# Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals

arXiv: https://arxiv.org/abs/2304.04451 · HTML: https://mlsanity.com/paper/2304.04451/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2023.

## Built on by (most cited first, 13 of 13)

- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [Robust Sublinear Convergence Rates for Iterative Bregman Projections](https://mlsanity.com/paper/2602.01372/) (2026)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Thermodynamic structure of the Sinkhorn flow](https://mlsanity.com/paper/2510.12639/) (2025)
- [Schr"odinger bridge problem via empirical risk minimization](https://mlsanity.com/paper/2602.08374/) (2026)

## Abstract (arXiv)

We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schr"odinger potentials, solutions of the quadratic Entropic Optimal Transport problem on ℝ^ d. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.