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Tensor Data Analysis In High Dimensions
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Download or read book Tensor Regression written by Jiani Liu and published by . This book was released on 2021-09-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor Regression is the first thorough overview of the fundamentals, motivations, popular algorithms, strategies for efficient implementation, related applications, available datasets, and software resources for tensor-based regression analysis.
Book Synopsis Tensor Computation for Data Analysis by : Yipeng Liu
Download or read book Tensor Computation for Data Analysis written by Yipeng Liu and published by Springer Nature. This book was released on 2021-08-31 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensor is a natural representation for multi-dimensional data, and tensor computation can avoid possible multi-linear data structure loss in classical matrix computation-based data analysis. This book is intended to provide non-specialists an overall understanding of tensor computation and its applications in data analysis, and benefits researchers, engineers, and students with theoretical, computational, technical and experimental details. It presents a systematic and up-to-date overview of tensor decompositions from the engineer's point of view, and comprehensive coverage of tensor computation based data analysis techniques. In addition, some practical examples in machine learning, signal processing, data mining, computer vision, remote sensing, and biomedical engineering are also presented for easy understanding and implementation. These data analysis techniques may be further applied in other applications on neuroscience, communication, psychometrics, chemometrics, biometrics, quantum physics, quantum chemistry, etc. The discussion begins with basic coverage of notations, preliminary operations in tensor computations, main tensor decompositions and their properties. Based on them, a series of tensor-based data analysis techniques are presented as the tensor extensions of their classical matrix counterparts, including tensor dictionary learning, low rank tensor recovery, tensor completion, coupled tensor analysis, robust principal tensor component analysis, tensor regression, logistical tensor regression, support tensor machine, multilinear discriminate analysis, tensor subspace clustering, tensor-based deep learning, tensor graphical model and tensor sketch. The discussion also includes a number of typical applications with experimental results, such as image reconstruction, image enhancement, data fusion, signal recovery, recommendation system, knowledge graph acquisition, traffic flow prediction, link prediction, environmental prediction, weather forecasting, background extraction, human pose estimation, cognitive state classification from fMRI, infrared small target detection, heterogeneous information networks clustering, multi-view image clustering, and deep neural network compression.
Book Synopsis Multimodal and Tensor Data Analytics for Industrial Systems Improvement by : Nathan Gaw
Download or read book Multimodal and Tensor Data Analytics for Industrial Systems Improvement written by Nathan Gaw and published by Springer Nature. This book was released on with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Tensor Data Analysis in High Dimensions by : Keqian Min
Download or read book Tensor Data Analysis in High Dimensions written by Keqian Min and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large number of tensor datasets have been appearing in modern scientific research, attracting much attention to the analysis of such datasets. Tensor data often have high dimensionality and tensor structure that contains extra information. Handling the high dimensionality and utilizing the structural information is essential for analyzing tensor data. Feature screening is a popular method to deal with high dimensionality. In the first part of this dissertation, we study the smoothness structure of tensors and propose a general framework for tensor screening called smoothed tensor screening (STS). We establish the SURE screening property for STS under mild conditions. In the second part, we study the tensor Gaussian graphical model, which reveals the conditional independence structure within tensor data. With normally distributed $M$-way tensors, the key to high-dimensional tensor graphical models becomes the sparse estimation of the $M$ inverse covariance matrices. To overcome the high computational cost of the existing cyclic approaches, we propose a separable and parallel estimation scheme. We provide numerical studies to demonstrate its performance. In the third part, we study the optimality theory of tensor discriminant analysis (TDA) in high dimensions. We provide a systematic investigation on the theoretical properties of TDA. We obtain the minimax lower bound for both coefficient estimation and misclassification risk. We further show that one existing high-dimensional tensor discriminant analysis estimator is minimax optimal.
Book Synopsis Efficient Analysis of High Dimensional Data in Tensor Formats by : Mike Espig
Download or read book Efficient Analysis of High Dimensional Data in Tensor Formats written by Mike Espig and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Applied Matrix and Tensor Variate Data Analysis by : Toshio Sakata
Download or read book Applied Matrix and Tensor Variate Data Analysis written by Toshio Sakata and published by Springer. This book was released on 2016-02-02 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides comprehensive reviews of recent progress in matrix variate and tensor variate data analysis from applied points of view. Matrix and tensor approaches for data analysis are known to be extremely useful for recently emerging complex and high-dimensional data in various applied fields. The reviews contained herein cover recent applications of these methods in psychology (Chap. 1), audio signals (Chap. 2) , image analysis from tensor principal component analysis (Chap. 3), and image analysis from decomposition (Chap. 4), and genetic data (Chap. 5) . Readers will be able to understand the present status of these techniques as applicable to their own fields. In Chapter 5 especially, a theory of tensor normal distributions, which is a basic in statistical inference, is developed, and multi-way regression, classification, clustering, and principal component analysis are exemplified under tensor normal distributions. Chapter 6 treats one-sided tests under matrix variate and tensor variate normal distributions, whose theory under multivariate normal distributions has been a popular topic in statistics since the books of Barlow et al. (1972) and Robertson et al. (1988). Chapters 1, 5, and 6 distinguish this book from ordinary engineering books on these topics.
Book Synopsis High-Dimensional Data Analysis with Low-Dimensional Models by : John Wright
Download or read book High-Dimensional Data Analysis with Low-Dimensional Models written by John Wright and published by Cambridge University Press. This book was released on 2022-01-13 with total page 717 pages. Available in PDF, EPUB and Kindle. Book excerpt: Connects fundamental mathematical theory with real-world problems, through efficient and scalable optimization algorithms.
Book Synopsis Large Dimensional Data Analysis Using Orthogonally Decomposable Tensors by : Arnab Auddy
Download or read book Large Dimensional Data Analysis Using Orthogonally Decomposable Tensors written by Arnab Auddy and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern data analysis requires the study of tensors, or multi-way arrays. We consider the case where the dimension d is large and the order p is fixed. For dimension reduction and for interpretability, one considers tensor decompositions, where a tensor T can be decomposed into a sum of rank one tensors. In this thesis, I will describe some recent work that illustrate why and how to use decompositions for orthogonally decomposable tensors. Our developments are motivated by statistical applications where the data dimension is large. The estimation procedures will therefore aim to be computationally tractable while providing error rates that depend optimally on the dimension. A tensor is said to be orthogonally decomposable if it can be decomposed into rank one tensors whose component vectors are orthogonal. A number of data analysis tasks can be recast as the problem of estimating the component vectors from a noisy observation of an orthogonally decomposable tensor. In our first set of results, we study this decompositionproblem and derive perturbation bounds.
Book Synopsis Tensor Methods in Statistics by : Peter McCullagh
Download or read book Tensor Methods in Statistics written by Peter McCullagh and published by Courier Dover Publications. This book was released on 2018-07-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: A pioneering monograph on tensor methods applied to distributional problems arising in statistics, this work begins with the study of multivariate moments and cumulants. An invaluable reference for graduate students and professional statisticians. 1987 edition.
Book Synopsis Tensors for Data Processing by : Yipeng Liu
Download or read book Tensors for Data Processing written by Yipeng Liu and published by Academic Press. This book was released on 2021-10-21 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods. As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry. Provides a complete reference on classical and state-of-the-art tensor-based methods for data processing Includes a wide range of applications from different disciplines Gives guidance for their application
Book Synopsis High-Dimensional Methodologies for Sufficient Dimension Reduction, Discriminant Analysis, and Tensor Data by : Jing Zeng
Download or read book High-Dimensional Methodologies for Sufficient Dimension Reduction, Discriminant Analysis, and Tensor Data written by Jing Zeng and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thanks to the advancement of data-collecting technology in brain imaging, genomics, financial econometrics, and machine learning, scientific data tend to grow in both size and structural complexity, which are not amenable to traditional statistical analysis. In this dissertation, we developed novel high-dimensional methodologies for dimension reduction, discriminant analysis, and tensor data. In the first chapter, we proposed a unified framework, called subspace estimation with automatic dimension and variable selection (SEAS), to extend many existing low-dimensional sufficient dimension reduction (SDR) methods to the high-dimensional setting. The flexibility of SEAS considerably widens the application scope of many SDR methods. Our proposal only relies on a double-penalized convex formulation, which can be solved efficiently. From the theoretical perspective, we established a satisfactory convergence rate for our proposal, which is optimal in a minimax sense. In the second chapter, we established a population model for the reduced-rank linear discriminant analysis (LDA) problem, which arises naturally in many scenarios. We also developed an efficient algorithm and derived the non-asymptotic results in the high-dimensional setting. In the last chapter, we studied how two data modalities associate and interact with each other given a third modality, which is a crucial problem in multimodal integrative analysis but has no available statistical solution. We formulated this problem as a tensor decomposition problem and proposed a novel generalized liquid association analysis (GLAA) method. A high-order orthogonal iteration algorithm is provided accordingly. Furthermore, we established the non-asymptotic results for the proposed estimators.
Book Synopsis Tensor Networks for Dimensionality Reduction and Large-scale Optimization by : Andrzej Cichocki
Download or read book Tensor Networks for Dimensionality Reduction and Large-scale Optimization written by Andrzej Cichocki and published by . This book was released on 2016 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern applications in engineering and data science are increasingly based on multidimensional data of exceedingly high volume, variety, and structural richness. However, standard machine learning algorithms typically scale exponentially with data volume and complexity of cross-modal couplings - the so called curse of dimensionality - which is prohibitive to the analysis of large-scale, multi-modal and multi-relational datasets. Given that such data are often efficiently represented as multiway arrays or tensors, it is therefore timely and valuable for the multidisciplinary machine learning and data analytic communities to review low-rank tensor decompositions and tensor networks as emerging tools for dimensionality reduction and large scale optimization problems. Our particular emphasis is on elucidating that, by virtue of the underlying low-rank approximations, tensor networks have the ability to alleviate the curse of dimensionality in a number of applied areas. In Part 1 of this monograph we provide innovative solutions to low-rank tensor network decompositions and easy to interpret graphical representations of the mathematical operations on tensor networks. Such a conceptual insight allows for seamless migration of ideas from the flat-view matrices to tensor network operations and vice versa, and provides a platform for further developments, practical applications, and non-Euclidean extensions. It also permits the introduction of various tensor network operations without an explicit notion of mathematical expressions, which may be beneficial for many research communities that do not directly rely on multilinear algebra. Our focus is on the Tucker and tensor train (TT) decompositions and their extensions, and on demonstrating the ability of tensor networks to provide linearly or even super-linearly (e.g., logarithmically) scalable solutions, as illustrated in detail in Part 2 of this monograph.
Book Synopsis Tensor Regression and Tensor Time Series Analyses for High Dimensional Data by : Herath Mudiyanselage Wiranthe Bandara Herath
Download or read book Tensor Regression and Tensor Time Series Analyses for High Dimensional Data written by Herath Mudiyanselage Wiranthe Bandara Herath and published by . This book was released on 2019 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many real data are naturally represented as a multidimensional array called a tensor. In classical regression and time series models, the predictors and covariate variables are considered as a vector. However, due to high dimensionality of predictor variables, these types of models are inefficient for analyzing multidimensional data. In contrast, tensor structured models use predictors and covariate variables in a tensor format. Tensor regression and tensor time series models can reduce high dimensional data to a low dimensional framework and lead to efficient estimation and prediction. In this thesis, we discuss the modeling and estimation procedures for both tensor regression models and tensor time series models. The results of simulation studies and a numerical analysis are provided.
Book Synopsis Tensor Spaces and Numerical Tensor Calculus by : Wolfgang Hackbusch
Download or read book Tensor Spaces and Numerical Tensor Calculus written by Wolfgang Hackbusch and published by Springer Nature. This book was released on 2019-12-16 with total page 605 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Introduction to Tensor Analysis by : Leonard Lovering Barrett
Download or read book An Introduction to Tensor Analysis written by Leonard Lovering Barrett and published by . This book was released on 2012-05-01 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data by : Carl-Fredrik Westin
Download or read book Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data written by Carl-Fredrik Westin and published by Springer. This book was released on 2014-07-17 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI). A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research. The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.
Book Synopsis High-Performance Tensor Computations in Scientific Computing and Data Science by : Edoardo Angelo Di Napoli
Download or read book High-Performance Tensor Computations in Scientific Computing and Data Science written by Edoardo Angelo Di Napoli and published by Frontiers Media SA. This book was released on 2022-11-08 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:
