On the Convergence of Alternating Least Squares Optimisation in Tensor Format Representations

TL;DR

This paper analyzes the convergence properties of the alternating least squares algorithm for tensor approximation across various tensor formats, emphasizing the role of multilinearity in ensuring convergence.

Contribution

It provides a general convergence analysis for ALS in arbitrary tensor formats based on multilinearity, extending understanding of its theoretical foundations.

Findings

Convergence of ALS is established for general tensor formats.

Multilinearity plays a key role in the convergence analysis.

The study enhances theoretical understanding of tensor approximation methods.

Abstract

The approximation of tensors is important for the efficient numerical treatment of high dimensional problems, but it remains an extremely challenging task. One of the most popular approach to tensor approximation is the alternating least squares method. In our study, the convergence of the alternating least squares algorithm is considered. The analysis is done for arbitrary tensor format representations and based on the multiliearity of the tensor format. In tensor format representation techniques, tensors are approximated by multilinear combinations of objects lower dimensionality. The resulting reduction of dimensionality not only reduces the amount of required storage but also the computational effort.