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Low-Rank Matrix Completion using Nuclear Norm 1 with Facial Reduction ∗

5 Minimization of the nuclear norm is often used as a surrogate, convex relaxation, for finding 6 the minimum rank completion (recovery) of a partial matrix. The minimum nuclear norm 7 problem can be solved as a trace minimization semidefinite programming problem (SDP ). 8 The SDP and its dual are regular in the sense that they both satisfy strict feasibility. Interior 9 point algorithms are the current methods of choice for these problems. This means that it is 10 difficult to solve large scale problems and difficult to get high accuracy solutions. 11 In this paper we take advantage of the structure at optimality for the minimum nuclear norm 12 problem. We show that even though strict feasibility holds, the facial reduction framework can 13 be successfully applied to obtain a proper face that contains the optimal set, and thus can 14 dramatically reduce the size of the final nuclear norm problem while guaranteeing a low-rank 15 solution. We include numerical tests for both exact and noisy cases. In all cases we assume that 16 knowledge of a target rank is available. 17