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].
【Author】 Jinhua Wang, Xiangmei Wang, Chong Li, Jen-Chih Yao
【Keywords】 Riemannian manifold,sectional curvature,gradient algorithm,local convergence,global convergence,linear convergence,Riemannian center of mass,Primary,53C20; Secondary,53C22
【Journal】 SIAM Journal on Optimization(IF：2.9) Time：2021-02-23