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Pré-Publication, Document De Travail Année : 2019

Density estimation on an unknown submanifold

Résumé

We investigate density estimation from a $n$-sample in the Euclidean space $\mathbb R^D$, when the data is supported by an unknown submanifold $M$ of possibly unknown dimension $d < D$ under a reach condition. We study nonparametric kernel methods for pointwise and integrated loss, with data-driven bandwidths that incorporate some learning of the geometry via a local dimension estimator. When $f$ has H\"older smoothness $\beta$ and $M$ has regularity $\alpha$ in a sense to be defined, our estimator achieves the rate $n^{-\alpha \wedge \beta/(2\alpha \wedge \beta+d)}$ and does not depend on the ambient dimension $D$ and is asymptotically minimax for $\alpha \geq \beta$. Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case $\alpha \leq \beta$: by estimating in some sense the underlying geometry of $M$, we establish in dimension $d=1$ that the minimax rate is $n^{-\beta/(2\beta+1)}$ proving in particular that it does not depend on the regularity of $M$. Finally, a numerical implementation is conducted on some case studies in order to confirm the practical feasibility of our estimators.
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Dates et versions

hal-02329703 , version 1 (23-10-2019)

Identifiants

Citer

Clément Berenfeld, Marc Hoffmann. Density estimation on an unknown submanifold. 2019. ⟨hal-02329703⟩
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