Asymptotics of Constrained Quantization for Compactly Supported Measures

TL;DR

This paper develops a comprehensive asymptotic theory for high-rate constrained quantization errors of compactly supported measures, extending classical results to scenarios with geometric constraints on quantizers.

Contribution

It introduces a novel dimension comparison formula for constrained quantization, unifying upper and lower bounds under mild geometric conditions.

Findings

Error decay rate matches the reciprocal of the d-th root of n for Ahlfors regular sets.

The dimension of the quantization error equals the measure's Hausdorff dimension.

Provides the first complete dimension comparison formula for constrained quantization.

Abstract

We investigated the asymptotics of high-rate constrained quantization errors for a compactly supported probability measure P on Euclidean spaces whose quantizers are confined to a closed set S. The key tool is the metric projection of K onto S that assigns each source point to its nearest neighbor in S, allowing the errors to be transferred to the projection, where K = supp P. For the upper estimate, we establish a projection pull-back inequality that bounds the errors by the classical covering radius of the projection. For the lower estimate, a weighted distance function enables us to perturb any quantizer element lying on the projection slightly into the complement in S without enlarging the error, provided the projection is nowhere dense (automatically true when S and K are disjoint). Under mild conditions on the pushforward measure of P by T, obtained via a measurable selector T, we derive a uniform lower bound. If this set is Ahlfors regular of dimension d, the error decays like the reciprocal of the d-th root of n and every constrained quantization dimension equals d. The two estimates coincide, giving the first complete dimension comparison formula for constrained quantization and closing the gap left by earlier self-similar examples by Pandey-Roychowdhury while extending classical unconstrained theory to closed constraints under mild geometric assumptions.