Pair Correlation Factor and the Sample Complexity of Gaussian Mixtures
This work addresses the sample complexity of Gaussian Mixture Models for machine learning practitioners, providing a more accurate geometric understanding that can lead to better algorithms, though it is incremental in refining existing theory.
The paper tackles the problem of learning Gaussian Mixture Models by identifying that the minimum pairwise separation between components is insufficient to explain sample complexity, and introduces the Pair Correlation Factor (PCF) as a better geometric measure, leading to an algorithm with improved sample complexity bounds in the uniform spherical case.
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 $ε^{-2}$ samples are necessary.