As indicated in the preface to volume 1, this volume is suitable for a onesemester course on vector and tensor analysis. Independent component analysis is a standard tool in modern data analysis and numerous different techniques for applying it exist. Independent component analysis computer science university. Thanks for contributing an answer to mathematics stack exchange. Consider a sample data log of network tra c messages. Multilinear tensor ica and dimensionality reduction springerlink. A secondorder tensor t is defined as a bilinear function from two copies of a vector space v into the space of real numbers. The standard methods however quickly lose their effectiveness. Independent component analysis for tensorvalued data. In short, this book is the best introduction to tensor analysis for beginning students. What are the best books to learn tensors from for the. December 2017 when solving physical problems, one must often choose between writing formulas in a coordinate independent form, or a form in which calculations are transparent. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo sition.
Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Common fe methods, such as principal component analysis pca, independent component analysis ica, etc. It is mathematicalsorrybut it gives the barebones definitions that are needed to do differential geometry. Nonnegative matrix and tensor factorizations guide books. Feng1 1physics department, the university of texas at austin dated. Tensor principal component analysis via convex optimization. Introduction to tensor calculus a scalar eld describes a onetoone correspondence between a single scalar number and a point. An improved cumulant based method for independent component. It is based on comons algorithm 1 but it takes third and fourthorder. More precisely, these methods all require an eigen. Deterministic independent component analysis eral tensor perspective anandkumar et al. Contrasted with the infinitesimal calculus, tensor calculus allows.
Pdf an improved cumulant based method for independent. Introduction to tensor calculus and continuum mechanics. More details on these techniques can be found in tutorial books on ica, such as 11,12. An improved method for independent component analysis based on the diagonalization of cumulant tensors is proposed. The aim of this book is to present a selfcontained, reasonably modern account of tensor analysis and the calculus of exterior differential forms, adapted to the needs of physicists, engineers, and applied mathematicians. In this paper we tackle this problem from the viewpoint of independent component analysis. Unsupervised contentbased identification of fake news. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler. One such approach is independent component analysis ica which has become quite popular in both eeg and fmri analysis 20 21 22. Handbook of blind source separation, independent component.
Independent component analysis and signal separation is one of the most exciting current areas of research in statistical signal processing and unsup vised machine learning. Directional independent component analysis with tensor. This is the first book to provide a comprehensive introduction to this new. I do not personally know the following book, but try tensor analysis for physicists by j. This is typically done by means of an eigenvalue decomposition evd or a singular value decomposition svd. Tensor independent component analysis and tensor non. If the op has a decent math background, i would suggest abraham, marsden, and ratius manifolds, tensor analysis, and applications. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and no background in tensor analysis.
This chapter discusses the four prewhiteningbased algebraic algorithms for independent component analysis ica. Im a fan of bishop and goldbergs tensor analysis on manifolds. Im sure there are better books out there, but this one is nice for its niche. An ndimensional vector eld is described by a onetoone correspondence between nnumbers and a point. Update rules for independent vector analysis based on. The area has received attention from several research communities including machine learning, neural networks, statistical signal p cessing and bayesian modeling. Vector a a physical quantity that has both direction and. The appendix, newly revised and enlarged for the dover edition, presents a reformulation of the principal concepts of the main text within the terminology of current global differential geometry, thus bridging the gap between classical tensor analysis and the fundamentals of more recent global theories. Linear dimension reduction based on the fourthorder. A novel fixedpoint algorithm for constrained independent component analysis. Tensor analysis is an essential tool in any science e. The fact that a book of nearly 500 pages can be written on this, and noting the authors comment that it is certain that i have missed some topics, and my coverage of others will be too brief for the taste of some. It is important to realize that the object one describes is independent of the coordi nate system i.
Representing multidimensional data as tensors tensors aka multidimensional arrays are a natural t for representing data with multiple associated attributes such as network tra c data. On occasions when we have taught a one semester course, we. The mixing matrix a transforms source signals as follows. Objects that tensors may map between include vectors which are often, but not always, understood as arrows with length that point in a direction and scalars which are often familiar numbers such as the real numbers, and, recursively, even.
An introduction for mathematicians and physicists by paul renteln. Tensor independent component analysis and tensor nonnegative factorization. Fundamentals of tensor analysis concepts of scalar, vector, and tensor scalar. Us20080247608a1 method, system, storage medium, and data. In this chapter, we describe two tensorbased subspace analysis approaches tensor ica and tensor nmf that can be used in many fields like face recognition. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Principal component analysis is the empirical manifestation of the eigen valuedecomposition of a correlation or covariance matrix.
A basic knowledge of vectors, matrices, and physics is assumed. Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3. This leads to the loss of the spatial information of the polsar data. Independent component analysis for tensor valued data. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. A numerical study demonstrates the validity of our method and its usefulness for extracting subgaussian structures. Decompositions of a higherorder tensor in block termspart. Candecompparafac cp decomposes a tensor as a sum of rankone tensors, and the tucker decomposition is a higherorder form of principal components analysis. Ouedraogo w, souloumiac a and jutten c nonnegative independent component analysis algorithm based on 2d givens rotations and a newton optimization proceedings of the 9th international conference on latent variable analysis and signal separation, 522529. An efficient learning procedure is presented via a novel tensor analysis, called modek directional independent component analysis with tensor representation lei zhang1, quanxue gao1,2 and david zhang1. Tensorial independent component analysisbased feature.
Extending independent component analysis to tensors has also attracted some attention but, to our knowledge, no modelbased treatise has been given. In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space typically a euclidean space or manifold. The first hurdle most students confront when studying tensors is what covariance and contravariance are. Tensors, differential forms, and variational principles. Ica for tensor data has been studied especially in the case of neuroimaging 14, 15 and. What are the best books to learn tensors from for the first. For each message, let us assume that the log records. As a tensor is a generalization of a scalar a pure number. Us11571,341 200401 200501 method, system, storage medium, and data structure for image recognition using multilinear independent component analysis active 20250810 us7693299b2 en priority applications 3. Dong p and wang j customer satisfaction evaluation model of ecommerce website based on tensor analysis proceedings of the 8th international conference on ebusiness, management and economics, 610. Abstract pdf 358 kb 2011 odlog nquantics approximation of nd tensors in highdimensional numerical modeling. Siam journal on matrix analysis and applications 32.
These representations are extremely useful as they are independent of the coordinate systems considered. Ica defines a generative model for the observed multivariate data, which is typically given as a large database of samples. Ica by tensorial methods independent component analysis. The ica problem for tensor data is discussed in 44, 55, where it is proposed to unmix each of the modes separately by m flattening the data tensor and subjecting the matrix of m mode. Perturbation theory and optimality conditions for the best multilinear rank approximation of a tensor. Source separation and machine learning sciencedirect. Enhancing network visibility and security through tensor analysis.
There are many other tensor decompositions, including. The expression of its component is independent of the choice of the coordinate system. A semiintuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higherorder vector products. As we shall show in section iv, the indeterminacies of matrix decompositions have required additional assumptions in order to obtain well defined models tuned to the data. Introduction to vectors and tensors vector and tensor analysis volume 2 ray m. This chapter discusses the four prewhitening based algebraic algorithms for independent component analysis ica. Because of these useful properties, we can use tensors to represent various fundamental laws occurring in physics, engineering, science and mathematics.
Given the ops background and interest in physicsengineering, i doubt most of these suggestions are good choices. Decompositions of a higherorder tensor in block terms. Tensorial independent component analysisbased feature extraction for polarimetric sar data classification. Part of the lecture notes in computer science book series lncs, volume 4666. Extending independent component analysis to tensors has also attracted some attention but, to our knowledge, no model based treatise has been given. Pdf independent component analysis ica is a widelyused blind source separation. Multilinear independent components analysis mit alumni. On a related note, walds general relativity is a great book that has a couple good chapters on tensor calculus in the beginning. To improve the results, tfidf term frequencyinverse document frequency has been widely proposed as termweighting schemes to enhance the eectiveness of words. Independent component analysis final version of 7 march 2001.
Independent component analysis ica is a statistical and computational technique for revealing hidden factors that underlie sets of random variables, measurements, or signals. The central principle of tensor analysis lies in the simple, almost trivial fact that scalars are unaffected by. Independent component analysis ica is a method for finding underlying factors or. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space.
This chapter provides a brief introduction to individual model based paradigms for source separation, including independent component analysis ica, nonnegative matrix factorization nmf, nonnegative tensor factorization ntf, deep neural network dnn, and recurrent neural network rnn. Unsupervised contentbased identification of fake news articles with tensor decomposition ensembles seyedmehdi hosseinimotlagh university of california riverside. The reader must be prepared to do some mathematics and to think. In that case, the tensor can be embedded into a symmetric matrix. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and. By modifying the contrast function of jade algorithm for independent component analysis, we propose a new linear dimension reduction method to identify the nongaussian subspace based on the fourthorder cumulant tensor. Sparse component analysis along the straight line passing through the origin and directed by vector an. Independent components analysis ica maximizes the statistical. The prewhitening takes into account the structure of the covariance matrix of the observed data. This concise text offers a straightforward treatment ofthe subject suitable for the student or practicing engineer. Independent component analysis request pdf researchgate. This survey provides an overview of higherorder tensor decompositions, their applications, and available software. Independent components analysis ica maximizes the statistical independence of the representational components of a training image ensemble, but it cannot distinguish between the different factors, or modes, inherent to image formation, including scene structure, illumination, and imaging.
Independent component analysis an overview sciencedirect. But avoid asking for help, clarification, or responding to other answers. A common problem faced by these methods is a minimal gap of the eigenvalues, which may result in an exponential dependence on the number of source signals d. Independent component analysis and signal separation. Tensor, dyadic, and engineering approaches by pei chi chou, vector and tensor analysis with applications by. In mathematics, tensor calculus, tensor analysis, or ricci calculus is an extension of vector calculus to tensor fields tensors that may vary over a manifold, e. A coefficient vector may be extracted from the image response fig. Tensor analysis and fusion of multimodal brain images. Enhancing network visibility and security through tensor. We introduce a nonlinear, multifactor model that generalizes ica. Linear dimension reduction based on the fourthorder cumulant.
Developed by gregorio riccicurbastro and his student tullio levicivita, 1 it was used by albert einstein to develop his theory of general relativity. I really, really love manifolds, tensors, and forms. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. Souloumiac a and jutten c nonnegative independent component analysis algorithm based on 2d. System identification based on mutual information criteria. The definition of a tensor as a linear operator is prevalent in physics. Svd 7 and independent component analysis ica 19 focus on frequencies of terms words in documents. However, it is likely that teachers will wish to generate additional exercises. An introduction to tensors for students of physics and.
This concise text offers a straightforward treatment of the subject suitable for the student or practicing engineer. In this chapter, we describe two tensor based subspace analysis approaches tensor ica and tensor nmf that can be used in many fields like face recognition. Request pdf independent component analysis a comprehensive introduction to ica. The evolution of different solutions is illustrated. Siam journal on matrix analysis and applications 37. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. A method, system, computerreadable medium and data structure are provided for processing image data in connection with image recognition. A physical quantity that can be completely described by a real number.
Signal processing independent component analysis, a new. This book is also suitable for a graduate level university course on ica, which is. Tensor independent component analysis and tensor nonnegative. Its fairly physically motivated while still keeping the math rigorous. This is equivalent to opti mizing a cumulantbased criterion 55, and is then.
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