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Riemannian Optimization And Its Applications
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Book Synopsis Riemannian Optimization and Its Applications by : Hiroyuki Sato
Download or read book Riemannian Optimization and Its Applications written by Hiroyuki Sato and published by Springer Nature. This book was released on 2021-02-17 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Convex Functions and Optimization Methods on Riemannian Manifolds by : C. Udriste
Download or read book Convex Functions and Optimization Methods on Riemannian Manifolds written by C. Udriste and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.
Book Synopsis Algorithmic Advances in Riemannian Geometry and Applications by : Hà Quang Minh
Download or read book Algorithmic Advances in Riemannian Geometry and Applications written by Hà Quang Minh and published by Springer. This book was released on 2016-10-05 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Optimization Algorithms on Matrix Manifolds by : P.-A. Absil
Download or read book Optimization Algorithms on Matrix Manifolds written by P.-A. Absil and published by Princeton University Press. This book was released on 2009-04-11 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Recent Advances in Optimization and its Applications in Engineering by : Moritz Diehl
Download or read book Recent Advances in Optimization and its Applications in Engineering written by Moritz Diehl and published by Springer Science & Business Media. This book was released on 2010-09-21 with total page 535 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Nonsmooth Optimization and Its Applications by : Seyedehsomayeh Hosseini
Download or read book Nonsmooth Optimization and Its Applications written by Seyedehsomayeh Hosseini and published by Springer. This book was released on 2019-03-29 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since nonsmooth optimization problems arise in a diverse range of real-world applications, the potential impact of efficient methods for solving such problems is undeniable. Even solving difficult smooth problems sometimes requires the use of nonsmooth optimization methods, in order to either reduce the problem’s scale or simplify its structure. Accordingly, the field of nonsmooth optimization is an important area of mathematical programming that is based on by now classical concepts of variational analysis and generalized derivatives, and has developed a rich and sophisticated set of mathematical tools at the intersection of theory and practice. This volume of ISNM is an outcome of the workshop "Nonsmooth Optimization and its Applications," which was held from May 15 to 19, 2017 at the Hausdorff Center for Mathematics, University of Bonn. The six research articles gathered here focus on recent results that highlight different aspects of nonsmooth and variational analysis, optimization methods, their convergence theory and applications.
Book Synopsis Riemannian Computing in Computer Vision by : Pavan K. Turaga
Download or read book Riemannian Computing in Computer Vision written by Pavan K. Turaga and published by Springer. This book was released on 2015-11-09 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).
Book Synopsis Introduction to Smooth Manifolds by : John M. Lee
Download or read book Introduction to Smooth Manifolds written by John M. Lee and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis An Introduction to Optimization on Smooth Manifolds by : Nicolas Boumal
Download or read book An Introduction to Optimization on Smooth Manifolds written by Nicolas Boumal and published by Cambridge University Press. This book was released on 2023-03-16 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.
Book Synopsis Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces by : Michael Ulbrich
Download or read book Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces written by Michael Ulbrich and published by SIAM. This book was released on 2011-07-28 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive treatment of semismooth Newton methods in function spaces: from their foundations to recent progress in the field. This book is appropriate for researchers and practitioners in PDE-constrained optimization, nonlinear optimization and numerical analysis, as well as engineers interested in the current theory and methods for solving variational inequalities.
Book Synopsis First Steps in Differential Geometry by : Andrew McInerney
Download or read book First Steps in Differential Geometry written by Andrew McInerney and published by Springer Science & Business Media. This book was released on 2013-07-09 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. Today it is possible to describe differential geometry as "the study of structures on the tangent space," and this text develops this point of view. This book, unlike other introductory texts in differential geometry, develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. The main goal of this book is to bring the undergraduate student who already has a solid foundation in the standard mathematics curriculum into contact with the beauty of higher mathematics. In particular, the presentation here emphasizes the consequences of a definition and the careful use of examples and constructions in order to explore those consequences.
Book Synopsis Introduction to Nonlinear Optimization by : Amir Beck
Download or read book Introduction to Nonlinear Optimization written by Amir Beck and published by SIAM. This book was released on 2014-10-27 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the foundations of the theory of nonlinear optimization as well as some related algorithms and presents a variety of applications from diverse areas of applied sciences. The author combines three pillars of optimization?theoretical and algorithmic foundation, familiarity with various applications, and the ability to apply the theory and algorithms on actual problems?and rigorously and gradually builds the connection between theory, algorithms, applications, and implementation. Readers will find more than 170 theoretical, algorithmic, and numerical exercises that deepen and enhance the reader's understanding of the topics. The author includes offers several subjects not typically found in optimization books?for example, optimality conditions in sparsity-constrained optimization, hidden convexity, and total least squares. The book also offers a large number of applications discussed theoretically and algorithmically, such as circle fitting, Chebyshev center, the Fermat?Weber problem, denoising, clustering, total least squares, and orthogonal regression and theoretical and algorithmic topics demonstrated by the MATLAB? toolbox CVX and a package of m-files that is posted on the book?s web site.
Book Synopsis Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control by : Marko M Makela
Download or read book Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control written by Marko M Makela and published by World Scientific. This book was released on 1992-05-07 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization. The second part is devoted to the methods of nonsmooth optimization and their development. A proximal bundle method for nonsmooth nonconvex optimization subject to nonsmooth constraints is constructed. In the last part nonsmooth optimization is applied to problems arising from optimal control of systems covered by partial differential equations. Several practical problems, like process control and optimal shape design problems are considered.
Book Synopsis Symplectic Geometry and Quantum Mechanics by : Maurice A. de Gosson
Download or read book Symplectic Geometry and Quantum Mechanics written by Maurice A. de Gosson and published by Springer Science & Business Media. This book was released on 2006-08-06 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.
Book Synopsis Population-Based Optimization on Riemannian Manifolds by : Robert Simon Fong
Download or read book Population-Based Optimization on Riemannian Manifolds written by Robert Simon Fong and published by Springer Nature. This book was released on 2022-05-17 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifold optimization is an emerging field of contemporary optimization that constructs efficient and robust algorithms by exploiting the specific geometrical structure of the search space. In our case the search space takes the form of a manifold. Manifold optimization methods mainly focus on adapting existing optimization methods from the usual “easy-to-deal-with” Euclidean search spaces to manifolds whose local geometry can be defined e.g. by a Riemannian structure. In this way the form of the adapted algorithms can stay unchanged. However, to accommodate the adaptation process, assumptions on the search space manifold often have to be made. In addition, the computations and estimations are confined by the local geometry. This book presents a framework for population-based optimization on Riemannian manifolds that overcomes both the constraints of locality and additional assumptions. Multi-modal, black-box manifold optimization problems on Riemannian manifolds can be tackled using zero-order stochastic optimization methods from a geometrical perspective, utilizing both the statistical geometry of the decision space and Riemannian geometry of the search space. This monograph presents in a self-contained manner both theoretical and empirical aspects of stochastic population-based optimization on abstract Riemannian manifolds.
Book Synopsis Algorithms for Convex Optimization by : Nisheeth K. Vishnoi
Download or read book Algorithms for Convex Optimization written by Nisheeth K. Vishnoi and published by Cambridge University Press. This book was released on 2021-10-07 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.
Book Synopsis Pseudolinear Functions and Optimization by : Shashi Kant Mishra
Download or read book Pseudolinear Functions and Optimization written by Shashi Kant Mishra and published by CRC Press. This book was released on 2014-12-18 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pseudolinear Functions and Optimization is the first book to focus exclusively on pseudolinear functions, a class of generalized convex functions. It discusses the properties, characterizations, and applications of pseudolinear functions in nonlinear optimization problems.The book describes the characterizations of solution sets of various optimiza
