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, computer graphics, and psychometrics.
Notation A scalar variable is denoted by lower case italic letters, a {\displaystyle a} and an upper bound scalar is denoted by an upper case italic letter, A {\displaystyle A} . Indices are denoted by a combination of lowercase and upper case italic letters, 1 ≤ i ≤ I {\displaystyle 1\leq i\leq I} . Multiple indices that one might encounter when referring to the multiple modes of a tensor are conveniently denoted by 1 ≤ i m ≤ I m {\displaystyle 1\leq i_{m}\leq I_{m}} where 1 ≤ m ≤ M {\displaystyle 1\leq m\leq M} . A vector is denoted by a lower case bold roman, a {\displaystyle \mathbf {a} } and a matrix is denoted by bold upper case letters A {\displaystyle \mathbf {A} } . A higher order tensor is denoted by calligraphic letters, A {\displaystyle {\mathcal {A}}} . An element of an M {\displaystyle M} -order tensor A ∈ C I 1 × I 2 × … I m × … I M {\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \dots I_{m}\times \dots I_{M}}} is denoted by a i 1 , i 2 , … , i m , … i M {\displaystyle a_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}} or A i 1 , i 2 , … , i m , … i M {\displaystyle {\mathcal {A}}_{i_{1},i_{2},\dots ,i_{m},\dots i_{M}}} .
Definition A data tensor A ∈ F I 0 × I 1 × … × I C {\displaystyle {\mathcal {A}}\in {\mathbb {F} }^{I_{0}\times I_{1}\times \ldots \times I_{C}}} 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 R {\displaystyle R} as a linear combination of R {\displaystyle R} rank-1 tensors:
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