Asymptotics of Constrained Quantization for Compactly Supported Measures

TL;DR

This work addresses high-rate constrained quantization for a compactly supported measure $P$ when the quantizers are restricted to a closed constraint set $S$. It introduces a projection-based framework using the metric projection $ ext{pi}_S$ and a measurable selector $T$ to transfer the problem onto the projected set $ ext{pi}_S(K)$ and its pushforward $T_*P$, enabling sharp upper and lower dimension bounds. Under mild regularity, particularly when $ ext{pi}_S(K)$ is governed by an Ahlfors-regular pushforward of dimension $d$, the constrained quantization error decays as $e_{n,r}(P;S)-e_{ty,r}(P;S) hicksim n^{-1/d}$ and all constrained quantization dimensions equal $d$, yielding a complete dimension comparison formula that extends classical unconstrained results. The framework blends geometric measure theory, projection geometry, and quantization theory, and is illustrated with concrete examples such as circles andCantor-like sets to demonstrate sharpness and alignment with known unconstrained behavior.

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.

Figures (4)

• Figure 1: Illustration of $K$, $S$, and $\pi_S(K)$ in Example \ref{['example:quantizers_out_pi_s_k']}
• Figure 2: Illustration of $K$, $S$, and $\pi_S(K)$ in Example \ref{['example:cond_a_fails']}.
• Figure 3: Illustration of $K$, $S$, and $\pi_S(K)$ in Example \ref{['example:dirac_measure']}
• Figure 4: Illustration of $K$, $S$, and $\pi_S(K)$ in Example \ref{['example:uniform_unit_circle']}

Theorems & Definitions (70)

• Theorem 1: Upper Asymptotics
• Theorem 2: Lower Asymptotics
• Corollary 3
• Proposition 4
• Remark 1
• Lemma 5: Existence of $C_{n,r}(P;S)$
• proof
• Lemma 6
• proof
• Lemma 7