# A gradient descent perspective on Sinkhorn

arXiv: https://arxiv.org/abs/2002.03758 · HTML: https://mlsanity.com/paper/2002.03758/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

## Built on by (most cited first, 24 of 24)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Flow Density Control: Generative Optimization Beyond Entropy-Regularized Fine-Tuning](https://mlsanity.com/paper/2511.22640/) (2025)
- [Provable Maximum Entropy Manifold Exploration via Diffusion Models](https://mlsanity.com/paper/2506.15385/) (2025)
- [Accelerating Sinkhorn Algorithm with Sparse Newton Iterations](https://mlsanity.com/paper/2401.12253/) (2024)
- [Momentum Multi-Marginal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.10168/) (2025)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Mirror Sinkhorn: Fast Online Optimization on Transport Polytopes](https://mlsanity.com/paper/2211.10420/) (2022)
- [Quantum Doubly Stochastic Transformers](https://mlsanity.com/paper/2504.16275/) (2025)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Inferring Dynamic Networks from Marginals with Iterative Proportional Fitting](https://mlsanity.com/paper/2402.18697/) (2024)
- [Variational Online Mirror Descent for Robust Learning in Schr"odinger Bridge](https://mlsanity.com/paper/2504.02618/) (2025)
- [A Unified Density Operator View of Flow Control and Merging](https://mlsanity.com/paper/2602.08012/) (2026)
- [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)
- [New Trends in the Stability of Sinkhorn Semigroups](https://mlsanity.com/paper/2601.12633/) (2026)
- [Robust Sublinear Convergence Rates for Iterative Bregman Projections](https://mlsanity.com/paper/2602.01372/) (2026)
- [Generalized Schr"odinger Bridge on Graphs](https://mlsanity.com/paper/2602.04675/) (2026)
- [Metric Distribution to Vector: Constructing Data Representation via Broad-Scale Discrepancies](https://mlsanity.com/paper/2210.00415/) (2022)
- [Linear Bandits beyond Inner Product Spaces, the case of Bandit Optimal Transport](https://mlsanity.com/paper/2502.07397/) (2025)
- [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 present a new perspective on the popular Sinkhorn algorithm, showing that it can be seen as a Bregman gradient descent (mirror descent) of a relative entropy (Kullback-Leibler divergence). This viewpoint implies a new sublinear convergence rate with a robust constant.