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 ener- gies. Here, we propose a principled probabilistic model to model the energy- correlations 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 fea- ture 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 two- stage model enables estimation of both the linear mixing and the interactions re- lated 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.