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

TL;DR

The paper builds a bridge between rigidity theory and linear-algebraic matroids by establishing dualities among $ S_n(d)$, $ W_n(d)$, and $ T_{m,n}(s,r,p)$ with their rigidity counterparts $ S_n(d)$, $ H_n(d)$, and $ B_{m,n}(a,b)$. It proves a central equivalence that connects correctable erasure patterns in maximally recoverable tensor codes to independent sets in the bipartite rigidity matroid $ B_{m,n}(a,b)$, and it provides a complete, characteristic-independent description of independence for the tensor matroid when $s\le 3$, yielding concrete criteria and Laman-type circuit analysis. The work also analyzes the dependence on field characteristic, giving several regimes where independence is characteristic-free and discussing separability in function fields to explain when this fails. It concludes with a conjectural, Bernstein-coloring based description of the bipartite rigidity matroid that would generalize known results and imply characteristic-independence of broader matroids, offering a unifying perspective and potential algorithmic consequences for matrix completion and tensor-code design.

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.

Figures (2)

• Figure 1: A circuit of $\mathcal{B}_{5,5}(2,2)$ which is not a Laman circuit. The squares of the $5 \times 5$ grid represent the ground set of the matroid, with the blue $\star$ squares representing the circuit elements. The red $\diamond$ squares form the corresponding circuit in $\mathrm{T}_{5,5}(3,3,p)$ for any $p$.
• Figure 2: (red $\diamond$) A circuit of $\mathrm{T}_{7,7}(4,4,p)$. (blue $\star$) The corresponding circuit in $\mathcal{B}_{7,7}(3,3)$. See Figure \ref{['fig:nonlaman']} for how to interpret.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Example 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 1.8
• Proposition 2.1
• Definition 2.2