g G. Thiscanbeshownasfollows: K1g S(W)Lg=K1g 1 |G| X h G K1 h WLh! LgsubstituteS(W)= K1g S(W)Lg (3) = 1 |G| X

W (13) =W (14) This means that the symmetrizer leaves the equivariant subspace invariant. The Idempotence Property Here we show that the symmetrizerS(W) from Equation 16 is idempotent,S(S(W)). For each experiment we first identified the groupG of transformations. Tosee that this is the case, consider the following example. Now, Kg[f(x)]=Kg[f2(f1(x))] (28) = f2(Pg[f1(x)] (f2 equivarianceconstraint) (29) = f2(f1(Lg[x])) (f1 equivarianceconstraint) (30) =f(Lg[x]) (31) and so the whole networkf is equivariant with regards to the input transformationLg and the outputtransformation Kg.