Minimax manifold estimation

Authors:
Christopher R. Genovese;Marco Perone-Pacifico;Isabella Verdinelli;Larry Wasserman
Affiliations:
Department of Statistics, Carnegie Mellon University, Pittsburgh, PA;Dipartimento di Scienze Statistiche, Sapienza University of Rome, Roma, Italy;Department of Statistics, Carnegie Mellon University, Pittsburgh, PA and Department of Statistical Sciences, Sapienza University of Rome, Italy;Department of Statistics, Carnegie Mellon University, Pittsburgh, PA and Machine Learning Department, Carnegie Mellon University
Venue:
The Journal of Machine Learning Research
Year:
2012

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Abstract

We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in RD 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.