a mixture example to evaluate the accuracy of different approaches is helpful to the community.\n\n2. The light (low rank) and alpha smoothing do have some potential to accelerate the algorithm.\n\n3. IMF is more appealing and interesting than the IPF style of algorithms.", "weaknesses": "1. Even the \"expensive\" version of the CSBM algorithm is evaluated on real-world image experiments. The light and alpha-smoothed algorithm is only evaluated on the mixture benchmark. This is far from sufficient.\n\n2. The writing is not good. The current draft is a mixture of lots of components, like a baby mixture benchmark, 2 light algorithms (without sufficient discussions on theoretical properties or empirical evaluations). The scope is ambitious, but for each contribution itself, it is a bit minor.\n\n3. Although low0", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)