A deterministic theory of low rank matrix completion

The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. Yet, a characterization of missing patterns that allow completion has remained an open question. The main result of this paper gives such a characterization in the language of graph limit theory. It is then shown that a modification of the Candes-Recht nuclear norm minimization algorithm succeeds in completing the matrix whenever completion is possible. The theory is fully deterministic, with no assumption of randomness.