The Vapnik-Chervonenkis Dimension of Norms on $\mathbb{R}^d$
Christian J. J. Despres · arXiv (Cornell University) · 2014
The Vapnik-Chervonenkis dimension of a collection of subsets of a set is an important combinatorial parameter in machine learning. In this paper we show that the VC dimension of the family of d-dimensional cubes in $\mathbb{R}^d$ (that is, the closed balls according to the $\ell^\infty$ norm) is $\lfloor (3d+1)/2 \rfloor$. We also prove that the VC dimension of certain families of convex sets in $\mathbb{R}^2$ (including the balls of all norms) is at most 3, and that there is a norm in $\mathbb{R}^3$ the collection of whose balls has infinite VC dimension.