Stephan Eckstein (Tübingen University)
https://sites.google.com/view/stephan-eckstein/
Title: Exponential convergence of Sinkhorn's algorithm and Hilbert's projective metric for unbounded functions
Date: April 5, 2024, at 14:30, 2AB40
Abstract: Entropic regularization of optimal transport has found numerous applications in various fields recently. A major reason of this surge in applications is the popularization of Sinkhorn's algorithm to efficiently solve the entropic optimal transport problem numerically. The topic of this talk is the (rate of) convergence of Sinkhorn's algorithm. In bounded settings, it is known that Sinkhorn's algorithm converges with exponential rate, which is a consequence of applying a commonly used version of Hilbert's metric corresponding to the cone of all non-negative functions. This talk shows how to define versions of Hilbert's metric so that we can show exponential convergence of Sinkhorn's algorithm even in unbounded settings. This is done through the use of cones which are a relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. Along the way, we establish that kernel integral operators are contractions with respect to suitable versions of Hilbert’s metric, even if the kernel functions are not bounded away from zero.