Joint Independent Subspace Analysis by Coupled Block Decomposition: Non-Identifiable Cases

This paper deals with the identifiability of joint independent subspace analysis of real-valued Gaussian stationary data with uncorrelated samples. This model is not identifiable when each mixture is considered individually. Algebraically, this model amounts to coupled block decomposition of several matrices. In previous work, we showed that if all the cross-correlations in this model were square matrices, the model was generally identifiable. In this paper, we show that this does not necessarily hold when the cross-correlation matrices are rectangular. In this current contribution, we first show that, in certain cases, the balance of degrees of freedom (d.o.f.) between model and observations does not allow identifiability; this situation never occurs in the square case. Second, we explain why for certain block sizes, even if the balance of d.o.f. seems adequate, the model is never identifiable.