Equivariant Symmetry Breaking Sets

TL;DR

This work proposes a novel symmetry breaking framework that is fully equivariant and is the first which fully addresses spontaneous symmetry breaking and proves that minimizing these sets translates to a well studied group theory problem, and tabulate solutions to this problem for the point groups.

Abstract

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, 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 and is the first which fully addresses spontaneous symmetry breaking. 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. The code for these examples is available at \url{https://github.com/atomicarchitects/equivariant-SBS}.

Figures (19)

• Figure 1: A classic double well potential of $x^4-2x^2+1$. Clearly this potential has reflection symmetry. The two minima are $x=1$ and $x=-1$.
• Figure 2: (a) Example of explicit symmetry breaking. Magnetic moment in domains align with a strong external magnetic field $\mathbf{B}$. The external field explicitly breaks symmetry of the system. (b) Example of spontaneous symmetry breaking. Presence of a moment in each domain breaks rotational symmetry. However, there is no magnetic field so governing laws of the system are symmetric. Consequently the observed moments are uniformly random in orientation.
• Figure 3: (a) Naive way to break symmetry in a triangular prism where one vector points to a vertex of a triangle and a second vector points to the lower or outer triangle. (b) A rotated version of the triangular prism in. Note that the same symmetry breaking objects now point to edges of the triangle rather than vertices. However, both prisms have the exact same symmetry elements.
• Figure 4: Diagram of how we might structure our symmetry breaking scheme. From our data $x$, we may obtain its symmetry $S$. This $S$ is then fed into a function $\sigma$ which gives us the set of symmetry breaking objects needed. We sample a $b$ from this set breaking the symmetry of our input and feed this $b$ along with the input $x$ into our equivariant function $f$. Finally we obtain an output $y$ which has lower symmetry than the input $x$.
• Figure 5: Diagram of how we break symmetry, but now we keep all possible outputs.

Theorems & Definitions (57)

• Lemma 2.1
• Definition 2.2: Symmetry breaking sample
• Definition 2.3: Explicit symmetry breaking
• Definition 2.4: Spontaneous symmetry breaking (SSB) function
• Definition 3.1: Symmetry breaking set
• Definition 3.2: Equivariant symmetry breaking sets
• Theorem 3.3
• Remark 3.4
• Definition 3.5: Degeneracy
• Corollary 3.6