Minimax adaptive estimation in manifold inference

We focus on the problem of manifold estimation: given a set observations sampled close to some unknown submanifold M, one wants recover information about geometry M. Minimax estimators which have been proposed so far all depend crucially priori knowledge parameters quantifying underlying distribution generating sample (such as bounds its density), whereas those quantities will be in practice. Our contribution matter is twofold. First, we introduce one-parameter family (Mˆt)t≥0 based localized version convex hulls, and show that for choice t, corresponding estimator minimax class models C2 manifolds introduced [21]. Second, propose completely data-driven selection procedure parameter leading adaptive this models. This actually allows us Hausdorff distance between can therefore used scale other settings, such tangent space estimation.