On the Vapnik-Chervonenkis dimension of products of intervals in $\mathbb{R}^d$

TL;DR

This paper investigates the Vapnik-Chervonenkis dimension of product classes of intervals in Euclidean space, revealing that the VC dimension of balls in the sup norm space is approximately 1.5 times the dimension.

Contribution

It provides new bounds on the VC dimension of product classes of intervals and precisely determines the VC dimension of balls in ll_^d with the sup norm.

Findings

VC dimension of product classes of intervals analyzed

VC dimension of ll_^d balls in sup norm space determined as loor((3d+1)/2)

Results contribute to understanding the combinatorial complexity in VC geometry

Abstract

We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$ equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.