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

A singular k-tuple of a tensor T of format (n1, ..., nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1, ..., nk) 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 ni, then the number of singular k-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable nk, that stabilizes when (n1, ..., nk) 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, ... , 2, n). As a consequence, if again we fix the first k - 1 dimensions ni and let nk 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.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

Automated summary

This paper studies the linear span of singular k-tuples of a specific format tensor and proves that the dimension of this linear span is stable in certain formats.