Geometric Structures Of Information
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Geometric Structures Of Information
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Author : Frank Nielsen
language : en
Publisher: Springer
Release Date : 2018-11-19
Geometric Structures Of Information written by Frank Nielsen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-19 with Technology & Engineering categories.
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.
Geometric Structures Of Statistical Physics Information Geometry And Learning
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Author : Frédéric Barbaresco
language : en
Publisher: Springer Nature
Release Date : 2021-06-27
Geometric Structures Of Statistical Physics Information Geometry And Learning written by Frédéric Barbaresco and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-27 with Mathematics categories.
Machine learning and artificial intelligence increasingly use methodological tools rooted in statistical physics. Conversely, limitations and pitfalls encountered in AI question the very foundations of statistical physics. This interplay between AI and statistical physics has been attested since the birth of AI, and principles underpinning statistical physics can shed new light on the conceptual basis of AI. During the last fifty years, statistical physics has been investigated through new geometric structures allowing covariant formalization of the thermodynamics. Inference methods in machine learning have begun to adapt these new geometric structures to process data in more abstract representation spaces. This volume collects selected contributions on the interplay of statistical physics and artificial intelligence. The aim is to provide a constructive dialogue around a common foundation to allow the establishment of new principles and laws governing these two disciplines in a unified manner. The contributions were presented at the workshop on the Joint Structures and Common Foundation of Statistical Physics, Information Geometry and Inference for Learning which was held in Les Houches in July 2020. The various theoretical approaches are discussed in the context of potential applications in cognitive systems, machine learning, signal processing.
Modern Geometric Structures And Fields
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Author : Сергей Петрович Новиков
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Modern Geometric Structures And Fields written by Сергей Петрович Новиков and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
Presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the important structures on them. This book shows that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications.
Information Geometry And Population Genetics
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Author : Julian Hofrichter
language : en
Publisher: Springer
Release Date : 2017-02-23
Information Geometry And Population Genetics written by Julian Hofrichter and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-23 with Mathematics categories.
The present monograph develops a versatile and profound mathematical perspective of the Wright--Fisher model of population genetics. This well-known and intensively studied model carries a rich and beautiful mathematical structure, which is uncovered here in a systematic manner. In addition to approaches by means of analysis, combinatorics and PDE, a geometric perspective is brought in through Amari's and Chentsov's information geometry. This concept allows us to calculate many quantities of interest systematically; likewise, the employed global perspective elucidates the stratification of the model in an unprecedented manner. Furthermore, the links to statistical mechanics and large deviation theory are explored and developed into powerful tools. Altogether, the manuscript provides a solid and broad working basis for graduate students and researchers interested in this field.
Information Geometry
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Author : Nihat Ay
language : en
Publisher: Springer
Release Date : 2017-08-25
Information Geometry written by Nihat Ay and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-08-25 with Mathematics categories.
The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.
Handbook Of Geometric Topology
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Author : R.B. Sher
language : en
Publisher: Elsevier
Release Date : 2001-12-20
Handbook Of Geometric Topology written by R.B. Sher and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-12-20 with Mathematics categories.
Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys written by leading experts and covering active areas of current research. They provide the reader with an up-to-date overview of this flourishing branch of mathematics.
Transformation Groups In Differential Geometry
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Author : Shoshichi Kobayashi
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Transformation Groups In Differential Geometry written by Shoshichi Kobayashi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc tures. All geometric structures are not created equal; some are creations of ~ods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo metric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec tures I gave in Tokyo and Berkeley in 1965.
Information Geometry And Its Applications
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Author : Shun-ichi Amari
language : en
Publisher: Springer
Release Date : 2016-02-02
Information Geometry And Its Applications written by Shun-ichi Amari and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-02-02 with Mathematics categories.
This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields.
Geometry And Statistics
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Author :
language : en
Publisher: Academic Press
Release Date : 2022-07-15
Geometry And Statistics written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-15 with Mathematics categories.
Geometry and Statistics, Volume 46 in the Handbook of Statistics series, highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. - Provides the authority and expertise of leading contributors from an international board of authors - Presents the latest release in the Handbook of Statistics series - Updated release includes the latest information on Geometry and Statistics