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

Abstract

A singular <i>k</i>-tuple of a tensor T of format (<i>n₁</i>, ..., <i>n</i>_<i><k></i>) is essentially a complex critical point of the distance function from <i>T</i> constrained to the cone of tensors of format (<i>n</i>₁, ..., <i>n_k</i>) of rank at most one. A generic tensor has finitely many complex singular <i>k</i>-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first <i>k</i> - 1 dimensions <i>n</i>_i</i>, then the number of singular <i>k</i>-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable <i>n>sub>k</sub></i>, that stabilizes when (<i>n₁, ..., n_k</i>) reaches a boundary format.<p> <p>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-<i>k</i> tensors of format (2, ..., 2, <i>n</i>). . As a consequence, if again we fix the first <i>k</i> - 1 dimensions <i>n</i> > 3 and let <i>n_k</i> 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 <i>k</i>>3. . Finally, we provide equations for the linear span of singular triples of a generic order three tensor <i>T</i> of some special non-sub-boundary format. From these equations, we conclude that <i>T</i> belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.