# Pair Correlation Factor and the Sample Complexity of Gaussian Mixtures

arXiv: https://arxiv.org/abs/2508.03633 · HTML: https://mlsanity.com/paper/2508.03633/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 11 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 3 of 3)

- [Tight bounds for learning a mixture of two gaussians](https://mlsanity.com/paper/1404.4997/) (2014)
- [Clustering Mixtures with Almost Optimal Separation in Polynomial Time](https://mlsanity.com/paper/2112.00706/) (2021)
- [A Fourier Approach to Mixture Learning](https://mlsanity.com/paper/2210.02415/) (2022)

## Abstract (arXiv)

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 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.