Résumé

Information/Contact Geometries and Koszul Entropy

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Optimal matching between curves in a manifold
Drone Tracking Using an Innovative UKF
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
Session Geometrical Structures of Thermodynamics (chaired by Frédéric Barbaresco, François Gay-Balmaz)
Opening and closing sessions (chaired by Frédéric Barbaresco, Frank Nielsen, Silvère Bonnabel)
GSI'17-Closing session
GSI'17 Opening session
Démonstrateur franco-britannique "IRM" : gestion intelligente et homéostatique des radars multifonctions
Principes & applications de la conjugaison de phase en radar : vers les antennes autodirectives
Nouvelles formes d'ondes agiles en imagerie, localisation et communication
Compréhension et maîtrise des tourbillons de sillage
Wake vortex detection, prediction and decision support tools
Ordonnancement des tâches pour radar multifonction avec contrainte en temps dur et priorité
Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics
Reparameterization invariant metric on the space of curves
Probability density estimation on the hyperbolic space applied to radar processing
SEE-GSI'15 Opening session
Lie Groups and Geometric Mechanics/Thermodynamics (chaired by Frédéric Barbaresco, Géry de Saxcé)
Opening Session (chaired by Frédéric Barbaresco)
Invited speaker Charles-Michel Marle (chaired by Frédéric Barbaresco)
Koszul Information Geometry & Souriau Lie Group 4Thermodynamics
MaxEnt’14, The 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
Koszul Information Geometry & Souriau Lie Group Thermodynamics
Robust Burg Estimation of stationary autoregressive mixtures covariance
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Oral session 7 Quantum physics (Steeve Zozor, Jean-François Bercher, Frédéric Barbaresco)
Opening session (Ali Mohammad-Djafari, Frédéric Barbaresco)
Tutorial session 1 (Ali Mohammad-Djafari, Frédéric Barbaresco, John Skilling)
Prix Thévenin 2014
SEE/SMF GSI’13 : 1 ère conférence internationale sur les Sciences  Géométriques de l’Information à l’Ecole des Mines de Paris
Synthèse (Frédéric Barbaresco)
POSTER SESSION (Frédéric Barbaresco)
ORAL SESSION 16 Hessian Information Geometry II (Frédéric Barbaresco)
Information/Contact Geometries and Koszul Entropy
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Geometric Science of Information - GSI 2013 Proceedings
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www.thalesgroup.com Information/Contact Geometries and Koszul Entropy F. Barbaresco 2 /2 / Outlines  Overview of Involved Geometries  Koszul-Vinberg Characteristic Function, Entropy & Density  Koszul-Vinberg Characteristic Function, Entropy & Density for Symmetric/Hermitian Positive Definite Matrices  Legendre Duality and Projective Duality  Legendre Duality in Information Geometry  Legrendre Duality in Thermodynamic (Massieu-Gibbs-Duhen Potentials)  Legendre Duality in Mechanics (Poincaré-Cartan Integral Invariant)  Legendre Transform and Contact Geometry  Extension of Metric for Toeplitz and Toeplitz-Block-ToeplitzHermitian Positive Definites Matrices (HPD) www.thalesgroup.com Overview of Involved Geometries 4 /4 / Geometric Science of Information Takeshi SASAKI W. BLASCHKE Eugenio CALABI Calyampudi R. RAO Nikolai N. CHENTSOV Hirohiko SHIMA Jean-Louis KOSZUL Von Thomas FRIEDRICH Y. SHISHIDO Homogeneous Convex Cones G. E.B. VINBERG Jean-Louis KOSZUL Homogeneous Symmetric Bounded Domains G. Elie CARTAN Carl Ludwig SIEGEL Probability in Metric Space Maurice R. FRECHET Information Theory Nicolas .L. BRILLOUIN Claude. SHANNON Probability on Riemannian Manifold Michel EMERY Marc ARNAUDON Geometric Science of Information KOSZUL-VINBERG METRIC (KOSZUL-VINBERG CHARACTERISTIC FUNCTION) FISHER METRIC Probability/G. on structures Y. OLLIVIER M. GROMOV Contact G. Vladimir ARNOLD www.thalesgroup.com Koszul-Vinberg Characteristic Function, Entropy & Density 6 /6 / Koszul-Vinberg Characteristic Function/Metric of convex cone  J.L. Koszul [and E. Vinberg have introduced an affinely invariant Hessian metric on a sharp convex cone through its characteristic function.  is a sharp open convex cone in a vector space of finite dimension on (a convex cone is sharp if it does not contain any full straight line).  is the dual cone of and is a sharp open convex cone.  Let the Lebesgue measure on dual space of , the following integral: is called the Koszul-Vinberg characteristic function  E R *   d * E E      xdex x )( * ,   7 /7 / Koszul-Vinberg Characteristic Function/Metric of convex cone  is analytic function defined on the interior of and as  If then  is logarithmically strictly convex, the function is strictly convex  Koszul 1-form : The differential 1-form is invariant by . If and then and  Koszul 2-form  : 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 structure invariant by and then the Riemanian metric      xdex x )( * ,     )(x x   Autg   )(det 1 xggx        )(log)( xx       /log ddd   AutG x Eu      * , .,,   deuu x x * x   log2 dD E   AutG        * ,2 ,,,log   devuvud u D  Aut  log2 dg 8 /8 / Koszul-Vinberg Characteristic Function/Metric of convex cone  Koszul-Vinberg Metric :  We can define a diffeomorphism by: with  When the cone is symmetric, the map is a bijection 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 :  log2 dg    loglog 2 1log log)(log 2 u 2 22        dudv dudvdd du dud dudxd vu vuv u uu u      )(log* xdx x   )()(),( 0 tuxf dt d xfDuxdf t u    xx * xx ** )( nxx * , cstexx  )()( * * * x  nyxyyx  ,,/)(minarg **  * x  nyxy  ,,* *        ** ,,* /.   dedex xx 9 /9 / Koszul Entropy via Legendre Transform  we can deduce “Koszul Entropy” defined as Legendre Transform of minus logarithm of Koszul-Vinberg characteristic function : with and where  Demonstration: we set Using and we can write: and )(,)( *** xxxx   xDx* * *  x Dx )(log)( xx        xdex x )( * ,         ** ,,* /.   dedex xx        ** ,,* /,)(log,   dedehxdhx xx h       ** ,,,* /.log,   dedeexx xxx                                    **** *** ,,,,,** ,,,,** /.loglog.)( log/.log)(     dedeededex dededeex xxxxx xxxx 10 /10 / Koszul Entropy via Legendre Transform  We can then consider this Legendre transform as an entropy, that we could named “Koszul Entropy”: With and                            **** ,,,,,** /.loglog.)(   dedeededex xxxxx       dppd de e de e xxx x x x              * * * * )(log)(log , , , , * Φ(x)x,ξ dξex,ξ Ω ξ,xξ,x x eedξeep *Ω ξ,x *       log /)(    dedξ.edpΦDx ξΦ Ω Φ(x)x,ξ xx * *       ** .)(.* 11 /11 / Koszul-Vey Theorem  J.L. Koszul and J. Vey have proved the following theorem:  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 Scuola Normale 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 Hessian metric . Suppose that admits a closed 1-form such that 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 real affine space not containing any full straight line. - If is compact, then is a sharp convex cone. M g  gD  G M  GM / M GM /  [] 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 Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tome 24,n°4, p.641-665, 1970 12 /12 /  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  )(/  Xd 4 1   D  0/  yiyxzV dx d yX  dy d yY                 YJX YYX ZYZYad , ,.               0 2 YJadJYadTr XJadJXadTr dydx y  2 1   2 22 2 2 2 2 1 4 1 2 y dydx ds y dydx d y dx X      13 /13 /  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 I J D BA S BDDBIDA DBAZSZ TTT -                              00 0 , 0 0 baseandwith 0 jiij ij ji ij ijT T ee e e ad bb d ba                                           ZddZYYTr p ds ZdYdZYTr P d dXYTr p idYdX pijij ij 112 11 1 8 13 8 13 4 1 2 13 13 0   14 /14 / 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 que tous les domaines homogènes dont la forme est définie positive sont symétriques, toute la théorie des domaines bornés homogènes serait élucidée. C’est là un problème de géométrie hermitienne certainement très intéressant » Dernière phrase de Elie Cartan, dans « Sur les domaines bornés de l'espace de n variables complexes », Abh. Math. Seminar Hamburg, 1935   ji ji ji zddz zz zzK     , 2 ,log  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) www.thalesgroup.com Koszul-Vinberg Characteristic Function, Entropy & Density for Symmetric/Hermitian Positive Definite Matrices 16 /16 / Koszul-Vinberg Characteristic Function/Metric of convex cone  Let v be the volume element of g. We define a closed 1-form  and  a symmetric bilinear form by: and  The forms  and  are called the first Koszul form and the second Koszul 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 closed 1-form such that is positive definite, whereupon 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 such that . vXvDX )(  D                         ji kl j i ij ijii n/ ij xx g x vg x α ..dxdxgv detlog 2 1 detlog det 2 2 1 121    D D  Dg  17 /17 / Koszul-Vinberg Characteristic Function/Metric of convex cone  We can apply this Koszul theory for Symmetric Positive 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 positive 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  * )(det)( 2 1 dual-self )(, , * * n n xyTryx x Ixdex            xd n dg detlog 2 1 log 22    1* 2 1 detlog 2 1 log       x n xd n dx  )detlog,( xDdD  xdet )detlog,( xDdg    1 .det det with     xx x x 18 /18 / Koszul Density: Application for SPD matrices  We can then named this new density as “Koszul Density”: With Φ(x)x,ξ dξex,ξ Ω ξ,xξ,x x eedξeep *Ω ξ,x *       log /)(    dedξ.edpΦDx ξΦ Ω Φ(x)x,ξ xx * *       ** .)(.*               * 1 ).(.withdet)( 1 detlog 2 1   dpeep x Tr x n xTr x                             1* 2 1 dual-self )(, , 2 1 detlog 2 1 log )(det)( )(,,, * * x n xd n dx Ixdex RSymyxxyTryx n n xyTryx x n    19 /19 / Link with Maximum Entropy solution  Reference:  Christian Soize, “A Nonparametric Model of Random Uncertainties for Reduced Matrix Models in Structural Dynamics”, "Probabilistic Engineering Mechanics 15, 3 , pp. 277-294“, 2000  Construction of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle  Let x be a random matrix with values in Sym+(n) whose probability distribution is defined by a probability density function .  This probability density function is such that:  Available information for construction of the probability model   xdxp ~  xp   1 ~ 0 x xdxp     xxdxpxxE x  0 ~ .          F xEvvxE 1 withdetlog   1 ~ 0 x xdxp         vvxdxpxxE x with ~ .detlogdetlog 0 20 /20 / Link with Maximum Entropy solution  Probability model using the maximum entropy principle : construct probability density function and characteristic function of random matrix x using the maximum entropy principle and the available information.  In order to perform this calculation, we need results concerning the Siegel integral for a positive-definite symmetric real matrix  [2] C L Siegel, "Über der analytische Theorie der quadratischen Formen", Ann. Math. 36, p. 527-606, 1935                        vxdxpxxdxpxdxpxppL xxx 00 0 0 ~ )(.detlog.1 ~ )(.1 ~ .)(log)()(    x p excxppLMaxArgp ,1 0 det.)()(          0 1, ~ det, x x xdxeJ                                1 2 )21( 1 2 1 det 2 2 2, dtetΓ(λ) x λlM ΓπJ t MM l )M(M    21 /21 / Link with Maximum Entropy solution  Characteristic function of x is given by:  Considering the analytic extension of the mapping by writing . Taking , we deduce that:  Finally, characteristic function of x is given by:       0 ~ x θ,xiθ,xi x xdp(x).eeEθΦ        2 21 1 det λM Mx θiξIθΦ     wJw , iθ(w)w  Re 0Re (w)    iθλ,ξ.JcθΦx  0   ),( 1 10 0 J cΦx        1 ,.   Jiθλ,ξJθΦx 22 /22 / Link with Maximum Entropy solution  First and second moments derived from characteristic function  Using: and  We deduce first two moments:       0 2         jk x jk jk θΦi xE        0'' 2 '' '' 22         kjjk x kjjk kjjk θΦi xxE    2 0)(1det F hhtrhI    1 .det det     bb b b jk     1 . 2 21   jkjk M xE                   '''''''' 21 1 kkjjjkkjkjjkkjjk xExExExE M xExExxE     23 /23 / Link with Maximum Entropy solution  First moment derived from characteristic function:  Characteristic function and probability density function of positive-definite random matrix   1 2 21   x M         2 21 21 2 det λM Mx θx λM i IθΦ                                   2 21 1 2 21 4 )1( 2 21 1, 2 21 1 det. 2 2 2 21 2 with .det..det.)( 11                                            MM l MM MM x xxTr M x xx M x x lM M c excexcxp 24 /24 / Link between Maximum Entropy & Koszul Density  If we put “Koszul Density”:  If we give Maximum Entropy Density with     2 1 K 2 1 2 1 1 detlog 2 1 , 2 1 withdet.)( 1                n Tr nn K x n x x n ecep                                                           n l nn nn ME Tr nn MEx ln n ep 1 2 1 4 )1( 2 1 2 1 1 2 2 2 1 2 withdet.)( 1    1                                n l )n(n Tr n x ln Γπc n cedp 1 2 1 1 11 0 2 1 2 2 2 2 1 withdet1 ~ )(       www.thalesgroup.com Legendre Duality & Projective Duality 26 /26 / DUALITY Concept in Mathematic/Mechanic/Thermodynamic (projective) LEGENDRE DUALITY LEGENDRE TRANSFORM (between Dual Space) CONTACT/SYMPLECTIC GEOMETRIES (Legendre mapping, fibration,…) (Analytic) FOURIER DUALITY FOURIER TRANSFORM (Time-/Frequency Dual Spaces) LINEAR ALGEBRA (Linear Signal Processing)  )(,)()( * xyxMaxArgyyL y  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 !!!  22 log ddgINFORMATION GEOMETRY METRIC Sddg 2*2*      * , log)(log)( dyexx yx )(,)( *** xxxx   dpp xx   * )(log)(* Φ(x)x,ξ Ω ξ,xξ,x x edξeep *   /)(    * )(.*  dpx x  ),,(.),,( tqqLqpSuptqpH q      STES k k    /)(with)( )( .     27 /27 / Duality in Projective Geometry: Pascal’s Mystic Hexagram In projective geometry, Pascal's theorem (the Hexagrammum 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. www.thalesgroup.com Legendre Duality in Information Geometry 29 /29 / Duality of Information Geometry: Multivariate Gaussian Law Dual Coordinates systems & Potential functions  Potential Functions are Dual and related by Legendre transformation :                                      enΗ)(ηΗηΗΦ nTrΘΨ mmmη,ΗΗ Σ)m,(Σθ,ΘΘ T T T   2log2detlog21log2 ~~ )log(2detlog22 ~~ , ~ 2 ~ scoordinateDual 1111 1112 11                          Θ Η Φ θ η Φ Η Θ Ψ η θ Ψ ~ ~ and~ ~  TT ΘΗTrH,Θ ΨH,ΘΦ    ~~ with ~~~~    pEΗΦ log ~~  Entropy ji * ij ji ij ΗΗ Φ g ΘΘ Ψ g       ~ and ~ 22  Hessians are convexe and define Riemannian metrics : 30 /30 / 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 mxp mxp,Σm,N,ΣmNDiv H,ΘΗΦΘΨ,Σm,N,ΣmNDiv H,ΘΗΦΘΨ,Σm,N,ΣmNDiv       22 11 111122 21211122 * 12121122 ,/ ,/ log,/ 0 ~~~~~~ 0 ~~~~~~ Riemannian Metric  As Potential are convexe, their Hessians define Riemannian Metrics :                             T mmmmTr Tr ,Σm,N,ΣmNDiv 2121 1 2 1 1 1 21 1 21 1122 detlog 2 1    332 2 1 2 1 iji ij ijiij dΗOdΗdΗgdΘOdΘdΘgds  www.thalesgroup.com Legendre Duality in Thermodynamic 32 /32 / 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 thermal properties of physical and chemical systems could be derived from 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 identified separately and are noted (function accounted for the sum of mechanical and thermal effects by equivalence between heat and work). The external work is thermally equivalent to (with the conversion factor between mechanical and thermal measures). The first principle provides . For a closed reversible cycle (Joule/Carnot principles) that is the complete differential of a function of . [] Massieu F., Thermodynamique: Mémoire sur les fonctions caractéristiques des divers fluides et sur la théorie des vapeurs, 92-pgs, Académie des Sciences, 1876 [] Duhem P., Sur les équations générales de la Thermodynamique, Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 1891 [] Arnold V.I., Contact geometry: the geometrical method of Gibbs’s thermodynamics, Pro-ceedings of the Gibbs Symposium, p.163–179, Amer. Math. Soc., Providence, RI, 1990 dQ dE E dVP. dVPA .. A dVPAdEdQ .. 0/  TdQ dS S TdQdS / 33 /33 / 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 function: , and  If we select pression P and temperature T as independent variables: Massieu characteristic function is then given by and we can deduce: and and inner energy:   dVPAdTSdETSddVPAdEdSTdQdST .......  ETSH  dV V H dT T H dVPAdTSdH .....       H T H S    V H A P    1 H T H THTSE     VAPHH .'  dP P H dT T H dPAVdTSdPAVdVAPdHdH . ' . ' ....'       T H S    ' P H A V    '1 P H PH T H TEVAPHTSHTSE       ' .' ' .' 34 /34 / Legendre Duality in Thermodynamic: Massieu Potentials  Deriving all body properties dealing with thermodynamics 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 ce corps»  In thermodynamics, the Massieu potential is the Legendre 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 TESFk /.)(   TSEF                                                k T T F kF k F kF k Fk k V       ETSTSE k S F k SkF k             )( 1 )( 2     E S       SEFkFkEk k kL         .).()( )( . 35 /35 / 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. Petersburg 1885 WTSEG  )( www.thalesgroup.com Legendre Duality in Mechanics (Poincaré-Cartan Integral Invariant) 37 /37 / Legendre Duality in Mechanics  In Mechanics, Legendre Duality gives the relation between:  the variational Euler-Lagrange  the symplectic Hamilton-Jacobi formulations of the equations of motion  As described by Vladimir Arnold, in the general case, 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 holds : with equality if and only if  ),,(.),,( tqqLqpSuptqpH q    ),,(.),,( tqqLqptqpH   q ),,( tqqLp q  ),,( tqpHq p pqtq ,,,  ),,(),,(. tqpHtqqLqp   ),,( tqqLp q  38 /38 / Legendre Duality in Mechanics  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 39 /39 / Legendre Duality in Mechanics  ),,(.),,( tqqLqpSuptqpH q    ),,( tqqLp q  ),,( tqpHq p 40 /40 / 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 among all forms the form is the only one satisfying , is a particular case of more general T. Lepage congruence related to transversality condition. dtHdqp ..     LdtLdtLqLdqL qqq    .... dtqdq . Lp q LqpH  .  mod.dtL dtHdqp ..   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. Volume 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 41 /41 / Pfaffian Form and Poincaré-Cartan Integral Invariant www.thalesgroup.com Legendre Transform and Contact Geometry 43 /43 / 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 dimensional contact manifold are called Legendre submanifolds.  A smooth fibration of a contact manifold, all of whose are Legendre, is called a Legendre Fibration.  In the neighbourhood of each point of the total space of a Legendre Fibration there exist contact Darboux coordinates (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 embedding of a smooth manifold as a Legendre submanifold in the total space of a Legendre fibration, and the projection of the total space of the Legendre fibration onto the base. 44 /44 / Legendre Transform and Contact Geometry  Let us consider the two Legendre fibrations of the standard 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 graph of a convex function, the Legendre transform of the function S: 12 n R n R ),(),,( quqpu  ),.(),,( puqpqpu  )(qSu             q S qS q S qq ),(  ppS ),(* 45 /45 / Legendre Duality & Contact Geometry  Symplectic geometry of even-dimensional phase spaces has an odd-dimensional twin: contact geometry.  The relation between contact geometry and symplectic geometry is similar to the relation between linear algebra and projective geometry. Any fact in symplectic geometry can be formulated as a contact geometry fact and vice versa. The calculations are simpler in the symplectic setting, but their geometric content is better seen in the contact version.  The functions and vector fields of symplectic geometry are replaced by hypersurfaces and line fields in contact geometry.  Each contact manifold has a symplectization, which is a symplectic manifold whose dimension exceeds that of the contact manifold by one.  Symplectic manifolds have contactizations whose dimensions exceed their own dimensions by one.  If a manifold has a serious reason to be odd dimensional it usually carries a natural contact structure.  one might now say ‘‘symplectic geometry is all geometry,’’ but I prefer to formulate it in a more geometrical form: contact geometry is all geometry. [] Arnold, V.I., Givental, A.G., Symplectic geometry, Encyclopedia of mathematical science, vol. 4., Springer Verlag (translated from Russian) 46 /46 / 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 locally equivalent. 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 by 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 ),,(   n Rpyx ii xfp  /   ),(/),(, 1 RVJxfpxfyx n  dxpdy . pdVTdSdE  Gibbs contact structure In Thermodynamics www.thalesgroup.com Extension of Metric for Toeplitz and Toeplitz-Block-ToeplitzH ermitian Positive Definites Matrices (HPD) 48 /48 / Trench/Verblunsky Theorem & Partial Iwasawa Parameterization  All Toeplitz Hermitian Positive Definite Matrix can be parameterized by Reflection/Verblunsky Coefficients:  Block structure of covariance matrix & Verblunsky Parameterization:  Verblunsky/Trench Theorem: Exitence of diffeomorphism                111 1 111 1111 ... . nnnnnn nnn n AARA A R                    111 11111 1 1 . ... nnn nnnnnn n RAR RAARA R    *)( )( 11 1- 00 1 1 21 . 001 00 100 where 1 . 0 and.1with VV AA A P n n n n nnn                                    1/with ,...,, : 110 1*      zCzD PR DRTHPD nn n n    S. Verblunsky (PhD student of Littlewood) Iwasawa (Lie Group Theory) 49 /49 / Preservation of Toeplitz Structure by Verblunsky Coefficients  Conformal Information Geometry metric (metric = Hessian of Entropy):  Entropy  as Kähler potential:  Conformal metric on Verblunsky parameterization:  T n n P 110 )(    )*()( 2 ~ n j n i ijg            ..ln.1ln).(.logdetlog, ~ 0 1 1 21 0 PenkneR)P(R n k knn       with                1 1 22 22 0 0)()(2 1 )(. n i i in ij n n d in P dP ndgdds    E. Kähler 1t withcoefficienVerblunsky: kk  50 /50 / Extension to Toeplitz Block Toeplitz Hermitian PD Matrices  Previous results can be extented to Block-Toeplitz Matrices :                         0 , 01 1 01 10 1, ~ ~ RR RR RRR R RR RRR R n nnp n n np     * 1 ~            n n R R VR                 00 0 0 00     p p p J J J Vwith 51 /51 / Extension of Trench/Verblunsky Theorem for TBTHPD Matrices  Every Toeplitz-Block-Toeplitz HPD matrix can be parametrized by Matrix Verblunsky Coefficients:  Extension of Trench/Verblunsky Theorem: Existence of Diffeomorphism                nnnnpnn nnn np AARA A R   ... . 1 , 1 1,                npnnp npnnnpnn np RAR RAARA R ,, ,, 1 1, . ...                                                 p p n p p n np n n p n n n n - n n n n nn I JAJ JAJ A A A A A RαAA *1 1 *1 1 1 1 1 0 1 0 1 1 1 . 0 and ,.1with         n n n n nnn IZZnHermZD AARR SDTHPDTBTHPD        /)(Swith ,...,, : 1 1 1 10 1   52 /52 / Information Geometry Metric : Entropic Kähler Potential  Kähler potential defined by Hessian of multi-channel/Multi-variate entropy :         npji npnpnp RHessg csteµRTrcsteRRΦ , ,,, logdetlog ~        0 1 1 , det..log.detlog).( ~ RenAAIknR n k k k k knnp                   1 1 112 0 1 0 2 )(. n k k k k k k kn k k k k k kn dAAAIdAAAITrkndRRTrnds We recover the previous metric for THPD matrix !! 53 /53 / Toeplitz Hermitian PD Matrices: Simple case n=2 DefinitePositiveHermitianToeplitz 0det 0 222 bah hiba ibah           Hadamard Compactification   DiskUnitPoincaré: 1/ 0 1 *1 . * D zzD h iba Rh h                   1/ )log( with),log(     zzD Rh h iba h   Scale Parameter Shape Parameter a b h >90° Ambligone Triangle h2 >a2 +b2    222 22 2222 0 02 0det bah bah bahh hiba ibah                 54 /54 / General Diffeomorphism for TBTHPD Matrices         1/ ,...,,log / ,...,, * 1 1100 11 1 1 10          zzzD DRPR IZZZSD SDTHPDAAR m m m n m n n  11 :bycodedStateDoppler-Spatio   nm SDDR             1 1 112 0 1 0 2 )(. n k k k k k k kn k k k k k kn dAAAIdAAAITrkndRRTrnds             1 1 22 2 2 0 )()(2 1 )(log. m i i im ij m m d imPdmdgdds   