Grassmann流形上的不等维小球与quantization
Unequal dimensional small balls and quantization on Grassmann Manifolds
摘要：
格拉斯曼流形G_{n,p}（L）是n维欧氏空间L^{n}中所有p维平面（通过原点）的集合，其中L是R或C。本文考虑了一种不等维量化，其中G_{n,p}（L）中的信源通过G_{n,q}（L）中的码进行量化，其中p和q不一定相同。它不同于大多数文学作品中p \equiv q.不等维量化的分析是基于G_{n,p}（L）中的度量球的体积，该度量球的中心在G_{n,q}（L）中。我们的主要结果是一个封闭的形式公式的体积公制球时，半径是足够小的。这个体积公式适用于任意n，p，q和L的格拉斯曼流形，而以前的结果只适用于某些特殊情况。基于这个体积公式，推导出假设量化率足够高的率失真折衷的几个界限。失真率函数的上下界是渐近相同的，因此精确地量化了渐近率失真折衷。我们还表明，随机码是渐近最优的意义上，他们实现了最小的可实现的失真与概率为n和码率接近无穷线性。最后，我们讨论了一些应用所得到的结果通信理论。提出了一种基于格拉斯曼流形的加性高斯白噪声信道容量计算的几何解释。此外，导出的失真率函数有利于表征多天线通信中波束成形矩阵选择的效果。
英文摘要：
The Grassmann manifold G_{n,p}(L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space L^{n}, where L is either R or C. This paper considers an unequal dimensional quantization in which a source in G_{n,p}(L) is quantized through a code in G_{n,q}(L), where p and q are not necessarily the same. It is different from most works in literature where p\equiv q. The analysis for unequal dimensional quantization is based on the volume of a metric ball in G_{n,p}(L) whose center is in G_{n,q}(L). Our chief result is a closed-form formula for the volume of a metric ball when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary n, p, q and L, while previous results pertained only to some special cases. Based on this volume formula, several bounds are derived for the rate distortion tradeoff assuming the quantization rate is sufficiently high. The lower and upper bounds on the distortion rate function are asymptotically identical, and so precisely quantify the asymptotic rate distortion tradeoff. We also show that random codes are asymptotically optimal in the sense that they achieve the minimum achievable distortion with probability one as n and the code rate approach infinity linearly. Finally, we discuss some applications of the derived results to communication theory. A geometric interpretation in the Grassmann manifold is developed for capacity calculation of additive white Gaussian noise channel. Further, the derived distortion rate function is beneficial to characterizing the effect of beamforming matrix selection in multi-antenna communications.
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