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Convergence of fixed-point continuation algorithms for matrix rank minimization

Donald Goldfarb (goldfarb***at***columbia.edu)
Shiqian Ma (sm2756***at***columbia.edu)

Abstract: The matrix rank minimization problem has applications in many fields such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem. By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.

Keywords: Matrix Rank Minimization, Matrix Completion, Greedy Algorithm, Fixed-Point Method, Singular Value Decomposition, Restricted Isometry Property

Category 1: Convex and Nonsmooth Optimization

Citation: Technical Report, Department of IEOR, Columbia University, 2009

Download: [PDF]

Entry Submitted: 06/18/2009
Entry Accepted: 06/18/2009
Entry Last Modified: 12/29/2010

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