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 ϕW : ℝM → ℝN, we show that ϕW is equivariant with respect to G-action iff G explains the symmetries of the network parameters W. Inspired by this observation, we then propose two parameter-sharing schemes to induce the desirable symmetry on W. Our procedure for tying the parameters achieves G-equivariance and, under some conditions on the action of G, it guarantees sensitivity to all other permutation groups outside G.