## Kruskal’s permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints

Venue: | IEEE Trans. Signal Process |

Citations: | 29 - 6 self |

### BibTeX

@ARTICLE{Jiang_kruskal’spermutation,

author = {Tao Jiang and Student Member and Nicholas D. Sidiropoulos and Senior Member},

title = {Kruskal’s permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints},

journal = {IEEE Trans. Signal Process},

year = {},

pages = {2625--2636}

}

### Years of Citing Articles

### OpenURL

### Abstract

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. Index Terms—CANDECOMP, constant modulus, identifiablity, PARAFAC, SVD, three-way array analysis, uniqueness. I.

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(Show Context)
Citation Context ...tified from up to column permutation and scaling. The model in (3) was independently introduced in 1970 by two different groups as CANonical DECOMPosition [3] (CANDECOMP) and PARAllel FACtor analysis =-=[4]-=-(PARAFAC), (1) (2) (3) 1053-587X/04$20.00 © 2004 IEEE2626 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 9, SEPTEMBER 2004 respectively. Nowadays, the term CANDECOMP/PARAFAC (CP) is often used ... |

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Citation Context ...es that for all . In particular, for . Since which is nothing but (13) Further simplifying (13), we obtain (14), shown at the bottom of the page. Equation (14) can be written as invoking the identity =-=[1]-=- vec diag ,we have diag diag (15) Each quadruple , , , , gives (14)2630 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 9, SEPTEMBER 2004 rise to an equation as in (15). Each such equation can b... |

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Citation Context ...o known as two-way arrays because they are indexed by two independent variables), low-rank decomposition of three-way arrays (also known as tensors) is unique under certain relatively mild conditions =-=[8]-=-. Let be a rank- three way array of order with typical element , and consider the component trilinear decomposition Let ( ) be the typical element of matrix (resp. , ). It has been shown by Kruskal [8... |

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Citation Context ...ng for communications. Three-way methods are often naturally applicable for the analysis of multidimensional data sets encountered in blind identification [13], multiuser signal separation [6], [11], =-=[14]-=-, and diversity systems [7]. For example, in the context of uplink reception for narrowband cellular DS-CDMA systems with symbol-periodic spreading and a base station antenna array, the received baseb... |

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Citation Context ...ometrics, and, more recently, signal processing for communications. Three-way methods are often naturally applicable for the analysis of multidimensional data sets encountered in blind identification =-=[13]-=-, multiuser signal separation [6], [11], [14], and diversity systems [7]. For example, in the context of uplink reception for narrowband cellular DS-CDMA systems with symbol-periodic spreading and a b... |

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Citation Context ...unications signals often exhibit FA or CM properties. As mentioned earlier, although bilinear decomposition is not unique in general, bilinear decomposition under FA/CM constraints can be unique [9], =-=[15]-=-, [17]. We show that the necessary and sufficient condition for unique bilinear decomposition under CM constraints is strikingly similar to the one for uniqueness of certain CP models. The impact of a... |

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Citation Context ...mponent matrices , , and can be uniquely identified from up to column permutation and scaling. The model in (3) was independently introduced in 1970 by two different groups as CANonical DECOMPosition =-=[3]-=- (CANDECOMP) and PARAllel FACtor analysis [4](PARAFAC), (1) (2) (3) 1053-587X/04$20.00 © 2004 IEEE2626 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 9, SEPTEMBER 2004 respectively. Nowadays, t... |

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(Show Context)
Citation Context ...e worth recounting at this point. One is that neither , nor , nor has a pair of proportional columns [5]. Another is that the Khatri–Rao product of any two component matrices must be full column rank =-=[10]-=-. In hindsight, the proof for uniqueness of CP decomposition can be decoupled into three separate steps. Given , the first step is to show that , the second step is to show that , and the last step is... |

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Citation Context .... Note that (22) or . (21) B. Bilinear Decomposition Under CM Constraints Although bilinear decomposition is not unique in general, bilinear decomposition with CM constraints can be unique [9], [15], =-=[17]-=-. Interestingly, as pointed out next, the identification condition on bilinear decomposition with CM constraints is very similar to Condition B derived herein for the identification of restricted CP m... |

4 |
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Citation Context ...lumn permutation and scaling. Therefore, Kruskal’s Permutation Lemma can again be taken as the cornerstone for uniqueness. Earlier work on the identification of bilinear mixtures under CM constraints =-=[9]-=-, [15] has yielded sufficient conditions, but necessity has been left open to the best of our knowledge. Equipped with Kruskal’s Permutation Lemma, we are ready to give a necessary and sufficient cond... |

3 |
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Citation Context ...ocessing for communications. Three-way methods are often naturally applicable for the analysis of multidimensional data sets encountered in blind identification [13], multiuser signal separation [6], =-=[11]-=-, [14], and diversity systems [7]. For example, in the context of uplink reception for narrowband cellular DS-CDMA systems with symbol-periodic spreading and a base station antenna array, the received... |

1 |
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Citation Context ...hence, the aforementioned attempt is unlikely to succeed. Necessary conditions for CP uniqueness are worth recounting at this point. One is that neither , nor , nor has a pair of proportional columns =-=[5]-=-. Another is that the Khatri–Rao product of any two component matrices must be full column rank [10]. In hindsight, the proof for uniqueness of CP decomposition can be decoupled into three separate st... |

1 |
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(Show Context)
Citation Context ...al processing for communications. Three-way methods are often naturally applicable for the analysis of multidimensional data sets encountered in blind identification [13], multiuser signal separation =-=[6]-=-, [11], [14], and diversity systems [7]. For example, in the context of uplink reception for narrowband cellular DS-CDMA systems with symbol-periodic spreading and a base station antenna array, the re... |

1 |
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Citation Context ...-way methods are often naturally applicable for the analysis of multidimensional data sets encountered in blind identification [13], multiuser signal separation [6], [11], [14], and diversity systems =-=[7]-=-. For example, in the context of uplink reception for narrowband cellular DS-CDMA systems with symbol-periodic spreading and a base station antenna array, the received baseband-equivalent data constit... |