A new perspective on low-rank optimization

TL;DR

This paper introduces a novel matrix perspective reformulation technique that characterizes convex hulls of low-rank sets, enabling strong, tractable convex relaxations for various low-rank problems in optimization and machine learning.

Contribution

It develops a matrix perspective reformulation method that generalizes existing techniques, providing explicit convex hull characterizations and semidefinite relaxations for low-rank problems.

Findings

Explicit convex hull characterizations for low-rank sets.

Strong relaxations for problems like reduced rank regression and NMF.

Relaxations can be modeled via semidefinite constraints.

Abstract

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 convex relaxations. We invoke the matrix perspective function - the matrix analog of the perspective function - and 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.