Multiple typical ranks in matrix completion

TL;DR

Problem: characterize real partial matrix patterns whose completions admit multiple typical ranks and understand how typical ranks grow in circulant graphs. Approach: combine algebraic geometry, rigidity theory, and graph-theoretic constructions to translate entry patterns into bipartite graphs and apply Schur-complement/kin to deduce completability properties and rank sets. Key contributions: complete characterization of when $n-1$ is a typical rank, full determination of typical ranks for $G(n,1)$ with $n<9$, and asymptotic bounds showing thresholds $c_r=(4^r-1)/3$ with extensions to $G(n,k)$; examples of families with paired typical ranks like $\\{n,n+1\\}$. Significance: advances understanding of real versus complex matrix completion, informs design of patterns with controlled rank behavior, and links low-rank recovery to algebraic geometry and graph theory.

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.

Figures (6)

• Figure 1: Black and white dots represent specified and unspecified entries respectively in a $6\times 6$ matrix. Each case starts with a generic rank 2 matrix and a partially filled row or column is added one at a time with exactly 2 specified entries. Both sets of 16 unspecified entries are generically uniquely completable to corank $4 = \sqrt{16}$.
• Figure 2: Pictured is $G(c_{r+1},1)^\mathsf{c}$ for the case $r = 2$, $c_r = 5$ and $c_{r+1} = 21$. The submatrix of the last $2c_r+1$ rows can be uniquely completed to rank $2c_r$. Removing the last row and applying Schur complement, the problem can be reduced to completing the top-left $2c_r\times (2c_r+1)$ submatrix. The last $c_r+1$ columns can be uniquely completed to rank $c_r$. Removing the last column and applying Schur complement, the problem can be reduced to completing the top-left $c_r\times c_r$ submatrix. This submatrix, $G(c_r,1)^\mathsf{c}$, is completable to corank $r$ so the entire matrix is completable to corank $r+1$.
• Figure 3: Bipartite graph $G(7,3)$ and its matrix entries.
• Figure 4: Bipartite graph $G'(7,3)$ and its matrix entries.
• Figure 5: The proof of \ref{['kcirculant']} for $G'(n,k)$ follows that of \ref{['1circulant']} for $G(n,1)$, except that $k$ rows, and then $k$ columns are eliminated in the process instead of 1. The polygonal areas represent the unspecified parts of the matrices.

Theorems & Definitions (45)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Example 2.6
• Definition 2.7
• Example 2.8
• Lemma 3.1