Continuation methods for Riemannian Optimization

TL;DR

This paper introduces a continuation method for Riemannian optimization that improves convergence by gradually transforming a simple problem into a complex one on manifolds, demonstrated on Karcher mean and matrix completion.

Contribution

It extends numerical continuation techniques to Riemannian optimization, providing a new algorithm for solving difficult manifold-based problems.

Findings

Improved convergence on challenging Riemannian problems

Effective for Karcher mean computation

Enhances low-rank matrix completion methods

Abstract

Numerical continuation in the context of optimization can be used to mitigate convergence issues due to a poor initial guess. In this work, we extend this idea to Riemannian optimization problems, that is, the minimization of a target function on a Riemannian manifold. For this purpose, a suitable homotopy is constructed between the original problem and a problem that admits an easy solution. We develop and analyze a path-following numerical continuation algorithm on manifolds for solving the resulting parameter-dependent equation. To illustrate our developments, we consider two typical classical applications of Riemannian optimization: the computation of the Karcher mean and low-rank matrix completion. We demonstrate that numerical continuation can yield improvements for challenging instances of both problems.