Dot products in ${\Bbb F}_q^3$ and the Vapnik-Chervonenkis dimension

TL;DR

This paper investigates the Vapnik-Chervonenkis dimension of a class of classifiers based on dot products in finite fields, showing that large enough subsets of ${Bbb F}_q^3$ have VC dimension equal to 3, matching the full space.

Contribution

The paper proves that for sufficiently large subsets of ${Bbb F}_q^3$, the VC dimension of classifiers defined by dot products reaches the maximum of 3, extending understanding of VC dimension in finite field geometries.

Findings

VC dimension equals 3 for large subsets of ${Bbb F}_q^3$

Threshold size for subset: at least $Cq^{11/4}$ elements

Results connect finite field geometry with VC theory.

Abstract

Given a set $E \subset {\Bbb F}_q^3$, where ${\Bbb F}_q$ is the field with $q$ elements. Consider a set of "classifiers" ${\mathcal H}^3_t(E)=\{h_y: y \in E\}$, where $h_y(x)=1$ if $x \cdot y=t$, $x \in E$, and $0$ otherwise. We are going to prove that if $|E| \ge Cq^{\frac{11}{4}}$, with a sufficiently large constant $C>0$, then the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is equal to $3$. In particular, this means that for sufficiently large subsets of ${\Bbb F}_q^3$, the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t(E)$ is the same as the Vapnik-Chervonenkis dimension of ${\mathcal H}^3_t({\Bbb F}_q^3)$. In some sense the proof leads us to consider the most complicated possible configuration that can always be embedded in subsets of ${\Bbb F}_q^3$ of size $\ge Cq^{\frac{11}{4}}$.