Sharp Rate for the Dual Quantization Problem

Abstract

In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was introduced in Pagès and Wilbertz (SIAM J Numer Anal 50(2):747–780, 2012). Dual quantizers, at least in a Euclidean setting, are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which “regular” quantization relies. This new approach to quantization 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. On the way we establish an extension of the so-called Pierce Lemma by a random quantization argument. Numerical results confirm our choices.