Dimension reduction on Riemannian manifolds,
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,
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.
The technical program of GSI2015 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 peer-reviewed by two or three experts before, and revised before acceptance.
The GSI15 program includes the renown invited speaker Professor Charles-Michel 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 Euler-Poincaré 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:
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.