Shifted and extrapolated power methods for tensor ℓp-eigenpairs

Cipolla, Stefano, Redivo-Zaglia, Michela and Tudisco, Francesco (2020) Shifted and extrapolated power methods for tensor ℓ^p-eigenpairs. Electronic Transactions on Numerical Analysis, 53, 1-27. (doi:10.1553/etna_vol53s1).

Record type: Article

Abstract

This work is concerned with the computation of `^p-eigenvalues and eigenvectors of square tensors with d modes. In the first part we propose two possible shifted variants of the popular (higher-order) power method, and, when the tensor is entry-wise nonnegative with a possibly reducible pattern and p is strictly larger than the number of modes, we prove convergence of both schemes to the Perron `^p-eigenvector and to the maximal corresponding `^p-eigenvalue of the tensor. Then, in the second part, motivated by the slow rate of convergence that the proposed methods achieve for certain real-world tensors when p ≈ d, the number of modes, we introduce an extrapolation framework based on the simplified topological ε-algorithm to efficiently accelerate the shifted power sequences. Numerical results for synthetic and real world problems show the improvements gained by the introduction of the shifting parameter and the efficiency of the acceleration technique.

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Published date: 29 January 2020

Additional Information: Funding Information: The work of M. R.-Z. was partially supported by the University of Padua, Project no. DOR1903575/19. The work of S. C. was partially supported by the GNCS - INdAM project “Efficient Methods for large scale problems with applications to data analysis and preconditioning”. The work F. T. was funded by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie individual fellowship “MAGNET” No 744014. All the authors are members of the INdAM Research group GNCS. Funding Information: Acknowledgements. The work of M. R.-Z. was partially supported by the University of Padua, Project no. DOR1903575/19. The work of S. C. was partially supported by the GNCS – INdAM project “Efficient Methods for large scale problems with applications to data analysis and preconditioning”. The work F. T. was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie individual fellowship “MAGNET” No 744014. All the authors are members of the INdAM Research group GNCS. Publisher Copyright: © 2020 Kent State University. All rights reserved.

Keywords: Extrapolation methods, Shanks transformations, Shifted higher-order power method, Tensors, ε-algorithms, ℓ-eigenvalues