Gauss-Southwell type descent methods for low-rank matrix optimization

TL;DR

This paper introduces Gauss-Southwell descent methods for low-rank matrix optimization, comparing two gradient-based strategies and demonstrating their convergence and robustness through theoretical analysis and numerical experiments.

Contribution

It proposes and analyzes two Gauss-Southwell type descent methods for low-rank matrix optimization, including new convergence results for alternating least squares.

Findings

Riemannian gradient-based method is more robust to small singular values.

Both methods have similar gradient convergence guarantees.

Numerical experiments validate the robustness of the Riemannian approach.

Abstract

We consider gradient-related methods for low-rank matrix optimization with a smooth cost function. The methods operate on single factors of the low-rank factorization and share aspects of both alternating and Riemannian optimization. Two possible choices for the search directions based on Gauss-Southwell type selection rules are compared: one using the gradient of a factorized non-convex formulation, the other using the Riemannian gradient. While both methods provide gradient convergence guarantees that are similar to the unconstrained case, numerical experiments on a quadratic cost function indicate that the version based on the Riemannian gradient is significantly more robust with respect to small singular values and the condition number of the cost function. As a side result of our approach, we also obtain new convergence results for the alternating least squares method.