Consistency and asymptotic normality of FastICA and bootstrap FastICA

Independent component analysis (ICA) is possibly the most widespread approach to solve the blind source separation problem. Many different algorithms have been proposed, together with several highly successful applications. There is also an extensive body of work on the theoretical foundations and limits of the ICA methodology. One practical concern about the use of ICA with real world data is the robustness of its estimates. Slight variations in the estimates, may stem from the inherent stochastic nature of the algorithms used or some deviations from the theoretical assumptions. To overcome this problem, different approaches have been proposed, most of which are based on the use of multiple runs of ICA algorithms with bootstrap. Here we show the consistency and asymptotic normality of FastICA and bootstrap FastICA, based on empirical process theory, including Z-estimators and Hoeffding's inequality. These results give theoretical grounds for the robust use of FastICA, in a multiple run, bootstrap and randomly initialized manner. In this framework, it is also possible to assess the convergence of the algorithm through a normality test.