Information Geometry of Covariance Matrix: Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger Fibration and Fréchet Median

Abstract

Information Geometry has been introduced by Rao, and axiomatized by Chentsov, to define a distance between statistical distributions that is invariant to non-singular parameterization transformations. For Doppler/Array/STAP Radar Processing, Information Geometry Approach will give key role to Homogenous Symmetric bounded domains geometry. For Radar, we will observe that Information Geometry metric could be related to Kahler metric, given by Hessian of Kahler potential (Entropy of Radar Signal given by \(-log[det(R)]\)). To take into account Toeplitz structure of Time/Space Covariance Matrix or Toeplitz-Block-Toeplitz structure of Space-Time Covariance matrix, Parameterization known as Partial Iwasawa Decomposition could be applied through Complex Autoregressive Model or Multi-channel Autoregressive Model. Then, Hyperbolic Geometry of Poincare Unit Disk or Symplectic Geometry of Siegel Unit Disk will be used as natural space to compute “p-mean” (\(p=2\) for “mean”, \(p=1\) for “median”) of covariance matrices via Karcher flow derived from Weiszfeld algorithm extension on Cartan-Hadamard manifold. This new mathematical framework will allow development of Ordered Statistic (OS) concept for Hermitian Positive Definite Covariance Space/Time Toeplitz matrices or for Space-Time Toeplitz-Block-Toeplitz matrices. We will define Ordered Statistic High Doppler Resolution CFAR (OS-HDR-CFAR) and Ordered Statistic Space-Time Adaptive Processing (OS-STAP).