We investigate the estimation of a density f from a n-sample on an Euclidean space 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 datadriven 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.

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 close in spirit to (Aamari and Levrard, 2019), 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.

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. 47(1), 177–204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a $${\mathscr {C}}^k$$ C k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the $${\mathscr {C}}^3$$ C 3 and $${\mathscr {C}}^4$$ C 4 cases, with a gap given by a logarithmic factor.