Optimal performance of second-order multidimensional ICA

Independent component analysis (ICA) and blind source separation (BSS) deal with extracting mutually-independent elements from their observed mixtures. In "classical" ICA, each component is one- dimensional in the sense that it is proportional to a column of the mixing matrix. However, this paper considers a more general setup, of multidimensional components. In terms of the underlying sources, this means that the source covariance matrix is block-diagonal rather than diagonal, so that sources belonging to the same block are correlated whereas sources belonging to different blocks are uncorrelated. These two points of view -correlated sources vs. multidimensional components- are considered in this paper. The latter offers the benefit of providing a unique decomposition. We present a novel, closed-form expression for the optimal performance of second-order ICA in the case of multidimensional elements. Our analysis is verified through numerical experiments.