Representing and Learning Functions Invariant Under Crystallographic Groups
This work addresses the challenge of modeling symmetric structures in crystals and repetitive patterns, which is important for materials science and related fields, but it is incremental as it builds on existing Fourier and group theory concepts.
The paper tackles the problem of representing and learning functions that are smooth and invariant under crystallographic groups, such as wallpaper and space groups, by deriving both linear and nonlinear representations. It shows that a Fourier-like basis exists for each group, constructs algorithms to compute these representations, and demonstrates their application in neural networks, kernel machines, and Gaussian processes.
Crystallographic groups describe the symmetries of crystals and other repetitive structures encountered in nature and the sciences. These groups include the wallpaper and space groups. We derive linear and nonlinear representations of functions that are (1) smooth and (2) invariant under such a group. The linear representation generalizes the Fourier basis to crystallographically invariant basis functions. We show that such a basis exists for each crystallographic group, that it is orthonormal in the relevant $L_2$ space, and recover the standard Fourier basis as a special case for pure shift groups. The nonlinear representation embeds the orbit space of the group into a finite-dimensional Euclidean space. We show that such an embedding exists for every crystallographic group, and that it factors functions through a generalization of a manifold called an orbifold. We describe algorithms that, given a standardized description of the group, compute the Fourier basis and an embedding map. As examples, we construct crystallographically invariant neural networks, kernel machines, and Gaussian processes.