  Discrete spatial patterns and their continuous transformations are two
important regularities contained in natural signals. Lie groups and
representation theory are mathematical tools that have been used in previous
works to model continuous image transformations. On the other hand, sparse
coding is an important tool for learning dictionaries of patterns in natural
signals. In this paper, we combine these ideas in a Bayesian generative model
that learns to disentangle spatial patterns and their continuous
transformations in a completely unsupervised manner. Images are modeled as a
sparse superposition of shape components followed by a transformation that is
parameterized by n continuous variables. The shape components and
transformations are not predefined, but are instead adapted to learn the
symmetries in the data, with the constraint that the transformations form a
representation of an n-dimensional torus. Training the model on a dataset
consisting of controlled geometric transformations of specific MNIST digits
shows that it can recover these transformations along with the digits. Training
on the full MNIST dataset shows that it can learn both the basic digit shapes
and the natural transformations such as shearing and stretching that are
contained in this data.
