Tensor rank decomposition

In multilinear algebra, the tensor rank decomposition or the ${\displaystyle rank-R}$ decomposition of a tensor is the decomposition of a tensor in terms of a sum of minimum ${\displaystyle R}$ ${\displaystyle rank-1}$ tensors. This is an open problem.

Canonical polyadic decomposition (CPD) is a variant of the rank decomposition which computes the best fitting ${\displaystyle K}$ ${\displaystyle rank-1}$ terms for a user specified ${\displaystyle K}$. 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). 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, psychometrics.