Title: Equivariance Through Parameter-Sharing,
Abstract: We propose to study equivariance in deep neural networks through parameter
symmetries. In particular, given a group $\mathcal{G}$ that acts discretely on
the input and output of a standard neural network layer $\phi_{W}: \Re^{M} \to
\Re^{N}$, we show that $\phi_{W}$ is equivariant with respect to
$\mathcal{G}$-action iff $\mathcal{G}$ 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 $\mathcal{G}$-equivariance and,
under some conditions on the action of $\mathcal{G}$, they guarantee
sensitivity to all other permutation groups outside $\mathcal{G}$.