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 $e_{n,r}\downarrow L_r=\Vert \mathrm{dist}(\cdot ,S){\Vert }_{L^r(P)}$. Set ${\Delta }_{n,r}=e_{n,r}^r-L_r^r$ for finite $r$. A Borel nearest-point selector $T$ induces $\mu =T_{\ast }P$. Writing $q_{n,s}$ for the classical $L^s$ quantization error, we obtain $e_{n,r}-L_r\le 2q_{n,r}(\mu )$ and ${\Delta }_{n,r}\le C_r{q}_{n,1}(\mu )$ for $1\le r<\infty$, with $C_r$ independent of $n$, and ${\Delta }_{n,r}\le 2^r{q}_{n,r}(\mu {)}^r$ for $0<r<1$. Also, $e_{n,\infty }-L_{\infty }\le 2c_n(E)$, where $E$ is the nearest-point image of $\mathrm{supp}P$ and $c_n(E)$ is its covering radius. For $1\le r<\infty$, a weighted power-growth condition with exponent $\beta >0$ on a fixed source set $F$ bounds ${\Delta }_{n,r}$ below by the classical $\beta$-power quantization cost of ${\nu }_F=T_{\ast }(wP{|}_F)$. If $d{\nu }_F=\hat{w}d\eta$ for a compactly supported reference probability $\eta$ and $\hat{w}\in A_p(\eta )$, $1<p<\infty$, then ${\Delta }_{n,r}\ge {\nu }_F({\mathbb{R}}^D)[\hat{w}{]}_{A_p(\eta )}^{-1}{q}_{n,\beta /p}(\eta {)}^{\beta }$. These comparisons yield bounds for the quantization dimensions defined by $e_{n,r}-L_r$ and ${\Delta }_{n,r}^{1/r}$, 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.