Identifiability and Separability of Linear Ica Models Revisited
Jan Eriksson, Visa Koivunen · 2003
We prove theorems that ensure identifiability, separability and uniqueness of linear ICA models. The currently used conditions in ICA community are hence extended to wider class of mixing models and source distributions. Examples illustrating the above concepts are presented as well. In this paper we are concerned with conditions ensuring that the mixing system may be identified and sources separated in a linear instantaneous ICA (Independent Component Analysis) model. The fundamental problems of identifiability, separability and uniqueness of ICA models are addressed. We give formal definitions to each of these concepts. A theorem is given for each of the above conditions and proofs are constructed. These theorems generalize the results given in earlier presented theorems addressing the same problems [Com94, CL96, Car98, DLDMV99, TJ99]. In particular, some of the restricting assumptions made in those theorems may be relaxed. As a result, the identifiability, separability or uniqueness can be ensured for wider class of ICA models and source distributions. The proofs given in this paper stem from the results in [KLR73, Com94, TJ99]. This paper is organized as follows. In section 2, the concepts of identifiability, separability and uniqueness are discussed. Various representations for the linear ICA model are considered as well. In section 3, a theorem for ensuring the identifiability of mixing system in the linear ICA model is given. In section 4, theorem on separability is presented. Separability ensures that the source signals may be recovered up to some ambiguities. In section 5, the uniqueness of the ICA model is addressed. It is a very relevant concept, in particular for underdetermined source separation problems. Finally, section 6 concludes the paper and proofs of the theorems are given in an Appendix.