Matrix Completion With Deterministic Pattern: A Geometric Perspective

Abstract

We consider the matrix completion problem with a deterministic pattern of observed entries. In this setting, we aim to answer the question: Under what conditions will there be (at least locally) unique solution to the matrix completion problem, i.e., the underlying true matrix is identifiable? We answer the question from a certain point of view and outline a geometric perspective. We give an algebraically verifiable sufficient condition, which we call the well-posedness condition, for the local uniqueness of minimum rank matrix completion (MRMC) solutions. It appears that this condition is basic for the analysis of MRMC, and we show that, in a sense, the condition is generic. We also argue that the low-rank approximation approaches are more stable than MRMC and further propose a sequential statistical testing procedure to determine the “true” rank from observed entries. Finally, we provide numerical examples aimed at verifying validity of the presented theory.