Unequal dimensional small balls and quantization on Grassmann Manifolds

—The Grassmann manifold Gn,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 Gn,p (L) is quantized through a code in Gn,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 Gn,p (L) whose center is in Gn,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 asymptotica...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Wei Dai, Brian Rider, Youjian Liu
• Keywords: CORR 2007, Distortion Rate Function, Education, Rate Distortion Tradeoff, Unequal Dimensional Quantization