Accelerating the Computation of Tensor $Z$-eigenvalues

TL;DR

This paper presents an extrapolation method to accelerate the shifted symmetric higher order power method for tensor Z-eigenvalue problems, with theoretical analysis and numerical validation showing improved convergence.

Contribution

It introduces a new extrapolation technique with an automated parameter selection for faster tensor Z-eigenvalue computations.

Findings

The method accelerates convergence of the power method.

Theoretical analysis determines optimal extrapolation parameters.

Numerical experiments confirm the predicted acceleration.

Abstract

Efficient solvers for tensor eigenvalue problems are important tools for the analysis of higher-order data sets. Here we introduce, analyze and demonstrate an extrapolation method to accelerate the widely used shifted symmetric higher order power method for tensor $Z$-eigenvalue problems. We analyze the asymptotic convergence of the method, determining the range of extrapolation parameters that induce acceleration, as well as the parameter that gives the optimal convergence rate. We then introduce an automated method to dynamically approximate the optimal parameter, and demonstrate it's efficiency when the base iteration is run with either static or adaptively set shifts. Our numerical results on both even and odd order tensors demonstrate the theory and show we achieve our theoretically predicted acceleration.