On the learning of nonlinear visual features from natural images by optimizing response energies

The operation of V1 simple cells in primates has been traditionally modelled with linear models resembling Gabor filters, whereas the functionality of subsequent visual cortical areas is less well understood. Here we explore the learning of mechanisms for further nonlinear processing by assuming a functional form of a product of two linear filter responses, and estimating a basis for the given visual data by optimizing for robust alternative of variance of the nonlinear model outputs. By a simple transformation of the learned model, we demonstrate that on natural images, both minimization and maximization in our setting lead to oriented, band-pass and localized linear filters whose responses are then nonlinearly combined. In minimization, the method learns to multiply the responses of two Gabor-like filters, whereas in maximization it learns to subtract the response magnitudes of two Gabor-like filters. Empirically, these learned nonlinear filters appear to function as conjunction detectors and as opponent orientation filters, respectively. We provide a preliminary explanation for our results in terms of filter energy correlations and fourth power optimization.