Riemannian Optimization And Its Applications
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Riemannian Optimization And Its Applications
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Author : Hiroyuki Sato
language : en
Publisher: Springer Nature
Release Date : 2021-02-17
Riemannian Optimization And Its Applications written by Hiroyuki Sato and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-17 with Technology & Engineering categories.
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.
Convex Functions And Optimization Methods On Riemannian Manifolds
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Author : C. Udriste
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11
Convex Functions And Optimization Methods On Riemannian Manifolds written by C. Udriste and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
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.
Nonsmooth Optimization And Its Applications
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Author : Seyedehsomayeh Hosseini
language : en
Publisher: Springer
Release Date : 2019-03-29
Nonsmooth Optimization And Its Applications written by Seyedehsomayeh Hosseini and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-29 with Mathematics categories.
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.
Recent Advances In Optimization And Its Applications In Engineering
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Author : Moritz Diehl
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-09-21
Recent Advances In Optimization And Its Applications In Engineering written by Moritz Diehl and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-09-21 with Technology & Engineering categories.
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.
An Introduction To Optimization On Smooth Manifolds
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Author : Nicolas Boumal
language : en
Publisher: Cambridge University Press
Release Date : 2023-03-16
An Introduction To Optimization On Smooth Manifolds written by Nicolas Boumal and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-16 with Mathematics categories.
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.
Optimization Algorithms On Matrix Manifolds
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Author : P.-A. Absil
language : en
Publisher: Princeton University Press
Release Date : 2007-12-23
Optimization Algorithms On Matrix Manifolds written by P.-A. Absil and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-12-23 with Mathematics categories.
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 Computing In Computer Vision
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Author : Pavan K. Turaga
language : en
Publisher: Springer
Release Date : 2015-11-09
Riemannian Computing In Computer Vision written by Pavan K. Turaga and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-09 with Technology & Engineering categories.
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).
Frontiers Of Algorithmics
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Author : Bo Li
language : en
Publisher: Springer Nature
Release Date : 2024-12-28
Frontiers Of Algorithmics written by Bo Li and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-28 with Computers categories.
This book constitutes the refereed proceedings of the 18th International Joint Conference on Theoretical Computer Science-Frontier of Algorithmic Wisdom (IJTCS-FAW 2024), consisting of the 18th International Conference on Frontier of Algorithmic Wisdom (FAW) and the 5th International Joint Conference on Theoretical Computer Science (IJTCS), held in Hong Kong, SAR, China, during July 29-31, 2024. FAW started as the Frontiers of Algorithmic Workshop in 2007 at Lanzhou, China, and was held annually from 2007 to 2021 and published archival proceedings. IJTCS, the International joint theoretical Computer Science Conference, started in 2020, aimed to bring in presentations covering active topics in selected tracks in theoretical computer science. To accommodate the diversified new research directions in theoretical computer science, FAW and IJTCS joined their forces together to organize an event for information exchange of new findings and work of enduring value in the field. The 20 full papers and 3 short papers included in this book were carefully reviewed and selected from 43 submissions. They focused tracks on algorithms, blockchain theory, computational economics and algorithmic game theory.
Population Based Optimization On Riemannian Manifolds
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Author : Robert Simon Fong
language : en
Publisher: Springer Nature
Release Date : 2022-05-17
Population Based Optimization On Riemannian Manifolds written by Robert Simon Fong and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-17 with Technology & Engineering categories.
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.
Introduction To Smooth Manifolds
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Author : John M. Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Introduction To Smooth Manifolds written by John M. Lee and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma trices, as easily as we think about the familiar 2-dimensional sphere in ]R3.