Equivariant symmetry breaking sets

Abstract

Equivariant neural networks (ENNs) have been shown to be extremely useful in many applications involving some underlying symmetries. However, equivariant networks are unable to produce lower symmetry outputs given a high symmetry input. Spontaneous symmetry breaking occurs in many physical systems where we have a less symmetric stable state from an initial highly symmetric one. Hence, it is imperative that we understand how to systematically break symmetry for equivariant neural networks. In this work, we propose the first symmetry breaking framework that is fully equivariant. Our approach is general and applicable to equivariance under any group. To achieve this, we introduce the idea of symmetry breaking sets (SBS). Rather than redesign existing networks to output symmetrically degenerate sets, we design sets of symmetry breaking objects which we feed into our network based on the symmetry of our input. We show there is a natural way to define equivariance on these sets which gives an additional constraint. Minimizing the size of these sets equates to data efficiency. We show that bounding the size of these sets translates to the well studied group theory problem of finding complements of normal subgroups. We tabulate solutions to this problem for the point groups. Finally, we provide some examples of symmetry breaking to demonstrate how our approach works in practice.