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On the number of modes of Gaussian kernel density estimators
Borjan Geshkovski, Philippe Rigollet, Yihang Sun · 2024-12-12 · via stat.ML updates on arXiv.org

We consider the Gaussian kernel density estimator with bandwidth $β^{-\frac12}$ 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 $Θ(\sqrt{β\logβ})$ as $β,n\to\infty$ provided $n^c\lesssim β\lesssim n^{2-c}$ 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.

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