# Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity

arXiv: https://arxiv.org/abs/2504.11133 · HTML: https://mlsanity.com/paper/2504.11133/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 56 references that are in the map.
- First posted on arXiv 15 Apr 2025.

## Builds on (most cited first, 30 of 33)

- [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)
- [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)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [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)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains](https://mlsanity.com/paper/2203.04163/) (2022)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [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)
- [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)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Stability of optimal transport maps on Riemannian manifolds](https://mlsanity.com/paper/2504.05412/) (2025)
- [On the contraction properties of Sinkhorn semigroups](https://mlsanity.com/paper/2503.09887/) (2025)
- [Regularity and stability for the Gibbs conditioning principle on path space via McKean-Vlasov control](https://mlsanity.com/paper/2410.23016/) (2024)

## Abstract (arXiv)

In this paper we determine quantitative stability bounds for the Hessian of entropic potentials, ie, the dual solution to the entropic optimal transport problem. To the authors' knowledge this is the first work addressing this second-order quantitative stability estimate in general unbounded settings. Our proof strategy relies on semiconcavity properties of entropic potentials and on the representation of entropic transport plans as laws of forward and backward diffusion processes, known as Schr"odinger bridges. Moreover, our approach allows to deduce a stochastic proof of quantitative stability estimates for entropic transport plans and for gradients of entropic potentials as well. Finally, as a direct consequence of these stability bounds, we deduce exponential convergence rates for gradient and Hessian of Sinkhorn iterates along Sinkhorn's algorithm, a problem that was still open in unbounded settings. Our rates have a polynomial dependence on the regularization parameter.