UNIVERSAL BOUNDS FOR FIXED POINT ITERATIONS VIA OPTIMAL TRANSPORT METRICS

We present a self-contained analysis of a particular family of metrics over the set of non-negative integers. We show that these metrics, which are defined through a nested sequence of optimal transport problems, provide tight estimates for general Krasnosel’skii-Mann fixed point iterations for non-expansive maps. We also describe some of their special properties, including their monotonicity and the so-called convex quadrangle inequality that yields a greedy algorithm for computing them efficiently.