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 $Ω_ε(m)$.

翻译：我们针对ε-近似符号秩（由Chornomaz、Moran和Waknine在STOC 2025中引入的符号秩的松弛概念）证明了新的上界和下界。我们证明：任意一个具有近似符号秩d的m×n符号矩阵包含一个大小为d^{-O(d)}m × d^{-O(d^2)}n的单色矩形，这一结果与精确符号秩的经典结论相平行。作为应用，我们为大间隔d维半空间建立了ε-近似符号秩的下界Ω(√(d/log d))。在我们的工作之前，已知的近似符号秩唯一通用下界技术仅能给出强度为ε^{-1} - 1的界，对于固定ε而言该值为常数。一个关键要素是关于超平面回避的新的几何定理：对于ℝ^d中处于一般位置的任意n个点，存在d个子集，每个大小为d^{-O(d)}n，使得没有任何超平面能同时分割所有这些子集。该证明结合了Forster-Barthe各向同性位置定理与Bourgain-Tzafriri受限可逆性原理。我们还研究了近似符号秩与VC维之间的关系。我们以VC维为参数证明了近似符号秩的下界，并展示了VC维为2却具有大近似符号秩的概念类。最后，我们研究了2^m × 2^m Hadamard矩阵H_m的近似符号秩。由Forster的经典定理可知，H_m的符号秩为Ω(√(2^m))。与此形成对比，我们改编了Alman和Williams的论证，证明H_m的近似符号秩至多为m^{O(√m log(1/ε))}，因此Hadamard矩阵无法为近似符号秩提供多项式强度的下界。利用我们的VC维界限，我们证明H_m的近似符号秩至少为Ω_ε(m)。