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Low-rank matrix approximation (LRMA) has been arisen in many applications,such as dynamic MRI, recommendation system and so on. The alternating directionmethod of multipliers (ADMM) has been designed for the nuclear norm regularizedleast squares problem and shows a good performance. However, due to the lack ofguarantees for the convergence, there are few ADMM algorithms designed directlyfor the rank-constrained matrix sensing problem (RCMS). Therefore, in thispaper, we propose an ADMM-based algorithm for the RCMS. Based on theKurdyka-Lojasiewicz (KL) property, we prove that the proposed algorithmglobally converges. And we discuss a specific case: the rank-constrained matrixcompletion problem (RCMC). Numerical experiments show that specialized for thematrix completion, the proposed algorithm performs better when the samplingrate is really low in noisy case, which is the key for the matrix completion.

Nonconvex ADMM for Rank-Constrained Matrix Sensing Problem