The local quantization behavior of absolutely continuous probabilities

TL;DR

This paper investigates the local behavior of optimal quantization for absolutely continuous probability measures, showing how local probabilities and errors scale with the number of codepoints under certain conditions.

Contribution

It establishes the asymptotic local probabilities and quantization errors for $L^r$-optimal codebooks in a broad class of absolutely continuous distributions.

Findings

Local probabilities scale as approximately n^{-1}.

Local $L^r$-quantization errors scale as approximately n^{-(1+r/d)}.

Results hold for codebooks intersecting a fixed interior compact set.

Abstract

For a large class of absolutely continuous probabilities $P$ it is shown that, for $r>0$, for $n$-optimal $L^r(P)$-codebooks $\alpha_n$, and any Voronoi partition $V_{n,a}$ with respect to $\alpha_n$ the local probabilities $P(V_{n,a})$ satisfy $P(V_{a,n})\approx n^{-1}$ while the local $L^r$-quantization errors satisfy $\int_{V_{n,a}}|x-a|^r dP(x)\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$.