We describe a directed bilinear model that learns higher-order groupings among features of natural images. The model represents images in terms of two sets of latent variables: one set of variables represents which feature groups are active, while the other specifies the relative activity within groups. Such a factorized representation is beneficial because it is stable in response to small variations in the placement of features while still preserving information about relative spatial relationships. When trained on MNIST digits, the resulting representation provides state of the art performance in classification using a simple classifier. When trained on natural images, the model learns to group features according to proximity in position, orientation, and scale. The model achieves high log-likelihood (-94 nats), surpassing the current state of the art for natural images achievable with an mcRBM model.

We describe a method for learning a group of continuous transformation operators to traverse smooth nonlinear manifolds. The method is applied to model how natural images change over time and scale. The group of continuous transform operators is represented by a basis that is adapted to the statistics of the data so that the infinitesimal generator for a measurement orbit can be produced by a linear combination of a few basis elements. We illustrate how the method can be used to efficiently code time-varying images by describing changes across time and scale in terms of the learned operators.