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PMICA identifiability in the overcomplete (different dimensions) setting

Extend identifiability results for Pairwise Mean Independent Component Analysis to the overcomplete case where the observation x ∈ R^m and the source vector s ∈ R^n have different dimensions (m ≠ n), determining conditions under which the mixing matrix A ∈ R^{m×n} is identifiable from cumulant restrictions consistent with pairwise mean independence.

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Background

The main results assume a square, invertible mixing matrix A with equal dimensions for x and s. Overcomplete ICA (with m ≠ n) has been studied under full independence, but pairwise mean independence introduces a broader class of cumulant structures.

Extending PMICA to rectangular A would generalize the identifiability theory beyond the square setting and align it with recent advances in overcomplete ICA.

References

It is also an open problem to extend to the case where x and s have different dimensions, following the study of full independence in the overcomplete setting.

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