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## 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

**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 Decomposition

- Author : Christian Jutten
- Publisher :Unknown
- Release Date :2021
- Total pages :229
- ISBN : 0128157607

**Summary :**

## 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

**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 Decompositions in Signal Processing

- Author : Favier
- Publisher :Unknown
- Release Date :2018-11-20
- Total pages :200
- ISBN : 1786301555

**Summary :**

## Algorithmic Aspects of Machine Learning

- Author : Ankur Moitra
- Publisher :Unknown
- Release Date :2018-09-27
- Total pages :176
- ISBN : 9781107184589

**Summary :** Introduces cutting-edge research on machine learning theory and practice, providing an accessible, modern algorithmic toolkit.

## 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

**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.

## Tensor Decomposition Meets Approximation Theory

- Author : Ferre Knaepkens
- Publisher :Unknown
- Release Date :2017
- Total pages :229
- ISBN : OCLC:1050023083

**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.

## 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

**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.

## Sketching as a Tool for Numerical Linear Algebra

- Author : David P. Woodruff
- Publisher :Unknown
- Release Date :2014-11-14
- Total pages :168
- ISBN : 168083004X

**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.

## Large Scale Eigenvalue Problems

- Author : J. Cullum,R.A. Willoughby
- Publisher :Unknown
- Release Date :1986-01-01
- Total pages :329
- ISBN : 0080872387

**Summary :** Results of research into large scale eigenvalue problems are presented in this volume. The papers fall into four principal categories: novel algorithms for solving large eigenvalue problems, novel computer architectures, computationally-relevant theoretical analyses, and problems where large scale eigenelement computations have provided new insight.

## Low Rank Tensor Decomposition for Feature Extraction and Tensor Recovery

- Author : Qiquan Shi
- Publisher :Unknown
- Release Date :2018
- Total pages :218
- ISBN : OCLC:1066258974

**Summary :** 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.

## Decomposability of Tensors

- Author : Luca Chiantini
- Publisher :Unknown
- Release Date :2019-02-15
- Total pages :160
- ISBN : 9783038975908

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

## Tensors

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

**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.

## Tensor Network Contractions

- Author : Shi-Ju Ran
- Publisher :Unknown
- Release Date :2020-01-01
- Total pages :150
- ISBN : 9783030344894

**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.

## Higher-order Kronecker Products and Tensor Decompositions

- Author : Carla Dee Martin
- Publisher :Unknown
- Release Date :2005
- Total pages :482
- ISBN : CORNELL:31924104143064

**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.

## Three-mode Principal Component Analysis

- Author : Pieter M. Kroonenberg
- Publisher :Unknown
- Release Date :1983
- Total pages :398
- ISBN : 9789066950023

**Summary :**

## New Frontiers in Applied Data Mining

- Author : Longbing Cao,Joshua Zhexue Huang,James Bailey,Yun Sing Koh,Jun Luo
- Publisher :Unknown
- Release Date :2012-02-21
- Total pages :508
- ISBN : 9783642283208

**Summary :** This book constitutes the thoroughly refereed post-conference proceedings of five international workshops held in conjunction with PAKDD 2011 in Shenzhen, China, in May 2011: the International Workshop on Behavior Informatics (BI 2011), the Workshop on Quality Issues, Measures of Interestingness and Evaluation of Data Mining Models (QIMIE 2011), the Workshop on Biologically Inspired Techniques for Data Mining (BDM 2011), the Workshop on Advances and Issues in Traditional Chinese Medicine Clinical Data Mining (AI-TCM 2011), and the Second Workshop on Data Mining for Healthcare Management (DMGHM 2011). The book also includes papers from the First PAKDD Doctoral Symposium on Data Mining (DSDM 2011). The 42 papers were carefully reviewed and selected from numerous submissions. The papers cover a wide range of topics discussing emerging techniques in the field of knowledge discovery in databases and their application domains extending to previously unexplored areas such as data mining based on optimization techniques from biological behavior of animals and applications in Traditional Chinese Medicine clinical research and health care management.

## Tensor Analysis

- Author : Liqun Qi,Ziyan Luo
- Publisher :Unknown
- Release Date :2017-04-19
- Total pages :318
- ISBN : 9781611974751

**Summary :** Tensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory?some of which are nontrivial?have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors. Tensor Analysis: Spectral Theory and Special Tensors is unique in that it is the first book on these three subject areas: spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors.

## A Multilingual Exploration of Semantics in the Brain Using Tensor Decomposition

- Author : Sharmistha Bardhan
- Publisher :Unknown
- Release Date :2018
- Total pages :85
- ISBN : OCLC:1104295815

**Summary :** 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.

## Tensor Networks for Dimensionality Reduction and Large-Scale Optimization

- Author : Andrzej Cichocki,Namgil Lee,Ivan Oseledets,Anh-Huy Phan,Qibin Zhao,Danilo P. Mandic
- Publisher :Unknown
- Release Date :2016-12-19
- Total pages :196
- ISBN : 1680832220

**Summary :** This monograph provides a systematic and example-rich guide to the basic properties and applications of tensor network methodologies, and demonstrates their promise as a tool for the analysis of extreme-scale multidimensional data. It demonstrates the ability of tensor networks to provide linearly or even super-linearly, scalable solutions.

## Signal Processing in Neuroscience

- Author : Xiaoli Li
- Publisher :Unknown
- Release Date :2016-08-31
- Total pages :288
- ISBN : 9789811018220

**Summary :** This book reviews cutting-edge developments in neural signalling processing (NSP), systematically introducing readers to various models and methods in the context of NSP. Neuronal Signal Processing is a comparatively new field in computer sciences and neuroscience, and is rapidly establishing itself as an important tool, one that offers an ideal opportunity to forge stronger links between experimentalists and computer scientists. This new signal-processing tool can be used in conjunction with existing computational tools to analyse neural activity, which is monitored through different sensors such as spike trains, local filed potentials and EEG. The analysis of neural activity can yield vital insights into the function of the brain. This book highlights the contribution of signal processing in the area of computational neuroscience by providing a forum for researchers in this field to share their experiences to date.