# Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure

arXiv: https://arxiv.org/abs/2108.08129 · HTML: https://mlsanity.com/paper/2108.08129/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 18 Aug 2021.

## Builds on (most cited first, 13 of 13)

- [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)
- [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)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Differentiable Particle Filtering via Entropy-Regularized Optimal Transport](https://mlsanity.com/paper/2102.07850/) (2021)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)

## Built on by (most cited first, 11 of 11)

- [Rethinking Initialization of the Sinkhorn Algorithm](https://mlsanity.com/paper/2206.07630/) (2022)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm](https://mlsanity.com/paper/2307.16421/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Soft-constrained Schrodinger Bridge: a Stochastic Control Approach](https://mlsanity.com/paper/2403.01717/) (2024)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (2024)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [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)
- [Low-Budget Active Learning through Entropic Optimal Transport](https://mlsanity.com/paper/2610.01199/) (2026)

## Abstract (arXiv)

We establish the uniform in time stability, w.r.t. the marginals, of the Iterative Proportional Fitting Procedure, also known as Sinkhorn algorithm, used to solve entropy-regularised Optimal Transport problems. Our result is quantitative and stated in terms of the 1-Wasserstein metric. As a corollary we establish a quantitative stability result for Schr"odinger bridges.