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

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 [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230--2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977--992].