Exponential convergence of Sinkhorn iterates for statistical mixture models

Establish exponential convergence rates for the Sinkhorn iterative proportional fitting procedure (Sinkhorn iterates) when the prescribed marginals are statistical mixture distributions (e.g., finite mixture models such as Gaussian mixtures), specifying conditions under which such convergence holds on non-compact spaces and for potentially unbounded cost functions.

Background

Most prior exponential convergence results for Sinkhorn iterations concern finite state spaces, compact spaces, or settings with bounded cost functions, often via Hilbert projective metric techniques. Recent advances have addressed certain non-compact settings (e.g., log-concave targets or Gaussian models) under additional structure.

However, many practical applications in statistics and machine learning employ finite mixture distributions, including Gaussian mixtures and kernel density estimates. For these statistical mixture models, a general theory guaranteeing exponential convergence of Sinkhorn iterates has not been established in the literature and was highlighted as an outstanding question motivating the development of Lyapunov-based contraction techniques in this work.