Pair Correlation Factor and the Sample Complexity of Gaussian Mixtures

TL;DR

This paper introduces the Pair Correlation Factor (PCF), a new geometric measure that better predicts the sample complexity of learning Gaussian Mixture Models than previous gap-based metrics.

Contribution

The paper proposes the PCF as a novel geometric property influencing GMM sample complexity and provides an improved algorithm with tighter bounds in the spherical case.

Findings

PCF more accurately predicts sample complexity than minimum gap

New algorithm with improved sample complexity bounds in spherical GMMs

More than  samples needed depending on PCF

Abstract

We study the problem of learning Gaussian Mixture Models (GMMs) and ask: which structural properties govern their sample complexity? Prior work has largely tied this complexity to the minimum pairwise separation between components, but we demonstrate this view is incomplete. We introduce the \emph{Pair Correlation Factor} (PCF), a geometric quantity capturing the clustering of component means. Unlike the minimum gap, the PCF more accurately dictates the difficulty of parameter recovery. In the uniform spherical case, we give an algorithm with improved sample complexity bounds, showing when more than the usual $\epsilon^{-2}$ samples are necessary.