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Mathematics of Distances and Applications Michel Deza, Michel Petitjean, Krassimir Markov (eds.) Mathematics of Distances and Applications I T H E A SOFIA 2012 Michel Deza, Michel Petitjean, Krassimir Markov (eds.) Mathematics of Distances and Applications ITHEA® ISBN 978-954-16-0063-4 (printed); ISBN 978-954-16-0064-1 (online) ITHEA IBS ISC No.: 25 Sofia, Bulgaria, 2012 First edition Printed in Bulgaria Recommended for publication by The Scientific Council of the Institute of Information Theories and Applications FOI ITHEA The papers accepted and presented at the International Conference “Mathematics of Distances and Applications”, July 02-05, 2012, Varna, Bulgaria, are published in this book. The concept of distance is basic to human experience. In everyday life it usually means some degree of closeness of two physical objects or ideas, i.e., length, time interval, gap, rank difference, coolness or remoteness, while the term metric is often used as a standard for a measurement. Except for the last two topics, the mathematical meaning of those terms, which is an abstraction of measurement, is considered. Distance metrics and distances have now become an essential tool in many areas of Mathematics and its applications including Geometry, Probability, Statistics, Coding/Graph Theory, Clustering, Data Analysis, Pattern Recognition, Networks, Engineering, Computer Graphics/Vision, Astronomy, Cosmology, Molecular Biology, and many other areas of science. Devising the most suitable distance metrics and similarities, to quantify the proximity between objects, has become a standard task for many researchers. Especially intense ongoing search for such distances occurs, for example, in Computational Biology, Image Analysis, Speech Recognition, and Information Retrieval. For experts in the field of mathematics, information technologies as well as for practical users. © All rights reserved. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Copyright © 2012 © 2012 ITHEA® – Publisher; Sofia, 1000, P.O.B. 775, Bulgaria. www.ithea.org ; e-mail: [email protected] © 2012 Michel Deza, Michel Petitjean, Krassimir Markov – Editors © 2012 Krassimira B. Ivanova – Technical editor © 2012 For all authors in the book. ® ITHEA is a registered trade mark of FOI-COMMERCE Co. IOI-978954/16/00634/25/1 (printed); IOI-978954/16/00641/25/1 (online) ISBN: 978-954-16-0063-4 (printed) ISBN: 978-954-16-0064-1 (online) C\o Jusautor, Sofia, 2012 Mathematics of Distances and Applications 3 Foreword This book consist of papers presented at the MDA 2012 International Conference on "Mathematics of Distances and Applications" held at Varna, Bulgaria, July 2-5, 2012. http://www.foibg.com/conf/ITA2012/2012mda.htm Several sessions were organized: • Geometric complexity in aperiodic order, chaired by Michael Baake and Uwe Grimm • Distances in probability, statistics and data analysis, chaired by Ernst Gabidulin • Distances in behavioral, life, and physical sciences, chaired by Ehtibar Dzhafarov and Michel Petitjean • Discrete geometry, combinatorics, and distances, chaired by Norman Johnson and Asia Weiss • Distances in graphs, chaired by Egon Schulte • Coding theory, chaired by Nina Pilipchuk Although a wide diversity of distances related topics were considered, the set of papers consists of approxima- tively equal two parts with mostly theoretical and mostly applied papers, even if the division is approximative also. The papers were submitted to an anonymous peer review process. The managing of this process was done with the session organizers and with the kind help of the members of the Programme Committee: Tetsuo Asano, Michael Baake, Stefan Dodunekov, Ehtibar Dzhafarov, Ernst Gabidulin, Uwe Grimm, Vladimir Gurvich, Sandi Klavzar, Jacobus Koolen, Svetlozar Rachev, Egon Schulte, Sergey Shpectorov, Kokichi Sugihara, Koen Vanhoof, Cedric Villani. We are very grateful to the following organizers: ITHEA International Scientific Society, Institute of Information Theories and Applications FOI ITHEA (Bulgaria), Institute of Mathematics and Informatics, BAS (Bulgaria). We are highly thankful to our general sponsor FOI Bulgaria for supporting the MDA 2012 Conference. Michel Deza (France) Michel Petitjean (France) Krassimir Markov (Bulgaria) 4 Mathematics of Distances and Applications Mathematics of Distances and Applications 5 Table of contents Part 1: Theoretical Aspects of Mathematics of Distances Differential geometry derived from divergence functions: information geometry approach Shun-ichi Amari ...................................................................... 9 Classification results for (v; k; 1) cyclic difference families with small parameters Tsonka Baicheva, Svetlana Topalova ..................................................... 24 Logarithmic distances in graphs Pavel Chebotarev .................................................................... 31 Cones of weighted quasimetrics, weighted quasihypermetrics and of oriented cuts Michel Deza, Vyacheslav P.Grishukhin, Elena Deza ........................................ 33 Cube-like polytopes and complexes Andrew Duke, Egon Schulte ............................................................ 54 A brief survey of metrics in coding theory Ernst Gabidulin ...................................................................... 66 Regular inversive polytopes Norman W. Johnson .................................................................. 85 Construction of geometric divergence on q-exponential family Hiroshi Matsuzoe ..................................................................... 96 6 Mathematics of Distances and Applications Table of contents Part 2: Applications of Mathematics of Distances Metric based recommender systems Ali Akhtarzada, Cristian S. Calude, John Hosking ........................................... 103 Rank codes over Gaussian integers and space time block codes Hafiz M. Asif, Ernst M. Gabidulin, Bahram Honary .......................................... 118 Ultrametric Fechnerian scaling of discrete object sets Hans Colonius, Ehtibar N. Dzhafarov ..................................................... 129 Symmetrization: ranking and clustering in protein interfaces Giovanni Feverati, Claire Lesieur, Laurent Vuillon ........................................... 133 Intervals as ultrametric approximations according to the supremum norm Bernard Fichet ....................................................................... 147 Distances on antimatroids Yulia Kempner, Vadim E. Levit .......................................................... 148 Distance between objects described by predicate formulas Tatiana Kosovskaya ................................................................... 153 On the discretization of distance geometry problems Antonio Mucherino, Carlile Lavor, Leo Liberti, Nelson Maculan ................................ 160 Geometrical tools for alpha-Voronoi partitions Atsumi Ohara and Yuya Nagatani ........................................................ 169 Metric tensor as degree of coherence in the dynamical organization of the central nervous system Sisir Roy, Rodolfo Llinás ............................................................... 174 Hardware implementation of rank codec Igor Y. Sysoev, Ernst M. Gabidulin ....................................................... 181 Mathematics of Distances and Applications 7 PART 1 Theoretical Aspects of Mathematics of Distances 8 Mathematics of Distances and Applications Mathematics of Distances and Applications 9 DIFFERENTIAL GEOMETRY DERIVED FROM DIVERGENCE FUNCTIONS: INFORMATION GEOMETRY APPROACH Shun-ichi Amari Abstract: We study differential-geometrical structure of an information manifold equipped with a divergence function. A divergence function generates a Riemannian metric and furthermore it provides a symmetric third-order tensor, when the divergence is asymmetric. This induces a pair of affine connections dually coupled to each other with respect to the Riemannian metric. This is the arising emerged from information geometry. When a manifold is dually flat (it may be curved in the sense of the Levi-Civita connection), we have a canonical divergence and a pair of convex functions from which the original dual geometry is reconstructed. The generalized Pythagorean theorem and projection theorem hold in such a manifold. This structure has lots of applications in information sciences including statistics, machine learning, optimization, computer vision and Tsallis statistical mechanics. The present article reviews the structure of information geometry and its relation to the divergence function. We further consider the conformal structure given rise to by the generalized statistical model in relation to the power law. Keywords: divergence, information geometry, dual affine connection, Bregman divergence, generalized Pythagorean theorem ACM Classification Keywords: G.3 PROBABILITY AND STATISTICS MSC: 52, 53, 60 Introduction A divergence function D[P : Q] between two points P and Q in a manifold M plays a fundamental role in many engineering problems, including information theory, statistics,
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