  A fundamental problem faced by object recognition systems is that objects and
their features can appear in different locations, scales and orientations.
Current deep learning methods attempt to achieve invariance to local
translations via pooling, discarding the locations of features in the process.
Other approaches explicitly learn transformed versions of the same feature,
leading to representations that quickly explode in size. Instead of discarding
the rich and useful information about feature transformations to achieve
invariance, we argue that models should learn object features conjointly with
their transformations to achieve equivariance. We propose a new model of
unsupervised learning based on sparse coding that can learn object features
jointly with their affine transformations directly from images. Results based
on learning from natural images indicate that our approach matches the
reconstruction quality of traditional sparse coding but with significantly
fewer degrees of freedom while simultaneously learning transformations from
data. These results open the door to scaling up unsupervised learning to allow
deep feature+transformation learning in a manner consistent with the
ventral+dorsal stream architecture of the primate visual cortex.
