  Independent component analysis (ICA) is a cornerstone of modern data
analysis. Its goal is to recover a latent random vector S with independent
components from samples of X=AS where A is an unknown mixing matrix.
Critically, all existing methods for ICA rely on and exploit strongly the
assumption that S is not Gaussian as otherwise A becomes unidentifiable. In
this paper, we show that in fact one can handle the case of Gaussian components
by imposing structure on the matrix A. Specifically, we assume that A is sparse
and generic in the sense that it is generated from a sparse Bernoulli-Gaussian
ensemble. Under this condition, we give an efficient algorithm to recover the
columns of A given only the covariance matrix of X as input even when S has
several Gaussian components.
