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In applications where the tensor rank decomposition arises, one often relieson its identifiability properties for interpreting the individual rank-$1$terms appearing in the decomposition. Several criteria for identifiability havebeen proposed in the literature, however few results exist on how frequentlythey are satisfied. We propose to call a criterion effective if it is satisfiedon a dense, open subset of the smallest semi-algebraic set enclosing the set ofrank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when itis combined with reshaping. It is proved that this criterion is effective forboth real and complex tensors in its entire range of applicability, which isusually much smaller than the smallest typical rank. Our proof explains whenreshaping-based algorithms for computing tensor rank decompositions may beexpected to recover the decomposition. Specializing the analysis to symmetrictensors or forms reveals that the reshaped Kruskal criterion may even beeffective up to the smallest typical rank for some third, fourth and sixthorder symmetric tensors of small dimension as well as for binary forms ofdegree at least three. We extended this result to $4 \times 4 \times 4 \times4$ symmetric tensors by analyzing the Hilbert function, resulting in acriterion for symmetric identifiability that is effective up to symmetric rank$8$, which is optimal.

Effective criteria for specific identifiability of tensors and forms