VC-Dimension and Distance Chains in $\mathbb{F}_q^d$

TL;DR

This paper investigates the VC-dimension of classes of functions related to spheres in finite fields, establishing size thresholds for subsets to achieve maximal VC-dimension across various dimensions.

Contribution

It extends previous work by determining size conditions on subsets of finite fields that guarantee maximal VC-dimension for sphere-based hypothesis classes in all dimensions.

Findings

VC-dimension reaches maximum when |E| ≥ C_d q^{d-1/(d-1)} for d ≥ 3

Improved bounds for d=3: |E| ≥ C_3 q^{7/3}

For d=2, the threshold is |E| ≥ C_2 q^{7/4}

Abstract

Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.