Optimization over bounded-rank matrices through a desingularization enables joint global and local guarantees

TL;DR

This paper introduces a desingularization-based Riemannian approach for optimization over bounded-rank matrices, achieving both global convergence guarantees and fast local convergence, with promising results in matrix completion.

Contribution

It develops a Riemannian geometry framework for desingularization of bounded-rank matrices, providing combined global and local convergence guarantees.

Findings

Comparable performance to existing methods on matrix completion

Better theoretical guarantees for general-purpose optimization

Achieves both global convergence and fast local rates

Abstract

Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a non-smooth and non-convex algebraic variety. Existing techniques include projected gradient descent, fixed-rank optimization (over the maximal-rank stratum), and the LR parameterization. They all lack either global guarantees (the ability to accumulate only at critical points) or fast local convergence (e.g., if the limit has non-maximal rank). We seek optimization algorithms that enjoy both. Khrulkov and Oseledets [2018] parameterize the bounded-rank variety via a desingularization to recast the optimization problem onto a smooth manifold. Building on their ideas, we develop a Riemannian geometry for this desingularization, also with care for numerical considerations. We use it to secure global convergence to critical points with fast local rates, for a large range of algorithms. On matrix completion tasks, we find that this approach is comparable to others, while enjoying better general-purpose theoretical guarantees.