Structural equations and divisive normalization for energy-dependent component analysis

Components estimated by independent component analysis and related methods are typically not independent in real data. A very common form of nonlinear dependency between the components is correlations in their variances or energies. Here, we propose a principled probabilistic model to model the energycorrelations between the latent variables. Our two-stage model includes a linear mixing of latent signals into the observed ones like in ICA. The main new feature is a model of the energy-correlations based on the structural equation model (SEM), in particular, a Linear Non-Gaussian SEM. The SEM is closely related to divisive normalization which effectively reduces energy correlation. Our new twostage model enables estimation of both the linear mixing and the interactions related to energy-correlations, without resorting to approximations of the likelihood function or other non-principled approaches. We demonstrate the applicability of our method with synthetic dataset, natural images and brain signals.