Geometry of Structured …indico.ictp.it/event/a12193/session/33/contribution/27/...Geometry of Structured Matrices based on Information, Koszul and Contact Geometries F. Barbaresco

www.thalesgroup.com

Geometry of Structured Matrices based on

Information, Koszul and Contact Geometries

F. Barbaresco

2 /2 / Outlines

General Framework of Matrix Geomety: Information/Co ntact & Koszul Geometries

Probability in Metric Spaces by Fréchet

Metric Geometry of Hermitian Positive Definites Mat rices (HPD)

View point of geometers: Cartan-Siegel Symmetric Bounded Domains

View point of probabilists: Information Geometry and Fisher Information metric

Karcher Flows for HPD matrices

Basic Karcher flow to compute « Mean » Barycenter of N HPD matrices

Weiszfeld-Karcher flow to compute « Median » Barycenter of N HPD matrices

Karcher Flows for Toeplitz HPD matrices (THPD)

Trench/Verblunsky Theorems: Partial Iwasawa Decomposition and CAR model

Karcher Flow for Hessian metric/Information Geometry metric of Complex Autoregressive Model

New « Ordered Statistic High Doppler Resolution CFAR » (OS-HDR-CFAR) OS-HDR-CFAR Processing Chain & results on real recorder data

Karcher Flows for Toeplitz-Block-Toeplitz HPD matrice s (TBTHPD) Matrix Extension of Trench/Verblunsky Theorem: Multivariate CAR parametrization

Karcher Flow in Siegel Disk with Mostow Decomposition/Berger Fibration

New « Ordered Statistic Space-Time Adaptive Processing » (OS-STAP)

Miscellaneous: Quantization of Complex Symmetric Sp aces, Analysis on Symmetric Cone, Homogeneous Hyperbolic Affine Hyper spheres

www.thalesgroup.com

Introduction

4 /4 / Geometric Science of Information

Takeshi SASAKIW. BLASCHKE

Eugenio CALABI

Calyampudi R. RAONikolai N. CHENTSOV

Hirohiko SHIMAJean-Louis KOSZUL

Von Thomas FRIEDRICHY. SHISHIDO

HomogeneousConvex Cones G.

E.B. VINBERGJean-Louis KOSZUL

HomogeneousSymmetric Bounded

Domains G.

Elie CARTANCarl Ludwig SIEGEL

Probability in Metric Space

Maurice R. FRECHET

Information Theory Nicolas .L. BRILLOUINClaude. SHANNON

Probability on Riemannian Manifold

Michel EMERYMarc ARNAUDON

GeometricScience of Information

KOSZUL-VINBERG METRIC

(KOSZUL-VINBERG CHARACTERISTIC

FUNCTION)

FISHER METRIC

Probability/G. on structures

Y. OLLIVIERM. GROMOV

Contact G.Vladimir ARNOLD

5 /5 /

Thales Air Systems Date

Probability on Riemannian Manifold

M. Arnaudon, L. Miclo, ” Means in complete manifolds: uniqueness and approximation ”, http://arxiv.org/abs/1207.3232M. Arnaudon, C. Dombry, A. Phan, Le Yang, « Stochast ic algorithms for computing means of probability measu res. », Stochastic Processes and their Applications122, pp. 1437-1455, 2012M. Arnaudon, A. Thalmaier, “Brownian motion and nega tive curvature”, Boundaries and Spectra of Random Walks, Progress in Probability, Vol. 64, 145--163, Springer Basel , 2011M. Arnaudon, F. Barbaresco, Le Yang, ” Medians and means in Riemannian geometry: existence, uniqueness and comp utation ”Matrix Information Geometry, Nielsen, Frank; Bhatia , Rajendra (Eds.), Springer, http://arxiv.org/pdf/1111.3120v1Le Yang, « Médianes de mesures de probabilité dans l es variétés riemanniennes et applications à la détecti on de cibles radar », PhD with advisors M. arnaudon & F. Barbares cohttp://tel.archives-ouvertes.fr/docs/00/66/41/88/PD F/Dissertation-Le_YANG.pdf , THALES PhD Award 2012

Marc ArnaudonPhD with M. Emery, Bordeaux University

P-Means Computation on Riemannian ManifoldStochastic Flow on Riemannian Manifold

Michel EmeryIRMA Lab, Strasbourg University

Probability on Riemannian ManifoldStochastc Calculus on Manifolds

M. Émery, G. Mokobodzki, « Sur le barycentre d'une probabilité dans une variété », Séminaire de probabilités de Strasbourg, 25 , p. 220-233, 1991http://archive.numdam.org/article/SPS_1991__25__220_0.pdfM. Émery, P.A. Meyer, « Stochastic calculus in manifolds », Springer 1989M. Émery, W. Zheng, « Fonctions convexes et semimartingales dans une variété », Séminaire de Probabilités XVIII, Lecture Notes in Mathematics 10 59, Springer 1984M. Fréchet, « L’intégrale abstraite d’une fonction abstraite et son application à la moyenne d’un élément aléatoire de nature quelconque », Revue Scientifique, 483-512, 1944M. Fréchet, « Les éléments aléatoires de nature quelconque dans un espace distancié ». Annales IHP, 10 no. 4, p. 215-310, 1948 http://archive.numdam.org/article/AIHP_1948__10_4_215_0.pdf

[ ] functionconvex continuous , )()( ϕϕϕ xEx ≤

[ ] ( )( ) 0)(exp)( 1 =⇒= ∫− dxPxgxgEb

M

b

6 /6 /


Probability on structures: Gromov/Ollivier work on Fisher Metric

Y. Ollivier, « Probabilités sur les espaces de confi guration d'origine géométrique », PhD with advisers M. Gromov & P . Pansuhttp://www.yann-ollivier.org/rech/publs/these.pdfY. Ollivier, « Le Hasard et la Courbure(Randomness and Curvature) », habilitation:http://www.yann-ollivier.org/rech/publs/hdr_intro.p df

Y. Ollivier & Youhei Akimoto, « Objective improvement in information-geometric optimization », FOGA 2013, preprint2012http://www.yann-ollivier.org/rech/publs/deeptrain.p dfYann Ollivier, Ludovic Arnold, Anne Auger, and Niko lausHansen, « Information-geometric optimization : A unifyingpicture via invariance principles », arXiv:1106.3708v 1, 2011http://www.yann-ollivier.org/rech/publs/quantile_ig o.pdfVideo « seminaire Brillouin »: http://archiprod-externe.ircam.fr/video/VI02023900-226.mp4

Yann Ollivier , Paris-Sud University, LRI Dept.CNRS Bronze Medal 2011

Introduction of probabilistic models on structured objectsInterplay between probability and differential geom etryNatural Gradient by Fisher Information Matrix (IGO)

Misha Gromov , IHESAbel Prize 2009

Mathematics about "interesting structures“Category-theoretic approach of Fisher MetricM. Gromov, « In a Search for a Structure, Part 1: O n Entropy », preprint, July 2012http://www.ihes.fr/~gromov/PDF/structre-serch-entropy-july5-2012.pdfChap 2. « Fisher Metric and Von Neumann Entropy»Such a rescaling, being a non-trivial symmetry, is a significant structure in its own right; for example , the group of families of such "rescalings" leads th e amazing orthogonal symmetry of the Fisher metric

M. Gromov, « Convex sets and Kähler manifolds »,in Advances in J. Differential Geom., F. Tricerri e d., World Sci., Singapore, pp. 1-38, 1990http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf

[ ] ∑=+=

∂∂

∂∂=

∂∂∂−=

jijiij

jijiij

ddgdxpxpKds

xpxpE

xpEg

,

2

2

),(),,(

),(log),(log),(log

θθθθθ

θθ

θθ

θθθ

7 /7 /


Koszul forms, characteristic function & metric: Hessian structure

H. Shima, “The Geometry of Hessian Structures”, World Scientific, 2007http://www.worldscientific.com/worldscibooks/10.114 2/6241dedicated to Prof. Jean-Louis Koszul (« the content o f the present book finds their origin in his studies »)H. Shima,Symmetric spaces with invariant locally He ssian structures, J. Math. Soc. Japan,, pp. 581-589., 197 7H. Shima, « Homogeneous Hessian manifolds », Ann. Ins t. Fourier, Grenoble, pp. 91-128., 1980H. Shima, « Vanishing theorems for compact Hessian manifolds », Ann. Inst. Fourier, Grenoble, pp.183-20 5., 1986H. Shima, « Harmonicity of gradient mappings of leve l surfaces in a real a ffiffiffiffine space », Geometriae Dedicata, pp. 177-184., 1995H. Shima, « Hessian manifolds of constant Hessian se ctional curvature », J. Math. Soc. Japan, pp. 735-753., 1995H. Shima, « Homogeneous spaces with invariant projec tively flat affiffiffiffine connections », Trans. Amer. Math. Soc., pp. 4713- 4726, 1999

Hirohiko Shima , Emeritus Professor of Yamaguchi Univ.PhD from Osaka University

Interplay between the Geometry of Hessian Structure s and Information Geometry

Jean-Louis Koszul , French Sciences AcademyPhD student of Henri Cartan, Bourbaki memberIntroduction of Koszul forms, Koszul-Vinbergcharacteristic function & metricJ.L. Koszul, « Sur la forme hermitienne canonique des espaces homogènes », complexes, Canad. J. Math. 7, pp. 562-576., 1955J.L. Koszul, « Domaines bornées homogènes et orbites de groupes de transformations affines », Bul l. Soc. Math. France 89, pp. 515-533., 1961J.L. Koszul, « Ouverts convexes homogènes des espaces affines », Math. Z. 79, pp. 254-259., 1962J.L. Koszul, « Variétés localement plates et convexité », Osaka J. Maht. 2, pp. 285-290., 1965J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968See: M. N. Boyom, « Convexité locale dans l’espace des connexions symétriques. Critère de comparaison des modèles statistiques »,, March 2012, IHP, Parishttp://www.ceremade.dauphine.fr/~peyre/mspc/mspc-thales-12/« Les connexions symétriques est un sous-ensemble co nvexe contenant le sous-ensemble des connexions localemen t plates »

( ) ( ) ψψψ

ψψ

logdetlogloglog

0log,)(

connectionflat cones, dual and ,

*

**,

*

dsdsd

Ddgdxex

Dxxx

=−=

>==

ΩΩΩ∈

∫Ω

−

o

Koszul-VinbergCharacteristic Function &

metric of regularconvex cone ΩΩΩΩ

8 /8 / The Geometry of Hessian Structures

Hessian Geometry and J.L. Koszul Works

Hirohiko Shima Book, « Geometry of Hessian Structures », world Scientific Publishing 2007, dedicated to Jean-Louis Koszul J.L. Koszul

9 /9 / The Geometry of Hessian Structures, Hirohiko Shima

H. Shima, “The Geometry of Hessian Structures”, Wor ld Scientific, 2007http://www.worldscientific.com/worldscibooks/10.114 2/6241dedicated to Prof. Jean-Louis Koszul (« the content of the present book finds their origin in his studi es »)

A pair (D; g) of a flat connection D and a Hessian metric g is called a Hessian structure. J.L. Koszul studied a flat manifold endowed with a closed 1-form αααα such that Dαααα is positive definite,

whereupon Dαααα is a Hessian metric. This is the ultimate origin of the notion of Hessian structures

A Hessian structure (D; g) is said to be of Koszul type , if there exists a closed 1-form α α α α such that g = Dαααα

The second Koszul form ββββ plays an important role similar to the Ricci tensor for a Kählerian metric

10 /10 / GSI’13 Geometric Sciences of Information, Ecole des Mines de Paris

Special sessions

Computational Information Geometry

Hessian Information Geometry

Symplectic Information Geometry

Optimization on Matrix Manifolds

Probability on Manifolds

Divergence Geometry & Ancillarity

Machine/Manifold/Topology Learning

Optimal Transport Geometry

Mathematical Morphology for Tensor/Matrix-Valued Im ages

Geometry of Audio Processing

Shape Spaces: Geometry and Statistic

Geometry of Shape Variability

Geometric Inverse Problems: Filtering/Estimation on Manifolds

Differential Geometry in Signal Processing

Relational Metric

Finite Metric Spaces

http://www.gsi2013.com

28-30th

August 2013 Paris


Yann OLLIVIER (Paris-Sud University, France ): « Information-geometric optimization: The interest of information theory for discrete and continuous optimization »

Hirohiko SHIMA (Yamaguchi University, Japan ): « Geometry of Hessian Structures »

Giovanni PISTONE (Collegio Carlo Alberto, Italy ): « Non-Parametric Information Geometry »


13 /13 / Séminaire Léon Nicolas Brillouin

Corpus thématique : « Science géométrique de l’informationScience géométrique de l’informationScience géométrique de l’informationScience géométrique de l’information »

Laboratoire d’accueil : IRCAM, Salle Igor Stravinsky

Projet INRIA/CNRS/Ircam MuSync (Arshia Cont)

Animation : Arshia Cont (IRCAM & INRIA), Frank Nielse n (Sony Research), F. Barbaresco (Thales)

Les laboratoires IRCAM (Arshia Cont, Gerard Assayag, Arnaud Dessein)

Polytechnique (Olivier Schwanger,F.Nielsen)

Mines ParisTech (Silvere Bonnabel, Jesus Angulo)

UTT Troyes (Hichem Snoussi)

Univ. Poitiers (Marc Arnaudon, Le Yang)

Univ. Montpellier (Michel Boyom, Paul Bryand)

INRIA (Xavier Pennec, Arshia cont)

Thales (Frédéric Barbaresco, Alexis Decurninge)

Sony Resarch (Frank Nielsen)

Site web: http://repmus.ircam.fr/brillouin/past-events

http://repmus.ircam.fr/brillouin/home

14 /14 / French-Indian CEFIPRA « Matrix Information Geometry» Thales Research & Technology and Ecole Polytechnique

With Prof. Rajendra Bhatia (New Dehli Indian Instit ut of Statistics)

15 /15 / SPRINGER BOOK: « MATRIX INFORMATION GEOMETRY » (440 pages)

http://www.informationgeometry.org/MIG/MIGBOOKWEB/

16 /16 / « La Messe est dite » by Marcel Berger

• Premier Miracle :

La théorie des espaces symétriques peut être consid érée comme le premier miracle de la géométrie

riemannienne, en fait comme un nœud de forte densit é dans l’arbre de toutes les mathématiques …

On doit à Elie Cartan dans les années 1926 d’avoir découvert que ces géométries sont , dans une

dimension donnée, en nombre fini, et en outre toute s classées.

• Second Miracle :

Entre les variétés localement symétriques et les va riétés riemanniennes générales, il existe une

catégorie intermédiaire, celle des variétés kähléri ennes. … On a alors affaire pour décrire le panorama

des métriques kählériennes sur notre variété, non p as à un espace de formes différentielles

quadratiques, très lourd, mais à un espace vectorie l de fonctions numériques [le potentiel de Kähler].

… La richesse Kählérienne fait dire à certains que l a géométrie kählérienne est plus importante que la

géométrie riemannienne.

• Pas d’espoir d’autre miracle :

Ne cherchez pas d’autres miracles du genre des espac es (localement) symétriques et des variétés

kählériennes. En effet, c’est un fait depuis 1953 q ue les seules variétés riemanniennes irréductibles

qui admettent un invariant par transport parallèle autre que g elle-même (et sa forme volume) sont les

espaces localement symétriques, les variétés kählér iennes, les variétés kählérienne de Calabi-Yau, et

les variétés hyperkählériennes.

Marcel Berger (IHES), « 150 ans de Géométrie Riemann ienne »,

Géométrie au 20 ième siècle, Histoire et horizons, Hermann

Éditeur, 2005

Read More : Marcel Berger, « A Panoramic View of Riemannian Geom etry », Springer 2003

M. Berger

Berger, Marcel, « Les espaces symétriques noncompact s ». Annales scientifiques de l'École Normale Supérieure, Sér. 3, 74 no. 2 (1957), p. 85-177

www.thalesgroup.com

General Framework of Matrix Geometry: Information/Contact Geometries & Koszul Geometry

18 /18 / DUALITY Concept in Mathematic/Mechanic/Thermodynamic

(projective) LEGENDRE DUALITYLEGENDRE TRANSFORM

(between Dual Space)

CONTACT/SYMPLECTICGEOMETRIES

(Legendre mapping, fibration,…)

(Analytic) FOURIER DUALITYFOURIER TRANSFORM

(Time-/Frequency Dual Spaces)

LINEAR ALGEBRA(Linear Signal Processing)

)(,)()( * xyxMaxArgyyLy

Ψ−=Ψ=Φ

Duality of Lagrangian/Hamiltonian in Physics

Massieu-Duhem Dual Potentials in Thermodynamics

Dual Potentials in Information Geometry

Legendre Transform of minus logarithm of

characteristic function (Fourier transform) =

ENTROPY !!!

Ψ=Φ−= 22 log ddgINFORMATION GEOMETRY METRIC Sddg 2*2* =Ψ=

∫Ω

−−=Φ−=Ψ*

,log)(log)( dyexx yx

)(,)( *** xxxx Ψ−=Ψ

ξξξ dpp xx∫Ω

−=Ψ*

)(log)(*

Φ(x)x,ξ

Ω

ξ,xξ,xx edξeep

*

+−−− == ∫/)(ξ ∫Ω

=*

)(.* ξξξ dpx x

( )),,(.),,( tqqLqpSuptqpHq

&&&

−=( ) STESk

k −==−∂∂

/)( with )()(

. ρφρφρρφρ

19 /19 / Koszul-Vinberg Characteristic Function/Metric of convex cone

J.L. Koszul [and E. Vinberg have introduced an affinel y invariant Hessian metric on a sharp convex cone through its c haracteristic function.

is a sharp open convex cone in a vector space of finite dimension on (a convex cone is sharp if it does n ot contain any full straight line).

is the dual cone of and is a sharp open conv ex cone.

Let the Lebesgue measure on dual spa ce of , the following integral:

is called the Koszul-Vinberg characteristic function

Ω E R

*Ω Ωξd *E E

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ


is analytic function defined on the interior of and

as

If then

is logarithmically strictly convex, the function is strictly convex

Koszul 1-form αααα: The differential 1-form

is invariant by . If and then

and

Koszul 2-form ββββ : The symmetric differential 2-form

is a positive definite symmetric bilinear form on invariant under

(from Schwarz inequality and )

Koszul-Vinberg Metric: defines a Riemanian s tructure invariant by and then the Riemanian metric

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ

Ωψ +∞→Ω )(xψΩΩ∂→x( )Ω∈ Autg ( ) )(det

1xggx Ω

−Ω = ψψ

Ωψ ( ))(log)( xx ΩΩ = ψφ

ΩΩΩΩ === ψψψφα /log ddd( )Ω= AutG Ω∈x Eu∈

∫Ω

−−=*

,.,, ξξα ξ deuu xx

*Ω−∈xα

Ω== ψαβ log2dDE ( )Ω= AutG

( ) ∫Ω

−Ω =

*

,2 ,,,log ξξξψ ξ devuvud u

αD ( )ΩAut

Ω= ψlog2dg


Koszul-Vinberg Metric :

We can define a diffeomorphism by:

with

When the cone is symmetric, the map is a biject ion and an isometry with a unique fixed point (the manifold is a Riemannian Symmetric Space given by this isometry):

, and

is characterized by

is the center of gravity of the cross section of :

Ω= ψlog2dg

[ ] ( )

loglog

2

1loglog)(log

2u

2

22

∫∫∫∫

∫∫

∫−

+==dudv

dudvdd

du

duddudxd

vu

vuv

u

uu

u ψψ

ψψψψ

ψ

ψψψψ

)(log* xdx x Ω−=−= ψα

)()(),(0

tuxfdt

dxfDuxdf

tu +==

=

Ω xx α−=*

xx =** )( nxx =*, cstexx =ΩΩ )()( *

*ψψ

*x nyxyyx =Ω∈= ,,/)(minarg ** ψ

*x nyxy =Ω∈ ,,* *Ω

∫∫Ω

−

Ω

−=**

,,* /. ξξξ ξξ dedex xx

22 /22 / Koszul Entropy via Legendre Transform

we can deduce “Koszul Entropy” defined as Legendre T ransform of minus logarithm of Koszul-Vinberg characteristic function :

with and where

Demonstration: we set

Using

and

we can write:

and

)(,)( *** xxxx Φ−=Φ Φ= xDx* ** Φ=

xDx

)(log)( xx Ω−=Φ ψ

Ω∈∀= ∫Ω

−Ω xdex x )(

*

, ξψ ξ

∫∫Ω

−

Ω

−=**

,,* /. ξξξ ξξ dedex xx

∫∫Ω

−

Ω

−Ω −==−

**

,,* /,)(log, ξξξψ ξξ dedehxdhx xxh

∫∫Ω

−

Ω

−−=−**

,,,* /.log, ξξ ξξξ dedeexx xxx

∫∫∫∫

∫∫∫

Ω

−

Ω

−−

Ω

−

Ω

−

Ω

−

Ω

−

Ω

−−

−

=Φ

+−=Φ

****

***

,,,,,**

,,,,**

/.loglog.)(

log/.log)(

ξξξξ

ξξξ

ξξξξξ

ξξξξ

dedeededex

dededeex

xxxxx

xxxx

23 /23 / Koszul Entropy via Legendre Transform

We can then consider this Legendre transform as an entropy, that we could named “ Koszul Entropy ”:

With

and

∫∫∫∫Ω

−

Ω

−−

Ω

−

Ω

−

−

=Φ

****

,,,,,** /.loglog.)( ξξξξ ξξξξξ dedeededex xxxxx

ξξξξξξ ξ

ξ

ξ

ξ

dppdde

e

de

exxx

x

x

x

∫∫

∫∫ Ω

Ω

−

−

ΩΩ

−

−

−=−=Φ*

*

*

*

)(log)(log,

,

,

,*

Φ(x)x,ξdξex,ξ

Ω

ξ,xξ,xx eedξeep *Ω

ξ,x

*

+−∫−−

−− ===−

∫log

/)(ξ

( )ξξξξξξ dedξ.edpΦDx

ξΦ

Ω

Φ(x)x,ξxx

*

*

−

Ω

+−

Ω∫∫∫ ====

**

.)(.*

24 /24 / Sasaki Works on Affine Geometry

If we denote by the level surface of :

which is a non-compact submanifold in , and by

the induced metric of on , then assuming th at

the cone is homogeneous under , Sasaki proved tha t

is a homogeneous hyperbolic affine hypersphere and e very

such hyperspheres can be obtained in this way .

Sasaki also remarks that is identified with the a ffine metric

and is a global Riemannian symmetric space w hen

is a self-dual cone.

Let be a regular convex cone and let be th e

canonical Hessian metric, then each level surface o f the

characteristic function is a minimal surface of the

Riemannian manifold .

cS Ωψ cxSc == Ω )(ψΩ cω

Ωψlog2d cS

Ω )(ΩG

cωcS Ω

Ω= ψlog2dgΩ

Ωψ),( gΩ


Let v be the volume element of g. We define a closed 1-form αααα and ββββ a symmetric bilinear form by:

and

The forms αααα and ββββ are called the first Koszul form and the second Kos zul form for a Hessian structure (D; g) :

A pair (D; g) of a flat connection D and a Hessian metric g is called a Hessian structure.

J.L. Koszul studied a flat manifold endowed with a c losed 1-form such that

is positive definite, whereupon is a Hess ian metric.

This is the ultimate origin of the notion of Hessia n structures. A Hessian structure (D; g) is said to be of Koszul type, if there exists a clos ed 1-form such that .

vXvDX )(α= αβ D=

[ ]( )[ ]( )

[ ]

∂∂∂=

∂∂=

∂∂=

⇒∧=

jikl

ji

ij

ijiin/

ij

xx

g

x

vgx

α

..dxdxgvdetlog

2

1

detlogdet

2

2

1

121

αβ

ααD αD

ααDg =


We can apply this Koszul theory for Symmetric Posit ive Definite Matrices.

Let the inner product

be the set of symmetric positive definite matrices is an open convex cone and is self-dual :

Let be the regular convex cone consisting of all p ositive definite symmetric matrices of degree n. Then is a Hessian structure on , and each level surface of is a minimal surface of the Riemannian manifold

( ) )(,,, RSymyxxyTryx n∈∀=Ω

Ω=Ω*

)(det)( 2

1

dual-self )(,

,

**

n

n

xyTryx

x Ixdex ψξψ ξ+−

Ω=Ω=

Ω

−Ω == ∫

xdn

dg detlog2

1log 22 +−== Ωψ

1*

2

1detlog

2

1log −

Ω+=+=−= x

nxd

ndx ψ

)detlog,( xDdD Ωxdet

)detlog,( xDdg −=Ω

27 /27 / Koszul-Vey Theorem

J.L. Koszul and J. Vey have proved the following theo rem: Koszul J.L., Variétés localement plates et convexité, Osaka J. Math. , n°2, p.285-290, 1965

Vey J., Sur les automorphismes affines des ouverts convexes saillants, Annali della ScuolaNormale Superiore di Pisa, Classe di Science, 3e série, tome 24,n°4, p.641-665, 1970

Koszul-Vey Theorem:

Let be a connected Hessian manifold with He ssian metric .

Suppose that admits a closed 1-form such tha t and

there exists a group of affine automorphisms of preserving :

- If is quasi- compact, then the universal covering manifold of

is affinely isomorphic to a convex domain rea l affine space not

containing any full straight line.

- If is compact, then is a s harp convex cone.

M gα gD =α

G M αGM / M

ΩGM /

Ω

[] Koszul J.L., Variétés localement plates et conve xité, Osaka J. Math. , n °2, p.285-290, 1965[] Vey J., Sur les automorphismes affines des ouver ts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tom e 24,n°4, p.641-665, 1970

28 /28 / Characteristic function for probability measure

Let µµµµ be a positive Borel Measure on euclidean space V. Assume thatthe following integral is finite for all x in an open set :

P. Levy has made a systematic use of characteristic function, and he claimed that he was influenced by 1912, 2nd edition of Poincaré’s book “Calcul des probabilités” where Poincaré designated for the first time the term “fonction caractéristique”. Poincaré used char acteristic function, for discuting exceptions to the Gaussian l aw. My opinion is that Poincaré used this name because of Massieu work used by him.

For the Koszul-Vinberg characteristic function, if we replace the Lebesgue measure by Borel measure, then we recover th e classical definition of characteristic function in Probabilit y, and the previous KVCF could be compared by analogy with:

V⊂Ω

Ω∈x ∫−= )()( , xdey xy

x µψ ( ) )()(

1, , xde

ydxyp xy

x

µψ

−=

)(log),()( ydxyxpym xψ−∇== ∫)(log),(),(),(,)( yDDdxypvymxuymxvuyV xvu ψ=−−= ∫

[ ]izxizxizxX eEdxxpexdFez === ∫∫

+∞

∞−

+∞

∞−

).()()(ψ

29 /29 / Laplace Principle of Large Deviation

Large deviation principle : Let be a sequence of random variables indexed by the positive integer n, and le t

denote the probability measure

associated with these random variables. We say that or

satisfies a large deviation principle if the limit

exists.

The function defined by this limit is ca lled the rate function;

the parameter n of decay is called in large deviation theory the

speed. The existence of a large deviation principle for means

concretely that the dominant behavior of is a

decaying exponential with n, with rate :

or

nΣ

( ) [ ]( )ςςςς dPdP nn +∈Σ=∈Σ ,

nΣ( )ςdP n ∈Σ

( ) ( )ςς dPn

LimI nn

∈Σ−=∞→

log1

( )ςI

nΣ( )ςdP n ∈Σ

( )ςI( ) ( ) ςς ς dedP In

n.−≈∈Σ ( ) ( )ςς In

n ep .−≈=Σ


If we write the Legendre-Fenchel transform of a func tion

defined by , we can express the following

principle and Laplace’s method:

Laplace Principle : Let be a sequence of

probability spaces, a complete separable metric s pace,

a sequence of random variables such that

maps into , and a rate function on . Then,

satisfies the Laplace principle on with rate func tion

if for all bounded, continuous functions mapping into R:

)(xh )(.)( xhxkSupkg

x−=

( ) NnPF nnn ∈Ω ,,,Θ

NnYn ∈, nY

nΩ Θ I Θ nYΘ I

f

( )[ ] )()(log1 . xIxfSupeEn

Limx

YfnP

n

n

n−=

Θ∈∞→


Laplace Principle :

Proof:

If satisfies the large deviation principle on with rate function

, then :

The asymptotic behavior of the last integral is det ermined by the largest value of the integrand :


Limx

YfnP

n

n

n−=

Θ∈∞→

nY Θ I( ) dxedxYP xIn

nn)(.−≈∈

( )[ ] ( ) ( )

[ ]∫∫

∫∫

Θ

−

∞→Θ

−

∞→

Θ∞→

Ω∞→∞→

=≈

∈==

dxen

Limdxeen

Lim

dxYPen

LimdPen

LimeEn

Lim

xIxfn

n

xInxfn

n

nnxfn

nn

Ynf

n

YfnP

nn

nn

n

.log1

.log1

log1

.log1

log1

)()(.)(.)(.

)(..

( )[ ] [ ] )()(log1

log1 )()(.. xIxfSupeSup

neE

nLim

x

xIxfn

x

YfnP

n

n

n−=

=

Θ∈

−

Θ∈∞→

Laplace P.S., Mémoire sur la probabilité des causes sur les évènements. Mémoires de Mathématique et de Physique, Tome Sixiè me, 1774


The generating function of is defined as:

In terms of the density , we have instead

In both expressions, the integral is over the domai n of .

The function defined by the limit:

is called the scaled cumulant generating function or the log-

generating function of .

The existence of this limit is equivalent to writin g


Limx

YfnP

n

n

n−=

Θ∈∞→

nΣ

( ) [ ] ( )∫ ∈Σ== Σ ςψ ς dPeeEx nnxnx

nn

( )np Σ( ) ( )∫ =Σ= ςςψ ς dpex n

nxn

nΣ( )xφ

( ) )(log1

xn

Limx nn

ψφ∞→

=

nΣ( ) )(. xn

n ex φψ ≈


Gärtner-Ellis Theorem:

If is differentiable, then satisfies a large deviation principle

with rate function by the Legendre-Fenchel transfo rm of :

Varadhan's Theorem:

If satisfies a large deviation principle wi th rate function ,

then its scaled cumulant generating function is th e Legendre-

Fenchel transform of :

Properties of : statistical moments of are given by derivatives

of :

)(xφ nΣ

nΣnΣ

( )ςI )(xφ )(.)( xxSupI

xφςς −=

nΣ ( )ςI)(xφ

( )ςI

)(.)( ςςφς

IxSupx −=

)(xφ nΣ)(xφ

[ ]nn

x

ELimdx

xd Σ=∞→

=0

)(φ [ ]nn

x

VarLimdx

xd Σ=∞→

=02

2 )(φ

34 /34 / Legendre Duality in Thermodynamic

In 1869, François Jacques Dominique MASSIEU, French Engineer from Corps des Mines, has presented two papers to French Science Academy on « Characteristic function » in Thermodynamic.

Massieu has demonstrated that some mechanical and t hermal properties of physical and chemical systems could be derived f rom two potentials called “ characteristic functions ”.

The infinitesimal amount of heat received by a body produces external work of dilatation, internal work, and an increase of body sensible heat. The last two effects could not be id entified separately and are noted (function accounted for the s um of mechanical and thermal effects by equivalence between heat and wor k). The external work is thermally equivalen t to (with the conversion factor between mechanical and thermal me asures). The first principle provides . For a closed reversible cycle (Joule/Carnot principles) th at is the complete differential of a function of .

[] Massieu F., Thermodynamique: Mémoire sur les fon ctions caractéristiques des divers fluides et sur l a théorie des vapeurs, 92-pgs, Académie des Sciences, 1876[] Duhem P., Sur les équations générales de la Ther modynamique, Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 18 91[] Arnold V.I., Contact geometry: the geometrical m ethod of Gibbs’s thermodynamics, Pro-ceedings of th e Gibbs Symposium, p.163–179, Amer. Math. Soc., Provi dence, RI, 1990

dQ

dE E

dVP. dVPA .. A

dVPAdEdQ ..+=0/ =∫ TdQ dS

S TdQdS /=

35 /35 / Legendre Duality in Thermodynamic: Massieu Potentials

If we select volume V and temperature T as independent variables:

If we set , then we have

Massieu has called the “characteristic function ” because all

body characteristics could be deduced of this funct ion:

, and

If we select pression P and temperature T as indepen dent variables:

Massieu characteristic function is then given by

and we can deduce: and

and inner energy:

( ) dVPAdTSdETSddVPAdEdSTdQdST ....... +=−⇒=−⇒=ETSH −=

dVV

HdT

T

HdVPAdTSdH .....

∂∂+

∂∂=+=

H

T

HS

∂∂=

V

H

AP

∂∂= 1

HT

HTHTSE −

∂∂=−=

VAPHH .' −=dP

P

HdT

T

HdPAVdTSdPAVdVAPdHdH .

'.

'....'

∂∂+

∂∂=−=−−=

T

HS

∂∂= '

P

H

AV

∂∂−= '1

P

HPH

T

HTEVAPHTSHTSE

∂∂+−

∂∂=⇒−−=−= '

.''

.'

36 /36 / Legendre Duality in Thermodynamic: Massieu Potentials

Deriving all body properties dealing with thermodyn amics from Massieu characteristic function and its derivatives :

« je montre, dans ce mémoire, que toutes les propriétés d’un corps peuvent se déduire d’une fonction unique, que j’appelle la fonction caractéristique de cecorps»

In thermodynamics, the Massieu potential is the Lege ndre transform of the Entropy, and depends on :

where F is the Free Energy and inner energy

The Legendre transform of the Massieu potential gives Entropy

kT/1=ρTESFk /.)( −=−= ρρφ

TSEF −=( )

( )( )( ) ( ) ( )

∂∂

∂∂+=

∂∂+=

∂∂=

∂∂−

ρρ

ρρ

ρρ

ρρφ

k

T

T

FkF

k

FkF

k

Fk

k V

( )( ) ( )

ETSTSEk

SF

kSkF

k=+−=+=

−−+=

∂∂− )(

1)(

2 ρρρ

ρρφ

E

S( ) ( ) ( ) SEFkFkEk

kkL −=−=−−=−

∂∂= ρρρρφ

ρρφρφ .).()(

)(.

37 /37 /Legendre Duality in Thermodynamic: Duhem-Massieu Potentials

Pierre Duhem Thermodynamic Potentials

Duhem P., « Sur les équations générales de la Thermodynamique », Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 1891

“Nous avons fait de la Dynamique un cas particulier de la Thermodynamique, une Science qui embrasse dans des principes communs tous les changements d’état des corps, aussi bien les changements de lieu que les changements de qualités physiques “

four scientists were credited by Duhem with having carried out “the most important researches on that subject”:

F. Massieu had managed to derive Thermodynamics from a “characteristic function and its partial derivatives”

J.W. Gibbs had shown that Massieu’s functions “could play the role of potentials in the determination of the states of equilibrium” in a given system.

H. von Helmholtz had put forward “similar ideas”

A. von Oettingen had given “an exposition of Thermodynamics of remarkable generality” based on general duality concept in “Die thermodynamischen Beziehungen antithetisch entwickelt“, St. Petersbur g 1885

WTSEG +−=Ω )(

38 /38 / Duality of Information Geometry: Multivariate Gaussian Law

Dual Coordinates systems & Potential functions

Potential Functions are Dual and related by Legendre transformation :

( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

−−−+−=

+Θ−Θ=⇒

+Σ−==

==

−−−−

−−−−

−−

enΗ)(ηΗηΗΦ

nTrΘΨ

mmmη,ΗΗ

Σ)m,(Σθ,ΘΘ

T

T

T

ππθθ

2log2detlog21log2~~

)log(2detlog22~~

,~

2~

scoordinate Dual

1111

1112

11

=∂∂

=∂∂

=∂∂

=∂∂

ΘΗ

Φ

θη

Φ

ΗΘ

Ψ

ηθ

Ψ

~

~

and~

~

( )TT ΘΗTrH,Θ

ΨH,ΘΦ

+=

−≡

θη~~with

~~~~

( ) [ ]pEΗΦ log~~ = Entropy

ji

*ij

jiij

ΗΗ

Φg

ΘΘ

Ψg

∂∂∂≡

∂∂∂=

~ and

~ 22

Hessians are convexe and define Riemannian metrics :

39 /39 / Duality of Information Geometry: Multivariate Gaussian Law

Link with Kullback Divergence

For each Dual Geometry, we can built a divergence that is directly related to Kulback-Leibler Divergence :

( ) ( )[ ] ( ) ( )( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )( )dx

mxp

mxpmxp,Σm,N,ΣmNDiv

H,ΘΗΦΘΨ,Σm,N,ΣmNDiv


∫ ΣΣΣ=

≥−+≡

≥−+≡

22

11111122

21211122*

12121122

,/

,/log,/

0~~~~~~

0~~~~~~

Riemannian Metric As Potential are convexe, their Hessians define Riemannian Metrics :

( ) ( )( ) ( )( ) ( )( )( )( )( )

−−Σ+

Σ−ΣΣ+ΣΣ−=

−

−−−

TmmmmTr

Tr,Σm,N,ΣmNDiv

21211

2

11

121

121

1122

detlog

2

1

( ) ( )332

2

1

2

1iji

ijijiij dΗOdΗdΗgdΘOdΘdΘgds +=+=

40 /40 / Legendre Duality in Mechanics

In Mechanics, Legendre Duality gives the relation b etween:

the variational Euler-Lagrange

the symplectic Hamilton-Jacobi formulations

of the equations of motion

As described by Vladimir Arnold, in the general cas e, we can define the Hamiltonian H as the fiberwise Legendre transformation of the Lagrangian L:

Due to strict convexity,

supremum is reached in a unique point such that :

and

Young-Fenchel inequality. For all ,the following hol ds :

with equality if and only if


&&&

−=

),,(.),,( tqqLqptqpH && −=q&

),,( tqqLp q &&∂= ),,( tqpHq p∂=&

pqtq ,,, &

),,(),,(. tqpHtqqLqp +≤ &&

),,( tqqLp q &&∂=


If we consider total differential of Hamiltonian:

Euler-Lagrange equation

with and

provides the 2nd Hamilton equation

with

in Darboux coordinates.

∂+∂+∂=

∂−∂−=∂−∂−∂−+=

HdtHdqHdpdH

LdtLdqdpqLdtqLdLdqqpddpqdH

tqp

tqtqq &&&&&

∂=∂−

∂=⇒

HL

Hq

qq

p&

0=∂−∂∂ LL qqt &

Lp q&∂= HL qq ∂=∂−Hp q−∂=&

Hq p∂=&



&&&

−=

),,( tqqLp q &&∂= ),,( tqpHq p∂=&

43 /43 / Pfaffian Form and Poincaré-Cartan Integral Invariant

Considering Pfaffian form

related to Poincaré-Cartan integral invariant , based on:

and

we can deduce:

with

P. Dedecker has observed, that the property that amo ng all forms

the form is the onl y one

satisfying , is a parti cular case of more general T.

Lepage congruence related to transversality condition .

dtHdqp .. −=ω

( ) ϖω LdtLdtLqLdqL qqq &&&& ∂+=−∂−∂= ....

dtqdq .&−=ϖ

Lp q&∂= LqpH −= &.

ϖθ mod.dtL≡ dtHdqp .. −=ωϖθ mod0≡d

[] Cartan E., Leçons sur les invariants intégraux, Hermann, Paris, 1922[] Dedecker P., A property of differential forms in the calculus of variations, Pacific J. Math. Volum e 7, Number 4,p. 1545-1549, 1957[] Lepage T., Sur les champs géodésiques du calcul des variations, Bull. Acad. Roy. Belg., CL. Sci.27, p.716-729, pp. 1036-1046, 1936

44 /44 / Pfaffian Form and Poincaré-Cartan Integral Invariant

45 /45 / Legendre Transform and Contact Geometry

Legendre transform and contact geometry where used in Mechanic and in Thermodynamic.

Integral submanifolds of dimension n in 2n+1 dimens ional contact manifold are called Legendre submanifolds.

A smooth fibration of a contact manifold, all of wh ose are Legendre, is called a Legendre Fibration.

In the neighbourhood of each point of the total spa ce of a Legendre Fibration there exist contact Darboux coor dinates (z, q, p) in which the fibration is given by the projection (z, q, p) =>(z, q).

Indeed, the fibres (z, q) = cst are Legendre subspaces of the standard contact space.

A Legendre mapping is a diagram consisting of an em bedding of a smooth manifold as a Legendre submanifold in the to tal space of a Legendre fibration, and the projection of the total space of the Legendre fibration onto the base.

46 /46 / Legendre Transform and Contact Geometry

Let us consider the two Legendre fibrations of the s tandard contact space of 1 -jets of functions on :

and

the projection of the 1-graph of a function onto the base

of the second fibration gives a Legendre mapping:

If S is convex, the front of this mapping is the gr aph of a convex function, the Legendre transform of the function S:

12 +nR nR

),(),,( quqpu a

),.(),,( puqpqpu −a

)(qSu =

∂∂−

∂∂

q

SqS

q

Sqq ),(a

( )ppS ),(*

47 /47 / Legendre Duality & Contact Geometry

Symplectic geometry of even-dimensional phase space s has an odd-dimensional twin: contact geometry.

The relation between contact geometry and symplecti c geometry is similar to the relation between linear algebra and projective geometry. Any fact in symplectic geometry can be formulated a s a contact geometry fact and vice versa. The calculations are simpler i n the symplectic setting, but their geometric content is better seen in the c ontact version.

The functions and vector fields of symplectic geome try are replaced by hypersurfaces and line fields in contact geometry.

Each contact manifold has a symplectization, which i s a symplectic manifold whose dimension exceeds that of the contac t manifold by one.

Symplectic manifolds have contactizations whose dime nsions exceed their own dimensions by one.

If a manifold has a serious reason to be odd dimens ional it usually carries a natural contact structure.

one might now say ‘‘symplectic geometry is all geom etry,’’ but I prefer to formulate it in a more geometrical form: contact geometry is all geometry .

[] Arnold, V.I., Givental, A.G., Symplectic geometr y, Encyclopedia of mathematical science, vol. 4., Springer Verlag (translated from Russian)

48 /48 / Legendre Duality & Contact Geometry

Contact structures and Legendre submanifolds :

A contact structure on an odd-dimensional manifold M2n+1 is a field of hyperplanes(of linear subspaces of codimension 1) in the tangent spaces to M at all its points.

All the generic fields of hyperplanes of a manifold of a fixed dimension are locallyequivalent. They define the (local) contact structures.

Example : A 1-jet of a function at point x of

manifold Vn is defined by the point where

The natural contact structure of this space is defined b y the following

condition: the 1-graphs of all

the functions on V should be the tangent structure hyperplane at every

point. In coordinates, this conditions means that the 1-form

should vanish on the hyperplanes of the contact field.

),...,,( 21 nxxxfy =12),,( +∈ nRpyx ii xfp ∂∂= /

),(/),(, 1 RVJxfpxfyx n⊂∂∂==

dxpdy .−

pdVTdSdE −=

Gibbs contact structureIn Thermodynamics

49 /49 / Projective duality and Legendre transformation

A contact structure on a manifold is a nondegenerat e field of tangent hyperplanes

The manifold of contact elements in projective spac e coincides with the manifold of contact elements of the dual projec tive space

A contact element in projective space is a pair, consisting of a point of the space and of a hyperplane containing this point. The hyperplane is a point of the dual projective space and the point of the original space defines a contact element of the dual space.

The manifold of contact elements of the projective space has two natural contact structures:

The first is the natural contact structure of the manifold of contact elements of the original projective space.

The second is the natural contact structure of the manifold of contact elements of the dual projective space

The dual of the dual hypersurface is the initial hy persurface (at least if both are smooth for instance for the boundaries of convex bodies)

The affine or coordinate version of the projective duality is called the Legendre transformation . Thus contact geometry is the geometrical base of the theory of Legendre transformation .

50 /50 / Duality in Projective Geometry: Pascal’s Mystic Hexagram

In projective geometry, Pascal's theorem (the Hexag rammum Mysticum Theorem) states that:

if an arbitrary six points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon

The dual of Pascal’s Theorem is known as Brianchon’ s Theorem:

Pascal’s Theorem: If the vertices of a simple hexagon are points of a point conic, then its diagonal points are collinear.

Brianchon’s Theorem: If the sides of a simple hexagon are lines of a line conic, then the diagonal lines are concurrent.

51 /51 / History of Contact Geometry

Huygens Principle : the wave front Fqo(to + t) is the envelope of the fronts Fq(t)

1872, Sophus Lie: the notion of contact transformat ion (Beriihrungstransformation) as a geometric tool to study systems of differential equations.

19th century and the beginning of the 20th century, F. Engel, H. Poincaré, E. Goursat and E. Cartan

www.thalesgroup.com

Probability in Metric Spaces by Fréchet

53 /53 / Maurice René Fréchet: Probability in Metric Spaces

Maurice René Fréchet

1948 (Annales de l’IHP)Les éléments aléatoires de nature quelconque

dans un espace distanciéExtension of Probability/Statistic in abstract/Metric space

1928 Book from Fréchet PhDAbstract Spaces

Invention of Metric Space 1928

Jacques Hadamard

PhD Supervisor

54 /54 /« Mean » of structured data: Fréchet Barycenter in Metric Space

Right Triangle a,b,h

h2=a2+b2

a

b

h

2221 with ,, iii

N

iiii bahhba +==

( )HBAhbadMinHBA iii

N

i

pgeodesique

HBA,,,,,arg,,

1,,∑

=

=

a

b

h>90°

222 bah +>Inside the cone(bounded domain)

Ambligone Triangle

p=2 : Mean, p=1 Median

55 /55 / Toeplitz Hermitian PD Matrices: Simple case n=2

Definite PositiveHermitian Toeplitz

0det

0

222 bah

hiba

ibah

+>⇔>Ω

>

+−

=Ω

Hadamard Compactification

Disk Unit Poincaré :

1/

01

*1.

*

D

zzDh

iba

Rh

h

<=∈+=

∈

>

=Ω

+

µ

µµ

1/

)log(

with ),log(

<=∈∈

+=

zzD

Rhh

ibah

µ

µµScale

ParameterShape

Parameter

a

b

h>90°

AmbligoneTriangleh2>a2+b2

( )( )

222

22

2222

0

02

0det

bah

bah

bahh

hiba

ibah

+>⇔>+±=⇒

=+−+−⇒

=

−+−−

λλ

λλλ

λ

56 /56 / Short Summary of the approach

matrix HPD Toeplitz

+−

=Ωhiba

ibah

=Ω⇒

)log(

*)log(

.

.

,)log(

mean

mean

hmeanmean

meanmeanh

Mean

meanMean

eh

he

h

µµ

µ

DRh ×∈µ),log(

:cationCompactifi Hadamard

[ ] [ ]

−

=

−+

=+

µµµµ

µd

hd

d

hdd

h

dhds

)log(

1

10

01)log(

122

22

222

1/

01

1.

,zation parameteri New*

*

<∈=∈+=⇒

>

=Ω

×∈ +

zCzDh

iba

h

DRh

µ

µµ

µ

( )µµ

µ

µ),log(,),log(arg

,)log(

1),log(

hhdMin

h

ii

N

i

pgeodesique

h

meanMean

∑=

=

FRECHET Barycenter in Metric Space

57 /57 / Maurice Fréchet : Random Elements on Abstract Spaces

M. Fréchet, “les éléments aléatoires de nature quelconque dans un espace distancié”, Annales de l’IHP, t.10, n°4, p.215-310, 1948( voir sur Numdam)

Il constitue la mise au point de certains des cours que nous avons fait à la Sorbonne en 1945 et en 1946

Formidable extension du Calcul [des probabilités qui] résulte de la simple introduction de la notion de distance de deux éléments aléatoire s

La nature, la science et la technique offrent de no mbreux exemples d’éléments aléatoires qui ne sont, ni des nombres, ni des séries, ni des vecteurs, ni des fonctions. Telles sont par exemple, la forme d’un fil jeté au hasard sur une table, la forme d’un oeuf pris au hasard dans un panier d’oeufs . On a ainsi une courbe aléatoire, une surface aléatoire . On peut aussi considérer d’autres éléments mathématiqu es aléatoires : par exemple des transformations aléatoires de courbe en courbe .

Il paraît certain que l’urbanisme conduira à étudie r des éléments aléatoires tels que la forme d’une ville prise au hasard , vérifier par exemple, d’une manière scientifique l’hypothèse de la tendance au développement des vil les vers l’Ouest.

Parmi les premières notions à généraliser figurent celles de loi de probabilité, de valeurs typiques (moyenne, équiprobable, etc.), de dispersi on, de convergence stochastique, que nous allons d’abord examiner.

Nous indiquerons plus loin en Note additionnelle, u n moyen très général d’étendre la notion de fonction de répartition à des éléments al éatoires de nature quelconque.

Appartement ou vécu Fréchet

12 Square Desnouettes, 15ème, Paris

58 /58 / Maurice Fréchet : Random Elements on Abstract Space

Nous avons entrepris une étude expérimentale de que lques exemples particuliers. Ceux-ci sont actuellement : la forme aléatoire d’un fil jeté sur une table, la forme aléatoire d’une section horizontale (prise à un niveau anatom ique déterminé) d’un crâne humainchoisi au hasard.

Pour ce second exemple, notre choix s’est porté sur deux collections de contours obtenus au “conformateur” d’une part sur des person nes vivantes (mais anonymes), contours fournis gracieusement par la grande maison parisienne de chapellerie Sools , d’autre part sur une collection ethnographique de crânes d’individus morts, collection mise aimablement à notre disposition par M. Lester, Directeur du laboratoire d’Ethnographie.

L’étude statistique d’une collection de contours crâniens préhistoriques .

Les fonctions représentatives sont développées en série de fourier et l’on procédera à une analyse statistique de ces séries . On déterminera les lois de fréquences des premiers coefficients pris isolément et l’on étudie ra d’autre part leur corrélation au moyen de divers indices de corrélation et en partic ulier l’indice diagonal que nous avons défini récemment.

Le laboratoire de Calcul de l’Institut Henri Poinca ré opérant sur deux analyseurs harmoniques.

Pour les fils, environ 120 coefficients (nous dispo sons de 100 de ces formes). Pour les contours crâniens dont les formes sont infiniment m oins variées, nous nous sommes contentés de 12 harmoniques soient 25 coefficients pour chaque crâne.

Dévelopement en série de Fourier de l’équation en c oordonnées polaires d’un contour crânien

59 /59 / Fond Fréchet : French Science Academy Archive

M. Fréchet collaboration avec M. Ozil

M. Ozil a dès 1938 émis l’idée qu’on pourrait aider à la classification des races au moyen de l’analyse harmonique des profils humain s. Les tableaux de M. Ozil ne correspondent pour des formes en fait assez c ompliqués qu’à 9 harmoniques et représentent seulement la moitié exp ressive du profil, c’est à dire une fonction non périodique.

Documents numérisés à partir du Fond Fréchet

des Archives de l’Académie des

Sciences:Correspondance entre M.

Fréchet et M. Ozil

www.thalesgroup.com

Metric Geometry of Hermitian Positive Definites

Matrices (HPD):What is the good metric ?

61 /61 / Probability Metrics: Distance between 2 random variables

Question in Probability and Statistic: Could we define distance between 2 random variables ?

Or equivalently, could we define distance between 2 probability densities of these 2 random variables ?

=

−)1(

)1(1

)1(2

)1(1

)1(

n

n

z

z

z

z

Z M

=

−)2(

)2(1

)2(2

)2(1

)2(

n

n

z

z

z

z

Z M( ) ???, )2()1( =ZZd

( ))1()1( /θZp ( ))2()2( /θZp( ) ???, )2()1( =θθd

62 /62 / Point of views of Probabilists: Information Geometry of Rao

Information Geometry:

Cramer-Rao-Fréchet-Darmois Bound and Fisher Informati on Matrix

Kulback-Leibler Divergence (variational definition by Donsker/Varadhan) :

Rao-Chentsov Metric (invariance by non-singular change of parameterization)

Invariance:

1945 1943( IHP Lecture

1939)

( )( ) ( ) 1ˆˆ −+≥

−− θθθθθ IE

CRFD Bound

[ ] ( )

∂∂∂−=

*

2

,

/ln)(

jiji

XpEI

θθθθ

Fisher Information Matrix R.A. Fisher

( ) ( )[ ] ∑==+= +

jijiji ddgdIddXpXpKds

,

*,

2 )(/,/ θθθθθθθθ

( ) ( )[ ] ( ) ( )( ) dxxq

xpxpeEESupqpK qp

=−= ∫ θ

θθφ φ

φ /

/ln/ln),(

N. N. Chentsov )()()( 22 θθ dswdsWw =⇒=

63 /63 / Combinatorial/Variational Foundation of Kullback Divergence

Combinatorial Fundation of Kullback Divergence

Kullback Divergence can be naturally introduced by combinatorial elements and stirling formula :

( )[ ])(log),( φ

φφ eEESupqpK qp −=

in

( ) ∏=

=M

i i

ni

MMM n

qNqqnnnP

i

1121 !

!,...,/,...,,

iq ∑=

=M

ii Nn

1 N

np i

i =

+∞→≈ − n when ..2..! nenn nn π

[ ] ),(log.log1

1

qpKq

ppP

NLim

M

i i

iiM

N=

=∑

=+∞→

Let multinomial Law of N elements spread on M leve ls

with priors , and

Stirling formula gives :

Variational Foundation of Kullback Divergence

Donsker and Varadhan have proposed a variational definition of Kullback divergence :

64 /64 /

This proves that the supremum over all is no smaller than the divergence

Kullback Divergence & VARADHAN’s Variational Approach

( )[ ])(log),( φ

φφ eEESupqpK qp −=

Donsker and Varadhan have proposed a variational definition of Kullback divergence :

( ) ),()1ln(),()(

)()(ln

)(

)(ln)()(ln)(

)(

)(ln)( :Consider

qpKqpKq

pq

q

ppeEE

q

p

qp =−=

−

=−⇒

=

∑∑ωω

φ

ωωω

ωωωφ

ωωωφ

( )

( )[ ] 0)(

)(ln)()(ln)(),(

)(

)()(with

)(

)(ln)(

)(ln)(ln)(

)(

)(

≥

=−−⇒=

=

=−

∑∑

∑

ωφ

φ

θ

θφ

ωφφ

ω

φ

φ

φφ

ωωωφ

θωω

ωωωφ

q

ppeEEqpK

eq

eqq

q

qp

eE

eEeEE

qp

qpqp

φ

Using the divergence inequality,

Link with « Large Deviation Theory » & Fenchel-Legendre Transform which gives that logarithm of generating function are dual to Kullback Divergence :

[ ] [ ][ ][ ]

[ ][ ])(

(.)

)(

(.)

)(

log)(),(

)(log)()(),(

),()()()(log

xVqp

V

xV

V

p

xV

eEVESupqpK

dxxqedxxpxVSupqpK

qpKdxxpxVSupdxxqe

−=⇔

−=⇔

−=

∫∫

∫∫Varadhan

Abel Prize 2007

65 /65 / Dual Information Geometry: Multivariate Gaussian Law

Dual Coordinates systems & Potential functions

Potential Functions are Dual and related by Legendre transformation :

( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )

−−−+−=

+Θ−Θ=⇒

+Σ−==

==

−−−−

−−−−

−−

enΗ)(ηΗηΗΦ

nTrΘΨ

mmmη,ΗΗ

Σ)m,(Σθ,ΘΘ

T

T

T

ππθθ

2log2detlog21log2~~

)log(2detlog22~~

,~

2~

scoordinate Dual

1111

1112

11

=∂∂

=∂∂

=∂∂

=∂∂

ΘΗ

Φ

θη

Φ

ΗΘ

Ψ

ηθ

Ψ

~

~

and~

~

( )TT ΘΗTrH,Θ

ΨH,ΘΦ

+=

−≡

θη~~with

~~~~

( ) [ ]pEΗΦ log~~ = Entropy

ji

*ij

jiij

ΗΗ

Φg

ΘΘ

Ψg

∂∂∂≡

∂∂∂=

~ and

~ 22

Hessians are convexe and define Riemannian metrics :

66 /66 / Dual Differential Geometry

Link with Kullback Divergence

For each Dual Geometry, we can built a divergence that is directly related to Kulback-Leibler Divergence :

( ) ( )[ ] ( ) ( )( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )( )dx

mxp

mxpmxp,Σm,N,ΣmNDiv



∫ ΣΣΣ=

≥−+≡

≥−+≡

22

11111122

21211122*

12121122

,/

,/log,/

0~~~~~~

0~~~~~~

Riemannian Metric As Potential are convexe, their Hessians define Riemannian Metrics :

( ) ( )( ) ( )( ) ( )( )( )( )( )

−−Σ+

Σ−ΣΣ+ΣΣ−=

−

−−−

TmmmmTr

Tr,Σm,N,ΣmNDiv

21211

2

11

121

121

1122

detlog

2

1

( ) ( )332

2

1

2

1iji

ijijiij dΗOdΗdΗgdΘOdΘdΘgds +=+=

67 /67 / Information Geometry of Multivariate Gaussian Density

Multivariate Gaussian Distribution of zero mean

Riemannian Information metric is given by Rao Metric

( )[ ]( )TABTrBAAAA

ddTrds

==

ΣΣΣ=ΣΣ= −−−

,et ,with

2

1

2

1

2

22/12/1212

Rao Metric

Multivariate Gaussian of non zero mean Isometries are given by following homeomorphisms :

( )[ ]2112

2

1 ΣΣ+Σ= −− dTrdmdmds T

( ) ( ) ( )( ) ),(x, with '

isometry ,',',222 RnGLRAadsdsds

AAamAmmn

TT

∈=Σ+=Σ→Σ

a

22

111

1then

As

−ΣΣ

−−−

=ΣΣ−=ΣΣ⇒=ΣΣ

dsds

ddI Invariance by inversion

( ) ( )12

1121 ,, −− ΣΣ=ΣΣ DD

68 /68 / Canonical Exemple: Monovariate Gaussian Law

Gauss-Laplace Law

Fisher Information Matrix for Gauss-Laplace Gaussian Law:

( )( )

=≥

−−

= −−

σθθθθθθσθ

mIEI

T and )(ˆˆ with

20

01.)( 12

( ) ( )

+

=+== − 22

22

2

2

22

2.2.2.. σσ

σσ

σθθθ d

dmddmdIdds T

Rao-Chentsov Metric of Information Geometry

H. Poincaré

.2

σim

z += ( )1 <+−= ωω

iz

iz

( )22

2

2

1.8

ω

ω

−=⇒

dds

Fisher Metric is equal to Poincaré Metric for Gaussian Laws:

( )

)*2()1(

)2()1()2()1(

2

)2()1(

)2()1(

22112

1),(with

),(1

),(1log.2,,,

ωωωωωωδ

ωωδωωδσσ

−−=

−+=mmd

69 /69 /Point of View of Geometers: Cartan-Siegel Domains Metric

Henri Poincaré(upper-half plane

model of hyperbolic geometry)

n=1

Elie Cartan(classification in 6 types of symmetric

homogeneous bounded domains )

n<=3

Ernest Vinberg(link with homogeneous convexe cones,

Siegel domains of 2nd kind)

Fine structure of Information Geometry(Hessian Geometry, Kählerian Geometry)

« Il est clair que si l’on parvenait à démontrer qu e tous les domaines homogènes dont la forme

est définie positive sont symétriques, toute la thé orie des domaines bornés homogènes serait élucidée.

C’est là un problème de géométrie hermitienne certainement très intéressant »

Dernière phrase de Elie Cartan, dans « Sur les domaines bornés de l'espace de n variables complexe s

», Abh. Math. Seminar Hamburg, 1935

( )ji

ji ji

zddzzz

zzK∑ ∂∂

∂=,

2 ,logΦ

Jean-Louis Koszul(canonical hermitian form of

complex homogeneous spaces, a complex homogeneous space with positive definite canonical hermitian form is isomorphic to

a bounded domain,,Study of Affine Transform Groups of

locally flat manifolds)

Carl Ludwig Siegel(Siegel domains in framework

of Symplectic Geometry)

Lookeng Hua(Bergman, Cauchy and Poisson

Kernels in Siegel domains)

70 /70 /Cartan-Siegel Symmetric Homogeneous Bounded Domains

( ) ( )2*

**11111

1

1,,1/

zwwzKzzCzΩΩΩΩ IVIIIIII

,−

=<∈====

Henri Poincaré(n=1)

( ) ( ) ( )

( ) ( ) ( )( ) domain. theof volumeeuclidean is where

, : IV Typefor 2ZZ11

,

1 , : III Type

1 , : II Type

, : I Type

for det1

,

-**t*

*

Ωµ

νΩµ

ν

ν

ν

Ωµ

ν

ν

nΩZWWWWZK

pΩ

pΩ

qpΩ

ZWIWZK

IVn

IIIp

IIp

Ip,q

=−+=

−=

+=

+=

−=

+

−+

Lookeng Hua

02ZZ1,1ZZ

:line 1 and rowsn with matricescomplex : IV Type

porder of matrices symmetric skewcomplex : III Type

porder of matrices symmetriccomplex : II Type

rows q and lines p with matricescomplex : I Type

):(

Matrixr RectangulaComplex :

2tt >−+<

−<

+

++

ZZ

Ω

Ω

Ω

Ω

conjugatetransposedIZZ

Z

IVn

IIIp

IIp

Ip,q

Carl Ludwig Siegel

Elie Cartan(n<=3)

71 /71 / Hyperbolic Poincaré Space

Action of SLAction of SLAction of SLAction of SL2222R Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré Plan

Möbius Transform is a transitive action that transforms upper half-plane to itself (homogeneous space) :

M and –M have same action then we consider the quotient Group :

Complex unit disk is link to upper half plane by Cayley transform :

0)Im(: and )( , 2 >∈=∈++=∈

= zCzHz

dcz

bazzMRSL

dc

baM

222 / IRSLRPSL ±=

1

1et

1:

z

ziz

HD

iz

izz

DH

zCzD

−+

→

+−

→

<∈=

aa

1- 1

i

- i

2

2

2

222

y

dz

y

dydxds =+=

x

y

H D

( )22

2

2

1

4

z

dzds

−=

Carl Ludwig Siegel Contribution in his seminal book « Symplectic Geometry » : a

generalization of Poincaré Space

72 /72 /

Siegel Metric for the Siegel Upper-Half Plane:

Upper-Half Plane :

Isometries of are given by the quotient group:

with the Symplectic Group:

Unique Metric invariante by :

Invariance by Automorphisms: Siegel Metric of SHn

nSH nIRnSpRnPSp 2/),(),( ±≡ ),( FnSp

( )( ) 1)( −++=⇒

= DCZBAZZM

DC

BAM

=−⇔∈

=

nTT

TT

IBCDA

DBCAFnSp

DC

BAM

symmetric et ),(

),2(0

0 , /),2(),( RnSL

I

IJJJMMFnGLMFnSp

n

nT ∈

−==∈≡

)(ZM

( ) ( )( )ZdYdZYTrdsSiegel112 −−=

==

nRY

X 0 ( )( )( )212nn dRRTrds −=

iYXZ +=

0Im/),( >=∈+== Y(Z)CnSymiYXZSHn

73 /73 / Siegel Uppel-Half Plane Geometry


Siegel Upper Half Space

X

0>Y

( ) ( )( )ZdYdZYTrds 112 −−=

( ) ( ) ( )( )∑=

−+=n

iiiZZd

1

221

2 1/1log, λλ

kkk YiXZ .+=

1=k

2=k

( )[ ]212 dRRTraceds −=

( )kkkk RNWRiZ ,0 if . ≡=

1=k

2=k

( ) ( )∑=

=n

kkRRd

1

221

2 log, λ

( ) ( )( ) ( )( ) 1

2121

1

212121,−− −−−−= ZZZZZZZZZZR

( ) 0...det 2/112

2/11 =−−− IRRR λ

( )( ) 0.,det 21 =− IZZR λ

74 /74 / QUANTIZATION IN COMPLEX SYMMETRIC SPACES

F. Berezin

( )( )( ) ( )

( ) ( )( ) ( ) ( )BBItraceBBIgFABg

ZFZgFZBAtraceZFZgF

ZZItraceZZIzF

AZBBAZZgBAAB

IBBAA

I

IJJJgg

BB

BAg

IZZZSD

t

t

t

n

++−

++

−

+

+

+

+=+=⇒=

∂∂=∂∂⇒++=−−=−−=

++=

=−=−

−==

=

<=

logdetlog))0(()0(

)()(logRe2)())((

logdetlog)( : potentialKähler

)( with 0

where

0

0 with and with

/

1*

****

1**

*

*

**

Z

gZZgjZZKZgjzgjgZgZKZZd

K

ZZKhc

ZZdK

ZZKZgZfhcgf

h

h

∂∂==

=

=

∫

∫−

−

−

),(, ),(),(),(),( with ),()0,0(

),()(

),()0,0(

),()()()(,

****

/1*1

*

/1*

µ

µ

( ) ( ) ( )

( ) ( ) νβααβ

βα

βαβα

πµµ

−++=∂∂

∂−==

=

∑ WWIWWFWW

WWFgdWdWgds

WWdWWFWWd

nL

det),( where,log

with

,,,

**

*2

,

*,

2

***

75 /75 /

Information Geometry for Multivariate Gaussian Law of zero Mean and intrinsec Geometry of Hermitian Positive Defini te Matrices (particular case of Siegel Upper-Half Plane) provid e the same metric


Geometry of Siegel Upper-Half Plane:

( )( )( )212nn dRRTrds −=

[ ]

( )( )[ ] nn

nnnnn

RRTrn

nnn

RRE

mZmZR

eRRZp nn

=

−−=

=+

−−− −

ˆ and

.ˆwith

..)()/(1.ˆ1π

−=

*

2

.

)/(ln)(

ji

nnij

ZpEg

∂θ∂θθ∂θ

0=nm

with

Siegel HPD Matrix Metric = IG Metric of Multiv. Gaussian Law


( ) ( )( ) iYXZZdYdZYTrdsSiegel +== −− with 112

==

nRY

X 0 ( )( )( )212nn dRRTrds −=

76 /76 /

Siegel Distance:

Particular Case (X=0) and General Case:

Particular Case (pure imaginary axis) :

General Case of Siegel Upper-Half Plane Distance:

Distance between HPD Matrices: particular case of Siegel

( ) ( ) ( )∑=

−− ==n

kkRRRRRd

1

222/112

2/1121

2 log..log, λ

( ) 0det 12 =− RR λwith

0 with ≠∈+= XSHiYXZ n

( ) n

n

k k

kSiegel SHZZZZd ∈

−+

= ∑=

211

221

2 , with 1

1log,

λλ

with ( ) 0.),(det 21 =− IZZR λ

( ) ( )( ) ( )( ) 1

2121

1


0 avec >= RiRZ

77 /77 /

Siegel has deduced an other distance from :

Another distance deduced from Siegel Work

( )[ ]212 .ds −ΣΣ= dTr

( )( ) 12122

12

112

11

12

1112 .T=R and . TTT +−−−−− =Σ+ΣΣ−Σ=

R is hermitian positive definite matrix with eigen-values :

)..( and )(r with 1

1 2/1)1(1)2(2/1)1(kk

2

nnnk

kk RRRRr −==

+−= σλσ

λλ

We deduce that : ( ) ∑=

−+=

−+=ΣΣ

n

k k

k

n

n

r

r

RI

RITrd

12/1

2/12

2/1

2/12

212

1

1lnln,

because :

[ ] [ ] ∑∑=

∞

=

− =

+==

−+ n

k

jk

k

k

n

n rak

RRR

RI

RI

1

j

2

0

22/112/1

2/12 RTr nd

1.2..4tanh.4ln

78 /78 / HPD Matrices Geometry

Geometry of Hermitian Positive Definite Matrices given by:

Geodesic :

Properties of this space

Symetric Space as studied by Elie Cartan : Existence of bijective geodesic isometry

Bruhat-Tits Space : semi-parallelogram inequality

Cartan-Hadamard Space (Complete, simply connected wi th negative sectional curvature Manifold)

[ ]10 with ),(.))(,( ,tRRdttRd YXX ∈=γ( ) ( )

YXYX

X

t

XYXXXRRRt

X

RRRR

RRRRRReRt XYX

o===== −−−−

)2/1( and )1( , )0(

)( 2/12/12/12/12/1log2/1 2/12/1

γγγγ

( ) ( ) 2/12/12/12/12/11-),( avec )(X ABAAABABABAXG BA

−−== ooo

Xx)d(x,x)d(x,xd(x,z)),xd(x

xzxx

∈∀+≤+

∀∃∀

224

que tel ,2

22

122

21

21

79 /79 /

x1x

2x

z

U

VU + VU −V

Cartan’s Symmetric Space

Symetric Space as studied by Elie Cartan : Existence bijective geodesic isometry


222222 VUVUVU +=++−

),( AGB BA=

BGA BA ),(= ( ) )(X-1),( BABAXG BA oo=

X

( ) 2/1/212/12/12/1 ABAAABA −−=o

( )[ ]1,0

)( 2/12/12/12/1

∈= −−

t

ABAAAttγ

A=)0(γ

B=)1(γ

22

21

2221 224 )d(x,x)d(x,xd(x,z)),xd(x +≤+

E. J. Cartan

M. Berger

J. Tits

80 /80 /

This isometry for metric space :

Is an extension of this one :

To be compared with Euclidean « symmetric » space

Symmetric Space

( )( )

( ) ( )

=

=•••=

−−

−−

22/12/12

2/12/12/12/12/1

1-),(

log, with )(X

BAABA

ABAAABABABAXG BA

δ

( ) ( )( ) ( )

=

==

− 212

1

log, with

abba

abbabaxbaxG -

(a,b)δ

ooo

( )

−=

+=

++

+=22 ,

2 with 22

baba

bababa

-xba

xG(a,b)

δ

o

81 /81 /

is the only fixed point because :

due to trace property of :

Unicity of fixed point

( ) 2/12/12/12/12/1 ABAAA −−

( )( )CXICXX

ICXXCXXXCCX

=⇒=⇒

=⇒=−−

−−−−−

2/12/1

root square ofunicity

2/12/12/12/11

( )( ) ( )( ) ( ))(,2,)(X ),(X 1-1-1 BAXdIXBABAdXBABAd •=••=•• −

11

1

2

)( of seigenvalue with

log)(

−=

=

•

=• ∑

XBA

)BAd(X,

n

ii

n

ki

λ

λ

82 /82 /

For space of Symmetric Positive Definite matrices, the analogue of

is :

It constitutes a natural framework for a generalized the ory of convexity, where the role of arithmetic mean is played by a midpoint pairing :

is called -convex if

For space of symmetric positive definite matrices, an affine function isgiven by :

and is closely related to entropy :

Convexity

ba )1( λλ −+ ( ) 2/12/12/12/1 ABAAABAλ

λ−−=•

BA λ•

BAA 0•= BAA 1•=

f ( )21,••

( ) ( ) ( )YfXfYXf 21 •≤•

detlog=f

( ) csteREntropy +−= detlog

83 /83 / Bruhat-Tits Space & Semi-parallelogram Law

Bruhat-Tits Space :

Space (Sym++,δ) is called Bruhat-Tits Space, where, according to δmetric, point at equal distance of 2 points is unique and given by geometric mean:

( )

( ) ),(),(,

),(2),(2),(

0det with log),(

definite positive ),(,

11

1

1

2

−−

−

=

==

==

=−

=

∈∀

∑

BABABCCACC

BABBAABA

IABBA

RnSymBA

TT

n

ii

δδδδδδ

λλδ

oo

Bruhat-Tits Space is a complete metric space that verifies semi-parallelogram law :

Xx)δ(x,x)δ(x,xδ(x,z)),xδ(x

zxx

∈+≤+

∃∈∀

allfor 224

:such that X,2

22

122

21

21

Bruhat-Tits Space are particular cases of Cartan-Hadamard Manifold

[1] F. Bruhat & J. Tits, « Groupes réductifs sur un corps local », IHES, n °41, pp.5-251, 1972

« Bruhat-TitsSpace»

(Metric Space)

84 /84 / Bruhat-Tits Space & Semi-Parallelogram Law

Bruhat-Tits Space :

Semi-parallelogram Law :

Xx)δ(x,x)δ(x,xδ(x,z)),xδ(x

zxx

∈+≤+

∃∈∀

allfor 224

:such that X,2

22

122

21

21

x1x

2x

z

U

VU + VU −

222222 VUVUVU +=++−

Deduced from parallelogram law :

V

222

212

232

2212 22 )(x,xd)(x,xd)(x,xd),x(xd +=+

222

212

22

2212322 224.2 )(x,xd)(x,xd(x,z)d),x(xd)(x,xd(x,z)d +=+⇒=

3x

85 /85 /

Geodesic between matrix P and Geodesic between Q & R :

The geodesic projection is a contraction :

Geodesic Projection

Q

R

P

P

Q

( ) ( )[ ] 2/1

2/12/1 2/12/12/12/12/1,)( PPQRQQQPPtstts

s−−−−== σσ

( ) ( )QPdistQPdist MM ,)(),( ≤ΠΠ

M : geodesically convex closed submanifold of

Hadamard Manifold ( )SPdistP

MSM ,minarg)(

∈=Π

ΠΠΠΠM(P)

ΠΠΠΠM(Q)

P

Q

86 /86 / Kähler Potential & Bergman Kernel

We can find Bergman Kernel of Kähler geometry for S iegel Upper-half space by mean of Siegel Unit Disk :

Analogy with Bergman Kernel for Poincaré unit disk :

),(ln ZZKΦ B−= ( ) 0 / on ),( >−= ZI-ZZWZIWZKB

( ) ( )

( ) ( )

( )22

222

2222

22

2

2

2

1

1

1

1

)(1

11ln

1/on 1),( with ),(ln

z

dzzdzd

zzzgdzdds

zz

zzz

zzg

z

z

zz

zzwzwzKzzK BB

−=

∂∂Φ∂==

−=

−

−−−=

∂∂Φ∂=⇒

−=

∂Φ∂

⇒−−=Φ

<−=−=Φ

( )( ) 1−+−

→Ψ

nn

nn

iIZiIZZ

SD:SH

a ZZZIZSym(n,C)/ZSDn =<∈=

22 with

With Bergman Kernel

( ) ( )( )ZdZZIdZZZITrZdZdZZ

ds11

22 −− −−=

∂∂Φ∂=

87 /87 / Upper-Half Planes & Disks of H. Poincaré and C.L. Siegel

( )( ) ( ) *1*1*2

22

2

2

11

1

dwwwdwwwds

w

dwds

−− −−=

−=

iz

izw

+−=

( )( ) 1−+−= iIZiIZW

Upper-Half Plane of H. Poincaré Unit Disk of Poincaré

Siegel Upper-Half Plane Unit Siegel Disk

( )

),( and ),(

with

112

CnHPDYCnHermX

iYXZ

ZddZYYTrds

∈∈+=

= −−

0 and with

*112

2

2

2

222

>+==

=+=

−−

yiyxz

dzdzyyds

y

dz

y

dydxds

( ) ( )[ ]+−+−+ −−= dWWWIdWWWITrds112

88 /88 /

Koszul Forms

1st Koszul Form :

2nd Koszul Form:

Application for Poincaré Upper-Hal Plane:

With and

We can deduce that

Jean-Louis Koszul Forms

( ) ( )[ ] gXXJadJXadTrX bg ∈∀−=Ψ )(/

( )XdΨ−=4

1α

αβ D=

0 / >+== yiyxzV

dx

dyX =

dy

dyY =

( ) [ ][ ]

=−==

⇒

YJX

YYX

ZYZYad

,

,.

( ) ( )[ ]( ) ( )[ ]

=−=−0

2

YJadJYadTr

XJadJXadTr

dydxy

∧=Ω2

1

( )

2

222

2

2

2

1

4

12

y

dydxds

y

dydxd

y

dxX

+=⇒

∧−=−=⇒= ΨαΨ

89 /89 /

Koszul form for Siegel Upper-Half Plane:

Symplectic Group :

Associated Lie Algebra:

Jean-Louis Koszul Forms for Siegel Upper-Half Plane

0 / >+== YiYXZV

( )

−=

=

==+=

0

0 and

0 with

,

1

I

IJ

D

BAS

BDDBIDA

DBAZSZTTT

-

+=

−=

−==

00

0 ,

0

0 base and with

0jiij

ijji

ij

ijT

T ee

e

e

ad

bb

d

baβα

( )( ) ( )

( ) ( )

( )( ) ( )

+=

∧+=Ψ−=Ω

+=+Ψ

⇒

+=Ψ

=Ψ

−−

−−

−

ZddZYYTrp

ds

ZdYdZYTrP

d

dXYTrp

idYdX

pijij

ij

112

11

1

8

138

13

4

12

13

13

0

δβα

www.thalesgroup.com

Geometry of Optimal Transportand 4th « Forgotten » distance of

Maurice Fréchet

91 /91 / Optimal Transport Theory: Wasserstein distance

Wasserstein Distance

Classical form of Wasserstein distance in Optimal t ransport

Particular case of Wasserstein distance

Case n = 1 : Monge-Kantorovich-Rubinstein Distance

Case n=2 : distance de Levy-Fréchet Distance

( )

( ) ( )[ ]( ) νµνµ

πνµνµΠπ

===

= ∫

℘∈

)(,)(,,,

),(),(,

/1

/1

),(

YlawXlawYXdEInfW

yxdyxdInfW

nnn

n

nn

( ) [ ]YXEInfQPdWYX

−==,

1 ,),( νµ

( ) [ ]2

,2 ,),( YXEInfQPdW

YX

−==νµ

92 /92 / Optimal Transport: Fréchet-Wasserstein Distance

Fréchet-Wasserstein Distance for Multivariate

Gaussian LAw

Fréchet-Wasserstein Distance

Proof:

Let

Solution is given by:

[ ] ( ) ( )( )[ ][ ] [ ] [ ] [ ]YXYX RTrRTrRTrYEXEYXE

YXYXTrEYXE

,

22

2

2)()( −++−=−

−−=− +

0≥

=⇒

= +

Y

XW R

RR

Y

XW

ΨΨ

( ) ( ) 2/12/12/12/12/1

0

..1

−+

≥−=⇒+

+−XXYXX

RR

RRRRRTrSupYX

ΨΨΨΨΨ

( ) ( )[ ]2/12/12/1 ...2 XYX RRRTrTr =+ +ΨΨ

( ) XRRRRRY XXYXX ... 2/12/12/12/12/1 −=


Wasserstein Distance for Multivariate Gaussian Law with non-zero mean

Fréchet-Wasserstein distance

Geodesic:

If we let:

Optimal Transport :

transport from to

( ) ( )( ) [ ] [ ] ( )[ ]2/12/12/122 .2,,, XYXYXYXYYXX RRRTrRTrRTrmmRmRmd −++−=

( ) 12/12/12/12/12/1,

−− == YXXXYXXYX RRRRRRRD o

( ) ( )

+−+−=

+−=

YXkYYXkt

XYt

DtItRDtItR

mtmtm

,,)(

)(

.)1(.)1(

.)1(

( ) ( )[ ] ( ) ( )[ ])1()1()0()0(2)()()()(2 ,,,).(,,, RmRmWstRmRmW ttss −≤

( ) ( )YYk RmNI ,, #ψ∇ ( )YY RmN , ( )XX RmN ,

( ) XYYX mmyDyx +−=∇= ,)(ψ( ) ( ) ( )νψ −+−−= +

XYYXY mmvDmvv ,2

1)(

( ) ( ) ( ) ( )YYYXXx myRmymxRmx −−=−− −+−+ 11

[ ] [ ] ( )[ ] [ ] ( )[ ] 2/12/12/12/12/12/12/12/12/1 with .2 YXXYXXXYXYX RIRRRRRMMMTrRRRTrRTrRTr −==−+ −+


Characterization of this space (cas m=0)

Wasserstein Metric :

Tangent Space & Exponential Map:


Alexandrov Space

Space of Positive sectionnal curvature (geodesicall y convexe and simply connected). Sectional Curvature is given by:

Where matrix S is a symetric matrix such that:

is anti -symmetric.

( )YRXTrYXg YRN Y..),(),0( =

( ) ( ))(,,0 ,0).(exp tRRN YYRNXt =

( ) ( )XtItRXtItR kYktRY.)1(.)1()(, +−+−=

( )VTYRN ,0

( ) ).(exp ,0 XtYRN

( )YRRN ,0

( ) ( ) ( ) ( )2222 )1(),0().1()(,.)0(,).1()(, γγγαγαγα dtttdtdttd −−+−≥

( ) [ ]( ) [ ]( )( )TVN SXYVSXYTrYXK −−= ,,.4/3),()(

[ ]( )VSXY −,


Wasserstein Barycenter of multivariate gaussian laws (Cas m=0)

Wasserstein Barycenter

Solution of Wasserstein Barycenter for N Mulitvaria te gaussian laws of zero mean :

Constrained for the barycenter:

Iterative solution (paris-dauphine university)

Fixed point (convergence for d=2, d>3 conditional convergence to initiation)

[ ]2

,2

1

22 ),( with ),( YXEInfWWInf

YX

N

kk −=∑

=

νµνµµ

( ) N

kkRN 1,0 =

( ) RRRRN

kk =∑

=1

2/12/12/1

( ) with 2/1

2/1

1

2/1)(2)()1(ii

N

k

nk

nn RKKKKK =

= ∑=

+


Levy-Fréchet Distance

In 1957, Maurice René Fréchet introduced this dista nce, based on a previous Paul Levy’s paper from 1950

( ) [ ] ∫∫ −=−= ),()(, 22

,

2 yxHddyxYXEInfGFd yxYX


Levy-Fréchet Distance

1957 Fréchet paper in CRAS :

Letter from Paul Levy to Maurice Fréchet (2 April 1 958) … J’ai ainsi pu apprécier ce que vous aviez fait, en prenant comme point de départ de votre

mémoire ce que vous appelez ma première définition de la distance de deux lois de probabilité (en fait ce n’était pas la première). Vous l’avez d’ailleurs généralisée, en ce sens que je ne l’avais associée qu’à une de vos définitions de deux variables aléatoires. Et j’ai beaucoup admiré comment avec votre quatrième définition, vous arrivez à faire quelque chose de maniable d’une idée qui pour moi était surtout théorique, vu la difficulté de déterminer le minimum de la distance de deux variables aléatoires ayant les répartitions marginales données.

98 /98 / 4th « Forgotten » distance of Fréchet : Frechet Extreme Copulas

4th distance of Fréchet

Extreme Copulas of Fréchet

4th distance of Fréchet

( ) ( )[ ])(),(),(with

),(,

1

122

yGxFMinyxH

yxHddyxGFd yx

=

−= ∫∫

[ ][ ])(),(),(

0,1)()(),(

with

),(),(),(

1

0

01

yGxFMinyxH

yGxFMaxyxH

yxHyxHyxH

=−+=

≤≤

Extreme Fréchet Copulas

),(1 yxH

),(0 yxH

),( yxH

≥≥≥≥

≥≥≥≥

99 /99 /

G. Dall’Aglio (Université La Sapienza à Rome), « Fréchet Classes: the Beginnings », Advanced in Probability Distributions with Given Marginals Beyond the Copulas, Rome 1991, Kluw er

With help of Giuseppe Pompilj, G. Dall’Aglio rmet Maurice Fréchet, 80 years old, in 1956 in Roma, during visit to l’Istituto di Calcolo delle Probabilita

Dall’Aglio met a 2nd time Fréchet in Paris and is in contact with Paul Levy

Dall’Aglio write : « Levy also showed some interest in distributions wit h given marginals. In a note in 1960 he refers to a f ormula by Poincaré for utilization in n-dimensional distributions with giv ens marginals, without developping the idea »

Paul Levy, « Sur les conditions de compatibilité des données marginales relatives aux lois de probabilité » CRAS, t. 250 pp.2507-2509, 1960, note présentée par M. Jacques Hadamard

4th distance of Fréchet & Romane School of statistic (Pompilj, Leti, Rizzi, Dall’Aglio,Salvemini, Gini…)

100 /100 /

G. Dall’Aglio (Université La Sapienza à Rome), « Fréchet Classes: the Beginnings », Advanced in Probability Distributions with Given Marginals Beyond the Copulas, Rome 1991, Kluw er

By integration by parts:

assigns to all sub-sets of diagonal y=x, maximum of probability compliant with marginals: S. Bertino, « Su di una sottoclasse della classe di Fréchet », Statistica 25, pp. 511-542, 1968

Gini Work : in 1914, Gini introduced « linear dissimilarity parameter » (solution in discret case for α = 1, 2)

Tommaso Salvemini (+ Leti) Work : construction of « tabella di cograduazione » and « di contrograduazione » (equivalent to Fréchet Extreme Repartition Functions )

Generalization to multivariate by Rizzi in 1957, Dall’Aglio (1960)

4th distance of Fréchet & Romane School of statistic (Pompilj, Leti, Rizzi, Dall’Aglio,Salvemini, Gini…)

[ ] ( )

( )

( )

≥

−−

≤

−−

=

−+=−

≤≤

≤≤

∫

yxzFzGInfyG

yxzGzFInfxF

yxH

dzzzHzGzFYXE

xzy

yzx

R

if 0,)()(max)(

if 0,)()(max)(

),(

:function n Repartitio gMinimlizin

),()()(

*

[ ] ( ) ( )

( )

)(),(),(

:Function n Repartitio Minimizing

)(),()()1(

)(),()(1

:1For

1

2

2

yGxFMinyxH

dudvuvvuFuF

dudvvuvuHvGYXE

vu

vu

=

−−−

+−−−=−

>

∫

∫

<

−

>

−

α

αα

αα

αα

α

),(* yxH

101 /101 / Comparison Information Geometry versus Optimal Transport

For transport optimal:A. Takatsu. On Wasserstein Geometry of the Space of Gaussian Measures, to appear in Osaka J. Math., 2011, http://arxiv.org/abs/0801.2250

www.thalesgroup.com

Karcher Flows for HPD matrices:

Mean & Median Computation

103 /103 / Geometric Mean Extension for Symmetric Positive Definite matrices

Geometric Mean Extension for Symmetric Positive Definite Matrices :Geometric Mean Extension for Symmetric Positive Definite Matrices :Geometric Mean Extension for Symmetric Positive Definite Matrices :Geometric Mean Extension for Symmetric Positive Definite Matrices :

Cauchy series given by harmonic and arithmetic mean :

( ) ( ) 2/ and 2

, with

1

1111

00

nnnnnn

nn

nn

BABBAA

BBAABLimALimBA

+=+=

====

+−−−

+

∞→∞→o

Solution of the following ODE :

∈

−==∈∀

++

−

∞→

++

SymX

XXABdt

dX

tXLimBASymBAt

)0(

with )(,,1

o

Solution to the inequality :

0 s.t. (Loewner)matrix higher ,, >

∈∀ ++

BX

XABASymBA o

Solution of the equation :

)det(log)( with ))(('' 11 XXFBAXXAXF −=== −−

104 /104 / Geometric Mean by Ricatti Equation

bxxaabxabx =⇔=⇔= −12

In scalar case, geometric mean of a,b>0 satisfies for x>=0 :

The last « symmetrized » version is suitable for generalization to noncommutative settings : the geometric mean is the unique solution of , if the unique solution exists.

We define a group equipped with the operation :

bxxa =−1

xxyyx 1−=•baxbaxbxxa o=⇔=•⇔=−1

To solve , we need to have a unique solution. Every element must have a unique square root :

aex =• exxex == 2

2/1aae =o

Extension for Matrix Geometric Mean :

( )( ) BAABAAAX

BAAXAA

BAAXAAXAA

BAAAXAXAABXXA

o==⇒

=⇒

=⇒

=⇒=

−−

−−−−

−−−−−−

−−−−−−−

2/12/12/12/12/1

2/12/12/12/12/1

2/12/12/12/12/12/1

2/12/12/12/12/12/11

))((

)(

105 /105 /

Iterative Method for Square Root Computation

Denman-Beabers Iteration :

Schultz Iteration (without matrix inversion) :

Björk Method by Schur Decomposition :

Key Enablers : Matrix Square Root

+

=

= −

−

+

++

0

0

0

0

2

10

01

1

1

11

k

k

k

k

k

kk

Y

Z

Z

Y

Z

YX

=

0

00 I

AXwith then

=

−∞→ 0

02/1

2/1

A

AXLim k

k

)3(2

1

0

0 2

1

11 kk

k

kk XIX

Z

YX −=

=

+

++

TAQQ =+ with T Upper triangular matrix

By recurrence, we compute : +=⇒= QUQATU 2/12

106 /106 / The arithmetic-geometric mean : C.F. Gauss result

Gauss (20 years old) & Lagrange independently prove d that

Arithmetic-geometric mean of a and b :

Is related to elliptic integral :

Landen transformation gives also :

( ) ( )

aaaabbbbn

baba

baba

bab

baa

bababa

nnnn

nnnn

nnnn

nnn

nnn

=≤≤≤≤≤≤≤=≥∀⇒

≥

−=−

+=−

=

+==>≥∀

++

++

+

+

0110

222

12

1

1

1

00

...... 0

022

2 and ,, , 0

( )

+=

==

=∫

∞→∞→

2/

02222 sincos

),(

),(

with ),(

2/, π

π

tbta

dtbaI

bLimaLimbaM

baMbaI

nn

nn

( ) ( )

( ) ( ) ( )( )

+=

+=

+=

=

+=∫

ta

bArctu

dttbtabaJ

baabIbaJbaJ

baIabba

IbaI

tantan : nsf.Landen tra

sincos, with

,,,2

,,2

, 2/

0

2222

11

11π

4.J(a,b) : Perimeter of Ellipse

of half-axis a & b

107 /107 /

In Rn, the center of mass is defined for finite set of points

Arithmetic mean:

This point minimizes the function of distances:

The median (Fermat-Weber Point) minimizes :

∑=

=M

ii

xcenter xxdMinx

1

2 ),(arg

1x

2x

3x

4x

5x

∑=

→=

M

ii

xcenter xxMinx

1

arg

∑=

=M

iicenter x

Mx

1

1

Miix ,...,1=

∑=

=M

iicenter x

Mx

1

1

∑=

=M

ii

xcenter xxdMinx

1

2 ),(arg

Center of Mass : Arithmetic Mean and Median in Rn

∑=

=M

ii

xmedian xxdMinx

1

),(arg

1x

2x

3x

4x

5x

∑=

→

→

=M

ii

i

xmedian

xx

xxMinx

1

arg

[ ]mxEMinmm

median −=[ ]2mxEMinm

mmean −=

∑=

=M

ii

xmedian xxdMinx

1

),(arg

∑=

→=

M

iicenterxx

1

0 ∑=

→

→

=M

iicenter

icenter

xx

xx

1

0

108 /108 / Cartan Center of Mass and Karcher Flow

Cartan Center of Mass

Elie Cartan has proved that the following functiona l :

is strictly convexe and has only one minimum (cent er of mass of A for distribution da) for a manifold of negative curvature

Karcher Flow

Hermann Karcher has proved the convergence of the following flow to the Center of Mass :

E. J. Cartan

H. Karcher

∫∈A

daamdmf ),(: 2aΜ

( ) )()0( avec )(.exp)(1 nnnnmnnn mfmfttmn

−∇=∇−==+ γγ &

)(exp 1∫

−−=∇A

m daaf

109 /109 / probability on a manifold : Emery’s exponential barycenter

Maurice René Fréchet, inventor of Cramer-Rao bound in 1939, has also introduced the entire concept of Metric Spaces Geometry and functional theory on this space (any normed vector space is a metric space by defining but not the contrary). On this base, Fréchet has then extended probability in abstract spaces .

In this framework, expectation of an abstract probabilistic variable where lies on a manifold is introduced by Emery as an exponential barycenter :

In Classical Euclidean space, we recover classical definition of Expectation E[.] :

xyyxd −=),(M. R. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace

distancié”, Annales de l’Institut Henri Poincaré, n°10, pp.215-310, 1948

( )[ ]xgEb =)(xg x

( )( ) 0)(exp 1 =∫− dxPxg

M

b

[ ] ∫∫ ==⇒−=⇒∈ −

nn R

X

R

pn dxxpxgdxPxgxgEpqqRqp )()()()()()(exp, 1

M. Emery & G. Mokobodzki, “Sur le barycentre d’une probabilité sur une variété”, Séminaire de Proba. XXV, Lectures note in Math. 1485, pp.220-233, Springer, 1991

110 /110 / Mean & Median of N matrices HPD(n)

Mean of N Hermitian Positive Definite Matrices HPD(n)

Solution given by Karcher Flow with Information Geo metry metric

Median (Fermat-Weber Point) of N matrices HPD(n)

PhD Yang Le supervised by Marc Arnaudon (Univ. Poit iers/Thales)

( ) ( )∑∑=

−−

=

==N

kk

N

kk

Xmoyenne XBXBXdMinArgX

1

22/12/1

1

2 ..log,

( )2/1

log2/1

11

2/12/1

n

XBX

nn XeXX

N

knkn∑

= =

−−

+

ε

( ) ( )∑∑=

−−

=

==N

kk

N

kk

Xmediane XBXBXdMinArgX

1

2/12/1

1

..log,

( )( ) knnn

XBX

XBX

nn BXkSXeXX nSk nkn

nkn

≠=∑

= ∈−−

−−

+ / with 2/1log

log

2/11

2/12/1

2/12/1

ε

111 /111 / Median on a Manifold: Karcher-Cartan-Fréchet

Gradient flow on Surface/Manifold

Gradient Flow :Pushed by Sum of Normalized Tangent vectors of

Geodesics

Fréchet-Karcher Barycenter : Sum of Normalized Tangent vectors

of Geodesics is equal to zero

∫∫ −

−

−=∇⇒Μ∈A m

m

MinA

daa

ahdaamdmh

)(exp

)(exp),(

2

1:

1

1

a

Sum of Normalized Tangent Vectors( )

)(exp

exp.exp

11

1

1

= ∑

=−

−

+

M

k km

kmmn

x

xtm

n

n

n

Compute point on the surface In the direction of sum of Normalized

Tangent Vector

112 /112 / Mean : Karcher Barycenter

( ) ( ) 2/1log2/12/1 2/12/12/1 2/12/1

)( XeXXXBXXt XBXtt

kkk

−−

== −−γ

( ) 2/12/12/12/10 log

)(XXBXX

dt

tdkt

k −−= =γ

( )∑ ∑= =

−−= =

=N

k

N

kkt

k XXBXXdt

td

1

2/1

1

2/12/12/10 0log

)(γ

1B 2B

3B

4B

5B

6B

X

113 /113 / Gradient Flow & Toeplitz structure

The following structure of a Matrix M is invariant by this flow :

where J is antidiagonal matrix. We can prove that :

is a close submanifold of (Hermitian Positive Definite Matrix)and sub-group of

For Siegel metric, this set is convex in a“ Geodesic” meaning (all geodesicsbetween two elements ofE are in E). Then barycenter of N elements ofE isin E.

( )2/1

log2/1

11

2/12/1

t

ABA

tt AeAAN

ktkt∑−

+=

−−

=ε

MJMJ =

MJMJCHPDME n =∈= /)(

)(CHPD n)(CGL n

MMJMJJMJ eJJeee === −− 11

www.thalesgroup.com

Karcher Flows for Toeplitz HPD matrices (THPD):

Trench/Verblunsky Theorem

115 /115 /

Covariances matrices are structured matrices that verify the following constraints :

Toeplitz Structure (for stationary signal) :

Hermitian structure :

Positive Definite Structure :

How to built a flow that preserves the Toeplitz stru cture ?

=

=

−

−

−

−

0121

*10

*212

*101

*1

*2

*10

0121

*10

*212

*101

*1

*2

*10

rrrr

rr

rrr

rrr

rrrr

rrrr

rr

rrr

rrr

rrrr

R

n

n

n

n

n

L

OOM

OO

MO

L

L

OOM

OO

MO

L

1−nR

conjuguate and d transpose: with +=+nn RR

( ) 00det such that ,0, >⇒=−∀>∈∀ + λλ nnnn λ.IRZRZCZ

Additional Constraint : Toeplitz Structure

[ ] kknn rzzEn =∀ −* ,

If you remember first example on « Right Triangle », in this case Pythagore constraint is replaced by HPD & Toeplitz constraints

116 /116 / Complex Circular Multivariate Gaussian Model

Radar Signal Model :

Complex Circular Multivariate Model :

Radar Model :

Complex autoregressive model :

Close Link with Issai Schur’s algorithm (1875-1941)

[Alpay] D. Alpay, « Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes », Panoramas et synthèse, n °6, Société Mathématique de France, 1998

[ ]

( )( ) [ ] nnnnnnn

RRTrn

nnn

RREmYmYR

eRRYp nn

=−−=

=+

−−− −

ˆet .ˆwith

..)()/(1.ˆ1π

[ ]

kikkkk

n

n

nnn

n

eibay

y

y

Y

YYER

m

φρ=+=

=

=

=+

with

Positive DefiniteHermitian Toeplitz

processmean zero 0

1

M

[ ] [ ]TNN

NNkknn

N

knkn

Nkn aaAbbEbyay )()(

12

0,*

1

)( ... and with ==+−= −=

−∑ σδ

117 /117 / Block matrix structure of CAR model

We can exploit Block structure of covariance matrix to compute Rao metric for CAR model :

Rao metric for Complex Multivariate Gaussian model of zero mean :

Block structure of covariance matrix for CAR model :

( )( ) ( )TABTrBAAAAdRRRdRRTrds ==== −−− ,et , with 222/12/1212

( ) ( ) ( ) ( ) 0detet loglog, 2/112

2/11

1

222/112

2/1121

2 =−== −−

=

−− ∑ IRRRRRRRRDn

kk λλ

+= +

−−−−−−−

+−−−−

1111111

1111

...

.

nnnnnn

nnnn

AARA

AR

αααα

−−+

=−−−

−+−−−

+−

−−

111

1111111

.

...

nnn

nnnnnnn

RAR

RAARAR

α

[ ] *)()(111

1

21 .

001

00

100

where1

.0

and .1with VVAA

A nn

nnnnn

=

+

=−= −

−−−−

−−

Nµαµα

118 /118 / Block matrix structure & extended eigen-values

Matrix or that is used for Rao Metric computation has same block structure

Invariance of Block structure :

This block structure to compute extended eigen-values :

2/1)1(1)2(2/1)1( .. nnn RRR −+2/1)1()2(2/1)1( .. −+−nnn RRR

[ ])1(1

)2(1

1-n)1(1

)2(1

2/1)1(1

)1(11

111111

1112/1)1(1)2(2/1)1(

and ..with

...

...

−

−−−

+−−

+−−−−−−

+−−−−+

=−=

+Ω==Ω

n

nnnnnn-

nnnnnn

nnnnnnn

AARW

WWW

WRRR

ααβα

ββββ

( ) ( )

( )

−Λ−=

=−

+−=

−+−

−−−−

−

=−

−+−

−− ∑

11

1

1)(

11)()(

1,

)(

1

1)()1(

2)1(1)(

11)()()(

.....

1

0.

..

nnnn

knnn

kn

k

nk

n

in

kn

i

ninn

knnn

kn

kn

WUIUX

X

XWF

λλ

λλλββλλ

119 /119 /

We can deduce an algorithm that could be parallelize d according to CAR model order for each extended eigen-values :

( ) ( )

( )

−Λ−=

=−

+−=

−+−

−−−−

−

=−

−+−

−− ∑

11

1

1)(

11)()(

1,

)(

1

1)()1(

2)1(1)(

11)()()(

.....

1

0.

..

nnnn

knnn

kn

k

nk

n

in

kn

i

ninn

knnn

kn

kn

WUIUX

X

XWF

λλ

λλλββλλ

( ) ..

.1)(

1,

)(2/1

1

12)()1(

2)1(12)(

)(1,

)(

nk

nk

n

in

kn

i

ninn

knk

nk

X

XXW

X

X−

−

=−

−+−

−+→ ∑

λλλ

( )( )∑

−

=−

−+−

−

−− >

−+==

1

12)1(

2)1(1

)1(

12)(1,

)(

1)(

1..

.1.

n

kn

k

nkn

nk

nn

kn

k

nn XW

X

F

ηη

ηβ

ηβ

∂ηη∂

[ ]nnnnn

nn

nn

n Tr Ω<<<<<<<< −−

−−

)(1

)1(1

)(2

)(1

)1(1

)( ...0 λλλλλλ

[ ] [ ]

+Ω=Ω

−

+

−−−

1111

11.

nnnnn WW

TrTr β

)()4( λF

stricly increasing curve on each interval :

)3(1λ )3(

2λ )3(3λ

[ ]4ΩTr

)4(1λ

Renormalisation at each iteration :

avec

)4(3λ)4(

2λ )4(4λ

( )

( )

.A

1

1

n

1=k)(

1,

)(1)()(

1-n

1

1)()(

=

=

∑

∑−

=

−

nk

nk

nk

n

n

k

nk

n

X

XF

F

∂ηλ∂

∂ηλ∂

Interpretation in term of projection of CAR vector :

0

Block matrix structure & extended eigen-values

120 /120 / Recursive Expression for Rao Metric

If we use block structure matrix, we can compute Rao metric at model order n from model order at n-1

Recursive Expression of Rao Metric:

We can observe that :

This could be interpreted with Information Geometry & Fisher matrix

( )[ ] 1111

2

1

121

212 .... −−+−−

−

−−

− +

+== nnnn

n

nnnnn dARdA

ddsdRRTrds α

αα

=+

−

−

−−

+

−

−−−

+−−

−

−

1

1

111

11111

2

1

1log

0

01.

log...

n

n

nnn

nnnnn

n

n

Ad

RAddARdA

d αα

αα

αα

( )

+

= −

−−

+

−

−

−

−1

111

1

1

1

0

01loglog

nnn

n

n

n

Ri

AAZ

ααα

[ ]

⊥

==

−−

−−−−

11

1111

log

ˆ

nn

nnn

A

RAIRnA

α

α

121 /121 / Non-Symmetric Square Root of Siegel Group

If we consider Cholesky decomposition of covariance matrix :

Cholesky decomposition (Goldberg inversion algorithm) :

All distribution of n-dimensionnal variable is associated with Affine Group. It is the element such that its action on vector

Is transformed to random vector :

This representation of Affine Group elements could be considered as non symmetric square root of Siegel Group element :

( ) ( )+

−−−−−

+−−−−

+−+−

ΩΩ=Ω

Ω−=

+Ω−===Ω

2/11

2/1112/1

11

2

n

1111

121

. and 01

1with

.

1.1..

nnnnn

n

nnnn

nnnnnnn

AW

AAA

AWWR

µ

µα

),0(~ nn INZ),(~ nnn ANX Ω

=

+Ω=

Ω −−−− XAZZA nnnn

1

.

11.

01

12/11

2/111

+Ω +−−−−

+−

1111

1

.

1

nnnn

n

AAA

A

122 /122 / Recursive Expression of Kullback Divergence

We consider expression of Kullback Divergence for M ultivariante Gaussian Law of zero mean :

Kullback divergence is given by :

If we consider recursive relation :

Kullback Divergence is given recursively by :

( ) [ ][ ]nTrRRK nnnn −Σ−Σ= ln2

1, )2()1(

[ ]

−

−=

==−=

+Σ==Σ

−−

+−

+

−

−−−

+−

+−−−−−−

+−−−−+

)1(11

)1(1

2)1(

)1(

)1()1()1()1()1(1

)2(1

1-n)1(1

)2(1

)1(11-n

111111

1112/1)1(1)2(2/1)1(

0

01

0

1

1

1

.. with and .V where

...

...

nn

n

n

n

nnnnn

nnnn

nnnnnn

nnnnnnn

FI

AF

FFRAAF

VVV

VRRR

µ

αααβ

ββββ

11. −− Σ=Σ nnn β [ ] [ ] [ ]1111 .1. −+−−− ++Σ=Σ nnnnn VVTrTr β

( ) ( ) ( ) ( )[ ]11111)2(1

)1(1

)2()1( ln..12

1,, −−

+−−−−− −+−+= nnnnnnnnn VVRRKRRK βββ

123 /123 /Trench/Verblunsky Theorem & Partial Iwasawa Parameterization

All Toeplitz Hermitian Positive Definite Matrix can beparameterized by Reflection/Verblunsky Coefficients:

Block structure of covariance matrix & Verblunsky Parameterization:

Verblunsky/Trench Theorem: Exitence of diffeomorphism ϕ

+= +

−−−−−−−

+−−−−

1111111

1111

...

.

nnnnnn

nnnn

AARA

AR

αααα

−−+

=−−−

−+−−−

+−

−−

111

1111111

.

...

nnn

nnnnnnn

RAR

RAARAR

α

[ ]*)(

)(11

1-00

11

21

.

001

00

100

where1

.0

and .1with

VVAA

A

P

nn

nn

nnn

=

+

=

=−=

−−−−

−−

−

Nµ

ααµα

( ) 1/with

,...,,

:

110

1*

<∈=

×→

−

−+

zCzD

PR

DRTHPD

nn

nn

µµϕ

a

S. Verblunsky(PhD student of Littlewood)

Iwasawa(Lie Group

Theory)

124 /124 / Preservation of Toeplitz Structure by Verblunsky Coefficients

Conformal Information Geometry metric (metric = Hes sian of Entropy):

Entropy ΦΦΦΦ as Kähler potential:

Conformal metric on Verblunsky parameterization:

[ ]Tnn P 110

)(−= µµθ L

)*()(

2 ~

nj

ni

ijgθθ ∂∂Φ∂≡

( ) ( ) [ ] [ ] ..ln.1ln).(.logdetlog,~

0

1

1

210 PenkneR)P(R

n

kknn πµπ −−−−=−=Φ ∑

−

=

−

with

[ ] ( )∑−

=

+

−−+

==

1

122

22

0

0)()(2

1)(.

n

ii

inij

nn

din

P

dPndgdds

µ

µθθ

E. Kähler

1t with coefficien Verblunsky : <kk µµ

125 /125 /

Erich Kähler…

Entropy U=-logdet[R] , for Complex Autoregressive model, can be considered as a Kähler Potential that depends on reflection coefficient. This expression is exactly the same than the first potential proposed by E. Kähler

« Kähler Erich, Mathematical Works », Edited by R. Berndt and O. Riemenschneider, Berlin, Walter de Gruyter, ix, 2003

Seminal Erich Kähler Paper, 1932, Hamburg

126 /126 / Regularized Verblunsky Coefficients: Regularized Burg Algorithm

• Regularized Burg Algorithm (THALES Patent)

+−=−+=

=

+=

=

−=+−+

−

+−−−=

==

=

==

−−

−−

−−

−

+=

−

=

−−−

+=

−

=

−−

−−−

=

∑ ∑

∑ ∑

∑

)(.)1()(

)1(.)()(

11 , .

1

).()2.( with ..2)1()(

1

...2)1().(2

to1For : (n) tep .

1

)(.1

ech.) nb. : (N 1 , )()()(f

:tion Initialisa .

1*

1

11

)(

)*1()1()(

)(0

221

)(

1

1

0

2)1()(2

1

2

1

1

1

1

)1()1()(*11

)0(0

1

2

0

00

kfkbkb

kbkfkf

a

,...,n-k=aaa

a

nkakbkf

nN

aakbkfnN

MnS

a

kzN

P

,...,N k=kzkbk

nnnn

nnnn

nn

n

nknn

nk

nk

n

nkN

nk

n

k

nk

nknn

N

nk

n

k

nkn

nk

nknn

n

N

k

µµ

µµ

πγββ

βµ

Regularization

127 /127 / Median by Median Reflection/Verblunsky Coefficients µµµµk

n,3µ

n,1µn,2µ

nmedian,µ

Classical Karcher Flow in Unit Disk

1,3 +nµ

1,1 +nµ

Duale Karcher Flow

εµl/ avec µ

µγw l,n

m

lkk k,n

k,nnn <= ∑

≠=1

nnmedian w=+1,µ

*nk,n

nk,nk,n .wµ

wµµ

−−

=+ 11

1,2 +nµ

*nmedian,n

nmedian,nmedian,n wµ

wµµ

++

=+ 11

128 /128 / Median by Median Reflection/Verblunsky Coefficients µµµµk

129 /129 / Inductive Mean (K.T. Sturm)

A natural way to define the « expectation » E(Y) of a random variable Y is to use generalizations of the law of large numbers.

Given any sequence of points, we defi ne a new sequence of points by induction on n as follow (inductive mean) :

strongly depend on permutations but

Extension of scalar case :

( ) 2/11

/12/11

2/11

2/11/1111 and −

−−

−−−− =•== n

n

nnnnnnnn SSBSSBSSBS

n

kkB 1= n

kkS 1=

nnnnnn bn

sn

bssbs11

1 and 1/1111 +

−=•== −−

( )( ) ( )112 1

),( BVn

SBEE n ≤δ

130 /130 / Stochastic Approach: Arnaudon Flow

Random selection at each iteration of 1 point in the disk + driven of the flow

along the geodesic

nnrand

nnrandnn γw

),(

),(

µµ

=

( )

=

−

−

+)(exp

exp.exp

)(1

)(1

1

nrandm

nrandm

nmnx

xtm

n

n

n

M. Arnaudon, C. Dombry, A.Phan, L.Yang, ”Stochastic algorithms for computing means of probability measures”, http://hal.archives -ouvertes.fr/hal-00540623

131 /131 /

nnrand

nnrandnn γw

),(

),(

µµ

=

( )

=

−

−

+)(exp

exp.exp

)(1

)(1

1

nrandm

nrandm

nmnx

xtm

n

n

n

Arnaudon Stochastic Flow

132 /132 / Arnaudon Flow extended on compact space

New Arnaudon Stochastic Flow on the sphere

M. Arnaudon, L. Miclo, ”Means in complete manifolds: uniqueness and approximation”, http://arxiv.org/abs/1207.3232

133 /133 /

Case of Poincaré unit disk:

G+ sub-group of automorphisms preserving orientation

avec et

G operate on of probability measure of

Principe : assign to all probability measure on a point such that is conformal and verify:

There is only one conformal way to assign to each probability measure on one vectors fields on D such that:

with

Conformal Busemann Barycenter (Douady-Earle,…)

1 / <∈= zCzD

za

azz

*1−−λa 1=z 1<a

( )1SP 1 / 1 =∈= zCzSµ 1S ( ) DB ∈µ( )µµ Ba

0)(.0)(1

=⇔= ∫S

dB ζµζµ

µ 1Sµξ 0)(.)0(

1

== ∫S

d ζµζξµ

( ) ( ) ( ) 0)(*1

1)0()('

1)(

1

2

)(* =

−−−== ∫

S

gw

dw

ww

wgw

wζµ

ζζξξ µµ

zw

wzzgw *1)(

−−=

134 /134 /

Case of Poincaré Disk

could be described according to that is the tangent vector of the geodesic in pointing to :

In Poincaré Geometry of unit disk, vectors field is the gradient of a function whose the level sets are the horocycles tangents to in :

Median Fréchet Barycenter = Busemann Barycenter

)(zµξ )(zζξDz∈ 1S∈ζ

∫=1

)()()(S

dzz ζµξξ ζµ

ζξ ζh1S 1S∈ζ

[ ]∫∫

∫

−=

−

−==

∇=

−→11

1

)(),(),0()(1

log2

1)(

)()(: avec

12

2

Sr

S

S

drzdrdLimdz

zzh

dzhzhh

ζµζζµζ

ζµξ

µ

ζµµµ a

135 /135 / Median Fréchet Barycenter = Busemann Barycenter

136 /136 / Median Fréchet Barycenter = Busemann Barycenter

137 /137 /

From Cartan Center of Mass to Busemann Barycenter

Elie Cartan has proved for simply connected riemannian manifolds with negative or null curvature, existence and unicity of the minimum of:

Following function, is simply connexe and reachs its minimum:

when goes to infinity converging to a point of , the normalized distance function to converges : Busemann function

Fixing an origine . Let finite measure on with a support of more than 2 points. The following strictly convexe function reachs its minimum at a unique point, independant of O choice and is called Busemann Barycenter :

Busemann Barycenter on Cartan-Hadamard Manifold

( )∫nH

zdzxdx )(, 2 νa

( )∫nH

zdzxdx )(, νa

z θ nH∂z

),(),(lim)( zOdzxdxBz

−=→θθ

nHO∈ ν nH∂

∫∂

=nH

dBx )()( θνβ θν

138 /138 / Comparison of Mean & Median Doppler Spectrum

Raw Doppler Spectrum

Mean Doppler Spectrum

Median Doppler Spectrum

Range axis

Doppler axis

No preservation of discontinuities

Perturbation by outlier

139 /139 /

Diffusion Fourier Equation on 1D graph (scalar case )

Approximation par un Laplacien discret :

with arithmetic mean of adjacent points :

Dicrete Fourier Heat Equation for Scalar Values in 1D :

By Analogy, we can define Fourier Heat Equation in 1D grapf of HDP(n) matrices :

( )nnnnnn uu

xx

uu

x

uu

xt

u

x

u

t

u −∇

=

∇−−

∇−

∇=

∂∂

⇒∂∂=

∂∂ −+ ˆ

212

112

2

( ) 2/ˆ 11 −+ += nnn uuu

J. Fourier

Other approach to smooth spectrum : Fourier Heat Equation

2,,1,

.2 with ˆ.).1(

x

tuuu tntntn ∇

∇=+−=+ ρρρ

( ) ( )( ) tntntntntntntntn

tntntntntntnXXX

tntn

XXXXXXXX

XXXXXXeXX tntntn

,12/1,12/1,1

2/12/1,1,1

2/1,1

2/1,1,

2/1,

2/1,,

2/1,

2/1,

2/1,

ˆlog2/1,1,

ˆwith

ˆ2/1,,

2/1,

−++−+−

−++

−−+

==

==−−

o

ρρ

140 /140 / Isotropic Diffusion of Doppler Spectrum

141 /141 / Anisotropic Diffusion of Doppler Spectrum

142 /142 / Canonical Exemple: Monovariate Gaussian Law

Gauss-Laplace Law

Fisher Information Matrix for Gauss-Laplace Gaussian Law:

( )( )

=≥

−−

= −−

σθθθθθθσθ

mIEI

T and )(ˆˆ with

20

01.)( 12

( ) ( )

+

=+== − 22

22

2

2

22

2.2.2.. σσ

σσ

σθθθ d

dmddmdIdds T

Rao-Chentsov Metric of Information Geometry

H. Poincaré

.2

σim

z += ( )1 <+−= ωω

iz

iz

( )22

2

2

1.8

ω

ω

−=⇒

dds

Fisher Metric is equal to Poincaré Metric for Gaussian Laws:

( )

)*2()1(

)2()1()2()1(

2

)2()1(

)2()1(

22112

1),(with

),(1

),(1log.2,,,

ωωωωωωδ

ωωδωωδσσ

−−=

−+=mmd

143 /143 / Median of Gaussian laws

144 /144 /

Travaux de Jacob Burbea (1984)

On considère une densité de probabilité pour une variable de Cn

Identification avec le carré du module d’une fonction normalisé:

On considère la pseudo-distance de Skwarczynski :

On en déduit l’équivalence entre métrique de Fisher et métrique Hessienne :

Lien entre géométrie Hessienne et de l’information

( ) ( ) ( ) ( ) ( ) ( )∫∫ ∂

∂∂

∂=∂

∂∂

∂= − )(/log/log

/)(//

/)(**

1 tdz

ztp

z

ztpztptd

z

ztp

z

ztpztpzg

jijiji µµ

22/1*

1

2 log)(µ

ppdzdzgzds ji

N

iji

∂==∑= ( )

1)/(. avec

)/(/2

2

=

=

µψ

ψ

z

ztztp

( ) ( ) )/(.,/ * ztzzKztg ψ= ( ) µ)/(),/(, * wtgztgwzK =

( ) ( ) µψψµψψλ )/(,/1)()/()/(1, wtzttdwtztwz −=−= ∫

[ ] ( ) *

1,

*22

2222 ,loglog,),( ki

N

ji kizwBergman dzdz

zz

zzKKKKKKddwzdds ∑

=

−

= ∂∂∂=∂∂=∂−∂∂=−== µµ ψψψλ

( ) ( )),(log)/(log)/(log)/(log

)/(.*,/

)/()/(**

2

zzKztgztgztpztzzKztg

ztztp−+=⇒

=

=

ψ

ψ

*

*2

*

2

*

*2

*

2 ),(log)/(log),(log)/(log

jijiji

jiji zz

zzK

zz

ztpEg

zz

zzK

zz

ztp

∂∂∂=

∂∂∂−=⇒

∂∂∂−=

∂∂∂

www.thalesgroup.com

New « Ordered Statistic High Doppler Resolution CFAR »

(OS-HDR-CFAR)

146 /146 / Radar Data Measurement

Radar estimates Electromagnetic fluctuations of :

Reflectivity : Amplitude of EM Wave reflected energ y

Phases variations due to Doppler-Fizeau

Polarization of EM Wave

147 /147 /

Radar estimates Electromagnetic fluctuations of :

Reflectivity : Amplitude of EM Wave reflected energ y

Phases variations due to Doppler-Fizeau

Polarization of EM Wave

Radar Data Measurement

2

arctan1

2

=s

s

φ

2

arctan22

21

3

+=

ss

s

τ

( )( )

=

−+

=

=

τφτφτ

2sin

2sin2cos

2cos2cos

Im2

Re2

0

0

0

0

*12

*12

2

2

2

1

2

2

2

1

3

2

1

0

s

s

s

s

zz

zz

zz

zz

s

s

s

s

Sr

∑=

=⇒

=n

kktéreflectivi

n

zn

R

z

z

Z0

21

1M

( ) ϕππ

π

λ

itffi

tf

r

cVr

rDoppler

eec

RtuAtz

etuts

Vf

cV

cVf

Doppler

r

22

2

1/0

0

0

2.)( : Receive

).()( :Transmit

.2/1

/2

+

<<

−≈

=

≈−

=

ChristianDoppler

Toeplitz Hermitian Positive DefiniteCovariance matrix

=

=

−

−+

011

1

01

11011

ccc

c

cc

ccc

z

z

z

z

ER

n

n

nnL

OOM

MO

L

MM

148 /148 / Radar Processing based on Covariance Matrix

Doppler Processing(Time Covariance Matrix)

Antenna Processing(Space Covariance Matrix)

STAP Processing(Space-Time Covariance Matrix)

Polarimetric Radar Processing(Polarimetry Covariance Matrix)

149 /149 /

Modify or Hide in the header / footer properties : Date

Military Air Defense Application

Detection of slow/stealth targets in inhomogeneous Clutter

New requirements in Air Defense to detect low altit ude or surface targets at low elevation. Target Doppler is very cl ose to fluctuating Clutter Doppler (Ground & Sea Clutters)

Detection of asymmetric & stealth targets in Ground Clutter Microlight Airplane

General Aviation

UAV & Micro UAV

Micro Helicopter

Detection of small targets in Sea Clutter Wooden & inflatable canoe

Jetski

Unmanned Boat

Naval Micro Helicopter

Periscope

Detection of low RCS targets

Detection of teneous Doppler Signal

Increase Range & Reactivity

150 /150 /


Civil Air Traffic Control Application

Monitoring of turbulences: New Requirement in ATC (SESAR)

In Turbulences, signal is no longer characterized by Doppler velocity Mean but by Doppler Spectrum Width & “shape”

Atmospheric Air turbulences

Eddy Dissipation Rate

Turbulent Kinetic Energy

Airplane Wake-Vortex turbulences (A380, B747-8)

Circulation

Decay Rate

Windshear in Final Approach

Headwind

Crosswind

Improve Safety by mitigating weather hazards

Increase Capacity by reducing safe separations

151 /151 / Challenges of Doppler Radar Processing: Detection on Ground

Detection of asymmetric & stealth targets in Inhomogenesous Ground Clutter

Classical Doppler Filter Banks (or FFT) are not efficient with very short bursts (<16 pulses) :

Low Resolution of Doppler Filters with short Bursts (Low sidelobes / high loss, wide filter)

If Target Doppler is between two Doppler filters, energy is spread on adjacent filters. Gain between 2 filters is lower than gain at filter center ("Straddling loss")

Ground Clutter Energy is not limited to zero-Doppler filter but pollution is spread over all filters due to poor Filter-Banks Resolution & Doppler side lobes in case of very short Bursts.

Filter 0 Filter 1 Filter 7Filter 2 Filtre 6Filter 3 Filter 4 Filter 5

Pollution of all Doppler Filters in case of Burst w ith low number of pulses

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10

Frequence normalisee (V/Va)

Ga

in

e

n d

B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10


Ga

in e

n d

B

S/N gain [dB]

0-10-20-30-40-50-60 V/ Vamb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10


Gain e

n dB

152 /152 /


Challenges of Doppler Radar Processing:Detection in Sea

Detection of slow targets in

Sea Clutter

Sea Clutter is highly inhomogeneous

Doppler fluctuation

Time/space Fluctuation

Sea Clutter is dependant of Sea current

Surface wind

fetch

Bathymetry

Sea Clutter is corrupted by Spikes due to breaking waves

“Moutonement”

Offshore Sea Doppler Spectrum

Close to the Shore Sea Doppler Spectrum

(Breaking waves)

153 /153 / Challenges of Doppler Radar Processing: Atm. turbulences

Monitoring of atmospheric turbulences

Air turbulence is characterized by Spread of fluctuating speeds

This composition of different Doppler speeds in Radar cells generates a Widen Doppler Spectrum

Speed variance of Doppler Spectrum Width are related to 2 measures of turbulence :

EDR: Eddy Dissipation Rate

TBE: Turbulent Kinetic Energy

154 /154 /

Monitoring of Wake-Vortex Turbulences

Wake vortex generate to contra-rotative roll-up spirals

Mean speed depends on cross-wind

Wake-Vortex has spiral geometry with increasing speed in the core and decreasing speed outside the core

Wake-Vortex Speed and structure depend on Wake-Vortex age/decay phase

Wake-Vortex Strength is characterized by Circulation in m 2/s

Monitoring of Windshear

Inversion of speed in range or in altitude

Microburst in the same radar cell

Challenges of Doppler Radar Processing: Wake Turbulences

Wake-Vortex Doppler Spectrum

Wind-Shear Doppler Spectrum

155 /155 /

For detection of Slow & Stealth/Small Target in inhomogeneous clutter, we need simultaneously :

High Doppler Resolution with short Bursts

Robust CFAR in inhomogeneous clutter & closely sepa rated targets

Proposed Solution : OS-HR-Doppler-CFAR

Avoid drawbacks of Doppler Filters / FFT in case of short bursts

Take advantages of Robust Ordered Statistic of OS-C FAR (Ordered Statistic CFAR, Median-CFAR)

Challenges to define OS-HR-Doppler-CFAR :

Can we order « Doppler Spectrums » : NO there is no total order of covariance matrices R1>R2>…>Rn

There is only Partial « Lowner » order : R1>R2 if and only if R1-R2 Positive Definite

Can we define « Median » of « Doppler Spectrums » : YES !!! In a « Metric Space », the median is defined as the point that minimizes the « geodesic »

distance to each point (compared to the mean that minimizes the square distance to each point)

We can define a deterministic or stochastic gradient flow that converges fastly to « median spectrum » (Modified Karcher Flow : THALES Patent )

Transport Optimal : barycentre de Fréchet-WassersteinOS-HD-Doppler-CFAR

156 /156 /

Median Mean

x1

x2

x3

x1

x2

x3

Median Mean

x1

x2

x3

x1

x2

x3

x’2

x’2

outliers outliers

MEDIAN MEAN

Sensitivity to outliers : Median versus Mean

157 /157 / Classical CFAR Chain versus OS-HDR-CFAR

I&Q (z1,…zn)

DopplerFilters

Log|.|

CA-CFAR

OS-CFAR >Seuil

OU

OU

Cases distances

>SOS-HDR-CFAR

(Doppler Spectrum Median)

ClassicalCFAR Chain

AdvancedOS-HDR-CFAR Chain

Casek-1

Casek

Casek+1

Log|.|CA-CFAR

OS-CFAR >Seuil

OU

>S

I&Q (z1,…zn)

High Doppler Resolution Coefficients

(P0,k,µ1,k,… µm,k)

Casek-2

Casek-1

Casek

Casek+1

Casek-2

∑= −

−−+=

m

n kn

mediannkn

k

medianmedianmmedianmediankmkk

mediann

thnmP

PmPPdist

1

2*

,

,,2

,0

,02,,1,0,,1,0 )]

1(arg)[()ln()),...,,(),,...,,((

,µµ

µµµµµµ

158 /158 / OS-HDR-CFAR on reflexion coefficients

Médiane

Cases distance

Cellule sous test

Cellules ignorées

Cellules ambiance

µ1 µ2

Calcul de la distance

>S

OS-HDR-CFAR

Valeur médiane (OS) pour ambiance

159 /159 / Reflection coefficients of only Ground Clutter

µ1µ2

µ3

160 /160 / Real Data Records of Ground Clutter

I&Q Data Records of Ground Clutter

12 and 10 scans

161 /161 / Context of the detection problem

Azimuth

Ran

ge c

ases

162 /162 / OS-HDR-CFAR Performance Analysis Methodology

Insertion of synthetic targets in real ground clutter

Normalized Doppler/speed distance from target to clu tter : α*Variance with Variance = Capon Spectrum Interval > X dB

Power of inserted targets : P target

Capon Spectrum before target insertion ( α=1; SNR=13dB)

Capon Spectrum before target insertion

163 /163 / Methodology

Capon Spectrum insertion before target insertion ( α=1; SNR=13dB)

164 /164 / Tests on real recorded I&Q data of Ground Clutter

Low Elevation Beam Recording

Perfo HDR MedianPerfo HDR mean

OS-HDR-CFARBF+OS/CA-CFAR

OS-HDR-CFARBF+OS/CA-CFAR

165 /165 / COR Curve

166 /166 / Comparison Frechet Mean / Frechet Median

167 /167 / Simulation

Robustness to multi-target case for a Weibull texture with

a shape parameter θ=0.1

30 target samples in the reference cells

168 /168 /Pd versus SNR for Pfa = 2.1*10-4

α = 1

α = 0.25 α = 0.5

169 /169 / Rao’s L1 approach

170 /170 / Rao’s L1 approach

Result on ground clutter :

The estimator of Γ is non-parametric : we should incorporate the information of stationarity

www.thalesgroup.com

Karcher Flows for Toeplitz-Block-Toeplitz HPD matrices

(TBTHPD)

172 /172 / Extension to Toeplitz Block Toeplitz Hermitian PD Matrices

Previous results can be extented to Block-Toeplitz M atrices :

=

=+

++

+

+0

,

01

1

01

10

1, ~

~

RR

RR

RRR

R

RR

RRR

Rn

nnp

n

n

np

L

OOM

MO

L

*

1~

=

n

n

R

R

VR M

=

00

0

0

00

L

MN

NNM

L

p

p

p

J

J

J

Vwith

173 /173 / Extension of Trench/Verblunsky Theorem for TBTHPD Matrices

Every Toeplitz-Block-Toeplitz HPD matrix can be param etrized by Matrix Verblunsky Coefficients:

Extension of Trench/Verblunsky Theorem: Existence o f Diffeomorphism ϕϕϕϕ

+= +−

+−

+nnnnpnn

nnnnp AARA

AR )))

)

...

.1,

11, αα

αα

−−+

=++−

+npnnp

npnnnpnnnp RAR

RAARAR

,,

,,1

1, .

...)

)))α

[ ]

+

=

=

=−=

−

−−

−

−−

+−

p

pn

p

pnnp

nn

p

n

nn

n

-n

nn

nnn

I

JAJ

JAJ

AA

A

A

A

RαAA

*11

*11

1

11

01

011

1

.0

and

,.1with

M)

M)

αα

( ) n

nn

nnnn

IZZnHermZD

AARR

SDTHPDTBTHPD

<∈=

×→

+

−−

−×

/)(Swith

,...,,

:11

110

1

a

ϕ

174 /174 / Information Geometry Metric : Entropic Kähler Potential

Kähler potential defined by Hessian of multi-channe l/Multi-variate entropy :

( ) ( ) ( )( )[ ]npji

npnpnp

RHessg

csteµRTrcsteRRΦ

,

,,, logdetlog~

φ=⇒

+−=+−=

( ) [ ] [ ]0

1

1, det..log.detlog).(

~RenAAIknR

n

k

kk

kknnp πΦ +−−=∑

−

=

+

( )[ ] ( ) ( )[ ]∑−

=

+−+−+− −−−+=1

1

112

01

02 )(.

n

k

kk

kk

kkn

kk

kk

kkn dAAAIdAAAITrkndRRTrnds

We recover the previous metric for THPD matrix !!

175 /175 / Multi-Channel Burg Algorithm: MOPUC

Multi-Channel Burg Algorithm :

[ ] [ ] [ ]

[ ] [ ][ ]

[ ][ ][ ]

=

=

−=

+

−−=

++−=⇒+

==

+−=

−+=

=

=

+−=

−=

+=

+

+

+

−+

=

++

=

++

=

+++

−++

+++

+++

=

=

−−−

−−

−−

−−

∑∑∑

∑

∑

++

)()(

)()(

)1()(

with

)()()()()1()(2

2

)()()( with )()1()(

)1()()(

and

001

0

01

100

with

)()()(

)()()(

,,...,,0,,...,,,...,,

1

111

11

1*11

*

00*111

111

0

0

*

0

*11

*12

*11

11

12

1121

11

kkEP

kkEP

kkEP

kkkkkkA

JJPPJJPPAJJPPTrMin

kZkkkJJAkk

kAkk

IAJ

lnkJZkJAk

lkZkAk

IJJAJJAJJAAAAAAAA

bn

bn

bn

fn

fn

fn

bn

fn

fbn

nN

k

bn

bn

nN

k

fn

fn

nN

k

bn

fn

nn

fn

bn

fbTn

fbn

nn

bn

fn

A

bf

fn

nn

bn

fn

bn

nn

fn

fn

n

n

l

nl

bn

n

l

nl

fn

nnn

nn

nn

nn

nnnn

nn

nn

εεεεεε

εεεεεε

εεεεε

εεε

ε

ε

L

MNN

NM

L

( ) Siegelnnn

nn

nn DiskAIAA ∈⇒< +

++++

+++

111

11

11.

176 /176 / Mostow Decomposition (& Berger Fibration)

Mostow Theorem :

Every matrix of can be decomp osed :

where

is unitary

is real antisymmetric

is real symmetric

Can be deduce from

Lemma : Let and two positive definite hermit ian matrices, there exist a unique positive definite hermitian ma trix such that:

Corollary : if is Hermitian Positive Definite, there exist a unique real symmetric matrix such that :

M ( )CnGL ,SiAeUeM =

UAS

A BX

BXAX =M

SSS eMeM 1* −=

177 /177 /

( )( )( ) MMPPPPPPS

PPPPPeePeP SSS

+−−

−−−

==⇒

=⇒=+

with log.2

1

:corrolary Lemma

2/12/12/1*2/12/1

2/12/12/1*2/12/12212*

Mostow Decomposition

Mostow Theorem :Mostow Theorem :Mostow Theorem :Mostow Theorem :

All matrix of can be decompos ed in :

is unitary, is real antisymmetric réelle and is real symmetric

Proof :

M ( )CnGL ,SiAeUeM =

U A S

( )SS

SSiASSSiAS

SiASSiA

ePeP

eeeeeeeeP

eeeMMPeUeM

212*

2222*

2

−

−−−−

+

=⇒

==⇒

==⇒=

( ) MMPPeei

A

Peee

SS

SSiA

+−−

−−

==⇒

=

with log2

1

:y injectivit lleexponentie 2

iASeMeU −−=

178 /178 / Siegel Disc Automorphism

Automorphism of Siegel Disc given by :

All automorphisms given by :

Distance given by:

Inverse automorphism given by :

nSD

( ) tZn UZUZCnUUSDAut )()(/,),(

0Φ=Ψ∈∃∈Ψ∀

( )

−+

=∈∀)(Φ1

)(Φ1log

2

1,,,

W

WWZdSDWZ

Z

Zn

( ) ( )( ) ( ) 2/1

00

1

00

2/1

00)(0

ZZIZZIZZZZIZZ+−+−+ −−−−== ΦΣ

( ) ( ) ( )( )( )

( ) ( )

−−=

++==⇒

−−=−−=

−++

+−+−

−+−++

2/1

00

2/1

00

01

01

1

00

2/1

00

2/1

00

with

)()(0

ZZIZZIG

ZGIGZZ

ZZIZZZZIZZIG

Z

Σ

ΣΦ

Σ

179 /179 / Iterated Computation of Median in Siegel disk

( )( )

( )( ) ( )( ) ( )

( ) ( ) ( ) ( ) 2/1

,

2/11

1

2/112/1

1

1

,1

,

,,,2/1

,

2/12/1,

*,

2/1,

2/1,,

2,

2,,,,

10010

with

then1For

with

with log.2/1

: w

until on Iterate

0

,,

,,,,

−++−++

+−+−++

+

≠=

+−−

−−−

−−=++=

−−−−=

==

<=

==

===⇒=

<

==

∑

nnnmediannnnnmedian,n

nnk,nnnk,nnnk,n

k,nGk,n

Fnk

m

lkk

nknn

nknknknknknknknknk

S

nk

SS

nkiA

nkk,nSiA

nknk

Fn

mm,,median,

GGIWGGIGGGIGGW

GGIWGIGWGGIW

W W,...,m k

εHl/ HγG

WWPPPPPPS

ith eWeeWeUHeeUW

Gn

,...,WW,...,WW and Wtion :Initialisa

n

nknk

nknknknk

Φ

ε

www.thalesgroup.com

New « Ordered Statistic Space-Time Adaptive

Processing » (OS-STAP)

181 /181 / Secondary Data Covariance Matrix Estimation

( ) ( ) ∑−

=

−−

−

++=

1

1 ,

,222/10,0

2/10

)(Φ1

)(Φ1loglog,

n

kk

mediankA

kmediankA

medianmedianA

ARRRRRd

kk

kk

( ) ( )( ) ( ) 2/112/1)( ZZIWZIZWZZIWZ

+−+−+ −−−−=Φ

1,1

1,1,0

/1

1,1

1,1,0 ,...,,,...,, −

−=−− → n

mediannmedianmedianBergerMostow

M

p

npnpp AARAAR

182

A Riemannian approach for training data

selection in STAP processing

J-Fr. DEGURSE1,2, L. SAVY1, J-Ph. MOLINIE, Prof. S. MARCOS2

1ONERA, Electromagnetism and Radar Department (DEMR)2Laboratoire des Signaux et des Systèmes (L2S SUPELEC-CNRS-Univ-Paris-Sud)

Airborne detection of slow moving targets in non stationary environments

183183

Speed (or Doppler frequency)

Ran

ge

Hidden area (clutter)

Fast targets(exo clutter)

Slow targets(endo clutter)

Need for STAP Processing !

Space Time Adaptive Processing (STAP)

184

• Doppler and receiving angles are coupled for clutter echoes

• Monodimensionnal clutter in a bidimensionnal space

Ground echo (clutter)

Doppler Freq. Or Speed

Doppler Frequency Fd (or speed)

Ang

le (

sin θ)

Objective

Which secondary data to choose?

185

Secondary data CUT*

Guard cells

Range gates

Secondary data

* Cell under test

Guard cells

• One covariance matrix is estimated for each range gate

• Clutter from secondary data must be homogenous with clutter from CUT

186186

Speed (or Doppler frequency)

Ran

ge

Hidden area (clutter)Clutter type

Gaussian clutter

level c0

Gaussian clutter

level c0/10

Gaussian clutter

level c0

Spiky clutter

Gaussian clutter

level c0

Sidelobes clutter A

Area

B

C

B

D

B

Detection of slow moving targets in non stationnary environements

187

Classical secondary data selection

Use Euclidean distance between covariance matrices:

Less powerfulclutter area

Main lobe clutter energy

Only detects power changing regions

188

Physical criterion for secondary data selection

is the weights vector obtained with from the range cell

then applied to

matchedfilter

whitenI+N

Output

• if is the optimal weights vector, is “white”, hence

we want

• that leads to the optimal criterion or “distance” but this is NOT a distance:

189

Secondary data selection: physical POV

Optimal criterion between covariance matrices:

Clutter power change

Heterogeneous area (spiky clutter)

No detection of the highto low power clutterchange

190

Secondary data selection: physical POV

Symetrized physical criterion between covariance matric es:



Link to the Riemannian geometry

191

• For positive definite Hermitian covariance matrix, th e right metric is defined by:

and the distance:

• From the following serie :

first order approximation:

with

This approximation is the physical criterion

192

Secondary data selection using Riemannian metric distance

Riemannian distance between covariance matrices:



Real data performance: SAREX

193

Speed (m/s)

Dis

tanc

e ce

ll

Targets

RAMSES RADAR

• 4 channels antenna

• X-band, side looking

• PRF = 3125 Hz

• Va = 85 m/s

Real data performance: SAREX (Euclidean Geometry)

Selection strategy: guardcells = 2, we pick the 10 closest matrices out of 20a maximum distance value can be set so the number of chosen matrices could be < 10

Threshold is set to 10dB

Real data performance: SAREX (Riemannian Geometry)

Selection strategy: guardcells = 2, we pick the 10 closest matrices out of 20a maximum distance value can be set so the number of chosen matrices could be < 10

• Clutter rejection is much better

• Small signal loss (1 dB) on target 3

• Small signal gain (0.5dB) on target 1

Threshold is set to 10dB

196

Way forward

Use geometric mean in Riemannian geometry instead o f classical arithmetic mean

Algorithm implementation on GPU to increase calcula tion speed• massively parallel algorithms, should run very fast !

Application to air-to-air radar mode: high heteroge neous sidelobesclutter + continuous mainlobe clutter

www.thalesgroup.com

Extension to Non-Stationary Time-Doppler Signal

198 /198 / Operational Need

Non-Stationary Doppler Processing

New challenge in Radar is the processing of non-sta tionary signal

fast time variation of Doppler Spectrum in one burst

Most of processing chains implemented in radars mak e the assumption of

Doppler stationary signal during the burst waveform duration.

This assumption is not always true, especially in c ase of:

high speed Doppler fluctuation

abrupt Doppler changes

of target or clutter signal

We can observe non-stationarity for

high speed or abrupt Doppler variations of clutter or target signal but also in case of target migration during the burst duration due to high range resolution.

199 /199 / Non-Stationary Doppler: Target Signal

Detection of Target Non-Stationary Doppler Signal

High speed Doppler fluctuation:

Rotor and blades of helicopter

Abrupt Doppler changes:

helicopter pop-up or low altitude target demasking behind a creast line

rocket with splitting events (multi-headed rocket, decoys, debris,...), Multiple Reentry vehicle payload for a ballistic missile that deploys multiple warheads

missile firing by an airborne platform

Target Range Migration in High Range Resolution Mod e (UWB)

Applications for other MFR radar functions

Kill Assessment for staring antenna (long burst/waf eform)

Manoeuver Detection for Tracking (coupled with Rada r Res. Mgt)

Advanced NCTR (Non-Cooperative Target Recognition)

Robust target recognition with time-scale invariance properties

200 /200 / Non-Stationary Doppler: Clutter Signal

Non-Stationary Doppler Signal of Non-homogeneous Clutter (amplitude/Doppler/Polar fluctuations)

Sea Clutter

Littoral Environment

Breaking Waves

Spikes

Atmospheric clutter

Turbulent Atmosphere (high Eddy Dissipation Rate)

Wind-Shear, Micro-Burst, Downdraft

Wake-Vortex of all airborne platforms

Wake-Vortex of Civil Commercial Aircrafts on airpor t (European SESAR, US Nexten in US, Japanese CARATS , NTU ATM R I « Research Institute » in Singapore)

Wake-Vortex of helicopters in Non Light of Sight

Wake-Vortex of fighters / Delta Wing UCAV in Rain

201 /201 / Non-Stationary Time-Doppler Signal in Radar

Time-Doppler Signature of BEL206 helico on one rang e cell staring antenna

Doppler signature of Sea Clutter

Time-Doppler signature of wind farm turbine blades

Time-Doppler signature of Helco Blades Wake-Vortex Doppler signature of aircraft wake-vortex

Bande UHF S X

Fmean 435 MHz 3.15 GHz10.00

GHz

“UWB” ≥ 87 MHz ≥ 630 MHz

≥ 2 GHz

Range

resolution≤ 1.72 m ≤ 23.8

cm≤ 7.5 cm

PRIN.

r∆

Target trajectory

PRIN

rV cible

rad .

∆>

)/ 36( / 10

25

300:

)/ 212( 24.

:

hkmsmV

cmr

kmPortée

smmsPRIN

Exemple

ciblerad >

=∆

×=

www.thalesgroup.com

NEW APPROACH AND MODEL FOR NON-STATIONNARY

DOPPLER SIGNAL: LOCAL STATIONARITY MODEL OF

NON-STATIONARY TIME SERIES

203 /203 / State-of-the-Art: Active Research

Non-Stationnary Time Series Analysis

Based on past work : Y. Grenier, “Time dependent ARMA modelling of nonstationary signals”. IEEE, Trans. Acoust. Speech S ignal Process. 31, 899–911, 1983

New active Research: Workshop 2012 : SFdS (Société Française de statistique)

Time series do not benefit from generous properties of stationarityof ergodicity which allow to recover the law of a pr ocess by the only observation. Various alternatives are envisaged:

the Local Stationarity allows to take into account regular variations of behavior

The Isotonique Regression gives one simple model in which a test of stationarityis even possible

The Hidden Markov Chains models the times of these breaks

204 /204 / State-of-The-Art: LOCAL STATIONARITY

Heidelberg University, 2012 : R. Dahlhaus, « Locally Stationary Processes », Handbook of Statistics, 30, 2012, http://arxiv.org/abs/1109.4174

If one restrict to linear processes or even more to Gaussian processes then amuch more general theory is possible. We give a general definition for linearprocesses and discuss time varying spectral densities in detail. Study containsthe Gaussian likelihood theory for locally stationary processes.

Mannheim University, 2011: Vogt, M. “Nonparametric regress ion forlocally stationary time series”. PhD, 2011

we study nonparametric models allowing for locally stationary regressors and aregression function that changes smoothly over time. These models are a naturalextension of time series models with time-varying coefficients. we show that themain conditions of the theory are satisfied for a large class of nonlinearautoregressive processes with time-varying regression function.

Télécom ParisTech: E. Moulines, P. Priouret, et F. Roueff., “Onrecursive estimation for locally stationary time varying a utoregressiveprocesses”, Ann. Statist. 33, 2610–2654, 2005

the properties of recursive estimates of tvAR-processes have been investigated inthe framework of locally stationary processes. The asymptotic properties of theestimator have been proved including a minimax result.

www.thalesgroup.com

INFORMATION GEOMETRY MODEL OF STATIONARY TIME SERIES

(Duality Concept): SIEGEL METRIC FOR HERMITIAN

POSITIVE DEFINITE (HPD) MATRIX

206 /206 /

Information Geometry for Multivariate Gaussian Law of zero Mean and intrinsec Geometry of Hermitian Positive Defini te Matrices (particular case of Siegel Upper-Half Plane) provid e the same metric


Geometry of Siegel Upper-Half Plane:

( )( )( )212nn dRRTrds −=

[ ]

( )( )[ ] nn

nnnnn

RRTrn

nnn

RRE

mZmZR

eRRZp nn

=

−−=

=+

−−− −

ˆ and

.ˆwith

..)()/(1.ˆ1π

−=

*

2

.

)/(ln)(

ji

nnij

ZpEg

∂θ∂θθ∂θ

0=nm

with

Siegel HPD Matrix Metric = IG Metric of Multiv. Gaussian Law


( ) ( )( ) iYXZZdYdZYTrdsSiegel +== −− with 112

==

nRY

X 0 ( )( )( )212nn dRRTrds −=

207 /207 /

Siegel Distance:

Particular Case (X=0) and General Case:

Particular Case (pure imaginary axis) :

General Case of Siegel Upper-Half Plane Distance:

Distance between HPD Matrices: particular case of Siegel

0 with ≠∈+= XSHiYXZ n

( ) n

n

k k

kSiegel SHZZZZd ∈

−+

= ∑=

211

221

2 , with 1

1log,

λλ

with ( ) 0.),(det 21 =− IZZR λ

( ) ( )( ) ( )( ) 1

2121

1


( ) ( ) ( )∑=

−− ==n

kkRRRRRd

1

222/112

2/1121

2 log..log, λ

( ) 0det 12 =− RR λwith

0 avec >= RiRZ

208 /208 / HPD Matrices Geometry

Geometry of Hermitian Positive Definite Matrices given by:

Geodesic :


Symetric Space as studied by Elie Cartan : Existence of bijective geodesic isometry


Cartan-Hadamard Space (Complete, simply connected wi th negative sectional curvature Manifold)

[ ]10 with ),(.))(,( ,tRRdttRd YXX ∈=γ( ) ( )

YXYX

X

t

XYXXXRRRt

X

RRRR

RRRRRReRt XYX

o===== −−−−

)2/1( and )1( , )0(

)( 2/12/12/12/12/1log2/1 2/12/1

γγγγ

( ) ( ) 2/12/12/12/12/11-),( avec )(X ABAAABABABAXG BA

−−== ooo

Xx)d(x,x)d(x,xd(x,z)),xd(x

xzxx

∈∀+≤+

∀∃∀

224

que tel ,2

22

122

21

21

www.thalesgroup.com

MODEL NON-STATIONARY TIME SERIESAS GEODESIC PATH OF LOCALLY

STATIONARY SIGNAL ON THPD MATRIX MANIFOLD

210 /210 /NS TIME SERIES = GEODESIC PATH OF LS TIME SERIES ON MANIFOLD

Locally Stationarity Assumption

We will assume that each non-stationary signal in o ne burst can be split into several short signals

with less Doppler resolution

but locally stationary,

represented by

time sequence of stationary covariance matrices (Toeplitz Hermitian PD matrix)

a geodesic path/polygon on covariance matrix manifold (THPD Matrix Manifold)

we will consider Time-Doppler signature:

as a geodesic path on an Information Geometry manifold (of covariance matrices)

by analyzing the time series in the burst with a short sliding window

211 /211 /NS TIME SERIES = GEODESIC PATH OF LS TIME SERIES ON MANIFOLD

( ) ( ) ( )∑=

−− ==n

kkRRRRRd

1

222/112

2/1121

2 log..log, λ

( ) 0det 12 =− RR λ

( ) ( )YXYX

X

t

XYXXXRRRt

X

RRRR

RRRRRReRt XYX

o===== −−−−

)2/1(et )1( , )0(

)( 2/12/12/12/12/1log2/1 2/12/1

γγγγ

EACH TIME SERIE IS A PATH ON COVARIANCE MATRIX MANIFOLD

)( 1tR)( 2tR)( 3tR)( 4tR

www.thalesgroup.com

FRECHET DISTANCE BETWEEN CURVES AND GEODESIC EXTENSION ON

MANIFOLD

213 /213 /FRECHET Distance between Curves/Paths in Euclidean Space Rn

Classical Frechet/Hausdorff distances between Curves

Classically, Hausdorff distance, that is the maximu m distance between a point on one curve and its nearest neighb or on the other curve, does not take into account the flow of the c urves

The Fréchet distance between two curves is defined a s:

The minimum length of a leash required to connect a dog and its owner as they walk without backtracking along t heir respective curves from one endpoint to the other.

Let P and Q be two given curves, the Fréchet distance between Pand Q is defined as the infimum over all reparameterizationsand of of the maximum over all of the distance in between and . In ma thematical notation:

αβ [ ]1,0 [ ]1,0∈t

))(( tP α ))(( tQ β( )

[ ]( ) ))(()),((,

1,0,tQtPdMaxInfQPd

tFréchet βα

βα ∈=

214 /214 /FRECHET Distance between Curves/Paths in Euclidean Space Rn

( )[ ]

( )

[ ] [ ]

→

=∈

surjective and ingNondecreas 1,01,0: and

))(()),((,1,0,

βα

βαβα

tQtPdMaxInfQPdt

Fréchet

215 /215 / FRECHET Distance: Free-Space Diagram

Polynomial Time Algorithm: Free-Space Diagram

Alt and Godau have introduced a polynomial-time algo rithm to compute the Fréchet distance between two polygonal c urves in Euclidean space.

For two polygonal curves with m and n segments, the computation time is

Alt and Godau have defined the free-space diagram between two curves for a given distance threshold ε is a two-dimensional region in the parameter space that consist of all point pa irs on the two curves at distance at most ε:

The Fréchet distance is at most ε if and only if the free-space diagram contains a p ath which from the lower left corner to the upper right corner which i s monotone both in the horizontal and in the vertical direction.

( ))log(mnmnO

( ) [ ] ( ) εβαβαε ≤∈= ))(()),((/1,0,),( 2 tQtPdQPD Fréchet

( )QPdFréchet ,),( QPDε

216 /216 / FRECHET Distance: Free-Space Diagram

Polynomial Time Algorithm: Free-Space Diagram

In an n m free-space diagram, shown in following fig ure, the horizontal and vertical directions of the diagram co rrespond to the natural parametrizations of P and Q respectively.

Therefore, if there is a monotone increasing curve from the lower left to the upper right corner of the diagram (corr esponding to a monotone mapping), it generates a monotonic path th at defines a matching between point-sets P and Q

www.thalesgroup.com

FRECHET DISTANCE OF GEODESIC CURVES REPRESENTATIVE OF TIME-

DOPPLER SIGNATURE ON INFORMATION GEOMETRY MANIFOLD

218 /218 /

Thales Land & Air Systems Date

FRECHET Distance of Geodesice Curves in Information Geometry

Fréchet Distance between 2 geodesic paths

We extend previous Fréchet distance between paths i n Rn to Information Geometry Manifold.

When the two curves are embedded in a more complex metric space, the distance between two points on the curve s is most naturally defined as the geodesic length of the sho rtest path between them.

If we consider N subsets of M Radar pulses in the burst, the Doppler burst can then be described by a poly-geode sic lines on Information Geometry Manifold.

The set of N covariance matrices time serie describe a discrete ”polygonal” geodesic path on Info rmation Geometry Manifold, and we can extend previous Frech et Distance but with Geodesic distance:

( ) ( ) ( ) NtRtRtR ,...,, 21

( )[ ]

( )

( ) ( )

=

=

−−

∈

tRtRtR))t((t)),R(Rd

tRtRdMaxInfRRd

geo

geot

Fréchet

22/112

2/1121

2

211,0,

21

))(())(())((log)(with

))(()),((,

αβαβα

βαβα

219 /219 / Extended FRECHET Distance of Geodesice Curves

Extended Fréchet Distance between 2 geodesic paths

As classical Fréchet distance doesn’t take into acco unt with Inf[Max] close dependence of elements between points of time series paths, we propose to define a new distance g iven by:

We have then to find the solution for computing the geodesic minimal path on the Fréchet free-space diagram. The length of the path is not given by euclidean metric where )

but geodesic metric weighted by d(.,.) of the free- space diagram :

( ) ( )

= ∫−

1

0

21,

21 ))(()),((, dttRtRdInfRRd geopathgeo βαβα

22 dtds = ∫=L

dsL

∫∫ ==L

g

L

g dsdsgL . with ( )dttRtRddsg .))((),(( 21 βα=

220 /220 / Extended FRECHET Distance of Geodesice Curves

Extended Fréchet Distance between 2 geodesic paths computable by « fast marching / shortest path » algorithms

( ) ( )

= ∫−

1

0

21,

21 ))(()),((, dttRtRdInfRRd geopathgeo βαβα

« Fast Marching » Algorithm

221 /221 /Computation of « Median Curves » in Covariance Matrix Manifold

Ordered CFAR for Non-Stationary Signal:

« Median Curve/Curve » on Matrix Manifold should Minimi ze :

Non-Stationary Ordered-Statistic CFAR:

( ) ThresholdRRd SerieTimetestundercell

SerieTimeMedianpathgeo ≥−−

− __,

( )∑=

−−−

−−

=n

k

SerieTimek

SerieTimepathgeo

R

SerieTime RRdMinArgRSerieTimeMedian

1

,

PhD Thesis 2013-2016

Approximated solution: take Path of the set that minimizes this functional

( ) ( )

= ∫−−

−

1

0

21,

21 ))(()),((, dttRtRdInfRRd geoSerieTimeSerieTime

pathgeo βαβα

222 /222 / Modified Fréchet distance between paths on Manifold

( ) ( ) ( )∑=

−− ==n

kkRRRRRd

1

222/112

2/1121

2 log..log, λ

( ) 0det 12 =− RR λ

( ) ( )YXYX

X

t

XYXXXRRRt

X

RRRR

RRRRRReRt XYX

o===== −−−−

)2/1(et )1( , )0(

)( 2/12/12/12/12/1log2/1 2/12/1

γγγγ

( )[ ]

( ) ( )( )

[ ] [ ] tedécroissannon surjection 1,01,0:et

)(,)(,1,0,

→

=∈

βα

βαβα

tQtPdMaxInfQPdt

Fréchet

Statistique de signature temps-Doppler: statistiques de

chemins géodésiques sur une variété ou un espace métrique

( ) ( ) ( )( )

[ ] [ ] tedécroissannon surjection 1,01,0:et

)(,)(,1

0,

→

= ∫

βα

βαβα

dttQtPdInfQPdTHALES

Utilisation de techniques de « Fast Marching »

)(tα

)(tβ)(tα

)(tβ

)( 1tR)( 2tR)( 3tR)( 4tR

223 /223 / References

[1] M. Fréchet, “Sur quelques points du calcul fonct ionnel », Rendiconti del Circolo Mathematico di Palermo, n °22, p1-74, 1906

[2] M. Fréchet, « L’espace dont chaque élément est une courbe n’est qu’un semi-espace de Banach », Annales scientifique s de l’ENS, 3ème série, tome 78, n °3, p.241-272, 1961

[3] M. Bauer, P. Harms, P.W. Michor, « Almost local metrics on shape space of hypersurfaces in n-spaces », arXiv:1001.071 7v4, 27 Jan 2011

[4] A.Chouakria-Douzal and P.N. Nagabhushan, « Improv ed Fréchet distance for time series », in Data Sciences and Cla ssiffication, Springer 2006, pp 13-20

[5] P.W. Michor and D. Mumford, « An overview of the Riemannian metrics on spaces of curves using the Hamiltonnian approach, » Appl. Comput. Harm. Analysis 23 (1): 74-113, 2007

[6] M. Bauer, M. Bruveris, C. Cotter, S. Marsland, P. W. Michor, “Constructing reparametrization invariant metrics on spaces of plane curves”, preprint http://arxiv.org/abs/1207.5965

[7] R. Dahlhaus, « Locally Stationary Processes », Handbo ok of Statistics, 30, 2012 , http://arxiv.org/abs/1109.4174

www.thalesgroup.com

Synthesis of theoretical framework of Structured

Covariance Matrices Geometries study

225 /225 / Synthesis

Covariance Matrix

Space or Time Covariance Matrix

Space-Time Covariance Matrix

Entropy Hessian metric

Karcher/Fréchetbarycenter/Median

Partial Iwasawa Decomposition

Isobarycentric & Fourier Flow

Median inPoincaré’s disk

Hermitian Positive Definite Matrix Group : HPD(n) nICnSpCnPSp 2/),(),( ±≡

Symplectic Quotient Group :

Toeplitz structure

Multi-variate Entropy Hessian metric

Partial Iwasawa Decomposition

Median inSiegel’s disk

Block-Toeplitzstructure

Polar Fibration in Poincaré’s disk

Mostow Fibration of Siegel’s disk

INFORMATION GEOMETRY

Maslov-Leray index in Siegel

Disk

Maslov-Leray Index in

Poincaré Disk

( ) ( ) ( )enRRΦ πlogdetlog~ −−=

( )ji

ji

RΦg

θθ∂∂=

~2

Robust Space-Time Processing

Robust Doppler Processing

( )[ ] ( ) ( )[ ]∑−

=

+−+−+− −−−+=1

1

112

01

02 )(.

n

k

kk

kk

kkn

kk

kk

kkn dAAAIdAAAITrkndRRTrnds( )∑

−

= −−+

=

1

122

22

0

02

1)(.

n

ii

in

din

P

dPnds

µ

µ

CARTAN-SIEGEL SYMMETRIC GEOMETRY

TRANSPORT GEOMETRY

Berezin Quantification

Douady-Earle Conf. Barycenter

226 /226 / General Diffeomorphism for TBTHPD Matrices

( )

( )( ) 1/

,...,,log

/

,...,,

*

11100

111

110

<=

×∈→

<=

×∈

−−

+

−−−

zzzD

DRPR

IZZZSD

SDTHPDAAR

mm

m

nm

nn

µµ

11 :by coded StateDoppler -Spatio −− ××∈ nm SDDR

( )[ ] ( ) ( )[ ]∑−

=

+−+−+− −−−+=1

1

112

01

02 )(.

n

k

kk

kk

kkn

kk

kk

kkn dAAAIdAAAITrkndRRTrnds

[ ] ( )( ) ( )∑−

=

+

−−+==

1

122

22

0)()(2

1)(log.

m

ii

imij

mm

dimPdmdgdds

µ

µθθ

227 /227 / Main Theorem : Poincaré Everywhere

( )( )

( ) ( )( ) ( )

( ) ( ) τφ

τφ

τφτφτ

,, :Model Poincaréy Polarimetr2

/arctan and

2

/arctanwith

)2sin(

2sin.2cos

2cos.2cos

1

.

Im.2

Re.2

0

22

21312

0

*

*

22

22

3

2

1

0

s

sssss

s

zz

zz

zz

zz

s

s

s

s

Sz

zE

HV

HV

VH

VH

V

H

+==

=

−+

=

=⇒

=

rr

( )

( )( ) 1/

,...,,log

/

,...,,

*

11100

111

110

<=

×∈→

<=

×∈

−−

+

−−−

zzzD

DRPR

IZZZSD

SDTHPDAAR

mm

m

nm

nn

µµ

Oriented Ellipse

Compact Hadamard Space:R x Poly-Poincaré Disk x Poly-Siegel Disk

Polarimetry Poincaré Model: R+*xS1

11 :by coded StateDoppler -Spatio −− ××∈ nm SDDR

1 :by coded StatePolar SR×∈

111 :by coded State EM −− ×××∈ nm SDDSR

228 /228 / Hadamard Compactification

From the cylindrical representation ofwe contract the caps and the side,

and blow-up the two circular corners

Géométrie des bords :compactifications différentiables

et remplissages holomorphesBenoît Kloeckner

UMPA, École normale supérieure de Lyon

229 /229 / Toeplitz Hermitian PD Matrices: Simple case n=2

Definite PositiveHermitian Toeplitz

0det

0

222 bah

hiba

ibah

+>⇔>Ω

>

+−

=Ω

Hadamard Compactification

Disk Unit Poincaré :

1/

01

*1.

*

D

zzDh

iba

Rh

h

<=∈+=

∈

>

=Ω

+

µ

µµ

1/

)log(

with ),log(

<=∈∈

+=

zzD

Rhh

ibah

µ

µµScale

ParameterShape

Parameter

a

b

h>90°

AmbligoneTriangleh2>a2+b2

( )( )

222

22

2222

0

02

0det

bah

bah

bahh

hiba

ibah

+>⇔>+±=⇒

=+−+−⇒

=

−+−−

λλ

λλλ

λ

230 /230 / Short Summary of the approach

matrix HPD Toeplitz

+−

=Ωhiba

ibah

=Ω⇒

)log(

*)log(

.

.

,)log(

mean

mean

hmeanmean

meanmeanh

Mean

meanMean

eh

he

h

µµ

µ

DRh ×∈µ),log(

:cationCompactifi Hadamard

[ ] [ ]

−

=

−+

=+

µµµµ

µd

hd

d

hdd

h

dhds

)log(

1

10

01)log(

122

22

222

1/

01

1.

,zation parameteri New*

*

<∈=∈+=⇒

>

=Ω

×∈ +

zCzDh

iba

h

DRh

µ

µµ

µ

( )µµ

µ

µ),log(,),log(arg

,)log(

1),log(

hhdMin

h

ii

N

i

pgeodesique

h

meanMean

∑=

=

FRECHET Barycenter in Metric Space

www.thalesgroup.com

Miscellaneous

www.thalesgroup.com

Quantization of Complex Symmetric Spaces by Berezin

233 /233 / Classical Symmetric Bounded Domains

Symmetric bounded domains in Cn are particular cases of symmetric

spaces of noncompact type.

Elie Cartan has proved that there is :

4 types of classical symmetric bounded domains

2 exceptional types (group of motion E6 and E7)

Classical Symmetric Bounded Domains (extension of Poincaré Disk)

>−+

<

−<

+

++

021

1ZZ

:such that row 1 and columnsn with matricesComplex : IV Type

porder of matrices symmetric-skewComplex : III Type

porder of matrices symmetricComplex : II Type

columns q and rows p with matricesComplex : I Type

):(

MatrixComplex r rectangula:

2

t

ZZZZ

Ω

Ω

Ω

Ω

conjugatetransposedIZZ

Z

t

IVn

IIIp

IIp

Ip,q

234 /234 / Kernel function of Symmetric Bounded Domains

Luo-Geng Hua has computed the kernel functions for all classical domains :

Particular case (p=q=n=1) : Poincaré Unit Disk

( ) ( ) ( )

( ) ( ) ( )( ) domain theof volumeeuclidean theis where

, : IV Typefor 2ZZ11

,

1 , : III Type

1 , : II Type

, : I Type

for det1

,

-**t*

*

Ωµ

νΩµ

ν

ν

ν

Ωµ

ν

ν

nΩZWWWWZK

pΩ

pΩ

qpΩ

ZWIWZK

IVn

IIIp

IIp

Ip,q

=−+=

−=

+=

+=

−=

+

−+

( ) ( )2*

*

*11111

1

1,

1/

zwwzK

zzCzΩΩΩΩ IVIIIIII,

−=

<∈====

235 /235 / Analytic automorphisms of Bounded Domains

Groups of analytic automorphisms of these domains a re locally isomorphic to the group of matrices which preserve following forms:

The group consits of block matrices A (generalizatio n of the fractional linear transformation). For the first 3 types :

−===

=

−===

−=

−===

=

−==

n

tIVn

p

p

p

ptIIIp

p

p

p

ptIIp

p

pIp,q

I

IHHAHAHAHAΩ

I

IL

I

IHLALAHAHAΩ

I

IK

I

IHKAKAHAHAΩ

AI

IHHAHAΩ

0

0,, , : IV Type

0

0,

0

0,, , : III Type

0

0,

0

0,, , : II Type

1det,0

0, , : I Type

2*

*

*

*

( )( ) 122211211

2221

1211 −++=⇒

= AZAAZAgZ

AA

AAA

236 /236 / Analytic automorphisms of Bounded Domains

All classical domains are circular following from C artan’s general theory, and the point 0 is distinguished for the potential :

Berezin quantization is based on the construction of the Hilbert Space of functions analytic in ΩΩΩΩ :

( ) ( )( ) ( ) νΦ −+−=

= ZZI

K

ZZKZZ detlog

0,0

,ln,

**

Z

gZZgjZZKZgjzgjgZgZK

ZZdK

ZZKhc

ZZdK

ZZKZgZfhcgf

h

h

∂∂==

=

=

∫

∫−

−

−

),( with ),(),(),(),(

),()0,0(

),()(

),()0,0(

),()()()(,

***

*

/1*1

*

/1*

µ

µ

237 /237 / Berezin Quantification of Poincaré Unit Disk

The most elementary example of Berezian quantificati on is, in the case of complex dimension 1, given by the Poincaré unit Disk with volume element :

Map from path on D to automorphy factor :

*22)1.(2/1 dzdzzi ∧− −

( ) ( )

*

2

*

2

**2

22

**

1

)()(

)(lnRe2)(1ln)( : potentialKähler

1 where with )1,1(

/)1,1(1/

zz

zF

zz

gzF

zFazbgzFzzF

baab

bagSUg

SSUzCzD

∂∂∂=

∂∂∂

⇒

++=⇒−−=

=−

=∈

=<∈=

( ) ( ) ( ))()(

1ln1ln))0(()0(

1

*

*1

2

1

21*1*

22

gFgFab

bag

babgFabgba

=⇒

−−

=

+=

−−=⇒=

−−

=−

−−

238 /238 /

Extension for Siegel Unit Disk :

The orbit of the matrix Z=0 is the space of matrice s of the form :

( )( )( ) ( )

( )( ) )()(

lnRe2)())((

lndetlog)( : potentialKähler

)(

0 where

0

0 with and with

/

**

**

1**

*

*

**

ZFZgF

ZBAtraceZFZgF

ZZItraceZZIzF

AZBBAZZg

BAAB

IBBAA

I

IJJJgg

BB

BAg

IZZZSD

t

t

t

n

∂∂=∂∂

++=−−=−−=

++=

=−=−

−==

=

<=

++

−

+

+

+

Berezin Quantification of Siegel Unit Disk

( ) ( ) ( )BBItraceBBIgFABg ++− +=+=⇒= lndetln))0(()0(1*

239 /239 / Special Berezin Coordinates

For every symmetric Riemannian space, there exist a dual space being compact. The isometry groups of all the compa ct symmetric spaces are described by block matrices (the action of the group in terms of special coordinates is described by the sa me formula as the action of the group of motions of the dual domain).

Berezin coordinates for Siegel domain :

===

=

=

+

+

+−

0

0 with , :ly equivalentor

, 1**

I

ILLLI

AB

BA

AB

BA

t

t

t

ΓΓΓΓ

ΓΓ

( ) ( ) ( )++− +=+=⇒= BBItraceBBIgFABg lndetln))0(()0(1*

( )( )

==

++=⇒

=

−

−

IiI

iIICCC

AWAAWAWAA

AA

2

1 with :Isometry

)(

1

122211211

2221

1211

ΓΓ

ΓΓ

240 /240 / Invariant metric in Special Berezin Coordinates

Let M be a classical complex compact symmetric spac e. The invariant volume and invariant metric in terms of special Ber ezin coordinates have the form :

Link with : For arbitrary Kählerian homogeneous spa ce, the logarithm of the density for the invariant measure is the potent ial of the metric

( ) ( ) ( )

( )

( ) ν

βααββα

βαβα

πµµ

−++=

∂∂∂−==

=

∑

WWIWWF

WW

WWFgdWdWgds

WWdWWFWWd

nL

det),( where

,ln with

,,,

*

*

*2

,

*,

2

***

www.thalesgroup.com

Analysis on Symmetric Coneby Faraut

242 /242 / Link with Analysis on Symmetric Cones (J. Faraut)

Jacques Faraut has published 2 books on Analysis on Symmetric Cones & on Lie Group :

[1] J. Faraut & A. Koranyi, « Analysis on Symmetric cones », Clarendon Press, Oxford, 1994

[2] J. Faraut, « Analyse sur les groupes de Lie », Calvage & Mounet, Paris, 2006

Harmonic Analysis in the special case of the cone o f positive definite matrices in the vector space of all real symmetric matrices plays a fundamental role in :

Number Theory (Minkowski, siegel, Maas,…)

Statistics (Wishart, Constantine, James, Muirhead)

Physic (study of Lorentz cone)

General Case has been studied by : Gindikin

Vinberg

Symmetric Cones & Tubes (over them) are example of Riemannian Symmetric Spaces


Convex cones :

Let V be a finite dimensional real Euclidean Space. A sub set C of V is said to be a cone if :

The closed dual cone of any cone C is defined by :

The automorphism group of an open convex cone is defined by :

is a closed subgroup of , and hence i s a Lie Group. The open cone is said to be homogeneous if acts on it transitively. The open cone is said to be symmetric if it is homo geneous and self-dual :

CxCx

∈⇒

>∈

λλ 0

( ) ( ) CC

CxyxVyC

=

∈∀≥∈=##

# ,0/

( ) ( ) Ω=Ω∈=Ω gVGLgG /( )ΩG Ω

( )ΩG ( )ΩGLΩ ( )ΩG

( ) ( )( )Ω∈

=

GLgg

ygxygx

ofelement an ofadjoint the:*

*


Convex cones :

For any proper open convex cone :

characterizes the symmetric cones

Characteristic Function of a Cone :

Let be a proper open convex cone, its charact eristic function is :

dy Euclidean Measure on V

The second derivative is po sitive definite at each point

Ω( ) ( )

( ) ( )Ω∈⇒

Ω∈Ω=Ω

Ω=Ω

GgGg

GG

**

**

if

( ) ( )Ω=Ω GG *

Ω( )

∫Ω

−=*

)( dyex yxϕ

( ) )(det)( , 1

xggxGg ϕϕ −=Ω∈∀)()( , 0 , xxλxgx nϕλλϕλ −=>=

0),()()(

)(log),(

0

>⇒

+=

=

=

uuGtux

dt

dxD

xDDvuG

x

tu

vux

φφ

ϕ)(log2 xD ϕ



Riemannian structure g is given by

Ω

[ ]( )

loglog

2

1log

log)(log

with)(log

2u

2

22

2

∫∫∫∫

∫∫

∫−

+=

=

=

dudv

dudvdd

du

dud

dudxd

xdg

vu

vuv

u

uu

u

ϕϕ

ϕϕϕϕ

ϕ

ϕϕ

ϕϕ

ϕ



The adjoint :

The map is a bijecti on : and has unique fixed point

Symmetric Cone as Riemannian Symmetric Space :

The bilinear form is positive definite, therefore it defines a Riemannian metric on

The cone equiped with this metric is a Riema nnian Manifold

Since the cone is symmetric, the map is a bijection and an isometry (the manifold is a Riem annian Symmetric Space given by this isometry)

( ) )()( with )(log* xfDuxfxx u=∇−∇= ϕ** Ω∈Ω∈ xx a ( ) nxx =*

)(log),( xDDvuG vux ϕ=Ω

ΩΩ )(log* xxx ϕ−∇=a

( )constxx

xx

=

=

)()( *

**

ϕϕ


The cone of Positive Definite Symmetric matrices :

Inner Product & quadratic form:

Let be the set of positive definite symmetric matrices. The set is an open conex cone and is self-dual :

is homegeneous :

Characteristic function :

( ) ( ) ( )( ) 0)( , 0 ,

, ,,

>=≠∈∀

=∈Tn xQR

xyTryxRnSymyx

ξξξξξ)(RnΠ=Ω

Ω=Ω*

)(RnΠ=Ω

=

==⇒Ω∈

T

nT

gxgxg

Igggxx

)(

)(

ρρ

( ) ( )nn IgxIgx ϕρϕρ 1)(det)()(

−=⇒=( )

( )[ ] ( )n

n

nIxx

gg

gxϕϕ

ρ2

1

1

2

)det()()det()(det

)det()det( +−+ =⇒

=

=

( ) ( )nIxnx ϕϕ logdetlog)1(2

1)(log ++−=

1*1 )1(2

1)det(log −− +=⇒=∇ xnxxx


Symmetric Cone & Exponential Familly of probability measure

Let µµµµ be a positive Borel Measure on euclidean space V. Assume that the following integral is finite for all x in an open set :

For , consider the probability measure (exponential familly) :

Then

V⊂Ω( )

∫−= )()( ydex yx µϕ

Ω∈x

( ) ( ) )()(

1, yde

xdyxp yx µ

ϕ−=

)(log),()( xdyxypxm ϕ−∇== ∫

( ) ( )( ) )(log),()()()( xDDdyxpvxmyuxmyvuxV vu ϕ=−−= ∫

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Homogeneous Hyperbolic Affine Hyperspheres

by SasakiF. Barbaresco

250 /250 / Affine Hyperbolic Hypersphere

Affine hyperbolic hypersphere

A locally strongly convex hypersurface in the affine space Rn+1 is called an affine hyperbolic hypersphere if the affin e normalsthrough each point of the hypersurface either all in tersect at one point, called its center, that is on the convex sid e.

This class of hypersurfaces was first studied system atically by W. Blaschke in the frame of affine geometry.

E. Calabi raised a conjecture that:

these hypersurfaces are asymptotic to the boundary of a convex cone

every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature.

He proved this conjecture for homogeneous convex cones under some conditions on the action of the automorphism group of the cone.

251 /251 / Homogeneous Affine Hyperbolic Hypersphere

Homogeneous Affine hyperbolic hypersphere

The theory of homogeneous convex cones plays a cent ral role.

Let V be a nondegenerate open convex cone in Rn+1(x) and V’ be its dual. A(V) means the group of all linear transforma tions which leave V invariant. The characteristic function of V , is given by the equation:

with the value of the linear functional ξ at x

We denote by Sc the level surface of w hich is a noncompact submanifold in V, and by ω the induced metric on Sc

The Hessian defines the metric on V.

Assuming the cone V is homogeneous under A(V), Sasa ki proved that Sc is a homogeneous hyperbolic affine hypersphere and every such hyperspheres can be obtained in this way

Sasaki remarks that ω is identified with the affine metric and S c is a global Riemannian symmetric space when V is a sel f-dual cone.

VxdexV

xV ∈= ∫

− , )('

, ξφ ξ

ξ,x cxVV =)(: φφ

Vd φlog2

KOSZUL-VINBERG CHARACTERISTIC

FUNCTION

252 /252 / Proper Affine Hyperbolic Hypersphere

Proper Affine hyperbolic hypersphere

Let S be a hypersurface in Rn+1 and be the imbedding of S. The imbedding defines a volume bundle va lued quadratic form G on S by the equation:

in terms of local coordinates of S. This is invariant under unimodular affine transformations in Rn+1. If this quadratic form is supposed to be non-degenerate, it defines a pseudoriemannian structure tensor g with corresponding volume element dv(g) , uniquely defined by the equation:

∑ ∧∧⊗

∂∂

∂∂

∂∂∂=

ji

njinji

dydydydyy

f

y

f

yy

fG

,

11

2

...,...,,det

1: +→ nRSff

( )nyy ,...,1

)(gdvgG ⊗=


Proper Affine hyperbolic hypersphere

we assume that the set S is locally strongly convex . Then the tensor g can be chosen to be positive definite choosing the orientation of S so that G is positive valued.

With this Riemannian metric, called the affine metr ic, the affine normal is defined to be the vector where ∆ is the Laplace-Beltrami operator with respect to g.

For an affine hyperbolic hypersphere with the center at the origin, n satisfies the equation:

where H, called the affine mean curvature , is a nonzero constant.

Calabi proved that the hypersurface S is a proper affine hypersphere with its center at the origin and the af fine mean curvature H if u satisfies the equation:

fnn ∆= )/1(r

Hfn −=r

( ) 22

)(det −−=

∂∂∂ n

ji

Huu ξξξ


Non-parametric characterization of an affine hypersphere

Let be a linear coordina te system of Rn+1 and

be the representation of S as the graph of a

locally strongly convex function for ranging

in a domain

Let be the i mage of D under the locally

invertible mapping

We define the function by the equation

where is the pairing giving the canonical duality.

u is the Legendre transform of and also the domai n Ω the Legendre transform of S with respect to the coordinates

( )11,..., +nxx( ) nn xxfx ,...,11 =+

f ( )nxxx ,...,1=nRD ⊂

( )nnR ξξξ ,...,, 21⊂Ω

( )iin x

ffffgradf

∂∂=== where,...,1ξ

( ) Ω∈nu ξξ ,...,1

( ) )(,)()( xgradfxxfxgradfu +−=.,.

f ( )11,..., +nxx

255 /255 / Construction of hyperbolic affine hypersphere

Theorem [Wu and Sacksteder]

Let S be a closed hyperbolic affine hypersphere with center at the origin and the affine mean curvature H . the hypersurface S is complete (with respect to the affine metric and with respect to the induced metric of the Riemannian metric of Rn+1), noncompact, orientable, smooth and locally strongly convex. In this situation we have:

Such a surface is the full boundary of some closed convex body and is the graph of a non-negative smooth strictly convex function defined in some hyperplane.

By this theorem the hypersurface S can be written globally as the set , where is a positive smooth strictly convex function on

The tangent plane at any point of S cannot contain the origin. In other words the affine normal is not tangent to S:

the normal vector in Euclidean sense at one point in S is proportional to:

with the coordinate of the Legendre transformation

The affine normal at that point is

with

Hence:

( ) nn xxfx ,...,11 =+ f 01 =+nx

( )1,,...,1 −= nEn ξξriξ

∂∂+

∂∂

∂∂= ∑

=

n

ii

in

n111

2,,...,

1 ξξρρ

ξρ

ξρ

ρr( ) 2/1det +−= n

ijfρ0/1, ≠−= ρEnn

rr

256 /256 / Asymptotic to boundary of convex cone

S is asymptotic to the boundary of a convex cone when the boundary is equal to the set of all asymptotic line s of S:

Let then and

is an open non-degenerate convex cone.

S is asymptotic to the boundary of V

Theorem:

Every closed hyperbolic affίne hypersphere is asymptotic to the boundary of a convex cone. Conversely, every non-degenerate cone V determines a hyperbolic affine hypersphere asymptotic to the boundary of V, and uniquely determined by the value of its mean curvature.

Legendre transformation is an isometry with respect to the LN-metric ττττ (introduced by Loewner & Nirenberg) and the affine m etric g:

For a negative convex solution u of

Loewner & Nirenberg defined the metric on a bounded convex domain Ω in

SpRkpS nk ∈∈= + /1 'for 0' kkSS kk ≠=I

U0>

=k

kSV

( ) 22

)(det −−=

∂∂∂ n

ji

Huu ξξξ

udHu

21=τ)(ξnR

257 /257 / Loewner-Nirenberg Metric

LN-metric ττττ (introduced by Loewner & Nirenberg)

This metric has the projective invariance in the fo llowing sense:

Let be a projective transformation which sends Ω onto AΩ. Then A is an isometry with respect to LN-metrics of Ω and AΩ.

Legendre transformation is an isometry with respect to the LN-metric τ τ τ τ and the affine metric g

Let be an equation of a hyperbolic affine hypersphere.

Since the affine metric is written as

but at the corresponding

points x and ξ by the Legendre transformation. Hence:

( ) 22

)(det −−=

∂∂∂ n

ji

Huu ξξξ

udHu

21=τ with

( )RnSLA ,1+∈

( )nn xxfx ,...,11 =+

( ) ijnji ffff =,...,,det 1,( )( ) fdfg nij

22/1det +−= )()( 22 ξudxfd =

gududHu

=== 221 ρτ ( ) 2/1det +−= nijfρwith

258 /258 / Homogeneous hyperbolic affine hyperspheres

Let V be a non-degenerate convex cone in Rn+1. First we recall some properties of the characteristic function:

tends to infinity when x approaches to any point of the boundary of V

The measure is invariant under :

is convex on V. Hence defines a metric on V

The level surface of is a noncompact submanifold in V

called the characteristic surface of V.

We denote by the induced metric of on

VxdexV

xV ∈= ∫

− , )('

, ξφ ξ

)(xVφdxxV )(φ )(VA

)(for )det(/)()( VAAAxAx VV ∈= φφVφlog Vd φlog2

cxS VcV == )(: φφ

cω Vd φlog2cS

KOSZUL-VINBERG CHARACTERISTIC

FUNCTION


Assuming V is affinely homogeneous. The characteristic surface Sc is obviously homogeneous with respect to unimodular elements of A(V):

THEOREM (SASAKI): Every characteristic surface Sc is a complete hyperbolic affine hypersphere with mean curvature ac2/n+2 where a is a negative constant depending only on V.

Proof: Let . . can be writtenlocally as by a s mooth function , the coordinate being chosen such that . Let the Legendre transform of . Since is constant on we have on . By the definition: But

Hence

Therefore

)(log)( xx Vφψ = cSc log== ψ),...,( 11 nn xxfx =+ f( )11,..., +nxx 01 ≠+nψ

u f ψcS

1/ += niif ψψ cS1

11 / +

=+

+−= ∑ n

n

ii

in xfu ψψψ

0, , log)1()()( >∀∈∀+−= kVxknxkx ψψ

∑+

=

+−=1

1

)1()(n

nxxα

ααψ

1/)1( ++= nnu ψ


Consider

Set

We have

Hence by the homogenei ty for some constant bwhich depends only on V itself. This means

Since , then

( )

( )0

1det

,1 ,detdet

1

1111

1

21

1

21

11

1

111

+

++++

+

++

+

++

+

+++

=−

≤≤

+

+−=−

nj

nnnjn

iinij

nijn

n

nnji

n

jnijinijijn

f

njif

ψψψψψψψψ

ψψ

ψψψψ

ψψψψψ

ψψ

1,1 , 0

)( +≤≤=Φ nx βαψ

ψψ

β

ααβ

( ) )( , )(det)( 2 VAAAxAx ∈Φ=Φ

( )1

2

detdet

−

∂∂∂=

jiij

uf

ξξ

)()( 2 xbx φ=Φ

( ) cijnn Sbcxf on )(det 21

21 =Φ=− ++ ψψ

2

22 11det

bcu

nun

ji

+

+−=

∂∂∂

ξξ

1

1

+

+=n

nu

ψRemark:


THEOREM: Let S a complete hyperbolic affine hypersphere with its center at the origin which is homogeneous under the subgroup G of the unimodular group. Let V, the convex cone to w hose boundary the hypersurface S is asymptotic: .

Let , The element acts on V by

V is homogeneous under .

Then S is a characteristic surface of V.

Proof:

Let function on V by the equation

Since , is well defined.

Then for A e G.

Therefore, by the homogeneity, f or some nonzero constant b.

U 0>=

k kSV+×= RGG

~ ( ) Gtgg~

, ∈= )(. xgtg =G~

kn Sxkx ∈= −− for )( 1γγ

'for ' kkSS kk ≠=∩ φ γ)det(/)()( AxAx γγ =

Vbφγ =


2 previous Sasaki ‘s Theorems prove that the classi fication of homogeneous hyperbolic affine hyperspheres is reduce d to the classification of homogeneous convex cones:

Rothaus, O. S., The construction of homogeneous convex cones, Ann. of Math., 83 , pp. 358-376., 1966

Vinberg, E. B., The theory of convex homogeneous cones, Trans. Moscow Math. Soc, 12 (1963), 340-403

E.B. Vinberg has defined an inductive method produci ng all homogeneous convex cones :

From a given convex cone Vi in Rn+1(x), one can construct another homogeneous cone V in R(t) X Rm(y) X Rn+1(x) by the equation :

where h is a linear mapping on Rn+1 whose values are real symmetric positive-definite matrices of order m and, corresponding to each element B of some transitive subgroup of AV1), there exists a matrix such that

. This method can be transposed to obtain all projectivelyhomogeneous bounded convex domains or all homogeneous hyperbolic aίfinehyperspheres

yxhytxytV t )(/),,( 1−>=

),( RmGLA∈( )BxhAxhAt =)(


THEOREM: Suppose V is homogeneous. Then the metric ωc is identified with the affine metric g up to a constant factor.

Proof :

Let . can be written locally as

by a smooth function , the coordinate

being chosen such that

Since

we have

But . Therefore

The assumption needed is not the homogeneity of V b ut the condition that the level surface is an affine hyper sphere.

)(log)( xx Vφψ = cSc log== ψ),...,( 11 nn xxfx =+ ( )11,..., +nxxf

01 ≠+nψ

∑= +

+ −=n

i

i

n

in dxdx1 1

1

ψψ

∑≤≤

+−==nji

jiijnSc dxdxfd

c,1

12 ψψω

unn /)1(1 +=+ψ Hgnc )1( +−=ω


Remark: The solution u of the equation

is given as a first order logarithmic derivative of the characteristic

function, i.e.

Since the characteristic function of the produc t of convex cones

V and W is equal to , the derivative of is wr itten using the

derivatives of and . This gives the composit ion formula for

reducible cones.

WVφφ

( ) 22

)(det −−=

∂∂∂ n

ji

Huu ξξξ

1/)1( ++= nnu ψφ

φVφ Wφ


Considering the self-dual cone, to formulate anoth er description of the Legendre transformation we introduce the *-m apping due to Koecher , It is a mapping from V to its dual V defi ned by the equation:

This mapping * has the following properties:

* sets up a one to one correspondence between V and V and holds for every

If V is homogeneous, then V’ is also homogeneous and is constant for all x. We denote this constant by

In homogeneous case, the *-image of Sc is also a characteristic surface of V which we denote by S'c. Taking a hyperplane H such that is a non-empty bounded convex domain, we correspond, for every point the inter section point of the line through the origin and x* with U. Thus we have a mapping from S'c to U.

LEMMA: The mapping * followed by this mapping is de fined on Scand projectively equivalent to every Legendre trans formation of Sc.

( ) Vxxxx n ∈−== + for )(),...,( 11* ψψξ

( ) *1*)( xAAx t −=)(VAA∈

)()( *' xx VV φφ

2κ

'VHU ∩='*cSx ∈


PROPOSITION: Suppose V is homogeneous. Then * is an isometry with respect to the metrics and

Proof:Vd φlog2

'2 log Vd φ

( ) ( ) ( )

V

lk

lkjilj

Vki

V

ji

V

jiji

ji

VV

ki

j kj

Vji

Vd

kk

kj

Vj

j

jji

Vi

VxVVVV

d

dxdxxxx

xxx

x

ddd

xxxx

I

ddxdxxxx

xd

gradx

ψ

ψψξξ

ψ

ξξξξ

ψψ

δξξ

ψψ

ξξξξ

ψξψξ

ψφψφψ

2

,,,

22*'

2

,

'2

**'

2**

*'22

**

'2

*2

*

*''

)()()(

)().(

)()(,)()(

,log,log

=

∂∂∂

∂∂∂

∂∂∂=

∂∂∂=

=∂∂

∂∂∂

∂⇒=

∂∂∂−=

∂∂∂−=

−===

∑

∑

∑

∑∑


Now assume V is a self-dual cone. Then is an automorphism of V and, moreover,

Let , * / S is an involutive automorphism of S. Since , we have

.Therefore K(x) is an automorphism of S

PROPOSITION: If the homogeneous cone V is self-dual, then the characteristic surface Sc is a globally symmetric space.

Proof: For one point define an automorp hism s of S by

. Since K(x) is symmetric, s is an involution by

By

For a general Sc, it is enough to translate the symmetry of S to Sc

by the mapping

xxKx )(* =xxxK ∂−∂= /)( *

*KSS =

( ) KxxxKx VVV det)()()( * φφφ ==SxxK ∈= for 1))(det(

Sx ∈0*1

0)()( xxKxs −=

dIxx

xxKxxxs −=

∂∂=

∂∂= − )()()(x

s and )( 0

*1

0000

Sxc

Sx x ∈→∈ κ


Let H+(n, K) be the cone of positive-definite hermitian symmetri c matrices over K = fields R, C, H (quaternions) or the Cayley algebra Ca. Then the followings are the list of all irreducib le self-dual cones V and the corresponding globally symmetric spaces S

EIV typeof space the),2(

)1(/)1,1()(

)(/)2(),(

)(/),(),(

)(/),(),(

*

*

=⇒=

−−=⇒=

=⇒=

=⇒==⇒=

+

+

+

SCaHV

nSOnSOSnCV

nSnSUSHnHV

nSUCnSLSRnHV

nSORnSLSRnHV

p


Corollary: Let be a regular convex cone and let be the canonical Hessian metric. Then each level surfa ce of the characteristic function is a minimal surface o f the Riemannian manifold

Example: Let be a regular convex cone consisting of all positive definite symmetric matrices of degree n. Then is a Hessian structure on , and each level sur face of is a minimal surface of the Riemannian manifold:

ψlogDdg =

ψ( )g,Ω

( )xDdgD detlog, −=Ω xdet

( )xDdg detlog, −=Ω

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Kähler-Ricci Flow

271 /271 / Complex Autoregressive Model & Kähler Geometry

Erich Kähler Geometry is given by :

complex Manifold of n dimensions, compact or not, with Kählerian metric, that could be locally given by positive definite Riemannian Form :

Kähler condition : Local Existence of Kähler potential function , (and Pluri-harmonic equivalent) such that :

Ricci Tensor is given by remarkable Expression [Erich Kähler] :

And scalar curvature :

∑=

=n

ji

jiji dzdzgds

1,

2 .2

jiji zzg

∂∂Φ∂=

2

( )ji

lkji zz

gR

∂∂∂

−=detlog2

∑=

=n

lklk

lk RgR1,

.

Φ

272 /272 / Kähler metric for Complex Autoregressive Model

Inthe framework of Affine Information Geometry, met ric is given by Hessian of Entropy :

Entropy for Multivariate Gaussian model of zero mean :

In cas of Compex Autoregressive Model of order n, Entropy could be expressed by reflection coefficients :

( ) ( ) ( )enRRΦ πlogdetlog~ −−= -RH

ΗΗ

Φg

jiij =

∂∂∂≡ and

~2

[ ] [ ]10

1

1

2..ln.1ln).(

~ −−

=

+−−=∑ απµ enkn)(RΦn

kkn

[ ]∑

=

−

−−

−

==

−=n

kk

nnn

xn

P1

2

01

0

11

21

1with

.1

α

αµα( ) [ ]∏∏

−

=

−−−

=

− −==1

1

2

0

1

0

1 1detn

k

kn

kn

n

kknR µαα

273 /273 /

We define « Doppler » metric in case of Complex Autor egressive Model by Hessian of Kähler Potential, where Potenti al is given as in Information Affine Geometry by Entropy :

Kähler Potential is given by Entropy parametrized by reflection coefficients :

Metric can be explicitly computed :

[ ] [ ]10

1

1

2..ln.1ln).(

~ −−

=

+−−=∑ απµ enkn)(RΦn

kkn

[ ] [ ]Tnn

nTn

n P )()(1110

)( θθµµθ LL == −

20

2011

−== nPng α ( )221

).(

i

ijij

ing

µ

δ

−

−=

( )∑−

= −−+

=

1

122

22

0

02

1)(.

n

ii

in

din

P

dPnds

µ

µ

Kähler metric for Complex Autoregressive Model

274 /274 / Scalar Curvature of Complex Autoregressive Model

We use Ricci Tensor expression given by Erich Kähle r in framework of Kähler Geometry

In Kähler Geometry, Ricci Tensor is given by :

We can compute Ricci tensor for Complex Autoregressive Model :

Its negative scalar curvature is given by :

( )ji

lkji zz

gR

∂∂∂

−=detlog2

( )

−=−

−=

−=

1,...,2for 1

2

12

22

20

11

nkR

PR

k

kllk

µ

δ

∑=lk

lklk RgR

,

. ∞−→

−−=

∞→

−

=∑

n

n

j jnR

1

0 )(

1.2

275 /275 / Autoregressive Model # Kähler-Einstein Metric

Previous metric is not a Kähler-Einstein metric, but a close matrix structure

A metric is called Kähler-Einstein metric if its Ricci tensor is proportional to the metric :

In case of Kähler-Einstein Metric, Kähler Potential is solution Monge-Ampère Equation :

For Complex Autoregressive Model, we have :

[ ] [ ] [ ]( ) ,....,2 whereand

)(

1.2 with

1)(

1

0

)()(

−

−

=

−−=

−−=== ∑

indiagB

jnBTrRgBR

n

n

j

nij

nij

( )ji

ji

lklkji zz

kzz

gkgkR

∂∂Φ∂=

∂∂∂

−⇒=2

0

2

00 .detlog

constant : with .

= Φ−

function holomorphe

PotentialKähler with )det( 0

2

ψ :

Φ : eg k

lk ψ

276 /276 / Study of Calabi & Kähler-Ricci Flows for CAR Model

Intrinsic Geometric Flow are of fundamental interes t in Mathematic & Physics based on variational Approach :

Kähler-Ricci Flow :

Calabi Flow :

Inegality between both functionals

Hilbert Action Kähler-Ricci Flow

∫

∫=

M

M

gdV

gdVgR

R)(

)()(

∫M

gdVgR )()(jiji

ji gn

RR

t

g+−=

∂∂

Calabi Action Calabi Flow ( )*

*2 ,

jiji zz

zzg

∂∂∂= φ Calabi Flow

∫=M

gdVgRgS )()()( 2

*

2

ji

ji

zz

R

t

g

∂∂∂=

∂∂

RRt

−=∂∂φ

∫∫

≥

MM

gdVgdVgRgS )(/)()()(

2

277 /277 /

In 2 dimension of the complex variable, Calabi & Kähler-Ricci flows are closely related :

Calabi & Kähler-Ricci Flows can be related by :

If second derivative according to Kähler-Ricci flow time is identificated with opposite of second derivative according to Calabi flow time , both flows are equivalent:

I. Bakas : Relation between Calabi & Kähler-Ricci Flows

( )

( )

( )

−=−=∂

∂

∂∂

∂∂∂=

∂∂

⇒

∂∂∂−=

∂∂

−=∂

∂

⇒−=∂

∂

∑ RRgt

g

t

g

zzt

g

zz

gR

t

R

t

g

Rt

g

jiji

ji

ji

ji

jiji

jiji

jiji

,

*

2

2

2

*

2

2

2

)det(log

)det(log

)det(log

CR t

t

t ∂∂−=

∂∂

2

2

*

2

2

2

ji

ji

zz

R

t

g

∂∂∂−=

∂∂

⇒

278 /278 /

Given Kähler metric :

Ionnas Bakas : Flows action on Kähler Potential

*),,(2 .2*

dzdzeds tzzΦ=

*

2

*

zzR

zz ∂∂Φ∂−=

( ) and

)det(log** *

2Φ=

∂∂∂−= eg

zz

gR

zzzz

*

2

*

2

*

2

.. with

zze

t

zzte

zzt

eR

t

gji

ji

∂∂∂=∆∆Φ=

∂Φ∂

⇒

∂∂Φ∂=

∂Φ∂

⇒∂∂Φ∂=

∂∂

⇒−=∂

∂

Φ−

ΦΦ

Kähler-Ricci Flow

∆∆Φ−=∂Φ∂

⇒

∂∂∆Φ∂−=

∂∂

⇒

∆Φ−=∂∂Φ∂−==

∂∂∂=

∂∂

Φ

Φ−∑

t

zzt

e

zzeRgR

zz

R

t

g

jiji

jiji

*

2

,*

2

*

2

and

Calabi Flow

279 /279 /

Based on previous Rao metric computation for Autoregressive Model :

Kähler-Ricci Flow on Reflection Coefficients :

Calabi Flow on Reflection Coefficients :

Kähler-Ricci and Calabi Flows for CAR Model

( ) 222

00)(

222

)0()( and .)0(1)(1 n

t

in

t

ii ePtPet−

−−

=−=− µµ

( )∑−

=

−

−−+

=

1

1

)(

4

22

242

0

02 22

)0(1)(

)0()(

n

i

in

t

i

in

t

n ed

ineP

dPntds

µ

µ

( ) n

tin

t

ii ePtPet−−

−=−=− )0()( and .)0(1)(1 00

)(22 µµ

( )∑−

=

−

−−+

=

1

1

)(

2

22

222

0

02

)0(1)(

)0()(

n

i

in

t

i

in

t

n ed

ineP

dPntds

µ

µ