全维高斯混合模型 NPMLE 的自正则化与结构自适应
Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions
Yanjun Han, Yandi Shen, Yihong Wu
AI 速览：作者证明：均值位于超立方体内的 d 维高斯混合模型，其 NPMLE 能以高概率自动拟合出 O((log n/log log n)^d) 个分量，并在有限混合数据下使拟合分量数依概率有界。
We establish structural properties of the nonparametric maximum likelihood estimator (NPMLE) for $d$-dimensional Gaussian mixtures whose means lie in a hypercube. We show that, with high probability, the NPMLE based on $n$ independent observations self-regularizes by fitting a mixture model with $O((\frac{\log n}{\log\log n})^d)$ components, achieving the optimal model complexity without explicit model selection. Furthermore, the NPMLE exhibits a structural adaptation property: For data drawn from a finite mixture, the number of fitted components is bounded in probability as $n\to\infty$. Our proof applies a multivariate Jensen-type bound to control the typical number of critical points of the KKT function associated with the first-order optimality conditions. The key step is to lower bound its gradient norm uniformly on complex spheres. We establish this lower bound by decomposing the KKT function into an empirical fluctuation term and a relative density estimation error term, combining small-ball probability estimates for the former with polynomial approximation and Hellinger error bounds for the latter.