Riemannian Levenberg-Marquardt Method with Global and Local Convergence Properties

TL;DR

This paper extends the Levenberg-Marquardt method to Riemannian manifolds, providing theoretical convergence guarantees and an efficient parameter update strategy, supported by numerical experiments.

Contribution

It introduces a Riemannian Levenberg-Marquardt method with proven global and local convergence properties and a novel trust-region-like parameter update approach.

Findings

Proven global convergence to stationary points.

Established local convergence under error-bound conditions.

Numerical experiments demonstrate algorithm efficiency.

Abstract

We extend the Levenberg-Marquardt method on Euclidean spaces to Riemannian manifolds. Although a Riemannian Levenberg-Marquardt (RLM) method was produced by Peeters in 1993, to the best of our knowledge, there has been no analysis of theoretical guarantees for global and local convergence properties. As with the Euclidean LM method, how to update a specific parameter known as the damping parameter has significant effects on its performances. We propose a trust-region-like approach for determining the parameter. We evaluate the worst-case iteration complexity to reach an epsilon-stationary point, and also prove that it has desirable local convergence properties under the local error-bound condition. Finally, we demonstrate the efficiency of our proposed algorithm by numerical experiments.