Identifiability and Kruskal conditions

Let A, B, and C be real factor matrices with R nonzero columns, forming a three-way CP decomposition. Reordering the columns simultaneously leaves the tensor unchanged. Scaling a component’s three factors by nonzero scalars whose product is one also leaves its pure product unchanged. Essential uniqueness means that all minimal decompositions differ only through these shared permutations and reciprocal component rescalings.

For a matrix with R columns, its Kruskal rank k is the largest integer for which every set of k columns is linearly independent. This is stronger than ordinary matrix rank, which requires only some independent set of that size. A repeated or proportional nonzero column makes Kruskal rank one, even if other columns span a larger space. Identifiability tests using ordinary ranks in place of Kruskal ranks can therefore certify a false conclusion.

For a three-way decomposition with R ≥ 2 nonzero terms, Kruskal’s condition k_A + k_B + k_C ≥ 2R + 2 guarantees that the tensor has CP rank R and that the R-term decomposition is essentially unique. The proof uses structured dependence relations among slices and column subspaces; checking the inequality is not a substitute for the theorem’s proof. A one-term nonzero tensor is separately unique up to its factor scalings, although the displayed inequality cannot hold when R = 1.

The criterion is sufficient, not a classification of all identifiable tensors. If the sum is too small, the correct conclusion is that this test is inconclusive, rather than that a competing decomposition exists. Stronger size-specific or generic identifiability theorems have their own hypotheses. The guarantee here also concerns exact full tensors, not completion from a subset of entries or reliable recovery in the presence of noise.

Two columns may remain independent while becoming nearly proportional. Their Kruskal rank stays two until exact dependence occurs, but small perturbations can then cause large changes in a factor recovery calculation. Thus identifiability and stability must be analyzed separately. Singular values of relevant column submatrices or the derivative of the parameter-to-tensor map provide quantitative information that the integer Kruskal ranks omit.

Normalizing factors removes some scale freedom but does not eliminate nearly dependent components or CP degeneracy. When interpreting latent variables, report the ambiguity conventions, the exact certificate used, and a perturbation or conditioning assessment appropriate to the data. A unique exact decomposition can coexist with wide uncertainty in finite-precision estimates. Conversely, several equivalent scaled factor triples should not be mistaken for distinct latent explanations.