CT GSI Geometric Sciences of Information

Frédéric Barbaresco Contacter cette personne

Prochaines manifestations

This workshop is devoted to František Matúš, who passed away on May 17, 2018. His research interests reached several mathematical fields: he was involved in information theory, probability theory, statistics, geometry, algebra and matroid theory.
GSI’19 SUBMISSION DEADLINE POSTPONED UNTIL 11th March As for GSI’13, GSI’15 and GSI’17, the objective of this SEE GSI’19 conference, h
Aug 27 – Aug 29 Geometric Science of Information (GSI 19) Aug 30 – Sep 6 Geometric Statistics Sep 29 – Sep 4 Topology for Learning and Data Analysis Oct 14 – Oct 19 Information Geometry Nov 6 – Nov 8 Computational Aspects of Geometry


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.


Jean-Louis Koszul passed away January 12th 2018. This tribute is a scientific exegesis and admiration of Jean-Louis Koszul’s works on homogeneous bounded domains that have appeared over time as elementary structures of Information Geometry. Koszul has introduced fundamental tools to characterize the geometry of sharp convex cones, as Koszul-Vinberg characteristic Function, Koszul Forms, and affine representation of Lie Algebra and Lie Group. The 2nd Koszul form is an extension of classical Fisher metric. Koszul theory of hessian structures and Koszul forms could be considered as main foundation and pillars of Information Geometry.

The community of “Geometric Science of Information” (GSI) has lost a mathematician of great value, who informed his views by the depth of his knowledge of the elementary structures of hessian geometry and bounded homogeneous domains. His modesty was inversely proportional to his talent. Professor Koszul built in over 60 years of mathematical career, in the silence of his passions, an immense work, which makes him one of the great mathematicians of the XX’s century, whose importance will only affirm with the time. In this troubled time and rapid transformation of society and science, the example of Professor Koszul must be regarded as a model for future generations, to avoid them the trap of fleeting glories and recognitions too fast acquired. The work of Professor Koszul is also a proof of fidelity to his masters and in the first place to Prof. Elie Cartan, who inspired him throughout his life. Henri Cartan writes on this subject “I do not forget the homage he paid to Elie Cartan’s work in Differential Geometry during the celebration, in Bucharest, in 1969, of the centenary of his birth. It is not a coincidence that this centenary was also celebrated in Grenoble the same year. As always, Koszul spoke with the discretion and tact that we know him, and that we love so much at home”.

We will conclude by quoting Jorge Luis Borges “both forgetfulness and memory are apt to be inventive” (Doctor Brodie’s report). Our generation and previous one have forgotten or misunderstood the depth of the work of Jean-Louis Koszul and Elie Cartan on the study of bounded homogeneous domains. It is our responsibility to correct this omission, and to make it the new inspiration for the Geometric Science of Information. We will invite readers  to listen to the last interview of Jean-Louis Koszul for 50th birthday of Joseph Fourier Institute https://www.youtube.com/watch?v=AzK5K7Q05sw

Frédéric Barbaresco & Michel Boyom


[1] Selected Papers of J L Koszul, Series in Pure Mathematics, Volume 17, World Scientific Publishing, 1994
[2] Cartan H., Allocution de Monsieur Henri Cartan, colloque Jean-Louis Koszul, Annales de l’Institut Fourier, tome 37, 4, pp.1-4, 1987
[3] Koszul J.L., L’oeuvre d’élie Cartan en géométrie différentielle, in élie Cartan, 1869-1951. Hommage de l’Académie de la République Socialiste de Roumanie à l’occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d’Expression Latine, tenu à Bucarest en 1969 (Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), pp. 39-45.
[4] dernière interview de J.L. Koszul pour le laboratoire mathématique de l’Institut Fourier en 2016 : vidéo : https ://www.youtube.com/watch ?v=AzK5K7Q05sw
[5] Koszul J.L. , Sur la forme hermitienne canonique des espaces homogènes complexes. Can. J. Math., 7, 562–576, 1955
[6] Koszul J.L. , Exposés sur les Espaces Homogènes Symétriques ; Publicação da Sociedade de Matematica de São Paulo : São Paulo, Brazil, 1959
[7] Koszul J.L., Domaines bornées homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, pp. 515-533., 1961
[8] Koszul J.L. , Ouverts convexes homogènes des espaces affines. Math. Z., 79, 254–259, 1962
[9] Koszul J.L. , Variétés localement plates et convexité. Osaka. J. Math., 2, 285–290, 1965
[10] Koszul J.L, Lectures on Groups of Transformations, Tata Institute of Fundamental Research, Bombay, 1965
[11] Koszul J.L., Déformations des variétés localement plates, .Ann Inst Fourier, 18 , 103-114, 1968
[12] Koszul J.L., Trajectoires Convexes de Groupes Affines Unimodulaires. In Essays on Topology and Related Topics ; Springer : Berlin, Germany, pp. 105–110, 1970
[13] Koszul J.L., ZOU Y. Introduction to Symplectic Geometry Springer: Berlin, Germany, 2019 https://www.springer.com/la/book/9789811339868
[14] Vey J., Sur une notion d’hyperbolicité des variétés localement plates, Thèse de troisième cycle de mathématiques pures, Faculté des sciences de l’université de Grenoble, 1969
[15] Vey J., Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tome 24, 4, p.641-665, 1970
[16] Alekseevsky D., Vinberg’s theory of homogeneous convex cones : developments and applications, Transformation groups 2017. Conference dedicated to Prof. Ernest B. Vinberg on the occasion of his 80th birthday, Moscou, December, 2017, https ://www.mccme.ru/tg2017/slides/alexeevsky.pdf  ,vidéo : http ://www.mathnet.ru/present19121
[17] Vinberg E.B., Homogeneous cones, Dokl. Akad. Nauk SSSR., 133, pp. 9–12, 1960 ; Soviet Math. Dokl., 1, pp. 787–790, 1961
[18] Vinberg E.B., The structure of the group of automorphisms of a convex cone, Trudy Moscov. Mat. Obshch., 13, pp.56–83, 1964 ; Trans. Moscow Math. Soc., 13, 1964
[19] Shima H., The Geometry of Hessian Structures, World Scientific, 2007
[20] Shima H., Geometry of Hessian Structures, Springer Lecture Notes in Computer Science, Vol. 8085, F. Nielsen, & Barbaresco, Frederic (Eds.), pp.37-55, 2013
[21] Malgrange B., Quelques souvenirs de Jean-Louis KOSZUL, Gazette des Mathématiciens - 156, pp. 63-64, Avril 2018
[22] Cartier P., In memoriam Jean-Louis KOSZUL, Gazette des Mathématiciens - 156, pp. 64-66, Avril 2018
[23] Nguiffo Boyom M., Transversally Hessian foliations and information geometry I. Am. Inst. Phys. Proc. , 1641, pp. 82–89, 2014
[24] Nguiffo Boyom M, Wolak, R., Transverse Hessian metrics information geometry MaxEnt 2014. AIP. Conf. Proc. Am. Inst. Phys. 2015
[25] Barbaresco, F. Jean-Louis Koszul and the elementary structures of Information Geometry. In Geometric Structures of Information Geometry; Nielsen, F.; Ed.; Springer: Berlin, Germany, 2018;  https://link.springer.com/chapter/10.1007/978-3-030-02520-5_12
[26] Barbaresco, F. Koszul Contemporaneous Lectures: Elementary Structures of Information Geometry and Geometric Heat Theory. In Introduction to Symplectic Geometry; Koszul, J.L., Ed.; Springer: Berlin, Germany, 2018.;  https://www.springer.com/la/book/9789811339868
[27] Barbaresco, F. Jean-Louis Koszul et les Structures Elémentaires de la Géométrie de l’Information; Revue SMAI Matapli; SMAI Editor; Volume 116, pp.71-84, Novembre 2018 ; http://smai.emath.fr/IMG/pdf/Matapli116.pdf (long version: http://forum.cs-dc.org/uploads/files/1520499744825-jean-louis-koszul-et-les-structures-%C3%A9l%C3%A9mentaires-de-la-g%C3%A9om%C3%A9trie-de-l-x27-information-final-revc-1.pdf )
[28] Barbaresco, F. Les densités de probabilité « distinguées » et l'équation d'Alexis Clairaut: regards croisés de Maurice Fréchet et de Jean-Louis Koszul, Conférence Histoire de la discipline, GRETSI'17 , Juan-Les-Pins, Septembre 2017 ; http://gretsi.fr/colloque2017/myGretsi/programme.php ;
[29] FGSI’19 Cartan-Koszul-Souriau « Foundations of Geometric Structure of Information », 4-6 Février 2019, IMAG Montpellier ; https://fgsi2019.sciencesconf.org/
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 : 2019-03-19
Format : 2019-03-19


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


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)
Author: Frédéric Barbaresco, Ali Mohammad-Djafari
Publisher: MDPI (2015), Binding: Paperback, 542 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.


This workshop is devoted to František Matúš, who passed away on May 17, 2018. His research interests reached several mathematical fields: he was involved in information theory, probability theory, statistics, geometry, algebra and matroid theory. Thus, the workshop to commemorate him is intended to be multidisciplinary, involving those fields in which František worked or the areas close to his interests. We particularly welcome contributions devoted to information geometry, entropic regions, information inequalities, cryptography, polymatroids, optimization of convex integral functionals, discrete Markovian random sequences, conditional independence, semigraphoids, graphical models, exponential families and algebraic statistics.
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