Density estimation on an unknown submanifold

We investigate the estimation of a density f from n-sample on an Euclidean space RD, when data are supported by unknown submanifold M possibly dimension d<D, under reach condition. several nonparametric kernel methods, with data-driven bandwidths that incorporate some learning geometry via local estimator. When has Hölder smoothness ? and regularity ?, our estimator achieves rate n?????(2???+d) for pointwise loss. The does not depend ambient D we establish procedure is asymptotically minimax ???. Following Lepski’s principle, bandwidth selection rule shown to achieve adaptation. also case ???: estimating in sense underlying M, d=1 n???(2?+1) proving particular it M. Finally, numerical implementation conducted studies order confirm practical feasibility estimators.