Historical background
As for the first edition of GSI (2013) and in past publications, GSI2015 addresses inter-relations 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, tensor-valued morphology, optimal transport theory, manifold & topology learning, ... and applications like geometries of audio-processing, 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 co-homological 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 inter-relations 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 Levi-Civita 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.”.
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.
Website SEE: https://www.see.asso.fr/en/gsi2015
Video of the SEE-GSI2015 opening session - Presentation by Frédéric Barbaresco and François Gerin (SEE president):
https://www.youtube.com/channel/UC5HHo1jbQXusNQzU1iekaGA
https://www.youtube.com/watch?v=RZIy7pMPgqE
Introduction Slides : SEE-GSI'15-Opening-session.pdf
Presentation: https://www.see.asso.fr/node/14256
POSTER OF THE CONFERENCE
Organizers:
Organization and communication team:
Jean VIEILLE
http://www.syntropicfactory.com/
Industrial systems control expert : SEE comunication, SyntropicFactory, Interaxys, Control Chain Group
Valérie ALIDOR
SEE France.
Flore Manier
SEE France.
COM-1FILM -Julien Schmitt & co. Video production and capture.
CS-DC: Pierre Baudot, Julien Baudry, Paul Bourgine, Pierre Parrend, Clément Schreiner.
Partners:
Acknowledgements:
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.
We would like to acknowledge all the Organizing and Scientific Committee members for their hard work, in evaluating submissions: PierreAntoine Absil, Bijan Afsari, Stéphanie Allassonniére, 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, Jean-François Marcotorchino, Bertrand Maury, Ali Mohammad-Djafari, 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, Charles-Michel Marle, Mathilde Marcolli, Jean-Philippe Ovarlez, JeanPhilippe Vert, Allessandro Sarti, Jean-Paul Gauthier, Wen Huang, Antonin Chambolle, Jean-Franc¸ois Bercher, Bruno Pelletier, Stephan Weis, Gilles Celeux, Jean-Michel 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, François-Xavier Viallard, Guido Francisco Montufar, Emmanuel Chevallier, Christian Leonard, Nikolaus Hansen, Laurent Younes, Sylvain Arguillère, Shun-Ichi Amari, Julien Rabin, Dena Asta, Pierre-Yves Gousenbourger, Nicolas Boumal, Jun Zhang, Jan Naudts, Alexis Decurninge, Roman Belavkin, Hugo Boscain, Eric Moulines, Udo Von Toussaint, Jean-Philippe 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 co-authors, 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 peer-reviewed papers presented in this LNCS proceedings will be a valuable resource for researchers working in the field of information geometry, and for graduate students.