Volume of metric balls in high-dimensional complex Grassmann manifolds

Renaud Alexandre Pitaval*, Lu Wei, Olav Tirkkonen, Jukka Corander

*Corresponding author for this work

Research output: Contribution to journalConference articleScientificpeer-review

2 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)5105-5116
Number of pages12
JournalIEEE Transactions on Information Theory
Volume62
Issue number9
DOIs
Publication statusPublished - 1 Sep 2016
MoE publication typeA4 Article in a conference publication
EventIEEE Information Theory Workshop - Jeju Island, Korea, Republic of
Duration: 11 Oct 201515 Oct 2015

Keywords

  • Grassmann manifold
  • high-dimension
  • metric ball
  • rate-distortion analysis
  • source coding
  • volume

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