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Author: Gérard Favier Publisher: John Wiley & Sons ISBN: 1786301547 Category : Technology & Engineering Languages : en Pages : 324
Book Description
Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.
Author: Gérard Favier Publisher: John Wiley & Sons ISBN: 1786301547 Category : Technology & Engineering Languages : en Pages : 324
Book Description
Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.
Author: Ray M. Bowen Publisher: Springer ISBN: Category : Mathematics Languages : en Pages : 224
Book Description
To Volume 1 This work represents our effort to present the basic concepts of vector and tensor analysis. Volume 1 begins with a brief discussion of algebraic structures followed by a rather detailed discussion of the algebra of vectors and tensors. Volume 2 begins with a discussion of Euclidean manifolds, which leads to a development of the analytical and geometrical aspects of vector and tensor fields. We have not included a discussion of general differentiable manifolds. However, we have included a chapter on vector and tensor fields defined on hypersurfaces in a Euclidean manifold. In preparing this two-volume work, our intention was to present to engineering and science students a modern introduction to vectors and tensors. Traditional courses on applied mathematics have emphasized problem-solving techniques rather than the systematic development of concepts. As a result, it is possible for such courses to become terminal mathematics courses rather than courses which equip the student to develop his or her understanding further.
Author: Gérard Favier Publisher: John Wiley & Sons ISBN: 1119700965 Category : Technology & Engineering Languages : en Pages : 386
Book Description
The second volume will deal with a presentation of the main matrix and tensor decompositions and their properties of uniqueness, as well as very useful tensor networks for the analysis of massive data. Parametric estimation algorithms will be presented for the identification of the main tensor decompositions. After a brief historical review of the compressed sampling methods, an overview of the main methods of retrieving matrices and tensors with missing data will be performed under the low rank hypothesis. Illustrative examples will be provided.
Author: Juan R. Ruiz-Tolosa Publisher: Springer Science & Business Media ISBN: 3540270663 Category : Computers Languages : en Pages : 675
Book Description
This textbook deals with tensors that are treated as vectors. Coverage details such new tensor concepts as the rotation of tensors, the transposer tensor, the eigentensors, and the permutation tensor structure. The book covers an existing gap between the classic theory of tensors and the possibility of solving tensor problems with a computer. A complementary computer package, written in Mathematica, is available through the Internet.
Author: N.B. Singh Publisher: N.B. Singh ISBN: Category : Mathematics Languages : en Pages : 141
Book Description
"Abstract Algebra: Tensor Products" provides a comprehensive exploration of tensor products within the framework of abstract algebra. Beginning with foundational definitions and universal properties, the book progresses to elucidate their applications across diverse algebraic structures such as modules, vector spaces, and rings. Emphasizing clarity and depth, it navigates through advanced topics including categorical perspectives, functorial properties, and their relevance in fields like quantum mechanics and topology. Through numerous examples, and theoretical insights, this book equips readers with the tools to understand and leverage tensor products as powerful algebraic tools, fostering a deeper appreciation for their role in modern mathematics.
Author: Mathias Drton Publisher: Springer Science & Business Media ISBN: 3764389052 Category : Mathematics Languages : en Pages : 177
Book Description
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
Author: Robert Wasserman Publisher: Oxford University Press, USA ISBN: 9780198510598 Category : Language Arts & Disciplines Languages : en Pages : 468
Book Description
This book sets forth the basic principles of tensors and manifolds and describes how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics.
Author: Gérard Favier
Author: Gérard Favier
Author: Ray M. Bowen
Author: Gérard Favier
Author: Juan R. Ruiz-Tolosa
Author: Richard L. Bishop
Author: N.B. Singh
Author: Michael Fey
Author: Mathias Drton
Author: Rolf Jeltsch
Author: Robert Wasserman