The Intrinsic Riemannian Proximal Gradient Method for Convex Optimization

TL;DR

This paper introduces an intrinsic Riemannian proximal gradient method for convex optimization on Hadamard manifolds, achieving efficient convergence without embedding, with demonstrated computational benefits on hyperbolic and positive definite matrix manifolds.

Contribution

It proposes a novel intrinsic CRPG method that operates directly on manifolds, providing convergence guarantees and outperforming existing methods in experiments.

Findings

Sublinear convergence for convex problems.

Linear convergence for strongly convex problems.

Significant computational advantages demonstrated in experiments.

Abstract

We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex Riemannian proximal gradient (CRPG) method that employs the manifold proximal map for the nonsmooth step, without operating in the embedding or tangent space. A sublinear convergence rate for convex problems and a linear convergence rate for strongly convex problems is established, and we derive fundamental proximal gradient inequalities that generalize the Euclidean case. Our numerical experiments on hyperbolic spaces and manifolds of symmetric positive definite matrices demonstrate substantial computational advantages over existing methods.