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Blocky decomposition of Boolean matrices with small γ₂-norm

Establish that for every Boolean matrix F with γ₂-factorization norm at most c, there exists an integer function ℓ(c) and blocky matrices B_1, …, B_L with L ≤ ℓ(c) such that F = ∑_{i=1}^L ± B_i.

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Background

The γ₂-factorization norm is equivalent (up to constants) to the μ-norm lower bound on deterministic communication complexity and enjoys multiplicativity under Schur product. Boolean matrices with γ₂-norm exactly 1 are precisely the blocky matrices. This motivates a structural conjecture that bounded γ₂-norm forces a bounded complexity representation as a signed sum of blocky matrices.

The conjecture is inspired by Cohen’s idempotent theorem and is known to hold for group-lifts (e.g., XOR-lifts) via quantitative versions of Cohen’s theorem; a resolution for general Boolean matrices would unify these structural results.

References

In , we conjectured that Boolean matrices of small $\gamma_2$-norm are precisely those of the form \cref{eq:blockyfuncrank}. Suppose that $F$ is a Boolean matrix with ${F}{\gamma_2} \le c$. Then we may write F = \sum{i=1}L \pm B_i, where $B_i$ are blocky matrices and $L \le \ell(c)$ for some integer $\ell(c)$ depending only on $c$.

— Structure in Communication Complexity and Constant-Cost Complexity Classes (2401.14623 - Hatami et al., 26 Jan 2024) in Conjecture 3.1 (after Proposition 3.3), Section 3.1