A nullspace analysis of the nuclear norm heuristic for rank minimization

The problem of minimizing the rank of a matrix subject to linear equality constraints arises in applications in machine learning, dimensionality reduction, and control theory, and is known to be NP-hard. A popular heuristic minimizes the nuclear norm (sum of the singular values) of the matrix instead of the rank, and was recently shown to give an exact solution in several scenarios. In this paper, we present a new analysis for this heuristic based on a property of the nullspace of the operator defining the constraints, called the spherical section property. We give conditions for the exact recovery of all matrices up to a certain rank, and show that these conditions hold with high probability for operators generated from random Gaussian ensembles. Our analysis provides simpler proofs than existing isometry-based methods, as well as robust recovery results when the matrix is not exactly low-rank.