Algebraic Geometry and Statistical Learning Theory

Author: Sumio Watanabe
Publisher: Cambridge University Press
ISBN: 0521864674
Format: PDF, ePub, Docs
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Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

Methods of Information Geometry

Author: Shun-ichi Amari
Publisher: American Mathematical Soc.
ISBN: 9780821843024
Format: PDF
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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.

Geometrical Foundations of Asymptotic Inference

Author: Robert E. Kass
Publisher: John Wiley & Sons
ISBN: 1118165977
Format: PDF, ePub, Mobi
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Differential geometry provides an aesthetically appealing and often revealing view of statistical inference. Beginning with an elementary treatment of one-parameter statistical models and ending with an overview of recent developments, this is the first book to provide an introduction to the subject that is largely accessible to readers not already familiar with differential geometry. It also gives a streamlined entry into the field to readers with richer mathematical backgrounds. Much space is devoted to curved exponential families, which are of interest not only because they may be studied geometrically but also because they are analytically convenient, so that results may be derived rigorously. In addition, several appendices provide useful mathematical material on basic concepts in differential geometry. Topics covered include the following: * Basic properties of curved exponential families * Elements of second-order, asymptotic theory * The Fisher-Efron-Amari theory of information loss and recovery * Jeffreys-Rao information-metric Riemannian geometry * Curvature measures of nonlinearity * Geometrically motivated diagnostics for exponential family regression * Geometrical theory of divergence functions * A classification of and introduction to additional work in the field

Random Fields and Geometry

Author: R. J. Adler
Publisher: Springer Science & Business Media
ISBN: 9780387481166
Format: PDF
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This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory.

Information Geometry

Author: Khadiga Arwini
Publisher: Springer
ISBN: 3540693939
Format: PDF, Mobi
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This volume uses information geometry to give a common differential geometric framework for a wide range of illustrative applications including amino acid sequence spacings, cryptology studies, clustering of communications and galaxies, and cosmological voids.

Algebraic Statistics for Computational Biology

Author: L. Pachter
Publisher: Cambridge University Press
ISBN: 9780521857000
Format: PDF, ePub, Mobi
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This book, first published in 2005, offers an introduction to the application of algebraic statistics to computational biology.

Algebraic and Geometric Methods in Statistics

Author: Paolo Gibilisco
Publisher: Cambridge University Press
ISBN: 0521896193
Format: PDF, ePub, Docs
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An up-to-date account of algebraic statistics and information geometry, which also explores the emerging connections between these two disciplines.

Greedy Approximation

Author: Vladimir Temlyakov
Publisher: Cambridge University Press
ISBN: 1139502808
Format: PDF, Mobi
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This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks. The fundamental question of nonlinear approximation is how to devise good constructive methods (algorithms) and recent results have established that greedy type algorithms may be the solution. The author has drawn on his own teaching experience to write a book ideally suited to graduate courses. The reader does not require a broad background to understand the material. Important open problems are included to give students and professionals alike ideas for further research.