Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs

Stefano  Cipolla; Michela  Redivo-Zaglia; Francesco  Tudisco

doi:doi:10.1553/etna_vol53s1

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs

Stefano Cipolla,

Michela Redivo-Zaglia,

Francesco Tudisco

ETNA - Electronic Transactions on Numerical Analysis, pp. 1-27, 2020/01/29

doi: 10.1553/etna_vol53s1

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doi:10.1553/etna_vol53s1

Abstract

This work is concerned with the computation of $\\ell^p$-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-wisenonnegative with a possibly reducible pattern and $p$ is strictly larger than the number of modes, we prove convergence of both schemes to thePerron $\\ell^p$-eigenvector and to the maximal corresponding $\\ell^p$-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$p\\approx d$, the number of modes, we introduce an extrapolation framework based on the simplified topological $\varepsilon$-algorithm to efficientlyaccelerate 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.

Keywords: $\ell^p$-eigenvalues, tensors, shifted higher-order power method, extrapolation methods, Shanks transformations, ε-algorithms

ETNA

Published Online: 2020/01/29 08:40:43

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs

Abstract