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 dimensions, and a generalized chordal distance for unequal dimensional subspaces is used. First, the volume is reduced to a 1-D 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. Finally, the pertinence of the derived results is shown by rate-distortion analysis of source coding on Grassmann manifolds.

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ratedistortion theory determinant asymptotic volume 1d integral ratedistortion analysis of source coding on grassmann manifolds momentmatching techniques characteristic function of the jacobi ensemble packing bounds on codes generalized chordal distance finitesize regimes tighter approximation

Olav Tirkkonen, Jukka Corander, Lu Wei,Renaud-Alexandre Pitaval,.Volume of Metric Balls in High-Dimensional Complex Grassmann Manifolds. 62 (9),5105-5116.

Olav Tirkkonen, et al."Volume of Metric Balls in High-Dimensional Complex Grassmann Manifolds" 62

Olav Tirkkonen, Jukka Corander, Lu Wei,Renaud-Alexandre Pitaval,"Volume of Metric Balls in High-Dimensional Complex Grassmann Manifolds"