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Tensor Decomposition
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generalize some matrix decompositions. Tensors are generalizations of matrices to higher dimensions and can consequently be treated as multidimensional fields. The main tensor decompositions are: * Tensor rank decomposition; * Higher-order singular value decomposition; * Tucker decomposition; * matrix product states, and operators or tensor trains; * Online Tensor Decompositions * hierarchical Tucker decomposition A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important ...; * block term decomposition Preliminary Definitions and Notation This section introduces basic notations a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p''-vectors and multivectors with Grassmann algebras. Origin In a vector space of dimension ''n'', normally only vectors are used. However, according to Hermann Grassmann and others, this presumption misses the complexity of considering the structures of pairs, triplets, and general multi-vectors. With several combinatorial possibilities, the space of multi-vectors has 2''n'' dimensions. The abstract formulation of the determinant is the most immediate application. Multilinear algebra also has applications in the mechanical study of material response to stress and strain with various moduli of elasticity. This practical reference led to the use of the word tensor, to describe the elements of the multilinear space. The extra structure in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A \mathbf = \mathbf, the matrix ''A'' can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix ''L'' and an upper triangular matrix ''U''. The systems L(U \mathbf) = \mathbf and U \mathbf = L^ \mathbf require fewer additions and multiplications to solve, compared with the original system A \mathbf = \mathbf, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses ''A'' as ''QR'' with ''Q'' an orthogonal matrix and ''R'' an upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Rank Decomposition
In multilinear algebra, the tensor rank decomposition or the rank-R decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum R rank-1 tensors. This is an open problem. Canonical polyadic decomposition (CPD) is a variant of the rank decomposition which computes the best fitting K rank-1 terms for a user specified K. The CP decomposition has found some applications in linguistics and chemometrics. The CP rank was introduced by Frank Lauren Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). Another popular generalization of the matrix SVD known as the higher-order singular value decomposition computes orthonormal mode matrices and has found applications in econometrics, signal processing, computer vision, computer graphics, psychometrics. Notation A scalar variable is denoted by lower case italic letters, a and an upper bound scalar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 L. De Lathauwer ''et al.'' in their Multilinear SVD work that employs the power method, and advocated by Vasilescu and Terzopoulos that developed M-mode SVD. The term HOSVD was coined by Lieven DeLathauwer, but the algorithm referred to commonly in the literature as the HOSVD and attributed to either Tucker or DeLathauwer was developed by Vasilescu and Terzopoulos.M. A. O. Vasilescu, D. Terzopoulos (2002) with the name M-mode SVD. It is a particul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tucker Decomposition
In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD). It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Product State
Matrix product state (MPS) is a quantum state of many particles (in N sites), written in the following form: : , \Psi\rangle = \sum_ \operatorname\left _1^ A_2^ \cdots A_N^\right, s_1 s_2 \ldots s_N\rangle, where A_i^ are complex, square matrices of order \chi (this dimension is called local dimension). Indices s_i go over states in the computational basis. For qubits, it is s_i\in \. For qudits (d-level systems), it is s_i\in \. It is particularly useful for dealing with ground states of one-dimensional quantum spin models (e.g. Heisenberg model (quantum)). The parameter \chi is related to the entanglement between particles. In particular, if the state is a product state (i.e. not entangled at all), it can be described as a matrix product state with \chi = 1. For states that are translationally symmetric, we can choose: : A_1^ = A_2^ = \cdots = A_N^ \equiv A^. In general, every state can be written in the MPS form (with \chi growing exponentially with the particle number ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Online Tensor Decompositions
In computer technology and telecommunications, online indicates a state of connectivity and offline indicates a disconnected state. In modern terminology, this usually refers to an Internet connection, but (especially when expressed "on line" or "on the line") could refer to any piece of equipment or functional unit that is connected to a larger system. Being online means that the equipment or subsystem is connected, or that it is ready for use. "Online" has come to describe activities performed on and data available on the Internet, for example: "online identity", "online predator", "online gambling", "online game", "online shopping", "online banking", and "online learning". Similar meaning is also given by the prefixes "cyber" and "e", as in the words "cyberspace", "cybercrime", "email", and "ecommerce". In contrast, "offline" can refer to either computing activities performed while disconnected from the Internet, or alternatives to Internet activities (such as shopping in bri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hierarchical Tucker Decomposition
A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as architecture, philosophy, design, mathematics, computer science, organizational theory, systems theory, systematic biology, and the social sciences (especially political philosophy). A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that are not linked vertically to one a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Term Decomposition
Block or blocked may refer to: Arts, entertainment and media Broadcasting * Block programming, the result of a programming strategy in broadcasting * W242BX, a radio station licensed to Greenville, South Carolina, United States known as ''96.3 the Block '' * WFNZ-FM, a radio station licensed to Harrisburg, North Carolina, United States, branded as ''92.7 The Block'' * Blocked (''The Flash''), an episode of the television series ''The Flash'' Music * Block Entertainment, a record label * Blocks Recording Club, a record label * Woodblock (instrument), a small piece of slit drum made from one piece of wood and used as a percussion instrument * "Blocks", by C418 from '' Minecraft - Volume Beta'', 2013 Toys * Toy block, one of a set of wooden or plastic pieces, of various shapes * Unit block, a type of standardized wooden toy block for children Video game * Blocked (video game), a puzzle game for the iPhone and iPod Touch Building and construction * Breeze block, cinder block o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
