Minimax Manifold Estimation

Christopher Genovese, Marco Perone-Pacifico, Isabella Verdinelli, Larry Wasserman.

Year: 2012, Volume: 13, Issue: 43, Pages: 1263−1291


We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in D given a noisy sample from the manifold. Under certain conditions, we show that the optimal rate of convergence is n-2/(2+d). Thus, the minimax rate depends only on the dimension of the manifold, not on the dimension of the space in which M is embedded.