Optimization Algorithms On Matrix Manifolds

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Optimization Algorithms on Matrix Manifolds

P.-A. Absil,R. Mahony,R. Sepulchre
Optimization Algorithms on Matrix Manifolds Book PDF

Author : P.-A. Absil,R. Mahony,R. Sepulchre
Publisher : Princeton University Press
Page : 240 pages
File Size : 44,9 Mb
Release : 2009-04-11
Category : Mathematics
ISBN : 1400830249

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Optimization Algorithms on Matrix Manifolds by P.-A. Absil,R. Mahony,R. Sepulchre Pdf

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

Optimization Algorithms on Matrix Manifolds

P.-A. Absil,R. Mahony,R. Sepulchre
Optimization Algorithms on Matrix Manifolds Book PDF

Author : P.-A. Absil,R. Mahony,R. Sepulchre
Publisher : Princeton University Press
Page : 240 pages
File Size : 45,8 Mb
Release : 2007-12-23
Category : Mathematics
ISBN : 0691132984

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Optimization Algorithms on Matrix Manifolds by P.-A. Absil,R. Mahony,R. Sepulchre Pdf

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

Riemannian Optimization and Its Applications

Hiroyuki Sato
Riemannian Optimization and Its Applications Book PDF

Author : Hiroyuki Sato
Publisher : Springer Nature
Page : 129 pages
File Size : 47,9 Mb
Release : 2021-02-17
Category : Technology & Engineering
ISBN : 9783030623913

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Riemannian Optimization and Its Applications by Hiroyuki Sato Pdf

This brief describes the basics of Riemannian optimization—optimization on Riemannian manifolds—introduces algorithms for Riemannian optimization problems, discusses the theoretical properties of these algorithms, and suggests possible applications of Riemannian optimization to problems in other fields. To provide the reader with a smooth introduction to Riemannian optimization, brief reviews of mathematical optimization in Euclidean spaces and Riemannian geometry are included. Riemannian optimization is then introduced by merging these concepts. In particular, the Euclidean and Riemannian conjugate gradient methods are discussed in detail. A brief review of recent developments in Riemannian optimization is also provided. Riemannian optimization methods are applicable to many problems in various fields. This brief discusses some important applications including the eigenvalue and singular value decompositions in numerical linear algebra, optimal model reduction in control engineering, and canonical correlation analysis in statistics.

Handbook of Variational Methods for Nonlinear Geometric Data

Philipp Grohs,Martin Holler,Andreas Weinmann
Handbook of Variational Methods for Nonlinear Geometric Data Book PDF

Author : Philipp Grohs,Martin Holler,Andreas Weinmann
Publisher : Springer Nature
Page : 701 pages
File Size : 45,9 Mb
Release : 2020-04-03
Category : Mathematics
ISBN : 9783030313517

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Handbook of Variational Methods for Nonlinear Geometric Data by Philipp Grohs,Martin Holler,Andreas Weinmann Pdf

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.

Multivariate Data Analysis on Matrix Manifolds

Nickolay Trendafilov,Michele Gallo
Multivariate Data Analysis on Matrix Manifolds Book PDF

Author : Nickolay Trendafilov,Michele Gallo
Publisher : Springer Nature
Page : 467 pages
File Size : 42,7 Mb
Release : 2021-09-15
Category : Mathematics
ISBN : 9783030769741

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Multivariate Data Analysis on Matrix Manifolds by Nickolay Trendafilov,Michele Gallo Pdf

This graduate-level textbook aims to give a unified presentation and solution of several commonly used techniques for multivariate data analysis (MDA). Unlike similar texts, it treats the MDA problems as optimization problems on matrix manifolds defined by the MDA model parameters, allowing them to be solved using (free) optimization software Manopt. The book includes numerous in-text examples as well as Manopt codes and software guides, which can be applied directly or used as templates for solving similar and new problems. The first two chapters provide an overview and essential background for studying MDA, giving basic information and notations. Next, it considers several sets of matrices routinely used in MDA as parameter spaces, along with their basic topological properties. A brief introduction to matrix (Riemannian) manifolds and optimization methods on them with Manopt complete the MDA prerequisite. The remaining chapters study individual MDA techniques in depth. The number of exercises complement the main text with additional information and occasionally involve open and/or challenging research questions. Suitable fields include computational statistics, data analysis, data mining and data science, as well as theoretical computer science, machine learning and optimization. It is assumed that the readers have some familiarity with MDA and some experience with matrix analysis, computing, and optimization.

Recent Advances in Optimization and its Applications in Engineering

Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels
Recent Advances in Optimization and its Applications in Engineering Book PDF

Author : Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels
Publisher : Springer Science & Business Media
Page : 535 pages
File Size : 46,8 Mb
Release : 2010-09-21
Category : Technology & Engineering
ISBN : 9783642125980

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Recent Advances in Optimization and its Applications in Engineering by Moritz Diehl,Francois Glineur,Elias Jarlebring,Wim Michiels Pdf

Mathematical optimization encompasses both a rich and rapidly evolving body of fundamental theory, and a variety of exciting applications in science and engineering. The present book contains a careful selection of articles on recent advances in optimization theory, numerical methods, and their applications in engineering. It features in particular new methods and applications in the fields of optimal control, PDE-constrained optimization, nonlinear optimization, and convex optimization. The authors of this volume took part in the 14th Belgian-French-German Conference on Optimization (BFG09) organized in Leuven, Belgium, on September 14-18, 2009. The volume contains a selection of reviewed articles contributed by the conference speakers as well as three survey articles by plenary speakers and two papers authored by the winners of the best talk and best poster prizes awarded at BFG09. Researchers and graduate students in applied mathematics, computer science, and many branches of engineering will find in this book an interesting and useful collection of recent ideas on the methods and applications of optimization.

Introduction to Smooth Manifolds

John M. Lee
Introduction to Smooth Manifolds Book PDF

Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 646 pages
File Size : 47,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9780387217529

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Introduction to Smooth Manifolds by John M. Lee Pdf

Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why

Introduction to Riemannian Manifolds

John M. Lee
Introduction to Riemannian Manifolds Book PDF

Author : John M. Lee
Publisher : Springer
Page : 437 pages
File Size : 51,7 Mb
Release : 2019-01-02
Category : Mathematics
ISBN : 9783319917559

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Introduction to Riemannian Manifolds by John M. Lee Pdf

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

Algorithmic Advances in Riemannian Geometry and Applications

Hà Quang Minh,Vittorio Murino
Algorithmic Advances in Riemannian Geometry and Applications Book PDF

Author : Hà Quang Minh,Vittorio Murino
Publisher : Springer
Page : 208 pages
File Size : 45,5 Mb
Release : 2016-10-05
Category : Computers
ISBN : 9783319450261

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Algorithmic Advances in Riemannian Geometry and Applications by Hà Quang Minh,Vittorio Murino Pdf

This book presents a selection of the most recent algorithmic advances in Riemannian geometry in the context of machine learning, statistics, optimization, computer vision, and related fields. The unifying theme of the different chapters in the book is the exploitation of the geometry of data using the mathematical machinery of Riemannian geometry. As demonstrated by all the chapters in the book, when the data is intrinsically non-Euclidean, the utilization of this geometrical information can lead to better algorithms that can capture more accurately the structures inherent in the data, leading ultimately to better empirical performance. This book is not intended to be an encyclopedic compilation of the applications of Riemannian geometry. Instead, it focuses on several important research directions that are currently actively pursued by researchers in the field. These include statistical modeling and analysis on manifolds,optimization on manifolds, Riemannian manifolds and kernel methods, and dictionary learning and sparse coding on manifolds. Examples of applications include novel algorithms for Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis,image classification, action recognition, and motion tracking.

Parallel Optimization

Department of Mathematics and Computer Science Yair Censor,Yair Censor,Stavros Andrea Zenios,Department of Public and Business Administration Stavros A Zenios
Parallel Optimization Book PDF

Author : Department of Mathematics and Computer Science Yair Censor,Yair Censor,Stavros Andrea Zenios,Department of Public and Business Administration Stavros A Zenios
Publisher : Oxford University Press on Demand
Page : 574 pages
File Size : 40,9 Mb
Release : 1997
Category : Mathematics
ISBN : 019510062X

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Parallel Optimization by Department of Mathematics and Computer Science Yair Censor,Yair Censor,Stavros Andrea Zenios,Department of Public and Business Administration Stavros A Zenios Pdf

This book offers a unique pathway to methods of parallel optimization by introducing parallel computing ideas into both optimization theory and into some numerical algorithms for large-scale optimization problems. The three parts of the book bring together relevant theory, careful study of algorithms, and modeling of significant real world problems such as image reconstruction, radiation therapy treatment planning, financial planning, transportation and multi-commodity network flow problems, planning under uncertainty, and matrix balancing problems.

Statistics on Special Manifolds

Yasuko Chikuse
Statistics on Special Manifolds Book PDF

Author : Yasuko Chikuse
Publisher : Springer Science & Business Media
Page : 425 pages
File Size : 47,5 Mb
Release : 2012-11-12
Category : Mathematics
ISBN : 9780387215402

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Statistics on Special Manifolds by Yasuko Chikuse Pdf

Covering statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, this book is designed as a reference for both theoretical and applied statisticians. It will also be used as a textbook for a graduate course in multivariate analysis. It is assumed that the reader is familiar with the usual theory of univariate statistics and a thorough background in mathematics, in particular, knowledge of multivariate calculation techniques.

Derivative-Free and Blackbox Optimization

Charles Audet,Warren Hare
Derivative Free and Blackbox Optimization Book PDF

Author : Charles Audet,Warren Hare
Publisher : Springer
Page : 302 pages
File Size : 40,5 Mb
Release : 2017-12-02
Category : Mathematics
ISBN : 9783319689135

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Derivative-Free and Blackbox Optimization by Charles Audet,Warren Hare Pdf

This book is designed as a textbook, suitable for self-learning or for teaching an upper-year university course on derivative-free and blackbox optimization. The book is split into 5 parts and is designed to be modular; any individual part depends only on the material in Part I. Part I of the book discusses what is meant by Derivative-Free and Blackbox Optimization, provides background material, and early basics while Part II focuses on heuristic methods (Genetic Algorithms and Nelder-Mead). Part III presents direct search methods (Generalized Pattern Search and Mesh Adaptive Direct Search) and Part IV focuses on model-based methods (Simplex Gradient and Trust Region). Part V discusses dealing with constraints, using surrogates, and bi-objective optimization. End of chapter exercises are included throughout as well as 15 end of chapter projects and over 40 figures. Benchmarking techniques are also presented in the appendix.

Optimization and Dynamical Systems

Uwe Helmke,John B. Moore
Optimization and Dynamical Systems Book PDF

Author : Uwe Helmke,John B. Moore
Publisher : Springer Science & Business Media
Page : 409 pages
File Size : 43,6 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781447134671

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Optimization and Dynamical Systems by Uwe Helmke,John B. Moore Pdf

This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys tems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods.

Riemannian Computing in Computer Vision

Pavan K. Turaga,Anuj Srivastava
Riemannian Computing in Computer Vision Book PDF

Author : Pavan K. Turaga,Anuj Srivastava
Publisher : Springer
Page : 391 pages
File Size : 53,6 Mb
Release : 2015-11-09
Category : Technology & Engineering
ISBN : 9783319229577

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Riemannian Computing in Computer Vision by Pavan K. Turaga,Anuj Srivastava Pdf

This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours).

Proximal Algorithms

Neal Parikh,Stephen Boyd
Proximal Algorithms Book PDF

Author : Neal Parikh,Stephen Boyd
Publisher : Now Pub
Page : 130 pages
File Size : 52,9 Mb
Release : 2013-11
Category : Mathematics
ISBN : 1601987161

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Proximal Algorithms by Neal Parikh,Stephen Boyd Pdf

Proximal Algorithms discusses proximal operators and proximal algorithms, and illustrates their applicability to standard and distributed convex optimization in general and many applications of recent interest in particular. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Proximal Algorithms discusses different interpretations of proximal operators and algorithms, looks at their connections to many other topics in optimization and applied mathematics, surveys some popular algorithms, and provides a large number of examples of proximal operators that commonly arise in practice.