Using the simultaneous generalized Schur decomposition as a Candecomp/Parafac algorithm for ill-conditioned data

摘要

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of outer product 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 outer product arrays become highly correlated in all three modes and some elements of the outer product arrays become very large in magnitude. It is known that for I x J x 2 arrays, fitting a simultaneous generalized Schur decomposition (SGSD) avoids the problems of 'degeneracy' due to the non-existence of an optimal CP solution. In this paper, we consider the application of the SGSD method also for other array formats, when the array has a best fitting CP decomposition with ill-conditioned component matrices, in particular such that it resembles the pattern of a 'degeneracy'. For these cases, we compare the performance of two SGSD algorithms and the alternating least squares (ALS) CP algorithm in a series of numerical experiments.