A deterministic theory of low rank matrix completion
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.
Cite
@article{arxiv.1910.01079,
title = {A deterministic theory of low rank matrix completion},
author = {Sourav Chatterjee},
journal= {arXiv preprint arXiv:1910.01079},
year = {2021}
}
Comments
22 pages. The numberings of sections, theorems, lemmas, definitions and equations have been changed to correspond to the version published in IEEE Trans. Inf. Theory