CP decomposition and low-rank approximation of antisymmetric tensors

TL;DR

This paper introduces a novel low-rank approximation method for antisymmetric tensors using three vectors, with an efficient algorithm and implementation in Julia, advancing tensor decomposition techniques.

Contribution

It presents a new low-rank format for antisymmetric tensors and an alternating least squares algorithm, linking the approximation problem to multilinear and unstructured rank-1 approximations.

Findings

Algorithm successfully computes low-rank antisymmetric tensor approximations.

Implementation in Julia demonstrates practical efficiency.

The approach extends to partially antisymmetric tensors.

Abstract

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-$1$ approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.