# A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm

arXiv: https://arxiv.org/abs/2412.09235 · HTML: https://mlsanity.com/paper/2412.09235/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 63 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 27 of 27)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [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)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [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)
- [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)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Displacement smoothness of entropic optimal transport](https://mlsanity.com/paper/2210.00225/) (2022)
- [Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals](https://mlsanity.com/paper/2304.04451/) (2023)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [A proof of the Caffarelli contraction theorem via entropic regularization](https://mlsanity.com/paper/1904.06053/) (2019)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Sharper Exponential Convergence Rates for Sinkhorn's Algorithm in Continuous Settings](https://mlsanity.com/paper/2407.01202/) (2024)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy](https://mlsanity.com/paper/2306.06940/) (2023)
- [Gradient descent with a general cost](https://mlsanity.com/paper/2305.04917/) (2023)
- [Strong c-concavity and stability in optimal transport](https://mlsanity.com/paper/2207.11042/) (2022)

## Built on by (most cited first, 4 of 4)

- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Mirror Langevin diffusions: Convergence rates and Markov chain approximations](https://mlsanity.com/paper/2607.22892/) (2026)
- [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 study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter, based on semiconcavity propagation results. Moreover, the convergence rate has a linear dependence on the regularization: this behavior is sharp and had only been previously obtained for compact distributions arXiv:2407.01202. These optimal rates are also established in situations where one of the two marginals does not have subgaussian tails. Other interesting new applications include subspace elastic costs, weakly log-concave marginals, marginals with light tails (where, under reinforced assumptions, we manage to improve the rates obtained in arXiv:2311.04041), the case of Lipschitz costs with bounded Hessian, and compact Riemannian manifolds.