Methods Of Information Geometry
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Methods Of Information Geometry
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Author : Shun-ichi Amari
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Methods Of Information Geometry written by Shun-ichi Amari 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 2000 with Mathematics categories.
Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the $\alpha$-connections. The duality between the $\alpha$-connection and the $(-\alpha)$-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections.The second half of the text provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book can serve as a suitable text for a topics course for advanced undergraduates and graduate students.
Methods Of Information Geometry
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Author : Shun'ichi Amari
language : en
Publisher:
Release Date : 2000
Methods Of Information Geometry written by Shun'ichi Amari and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Geometry, Differential categories.
Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the \alpha-connections. The duality between the \alpha-connection and the ( -\alpha)-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquit.
Advanced Mean Field Methods
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Author : Manfred Opper
language : en
Publisher: MIT Press
Release Date : 2001
Advanced Mean Field Methods written by Manfred Opper and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Computers categories.
This book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling. A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations. One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. Leading approaches include the variational approach, which goes beyond factorizable distributions to achieve systematic improvements; the TAP (Thouless-Anderson-Palmer) approach, which incorporates correlations by including effective reaction terms in the mean field theory; and the more general methods of graphical models. Bringing together ideas and techniques from these diverse disciplines, this book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.
Information Geometry
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Author : Geert Verdoolaege
language : en
Publisher: MDPI
Release Date : 2019-04-04
Information Geometry written by Geert Verdoolaege and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-04-04 with Juvenile Nonfiction categories.
This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience.
Handbook Of Variational Methods For Nonlinear Geometric Data
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Author : Philipp Grohs
language : en
Publisher: Springer Nature
Release Date : 2020-04-03
Handbook Of Variational Methods For Nonlinear Geometric Data written by Philipp Grohs and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Mathematics categories.
This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.
Differential Geometrical Theory Of Statistics
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Author : Frédéric Barbaresco
language : en
Publisher: MDPI
Release Date : 2018-04-06
Differential Geometrical Theory Of Statistics written by Frédéric Barbaresco and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-04-06 with Computers categories.
This book is a printed edition of the Special Issue "Differential Geometrical Theory of Statistics" that was published in Entropy
Handbook Of Geometric Computing
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Author : Eduardo Bayro Corrochano
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-12-06
Handbook Of Geometric Computing written by Eduardo Bayro Corrochano 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 2005-12-06 with Computers categories.
Many computer scientists, engineers, applied mathematicians, and physicists use geometry theory and geometric computing methods in the design of perception-action systems, intelligent autonomous systems, and man-machine interfaces. This handbook brings together the most recent advances in the application of geometric computing for building such systems, with contributions from leading experts in the important fields of neuroscience, neural networks, image processing, pattern recognition, computer vision, uncertainty in geometric computations, conformal computational geometry, computer graphics and visualization, medical imagery, geometry and robotics, and reaching and motion planning. For the first time, the various methods are presented in a comprehensive, unified manner. This handbook is highly recommended for postgraduate students and researchers working on applications such as automated learning; geometric and fuzzy reasoning; human-like artificial vision; tele-operation; space maneuvering; haptics; rescue robots; man-machine interfaces; tele-immersion; computer- and robotics-aided neurosurgery or orthopedics; the assembly and design of humanoids; and systems for metalevel reasoning.
Algebraic And Geometric Methods In Statistics
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Author : Paolo Gibilisco
language : en
Publisher: Cambridge University Press
Release Date : 2010
Algebraic And Geometric Methods In Statistics written by Paolo Gibilisco and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
An up-to-date account of algebraic statistics and information geometry, which also explores the emerging connections between these two disciplines.
Handbook Of Research On Computational Science And Engineering Theory And Practice
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Author : Leng, J.
language : en
Publisher: IGI Global
Release Date : 2011-10-31
Handbook Of Research On Computational Science And Engineering Theory And Practice written by Leng, J. and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-31 with Technology & Engineering categories.
By using computer simulations in research and development, computational science and engineering (CSE) allows empirical inquiry where traditional experimentation and methods of inquiry are difficult, inefficient, or prohibitively expensive. The Handbook of Research on Computational Science and Engineering: Theory and Practice is a reference for interested researchers and decision-makers who want a timely introduction to the possibilities in CSE to advance their ongoing research and applications or to discover new resources and cutting edge developments. Rather than reporting results obtained using CSE models, this comprehensive survey captures the architecture of the cross-disciplinary field, explores the long term implications of technology choices, alerts readers to the hurdles facing CSE, and identifies trends in future development.
Fault Diagnosis And Prognostics Based On Cognitive Computing And Geometric Space Transformation
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Author : Chen Lu
language : en
Publisher: Springer Nature
Release Date : 2025-01-02
Fault Diagnosis And Prognostics Based On Cognitive Computing And Geometric Space Transformation written by Chen Lu and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-02 with Computers categories.
This monograph introduces readers to new theories and methods applying cognitive computing and geometric space transformation to the field of fault diagnosis and prognostics. It summarizes the basic concepts and technical aspects of fault diagnosis and prognostics technology. Existing bottleneck problems are examined, and the advantages of applying cognitive computing and geometric space transformation are explained. In turn, the book highlights fault diagnosis, prognostic, and health assessment technologies based on cognitive computing methods, including deep learning, transfer learning, visual cognition, and compressed sensing. Lastly, it covers technologies based on differential geometry, space transformation, and pattern recognition.