Comon's Conjecture, Rank Decomposition, and Symmetric Rank Decomposition of Symmetric Tensors

Abstract

Comon's Conjecture claims that for a symmetric tensor, its rank and its symmetric rank coincide. We show that this conjecture is true under an additional assumption that the rank of that tensor is not larger than its order. Moreover, if its rank is less than its order, then all rank decompositions are necessarily symmetric rank decompositions.