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Convergence of fixed-point continuation algorithms for matrix rank minimization
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.Comment: Conditions on RIP constant for an approximate recovery are improve
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