On the number of modes of Gaussian kernel density estimators

We consider the Gaussian kernel density estimator with bandwidth ${\beta }^{-\frac{1}{2}}$ 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 $\mathrm{\Theta }(\sqrt{\beta \log \beta })$ as $\beta ,n\to \mathrm{\infty }$ 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.