# Light and Optimal Schr"odinger Bridge Matching

arXiv: https://arxiv.org/abs/2402.03207 · HTML: https://mlsanity.com/paper/2402.03207/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Diffusion-GAN: Training GANs with Diffusion](https://mlsanity.com/paper/2206.02262/) (2022)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Adversarial Latent Autoencoders](https://mlsanity.com/paper/2004.04467/) (2020)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)

## Built on by (most cited first, 30 of 45)

- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Adversarial Schr"odinger Bridge Matching](https://mlsanity.com/paper/2405.14449/) (2024)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Distribution Prototype Diffusion Learning for Open-set Supervised Anomaly Detection](https://mlsanity.com/paper/2502.20981/) (2025)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Finding Local Diffusion Schr"odinger Bridge using Kolmogorov-Arnold Network](https://mlsanity.com/paper/2502.19754/) (2025)
- [InfoBridge: Mutual Information estimation via Bridge Matching](https://mlsanity.com/paper/2502.01383/) (2025)
- [Go With the Flow: Fast Diffusion for Gaussian Mixture Models](https://mlsanity.com/paper/2412.09059/) (2024)
- [Categorical Schr"odinger Bridge Matching](https://mlsanity.com/paper/2502.01416/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [LightSBB-M: Bridging Schr"odinger and Bass for Generative Diffusion Modeling](https://mlsanity.com/paper/2601.19312/) (2026)
- [BM2: Coupled Schr"odinger Bridge Matching](https://mlsanity.com/paper/2409.09376/) (2024)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matching](https://mlsanity.com/paper/2602.15396/) (2026)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Variational Entropic Optimal Transport](https://mlsanity.com/paper/2602.02241/) (2026)
- [Generalized Schr"odinger Bridge on Graphs](https://mlsanity.com/paper/2602.04675/) (2026)
- [Discrete Bridges for Mutual Information Estimation](https://mlsanity.com/paper/2602.08894/) (2026)
- [Rectified Schr"odinger Bridge Matching for Few-Step Visual Navigation](https://mlsanity.com/paper/2604.05673/) (2026)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Tilted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2609.34642/) (2026)
- [Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization](https://mlsanity.com/paper/2410.02628/) (2024)
- [Entering the Era of Discrete Diffusion Models: A Benchmark for Schr"odinger Bridges and Entropic Optimal Transport](https://mlsanity.com/paper/2509.23348/) (2025)

## Abstract (arXiv)

Schr"odinger Bridges (SB) have recently gained the attention of the ML community as a promising extension of classic diffusion models which is also interconnected to the Entropic Optimal Transport (EOT). Recent solvers for SB exploit the pervasive bridge matching procedures. Such procedures aim to recover a stochastic process transporting the mass between distributions given only a transport plan between them. In particular, given the EOT plan, these procedures can be adapted to solve SB. This fact is heavily exploited by recent works giving rise to matching-based SB solvers. The cornerstone here is recovering the EOT plan: recent works either use heuristical approximations (e.g., the minibatch OT) or establish iterative matching procedures which by the design accumulate the error during the training. We address these limitations and propose a novel procedure to learn SB which we call the optimal Schr"odinger bridge matching. It exploits the optimal parameterization of the diffusion process and provably recovers the SB process (a) with a single bridge matching step and (b) with arbitrary transport plan as the input. Furthermore, we show that the optimal bridge matching objective coincides with the recently discovered energy-based modeling (EBM) objectives to learn EOT/SB. Inspired by this observation, we develop a light solver (which we call LightSB-M) to implement optimal matching in practice using the Gaussian mixture parameterization of the adjusted Schr"odinger potential. We experimentally showcase the performance of our solver in a range of practical tasks. The code for our solver can be found at https://github.com/SKholkin/LightSB-Matching.