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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.

Tensor Decomposition Meets Approximation Theory

Tensor Decomposition Meets Approximation Theory Book
Author : Ferre Knaepkens
Publisher :
Release : 2017
ISBN :
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 :
Release : 2018
ISBN :
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 :
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 :
Release : 2005
ISBN :
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.

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 :
Release : 2018
ISBN : 9780438430440
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.

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.

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.

Decomposability of Tensors

Decomposability of Tensors Book
Author : Luca Chiantini
Publisher : MDPI
Release : 2019-02-15
ISBN : 3038975907
Language : En, Es, Fr & De

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Book Description :

This book is a printed edition of the Special Issue "Decomposability of Tensors" that was published in Mathematics

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 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 : N.A
Publisher :
Release : 2013
ISBN :
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.

Multimodal Analytics for Next Generation Big Data Technologies and Applications

Multimodal Analytics for Next Generation Big Data Technologies and Applications Book
Author : Kah Phooi Seng,Li-minn Ang,Alan Wee-Chung Liew,Junbin Gao
Publisher : Springer
Release : 2019-07-18
ISBN : 3319975986
Language : En, Es, Fr & De

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Book Description :

This edited book will serve as a source of reference for technologies and applications for multimodality data analytics in big data environments. After an introduction, the editors organize the book into four main parts on sentiment, affect and emotion analytics for big multimodal data; unsupervised learning strategies for big multimodal data; supervised learning strategies for big multimodal data; and multimodal big data processing and applications. The book will be of value to researchers, professionals and students in engineering and computer science, particularly those engaged with image and speech processing, multimodal information processing, data science, and artificial intelligence.

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.

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. ​

Matrix Computations

Matrix Computations Book
Author : Gene H. Golub,Charles F. Van Loan
Publisher : JHU Press
Release : 2013-02-15
ISBN : 1421408597
Language : En, Es, Fr & De

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Book Description :

The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool. This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on • fast transforms • parallel LU • discrete Poisson solvers • pseudospectra • structured linear equation problems • structured eigenvalue problems • large-scale SVD methods • polynomial eigenvalue problems Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature—everything needed to become a matrix-savvy developer of numerical methods and software. The second most cited math book of 2012 according to MathSciNet, the book has placed in the top 10 for since 2005.

Architecture aware Algorithm Design of Sparse Tensor matrix Primitives for GPUs

Architecture aware Algorithm Design of Sparse Tensor matrix Primitives for GPUs Book
Author : Israt J. Nisa
Publisher :
Release : 2019
ISBN :
Language : En, Es, Fr & De

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Book Description :

parse matrix/tensor operations have been a common computational motif in a wide spectrum of domains - numerical linear algebra, graph analytics, machine larning, health-care, etc. Sparse kernels play a key role in numerous machine learning algorithms and the rising popularity of this domain increases the significance of the primitives like SpMV (Sparse Matrix-Vector Multiplication), SDDMM (Sampled Dense-Dense Matrix Multiplication), MF/TF(Sparse Matrix/Tensor Factorization), etc. These primitives are data-parallel and highly suitable for GPU-like architectures that provide massive parallelism. Real-world matrices and tensors are large-scale and have millions of data points, which is sufficient to utilize all the cores of a GPU. Yet, a data-parallel algorithm can become the bottleneck of an application and perform way below than the upper bound of the roofline model. Some common reasons are frequent irregular global memory access, low data reuse, and imbalanced work distribution. However, efficient utilization of GPU memory hierarchy, reduced thread communication, increased data locality, and an even workload distribution can provide ample opportunities for significant performance improvement. The challenge lies in utilizing the techniques across applications and achieve an even performance in spite of the irregularity of the input matrices or tensors. In this work, we systematically identify the performance bottlenecks of the important sparse algorithms and provide optimized and high performing solutions.

Multilinear Operators for Higher order Decompositions

Multilinear Operators for Higher order Decompositions Book
Author : Tamara Gibson Kolda
Publisher :
Release : 2006
ISBN :
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.