The key contributions of this work are:
Introduction of a novel framework for learning group equivariant representations, where the latent representation is separated into an invariant and an equivariant component.
Characterization of the mathematical conditions for the group action function component and proposal of an explicit construction suitable for any group G. This is the first method for unsupervised learning of separated invariant-equivariant representations valid for any group.
Experimental validation of the framework on diverse data types (MNIST, sets of digits, point clouds, molecular conformations) and different network architectures, demonstrating the flexibility and validity of the approach.
The proposed framework learns to encode data into a group-invariant latent code and a group action. By separating the embedding into an invariant and an equivariant part, the method can learn expressive low-dimensional group-invariant representations using the power of autoencoders.
The key idea is that the network learns to encode and decode data to and from a group-invariant representation by additionally learning to predict the appropriate group action to align input and output. The authors derive the necessary conditions on the equivariant encoder and present a construction valid for any group G, both discrete and continuous.
The experiments demonstrate the effectiveness of the approach. For example, on rotated MNIST, the model can reconstruct rotated versions of digits by predicting the appropriate rotation. On sets of digits, the model can compress the set information into a much lower-dimensional representation compared to a non-invariant autoencoder. Similarly, on point cloud and molecular conformation data, the model learns representations that are invariant to translations, rotations and permutations.
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