发表机构
Laboratoire I3S, CNRS-Université Côte d’Azur; School of Mathematics, Cardiff University（I3S实验室，法国蔚蓝海岸大学; 卡迪夫大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对球对称分布，本文证明其量化优化为凸问题，提出等价定理与构造性算法，发现中等n时合适半径球面上的均匀随机量化器表现优异，且高维下失真方差趋于零，s=2时最优半径与失真有精确表达式。
AI 中文摘要
Zador定理是最优量化的基石，它确立了R^d中n点最优量化器经验分布的弱极限，以及对应的L_s-平均量化误差的衰减率。然而在高维场景下，要观测到这种渐近行为需要极大的样本量。我们证明，对于球对称目标分布，对所有球对称分布的优化是一个凸问题，并推导出等价定理，该定理既刻画了全局最优性，又给出了构造性算法。我们表明，对于中等规模的n，均匀分布在半径为R的合适球面上的随机量化器表现极佳，且在广泛的n取值范围内，经数值验证其在所有随机量化器中是最优的。它们的期望失真具有显式积分表示，可被计算到任意精度，且我们证明了随机量化器的集中性：对于固定的d，当n→∞时，失真方差趋于零。当s=2时，最优半径和对应的最小期望失真均有精确表达式；对于一般的s，最优半径可高效确定，且当n随d增长时，极值理论能提供有用的近似。根据该增长率，R要么收敛到零，要么趋近于与s无关的正极限。
英文摘要
Zador's celebrated theorem is a cornerstone of optimal quantisation: it establishes both the weak limit of the empirical distribution of an optimal $n$-point quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. In large dimension, however, observing this asymptotic behaviour requires an astronomically large sample size. We prove that, for spherically symmetric target distributions, optimisation over all spherically symmetric distributions is a convex problem and derive an equivalence theorem that both characterises global optimality and yields a constructive algorithm. We show that, for moderate $n$, random quantisers uniformly distributed on a sphere of suitably chosen radius $R$ perform exceptionally well and, over a broad range of values of $n$, are numerically certified to be optimal among all random quantisers. Their expected distortion has an explicit integral representation that can be evaluated to arbitrary precision, and we prove concentration across random quantisers: the distortion variance tends to zero as $n\to\infty$ for fixed $d$. For $s=2$, both the optimal radius and the associated minimum expected distortion admit exact expressions. For general $s$, the optimal radius can be determined efficiently, and extreme-value theory provides useful approximations when $n$ grows with $d$. Depending on this growth rate, $R$ either converges to zero or approaches a positive limit that is independent of $s$.
