Littlestone and VC-dimension of families of zero sets

TL;DR

This paper establishes the VC-dimension and Littlestone dimension of zero set families derived from linearly independent functions, providing characterizations of their maximality conditions.

Contribution

It proves the dimensions for zero set families of linearly independent functions and characterizes their maximality conditions for VC-dimension and Littlestone dimension.

Findings

VC-dimension and Littlestone dimension are both d-1 for these families

Characterization of maximal families with VC-dimension d-1

Sufficient conditions for maximal Littlestone dimension d-1

Abstract

We prove that, for any $d$ linearly independent functions from some set into a $d$-dimensional vector space over any field, the family of zero sets of all non-trivial linear combination of these functions has VC-dimension and Littlestone dimension $d-1$. Additionally, we characterize when such families are maximal of VC-dimension $d-1$ and give a sufficient condition for when they are maximal of Littlestone dimension $d-1$.