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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 (or rather to higher orders, i.e. the higher number of dimensions) and can consequently be treated as multidimensional fields. The main tensor decompositions are: * Tensor rank decomposition; * 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 application ...; * Tucker decomposition; * matrix product states, and operators or tensor trains; * Online Tensor Decompositions * hierarchical Tucker decomposition; * block term decomposition Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Tensor (machine Learning)
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a array (data type), multidimensional array (''M''-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector space. Observations, such as images, movies, volumes, sounds, and relationships among words and concepts, stored in an ''M''-way array ("data tensor"), may be analyzed either by Neural network (machine learning), artificial neural networks or tensor decomposition, tensor methods. Tensor decomposition factorizes data tensors into smaller tensors. Operations on data tensors can be expressed in terms of matrix multiplication and the Kronecker product. The computation of gradients, a crucial aspect of backpropagation, can be performed using Library (computing), software lib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 example, 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] [Amazon] |
Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with 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; those components form an array, which can be thought of as a high-dimensional matrix. 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, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tensor Rank Decomposition
In multilinear algebra, the tensor rank decomposition or rank-''R'' decomposition is the decomposition of a tensor as a sum of ''R'' rank-1 tensors, where ''R'' is minimal. Computing this decomposition is an open problem. Canonical polyadic decomposition (CPD) is a variant of the tensor rank decomposition, in which the tensor is approximated as a sum of ''K'' rank-1 tensors for a user-specified ''K''. The CP decomposition has found some applications in linguistics and chemometrics. It 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). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition. 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, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
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) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Matrix Product State
A matrix product state (MPS) is a representation of a quantum many-body state. It is at the core of one of the most effective algorithms for solving one dimensional strongly correlated quantum systems – the density matrix renormalization group (DMRG) algorithm. For a system of N spins of dimension d, the general form of the MPS for periodic boundary conditions (PBC) can be written in the following form: , \Psi\rangle = \sum_ \operatorname\left[A_1^ A_2^ \cdots A_N^\right] , s_1 s_2 \ldots s_N\rangle. For open boundary conditions (OBC), , \Psi\rangle takes the form , \Psi\rangle = \sum_ A_1^ A_2^ \cdots A_N^ , s_1 s_2 \ldots s_N\rangle. Here A_i^ are the D_i\times D_ matrices (D is the dimension of the virtual subsystems) and , s_i\rangle are the single-site basis states. For periodic boundary conditions, we consider D_=D_1, and for open boundary conditions D_1=1. The parameter D is related to the Quantum entanglement, entanglement between particles. In particular, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 as "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 and concepts that take place on the Internet, such as online identity, online predator and online shop. A similar meaning is also given by the prefixes cyber and e, as in words ''cyberspace'', ''cybercrime'', ''email'', and ''e-commerce''. In contrast, "offline" can refer to either computing activities performed while disconnected from the Internet, or alternatives to Internet activities (such as shopping in brick-and-mortar stores). The term "offline" is sometimes used interchangeably with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 science). 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 anothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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", 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 games * Blocked (video game), a puzzle game for the iPhone and iPod Touch Building and construction * Concrete block, cinder block or cement bloc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
