Necessary and Sufficient Null Space Condition for Nuclear Norm Minimization in Low-Rank Matrix Recovery

TL;DR

This paper clarifies the null space conditions necessary and sufficient for successful low-rank matrix recovery via nuclear norm minimization, correcting previous misconceptions and providing a precise criterion.

Contribution

It establishes a new weak null space condition that is both necessary and sufficient for nuclear norm minimization success in low-rank matrix recovery.

Findings

Weak null space condition is only sufficient, not necessary, as previously claimed.

New necessary and sufficient null space condition derived.

Provides an inequality for nuclear norms of block matrices.

Abstract

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is the nuclear norm minimization. In [12,14,15], the authors established the necessary and sufficient null space conditions for nuclear norm minimization to recover every possible low-rank matrix with rank at most r (the strong null space condition). In addition, in [12], Oymak et al. established a null space condition for successful recovery of a given low-rank matrix (the weak null space condition) using nuclear norm minimization, and derived the phase transition for the nuclear norm minimization. In this paper, we show that the weak null space condition in [12] is only a sufficient condition for successful matrix recovery using nuclear norm minimization, and is not a necessary condition as claimed in [12]. In this paper, we further give a weak null space condition for low-rank matrix recovery, which is both necessary and sufficient for the success of nuclear norm minimization. At the core of our derivation are an inequality for characterizing the nuclear norms of block matrices, and the conditions for equality to hold in that inequality.