On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

Abstract

The Schmidt--Eckart--Young theorem for matrices states that the optimal rank-$r$ approximation of a matrix is obtained by retaining the first $r$ terms from the singular value decomposition of that matrix. This paper considers a generalization of this optimal truncation property to the rank decomposition (Candecomp/Parafac) of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved, moreover, that for complex tensors of small rank this condition can be satisfied only by a set of tensors of Lebesgue measure zero. Finally, we demonstrate that generic tensors in cubic tensor spaces are not optimally truncatable.