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 tightest 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 [33]. 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...

• Type: Journal
• Year: 2011
• Where: FOCM
• Authors: Donald Goldfarb, Shiqian Ma
• Keywords: Algorithm, FOCM 2011, Matrix, Minimization Problems