Sharp rate for the dual quantization problem

TL;DR

This paper proves the optimal rate of convergence for dual quantization, a new approach based on Delaunay triangulation, extending classical quantization results with applications in numerical methods.

Contribution

It establishes the sharp asymptotic rate for dual quantization error, extending Zador's theorem to this new framework and utilizing an extended Pierce Lemma.

Findings

Established the sharp rate of dual quantization error asymptotics.

Linked dual quantization to Delaunay triangulation and stationarity properties.

Extended classical quantization theorems to the dual setting.

Abstract

In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was recently introduced in the paper [8], where it was shown that, at least in an Euclidean setting, dual quantizers are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which "regular" quantization relies. Moreover, this new approach shares an intrinsic stationarity property, which makes it very valuable for numerical applications. We establish in this paper the counterpart for dual quantization of the celebrated Zador theorem, which describes the sharp asymptotics for the quantization error when the quantizer size tends to infinity. The proof of this theorem relies among others on an extension of the so-called Pierce Lemma by means of a random quantization argument.