Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction

TL;DR

This paper investigates how dilating and translating $L^r$-optimal quantizers affects their $L^s$-rate-optimality across various distributions, demonstrating conditions under which the transformed quantizers remain optimal.

Contribution

It establishes that dilated and translated $L^r$-optimal quantizers are $L^s$-rate-optimal for many distributions, and provides specific parameter choices for Gaussian and exponential cases.

Findings

Dilated and translated quantizers are $L^s$-rate-optimal for many distributions.

Explicit parameter choices ensure empirical measure theorem satisfaction for Gaussian and exponential distributions.

The approach broadens understanding of quantizer transformations and their asymptotic optimality.

Abstract

Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0$, one defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $\forall n \geq 1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu + \theta(a- \mu), a \in \alpha_n \} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal.