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Author: C. Udriste Publisher: Springer Science & Business Media ISBN: 9401583900 Category : Mathematics Languages : en Pages : 365
Book Description
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.
Author: C. Udriste Publisher: Springer Science & Business Media ISBN: 9401583900 Category : Mathematics Languages : en Pages : 365
Book Description
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.
Author: Constantin Udriste Publisher: Springer ISBN: 9789401583916 Category : Mathematics Languages : en Pages : 350
Book Description
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.
Author: Miroslav Bacak Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110391082 Category : Mathematics Languages : en Pages : 217
Book Description
In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.
Author: Seyedehsomayeh Hosseini Publisher: Springer ISBN: 3030113701 Category : Mathematics Languages : en Pages : 149
Book Description
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.
Author: Philipp Grohs Publisher: Springer Nature ISBN: 3030313514 Category : Mathematics Languages : en Pages : 701
Book Description
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.
Author: Hiroyuki Sato Publisher: Springer Nature ISBN: 3030623912 Category : Technology & Engineering Languages : en Pages : 129
Book Description
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.
Author: Jan Valdman Publisher: BoD – Books on Demand ISBN: 1789236762 Category : Mathematics Languages : en Pages : 148
Book Description
This book presents examples of modern optimization algorithms. The focus is on a clear understanding of underlying studied problems, understanding described algorithms by a broad range of scientists and providing (computational) examples that a reader can easily repeat.
Author: Dewei Zhang (Ph. D. in systems engineering) Publisher: ISBN: Category : Mathematical optimization Languages : en Pages : 0
Book Description
Optimization methods have been extensively studied given their broad applications in areas such as applied mathematics, statistics, engineering, healthcare, business, and finance. In the past two decades, the fast growth of machine learning and artificial intelligence and their increasing applications in different industries have resulted in various optimization challenges related to scalability, uncertainty, or requirement to satisfy certain constraints. This dissertation mainly looks into the optimization problems where their solutions are required to satisfy certain (possibly nonlinear) constraints, \emph{namely} Riemannian manifold constraints or they should satisfy certain sparsity structures in conformance with directed acyclic graphs. More specifically, this dissertation explores the following research directions. \begin{enumerate} \item To optimize objective functions in form of finite-sum over Riemannian manifolds, the dissertation proposes a stochastic variance-reduced cubic regularized Newton algorithm in Chapter~\ref{chapter2:cubic}. The proposed algorithm requires a full gradient and Hessian updates at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of the algorithm to obtain an $(\epsilon,\sqrt{\epsilon})$-second order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $\epsilon$ and minimum eigenvalue of Riemannian Hessian eigenvalue lower bounded by $-\sqrt{\epsilon}$, is shown to be $O(\epsilon^{-3/2})$. Furthermore, this dissertation proposes a computationally more appealing extension of the algorithm which only requires an \emph{inexact} solution of the cubic regularized Newton subproblem with the same iteration complexity. \item To optimize the nested composition of two or more functions containing expectations over Riemannian manifolds, this dissertation proposes multi-level stochastic compositional algorithms in Chpter~\ref{chapter3:compositional}. For two-level compositional optimization, the dissertation presents a Riemannian Stochastic Compositional Gradient Descent (R-SCGD) method that finds an approximate stationary point, with expected squared Riemannian gradient smaller than $\epsilon$, in $\cO(\epsilon^{-2})$ calls to the stochastic gradient oracle of the outer function and stochastic function and gradient oracles of the inner function. Furthermore, this dissertation generalizes the R-SCGD algorithms for problems with multi-level nested compositional structures, with the same complexity of $\cO(\epsilon^{-2})$ for first-order stochastic oracles. \item In many statistical learning problems, it is desired that the optimal solution conforms to an a priori known sparsity structure represented by a directed acyclic graph. Inducing such structures by means of convex regularizers requires nonsmooth penalty functions that exploit group overlapping. Chapter~\ref{chap4_HSS} investigates evaluating the proximal operator of the Latent Overlapping Group lasso through an optimization algorithm with parallelizable subproblems. This dissertation implements an Alternating Direction Method of Multiplier with a sharing scheme to solve large-scale instances of the underlying optimization problem efficiently. In the absence of strong convexity, global linear convergence of the algorithm is established using the error bound theory. More specifically, this work also contributes to establishing primal and dual error bounds when the nonsmooth component in the objective function \emph{does not have a polyhedral epigraph}. \end{enumerate} The theoretical results established in each chapter are numerically verified through carefully designed simulation studies and also implemented on real applications with real data sets.
Author: Vladik Kreinovich Publisher: Springer ISBN: 3030042006 Category : Technology & Engineering Languages : en Pages : 1157
Book Description
This book presents recent research on probabilistic methods in economics, from machine learning to statistical analysis. Economics is a very important – and at the same a very difficult discipline. It is not easy to predict how an economy will evolve or to identify the measures needed to make an economy prosper. One of the main reasons for this is the high level of uncertainty: different difficult-to-predict events can influence the future economic behavior. To make good predictions and reasonable recommendations, this uncertainty has to be taken into account. In the past, most related research results were based on using traditional techniques from probability and statistics, such as p-value-based hypothesis testing. These techniques led to numerous successful applications, but in the last decades, several examples have emerged showing that these techniques often lead to unreliable and inaccurate predictions. It is therefore necessary to come up with new techniques for processing the corresponding uncertainty that go beyond the traditional probabilistic techniques. This book focuses on such techniques, their economic applications and the remaining challenges, presenting both related theoretical developments and their practical applications.
Author: C. Udriste
Author: C. Udriste
Author: Constantin Udriste
Author: Miroslav Bacak
Author: Seyedehsomayeh Hosseini
Author: Philipp Grohs
Author: Hiroyuki Sato
Author: Jan Valdman
Author: Dewei Zhang (Ph. D. in systems engineering)
Author: Vladik Kreinovich
Author: