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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 (r) ij k = a (r) i b (r) j c (r) k. These vectors make up the R columns of the… (More)

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP… (More)

Is has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Can-decomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank-1 approximations. The reason… (More)

- Alwin Stegeman
- 2007

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and… (More)