几何均值量化：基于自适应逼近
Geometric mean quantization via adaptive approximation
- University of Bremen（不来梅大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对紧支撑测度提出几何均值量化维数的质量阈值刻画，证明其存在性等价于特定级数收敛，并建立谱区间、依分布收敛与维数相等性的若干结果。
AI中文摘要：
设 $\nu$ 为 $\mathbb R^{d}$ 上具有紧支撑的 Borel 概率测度，且对某个 $a>0$ 满足 $\nu(B(x,r))\leq Cr^{a}$。当二进立方体的质量至少为 $t$ 时对其进行细化，并令 $\mathcal{L}_\nu(t)$ 为该细化停止时的平均深度。我们证明 $\nu$ 的下、上几何均值量化维数分别是 $\log(1/t)/\mathcal{L}_\nu(t)$ 的下、上极限。该维数存在的充要条件是当 $q\downarrow1$ 时，对所有二进立方体求和的 $(q-1)\sum_{Q}\nu(Q)^{q}$ 收敛，且此时维数由该极限确定。质量阈值公式给出了用局部维数表示的调和积分界，并将熵维数与量化维数包含在同一谱区间内。局部信息率的依分布收敛等价于在 $q=1$ 附近宽度为 $1/k$ 的窗口内重标度谱的收敛；此时两个维数分别是极限分布的算术平均与调和平均，我们通过尖锐界和方差恒等式量化了它们的差异。在没有任何收敛假设的情况下，阈值方差的消失仍强制对应的下、上维数相等。伯努利混合实现了 $(0,1]$ 内具有紧支撑的任意局部维数分布，而一个机制切换的例子将依分布收敛与几乎处处收敛区分开来。
英文摘要：
Let $ν$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $ν(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_ν(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of $ν$ are the lower and upper limits of $\log(1/t)/\mathcal{L}_ν(t)$. The dimension exists precisely when $(q-1)\sum_{Q}ν(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width $1/k$ around $q=1$; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in $(0,1]$, and a regime-switching example separates convergence in law from almost-everywhere convergence.
