Sets computing the symmetric tensor rank

TL;DR

This paper classifies symmetric tensors with rank less than 1.5 times the degree and explores the structure of their computing subsets, showing that certain sets have no isolated points.

Contribution

It provides a complete classification of symmetric tensors with low rank and multiple computing subsets, revealing the topological structure of these sets.

Findings

Classifies tensors with sr(P) < 3d/2 and multiple computing subsets.

Proves that the set of computing subsets S(P) has no isolated points for these tensors.

Enhances understanding of the geometry of symmetric tensor rank and its computational aspects.

Abstract

Let n_d denote the degree d Veronese embedding of a projective space P^r. For any symmetric tensor P, the 'symmetric tensor rank' sr(P) is the minimal cardinality of a subset A of P^r, such that n_d(A) spans P. Let S(P) be the space of all subsets A of P^r, such that n_d(A) computes sr(P). Here we classify all P in P^n such that sr(P) < 3d/2 and sr(P) is computed by at least two subsets. For such tensors P, we prove that S(P) has no isolated points.