Does large margin imply constant sign-rank?

Establish whether there exists a universal function η such that for every N×N boolean matrix M, the sign-rank satisfies signrank(M) ≤ η(1/margin(M)). Equivalently, determine whether matrices of large (constant) margin necessarily have constant sign-rank, where margin(M) is the supremum over unit-vector assignments of the minimum absolute inner product realizing the sign pattern of M.

Background

The paper studies the relationship between sign-rank (dimension complexity) and margin (or discrepancy) of boolean matrices, a central theme in communication complexity and learning theory. While small sign-rank does not imply large margin, the converse direction—whether large margin forces constant sign-rank—remains a fundamental question with broad implications.

They connect this question to communication complexity: constant bounded-error randomized communication (Rand(M)=O(1)) is equivalent to constant margin, and constant unbounded-error communication (U(M)=O(1)) is equivalent to constant sign-rank. A positive resolution would imply that any problem learnable with constant bounded-error randomized communication is also easy in the unbounded-error model.