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.