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Convexity and Concentration

Author: Eric Carlen

Publisher: Springer

Published: 2017-04-20

Total Pages: 626

ISBN-13: 1493970054

This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute of Mathematics and its Applications during the Spring 2015 where geometric analysis, convex geometry and concentration phenomena were the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The volume is organized into two parts. Part I contains those contributions that focus primarily on problems motivated by probability theory, while Part II contains those contributions that focus primarily on problems motivated by convex geometry and geometric analysis. This book will be of use to those who research convex geometry, geometric analysis and probability directly or apply such methods in other fields.

Convex Optimization

Author: Stephen P. Boyd

Publisher: Cambridge University Press

Published: 2004-03-08

Total Pages: 744

ISBN-13: 9780521833783

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Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.

Geometry of Isotropic Convex Bodies

Author: Silouanos Brazitikos

Publisher: American Mathematical Soc.

Published: 2014-04-24

Total Pages: 618

ISBN-13: 1470414562

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The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

Lectures on Discrete Geometry

Author: Jiri Matousek

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 491

ISBN-13: 1461300398

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The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Convex Geometric Analysis

Author: Keith M. Ball

Publisher: Cambridge University Press

Published: 1999-01-28

Total Pages: 260

ISBN-13: 9780521642590

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Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.

Harmonic Analysis and Convexity

Author: Alexander Koldobsky

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2023-07-24

Total Pages: 480

ISBN-13: 3110775387

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In recent years, the interaction between harmonic analysis and convex geometry has increased which has resulted in solutions to several long-standing problems. This collection is based on the topics discussed during the Research Semester on Harmonic Analysis and Convexity at the Institute for Computational and Experimental Research in Mathematics in Providence RI in Fall 2022. The volume brings together experts working in related fields to report on the status of major problems in the area including the isomorphic Busemann-Petty and slicing problems for arbitrary measures, extremal problems for Fourier extension and extremal problems for classical singular integrals of martingale type, among others.

High-Dimensional Probability

Author: Roman Vershynin

Publisher: Cambridge University Press

Published: 2018-09-27

Total Pages: 299

ISBN-13: 1108415199

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An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Random Matrix Methods for Machine Learning

Author: Romain Couillet

Publisher: Cambridge University Press

Published: 2022-07-21

Total Pages: 412

ISBN-13: 1009301896

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This book presents a unified theory of random matrices for applications in machine learning, offering a large-dimensional data vision that exploits concentration and universality phenomena. This enables a precise understanding, and possible improvements, of the core mechanisms at play in real-world machine learning algorithms. The book opens with a thorough introduction to the theoretical basics of random matrices, which serves as a support to a wide scope of applications ranging from SVMs, through semi-supervised learning, unsupervised spectral clustering, and graph methods, to neural networks and deep learning. For each application, the authors discuss small- versus large-dimensional intuitions of the problem, followed by a systematic random matrix analysis of the resulting performance and possible improvements. All concepts, applications, and variations are illustrated numerically on synthetic as well as real-world data, with MATLAB and Python code provided on the accompanying website.

The Proceedings of the Coastal Sediments 2011

Author: Ping Wang

Publisher: World Scientific

Published: 2011

Total Pages: 2751

ISBN-13: 9814355534

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This proceedings contains nearly 200 papers on cutting-edge research presented at the seventh international Symposium on Coastal Engineering and Science of Coastal Sediment Processes, held May 2OCo6, 2011, in Miami, Florida, USA. This technical specialty conference was devoted to promoting an interdisciplinary exchange of state-of-the-art knowledge among researchers in the fields of coastal engineering, geology, oceanography, and related disciplines, with a theme of bringing together theory and practice. Focusing on the physical aspects of sediment processes in various coastal environments, this three-volume conference proceedings provides findings from the latest research and newest engineering applications. Session topics cover a wide range including barrier-island morphodynamics and evolution, beach nourishment and shore protection, coastal dunes, cohesive sediment transport, field and laboratory measurements of sediment transport processes and numerical modeling, gravel transport, large-scale and long-term coastal changes, LiDAR and remote sensing, longshore and cross-shore sediment transport, marsh and wetlands, regional sediment management, river deltas, sea-level changes, shelf and sand bodies, shoreline changes, tidal inlets and navigation channels. A special session on recent research findings at the Northern Gulf of Mexico is also included."

Proceedings Of The Coastal Sediments 2011, The (In 3 Volumes)

Author: Roberts Tiffany M

Publisher: World Scientific

Published: 2011-04-21

Total Pages: 2768

ISBN-13: 981445852X

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This proceedings contains nearly 200 papers on cutting-edge research presented at the seventh international Symposium on Coastal Engineering and Science of Coastal Sediment Processes, held May 2-6, 2011, in Miami, Florida, USA. This technical specialty conference was devoted to promoting an interdisciplinary exchange of state-of-the-art knowledge among researchers in the fields of coastal engineering, geology, oceanography, and related disciplines, with a theme of bringing together theory and practice.Focusing on the physical aspects of sediment processes in various coastal environments, this three-volume conference proceedings provides findings from the latest research and newest engineering applications. Session topics cover a wide range including barrier-island morphodynamics and evolution, beach nourishment and shore protection, coastal dunes, cohesive sediment transport, field and laboratory measurements of sediment transport processes and numerical modeling, gravel transport, large-scale and long-term coastal changes, LiDAR and remote sensing, longshore and cross-shore sediment transport, marsh and wetlands, regional sediment management, river deltas, sea-level changes, shelf and sand bodies, shoreline changes, tidal inlets and navigation channels. A special session on recent research findings at the Northern Gulf of Mexico is also included.