On the exact volume of metric balls in complex Grassmann manifolds

Abstract

We evaluate the volume of metric balls in complex Grassmann manifolds. The ball is defined as a set of hyperplanes of a fixed dimension with reference to a center of not necessarily the same dimension. The normalized volume of balls corresponds to the cumulative distribution of quantization error for uniformly-distributed sources, a problem notably related to rate-distortion analysis, and to packing bounds. A generalized chordal distance for unequal dimensional subspaces is used. First, a symmetry property between complementary balls is presented, extending previous small ball results to larger radius. Then, the volume is shown to be reducible to a one-dimensional integral representation, valid for any radius. Accordingly, the overall problem boils down to evaluating a determinant of a matrix of same size than the subspace dimensionality. Examples of explicit polynomial expressions emanating from the integral formulation are given.