Lectures on Gaussian Processes

Lectures on Gaussian Processes
Author: Mikhail Lifshits
Publisher: Springer Science & Business Media
Total Pages: 129
Release: 2012-01-11
Genre: Mathematics
ISBN: 3642249396

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Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of Gaussian processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.​

Twenty Lectures About Gaussian Processes

Twenty Lectures About Gaussian Processes
Author: Vladimir Ilich Piterbarg
Publisher:
Total Pages: 182
Release: 2015-03-18
Genre: Mathematics
ISBN: 9780984422197

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"Twenty Lectures ..." is based on a course that Professor Piterbarg, a founder of the asymptotic theory of Gaussian processes and fields, teaches to higher-level undergraduate and graduate students at the Faculty of Mechanics and Mathematics, Lomonosov Moscow State University. Written in a clear and succinct style, the book provides a wide-ranging introduction to the field. The first half of the book is devoted to the general theory of Gaussian distributions in both finite- and infinite-dimensional vector spaces. Fundamental results, such as Slepian's, Fernique-Sudakov's and Berman's inequalities, among many others, are clearly explained from a modern, unified point of view. The second half of the book focuses on asymptotic methods, in particular on distributions of high extrema of Gaussian processes and fields. Foundational tools such as the Double Sum Method, the Method of Moments, and the Comparison Method, invented and popularized by the author, are prominently featured. This part adapts material from Professor Piterbarg's famous monograph to make it more accessible to a wider audience. No previous knowledge of stochastic processes is assumed, as all results are derived from a few basic facts of calculus and functional analysis. Written by a world-renowned expert in the field, "Twenty Lectures ..." is a must-read for students and experienced researchers alike - or anyone with an interest in Gaussian processes and fields. The text provides an excellent basis for a full-length graduate course. Albert N. Shiryaev, Member of the Russian Academy of Sciences, Chair of the Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, says: "Professor Piterbarg's lectures are finally available in English and there is simply no other book on the subject that compares. Having contributed so much to the development of the asymptotic theory of Gaussian processes, the author manages to keep his lectures accessible yet rigorous. The lectures cover such a wide range of results and tools that this book is absolutely indispensable to anyone with an interest in the subject."

Gaussian Processes for Machine Learning

Gaussian Processes for Machine Learning
Author: Carl Edward Rasmussen
Publisher: MIT Press
Total Pages: 266
Release: 2005-11-23
Genre: Computers
ISBN: 026218253X

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A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

Advanced Lectures on Machine Learning

Advanced Lectures on Machine Learning
Author: Olivier Bousquet
Publisher: Springer
Total Pages: 249
Release: 2011-03-22
Genre: Computers
ISBN: 3540286500

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Machine Learning has become a key enabling technology for many engineering applications, investigating scientific questions and theoretical problems alike. To stimulate discussions and to disseminate new results, a summer school series was started in February 2002, the documentation of which is published as LNAI 2600. This book presents revised lectures of two subsequent summer schools held in 2003 in Canberra, Australia, and in Tübingen, Germany. The tutorial lectures included are devoted to statistical learning theory, unsupervised learning, Bayesian inference, and applications in pattern recognition; they provide in-depth overviews of exciting new developments and contain a large number of references. Graduate students, lecturers, researchers and professionals alike will find this book a useful resource in learning and teaching machine learning.

Zeros of Gaussian Analytic Functions and Determinantal Point Processes

Zeros of Gaussian Analytic Functions and Determinantal Point Processes
Author: John Ben Hough
Publisher: American Mathematical Soc.
Total Pages: 170
Release: 2009
Genre: Mathematics
ISBN: 0821843737

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Examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients. This title presents a primer on modern techniques on the interface of probability and analysis.

An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes

An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes
Author: Robert J. Adler
Publisher:
Total Pages: 160
Release: 2008*
Genre: Gaussian processes
ISBN:

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This e-book is the product of Project Euclid and its mission to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Project Euclid was developed and deployed by the Cornell University Library and is jointly managed by Cornell and the Duke University Press.

Switching and Learning in Feedback Systems

Switching and Learning in Feedback Systems
Author: Roderick Murray-Smith
Publisher: Springer
Total Pages: 353
Release: 2005-01-27
Genre: Computers
ISBN: 3540305602

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A central theme in the study of dynamic systems is the modelling and control of uncertain systems. While ‘uncertainty’ has long been a strong motivating factor behind many techniques developed in the modelling, control, statistics and mathematics communities, the past decade, in particular, has witnessed remarkable progress in this area with the emergence of a number of powerful newmethodsforbothmodellingandcontrollinguncertaindynamicsystems. The speci?c objective of this book is to describe and review some of these exciting new approaches within a single volume. Our approach was to invite some of the leading researchers in this area to contribute to this book by submitting both tutorial papers on their speci?c area of research, and to submit more focussed research papers to document some of the latest results in the area. We feel that collecting some of the main results together in this manner is particularly important as many of the important ideas that emerged in the past decade were derived in a variety of academic disciplines. By providing both tutorial and researchpaperswehopetobeabletoprovidetheinterestedreaderwithsu?cient background to appreciate some of the main concepts from a variety of related, but nevertheless distinct ?elds, and to provide a ?avor of how these results are currently being used to cope with ‘uncertainty. ’ It is our sincere hope that the availability of these results within a single volume will lead to further cro- fertilization of ideas and act as a spark for further research in this important area of applied mathematics.

Lectures on White Noise Functionals

Lectures on White Noise Functionals
Author: Takeyuki Hida
Publisher: World Scientific
Total Pages: 281
Release: 2008
Genre: Technology & Engineering
ISBN: 9812560521

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White noise analysis is an advanced stochastic calculus that has developed extensively since three decades ago. It has two main characteristics. One is the notion of generalized white noise functionals, the introduction of which is oriented by the line of advanced analysis, and they have made much contribution to the fields in science enormously. The other characteristic is that the white noise analysis has an aspect of infinite dimensional harmonic analysis arising from the infinite dimensional rotation group. With the help of this rotation group, the white noise analysis has explored new areas of mathematics and has extended the fields of applications.