Joint Independent Subspace Analysis Using Second-Order Statistics
Résumé
This paper deals with a novel generalization of classical blind source separation (BSS)/independent component analysis (ICA) in two directions. First, we relax the constraint that the latent sources must be statistically independent. This generalization is well-known and often termed independent subspace analysis (ISA). Second, we deal with joint analysis of several such ISA problems, where the link between mixtures is formed by statistical dependence across corresponding sources in different mixtures. For the case that the data is one-dimensional, i.e., multiple ICA problems, this model, termed independent vector analysis (IVA), is well-known and has already been studied. Therefore, in this work, we generalize IVA to multidimensional components and term this new model joint ISA (JISA). We provide full performance analysis of this new model, including closed-form expressions for minimal mean square error (MSE), Fisher information matrix (FIM) and Cramér-Rao lower bound (CRLB) in the separation of Gaussian data. We prove in theory and validate in numerical simulations that this analysis predicts the MSE also for non-Gaussian data, when second-order statistics (SOS) are used. We present a Newton-based algorithm that converges in a significantly smaller number of iterations than the previously proposed relative gradient (RG) approach. We show that all our results indeed generalize previously-known results on IVA via SOS, including the ability to resolve static mixtures of Gaussian stationary data and the individual arbitrary permutation. Finally, we discuss some links between this model and BSS of non-stationary multidimensional data.
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