  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
Holographic-(Variational) Auto Encoder (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 in 3D. H-(V)AE is trained to reconstruct the spherical Fourier encoding
of data, learning in the process a low-dimensional representation of the data
(i.e., a latent space) with a maximally informative rotationally invariant
embedding alongside an equivariant frame describing the orientation of the
data. We extensively test the performance of H-(V)AE on diverse datasets. We
show that the learned latent space efficiently encodes the categorical features
of spherical images. Moreover, H-(V)AE's latent space can be used to extract
compact embeddings for protein structure microenvironments, and when paired
with a Random Forest Regressor, it enables state-of-the-art predictions of
protein-ligand binding affinity.
