E$(n)$-Equivariant Spherical Decision Surfaces

Pavlo Melnyk, Michael Felsberg, Kostas Daniilidis

Track: long paper (up to 8 pages)

Keywords: Euclidean group, equivariance, spheres, decision surfaces

TL;DR: We make a neuron with a spherical decision surface E(n)-equivariant.

Abstract: 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.

Anonymization: This submission has been anonymized for double-blind review via the removal of identifying information such as names, affiliations, and identifying URLs.

Submission Number: 33