Kruskal's Permutation Lemma and the Identification of CANDECOMP/PARAFAC and Bilinear Models with Constant Modulus Constraints
- 16 August 2004
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Signal Processing
- Vol. 52 (9), 2625-2636
- https://doi.org/10.1109/tsp.2004.832022
Abstract
CANDECOMP/PARAFAC (CP) analysis is an extension of low-rank matrix decomposition to higher-way arrays, which are also referred to as tensors. CP extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angle-carrier estimation to blind multiuser detection. The uniqueness of CP decomposition is not fully understood yet, despite its theoretical and practical significance. Toward this end, we first revisit Kruskal's permutation lemma, which is a cornerstone result in the area, using an accessible basic linear algebra and induction approach. The new proof highlights the nature and limits of the identification process. We then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank. These new conditions explain a curious example provided recently in a previous paper by Sidiropoulos, who showed that Kruskal's condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the (possibly pretransformed) component matrices. As another interesting application of the permutation lemma, we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus (CM) constraints, thus providing an interesting link to (and unification with) CP.Keywords
This publication has 13 references indexed in Scilit:
- Partial uniqueness in CANDECOMP/PARAFACJournal of Chemometrics, 2004
- Blind identification of out-of-cell users in DS-CDMA: An algebraic approachPublished by Institute of Electrical and Electronics Engineers (IEEE) ,2002
- Cramer-Rao lower bounds for low-rank decomposition of multidimensional arraysIEEE Transactions on Signal Processing, 2001
- Asymptotic properties of the algebraic constant modulus algorithmIEEE Transactions on Signal Processing, 2001
- Parallel factor analysis in sensor array processingIEEE Transactions on Signal Processing, 2000
- On the uniqueness of multilinear decomposition ofN-way arraysJournal of Chemometrics, 2000
- Blind PARAFAC receivers for DS-CDMA systemsIEEE Transactions on Signal Processing, 2000
- Blind separation of synchronous co-channel digital signals using an antenna array. I. AlgorithmsIEEE Transactions on Signal Processing, 1996
- Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statisticsLinear Algebra and its Applications, 1977
- Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decompositionPsychometrika, 1970