发表机构
Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文指出EDEN随机旋转量化中的两种重建尺度对应经典CDEF的维纳与无偏系数，并证明CDEF +1勾股关系在有限维逐点成立但平均后失效，且EDEN的Haar旋转保证更强的条件无偏性。
AI 中文摘要
基于随机旋转的量化方案近期重新受到关注，包括MMSE和无偏重建缩放的作用。在本文中，我们指出其与统计信号处理和通信理论中经典结果的联系。具体而言，EDEN系列工作中使用的两种重建尺度可自然解释为经典CDEF公式中维纳系数和无偏系数的有限维、依赖于实现的对等物。在有限块长下，CDEF +1关系对每个旋转实现逐点成立，作为精确的几何（勾股）恒等式，但在对旋转上的失真进行平均后不再成立。经典信噪比关系 $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ 在 $d\to\infty$ 时恢复：一旦整体尺度被单独处理，随机旋转向量的经验坐标统计趋近于其独立同分布高斯对应物，且依赖于旋转的量发生集中。重要的是，EDEN超越了这种经典对应：对于每个有限 $d$，其Haar旋转公式保证了精确的条件无偏性，这是比CDEF中的二阶无偏性概念更强的性质。我们进一步评论随机旋转在量化中扮演的两个不同角色：一是坐标的近似高斯化；二是跨量化分支的重建误差去相关。
英文摘要
Quantization schemes based on randomized rotations have recently received renewed attention, including the roles of MMSE and unbiased reconstruction scalings. In this note, we point out the connection to classical results in statistical signal processing and communication theory. Specifically, the two reconstruction scales used in the EDEN line of work admit a natural interpretation as finite-dimensional, realization-dependent counterparts of the Wiener and unbiased coefficients in the classical CDEF formulation. At finite blocklength, the CDEF +1 relation holds pointwise for each rotation realization as an exact geometric (Pythagorean) identity, but does not hold after averaging the distortions over the rotation. The classical SNR relation $\sf{SNR}_{\rm MMSE}=\sf{SNR}_{\rm MMSE,U}+1$ is recovered as $d\to\infty$: once the overall scale is handled separately, the empirical coordinate statistics of a randomly rotated vector approach their i.i.d. Gaussian counterparts, and the rotation-dependent quantities concentrate. Importantly, EDEN goes beyond this classical correspondence: for every finite $d$, its Haar-rotation formulation guarantees exact conditional unbiasedness, a stronger property than the second-order notion of unbiasedness in CDEF. We further comment on two distinct roles random rotations play in quantization: one is approximate Gaussianization of the coordinates; the other is decorrelation of reconstruction errors across quantization branches.
