Rank bounds for design matrices with block entries and geometric applications

TL;DR

This paper establishes lower bounds on the rank of block-entry design matrices, extending previous scalar bounds, and applies these results to problems in combinatorial geometry such as rigidity, subspace arrangements, and incidence bounds.

Contribution

It introduces a rank bound for block-entry design matrices and extends matrix scaling techniques to this setting, enabling new geometric and combinatorial applications.

Findings

Extended rank bounds for block matrices.

Improved Sylvester-Gallai type results for subspace arrangements.

New incidence bounds for high-dimensional geometric configurations.

Abstract

Design matrices are sparse matrices in which the supports of different columns intersect in a few positions. Such matrices come up naturally when studying problems involving point sets with many collinear triples. In this work we consider design matrices with block (or matrix) entries. Our main result is a lower bound on the rank of such matrices, extending the bounds proved in {BDWY12,DSW12} for the scalar case. As a result we obtain several applications in combinatorial geometry. The first application involves extending the notion of structural rigidity (or graph rigidity) to the setting where we wish to bound the number of `degrees of freedom' in perturbing a set of points under collinearity constraints (keeping some family of triples collinear). Other applications are an asymptotically tight Sylvester-Gallai type result for arrangements of subspaces (improving {DH16}) and a new incidence bound for high dimensional line/curve arrangements. The main technical tool in the proof of the rank bound is an extension of the technique of matrix scaling to the setting of block matrices. We generalize the definition of doubly stochastic matrices to matrices with block entries and derive sufficient conditions for a doubly stochastic scaling to exist.