Independent component analysis by lp-norm optimization

Abstract

In this paper, a couple of new algorithms for independent component analysis (ICA) are proposed. In the proposed methods, the independent sources are assumed to follow a predefined distribution of the form f (s) = alpha exp(-beta vertical bar s vertical bar(p)) and a maximum likelihood estimation is used to separate the sources. In the first method, a gradient ascent method is used for the maximum likelihood estimation, while in the second, a non-iterative algorithm is proposed based on the relaxation of the problem. The maximization of the log-likelihood of the estimated source X(T)w given the parameter p and the data X is shown to be equivalent to the minimization of l(p)-norm of the projected data X(T)w. This formulation of ICA has a very close relationship with the Lp-PCA where the maximization of the same objective function is solved. The proposed algorithm solves an approximation of the l(p)-norm minimization problem for both super-(p < 2) and sub-Gaussian (p > 2) cases and shows superior performance in separating independent sources than the state of the art algorithms for ICA computation.