A two-phase rank-based algorithm for low-rank matrix completion

TL;DR

This paper introduces a two-phase algorithm for low-rank matrix completion that combines a rank-based heuristic with an accelerated Soft-Impute method, achieving faster and accurate recovery of low-rank matrices.

Contribution

It proposes a novel two-phase approach that uses a rank-based heuristic as a warm-start for an accelerated matrix completion algorithm, improving efficiency.

Findings

Faster recovery of low-rank matrices compared to existing methods.

Effective in both synthetic and real data scenarios.

Provides high-precision matrix completion results.

Abstract

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance. In this paper, first we revisit a recently proposed rank-based heuristic for "known-rank" matrix completion and establish a condition under which the generated sequence is quasi-Fej\'er convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of such heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase, providing a suitable estimate for the regularization parameter and a good starting-point, to an accelerated Soft-Impute algorithm. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.