Low-Rank Approximation of Generic $p \timesq \times2$ Arrays and Diverging Components in the Candecomp/Parafac Model

Abstract

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Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

“…It has been proven that in some cases cost function (18) (or any other measure of the difference between and its approximation) only has an infimum, and not a minimum [23], [41], [51], [60], [61]. However, this did not seem to pose major problems in our simulations.…”

On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

“…Nonexistence of a best rank-2 approximation of Z ∈ R I×J×K holds on a set of positive volume [14]. Nonexistence of a best rank-R approximation holds on a set of positive volume or even almost everywhere for certain classes of Z ∈ R I×J×2 [23,24]. Note that a best rank-1 approximation always exists, since S 1 (I, J, K) is closed [14].…”

“…For S R (I, J, 2) and R ≤ min(I, J) this can be done via the Generalized Schur Decomposition (GSD) [25,26]. Results on the existence of best rank-R approximations for generic Z ∈ R I×J×2 can be found in [23,24]. For S 2 (I, J, K) the boundary points are described in [14] and an algorithm is developed in [27] by means of finding a best rank-(2,2,2) approximation with several zero restrictions on the 2 × 2 × 2 core tensor.…”

A Brief Survey of Asymmetric MDS and Some Open Problems

“…As reviewed extensively in Stegeman and Comon (2010), there has already been an extensive literature on this problem and related topics, and several psychometricians have partly contributed to them (e.g., Krijnen, Dijkstra, & Stegeman, 2008;Kruskal, 1989;Stegeman, 2006Stegeman, , 2007Stegeman, , 2008Stegeman & Lathauwer, 2009;Ten Berge & Kiers, 1999;Ten Berge, Kiers, & De Leeuw, 1988;Ten Berge, Sidiropoulos, & Rocci, 2004).…”