Separation theorem for independent subspace analysis and its consequences

Independent component analysis (ICA) – the theory of mixed, independent, non-Gaussian sources – has a central role in signal processing, computer vision and pattern recognition. One of the most fundamental conjectures of this research field is that independent subspace analysis (ISA) – the extension of the ICA problem, where groups of sources are independent – can be solved by traditional ICA followed by grouping the ICA components. The conjecture, called ISA separation principle, (i) has been rigorously proven for some distribution types recently, (ii) forms the basis of the state-of-the-art ISA solvers, (iii) enables one to estimate the unknown number and the dimensions of the sources efficiently, and (iv) can be extended to generalizations of the ISA task, such as different linear-, controlled-, post nonlinear-, complex valued-, partially observed problems, as well as to problems dealing with nonparametric source dynamics. Here, we shall review the advances on this field.