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Low-Rank Matrices and Nuclear Norm Minimization

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Speaker: Pablo A. Parrilo, Massachusetts Institute of Technology
Location: 2112 MSB
Start time: Thu, Jun 4 2009, 5:10PM

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. We show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm (the sum of the singular values) over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. Time permitting, we will describe some recent extensions to the case of combined sparsity/low-rank, and underlying uncertainty principles.