Unequal dimensional small balls and quantization on Grassmann Manifolds

The Grassmann manifold G_n,p (L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space Lⁿ, 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 = 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.