Density estimation on an unknown submanifold

Abstract

We investigate the estimation of a density f from a n-sample on an Euclidean space ${\mathbb{R}}^D$, when the data are supported by an unknown submanifold M of possibly unknown dimension $d<D$, under a reach condition. We investigate several nonparametric kernel methods, 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 $n^{-\mathit{\alpha }\wedge \mathit{\beta }∕(2\mathit{\alpha }\wedge \mathit{\beta }+d)}$ for a pointwise loss. The rate does not depend on the ambient dimension D and we establish that our procedure is asymptotically minimax for $\mathit{\alpha }\ge \mathit{\beta }$. Following Lepski’s principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case $\mathit{\alpha }\le \mathit{\beta }$: by estimating in some sense the underlying geometry of M, we establish in dimension $d=1$ that the minimax rate is $n^{-\mathrm{\beta }∕(2\mathrm{\beta }+1)}$ 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.