The Vapnik-Chervonenkis dimension of cubes in $\mathbb{R}^d$

TL;DR

This paper determines the VC dimension of the family of d-dimensional cubes in real d-space, establishing it as loor((3d+1)/2), which advances understanding in discrete geometry and machine learning.

Contribution

It provides a precise calculation of the VC dimension for d-dimensional cubes, a key combinatorial measure in geometry and learning theory.

Findings

VC dimension of d-cubes is loor((3d+1)/2)

The result applies to discrete geometry and machine learning contexts

Enhances understanding of the complexity of cube-shaped classifiers

Abstract

The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$.