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## DOI: 10.1007/S11336-007-9022-3 DEGENERACY IN CANDECOMP/PARAFAC AND INDSCAL EXPLAINED FOR SEVERAL THREE-SLICED ARRAYS WITH A TWO-VALUED TYPICAL RANK (2007)

### Citations

591 |
Individual differences and multidimensional scaling. In:
- Carroll
- 1972
(Show Context)
Citation Context ...as described in Section 1. This completes the proof of (II). □ The Indscal model can be understood as CP for an I × I × K array with symmetric I × I slices, with the additional restriction A = B (see =-=Carroll & Chang, 1970-=-). Since the CP solution (4.6) is essentially unique and features A = B, the CP model is equivalent to the Indscal model in this case. Hence, Theorem 4.2 also explains the occurrence of degenerate seq... |

540 | Foundation of the PARAFAC procedure: model and conditions for an explanatory mutil-mode factor analysis,” - Harshman - 1972 |

310 | Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra and its - Kruskal - 1977 |

190 | Tensor rank and the ill-posedness of the best low-rank approximation problem. - Lim, Silva - 2008 |

72 |
A comparison of algorithms for fitting the PARAFAC model
- Tomasi, Bro
- 2006
(Show Context)
Citation Context ...i.e., y (r) ij k = a (r) i b(r) j c(r) k . For fixed R, the CP decomposition (1.1) is found by minimizing the sum of squares of E. Usually, an iterative algorithm is used for this purpose (see, e.g., =-=Tomasi & Bro, 2006-=-). In this paper we will denote column vectors as x, matrices as X, and three-way arrays as X. We consider the real-valued CP model, i.e., we assume the array X and the component matrices A, B, and C ... |

56 |
Multi-way analysis: applications in the chemical sciences
- Smilde, Bro, et al.
- 2004
(Show Context)
Citation Context ... we assume the array X and the component matrices A, B, and C to be real-valued. The real-valued CP model is used in a majority of applications in psychology and chemistry (see Kroonenberg, 1983; and =-=Smilde, Bro, & Geladi, 2004-=-). Complex-valued applications of CP occur in, e.g., signal processing and telecommunications research (see Sidiropoulos, 2004). The concept of rank is the same for matrices and three-way arrays. The ... |

38 |
Data preprocessing and the extended Parafac model, in Research Methods for Multimode Data Analysis
- Harshman, Lundy
- 1984
(Show Context)
Citation Context ...r sum Y (s) + Y (t) still contributes to a better fit of the CP decomposition. Degenerate sequences of CP solutions can be avoided by imposing orthogonality constraints on the component matrices (see =-=Harshman & Lundy, 1984-=-). Of course, this will result in some loss of fit. Lim (2005) shows that if X is nonnegative and (A, B, C) are required to be nonnegative, then degeneracy does not occur. Analogous to two-factor dege... |

34 | The Multilinear engine-A table-driven, least squares program for solving multilinear problems, including the n-way parallel factor analysis model, - Paatero - 1999 |

32 | Construction and analysis of degenerate PARAFAC models - Paatero |

24 |
Slowly converging PARAFAC sequences: Swamps and two-factor degeneracies
- Mitchell, Burdick
- 1994
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Citation Context ..., is sometimes complicated by the occurrence of so-called degenerate sequences of CP solutions. In such cases, convergence of the CP algorithm is extremely slow (it seems to be caught in a swamp, see =-=Mitchell & Burdick, 1994-=-) and some components of the CP solution become more and more correlated as the CP algorithm runs. DegenerateALWIN STEGEMAN 603 sequences of CP solutions were first reported in Harshman and Lundy (19... |

23 |
Degeneracy in Candecomp/Parafac explained for p× p× 2 arrays of rank p+1 or higher
- Stegeman
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Citation Context ... D are those arrays for which P has four real roots, but not all distinct. Such arrays can be approximated arbitrarily closely from D, but also by arrays for which P has complex roots (see Lemma 2 in =-=Stegeman, 2006-=-). The boundary points of D do not lie in D itself and, hence, D is an open subset of R.Thisproves(iv). □ We consider the CP problem (3.1) where D is given by (4.4). The following theorem states our r... |

22 |
Three-Mode Principal Component Analysis
- Kroonenberg
- 1983
(Show Context)
Citation Context ...-valued CP model, i.e., we assume the array X and the component matrices A, B, and C to be real-valued. The real-valued CP model is used in a majority of applications in psychology and chemistry (see =-=Kroonenberg, 1983-=-; and Smilde, Bro, & Geladi, 2004). Complex-valued applications of CP occur in, e.g., signal processing and telecommunications research (see Sidiropoulos, 2004). The concept of rank is the same for ma... |

22 | Simplicity of core arrays in threeway principal component analysis and the typical rank of p × q × 2 arrays - Berge, Kiers - 1999 |

19 |
How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships
- Kruskal, Harshman, et al.
- 1989
(Show Context)
Citation Context ... of these sign changes being −1. • The magnitudes of the elements of columns s and t in the unrestricted component matrix become arbitrarily large. This pattern is called a two-factor degeneracy (see =-=Kruskal, Harshman, & Lundy, 1989-=-). The contributions of Y (s) and Y (t) diverge in nearly opposite directions. However, their sum Y (s) + Y (t) still contributes to a better fit of the CP decomposition. Degenerate sequences of CP so... |

15 | The typical rank of tall three-way arrays - Berge |

14 |
Typical rank and INDSCAL dimensionality for symmetric three-way arrays of order Ix2x2 or Ix3x3
- Berge, Sidiropoulos, et al.
(Show Context)
Citation Context ...). Case 2 is completely analogous, since the criterion to distinguish p × p × 2 arrays of rank p from those of rank p + 1 does not depend on whether the two p × p slices are symmetric or not (see Ten =-=Berge et al., 2004-=-). Using the tools developed in Stegeman (2006), we will explain the occurrence of degenerate sequences of CP solutions in Cases 3, 4, 5, and 6 of Table 1. It will be shown that the twovalued typical ... |

12 |
Two-Factor Degeneracies and a Stabilization of PARAFAC
- Rayens, Mitchell
- 1997
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Citation Context ... DR which does not belong to DR, then it becomes degenerate. Hence, in this case, modified CP algorithms designed to avoid degenerate solutions, yet still trying to find an optimal CP solution (e.g., =-=Rayens & Mitchell, 1997-=-; Cao, Chen, Mo, & Yu, 2000) are no remedy. This is true for all degeneracies that are not bounded. In this paper we further investigate how a two-valued typical rank is related to the occurrence of d... |

12 | Low-rank decomposition of multi-way arrays: A signal processing perspective
- Sidiropoulos
(Show Context)
Citation Context ...ications in psychology and chemistry (see Kroonenberg, 1983; and Smilde, Bro, & Geladi, 2004). Complex-valued applications of CP occur in, e.g., signal processing and telecommunications research (see =-=Sidiropoulos, 2004-=-). The concept of rank is the same for matrices and three-way arrays. The three-way rank of X is defined as the smallest number of rank-1 arrays whose sum equals X. A three-way array has The author is... |

12 | Partial uniqueness in CANDECOMP/PARAFAC - Berge |

8 | Low-rank approximation of generic p× q × 2 arrays and diverging components in the Candecomp/Parafac model - Stegeman |

6 | Optimal solutions to non-negative parafac/multilinear nmf always exist - Lim |

4 | A PARAFAC algorithm using penalty diagonalization error (PDE) for three-way data array resolution. Analyst 125 - Cao, Chen, et al. - 2000 |

2 | Degenerate solutions obtained from several variants of factor analysis - Zijlstra, Kiers - 2002 |