Independent subspace analysis can cope with the 'curse of dimensionality'

We search for hidden independent components, in particular we consider the independent subspace analysis (ISA) task. Earlier ISA procedures assume that the dimensions of the components are known. Here we show a method that enables the non-combinatorial estimation of the components. We make use of a decomposition principle called the ISA separation theorem. According to this separation theorem the ISA task can be reduced to the independent component analysis (ICA) task that assumes one-dimensional components and then to a grouping procedure that collects the respective non-independent elements into independent groups. We show that non-combinatorial grouping is feasible by means of the non-linear f-correlation matrices between the estimated components

Complex independent process analysis

We present a general framework for the search of hidden independent processes in the complex domain. The task is to estimate the hidden independent multidimensional complex-valued components observing only the mixture of the processes driven by them. In our model (i) the hidden independent processes can be multidimensional, they may be subject to (ii) moving averaging, or may evolve in an autoregressive manner, or (iii) they can be non-stationary. These assumptions are covered by integrated autoregressive moving average processes and thus our task is to solve their complex extensions. We show how to reduce the undercomplete version of complex integrated autoregressive moving average processes to real independent subspace analysis that we can solve. Simulations illustrate the working of the algorithm

Separation Theorem for Independent Subspace Analysis

Here, a separation theorem about Independent Subspace Analysis (ISA), a generalization of Independent Component Analysis (ICA) is proven. According to the theorem, ISA estimation can be executed in two steps under certain conditions. In the first step, 1-dimensional ICA estimation is executed. In the second step, optimal permutation of the ICA elements is searched for. We shall show that elliptically symmetric sources, among others, satisfy the conditions of the theorem

Auto-Regressive Independent Process Analysis without Combinatorial Efforts

We treat the problem of searching for hidden multi-dimensional independent auto-regressive processes (Auto-Regressive Independent Process Analysis, AR-IPA). Independent Subspace Analysis (ISA) can be used to solve the AR-IPA task. The so-called separation theorem simplifies the ISA task considerably: the theorem enables one to reduce the task to 1-dimensional Blind Source Separation (BSS) task followed by the grouping of the coordinates. However, the grouping of the coordinates still involves 2 types of combinatorial problems: (i) the number of the independent subspaces and their dimensions, and then (ii) the permutation of the estimated coordinates are to be determined. Here, we generalize the separation theorem. We also show a non-combinatorial procedure, which under certain conditions can treat these 2 combinatorial problems. Numerical simulations have been conducted. We investigate problems that fulfill sufficient conditions of the theory and also others that do not. The success of the numerical simulations indicates that further generalizations of the separation theorem may be feasible