Chairmen Welcome
Welcome to GSI’15!
Welcome to the LIX Colloquium 2015 of Ecole Polytechnique!´
On behalf of both the organizing and the scientific committees, it is our great pleasure to welcome all delegates, representatives and participants from around the world to the second International SEE conference on “Geometric Science of Information” (GSI’15), hosted by Ecole´ Polytechnique (Palaiseau, France), from 28th to 30th of October 2015 (http://www.gsi2015.org/).
GSI’15 benefits from scientific sponsor of Société de Mathématique Appliquées et Industrielles (SMAI), and financial sponsor by:
GSI’15 is also supported by CNRS Federative Networks MIA and ISIS.
The 3day conference is also organized in the frame of the relations set up between SEE and scientific institutions or academic laboratories: Ecole Polytechnique, Ecole des Mines de Paris, INRIA, Supélec, Université ParisSud, Institut Mathématique de Bordeaux, Sony Computer Science Laboratories, Telecom SudParis, and Telecom ParisTech.
We would like to express all our thanks to the Computer Science Department LIX of Ecole Polytechnique for hosting this second scientific´ event at the interface between Geometry, Probability and Information Geometry. In particular, we warmly thank Evelyne Rayssac of LIX, Ecole Polytechnique for her kind administrative support that helped us book the auditorium and various ressources at Ecole Polytechnique, and Olivier Bournez (LIX Director) for providing financial support.
The GSI conference cycle has been initiated by the Brillouin Seminar Team. The GSI’15 event has been motivated in the continuity of first initiatives launched in 2013 (see LNCS proceedings 8085). We mention that in 2011, we organized an indofrench workshop on “Matrix Information Geometry” that yielded an edited book in 2013.
The technical program of GSI’15 covers all the main topics and highlights in the domain of “Geometric Science of Information” including Information Geometry Manifolds of structured data/information and their advanced applications. This proceedings consists solely of original research papers that have been carefully peerreviewed by two or three experts before, and revised before acceptance.
The GSI15 program includes the renown invited speaker Professor CharlesMichel Marle (UPMC, Université Pierre et Marie Curie, Paris, France) that gives a talk on “Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds”, and three (3) keynote distinguished speakers:
 Professor Marc Arnaudon (Bordeaux University, France): “Stocastic EulerPoincaré reduction,”
 Professor Tudor Ratiu (EPFL, Switzerland): “Symetry methods in geometric mechanics,”
 Professor Matilde Marcolli (Caltech, US): “From Geometry and Physics to Computational Linguistics”,
and a short course given by Professor Dominique Spehner (Grenoble University, France) on the “Geometry on the set of quantum states and quantum correlations” chaired by Roger Balian (CEA, France).
The collection of papers have been arranged into the following seventeen (17) thematic sessions that illustrates the richness and versatility of the field:
 Dimension reduction on Riemannian manifolds,
 Optimal Transport,
 Optimal Transport and applications in Imagery/Statistics,
 Shape Space & Diffeomorphic mappings,
 Random Geometry & Homology,
 Hessian Information Geometry,
 Topological forms and Information,
 Information Geometry Optimization,
 Information Geometry in Image Analysis,
 Divergence Geometry,
 Optimization on Manifold,
 Lie Groups and Geometric Mechanics/Thermodynamics,
 Computational Information Geometry,
 Lie Groups: Novel Statistical and Computational Frontiers,
 Geometry of Time Series and Linear Dynamical systems,
 Bayesian and Information Geometry for Inverse Problems,
 Probability Density Estimation.
Historical background
As for the first edition of GSI (2013) and in past publications, GSI’15 addresses interrelations between different mathematical domains like shape spaces (geometric statistics on manifolds and Lie groups, deformations in shape space, ...), probability/optimization & algorithms on manifolds (structured matrix manifold, structured data/Information, ...), relational and discrete metric spaces (graph metrics, distance geometry, relational analysis,...), computational and Hessian information geometry, algebraic/infinite dimensional/Banach information manifolds, divergence geometry, tensorvalued morphology, optimal transport theory, manifold & topology learning, ... and applications like geometries of audioprocessing, inverse problems and signal processing.
At the turn of the century, new and fruitful interactions were discovered between several branches of science: Information Science (information theory, digital communications, statistical signal processing,), Mathematics (group theory, geometry and topology, probability, statistics,...) and Physics (geometric mechanics, thermodynamics, statistical physics, quantum mechanics, ...).
From Probability to Geometry
Probability is again the subject of a new foundation to apprehend new structures and generalize the theory to more abstract spaces (metric spaces, shape space, homogeneous manifolds, graphs ....). A first attempt to probability generalization in metric spaces was developed by Maurice Fréchet in the middle of last century, in the framework of abstract spaces topologically affine and “distance space” (“espace distancié”). More recently, Misha Gromov, at IHES (Institute of Advanced Scientific Studies), indicates possibilities for (non)homological linearization of basic notions of the probability theory and also the replacement of the real numbers as values of probabilities by objects of suitable combinatorial categories. In parallel, Daniel Bennequin, from Institut mathématique de Jussieu, observes that entropy is a universal cohomological class in a theory associated to a family of observable quantities and a family of probability distributions.
From Groups Theory to Geometry
As observed by Gaston Bachelard, “the group provides evidence of a mathematic closed on itself. Its discovery closes the era of conventions, more or less independent, more or less coherent”. About Elie Cartan’s work on Group Theory, Henri Poincaré said that “the problems addressed by Elie Cartan are among the most important, most abstract and most general dealing with Mathematics; group theory is, so to speak, the whole Mathematics, stripped of its material and reduced to pure form. This extreme level of abstraction has probably made my presentation a little dry; to assess each of the results, I would have had virtually render him the material which he had been stripped; but this refund can be made in a thousand different ways; and this is the only form that can be found as well as a host of various Garments, which is the common link between mathematical theories that are often surprised to find so near”.
From Mechanics to Geometry
The last elaboration of geometric structure on information is emerging at the interrelations between “Geometric Mechanics” and ”Information Theory” that will be largely debated at GSI15 conference with invited speakers as C. M. Marle, T. Ratiu and M. Arnaudon. Elie Cartan, the master of Geometry during the last century, said ”distinguished service that has rendered and will make even the absolute differential calculus of Ricci and LeviCivita should not prevent us to avoid too exclusively formal calculations, where debauchery indices often mask a very simple geometric fact. It is this reality that I have sought to put in evidence everywhere.”.
Fig. 1. Into the Flaming Forge of Vulcan, into the Ninth Sphere, Mars descends in order to retemper his aming sword and conquer the heart of Venus (Diego Velazquez, Museo Nacional del Prado)"
Public domain image, courtesy of https://en.wikipedia.org/wiki/Apollo_in_the_Forge_of_Vulcan
For the anecdote, Elie Cartan, was the son of Joseph Cartan who was the village blacksmith, and Elie recalled that his childhood had passed under ”blows of the anvil, which started every morning from dawn”. One can imagine that the hammer blows given by Joseph on the anvil, giving shape and CURVATURE to the metal, insidiously influencing Elie’s mind with germinal intuition of fundamental geometric concepts. Alliance of Geometry and Mechanics is beautifully illustrated by this image of Forge, in this painting of Velasquez about Vulcan God (see Figure 1). This concordance of meaning is also confirmed by etymology of word “Forge”, that comes from late XIV century, “a smithy,” from Old French forge “forge, smithy” (XII century), earlier faverge, from Latin fabrica “workshop, smith’s shop”, from faber (genitive fabri) “workman in hard materials, smith”.
As Henri Bergson said in book “The Creative Evolution” in 1907: “As regards human intelligence, there is not enough noticed that mechanical invention was first its essential approach ... we should say perhaps not Homo sapiens, but Homo faber. In short, intelligence, considered in what seems to be its original feature, is the faculty of manufacturing artificial objects, especially tools to make tools, and of indefinitely varying the manufacture.”
Geometric Science of Information: a new Grammar of Sciences
Henri Poincaré said that “Mathematics is the art of giving the same name to different things” (“La mathématique est l’art de donner le même nom `a des choses différentes.” in “Science et méthode”, 1908). By paraphrasing Henri Poincaré, we could claim that “Geometric Science of Information” is the art of giving the same name to different sciences. The rules and the Structures developed in GSI15 conference is a kind of new Grammar for Sciences.
Acknowledgements
We would like to acknowledge all the Organizing and Scientific Committee members for their hard work, in evaluating submissions: PierreAntoine Absil, Bijan Afsari, Stéphanie Allassonni`ere, Jésus Angulo, Marc Arnaudon, Michael Aupetit, Roger Balian, Barbara Trivellato, Pierre Baudot, Daniel Bennequin, Yannick Berthoumieu, Jérémie Bigot, Silvère Bonnabel, Michel Boyom, Michel Broniatowski, Martins Bruveris, Charles Cavalcante, Frédéric Chazal, Arshia Cont, Gery de Saxcé, Laurent Decreusefond, Michel Deza, Stanley Durrleman, Patrizio Frosini, Alfred Galichon, Alexander Ivanov, Jérémie Jakubowicz, Hongvan Le, Nicolas Le Bihan, Luigi Malago, Jonathan Manton, JeanFrançois Marcotorchino, Bertrand Maury, Ali MohammadDjafari, Richard Nock, Yann Ollivier, Xavier Pennec, Michel Petitjean, Gabriel Peyré, Giovanni Pistone, Olivier Rioul, Said Salem, Olivier Schwander, Rodolphe Sepulchre, Hichem Snoussi, Alain Trouvé, Claude Vallée, Geert Verdoolaege, Rui Vigelis, Susan Holmes, Martin Kleinsteuber, Shiro Ikeda, Martin Bauer, CharlesMichel Marle, Mathilde Marcolli, JeanPhilippe Ovarlez, JeanPhilippe Vert, Allessandro Sarti, JeanPaul Gauthier, Wen Huang, Antonin Chambolle, JeanFranc¸ois Bercher, Bruno Pelletier, Stephan Weis, Gilles Celeux, JeanMichel Loubes, Anuj Srivastana, Johannes Rauh, Joan Alexis Glaunes, Quentin Mérigot, K. S. Subrahamanian Moosath, K.V. Harsha, Emmanuel Trelat, Lionel Bombrun, Olivier Cappé, Stephan Huckemann, Piotr Graczyk, Fernand Meyer, Corinne Vachier, Tudor Ratiu, Klas Modin, Herve Lombaert, Michèle Basseville, Juliette Matiolli, Peter D. Grünwald, FrançoisXavier Viallard, Guido Francisco Montufar, Emmanuel Chevallier, Christian Leonard, Nikolaus Hansen, Laurent Younes, Sylvain Arguillère, ShunIchi Amari, Julien Rabin, Dena Asta, PierreYves Gousenbourger, Nicolas Boumal, Jun Zhang, Jan Naudts, Alexis Decurninge, Roman Belavkin, Hugo Boscain, Eric Moulines, Udo Von Toussaint, JeanPhilippe Anker, Charles Bouveyron, Michael Blum, Sylvain Chevallier, Jeremy Bensadon, Philippe Cuvillier, Hervé Lombaert, Frédéric Barbaresco and Frank Nielsen.
We also give our thanks to authors and coauthors, for their tremendous effort and scientific contribution.
As for GSI’13, a selected number of contributions focusing on a core topic have been invited to contribute to a chapter without page restriction of a collective book: This yielded the edited book “Geometric Theory of Information” in 2014. Similarly, for GSI’15, we invite prospective authors to submit their original work to a special issue on “ADVANCES IN DIFFERENTIAL GEOMETRICAL THEORY OF STATISTICS” of the MDPI Entropy journal.
It is our hope that the fine collection of peerreviewed papers presented in this LNCS proceedings will be a valuable resource for researchers working in the field of information geometry, and for graduate students.
July 2015.
Frank NIELSEN, cochair
Ecole Polytechnique, Palaiseau, France
Sony Computer Science Laboratories, Tokyo, Japan


Frédéric BARBARESCO, cochair
President of SEE ISIC Club (Ingéniérie des Systèmes d'Information et de Communications) https://www.see.asso.fr/ctisic
Senior Scientist & Advanced Studied Manager, Thales Land & Air Systems, France
