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Concept
Tensor rank decomposition
Summary
In multilinear algebra, the tensor rank decomposition or the decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum tensors. This is an open problem.
Canonical polyadic decomposition (CPD) is a variant of the rank decomposition which computes the best fitting terms for a user specified . 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). PARAFAC2 rank decomposition is yet to explore.
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.
A scalar variable is denoted by lower case italic letters, and an upper bound scalar is denoted by an upper case italic letter, .
Indices are denoted by a combination of lowercase and upper case italic letters, . Multiple indices that one might encounter when referring to the multiple modes of a tensor are conveniently denoted by where .
A vector is denoted by a lower case bold Times Roman, and a matrix is denoted by bold upper case letters .
A higher order tensor is denoted by calligraphic letters,. An element of an -order tensor is denoted by or .
A data tensor is a collection of multivariate observations organized into a M-way array where M=C+1. Every tensor may be represented with a suitably large as a linear combination of rank-1 tensors:
where and where . When the number of terms is minimal in the above expression, then is called the rank of the tensor, and the decomposition is often referred to as a (tensor) rank decomposition, minimal CP decomposition, or Canonical Polyadic Decomposition (CPD). If the number of terms is not minimal, then the above decomposition is often referred to as CANDECOMP/PARAFAC, Polyadic decomposition'.
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