A deterministic theory of low rank matrix completion

TL;DR

This paper presents a deterministic, graph-theoretic framework for understanding when large low-rank matrices can be completed from partial data, providing necessary and sufficient conditions without relying on randomness.

Contribution

It introduces a deterministic theory using graph limit concepts to characterize asymptotic solvability in low-rank matrix completion problems.

Findings

Provides necessary and sufficient conditions for asymptotic solvability

Shows a modified nuclear norm minimization algorithm achieves solutions under these conditions

Develops a fully deterministic framework without randomness assumptions

Abstract

The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. The main result of this paper gives a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. It is then shown that a small modification of the Cand\`es-Recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable. The theory is fully deterministic, with no assumption of randomness. A number of open questions are listed.