New Trends in the Stability of Sinkhorn Semigroups
Pierre Del Moral, Ajay Jasra
TL;DR
The paper develops a unified, operator-theoretic framework for entropic optimal transport via Sinkhorn/Schrödinger bridges, recasting the iterative scheme as a time-inhomogeneous Markov semigroup. Using contraction coefficients, Kantorovich semi-distances, and Lyapunov methods, it provides quantitative stability and convergence results for a broad class of models, including bounded, Gaussian, and log-concave-at-infinity settings. Central contributions include new contraction estimates w.r.t. generalized $\phi$-divergences and Wasserstein-type distances, illustrated through Riccati-equation connections in Gaussian models and through Lyapunov-based bounds in more general settings. The results enable robust analysis of Sinkhorn iterations, including uniform contraction rates and entropy decay, with potential implications for numerical schemes and diffusion-based generative models. Overall, the work offers a cohesive semigroup perspective that sharpens convergence guarantees and unifies disparate methods in entropic transport stability.
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), $φ$-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 $φ$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances.