Some new maximum VC classes

TL;DR

This paper expands the class of maximum VC classes by showing that the sets of positivity for any linear combination of real analytic functions are maximum on points in general position, broadening previous results.

Contribution

It generalizes Floyd's lemma to include linear combinations of real analytic functions, encompassing multivariate polynomials as maximum VC classes.

Findings

Sets of positivity for linear combinations of real analytic functions are maximum.

Includes multivariate polynomials as maximum VC classes.

Floyd's conditions are applicable to a wider class of functions.

Abstract

Set systems of finite VC dimension are frequently used in applications relating to machine learning theory and statistics. Two simple types of VC classes which have been widely studied are the maximum classes (those which are extremal with respect to Sauer's lemma) and so-called Dudley classes, which arise as sets of positivity for linearly parameterized functions. These two types of VC class were related by Floyd, who gave sufficient conditions for when a Dudley class is maximum. It is widely known that Floyd's condition applies to positive Euclidean half-spaces and certain other classes, such as sets of positivity for univariate polynomials. In this paper we show that Floyd's lemma applies to a wider class of linearly parameterized functions than has been formally recognized to date. In particular we show that, modulo some minor technicalities, the sets of positivity for any linear combination of real analytic functions is maximum on points in general position. This includes sets of positivity for multivariate polynomials as a special case.