We consider the Gaussian kernel density estimator with bandwidth $\beta^{-\frac12}$ of $n$ iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as $\Theta(\sqrt{\beta\log\beta})$ as $\beta,n\to\infty$ provided $n^c\lesssim \beta\lesssim n^{2-c}$ for some constant $c>0$. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.

翻译：我们考虑对n个独立同分布高斯样本采用带宽为$\\beta^{-\\frac12}$的高斯核密度估计器。利用Kac-Rice公式和Edgeworth展开，我们证明了当$\\beta,n\\to\\infty$时，实轴上期望模态数量以$\\Theta(\\sqrt{\\beta\\log\\beta})$的尺度增长，其中$n^c\\lesssim \\beta\\lesssim n^{2-c}$（$c>0$为常数）。本研究的一个动机在于确定Transformer模型在亚稳态中被吸引到的聚类数量。