Uniqueness of linear factorizations into independent subspaces

Given a random vector X, we address the question of linear separability of X, that is, the task of finding a linear operator W such that we have (S1, . . . ,SM ) = (WX) with statistically independent random vectors Si. As this requirement alone is already fulfilled trivially by X being independent of the empty rest, we require that the components be not further decomposable. We show that if X has finite covariance, such a representation is unique up to trivial indeterminacies. Related algorithms, however with fixed dimensionality of the subspaces, have already been successfully employed in biomedical applications, such as separation of fMRI recorded data. Based on the presented uniqueness result, it is now clear that also subspace dimensions can be determined in a unique and therefore meaningful fashion, which shows the advantages of Independent Subspace Analysis in contrast to methods like Principal Component Analysis.