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Recent advances in imaging from celestial objects in astronomy visualized viaoptical and radio telescopes to atoms and molecules resolved via electron andprobe microscopes are generating immense volumes of imaging data, containinginformation about the structure of the universe from atomic to astronomiclevels. The classical deep convolutional neural network architecturestraditionally perform poorly on the data sets having a significantorientational disorder, that is, having multiple copies of the same or similarobject in arbitrary orientation in the image plane. Similarly, while clusteringmethods are well suited for classification into discrete classes and manifoldlearning and variational autoencoders methods can disentangle representationsof the data, the combined problem is ill-suited to a classical non-supervisedlearning paradigm. Here we introduce a joint rotationally (and translationally)invariant variational autoencoder (j-trVAE) that is ideally suited to thesolution of such a problem. The performance of this method is validated onseveral synthetic data sets and extended to high-resolution imaging data ofelectron and scanning probe microscopy. We show that latent space behaviorsdirectly comport to the known physics of ferroelectric materials and quantumsystems. We further note that the engineering of the latent space structure viaimposed topological structure or directed graph relationship allows forapplications in topological discovery and causal physical learning.

Robust Feature Disentanglement in Imaging Data via Joint Invariant Variational Autoencoders: from Cards to Atoms