Density estimation on an unknown submanifold

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 Hölder smoothness $β$ and $M$ has regularity $α$, our estimator achieves the rate $n^{-α\wedge β/(2α\wedge β+d)}$ and does not depend on the ambient dimension $D$ and is asymptotically minimax for $α\geq β$. Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case $α\leq β$: by estimating in some sense the underlying geometry of $M$, we establish in dimension $d=1$ that the minimax rate is $n^{-β/(2β+1)}$ 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.