Density estimation on an unknown submanifold

We investigate the estimation of a density f from a n-sample on an Euclidean space RD, 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−α∧β∕(2α∧β+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 α≥β. Following Lepski’s principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case α≤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.