On the Geometry of Separation in Finite Gaussian Mixtures

2026-06-14

AI Summary

This study investigates the influence of the minimal component separation on the convergence rate of parameter estimation in finite Gaussian mixture models. By constructing a unified geometric framework, the authors establish an explicit link between the Hellinger distance of mixture densities and the Wasserstein distance between their underlying mixing measures via a novel lower bound. They develop moment-extracting test functions using interpolation polynomials and divided differences to characterize separation complexity under varying component configurations. The work reveals a localization phenomenon induced by the spatial arrangement of components and clarifies the fundamental distinction in separation complexity between settings with known and unknown numbers of components. It establishes a continuum of convergence rates bridging pointwise and uniform estimation, providing the first explicit convergence guarantees that depend directly on both the minimal separation and minimal mixing weights, thereby fully characterizing the fundamental limits of parameter recovery.

Abstract

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

Problem

• Gaussian mixtures
• parameter estimation
• convergence rates
• component separation
• Wasserstein distance

Innovation

• Gaussian mixtures
• Wasserstein distance
• Hellinger lower bounds
• separation complexity
• interpolation polynomials