A new perspective on low-rank optimization

• 成果类型: Article
• 署名作者: Bertsimas, Dimitris; Cory-Wright, Ryan; Pauphilet, Jean
• 署名单位: Massachusetts Institute of Technology (MIT); Imperial College London; International Business Machines (IBM); IBM USA; University of London; London Business School
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01933-9
• 发表日期: 2023
• 页码: 47-92
• 关键词: generating convex-functions; scalable algorithms; semidefinite programs; LARGEST EIGENVALUES; variable selection; REPRESENTATION; approximations; matrices

摘要

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable relaxations. We invoke the matrix perspective function-the matrix analog of the perspective function-to characterize explicitly the convex hull of epigraphs of simple matrix convex functions under low-rank constraints. Further, we combine the matrix perspective function with orthogonal projection matrices-the matrix analog of binary variables which capture the row-space of a matrix-to develop a matrix perspective reformulation technique that reliably obtains strong relaxations for a variety of low-rank problems, including reduced rank regression, non-negative matrix factorization, and factor analysis. Moreover, we establish that these relaxations can be modeled via semidefinite constraints and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization and leads to new relaxations for a broad class of problems.