Contraction and entropy production in continuous-time Sinkhorn dynamics

2025-10-14

AI Summary

This work investigates the contraction properties and entropy production mechanisms of continuous-time Sinkhorn dynamics. Method: Leveraging the mirror descent framework, Onsager gradient flow theory, and nonlocal Dirichlet form analysis, we derive an exact analytical expression for the entropy production rate of the Sinkhorn flow and establish its connection to reversible Markov dynamics on the target marginal distributions. We further develop a functional inequality framework analogous to diffusion processes, linking exponential entropy decay to the log-Sobolev inequality under a positive spectral gap (ε > 0). Contribution/Results: Our analysis provides the first rigorous characterization of entropy dissipation in continuous-time Sinkhorn dynamics, proving that exponential entropy decay occurs if and only if the log-Sobolev inequality holds. These theoretical advances yield principled stability guarantees for training generative models and lead to a practical stopping criterion for discrete Sinkhorn algorithms.

Technology Category

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

Application Category

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

Abstract

Recently, the vanishing-step-size limit of the Sinkhorn algorithm at finite regularization parameter $varepsilon$ was shown to be a mirror descent in the space of probability measures. We give $L^2$ contraction criteria in two time-dependent metrics induced by the mirror Hessian, which reduce to the coercivity of certain conditional expectation operators. We then give an exact identity for the entropy production rate of the Sinkhorn flow, which was previously known only to be nonpositive. Examining this rate shows that the standard semigroup analysis of diffusion processes extends systematically to the Sinkhorn flow. We show that the flow induces a reversible Markov dynamics on the target marginal as an Onsager gradient flow. We define the Dirichlet form associated to its (nonlocal) infinitesimal generator, prove a Poincaré inequality for it, and show that the spectral gap is strictly positive along the Sinkhorn flow whenever $varepsilon > 0$. Lastly, we show that the entropy decay is exponential if and only if a logarithmic Sobolev inequality (LSI) holds. We give for illustration two immediate practical use-cases for the Sinkhorn LSI: as a design principle for the latent space in which generative models are trained, and as a stopping heuristic for discrete-time algorithms.

Problem

• Analyzing contraction properties of continuous-time Sinkhorn dynamics
• Characterizing entropy production rate in Sinkhorn flow evolution
• Establishing functional inequalities for Sinkhorn algorithm convergence

Innovation

• Mirror descent in probability measures space
• Entropy production rate identity for Sinkhorn flow
• Reversible Markov dynamics as Onsager gradient flow