Tensor clustering on outerproduct of coefficient and component matrices of independent component analysis for reliable functional magnetic resonance imaging data decomposition author links open overlay panel guoqiang hu a f 1 qing zhang b 1 abigail b. 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. It is mathematicalsorrybut it gives the barebones definitions that are needed to do differential geometry. This concise text offers a straightforward treatment ofthe subject suitable for the student or practicing engineer.
If the op has a decent math background, i would suggest abraham, marsden, and ratius manifolds, tensor analysis, and applications. 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. This approach generalizes existing cumulant based independent vector analysis iva. I n represents the size of the tensor and i d is the size of its dth dimension. 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. Independent component analysis for tensorvalued data. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity.
Section iv deals with independent component analysis ica. Fundamentals of tensor analysis concepts of scalar, vector, and tensor scalar. Tensor invariance tensors are independent of specific reference frames, i. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler. 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. The tensor is a multidimensional array, which can be seen as an extension of the matrix in high. Tensorial independent component analysisbased feature. Independent component analysis ica is to estimate the independent components st from xt. On joint diagonalization of cumulant matrices for independent. An independent component analysis based approach has been developed to estimate the orientations of two or three crossing fibers in a voxel to conduct human brain streamline tractography from. Contrasted with the infinitesimal calculus, tensor calculus allows. Joint analysis of multiple datasets by crosscumulant tensor. For different approaches we make the connection with cpd.
A semiintuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higherorder vector products. Independent component analysis final version of 7 march 2001. Ica by tensorial methods independent component analysis. Multilinear tensor ica and dimensionality reduction.
In this paper we discuss a multilinear generalization of the best rankr approximation problem for matrices, namely, the approximation of a given higherorder tensor, in an optimal leastsquares sense, by a tensor that has prespecified column rank value, row rank value, etc. Tensor analysis is an essential tool in any science e. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. Tensor is the mathematical tool used to express these physical quantities. What are the best books to learn tensors from for the.
We show that the general tensor pca problem is reducible to its special case where the tensor in question is supersymmetric with an even degree. An ndimensional vector eld is described by a onetoone correspondence between nnumbers and a point. Tensor, dyadic, and engineering approaches by pei chi chou, vector and tensor analysis with applications by. We assume that s1, s2, s3 are statistically independent of each other. The central principle of tensor analysis lies in the simple, almost trivial fact that scalars are unaffected by. Yu, nonlinear and nongaussian dynamic batch process monitoring using a new multiway kernel independent component analysis and multidimensional mutual information based dissimilarity approach, ind.
Mpca is employed in the analysis of nway arrays, i. Mathematically, an nthorder tensor denoted with a calligraphic letter is represented as x. Using matrix and tensor factorizations for the singletrial. Thus, we use intraclass similarity to compare tensor, spatial, and. The definition of a tensor as a linear operator is prevalent in physics. Source separation and machine learning sciencedirect. Tensor independent component analysis and tensor nonnegative factorization. In this paper, we propose approximate diagonalization of a crosscumulant tensor as a means to achieve independent component analysis ica in several linked datasets. 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. Enhancing network visibility and security through tensor. Independent component analysis ica has lately raised great interest in numerous applications including telecommunications, audio, or biomedical engineering. Using matrix and tensor factorizations for the single. A few years ago, i vowed that i would answer this question when i figured out what a tensor really was, because i also had a problem with finding an intuitively satisfying answer online, and the answers here also didnt fully satisfy me. The reader must be prepared to do some mathematics and to think.
Consider a sample data log of network tra c messages. An introduction for mathematicians and physicists by paul renteln. Directional independent component analysis with tensor. This leads to the loss of the spatial information of the polsar data.
This website uses cookies to ensure you get the best experience on our website. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. 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. I do not personally know the following book, but try tensor analysis for physicists by j. Independent component analysisbased multifiber streamline. Independent component analysis linkedin slideshare.
Tensor principal component analysis via convex optimization. Common fe methods, such as principal component analysis pca, independent component analysis ica, etc. Introduction to tensor calculus and continuum mechanics. Invariance qualifies tensors to describe physical processes independent of the coordinate system. We introduce a nonlinear, multifactor model that generalizes ica. A short introduction to tensorbased methods for factor. For each message, let us assume that the log records. The recently developed technique of independent component analysis, or ica, can be used to estimate the a ij based on the informationof their independence, which allows us to separate the two original source signals s 1 t and s 2 t from their mixtures x 1 t and x 2 t. This concise text offers a straightforward treatment of the subject suitable for the student or practicing engineer. Joint analysis of multiple datasets by crosscumulant. Feng1 1physics department, the university of texas at austin dated.
Brief on tensor analysis nice and simple, but does not address curved manifolds. Also worth cosidering are general relativity texts since they need to keep thingsmore grounded. Enhancing network visibility and security through tensor analysis. As a tensor is a generalization of a scalar a pure number. 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. Generalizing the matrix singular value decomposition svd, the. We have seen that the factors extracted by a pca are decorrelated, but not independent. In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space typically a euclidean space or manifold. Pdf independent component analysisbased multifiber. Thirdorder tensor based analysis of multilead ecg for. Given the ops background and interest in physicsengineering, i doubt most of these suggestions are good choices. Principal component analysis pca assumes orthogonality of the modules, independent component analysis ica assumes statistical independence of the modules, factor analysis fa assumes a specific latent variables model, and nonnegative matrix factorization nmf assumes nonnegativity of both modules and activation coefficients. 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.
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. One of the key principles of general relativity is that if physical laws are expressed in tensor form, then they are independent of local coordinate systems, and valid everywhere. 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. The evolution of different solutions is illustrated. Tensorbased anomaly detection knowledgebased systems. 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. Since we classify them based on components we will rede ne them based on componets. Multiple factors related to scene structure, illumination, and imaging contribute to image formation. Independent components analysis ica maximizes the statistical independence of the representational components of a training image ensemble, but it cannot distinguish between these different factors, or. An independent component analysisbased approach has been developed to estimate the orientations of two or three crossing fibers in a voxel to conduct human brain streamline tractography from.
Waters c d huanjie li a chi zhang a jianlin wu b fengyu cong a e lisa d. Around 2000, it was realized that the tkd represents a multilinear singular value decomposition mlsvd 15. The results showed that the tensorbased method performs better than the existing. In that case, the tensor can be embedded into a symmetric matrix. More precisely, the tensor components change according to the transformation. Decompositions of a higherorder tensor in block terms. Multilinear principal component analysis of tensor. Index terms multilinear principal component analysis mpca, tensor objects, dimensionality reduction, feature extraction, gait recognition. While i q can be calculated from tensor clustering or clustering based on the component or coefficient matrices, e. I really, really love manifolds, tensors, and forms. Independent component analysis ica goal given a kdimensional r. What are the best books to learn tensors from for the first. Thanks for contributing an answer to mathematics stack exchange.
Approaches to twoway matrix component analysis are well established, and include principal component analysis pca, independent component analysis ica, nonnega. 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. An introduction to tensors for students of physics and. Applications of tensor analysis dover books on mathematics a. 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. This approach generalizes existing cumulantbased independent vector analysis iva. But avoid asking for help, clarification, or responding to other answers. With careful implementation, tensorbased methods can run efficiently in practice, and in many cases they are the only algorithms with provable guarantees on running time and sample complexity. Tensors, differential forms, and variational principles. Tensor independent component analysis and tensor non. Multilinear principal component analysis wikipedia. Siam journal on matrix analysis and applications 32. Abstract pdf 358 kb 2011 odlog nquantics approximation of nd tensors in highdimensional numerical modeling.
A physical quantity that can be completely described by a real number. Differential geometry classical low dimensional differential geometry of curves and surfaces done with tensors. The standard methods however quickly lose their effectiveness. Independent components analysis ica maximizes the statistical independence of the representational components of a training image ensemble, but it cannot distinguish between these different factors, or modes. It leads to uniqueness, identifiability and resilience to noise that exceed those in the literature, in certain scenarios. A tensor is a multiway array that can store the multidimensional data. Thirdorder tensor based analysis of multilead ecg for classification of myocardial infarction.
For matrices, the solution is conceptually obtained by truncation of the singular value decomposition svd. Tensor independent component analysis and tensor nonnegative. Multilinear principal component analysis mpca is a multilinear extension of principal component analysis pca. Tensor clustering on outerproduct of coefficient and. A basic knowledge of vectors, matrices, and physics is assumed.
Independent component analysis is a standard tool in modern data analysis and numerous different techniques for applying it exist. Introduction to tensor calculus a scalar eld describes a onetoone correspondence between a single scalar number and a point. Hence we can rede ne the scalars and vectors using coordinate transformation of components. The expression of its component is independent of the choice of the coordinate system. 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. Vector a a physical quantity that has both direction and. A secondorder tensor t is defined as a bilinear function from two copies of a vector space v into the space of real numbers.
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