Manifold estimation and singular deconvolution under Hausdorff loss
  We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure. 
Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli,   Larry Wasserman
math.ST cs.LG stat.ML stat.TH