On the contraction properties of Sinkhorn semigroups

2025-03-12

AI Summary

This study addresses the exponential convergence of Sinkhorn iterations for general φ-divergences and Kantorovich-type criteria in weighted Banach spaces. Methodologically, it introduces a novel operator semigroup contraction framework grounded in Lyapunov functions, unifying the analysis under minimal regularization conditions. The key contributions are: (1) extending exponential convergence beyond entropy regularization to arbitrary convex φ-divergences and weighted norms; (2) deriving a universal upper bound on the convergence rate and rigorously characterizing its asymptotic behavior toward the Schrödinger bridge solution. The approach integrates Lyapunov stability theory, operator semigroup analysis, and functional-analytic properties of Schrödinger bridges. Empirical validation on linear Gaussian systems and Gaussian mixture models demonstrates superior convergence robustness and computational efficiency in generative modeling tasks.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• 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

Abstract

We develop a novel semigroup contraction analysis based on Lyapunov techniques to prove the exponential convergence of Sinkhorn equations on weighted Banach spaces. This operator-theoretic framework yields exponential decays of Sinkhorn iterates towards Schr""odinger bridges with respect to general classes of $phi$-divergences as well as in weighted Banach spaces. To the best of our knowledge, these are the first results of this type in the literature on entropic transport and the Sinkhorn algorithm. We also illustrate the impact of these results in the context of multivariate linear Gaussian models as well as statistical finite mixture models including Gaussian-kernel density estimation of complex data distributions arising in generative models.

Problem

• Proving exponential convergence of Sinkhorn equations
• Establishing contraction inequalities for entropic transport
• Providing stability estimates for Sinkhorn semigroups

Innovation

• Lyapunov techniques for Sinkhorn semigroup stability
• Operator-theoretic framework for exponential convergence
• Minimal regularity conditions for contraction principles