Equivariant Symmetry Breaking Sets
Authors: YuQing Xie
Tess Smidt
Venue: Accepted by TMLR ()

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}.

Published in Transactions on Machine Learning Research (10/2024)
Equivariant Symmetry Breaking Sets
YuQing Xic
syuqinng@mitlcco edu
Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology
Tess Smidt
tsmidt tmidtt@mit.cmm edu
Department of Electrical Engineering and Computer Science
Massachuselts Institute of Technology
Reviewed on OpenReview: htips: // openreview. net/ met/ forum? id- tHKH4DNSRS
Abstract
Equivariant neural networks (ENNs) have been shown be extremely effective in applica- tions involving underlying symmetries. By construction ENNs cannot produce lower symme- try outputs given higher symmetry input. However, symmetry breaking occurs in many physical systems and we may obtain less symmetric stable state from an initial highly symmetric one. Hence, it is imperative that we understand how to systematically break symmelry in ENNs. In this work, we propose a novel symmetry breaking framework that fully equivariant and is the first which fully addresses spontaneous symmetry breaking. We emphasize that our approach general and applicable to equivariance under any group. To achieve this, we introduce the idea of symmetry breaking sets (SBS). Rather than design 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 natural way to define equivariance on these sets, which gives an additional constraint. Minimiz- ing the size of these sets equates to data elficiency. 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 demon- strate how our approach works in practice. The code for these examples available at https://github.. http://hww...eoeeoocc/cccc/tttt/
1 Introduction
Equivariant neural networks have emerged as a promising class of models for domains with latent symmetry (Wang et al.] 2022al). This is especially useful for scientific and geometric data, where the underlying symmetries often well known. For example, the coordinates of molecule may be different under rotations and translations, but the molecule remains the same. Traditional neural networks must learn this symmetry through data augmentation or other training schemes. In contrast, equivariant neural networks already incorporate these symmetrics and can focus on the underlying physics. ENNs have achieved state-of- the-art results on numerous tasks including molecular dynamics, molecular generation, and protein folding
Batatii et al.] 2022
Batzner et al. 2022
Daigavane et al. 2023 Ganea et al. 2021 Hoogeboom et al.
2022
Jia et al.
2020
Jumper et al. 2021
Liao & Smidi 2022).
A consequence of the symmetries built-in to ENNs is that their outputs must have equal or higher symmetry than their inputs (Smidt et al. 2021). However, we frequently encounter situations where we desire a lower symmetry output given a higher symmetry input. We refer to an inpul-oulput pair where the output has lower symmetry as a symmetry breaking sample. Such symmetry breaking samples occur frequently in physical systems. physics, these are classified into two types: explicit symmetry breaking and spontancous symmetry breaking (Castellani et al. (2003). In explicit symmetry breaking, the governing laws are manifestly asymmetric explaining the symmetry breaking samples while in spontaneous symmetry breaking the laws

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are symmetric but we still observe symmetry breaking samples. Notably, in spontaneous symmetry breaking there are multiple correct outputs for given input and we have modified form of equivariance. We expand on this in Section 2
One class of approaches conducive for explicit symmetry breaking is learning to break symmetry. For example, (midd et al. (2021) showed that the gradients of the loss function can be used to learn a symmetry breaking order parameter. This lets us identify what type of missing asymmetry is needed to correctly model the results. Another related approach is approximate and relaxed equivariant networks (Huang et all, 2023 van der Ouderaa et al., 2022 Wang et al., 2023 2022b). These networks have similar architectures to equivariant models, but allow nontrivial weights in the layers to break equivariance. Hence, they can learn how much symmetry to preserve to fit the data distribution. However, since these methods explicitly break equivariance, they are not appropriate for spontaneous symmetry breaking. If all symmetrically related lower symmetry outputs are equally likely in the data, then relaxed networks will see the distribution as symmotric and fail to break symmetry. Further, since oquivariance is broken, the method is not guaranteod to behave properly when shown data transformed under the group.
In the case of spontaneous symmetry breaking, there exist some works which partially solve the problem. Balachandar et al. (2022) design symmetry detection algorithm and an orientation aware linear layer for mirror plane symmetries. However, the scope of their methods are specific to mirror symmetries and point cloud data. Finally, Kaba & Ravanbakhsh (2023) take the approach of defining a relaxed version of equivariance which takes input symmetry into account. They derive modified constraints linear layers would need to satisfy for this relaxed equivariance and argue such models can give lower symmetry outputs. However, they mention these conditions do not reduce easily as for the usual equivariant linear layers. Further this method still does not provide a mechanism to sample all possible outputs.
Our work focuses on the spontaneous symmetry breaking case. This case is extremely important as sponta- ncous symmetry breaking occurs in many physical phenomena (Beekman et al. 2019). Hence the widespread adoption of machine learning techniques for scientific applications will inevitably run into symmetry breaking issues. Examples of existing applications which may run into difficulties include crystal distortion, predict- ing ground state solutions from Hamiltonians, and solving PDEs (Jafary-
al.) [019 Lewis et all, 2024 Lino et al. 2022). In the crystal distortion case, there are distortions from high symmetry to low symmetry structures (Kay & Vousden 1949). The lowest energy states of physical systems (ground states of the Hamiltonian) are often low symmetry and famously explains the Higgs mechanism for giving particles mass (Higgs 1964). Finally, Karman vortex sheets are a well known example of symmetry breaking in fluid simulations Tang & Aubry 1997).
In this work, we provide the first general solution for the spontancous symmetry breaking problem in equiv- ariant networks. We identify that the key difficully is how to allow equivariant networks output set of valid lower symmetry outputs. Similar to Smidt et al. (2021), we use the natural idea of providing additional symmetry breaking parameters as input to the model. However, rather than learning these parameters, we show that we can sample them from a symmetry breaking set (SBS) that we design based only on the input and out put symmetries. In particular, we prove that minimizing the size of the equivariant SBSs is equivalent to a fundamental group theory question. We emphasize that this fully characterizes how to efficiently break symmotry with cquivariant SBSs for any group. Counter-intuitively we find that it is sometimes beneficial to break more symmetry than needed.
The main features of our method are the following:
1. Equivariance: Our framework is fully equivariant. That we can achieve this is a key point of this work and allows simulation of spontancous symmetry breaking.
2. Simple to implement: Our approach only requires designing a set of additional inputs into an equivariant network. We have fully characterized the design of such sets.
3. Generalizability: We emphasize that our characterization of SBSs applies to any groups
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We would like to point out that to achieve our results, assume we can detect the symmetry of our imput and outputs. Further, there is the more general problem of treating symmetrically related outputs as the same in our loss. We discuss these limitations in Appendix D
The rest of this paper is organized as follows. In Section 2 we formalize the symmetry breaking problem, the distinction between explicit and spontaneous symmetry breaking, and the type of task performed in the spontaneous symmetry breaking setting. In Section 3 we examine the case where we break all symmetries of our imput. We motivate the idea of SBS and show that imposing equivariance leads to an additional constraint of closure under the normalizer. The intuition is that the normalizer characterizes all orientations of our data which do not change its symmetry. Next, we translate bounds on the size of the equivariant SBSs into the purely group theoretical problem of finding complements. We have tabulated these complements in Appendix [F] for the point groups. In Section 4 we generalize to the case where we may still share some symmetries with our input. In Section [ we describe how to construct SBSs in an actual implementation. Having fully described our framework, we then highlight how our method relates to prior symmotry breaking works in Section B Finally, in Section we introduce examples of symmetry breaking and demonstrate how
our method works in practice.
Notation and background: An overview of the notation and common symbols used can be found in Appendix A A brief overview of mathematical concepts needed in the paper can be found in Appendix B and an overview of ENNs can be found in Appendix C
2 Symmetry breaking problem
Here, we make precise what we mean by a symmetry breaking and the issue it poses for equivariant neural networks. We begin with the following observation first made in Smidt et al. (2021).
Lemma 2.1. Let X and Y be spaces equipped with a group action of G. We can choose an equivoriant f: -Y such that f(x) = y only if Staba (y) 2 Stabo(z).
See Appendix E.1 for a generalization of this lemma and a proof.
Hence, the output of an equivariant function must have at least the symmetry of the input. This motivates the following definition of symmetry breaking at the individual sample level.
Definition 2.2 (Symmetry breaking sample). Let G be group. A sample with input x and output y is symmetry breaking with respect to G if Staba(y) ž Staba(x).
Lemma 2.1 tells us symmetry breaking sample with respect G can never be perfectly modeled by a G-equivariant function. In experimental samples, may have random noise in our observations of our outputs which causes samples to be symmetry breaking. In such cases equivariance is beneficial since it can help remove the noisc. However, in some cases we truly have a symmetry breaking sample even with a perfect measurement. In physics this is classified into two cases: explicit symmetry breaking and spontaneous symmetry breaking. In the following discussion, it is useful to view the underlying model as a set valued function. A typical function f X >> Y can be thought of as a set valued function F :: X + P(Y) defined as F(x)= {f())) Note that if there is an action of G on Y, one can naturally define an action on P(Y) such that U CCY Y transforms gU- {gu: a E U). Hence there is natural way to define equivariance of set valued functions.
In explicit symmetry breaking, the underlying physics of the system is asymmetric. For example, there may be an unknown electric field which breaks rotation symmetry.
Definition 2.3 (Explicit symmetry breaking). Let G be group. A function F: X P(Y) which is not G-equivariant explicitly symmetry breaks G.
In such cases using an equivariant function actually prevents us from learning the true function.
In spontancous symmetry breaking, the underlying physics of the system is symmetric, however there is a set of stable lower symmetry outputs which equally likely.
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Definition 2.4 (Spontaneous symmetry breaking (SSB) function). Let G be group with actions defined spaces X, Y. Let F: X - P(Y) be G-equivariant. We say F spontaneous symmetry breaks at æ if there is some y E F()) such that Staba(r) > Staba(y).
The key things here are that the set valued function F is oquivariant even though individual observed samples (z. y) for y E F(x) may be symmetry breaking. Hence, our problem becomes the following.
Problem: How does create equivariant architecture which output set of possibly lower sym- metry outputs?
2.1
Examples of explicit and spontaneous symmetry breaking
2.1.1
Double well potential
The double well potential a classic example of symmetry breaking in physics. Consider a I-dimensional energy potential defined as U(x) -+-22" + Note that this system has reflection symmetry about x since substituting + x does not change the potential
2z
U(-x) (-x)* -(-x)) +1 x 1-x2222 + U(x).
2.5
2.0
1.5
10
0.5
-1.0
0.5
Figure 1: A classic double well potential of x² 2x + 1. Clearly this potential has reflection symmetry. The two minima are x 1 and x -1.
In physics often care about finding the ground state solutions which corresponds to the value of æ minimizing any given potential. In the potential above, we see that
U(x) =² 2x. 11 (x² - 1)² = (1-)
there two global minima at = +1. However, individually these minima break reflection symmetry since 1 * -1. Hence if our sample consists of input potential U(x) (perhaps encoded using polynomial coefficients) and output æ = 1, then it is a symmetry breaking sample with respect to reflection. Similarly the input-output pair U(x, = -1 would also be a symmetry breaking sample.
Note that by Lemma 2.1 network equivariant reflection across x = would unable to directly output the minima. Such a network realizes that 1 and -1 are symmetric and would return the average of 0.
Perhaps in the reflection symmetry and methods such as finding order parameters and relaxed equivariance learn to break such symmetry HHaang et al. (2023); van der Ouderaa et al. [2022] Smidt et al. (2021); Wang et al. 2022bb 2023). Since these learned functions are not equivariant, they are explicitly symmetry breaking by
cases, we decide to only consider positive solutions. In this case explicitly break
Definition 2.3
However, in many other cases both solutions are valid. In this case, the set f1, -13 of minima actually invariant under reflection but the individual solutions want are not. We would satisfy Defiiiiion 2.4 since
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for set-valued function returning both minima remain equivariant, while picking either of the individual minima gives symmetry breaking sample.
2.1.2 Ferromagnetism
A good physical example of symmetry breaking is ferromagnctism. Individual atoms in such materials have magnetic dipole moment and due to strong interaction between neighboring atoms, the moments tend to align at cool enough temperatures Aharoni (2000). Hence, if we heat and cool ferromagnetic material, we obtain regions where the dipole moments of individual atoms align. These regions are known as magnetic domains. Because the dipoles are aligned, there is a net magnetic moment in the domain which breaks rolational symmelry.
In the presence of a strong external magnetic field B, the moments of the magnetic domains tend align with the external field. This is an example of explicit symmetry breaking since the external field explicitly breaks rotational symmetry. From the perspective of Definition 2.3 we have non-equivariant function since we prefer the direction which aligns with the external field.
However, if there is no external magnetic field, the direction of the magnetic moment in each domain is uniformly random. This an example of spontaneous symmetry breaking, the governing laws are symmetric yet we observe asymmetry the individual domains. From the perspective of Definition 2.4 we have a. set of vectors in all orientations valid moments and do not need to break equivariance. However choosing particular moment would give symmetry breaking sample.
These two cases are depicted in Figure 2
B
B = 0
(a) Explicit symmetry breaking
(b) Spontaneous symmetry breaking
Figure (a) Example of explicit symmetry breaking. Magnetic moment in domains align with a strong external magnetic field B. The external field explicitly breaks symmetry of the system. (b) Example of spontaneous symmetry breaking. Presence of 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.
3 Fully broken symmetry
First, we consider the case where we break all symmetry of our input. Here, our desired outputs y share no symmetry with æ. In other words Stabs(y) (e). This will lay the foundation for analyzing the general case of partially broken symmetry. Let the symmetry group of data æ be S.
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3.1 Symmetry breaking set (SBS)
When there symmetry breaking there are multiple equally valid symmetrically related outputs. The purpose of symmetry breaking object is allow an equivariant network to pick one of them. In principle, want all symmetrically related outputs to be equally likely it makes sense to think of set B of symmetry breaking objects we sample from.
For any E S and b E B, since sh is symmetrically related to it is natural to also include it in B. Hence, acting with s on the elements of B should leave the set unchanged. Further, for any be B, the stabilizer (b) must. be trivial since we want break all symmetries of our input. This is exactly the definition of free group action of S on B. Hence, we define a symmetry breaking set as follows.
Definition 3.1 (Symmetry breaking set). Let S be symmetry group. Let B be set of elements which S acts on. Then B is a symmetry breaking set (SBS) if the action of S on B is a free action.
3.2 Equivariant SBS
However, the above definition an SBS insufficient when considering equivariance. Here, we show that we nood the stronger constraint of closure under the normalizer.
To illustrate the problem, consider a network which is SO(3) equivariant and triangular prism aligned that the triangular faces are parallel to the zy plane. Suppose our task was just to pick point of the prism. A naive way to break the symmetry is to have an ordered pair of unit vectors. The first vector is in the zy plane and points towards one of the triangle vertices. The second vector points up or down in the a direction, corresponding to the upper or lower triangle.
(a)
(b)
Figure 3: (a) Naive way to break symmetry a triangular prism where one vector points to a verlex 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 symmctry breaking objects now point to edges of the triangle rather than vertices. However, both prisms have the exact same symmetry elements.
However, consider the same prism but rotated 180 around z. We can check that the symmetry groups are exactly the same we want the same SBS. But the symmetry breaking objects are related differently. In the second prism, the first vector points to an edge rather than vertex. For equivariance, our symmetry breaking objects should be related to both prisms in the same way. So our choice of SBS was not equivariant.
Here, one may simply choose a canonical orientation and decide that we will rotate the original SBS by 180º in the latter case. However, our input may be arbitrarily complicated, and it may be hard to decide on a canonicalization. Further, canonicalization may introduce discontinuitics. Hence, we would like construct SBSs to be only dependent on the symmetry of our data, not how our data is represented. To understand exactly what additional condition is necessary, need to carefully investigate how the symmetry breaking schome works.
Let ʃ be our G-equivariant function and æ be our input data. Suppose we know the symmetry S of our input. Let B be some set with a group action symmetry breaking objects based on just information about the input symmetry. So suppose we have a function Sub(G) + P(B) that does so. This function takes in subgroup symmetry and gives a SBS composed of elements from B. Then the symmetry breaking step happens when we take random sample
defined on it. We would like to obtain our set of
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S
0
B
f y
Figure 4: Diagram of how might structure our symmetry breaking scheme. From our data x; we may obtain its symmetry 8. This S is then fed into function which gives us the set of symmetry breaking objects needed. We sample b from this set breaking the symmelry of our input and feed this along with the input x into our equivariant function f. Finally we obtain an output which has lower symmetry than the input x.
from the SBS. This symmetry breaking object is then fod into our equivariant function, allowing it to break symmetry. A diagram of this process is shown in Figure 4
Certainly, since we break the symmetry of our imput data, we break equivariance. However, imagine we give our function all possible symmetry breaking objects and collect at the end the set Y of all outputs our model gives. This process shown Figure
g would then not need to break any symmetry. This is because all outputs has the same symmetry as the input. Hence we can impose equivariance on our process.
S
> B
f Y
Figure 5: Diagram of how we break symmetry, but now we keep all possible outputs.
It is well known that the composition of equivariant functions remains equivariant. Hence, we just need to impose equivariance on a. In order to do so, must understand how the input and output transform. Suppose we act on our data with some group element g E G. Then it becomes gz. Since S is the symmetry of our original data, we find gr = gst = (gsg (gZ) for any 8 E S. So the symmetry of the transformed data is gS Hence, the input of transforms as conjugation. Next, recall the output of a is some subset B of clements of B. Since there a group action for G defined on B, we can define an action on B by just acting on its elements and forming new set. Figure shows how our procedure would change if it were equivariant, and act input by some group element g.
gSg 1
9B
σ
g.c
f
gY
Figure 6: Diagram of what happens when act on the input with some group element g.
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We can now clearly see the issue. The imput into a transforms under conjugation, and the stabilizer of subgroup under conjugation is precisely the definition of the normalizer Na(S). However, in many cases (S) is a supergroup of S. Therefore by Lemma E.1 our SBS not only needs to be invariant under S, but also be invariant under Na(S). See Appendix E.2 for a more formal justification.
Hence, we will call a SBS equivariant if it can be the output of an equivariant a. The intuition above tells this amounts to closure under the normalizer (a(S).
Definition 3.2 (Equivariant symmetry breaking scts). Let be a subgroup symmetry of a group G and B be a set with an action of G defined on it. Let B B be SBS. Then B is G-equivariant if Vy E NN((S) we have B = gB.
3.3 Ideal case and complement of normal subgroups
Now that we know how to equivariantly break symmetry, we would like to understand how to do efficiently. Intuitively, we expect smaller SBS to be better. If we have a larger SBS, multiple symmetry breaking objects map to the same output so the network needs to learn that these are the same. Reducing the SBS would decrease the equivalences our network needs to learn. In the ideal case, exactly one symmetry breaking parameter corresponds to each output. Since our outputs are related by symmetry transformations (transitive) under S, this correspouds to the equivariant SBS being transitive under S. It turns out, we can equate constructing ideal equivariant SBSs to the constructing complements of normal subgroups. slightly weaker version of this statement can be found in Theorem 3.1.4 of Kurzweil & Stellmacher (2004).
Theorem 3.3. Let G be group and S subgroup. Let B be G-equvvriant SBS for S. Then it is possible to choose an ideal B if and only has complement in (S(S).
If is element where (t) (b) is complement of S in Na(S), then Orbg(b) is an ideal G-equivoriant SBS.
Remark 3.4. It turns out the complement if it exists is isomorphic to Na(S)/S. We can intuitively think of Na(S)/S as giving all possible orientations of our data such that its symmetry remains unchanged.
The proof of this theorem is in Appendix E.3 Finding complements of normal subgroups is well studied group theory problem Kurzweil & Stellmacher (2004). For the point groups, which are the finite subgroups of (3)), we have tabulated the complements if they exist in Appendix F
The intuition for this theorem comes from the following observation. Essentially equivariant SBSs consist of orbits of NG) By the orbit-stabilizer theorem, we can reduce the size of an orbit by making an element generating the orbit more symmetric. However, must still fully break the symmetry of S. The complement, if it exists, is essentially the maximal symmetry b can have while still breaking the symmetry of S.
3.4 Nonideal equivariant SBSs
In the case where we cannot achieve an idcal equivariant SBS, we would still like to characterize how efficient it is. To do this, we define what we call the degeneracy of an equivariant SBS. In general, each orbit under S gives SBS which be matched one to one to outputs.
Definition 3.5 (Degeneracy). Let B be a G-equivariant. SBS for S. We define the degeneracy to be
(Bg IB/S).
Note that an ideal equivariant SBS Bideal (if it exists) has exactly 1 orbit of S, so Degs (Bdeat) = 1. We would also like to understand how small we can make the degeneracy if we cannot make it 1. It turns out Theorem 3.3 allows us to convert this to a group theory problem.
Degs(Bideat)
Corollary 3.6. Let G be a group and S a subgroup. Let M be such that S < M S No(S). Let B be a G-equivoriant SBS for S which trunsitive under No(S). Then it possible to choose B such that every S-orbit is also a M -obitt if and only if has complement in M. In particular,
Degs (B) S (No(S)/M)
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See Appendix E.4 for proof. In the ideal case we make M to be Na(() so the above formula gives degeneracy of as expected.
4 Partially broken symmetry
We can now use our framework for full symmetry breaking to understand the case of partial symnetry breaking. In this case, our desired output may share some nontrivial subgroup symmetry K I S with our imput. Note the case of K = corresponds to full symmetry breaking and K = corresponds to no symmetry breaking.
4.1 Partial SBS
Similar the full symmetry breaking case, would like to create of objects which we can use break our symmetry. Now relax the restriction of free action. Intuitively, can allow symmetry breaking objects to share symmetry with impul, as long it is lower symmelry than However, the symmetrically related outputs may be invariant under different subgroups of S. Recall that if some element y gets transformed to sy, its stabilizer K gets transformed to sKs- Hence, the stabilizers of the outputs are the subgroups conjugate to K under S denoted as Cls(K). Basod on this intuition, we can define partial SBS as follows.
outpuis.
Definition 4.1 (Partial SBSs). Let S be a symmetry group and K a subgroup of S. Let P be a set of clements with an action by S. Then is a K-partial SBS if for any p E P, there exists some K' E Cls(K) such that K' 2 Staby(p).
Certainly, a full SBS is a partial one as well since the stabilizers of all its elements under S is the trivial group. In general, we can always break more symmetry than neoded and still obtain our desired output. However, useful to consider the case where we only break the necessary symmetries. Counter intutirlntuiiivell, we discuss Section that this turns out not always be optimal.
Definition 4.2 (Exact partial SBS). Let S be a group and K a subgroup of S. Let P be a K K-partial SBS for S. We say P is exact if for all E P, we have Stabs(p) E (ls(
4.2 Equivariant partial SBS
Similar to before, we define equivariant partial SBSs. The idea is the same, but now we need to identify the symmotry of the input and the set of conjugate symmetrics for the output. Definc
SubCI(G) = {(S, Cls(K)) S Sub(G), K S S).
Let P a set with a group action of G defined on it. As before, the idea is that we have an function . : SubCI(G) + P(P) which outputs our partial SBS. The condition of equivariance for our partial SBS is imposing equivariance
S, Cls(s(K)
π
P
f Y
Figure Diagram how we perform partial symmetry breaking. Here, we need specify not just the symmetry of our input but also the symmetries of output. Since any of our outputs are equally valid, it only makes sense to specify the set of conjugate subgroups Cls(K) our outputs are symmetric under.

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The symmetry breaking scheme is depicted in Figure 7
impose equivariance this diagram. We need to know how Cls(K) transforms. Note that if our input gets acted by 9, expect the outputs to also get acted by Since K is the stabilizer of one of the outputs, we expect K to transform to gKg-'. Hence we have the transformation
As before,
Cls(K) - (g
Similar to before, by Lemma we need the output of x to also be invariant under the stabilizer of the input. Noting that the normalizer defined as the stabilizer of under conjugation, we can define generalized normalizer as the stabilizer of S, Cls(K).
gSg , -1 (gKg
gP P
g
f
gY
Figure Diagram of how our symmetry scheme changes when we transform our input by some group element g G.
Definition 4.3 (Generalized normalizer). Define the generalized normalizer No(S, K) to be
NG(S, K) = {g gKg- E Cls(K),g E Na(S)).
We can now define equivariant partial SBSs using closure under this generalized normalizer. See Ap- pendix D5 for a more formal justification.
Definition 4.4 (Equivariant partial SBSs). Let S be a subgroup symmetry a group G. Let P be a K-partial SBS. Then P breaks the symmetry G-equivariantly if Ag E No(S, K) have P= PggP.
Note that closure under Nc(S, K) weaker condition than closure under Nc(S). Hence any equivariant full SBS also equivariant K-partial SBS for any K.
4.3 Ideal equivariant partial SBS
Similar to the full symmetry breaking case, we ideally would like to have a one one correspondence between elements in our equivariant SBS and our symmetrically related outputs. For this to happen, clearly need our SBS to be exact and for our SBS to be transitive under S. We can generalize Theorem 3.3 to obtain a necessary and sufficient condition to have an idcal equivariant partial SBS.
Theorem 4.5. Let G be group and S and K be subgroups K < S < G. Let P. be G-equivariont K- partial SBS. Then we can choose an ideal P (exact and transitive under S) if and only if Ns(K)/K has complement in (Nc(S.K)(K//K.
If p is an element such that (tak is complement of Ns(K)/K in (K)/KK then Orbs(p) is an ideal G-equivariant K -partial SBS
See Appendix
F.6 for a proof.
4.4 Nonideal equivariant partial SBS
Similar to the full symmetry breaking case, when we cannot achieve an ideal equivariant partial SBS we want characterize how efficient our nonidcal partial SBS is. Again, the idca that in the nonideal case, our network needs map multiple symmetry breaking objects to the same output. We define the degeneracy of P to quantify this multiplicity.
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Definition 4.6 (Degeneracy). Let G be group, S be subgroup, and K a subgroup of S. Let P be G-equivariant K-partial SBS for S. Let T be transversal of S/K. Let Pt be such that every E P is uniquely written as p = tpr for some t E T"' and Ps E Pt. Then we define
Degs, = PPrl-
The intuition for this definition is that Pt is the set of objects which together with our imput may get mapped to some output y by our equivariant network. In other words, we have fxx Pt) = {v} for equivariant f and all other P get mapped to different symmetrically related outputs. Without loss of generality assume has (y) = K. Then for any symmetrically related output by (where E ?), we can see from equivariance of that f(x,tP)) = {ty). It now clear that the size Pt counts how many symmetry breaking objects must be mapped to the same output.
Note that in the case K 1, S/K = S Pt just consists of representatives from P/S. So this reduces to the degeneracy defined for full SBSS Also, note that in the ideal case, there is exactly symmetry breaking object for each output. So degeneracy is in that case.
We would like to derive bounds on the degeneracy of our equivariant partial SBSs. Similar to the full SBS case, we use Theorem 4.5 to convert this into a group theory question.
Corollary 4.7. Let G be group, S a subgroup, and K subgroup of S. Let K' be a subgroup of K and M a subgroup of NG(S, K) iinNe Na (') which contains S. Suppose P is a G-equivoriant K K-pprtial SBS for S which is transitive under (S, K). We can choose P such that Sta E C)No(S.K)( (K") for all p and all S S-orbits in P are also M Morriiss if and only if Ns(K')/K has a complement in Nas (K")/K Further, such P has
Degs. (PS |K/R| [No(S, K)/M
See Appendix
E.7 for proof.
4.5 Optimality of exact partial symmetry breaking
Note that in the previous section, we have been very careful to allow our partial SBS to break more symmetry than ncedod. Intuitively, we would like say that always optimal break down exactly the symmetry of our output. That is, we only need to consider exact partial SBSs.
Certainly, ignoring any equivariance constraints, given any non-exact K- K-partial SBS, we construct an exact K- K-partial SBS by picking an clement b with Stabs(b) < K and identifying its orbit under K together as one partial symmetry breaking object p Kb. We construct the orbit of under action by S as our K-partial SBS.
We might expect that some modification of this construction can convert any non-exact equivariant K- partial SBS into an exact equivariant K-partial symmetry breaking one. Naively, we just take the orbit of the elements in the construction above under No K) to obtain G-equivariance. However, in Appendix G come up with explicit example where no exact equivariant K-partial SBS is smaller than the best equivariant full SBS.
5
Constructing SBSs
Now that have fully characterized what equivariant symmetry breaking sets are, show how to construct them. We focus G O()) in this section though the ideas here applicable for other groups as well.
5.1
Expressing subgroups
First, important to have a way of expressing subgroup of G, the group we want to be equivariant under. The subgroups of (33) well studied and have been completely classified Hahn et al. (183). In particular,
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there are 7 infinite axial families of finite point groups and additional finite ones. There only 5 infinite subgroups which are closed. However, the names of these subgroups do not specify how they are "oriented" in ((3). Hence, we propose to represent the subgroups in the following way. We first choose a canonical orientation of the classified point groups.
Figure 9: Two identical triangular prisms differing by a rotation. Both have symmetry Doh by name, however the actual symmetry axes differ.
A standard choice inspired by the Hermann-Na Marnann naming scheme for point groups. For the 7 infinito series of axial groups, we choose to align the high order symmetry axis along the z-axis. in addition there are 2-fold rotations, we choose one of them to be aloug the r-axis. If there are no 2-fold rotations but there are mirror plane parallel to the z-axis, we choose the yи-plane to be in the group. Of the remaining 7 point groups, 5 are cubic groups. For these we can choose the cube they leave invariant to have sides perpendicular to one of the v, 2 axes. Finally, the remaining 2 point groups are the icosahedral groups with and without inversion. For these we can choose to align a 5-fold axis with the z-axis and a 3-fold axis with the r-axis.
Next, for any point group S with arbitrary alignment, there is always some E 0(3) (in fact we can always pick g E S0(3)) such that Sg brings it to the canonical orientations defined above. Hence, we can always express an arbitrary point group S a pair g. name(S) of rotation and the name of the group.
To generalize to arbitrary groups, note that the point groups classify the classes of conjugate subgroups Clor3) (8) of 0(3). Hence for general G, we need to classify the corresponding classes of conjugate subgroups Cla(S) and choose a canonical representative for each class in order to apply schome outlined in this section.
5.2 Representing a set of conjugate subgroups
In the partial symnmetry breaking case, we also provide set of conjugate subgroups. Similar to our notation Cls(K), we specify set of conjugate subgroups with (S, K), where subgroups S and K are represented in the way described previously.
5.3 Representing a SBS
The idea is very simple. We start with some object which breaks enough symmetry for our task. To satisfy the equivariance condition of closure under the normalizer generalized normalized, we can simply take the orbit as our SBS or partial SBS. In principle SBS can consist of multiple such orbits, but can always only use one orbit as a SBS and multiple orbits increase the degeneracy. Hence we assume all our SBSs consist of one orbit and we can fully specify a SBS as a pair (b, N) where b is a symmetry breaking object and is a group over which we take the orbit of b.
In the case of finite group N, we can explicitly compute the elements in the orbit. However, if N is infinite then this does not work. In practice, it is usually enough that we can sample lower symmetry outputs. Hence, it suffices to be able to sample the SBS which we can do by sampling element from N.
5.4 Constructing an equivariant full SBS
We would like to construct a full SBS given an input symmetry S (g, name However, we want to do so in an equivariant way. One way to achieve this is first consider only any input group in its canonical orientation and construct SBS B for it. Then simply returning gB would guarantee that our construction
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is equivariant. Hence, we just need to construct canonical SBS for each possible point group. Note that this can be viewed as case of equivariance by canonicalization Kaba et al. (2023).
Note an equivariant full SBS needs closure under a normalizer. If we work with 0(3) equivariance, we simply look up the normalizer (S) from Table ! If we are equivariant under some subgroup G C O(3) then we note that (S) = (ora) nG which can be used compute the desired normalizer. All that remains is how to specify a canonical symmetry breaking object for each point group in canonical orientation. If have such an object, then we obtain the following algorithm for creating equivariant full SBSs. Let Normalizers be function which takes in name of point group and gives the corresponding normalizer classified in Table
Algorithm Equivariant full SBS
Input
5
Symmetry of input expressed as pair (g. nam)(S))
Canonical symmetry breaking object
Output B
Symmetry breaking set expressed pair (8, N)
(n, name(No (S))) + Norma ietaaliizeeeenaa (name(())
N < (gn, name (S))
return N)
In general, the choice of a canonical symmetry breaking object is flexible. To satisfy the definition, one just needs the corresponding object for name(S) to not share any symmetries with S (e, na)))) However, as discussed Scction 3.3, an ideal SBS should be more efficient than a nonideal one. Hence, if possible we would like to pick such an object that it generates conditions needed to choose ideal SBS. In particular, we would need the additional condition that b have the symmetry of a complement H while not having the symmetry of S. We fully characterized the relevant cases in Appendix
ideal SBS. Theorem 33 tells exactly the
5.5 Constructing equivariant partial SBS
In the partial symmetry breaking case, we want to obtain a partial SBS from the symmetry of the input and the set of conjugate subgroup symmetries of the outputs. Importantly, note that we want closure under No(S, K) rather than under NS) To compute Ns K), we use the following fact.
Lemma 5.1. We have the following formula
No
= = ((N((S) No(K))
See Appendix E.8 for proof.
Thus we get the following algorithm for computing an equivariant partial SBS.
We emphasize that for any K' < K, a K' K'-partial SBS can also serve as a K-partial SBS since we can always break more symmetry than nooded. In particular, a full SBS often suffices for simplicity.
Similar to the full SBS case, we have flexibility in choosing canonical object used to generate the partial SBS. To satisfy the definition, all we need is for Stabe(p) < K' for some K' CKs( An ideal partial SBS is desirable, especially if we wish to ensure we do not break any extra symmetry. The condition given by Theorem 4.5 is more complicated. However, if we have FindCompl ement function, we can automate the process of finding a symmetry an object which generates an ideal partial SBS should have. Here, let Quotient be a function which returns quotient group and a mapping from cosets to elements of the quotient group. We have the following algorithm which returns a pair of subgroups (H, K) such that if p has H < Stabo (p) and Spabs(p) K then p generates ideal partial SBS.
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Algorithm 2 Equivariant partial SBS from object
Input
S
Symmetry of imput expressed as pair (gs. name(S))
Cls(K)
Set of conjugate subgroups expressed as (S, K) ((gs, mame(S)), (gK, (aame()))
Canonical partial symmetry breaking object
Output P
Symmetry breaking set expressed pair (v/, N)
Ni + Normalizersjnanee
(n, name (N2)) < Normal iiemsiiiieeeree (mam(())) (K))
N2 (gs name
N N1 N
N (e, (aaa(m)))
N N55
return (gsp, N)
Algorithm 3 Ideal partial SBS generating object symmetry
Input
S
Symmetry of input expressed as pair (gs, nSm)
Cls(K)
Set of conjugate subgroups expressed as (S, K) ((gs; nSm) (9K, name K)))
Output
H
Symmetry needed for object to generate ideal partial SBS
N₁ Normaooollzziii pormalizerssnanee
(n, name (N2)) < Normal izers (name(K))
N2 (95 gKn, name(N))
N Ni N2
N' + (e.name(S)) n N₂
(Q1, Φ) + Quotient [N, (gg 9к. K)]
Q2 - d(N')
C + FindComplement/ [Q, Q21
if C exists then
(h, nam.mmm)H)) (C)
return (ggsh, name (H)), K)
else
return None
end if
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5.6 Using symmetry breaking object
In general, one has the freedom to decide how to incorporate symmetry breaking object into their model. This is often dependent on which equivariant architecture we apply our framework to and influences the choice of a canonical symmetry breaking object. For example, if we consider images and rotational symmetry, one natural choice is to add an additional image channel for the symmetry breaking object. In this case one would naturally choose to use an asymmetric image as the canonical symmetry breaking object. In the experiments section, we apply our framework to equivariant message passing graph neural networks (GNNs). In that case we can insert the symmelry breaking object an additional node feature and specify the representation it transforms
6
Relation to other works
Having fully described our framework, it is useful to briefly discuss how our method relates to existing symmetry breaking works.
6.1 Adaption of explicit symmetry breaking methods
In Smidt et al. [2021), an additional symmetry breaking object is learned from the data. Similarly in relaxed equivariant networks, we learn additional nontrivial weights which we can often interpret as an additional
symmetry breaking object (Huang et al. 2023 van der Ouderaa et al. 2022 Wang et al.] 2022a b 2023). It
is therefore natural to ask how these learned symmetry breaking objects relate to our framework.
In the case where we apply the method Smidt al. (2021) to single symmetry breaking sample x, y, then we oblain some additional input b which lets an equivariant model output y when given input x. As long there is an action of G on the learned symmetry breaking object b, we can adapt it to form equivariant SBS by taking an orbit under (S(S) (or only No(S, K)). In fact with a suitable group transformation we may simply adopt them as canonical symmetry breaking objects for Algorithuns anddd 2
However, these learned symmetry breaking objects will in general not generate an ideal SBS. The procedure of (Smidt et al. (2021) uses gradicnts of G-invariant loss. Since the loss depends on both x, y, we can view the obtained symmetry breaking object as coming from some G-equivariant function b = h(x, Hence, b will have the symmetry of Stabe (1:) n Stabe which is why it can help equivariant model break the symmetry of input In fact, this means that we always will obtain an exact SBS in the partial symmetry breaking case. However, our Theorem 3.3 and 4.5 tells us b may need additional symmetry to generate an ideal SBS which Smidt et al. (2021) does not guarantee.
6.2 Relation to spontaneous symmetry breaking methods
In the work of Balachandar et al. (0022), symmetry breaking is done through introducing nonequivariant components in their architecture. In particular, their symmetry detection algorithm generates vector perpendicular to the mirror plane and hence can be modified to construct. SBS by taking positive and negative values of the vector.
The framework outlined by Kaba & Ravanbakhsh 2023 where rather than f(gx) = (x) we instead just. have [gr)) = g' fxx for some g' E gStaba(>) = gS. Tn Definition 2.4 we arguc we should consider a sct-valued function F X + y and impose equivariance for group action on the set. By restricting a specific member of the output set, we would obtain function f X + Y so that f(x) E F(z). It is not hard to show that such an satisfies the relaxed equivariance of Kaba & Ravanbakhsh (2023) This can be implemented by modifying Algorithm so we pick a specific b E B rather than returning the entire SBS B.
advocates for a notion relaxed equivariance
Finally, a natural attempt for breaking symmetry is simply noise injection (Liu et al. 2019 Locatello et al, 2020). Noise which is isotropic with respect to group G can be viewed as SBS, however, we expect our network would need to learn to map multiple (possibly infinite) input, noise pairs to the same output. A motivation of our work is that by using knowledge of input and output symmetry, we can reduce this
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degeneracy and characterize exactly how to analyze this with Corollaries 3.6 and 1.7 In many cases, can even have the ideal degeneracy of 1. However, also let us prove we sometimes cannot do better than noise. For example, we can use Theorem F. and Corollary 3.6 to show that for G = (2),), any full symmetry breaking schome using only input symmetry must have inlinite degeneracy.
would like to point out that our theoretical results
7 Experiments
Here, we provide some example tasks where we apply our framework of full symmetry breaking and partial symmetry breaking to an equivariant message passing GNN. We consider the cases where we can find an ideal equivariant SBS or partial SBS. This section serves primarily as a proof of concept for how our approach works in practice. Note that while these examples only have a single type of input in training for simplicity, this is by no mcans a restriction of our framework. In general one simply uses the given symmetry of the inputs and oulputs in partial symmetry breaking in addition to the algorithms from Section to choose what specific symmetry breaking object to use. This lets us work with multiple types of imputs and input symmetries with the same model.
7.1
Full symmetry breaking: triangular prism
7.1.1 Task
For an example of full symmetry breaking, we consider the task of picking a vertex of a triangular prism, similar to that described in Section 3.2
Input: Graph with 6 nodes with edges given by the edges of the prism and position features at the nodes corresponding to the positions of the vertices. We add an additional pseudoscalar feature of magnitude 1.
Input symmetry: The prism with only the position features has Das symmetry, but the addition of pscudoscalar fcatures reduces this to D3 symmetry.
Output:
Vector which points from the center of the prism to one of the vertices. Note that there is a set of equally valid outputs.
Output symmetry: Nonzero vectors have C.cs symmetry where the high symmetry axis is aligned along the vector.
We note that the there is no shared symmetry between the output and input if we include the pseudoscalar feature on the prism. Hence, this is an example of a full symmetry breaking task. Further, this is a spontaneous symmetry breaking task because as we rotate the prism, the set of valid vectors rotate as well.
7.1.2 Architecture
We apply our framework to the default equivariant message passing GNN implemented in the e3nn pytorch library Geiger & Smidt (2022). Each layer consists of the equivariant 3D steerable convolutions followed by gated nonlinearities described in Weiler et al. (2018). Node features in this network are separated into 3 types, position fcatures, node features, and node attributes. The position features are usod for point convolutions and node altributes are permanent features which remain through all message layers. Naturally, we input the positions of the prism vertices as position features. We input the pseudoscalar feature as shared node attribute for all vertices. Similarly, to implement the input of a symmetry breaking object, we also add it as shared node attribute for all vertices.
We speicify the irreps of the hidden features in our model up to = 2 both parities and use up to = 4 spherical harmonics for point convolution filters. Further for the radial network, we use a 3 layer fully connected network with 16 hidden features in each layer. In the final layer, we specify odd node I (vector) feature. We take the sum of the final output vectors the model output.
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7.1.3 Symmetry breaking set
In this case, it turns out one can set the vector 1/22,1/2,00 as a canonical symmetry breaking object and it will generate an ideal SBS. See Appendix H.I.1 for details on why this choice works. Following Algorithm| []] we see that we will obtain a set of unit vectors parallel an edge of the triangular faces of the prism as our SBS.
7.1.4 Results
(a)
(b)
Figure 10: (a) Output (red) generated by our model and symmetry breaking object (blue) given. (b) The set of all the outputs generated by our model if we feod all symmetry breaking objects.
Since our framework is equivariant, we fix the orientation of the prism in training.
We first fix a choice of one symmetry breaking object from our equivariant SBS and one of the vertices of the triangular prism. We then give the chosen symmelry breaking object as the additional input to our model and train the model match the vector from the center of the prism the chosen vertex using MSE loss. An example of the result of this training is shown in Figure [0a] We also observe that no matter which pair of vertex and symmetry breaking object we pick, our equivariant network is able to learn the vector pointing that that vertex. In practice, this means that we can match any of our symmetry breaking objects with a single observed output.
Once trained on one pair of symmetry breaking object and vertex, the equivariance of our GNN means that inputting the other symmetry breaking objects our SBS gives the other symmetrically related outputs. This is shown in Figure [Ob
Further, rather than picking vertex, also tried modifying our loss so that compute the MSE loss for all choices of vertex and take the minimum. Hence, our network can learn which vertex to pair with each symmetry breaking object. In this prism example, our pairing is random. This method of taking the minimum loss is especially useful when we have multiple instances of symmetry breaking in our data.
Finally, Appendix H.1.2 we demonstrate that a non-equivariant SBS fails as described in Scction 3.2 and in Appendix H.1.3 we demonstrate the degeneracy of nonideal SBS compared to an ideal one as described
in Section 3.4
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7.2
Partial symmetry breaking: octagon to rectangle
7.2.1 Task
For an example of partial symmetry breaking, we consider the task of deforming an octagon to rectangle.
Graph with 8 nodes with edges corresponding to the edges of the octagon and position features nodes corresponding to positions of the vertices. We add an additional pseudoscalar feature of magnitude 1.
Input:
Input symmetry: The octagon with only the position features has Dsh symmetry, but the addition of pseudoscalar features reduces this to Dg symmetry.
Output: Graph with 8 nodes with edges corresponding to the edges of the octagon and position features at nodes corresponding to new positions of the vertices. The positions are such that 4 of the vertices which form the shape of rectangle remain unchanged and the remaining 4 vertices are shifted so their positions coincide with the nearest of the 4 vertices. Note that there is a set of equally valid outputs.
Output symmetry: Output graph has D24 symmetry.
We note that the shared symmetry between the input and output is D2 compared to the Dg symmetry of the imput. Hence, we may apply our partial SBS framework here. The reason for forcing Dg symmetry by adding chirality is demonstrate that each step in Algorithm in general nontrivial operation.
7.2.2 Architecture
We use the same default equivariant message passing GNN implemented as for the prism case. The symmetry breaking object is a different type of representation so we specify different irreps for the shared node attribute for all vertices to input the symmetry breaking object. In addition, we keep the final output irreps as a vector at all nodes, but rather than averaging them, we now interpret them as a displacement to generate new positions for the distorted octagon.
7.2.3 Symmetry breaking set
In this case, choosing (0, 0, for a -1-2 2 irrep canonical partial symmetry brcaking object generates an ideal partial symmelry breaking set. See Appendix H.2.1 The resulting SBS from Algorithm 2 is the set of -= harmonics "parallel" to the edges of the octagon.
7.2.4 Results
Similar to the prism case, try training by matching a specific symmetry breaking object to rectangle distortion. When the symmetry breaking object and rectangle are compatible (share the same symmetries). then our model has no problem learning to deform the octagon into the rectangle. This is shown in Figures [Ia and Iib An interesting failure case occurs when we try to match a symmetry breaking object and rectangle with incompatible symmetries. This is shown in Figure TTc Here, the D2 symmetry of the rectangle and of the symmetry breaking object are misaligned. As a result, our model predicts an output which has symmetry of D4 which the group generated when include the symmetry elements of both the target rectangle and the symmetry breaking object. Hence, the resulting shape is square.
As with the triangular prism case, we also tried letting the model choose which rectangle to deform to given a symmelry breaking object. In this case, our model computes loss separately for all 4 possible rectangle distortions and takes the minimum. We note that for a given symmetry breaking object, of the possible rectangles are symmetrically compatible while 2 are not. Over 200 random initializations, we find roughly 30% of the time our model attempts to match symmetrically incompatible symmetry breaking objects to a rectangle. This is better than the 50% we would expect if it matches pairs randomly.
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(b)
Figure 11: (a) Output (bluc) of our model when we match symmetry breaking object with compatible rectangle. (b) Output (blue) of our model when we match a symmetry breaking object with a different compatible rectangle. (c) Output (blue) when we match a symmetry breaking object with an incompatible rectangle (green). Note the square has symmetries of both the symmetry breaking object and the target rectangle.
7.3
BaTiO; phase transitions
7.3.1 Task
Finally, we demonstrate our framework on more realistic example. For this, we examine the crystal structure of barium titanate (BaTiO3). Specifically, as we decrease temperature, there is phase transition from high space-group symmetry P mmm state to lower space-group symmetry Pamm state at 403K Kay & Vousden 1949); Oliveira et al. (2020); Woodward (1997). The high and low symmetry states are shown in Figures 2a and 12b respectively. Note that the real distortions are rather small and hard to see visually, Table 1 provides some numerical quantities which help distinguish the two. In particular, there are 3 distinct Ti-0-Ti bond angles in a primitive coll, 2 which are distorted equally to 171.80 the low symmetry structure. This bent angle is shown more clearly the schematic Figure 12b
Input: Graph with 5 nodes corresponding to the atoms in primitive cell of BaTiO3, We have position features corresponding to the positions of the atoms in the cell. We fix the unit cell to be cube with sides 4A, close to the size of the real cells.
Input symmetry:
The high symmetry structure has a space group symmetry of Pm3m. Ignoring trans- lational symmetries, this corresponds to O₁ point group symmetry (denoted as m3m in Hermann-Maugh notation). Note the symmetry already listed in the materials project database Jain al. (2013).
Output:
Graph with 5 nodes corresponding to the atoms in primitive of BaTiO3, We have position features corresponding to the positions of the atoms in the cell. We fix the unit cell to be cube with sides 4A, close to the size of the real cells.
Output symmetry:
The high symmetry structure has a space group symmetry of P4mm. Ignoring translational symmetries, this corrsponds to C4. point group symmetry (denoted as 4mm in Hermann- Maugin notation). Note the symmetry is already listed in the materials project database Jain et al. (0013
The input and output share Dan symmetry compared the O₂ symmetry of the input. Hence may apply partial symmetry breaking framework here.
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(b)
Figure 12: (a) Initial high symmetry crystal structure of BaTiO,, Left is actual plot of the crystal structure and right is side-on schematic. (b) Target low symmetry crystal structure of BaTiOg, Left is an actual plot of the distorted crystal structure and right is a sidc-on schematic with exaggerated distortion. The angle of the bent bond is 171.80
7.3.2 Architecture
We use the same default equivariant message passing GNN implemented as for the previous examples. Here, we also input atom type as an additional node feature using one hot encoding. Similar to the oclagon example, have vector features at each node as output and interpret it as the displacement to generate the distorted structure. We can choose a SBS consisting of a single vector so we have an additional vector node attribute to input a symmetry breaking object. Finally, since crystals are periodic, we modified the network to include periodicity when computing relative displacement vectors.
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7.3.3 Symmetry breaking set
Note that O₁ has itself as normalizer in O(3) so the symmetry completely determines orientation. Hence any object sharing Cie symmetry works for generating an ideal equivariant partial SBS. simple choice consists of vectors (odd parity I ==1 object) pointing along the 4-fold rotation axes.
7.3.4 Results
As shown Figure 13 and Table ! our model is able to learn to distort the crystal structure appropriately when given an appropriate symmetry breaking object. Without such an imput the model cannot provide any distortions. In additional, we computed
additional invariant. quantities of the crystal structures and that these invariants for the output structure of the model with SB object matches the invariants for the target lower symmetry structure.
Figure 13:
Distorted crystal structure generated by our model when given symmetry breaking object shown the right in blue.
Table 1:
Values of various quantities which help distinguish the high symmetry and low symmetry structures. Our models here try to distort the high symmetry structure to the low symmetry one.
Structure
Bond length average
Bond length variance Ti-O-Ti
High symmetry
2
180
Low symmetry
2.003417
0.01392
171.80
Model (no SBS)
2
180º
Model (SB object 0),0,0)
2.003417
0.01392
171.80
8 Conclusion
We formalize the problem equivariant neural networks face in the spontancous symmetry breaking setting. We propose the idea of equivariant symmetry breaking sels which allows ENNs to sample or generate all possible symmetrically related outputs given a highly symmetric input. Importantly, we show that minimizing these sets intimately connected well studied group thcory problem, and tabulate solutions for the ideal case for the point groups. We then demonstrate how our symmetry breaking framework works in practice example problems.
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One future direction is to include translations and tabulate complements for the space groups in their re- spective normalizers. This would be particularly useful for crystallography applications. Another direction is to automate finding stabilizers for partial symmetry breaking objects. In addition, our method assumes we can efficiently detect the symmetry of our imput and outputs. Designing fast symmetry detection algo- rithmss would also be extremely beneficial. Finally, designing efficient loss functions which do not punish symmetrically related outputs would be useful for any network dealing with spontancous symmetry breaking.
Acknowledgments
We thank the helpful discussions with Robin Walters, Elyssa Hofgard, and Rui Wang for framing the different types of symmetry breaking.
YuQing Xie was supported by the MIT College of Computing fellowship and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1745302. Tess Smidt was supported by DOE ICDI grant DE-SC0022215.
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Table of Contents
A Notation and commonly used symbols
27
B Group theory
28
C Equivariant neural networks
30
D Limitations
31
D.1 Symmetry detection
31
D.2 Loss functions
31
E Proofs
32
Proof of Lemma 2.1
32
E.2
Formal justification of Definition 3.2
................................
E.3
Proof of Theorem 3.
32
E.4
Proof of Corollary 3.6
33
5
Justification of Defiiitinn
33
Proof of Theorem
33
E.7
Proof of Corollary
34
E.8
Proof of Lemma 5.1
35
F
Classification of full symmetry breaking cases for (
36
1
Normalizer: KM
36
2
Normalizer: Doch
36
3
Normalizer: D(2n)h
38
Normalizer:
39
5
Normalizer: On
39
G Equivariant full SBS better than exact partial SBS
40
H Experiments
41
H.1 Triangular prism
41
H.2 Octagon to rectangle
46
H.3 BaTiO, experiment)
47
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A
Notation and commonly used symbols
Here, we present the notation we use throughout this paper and the typical variable names.
Table 2: Notation used throughout this paper
(a)
Stabilizer an element a under a group G
No(S)
Normalizer of group S in group
Cla(S)
Set of groups obtained by conjugating group S with elements in
(rbe(z)
Orbit of an element under action by elements of group G
P(X)
Set of all subsets of X
G/S
When G is a group, this is the set of left cosets. If S is normal subgroup, this also
denotes the quotient group
X/S
When X is a set, this the equivalence classes induced by action of S on X If and G are groups, this denotes that S is a subgroup of G Function f with domain restricted to X
S G
flx
Table 3: Commonly used symbols
Group our network is equivariant under
1 Used to denote the trivial group
Identity element of group Input
y
Output
S
Symmetry of our input, more preciscly Staba
K
Symmetry of our output, more precisely Stabs(y)
B Full symmetry breaking set
Partial symmetry breaking set
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B
Group theory
Group theory is the mathematical language used to describe symmetrics. Here, we present brief overview of concepts from group theory we need to both define equivariance, and to understand our proposed symnetry breaking scheme. For a more comprehensive treatment of group theory, we refer to standard textbooks Dresselhaus et al. (2007); Dummit. Foote (2004); Kurzweil & Stellmacher (2004). We begin by defining what group is.
Desinition B.1 (Group). Let G be a nonempty set equipped with a binary operator :: G G - G. This is group if the follwing group axioms satsfied
Associativity; For all a, c G, we have (a (b)b). c a (b.c)
Identity element: There is an element E such that for all g E G we have e gg = g. -
3. Inverse element: For all g E G, there is an inverse g- E G such that g-g-tt 1 = g = e for identity e.
Some examples of groups include the group of rotation matrices with matrix multiplication as the group operation, the group of integers under addition, and the group of positive reals under multiplication. One very important group the group of automorphisms on a vector space. This group denoted GL((V and we can think of it as the group of invertible matrices.
While abstractly groups are interesting on their own, we care about using them to describe symmetries. Intuitively, the group elements abstractly represent the symmetry operations. In order to understand what these actions are, we need to define group action.
Definition B.2 (Group action). Let G be group and Q set. A group action is a function a : G x Q + such that ade, x) = x and alg, ah,, x)) = a(gh, for all g, h E and x E Ω.
Often, we may want to relate two groups to each other. This is done using group homomorphisms, a mapping which preserves the group structure.
Definition B.3 (Group homomorphism and isomorphism). Let G and H be groups. group homomorphism is a function f G + Il such that f(u-v) f(u) f(u) for all u, v G. A group homomorphism is an isomorphism if is bijection.
Because there many linear algebra tools for working with matrices, it is particular useful to relate arbitrary groups to groups consisting of matrices. Such a homomorphism together with the vector space the matrices act on is a group representation.
Definition B.4 (Group representation). Let G be a group and V a vector space over a field F. A group representation is a homomorphism p: G GL(V) taking elements of G to autmorphisms of V.
Given any representation, there are often orthogonal subspaces which do not interact with each other. If this the case, we can break our representation down into smaller pieces by restricting to these subspaces. Hence, is useful to consider the representations which cannot be broken down. These are known as the irreducible representations (irreps) and often form the building blocks of more complex representations.
Definition B.5 (Irreducible representationn. Let G be group, V vector space, and p G GL(V) representation. A representation is irreducible if there is nontrivial proper subspace W V such that Plw is represcntation of G over space W.
There has been much work on understanding the irreps of various groups and many equivariant neural network designs use this knowledge.
One natural question is whether there is a subset of group elements which themselves form a group under the same group operation. Such a subset called subgroup.
Definition B.6 (Subgroup). Let G be group and S G. If S together with the group operation of G satisfy the group axioms, then S is a subgroup of G which we denote S G.
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One particular feature of a subgroup is that we use them to decompose group into disjoint chunks called cosets.
Definition B.7 (Cosels). Let G be a group and a subgroup. The left cosets are sets obtained by multiplying S with some fixed element of G on the left. That is, the left cosets are for all g G
g$ (os SE.
We denole the set of left cosets as G/S. The right cosets are defined similarly except we multiply with a fixed element of G on the right. That is, the right cosets are for all g E G
Sg (sg SES).
We denote the set of right cosets as G
In general, the left and right cosets are not the same. However, for some subgroups they are the same. Those subgroups are called normal subgroups.
Definition B.8 (Normal subgroup). Let G be group and N a subgroup. Then N is a normal subgroup if for all g G, we have g.Vg N.
It turns out that given a normal subgroup, one can construct a group operation on the cosets. The resulting group is called a quotient group.
Definition B.9 (Quolient group). Let G be a group and N a normal subgroup. One can define a group operation on the cosets as aN- = (a-b)N. The resulting group is called the quotient group and is denoted G/N.
For subgroups which not normal in G, it is often useful to consider subgroup of G containing $ where S is in fact normal. The largest such subgroup is called the normalizer.
Definition B.10 (Normalizer). Let G be a group and S a subgroup. The normalizer of S in G is
No(S) {g 9Sg -)..
Similar to orthogonal vector spaces, one can imagine an analogous notion for groups. These are called complement subgroups.
Definition B.11 (Complement). Let G' be group and S subgroup. A subgroup H is complement of S' if for all g E G, we have sh for some s E S and h E H and S H {c).
It turns out that if a normal subgroup of G and H is a complement, then H isomorphic to the quotient group.
Finally, it is useful to define what mean by symmetry of an object. These are all group elements which leave the object unchanged and is called the stabilizer.
Definition B.12 (Stabilizer). Let G be a group, Ω some set with an action of G defined on it, and u E Ω. The stabilizer of u is all elements of G which leave u invariant. That is
Stabo = {g: gu 9 E G}.
One can check that the stabilizer indeed subgroup. Closely related to the stabilizer is the orbit. This is all the values we get when we act with our group on some object.
Definition B.13 (Orbit). Let G be group, Ω some set with an action of G defined on it, and u E Ω. The orbit of is the set all values obtained when we act with all elements of G on it. That is,
Orba(u) {gu: g G) Gu.
It turns out one can show that the stabilizer of elements in the orbit are related. This relation turns out to be conjugation which we define below.
Definition B.14 (Conjugate subgroups). Let S and S' be subgroups of G. We say S and S' are conjugate in G if there is some E G such that S = - We denote the set of all conjugate subgroups by Clo(S) (gSg e G).
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C Equivariant neural networks
Here, we give brief overview of equivariant neural networks. For more in depth coverage of the general theory and construction of equivariance, we refer to works such as Cohen et al. (2019); | Finzi et al. (2020);
Kondor & Trivedi (2018). We emphasize that the symmetry breaking techniques presented the paper apply to any equivariant architecture.
We first define equivariance.
Definition C.1 (Equivariance). Let G' be group with actions on spaces X and A function J: - is said to be equivariant if for all æ E X and g G we have
f(gz) = gf
Intuitively, we can interpret this as rolating the input giving the same result as just rolating the output. It is easy to check that the composition of equivariant functions is still an equivariant function. Hence, equivariant neural networks are dosigned using composition of equivariant layers.
There has been considerable study into how one should design equivariant layers. One approach is to modily convolutional filters by transforming them with the elements of our group Cohen & Welling (2116a). This approach known as group convolution and is based on the intuition that convolutional filters are translation equivariant. In group convolution, one interprets our data as signal over some domain. The first layer is lifting convolution which transforms our data into a signal over the group. The remaining layers then just convolve this signal with filters which are also signals over the group.
One can further group theory tools to break down the convolutional filters into irreps. This leads to steerable convolutional networks & Welling (2016b). These can be extended and used to paramcterize continuous filters which can be used for infinite groups Cohen et al. (0118). It turns out the irreps of the group are natural data types for equivariant networks. Further, we can express the convolutions as tensor products of irreps. We can think of equivariant operations as being composed of tensor products of irreps, lincar mixing of irreps, and scaling by invariant quantitics. Combining these, we get tensorficld networks which works point clouds and is rotation equivariant TThoma al. (2018). In this paper, we demonstrate our method using networks built from the e3nn framework for 0(3) equivariance Geiger & Smidt (2022).
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D Limitations
D.1 Symmetry detection
To use our procedure, we do assume knowledge of the symmetrics of the inputs and outputs to our network. In the full symmelry breaking framework, we only need the symmelry of the input. In the partial symmelry breaking framework, we need the symmetry of both the input and the output. However, we argue that this is not major concern.
First, symmetry detection is well studied problem and there many algorithms exist. for various types of data Bokeloh et al. (2009); ; Keller & Shkolnisky (2004;; Largent et al. (2012) Mitra et al. (2006). Further, sometimes the symmetrics of the inputs and outputs are already known. This is especially true for crystallo- graphic data Jain et al. (2013). In addition, because we only need the symmetry to design equivariant SBSs, we only need to perform symmetry detection once. This can simply be incorporated as preprocessing step for our data.
Further, we want emphasize that our framework can be used to prove whether knowledge of input sym- metry beneficial. We prove in Lemma F.1 that no finite subgroups of SO(2) have complement in their normalizer (which is also just SO(2)). Combined with Corollary 3.6 this actually implies the degeneracy of any (Qi))-equivaiiin SBS for cyclic groups is infinile. Hence, we cannol do much betler than a something like noise injection, which introduces asymmetry without knowledge of input symmetry.
D.2 Loss functions
While this work focuses on allowing equivariant networks to produce set of lower symmetry outcomes, it turns out another important problem is designing appropriate loss functions. Supposc we only have one example input output pair y where shares no symmetries with x. In this case there is no problem. We can any b E B and train to minimize a simple MAE loss [[f b) yll² for example.
However, suppose we observe z, y and y in the data where y' = sy and s E S Stabe(æ). Then suppose try to minimize MSE loss (x b) y|12 (1, s11 where V/ s'b. Then by equivariance, the second term in the loss is
llf( - = (1f(x, - v211² f|s'f(r,b)) syll = llf s'
So in fact we see must choose s' 8. So with multiple input output pairs and simple loss directly comparing outputs such as MSE, have problem pairing symmetry breaking objects with outputs.
However, suppose instead our loss was chosen such that loss(y, loss(v. v) is small if y sy for any E S. Then even if our network outputs sy instead of y when given some symmetry breaking object b, we do not punish it. Hence, this pairing problem would not exist. A simple version of such a loss would to compute the MSE for all possible symmetrically related outputs and take the closest one
loss( b), = min (I15(( -
sES
This is what we use for our experimental examples in this work. However, this can be inefficient for large infinite S and designing appropriate loss functions in such cases remains an open question.
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E Proofs
E.1 Proof of Lemma 2.1
Lemma E.1. Let X and be spaces equipped with group action of Suppose the action on X is transitive. We can choose a G-equivariant f :: such that f(u) if and only if Stabely) 2 (u) Further this uniquely defines f.
Proof. First suppose we did have [(u) = y- For any y E (tab we have by equivariance of f that gy f(gu) f())
So g E Staba
Next, suppose Stabo(y) 2 St) For any x E X, there is some r E G so that x = ru. Let us pick exactly one such r for each X and form set R. Hence any x is uniquely written as x = ru for E R. Define
f(z) f(ru) ry.
We claim f is equivariant. For any E G and x E X, let I = ru and gr = r'u for some r, E R. Then,
f(xx) f(gru) = s(r'u) = rryy.
But note that gr r'u implics r' gr r111 gru u. So r' gr E Stabo(u) s Staba Hence, also have r'-lgry = y. So,
f(gr) = r'y = r' gry g- gf(x).
Hence, f cquivariant.
Finally, for uniqueness, suppose f: are two equivariant functions such that f(u) ('(u) y. Then by equivariance, for any æ gu X we have
f(x) = f(gu) = gy = f' = f'x).
E.2 Formal justification of Definition 3.2
We can justify Definition 3.2 by characterizing exactly when a can be equivariant. This leads to the following proposition.
Proposition E.2. Let G be group and S be a subgroup of G. Let B P(B) be set where there is some group action of G defined B. Then there erists equivariant alcic(9) Cla(S) - P(B) such that. af o(C(o(S) = B if and only if nB = B for all E Nc(8).
Proof. Note that Clo(S) is set where action by conjugation is transitive one. Also note by definition that Sta for this action is preciscly the definition of a normalizer Nc(S). Then by Lemma E.1 we see such a function exists if and only B is also symmetric under No(S).
E.3 Proof of Theorem 3.3
Theorem 3.3. Let G be group and S subgroup. Let B be G-equivriaatt SBS for S. Then it is possible to choose an ideal if and only if S has a complement in No(S).
Proof. Suppose B is transitive under S and pick E B. Consider the slabilizer group Stab (b). For any g E NG(S), by transitivity under we must have gó = sb for some s S. So, s s gu = u implying that s-ig g (t)) So we find that we can write any g as g = sh for some s and (s)
No(S) S Stabno(s)(w).
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But note that since B is symmetry breaking, S Stabno(s) {e}. Hence, Stabwa(s)(u) is indeed complement.
For the converse, suppose H is a complement of S in (S).). We claim B = No(S)/H is the equivariant SBS we desire. Note that clearly by construction, this is closed under (S) so we satisfy the equivariance condition. Further, note that Stabs(H) = (ta (H) ns = H HnS {e}- Since B transitive under NG()), stabilizers of all other elements are obtained by conjugation and hence also trivial. Hence, it is indeed symmetry breaking. Finally, any 9 Na(()) is uniquely written as sh for some S, H gHl shH sH. So B is transitive under S as well.
]
E.4 Proof of Corollary 3.6
Corollary 3.6. Let G be a group and a subgroup. Let M be such that S & Nc((). Let be a G-equivoriant SBS for S which transitive under NG(S). Then it possible to choose B such that every M-orbil is also transilive under S if and only S has complement in M. In particulor, such a B has
Degs(B) S [Nc(S)/M).
Proof. Suppose we have such B and pick any b E B. By transitivity of the orbit under S, we have Mb Sb. Let B' Mb. We can check that this is in fact ideal M-equivariant SBS for S. That it is a symmetry breaker follows since B is symmetry breaking, That it is M-equivariant and transitive follows since Mb Sb and (S) M. By Theorem 3.3 this implics S has a complement in M.
Next, suppose we have a complement of S in M. By Theorem 3.3 we can construct B' which is an ideal M-equivariant SBS for S. We can lift this to a G-equivariant SBS for S by just taking = N'(S)
Finally, to compute the order, we note that every S-orbit also M orbit. Since B is transitive under (S( there at most [Na(S)/M number of M-orbits and hence only that many S-orbits. So
Degs(B) S [Nc(S)/M).
E.5 Justification of Definition 4.4
Similar to the full SBS case, we can justify Definition 1.4 by characterizing exactly when an equivariant æ can exist. This leads to the following proposition.
Proposition E.3. Let G be a group, S subgroup of G, and K subgroup S. S. Let P be a set with a group action of G defined on il and P C.P There erists on equivarianl 7|0rba (Is)))s(K)) Orbg((S, Cls(K))) P(P) such that (obbo((S.Css(()))(
(ls(K))) ((I P if and only if No(S, K) leaves P invariant.
Proof. By Lemma E.1 we need P to be closed under the stabilizer of the input. But the generalized normalizer No(S, K) is precisely this stabilizer.
E.6 Proof of Theorem 4.5
Theorem 4.5. Let. G be a group and S and K be subgroups K < S G. Let P be G-equivariant K- partial SBS. Then can choose ideal P (exact and transitive under S) if and only if Ns(K)/KK has complement in (K)),
Proof. Let P Su where u has symmetry Stabs(u) = K. We can define an action of any coset No K)/K on u as just the action of a coset representative on u. This is consistent since u is invariant under K. particular, note that K is normal subgroup of (K( so Ns(K)/K is quotient group. Let B' (K))K)) Since is in a K-partial SBS, we must have su # u for any S K. Hence, for any coset gK Ns(K)/K, K, gu if g & K. Therefore, B' must be SBS for Ns(K)/K: KK
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Next, consider any coset gK in NNo(s.K)(K)) (K)/K, Then we know gu E Su gu su for some S. Since K was a symmetry of u, gKg- - sKs-1 is a symmetry of gu su. So the stabilizer of su must be sKs- = K. Hence, must be in (s).). Therefore the action of gK on gives us an element of B' = (N))K))K)). Hence B' is (N KKquu/irriiiaa.nn
By Theorem 3.3 the existence of an ideal NNa(s.K) Kooo///aaravaiin SBS for Ns(K)/K implies that (s( K has a complement in (o (K))),))KK// (K)/K.
For the converse direction, suppose that A is complement of [Ns(K)K in (K)(), Note the elements of A are cosets of K so we can define a set of elements of Na K) as
H = CC CEA
Define p Or).(H). We claim that P transitive exact equivariant partial SBS. We first show that P is exact K-partial symmetry breaking. Consider E S. We can write
sll = SC= C
CEA CESA
Now we see if E K, then since K is the identity in the quotient group sA= A. Hence sHl H in this case. If s E Ns(K)- - K, then sK is not the identity in KK)K. But A is complement so SA o A implying sH * H. Finally, if & Ns(K) then sK & N (N K. So sH &. (N)(s..) But H C NNatss (K) so sH # H. Hence, Stabs(H) (H) = K and since the rest of P is just the orbit of H, stabilizers of the other elements are in Cls(K). Hence, P as we constructed is an exact K-partial SBS.
For equivariance consider any No(S, K) giving a coset
nH = U nC = U UCCUC.
CEA
CENA
If n E (K) then since is a complement, (nK) = (sK)(aK) for some sk E Ng(K)/K and aK E A, so nd (nK)A= = (sK)(aK) (sK)A sA for some s E Ns(K) C. Hence, nH = sHl for some E S. If (NG(S.,)((K) then there is some that nKn sKs- Therefore, s '''nKn" so 'n E (S) But we saw before that this means there is some s' such that I' s.H. Thus, nH ss' H and ss' S. So nH E P so P is indeed closed under action by (S,
K
E.7 Proof of Corollary 4.7
Corollary 4.7. Let G be group, S a subgroup, and K subgroup of S. Let K' be subgroup of K and M subgroup of Nc(S, No(S, which contains S. Suppose P is a G-equivoriant K K-partial SBS for S which is transitive under NG K). We can choose P such that (t) E ClNo(S,K) (K') for all and all M -orbits in P are transitive under S if and only if Ns(K))/KK K' has complement in (MKK))K Further, such a P has
Degs, (P) S |K/K| K))
Proof. Suppose we had such a P. Pick some e P such that Stabs(p) = K'. Since M-orbits are transitive under 5, we have Mp = Sp. Let P Mp. We can check then that this M-equivariant set. Further, since M NS,, K"), we see that this is an exact K' K'-partial symmetry breaking set.. Hence, it an ideal M M-equivariant K' K'-partial SBS. Also note that since M No(S, K'), we have M (S, K'). So by
Theorem 4.5 Ns'K')/K' must have a complement in (K')K',.
Conversely, suppose Ns'K')/K' has a complement in Nu (K')/". Again, we note M = (S, K') so by Theorem 4.5 we have an ideal M-equivariant, K'-partial SBS for S. We can lift this to G-equivariance by taking the orbit under Nc(5,K)
To see the order of such P. we consider the S-orbits. Let T be a transversal of S/K. For each S-orbit, we pick some p in that orbit so that Stabs(p) S K. We put the elements of K p in our Pt. Within
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each S-orbit, any p' can be written sp for some E S and any s is uniquely written th for some 7 and k E K. So p' lkp. However, note that any other s' where p' s'p sp can be written as s' = sk' for some k' E Stabs(p). Hence, p' is uniquely written as p' = t(kp) since kk'p = kp. So each S-orbit contributes K K/Stabb(p) elements to Pt. However, since (p) E (IN we must have (y) = |K/K),
Finally, we know each S-orbit also an M-rbbit, since P transilive under (S, K), there are most [N((S, K)/M] different S-orbits. So
Degs,
[P S |K/K| (No(S, K)/M).
E.8 Proof of Lemma 5.1
Lemma 5.1. We have the following formula
()) = S(NG(S)
Proof. For any E K), by definition we must have nKn = sKs- for some E S. Hence, s nn'nKn - = K so s E Nc(K). Further, clearly E Na(S) and n E No(S) so also s-in E No(S). Therefore, 8 s'n E Nc((S) So n nn) S(NG(S) n No(K)). Hence
K)(S,K)= S S(Na(S) n (K)))..
Next, consider any n' = 8n E (SN n Na(K)) where 8 E S,n E Na(S) n No(K). Since E Na(S), a e (S(S) clearly sn E NN(()). Further, we find
snK(sn) s(nKn=1)s"* sKs E Cls(K).
Therefore by definition sn E NS K). So also
(NNa())naala K).
Hence, we find that
N)( = S(Na(S) n (K())).
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F Classification of full symmetry breaking cases for 0(3)
Here we tabulate the cases for full symmetry breaking for the finite subgroups of 0(3). These are the point groups and the normalizers are tabulated in the International Tables for Crystallography in Her- mann Mauguin notation Koch & Fischer (2006). We have translated these Schönflies notation in Table 4
Table 4: Normalizers of the point groups in Schönflies notation. Note have the equivalences C₁ 1,
S C. Cıa C1 C., D1 C2, D1 C2v. Did C2h-
Normalizer:
Groups:
Kh
1, C2
Dooh
Cn San: Cub Vn 2 C.
D(2n)h
Cuus Dnd, Duh vn 2 2; Dn Vn 2
T
I, T
On
D2, D2n, T, Ta, Th, O, On
In the following subsections we do casework by normalizers. For each, subgroup with given normalizer, we give a valid complement by name if it exists. In some normalizers, the name a subgroup is not sufficient to identify it. This is because there are multiple copies subgroups with that name the normalizer. In such cases, we must identify which copy of the subgroup we care about. To do so, we give the normalizers in terms of group presentation found with the help of GAP Group presentations are essentially a set of generators and relations among the generators. We then specify any specific subgroups of the normalizer using the generators of the normalizer.
F.1 Normalizer: Kn.
All the groups with this normalizer do have complements. Note that in Schönflies notation, K₁ is just the entire group 0(3). The only subgroups with O(3) normalizer are the trivial group C1 and inversion C₁ Clearly for the trivial group the complement is 0(3). For inversion, the complement is just SO(3).
Table 5: Groups with normalizer K, = ((3) and their complements.
Group
Complement
I
K₁, ((3)
C,
K = 50(3)
F.2 Normalizer: Doch
Unfortunately, most of the groups this case have no complements. We provide a proof of this fact here. We begin by showing no nontrivial cyclic group has complement in Co (which SO(2) Schönflics notation).
Table 6: Groups with normalizer Dooh and their complements.
Group
Complement
C.
Cocor D
On, Syns Cuh Vn 2
None
Lemma F.1. Let Cn be a cyclic group of order n 2 which is embedded in Coc. Note that it is normal subgroup since all groups here abelian. Then Cn does not have complement in Coc.
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Proof. 1 Suppose there was complement H. By definition, for any g E Coc have 9 ch for unique h E H and c E Ca.
Next, note that Coo is divisible group. In particular, for any g E Coo, there exists some o' such that = (g')". Let o be uniquely written as h'd' for some E H and d' E Cn. Then we have
g = (chh'""('')(
(
where we noted (d')² = e. So is uniquely written as g = ch for c= = e and h = (h')". But this holds for all so all elements of Coc are just elements of This contradicts the fact that H CCC, = {e).
Proof. 2 For those familiar with exact sequences, one can consider the following alternative proof. Consider short exact sequence
C, Cx Cx/Cn 1.
We can check that Ca/Cn = Coc. Existence of complement for Cn implies the above sequence is split, which by the splitting lemma Hatcher 2002) implies Coo ~ Ci Cn. contradiction.
We can extend the lemma above to there being no complement of any cyclic group in Doo (which is ((2) in Schönflies notation).
Lemma F.2. Let Cn be cyclic group of order n 2 2 which is embedded in Doo such that the rolation aris aligns with the infinite rotation aris in Do.. Note that it is a normal subgroup since all groups here are abelian. Then On does not have a complement in Doo.
Proof. Suppose there was complement H. Consider = H Clearly, we have H' nCn = {e}. Next, for any E C.o, there is unique e E Cn and h E such that g = ch. But h = ceccloo g E Co so H'. So H' is a complement of Cn in Coo. But this contradicts Lemma F.1
Finally, we can prove that no subgroups except for C₂ in this case have complements. Note here that K₃ is just. 0(3) and K just 50(3) in Schönflies notation.
Theorem F.3. Consider any point group A which has normalizer Dach in Kn. If A has a nontrivial pure rotation, then has no complement in D
Proof. First, note that K2 the direct product of and inversion C.. Suppose had complement H.
We can split A = A. U A; where A, Ank K is the subgroup of pure rotations and A is coset consisting of elements with an inversion. Similarly, can split H He L H, and Doch Doo L D; into subgroups of pure rotations and coset of elements with inversions.
We claim the elements of G = A,He form a group. Consider any a,a' E A, and h, h' E Ha- Since A is a normal subgroup of Doch = AH, we have hA Ah so ha' = h for some a" E A. But since h, a' E K, we have h K so a" E K. Hence a" E Ac. Therefore,
(ah)(a'h') = aa(a()) a(a"h)i (ad" (aa")(hii)) E A.He.
So G A.He is a group.
Next, since H a complement of A, clearly A H. = fe}. Since G S AH Dock: any g == ah for unique A and H which by construction of G are a E A. and h He. So He is certainly complement of A, in G.
Now, we claim either G Doo or G C.o. For any g E Doo, since H is a complement, there is unique a E and E H such that = ah. In particular, we note we must either have a E Ar and h E H. or A.H, = D For the other
a E A, and & E H₃ to have the right inversion parity. One possibility is and consider any g' E Doo G. Then g = ah
possibility, suppose Dx -G nonempty. Fix g E Dxo and g' = a'h' where a A, and hh' E He. Now, note that a' the combination of elements with odd parity in so a ta E A Next, since is normal subgroup, we have h-1 A Ah-1 and in particular,
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h a''a-'d' d'h-! for some a" A. But since h has odd parity and a a= a' has parity in inversion, a" must have even parity in inversion. Hence, we have
g
(d't') = a d'l''
Since h, h'' both have odd parity, h" = h h1 has even parity so in fact g g = a" where a" E A. and h" He. Therefore, g"'gg E so Doo - = gG is just a G-coset of Dx We can similarly also show that Dx Gg. Now, we claim gG n Coo 6. Suppose not. Then there some c gG n C.s. Since Cso is a divisible group, there is some è' where gG n Cx Φ so G n Coo Co. To conclude, must have Doo Coo and it is clear gCcc would generate the remaining elements in Dxo So G = C in this casc.
(c)² But must have (c')2 G, contradiction. Hence
From Table we can see that Ag must in fact be nontrivial cyclic group. But then the above implies that H. is a complement of this cyclic group in either G = Doo or G = Coo; which contradict Lemma F.2 and Lemma respectively. So cannot have a complement Dooh-
F.3 Normalizer: D(2n)h
All groups with this normalizer do have complements. We list the subgroup and its complement in Table 7 One presentation of D(2n)h is
(a, b, mlmaa 2n 6², m² (ab)2 (am)?, (bm)²).
Figure 14 depicts an example of Dios object. The element correspond a 2 /10 rotation about the blue vertical axis, b corresponds to rotation about the red axis, and m corresponds to reflection across the mirror plane shown in orange.
Figure 14: Object with SymmetryDuur We can identify generator as the 10-fold rotation about the blue axis, generator as the 2-fold rotation about the red axis, and m as the reflection over the plane shown in orange.
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Table 7: Groups with normalizer (22k) and their complements.
Group
Generators of group
Complement
Generators of complement
Owe Dnd
a², m d² abm,
C2v
am, bm bm
C2
Duk
a² b. a², b
C,
Dn
C2v
am, bm
F.4 Normalizer: In.
This case is simple, we either have or In- Clearly we just need to add inversion to gel a complement in the former case and the latter case we can just take the trivial group.
Table 8: Groups with normalizer In and their complements.
Group
Complement
T
C
Ix
F.5 Normalizer: On
All subgroups in this case have complements well. One presentation of On is (a, bll", ba (aba)² (ab)², iaia -1 ibib ibib-"). Here, and b are 7/2 rotations about perpendicular axes and is just inversion.
Table 9: Groups with normalizer O₂ and their complements.
Group
Generators of group
Complement
Generators of complement ab, ba i ab. ba a b, o
D2
a² b2 a² 12, ab, ba ab, ba, ai ab, ba, a, b a, b,
Dad
D2n
D3
T
S
Ta
C2
Th
C2
a
O
C
On
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G Equivariant full SBS better than exact partial SBS
We provide an outline of the construction of the counterexample. It is casiest to explain this by introducing the concept of a wreath product on groups.
Definition G.1 (Wreath product). Let H be a group with group action some set Ω. Let A be another group. We can define a direct product group indexed by O as the set of sequences (aw)eeR where E A. The action of H on Ω induces a semidirect product by reindexing. In particular, for all E H and sequences in d we define
h (aw)wen = (an twenn
The resulting group is the unrestricted wreath product and denoted as A Wig H.
If rather than a direct product group A we restrict ourselves to direct sum where all but finitely many elements in our sequence is not the identity, then we get the restricted wreath product denoted as A wig H.
Note that the direct sum and direct product are the same for finite so the restricted and unrestricted wreath products also coincide in those cases.
Consider the space f2 = {1, -1] and an action of D4 on corresponding to the Ag representation. Intuitively, if we think of D4 as the rotational symmetries of a square in the xy-plane, this corresponds to how the = coordinate transforms by flipping signs. Define a group G' as G' C2 wIa Dz. This is group of order 32 and is SmallGroup (32,28) 28) in the Small Groups library GAP One presentation of this group
(a, cla", b*, (ab)*, e², beb- (ac))).
(1)
In this presentation, we can interpret a, b as generators of D4 and the generator one copy of Cz.
Consider the group G G' x G' defined using the direct product. We can write generators of G ai bi, Ct, d2 b2, 2 corresponding to two copies of those in the presentation given in equation I where gener- ators with different indices commute. Define S as the subgroup generated by a1 bf. c1, 02 bg 02 and K as the subgroup generated by 0102.
We can check that Na(S)= = K) = G. It is also not hard to check that a bi, azb2 generate a complement for S in G. Hence, by Theorem 3.3 we know that an ideal G-equivariant SBS is possible for S. Hence, we know the size of the equivariant full SBS is |S| 256.
Next, suppose we wanted G-equivariant exact partial SBS. We can always generate this partial SBS by taking the orbit of some element under action by No(S, K) G where Stabs(p) = K. We claim we must also have Staba(p) = K. Suppose not, then there must be some g E Stabe such that $ S. However, we can check through casework or brute force that for all such g, either gKg * or g² E S - K. But would mean that there are elements not in K which stabilize p so Stabs(p) + K, a contradiction. Hence, no such g can exist so Stabo(p) K. Finally, by Orbit-Stabilize theorem, this means the set must have size PP [O)ba(p]] G//[Stabo (p) = (82 222 512. This is larger than the equivariant full SBS.
In our supplementary material, we provide script in GAP which verifies our claims above. In particular it performs brute force check that Spabs(p) = K implies Stabe(p) = K for the group G, S, K described above.
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H
Experiments
We provide some additional details on our experiments here. We also provide our code for running these experiments in the supplementary material.
H.1
Triangular prism
H.1.1 Obtaining ideal SBS
(a)
(b)
Figure 15: (a) Triangular prism with D3 symmetry and patterned cylinder with Den symmetry. The gen- erators are b, m where is 2n /6 rotation about the blue axis, b is rotation about the red axis, and m is a reflection across the orange plane. (b) An ideal symmetry breaking set for the triangular prism. A complement of D3 in D&a is generated by the mirror planes shown here in yellow and purple. The vector in red is a symmetry breaking object with this complement as stabilizer. The orbit of this vector under the normalizer generates the other vectors shown in black.
Table tells D3 is generated by a² b and that a complement H/ is generated by am, bm. By Theorem 3.3 we know that if we can pick some object with stabilizer Stabp,, (v) = H, then the orbit of under D gives an ideal equivariant SBS. The symmetry axis for and b are depicted in Figure 15a and correspond to the 2 and x axes the canonical oriontation. The generators am, bm of the complement are depicted as mirror planes in yellow and purple in Figure 15b
In this case, we can see that the vector the intersection of the mirror planes has the symmetries of the complement. This is depicted the red vector in Figure 15b One can further check it shares symmetrics with the triangular prism. The her 5 arrows in black are the other symmetry breaking objects we obtain by taking the orbit of the red arrow under action by Des. In the canonical orientation, this red vector normalized to unit length is
(cos(T/6),,in((( (V332,))11
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H.1.2 Nonequivariant SBS
addition to using an equivariant SBS as presented the main paper, we also tried training with the non-equivariant SBS described Section 3.2 Recall that the symmetry breaking objects here are veclor pointing to one of the vertices of the triangle projected in the zy plane and a vector pointing up or down symmetry breaking object
corresponding to which triangle pick from. We fix one pair of vectors and train our equivariant model to match it with a verlex.
As shown in Figure 16a our model is able to complete this. However, rotating the prism by 180 and feeding this rotated prism along with our symmetry breaking object, we find that our model outputs a vector which does not. point to any vertex. This is shown in Figure 16b Contrast this with the equivariant case Figures 16c and 16d where our model still produces vector which points a vertex of the prism.
H.1.3 Nonideal SBS
We can modify the nonequivariant SBS into an equivariant one by adding the additional objects needed for closure under the normalizer. Doing so we see there are 2 S-orbits in this nonideal equivariant SBS shown in Figure 17 This corresponds to a degeneracy of 2 as defined in Scction H.1.3
we expect. this SBS to be less efficient than ideal We first. train using only symmetry breaking object from one of the S-orbits. We see that when given objects from that orbit, the network correctly outputs vectors pointing to vertices of the prism but fails when given objects from the other S-orbit. However, if we use objects from both orbits in training, the output vectors point to vertices when given objects from either S-orbit. Hence, we need to use at minimum 2 symmetry breaking objects during training to guarantee correct behavior for objects in the SBS compared to only needing one example to train for an ideal SBS.
From the discussion in Section 3.4
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(d)
Figure 16: (a) Output (red) generated by model and symmetry breaking object (blue) given that is chosen from a nou-equivariant SBS (b) Output (red) generated by our model and symmetry breaking object (blue) given when our prism is rotated by 180 (c) Output (red) generated by our model and symmetry breaking object (bluc) given that is chosen from an equivariant SBS (d) Output (red) generated by our model and symmetry breaking object (blue) given when our prism rotated by 180
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(a) S-orbit
(b) S-orbit 2
Figure 17: (a) S-orbit corresponding to the original nonequivariant SBS (b) Additional S-orbit needed which makes the set closed under the normalizer.
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Table 10: Results of training using a nonideal equivariant SBS. If only see one of the S-orbits in training, the network fails on the unseen orbits. If
both S-orbits then the network behaves correctly.
Seen in training S-orbit S-orbit 2
Outputs
S-orbit
S-orbit
Yes
No
No
Yes
Yes
Yes
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H.2 Octagon to rectangle
H.2.1
Obtaining an ideal partial SBS
In this scenario, we want G-equivariance for G ((3) and we have Dg and partial symmetry K D2.
(a)
(b)
Figure 18: (a) Octagon with Ds symmetry input and patterned cylinder with (G(S,K) Daa symmetry, The generators are a², b, m where a² is 2 2 rotation about the blue axis, b is rotation about the red axis, and m is reflection across the orange plane.
(b) A symmetry breaking object which generates an ideal partial SBS. Note that in addition to the D2 symmetry, this object is also symmetric under reflections across the purple and yellow planes.
We follow Algorithm Eoo understand what symmetry a canonical symmetry breaking object needs to generate an ideal partial SBS. First, we note S (e, Dz) and K (e, D2). The first few steps are straightforward and we can evaluate them directly.
Operation
Evaluated
N1 t- Normalizers (S)
Normalizers Dg (c, D16h)
(n, (N2)) Normalizerg [KK)
Normalizers (e, Dan)
N₂ <- (gs gkn, (N2))
(e, D4h)
N N2
Dien) n (e, Dan) (e,
N' + (e, (S)) n2
(s) n (e, Dan) (4)
The next step
(Q1, ②) +- Quotient[N, (g3 9K. (K)))
where we need to create quotient group. Here we note a presentation of N = (e, D4n) is (a, mla* 6² m² (ab)², (am)?, (bm)
The normal subgroup (05 gk, (K)) = (e, D2) then consists of e, a² b, bb. Hence we can set the cosets ab,
X {a,a² aab, a²b}
Y (m, a² m, bm, a² bm)
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and we check these generate the quotient group and we have relations X² = y (XY)² Hence we have
Q = (x, (XX)))
Finally, setting
((a) p(a²) a(ab) k(a²b) X
6(m) = (saa m) (bbm) Q(a bm))bm) Y
defines our homomorphism ø. The next step is
Q2 t ((N").
We note N' = (e, D4) which is generated by b.b. We can find that Φ(a) = and x(b) = o((a))ab))) = ((a)) = x² Hence,
Q2 = (X|xa).
The next step is
C t WindComplement [Q1, Q2l
which this case one can check to be generated by Y. This gives
C=
Since C' exists, the next step is
(h, (H)) + (C))
The elements of are e, and from the definition of o, we can see that Φ (C) consists of the elements
1 (e) = {e, a² bb, abb
(V) (m, a m, bm, a bmb).
We can check that the group consisting of these elements is in fact generated by a² b, m which is (e, D2R). Hence
(h, (H)) (e. D2n).
If an object has this symmetry then it can generate an idea partial SBS. In this case a simple choice is a = 2 object even parity. This is depicted in Figure [8b we align it the æ axis, this would correspond
(0, 0, 0, 0) using real spherical harmonic conventions. Applying Algorithm 2 we would obtain the set of
2 objects "parallel" to edge of the octagon.
H.3
BaTiO, experiment
H.3.1
Atom matching algorithm
We would like to predict atom distortions. However, our data consists of atom coordinates in the initial and target structures not necessarily in the same order. Hence, we need an algorithm to match similar atoms together so we know how much they are distorted. This process is complicated by the fact that we have periodic boundary conditions with periodicity determined by lattice vectors and that our atoms may be translated within the lattice. We assume our structures are given in the same rotational orientation.
For our algorithm, atoms. Hence, we can compute signature for an atom by taking the difference of the positions of that atom and all other atoms in the lattice. We call the set of position differences for atom a from all other atoms the signature aa of a. Note in our implementation, we also separate out the atom types in addition to the position differences.
the insight that the distorted atom should still have similar vectors to neighboring
Next, we need way to compare signatures. Suppose we had atom a from the initial structure and atom a' from the target structure. Certainly, if they are different atom types, assign cost of oo to this pairing.
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Otherwise, we look at their signatures. However, we actually need to optimally pair the other atoms to do For a pair of atoms b and B' from the initial and target structures, we can give cost of co if they are different atoms and [b [b] otherwise. If the atoms are similar, this should be small, but it may also be shifted by lattice parameters from the smallest it could be. So we just look at all small lattice shifts L and set the smallest. | a [b] + Lll² as the cost of pairing b, V. This gives a cost madrix M, With all the costs, we can run the matching algorithm in Crouse (2016) match the atoms. The cost of the assignment the difference in the signatures of a,a'.
match the atoms using the comparison of signatures. We create cost matrix where Ca,a' is the difference in signatures of a, a' from the initial and target structures. We then run another iteration of the algorithm from Crouse (2016) to find our matching.
Finally,
Because we only ever usee differences atoms, this matching algorithm is independent of translations.
H.3.2 Translation invariant loss
In the previous experiments, we could simply use MSE loss of the vector differences. However, for crystals we wanted to have loss which is invariant under translation. We realize that our matching algorithm in fact produces such a loss. By storing the matching information, we can effectively compute the same loss without having to run the matching algorithu every time. This is what we use to evaluate our model.
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H.3.3 Symmetrically related outputs
(a)
(b)
(c)
(d)
(e)
(f)
Figure 19: Distortions of a highly symmetric crystal structure of BaTiO, when provided with each of the possible symmetry breaking objects in our ideal equivariant partial SBS.
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Table 11: Values of various quantities which help distinguish the high symmetry and low symmetry struc- tures. Our models here try to distort the high symmetry structure the low symmetry one.
Structure
SB object
Bond length average
Bond length variance
Ti-O1-Ti
Ti-O2-Ti
Ti-O3-Ti
High symmetry
2
180
180
180"
Low symmetry
2.003417
0.01392
180
171.80"
171.80
Model
None
2
180
180º
180º
Model
(1,0,0)
2.003417
0.01392
180
171.80
171.80
Model
(-1,0,0)
2.003417
0.01392
180
171.80
171.80
Model
10,1,0)
2.003417
0.01392
171.80
180
171.80
Model
(0,-1,0)
2.003417
0.01392
171.80
180
171.80
Model
(0,0,1)
2.003417
0.01392
171.80º
171.80
180"
Model
-0,0,-1)
2.003417
0.01392
171.80
171.80"
180%
50


Reviews:
[{"review_id": "lOrgYjJEUV", "reviewer": "Reviewer_PWxe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n1. The paper is clearly written and explains well the motivation of spontaneous symmetry breaking\n\n2. The formulation of equivariant symmetry breaking sets (SBS) is interesting. The connection of constructing equivariant SBS to finding complements of normal subgroups is novel.\n\nWeaknesses\n\n1. Equivariant SBS proposed in this work is closely related to the learnable symmetry breaking approach in Smidt et al.(2021). However, a clear justification of why equivariant SBS is more desirable (in what settings) is lacking, and a empirical comparison of these two approaches can significantly strengthen the paper.\n\n2. The authors mentioned their framework allows \"efficiently\" symmetry breaking. However, it is not clear to me how efficient this framework is. The various discussions of efficiency (e.g. Sec 3.4, Sec 4.4) seem to only concern the characterization of non-ideal SBS. Given such characterizations, what can we infer about efficiency in terms of computational complexity? Or do the authors consider some other notion of efficiency?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Motivated from physical problems where the outputs have less symmetries than the inputs, this paper proposed a symmetry breaking framework for equivariant neural networks (ENNs). The key idea is to feed the symmetry breaking parameters as additional input to the ENNs. Such symmetry breaking parameters are obtained from a symmetry breaking set (SBS) constructed from the input and output symmetries. The authors provided characterizations of equivariant SBS under different settings: (1) full symmetry breaking (ideal versus non-ideal case); (2) partial symmetry breaking (ideal versus non-ideal). The authors also presented algorithms in constructing equivariant SBS for (1) and (2). This framework is illustrated via numerical experiments on synthetic examples and crystal structure deformation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Conceptual and empirical comparison between this approach and the approach in Smidt et al.(2021)\n\n2. Discussion or clarification of the \"efficiency\" claim.\n\n3. Most of the discussions and the numerical examples seem to concern with one input-output pair. How well can this framework be applied to a large dataset with input-output pairs having potentially different symmetries? This needs to be further elaborated.\n\n4. The current formulation relies on the input and output having different amounts of symmetries. Can this be alternatively characterized as the same group with different actions in the input and output space? It will be nice to add some discussions on this front.", "broader_impact_concerns": "No ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Oz9h4a0cba", "reviewer": "Reviewer_PoBz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths.\n\nThere has been recent interest in the community to symmetry breaking in neural network architectures. The paper recognizes an important destinction between different kinds of symmetry breaking in physics, namely asymmetries due to asymmetries in the underlying governing laws (explicit symmetry breaking) and asymmetry in observations while the underlying process is symmetry (spontaneous symmetry breaking). The paper provides a group theoretical treatment of different types of symmetry breaking and how such symmetry breaking can be incorporated in the architecture. This seems to be the main strength of the paper. The paper also provides some experiments to demonstrate the approach in practice.\n\n### Weaknesses.\n\nAlthough the reviewer appreciates the generality of stated results and group theoretical formalization of the proposed concepts, the effective methodological contribution from a machine learning perspective is very minimal. The method seems to boil down to adding additional inputs to the equivariant layers. In terms of method, there seems quite some overlap with Smidt et al. (2021). I do not find this too worrying, given there is value in a generalizing theory connecting this approach to other methods and puts it in a broad group theoretical framework. The experimental validation on the other hand is very hard to follow and lacks the details to really assess claims being made.\n\n> Analysis is limited to architecture, but claims suggest architectures are used in some training scheme.\n\nWhat confuses me about this work is that most of the analysis seems to be limited to the architecture. However, some of the claims imply that these architectures are used for predictions without ever mentioning how these architectures are used or trained and how predictions are made. If claims are made about using the architecture in practice, it should be made explicit how these architectures are trained or used to make predictions.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper Equivariant Symmetry Breaking Sets provides a mathematical analysis of symmetry breaking in machine learning architectures. It distinghuishes explicit symmetry breaking, in which the underlying phenomenon is asymmetric, and spontaneous symmetry breaking, in which underlying phenomenon being modelled is symmetric, but the observed data is not due to noise. There has been a recent surge of interest in relaxed equivariances and symmetry breaking in the community. This paper provides a formal treatment of different forms of symmetry breaking, and ways to incorporate symmetry breaking in neural network architectures through symmetry breaking sets (SBS). Lastly, some experiments are provided to demonstrate these techniques in practice.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, I think this is an interesting work that will definitely be of interest to members in the community. In particular, I value the formal group theoretical analysis of spontaneous symmetry breaking and results. I think this is the strongest part of the paper.\n\nI have two main concerns relating this work. The first is that it is not clear how architectures are being used, which makes it very hard or impossible or hard to assess how proposed symmetry breaking will actually be able to solve the formulated goals. The second concern is about the experimental validation, which lacks details to sufficiently assess the made claims.\n\n## Critical points\n\n> Make explicit how architectures are used.\n\nThe paper focuses on group theoretical analysis of the architecture. However, the stated goals, claims and experimental validation all imply that the architectures are somehow used to solve particular tasks. The paper does not mention or formalize how these architectures are used, which makes it hard to follow and verify some of the statements. The paper does mention that there is a \"distribution of data\" and imply some empirical fit on the data \"equally likely\". However, without specifying how this fit is made and what task is being performed it is hard to judge what properties (in terms of symmetry breaking) are desired in the architecture. I think this modelling context should be made explicit when claims are made about certain (symmetry breaking) properties being desirable in certain contexts.\n\n> Concerns about experimental validation\n\nAlthough I believe the method could be very useful, I don't find the experimental validation sufficient enough to support the made claims. This is in part because I found the description of the experiments to be very vague, which made it hard to assess and interpret the results. The authors do mention that the experiments are more of a proof of concept. However, I would suggest either clearly describing the experiments or leaving out the experimental valdiation all together. Currently, it is not clear what the training data is, what objectives are being optimized, and how results are evaluated.\n\n\n## Minor points\n\n\n> Difference between explicit / spontaneous symmetry breaking well explained, but Figure 1 very confusing\n\nThe destinction between explicit and spontaneous symmmetry breaking is a central subject of the paper. I find that the difference is quite well explained in the text, in whether the underlying governing laws are symmetric. Although this was clear, and further formalized in Sec. 2, I found Figure 1 really confusing and not helpful in its current form. It starts to mentions magnetic moments and external fields without ever introducing these. Also, \"B\" is never introduced.\n\n> Relation to other works on relaxed equivariance\n\nThe authors mentions some architectures that encorporate relaxed equivariant architectures (Huang et al., 2023; van der Ouderaa et al., 2022; Wang et al., 2023; 2022b). Can the authors elaborate what is meant by \"If all symmetrically related lower symmetry\noutputs are equally likely in the data, then relaxed networks will see the distribution as symmetric and fail\nto break symmetry \". Could authors elaborate on this and explain why this would not actually be desirable in such cases? From a generalization perspective, wouldn't it be desirable to have a symmetric architecture in such case? This comes back to the previous question that it is not clear how architectures relate to the task at hand or predictions being made.", "broader_impact_concerns": "No ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "E9lIMNDzkG", "reviewer": "Reviewer_A4Dc", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- The paper addresses an interesting and as far as I understand, potentially important problem of equivariant deep learning.\n- As far as I could check, the theoretical results are correct. The presented framework is self-contained.\n- The discussion of the limitations of the method is detailed and helpful.\n\n**Weaknesses**\n\n- I found that sometimes the explications of the paper are slightly unclear or imprecise. Some of the claims also require more justification. See Requested Changes.\n- The experiments are a bit lackluster as they do not showcase any real application. But this is acceptable for a mostly theoretical paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a method to address the problem of symmetry breaking in equivariant neural networks. The goal is to make it so that an equivariant neural network can have outputs of lower symmetry than its input. This is done by sampling elements from a set that depends on the symmetry of the input. The authors characterize these sets that can break the symmetry, the goal being to make them as small as possible using group theoretical notions. They also look at partial symmetry breaking, for which the goal is to obtain outputs that break some of the symmetry but not all. The authors also perform some toy experiments to show how the method works.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Major changes**\n- I think definition 3.2 is confusing. The set B which is defined as equivariant instead of sigma. Only functions can be equivariant not outputs of functions. I understand that this is motivated by the fact that if B satisfies definition 3.2, then an equivariant sigma exists, but I think this should be mentioned explicitly.\n- \"Intuitively, we expect a smaller SBS to be better and this is indeed true.\" The last part of the statement is too strong. I did not find the experiments convincing in supporting this statement (see below). This should be changed to \"we intuitively expect this to be true because...\" or something like that.\n- I don't understand the motivation behind the definition 4.1 of the partial (non-exact) SBS. I thought that the motivation for partial SBSs is that we might know that the output has some symmetry? Likewise, I don't understand the sentence, \"In general, we can always break more symmetry than needed and still obtain our desired output.\". If the desired output has a symmetry, we shouldn't be allowed to break it? Overall, I think that the necessity of this definition should be questioned it would be beneficial to the paper to remove it. If it can't be removed, it should be justified much more clearly.\n- I found that the experimental methods described in section 6 do not tie in clearly with the method presented in section 5. The method introduced seemed quite systematic, but then some unjustified choices are made in the experiments. It seems like I am reading about a completely different method in section 6. It is important to show if and how the method follows the algorithm introduced in section 5.\n- \"in Appendix H.1.3 we show that a nonideal SBS is less efficient than an ideal one.\" I think this statement is misleading. A notion of efficiency was introduced earlier in the paper, according to which an ideal SBS is by definition more efficient than an ideal one. If that is the notion this statement refers to, then it doesn't add anything. However, it seems like efficiency here is taken in the sense of better empirical performance. I don't find the evidence presented in Appendix H.13 against the nonideal SBS conclusive. If I understand correctly, the SBS are built to be nonideal (2 orbits) but in one version of the experiment the network is trained only on one orbit? That doesn't make sense to me, the network should be trained on the full SBS. Indeed when it is, it produces good results. I think the discussion is misleading, it seems like the non-ideal SBS is fine. Possibly a more challenging task would be necessary to see if nonideal SBSs are really worst.\n- One thing that is not clear is how the element of the symmetry breaking set should be in general be given as input to the equivariant neural network. This is worth elaborating in the methods section.\n\n**Minor changes**\n\n- Lemma 2.1: \n    - If the group action is transitive, there is only one orbit so it should not be necessary to restrict the domain to the orbit\n    - Why is the version 2.1 in the main text an \"only if\" and the version in the appendix an \"if and only if\"?\n- \"However, in some cases we truly have a symmetry breaking sample even if there is no noise.\": I don't understand this statement, spontaneous symmetry breaking requires noise, else what mechanism chooses how the symmetry is broken?\n- \"The intuition is that we want to maximize the symmetries of the symmetry breaking objects but only in the directions “orthogonal\" to S.\" I don't find the intuition obvious. If I understand correctly, given S we want to find the complement of S in N_G(S), K. If this complement exists, then by making the elements of the SBS symmetric with respect to K, but asymmetric with respect to S, we obtain an ideal SBS. If this is correct, I suggest fleshing out this intuition more, as it can help the reader understand, but the current version is not clear enough, especially what is meant by orthogonal.\n- I think the method you introduce for construction an Equivariant SBS in section 5.4 can be seen as a case of equivariance by canonicalization (Kaba, 2023) since the SBS is constructed only for the canonical subgroup S, and the SBS for any input is obtained by applying the canonicalizer to this SBS. I think this connection is worth mentioning.\n- \"Since we want this to be a full-symmetry breaking task, we require chirality in our prism.\" This seems quite ad hoc, I don't understand why this should be done. Isn't it sufficient that the group acts freely on the symmetry breaking objects to break all the symmetry as explained in the theory section?", "broader_impact_concerns": "This work doesn't pose specific concerns, but some discussion of impact areas (e.g. chemistry) could be considered.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: One of the reviewers noted that one comment of their's remained unanswered by the authors after the rebuttal. Namely, 

"The following question is not addressed by the authors yet. I would appreciate if some discussions can be added to the camera-ready version. “Regarding the comparison with Smidt et al. (2021): Based on the updated manuscript, I agree with the authors that the goals are different. However, both methods intend to find a symmetry-breaking object (symmetry-breaking set in this work, and symmetry-breaking parameters in Smidt et al.). Can the authors comment on if there are any explicit connections between them (e.g will the learnable parameters in Smidt et al. learn the SBS in this work, if set-up properly)? Perhaps there are fundamental distinctions between these two approaches that I am missing. In any case, a more detailed discussion would be great.” " 

I think the authors can and should address this in the camera ready version, hence the need for minor revision. 

Otherwise, all reviewers seemed satisfied with the state of the revised manuscript and the unanimous recommendation of acceptance led to my decision to accept.
Decision: Accept (poster)