# Thermodynamic structure of the Sinkhorn flow

arXiv: https://arxiv.org/abs/2510.12639 · HTML: https://mlsanity.com/paper/2510.12639/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 45 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Builds on (most cited first, 12 of 12)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Mirrorless Mirror Descent: A Natural Derivation of Mirror Descent](https://mlsanity.com/paper/2004.01025/) (2020)
- [Beyond Convexity -- Contraction and Global Convergence of Gradient Descent](https://mlsanity.com/paper/1806.06655/) (2018)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals](https://mlsanity.com/paper/2304.04451/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)

## Abstract (arXiv)

Entropy-regularized optimal transport, which has strong links to the Schr"odinger bridge problem in statistical mechanics, enjoys a variety of applications from trajectory inference to generative modeling. A major driver of renewed interest in this problem is the recent development of fast matrix-scaling algorithmstextemdash known as iterative proportional fitting or the Sinkhorn algorithmtextemdash for entropic optimal transport, which have favorable complexity over traditional approaches to the unregularized problem. Here, we take a perspective on this algorithm rooted in the thermodynamic origins of Schr"odinger's problem and inspired by the modern geometric theory of diffusion: is the Sinkhorn flow (viewed in continuous-time as a mirror descent by recent results) the gradient flow of entropy in a formal Riemannian geometry? We answer this question affirmatively, finding a nonlocal Wasserstein gradient structure in the dynamics of its free marginal. This offers a physical interpretation of the Sinkhorn flow as the stochastic dynamics of a particle with law evolving by the nonlocal diffusion of a chemical potential. Simultaneously, it brings a standard suite of functional inequalities characterizing Markov diffusion processes to bear upon its geometry and convergence. We prove an entropy-energy (de Bruijn) identity, a Poincar'e inequality, and a Bakry-'Emery-type condition under which a logarithmic Sobolev inequality (LSI) holds and implies exponential convergence of the Sinkhorn flow in entropy. We lastly discuss computational applications such as stopping heuristics and latent-space design criteria leveraging the LSI and, returning to the physical interpretation, the possibility of natural systems whose relaxation to equilibrium inherently solves entropic optimal transport or Schr"odinger bridge problems.