DLightSB-M: Dynamic Discrete Schrödinger Bridge Solver

• The paper introduces DLightSB-M, a dynamic solver that recovers full SB dynamics via a single optimal projection step aligned with closed-form benchmarks.
• DLightSB-M leverages CP-decomposition to parameterize intermediate transition kernels, enabling tractable high-dimensional discrete SB and EOT recovery.
• The method optimizes a KL-based loss to achieve accurate pathwise transitions, though it can be computationally intensive in high-dimensional settings.

DLightSB-M is a dynamic solver for discrete Schrödinger bridges (SB) and entropic optimal transport (EOT), introduced in "Entering the Era of Discrete Diffusion Models: A Benchmark for Schrödinger Bridges and Entropic Optimal Transport" (Carrasco et al., 27 Sep 2025). In the paper’s framing, it is the dynamic, discrete-space analogue of LightSB-M, built as one of the main solvers motivated by a new benchmark for discrete SB and EOT. It is not presented as a generic discrete diffusion model; rather, it is a solver whose parameterization is aligned with the benchmark’s analytically known SB structure, with the goal of recovering the full SB dynamics by optimizing a tractable projection objective. The key idea is a discrete-space optimal projection that recovers the SB through a single projection step.

1. Definition and motivation

DLightSB-M was introduced in response to two gaps identified for discrete SB and EOT methods: the absence of a benchmark with known ground-truth SB solutions, and the absence of practical solvers that can be compared fairly against those known solutions (Carrasco et al., 27 Sep 2025). The benchmark construction provides pairs of probability distributions for which the optimal SB is known in closed form. As a byproduct of that construction, the paper derives new solvers, including DLightSB and its dynamic extension, DLightSB-M.

The motivation for DLightSB-M is threefold. It is intended to provide a dynamic SB solver in discrete spaces, to mirror the continuous LightSB-M idea, and to use the benchmark’s analytically known SB structure to define a single projection step onto the SB family rather than the more indirect Markovian fitting used in earlier methods. The paper states: “Finally, we introduce DLightSB-M, a dynamic variant of DLightSB, which also uses our benchmark construction. Inspired by \citep{gushchin2024light}, we introduce a discrete-space optimal projection that recovers the SB through a single projection step” (Carrasco et al., 27 Sep 2025).

Within the paper’s solver taxonomy, DLightSB is the static solver, whereas DLightSB-M is the dynamic solver. DLightSB learns the conditional transport kernel $q_\theta(x_1\mid x_0)$ directly. DLightSB-M instead parameterizes the intermediate-time transition kernels of an SB process $q_\theta(x_{t_n}\mid x_{t_{n-1}})$. This distinguishes endpoint transport from full pathwise recovery of the bridge dynamics.

2. Discrete problem setting and benchmark construction

The discrete state space is

$X = S^D,$

where $S = \{0,1,\dots,S-1\}$ is the set of categories, $D$ is the dimensionality, and each sample $x \in X$ is a vector $x=(x^1,\dots,x^D)$ (Carrasco et al., 27 Sep 2025). Time is discretized as

$0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$

which induces a path space $X^{N+2}$. The intermediate states are collected in $x_{\text{in}}=(x_{t_1},\dots,x_{t_N})$.

The paper distinguishes $q_\theta(x_{t_n}\mid x_{t_{n-1}})$0, the set of all discrete-time stochastic processes on the path space, from $q_\theta(x_{t_n}\mid x_{t_{n-1}})$1, the set of Markov processes. A Markov process $q_\theta(x_{t_n}\mid x_{t_{n-1}})$2 admits both forward and backward decompositions, and the notation $q_\theta(x_{t_n}\mid x_{t_{n-1}})$3 is used for either conditional or transition distributions. The discrete generative EOT/SB task is to use samples from unknown $q_\theta(x_{t_n}\mid x_{t_{n-1}})$4 to learn a model $q_\theta(x_{t_n}\mid x_{t_{n-1}})$5 that transports $q_\theta(x_{t_n}\mid x_{t_{n-1}})$6 to $q_\theta(x_{t_n}\mid x_{t_{n-1}})$7 and supports out-of-sample generation $q_\theta(x_{t_n}\mid x_{t_{n-1}})$8.

The benchmark is based on an analytically controlled construction. If one chooses a function $q_\theta(x_{t_n}\mid x_{t_{n-1}})$9, then one can define a joint distribution $X = S^D,$0 with first marginal $X = S^D,$1 and conditional structure

$X = S^D,$2

The theorem given in the paper states that if $X = S^D,$3 is a given initial distribution on a discrete space $X = S^D,$4 and $X = S^D,$5 is a given scalar-valued function, then a joint distribution $X = S^D,$6 satisfying $X = S^D,$7 and $X = S^D,$8, together with its second marginal $X = S^D,$9, defines the discrete-space EOT/SB (Carrasco et al., 27 Sep 2025).

The normalized conditional is

$S = \{0,1,\dots,S-1\}$0

with

$S = \{0,1,\dots,S-1\}$1

This analytic form constitutes the ground truth against which DLightSB-M is evaluated.

3. Dynamic parameterization of the bridge

To make the benchmark feasible in high dimensions, the paper parameterizes $S = \{0,1,\dots,S-1\}$2 via a Canonical Polyadic (CP) decomposition,

$S = \{0,1,\dots,S-1\}$3

where $S = \{0,1,\dots,S-1\}$4 is the number of mixture components, $S = \{0,1,\dots,S-1\}$5 are mixture weights, and $S = \{0,1,\dots,S-1\}$6 are CP cores (Carrasco et al., 27 Sep 2025). This induces tractable expressions for the conditional distribution and its normalization constant: $S = \{0,1,\dots,S-1\}$7 and

$S = \{0,1,\dots,S-1\}$8

For the dynamic SB, DLightSB-M uses the standard Schrödinger bridge representation

$S = \{0,1,\dots,S-1\}$9

Using the CP structure, the paper derives

$D$0

where

$D$1

This is the transition model on which DLightSB-M relies. A plausible implication is that the method’s tractability depends not only on CP factorization of $D$2 but also on the factorizable structure of the reference process, because that is what turns otherwise intractable sums over $D$3 into products of one-dimensional computations.

4. Projection principle and optimization objective

DLightSB-M is introduced as a projection method. Instead of using the D-IMF alternating projections associated with CSBM, it projects a reciprocal process directly onto the set of all SBs (Carrasco et al., 27 Sep 2025). The SB set is defined in the paper as

$D$4

The paper’s theoretical support is a projection result: if $D$5 is a reciprocal process defined with a reference process $D$6 and a joint distribution $D$7, then the optimal projection of $D$8 onto the set of all SBs is the SB $D$9 between the desired marginals $x \in X$0 and $x \in X$1, namely

$x \in X$2

DLightSB-M parameterizes this SB using the same closed-form transition parameterization as DLightSB, with $x \in X$3, and optimizes the Markovian projection objective. The practical objective is

$x \in X$4

The paper states that this objective can be used for DLightSB-M with stochastic gradient descent. Because the reference process factorizes across dimensions, the KL and CE terms decompose as sums over dimensions, and the required per-dimension transitions can be computed by marginalizing out all other dimensions using the tractable parameterization. This is the mechanism that makes the SB projection feasible in high-dimensional discrete spaces.

5. Algorithmic role and relation to adjacent methods

The paper does not present DLightSB-M as a bespoke training loop but as a solver instantiated through four steps: construct a benchmark pair $x \in X$5 using the CP-parameterized $x \in X$6 and reference process $x \in X$7; parameterize the SB transitions $x \in X$8 using the CP-based closed form; optimize the SB projection objective using SGD on the KL-based Markovian loss; and sample trajectories or endpoint samples from the learned SB via ancestral sampling (Carrasco et al., 27 Sep 2025). For discrete-time dynamics, transitions are normalized in log-space for numerical stability, and the learned model is fully Markovian.

Relative to DLightSB, the distinction is static versus dynamic. DLightSB learns the endpoint conditional $x \in X$9, whereas DLightSB-M parameterizes the full pathwise dynamics through intermediate-time transition kernels. Relative to CSBM and $x=(x^1,\dots,x^D)$0-CSBM, the distinction is one of optimization geometry and inductive bias. CSBM uses Iterative Markovian Fitting (D-IMF), alternating projections between reciprocal and Markov classes. $x=(x^1,\dots,x^D)$1-CSBM uses an online or partial-update variant of IMF, reducing cost but remaining within the CSBM framework. DLightSB-M instead leverages the closed-form SB transition structure induced by the benchmark parameterization and optimizes a KL-based objective tied directly to SB projection.

The paper notes a limitation of CSBM’s transition parameterization: it is factorized across dimensions to avoid the full $x=(x^1,\dots,x^D)$2 space, which introduces approximation error. DLightSB-M also uses factorization, but in a manner consistent with the benchmark’s tractable SB form. The paper further references DDSBM and other discrete diffusion methods, emphasizing that they are not directly benchmarked as true SB solvers in the generative EOT/SB sense. Some such methods discretize time, match generators or transition kernels, and may reduce to similar training objectives, but the contribution here is the combination of exact benchmark instances with solvers that exploit the benchmark’s known SB structure.

6. Evaluation, empirical behavior, and limitations

The benchmark provides analytically known $x=(x^1,\dots,x^D)$3 and transition distributions, so DLightSB-M can be evaluated against true SB targets rather than proxy metrics (Carrasco et al., 27 Sep 2025). The reported metrics are Conditional Shape Score (SSM), Conditional Trend Score (TSM), Trajectory KL divergence, and unconditional versions in the appendix. The conditional metrics are emphasized as the best reflection of “how well the EOT/SB solver solves the underlying problem.”

The benchmark protocol uses independent samples $x=(x^1,\dots,x^D)$4 and $x=(x^1,\dots,x^D)$5 during training. Testing uses $x=(x^1,\dots,x^D)$6 precomputed benchmark pairs $x=(x^1,\dots,x^D)$7, and conditional metrics use $x=(x^1,\dots,x^D)$8 generated $x=(x^1,\dots,x^D)$9 per $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$0 unique $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$1. This evaluation design allows direct comparison to analytically known bridge targets rather than only to marginal sample quality.

The main empirical conclusions reported for DLightSB-M are that DLightSB is the strongest overall on the benchmark, while DLightSB-M is typically close behind and often competitive. The paper states that DLightSB-M attains comparable results with a slight drop in metrics, “likely due to the additional variance introduced by the KL minimization loss.” The paper also reports that CSBM and $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$2-CSBM are generally worse than DLightSB and DLightSB-M, that $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$3-CSBM is more efficient than CSBM while maintaining similar quality, that increasing the number of Markov steps $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$4 generally improves performance, and that KL loss performs better than MSE for these SB solvers, likely because MSE oversmooths and blurs modes.

Several limitations are made explicit. DLightSB-M is computationally demanding, especially in high dimensions. The CP parameterization may require a large number of components $0 = t_0 < t_1 < \dots < t_N < t_{N+1} = 1,$5 to represent complex structures. In the high-dimensional setting, DLightSB and DLightSB-M become memory-heavy. The paper also notes that because the benchmark is constructed to align with the CP-based solver family, DLightSB and DLightSB-M can be viewed as somewhat “oracle-like” on the forward benchmark, and thus not a fully neutral measure of solver generality. This suggests that the method’s strong results partly reflect an intentionally aligned inductive bias. The paper nevertheless treats that alignment as informative, because it reveals what a strong inductive bias can achieve under analytically controlled discrete SB structure.