arXiv：cs.LG（机器学习，全量分类）· Hansheng Jiang·· 9 小时前AI 评分37
高斯混合模型随机重加权 NPMLE 的多对数稀疏性
Polylogarithmic Sparsity of Randomly Reweighted NPMLEs for Gaussian Mixtures
AI 导读
研究表明，对高斯位置混合模型 NPMLE 的似然施加极小随机扰动，可得到精确的多对数稀疏性：重加权 NPMLE 以高概率唯一，在 d 维下仅含 O{(log n/log log n)^d + log n} 个原子，且几乎最大化普通似然，混合密度估计达到对数因子内参数化 Hellinger 速率。该稀疏性属于估计量本身而非近似，且无需支撑惩罚。
正文
Abstract:The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian location mixture maximizes the likelihood over the infinite-dimensional space of mixing distributions. The maximizing mixing distribution can be nonunique, and the classical bound on its number of atoms grows linearly with the sample size $n$. We show that a vanishingly small random perturbation of the likelihood yields exact polylogarithmic sparsity. The resulting randomly reweighted NPMLE maximizes a weighted likelihood whose independent weights, taken to be Gamma in our analysis, concentrate around one as $n$ grows. With high probability, it is unique, has $O\{(\log n/\log\log n)^d+\log n\}$ atoms in dimension $d$, nearly maximizes the ordinary likelihood, and estimates the mixture density at a Hellinger rate that is parametric up to logarithmic factors. This sparsity holds for the estimator itself, not for an approximation of it, and requires no support penalty. The proof rests on an effective-dimension principle for positive kernel mixtures: low-dimensional variation of the fitted values controls the support of every extreme point of the set of maximizers. Numerical illustrations verify that the reweighted NPMLE has Hellinger risk and support size comparable to those of the ordinary NPMLE.
| Subjects: | Methodology (stat.ME); Machine Learning (cs.LG); Statistics Theory (math.ST) |
| Cite as: | arXiv:2610.01088 [stat.ME] |
| (or arXiv:2610.01088v1 [stat.ME] for this version) | |
| https://doi.org/10.48550/arXiv.2610.01088 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Hansheng Jiang [view email]
[v1]
Thu, 1 Oct 2026 05:32:53 UTC (4,634 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org