One-unit contrast functions for independent component analysis: A statistical analysis

The author introduced previously a large family of one-unit contrast functions to be used in independent component analysis (ICA). In this paper, the family is analyzed mathematically in the case of a finite sample. Two aspects of the estimators obtained using such contrast functions are considered: asymptotic variance, and robustness against outliers. An expression for the contrast function that minimizes the asymptotic variance is obtained as a function of the probability densities of the independent components. Combined with robustness considerations, these results provide strong arguments in favor of the use of contrast functions based on slowly growing functions, and against the use of kurtosis, which is the classical contrast function.