On the generic and typical ranks of 3-tensors

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.Comment: 24 page

On the typical rank of real binary forms

We determine the rank of a general real binary form of degree d=4 and d=5. In the case d=5, the possible values of the rank of such general forms are 3,4,5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d.Comment: 12 pages, 2 figure

On the (non)existence of best low-rank approximations of generic IxJx2 arrays

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation

On complex and real identifiability of tensors

We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume 8, Number 2, 2017, pages 367-37

Effective criteria for specific identifiability of tensors and forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.Comment: 31 pages, 2 Macaulay2 code