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Convergence of fixed-point continuation algorithms for matrix rank minimization. (English) Zbl 1219.90195

Summary: 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, S. Ma, D. Goldfarb and L. Chen [Math. Program. 128, No. 1–2 (A), 321–353 (2011; Zbl 1221.65146)] 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.

MSC:

90C59 Approximation methods and heuristics in mathematical programming
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 1221.65146
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