An Inexact Semi-smooth Newton Method on Riemannian Manifolds with Application to Duality-based Total Variation Denoising

Abstract: We propose a higher-order method for solving non-smooth optimization problems on manifolds. In order to obtain superlinear convergence, we apply a Riemannian Semi-smooth Newton method to a non-smooth non-linear primal-dual optimality system based on a recent extension of Fenchel duality theory to Riemannian manifolds. We also propose an inexact version of the Riemannian Semi-smooth Newton method and prove conditions for local linear and superlinear convergence that hold independent of the sign of the curvature. Numerical experiments on l2-TV-like problems with dual regularization confirm superlinear convergence on manifolds with positive and negative curvature.