Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes

TL;DR

This paper explores the connections between rigidity theory and linear algebraic matroids, applying these insights to problems in matrix completion, tensor codes, and graph rigidity, with new combinatorial descriptions and applications.

Contribution

It establishes a novel link between rigidity theory and algebraic matroids, providing new combinatorial descriptions and characterizations for tensor codes and matrix patterns.

Findings

Characterization of correctable erasure patterns in maximally recoverable tensor codes.

New combinatorial descriptions of matroids related to symmetric products and rigidity.

First description of correctable patterns in an (m, n, 2, 2) tensor code.

Abstract

We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an (m, n, a=2, b=2) maximally recoverable tensor code.