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Recent computational advances in the accurate prediction of protein three-dimensional (3D) structures from amino acid sequences now present a unique opportunity to decipher the interrelationships between proteins. This task entails--but is not equivalent to--a problem of 3D structure comparison and classification. Historically, protein domain classification has been a largely manual and subjective activity, relying upon various heuristics. Databases such as CATH represent significant steps towards a more systematic (and automatable) approach, yet there still remains much room for the development of more scalable and quantitative classification methods, grounded in machine learning. We suspect that re-examining these relationships via a Deep Learning (DL) approach may entail a large-scale restructuring of classification schemes, improved with respect to the interpretability of distant relationships between proteins. Here, we describe our training of DL models on protein domain structures (and their associated physicochemical properties) in order to evaluate classification properties at CATH's "homologous superfamily" (SF) level. To achieve this, we have devised and applied an extension of image-classification methods and image segmentation techniques, utilizing a convolutional autoencoder model architecture. Our DL architecture allows models to learn structural features that, in a sense, 'define' different homologous SFs. We evaluate and quantify pairwise 'distances' between SFs by building one model per SF and comparing the loss functions of the models. Hierarchical clustering on these distance matrices provides a new view of protein interrelationships--a view that extends beyond simple structural/geometric similarity, and towards the realm of structure/function properties.