Multiple typical ranks in matrix completion

TL;DR

This paper investigates the phenomenon of multiple typical ranks in real matrix completion, providing techniques to identify when a set of entries has one or two typical ranks, with a focus on circulant graphs and asymptotic behavior.

Contribution

It introduces methods to determine the number of typical ranks for specific entry sets, characterizes when $n-1$ is typical, and fully determines typical ranks for certain circulant graph cases.

Findings

Sets with only one typical rank identified

Families with two typical ranks demonstrated, especially circulant graphs

Complete characterization of $n-1$ as a typical rank for certain matrices

Abstract

Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an $n\times n$ matrix such that $n-1$ is a typical rank and fully determine the typical ranks for entry set $G(n,1)$ for $n<9$. Moreover, we study the asymptotic behaviour of typical ranks and present results regarding unique matrix completions.