Quantitative Convergence of Proximal Splitting Iterations in Uniformly Convex Metric Spaces

TL;DR

This paper establishes conditions for the quantitative convergence of proximal splitting algorithms in various convex metric spaces, including Hadamard and CAT($7$) spaces, with applications to computing Fre9chet means.

Contribution

It extends convergence analysis of proximal splitting methods to general p-uniformly convex metric spaces without requiring common minima or vanishing step sizes.

Findings

Convergence conditions are derived for algorithms in uniformly convex metric spaces.

Results are applied to compute Fre9chet means in SPD matrices and spheres.

The theory covers spaces with curvature bounds, broadening applicability.

Abstract

We provide sufficient conditions for quantitative convergence of the iterates of proximal splitting algorithms for minimizing a sum of functions on a metric space. The theory does not assume that the functions have common minima, nor does it require vanishing proximal parameters or step sizes. Our results are stated for general $p$-uniformly convex spaces with curvature bounded above, and a corollary specializes the main theorem to Hadamard spaces, where many assumptions for the more general setting can be dropped. The theory is demonstrated with computation of Fr\'echet means in the space of SPD matrices with the affine invariant metric (a Hadamard space) and the sphere with the usual geodesic metric (a CAT($\kappa$) metric space).