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 Kähler metric, given by Hessian of Kähler 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 Poincaré 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).

Appendix: Iwasawa, Cartan and Hua Coordinates

 

• Cartan Decomposition on Poincaré Unit Disk

  $D=\left\{ {z/\left| z \right|<1} \right\}$

  $g\in SU(1,1)\;{\text{ with}}\;g=\left({{\begin{array}{ll} a&b\\ {b^{*}}&{a^{*}} \end{array} }}\right)\;{\text{ and}}\;g(z)=\frac{az+b}{b^{*}z+a^{*}}\;{\text{ where}}\;\left| a \right|^{2}-\left| b \right|^{2}=1$

  $\text{ Cartan} \text{ Decomposition:}\;g=u_\varPhi d_\tau u_\varPsi$

  ${\text{ with}}\; u_\varPhi =\left(\begin{array}{ll} e^{i\varPhi }&0\\ 0&e^{-i\varPhi } \end{array}\right)\;{\text{ and}}\;d_\tau =\left(\begin{array}{ll} ch(t)&sh(t)\\ sh(t)&ch(t) \end{array}\right)$

  ${\Rightarrow } \left\{ \begin{array}{l} a=e^{i\left({\varPhi +\varPsi }\right)}ch(\tau )\\ b=e^{i(\varPhi -\varPsi )}sh(\tau ) \end{array}\right.\Rightarrow z=b\left(a^{*}\right)^{-1}=th(\tau )e^{i2\varPhi }$

  $ds^{2}=8\left({d\tau ^{2}+sh^{2}(2\tau )d\theta ^{2}} \right)\Rightarrow \varDelta _{LB}\cdot =\frac{\partial ^{2}\cdot }{\partial \tau ^{2}}+\coth (2\tau )\frac{\partial \cdot }{\partial \tau }+\frac{1}{sh^{2}(\tau )}\frac{\partial ^{2}\cdot }{\partial \varPhi ^{2}}$

  $F(z)=-\ln \left({1-\left| z \right|^{2}} \right)=2\ln ch(\tau )$

• Iwasawa Decomposition on Poincaré Unit Disk (Lemma of Iwasawa for Radial Coordinates in Poincaré Disk)

  $D=\left\{ {z/\left| z \right|<1} \right\}$

  $g\in SU(1,1)\;{\text{ with}}\;g=\left({{\begin{array}{ll} a&b\\ {b^{*}}&{a^{*}} \end{array} }} \right)\;{\text{ and}}\;g(z)=\frac{az+b}{b^{*}z+a^{*}}\;{\text{ where}}\;\left| a \right|^{2}-\left| b \right|^{2}=1$

  $\text{ Iwasawa} \text{ Dec.:}\; g=h\left({K_\theta D_\tau N_\xi } \right)\;{\text{ with}}\;h(g)=CgC^{-1}, C=\frac{1}{\sqrt{2}}\left({{\begin{array}{ll} 1&{-i} \\ 1&i \end{array} }} \right)$

  $K_\theta =\left({{\begin{array}{ll} {\cos \left({\theta /2} \right)}&{\sin \left({\theta /2} \right)} \\ {-\sin \left({\theta /2} \right)}&{\cos \left({\theta /2} \right)} \end{array} }} \right), D_\tau =\left({{\begin{array}{ll} {ch(t)}&{sh(t)}\\ {sh(t)}&{ch(t)} \end{array}}} \right)\;\text{ and}\;N_\xi =\left({{\begin{array}{ll} 1&\xi \\ 0&1 \end{array} }} \right)$

  $\Rightarrow \left\{ {\begin{array}{l} a=e^{i\theta /2}\left({ch\left({\tau /2} \right)+i\frac{u}{2}e^{-\frac{\tau }{2}}} \right)\\ b=e^{i\theta /2}\left({sh\left({\tau /2} \right)-i\frac{u}{2}e^{-\frac{\tau }{2}}} \right) \end{array}} \right.\;{\text{ with}}\;u=\xi e^{\tau }$

• Hua-Cartan Decomposition on Siegel Unit Disk (Lemma of Hua for Radial Coordinates in Siegel Disk)

  $\tau =\left[ {\tau _1 }\; {\tau _2}\; \cdots \; {\tau _n } \right]\;{\text{ with} \text{0}}\;\le \tau _n \le \tau _{n-1} \le \cdots \le \tau _1$

  $A_0 (\tau )=diag\left[ {ch(\tau _1 )}\; {ch(\tau _2 )}\; \cdots \; {ch(\tau _n )} \right]$

  $B_0 (\tau )=diag\left[ {sh(\tau _1 )}\; {sh(\tau _2 )}\; \cdots \; {sh(\tau _n )} \right]$

  $g=\left[\begin{array}{cc} A&B \\ {B^{*}}&{A^{*}} \end{array} \right]\in Sp(n), g=\left[\begin{array}{ll} {U^{t}}&0 \\ 0&{U^{+}} \end{array}\right]\left[\begin{array}{ll} {A_0 }&{B_0 }\\ {B_0 }&{A_0 } \end{array}\right]\left[\begin{array}{ll} {V^{*}}&0 \\ 0&V \end{array} \right]$

  $\text{ there} \text{ exist}\;U\;{\text{ and}}\;V\;{\text{ unitary} \text{ complex} \text{ matrices} \text{ of} \text{ order} \text{ n}}\;$

  $\left[\begin{array}{l} A=U^{t}A_0 (\tau )V^{*} \\ B=U^{t}B_0 (\tau )V \end{array} \right].\Rightarrow \left(\begin{array}{ll} {A_0 (\tau )}&{B_0 (\tau )}\\ {B_0 (\tau )}&{A_0 (\tau )} \end{array}\right)=\exp \left(\begin{array}{ll} 0&{Z_2 (\tau )}\\ {Z_2 (\tau )}&0 \end{array}\right)$

  $\text{ with}\;Z_2 (\tau )=diag\left[\begin{array}{llll} {\tau _1 }&{\tau _2 }&\cdots&{\tau _n } \end{array} \right]$

  $\text{ Let}\;Z=B\left({A^{*}} \right)^{-1}=U^{t}PU, P^{2}=B_0^2 \left({A_0^{-1} } \right)^{2}=\;{\text{ diag}}\;\left[{eigen\left({ZZ^{+}} \right)} \right]$

  $P=diag\left[\begin{array}{llll} {th(\tau _1 )}&{th(\tau _2 )}&\cdots&{th(\tau _n )} \end{array} \right]$

• Iwasawa Decomposition on Siegel Unit Disk (Iwasawa Coordinates in Siegel Disk)

  $SD_n =\left\{ {Z/ZZ^{+}<I} \right\} \;{\text{ and}}\;g(Z)=\left({AZ+B} \right)\left({B^{*}Z+A^{*}} \right)^{-1}$

  $g=\left(\begin{array}{ll} A&B\\ {B^{*}}&{A^{*}} \end{array} \right), h(g)=CgC^{-1}\;{\text{ with}}\;C=\frac{1}{\sqrt{2}}\left(\begin{array}{ll} I&{-iI}\\ I&{iI} \end{array} \right)$

  $\begin{array}{l} K=\left\{ {g/h(g)=\left(\begin{array}{cc} U&0\\ 0&{U^{*}} \end{array}\right), U\;\text{ unitary} \text{ order} \text{ n}}\right\} \\ \quad \;\Rightarrow g=\frac{1}{2}\left(\begin{array}{cc} {U+U^{*}}&{-i\left({U-U^{*}} \right)}\\ {i\left({U-U^{*}} \right)}&{U+U^{*}} \end{array} \right)=C^{-1}\left(\begin{array}{cc} U&0\\ 0&{U^{*}} \end{array} \right)C \end{array}$

  $A=\left\{ {A=\left(\begin{array}{cc} {A_0 +B_0 }&0\\ 0&{A_0 -B_0 } \end{array}\right)=\left(\! \begin{array}{ll} diag\big [ {e^{\tau _1}}\; \cdots \; {e^{\tau _n }} \big ]&0\\ 0&diag\big [{e^{-\tau _1}}\; \cdots \;{e^{-\tau _n }}\big ] \end{array}\!\right)} \right\}$

  $N=\left\{ N/N=\left\{ \begin{array}{ll} I&S\\ 0&I \end{array} \right\} , S\;{\text{ real} \text{ matrix} \text{ of} \text{ order} \text{ n}} \right\}$

  $h(A)=\left(\begin{array}{cc} {A_0 }&{B_0 } \\ {B_0 }&{A_0 } \end{array} \right), h(N)=\left(\begin{array}{cc} {I+i/2 \cdot S}&{-i/2\cdot S}\\ {i/2 \cdot S}&{I-i/2\cdot S} \end{array} \right) \\ \Rightarrow h(KAN)=\left(\begin{array}{cc} A_{1}&B_{1}\\ B_{1}^{*}&A_{1}^{*} \end{array}\right)$

  $\text{ with}\;\left\{ \begin{array}{l} A_1 =U\left[{A_0 +i\left({A_0 +B_0 } \right)\frac{1}{2}S} \right]\\ B_1 =U\left[{B_0 -i\left({A_0 +B_0 } \right)\frac{1}{2}S} \right] \end{array} \right.$

• Iwasawa/Cartan Coordinates on Siegel Unit Disk (Iwasawa/Cartan Coordinates Relation in Siegel Disk)

  $M_S =\left(\begin{array}{cc} {I+i/2\cdot S}&{-i/2\cdot S}\\ {i/2\cdot S}&{I-i/2\cdot S} \end{array}\right),\left(\begin{array}{cc} {A_0 }&{B_0}\\ {B_0 }&{A_0} \end{array}\right)M_S =M_{\tilde{S}} \left(\begin{array}{cc} {A_0 }&{B_0}\\ {B_0 }&{A_0} \end{array}\right)$

  $\text{ with}\;\left({A_0 +B_0 } \right)S=\tilde{S}\left({A_0 -B_0 } \right)$

  $g=\left(\begin{array}{ll} A&B\\ {B^{*}}&{A^{*}} \end{array}\right)\Rightarrow \left\{ \begin{array}{l}\text{ Cartan:}\;\left\{ \begin{array}{l} A=U^{t}A_0 V^{*}\\ B=U^{t}B_0 V\\ \end{array}\right. \\ \text{ Iwasawa:}\;\left\{ \begin{array}{l} A=U_1 \left[{A_0 +i\left({A_0 +B_0 } \right)\frac{1}{2}S} \right]\\ B=U_1 \left[{B_0 -i\left({A_0 +B_0 } \right)\frac{1}{2}S} \right]\\ \end{array}\right.\end{array}\right.$

  $\begin{array}{l}Z=B\left({A^{*}} \right)^{-1}=U_1 HU_1^t =U^{t}PU\\ \text{ with}\;H=\left[{B_0 \left({A_0 +B_0 } \right)-\frac{i}{2}\tilde{S}} \right]\left[{A_0 \left({A_0 +B_0 } \right)-\frac{i}{2}\tilde{S}} \right]^{-1} \end{array}$

 

“Il est clair que si l’on parvenait à démontrer que tous les domaines homogènes dont la forme $\varPhi = \sum \nolimits _{i,j} \frac{\partial ^{2} \log K({z,\bar{z}})}{\partial z_{i} \partial \bar{z}_{j}}$ est définie positive sont symétriques, toute la théorie des domaines bornés homogènes serait élucidée. C’est là un problème de géométrie hermitienne certainement très intéressant”

Last sentence in Elie Cartan, “Sur les domaines bornés de l’espace de n variables complexes”, Abh. Math. Seminar, Hamburg, 1935