Adaptive Regularized Newton Method for Riemannian Optimization

TL;DR

This paper introduces an adaptive regularized Newton method for Riemannian optimization that guarantees global and superlinear local convergence, effectively handling manifold constraints with practical computational strategies.

Contribution

It proposes a novel adaptive regularized Newton algorithm for Riemannian optimization that combines second-order approximation with manifold constraints and demonstrates strong convergence properties.

Findings

Algorithm achieves global convergence and superlinear local convergence.

Computational experiments show superior performance over existing methods.

Method effectively handles manifold constraints with inexact subproblem solutions.

Abstract

Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive regularized Newton method which approximates the original objective function by the second-order Taylor expansion in Euclidean space but keeps the Riemannian manifold constraints. The regularization term in the objective function of the subproblem enables us to establish a Cauchy-point like condition as the standard trust-region method for proving global convergence. The subproblem can be solved inexactly either by first-order methods or a modified Riemannian Newton method. In the later case, it can further take advantage of negative curvature directions. Both global convergence and superlinear local convergence are guaranteed under mild conditions. Extensive computational experiments and comparisons with other state-of-the-art methods indicate that the proposed algorithm is very promising.