Density estimation on an unknown submanifold
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We investigate density estimation from a n-sample in the Euclidean space 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 Hölder smoothness β and M has regularity α in a sense to be defined, our estimator achieves the rate n^-α∧β/(2α∧β+d) and does not depend on the ambient dimension D and 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.
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