Distributional Convergence of Subspace Estimates in FastICA: A Bootstrap Study

Independent component analysis (ICA) is possibly the most widespread approach to solve the blind source separation (BSS) problem. Many different algorithms have been proposed, together with an extensive body of work on the theoretical foundations and limits of the methods. One practical concern about the use of ICA with real-world data is the reliability of its estimates. Variations of the estimates may stem from the inherent stochastic nature of the algorithm, or deviations from the theoretical assumptions. To overcome this problem, some approaches use bootstrapped estimates. The bootstrapping also allows identification of subspaces, since multiple separated components can share a common pattern of variation, when they belong to the same subspace. This is a desired ability, since real-world data often violates the strict independence assumption. Based on empirical process theory, it can be shown that FastICA and bootstrapped FastICA are consistent and asymptotically normal. In the context of subspace analysis, the normal convergence is not satisfied. This paper shows such limitation, and how to circumvent it, when one can estimate the canonical directions within the subspace.