Null space conditions and thresholds for rank minimization |
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Authors: | Benjamin Recht Weiyu Xu Babak Hassibi |
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Institution: | 1.Department of Computer Sciences,University of Wisconsin,Madison,USA;2.Electrical Engineering,California Institute of Technology,Pasadena,USA |
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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 equality constraints. We characterize properties of the
null space of the linear operator defining the constraint set that are necessary and sufficient for the heuristic to succeed.
We then analyze linear constraints sampled uniformly at random, and obtain dimension-free bounds under which our null space
properties hold almost surely 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 non-asymptotic scenarios. |
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