Information Geometry and Its Applications

450 Pages·2018·3.508 MB·English

by  Ay, Nihat

Springer Proceedings in Mathematics & Statistics
Nihat Ay · Paolo Gibilisco 
František Matúš    Editors
Information
Geometry and
Its Applications
IGAIA IV, Liblice, Czech Republic,
June 2016
Springer Proceedings in Mathematics & Statistics
Volume 252
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Nihat Ay Paolo Gibilisco
(cid:129)
š úš
Franti ek Mat
Editors
Information Geometry and Its
Applications
On the Occasion of
’
Shun-ichi Amari s 80th Birthday,
IGAIA IV, Liblice, Czech Republic, June 2016
123
Editors
NihatAy PaoloGibilisco
Information Theory ofCognitive Systems Department ofEconomics andFinance
MPIfor Mathematics inthe Sciences University of Rome“Tor Vergata”
Leipzig, Germany Rome, Italy
and FrantišekMatúš
Institute of Information Theory and
SantaFeInstitute Automation
SantaFe, NM, USA AcademyofSciencesoftheCzechRepublic
Prague,Czech Republic
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-319-97797-3 ISBN978-3-319-97798-0 (eBook)
https://doi.org/10.1007/978-3-319-97798-0
LibraryofCongressControlNumber:2018950804
Mathematics Subject Classification (2010): 60A10, 62B05, 62B10, 62G05, 53B21, 53B05, 46B20,
94A15,94A17,94B27
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Shun-ichiAmari
Contents
Part I Applications of Information Geometry
Geometry of Information Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Shun-ichi Amari, Naotsugu Tsuchiya and Masafumi Oizumi
Information Geometry and Game Theory . . . . . . . . . . . . . . . . . . . . . . . 19
Jürgen Jost, Nils Bertschinger, Eckehard Olbrich and David Wolpert
Higher Order Equivalence of Bayes Cross Validation
and WAIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Sumio Watanabe
Restricted Boltzmann Machines: Introduction and Review . . . . . . . . . . 75
Guido Montúfar
Part II Infinite-Dimensional Information Geometry
Information Geometry of the Gaussian Space . . . . . . . . . . . . . . . . . . . . 119
Giovanni Pistone
Congruent Families and Invariant Tensors . . . . . . . . . . . . . . . . . . . . . . 157
Lorenz Schwachhöfer, Nihat Ay, Jürgen Jost and Hông Vân Lê
Nonlinear Filtering and Information Geometry:
A Hilbert Manifold Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Nigel J. Newton
Infinite-Dimensional Log-Determinant Divergences III:
Log-Euclidean and Log-Hilbert–Schmidt Divergences . . . . . . . . . . . . . . 209
Hà Quang Minh
Part III Theoretical Aspects of Information Geometry
Entropy on Spin Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Peter Harremoës
vii
viii Contents
Information Geometry Associated with Generalized Means. . . . . . . . . . 279
Shinto Eguchi, Osamu Komori and Atsumi Ohara
Information Geometry with (Para-)Kähler Structures . . . . . . . . . . . . . . 297
Jun Zhang and Teng Fei
Doubly Autoparallel Structure on the Probability Simplex . . . . . . . . . . 323
Atsumi Ohara and Hideyuki Ishi
Complementing Chentsov’s Characterization. . . . . . . . . . . . . . . . . . . . . 335
Akio Fujiwara
Constant Curvature Connections On Statistical Models. . . . . . . . . . . . . 349
Alexander Rylov
Relation Between the Kantorovich–Wasserstein Metric
and the Kullback–Leibler Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Roman V. Belavkin
Part IV Quantum Information Geometry
Some Inequalities for Wigner–Yanase Skew Information. . . . . . . . . . . . 377
Shunlong Luo and Yuan Sun
Information Geometry of Quantum Resources. . . . . . . . . . . . . . . . . . . . 399
Davide Girolami
Characterising Two-Sided Quantum Correlations Beyond
Entanglement via Metric-Adjusted f–Correlations. . . . . . . . . . . . . . . . . 411
MarcoCianciaruso,IrénéeFrérot,TommasoTufarelliandGerardoAdesso
The Effects of Random Qubit-Qubit Quantum Channels
to Entropy Gain, Fidelity and Trace Distance . . . . . . . . . . . . . . . . . . . . 431
Attila Andai
Robertson-Type Uncertainty Principles and Generalized Symmetric
and Antisymmetric Covariances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Attila Lovas
Introduction
Thisvolumecontainscontributionsfromtheparticipantsofthefourthinternational
conference on Information Geometry and Its Applications (IGAIA IV) which took
placeattheLibliceCastleinCzechRepublic,abeautifulandideallocationforthis
very special event. Indeed, in the list of the IGAIA conferences (Pescara 2002,
Tokyo 2005, Leipzig 2010) this event was organised in order to honour the
numerous scientific achievements of Shun-ichi Amari on the occasion of his 80th
birthday. Amari has pioneered the field of information geometry (IG) and con-
tributed to a variety of its applications. Moreover, he is the one who found the
perfect and powerfully evocative name for this branch of mathematics.
The aim of the IGAIA series was, from the very beginning, to bring together
researchers working in the many different branches of IG, no matter if purely
mathematicalorapplied,withtheexplicitgoaltocreateunexpectedlinkswithinthe
IG community. IGAIA IV was no exception to this rule, which is reflected by the
variety of the contributions that the reader will find in this volume. As any mature
mathematical field,IGhasitsowninternalandpurelymathematicalproblems.But
itstillremainsverymuchlinkedtotheapplicationsinstatistics,informationtheory,
neurosciences,machinelearning,andquantumphysics,justtomentionsomeareas.
Correspondingly,wehaveinthisvolumetwoapplication-orientedparts(IandIV),
and two parts of more general nature (II and III).
The conference was financially supported by the Max Planck Institute for
MathematicsintheSciences(InformationTheoryofCognitiveSystemsGroup),the
InstituteofInformationTheoryandAutomationoftheCzechAcademyofSciences,
and the Department of Economics and Finance of the Università degli Studi di
Roma “Tor Vergata”.
We certainly want to thank all the participants for their contribution to create
such a beautiful and stimulating atmosphere during the conference. Furthermore,
we are grateful for the terrific work with the local organisation of the conference
ix
x Introduction
made by Václav Kratochvíl and the support that we received from Milan Studený.
Special thanks go to Antje Vandenberg who managed all the administrative works
in Leipzig.
July 2018 Nihat Ay
Paolo Gibilisco
František Matúš
Post Scriptum. While finalising the editorial work on this volume, our dear
colleague and friend František Matúš passed away. There are no words to express
this great loss of an exceptional person and a highly respected scientist. Our
thoughts are with his family. We will miss him and dedicate this book to his
memory.
Nihat Ay
Paolo Gibilisco