Decomposition loci of tensors

TL;DR

This paper characterizes the decomposition loci of tensors, especially those on tangential varieties, and explicitly computes these loci for tensors with finitely many group orbits, advancing understanding of tensor decompositions.

Contribution

It provides a detailed description of decomposition loci for tensors on tangential varieties and computes these loci for tensors with finitely many group orbits, offering new insights into tensor structure.

Findings

Decomposition locus of tensors on tangential varieties excludes the tangency point.

Explicit computation of decomposition loci for tensors with finitely many orbits.

Characterization of rank-one tensors in minimal decompositions for specific tensor classes.

Abstract

The decomposition locus of a tensor is the set of rank-one tensors appearing in a minimal tensor-rank decomposition of the tensor. For tensors lying on the tangential variety of any Segre variety, but not on the variety itself, we show that the decomposition locus consists of all rank-one tensors except the tangency point only. We also explicitly compute decomposition loci of all tensors belonging to tensor spaces with finitely many orbits with respect to the action of product of general linear groups.