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 criterion value to its infimum will exhibit the features of a so-called "degeneracy". That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.
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The CANDECOMP algorithm for the PARAFAC analysis of n × m × p three-way arrays is adapted to handle arrays in which n > rnp more efficiently. For such arrays, the adapted algorithm needs less memory space to store the data during the iterations, and uses less com- putation time than the original CANDECOMP algorithm. The size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units (n) in the data.
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