Minimax Manifold Estimation

2010/07/04 by Christopher R. Genovese, Genovese, Christopher, Marco Perone-Pacifico +5 · 6 citations
Mathematics · #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1007.0549

Abstract

We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R^D given a noisy sample from the manifold. We assume that the manifold satisfies a smoothness condition and that the noise distribution has compact support. 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.

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