Necessary and Sufficient Conditions for Success of the Nuclear Norm Heuristic for Rank Minimization

TL;DR

This paper establishes exact conditions under which the nuclear norm heuristic successfully solves rank minimization problems, supported by probabilistic analysis and empirical validation.

Contribution

It provides a necessary and sufficient condition for the success of the nuclear norm heuristic in rank minimization, along with probabilistic bounds and empirical evidence.

Findings

Conditions precisely characterize when the heuristic succeeds

Probabilistic bounds predict heuristic performance in practice

Empirical results align with theoretical predictions

Abstract

Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in control theory, machine learning, and discrete geometry. This class of optimization problems, known as rank minimization, is NP-HARD, and for most practical problems there are no efficient algorithms that yield exact solutions. A popular heuristic algorithm replaces the rank function with the nuclear norm--equal to the sum of the singular values--of the decision variable. In this paper, we provide a necessary and sufficient condition that quantifies when this heuristic successfully finds the minimum rank solution of a linear constraint set. We additionally provide a probability distribution over instances of the affine rank minimization problem such that instances sampled from this distribution satisfy our conditions for success with overwhelming probability provided the number of constraints is appropriately large. Finally, we give empirical evidence that these probabilistic bounds provide accurate predictions of the heuristic's performance in non-asymptotic scenarios.