Density estimation on an unknown submanifold

TL;DR

This paper develops a geometry-aware kernel density estimator for data supported on unknown submanifolds, achieving minimax rates that adapt to smoothness and dimension, with practical implementation demonstrated.

Contribution

It introduces a nonparametric density estimator that adapts to unknown submanifold geometry and achieves optimal convergence rates without depending on ambient dimension.

Findings

Estimator achieves rate $n^{-eta/(2eta+1)}$ in dimension 1.

Method is asymptotically minimax for certain smoothness conditions.

Numerical experiments confirm practical feasibility.

Abstract

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 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 $\alpha$, 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.