Maximum Non-Gaussianity Estimation Revisit: Uniqueness Analysis from the Perspective of Constrained Cost Function Optimization

Abstract

In independent component analysis (ICA) and its diverse algorithms, uniqueness is the most essential requirement and rationality problem compared with performances of existence, stability and convergence. For ICA’s maximum non-Gaussianity estimation (MNE), many achievements have been made in recent twenty years based on uniqueness assumption, which has been taken for granted all along except for some intuitive interpretation. From the perspective of constrained cost function optimization, the paper is to provide a mathematical proof for uniqueness principle in MNE. The research focuses on skewless assumption and kurtosis-based cost function with basic linear ICA model. The relationship between the Kuhn–Tucker (K–T) points of cost function and the local maxima of non-Gaussianity are derived with the help of constrained optimization theory, and then a conclusion is drawn that there is a one-to-one correspondence between independent components and the local maxima, i.e. maximum non-Gaussianity is the sufficient and necessary condition for independent sources recovery. Moreover, the result also leads to an alternative and straightforward approach to the proof of Xu’ one-bit-matching conjecture for the availability of multi-unit approaches.