Holographic-(V)AE: an end-to-end SO(3)-Equivariant (Variational) Autoencoder in Fourier Space

Group-equivariant neural networks have emerged as a data-efficient approach to solve classification and regression tasks, while respecting the relevant symmetries of the data. However, little work has been done to extend this paradigm to the unsupervised and generative domains. Here, we present \textit{Holographic}-(V)AE (H-(V)AE), a fully end-to-end SO(3)-equivariant (variational) autoencoder in Fourier space, suitable for unsupervised learning and generation of data distributed around a specified origin. H-(V)AE is trained to reconstruct the spherical Fourier encoding of data, learning in the process a latent space with a maximally informative invariant embedding alongside an equivariant frame describing the orientation of the data. We extensively test the performance of H-(V)AE on diverse datasets and show that its latent space efficiently encodes the categorical features of spherical images and structural features of protein atomic environments. Our work can further be seen as a case study for equivariant modeling of a data distribution by reconstructing its Fourier encoding.