The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization

TL;DR

This paper introduces an intrinsic Riemannian proximal gradient method tailored for nonconvex optimization on manifolds, avoiding embedding requirements and demonstrating promising convergence and numerical performance.

Contribution

It develops a novel intrinsic proximal gradient algorithm on Riemannian manifolds that handles nonconvex, nonembedded problems without relying on embeddings.

Findings

Convergence properties established for the proposed method.

Numerical experiments show effectiveness on nonconvex, nonembedded problems.

Method outperforms existing approaches in specific nonconvex scenarios.

Abstract

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses proximal maps defined on the manifold and therefore does not require or work in the embedding. We investigate its convergence properties and illustrate its numerical performance, particularly for nonconvex or nonembedded problems that are hence out of reach for other methods.