Overcomplete Independent Component Analysis via SDP.
TITLE: Overcomplete Independent Component Analysis via SDP.
AUTHORS: Anastasia Podosinnikova, Amelia Perry, Alexander Wein, Francis Bach, Alexandre d'Aspremont, David Sontag.
ABSTRACT: 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 semidefinite 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 CIFAR10 dataset of real images.
STATUS: AISTATS 2019
ArXiv PREPRINT: ArXiv:1901.08334
PAPER: Overcomplete Independent Component Analysis via SDP in pdf
