Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

TL;DR

This paper demonstrates that under certain conditions, the minimum rank solution to linear matrix equations can be reliably recovered via nuclear norm minimization, extending compressed sensing principles to matrix problems.

Contribution

It establishes that nuclear norm minimization can find minimum-rank solutions under a restricted isometry property, providing a convex approach to an NP-hard problem.

Findings

Nuclear norm minimization recovers minimum-rank solutions under specific conditions.

The restricted isometry property holds with high probability for certain random ensembles.

The approach generalizes compressed sensing concepts to matrix rank minimization.

Abstract

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization.