  We present a novel algorithm for overcomplete independent components analysis
(ICA), where the number of latent sources k exceeds the dimension p of observed
variables. Previous algorithms either suffer from high computational complexity
or make strong assumptions about the form of the mixing matrix. Our algorithm
does not make any sparsity assumption yet enjoys favorable computational and
theoretical properties. Our algorithm consists of two main steps: (a)
estimation of the Hessians of the cumulant generating function (as opposed to
the fourth and higher order cumulants used by most algorithms) and (b) a novel
semi-definite programming (SDP) relaxation for recovering a mixing component.
We show that this relaxation can be efficiently solved with a projected
accelerated gradient descent method, which makes the whole algorithm
computationally practical. Moreover, we conjecture that the proposed program
recovers a mixing component at the rate k < p^2/4 and prove that a mixing
component can be recovered with high probability when k < (2 - epsilon) p log p
when the original components are sampled uniformly at random on the hyper
sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of
real images.
