The local geometry of finite mixtures

TL;DR

This paper investigates the local geometric structure of finite mixture models, establishing bounds on their complexity and metric entropy, which are crucial for understanding their statistical properties.

Contribution

It provides a novel analysis of the local geometry of convex combinations of smooth densities, revealing bounds on their local doubling dimension and metric entropy.

Findings

Local doubling dimension proportional to the number of mixture components

Bounded bracketing metric entropy for mixture classes

Derived local entropy bounds through a slicing procedure

Abstract

We establish that for q>=1, the class of convex combinations of q translates of a smooth probability density has local doubling dimension proportional to q. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local entropy bound is deduced by a general slicing procedure.