# Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm

arXiv: https://arxiv.org/abs/2311.16706 · HTML: https://mlsanity.com/paper/2311.16706/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 57 references that are in the map.
- First posted on arXiv 28 Nov 2023.

## Builds on (most cited first, 26 of 26)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [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)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [A gradient descent perspective on Sinkhorn](https://mlsanity.com/paper/2002.03758/) (2020)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [Fast Stochastic Bregman Gradient Methods: Sharp Analysis and Variance Reduction](https://mlsanity.com/paper/2104.09813/) (2021)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Fastest Rates for Stochastic Mirror Descent Methods](https://mlsanity.com/paper/1803.07374/) (2018)
- [Schr"odinger Bridge Samplers](https://mlsanity.com/paper/1912.13170/) (2019)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Sinkhorn Algorithm as a Special Case of Stochastic Mirror Descent](https://mlsanity.com/paper/1909.06918/) (2019)
- [Mirror Sinkhorn: Fast Online Optimization on Transport Polytopes](https://mlsanity.com/paper/2211.10420/) (2022)
- [Variational Principles for Mirror Descent and Mirror Langevin Dynamics](https://mlsanity.com/paper/2303.09532/) (2023)
- [A Dynamical System View of Langevin-Based Non-Convex Sampling](https://mlsanity.com/paper/2210.13867/) (2022)

## Built on by (most cited first, 1 of 1)

- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)

## Abstract (arXiv)

Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics not only generalize but also offer a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of (Deb et al. 2023) or the "mean-field Schr"odinger equation" of (Claisse et al. 2023).