Block structure in boolean matrices of bounded factorization norm

TL;DR

This paper investigates the structure of boolean matrices with bounded gamma_2 factorization norm, showing they can be covered by disjoint monochromatic submatrices covering a significant fraction of their ones.

Contribution

It establishes a quantitative relationship between gamma_2 norm bounds and the existence of large disjoint monochromatic submatrix covers in boolean matrices.

Findings

Boolean matrices with gamma_2 norm ≤ λ can be covered by disjoint monochromatic submatrices.

Such covers include at least a 1/2^{2^{O(λ)}} fraction of the matrix's ones.

The results extend previous work by providing bounds based on the gamma_2 norm.

Abstract

A boolean matrix is blocky if its $1$-entries form a collection of 1-monochromatic submatrices that are disjoint in both rows and columns. Blocky matrices are precisely the set of boolean matrices with $\gamma_2$ factorization norm at most $1$. Building on recent work by Balla, Hambardzumyan, and Tomon, we show that for any boolean matrix with $\gamma_2$ norm at most $\lambda$, there exists a a collection of row- and column-disjoint 1-monochromatic submatrices that together cover a significant portion (at least a $1/2^{2^{O(\lambda)}}$ fraction) of its $1$-entries.