The Parametric Stability of Well-separated Spherical Gaussian Mixtures

TL;DR

This paper establishes explicit bounds on the stability of spherical Gaussian Mixture Models under small distribution perturbations, providing theoretical guarantees for model fitting with improved bounds.

Contribution

It derives the first explicit, distribution-free bounds linking distribution proximity to parameter proximity in spherical GMMs, tightening previous results.

Findings

Bounds depend only on the original distribution P

Results match known sharp thresholds asymptotically

Constants involved are well-defined and distribution-free

Abstract

We quantify the parameter stability of a spherical Gaussian Mixture Model (sGMM) under small perturbations in distribution space. Namely, we derive the first explicit bound to show that for a mixture of spherical Gaussian $P$ (sGMM) in a pre-defined model class, all other sGMM close to $P$ in this model class in total variation distance has a small parameter distance to $P$. Further, this upper bound only depends on $P$. The motivation for this work lies in providing guarantees for fitting Gaussian mixtures; with this aim in mind, all the constants involved are well defined and distribution free conditions for fitting mixtures of spherical Gaussians. Our results tighten considerably the existing computable bounds, and asymptotically match the known sharp thresholds for this problem.