E$(n)$-Equivariant Spherical Decision Surfaces

We present a constructive derivation of exactly E(n)-equivariant spherical decision surfaces by extending prior O(n)-equivariant hypersphere neurons to include translations. To achieve this, we present a decomposition of the features of the O(n)-equivariant neurons and provide explicit representations for translation and E(n)-transformations to fulfil the respective equivariance constraints. The resulting decision surfaces are exactly E(n)-equivariant without input centring or explicit pairwise differences, and admit explicit closed-form matrix representations. In addition, we numerically verify the correctness of the derivations and perform a downstream check of the resulting geometric primitives.