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Matrix And Tensor Decomposition

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Matrix and Tensor Factorization Techniques for Recommender Systems

Matrix and Tensor Factorization Techniques for Recommender Systems Book
Author : Panagiotis Symeonidis,Andreas Zioupos
Publisher : Springer
Release : 2017-01-29
ISBN : 3319413570
Language : En, Es, Fr & De

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Book Description :

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 Decomposition

Matrix and Tensor Decomposition Book
Author : Christian Jutten
Publisher : Unknown
Release : 2021-05-06
ISBN : 9780128157602
Language : En, Es, Fr & De

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Book Description :

Download Matrix and Tensor Decomposition book written by Christian Jutten, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Matrix and Tensor Decompositions in Signal Processing

Matrix and Tensor Decompositions in Signal Processing Book
Author : Favier
Publisher : Wiley-Blackwell
Release : 2018-11-20
ISBN : 9781786301550
Language : En, Es, Fr & De

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Book Description :

Download Matrix and Tensor Decompositions in Signal Processing book written by Favier, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Tensor Decomposition Meets Approximation Theory

Tensor Decomposition Meets Approximation Theory Book
Author : Ferre Knaepkens
Publisher : Unknown
Release : 2017
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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.

Low Rank Tensor Decomposition for Feature Extraction and Tensor Recovery

Low Rank Tensor Decomposition for Feature Extraction and Tensor Recovery Book
Author : Qiquan Shi
Publisher : Unknown
Release : 2018
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Feature extraction and tensor recovery problems are important yet challenging, particularly for multi-dimensional data with missing values and/or noise. Low-rank tensor decomposition approaches are widely used for solving these problems. This thesis focuses on three common tensor decompositions (CP, Tucker and t-SVD) and develops a set of decomposition-based approaches. The proposed methods aim to extract low-dimensional features from complete/incomplete data and recover tensors given partial and/or grossly corrupted observations.

Spectral Learning on Matrices and Tensors

Spectral Learning on Matrices and Tensors Book
Author : Majid Janzamin,Rong Ge,Jean Kossaifi,Anima Anandkumar
Publisher : Unknown
Release : 2019-11-25
ISBN : 9781680836400
Language : En, Es, Fr & De

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Book Description :

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.

Higher order Kronecker Products and Tensor Decompositions

Higher order Kronecker Products and Tensor Decompositions Book
Author : Carla Dee Martin
Publisher : Unknown
Release : 2005
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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.

Nonnegative Matrix and Tensor Factorizations

Nonnegative Matrix and Tensor Factorizations Book
Author : Andrzej Cichocki,Rafal Zdunek,Anh Huy Phan,Shun-ichi Amari
Publisher : John Wiley & Sons
Release : 2009-07-10
ISBN : 9780470747285
Language : En, Es, Fr & De

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Book Description :

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.

A Multilingual Exploration of Semantics in the Brain Using Tensor Decomposition

A Multilingual Exploration of Semantics in the Brain Using Tensor Decomposition Book
Author : Sharmistha Bardhan
Publisher : Unknown
Release : 2018
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

The semantic concept processing mechanism of the brain shows that different neural activity patterns occur for different semantic categories. Multivariate Pattern Analysis of the brain fMRI data shows promising results in identifying active brain regions for a specific semantic category. Unsupervised learning technique such as tensor decomposition discovers the hidden structure from the brain data and proved to be useful as well. However, the existing methods are used for analyzing data from subjects who speak in one language and do not consider the cultural effect on it. This thesis presents an exploratory analysis of the neuro-semantic problem in a new dimension. The brain fMRI tensors of subjects who speak in Chinese or Italian language are analyzed both individually and together to discover the hidden structure. The Chinese and Italian tensors are jointly analyzed by coupling them along the stimuli object mode to discover the cultural effect. Moreover, the joint analysis of semantic features and brain fMRI tensor using the Advanced Coupled Matrix Tensor Factorization (ACMTF) method finds latent variables that explain the correlation between them. The results of the joint analysis of the tensors support the preliminary predictive analysis and find meaningful clusters for the different categories of stimuli object. Moreover, for a rank 2 decomposition, the prediction of brain activation pattern given semantic features gives an accuracy of 71.43%. It is expected that, the proposed exploratory and predictive analysis will improve existing approaches of analyzing conceptual knowledge representation of brain and guide future research in this domain.

Urban Computing

Urban Computing Book
Author : Yu Zheng
Publisher : MIT Press
Release : 2019-02-05
ISBN : 0262039087
Language : En, Es, Fr & De

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Book Description :

An authoritative treatment of urban computing, offering an overview of the field, fundamental techniques, advanced models, and novel applications. Urban computing brings powerful computational techniques to bear on such urban challenges as pollution, energy consumption, and traffic congestion. Using today's large-scale computing infrastructure and data gathered from sensing technologies, urban computing combines computer science with urban planning, transportation, environmental science, sociology, and other areas of urban studies, tackling specific problems with concrete methodologies in a data-centric computing framework. This authoritative treatment of urban computing offers an overview of the field, fundamental techniques, advanced models, and novel applications. Each chapter acts as a tutorial that introduces readers to an important aspect of urban computing, with references to relevant research. The book outlines key concepts, sources of data, and typical applications; describes four paradigms of urban sensing in sensor-centric and human-centric categories; introduces data management for spatial and spatio-temporal data, from basic indexing and retrieval algorithms to cloud computing platforms; and covers beginning and advanced topics in mining knowledge from urban big data, beginning with fundamental data mining algorithms and progressing to advanced machine learning techniques. Urban Computing provides students, researchers, and application developers with an essential handbook to an evolving interdisciplinary field.

Advances in Knowledge Discovery and Data Mining

Advances in Knowledge Discovery and Data Mining Book
Author : Qiang Yang,Zhi-Hua Zhou,Zhiguo Gong,Min-Ling Zhang,Sheng-Jun Huang
Publisher : Springer
Release : 2019-05-20
ISBN : 303016148X
Language : En, Es, Fr & De

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Book Description :

The three-volume set LNAI 11439, 11440, and 11441 constitutes the thoroughly refereed proceedings of the 23rd Pacific-Asia Conference on Knowledge Discovery and Data Mining, PAKDD 2019, held in Macau, China, in April 2019. The 137 full papers presented were carefully reviewed and selected from 542 submissions. The papers present new ideas, original research results, and practical development experiences from all KDD related areas, including data mining, data warehousing, machine learning, artificial intelligence, databases, statistics, knowledge engineering, visualization, decision-making systems, and the emerging applications. They are organized in the following topical sections: classification and supervised learning; text and opinion mining; spatio-temporal and stream data mining; factor and tensor analysis; healthcare, bioinformatics and related topics; clustering and anomaly detection; deep learning models and applications; sequential pattern mining; weakly supervised learning; recommender system; social network and graph mining; data pre-processing and feature selection; representation learning and embedding; mining unstructured and semi-structured data; behavioral data mining; visual data mining; and knowledge graph and interpretable data mining.

Robust Statistics for Signal Processing

Robust Statistics for Signal Processing Book
Author : Abdelhak M. Zoubir,Visa Koivunen,Esa Ollila,Michael Muma
Publisher : Cambridge University Press
Release : 2018-10-31
ISBN : 1107017416
Language : En, Es, Fr & De

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Book Description :

Understand the benefits of robust statistics for signal processing using this unique and authoritative text.

From Algebraic Structures to Tensors

From Algebraic Structures to Tensors Book
Author : Gérard Favier
Publisher : John Wiley & Sons
Release : 2020-01-02
ISBN : 1786301547
Language : En, Es, Fr & De

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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.

Tensors

Tensors Book
Author : J. M. Landsberg
Publisher : American Mathematical Soc.
Release : 2011-12-14
ISBN : 0821869078
Language : En, Es, Fr & De

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Book Description :

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.

Tensor multidimensional Array Decomposition Regression and Software for Statistics and Machine Learning

Tensor  multidimensional Array  Decomposition  Regression and Software for Statistics and Machine Learning Book
Author : James Yi-Wei Li
Publisher : Unknown
Release : 2014
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

This thesis illustrates connections between statistical models for tensors, introduces a novel linear model for tensors with 3 modes, and implements tensor software in the form of an R package. Tensors, or multidimensional arrays, are a natural generalization of the vectors and matrices that are ubiquitous in statistical modeling. However, while matrix algebra has been well-studied and plays a crucial role in the interaction between data and the parameters of any given model, algebra of higher-order arrays has been relatively overlooked in data analysis and statistical theory. The emergence of multilinear datasets - where observations are vector-variate, matrix-variate, or even tensor-variate - only serve to emphasize the relative lack of statistical understanding around tensor data structures. In the first half of the thesis, we highlight classic tensor algebraic results and models used in image analysis, chemometrics, and psychometrics, as well as connect them to recent statistical models. The second half of the thesis features a linear model that is based off a recently introduced tensor multiplication. For this model, we prove some of the classic properties that we would expect from a 3-tensor generalization of the matrix ordinary least squares. We also apply our model to a functional dataset to demonstrate one possible usage. We conclude this thesis with an exposition of the software developed to facilitate tensor modeling and manipulation in R. This software implements many of the classic tensor decomposition models as well as our own linear model.

Tensor Representation Techniques in Post Hartree Fock Methods Matrix Product State Tensor Format

Tensor Representation Techniques in Post Hartree Fock Methods  Matrix Product State Tensor Format Book
Author : Anonim
Publisher : Unknown
Release : 2013
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

A approximation for post-Hartree Fock (HF) methods is presented applying tensor decomposition techniques in the matrix product state tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion of one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation.

Multilinear Operators for Higher order Decompositions

Multilinear Operators for Higher order Decompositions Book
Author : Tamara Gibson Kolda
Publisher : Unknown
Release : 2006
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to concisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very concise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.

The Matrix and Tensor Quarterly

The Matrix and Tensor Quarterly Book
Author : Anonim
Publisher : Unknown
Release : 1980
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download The Matrix and Tensor Quarterly book written by , available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Tensor Spaces and Numerical Tensor Calculus

Tensor Spaces and Numerical Tensor Calculus Book
Author : Wolfgang Hackbusch
Publisher : Springer Science & Business Media
Release : 2012-02-23
ISBN : 3642280277
Language : En, Es, Fr & De

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Book Description :

Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d 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. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with stochastic coefficients, etc. ​

Tensor Tools with Application in Signal Processing

Tensor Tools with Application in Signal Processing Book
Author : Mikael Sorensen
Publisher : Unknown
Release : 2010
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Several problems in signal processing can be put in a tensorial framework. In many cases, these problems lead to compute structured tensor decompositions. The main purpose of this thesis is the development of computational methods for structured tensor decomposition problems with application in signal processing. First, we propose methods for the computation of the CANDECOMP/PARAFAC (CP) decomposition with a semi-unitary matrix factor. Moreover, we develop methods to solve CP structured independent component analysis problems by addressing them as semi-unitary constrained CP decompositions problems. Second, we propose simultaneous (generalized) Schur decomposition methods to compute CP decompositions possibly with partial (Hermitian) symmetries. Moreover, a Jacobi-type sweeping procedure for the computation of a real CP decomposition will be considered. Third, we propose methods to compute tensor decompositions with banded or Hankel/Toeplitz matrix factors and possibly also with partial (Hermitian) symmetries. The methods will be applied to solve some cumulant based blind identification problems. Fourth, we propose a more efficient Jacobi method for blind equalization of paraunitary channels. Furthermore, we also derive an algebraic solution for the Jacobi method for likelihood based joint diagonalization of real positive definite matrices.