Equivariant Symmetry Breaking Sets
YuQing Xie Tess Smidt
Equivariant neural networks (ENNs) have been shown to be extremely effective
in applications involving underlying symmetries. By construction ENNs cannot
produce lower symmetry outputs given a higher symmetry input. However,
spontaneous symmetry breaking occurs in many physical systems and we may obtain
a less symmetric stable state from an initial highly symmetric one. Hence, it
is imperative that we understand how to systematically break symmetry in ENNs.
In this work, we propose a novel symmetry breaking framework that is fully
equivariant. We emphasize that 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, we design
sets of symmetry breaking objects which we feed into our network based on the
symmetry of our inputs and outputs. 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 prove that minimizing
these sets translates to a well studied group theory problem, and 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.