CT GSI Geometric Sciences of Information

Prochaines manifestations

As for GSI’13, GSI’15 and GSI’17, the objective of this SEE GSI’19 conference, hosted in Toulouse at ENAC, is to bring together pure/applied mathematicians and engineers, with common interest for Geometric tools and their applications for Information analysis.


The objective of this group is to bring together pure/applied mathematicians, physicist and engineers, with common interest for Geometric tools and their applications. It notably aim to organize conferences and to promote collaborative european and international research projects, and diffuse research results on the related domains. It aims to organise conferences, seminar, to promote collaborative local, european and international research project, and to diffuse research results in the the different related interested domains.

It emphasizes an active participation of engineers and researchers to develop emerging areas of collaborative research on “Information Geometry and Their Advanced Applications”. Current and ongoing uses of Information Geometry in applied mathematics are the following:

  • Thermodynamic, statistical physic.
  • Advanced Signal/Image/Video Processing, medical imaging
  • Complex Data Modeling and Analysis, Topological data analysis, dimension reduction, clustering, pattern detection
  • Information Ranking and Retrieval, Coding, Compression
  • Cognitive Systems, Artificial intelligence, Neural networks, Optimal Control, biological modelisation and computational morphology, Speech-sound recognition, natural language treatment.
  • Quantum information, correlations, coding
  • Statistics on Manifolds, Machine Learning, Manifold Learning
  • etc...

which are also substantially relevant for industry and current social challenges.


We describe the fundamental differential-geometric structures of information manifolds, state
the fundamental theorem of information geometry, and illustrate some uses of these information
manifolds in information sciences. The exposition is self-contained by concisely introducing the
necessary concepts of differential geometry with proofs omitted for brevity.
An elementary introduction to information geometry
    The "calculation of probabilities" began four years after the death of René Descartes [4][5][6][7][8][9][10], in 1654, in a correspondence between Blaise Pascal and Pierre Fermat [1][11][13][14][16][21][22][32]. They exchanged letters on elementary problems of gambling, in this case a problem of dice and a problem of "parties".  Pascal and Fermat were particularly interested by this problem and succeeded in "Party rule" by two different methods.  One understands the legitimate pride of Pascal in his address of the same year at the Académie Parisienne created by Mersenne, to which he presented, among "the ripe fruit of our Geometry" (“les fruits mûrs de notre Géométrie” in french) an entirely new treaty, of an absolutely unexplored matter, the distribution of chance in the games. In the same way, Pascal in his introduction to “Les Pensées” wrote that "under the influence of Méré, given to the game, he throws the bases of the calculation of probabilities and composes the Treatise of the Arithmetical Triangle. If Pascal appears at first sight as the initiator of the calculation of probabilities, watching a little closer, its role in the emergence of this theory is more complex. However, there is no trace of the word probabilities in Pascal's work. To designate what might resemble what we now call calculation of probabilities, one doesn’t even find the word in such a context. The only occurrences of probability are found in “Les Provinciales” where he referred to the doctrine of the Jesuits, or in “Les Pensées”. We do not find in Pascal's writings, the words of “Doctrine des chances”, or “Calcul des chances”, but only “Géométrie du hasard” (geometry of chance). In 1654, Blaise Pascal submitted a short paper to "Celeberrimae matheseos Academiae Parisiensi" (ancestor of the French Royal Academy of Sciences founded in 1666), with the title "Aleae Geometria” (Geometry of Chance) or “De compositione aleae in ludis ipsi subjectis", that was the seminal paper founding Probability as a new discipline in Science. In this paper, Pascal said “… et sic matheseos demonstrationes cum aleae incertitudine jugendo, et quae contraria videntur conciliando, ab utraque nominationem suam accipiens, stupendum hunc titulum jure sibi arrogat: Aleae Geometria” that we can translate as “By the union thus realized between the demonstrations of mathematics and the uncertainty of chance, and by the conciliation of apparent contradictions, it can derive its name from both sides and arrogate to itself this astonishing title: Geometry of Chance” (« … par l’union ainsi réalisée entre les démonstrations des mathématiques et l’incertitude du hasard, et par la conciliation entre les contraires apparents, elle peut tirer son nom de part et d’autre et s’arroger à bon droit ce titre étonnant: Géométrie du Hasard ». We can observe that Blaise Pascal attached a geometrical sense to probabilities in this seminal paper.  As Jacques Bernoulli, we can also give references to another Blaise Pascal document entitled “Art de penser” (the “Logique” of Port-Royal), published the year of his death (1662), with last chapters containing elements on the calculus of probabilities applied to history, to medicine, to miracles, to literary criticism, to events of life, etc.
   In “De l'esprit géométrique », the use of reason for knowledge is thought on a geometric model. In geometry, the first principles are given by the natural lights common to all men, and there is no need to define them. Other principles are clearly defined by definitions of names such that it is always possible to mentally substitute the definition for the defined [23][24][25]. These definitions of names are completely free, the only condition to be respected is univocity and invariability.  Judging his solution as one of his most important contributions to science, Pascal envisioned the drafting of a small treatise entitled “Géométrie du Hasard” (Geometry of Chance). He will never write it. Inspired by this, Christian Huygens wrote the first treatise on the calculation of chances, the “De ratiociniis in ludo aleae” ("On calculation in games of chance", 1657). We can conclude this preamble by observing that seminal work of Blaise Pascal on Probability was inspired by Geometry. The objective of GSI conference is to come back to this initial idea that we can geometrize statistics in a rigorous way.
   We can also make reference to Blaise Pascal for this GSI conference on computing geometrical statistics, because he was the inventor of computer with his “Pascaline” machine. The introduction of Pascaline marks the beginning of the development of mechanical calculus in Europe. This development, which will pass from the calculating machines to the electrical and electronic calculators of the following centuries, will culminate with the invention of the microprocessor. But it was also Charles Babbage who conceived his analytical machine from 1834 to 1837, a programmable calculating machine which was the ancestor of the computers of the 1940s, combining the inventions of Blaise Pascal and Jacquard’s machine, with instructions written on perforated cards, one of the descendants of the Pascaline, the first machine which supplied the intelligence of man.
[1] ABOUT, P.J., BOY, M., «La correspondance de Blaise Pascal et de Pierre de Fermat», Cahiers de Fontenay, n° 32, p. 59-73.
[2] BARBARESCO, F., “Les densités de probabilité « distinguées » et l’équation d’Alexis Clairaut:                                                regards croisés de Maurice Fréchet et de Jean-Louis Koszul », Colloque GRETSI’17, Juan-Les-Pins-September 2017
[3] BARBARESCO, F. & DJAFARI, A., ”Information, Entropy and Their Geometric Structures”, MDPI Entropy, September 2015; http://www.mdpi.com/books/pdfview/book/127
[4] BAYES, Th., «An essay towards solving a problem in the doctrine of chance», Philosophical Transactions of the Royal Society of London, 53 (1763), trad. J.-P. Cléro, Cahiers d'histoire et de philosophie des sciences, n° 18, 1988.
[5] BERNOULLI, J., Ars conjectandi (1713), die Werke von Jakob Bernoulli, 3 vols., Basel, 1969-1975.
[6] BYRNE, E., Probability and Opinion: A Study in the Medieval Pre-suppositions of Post-Medieval Theories of probability, La Haye, Martinus Nijhoff, 1968.
[7] CARDANO, De ludo aleae (ca. 1520), Opera Omnia, 10 vols., Stuttgart, 1966.
[8] CARDANO, The Book on Games of Chance, trad. S. H. Gould, New York, 1961.
[9] DASTON, L., Probability in the Enlightenment, Princeton, 1988.
[10] DAVID, F. N., Games, Gods and Gambling, A History of Probability and Statistical Ideas, London, Charles Griffin & Co, 1962.
[11] DAVIDSON, H. M., Pascal and the Arts of the Mind, Cambridge, Cambridge University Press, 1993.
[12] DE MOIVRE, A., The Doctrine of Chances, 3rd edition, London, 1756.
[13] EDWARDS, A. W. F., «Pascal and the Problem of Points», International Statistical Review, t. 51, 1983, p. 259-266.
[14] EDWARDS, A. W. F., «Pascal's Problem: The Gambler's Ruin», International Statistical Review, t. 50, 1982, p. 73-79.
[15] FRECHET M., Sur l’extension de certaines évaluations statistiques au cas de petits échantillons. Revue de l’Institut International de Statistique 1943, vol. 11, n° 3/4, pp. 182–205.
[16] GODFROY-GÉNIN A.S., Pascal la Géométrie du Hasard, Math. & Sci. hum., (38e année, n° 150, 2000, p. 7-39
[17] HACKING, I., The Emergence of Probability, Cambridge, 1975.
[18] KENDALL, M. G., PEARSON, E. S., (eds.)., Studies in the History of Statistics and Probability, 2 vols., London, 1970-1977.
[19] LEIBNIZ, G. W., «Nouveaux essais sur l'entendement humain», Sämtliche Schriften und Briefe, Berlin, 1962-1980, réed. Garnier-Flammarion, 1966.
[20] LEIBNIZ, G. W., Opuscules et fragments inédits, Couturat, ed., Paris, 1961.
[21] ORE, O., Cardano, the gambling scholar, Princeton, 1953.
[22] «Pascal et les probabilités», Cahiers Pédagogiques de philosophie et d'histoire des mathématiques, fascicule 4, IREM et CRDP de Rouen, 1993.
[23] PASCAL, B., Les Provinciales, Paris, Le Guern éd., 1987.
[24] PASCAL, B., Oeuvres complètes, J. Mesnard éd., 4 volumes publiés, 1964-1970.
[25] PASCAL, B., Pensées de Pascal, Paris, Ph. Sellier éd., 1991.
[26] PEARSON, K., The History of Statistics in the 17th and 18th Centuries, London, E.S. Pearson, ed., 1978.
[27] SCHNEIDER, I., «Why do we find the origin of a calculus of probabilities in the seventeenth century ?», Pisa Conference Proceedings, vol. 2, Dordrecht and Boston, J.Hintikka, D. Gruender, E. Agazzi eds., 1980.
[28] SCHNEIDER, I., Die Entwicklung des Wahrscheinlichkeitsbegriff in des Mathematik von Pascal bis Laplace, Munich, 1972.
[29] SHEYNIN, O., «On the early history of the law of large numbers», Studies in the History of Statistics and Probability, vol. 1, Paerson and Kendall eds., 1970.
[30] SHEYNIN, O., «On the prehistory of the theory of probability», Archives for History of Exact Sciences 12, 1974.
[31] STIGLER, S., The History of Statistics: The measurement of Uncertainety Before 1900, Cambridge (Mass.), The Belknap Press of Harvard University Press, 1986.

[32] TODHUNTER, I., A History of Mathematical Theory of Probability from the Time of Pascal to that of Laplace, Cambridge et Londres, Macmillan, 1865.



Frédéric Barbaresco


Pierre Baudot Frank Nielsen


Séminaire Léon Brillouin Logo


Format: 2018-12-10
Format: 2018-12-10


One of the most frequently used scientific words, is the word “Entropy”. The reason is that it is related to two main scientific domains: physics and information theory. Its origin goes back to the start of physics (thermodynamics), but since Shannon, it has become related to information theory.



An elementary introduction to information geometry Frank Nielsen
Détails de l'article
We describe the fundamental differential-geometric structures of information manifolds, state
the fundamental theorem of information geometry, and illustrate some uses of these information
manifolds in information sciences. The exposition is self-contained by concisely introducing the
necessary concepts of differential geometry with proofs omitted for brevity.
An elementary introduction to information geometry
Information geometry: Dualistic manifold structures and their uses Frank Nielsen
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information Frédéric Barbaresco


Image of Differential Geometrical Theory of Statistics
Publisher: Mdpi AG (2017)
Binding: Broché, 472 pages
This Special Issue "Differential Geometrical Theor y of Statistics" collates selected invited and contributed talks presented during the conference GSI'15 on "Geometric Science of Information" which was held at the Ecole Polytechnique, Paris-Saclay Campus, France, in October 2015 (Conference web site: http://www.see.asso.fr/gsi2015).
ISBN 978-3-03842-424-6 (print) • ISBN 978-3-03842-425-3 (electronic)
Image of Computational Information Geometry: For Image and Signal Processing
Publisher: Springer International Publishing AG (2016)
Binding: Relié, 299 pages
Author: Frédéric Barbaresco, Ali Mohammad-Djafari
Publisher: MDPI (2015), Binding: Paperback, 542 pages
Image of Geometric Structures of Information
Publisher: Springer Nature Switzerland AG (2018)
Binding: Relié, 392 pages

This book focuses on information geometry manifolds of structured data/information and their advanced applications featuring new and fruitful interactions between several branches of science: information science, mathematics and physics. It addresses interrelations between different mathematical domains like shape spaces, probability/optimization & algorithms on manifolds, relational and discrete metric spaces, 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.

The book collects the most important contributions to the conference GSI’2017 – Geometric Science of Information.


Orbituary: Profound sadness on the passing of Professeur Jean-Louis Koszul (1921-2018+), Friday January 12th 2018, Geometer, Henri Cartan's PhD student, member of Bourbaki, who honored us with his presence at GSI'13 (https://www.see.asso.fr/gsi2013) for Hirohiko Shima keynote on Geometry of Hessian Structures related to Koszul forms and Koszul-Vinberg Characteristic Function.
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information
Marcel Berger greatly contributed to mathematics, through his own publications, for example on holonomy groups, symmetric spaces, curvature pinching and the sphere theorem, spectral geometry or systolic geometry. His influence goes far beyond his research papers. His books and surveys have inspired not only his students, but a much broader audience. Important features of Marcel Berger's mathematical heritage are also his seminar and his influence on the round tables organized by his friend Arthur L. Besse. Marcel Berger's Riemannian geometry seminar held at the Universite Paris VII in the nineteen-seventies and eighties, hosted lectures by both reputable mathematicians and young researchers. For the participants, it was a unique place for lively and informal mathematical discussions and exchanges, as well as inspiration.
IHES event:
Marcel Berger is author of these Books :
Marcel Berger was a friend of Arthur L. Besse:
Michel Marie Deza died on  23 nov 2016 in an accidental fire in his apartment in Paris.
He was a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory.
He is a retired director of research at the French National Centre for Scientific Research (CNRS), the vice president of the European Academy of Sciences,a research professor at the Japan Advanced Institute of Science and Technology, and one of the three founding editors-in-chief of the European Journal of Combinatorics.
2012 video of Michel Marie Deza at IRCAM for Brillouin Seminar for « Geometric Science of Information »
Michel Marie Deza is author of the SPRINGER book:
Encyclopedia of Distances
Authors: Deza, Michel Marie, Deza, Elena
Michel was in the board of the GSI conferences since 2013.
Dear colleagues,
We are currently collecting papers for a special issue of Optimization Letters (OPTL, Springer) that is dedicated to the memory of Michel Deza (https://en.wikipedia.org/wiki/Michel_Deza). The title of the issue is "Applications of Distance Geometry".
Michel suddenly passed away on last November 2016. He is author of the Encyclopedia of Distances, that almost reached the threshold of 1000 citations, and which makes him an eminent member of the distance geometry community.
OPTL generally collects short contributions (about 10 pages long).
Submissions should be performed via the editorial system at the address:
by selecting the article type si:DGD16. The submission deadline is June 2nd, 2017.
All the best,
Antonio Mucherino
Carlile Lavor
OPTL guest co-editors
Call for Paper GSI17
The journal Information Geometry has taken up the challenge of how to think about and to look at mathematical science.
In principle, Information Geometry can connect various branches of mathematical sciences to allow for uncertainty from geometric thinking. There is still great potential for exploring new paradigms to break through conventional notions. The journal will publish papers on such research along with those on application of information geometry, broadly construed, emphasizing both theoretical and computational aspects. 
Topics of interests will include, but not be limited to, the Fisher–Rao metric, dual connections, divergence functions, entropy/cross-entropy, Hessian geometry, exponential/mixture geodesics and projections, Q-statistics, quantum statistical inference and computation, computational information geometry, algebraic statistics, optimal transportation problems, deep neural networks, and related topics. 
The authors and audience of the journal will be interdisciplinary, coming from mathematics, statistics, machine learning, statistical and quantum physics, information theory, control theory, neural computation, complex networks, cognitive science, and allied disciplines.
Annonce lancement Journal SPRINGER "Information Geometry"