Genericity conditions for PMI tensors across all orders

Determine, for every tensor order d, the complete genericity conditions under which a symmetric tensor T lying in the pairwise mean independence zero-pattern subspace V_pmi ⊂ S^d(R^n) has a unique orthogonal basis of eigenvectors (up to sign), thereby specifying when the mixing matrix in Pairwise Mean Independent Component Analysis is identifiable from a single d-th order cumulant.

Background

Pairwise Mean Independent Component Analysis (PMICA) reduces identifiability to uniqueness of an orthogonal basis of eigenvectors of the d-th order cumulant tensor that lies in the linear subspace V_pmi. The paper fully characterizes genericity conditions for d ≤ 4 and provides explicit polynomial criteria for some higher orders in the binary case, but does not offer a comprehensive solution for all orders.

Resolving these conditions for arbitrary d would complete the characterization of “sufficiently general” PMI cumulants needed for identifiability, generalizing the simple ICA rule about Gaussian sources to the broader PMI setting.

References

It is an open problem to resolve the genericity conditions for all d.

— Beyond independent component analysis: identifiability and algorithms  (2510.07525 - Ribot et al., 8 Oct 2025) in Remark “genericity-large-d”, Section 4 (Sufficiently general moments and cumulants)

Following the above remarks, we now present the conjecture below, which has been proved in case iii. Settling this conjecture in cases i and ii remains an open problem.

— Identifiability from Moments in Independent Component Analysis Mixture Models  (2610.23846 - Semnani et al., 16 Oct 2026) in Conjecture 1, Section 8, immediately before Section 9 (Discussion)