# Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm

arXiv: https://arxiv.org/abs/2311.04041 · HTML: https://mlsanity.com/paper/2311.04041/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 75 references that are in the map.
- First posted on arXiv 7 Nov 2023.

## Builds on (most cited first, 28 of 28)

- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [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)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [OTA: Optimal Transport Assignment for Object Detection](https://mlsanity.com/paper/2103.14259/) (2021)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2301.01766/) (2023)
- [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)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [On the Convergence Rate of Sinkhorn's Algorithm](https://mlsanity.com/paper/2212.06000/) (2022)
- [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021)
- [Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2201.10059/) (2022)
- [Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals](https://mlsanity.com/paper/2304.04451/) (2023)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Quantitative Uniform Stability of the Iterative Proportional Fitting Procedure](https://mlsanity.com/paper/2108.08129/) (2021)
- [Stochastic Optimization for Regularized Wasserstein Estimators](https://mlsanity.com/paper/2002.08695/) (2020)
- [Convergence rate of general entropic optimal transport costs](https://mlsanity.com/paper/2206.03347/) (2022)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Estimating the Rate-Distortion Function by Wasserstein Gradient Descent](https://mlsanity.com/paper/2310.18908/) (2023)
- [Convergence rate of entropy-regularized multi-marginal optimal transport costs](https://mlsanity.com/paper/2307.03023/) (2023)

## Built on by (most cited first, 8 of 8)

- [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)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Forward Reverse Kernel Regression for the Schr"odinger bridge problem](https://mlsanity.com/paper/2507.00640/) (2025)
- [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)
- [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)

## Abstract (arXiv)

Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.