One-Bit-Matching Conjecture for Independent Component Analysis

The one-bit-matching conjecture for independent component analysis (ICA) could be understood from different perspectives but is basically stated as "all the sources can be separated as long as there is a one-to-one same-sign-correspondence between the kurtosis signs of all source probability density functions (pdf's) and the kurtosis signs of all model pdf's" (Xu, Cheung, & Amari, 1998a). This conjecture has been widely believed in the ICA community and implicitly supported by many ICA studies, such as the Extended Infomax (Lee, Girolami, & Sejnowski, 1999) and the soft switching algorithm (Welling & Weber, 2001). However, there is no mathematical proof to confirm the conjecture theoretically. In this article, only skewness and kurtosis are considered, and such a mathematical proof is given under the assumption that the skewness of the model densities vanishes. Moreover, empirical experiments are demonstrated on the robustness of the conjecture as the vanishing skewness assumption breaks. As a by-product, we also show that the kurtosis maximization criterion (Moreau & Macchi, 1996) is actually a special case of the minimum mutual information criterion for ICA.