A closed-form solution for multilinear PARAFAC decompositions

In this paper we study the R-way Parallel Factor Analysis (also referred to as R-way PARAFAC) problem. This branch of multi-way signal processing has received increased attention recently which is due to the versatility of the model as well as the identifiability results demonstrating its superiority to matrix-only (2-way) approaches. In R-way PARAFAC analysis, the goal is to decompose an R-dimensional tensor into a minimal sum of rank-1 terms. So far, there exist sub-optimal closed-form solutions as well as iterative techniques for finding these decompositions. However, the latter often require many iterations to converge. In this contribution we demonstrate that the R-way PARAFAC decomposition can be reduced to a set of simultaneous matrix diagonalization problems. Exploiting the structure of the R-dimensional problem, we obtain several estimates for each of the factors and present a "best matching" scheme to select the best estimate for each factor. By means of computer simulations we compare our closed-form solution to an iterative technique and demonstrate the enhanced robustness in critical scenarios.