Continuous and Random Vapnik-Chervonenkis Classes

TL;DR

This paper extends the concept of Vapnik-Chervonenkis classes to continuous functions and demonstrates that dependent theories preserve their dependence under Keisler randomisation, using geometric growth rate characterizations.

Contribution

It introduces continuous VC classes for [0,1]-valued functions and characterizes them through convex compact growth rates, linking model theory and geometric analysis.

Findings

Dependent theories remain dependent after Keisler randomisation

Continuous VC classes characterized by convex compact growth rates

Generalization of VC theory to [0,1]-valued functions

Abstract

We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \emph{continuous} Vapnik-Chervonenkis class), and we characterise families of functions having this property via the growth rate of the mean width of an associated family of convex compacts.