$\alpha$-Firmly Nonexpansive Operators on Metric Spaces

TL;DR

This paper extends tools for fixed point iterations to p-uniformly convex metric spaces, establishing a calculus for $ ext{α}$-firmly nonexpansive mappings and analyzing convergence of splitting algorithms.

Contribution

It introduces a calculus for $ ext{α}$-firmly nonexpansive mappings in p-uniformly convex spaces, enabling convergence analysis of splitting algorithms in curved metric spaces.

Findings

Preservation of $ ext{α}$-firm nonexpansiveness under compositions and convex combinations.

Convergence of splitting algorithms in spaces with curvature bounded from above.

Quantitative convergence analysis using gauge metric subregularity.

Abstract

We extend to $p$-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise $\alpha$-averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying $\alpha$-firm nonexpansiveness of the individual components at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that only $\alpha$-firm nonexpansiveness with respect to fixed points is required. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.