New Trends in the Stability of Sinkhorn Semigroups

2026-01-19

AI Summary

This study investigates the convergence stability of the Sinkhorn algorithm as the number of iterations tends to infinity, with a focus on challenges arising in high-dimensional machine learning and generative modeling. By developing a unified theoretical framework grounded in semigroup analysis, the work introduces novel tools—including transport cost inequalities (such as log-Sobolev and Talagrand inequalities) and φ-divergences—and integrates Lyapunov operator theory, Kantorovich criteria, and Dobrushin coefficients on weighted Banach spaces to systematically characterize the contraction behavior of Sinkhorn bridges. This approach not only streamlines and unifies existing stability proofs but also yields new contraction estimates under generalized φ-entropies, weighted total variation norms, and Wasserstein distances, thereby substantially deepening the understanding of convergence mechanisms in high-dimensional optimal transport problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Sinkhorn algorithm
• entropic optimal transport
• stability
• Schrodinger bridges
• convergence

Innovation

• Sinkhorn semigroups
• contraction coefficients
• Lyapunov-type operator theory
• transportation cost inequalities
• φ-divergences