Lecture Notes in Computer Science 12829

Founding Editors

Gerhard GoosKarlsruhe Institute of Technology, Karlsruhe, Germany

Juris HartmanisCornell University, Ithaca, NY, USA

Editorial Board Members

Elisa BertinoPurdue University, West Lafayette, IN, USA

Wen GaoPeking University, Beijing, China

Bernhard SteffenTU Dortmund University, Dortmund, Germany

Gerhard WoegingerRWTH Aachen, Aachen, Germany

Moti YungColumbia University, New York, NY, USA

More information about this subseries at http://www.springer.com/series/7412

Frank Nielsen • Frédéric Barbaresco (Eds.)

Geometric Scienceof Information5th International Conference, GSI 2021Paris, France, July 21–23, 2021Proceedings

123

EditorsFrank NielsenSony Computer Science Laboratories Inc.Tokyo, Japan

Frédéric BarbarescoTHALES Land & Air SystemsMeudon, France

ISSN 0302-9743 ISSN 1611-3349 (electronic)Lecture Notes in Computer ScienceISBN 978-3-030-80208-0 ISBN 978-3-030-80209-7 (eBook)https://doi.org/10.1007/978-3-030-80209-7

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Preface

At the turn of the twenty-first century, new and fruitful interactions were discoveredbetween several branches of science: information sciences (information theory, digitalcommunications, statistical signal processing, etc.), mathematics (group theory,geometry and topology, probability, statistics, sheaves theory, etc.), and physics(geometric mechanics, thermodynamics, statistical physics, quantum mechanics, etc.).The aim of the Geometric Science of Information (GSI) biannual international con-ference is to discover mathematical structures common to all these disciplines byelaborating a “General Theory of Information” embracing physics, information science,and cognitive science in a global scheme.

As for GSI 2013, GSI 2015, GSI 2017, and GSI 2019 (https://franknielsen.github.io/GSI/), the objective of the 5th International SEE Conference on Geometric Science ofInformation (GSI 2021), hosted in Paris, was to bring together pure and appliedmathematicians and engineers with a common interest in geometric tools and theirapplications for information analysis. GSI emphasizes the active participation of youngresearchers to discuss emerging areas of collaborative research on the topic of“Geometric Science of Information and their Applications”. In 2021, the main themewas “LEARNING GEOMETRIC STRUCTURES”, and the conference took place atSorbonne University, France, during July 21–23, 2021, both in-person and virtually viavideo conferencing.

The GSI conference cycle was initiated by the Léon Brillouin Seminar team as earlyas 2009 (http://repmus.ircam.fr/brillouin/home). The GSI 2021 event was motivated inthe continuity of the first initiative launched in 2013 (https://www.see.asso.fr/gsi2013)at Mines ParisTech, consolidated in 2015 (https://www.see.asso.fr/gsi2015) at EcolePolytechnique, and opened to new communities in 2017 (https://www.see.asso.fr/gsi2017) at Mines ParisTech, and 2019 (https://www.see.asso.fr/gsi2019) at ENAC. In2011, we organized an Indo-French workshop on the topic of “Matrix InformationGeometry” that yielded an edited book in 2013, and in 2017, we collaborated with theCIRM seminar in Luminy, TGSI 2017 “Topological & Geometrical Structures ofInformation” (http://forum.cs-dc.org/category/94/tgsi2017).

GSI satellites events were organized in 2019 and 2020 as FGSI 2019 “Foundation ofGeometric Science of Information” in Montpellier, France (https://fgsi2019.sciencesconf.org/) and SPIGL 2020 “Joint Structures and Common Foundations ofStatistical Physics, Information Geometry and Inference for Learning” in Les Houches,France (https://www.springer.com/jp/book/9783030779566), respectively.

The technical program of GSI 2021 covered all the main topics and highlights in thedomain of the “Geometric Science of Information” including information geometrymanifolds of structured data/information and their advanced applications. This SpringerLNCS proceedings consists solely of original research papers that have been carefullypeer-reviewed by at least two or three experts. Accepted contributions were revisedbefore acceptance.

As with the previous GSI conferences, GSI 2021 addressed interrelations betweendifferent mathematical domains like shape spaces (geometric statistics on manifoldsand Lie groups, deformations in shape space, etc.), probability/optimization andalgorithms on manifolds (structured matrix manifold, structured data/information, etc.),relational and discrete metric spaces (graph metrics, distance geometry, relationalanalysis, etc.), computational and Hessian information geometry, geometric structuresin thermodynamics and statistical physics, algebraic/infinite dimensional/Banachinformation manifolds, divergence geometry, tensor-valued morphology, optimaltransport theory, manifold and topology learning, and applications like geometries ofaudio-processing, inverse problems, and signal/image processing. GSI 2021 topicswere enriched with contributions from Lie group machine learning, harmonic analysison Lie groups, geometric deep learning, geometry of Hamiltonian Monte Carlo, geo-metric and (poly)symplectic integrators, contact geometry and Hamiltonian control,geometric and structure preserving discretizations, probability density estimation andsampling in high dimension, geometry of graphs and networks, and geometry inneuroscience and cognitive sciences.

The GSI 2021 conference was structured in 22 sessions as follows:

• Probability and Statistics on Riemannian Manifolds - Chairs: Xavier Pennec,Cyrus Mostajeran

• Shapes Spaces - Chairs: Salem Said, Joan Glaunès• Geometric and Structure Preserving Discretizations - Chairs: Alessandro

Bravetti, Manuel de Leon• Lie Group Machine Learning - Chairs: Frédéric Barbaresco, Gery de Saxcé• Harmonic Analysis on Lie Groups - Chairs: Jean-Pierre Gazeau, Frédéric

Barbaresco• Geometric Mechanics - Chairs: Gery de Saxcé, Frédéric Barbaresco• Sub-Riemannian Geometry and Neuromathematics - Chairs: Alessandro Sarti,

Dario Prandi• Statistical Manifold and Hessian Information Geometry - Chairs: Noemie

Combe, Michel Nguiffo Boyom• Information Geometry in Physics - Chairs: Geert Verdoolaege, Jun Zhang• Geometric and Symplectic Methods for Hydrodynamical Models - Chairs:

Cesare Tronci, François Gay-Balmaz• Geometry of Quantum States - Chairs: Florio Maria Ciaglia, Michel Berthier• Deformed Entropy, Cross-Entropy, and Relative entropy - Chairs: Ting-Kam

Leonard Wong, Léonard Monsaingeon• Geometric Structures in Thermodynamics and Statistical Physics - Chairs:

Hiroaki Yoshimura, François Gay-Balmaz• Geometric Deep Learning - Chairs: Gabriel Peyré, Erik J. Bekkers• Computational Information Geometry 1 - Chairs: Frank Nielsen, Clément

Gauchy• Computational Information Geometry 2 - Chairs: Giovanni Pistone, Goffredo

Chirco• Optimal Transport and Learning - Chairs: Yaël Frégier, Nicolas Garcia Trillos

vi Preface

• Statistics, Information, and Topology - Chairs: Pierre Baudot, Michel NguiffoBoyom

• Topological and Geometrical Structures in Neurosciences - Chairs: PierreBaudot, Giovani Petri

• Manifolds and Optimization - Chairs: Stéphanie Jehan-Besson, Bin Gao• Divergence Statistics - Chairs: Michel Broniatowski, Wolfgang Stummer• Transport information geometry - Chairs: Wuchen Li, Philippe Jacquet

In addition, GSI 2021 has hosted 6 keynote speakers:

• Yvette Kosmann-Schwarzbach on “Structures of Poisson Geometry: old and new”• Max Welling on “Exploring Quantum Statistics for Machine Learning”• Michel Broniatowski on “Some insights on statistical divergences and choice of

models”• Maurice de Gosson on “Gaussian states from a symplectic geometry point of view”• Jean Petitot on “The primary visual cortex as a Cartan engine”• Giuseppe Longo on “Use and abuse of “digital information” in life sciences, is

Geometry of Information a way out?”

We would like to thank everyone involved in GSI 2021 for helping to make it suchan engaging and successful event.

June 2021 Frank NielsenFrédéric Barbaresco

Preface vii

Organization

Conference Co-chairs

Frank Nielsen Sony Computer Science Laboratories Inc., JapanFrédéric Barbaresco Thales Land and Air Systems, France

Local Organizing Committee

Gérard Biau Sorbonne Center for AI, FranceXavier Fresquet Sorbonne Center for AI, FranceMichel Broniatowski Sorbonne University, FranceJean-Pierre Françoise Sorbonne University, FranceOlivier Schwander Sorbonne University, FranceGabriel Peyré ELLIS Paris Unit and ENS Paris, France

Secretariat

Marianne Emorine SEE, France

Scientific Committee

Bijan Afsari Johns Hopkins University, USAPierre-Antoine Absil Université Catholique de Louvain, BelgiumStephanie Allasonnière University of Paris, FranceJesus Angulo Mines ParisTech, FranceMarc Arnaudon Bordeaux University, FranceJohn Armstrong King’s College London, UKAnne Auger Ecole Polytechnique, FranceNihat Ay Max Planck Institute, GermanyRoger Balian CEA, FranceFrédéric Barbaresco Thales Land and Air Systems, FrancePierre Baudot Median Technologies, FranceDaniel Bennequin University of Paris, FranceYannick Berthoumieu Bordeaux University, FranceJeremie Bigot Bordeaux University, FranceSilvere Bonnabel Mines ParisTech, FranceMichel Boyom Montpellier University, FranceMarius Buliga Simion Stoilow Institute of Mathematics

of the Romanian Academy, RomaniaGiovanna Citi Bologna University, ItalyLaurent Cohen Paris Dauphine University, FranceNicolas Couellan ENAC, France

Ana Bela Cruzeiro Universidade de Lisboa, PortugalRemco Duits Eindhoven University of Technology, NetherlandsStanley Durrleman Inria, FranceStephane Puechmorel ENAC, FranceFabrice Gamboa Institut Mathématiques de Toulouse, FranceJean-Pierre Gazeau University of Paris, FranceFrançois Gay-Balmaz Ecole Normale Supérieure, FranceMark Girolami Imperial College London, UKAudrey Giremus Bordeaux University, FranceHatem Hajri IRT SystemX, FranceSusan Holmes Stanford University, USAStephan Huckeman University of Göttingen, GermanyJean Lerbet University of Évry Val d’Essonne, FranceNicolas Le Bihan Grenoble Alpes University, FranceAlice Le Brigant Pantheon-Sorbonne University, FranceLuigi Malago Romanian Institute of Science and Technology,

RomaniaJonathan Manton University of Melbourne, AustraliaGaetan Marceau-Caron Mila, CanadaMatilde Marcolli CALTECH, USAJean-François

MarcotorchinoSorbonne University, France

Charles-Michel Marle Sorbonne University, FranceHiroshi Matsuzoe Nagoya Institute of Technology, JapanNina Miolane Stanford University, USAJean-Marie Mirebeau Paris-Sud University, FranceAli Mohammad-Djafari Centrale Supelec, FranceAntonio Mucherino IRISA, University of Rennes 1, FranceFlorence Nicol ENAC, FranceFrank Nielsen Sony Computer Science Laboratories Inc., JapanRichard Nock Data61, AustraliaYann Ollivier Facebook, FranceSteve Oudot Inria, FrancePierre Pansu Paris-Saclay University, FranceXavier Pennec Inria, FranceGiovanni Pistone Collegio Carlo Alberto, ItalyOlivier Rioul Telecom ParisTech, FranceGery de Saxcé Lille University, FranceSalem Said Bordeaux University, FranceAlessandro Sarti EHESS, Paris, FranceRodolphe Sepulchre Liège University, BelgiumOlivier Schwander Sorbonne University, FranceStefan Sommer Copenhagen University, DenmarkDominique Spehner Grenoble Alpes University, FranceWolfgang Stummer Friedrich-Alexander-Universität Erlangen, GermanyAlain Trouvé Ecole Normale Supérieure Paris-Saclay, France

x Organization

Geert Verdoolaege Ghent University, BelgiumRene Vidal Johns Hopkins University, USAJun Zhang University of Michigan, USAPierre-Antoine Absil Université Catholique de Louvain, BelgiumFlorio M. Ciaglia Max Planck Institute, GermanyManuel de Leon Real Academia de Ciencias, SpainGoffredo Chirco INFN, Naples, ItalyHông Vân Lê Institute of Mathematics of Czech Academy

of Sciences, Czech RepublicBruno Iannazzo Università degli Studi di Perugia, ItalyHideyuki Ishi Osaka City University, JapanKoichi Tojo RIKEN, Tokyo, JapanAna Bela Ferreira Cruzeiro Universidade de Lisboa, PortugalNina Miolane Stanford University, USACyrus Mostajeran University of Cambridge, UKRodolphe Sepulchre University of Cambridge, UKStefan Horst Sommer University of Copenhagen, DenmarkWolfgang Stummer University of Erlangen-Nürnberg, GermanyZdravko Terze University of Zagreb, CroatiaGeert Verdoolaege Ghent University, BelgiumWuchen Li UCLA, USAHiroaki Yoshimura Waseda University, JapanJean Claude Zambrini Universidade de Lisboa, PortugalJun Zhang University of Michigan, USA

Organization xi

Abstracts of Keynotes

Structures of Poisson Geometry: Old and New

Yvette Kosmann-Schwarzbach

Professeur des Universités honoraire, CNRS, Francehttp://www.cmls.polytechnique.fr/perso/kosmann/

Abstract: How did the brackets that Siméon-Denis Poisson introduce in 1809evolve into the Poisson geometry of the 1970’s? What are Poisson groups and,more generally, Poisson groupoids? In what sense does Dirac geometry gen-eralize Poisson geometry and why is it relevant for applications? I shall sketchthe definition of these structures and try to answer these questions.

References

. Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Pub-lishing Company (1987)

. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathe-matics, vol. 17, 2nd edn. Springer, New York. https://doi.org/10.1007/978-0-387-21792-5(1998)

. Laurent-Gengoux, C., Pichereau, and Vanhaecke, P.: Poisson Structures, Grundlehren dermathematischen Wissenschaften 347, Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-31090

. Kosmann-Schwarzbach, Y.: Multiplicativity from Lie groups to generalized geometry. In: Gra-bowska, K., et al., (eds.) Geometry of Jets and Fields, vol. 110. Banach Center Publications(2016)

. Alekseev, A., Cattaneo, A., Kosmann-Schwarzbach, Y., Ratiu, T.: Lett. Math. Phys. 90 (2009).Special volume of LMP on Poisson Geometry, guest editors

. Kosmann-Schwarzbach, Y., (éd.) Siméon-Denis Poisson : les Mathématiques au service de lascience, Editions de l’Ecole Polytechnique (2013)

. Kosmann-Schwarzbach, Y.: The Noether Theorems: Invariance and Conservation Laws in theTwentieth Century, Sources and Studies in the History of Mathematics and Physical Sciences.Springer, New York (2011). https://doi.org/10.1007/978-0-387-87868-3. translated by B. E.Schwarzbach

Exploring Quantum Statistics for MachineLearning

Max Welling

Informatics Institute, University of Amsterdam and Qualcomm Technologieshttps://staff.fnwi.uva.nl/m.welling/

ELLIS Board Member (European Laboratory for Learningand Intelligent Systems)https://ellis.eu/

Abstract: Quantum mechanics represents a rather bizarre theory of statistics thatis very different from the ordinary classical statistics that we are used to. In thistalk I will explore if there are ways that we can leverage this theory in devel-oping new machine learning tools: can we design better neural networks bythinking about entangled variables? Can we come up with better samplers byviewing them as observations in a quantum system? Can we generalize prob-ability distributions? We hope to develop better algorithms that can be simulatedefficiently on classical computers, but we will naturally also consider the pos-sibility of much faster implementations on future quantum computers. Finally, Ihope to discuss the role of symmetries in quantum theories.

Reference

. Bondesan, R., Welling, M.: Quantum Deformed Neural Networks, arXiv:2010.11189v1 [quant-ph], 21 October 2020. https://arxiv.org/abs/2010.11189

Some Insights on Statistical Divergencesand Choice of Models

Michel Broniatowski

Sorbonne Université, Paris

Abstract: Divergences between probability laws or more generally betweenmeasures define inferential criteria, or risk functions. Their estimation makes itpossible to deal with the questions of model choice and statistical inference, inconnection with the regularity of the models considered; depending on thenature of these models (parametric or semi-parametric), the nature of the criteriaand their estimation methods vary. Representations of these divergences as largedeviation rates for specific empirical measures allow their estimation in non-parametric or semi parametric models, by making use of information theoryresults (Sanov’s theorem and Gibbs principles), by Monte Carlo methods. Thequestion of the choice of divergence is wide open; an approach linking non-parametric Bayesian statistics and MAP estimators provides elements ofunderstanding of the specificities of the various divergences in the Ali-Silvey-Csiszar-Arimoto class in relation to the specific choices of the priordistributions.

References

. Broniatowski, M., Stummer, W.: Some universal insights on divergences for statistics, machinelearning and artificial intelligence. In: Nielsen, F., (eds.) Geometric Structures of Information.Signals and Communication Technology, pp. 149–211 (2019). Springer, Cham. https://doi.org/10.1007/978-3-030-02520-5_8

. Broniatowski, M.: Minimum divergence estimators, Maximum Likelihood and the generalizedbootstrap, to appear in “Divergence Measures: Mathematical Foundations and Applications inInformation-Theoretic and Statistical Problems” Entropy (2020)

. Csiszár, I., Gassiat, E.: MEM pixel correlated solutions for generalized moment and interpolationproblems. IEEE Trans. Inf. Theory 45 (7), 2253–2270 (1999)

. Liese, F., Vajda, I.: On divergences and informations in statistics and information theory. IEEETrans. Inf. Theory 52(10), 4394–4412 (2006)

Gaussian States from a Symplectic GeometryPoint of View

Maurice de Gosson

Faculty of Mathematics, NuHAG group, Senior Researcher at the Universityof Vienna

https://homepage.univie.ac.at/maurice.de.gosson

Abstract: Gaussian states play a ubiquitous role in quantum information theoryand in quantum optics because they are easy to manufacture in the laboratory,and have in addition important extremality properties. Of particular interest aretheir separability properties. Even if major advances have been made in theirstudy in recent years, the topic is still largely open. In this talk we will discussseparability questions for Gaussian states from a rigorous point of view usingsymplectic geometry, and present some new results and properties.

References

. de Gosson, M.: On the Disentanglement of Gaussian quantum states by symplectic rotations 358(4), 459–462 (2020). C.R. Acad. Sci. Paris

. de Gosson, M.A.: On Density operators with Gaussian weyl symbols. In: Boggiatto, P., et al.(eds.) Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Har-monic Analysis, pp. 191–206. Birkhäuser, Cham (2020). https://doi.org/10.1007/978-3-030-36138-9_12

. de Gosson, M.: Symplectic coarse-grained classical and semiclassical evolution of subsystems:new theoretical aspects. J. Math. Phys. 9, 092102 (2020)

. Cordero, E., de Gosson, M., Nicola, F.: On the positivity of trace class operators. in Advances inTheoretical and Mathematical Physics 23(8), 2061–2091 (2019). to appear

. Cordero, E., de Gosson, M., Nicola, F.: A characterization of modulation spaces by symplecticrotations. J. Funct. Anal. 278(11), 108474 (2020). to appear in

The Primary Visual Cortex as a Cartan Engine

Jean Petitot

CAMS, École des Hautes Études en Sciences Sociales, Francehttp://jean.petitot.pagesperso-orange.fr/

Abstract: Cortical visual neurons detect very local geometric cues as retinalpositions, local contrasts, local orientations of boundaries, etc. One of the maintheoretical problem of low level vision is to understand how these local cues canbe integrated so as to generate the global geometry of the images perceived, withall the well-known phenomena studied since Gestalt theory. It is an empiricalevidence that the visual brain is able to perform a lot of routines belonging todifferential geometry. But how such routines can be neurally implemented?Neurons are «point-like» processors and it seems impossible to do differentialgeometry with them. Since the 1990s, methods of “in vivo optical imagingbased on activity-dependent intrinsic signals” have made possible to visualizethe extremely special connectivity of the primary visual areas, their “functionalarchitectures.” What we called «Neurogeometry» is based on the discovery thatthese functional architectures implement structures such as the contact structureand the sub-Riemannian geometry of jet spaces of plane curves. For reasons ofprinciple, it is the geometrical reformulation of differential calculus from Pfaff toLie, Darboux, Frobenius, Cartan and Goursat which turns out to be suitable forneurogeometry.

References

. Agrachev, A., Barilari, D., Boscain, U.: A Comprehensive Introduction to Sub-RiemannianGeometry, Cambridge University Press (2020)

. Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space.J. Math. Imaging Vis. 24(3), 307–326 (2006). https://doi.org/10.1007/s10851-005-3630-2

. Petitot, J.: Neurogéométrie de la vision. Modèles mathématiques et physiques des architecturesfonctionnelles, Les Éditions de l’École Polytechnique, Distribution Ellipses, Paris (2008)

. Petitot J.: Landmarks for neurogeometry. In: Citti, G., Sarti, A., (eds.) Neuromathematics ofVision. LNM, pp. 1–85. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-34444-2_1

. Petitot, J.: Elements of Neurogeometry. Functional Architectures of Vision. LNM, Springer,2017. https://doi.org/10.1007/978-3-319-65591-8

. Prandi, D., Gauthier, J.-P.: A Semidiscrete Version of the Petitot Model as a Plausible Model forAnthropomorphic Image Reconstruction and Pattern Recognition, https://arxiv.org/abs/1704.03069v1 (2017)

Use and Abuse of “Digital Information”in Life Sciences, is Geometryof Information a Way Out?

Giuseppe Longo

Centre Cavaillès, CNRS & Ens Paris and School of Medicine,Tufts University, Boston

http://www.di.ens.fr/users/longo/

Abstract: Since WWII, the war of coding, and the understanding of thestructure of the DNA (1953), the latter has been considered as the digitalencoding of the Aristotelian Homunculus. Till now DNA is viewed as the“information carrier” of ontogenesis, the main or unique player and pilot ofphylogenesis. This heavily affected our understanding of life and reinforced amechanistic view of organisms and ecosystems, a component of our disruptiveattitude towards ecosystemic dynamics. A different insight into DNA as a majorconstraint to morphogenetic processes brings in a possible “geometry ofinformation” for biology, yet to be invented. One of the challenges is in the needto move from a classical analysis of morphogenesis, in physical terms, to a“heterogenesis” more proper to the historicity of biology.

References

. Islami, A., Longo, G.: Marriages of mathematics and physics: a challenge for biology,invited paper. In: Matsuno, K., et al. (eds.) The Necessary Western Conjunction tothe Eastern Philosophy of Exploring the Nature of Mind and Life, vol. 131,pp. 179¬192, December 2017. SpaceTimeIslamiLongo.pdf. Special Issue of Progressin Biophysics and Molecular Biology

. Longo, G.: How Future Depends on Past Histories and Rare Events in Systems of Life,Foundations of Science, (DOI), (2017). biolog-observ-history-future.pdf

. Longo, G.: Information and causality: mathematical reflections on cancer biology. InOrganisms. Journal of Biological Sciences, vo. 2, n. 1, 2018. BiologicalConseq-ofCompute.pdf

. Longo, G.: Information at the threshold of interpretation, science as human con-struction of sense. In: Bertolaso, M., Sterpetti, F. (eds.) A Critical Reflection onAutomated Science Will Science Remain Human? Springer, Dordrecht (2019).Information-Interpretation.pdf

. Longo, G., Mossio, M.: Geocentrism vs genocentrism: theories without metaphors,metaphors without theories. Interdisciplinary Sci. Rev. 45 (3), 380–405 (2020).Metaphors-geo-genocentrism.pdf

Contents

Probability and Statistics on Riemannian Manifolds

From Bayesian Inference to MCMC and Convex Optimisationin Hadamard Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Salem Said, Nicolas Le Bihan, and Jonathan H. Manton

Finite Sample Smeariness on Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Benjamin Eltzner, Shayan Hundrieser, and Stephan Huckemann

Gaussian Distributions on Riemannian Symmetric Spacesin the Large N Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Simon Heuveline, Salem Said, and Cyrus Mostajeran

Smeariness Begets Finite Sample Smeariness . . . . . . . . . . . . . . . . . . . . . . . 29Do Tran, Benjamin Eltzner, and Stephan Huckemann

Online Learning of Riemannian Hidden Markov Models in HomogeneousHadamard Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Quinten Tupker, Salem Said, and Cyrus Mostajeran

Sub-Riemannian Geometry and Neuromathematics

Submanifolds of Fixed Degree in Graded Manifoldsfor Perceptual Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Giovanna Citti, Gianmarco Giovannardi, Manuel Ritoré,and Alessandro Sarti

An Auditory Cortex Model for Sound Processing . . . . . . . . . . . . . . . . . . . . 56Rand Asswad, Ugo Boscain, Giuseppina Turco, Dario Prandi,and Ludovic Sacchelli

Conformal Model of Hypercolumns in V1 Cortex and the Möbius Group.Application to the Visual Stability Problem . . . . . . . . . . . . . . . . . . . . . . . . 65

Dmitri Alekseevsky

Extremal Controls for the Duits Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Alexey Mashtakov

Multi-shape Registration with Constrained Deformations . . . . . . . . . . . . . . . 82Rosa Kowalewski and Barbara Gris

Shapes Spaces

Geodesics and Curvature of the Quotient-Affine Metrics on Full-RankCorrelation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Yann Thanwerdas and Xavier Pennec

Parallel Transport on Kendall Shape Spaces . . . . . . . . . . . . . . . . . . . . . . . . 103Nicolas Guigui, Elodie Maignant, Alain Trouvé, and Xavier Pennec

Diffusion Means and Heat Kernel on Manifolds . . . . . . . . . . . . . . . . . . . . . 111Pernille Hansen, Benjamin Eltzner, and Stefan Sommer

A Reduced Parallel Transport Equation on Lie Groupswith a Left-Invariant Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Nicolas Guigui and Xavier Pennec

Currents and K-functions for Fiber Point Processes . . . . . . . . . . . . . . . . . . . 127Pernille E. H. Hansen, Rasmus Waagepetersen, Anne Marie Svane,Jon Sporring, Hans J. T. Stephensen, Stine Hasselholt,and Stefan Sommer

Geometry of Quantum States

Q-Information Geometry of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Christophe Corbier

Group Actions and Monotone Metric Tensors: The Qubit Case. . . . . . . . . . . 145Florio Maria Ciaglia and Fabio Di Nocera

Quantum Jensen-Shannon Divergences Between Infinite-DimensionalPositive Definite Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Hà Quang Minh

Towards a Geometrization of Quantum Complexity and Chaos . . . . . . . . . . . 163Davide Rattacaso, Patrizia Vitale, and Alioscia Hamma

Hunt’s Colorimetric Effect from a Quantum Measurement Viewpoint . . . . . . 172Michel Berthier and Edoardo Provenzi

Geometric and Structure Preserving Discretizations

The Herglotz Principle and Vakonomic Dynamics. . . . . . . . . . . . . . . . . . . . 183Manuel de León, Manuel Lainz, and Miguel C. Muñoz-Lecanda

Structure-Preserving Discretization of a Coupled Heat-Wave System,as Interconnected Port-Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . 191

Ghislain Haine and Denis Matignon

xxii Contents

Examples of Symbolic and Numerical Computation in Poisson Geometry . . . 200Miguel Evangelista–Alvarado, José Crispín Ruíz–Pantaleón,and Pablo Suárez–Serrato

New Directions for Contact Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Alessandro Bravetti, Marcello Seri, and Federico Zadra

Information Geometry in Physics

Space-Time Thermo-Mechanics for a Material Continuum . . . . . . . . . . . . . . 219Emmanuelle Rouhaud, Richard Kerner, Israa Choucair,Roula El Nahas, Alexandre Charles, and Benoit Panicaud

Entropic Dynamics Yields Reciprocal Relations . . . . . . . . . . . . . . . . . . . . . 227Pedro Pessoa

Lie Group Machine Learning

Gibbs States on Symplectic Manifolds with Symmetries . . . . . . . . . . . . . . . 237Charles-Michel Marle

Gaussian Distributions on the Space of Symmetric Positive DefiniteMatrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbitand Moment Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Frédéric Barbaresco

On Gaussian Group Convex Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Hideyuki Ishi

Exponential-Wrapped Distributions on SLð2;CÞ and the Möbius Group. . . . . 265Emmanuel Chevallier

Information Geometry and Hamiltonian Systems on Lie Groups . . . . . . . . . . 273Daisuke Tarama and Jean-Pierre Françoise

Geometric and Symplectic Methods for Hydrodynamical Models

Multisymplectic Variational Integrators for Fluid Models with Constraints . . . 283François Demoures and François Gay-Balmaz

Metriplectic Integrators for Dissipative Fluids . . . . . . . . . . . . . . . . . . . . . . . 292Michael Kraus

From Quantum Hydrodynamics to Koopman Wavefunctions I . . . . . . . . . . . 302François Gay-Balmaz and Cesare Tronci

Contents xxiii

From Quantum Hydrodynamics to Koopman Wavefunctions II . . . . . . . . . . . 311Cesare Tronci and François Gay-Balmaz

Harmonic Analysis on Lie Groups

The Fisher Information of Curved Exponential Familiesand the Elegant Kagan Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Gérard Letac

Continuous Wavelet Transforms for Vector-Valued Functions. . . . . . . . . . . . 331Hideyuki Ishi and Kazuhide Oshiro

Entropy Under Disintegrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Juan Pablo Vigneaux

Koszul Information Geometry, Liouville-Mineur Integrable Systemsand Moser Isospectral Deformation Method for HermitianPositive-Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

Frédéric Barbaresco

Flapping Wing Coupled Dynamics in Lie Group Setting . . . . . . . . . . . . . . . 360Zdravko Terze, Viktor Pandža, Marijan Andrić, and Dario Zlatar

Statistical Manifold and Hessian Information Geometry

Canonical Foliations of Statistical Manifolds with HyperbolicCompact Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Michel Boyom, Emmanuel Gnandi, and Stéphane Puechmorel

Open Problems in Global Analysis. Structured Foliationsand the Information Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Michel Boyom

Curvature Inequalities and Simons’ Type Formulasin Statistical Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Barbara Opozda

Harmonicity of Conformally-Projectively Equivalent Statistical Manifoldsand Conformal Statistical Submersions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

T. V. Mahesh and K. S. Subrahamanian Moosath

Algorithms for Approximating Means of Semi-infiniteQuasi-Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Dario A. Bini, Bruno Iannazzo, and Jie Meng

xxiv Contents

Geometric Mechanics

Archetypal Model of Entropy by Poisson Cohomology as Invariant CasimirFunction in Coadjoint Representation and Geometric FourierHeat Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Frédéric Barbaresco

Bridge Simulation and Metric Estimation on Lie Groups . . . . . . . . . . . . . . . 430Mathias Højgaard Jensen, Sarang Joshi, and Stefan Sommer

Constructing the Hamiltonian from the Behaviour of a Dynamical Systemby Proper Symplectic Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Nima Shirafkan, Pierre Gosselet, Franz Bamer, Abdelbacet Oueslati,Bernd Markert, and Géry de Saxcé

Non-relativistic Limits of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . 448Eric Bergshoeff, Johannes Lahnsteiner, Luca Romano, Jan Rosseel,and Ceyda Şimşek

Deformed Entropy, Cross-Entropy, and Relative Entropy

A Primer on Alpha-Information Theory with Application to Leakagein Secrecy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Olivier Rioul

Schrödinger Encounters Fisher and Rao: A Survey . . . . . . . . . . . . . . . . . . . 468Léonard Monsaingeon and Dmitry Vorotnikov

Projections with Logarithmic Divergences . . . . . . . . . . . . . . . . . . . . . . . . . 477Zhixu Tao and Ting-Kam Leonard Wong

Chernoff, Bhattacharyya, Rényi and Sharma-Mittal Divergence Analysisfor Gaussian Stationary ARMA Processes . . . . . . . . . . . . . . . . . . . . . . . . . 487

Eric Grivel

Transport Information Geometry

Wasserstein Statistics in One-Dimensional Location-Scale Models. . . . . . . . . 499Shun-ichi Amari and Takeru Matsuda

Traditional and Accelerated Gradient Descent for NeuralArchitecture Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Nicolás García Trillos, Félix Morales, and Javier Morales

Recent Developments on the MTW Tensor . . . . . . . . . . . . . . . . . . . . . . . . 515Gabriel Khan and Jun Zhang

Contents xxv

Wasserstein Proximal of GANs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524Alex Tong Lin, Wuchen Li, Stanley Osher, and Guido Montúfar

Statistics, Information and Topology

Information Cohomology of Classical Vector-Valued Observables. . . . . . . . . 537Juan Pablo Vigneaux

Belief Propagation as Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547Olivier Peltre

Towards a Functorial Description of Quantum Relative Entropy . . . . . . . . . . 557Arthur J. Parzygnat

Frobenius Statistical Manifolds and Geometric Invariants . . . . . . . . . . . . . . . 565Noemie Combe, Philippe Combe, and Hanna Nencka

Geometric Deep Learning

SU ð1; 1Þ Equivariant Neural Networks and Application to Robust ToeplitzHermitian Positive Definite Matrix Classification. . . . . . . . . . . . . . . . . . . . . 577

Pierre-Yves Lagrave, Yann Cabanes, and Frédéric Barbaresco

Iterative SE(3)-Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585Fabian B. Fuchs, Edward Wagstaff, Justas Dauparas,and Ingmar Posner

End-to-End Similarity Learning and Hierarchical Clustering for UnfixedSize Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

Leonardo Gigli, Beatriz Marcotegui, and Santiago Velasco-Forero

Information Theory and the Embedding Problemfor Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Govind Menon

cCorrGAN: Conditional Correlation GAN for Learning EmpiricalConditional Distributions in the Elliptope . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Gautier Marti, Victor Goubet, and Frank Nielsen

Topological and Geometrical Structures in Neurosciences

Topological Model of Neural Information Networks . . . . . . . . . . . . . . . . . . 623Matilde Marcolli

On Information Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634Pierre Baudot

xxvi Contents

Betti Curves of Rank One Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . 645Carina Curto, Joshua Paik, and Igor Rivin

Algebraic Homotopy Interleaving Distance. . . . . . . . . . . . . . . . . . . . . . . . . 656Nicolas Berkouk

A Python Hands-on Tutorial on Network and Topological Neuroscience . . . . 665Eduarda Gervini Zampieri Centeno, Giulia Moreni, Chris Vriend,Linda Douw, and Fernando Antônio Nóbrega Santos

Computational Information Geometry

Computing Statistical Divergences with Sigma Points . . . . . . . . . . . . . . . . . 677Frank Nielsen and Richard Nock

Remarks on Laplacian of Graphical Models in Various Graphs. . . . . . . . . . . 685Tomasz Skalski

Classification in the Siegel Space for Vectorial Autoregressive Data . . . . . . . 693Yann Cabanes and Frank Nielsen

Information Metrics for Phylogenetic Trees via Distributions of Discreteand Continuous Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

Maryam K. Garba, Tom M. W. Nye, Jonas Lueg,and Stephan F. Huckemann

Wald Space for Phylogenetic Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710Jonas Lueg, Maryam K. Garba, Tom M. W. Nye,and Stephan F. Huckemann

A Necessary Condition for Semiparametric Efficiencyof Experimental Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

Hisatoshi Tanaka

Parametrisation Independence of the Natural Gradient in OverparametrisedSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

Jesse van Oostrum and Nihat Ay

Properties of Nonlinear Diffusion Equations on Networks and TheirGeometric Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

Atsumi Ohara and Xiaoyan Zhang

Rényi Relative Entropy from Homogeneous Kullback-LeiblerDivergence Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

Goffredo Chirco

Statistical Bundle of the Transport Model. . . . . . . . . . . . . . . . . . . . . . . . . . 752Giovanni Pistone

Contents xxvii

Manifolds and Optimization

Efficient Quasi-Geodesics on the Stiefel Manifold . . . . . . . . . . . . . . . . . . . . 763Thomas Bendokat and Ralf Zimmermann

Optimization of a Shape Metric Based on Information Theory Appliedto Segmentation Fusion and Evaluation in Multimodal MRI for DIPGTumor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772

Stéphanie Jehan-Besson, Régis Clouard, Nathalie Boddaert,Jacques Grill, and Frédérique Frouin

Metamorphic Image Registration Using a Semi-lagrangian Scheme . . . . . . . . 781Anton François, Pietro Gori, and Joan Glaunès

Geometry of the Symplectic Stiefel Manifold Endowedwith the Euclidean Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

Bin Gao, Nguyen Thanh Son, P.-A. Absil, and Tatjana Stykel

Divergence Statistics

On f-divergences Between Cauchy Distributions . . . . . . . . . . . . . . . . . . . . . 799Frank Nielsen and Kazuki Okamura

Transport Information Hessian Distances . . . . . . . . . . . . . . . . . . . . . . . . . . 808Wuchen Li

Minimization with Respect to Divergences and Applications . . . . . . . . . . . . 818Pierre Bertrand, Michel Broniatowski,and Jean-François Marcotorchino

Optimal Transport with Some Directed Distances . . . . . . . . . . . . . . . . . . . . 829Wolfgang Stummer

Robust Empirical Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841Amor Keziou and Aida Toma

Optimal Transport and Learning

Mind2Mind: Transfer Learning for GANs . . . . . . . . . . . . . . . . . . . . . . . . . 851Yael Fregier and Jean-Baptiste Gouray

Fast and Asymptotic Steering to a Steady State for Networks Flows . . . . . . . 860Yongxin Chen, Tryphon Georgiou, and Michele Pavon

Geometry of Outdoor Virus Avoidance in Cities . . . . . . . . . . . . . . . . . . . . . 869Philippe Jacquet and Liubov Tupikina

xxviii Contents

A Particle-Evolving Method for Approximating the OptimalTransport Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878

Shu Liu, Haodong Sun, and Hongyuan Zha

Geometric Structures in Thermodynamics and Statistical Physics

Schrödinger Problem for Lattice Gases: A Heuristic Point of View . . . . . . . . 891Alberto Chiarini, Giovanni Conforti, and Luca Tamanini

A Variational Perspective on the Thermodynamics of Non-isothermalReacting Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900

François Gay-Balmaz and Hiroaki Yoshimura

On the Thermodynamic Interpretation of Deep Learning Systems . . . . . . . . . 909Rita Fioresi, Francesco Faglioni, Francesco Morri,and Lorenzo Squadrani

Dirac Structures in Thermodynamics of Non-simple Systems . . . . . . . . . . . . 918Hiroaki Yoshimura and François Gay-Balmaz

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927

Contents xxix