A Riemannian rank-adaptive method for low-rank matrix completion

TL;DR

This paper introduces a Riemannian rank-adaptive method for low-rank matrix completion that dynamically adjusts the rank during optimization, improving performance over fixed-rank approaches.

Contribution

It proposes a novel Riemannian rank-adaptive algorithm with rank increase and reduction steps for matrix completion, enhancing flexibility and effectiveness.

Findings

Outperforms state-of-the-art fixed-rank algorithms on synthetic and real data

Demonstrates the benefit of adaptive rank adjustment in matrix completion

Can improve existing algorithms by incorporating rank-adaptive components

Abstract

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.