The Procrustes Problem For Orthogonal Stiefel Matrices

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The Procrustes Problem For Orthogonal Stiefel Matrices

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Author : Adam Bojanczyk
language : en
Publisher:
Release Date : 1996

The Procrustes Problem For Orthogonal Stiefel Matrices written by Adam Bojanczyk and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Convergence categories.

Matrix Computations

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Author : Gene Howard Golub
language : en
Publisher: JHU Press
Release Date : 2013-02-15

Matrix Computations written by Gene Howard Golub and has been published by JHU Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-15 with Mathematics categories.

This revised edition provides the mathematical background and algorithmic skills required for the production of numerical software. It includes rewritten and clarified proofs and derivations, as well as new topics such as Arnoldi iteration, and domain decomposition methods.

Siam Journal On Scientific Computing

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Author :
language : en
Publisher:
Release Date : 2004

Siam Journal On Scientific Computing written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Electronic journals categories.

Convex Optimization Euclidean Distance Geometry

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Author : Jon Dattorro
language : en
Publisher: Meboo Publishing USA
Release Date : 2005

Convex Optimization Euclidean Distance Geometry written by Jon Dattorro and has been published by Meboo Publishing USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.

The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on "Geometry of convex functions",observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter "EDM cone", we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone."Semidefinite programming" is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In "EDM proximity"we explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho.

Pattern Recognition

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Author : Carl Edward Rasmussen
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-08-23

Pattern Recognition written by Carl Edward Rasmussen 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 2004-08-23 with Computers categories.

This book constitutes the refereed proceedings of the 26th Symposium of the German Association for Pattern Recognition, DAGM 2004, held in Tübingen, Germany in August/September 2004. The 22 revised papers and 48 revised poster papers presented were carefully reviewed and selected from 146 submissions. The papers are organized in topical sections on learning, Bayesian approaches, vision and faces, vision and motion, biologically motivated approaches, segmentation, object recognition, and object recognition and synthesis.

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.

Multivariate Data Analysis On Matrix Manifolds

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Author : Nickolay Trendafilov
language : en
Publisher: Springer Nature
Release Date : 2021-09-15

Multivariate Data Analysis On Matrix Manifolds written by Nickolay Trendafilov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-15 with Mathematics categories.

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.

Symmetry Nonlinear Bifurcation Analysis And Parallel Computation

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Author : James Christopher Wohlever
language : en
Publisher:
Release Date : 1996

Symmetry Nonlinear Bifurcation Analysis And Parallel Computation written by James Christopher Wohlever and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Bifurcation theory categories.

Pattern Recognition

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Author : Jesús Ariel Carrasco-Ochoa
language : en
Publisher: Springer
Release Date : 2017-05-31

Pattern Recognition written by Jesús Ariel Carrasco-Ochoa and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-05-31 with Computers categories.

This book constitutes the refereed proceedings of the 9th Mexican Conference on Pattern Recognition, MCPR 2017, held in Huatulco, Mexico, in June 2017. The 29 revised full papers presented were carefully reviewed and selected from 55 submissions. The papers are organized in topical sections on pattern recognition and artificial intelligence techniques, image processing and analysis, robotics and remote sensing, natural language processing and recognition, applications of pattern recognition.

Information Theoretic Radar Signal Processing

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Author : Yujie Gu
language : en
Publisher: John Wiley & Sons
Release Date : 2024-11-27

Information Theoretic Radar Signal Processing written by Yujie Gu and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-27 with Technology & Engineering categories.

A comprehensive introduction to the emerging research in information-theoretic radar signal processing Signal processing plays a pivotal role in radar systems to estimate, visualize, and leverage useful target information from noisy and distorted radar signals, harnessing their spatial characteristics, temporal features, and Doppler signatures. The burgeoning applications of information theory in radar signal processing provide a distinct perspective for tackling diverse challenges, including optimized waveform design, performance bound analysis, robust filtering, and target enumeration. Information-Theoretic Radar Signal Processing provides a comprehensive introduction to radar signal processing from an information theory perspective. Covering both fundamental principles and advanced techniques, the book facilitates the integration of information theory into radar signal processing, broadening the scope and improving the performance. Tailored to the needs of researchers and students alike, it serves as a valuable resource for comprehending the information-theoretic aspects of radar signal processing. Information-Theoretic Radar Signal Processing readers will also find: Presentation of alternative hypotheses in adaptive radar detection Detailed discussion of topics including resource management and power allocation Direction-of-arrival (DOA) estimation and integrated sensing and communications (ISAC) Information-Theoretic Radar Signal Processing is ideal for graduate students, scientists, researchers, and engineers, who work on the broad scope of radar and sonar applications, including target detection, estimation, imaging, tracking, and classification using radio frequency, ultrasonic, and acoustic methods.