A Unifying Objective Function of Independent Component Analysis for Ordering Sources by Non-Gaussianity

The independent component analysis (ICA) is a widely used method for solving blind separation problems. The ICA assumes that the sources are independent of each other and extracts them by maximizing their non-Gaussianity as the objective function. There are the two types of non-Gaussianity of the sources (the super-Gaussian type with the positive kurtosis and the sub-Gaussian one with the negative kurtosis). In this paper, we propose a new objective function unifying the two types of non-Gaussianity naturally, which is derived by applying the Gaussian approximation to the distribution of sources in the second-order polynomial feature space. The proposed objective function [called the adaptive ICA function (AIF)] is a simple form given as a summation of weighted fourth-order statistics, where the weights are adaptively estimated by the current kurtoses. The first practical advantage of the AIF is that it can extract the sources one by one in the descending order of the criterion of non-Gaussianity. It can solve the permutation ambiguity problem. The second and more important advantage is that it can estimate the number of non-Gaussian sources by the Akaike information criterion irrespective of the specific form of their distributions. In order to utilize the above-mentioned advantages of the AIF, we construct a new algorithm named the ordering ICA by extending the fast ICA. Experimental results verify that the ordering ICA can estimate the number of non-Gaussian sources correctly in both artificial and real data sets.