Infinite families of graphs and stable completion of arbitrary matrices, Part I

TL;DR

This paper introduces deterministic graph constructions that enable the exact and stable completion of low-rank matrices for any values, using specific patterns in the graph's structure, applicable to infinite families.

Contribution

It presents a novel method linking graph patterns to matrix completion, allowing the design of infinite graph families with guaranteed low-rank matrix completion capabilities.

Findings

Graphs with certain patterns enable unique low-rank matrix completion.

The construction supports stable completion for all fixed ranks.

Applicable to infinite graph families using sum-of-squares hierarchy.

Abstract

We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular unions of self-avoiding walks) in the subgraph of the lattice graph generated from the support of the bi-adjacency matrix. The construction makes it possible to design infinite families of graphs on which exact and stable completion is possible for every fixed rank matrix through the sum-of-squares hierarchy.