# Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark

arXiv: https://arxiv.org/abs/2306.10161 · HTML: https://mlsanity.com/paper/2306.10161/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 57 references that are in the map.
- First posted on arXiv 16 Jun 2023.

## Builds on (most cited first, 30 of 37)

- [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)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [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)
- [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)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Score-based Generative Neural Networks for Large-Scale Optimal Transport](https://mlsanity.com/paper/2110.03237/) (2021)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Neural Monge Map estimation and its applications](https://mlsanity.com/paper/2106.03812/) (2021)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)

## Built on by (most cited first, 19 of 19)

- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Adversarial Schr"odinger Bridge Matching](https://mlsanity.com/paper/2405.14449/) (2024)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Diffusion & Adversarial Schr"odinger Bridges via Iterative Proportional Markovian Fitting](https://mlsanity.com/paper/2410.02601/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [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)
- [BM2: Coupled Schr"odinger Bridge Matching](https://mlsanity.com/paper/2409.09376/) (2024)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [RadioFlow: Efficient Radio Map Construction Framework with Flow Matching](https://mlsanity.com/paper/2510.09314/) (2025)
- [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)
- [A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution](https://mlsanity.com/paper/2606.02515/) (2026)

## Abstract (arXiv)

Over the last several years, there has been significant progress in developing neural solvers for the Schr"odinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is interconnected with the practically well-performing diffusion models and theoretically grounded entropic optimal transport (EOT). Still, the area lacks non-trivial tests allowing a researcher to understand how well the methods solve SB or its equivalent continuous EOT problem. We fill this gap and propose a novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. Our methodology is generic and works for a wide range of OT formulations, in particular, it covers the EOT which is equivalent to SB (the main interest of our study). This development allows us to create continuous benchmark distributions with the known EOT and SB solutions on high-dimensional spaces such as spaces of images. As an illustration, we use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution. Our code for constructing benchmark pairs under different setups is available at: https://github.com/ngushchin/EntropicOTBenchmark.