Nonasymptotic bounds for vector quantization in Hilbert spaces

TL;DR

This paper extends nonasymptotic bounds for vector quantization to general Hilbert spaces, providing new insights into the distortion rate and its dependencies under a margin condition.

Contribution

It introduces a margin condition for Hilbert space distributions and derives nonasymptotic upper bounds on quantization distortion, generalizing previous finite-dimensional results.

Findings

Derived nonasymptotic upper bounds on expected distortion.

Established a minimax lower bound for distortion dependencies.

Extended quantization bounds to infinite-dimensional Hilbert spaces.

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.