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.