How 3-MFA data can cause degenerate parafac solutions, among other relationships
Multiway data analysis
Independent component analysis, a new concept?
Signal Processing - Special issue on higher order statistics
Tensor-based techniques for the blind separation of DS-CDMA signals
Signal Processing
SIAM Journal on Matrix Analysis and Applications
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Unsupervised Multiway Data Analysis: A Literature Survey
IEEE Transactions on Knowledge and Data Engineering
A comparison of algorithms for fitting the PARAFAC model
Computational Statistics & Data Analysis
SIAM Journal on Matrix Analysis and Applications
Tensor Decompositions and Applications
SIAM Review
Blind PARAFAC receivers for DS-CDMA systems
IEEE Transactions on Signal Processing
Fourth-Order Cumulant-Based Blind Identification of Underdetermined Mixtures
IEEE Transactions on Signal Processing
Parallel factor analysis in sensor array processing
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Cramer-Rao lower bounds for low-rank decomposition ofmultidimensional arrays
IEEE Transactions on Signal Processing
SIAM Journal on Matrix Analysis and Applications
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We study uniqueness of the decomposition of an $n$th order tensor (also called $n$-way array) into a sum of $R$ rank-1 terms (where each term is the outer product of $n$ vectors). This decomposition is also known as Parafac or Candecomp, and a general uniqueness condition for $n=3$ has been obtained by Kruskal in 1977 [Linear Algebra Appl., 18 (1977), pp. 95-138]. More recently, Kruskal's uniqueness condition has been generalized to $n\geq3$, and less restrictive uniqueness conditions have been obtained for the case where the vectors of the rank-1 terms are linearly independent in (at least) one of the $n$ modes. For this case, only $n=3$ and $n=4$ have been studied. We generalize these results by providing a framework of analysis for arbitrary $n\geq3$. Our results include necessary, sufficient, necessary and sufficient, and generic uniqueness conditions. For the sufficient uniqueness conditions, the rank of a matrix needs to be checked. The generic uniqueness conditions have the form of a bound on $R$ in terms of the dimensions of the tensor to be decomposed.