Equivariance Through Parameter-Sharing
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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 ϕ_W: ^M→^N, we show that ϕ_W 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 procedures for tying the parameters achieve G-equivariance and, under some conditions on the action of G, they guarantee sensitivity to all other permutation groups outside G.
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