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. 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.
Minimax Manifold Estimation / Genovese, C. R.; Perone Pacifico, Marco; Verdinelli, Isabella; Wasserman, L.. - In: JOURNAL OF MACHINE LEARNING RESEARCH. - ISSN 1532-4435. - 13:(2012), pp. 1263-1291.
Minimax Manifold Estimation
Marco Perone Pacifico;
2012
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. 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.
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