A primary task of biological vision is to maintain stable representations of objects and patterns as they undergo transformations such as translations, rotations, scaling, deformations, and changes in lighting. Despite such variations, organisms effortlessly and robustly recognize objects and parse complex scenes, demonstrating perceptual invariance. Moreover, organisms are able to explicitly model transformations in a scene—for example, extrapolating trajectories and recognizing novel poses of objects. What are the mathematics underlying computations in networks of neurons as they accomplish this feat?

In the natural world, variations are highly structured, arising from the symmetry and geometry of the space in which objects lie. Consequently, many of the transformations in natural data can be described in terms of the actions of groups. In this talk, I explore the hypothesis that computations performed by neural circuits reflect this group structure—the mathematical symmetries of the natural world. Here, I tie the mathematics of groups to biologically plausible computational mechanisms by way of generalized Fourier analysis and its roots in group representation theory. This perspective generalizes canonical models of receptive fields in visual cortex, explains extra-classical results, and makes novel empirical predictions. It also leads us to a natural and complete analytical solution to the perceptual invariance problem: the generalized, group-invariant bispectrum. Further, I demonstrate how the ansatz of this analytical form can be used to design artificial neural networks with improved robustness and invariance.

This lecture will be delivered in person in LSK 306 and also streamed via Zoom. If you do not receive IAM seminar announcements by email, please RSVP here to receive the Zoom link to this and other IAM seminars.