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：我们证明了关于$\epsilon$-近似符号秩的新的上下界，该概念是由Chornomaz、Moran和Waknine（STOC 2025）引入的符号秩的松弛形式。我们证明每个近似符号秩为$d$的$m \times n$符号矩阵包含一个大小为$d^{-O(d)}m \times d^{-O(d^2)}n$的单色矩形，这平行于精确符号秩的经典结果。作为应用，我们为大间隔$d$维半空间建立了$\epsilon$-近似符号秩的下界$\Omega(\sqrt{d/\log d})$。在我们工作之前，已知的近似符号秩的唯一通用下界技术只能产生强度为$\epsilon^{-1} - 1$的界，该界对于固定$\epsilon$是常数。其中一个关键要素是关于超平面回避的新几何定理：对于$\mathbb{R}^d$中处于一般位置的任意$n$个点集，存在$d$个子集，每个子集大小为$d^{-O(d)} n$，使得没有超平面能同时分割所有子集。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。我们根据VC维证明了近似符号秩的下界，并展示了VC维为$2$且具有大近似符号秩的概念类。最后，我们研究了$2^m \times 2^m$ Hadamard矩阵$H_m$的近似符号秩。根据Forster的经典定理，$H_m$的符号秩已知为$\Omega(\sqrt{2^m})$。与此对比，我们改编了Alman和Williams的一个论证，表明$H_m$的近似符号秩至多为$m^{O(\sqrt{m} \log(1/\epsilon))}$，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，我们证明$H_m$的近似符号秩至少为$\Omega_\epsilon(m)$。