• abotsi

    Author: Charles-Michel Marle
    Institution: Professeur honoraire à l'Université Pierre et Marie Curie, Institut Mathématique de Jussieu, Correspondant de l’Académie des Sciences, Paris, France.
    Website: http://charles-michel.marle.pagesperso-orange.fr/
    Video: http://www.youtube.com/watch?v=qdySpxlolrM
    Slides: Marle_Actions of Lie Groups algebras.pdf
    Presentation: https://www.see.asso.fr/node/13649
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.

    References :

    • Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Reading (1978).
    • Arnold, V.I.: Mathematical methods of Classical Mechanics, 2nd edn. Springer, New York (1978).
    • Ganghoffer, J.-F., Maldenov, E. (editors): Similarity and Symmetry Methods; Applications in Elasticity and mechanics of Materials. Lecture Notes in Applied and Computational Mechanics 73, Springer, Heidelberg (2014).
    • Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984).
    • Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson structures. Springer, Berlin (2013).
    • Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)

    Bio:
    Charles-Michel Marle was born in 1934; He studied at Ecole Polytechnique (1953--1955), Ecole Nationale Supérieure des Mines de Paris (1957--1958) and Ecole Nationale Supérieure du Pétrole et des Moteurs (1957--1958). He obtained a doctor's degree in Mathematics at the University of Paris in 1968. From 1959 to 1969 he worked as a research engineer at the Institut Français du Pétrole. He joined the Université de Besançon as Associate Professor in 1969, and the Université Pierre et Marie Curie, first as Associate Professor (1975) and then as full Professor (1981). His resarch works were first about fluid flows through porous media, then about Differential Geometry, Hamiltonian systems and applications in Mechanics and Mathematical Physics.

    Marle.jpg

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  • abotsi

    Author(s): Amor Keziou, Philippe Regnault
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_49
    Video: http://www.youtube.com/watch?v=VGucQ4FTMR4
    Slides: Regnault_Semi-parametric estimation.pdf
    Presentation: https://www.see.asso.fr/node/14348
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of mutual information associated to ϕ-divergences are derived through the dual representations of ϕ-divergences. The asymptotic properties of the estimates are established, including consistency, asymptotic distribution and large deviations principle. The related tests of independence are compared through their relative asymptotic Bahadur efficiency and numerical simulations.

    posted in Divergence Geometry read more
  • abotsi

    Author(s): Amor Keziou
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_48
    Video: No autorisation
    Slide: No autorisation
    Presentation: https://www.see.asso.fr/node/14347
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We introduce what we will call multivariate divergences between K, K_³ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a _-field (X,B), extending the well known divergences between two measures, a signed finite measure Q1 and a given probability distribution P. We investigate the Fenchel duality theory for the introduced multivariate divergences viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We obtain new dual representations of these criteria, which we will use to define new family of estimates and test statistics with multiple samples under multiple semiparametric density ratio models. This family contains the estimate and test statistic obtained through empirical likelihood. Moreover, the present approach allows obtaining the asymptotic properties of the estimates and test statistics both under the model and under misspecification. This leads to accurate approximations of the power function for any used criterion, including the empirical likelihood one, which is of its own interest. Moreover, the proposed multivariate divergences can be used, in the context of multiple samples in density ratio models, to define new criteria for model selection and multi-group classification.

    posted in Divergence Geometry read more
  • abotsi

    Author(s): Imre Csiszár, Michel Broniatowski, Thomas Breuer
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_47
    Video: not available
    Slides: Breuer_Information geometry finance.pdf
    Presentation: https://www.see.asso.fr/node/14346
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family _ of plausible risk factor distributions IP. We consider _ determined by a bound on a convex integral functional of the density of IP, thus _ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IPo. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP__), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When _ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

    posted in Divergence Geometry read more
  • abotsi

    Author(s): Ben Anthonis, Jan Naudts, Michel Broniatowski
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_46
    Video: http://www.youtube.com/watch?v=wnhAE9sVYZE
    Slides: Naudts_Extension of information geometry.pdf
    Presentation: https://www.see.asso.fr/node/14345
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

    posted in Divergence Geometry read more
  • abotsi

    Author: Diaa Al Mohamad, Michel Broniatowski
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_45
    Video: http://www.youtube.com/watch?v=eVrLiyB0M9A
    Slides: ALMohamad_generalized EM algo.pdf
    Presentation: https://www.see.asso.fr/node/14344
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Minimum divergence estimators are derived through the dual form of the divergence in parametric models. These estimators generalize the classical maximum likelihood ones. Models with unobserved data, as mixture models, can be estimated with EM algorithms, which are proved to converge to stationary points of the likelihood function under general assumptions. This paper presents an extension of the EM algorithm based on minimization of the dual approximation of the divergence between the empirical measure and the model using a proximaltype algorithm. The algorithm converges to the stationary points of the empirical criterion under general conditions pertaining to the divergence and the model. Robustness properties of this algorithm are also presented. We provide another proof of convergence of the EM algorithm in a two-component gaussian mixture. Simulations on Gaussian andWeibull mixtures are performed to compare the results with the MLE.

    posted in Divergence Geometry read more
  • abotsi

    Author: Eric Justh, P. S. Krishnaprasad
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_60
    Video: http://www.youtube.com/watch?v=leK1MggION0
    Slides: Justh_Enlargement geodesics collectives.pdf
    Presentation: https://www.see.asso.fr/node/14343
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

    posted in Lie Groups and Geometric Mechanics/Thermodynamics read more
  • abotsi

    Author: Fátima Leite, Krzysztof Krakowski, Luís Machado
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_59
    Video: http://www.youtube.com/watch?v=Dm1ht6tNJbo
    Slides: Machado_Rolling Symmetric Spaces.pdf
    Presentation: https://www.see.asso.fr/node/14342
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Graßmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

    posted in Lie Groups and Geometric Mechanics/Thermodynamics read more
  • abotsi

    Author: Danielle Fortune, Francois Dubois, Juan Antonio Rojas Quintero
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_58
    Video: http://www.youtube.com/watch?v=uiOKLhoh3IA
    Slides: Dubois Pontryagin calculus Riemannian.pdf
    Presentation: https://www.see.asso.fr/node/14340
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

    posted in Lie Groups and Geometric Mechanics/Thermodynamics read more
  • abotsi

    Author: Frédéric Barbaresco
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_57
    Video: http://www.youtube.com/watch?v=wTxIdmALzjo
    Slides: Barbaresco_symplectic structure.pdf
    Presentation: https://www.see.asso.fr/node/14339
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

    posted in Lie Groups and Geometric Mechanics/Thermodynamics read more

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