Asymptotics of Constrained Quantization for Compactly Supported Measures

Abstract

We study constrained quantization of a compactly supported probability $P$ on $\mathbb{R}^D$ by codebooks in a nonempty closed set $S$. For $0<r\le\infty$, the constrained error $e_{n,r}$ is attained and satisfies $e_{n,r}\downarrow L_r=\|\operatorname{dist}(\cdot,S)\|_{L^r(P)}$. Set $Δ_{n,r}=e_{n,r}^r-L_r^r$ for finite $r$. A Borel nearest-point selector $T$ induces $μ=T_*P$. Writing $q_{n,s}$ for the classical $L^s$ quantization error, we obtain $e_{n,r}-L_r\le2q_{n,r}(μ)$ and $Δ_{n,r}\le C_rq_{n,1}(μ)$ for $1\le r<\infty$, with $C_r$ independent of $n$, and $Δ_{n,r}\le2^rq_{n,r}(μ)^r$ for $0<r<1$. Also, $e_{n,\infty}-L_\infty\le2c_n(E)$, where $E$ is the nearest-point image of $\operatorname{supp}P$ and $c_n(E)$ is its covering radius. For $1\le r<\infty$, a weighted power-growth condition with exponent $β>0$ on a fixed source set $F$ bounds $Δ_{n,r}$ below by the classical $β$-power quantization cost of $ν_F=T_*(wP|_F)$. If $dν_F=\widehat w\,dη$ for a compactly supported reference probability $η$ and $\widehat w\in A_p(η)$, $1<p<\infty$, then $Δ_{n,r}\geν_F(\mathbb{R}^D)[\widehat w]_{A_p(η)}^{-1}q_{n,β/p}(η)^β$. These comparisons yield bounds for the quantization dimensions defined by $e_{n,r}-L_r$ and $Δ_{n,r}^{1/r}$, with distinct behavior at positive and zero baselines. Convex and uniformly prox-regular constraints supply the growth weights through their projection inequalities. Circle and lifted Cantor distributions give matching quadratic-excess rates, while rectifiable weighted projections give lower Zador estimates.