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)=αexp(−β|s|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 XTw given the parameter p and the data X is shown to be equivalent to the minimization of lp-norm of the projected data XTw. 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 lp-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.