## Null Space Conditions and Thresholds for Rank Minimization (2009)

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Citations: | 16 - 1 self |

### BibTeX

@MISC{Recht09nullspace,

author = {Benjamin Recht and Weiyu Xu and Babak Hassibi},

title = {Null Space Conditions and Thresholds for Rank Minimization},

year = {2009}

}

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### Abstract

Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in machine learning, control theory, 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 replaces the rank function with the nuclear norm—equal to the sum of the singular values—of the decision variable and has been shown to provide the optimal low rank solution in a variety of scenarios. In this paper, we assess the practical performance of this heuristic for finding the minimum rank matrix subject to linear constraints. Our starting point is the characterization of a necessary and sufficient condition that determines when this heuristic finds the minimum rank solution. We then obtain conditions, as a function of the matrix dimensions and rank and the number of constraints, such that our conditions for success are satisfied for almost all linear constraint sets as the matrix dimensions tend to infinity. Finally, we provide empirical evidence that these probabilistic bounds provide accurate predictions of the heuristic’s performance in nonasymptotic scenarios.