Equivariance Through Parameter-Sharing

We propose to study equivariance in deep neural networks through parameter symmetries. In particular, given a group G that acts discretely on the input and output of a standard neural network layer $\phi_W$, we show that equivariance of $\phi_W$ is linked to the symmetry group of network parameters W. We then propose a sparse parameter-sharing scheme to induce the desirable symmetry on W. Under some conditions on the action of G, our procedure for tying the parameters achieves G-equivariance and guarantee sensitivity to all other permutation groups outside G. We demonstrate the relation of our approach to recently-proposed "structured" neural layers such as group-convolution and graph-convolution which leads to new insights and improvement of these operations.