# Iterated Schr"odinger bridge approximation to Wasserstein Gradient Flows

arXiv: https://arxiv.org/abs/2406.10823 · HTML: https://mlsanity.com/paper/2406.10823/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 16 Jun 2024.

## Builds on (most cited first, 7 of 7)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)

## Built on by (most cited first, 5 of 5)

- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoders](https://mlsanity.com/paper/2506.15881/) (2025)
- [Mirror Bridges Between Probability Measures](https://mlsanity.com/paper/2410.07003/) (2024)
- [Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2505.07647/) (2025)
- [Variational Inference via Entropic Transport Descent](https://mlsanity.com/paper/2606.25265/) (2026)

## Abstract (arXiv)

We introduce a novel discretization scheme for Wasserstein gradient flows that involves successively computing Schr"odinger bridges with the same marginals. This is different from both the forward/geodesic approximation and the backward/Jordan-Kinderlehrer-Otto (JKO) approximations. The proposed scheme has two advantages: one, it avoids the use of the score function, and, two, it is amenable to particle-based approximations using the Sinkhorn algorithm. Our proof hinges upon showing that relative entropy between the Schr"odinger bridge with the same marginals at temperature ε and the joint distribution of a stationary Langevin diffusion at times zero and ε is of the order o(ε2) with an explicit dependence given by Fisher information. Owing to this inequality, we can show, using a triangular approximation argument, that the interpolated iterated application of the Schr"odinger bridge approximation converge to the Wasserstein gradient flow, for a class of gradient flows, including the heat flow. The results also provide a probabilistic and rigorous framework for the convergence of the self-attention mechanisms in transformer networks to the solutions of heat flows, first observed in the inspiring work SABP22 in machine learning research.