The span of singular tuples of a tensor beyond the boundary format

TL;DR

This paper investigates the linear span of singular tuples of generic tensors, revealing stabilization phenomena in their dimension and providing equations for specific tensor formats, with conjectures on broader applicability.

Contribution

It introduces the study of the linear span of singular tuples of tensors, showing stabilization in their dimension and deriving equations for certain formats, advancing understanding of tensor singularities.

Findings

Dimension of the linear span stabilizes in special tensor formats.

Equations for the linear span of singular triples in order-three tensors are provided.

Conjecture that all tensors belong to the span of their singular triples.

Abstract

A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic tensor has finitely many complex singular $k$-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first $k-1$ dimensions $n_i$, then the number of singular $k$-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable $n_k$, that stabilizes when $(n_1,\ldots,n_k)$ reaches a boundary format. In this paper, we study the linear span of singular $k$-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-$k$ tensors of format $(2,\ldots,2,n)$. As a consequence, if again we fix the first $k-1$ dimensions $n_i$ and let $n_k$ increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with $k>3$. Finally, we provide equations for the linear span of singular triples of a generic order three tensor $T$ of some special non-sub-boundary format. From these equations, we conclude that $T$ belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.