Structure-preserving low multilinear rank approximation of antisymmetric tensors

TL;DR

This paper investigates low multilinear rank approximations of antisymmetric tensors, focusing on rank constraints, algorithm adaptation for antisymmetry preservation, and efficient methods for special cases.

Contribution

It characterizes attainable ranks for antisymmetric tensors and adapts existing algorithms, like the Jacobi method, to maintain antisymmetry during approximation.

Findings

Certain ranks are shown to be attainable by antisymmetric tensors.

The Jacobi algorithm can be adapted to preserve antisymmetry.

The rank equal to the tensor order can be addressed with a rank-1 approximation.

Abstract

This paper is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi algorithm. Particular attention is paid to the important special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-$1$ approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies.