Universal bounds for fixed point iterations via optimal transport metrics

TL;DR

This paper introduces a family of metrics based on optimal transport over non-negative integers, providing tight bounds for fixed point iterations of non-expansive maps, with properties enabling efficient computation.

Contribution

It defines a new class of metrics via nested optimal transport problems that precisely estimate fixed point iteration behavior and can be computed efficiently.

Findings

Metrics provide tight bounds for fixed point iterations.

Metrics exhibit monotonicity and convex quadrangle inequality.

Efficient greedy algorithms can compute these metrics.

Abstract

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 very special properties, including their monotonicity and the so-called "convex quadrangle inequality" that yields a greedy algorithm to compute them efficiently.