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Decomposability Of Tensors
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Book Synopsis Decomposability of Tensors by : Luca Chiantini
Download or read book Decomposability of Tensors written by Luca Chiantini and published by MDPI. This book was released on 2019-02-15 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a printed edition of the Special Issue "Decomposability of Tensors" that was published in Mathematics
Book Synopsis Decomposability of Tensors by : Luca Chiantini
Download or read book Decomposability of Tensors written by Luca Chiantini and published by . This book was released on 2019 with total page 1 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor decomposition is a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures that are able to understand and efficiently handle the information that a tensor encodes. Recent advances were obtained with a systematic application of geometric methods: secant varieties, symmetries of special decompositions, and an analysis of the geometry of finite sets. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique have been introduced or significantly improved. New types of decompositions, whose elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions), are now systematically studied and produce deeper insights into this topic. The aim of this Special Issue is to collect papers that illustrate some directions in which recent researches move, as well as to provide a wide overview of several new approaches to the problem of tensor decomposition.
Book Synopsis Decomposing Matrices, Tensors, and Images by : Elina Robeva
Download or read book Decomposing Matrices, Tensors, and Images written by Elina Robeva and published by . This book was released on 2016 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we apply techniques from algebraic geometry to problems arising from optimization and statistics. In particular, we consider data that takes the form of a matrix, a tensor or an image, and we study how to decompose it so as to find additional and seemingly hidden information about its origin and formation. We show that the practical uses of such decompositions are complemented by appealing algebraic and geometric structure. In Chapter 2 of this thesis we focus on matrix shaped data. The singular value decompo- sition, which lies at the core of modern algorithms and can be found efficiently, is not always enough to capture the structure of the data. Often times the matrix at hand as well as the elements of its decomposition are required to have a certain positivity structure, and we need to design algorithms and theory to exploit this structure. Statistical mixture models, for instance, are based on finding a nonnegative decomposition of a nonnegative matrix. We study the algebraic and geometric properties of such decompositions in Section 2.1. Another type of decomposition of a nonnegative matrix, which is useful in convex optimization as well as quantum information theory, is positive semidefinite decomposition. Here we require the elements of the decomposition to be positive semidefinite matrices of a given size. We explore this notion in Section 2.2. One of the most appealing properties of a nonnegative matrix is that we can think of it in terms of a pair of nested polyhedra. We rely on this geometric interpretation when studying nonnegative and positive semidefinite decompositions. In Chapters 3 and 4 we turn our attention to data in the shape of a tensor. It is even more crucial in this case than in the matrix case to find a decomposition, not only because it provides hidden information about the data, but also because it allows us to store the tensor more concisely. However, one of the biggest obstacles in the field is that finding a decomposition of a general tensor is NP-hard. Inspired by the spectral theorem and the singular value decomposition for matrices, we study tensors whose decomposition consists of elements with an orthogonality structure. We call such tensors orthogonally decomposable, or odeco. One of their best properties is that, like matrices, odeco tensors can be decomposed efficiently. In Chapter 3 we study the spectral properties of such tensors. We give a formula for their eigenvectors and singular vector tuples. We note that computing these for a general tensor is hard both algebraically and computationally. In Chapter 4 we study the variety of orthogonally decomposable tensors, and we give polynomial equations that cut it out. We do this by showing that a tensor is orthogonally decomposable if and only if a given algebra that arises from it is associative, yet another appealing property of odeco tensors. Despite all of these appealing properties, odeco tensors constitute a very low-dimensional variety. This is why in Section 4.2 we conclude our study of tensors by generalizing the notion of orthogonally decomposable tensors to that of frame decomposable tensors, which now cover the space of all tensors. In Chapter 5 we study super-resolution imaging. The aim here is, given a low-resolution blurred image, to increase the resolution and remove the blur. This is achieved by decompos- ing the image into a sum of simpler images, one for each point source of light. We encode the locations of the point sources of light and their intensities in a discrete measure, and propose a convex optimization problem in the space of measures to find this unknown measure. We show that in the absence of noise and in the case of a one-dimensional image, the global optimum of this optimization problem recovers the true locations.
Book Synopsis Large Dimensional Data Analysis Using Orthogonally Decomposable Tensors by : Arnab Auddy
Download or read book Large Dimensional Data Analysis Using Orthogonally Decomposable Tensors written by Arnab Auddy and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern data analysis requires the study of tensors, or multi-way arrays. We consider the case where the dimension d is large and the order p is fixed. For dimension reduction and for interpretability, one considers tensor decompositions, where a tensor T can be decomposed into a sum of rank one tensors. In this thesis, I will describe some recent work that illustrate why and how to use decompositions for orthogonally decomposable tensors. Our developments are motivated by statistical applications where the data dimension is large. The estimation procedures will therefore aim to be computationally tractable while providing error rates that depend optimally on the dimension. A tensor is said to be orthogonally decomposable if it can be decomposed into rank one tensors whose component vectors are orthogonal. A number of data analysis tasks can be recast as the problem of estimating the component vectors from a noisy observation of an orthogonally decomposable tensor. In our first set of results, we study this decompositionproblem and derive perturbation bounds.
Book Synopsis Tensors: Geometry and Applications by : J. M. Landsberg
Download or read book Tensors: Geometry and Applications written by J. M. Landsberg and published by American Mathematical Soc.. This book was released on 2011-12-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.
Book Synopsis Equality of Symmetrized Decomposable Tensors by : John Paul Chollet
Download or read book Equality of Symmetrized Decomposable Tensors written by John Paul Chollet and published by . This book was released on 1979 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis What Are Tensors Exactly? by : Hongyu Guo
Download or read book What Are Tensors Exactly? written by Hongyu Guo and published by World Scientific. This book was released on 2021-06-16 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors have numerous applications in physics and engineering. There is often a fuzzy haze surrounding the concept of tensor that puzzles many students. The old-fashioned definition is difficult to understand because it is not rigorous; the modern definitions are difficult to understand because they are rigorous but at a cost of being more abstract and less intuitive.The goal of this book is to elucidate the concepts in an intuitive way but without loss of rigor, to help students gain deeper understanding. As a result, they will not need to recite those definitions in a parrot-like manner any more. This volume answers common questions and corrects many misconceptions about tensors. A large number of illuminating illustrations helps the reader to understand the concepts more easily.This unique reference text will benefit researchers, professionals, academics, graduate students and undergraduate students.
Book Synopsis Theory and Computation of Complex Tensors and its Applications by : Maolin Che
Download or read book Theory and Computation of Complex Tensors and its Applications written by Maolin Che and published by Springer Nature. This book was released on 2020-04-01 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides an introduction of very recent results about the tensors and mainly focuses on the authors' work and perspective. A systematic description about how to extend the numerical linear algebra to the numerical multi-linear algebra is also delivered in this book. The authors design the neural network model for the computation of the rank-one approximation of real tensors, a normalization algorithm to convert some nonnegative tensors to plane stochastic tensors and a probabilistic algorithm for locating a positive diagonal in a nonnegative tensors, adaptive randomized algorithms for computing the approximate tensor decompositions, and the QR type method for computing U-eigenpairs of complex tensors. This book could be used for the Graduate course, such as Introduction to Tensor. Researchers may also find it helpful as a reference in tensor research.
Book Synopsis Tensor Categories by : Pavel Etingof
Download or read book Tensor Categories written by Pavel Etingof and published by American Mathematical Soc.. This book was released on 2016-08-05 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.
Book Synopsis Tensor Decomposition Meets Approximation Theory by : Ferre Knaepkens
Download or read book Tensor Decomposition Meets Approximation Theory written by Ferre Knaepkens and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis studies three different subjects, namely tensors and tensor decomposition, sparse interpolation and Pad\'e or rational approximation theory. These problems find their origin in various fields within mathematics: on the one hand tensors originate from algebra and are of importance in computer science and knowledge technology, while on the other hand sparse interpolation and Pad\'e approximations stem from approximation theory. Although all three problems seem totally unrelated, they are deeply intertwined. The connection between them is exactly he goal of this thesis. These connections are of importance since they allow us to solve the symmetric tensor decomposition problem by means of a corresponding sparse interpolation problem or an appropriate Pad\'e approximant. The first section gives a short introduction on tensors. Here, starting from the points of view of matrices and vectors, a generalization is made to tensors. Also a link is made to other known concepts within matrix-algebra. Subsequently, three definitions of tensor rank are discussed. The first definition is the most general and is based on the decomposition by means of the outer product of vectors. The second definition is only applicable for symmetric tensors and is based on a decomposition by means of symmetric outer products of vectors. Finally, the last definition is also only applicable for symmetric tensors and is based o the decomposition of a related homogeneous polynomial. It can be shown that these last two definitions are equal and they are also the only definitions used in the continuation of the thesis. In particular, this last definition since it supplies the connection with approximation theory. Finally, a well-known method (ALS) to find these tensor decompositions is shortly discussed. However, ALS has some shortcomings en that is exactly the reason that the connections to approximation theory are of such importance. Sections two and three discuss the first problem of both within approximation theory, namely sparse interpolation. In the second section, The univariate problem is considered. This problem can be solved with Prony's method, which consists of finding the zeroes of a related polynomial or solving a generalized eigenvalue problem. The third section continues on the second since it discusses multivariate sparse interpolation. Prony's method for the univariate case is changed to also provide a solution for the multivariate problem. The fourth and fifth section have as subject Pad\'e or rational approximation theory. Like the name suggests, it consists of approximating a power series by a rational function. Section four first introduces univariate Pad\'e approximants and states some important properties of them. Here, shortly the connection is made with continued fraction to use this theory later on. Finally, some methods to find Pad\'e approximants are discussed, namely the Levinson algorithm, the determinant formulas and the qd-algorithm. Section five continues on section four and discusses multivariate Pad\'e approximation theory. It is shown that a shift of the univariate conditions occurs, however, despite this shift still a lot of the important properties of the univariate case remain true. Also an extension of the qd-algorithm for multivariate Pad\'e approximants is discussed. Section six bundles all previous sections to expose the connections between the three seemingly different problems. The discussion of these connections is done in two steps in the univariate case, first the tensor decomposition problem is rewritten as a sparse interpolation problem and subsequently, it is shown that the sparse interpolation problem can be solved by means of Pad\'e approximants. In the multivariate case, also the connection between tensor decomposition and sparse interpolation is discussed first. Subsequently, a parameterized approach is introduces, which converts the multivariate problem to a parameterized univariate problem such that the connections of the first part apply. This parameterized approach also lead to the connection between tensor decomposition, multivariate sparse interpolation and multivariate Pad\'e approximation theory. The last or seventh section consists of two examples, a univariate problem and a multivariate one. The techniques of previous sections are used to demonstrate the connections of section six. This section also serves as illustration of the methods of sections two until five to solve sparse interpolation and Pad\'e approximation problems.
Book Synopsis Tensors for Data Processing by : Yipeng Liu
Download or read book Tensors for Data Processing written by Yipeng Liu and published by Academic Press. This book was released on 2021-10-21 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods. As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry. Provides a complete reference on classical and state-of-the-art tensor-based methods for data processing Includes a wide range of applications from different disciplines Gives guidance for their application
Book Synopsis Brain-like Super Intelligence from Bio-electromagnetism by : Anirban Bandyopadhyay
Download or read book Brain-like Super Intelligence from Bio-electromagnetism written by Anirban Bandyopadhyay and published by Springer Nature. This book was released on with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan by : Josef Dick
Download or read book Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan written by Josef Dick and published by Springer. This book was released on 2018-05-23 with total page 1309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a tribute to Professor Ian Hugh Sloan on the occasion of his 80th birthday. It consists of nearly 60 articles written by international leaders in a diverse range of areas in contemporary computational mathematics. These papers highlight the impact and many achievements of Professor Sloan in his distinguished academic career. The book also presents state of the art knowledge in many computational fields such as quasi-Monte Carlo and Monte Carlo methods for multivariate integration, multi-level methods, finite element methods, uncertainty quantification, spherical designs and integration on the sphere, approximation and interpolation of multivariate functions, oscillatory integrals, and in general in information-based complexity and tractability, as well as in a range of other topics. The book also tells the life story of the renowned mathematician, family man, colleague and friend, who has been an inspiration to many of us. The reader may especially enjoy the story from the perspective of his family, his wife, his daughter and son, as well as grandchildren, who share their views of Ian. The clear message of the book is that Ian H. Sloan has been a role model in science and life.
Book Synopsis Tensor Computation for Data Analysis by : Yipeng Liu
Download or read book Tensor Computation for Data Analysis written by Yipeng Liu and published by Springer Nature. This book was released on 2021-08-31 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor is a natural representation for multi-dimensional data, and tensor computation can avoid possible multi-linear data structure loss in classical matrix computation-based data analysis. This book is intended to provide non-specialists an overall understanding of tensor computation and its applications in data analysis, and benefits researchers, engineers, and students with theoretical, computational, technical and experimental details. It presents a systematic and up-to-date overview of tensor decompositions from the engineer's point of view, and comprehensive coverage of tensor computation based data analysis techniques. In addition, some practical examples in machine learning, signal processing, data mining, computer vision, remote sensing, and biomedical engineering are also presented for easy understanding and implementation. These data analysis techniques may be further applied in other applications on neuroscience, communication, psychometrics, chemometrics, biometrics, quantum physics, quantum chemistry, etc. The discussion begins with basic coverage of notations, preliminary operations in tensor computations, main tensor decompositions and their properties. Based on them, a series of tensor-based data analysis techniques are presented as the tensor extensions of their classical matrix counterparts, including tensor dictionary learning, low rank tensor recovery, tensor completion, coupled tensor analysis, robust principal tensor component analysis, tensor regression, logistical tensor regression, support tensor machine, multilinear discriminate analysis, tensor subspace clustering, tensor-based deep learning, tensor graphical model and tensor sketch. The discussion also includes a number of typical applications with experimental results, such as image reconstruction, image enhancement, data fusion, signal recovery, recommendation system, knowledge graph acquisition, traffic flow prediction, link prediction, environmental prediction, weather forecasting, background extraction, human pose estimation, cognitive state classification from fMRI, infrared small target detection, heterogeneous information networks clustering, multi-view image clustering, and deep neural network compression.
Book Synopsis Hierarchical Singular Vlaue Decomposition of Tensors by : Lars Grasedyck
Download or read book Hierarchical Singular Vlaue Decomposition of Tensors written by Lars Grasedyck and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Introduction to Tensors and Group Theory for Physicists by : Nadir Jeevanjee
Download or read book An Introduction to Tensors and Group Theory for Physicists written by Nadir Jeevanjee and published by Birkhäuser. This book was released on 2015-03-11 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics. Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations. New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students. Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part I by introducing representation theory. Examples and exercises are provided in each chapter for good practice in applying the presented material and techniques. Prerequisites for this text include the standard lower-division mathematics and physics courses, though extensive references are provided for the motivated student who has not yet had these. Advanced undergraduate and beginning graduate students in physics and applied mathematics will find this textbook to be a clear, concise, and engaging introduction to tensors and groups. Reviews of the First Edition “[P]hysicist Nadir Jeevanjee has produced a masterly book that will help other physicists understand those subjects [tensors and groups] as mathematicians understand them... From the first pages, Jeevanjee shows amazing skill in finding fresh, compelling words to bring forward the insight that animates the modern mathematical view...[W]ith compelling force and clarity, he provides many carefully worked-out examples and well-chosen specific problems... Jeevanjee’s clear and forceful writing presents familiar cases with a freshness that will draw in and reassure even a fearful student. [This] is a masterpiece of exposition and explanation that would win credit for even a seasoned author.” —Physics Today "Jeevanjee’s [text] is a valuable piece of work on several counts, including its express pedagogical service rendered to fledgling physicists and the fact that it does indeed give pure mathematicians a way to come to terms with what physicists are saying with the same words we use, but with an ostensibly different meaning. The book is very easy to read, very user-friendly, full of examples...and exercises, and will do the job the author wants it to do with style.” —MAA Reviews
Book Synopsis Algorithmic Learning Theory by : Kamalika Chaudhuri
Download or read book Algorithmic Learning Theory written by Kamalika Chaudhuri and published by Springer. This book was released on 2015-10-04 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 26th International Conference on Algorithmic Learning Theory, ALT 2015, held in Banff, AB, Canada, in October 2015, and co-located with the 18th International Conference on Discovery Science, DS 2015. The 23 full papers presented in this volume were carefully reviewed and selected from 44 submissions. In addition the book contains 2 full papers summarizing the invited talks and 2 abstracts of invited talks. The papers are organized in topical sections named: inductive inference; learning from queries, teaching complexity; computational learning theory and algorithms; statistical learning theory and sample complexity; online learning, stochastic optimization; and Kolmogorov complexity, algorithmic information theory.
