## The Annals of Statistics

### Nonasymptotic bounds for vector quantization in Hilbert spaces

Clément Levrard

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 43, Number 2 (2015), 592-619.

Dates
First available in Project Euclid: 24 February 2015

https://projecteuclid.org/euclid.aos/1424787429

Digital Object Identifier
doi:10.1214/14-AOS1293

Mathematical Reviews number (MathSciNet)
MR3316191

Zentralblatt MATH identifier
1314.62143

#### Citation

Levrard, Clément. Nonasymptotic bounds for vector quantization in Hilbert spaces. Ann. Statist. 43 (2015), no. 2, 592--619. doi:10.1214/14-AOS1293. https://projecteuclid.org/euclid.aos/1424787429

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