Optimization Algorithms On Matrix Manifolds
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Optimization Algorithms On Matrix Manifolds
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Author : P.-A. Absil
language : en
Publisher: Princeton University Press
Release Date : 2009-04-11
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 2009-04-11 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 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.
Handbook Of Variational Methods For Nonlinear Geometric Data
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Author : Philipp Grohs
language : en
Publisher: Springer Nature
Release Date : 2020-04-03
Handbook Of Variational Methods For Nonlinear Geometric Data written by Philipp Grohs and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-03 with Mathematics categories.
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.
Algorithmic Advances In Riemannian Geometry And Applications
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Author : Hà Quang Minh
language : en
Publisher: Springer
Release Date : 2016-10-05
Algorithmic Advances In Riemannian Geometry And Applications written by Hà Quang Minh and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-10-05 with Computers categories.
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.
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.
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.
Geometric Optimization Methods For Adaptive Filtering
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Author : Steven Thomas Smith
language : en
Publisher:
Release Date : 1993
Geometric Optimization Methods For Adaptive Filtering written by Steven Thomas Smith and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Adaptive filters categories.
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.
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
Nonlinear Preconditioning Methods For Optimization And Parallel In Time Methods For 1d Scalar Hyperbolic Partial Differential Equations
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Author : Alexander Howse
language : en
Publisher:
Release Date : 2017
Nonlinear Preconditioning Methods For Optimization And Parallel In Time Methods For 1d Scalar Hyperbolic Partial Differential Equations written by Alexander Howse and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Burgers equation categories.
This thesis consists of two main parts, part one addressing problems from nonlinear optimization and part two based on solving systems of time dependent differential equations, with both parts describing strategies for accelerating the convergence of iterative methods. In part one we present a nonlinear preconditioning framework for use with nonlinear solvers applied to nonlinear optimization problems, motivated by a generalization of linear left preconditioning and linear preconditioning via a change of variables for minimizing quadratic objective functions. In the optimization context nonlinear preconditioning is used to generate a preconditioner direction that either replaces or supplements the gradient vector throughout the optimization algorithm. This framework is used to discuss previously developed nonlinearly preconditioned nonlinear GMRES and nonlinear conjugate gradients (NCG) algorithms, as well as to develop two new nonlinearly preconditioned quasi-Newton methods based on the limited memory Broyden and limited memory BFGS (L-BFGS) updates. We show how all of the above methods can be implemented in a manifold optimization context, with a particular emphasis on Grassmann matrix manifolds. These methods are compared by solving the optimization problems defining the canonical polyadic (CP) decomposition and Tucker higher order singular value decomposition (HOSVD) for tensors, which are formulated as minimizing approximation error in the Frobenius norm. Both of these decompositions have alternating least squares (ALS) type fixed point iterations derived from their optimization problem definitions. While these ALS type iterations may be slow to converge in practice, they can serve as efficient nonlinear preconditioners for the other optimization methods. As the Tucker HOSVD problem involves orthonormality constraints and lacks unique minimizers, the optimization algorithms are extended from Euclidean space to the manifold setting, where optimization on Grassmann manifolds can resolve both of the issues present in the HOSVD problem. The nonlinearly preconditioned methods are compared to the ALS type preconditioners and non-preconditioned NCG, L-BFGS, and a trust region algorithm using both synthetic and real life tensor data with varying noise level, the real data arising from applications in computer vision and handwritten digit recognition. Numerical results show that the nonlinearly preconditioned methods offer substantial improvements in terms of time-to-solution and robustness over state-of-the-art methods for large tensors, in cases where there are significant amounts of noise in the data, and when high accuracy results are required. In part two we apply a multigrid reduction-in-time (MGRIT) algorithm to scalar one-dimensional hyperbolic partial differential equations. This study is motivated by the observation that sequential time-stepping is an obvious computational bottleneck when attempting to implement highly concurrent algorithms, thus parallel-in-time methods are particularly desirable. Existing parallel-in-time methods have produced significant speedups for parabolic or sufficiently diffusive problems, but can have stability and convergence issues for hyperbolic or advection dominated problems. Being a multigrid method, MGRIT primarily uses temporal coarsening, but spatial coarsening can also be incorporated to produce cheaper multigrid cycles and to ensure stability conditions are satisfied on all levels for explicit time-stepping methods. We compare convergence results for the linear advection and diffusion equations, which illustrate the increased difficulty associated with solving hyperbolic problems via parallel-in-time methods. A particular issue that we address is the fact that uniform factor-two spatial coarsening may negatively affect the convergence rate for MGRIT, resulting in extremely slow convergence when the wave speed is near zero, even if only locally. This is due to a sort of anisotropy in the nodal connections, with small wave speeds resulting in spatial connections being weaker than temporal connections. Through the use of semi-algebraic mode analysis applied to the combined advection-diffusion equation we illustrate how the norm of the iteration matrix, and hence an upper bound on the rate of convergence, varies for different choices of wave speed, diffusivity coefficient, space-time grid spacing, and the inclusion or exclusion of spatial coarsening. The use of waveform relaxation multigrid on intermediate, temporally semi-coarsened grids is identified as a potential remedy for the issues introduced by spatial coarsening, with the downside of creating a more intrusive algorithm that cannot be easily combined with existing time-stepping routines for different problems. As a second, less intrusive, alternative we present an adaptive spatial coarsening strategy that prevents the slowdown observed for small local wave speeds, which is applicable for solving the variable coefficient linear advection equation and the inviscid Burgers equation using first-order explicit or implicit time-stepping methods. Serial numerical results show this method offers significant improvements over uniform coarsening and is convergent for inviscid Burgers' equation with and without shocks. Parallel scaling tests indicate that improvements over serial time-stepping strategies are possible when spatial parallelism alone saturates, and that scalability is robust for oscillatory solutions that change on the scale of the grid spacing.
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 Computers categories.
An invitation to optimization with Riemannian geometry for applied mathematics, computer science and engineering students and researchers.