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

  • Pierre B. Borckmans
  • Mariya Ishteva
  • Pierre-Antoine Absil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6234)

Abstract

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.

Keywords

Multi-Linear Rank Higher-Order Tensors Particle Swarm Optimization Grassmann Manifold Global Optimization 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Pierre B. Borckmans
    • 1
  • Mariya Ishteva
    • 1
  • Pierre-Antoine Absil
    • 1
  1. 1.Department of Mathematical EngineeringUniversité catholique de LouvainLouvain-la-NeuveBelgium

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