Tensor rank decomposition

In multilinear algebra, the tensor rank decomposition [1] 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.[clarification needed]

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[2] and later rediscovered several times, notably in psychometrics.[3][4] The CP decomposition is referred to as CANDECOMP,[3] PARAFAC,[4] or CANDECOMP/PARAFAC (CP). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition.[5]

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.

  1. ^ Papalexakis, Evangelos. "Automatic Unsupervised Tensor Mining with Quality Assessment" (PDF).
  2. ^ F. L. Hitchcock (1927). "The expression of a tensor or a polyadic as a sum of products". Journal of Mathematics and Physics. 6 (1–4): 164–189. doi:10.1002/sapm192761164.
  3. ^ a b Carroll, J. D.; Chang, J. (1970). "Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart–Young' decomposition". Psychometrika. 35 (3): 283–319. doi:10.1007/BF02310791. S2CID 50364581.
  4. ^ a b Harshman, Richard A. (1970). "Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis" (PDF). UCLA Working Papers in Phonetics. 16: 84. No. 10,085. Archived from the original (PDF) on October 10, 2004.
  5. ^ Gujral, Ekta. "Aptera: Automatic PARAFAC2 Tensor Analysis" (PDF). ASONAM 2022.