A flexible non-linearity and decorrelating manifold approach to ICA

Independent Components Analysis finds a linear transformation to variables which are maximally statistically independent. We examine ICA from the point of view of maximising the likelihood of the data. We elucidate how scaling of the unmixing matrix permits a "static" nonlinearity to adapt to various marginal densities. We demonstrate a new algorithm that uses generalised exponentials functions to model the marginal densities and is able to separate densities with light tails. We characterise decorrelating matrices and numerically show that the manifold of decorrelating matrices lies along the ridges of high-likelihood unmixing matrices in the space of all unmixing matrices. We show how to find the optimum ICA matrix on the manifold of decorrelating matrices.