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April 2015 Nonasymptotic bounds for vector quantization in Hilbert spaces
Clément Levrard
Ann. Statist. 43(2): 592-619 (April 2015). DOI: 10.1214/14-AOS1293


Recent results in quantization theory show that the mean-squared expected distortion can reach a rate of convergence of $\mathcal{O}(1/n)$, where $n$ is the sample size [see, e.g., IEEE Trans. Inform. Theory 60 (2014) 7279–7292 or Electron. J. Stat. 7 (2013) 1716–1746]. This rate is attained for the empirical risk minimizer strategy, if the source distribution satisfies some regularity conditions. However, the dependency of the average distortion on other parameters is not known, and these results are only valid for distributions over finite-dimensional Euclidean spaces.

This paper deals with the general case of distributions over separable, possibly infinite dimensional, Hilbert spaces. A condition is proposed, which may be thought of as a margin condition [see, e.g., Ann. Statist. 27 (1999) 1808–1829], under which a nonasymptotic upper bound on the expected distortion rate of the empirically optimal quantizer is derived. The dependency of the distortion on other parameters of distributions is then discussed, in particular through a minimax lower bound.


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Clément Levrard. "Nonasymptotic bounds for vector quantization in Hilbert spaces." Ann. Statist. 43 (2) 592 - 619, April 2015.


Published: April 2015
First available in Project Euclid: 24 February 2015

zbMATH: 1314.62143
MathSciNet: MR3316191
Digital Object Identifier: 10.1214/14-AOS1293

Primary: 62H30

Keywords: fast rates , Localization , margin conditions , quantization

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 2 • April 2015
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