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Group-equivariant neural networks have emerged as a data-efficient approachto solve classification and regression tasks, while respecting the relevantsymmetries of the data. However, little work has been done to extend thisparadigm to the unsupervised and generative domains. Here, we presentHolographic-(Variational) Auto Encoder (H-(V)AE), a fully end-to-endSO(3)-equivariant (variational) autoencoder in Fourier space, suitable forunsupervised learning and generation of data distributed around a specifiedorigin in 3D. H-(V)AE is trained to reconstruct the spherical Fourier encodingof data, learning in the process a low-dimensional representation of the data(i.e., a latent space) with a maximally informative rotationally invariantembedding alongside an equivariant frame describing the orientation of thedata. We extensively test the performance of H-(V)AE on diverse datasets. Weshow that the learned latent space efficiently encodes the categorical featuresof spherical images. Moreover, H-(V)AE's latent space can be used to extractcompact embeddings for protein structure microenvironments, and when pairedwith a Random Forest Regressor, it enables state-of-the-art predictions ofprotein-ligand binding affinity.

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