A chiseling algorithm for low-rank Grassmann decomposition of skew-symmetric tensors

TL;DR

This paper introduces a numerical algorithm for decomposing low-rank skew-symmetric tensors into elementary components using Grassmannian techniques, which is efficient and accurate for such tensors.

Contribution

The paper presents a novel algorithm for Grassmann decomposition of skew-symmetric tensors based on differential relations and kernel diagonalization.

Findings

Algorithm is computationally efficient.

High accuracy for low-rank tensors.

Effective in uncovering tensor structure.

Abstract

A numerical algorithm to decompose an exact low-rank skew-symmetric tensor into a sum of elementary (rank-$1$) skew-symmetric tensors is introduced. The algorithm uncovers this Grassmann decomposition based on linear relations that are encoded by the kernel of the differential of the natural action of the general linear group on the tensor, following the ideas of [Brooksbank, Kassabov, and Wilson, Detecting null patterns in tensor data, arXiv:2408.17425v2, 2025]. The Grassmann decomposition can be recovered, up to scale, from a diagonalization of a generic element in this kernel. Numerical experiments illustrate that the algorithm is computationally efficient and quite accurate for mathematically low-rank tensors.