The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Optimal Linear Representations of Images for Object Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Mesh Adaptive Direct Search Algorithms for Constrained Optimization
SIAM Journal on Optimization
Fundamentals of Computational Swarm Intelligence
Fundamentals of Computational Swarm Intelligence
Multiway analysis of epilepsy tensors
Bioinformatics
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
A sequential niching technique for particle swarm optimization
ICIC'05 Proceedings of the 2005 international conference on Advances in Intelligent Computing - Volume Part I
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The multilinear rank of a tensor is one of the possible generalizations for the concept of matrix rank. In this paper, we are interested in finding the best low multilinear rank approximation of a given tensor. This problem has been formulated as an optimization problem over the Grassmann manifold [14] and it has been shown that the objective function presents multiple minima [15]. In order to investigate the landscape of this cost function, we propose an adaptation of the Particle Swarm Optimization algorithm (PSO). The Guaranteed Convergence PSO, proposed by van den Bergh in [23], is modified, including a gradient 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 numerical experiments and we discuss the ability of the proposed method to address the multimodal aspects of the studied problem.