Analog of the ICA ‘at most one Gaussian’ rule using all cumulants

Characterize distributional assumptions on the source vector s that ensure genericity when all higher-order cumulant tensors are available, identifying necessary and sufficient conditions—analogous to the classical ICA rule that at most one source is Gaussian—that guarantee identifiability of the mixing matrix under pairwise mean independence.

Background

In classical ICA, identifiability is guaranteed by the condition that at most one source is Gaussian, which is equivalent to non-vanishing of some higher-order cumulants. PMICA requires more subtle genericity because non-Gaussianity alone does not suffice, and using a single fixed-order cumulant may be insufficient.

The paper establishes identifiability under PMI when a fixed-order cumulant is generic in V_pmi, but leaves open the distributional characterization when all cumulants are available, seeking an analog of the simple ICA condition.

References

It is an open question to characterize distributional assumptions for genericity when all cumulants are available—the analog of the 'at most one Gaussian source' condition in ICA.

— Beyond independent component analysis: identifiability and algorithms  (2510.07525 - Ribot et al., 8 Oct 2025) in Discussion

It remains an open question to investigate whether the values m_{\max}(n,d) or m{}_{\max}(n,d) are the optimal bounds, i.e. if algebraic/rational identifiability can be achieved with m_{\max}(n,d) or m{}_{\max}(n,d) components in general.

— Identifiability from Moments in Independent Component Analysis Mixture Models  (2610.23846 - Semnani et al., 16 Oct 2026) in Section 9, Discussion, final paragraph before the paragraph on rational identifiability