New Trends in the Stability of Sinkhorn Semigroups

TL;DR

This paper reviews recent advances in the stability analysis of Sinkhorn semigroups, highlighting new contraction estimates and unifying techniques based on operator theory and transportation inequalities in entropic optimal transport.

Contribution

It introduces a unified, semigroup-based framework for analyzing Sinkhorn algorithm stability using contraction coefficients and Lyapunov methods, simplifying prior approaches.

Findings

New contraction estimates for Sinkhorn semigroups.

Unified analysis framework using transportation inequalities.

Simplified proofs and broader applicability of stability results.

Abstract

Entropic optimal transport problems play an increasingly important role in machine learning and generative modelling. In contrast with optimal transport maps which often have limited applicability in high dimensions, Schrodinger bridges can be solved using the celebrated Sinkhorn's algorithm, a.k.a. the iterative proportional fitting procedure. The stability properties of Sinkhorn bridges when the number of iterations tends to infinity is a very active research area in applied probability and machine learning. Traditional proofs of convergence are mainly based on nonlinear versions of Perron-Frobenius theory and related Hilbert projective metric techniques, gradient descent, Bregman divergence techniques and Hamilton-Jacobi-Bellman equations, including propagation of convexity profiles based on coupling diffusions by reflection methods. The objective of this review article is to present, in a self-contained manner, recently developed Sinkhorn/Gibbs-type semigroup analysis based upon contraction coefficients and Lyapunov-type operator-theoretic techniques. These powerful, off-the-shelf semigroup methods are based upon transportation cost inequalities (e.g. log-Sobolev, Talagrand quadratic inequality, curvature estimates), $\phi$-divergences, Kantorovich-type criteria and Dobrushin contraction-type coefficients on weighted Banach spaces as well as Wasserstein distances. This novel semigroup analysis allows one to unify and simplify many arguments in the stability of Sinkhorn algorithm. It also yields new contraction estimates w.r.t. generalized $\phi$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances.