Computing symmetric rank for symmetric tensors

TL;DR

This paper introduces algorithms for computing the symmetric rank of symmetric tensors, especially for small tensors and border ranks, using algebraic geometry and describes related geometric structures.

Contribution

It provides novel algorithms for symmetric tensor rank computation and characterizes symmetric rank strata via secant varieties of Veronese varieties.

Findings

Algorithms for 2x...x2 tensors and small border rank tensors

Description of symmetric rank strata for certain secant varieties

Geometric interpretation of symmetric tensor rank

Abstract

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.