On best rank one approximation of tensors

TL;DR

This paper introduces a new algorithm called alternating singular value decomposition for computing the best rank one approximation of tensors, improving convergence and performance over existing methods.

Contribution

The paper proposes a novel algorithm based on singular value decomposition and modifications to ensure convergence to semi-maximal points, enhancing tensor approximation techniques.

Findings

The new method outperforms traditional alternating least squares in numerical tests.

Convergence to semi-maximal points is guaranteed with the proposed modifications.

Numerical examples demonstrate improved computational efficiency.

Abstract

In this paper we suggest a new algorithm for the computation of a best rank one approximation of tensors, called alternating singular value decomposition. This method is based on the computation of maximal singular values and the corresponding singular vectors of matrices. We also introduce a modification for this method and the alternating least squares method, which ensures that alternating iterations will always converge to a semi-maximal point. (A critical point in several vector variables is semi-maximal if it is maximal with respect to each vector variable, while other vector variables are kept fixed.) We present several numerical examples that illustrate the computational performance of the new method in comparison to the alternating least square method.