On generic nonexistence of the Schmidt-Eckart-Young decomposition for tensors

The Schmidt-Eckart-Young theorem for matrices states that the optimal rank-r approximation to a matrix in the Euclidean topology is obtained by retaining the first r terms from the singular value decomposition of that matrix. In this talk, we consider a generalization of this optimal truncation property to the CANDECOMP/PARAFAC decomposition of tensors and establish a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure. We prove, moreover, that for tensors of small rank this orthogonality condition can be satisfied on only a set of tensors of Lebesgue measure zero.status: publishe