Alwin Stegeman

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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 ijk = a i b (r) j c (r) k . These vectors make up the R columns of the component matrices(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)
A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a(More)
In this paper, three sufficient conditions are derived for the three-way CANDECOMP/PARAFAC (CP) model, which ensure uniqueness in one of the three modes (“uni-modeuniqueness”). Based on these conditions, a partial uniqueness condition is proposed which allows collinear loadings in only one mode. We prove that if there is uniqueness in one mode, then the(More)
We consider the low-rank approximation over the real field of generic p×q×2 arrays. For all possible combinations of p, q, and R, we present conjectures on the existence of a best rank-R approximation. Our conjectures are motivated by a detailed analysis of the boundary of the set of arrays with at most rank R. We link these results to the Candecomp/Parafac(More)
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)
Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew-normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew-normal factor are equivalent to those under a quadratic model up to(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 Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank-1(More)