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, we show that its equivariance is linked to the symmetry group of network parameters. We then propose two parameter-sharing scheme to induce the desirable symmetry on the parameters of the neural network. Under some conditions on the action of G, our procedure for tying the parameters achieves G-equivariance and guarantees sensitivity to all other permutation groups outside of G.