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Minimization of the nuclear norm is often used as a surrogate, convexrelaxation, for finding the minimum rank completion (recovery) of a partialmatrix. The minimum nuclear norm problem can be solved as a trace minimizationsemidefinite programming problem, (SDP). The SDP and its dual are regular inthe sense that they both satisfy strict feasibility. Interior point algorithmsare the current methods of choice for these problems. This means that it isdifficult to solve large scale problems and difficult to get high accuracysolutions.  In this paper we take advantage of the structure at optimality for theminimum nuclear norm problem. We show that even though strict feasibilityholds, the facial reduction framework can be successfully applied to obtain aproper face that contains the optimal set, and thus can dramatically reduce thesize of the final nuclear norm problem while guaranteeing a low-rank solution.We include numerical tests for both exact and noisy cases. In all cases weassume that knowledge of a target rank is available.

Low-Rank Matrix Completion using Nuclear Norm with Facial Reduction