Matrix completion with deterministic pattern - a geometric perspective
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We consider the matrix completion problem with a deterministic pattern of observed entries and aim to find conditions such that there will be (at least locally) unique solution to the non-convex Minimum Rank Matrix Completion (MRMC) formulation. We answer the question from a somewhat different point of view and to give a geometric perspective. We give a sufficient and "almost necessary" condition (which we call the well-posedness condition) for the local uniqueness of MRMC solutions and illustrate with some special cases where such condition can be verified. We also consider the convex relaxation and nuclear norm minimization formulations. Then we argue that the low-rank approximation approaches are more stable than MRMC and further propose a sequential statistical testing procedure to determine the rank of the matrix from observed entries. Finally, numerical examples verified the validity of our theory.
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