Effective criteria for specific identifiability of tensors and forms

Abstract: 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.