Conjugation Invariant Learning with Neural Networks

Machine learning under the constraint of symmetries, given by group invariances or equivariances, has emerged as a topic of active interest in recent years. Natural settings for such applications include the multi-reference alignment and cryo electron microscopy, multi-object tracking, spherical images, and so on. A fundamental paradigm among such symmetries is the action of a group by symmetries, which often pertains to change of basis or relabelling of objects in pure and applied mathematics. Thus, a naturally significant class of functions consists of those that are intrinsic to the problem, in the sense of being independent of such base change or relabelling; in other words invariant under the conjugation action by a group. In this work, we investigate such functions, known as class functions, leveraging tools from group representation theory. A fundamental ingredient in our approach are given by the so-called irreducible characters of the group, which are canonical tracial class functions related to its irreducible representations. Such functions form an orthogonal basis for the class functions, extending ideas from Fourier analysis to this domain, and accord a very explicit structure. Exploiting a tensorial structure on representations, which translates into a multiplicative algebra structure for irreducible characters, we propose to efficiently approximate class functions using polynomials in a small number of such characters. Thus, our approach provides a global, non-linear coordinate system to describe functions on the group that is intrinsic in nature, in the sense that it is independent of local charts, and can be easily computed in concrete models. We demonstrate that such non-linear approximation using a small dictionary can be effectively implemented using a deep neural network paradigm. This allows us to learn a class function efficiently from a dataset of its outputs.