Subspace Expanders and Matrix Rank Minimization

TL;DR

This paper introduces subspace expanders as a new measurement operator for matrix rank minimization, enabling faster recovery of low-rank matrices with fewer measurements compared to traditional methods.

Contribution

The work presents a novel class of measurement operators called subspace expanders and a faster recovery algorithm for matrix rank minimization, extending beyond Gaussian measurements.

Findings

Unique recovery of rank r PSD matrices from O(nr) measurements.

Recovery algorithm reduces to matrix inversion after initial steps.

Faster than nuclear norm minimization methods.

Abstract

Matrix rank minimization (RM) problems recently gained extensive attention due to numerous applications in machine learning, system identification and graphical models. In RM problem, one aims to find the matrix with the lowest rank that satisfies a set of linear constraints. The existing algorithms include nuclear norm minimization (NNM) and singular value thresholding. Thus far, most of the attention has been on i.i.d. Gaussian measurement operators. In this work, we introduce a new class of measurement operators, and a novel recovery algorithm, which is notably faster than NNM. The proposed operators are based on what we refer to as subspace expanders, which are inspired by the well known expander graphs based measurement matrices in compressed sensing. We show that given an $n\times n$ PSD matrix of rank $r$, it can be uniquely recovered from a minimal sampling of $O(nr)$ measurements using the proposed structures, and the recovery algorithm can be cast as matrix inversion after a few initial processing steps.