Finding a low-rank basis in a matrix subspace

TL;DR

This paper introduces a greedy algorithm to find low-rank bases in matrix subspaces, extending previous methods for rank-1 cases to higher ranks with applications in data compression, eigenvector computation, and image separation.

Contribution

It presents a novel greedy algorithm that estimates the minimum rank and computes low-rank matrices in higher rank subspaces, with theoretical convergence guarantees.

Findings

Algorithm effectively estimates minimum rank using nuclear norm relaxation.

Method outperforms alternative approaches in experiments.

Applications demonstrate improved data compression and eigenvector accuracy.

Abstract

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems.