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

TL;DR

This paper introduces a higher-order inexact semi-smooth Newton method for non-smooth optimization on Riemannian manifolds, demonstrating superlinear convergence in duality-based total variation denoising problems.

Contribution

It extends semi-smooth Newton methods to Riemannian manifolds and proves convergence properties for the inexact version, applicable to manifolds with various curvatures.

Findings

Superlinear convergence observed in numerical experiments.

Method effective on manifolds with positive and negative curvature.

Applicable to duality-based total variation denoising problems.

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.