Finite sample effects in higher order statistics contrast functions for sequential blind source separation

There is a large family of contrast (or cost) functions in blind source separation that can yield learning algorithms for extracting single source signals from linear mixtures. One of these families is based on higher order statistics (HOSs), which assumes the statistical independence of source signals and their non-Gaussianity (all except one) in order to successfully extract them one by one. In cases in which source signals exhibit unit variance and the mixing matrix is orthonormal, many HOS contrast functions are equivalent (e.g., kurtosis, fourth cumulant, and fourth moment). However, these contrast functions are estimated in practice from a finite data set, which introduces stochastic errors, so their equivalence has remained uncertain. Our letter introduces error bounds for several sample-based HOS contrast functions, which demonstrate their dependence upon different source signal statistics and, thus, more importantly, provide a foundation for comparing them in terms of accuracy.