Information Geometry and Its Applications PDF
Preview Information Geometry and Its Applications
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 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 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 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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