On the number of modes of Gaussian kernel density estimators

TL;DR

This paper analyzes how the expected number of modes in Gaussian kernel density estimators scales with bandwidth and sample size, providing asymptotic results using advanced mathematical tools.

Contribution

It establishes the asymptotic scaling law for the expected number of modes of Gaussian kernel density estimators on the real line.

Findings

Expected number of modes scales as Θ(√(β log β))

Results hold for bandwidth and sample size in specified ranges

Uses Kac-Rice formula and Edgeworth expansion

Abstract

We consider the Gaussian kernel density estimator with bandwidth $\beta^{-\frac12}$ of $n$ iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as $\Theta(\sqrt{\beta\log\beta})$ as $\beta,n\to\infty$ provided $n^c\lesssim \beta\lesssim n^{2-c}$ for some constant $c>0$. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.