Research map / Learning theory and algorithms / Optimal transport
Light Schrödinger Bridge
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Proposes LightSB, a simple simulation-free Schrödinger Bridge solver that parameterizes the adjusted Schrödinger potential as an unnormalized Gaussian mixture, yielding a closed-form KL objective optimized by minibatch SGD on CPU.
Problem
Existing continuous Schrödinger Bridge (SB) / entropic OT solvers are heavy: they need neural networks, max-min optimization, MCMC, or iterative proportional fitting with trajectory simulation, making them slow for moderate-dimensional problems.
Approach
Combine sum-exp (Gaussian mixture) parameterization of Schrödinger potentials with the energy-based view of log-potentials: the KL between the EOT plan and a Gaussian-mixture-based plan reduces to a closed-form, non-minimax objective with tractable normalizer, allowing SGD without MCMC or simulation; the learned plan induces an SB drift in closed form.
Key results
- On the EOT benchmark mixtures, LightSB reports conditional BW2^2-UVP of 0.03% (D=2, eps=0.1) to 0.62% (D=128, eps=1), far below the best competing solver (e.g., 1.94% and 11.4% at D=2 and D=128 for eps=0.1).
- On MSCI with DIM=1000, LightSB achieves energy distance 1.27 ± 0.19 in 146 s on 4 CPU cores, comparable to or better than GPU-based solvers such as Langevin-based (1.46) and minimax (1.32) that take 15-71 minutes.
- Unpaired male-to-female FFHQ translation in ALAE 512-dim latent space trains in under 1 minute on 4 CPU cores.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=0.1, D=2 | cBW₂²-UVP (%): 1.94 -> 0.03 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=0.1, D=16 | cBW₂²-UVP (%): 13.67 -> 0.08 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=0.1, D=64 | cBW₂²-UVP (%): 11.74 -> 0.28 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=0.1, D=128 | cBW₂²-UVP (%): 11.4 -> 0.60 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=1, D=2 | cBW₂²-UVP (%): 1.04 -> 0.05 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=1, D=16 | cBW₂²-UVP (%): 9.08 -> 0.09 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=1, D=64 | cBW₂²-UVP (%): 18.05 -> 0.24 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=1, D=128 | cBW₂²-UVP (%): 15.23 -> 0.62 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=10, D=2 | cBW₂²-UVP (%): 1.40 -> 0.07 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=10, D=16 | cBW₂²-UVP (%): 1.27 -> 0.11 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=10, D=64 | cBW₂²-UVP (%): 2.36 -> 0.21 | Helped |
| Use LightSB instead of the best reported solver | Mixtures benchmark · ε=10, D=128 | cBW₂²-UVP (%): 1.31 -> 0.37 | Helped |
Surprising findings
- Scalar covariance Gaussian mixtures suffice for universal approximation of SBs, justifying a simpler parameterization.
Limitations
- Designed for moderate dimensions; not intended for direct use in image space
- Generalization bound relies on bounded parameter classes and compactly supported distributions; approximation-estimation trade-off not analyzed
Experimental setup
Datasets: FFHQ MSCI (Open Problems - Multimodal Single-Cell Integration) Entropic OT benchmark mixtures
Benchmarks: Entropic OT benchmark (mixtures) MSCI single-cell energy distance
Metrics: cB W2^2-UVP energy distance training time
Baselines: Entropic OT benchmark best solver Sinkhorn Langevin-based EOT solver Minimax EOT solver IPF SB solver
Builds on: Entropic optimal transport Schrödinger bridge Gaussian mixture potentials from Gushchin et al. OT benchmark energy-based EOT (Mokrov et al.)
Closest papers by meaning
- Light and Optimal Schrödinger Bridge Matching (ICML 2024 · similarity 0.92)
- Schrödinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres (NeurIPS 2025 · similarity 0.88)
- Linear convergence of Sinkhorn's algorithm for generalized static Schrödinger bridge (ICML 2025 · similarity 0.87)
- Categorical Schrödinger Bridge Matching (ICML 2025 · similarity 0.87)
- Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.86)
- Finding Local Diffusion Schrodinger Bridge using Kolmogorov-Arnold Network (CVPR 2025 · similarity 0.86)
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