A Convergence and Asymptotic Analysis of the Generalized Symmetric FastICA Algorithm

This contribution deals with the FastICA algorithm in the domain of Independent Component Analysis (ICA). The focus is on the asymptotic behavior of the generalized symmetric variant of the algorithm. The latter has already been shown to possess the potential to achieve the Cramér-Rao Bound (CRB) by allowing the usage of different nonlinearity functions in its implementation. Although the FastICA algorithm along with its variants are among the most extensively studied methods in the domain of ICA, a rigorous study of the asymptotic distribution of the generalized symmetric FastICA algorithm is still missing. In fact, all the existing results exhibit certain limitations. Some ignores the impact of data standardization on the asymptotic statistics; others are only based on heuristic arguments. In this work, we aim at deriving general and rigorous results on the limiting distribution and the asymptotic statistics of the FastICA algorithm. We begin by showing that the generalized symmetric FastICA optimizes a function that is a sum of the contrast functions of traditional one-unit FastICA with a correction of the sign. Based on this characterization, we established the asymptotic normality and derived a closed-form analytic expression of the asymptotic covariance matrix of the generalized symmetric FastICA estimator using the method of estimating equation and M-estimator. Computer simulations are also provided, which support the theoretical results.