Densely Connected G-invariant Deep Neural Networks with Signed Permutation Representations

We introduce and investigate, for finite groups G , G-invariant deep neural network (G-DNN) architectures with ReLU activation that are densely connected--i.e., include all possible skip connections. In contrast to other G-invariant architectures in the literature, the preactivations of the G-DNNs presented here are able to transform by signed permutation representations (signed perm-reps) of G. Moreover, the individual layers of the G-DNNs are not required to be G-equivariant; instead, the preactivations are constrained to be G-equivariant functions of the network input in a way that couples weights across all layers. The result is a richer family of G-invariant architectures never seen previously. We derive an efficient implementation of G-DNNs after a reparameterization of weights, as well as necessary and sufficient conditions for an architecture to be "admissible"-- i.e., nondegenerate and inequivalent to smaller architectures. We include code that allows a user to build a G-DNN interactively layer-by-layer, with the final architecture guaranteed to be admissible. We show that there are far more admissible G-DNN architectures than those accessible with the "concatenated ReLU" activation function from the literature. Finally, we apply G-DNNs to two example problems---(1) multiplication in {-1,1} (with theoretical guarantees) and (2) 3D object classification---finding that the inclusion of signed perm-reps significantly boosts predictive performance compared to baselines with only ordinary (i.e., unsigned) perm-reps.