Quantitative Convergence of Proximal Splitting Iterations in Uniformly Convex Metric Spaces

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).