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Matrix and Tensor Factorization Techniques for Recommender Systems

Matrix and Tensor Factorization Techniques for Recommender Systems
  • Author : Panagiotis Symeonidis,Andreas Zioupos
  • Publisher :Unknown
  • Release Date :2016-09-25
  • Total pages :102
  • ISBN : 3319413562
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Summary : This book presents the algorithms used to provide recommendations by exploiting matrix factorization and tensor decomposition techniques. It highlights well-known decomposition methods for recommender systems, such as Singular Value Decomposition (SVD), UV-decomposition, Non-negative Matrix Factorization (NMF), etc. and describes in detail the pros and cons of each method for matrices and tensors. This book provides a detailed theoretical mathematical background of matrix/tensor factorization techniques and a step-by-step analysis of each method on the basis of an integrated toy example that runs throughout all its chapters and helps the reader to understand the key differences among methods. It also contains two chapters, where different matrix and tensor methods are compared experimentally on real data sets, such as Epinions, GeoSocialRec, Last.fm, BibSonomy, etc. and provides further insights into the advantages and disadvantages of each method. The book offers a rich blend of theory and practice, making it suitable for students, researchers and practitioners interested in both recommenders and factorization methods. Lecturers can also use it for classes on data mining, recommender systems and dimensionality reduction methods.

Matrix and Tensor Factorization Techniques for Recommender Systems

Matrix and Tensor Factorization Techniques for Recommender Systems
  • Author : Panagiotis Symeonidis,Andreas Zioupos
  • Publisher :Unknown
  • Release Date :2017-01-29
  • Total pages :102
  • ISBN : 9783319413570
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Summary : This book presents the algorithms used to provide recommendations by exploiting matrix factorization and tensor decomposition techniques. It highlights well-known decomposition methods for recommender systems, such as Singular Value Decomposition (SVD), UV-decomposition, Non-negative Matrix Factorization (NMF), etc. and describes in detail the pros and cons of each method for matrices and tensors. This book provides a detailed theoretical mathematical background of matrix/tensor factorization techniques and a step-by-step analysis of each method on the basis of an integrated toy example that runs throughout all its chapters and helps the reader to understand the key differences among methods. It also contains two chapters, where different matrix and tensor methods are compared experimentally on real data sets, such as Epinions, GeoSocialRec, Last.fm, BibSonomy, etc. and provides further insights into the advantages and disadvantages of each method. The book offers a rich blend of theory and practice, making it suitable for students, researchers and practitioners interested in both recommenders and factorization methods. Lecturers can also use it for classes on data mining, recommender systems and dimensionality reduction methods.

Matrix and Tensor Decompositions in Signal Processing

Matrix and Tensor Decompositions in Signal Processing
  • Author : Gérard Favier
  • Publisher :Unknown
  • Release Date :2021-09-15
  • Total pages :200
  • ISBN : 1786301555
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Summary : 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.

Algorithmic Aspects of Machine Learning

Algorithmic Aspects of Machine Learning
  • Author : Ankur Moitra
  • Publisher :Unknown
  • Release Date :2018-09-27
  • Total pages :176
  • ISBN : 9781107184589
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Summary : Introduces cutting-edge research on machine learning theory and practice, providing an accessible, modern algorithmic toolkit.

Nonnegative Matrix and Tensor Factorizations

Nonnegative Matrix and Tensor Factorizations
  • Author : Andrzej Cichocki,Rafal Zdunek,Anh Huy Phan,Shun-ichi Amari
  • Publisher :Unknown
  • Release Date :2009-07-10
  • Total pages :500
  • ISBN : 0470747285
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Summary : This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMF’s various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors’ own recently developed techniques in the subject area. Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms. Provides a comparative analysis of the different methods in order to identify approximation error and complexity. Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.

Tensor Decomposition Meets Approximation Theory

Tensor Decomposition Meets Approximation Theory
  • Author : Ferre Knaepkens
  • Publisher :Unknown
  • Release Date :2017
  • Total pages :229
  • ISBN : OCLC:1050023083
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Summary : 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.

Nonnegative Matrix and Tensor Factorizations

Nonnegative Matrix and Tensor Factorizations
  • Author : Andrzej Cichocki,Rafal Zdunek,Anh Huy Phan,Shun-ichi Amari
  • Publisher :Unknown
  • Release Date :2009-10-12
  • Total pages :500
  • ISBN : 0470746661
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Summary : This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMF’s various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors’ own recently developed techniques in the subject area. Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms. Provides a comparative analysis of the different methods in order to identify approximation error and complexity. Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.

Matrix and Tensor Decomposition

Matrix and Tensor Decomposition
  • Author : Christian Jutten
  • Publisher :Unknown
  • Release Date :2021
  • Total pages :229
  • ISBN : 0128157607
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Summary :

Spectral Learning on Matrices and Tensors

Spectral Learning on Matrices and Tensors
  • Author : Majid Janzamin,Rong Ge,Jean Kossaifi,Anima Anandkumar
  • Publisher :Unknown
  • Release Date :2019-11-25
  • Total pages :156
  • ISBN : 1680836404
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Summary : The authors of this monograph survey recent progress in using spectral methods including matrix and tensor decomposition techniques to learn many popular latent variable models. With careful implementation, tensor-based methods can run efficiently in practice, and in many cases they are the only algorithms with provable guarantees on running time and sample complexity. The focus is on a special type of tensor decomposition called CP decomposition, and the authors cover a wide range of algorithms to find the components of such tensor decomposition. They also discuss the usefulness of this decomposition by reviewing several probabilistic models that can be learned using such tensor methods. The second half of the monograph looks at practical applications. This includes using Tensorly, an efficient tensor algebra software package, which has a simple python interface for expressing tensor operations. It also has a flexible back-end system supporting NumPy, PyTorch, TensorFlow, and MXNet. Spectral Learning on Matrices and Tensors provides a theoretical and practical introduction to designing and deploying spectral learning on both matrices and tensors. It is of interest for all students, researchers and practitioners working on modern day machine learning problems.

Matrix and Tensor Decompositions in Signal Processing

Matrix and Tensor Decompositions in Signal Processing
  • Author : Gérard Favier
  • Publisher :Unknown
  • Release Date :2021-09-15
  • Total pages :200
  • ISBN : 9781786301550
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Summary : 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.

Decomposability of Tensors

Decomposability of Tensors
  • Author : Luca Chiantini
  • Publisher :Unknown
  • Release Date :2019-02-15
  • Total pages :160
  • ISBN : 9783038975908
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Summary : This book is a printed edition of the Special Issue "Decomposability of Tensors" that was published in Mathematics

Scalable Low-rank Matrix and Tensor Decomposition on Graphs

Scalable Low-rank Matrix and Tensor Decomposition on Graphs
  • Author : Nauman Shahid
  • Publisher :Unknown
  • Release Date :2017
  • Total pages :229
  • ISBN : OCLC:1018448374
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Summary : Mots-clés de l'auteur: Principal Component Analysis ; graphs ; low-rank and sparse decomposition ; clustering ; low-rank tensors.

Sketching as a Tool for Numerical Linear Algebra

Sketching as a Tool for Numerical Linear Algebra
  • Author : David P. Woodruff
  • Publisher :Unknown
  • Release Date :2014-11-14
  • Total pages :168
  • ISBN : 168083004X
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Summary : Sketching as a Tool for Numerical Linear Algebra highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compressed it to a much smaller matrix by multiplying it by a (usually) random matrix with certain properties. Much of the expensive computation can then be performed on the smaller matrix, thereby accelerating the solution for the original problem. It is an ideal primer for researchers and students of theoretical computer science interested in how sketching techniques can be used to speed up numerical linear algebra applications.

Tensors in Image Processing and Computer Vision

Tensors in Image Processing and Computer Vision
  • Author : Santiago Aja-Fernández,Rodrigo de Luis Garcia,Dacheng Tao,Xuelong Li
  • Publisher :Unknown
  • Release Date :2009-05-21
  • Total pages :470
  • ISBN : 9781848822993
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Summary : Tensor signal processing is an emerging field with important applications to computer vision and image processing. This book presents the state of the art in this new branch of signal processing, offering a great deal of research and discussions by leading experts in the area. The wide-ranging volume offers an overview into cutting-edge research into the newest tensor processing techniques and their application to different domains related to computer vision and image processing. This comprehensive text will prove to be an invaluable reference and resource for researchers, practitioners and advanced students working in the area of computer vision and image processing.

Machine Learning for Text

Machine Learning for Text
  • Author : Charu C. Aggarwal
  • Publisher :Unknown
  • Release Date :2018-03-19
  • Total pages :493
  • ISBN : 9783319735313
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Summary : Text analytics is a field that lies on the interface of information retrieval,machine learning, and natural language processing, and this textbook carefully covers a coherently organized framework drawn from these intersecting topics. The chapters of this textbook is organized into three categories: - Basic algorithms: Chapters 1 through 7 discuss the classical algorithms for machine learning from text such as preprocessing, similarity computation, topic modeling, matrix factorization, clustering, classification, regression, and ensemble analysis. - Domain-sensitive mining: Chapters 8 and 9 discuss the learning methods from text when combined with different domains such as multimedia and the Web. The problem of information retrieval and Web search is also discussed in the context of its relationship with ranking and machine learning methods. - Sequence-centric mining: Chapters 10 through 14 discuss various sequence-centric and natural language applications, such as feature engineering, neural language models, deep learning, text summarization, information extraction, opinion mining, text segmentation, and event detection. This textbook covers machine learning topics for text in detail. Since the coverage is extensive,multiple courses can be offered from the same book, depending on course level. Even though the presentation is text-centric, Chapters 3 to 7 cover machine learning algorithms that are often used indomains beyond text data. Therefore, the book can be used to offer courses not just in text analytics but also from the broader perspective of machine learning (with text as a backdrop). This textbook targets graduate students in computer science, as well as researchers, professors, and industrial practitioners working in these related fields. This textbook is accompanied with a solution manual for classroom teaching.

Tensor Network Contractions

Tensor Network Contractions
  • Author : Shi-Ju Ran
  • Publisher :Unknown
  • Release Date :2020-01-01
  • Total pages :150
  • ISBN : 9783030344894
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Summary : Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K.G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.

Decomposing Matrices, Tensors, and Images

Decomposing Matrices, Tensors, and Images
  • Author : Elina Robeva
  • Publisher :Unknown
  • Release Date :2016
  • Total pages :195
  • ISBN : OCLC:957714171
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Summary : 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.

Tensors

Tensors
  • Author : J. M. Landsberg
  • Publisher :Unknown
  • Release Date :2011-12-14
  • Total pages :439
  • ISBN : 9780821869079
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Summary : 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.

Higher-order Kronecker Products and Tensor Decompositions

Higher-order Kronecker Products and Tensor Decompositions
  • Author : Carla Dee Martin
  • Publisher :Unknown
  • Release Date :2005
  • Total pages :482
  • ISBN : CORNELL:31924104143064
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Summary : The second problem in this dissertation involves solving shifted linear systems of the form (A - lambdaI) x = b when A is a Kronecker product of matrices. The Schur decomposition is used to reduce the shifted Kronecker product system to a Kronecker product of quasi-triangular matrices. The system is solved using a recursive block procedure which circumvents formation of the explicit product.

Recent Trends in Learning From Data

Recent Trends in Learning From Data
  • Author : Luca Oneto,Nicolò Navarin,Alessandro Sperduti,Davide Anguita
  • Publisher :Unknown
  • Release Date :2020-04-03
  • Total pages :221
  • ISBN : 9783030438838
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Summary : This book offers a timely snapshot and extensive practical and theoretical insights into the topic of learning from data. Based on the tutorials presented at the INNS Big Data and Deep Learning Conference, INNSBDDL2019, held on April 16-18, 2019, in Sestri Levante, Italy, the respective chapters cover advanced neural networks, deep architectures, and supervised and reinforcement machine learning models. They describe important theoretical concepts, presenting in detail all the necessary mathematical formalizations, and offer essential guidance on their use in current big data research.

Tensor Spaces and Numerical Tensor Calculus

Tensor Spaces and Numerical Tensor Calculus
  • Author : Wolfgang Hackbusch
  • Publisher :Unknown
  • Release Date :2019-12-16
  • Total pages :605
  • ISBN : 9783030355548
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Summary : Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.