Gaussian mixtures and non-parametric likelihoods through the lens of statistical mechanics

2026-03-24

AI Summary

This work addresses the instability of Gaussian mixture models and the difficulty in controlling Kullback–Leibler (KL) divergence in nonparametric maximum likelihood estimation (NPMLE). Drawing on insights from statistical mechanics, the authors integrate tools from function class complexity, high-dimensional probability inequalities, and approximate optimization theory to establish, for the first time, a stability theory for NPMLE. They uncover deep connections between NPMLE and the chaotic, multi-valley structure of random energy landscapes as well as Langevin dynamics. The central contribution is a high-probability upper bound on the KL divergence between the true density and the NPMLE, given by $\min\left\{\frac{(\log n)^{d+2}}{n}, \frac{\log n}{\sqrt{n}}\right\}$, which is further extended to finite-time approximate solutions, substantially broadening the scope of existing theoretical guarantees.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

In this work, we investigate Gaussian Mixture Models ({\it abbrv} GMM) and the related problem of non parametric maximum likelihood estimation ({\it abbrv} NPMLE) from the perspective of statistical mechanics. In particular, we establish stability guarantees for the NPMLE procedure that extend well beyond the state of the art. Crucially, we obtain guarantees on the Kullback-Leibler divergence between NPMLE estimators and the ground truth, a type of result which has been known to be challenging in the literature on this problem. In particular, we provide high probability upper bounds on the KL divergence between the NPMLE and the true density that are of the order of $\min\big\{\frac{(\log n)^{d+2}}{n} , \frac{\log n}{\sqrt n}\big\}$, which cover a wide range of scenarios for the comparative sizes of $n$ and $d$. We obtain similar guarantees for approximate solutions to the NPMLE problem, addressing realistic situations wherein optimization algorithms need to be stopped in finite time, allowing access only to approximations to the true NPMLE. A technical cornerstone of our approach is an analysis of the function class complexity of logarithms of gaussian mixture densities, which is able to handle their unboundedness, and could be of wider interest. We also establish correspondences between stability phenomena in the NPMLE problem and concepts from chaos and multiple valleys in random energy landscapes of statistical mechanics models. We believe that these correspondences may be useful for a wide variety of random optimization problems in statistics and machine learning, especially the connections to the the technical ingredients of concentration phenomena and Langevin dynamics for these models.

Problem

• non-parametric maximum likelihood estimation
• Gaussian mixture models
• Kullback-Leibler divergence
• statistical stability
• density estimation

Innovation

• non-parametric MLE
• Gaussian mixture models
• statistical mechanics
• KL divergence bounds
• function class complexity