Abstract
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.
Highlights
Independent component analysis (ICA) [1, 2, 3] has received considerable attention in signal processing, computer vision and pattern recognition, e.g., in face representation and recognition [4, 5], information theoretical image matching [6], fMRI analysis [7], feature extraction of natural images [8], texture segmentation [9], artifact separation in MEG recordings, and the exploration of hidden factors in nancial data [10]
The separation task requires an extension of ICA, which is called multidimensional ICA [14], independent subspace analysis (ISA) [15], independent feature subspace analysis [16], subspace ICA
In what follows we briey review the generalization of the independent process analysis (IPA) problem to controlled (ARX-IPA, X stands for exogenous input) and partially observed problems, as well as to problems with nonparametric source dynamics
Summary
Independent component analysis (ICA) [1, 2, 3] has received considerable attention in signal processing, computer vision and pattern recognition, e.g., in face representation and recognition [4, 5], information theoretical image matching [6], fMRI analysis [7], feature extraction of natural images [8], texture segmentation [9], artifact separation in MEG recordings, and the exploration of hidden factors in nancial data [10]. While the extent of this conjecture, the ISA separation principle, is still an open issue, it has recently been rigorously proven for some distribution types [33], and for this reason we call it ISA Separation Theorem This principle (i) forms the basis of many state-of-the-art ISA algorithms, (ii) can be used to design algorithms that scale well and eciently estimate the dimensions of the hidden sources and (iii) can be extended to dierent linear-, controlled-, post nonlinear-, complex valued-, partially observed systems, as well as to systems with nonparametric source dynamics. Post nonlinear models: The linear mixing restriction of ICA can be relaxed by assuming that there is an unknown component-wise nonlinear function superimposed on the linear mixture This ICA generalization has many successful applications, e.g., in sensor array processing, data processing in biological systems, and satellite communications.
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