The VC-dimension and point configurations in $\mathbb{R}^d$

TL;DR

This work analyzes the VC-dimension of sphere-based distance classifiers in Euclidean spaces, connecting geometric configurations to learning-theoretic complexity. It introduces explicit Hausdorff-dimension thresholds, notably $s_d=\frac{29d+2-\sqrt{81d^2+116d-156}}{20}$, guaranteeing that for compact $E\subset {\mathbb R}^d$ with $\dim E>s_d$ there exists an interval of radii $t$ for which the VC-dimension of spheres centered at $E$ is at least $3$. The authors develop a framework of configuration integrals $\Lambda_{G,t,c}^\varepsilon\mu$ associated with finite graphs to count nondegenerate point configurations, and prove upper and lower bounds via harmonic-analytic tools (Fourier decay of sphere intersections, Frostman measures, energy estimates) and symmetry-based reductions. They also present an explicit $4$-cycle threshold in ${\mathbb R}^3$, extending prior results and linking to the Falconer-distance methodology. The results provide quantitative links between fractal geometry and learning-theoretic complexity of geometric classifiers, with potential implications for PAC learnability in geometric settings and for understanding patterns in high-dimensional fractal sets.

Abstract

Given a set $X$ and a collection ${\mathcal H}$ of functions from $X$ to $\{0,1\}$, the VC-dimension measures the complexity of the hypothesis class $\mathcal{H}$ in the context of PAC learning. In recent years, this has been connected to geometric configuration problems in vector spaces over finite fields. In particular, it is easy to show that the VC-dimension of the set of spheres of a given radius in $\mathbb{F}_q^d$ is equal to $d+1$, since this is how many points generically determine a sphere. It is known that for $E\subseteq \mathbb{F}_q^d$, $|E|\geq q^{d-\frac{1}{d-1}}$, the set of spheres centered at points in $E$, and intersected with the set $E$, has VC-dimension either $d$ or $d+1$. In this paper, we study a similar question over Euclidean space. We find an explicit dimensional threshold $s_d<d$ so that whenever $E\subseteq \mathbb{R}^d$, $d\geq 3$, and the Hausdorff dimension of $E$ is at least $s_d$, it follows that there exists an interval $I$ such that for any $t\in I$, the VC-dimension of the set of spheres of radius $t$ centered at points in $E$, and intersected with $E$, is at least $3$. In the process of proving this theorem, we also provide the first explicit dimensional threshold for a set $E\subseteq \mathbb{R}^3$ to contain a $4$-cycle, i.e. $x_1,x_2,x_3,x_4\in E$ satisfying $$ |x_1-x_2|=|x_2-x_3|=|x_3-x_4|=|x_4-x_1| $$

Figures (5)

• Figure 1: 4-cycle graph $\Gamma$
• Figure 2: $\text{dist}(u_3,A_3(u_1,u_2))>c$
• Figure 3: The graphs ${\mathcal{G}}_3, G, H$.
• Figure 4: The "book" graph $B$.
• Figure 5: The 2-shattering graph

Theorems & Definitions (44)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7: Maga, Theorem 2.3
• Corollary 1.8
• Theorem 1.9
• Remark 1.10