Finding a low-rank basis in a matrix subspace

• 成果类型: Article
• 署名作者: Nakatsukasa, Yuji; Soma, Tasuku; Uschmajew, Andre
• 署名单位: University of Oxford; University of Tokyo; University of Bonn; University of Bonn
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1042-2
• 发表日期: 2017
• 页码: 325-361
• 关键词: least-squares method; alternating projections; canonical decomposition; tensor decompositions; CONVERGENCE; approximation; minimization; algorithm; POWER

摘要

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.