Convergence Analysis of Gradient Algorithms on Riemannian Manifolds Without Curvature Constraints and Application to Riemannian Mass

TL;DR

This paper analyzes the convergence of gradient algorithms on general Riemannian manifolds without curvature restrictions, establishing local and global convergence results and applying them to Riemannian mass problems.

Contribution

It provides new convergence analysis for gradient methods on manifolds without curvature constraints and extends results for Riemannian center of mass computations.

Findings

Local and global convergence under convexity assumptions

Linear convergence with constant and Armijo step sizes

Improved convergence results for Riemannian mass problems

Abstract

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp. weak sharp minima), local/global convergence (resp. linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in \cite{Afsari2013}.