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