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(2017) GSI2017

Information Distances in Stochastic Resolution Analysis (slides) GSI2017
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics Geert Verdoolaege GSI2017
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We present a new regression method called geodesic least squares (GLS), which is particularly robust against data and model uncertainty. It is based on minimization of the Rao geodesic distance on a probabilistic manifold. We apply GLS to Tully-Fisher scaling of the total baryonic mass vs. the rotation velocity in disk galaxies and we show the excellent robustness properties of GLS for estimating the coefficients and the tightness of the scaling.
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics (slides) GSI2017
Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system? Frédéric Hélein, Joël Bensoam, Pierre Carré GSI2017
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Although acoustics is one of the disciplines of mechanics, its ”geometrization” is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. More precisely, the dynamics of the Reissner beam is formulated as a map over spacetime with values in a nonlinear manifold (a Lie group). Fortunatly, this multi-symplectic approach can be related to the study of harmonic maps for which two dimensional cases can be solved exactly. It allows us to identify, among the family of problems, a particular case where the system is completely integrable. Among almost explicit solutions of this fully nonlinear problem, it is tempting to identify solitons, and to test the known numerical methods on these solutions.
Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system?
Non Linear Propagation in Reissner Beams: an integrable system? (slides) GSI2017
Joint geometric and photometric visual tracking based on Lie group Chenxi Li, Tianci Liu, Yunpeng Liu, Zelin Shi GSI2017
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This paper presents a novel efficient and robust direct visual tracking method under illumination variations. In our approach, non-Euclidean Lie group characteristics of both geometric and photometric transformations are exploited. These transformations form Lie groups and are parameterized by their corre-sponding Lie algebras. By applying the efficient second-order minimization trick, we derive an efficient second-order optimization technique for jointly solving the geometric and photometric parameters. Our approach has a high convergence rate and low iterations. Moreover, our approach is almost not affected by linear illu-mination variations. The superiority of our proposed method over the existing direct methods, in terms of efficiency and robustness is demonstrated through experiments on synthetic and real data.
Joint geometric and photometric visual tracking based on Lie group
Joint geometric and photometric visual tracking based on Lie group GSI2017
Coordinate-wise transformation and Stein-type densities (slides) GSI2017
Inductive means and sequences applied to online classi cation of EEG Estelle Massart, Sylvain Chevallier GSI2017
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The translation of brain activity into user command, through Brain-Computer Interfaces (BCI), is a very active topic in machine learning and signal processing. As commercial applications and out-of-the-lab solutions are proposed, there is an increased pressure to provide online algorithms and real-time implementations. Electroencephalography (EEG) systems o er lightweight and wearable solutions, at the expense of signal quality. Approaches based on covariance matrices have demonstrated good robustness to noise and provide a suitable representation for classi cation tasks, relying on advances in Riemannian geometry. We propose to equip the minimum distance to mean (MDM) classi er with a new family of means, based on the inductive mean, for block-online classi cation tasks and to embed the inductive mean in an incremental learning algorithm for online classi cation of EEG.
Inductive means and sequences applied to online classication of EEG
Inductive means and sequences applied to online classi cation of EEG (slides) GSI2017
Riemannian metrics on Shape spaces of curves and surfaces Alice Barbara Tumpach GSI2017
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics Frédéric Barbaresco GSI2017
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We introduce poly-symplectic extension of Souriau Lie group Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics GSI2017
Noncommutative geometry and stochastic processes Marco Frasca GSI2017
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The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that reproduce in a proper way a stochastic process on a noncommutative geometry.
These processes are characterized by producing complex values and so, the corresponding Fokker–Planck equation resembles the Schr¨odinger equation. Indeed, by a direct numerical check, one can recover the kernel
of the Schr¨odinger equation starting by an ordinary Brownian motion.
This class of stochastic processes needs a Clifford algebra to exist.
Noncommutative geometry and stochastic processes
Coordinate-wise transformation and Stein-type densities Tomonari Sei GSI2017
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A Stein-type density function is de ned as a stationary point of the free-energy functional over a ber that consists of probability densities obtained by coordinate-wise transformations of a given density.
It is shown that under some conditions there exists a unique Stein-type density in each ber. An application to rating is discussed
Coordinate-wise transformation and Stein-type densities
Noncommutative geometry and stochastic processes (slides) GSI2017
Gender Diversity in industry : Overview of the situation, motivation and areas for action Valérie Archambault GSI2017
On affine immersions of the probability simplex and their conformal flattening (slides) GSI2017
On affine immersions of the probability simplex and their conformal flattening Atsumi Ohara GSI2017
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Embedding or representing functions play important roles in order to produce various information geometric structure. This paper investigates them from a viewpoint of affine differential geometry [2]. By restricting affine immersions to a certain class, the probability simplex is realized to be 1-conformally flat [3] statistical manifolds immersed in Rn+1. Using this fact, we introduce a concept of conformal flattening of such manifolds to obtain dually flat statistical (Hessian) ones with conformal divergences, and show explicit forms of potential functions, dual coordinates. Finally, we demonstrate applications of the conformal flattening to nonextensive statistical physics and certain replicator equations on the probability simplex.
On affine immersions of the probability simplex and their conformal flattening
Matrix realization of a homogeneous Hessian domain (slides) GSI2017
Matrix realization of a homogeneous Hessian domain Hideyuki Ishi GSI2017
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Extending previous results about matrix realization of a homogeneous cone by the author, we realize any homogeneous Hessian domain as a set of symmetric matrices with a speci c block decomposition.
A global potential function as well as a transitive affine group action preserving the Hessian structure is also expressed in terms of the matrix realization.
Matrix realization of a homogeneous Hessian domain
A family of anisotropic distributions on the hyperbolic plane Emmanuel Chevallier GSI2017
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Most of the parametric families of distributions on manifold are constituted of radial distributions. The main reason is that quantifying the anisotropy of a distribution on a manifold is not as straightforward as in vector spaces and usually leads to numerical computations.
Based on a simple de nition of the covariance on manifolds, this paper presents a way of constructing anisotropic distributions on the hyperbolic space whose covariance matrices are explicitly known. The approach remains valid on every manifold homeomorphic to vector spaces.
A family of anisotropic distributions on the hyperbolic plane
Global exponential attitude and gyro bias estimation from vector measurements Ioannis Sarras, Philippe Martin GSI2017
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We consider the classical problem of estimating the attitude and gyro biases of a rigid body from at least two vector measurements and a triaxial rate gyro. We propose a solution based on a dynamic nonlinear estimator designed without respecting the geometry of SO(3), which achieves uniform global exponential convergence. The convergence is established thanks to a dynamically scaled Lyapunov function.
Global exponential attitude and gyro bias estimation from vector measurements
k-Means Clustering with Hölder divergences Frank Nielsen, Ke Sun, Stéphane Marchand-Maillet GSI2017
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We introduced two novel classes of Hölder divergences and Hölder pseudo-divergences that are both invariant to rescaling, and that both encapsulate the Cauchy-Schwarz divergence and the skew Bhattacharyya divergences. We review the elementary concepts of those parametric divergences, and perform a clustering analysis on two synthetic datasets. It is shown experimentally that the symmetrized Hölder divergences consistently outperform signi cantly the Cauchy-Schwarz divergence in clustering tasks.
k-Means Clustering with Hölder divergences
Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics Elena Cordero, Fabio Nicola, Maurice A. de Gosson GSI2017
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The study of positivity properties of trace class operators is essential in the theory of quantum mechanical density matrices; the latter describe the “mixed states” of quantum mechanics and are essential in information theory. While a general theory for these positivity results is still lacking, we present some new results we have recently obtained and which generalize and extend the well-known conditions given in the 1970s by Kastler, Loupias, and Miracle-Sole, generalizing Bochner’s theorem on the Fourier transform of a probability measure. The tools we use are the theory of pseudodi¤erential operators, symplectic geometry, and Gabor frame theory. We also speculate about some consequences of a possibly varying Planck’s constant for the early universe.
Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics
Hamilton-Jacobi theory and Information Geometry Fabio Di Cosmo, Florio M. Ciaglia, Giuseppe Marmo GSI2017
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Recently, a method to dynamically de ne a divergence function D for a given statistical manifold (M; g ; T) by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function L on TM has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function D to be known and we look for a Lagrangian function L for which D is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.
Hamilton-Jacobi theory and Information Geometry
Hamilton-Jacobi theory and Information Geometry (slides) GSI2017
Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives Francesco Becattini GSI2017
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The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed in a quantum framework. The main building block of the proper relativistic extension of classical thermodynamics laws is the four-temperature vector.
The general relativistic thermodynamic equilibrium condition demands to be a Killing vector eld. A remarkable consequence of this condition is that all Lie derivatives of all physical observables along the four-temperature 
ow vanish.
Thermodynamic equilibrium and relativity: Killing vectors and Lie derivatives
Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator Gyu Jong Kim, Imsoon Jeong, Young Jin Suh GSI2017
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We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric Qm. The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a non-existence theorem of real hypersurfaces in Qm with normal Jacobi operator of Codazzi type.
Real hypersurfaces in the complex quadric with certain condition of normal Jacobi operator

(2017) ETTC 2017

Présentations ETTC 2017 ETTC 2017
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Présentations ETTC 2017