A Modified Particle Swarm Optimization Algorithm for the Best Low Multilinear Rank Approximation of Higher-Order Tensors

Abstract. The multilinear rank of a tensor is one of the possible gener-alizations for the concept of matrix rank. In this paper, we are interested in finding the best low multilinear rank approximation of a given ten-sor. This problem has been formulated as an optimization problem over the Grassmann manifold [14] and it has been shown that the objec-tive function presents multiple minima [15]. In order to investigate the landscape of this cost function, we propose an adaptation of the Parti-cle Swarm Optimization algorithm (PSO). The Guaranteed Convergence PSO, proposed by van den Bergh in [23], is modified, including a gradi-ent component, so as to search for optimal solutions over the Grassmann manifold. The operations involved in the PSO algorithm are redefined using concepts of differential geometry. We present some preliminary nu-merical experiments and we discuss the ability of the proposed method to address the multimodal aspects of the studied problem

Three-Mode Factor Analysis by Means of Candecomp/Parafac

<p>A three-mode covariance matrix contains covariances of N observations (e.g., subject scores) on J variables for K different occasions or conditions. We model such an JK×JK covariance matrix as the sum of a (common) covariance matrix having Candecomp/Parafac form, and a diagonal matrix of unique variances. The Candecomp/Parafac form is a generalization of the two-mode case under the assumption of parallel factors. We estimate the unique variances by Minimum Rank Factor Analysis. The factors can be chosen oblique or orthogonal. Our approach yields a model that is easy to estimate and easy to interpret. Moreover, the unique variances, the factor covariance matrix, and the communalities are guaranteed to be proper, a percentage of explained common variance can be obtained for each variable-condition combination, and the estimated model is rotationally unique under mild conditions. We apply our model to several datasets in the literature, and demonstrate our estimation procedure in a simulation study. © 2013 The Psychometric Society.</p>