Electronic Journal of Statistics

Tight minimax rates for manifold estimation under Hausdorff loss

Arlene K. H. Kim and Harrison H. Zhou

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This paper deals with minimax rates of convergence for manifold estimation. A new lower bound is obtained by a novel construction of two sets of manifolds and an application of convex hull testing method of Le Cam (1973). The minimax lower bound matches the upper bound up to a constant factor considered by Genovese et al. (2012b).

Article information

Electron. J. Statist., Volume 9, Number 1 (2015), 1562-1582.

Received: November 2013
First available in Project Euclid: 23 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C25: Compound decision problems 62G86: Nonparametric inference and fuzziness
Secondary: 65C50: Other computational problems in probability

minimax lower bound manifold estimation convex hull testing


Kim, Arlene K. H.; Zhou, Harrison H. Tight minimax rates for manifold estimation under Hausdorff loss. Electron. J. Statist. 9 (2015), no. 1, 1562--1582. doi:10.1214/15-EJS1039. https://projecteuclid.org/euclid.ejs/1437658703

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  • Cai, T. T. and M. G. Low (2011). Testing composite hypotheses, hermite polynomials and optimal estimation of a nonsmooth functional., Annals of Statistics 39(2), 1012–1041.
  • Chvátal, V. (1979). The tail of the hypergeometric distribution., Discrete Mathematics 25(3), 285–287.
  • Genovese, C., M. Perone-Pacifico, I. Verdinelli, and L. Wasserman (2012a). Minimax manifold estimation., Journal of machine learning research (3), 1263–1291.
  • Genovese, C. R., M. Perone-Pacifico, I. Verdinelli, and L. Wasserman (2012b). Manifold estimation and singular deconvolution under hausdorff loss., The Annals of Statistics 40(2), 941–963.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables., The Annals of Statistics 58(301), 13–30.
  • Kamath, A., R. Motwani, K. Palem, and P. Spirakis (1994). Tail bounds for occupancy and the satisfiability threshold conjecture. In, Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on, pp. 592–603.
  • Kolchin, V. F., B. A. Sevast’yanov, and V. P. Chistyakov (1978)., Random allocations. Washington, D.C.: V. H. Winston & Sons. Translated from the Russian, Translation edited by A. V. Balakrishnan, Scripta Series in Mathematics.
  • Le Cam, L. (1973). Convergence of estimates under dimensionality restrictions., The Annals of Statistics 1, 38–53.
  • Niyogi, P., S. Smale, and S. Weinberger (2006). Finding the homology of submanifolds with high confidence from random samples., Discrete and computational geometry 39, 419–441.
  • Tsybakov, A. B. (2009)., Introduction to Nonparametric Estimation. New York: Springer-Verlag.
  • Yu, B. (1997). Assouad, Fano, and Le Cam. In D. Pollard, E. Torgersen, and G. L. Yang (Eds.), A Festschrift for Lucien Le Cam, pp. 423–435. New York: Springer-Verlag.