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使用带宽为$\beta^{-\frac12}$的高斯核密度估计器。通过运用Kac-Rice公式和Edgeworth展开，我们证明当$\beta,n\to\infty$且满足$n^c\lesssim \beta\lesssim n^{2-c}$（$c>0$为常数）时，实轴上期望模态数量的渐近阶为$\Theta(\sqrt{\beta\log\beta})$。本研究的动机之一在于确定Transformer模型在亚稳态下被吸引至的聚类数量。