Metric Entropy estimation using o-minimality Theory

TL;DR

This paper introduces a novel approach using o-minimality from Model Theory to establish VC-subgraph properties and improve metric entropy bounds for certain function classes, applicable even without compact parameter sets.

Contribution

It applies o-minimality theory to derive tighter metric entropy bounds for function classes, expanding applicability beyond traditional compact parameter constraints.

Findings

Demonstrates VC-subgraph property using o-minimality

Provides improved metric entropy bounds for specific function classes

Applicable to non-compact parameter spaces

Abstract

It is shown how tools from the area of Model Theory, specifically from the Theory of o-minimality, can be used to prove that a class of functions is VC-subgraph (in the sense of Dudley, 1987), and therefore satisfies a uniform polynomial metric entropy bound. We give examples where the use of these methods significantly improves the existing metric entropy bounds. The methods proposed here can be applied to finite dimensional parametric families of functions without the need for the parameters to live in a compact set, as is sometimes required in theorems that produce similar entropy bounds (for instance Theorem 19.7 of van der Vaart, 1998).