Lower Bounds for Approximate Sign Rank

2026-05-01

AI Summary

This work investigates lower bounds on approximate sign rank, focusing on canonical structures such as large-margin halfspaces and Hadamard matrices. By synthesizing the hyperplane avoidance theorem from geometry, Forster–Barthe’s isotropic position theorem, and the Bourgain–Tzafriri restricted invertibility principle, the paper establishes—for the first time—lower bounds on approximate sign rank that depend explicitly on the ambient dimension $d$, rather than solely on the approximation error $\varepsilon$. It further demonstrates that function classes with VC dimension 2 can still exhibit large approximate sign rank. Key contributions include proving an $\Omega(\sqrt{d / \log d})$ lower bound for the $\varepsilon$-approximate sign rank of $d$-dimensional large-margin halfspaces, and showing that Hadamard matrices cannot refute polynomial lower bounds, with their approximate sign rank sandwiched between $\Omega_\varepsilon(m)$ and $m^{O(\sqrt{m} \log(1/\varepsilon))}$.

Abstract

We prove new upper and lower bounds on $ε$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \times n$ sign matrix with approximate sign-rank $d$ contains a monochromatic rectangle of size $d^{-O(d)}m \times d^{-O(d^2)}n$, paralleling classical results for exact sign-rank. As an application, we establish a lower bound of $Ω(\sqrt{d/\log d})$ on the $ε$-approximate sign-rank of large-margin $d$-dimensional half-spaces. Prior to our work, the only general lower bound technique known for approximate sign-rank yielded bounds of strength $ε^{-1} - 1$, which are constant for fixed $ε$. A key ingredient is a new geometric theorem on hyperplane avoidance: for any set of $n$ points in general position in $\mathbb{R}^d$, there exist $d$ subsets, each of size $d^{-O(d)} n$, such that no hyperplane simultaneously splits all of them. The proof combines the Forster-Barthe isotropic position theorem with the Bourgain-Tzafriri restricted invertibility principle. We also study the relationship between approximate sign-rank and VC dimension. We prove a lower bound on approximate sign-rank in terms of VC dimension, and exhibit concept classes of VC dimension $2$ with large approximate sign-rank. Finally, we study the approximate sign-rank of the $2^m \times 2^m$ Hadamard matrix $H_m$. The sign-rank of $H_m$ is known to be $Ω(\sqrt{2^m})$ by Forster's classic theorem. Contrasting this, we adapt an argument of Alman and Williams to show that the approximate sign-rank of $H_m$ is at most $m^{O(\sqrt{m} \log(1/ε))}$, and hence the Hadamard matrix does not witness polynomial-strength lower bounds for approximate sign-rank. Using our VC dimension bound, we prove that the approximate sign-rank of $H_m$ is at least $Ω_ε(m)$.

Problem

• approximate sign-rank
• lower bounds
• VC dimension
• Hadamard matrix
• large-margin half-spaces

Innovation

• approximate sign-rank
• hyperplane avoidance
• VC dimension
• restricted invertibility
• isotropic position