DLightSB: Discrete Schrödinger Bridge

• DLightSB is a method for discrete entropic optimal transport that focuses on the static conditional coupling rather than full dynamic interpolation.
• It employs a CP tensor decomposition to parameterize scaling functions, enabling efficient and scalable recovery of the SB solution in high-dimensional spaces.
• Benchmark results show DLightSB achieves high conditional Shape and Trend scores, outperforming traditional solvers in discrete generative transport tasks.

DLightSB, short for Discrete Light Schrödinger Bridge, is an algorithm for solving the discrete-state entropic optimal transport (EOT) or Schrödinger bridge (SB) problem in a generative setting. It was introduced in the benchmark paper "Entering the Era of Discrete Diffusion Models: A Benchmark for Schrödinger Bridges and Entropic Optimal Transport" (Carrasco et al., 27 Sep 2025), which studies SB methods on discrete spaces by constructing pairs of probability distributions with analytically known SB solutions. In the DLightSB formulation, one is given two discrete marginals $p_0(x_0)$ and $p_1(x_1)$ on $\mathcal{X}=S^D$, together with a reference Markov chain $q^{\mathrm{ref}}(x_1 \mid x_0)$, and seeks the joint law $q^*(x_0,x_1)\in\Pi(p_0,p_1)$ minimizing $\mathrm{KL}(q\|q^{\mathrm{ref}})$. The method focuses on the static conditional coupling $q^*(x_1\mid x_0)$ rather than directly parameterizing the full dynamic interpolation.

1. Problem formulation

DLightSB is defined against the static EOT or SB problem on a discrete product space. The admissible set is

$$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$

The optimization problem is

$$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$

where the reference transition $q^{\mathrm{ref}}(x_1\mid x_0)$ induces the cost

$p_1(x_1)$0

With this identification, the same problem can be written as

$p_1(x_1)$1

or equivalently

$p_1(x_1)$2

This places DLightSB at the intersection of entropic transport and generative modeling, with the entropy term $p_1(x_1)$3 acting as the regularizing component of the coupling objective (Carrasco et al., 27 Sep 2025).

In dynamic terms, the SB interpolates $p_1(x_1)$4 over multiple timesteps. DLightSB, however, is explicitly described as focusing on the static conditional coupling $p_1(x_1)$5. A common misconception is to read it as a direct solver for the entire time-indexed bridge process; the construction instead targets the endpoint coupling and derives a tractable conditional model for that object.

2. Closed-form SB structure and surrogate objective

The starting point of DLightSB is the closed-form structure of the static SB solution: $p_1(x_1)$6 where the unknown scaling function $p_1(x_1)$7 is chosen so that

$p_1(x_1)$8

Thus the problem reduces to recovering the scaling function that enforces the terminal marginal.

The paper derives DLightSB by substituting

$p_1(x_1)$9

into

$\mathcal{X}=S^D$0

After eliminating constants, one obtains the surrogate loss

$\mathcal{X}=S^D$1

The significance of this reduction is that it converts the constrained transport problem into an unconstrained optimization problem over a parameterized conditional model. The paper states that minimizing $\mathcal{X}=S^D$2 over $\mathcal{X}=S^D$3 drives $\mathcal{X}=S^D$4 via Proposition 3 (Carrasco et al., 27 Sep 2025).

This suggests that DLightSB operationalizes the SB constraint through the normalization structure embedded in $\mathcal{X}=S^D$5 and the representational capacity of $\mathcal{X}=S^D$6, rather than through an explicit projection onto $\mathcal{X}=S^D$7 at each optimization step.

3. CP parameterization

DLightSB parameterizes the scaling function with a CP, or rank-$\mathcal{X}=S^D$8, tensor decomposition: $\mathcal{X}=S^D$9 The corresponding normalizer is

$q^{\mathrm{ref}}(x_1 \mid x_0)$0

The parameters are $q^{\mathrm{ref}}(x_1 \mid x_0)$1.

This factorization is central to the method’s scalability in high-dimensional discrete spaces. Each rank component factorizes across coordinates, and the interaction with the reference chain appears only through one-dimensional sums over the $q^{\mathrm{ref}}(x_1 \mid x_0)$2 categories in each dimension. The paper states that computation of $q^{\mathrm{ref}}(x_1 \mid x_0)$3 and its gradient costs $q^{\mathrm{ref}}(x_1 \mid x_0)$4 per data point, since the evaluation requires sums over $q^{\mathrm{ref}}(x_1 \mid x_0)$5 categories in each dimension (Carrasco et al., 27 Sep 2025).

Theoretical guarantees are partial but explicit. By Proposition 3,

$q^{\mathrm{ref}}(x_1 \mid x_0)$6

so the global minimizer of $q^{\mathrm{ref}}(x_1 \mid x_0)$7 recovers the true SB coupling. At the same time, the paper notes that no explicit rates are proved for the nonconvex CP parameterization. It also states that, in principle, if $q^{\mathrm{ref}}(x_1 \mid x_0)$8 is large enough the approximation is arbitrarily accurate. A plausible implication is that approximation quality is controlled by the CP rank, whereas optimization quality is controlled by the nonconvex learning dynamics.

4. Optimization procedure and inference

The algorithm is specified through a minibatch stochastic optimization loop. The inputs are samples from $q^{\mathrm{ref}}(x_1 \mid x_0)$9 and $q^*(x_0,x_1)\in\Pi(p_0,p_1)$0 together with reference transitions $q^*(x_0,x_1)\in\Pi(p_0,p_1)$1. The hyperparameters are the CP rank $q^*(x_0,x_1)\in\Pi(p_0,p_1)$2, learning rate $q^*(x_0,x_1)\in\Pi(p_0,p_1)$3, batch size $q^*(x_0,x_1)\in\Pi(p_0,p_1)$4, and number of steps $q^*(x_0,x_1)\in\Pi(p_0,p_1)$5.

Initialization uses positive values for $q^*(x_0,x_1)\in\Pi(p_0,p_1)$6 and positive cores $q^*(x_0,x_1)\in\Pi(p_0,p_1)$7 for all $q^*(x_0,x_1)\in\Pi(p_0,p_1)$8. At each iteration, the procedure samples minibatches from $q^*(x_0,x_1)\in\Pi(p_0,p_1)$9 and $\mathrm{KL}(q\|q^{\mathrm{ref}})$0, computes $\mathrm{KL}(q\|q^{\mathrm{ref}})$1 for each sample from $\mathrm{KL}(q\|q^{\mathrm{ref}})$2, computes $\mathrm{KL}(q\|q^{\mathrm{ref}})$3 for each sample from $\mathrm{KL}(q\|q^{\mathrm{ref}})$4, forms the stochastic loss

$\mathrm{KL}(q\|q^{\mathrm{ref}})$5

and performs a gradient step on $\mathrm{KL}(q\|q^{\mathrm{ref}})$6. The pseudocode also allows optional projection to enforce nonnegativity on $\mathrm{KL}(q\|q^{\mathrm{ref}})$7 and $\mathrm{KL}(q\|q^{\mathrm{ref}})$8 (Carrasco et al., 27 Sep 2025).

Inference is ancestral. For a given $\mathrm{KL}(q\|q^{\mathrm{ref}})$9, one first samples a component

$q^*(x_1\mid x_0)$0

and then samples each dimension independently according to

$q^*(x_1\mid x_0)$1

This dimensionwise factorization is a direct consequence of the CP structure.

The paper provides concrete recommendations. The rank $q^*(x_1\mid x_0)$2 controls the expressivity of $q^*(x_1\mid x_0)$3. In experiments, $q^*(x_1\mid x_0)$4 achieved near-exact recovery for $q^*(x_1\mid x_0)$5 up to $q^*(x_1\mid x_0)$6. The learning rate used by the authors was $q^*(x_1\mid x_0)$7 with AdamW and $q^*(x_1\mid x_0)$8. The batch size $q^*(x_1\mid x_0)$9 should be large enough to stabilize estimates of $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$0 and $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$1, and the reported number of steps is approximately $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$2 gradient updates. The reference chain parameter $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$3 controls entropy regularization strength via $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$4.

5. Theoretical interpretation and scope

The theoretical core of DLightSB is the identity

$$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$5

which links optimization of the surrogate objective directly to recovery of the SB solution. This provides a precise interpretation of the training objective: minimizing $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$6 is not merely heuristic but is aligned with the KL divergence between the true coupling and the parameterized coupling (Carrasco et al., 27 Sep 2025).

At the same time, the method inherits the limitations of its parameterization. The CP model is nonconvex, and the paper explicitly states that no explicit convergence rates are proved for that parameterization. Therefore, theoretical exactness is attached to the global minimizer, not to any guaranteed optimization trajectory. A plausible implication is that empirical performance depends materially on initialization, optimizer choice, and rank selection, even though the underlying objective is well aligned with the target coupling.

Another important scope condition concerns what object is being approximated. DLightSB is presented as solving the discrete-state EOT or SB problem in a generative setting, but its direct target is the static conditional coupling $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$7. This matters when comparing it with methods centered on full path-space bridges or multi-time transport constructions. The dynamic SB interpretation remains the broader context, while DLightSB specializes to the endpoint conditional structure.

6. Benchmark results and comparative position

DLightSB was introduced as part of a benchmark for Schrödinger bridges on discrete spaces. The benchmark constructs pairs of probability distributions with analytically known SB solutions, enabling rigorous evaluation. Within that benchmark, the paper also introduces DLightSB-M and extends prior related work to construct the $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$8-CSBM algorithm. The code for the benchmark and associated experiments is available at https://github.com/gregkseno/catsbench (Carrasco et al., 27 Sep 2025).

Empirical evaluation is reported on high-dimensional Gaussian-mixture benchmarks with $$\Pi(p_0,p_1)=\left\{q(x_0,x_1):\sum_{x_1} q(x_0,x_1)=p_0(x_0),\ \sum_{x_0} q(x_0,x_1)=p_1(x_1)\right\}.$$9, $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$0, and two reference processes. DLightSB consistently achieved the highest conditional Shape and Trend scores, with values at least $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$1 in all settings, and outperformed the baselines “Independent,” “Reference,” and “Feature-wise SB,” as well as the other solvers CSBM and $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$2-CSBM. In a representative case with $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$3 under a Gaussian reference with $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$4, DLightSB obtained a conditional Shape Score of $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$5, compared with $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$6 for the best CSBM variant and $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$7 for $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$8-CSBM. Runtime was reported as under $$\min_{q\in\Pi(p_0,p_1)} \mathrm{KL}(q\|q^{\mathrm{ref}}),$$9 h on a single A100 GPU for $q^{\mathrm{ref}}(x_1\mid x_0)$0, whereas CSBM and $q^{\mathrm{ref}}(x_1\mid x_0)$1-CSBM took $q^{\mathrm{ref}}(x_1\mid x_0)$2 h and $q^{\mathrm{ref}}(x_1\mid x_0)$3 h, respectively.

These results position DLightSB as a solver whose design choices—single unconstrained KL minimization, low-dimensional expectations, and ancestral sampling—translate into strong empirical performance on the benchmark’s discrete generative transport tasks. Because the benchmark uses analytically known SB solutions, the reported scores are not only comparative but also tied to a controlled evaluation setting. This suggests that DLightSB’s contribution is methodological and evaluative at once: it is both a new solver and a vehicle for assessing how well discrete SB methods recover the target coupling.