Holographic-(V)AE: An end-to-end SO(3)-equivariant (variational) autoencoder in Fourier space

Group-equivariant neural networks have emerged as an efficient approach to model complex data, using generalized convolutions that respect the relevant symmetries of a system. These techniques have made advances in both the supervised learning tasks for classification and regression, and the unsupervised tasks to generate new data. However, little work has been done in leveraging the symmetry-aware expressive representations that could be extracted from these approaches. Here, we present holographic-(variational) autoencoder [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, the low-dimensional representations learned by H-VAE can be used for downstream data-scarce tasks. Specifically, we show that 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.