Biological Learning of Irreducible Representations of Commuting Transformations

A longstanding challenge in neuroscience is to understand neural mechanisms underlying the brain’s remarkable ability to learn and detect transformations of objects due to motion. Translations and rotations of images can be viewed as orthogonal transformations in the space of pixel intensity vectors. Every orthogonal transformation can be decomposed into rotations within irreducible two-dimensional subspaces (or representations). For sets of commuting transformations, known as toroidal groups, Cohen and Welling proposed a mathematical framework for learning the irreducible representations. We explore the possibility that the brain also learns irreducible representations using a biologically plausible learning mechanism. The first is based on SVD of the anti-symmetrized outer product of the vectors representing consecutive images and is implemented by a single-layer neural network. The second is based on PCA of the difference between consecutive frames and is implemented in a two-layer network but with greater biological plausibility. Both networks learn image rotations (replicating Cohen and Welling’s results) as well as translations. It would be interesting to search for the proposed networks in nascent connectomics and physiology datasets.