Subspace expanders and matrix rank minimization

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 × 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.

SNIG Property of Matrix Low-Rank Factorization Model

“…It is also arisen from many areas of scientific and engineering applications including matrix completion, principle component analysis (PCA) and others [21,1,8]. LMaFit [28], for instance, using a series of matrix factorization models with different k (the approximation of the optimal rank) to describe the matrix completion problem, turns out to be an efficient and robust alternative to the convex relaxation model [3,7,11,18] based on nuclear norm relaxation [4,5,6,12,19,25]. Matrix factorization is also used to tackle semidefinite programs (SDP) problems.…”