Renaud-Alexandre Pitaval, Lu Wei, O. Tirkkonen, J. Corander
{"title":"On the exact volume of metric balls in complex Grassmann manifolds","authors":"Renaud-Alexandre Pitaval, Lu Wei, O. Tirkkonen, J. Corander","doi":"10.1109/ITWF.2015.7360783","DOIUrl":null,"url":null,"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.","PeriodicalId":281890,"journal":{"name":"2015 IEEE Information Theory Workshop - Fall (ITW)","volume":"32 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2015 IEEE Information Theory Workshop - Fall (ITW)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ITWF.2015.7360783","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.