  We present a neural network architecture, Bispectral Neural Networks (BNNs)
for learning representations that are invariant to the actions of compact
commutative groups on the space over which a signal is defined. The model
incorporates the ansatz of the bispectrum, an analytically defined group
invariant that is complete -- that is, it preserves all signal structure while
removing only the variation due to group actions. Here, we demonstrate that
BNNs are able to simultaneously learn groups, their irreducible
representations, and corresponding equivariant and complete-invariant maps
purely from the symmetries implicit in data. Further, we demonstrate that the
completeness property endows these networks with strong invariance-based
adversarial robustness. This work establishes Bispectral Neural Networks as a
powerful computational primitive for robust invariant representation learning

Machine Learning