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.

When trying to fit a deep neural network (DNN) to a $G$-invariant target function with $G$ a group, it only makes sense to constrain the DNN to be $G$-invariant as well. However, there can be many different ways to do this, thus raising the problem of ``$G$-invariant neural architecture design'': What is the optimal $G$-invariant architecture for a given problem? Before we can consider the optimization problem itself, we must understand the search space, the architectures in it, and how they relate to one another. In this paper, we take a first step towards this goal; we prove a theorem that gives a classification of all $G$-invariant single-hidden-layer or ``shallow'' neural network ($G$-SNN) architectures with ReLU activation for any finite orthogonal group $G$, and we prove a second theorem that characterizes the inclusion maps or ``network morphisms'' between the architectures that can be leveraged during neural architecture search (NAS). The proof is based on a correspondence of every $G$-SNN to a signed permutation representation of $G$ acting on the hidden neurons; the classification is equivalently given in terms of the first cohomology classes of $G$, thus admitting a topological interpretation. The $G$-SNN architectures corresponding to nontrivial cohomology classes have, to our knowledge, never been explicitly identified in the literature previously. Using a code implementation, we enumerate the $G$-SNN architectures for some example groups $G$ and visualize their structure. Finally, we prove that architectures corresponding to inequivalent cohomology classes coincide in function space only when their weight matrices are zero, and we discuss the implications of this for NAS.

There has recently been much work on the "wide limit" of neural networks, where Bayesian neural networks (BNNs) are shown to converge to a Gaussian process (GP) as all hidden layers are sent to infinite width. However, these results do not apply to architectures that require one or more of the hidden layers to remain narrow. In this paper, we consider the wide limit of BNNs where some hidden layers, called "bottlenecks", are held at finite width. The result is a composition of GPs that we term a "bottleneck neural network Gaussian process" (bottleneck NNGP). Although intuitive, the subtlety of the proof is in showing that the wide limit of a composition of networks is in fact the composition of the limiting GPs. We also analyze theoretically a single-bottleneck NNGP, finding that the bottleneck induces dependence between the outputs of a multi-output network that persists through extreme post-bottleneck depths, and prevents the kernel of the network from losing discriminative power at extreme post-bottleneck depths.