On Finite Gaussian Mixtures: Finiteness of the Number of Modes and an Application to NPMLE

Abstract

We prove that every isotropic Gaussian mixture with finitely many components has finitely many modes. In one dimension, classical theory of Chebyshev systems gives the sharp bound of at most $n$ modes for an $n$-component mixture. In several dimensions, however, it has remained open whether every such mixture has finitely many modes. Our main result is the stronger statement that the entire critical set has finite cardinality, which is proven by combining real analytic curve selection theorem and Ax's functional-transcendence theorem. As an application, we show that, for Gaussian location mixtures, every nonparametric maximum likelihood estimator (NPMLE) based on a finite dataset is finitely supported. More specifically, all NPMLEs share the same finite set of allowable atom locations.