On the number of modes of Gaussian kernel density estimators
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911254079078400 |
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| author | Geshkovski, Borjan Rigollet, Philippe Sun, Yihang |
| author_facet | Geshkovski, Borjan Rigollet, Philippe Sun, Yihang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09080 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of modes of Gaussian kernel density estimators Geshkovski, Borjan Rigollet, Philippe Sun, Yihang Statistics Theory Machine Learning 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. |
| title | On the number of modes of Gaussian kernel density estimators |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2412.09080 |