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

TL;DR

This paper introduces shifted power methods and an extrapolation framework to efficiently compute and accelerate the convergence of $ ext{ell}^p$-eigenpairs of tensors, with proven convergence and improved performance on real data.

Contribution

It proposes two shifted variants of the power method with convergence guarantees and an extrapolation technique to speed up convergence for tensors with large $p$ values.

Findings

Convergence of shifted power methods to Perron $ ext{ell}^p$-eigenvectors.

Acceleration of convergence using topological $ ext{ε}$-algorithm.

Effective performance demonstrated on synthetic and real-world tensors.

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 %for the computation of $\ell^p$-eigenpairs proving the convergence of both the schemes to the Perron $\ell^p$-eigenvector of the tensor, and the maximal corresponding $\ell^p$-eigenvalue, when the tensor is entrywise nonnegative and $p$ is strictly larger than the number of modes. Then, 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, in the second part we introduce an extrapolation framework based on the simplified topological $\varepsilon$-algorithm to efficiently accelerate the shifted power sequences. Numerical results on synthetic and real world problems show the improvements gained by the introduction of the shifting parameter and the efficiency of the acceleration technique.