On the globalization of Riemannian Newton method

TL;DR

This paper introduces a new globalization strategy for the Riemannian Newton method that ensures global convergence without assumptions on singularities, demonstrated through various applications and numerical experiments.

Contribution

It generalizes existing damped Newton methods to Riemannian manifolds with a new globalization approach and proves its superlinear convergence without singularity hypotheses.

Findings

The method achieves global convergence with superlinear rate.

Numerical experiments show improved robustness over previous methods.

Applications include singular value problems and optimization on manifolds.

Abstract

In the present paper, in order to fnd a singularity of a vector field defined on Riemannian manifolds, we present a new globalization strategy of Newton method and establish its global convergence with superlinear rate. In particular, this globalization generalizes for a general retraction the existing damped Newton's method. The presented global convergence analysis does not require any hypotesesis on singularity of the vector field. We applied the proposed method to solve the truncated singular value problem on the product of two Stiefel manifolds, the dextrous hand grasping problem on the cone of symmetric positive definite matrices and the Rayleigh quotient on the sphere. Moreover, some academic problems are solved. Numerical experiments are presented showing that the proposed algorithm has better robustness compared with the aforementioned method.