R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

TL;DR

This paper introduces R3MC, a Riemannian conjugate-gradient algorithm specifically designed for low-rank matrix completion, leveraging a novel metric to improve robustness and performance on challenging datasets.

Contribution

The paper presents a new Riemannian optimization framework with a tailored metric for low-rank matrix completion, outperforming existing algorithms.

Findings

R3MC outperforms state-of-the-art algorithms in various scenarios.

The method is particularly effective with scarcely sampled and ill-conditioned data.

Numerical experiments validate the robustness and efficiency of R3MC.

Abstract

We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.