Volume of Metric Balls in High-Dimensional Complex Grassmann Manifolds

TL;DR

This paper derives an asymptotic formula for the volume of metric balls in complex Grassmann manifolds, which is useful for rate-distortion theory and code packing bounds, with refined finite-size approximations.

Contribution

It introduces a new integral representation for the volume of metric balls in complex Grassmann manifolds and analyzes their asymptotic behavior, including finite-size refinements.

Findings

Derived a one-dimensional integral for volume calculation

Established an asymptotic expression for large dimensions

Provided tighter finite-size volume approximations

Abstract

Volume of metric balls relates to rate-distortion theory and packing bounds on codes. In this paper, the volume of balls in complex Grassmann manifolds is evaluated for an arbitrary radius. The ball is defined as a set of hyperplanes of a fixed dimension with reference to a center of possibly different dimension, and a generalized chordal distance for unequal dimensional subspaces is used. First, the volume is reduced to one-dimensional integral representation. The overall problem boils down to evaluating a determinant of a matrix of the same size as the subspace dimensionality. Interpreting this determinant as a characteristic function of the Jacobi ensemble, an asymptotic analysis is carried out. The obtained asymptotic volume is moreover refined using moment-matching techniques to provide a tighter approximation in finite-size regimes. Lastly, the pertinence of the derived results is shown by rate-distortion analysis of source coding on Grassmann manifolds.