发表机构
UNSW Sydney; Istanbul Nişantaşı University（新南威尔士大学; 伊斯坦布尔尼尚塔西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有非负权重的\(L_1\)差异受维度诅咒问题，通过结合体积偏差概率测度变换与分数阶矩估计证明其倒数下界，此下界适用于等权重点集，论证利用权重非负性，不涉及任意有符号权重。
AI 中文摘要
我们证明了具有任意非负权重的\(L_1\)差异受维度诅咒影响。具体而言，对于每个\(\varepsilon\in(0,1)\)和\(d\in\mathbb{N}\)，\(L_1\)差异的倒数满足\[N_{1,+}(\varepsilon,d)\geq\frac{(1 - \varepsilon)^2}{1+\varepsilon}\left(\frac{3 + 2\sqrt{3}}{6}\right)^d,\]其中\((3 + 2\sqrt{3})/6 = 1.07735\ldots\)。证明结合了对体积偏差概率测度的变换和对归一化差异函数的分数阶矩估计。该下界尤其适用于等权重点集。论证本质上使用了权重的非负性，不涵盖任意有符号权重。
英文摘要
We prove that the normalized star, extreme, and periodic $L_1$-discrepancies with arbitrary nonnegative weights suffer from the curse of dimensionality. A single elementary lemma is used in all three cases: after normalizing the target and every one-dimensional kernel to probability densities, a uniform square-root-affinity gap tensorizes over the coordinates. For every $d\in \mathbb{N}$ and $0<\varepsilon<1$, the normalized information complexities have lower bounds of the form \[ \frac{(1-\varepsilon)^2}{1+\varepsilon}\,q^d, \] with \[ q_{\mathrm{star}}=\frac12+\frac1{\sqrt3},\qquad q_{\mathrm{ext}}=\frac{75(9+4\sqrt2)}{784},\qquad q_{\mathrm{per}}=\frac{25}{16}. \] All three constants are larger than one. The argument uses only nonnegativity, product integration, the elementary inequalities for the square root, and Cauchy--Schwarz.
Comments11 pages
