### Abstract

Let X be a real-valued three-way array. The Candecomp/Parafac (CP) decomposition is written as X = Y^{(1)} + ⋯ + Y^{(R)} + E, where Y^{(r)} are rank-1 arrays and E is a rest term. Each rank-1 array is defined by the outer product of three vectors a^{(r)}, b^{(r)} and c^{(r)}, i.e. y_{ijk}^{(r)} = a_{i}^{(r)} b_{j}^{(r)} c_{k}^{(r)}. These vectors make up the R columns of the component matrices A, B and C. If 2R + 2 is less than or equal to the sum of the k-ranks of A, B and C, then the fitted part of the decomposition is unique up to a change in the order of the rank-1 arrays and rescaling/counterscaling of each triplet of vectors (a^{(r)}, b^{(r)}, c^{(r)}) forming a rank-1 array. This classical result was shown by Kruskal. His proof is, however, rather inaccessible and does not seem intuitive. In order to contribute to a better understanding of CP uniqueness, this paper provides an accessible and intuitive proof of Kruskal's condition. The proof is both self-contained and compact and can easily be adapted for the complex-valued CP decomposition.

Original language | English (US) |
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Pages (from-to) | 540-552 |

Number of pages | 13 |

Journal | Linear Algebra and Its Applications |

Volume | 420 |

Issue number | 2-3 |

DOIs | |

State | Published - Jan 15 2007 |

### Keywords

- Candecomp
- Kruskal-rank condition
- Parafac
- Three-way arrays
- Uniqueness

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## Cite this

*Linear Algebra and Its Applications*,

*420*(2-3), 540-552. https://doi.org/10.1016/j.laa.2006.08.010