Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions
Abstract
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(((log n)/(loglog 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→∞. 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.
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BibTeX
@misc{han2026selfregularization,
title = {{Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions}},
author = {Yanjun Han and Yandi Shen and Yihong Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40566},
eprint = {2610.40566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40566},
}
RIS
TY - UNPB
TI - Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions
AU - Yanjun Han
AU - Yandi Shen
AU - Yihong Wu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40566
UR - https://arxiv.org/abs/2610.40566
AB - 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(((log n)/(loglog 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→∞. 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.
ER -