Finding a low-rank basis in a matrix subspace

• Published in: Mathematical programming Vol. 162; no. 1-2; pp. 325 - 361
• Main Authors: Nakatsukasa, Yuji, Soma, Tasuku, Uschmajew, André
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Data compression; Eigenvalues; Eigenvectors; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Matrices (mathematics); Matrix; Matrix methods; Numerical Analysis; Optimization; Sparsity; Studies; Subspaces; Tensors; Theoretical; Alternating projections; Matrix compression; 90C26 Nonconvex programming, global optimization; Low-rank matrix subspace; Singular value thresholding; relaxation
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-016-1042-2

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.