  We describe a model for capturing the statistical structure of local
amplitude and local spatial phase in natural images. The model is based on a
recently developed, factorized third-order Boltzmann machine that was shown to
be effective at capturing higher-order structure in images by modeling
dependencies among squared filter outputs (Ranzato and Hinton, 2010). Here, we
extend this model to $L_p$-spherically symmetric subspaces. In order to model
local amplitude and phase structure in images, we focus on the case of two
dimensional subspaces, and the $L_2$-norm. When trained on natural images the
model learns subspaces resembling quadrature-pair Gabor filters. We then
introduce an additional set of hidden units that model the dependencies among
subspace phases. These hidden units form a combinatorial mixture of phase
coupling distributions, concentrated in the sum and difference of phase pairs.
When adapted to natural images, these distributions capture local spatial phase
structure in natural images.
