# Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm

arXiv: https://arxiv.org/abs/2307.16421 · HTML: https://mlsanity.com/paper/2307.16421/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 31 Jul 2023.

## Builds on (most cited first, 11 of 11)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [A gradient descent perspective on Sinkhorn](https://mlsanity.com/paper/2002.03758/) (2020)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2201.10059/) (2022)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Convergence rate of general entropic optimal transport costs](https://mlsanity.com/paper/2206.03347/) (2022)

## Built on by (most cited first, 10 of 10)

- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach](https://mlsanity.com/paper/2401.08978/) (2024)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [Information Geometry for the Working Information Theorist](https://mlsanity.com/paper/2310.03884/) (2023)
- [Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm](https://mlsanity.com/paper/2311.16706/) (2023)
- [Mirror Langevin diffusions: Convergence rates and Markov chain approximations](https://mlsanity.com/paper/2607.22892/) (2026)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective](https://mlsanity.com/paper/2507.12246/) (2025)

## Abstract (arXiv)

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the 2-Wasserstein space, as the regularization parameter varepsilon goes to zero and the number of iterations is scaled as 1/varepsilon (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. An equivalent description of this flow is provided by the parabolic Monge-Amp`ere PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.