A closed-form solution for Parallel Factor (PARAFAC) Analysis

Parallel factor analysis (PARAFAC) is a branch of multi-way signal processing that has received increased attention recently. This is due to the large class of applications as well as the milestone identifiability results demonstrating the superiority to matrix (two-way) analysis approaches. A significant amount of research was dedicated to iterative methods to estimate the factors from noisy data. In many situations these require many iterations and are not guaranteed to converge to the global optimum. Therefore, suboptimal closed-form solutions were proposed as initializations. In this contribution we derive a closed-form solution to completely replace the iterative approach by transforming PARAFAC into several joint diagonalization problems. Thereby, we obtain several estimates for each of the factors and present a new "best matching" scheme to select the best estimate for each factor. In contrast to the techniques known from the literature, our closed-form solution can efficiently exploit symmetric as well as Hermitian symmetric models and solve the underdetermined case, if there are at least two modes that are non-degenerate and full rank. This closed-form solution achieves approximately the same performance as previously proposed iterative solutions and even outperforms them in critical scenarios.