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Higher-order Singular Value Decomposition
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing. Some aspects can be traced as far back as F. L. Hitchcock in 1928, but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, further advocated by Lieven De Lathauwer, L. De Lathauwer ''et al.'' , or advocated by Vasilescu and Terzopoulos. Although the term HOSVD was coined by De Lathauwer, the algorithm most commonly referred to as the Tucker or Higher-Order Singular Value Decomposition (HOSVD) in the literature was originally introduced by Vasilescu and Terzopoulos under the name M-mode SVD.M. A. O. Vasilescu, D. Terzopoulos (2002), "Multilinear Analysis of Image Ensembles: TensorFace ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concepts such as Matrix (mathematics), matrices, tensors, multivectors, System of linear equations, systems of linear equations, Higher-dimensional space, higher-dimensional spaces, Determinant, determinants, inner product, inner and outer product, outer products, and Dual space, dual spaces. It is a mathematical tool used in engineering, machine learning, physics, and mathematics. Origin While many theoretical concepts and applications involve Vector space, single vectors, mathematicians such as Hermann Grassmann considered structures involving pairs, triplets, and multivectors that generalize Vector (mathematics and physics), vectors. With multiple combinational possibilities, the space of multivectors expands to 2''n'' dimensions, where ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition#Matrix polar decomposition, polar decomposition. Specifically, the singular value decomposition of an m \times n complex matrix is a factorization of the form \mathbf = \mathbf, where is an complex unitary matrix, \mathbf \Sigma is an m \times n rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \mathbf V^* is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrix, orthogonal matrices; in such contexts, the SVD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics that maintain their properties even if the underlying distributional assumptions are incorrect. Robust Statistics, statistical methods have been developed for many common problems, such as estimating location parameter, location, scale parameter, scale, and regression coefficient, regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a Parametric statistics, parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TP Model Transformation In Control Theories
TP may refer to: Arts and entertainment Music * Test pressing, of a vinyl record * Tonic parallel (Tp and tP), in music theory * ''TP'' (Teddy Pendergrass album), 1980 * ''TP'' (Tony Parker album), 2007 * Tonus Peregrinus (vocal ensemble), a British group * Either of two R&B albums by R. Kelly: ** '' TP-2.com'', 2000 ** '' TP.3 Reloaded'', 2005 Other media * '' The Legend of Zelda: Twilight Princess'', a Nintendo video game * Test pattern or test card, a broadcast television signal * '' Tahanang Pinakamasaya'', a Filipino variety show * '' The Times-Picayune , The New Orleans Advocate'', an American daily newspaper * ''The Tomorrow People'', a British science fiction television series Businesses and organizations Technology brands and businesses * TP-Link, a global manufacturer of computer networking products * TP Vision, a subsidiary of TPV Technology, Amsterdam, Netherlands * Telekomunikacja Polska (now ''Orange Polska''), a Polish telecommunications provider * Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TP Model Transformation
In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or neural networks, fuzzy logic, etc.) into TP function form if such a transformation is possible. If an exact transformation is not possible, then the method determines a TP function that is an approximation of the given function. Hence, the TP model transformation can provide a trade-off between approximation accuracy and complexity. A free MATLAB implementation of the TP model transformation can be downloaded a or an old version of the toolbox is available at MATLAB Centra A key underpinning of the transformation is the higher-order singular value decomposition. Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridgin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disease Surveillance
Disease surveillance is an epidemiological practice by which the spread of disease is monitored in order to establish patterns of progression. The main role of disease surveillance is to predict, observe, and minimize the harm caused by outbreak, epidemic, and pandemic situations, as well as increase knowledge about which factors contribute to such circumstances. A key part of modern disease surveillance is the practice of disease case reporting. In modern times, reporting incidences of disease outbreaks has been transformed from manual record keeping, to instant worldwide internet communication. The number of cases could be gathered from hospitals – which would be expected to see most of the occurrences – collated, and eventually made public. With the advent of modern communication technology, this has changed dramatically. Organizations like the World Health Organization (WHO) and the Centers for Disease Control and Prevention (CDC) now can report cases and de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Preliminaries Given a field \ K\ of either real or complex numbers (or any complete subset thereof), let \ K^\ be the -vector space of matrices with m rows and n columns and entries in the field \ K ~. A matrix norm is a norm on \ K^~. Norms are often expressed with double vertical bars (like so: \ \, A\, \ ). Thus, the matrix norm is a function \ \, \cdot\, : K^ \to \R^\ that must satisfy the following properties: For all scalars \ \alpha \in K\ and matrices \ A, B \in K^\ , * \, A\, \ge 0\ (''positive-valued'') * \, A\, = 0 \iff A=0_ (''definite'') * \left\, \alpha\ A \right\, = \left, \alpha \\ \left\, A\right\, \ (''absolutely homogeneous'') * \, A + B \, \le \, A \, + \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mode-k Multiplication
In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication. Abstract definition Let F be a field of characteristic zero, such as \mathbb or \mathbb . Let V_k be a finite-dimensional vector space over F, and let \mathcal \in V_1 \otimes V_2 \otimes \cdots \otimes V_d be an order-d simple tensor, i.e., there exist some vectors \mathbf_k \in V_k such that \mathcal = \mathbf_1 \otimes \mathbf_2 \otimes \cdots \otimes \mathbf_d. If we are given a collection of linear maps A_k : V_k \to W_k, then the multilinear multiplication of \mathcal with (A_1, A_2, \ldots, A_d) is defined as the action on \mathcal of the tensor product of these linear maps, namely \begin A_1 \otimes A_2 \otimes \cdots \otimes A_d : V_1 \otimes V_2 \otimes \cdots \otimes V_d & \to W_1 \otimes W_2 \otimes \cdots \otimes W_d, \\ \mathbf_1 \otimes \mathbf_2 \otimes \cdots \otimes \mathbf_d & \mapsto A_1(\mathbf_1) \otimes A_2(\mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demetri Terzopoulos
Demetri Terzopoulos is a Greek-Canadian-American computer scientist and entrepreneur. He is currently a Distinguished Professor and Chancellor's Professor of Computer Science in the Henry Samueli School of Engineering and Applied Science at the University of California, Los Angeles, where he directs the UCLA Computer Graphics & Vision Laboratory. Education Terzopoulos was educated at McGill University where he was awarded an Honours Bachelor of Engineering degree with Distinction in 1978 and a Master of Engineering degree, advised bSteven W. Zucker in 1980, both in electrical engineering. He went on to study at the Massachusetts Institute of Technology, where he was awarded a PhD degree in Artificial Intelligence in 1984 for computer vision research on the computation of visible-surface representations, advised by Shimon Ullman and J. Michael Brady. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
