Volume of geodesic balls in the real Stiefel manifold

Abstract

Volume estimates of balls in Riemannian manifolds find many applications in information theory such as in the determination of the rate-distortion tradeoff for the quantization of a source randomly distributed over such surfaces. This paper computes the precise power series expansion of volume of small geodesic balls in a real Stiefel manifold of arbitrary dimension. An application of the result obtained is also demonstrated.

Distortion-rate tradeoff of a source uniformly distributed over positive semi-definite matrices

“…Similar approaches have been used in [1], [4] and [5] for the Grassmann and the real/complex Stiefel manifolds, respectively.…”

“…We extended the analysis to the more general case of the Stiefel manifold V n,k , the set of all n × k semi-orthogonal (or semi-unitary) matrices, by characterizing the distortion-rate tradeoff D * (K) of a source invariantly distributed over it -for real matrices in [4] and for complex matrices in [5]. In this paper, we consider the more general case of arbitrary covariance matrix feedback by relating it to quantization on P F (n), F = R or C, and by providing asymptotically tight bounds for D * (K) for uniform sources over them.…”

“…to many problems in coding and information theory (cf. [6], [7], [8]) and hence can also be viewed as a problem of independent interest [4]. We use the following notation in this paper.…”

“…Following the approach in [4], our calculation of the distortion-rate function involves the volume expansion of a geodesic ball of arbitrary radius in P F (n). A geodesic ball is a ball where distances are measured along paths of shortest lengths or geodesics.…”

Volume of ball and Hamming-type bounds for Stiefel manifold with Euclidean distance

“…In [10], a closed-form expression on the volume of a small ball in Grassmannians under the chordal distance was derived. Finally, a power series expansion of the (unnormalized) volume of small ball valid for any Riemann manifold is leveraged in [11], [12]. This provides a powerful tool-in order to obtain a normalized volume expansion, it suffices to divide by the overall volume of the manifold.…”