A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm

Understand

We study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem.

• In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals.
• When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded.
• By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results.