Low-Rank Matrix Completion Theory via Plüucker Coordinates

Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random patterns. Specifically, a fundamental yet largely open question is to describe patterns that allow for unique or finitely many completions. This paper provides three such families of patterns for any rank and any matrix size. A key to achieving this is a novel formulation of low-rank matrix completion in terms of Plüucker coordinates, the latter a traditional tool in computer vision. This connection is of potential significance to a wide family of matrix and subspace learning problems with incomplete data.