Block complexity and idempotent Schur multipliers

TL;DR

This paper investigates the structure of blocky matrices, a class of boolean matrices related to Schur multipliers, and proves bounds on their decompositions, advancing understanding of matrix norms and idempotent Schur multipliers.

Contribution

It establishes polylogarithmic bounds on the number of blocky matrices needed to express matrices with bounded Schur multiplier norm, generalizing Cohen's idempotent theorem.

Findings

Bound on L is polylogarithmic in matrix dimension

Decomposition of matrices into blocky matrices is possible with bounded L

Advances understanding of Schur multipliers and matrix norm structure

Abstract

We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column. Blocky matrices are precisely the boolean matrices that are contractive when considered as Schur multipliers. It is conjectured that any boolean matrix with Schur multiplier norm at most $\gamma$ is expressible as a signed sum \begin{equation*}A = \sum_{i=1}^L \pm B_i\end{equation*} for some blocky matrices $B_i$, where $L$ depends only on $\gamma$. This conjecture is an analogue of Green and Sanders's quantitative version of Cohen's idempotent theorem. In this paper, we prove bounds on $L$ that are polylogarithmic in the dimension of $A$. Concretely, if $A$ is an $n\times n$ matrix, we show that one may take $L = 2^{O(\gamma^7)} \log(n)^2$.