On the Geometry and Quantization of Manifolds of Positive Semi-Definite Matrices

Author

Krishnamachari, Rajesh T. ; Varanasi, Mahesh K.

Author_Institution

Dept. of Electr., Comput., & Energy Eng., Univ. of Colorado, Boulder, CO, USA

Volume

61

Issue

18

fYear

2013

fDate

Sept.15, 2013

Firstpage

4587

Lastpage

4599

Abstract

The geometry of different spaces of positive semi-definite matrices buffeted by rank and trace constraints is studied. In addition to revealing their Riemannian structure, we derive the normalized volume of a ball over these spaces. Further, we use the leading coefficient from the ball volume expansion to bound the quantization error incurred with finite-sized sphere-packing codebooks as well as random codebooks to represent sources distributed over general Riemannian manifolds.

Keywords

constraint theory; geometry; matrix algebra; quantisation (signal); Riemannian structure; ball volume expansion; finite-sized sphere-packing codebooks; general Riemannian manifolds; geometry; normalized volume; positive semidefinite matrices; quantization error; random codebooks; rank constraint; trace constraint; Approximation methods; Geometry; Jacobian matrices; Manifolds; Measurement; Monte Carlo methods; Quantization (signal); Positive semi-definite matrices; Riemannian geometry; quantization; random codes; sphere packing;

fLanguage

English

Journal_Title

Signal Processing, IEEE Transactions on

Publisher

ieee

ISSN

1053-587X

Type

jour

DOI

10.1109/TSP.2013.2272552

Filename

6557075