L De-gaussianization and Independent Component Analysis

Given an arbitrary standardized (zero mean and unit variance) probability density, we measure its departure from the standard normal density by the L distance between the two density functions. In particular, we consider three different L norms, each distinguished by their weight functions. We investigate the reciprocal Gaussian, uniform, and Gaussian weight functions, and present respective Hermite series representations of non-Gaussianity. We show that theL metric defined with the reciprocal Gaussian weight is directly related to the moment-based approximation of differential entropy. We argue that this is a non-robust measure of nonGaussianity, as the division by the Gaussian places heavy weights on the tails of the density. Improved robustness is achieved by using the uniform or Gaussian weight functions, both of which effectively suppresses the sensitivity to outliers in the estimations. We choose the L Euclidean metric to define a measure of nonGaussianity, and show how it leads to an L de-Gaussianization algorithm for independent component analysis.