AtlasD: Automatic Local Symmetry Discovery
Authors: Manu Bhat
Jonghyun Park
Jianke Yang
Nima Dehmamy
Robin Walters
Rose Yu
Venue: ICML 2025 poster (2025)
Keywords: Local symmetry discovery
symmetry discovery
equivariance
gauge equivariant neural network
Lie theory
Primary area: deep_learning->everything_else
TL;DR: A symmetry discovery method capable of learning local transformations.

Abstract: Existing symmetry discovery methods predominantly focus on global transformations across the entire system or space, but they fail to consider the symmetries in local neighborhoods. This may result in the reported symmetry group being a misrepresentation of the true symmetry. In this paper, we formalize the notion of local symmetry as atlas equivariance. Our proposed pipeline, automatic local symmetry discovery (AtlasD), recovers the local symmetries of a function by training local predictor networks and then learning a Lie group basis to which the predictors are equivariant. We demonstrate AtlasD is capable of discovering local symmetry groups with multiple connected components in top-quark tagging and partial differential equation experiments. The discovered local symmetry is shown to be a useful inductive bias that improves the performance of downstream tasks in climate segmentation and vision tasks. Our code is publicly available at https://github.com/Rose-STL-Lab/AtlasD.

AtlasD: Automatic Local Symmetry Discovery
Manu Bhat' Jonghyun Park
Jianke Yang'
Nima Dehmamy? 2
Robin Walters?
Rose Yu'
Abstract
Existing symmetry discovery methods predomi- nantly focus on global transformations across the entire system or space, but they fail to consider the symmetries in local neighborhoods. This may result in the reported symmetry group being a misrepresentation of the true symmetry. this paper, we formalize the notion of local symme- try atlas equivariance. Our proposed pipeline, automatic local symmetry discovery (At1 las), recovers the local symmetries of function by training local predictor networks and then learn- ing a Lie group basis to which the predictors are equivariant. We demonstrate AtlasD is capable of discovering local symmetry groups with multi- ple connected components in top-quark tagging and partial differential equation experiments. The discovered local symmetry is shown to be a use- ful inductive bias that improves the performance of downstream tasks in climate segmentation and vision tasks. Our code is publicly available http://github.com/ www.iiiuuuooooooooeeeeeeeeeeeeee Saaaa/oossSSLLLa/AAAAAAAA- STL-Lab/AtlasD.
Figure I. Global vs local transformations. Global transformations (left) alter the space in uniform manner, whereas local transfor- mations (right) only affect particular neighborhood.
networks require knowledge of the system's 's symmetries be- forehand. To adhere to this successful design principle even when the symmetry group is unknown a priori, many works have developed auxiliary neural networks to automatically identify symmetries (Benton al., 2020; Zhou et al., 2021; Dehmamy al., 2021; Moskalev et al., 2022; Yang et al., 2023; Gabel et al., 2023).
The aforementioned equivariant models and symmetry dis- covery pipelines focus on global symmetries, where a trans- formation applies across the entire space. However, arbi- trary manifolds generally do not have global symmetries to begin with (Gerken et al., 2023), preventing the use of globally equivariant networks. This begs the need to con- sider local borhoods-wihh are much more generalized (Figure 1), Indeed, some recent work considers local symmetry, such as Cohen et al. (2019), which develops gauge equivariant CNNs to take advantage of the gauge symmetries of arbi- trary manifolds. Construction of such networks once again requires knowledge of the symmetry beforehand. However, existing discovery methods of limited help they ignore such local symmetries. Hence, the need to develop local symmetry discovery pipeline is clear: it would generalize symmetry discovery to arbitrary manifolds and allow for downstream use in gauge equivariant networks.
1. Introduction
small neigh-
Equivariant neural networks (Bronstein et al., 2021), a fam- ily of models that exploit symmetry as an inductive bias for neural network architectures, have received increasing attention in deep learning due to their training efficiency and improved generalization (Krizhevsky et al., 2017; Wor- rall & Welling, 2019; Zaheer et al., 2017). The key idea behind these models is that many real-world situations ex- such as rotation,
hibit inherent tammmerres-ttrnsfommmaa translation, and scaling, which leave the essential properties of a system unchanged. This has enabled wide range of applications, leading to empirical success (Winkels Co- hen, 2018; Brown & Lunter, 2018; Cohen & Welling, 2016; Cohen et al., 201 8). Despite its achievements, equivariant
In this work, we define local symmetry around the notion of an atlas. An atlas is a collection of local regions, or charts, that cover a manifold. In short, the principle of atlas equiv- ariance states that when restricting function to a particular chart, the localized function must be equivariant. We de- velop a method based on deep learning that can discover the atlas equivariances of dataset the form a Lie group. To do so, we first model the task function localized to the
'University of Califoria San Diego 'IBM Research NNottheastem University. Correspondence to: Rose uuc...duu...d
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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AtlasD: Automatic Local Symmetry Discovery
various charts using neural networks. Then, we create Lie group basis and optimize it until the localized networks are equivariant with respect to the group. After discovery, we use the resulting symmetry as an inductive bias to cre- ate equivariant networks. Specifically, we experiment with top-quark tagging, synthetic partial differential equations, MNIST classification, and climate segmentation to test the validity of our discovery method and measure performance gains in downstream models.
and that using our discovered symmetry in gauge equiv- ariant CNN can lead to better performance and parameter efficiency.
To deal with situations where symmetries are only partially present in systems, some works introduce the notion of approximately equivariant neural networks. Implementing such networks can be done in various manners, including modification of convolutional layers (Wang et al., 2022; van der Ouderaa et al., 2022) and addition of non-equivariant residual terms (Finzi et al., 2021). Other works seek to match model and data equivariance error more closely by learning the degree of equivariance from the data (Veefkind & Cesa, 2024a; Romero & Lohit, 2022a; Veefkind & Cesa, 2024b; van der Ouderaa et al., 2023). Petrache & Trivedi (2023) gives results about the generalizability of approxi- mately equivariant networks, whereas Wang et al. (2023) provides a theoretical understanding of equivariant networks in systems with varying degrees of symmetry. While approx- imate equivariance is related to local symmetry in that they both deal with situations where the global symmetry may not be fully present, local symmetry is more generalized notion as can be applied arbitrary manifolds.
Our contributions can be summarized as follows:
We formalize the notion of local symmetry through the definition of atlas equivariance and establish theoreti- cal connections.
We develop a pipeline, automatic local symmetry discovery (AtlasD), to recover local symmetries from a dataset. AtlasD can learn both continuous and dis- crete symmetries.
AtlasD can discover atlas equivariances in cases where existing methods are not applicable, thereby proving the need to consider local symmetries.
Automatic Symmetry Discovery. Many works perform automatic symmetry discovery to identify the unknown sym- metry within a dataset. Attempts have been made to dis- cover general continuous symmetries with Lie theory, such as LieGG to discover symmetry from the polarization ma- trix (Moskalev al., 2022), L-conv (Dehmamy et al., 2021) to find group equivariant functions. and Forestano al. (2023) who discover closed Lie subalgebras from dataset. LieGAN (Yang et al., 2023) uses generator-discriinnator pattern to discover global symmetries in the form of both continuous Lie groups and discrete subgroups. Gabel et al. (2023) aim to find the symmetry group as well as quantify the exact distribution of transformations present in a dataset.
We show that incorporating the symmetries discovered by At AasD in a gauge equivariant CNN (Cohen al., 2019) can lead to better performance and parameter efficiency.
2. Related Work
Equivariant Neural Networks. Equivariant neural net- works use known symmetry as an inductive bias when fitting model. Group equivariant CNN extends the translational equivariance of a CNN to rotations and reflection using group theory (Cohen & Welling, 2016). Other works focus on designing networks that are equivariant to wider range of transformations; particularly, F222) transformations on the Euclidean plane (Weiler& & Cesa, 2019), rotations on the sphere (Cohen et al., 2018), Lorentz group transformations (Gong et al., 2022), and E((n) transformations in higher- dimensional spaces (Satorras et al., 2021). These works focus on global transformations and require prior knowl- edge of the symmetry. In contrast, our pipeline focuses on discovering local symmetry.
Symmetry discovery works also consider slightly varied problem settings. Some authors seek to find subset of possible symmetries (Benton et al., 2020; Romero & Lohit, 2022b). Others consider the case where the group acts on the latent space instead of the feature space (Yang et al., 2024; Gabel et al., 2023; Keurti et al., 2023; Koyama et al.,
2024). Rather than focusing solely on the symmetry group, van der Ouderaa et al. (2024); Hou et al. (2024) learn the conserved quantities of systems. Still, these papers mainly focus on global transformations. More relevant to local sym- metry discovery is the work by Decelle et al. (2019), which attempts to see if two datapoints are related by a particular local transformation. This is distinct from At lasa, where we characterize the local symmetry group in an interpretable manner.
Gauge equivariant neural networks extend the ideas of glob- ally equivariant neural networks by enforcing local sym- metries instead of global ones. Gauge equivariance has been applied in various contexts, including surface meshes (De Haan et al., 2020), lattice structures (Favoni et al., 2022), and general manifolds (Cohen et al., 2019). Once again, these networks require user intervention to determine the gauge group. On the other hand, we show that our defini- tion of local symmetry is connected to gauge equivariance
Comparison with State-of-the-Art, LieGAN, LieGG, and AtsasD are all capable of discovering continuous equiv-
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AtlasD: Automatic Local Symmetry Discovery
ariances, but LieGG requires significant memory and Lie- GAN uses adversarial methods, which can be complex to train. Moreover, while LieGG cannot discover any discrete symmetriie and LieGAN can only discover those with posi- tive determinants, At asD proves capable of discovering orientation--eevrriing discrete symmetries. The largest dif- ference, however, the implicit hypothesis space of each model. LieGG and LieGAN only consider global symme- tries, but At 1 asD can also discover local ones. The hypoth- esis space is a crucial component of a discovery method as it determines the set of symmetries that can be recovered.
4. AtlasD: Automatic Local Symmetry Discovery
Existing discovery methods are fundamentally limited in that they ignore local symmetries. To address this problem, we first formulate atlas equivariance as a definition of local symmetry. Then, we detail our methodology for discovering local symmetry in the form of a Lie algebra basis and com- ponent group. Finally, we highlight theoretical connections to existing work as well as implementation notes.
4.1. Atlas Equivariance
3. Background
To provide an intuition of local symmetry, we highlight the heat equation on a torus in Figure 2. Consider the time-stepping function that evolves the current state of the system for some fixed time interval. If we focus only neighborhood of the input feature field and its corresponding neighborhood in the output field, a local rotation in the input results in an identical local rotation in the output.
We provide background information on Lie groups, equiv- ariance, feature fields, and atlases. We assume some knowl- edge of group theory and otherwise refer readers to Artin (201 1); Weiler et al. (2021) as useful starting points.
Lie Groups. A Lie group is group that is also a differen- tiable manifold. Some examples include SO(2). O(3), and (3). The Lie algebra of a Lie group, denoted g. is the tangent space at the identity element. Being vector space, the Lie algebra is often simpler to work with than the group.
To define local symmetry formally, assume we have a func- tion map T that transforms input feature field Fi . + Rd to an output field Fou . - R Then, suppose A is an atlas on M given by finite collection of charts {UVer (c)] For any feature field F' on M. we can relo- cate the subset of F contained in the neighborhood U. to a flat Euclidean space by pulling back over yF
For matrix Lie groups, the matrix exponential exp(AA) pro- vides way to map elements of the Lie algebra to elements of the group's identity component Go. i.e. the connected component containing the identity element. The various connected components are cosets of Go and will also be plainly referred to as cosets in our work. In many cases, we can factor an arbitrary element of the group as a product, g C exp(A), for some coset representative C E G and Lie algebra element A E Thus, to understand a Lie group, it is often enough to enumerate all the cosets in the component group G/Gg, and identify a basis for its Lie algebra. For further information, see Kirillov (2008).
F(
if E (e(cc) else
((P=)) (z)
0
Note that the flattened feature field is trivially extended out- side (U.)). This is necessary for introducing atlas equivari- ance, where the group action may take a point p E (P,)U) outside this original domain. This O-padding can also be replaced with another value appropriate to the context.
Equivariance. A function f is said to be G-equivariant for some group G if the following holds:
We define local (atlas) equivariance for functions & where the output signal depends locally on the input signal. We formalize this as A locality. The intuitive notion of a local function is that, under an appropriate choice of atlas, we can fully reconstruct the output field in any individual chart Ue solely from the input field along the same chart. Thus, we say + is A atlas local if we are able decompose it into various de, where D. is map between the pullback of the cth chart of the input and output feature fields. We sometimes refer to (Φc} as the localized functions of 4. Formally,
(Vg E G) Here, g x and g f(x) denote (possibly different) group actions.
l'g x) = g f(x)
(1)
Feature Fields. A feature field identifies feature vector for each point in a manifold M. Specifically, a feature field is given as a map F .M- - Rd, where d is the dimension of the feature field.
Charts and Atlases. is not possible give consis- tent choice of coordinates across manifolds with non-trivial topology. We define local coordinates in terms of local charts. A chart is pair (U, ()) where U is an open subset of M and ip is a homeomorphism from U to an open subset of Euclidean space. An atlas is a set of charts that collectively cover a manifold M.
Definition 4.1 (Atlas Locality). & is A atlas local if for each chart in A and arbitrary F -M - Rda, there exists &. such that &e ((p=1)) F) ((==)* ((F) when restricted to (U.)).
Here, the restriction of the feature field to the subset (U.)) indicates that we are not particular about the output of $
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AtlasD: Automatic Local Symmetry Discovery
P3
(vg y
input charts
g
(1
Φ2
Fin
S
global map
9
1
input chart
output chart
output charts
local map
(h)
(a)
Figure 2. Atlas equivariance explained through the example of the heat equation. highlights how the task function & is function whose input and output are scalar feature fields on a torus. + then broken up into localized functions, i.e. the 4. Although we only highlight three &e for visual purposes, in reality there is one each chart. (b) commutative diagram that highlights the rotational equivariance of localized function and hence the rotational atlas equivariance of 4,
outside the projected chart. We also note that atlas locality is related to the idea of sheaf morphisms. In Appendix A, we show that the former is weaker condition than the latter.
vidual localized functions +. with neural networks or other differentiable oracles. The exact method unique to each task, but is generally a simple regression problem. Further details are available Appendix C. In contrast to global dis- covery techniques, emphasize that the individual neural networks are localized maps rather than functions over the entire manifold.
For these atlas local functions, it is possible consider the symmetry transformations that operate within local neigh- borhoods. We formalize this notion of local symmetry as follows.
Next, we find the symmetry group of the localized func- tions. Note that there are important differences between our procedure and global symmetry discovery. our setting, a group element must only act on a local chart, rather than the full space. Likewise, apply different transformations to different regions of the feature field. Care must be taken, therefore, that the actions on different charts are truly independent.
Definition 4.2 (Atlas Equivariance). & is A atlas equiv- ariant to some group G if d is A atlas local with local- ized functions { and all &. are globally G equivariant. Specifically, for the group action (p) = E(y -pp) where E' a feature field on the Euclidean space, we must have ( F)) = g. " FF) for arbitrary g G and feature fields F .M R.
tenet of local symmetry is that
A technical note is that the T may not be unique in that for given chart there are many potential localized maps that satisfy the condition specified in Definition 4.1. Therefore, is said to be atlas equivariant if, for each chart c, any potential & is G globally equivariant.
We aim to discover the maximal group of local symmetry, In practice, these symmetries often involve both discrete and continuous transformations. which motivates to Lie groups to describe the symmetries of interest. Specif- ically, we seek to discover both the Lie algebra which characterizes the continuous transformations, and the casets that describe the discrete actions the target group. Put together, these allow us to describe a wide variety of Lie groups. Algorithm outlines our overall procedure, with the details of the subroutines introduced the following subsections. We analyze the time and space complexity of AtlasD in Appendix E.
4.2. Atlas Equivariance Discovery
In our problem setup, given an unknown task function t X + y between feature fields on the same manifold, we have a dataset ((X,, H= C X y. We assume that suitable atlas A for the problem is known. We further assume that the dataset is large enough such that ground truth symmetry is represented in any chart (i.e., all members of any orbit are present). We aim to find the maximal matrix Lie group to which & is A atlas equivariant.
1.2.1. DISCOVERING INFINITESIMAL GENERATORS
To discover the Lie algebra, we view it as a vector space and learn its basis, also known as the infinitesimal generators (of the group). To enforce the local symmetry condition, we op-
We propose Atsas,, automatic local symmetry discovery, to tackle the problem. First, we must approximate the indi-

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AtlasD: Automatic Local Symmetry Discovery
Algorithm I Automatic Local Symmetry Discovery
absolute value and y is a positive weighting constant:
input Atlas A (U),
dataset D = {X; M :M + , .Y + Rd }im
Ssbe = Σ Σ
vec( voy() (vec(B,)]l vec
(3)
output Lie algebra {B.}, cosets {C²]
1. Train a predictor network @ for each chart (Ues sP) under loss L ( X;); (Y)))
We prove a result about the global minima of Lar under certain conditions in Appendix A. We also list additional regularizations and method for selecting the hyperparame- ter k in Appendix B.
2. Given the trained predictors {Re), discover the Lie algebra basis {
Set exp 1) B,) where 1) (0,
Optimize B under loss
4.2.2. DISCOVERING DISCRETE SYMMETRIES
[(g 2o((F=")) Xx),, Lolg (p)))
3. Given ( and (B:), find the coset representatives {Ct}
Dataset
Cosets
Set normalizziz((C) Optimize Ct under loss
Loas
(gg $e(( X.), Be(g (P: (),))) Filter duplicate cosets using B
CLK
normalize
Σ C( 2), +c(x))
Return (Ct
Figure 3. Discrete discovery training loop of At LasD. All K pur- ple and white squares depict representative of a discovered coset. The matrices are optimized so that their normalized forms become elements of the ground truth cosets.
timize the basis of vectors to minimize the atlas equivariance error of the map & on local charts.
Many symmetry discovery methods only discover a Lie algebra basis, limiting the results to connected Lie groups. In practice, groups such as O(2) and SO(1, have multiple connected components, which is a natural consequence of discrete symmetries such as reflections. In this subsection, we introduce a method for discovering discrete symmetries by identifying the Go-cosets in the component group G/ /Go
Specifically, we first create a trainable tensor B, consisting of & randomly initialized matrices of shape m x m where m = dim M. B represents the Lie algebra basis be discovered. Each basis vector is parametrized by a m m matrix because, after exponentiation, it linearly transforms the flattened local neighborhoods of M. In the training loop, we randomly sample an element from a dataset as well as a coefficient vector ~ (0, I). Using the coefficient vector and B, we have a group element: = exp(Ek= T): B:). The loss is the sum of L( z).g
The discovery of these cosets faces several challenges. For one, we cannot parameterize the search space through Lie algebra since there is no real-valued matrix that maps to an orientation reversing matrix via the exponential map. A discovery method operating on a Lie algebra is unable to realize both connected components of ((2) since it includes reflection. Moreover, even when the search space is set to
(x(x)) over all &., which measures the equivariance of each l. with respect to the group element g. In this case, C an error function appropriate the context.
(n), we observe an abundance of local minima. Unless the seeded matrix is already close to coset, it may fail to converge anything useful.
One problem with the given loss is that it often results in duplicate generators. Although cosine similarity is an es- tablish regularization technique to avoid this issue (Yang et al., 2023; Forestano et al., 2023), it is sensitive to initial conditions and fails to produce consistent generators on con- secutive runs. Therefore, we introduce the standard basis regularization instead, where one applies element-wise ab- solute value each generator before applying the cosine similarity function. This incentivizes different vectors to share as little non-zero positions as possible, thereby driving the basis into standard form. We observe more interpretable results that are consistent across runs, albeit with a higher rate of duplicate generators. The standard basis regular- ization is provided below, where [B] denotes element-wise
To narrow the search space, we first assume the target group contains a finite number of connected components, which applies to most dime--iimessional Lie groups of inter- est. This implies we only need to consider transformations whose determinant has absolute value
In the discovery process, we create trainable tensor C that contains representatives of Go-cosets. C' is initially set to K random matrices in m where K is chosen to be sig- nificantly larger than the expected number of cosets. Each Ce is then independently optimized according to the loss of C(#=.normalize (Ct) x),, normalize (C+) +2()) across
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AtlasD: Automatic Local Symmetry Discovery
all localized functions &e and all X. The normalize function scales matrix so that its determinant has absolute value 1.
Empirically, if the charts span the majority of M rather than fully covering M, our method is still able discover local symmetries within the given region.
After convergence. the top matrices in C' by loss value are taken to be the representatives of the ground truth cosets. We avoid duplicate cosets by comparing C; to C2 and checking if C,C, belongs to the identity component, specified by see if for threshold After applying the filtration process, the final list comprises unique representatives of each coset of the target Lie group.
5. Experiments
We experiment on the following tasks to validate our methodology and implementation: (1) top-quark tagging task for direct comparison with global symmetry discov- ery baselines; (2) synthetic partial differential equation to test our model's sensitivity to various atlases; (3) projected MNIST classification and ClimateNet weather segmentation tasks to highlight our success the discovery atlas equiv- ariances as well as the performance gains when discovered symmetries are incorporated into downstream models. Ad- ditional details about each experiment, such chart sizes and other hyperparameters, present in Appendix C. Ad- ditional experiments and ablations are in Appendix D.
the already discovered Lie algebra. In particular, min |c,C,l exp(E B2)ll2
4.3. Connection to Gauge Equivariant CNN
A related notion of local symmetry is introduced by Cohen et al. (2019) when defining gauge equivariant CNNs. In short, gauge equivariance implies that one should be able to arbitrarily orient the local coordinate systems used to define input features and compute convolutions. Hence, is property of the deep learning model, rather than the task function itself. This is notable difference compared to work, where the atlas equivariance group is intrinsic to the system and can be discovered.
5.1. Global Symmetry Comparison
The following theorem provides a concrete connection be- tween gauge equivariance and atlas equivariance (proof in Appendix A).
0.25
0.25
Theorem 4.3. Let M be a gauge equivariant CNN that (a) has linear gauge group G, (b) is A atlas local for some atlas A with trivial charts, and (c) operates Euclidean space. Then, M is A atlas equivariant to G.
0,00
6.00
-025
-05
0.03
Figure 4. At LasD discovers (+,3)) in the top tagging task. Here, each rod and blue heatmap denotes a Lie algebra basis element. For each generator, the value of its entries are depicted by the individual colors. When read in row-major order, generators 0, 1, correspond to SO(3) rotation and generators 4, 5, 6 indicate boosts. The pink and green heatmap in the bottom-right displays the computed invariant metric.
In practice, a gauge equivariant CNN is neither meant to operate on Euclidean space nor completely A atlas local. However, the result is approximately true for an arbitrary manifold manifolds are locally flat. This implies that if system is atlas equivariant for some group G, it is logical to set the gauge group of downstream gauge equivariant CNN to G. We employ this technique as application of our discovered symmetries below.
1.0
1.0
4.4. Implementation Notes
Due to issues such as discretization, noise, boundary con- ditions, or limited priori knowledge of perfect atlas, real-world datasets may only be approximately, not exactly, A atlas local. To mitigate this issue, we sometimes allow the Te predictors to look slightly outside of the associated chart, i.e. the radius of the input chart is higher than the radius of the output chart for any given &.. This provides the localized functions with additional context that may be missing from the unmodified input. Additionally, to avoid boundaries and awkward topologies (e.g. poles of a spheri- cal mesh), we partially deviate from the definition of an atlas and do not require that the charts fully cover the manifold.
-1.0
1.0
Figure 5. lasD discovers two cosets in the top tagging exper- iment: a representative from the identity and parity component. Each yellow and purple heatmap depicts coset's representative in matrix form, where the colors denote the values of that matrix's entries.
To directly compare our method to existing discovery pipelines that focus on global symmetries, we first attempt to learn global invariances the top quark tagging exper-
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AtlasD: Automatic Local Symmetry Discovery
5.2. Partial Differential Equation
Table Downstream results for top tagging task.
MODEL
ACCURACY
AUROC
LORENTZNET
0.941 10.0010
0.9862 10.0004
ATLASGNN
0.939 +0.0002
0.9852 =0.0001
b-05
LIEGNN
0.938 +0.0001
0.9849 =0.0001
LORENTZNET (W/o)
0.935 #0.001
0.9835
+0.0003
EGNN
0.925 10.0001
0.9799 10.0004
iment. Specifically, we compare our results with those of LieGAN (Yang al., 2023).
First, we fit a predictor network to the dataset using 3-layer MLP. Recall that at this point, we have no information about symmetries of the system and thus this is non-equivariant model. The trained predictor is assumed to be accurate enough such that its symmetries agree with those of the dataset. Hence, we now seek to find the symmetries of the predictor.
(a) Example input and output field
The goal is to classify between top quark and lighter quarks jets present in the Top Quark Tagging Reference Dataset (Kasieczka et al., 2019). The dataset contains 2M observa- tions, consisting of four-momentum of up to 200 particle jets. The classification invariant to the entire Lorentz group 3), which we will try discover.
(b) Charts in each atlas
Figure 6. Illustration PDE experiment. The top row depicts an example input and output scalar field for the heat experiment. The bottom row shows the two atlases used for experiment. Each blue or red square represents individual chart.
We use infinitesimal generator discovery pipeline to learn the invariances of the predictor. We seed our basis with 7 generators. In Figure 4, we show that the discovered basis matches closely with that of SO* (1, 3), the identity compo- nent of the Lorentz group. Moreover, computing the invari- tensor using the method from Yang et (2023), we find that the invariant tensor has a cosine correlation of 0.9996 with the ground truth Minkowski tensor diag(-10 -1 1, 1)- This is strong result that is slightly superior to LieGAN's cosine correlation of 0.9975.
Next, want to see if our method can indeed learn atlas equivariances and also measure its sensitivity to various atlas configurations. Specifically, we experiment if our model can discover the local symmetries of the heat equation 82 8y du al dz 0y in g (Figure 6a). The task function
0 in this case simulates heat flow for 0.5 seconds given an initial condition. In the simulation, we exclude a certain rectangular region and treat as a heat source, thereby breaking any global symmetry while keeping O(2) local symmetry sufficiently far from any boundary.
We then try to discover the various cosets of the symmetry group. We seed our discovery process with K 64 matri- ces. In the dataset, the time component of all momenta are positive, and hence it is difficult find the time-reversal generator of the Lorentz group. In the last two heatmaps of Figure 4, we discover a parity transformation. which means that the entire learned symmetry group is O+ (1, In contrast, LieGAN cannot discover orientation-reverte transformations and hence only reports the identity compo- nent S0 (1,3).
To test the sensitivity of our method to different atlases, perform our experiments with one atlas containing 19 charts and another containing 3 charts (Figure 6b). In either case, we seed the model with single infinitesimal generator and = 16 cosets and report the unique cosets from the top q = 8. Figure 7, we demonstrate that At lasD is able to accurately recover the O(2) atlas equivariance group in both situations. However, the first atlas does slightly outperform the second.
Finally, we use the computed invariant metric tensor as inductive bias construct well-performing classification model. Specifically, we create AtlasGNN by modifying LorentzNet (Gong et al., 2022) to use our discovered metric instead of the Minkowski tensor. In Table 1, we observe better accuracy and AUROC than many baselines and nearly match LorentzNet, which uses ground truth symmetry.
We also run the global symmetry discovery baseline LieGG in similar setup. The full details are deferred to Ap- pendix D.4. short, LieGG fails to discover any global symmetry due the heat source (Figure 8).

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AtlasD: Automatic Local Symmetry Discovery
3
0
3
S
(b)
(a)
Figure 9. MNIST experiment setup. (a) The input feature fields (top) are given by three digits rotated and then projected onto the equator of sphere. To construct the output feature fields (bottom), the model must label each pixel as either background or its numeric value if it is part of number. (b) We highlight two of the charts used in our atlas.
Figure 7. PDE discovered symmetry. The results for the first and second atlas are depicted in the top and bottom rows, respectively. In each case, the leftmost entry is the discovered infinitesimal generator, and the right columns the discovered cosets.
Polarization spectrum
CNNs. In the discovery process, we seed our model with a single infinitesimal generator. To demonstrate the benefit of considering local symmetry, we compare our results against modified LieGAN that represents global symmetries as subgroups of SO(3).
10a
10
With Heat Source
Without Heat Source
5 u
1
5
10
i-th singular value
5
Figure Singular values of generators discovered by LieGG in modified PDE experiment. We run twice: once without the rectan- gular heat source (blue) and once with the heat source (red). The without the heat source provides reference for singular values the case that there is global symmetry.
Figure 10. Local and global transformations on MNIST. the upper row, highlight element from the dataset in its projected form. In the middle row, we apply local transformation based upon a discovery. The last is the result after applying global rotation suggested hy LieGAN.
5.3. MNIST on Sphere
After running the discovery process, we find an approximate
To highlight the benefits of using our learned results in down- stream models, design projected MNIST segmentation task. In this experiment, we project three digits from the MNIST dataset onto a sphere (Figure 9). Before projection, each image is randomly rotated up to 60 degrees clockwise or counterclockwise. The goal of the model is to classify each pixel as either the background or its numeric value. Although the rotation of the digits adds a local symmetry, there is continuous global symmetry since the position of each of the three digits is fixed. For this problem, we construct an atlas by assigning a single chart to the region of each digit. This is an admittedly idealized setup, but our main purpose is to demonstrate the full pipeline.
-00.3
-1.00)
In Figure 10, we show that applying local transformation suggested by Atlast laSD leads to non-trivial change, but one that still preserves the form of the dataset. On the other hand, the global transfor- mation sampled from LieGAN's result clearly modifies the input out of distribution, suggesting the result a random rotation rather than an actual symmetry. This highlights a where considering local symmetry is more appropriate than searching for global symmetry.
(2) generator:
1.00
0.02
In addition, we construct gauge equivariant CNN using the discovered SO(2) group and compare it to a regular CNN. We train each model on a dataset where the digits are rotated +0 degrees, and test it on one where digits
We then train predictor for each of the three charts using
8

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AtlasD: Automatic Local Symmetry Discovery
scalle
-11
fiscale x)
fishear x)
firot
Figure/ 11. AtlasD discovers G2) in climate experiment.
are rotated +180 degrees. We observe that the inductive bias of the gauge equivariant network allows it to generalize outside of its training set, achieving an accuracy of 0.9381 compared to 0.6975 for a vanilla CNN.
scale fix)
shear fix)
rot fix)
5.4. ClimateNet
Figure 12. The inputs and outputs of ClimateNet predictor ʃ after applying various transformations. The plotted x: values are visual- izations of TMQ. In the outputs, purple indicates the background, and yellow represents the atmospheric river.
For our final experiment, we evaluate our method on a real- world dataset, ClimateNet, proposed by Prabhat et al. (2021). Each input in the dataset contains 16 atmospheric variables across the surface of the Earth, and the output is human label to determine whether each pixel is part of the back- ground, atmospheric river, or tropical cyclone. We aim to discover the atlas equivariance group.
Table 2. ClimateNet dataset accuracy results. We compute the IoU obtained for the baseline SO(2) gauge group model, our GL+(2) gauge group model, and between human experts. We include the last row to demonstrate that human labelers have a degree of disagreement, providing context for low IoU scores, See Appendix C for full results and details.
We use an atlas that has 4 charts spread through the surface of the Earth. When we seeded our model with 1,2, or infinitesimal generators, we find that the resultant basis is not similar across consecutive runs. This suggests that the symmetry group is actually 4-dimensional. To confirm this, we plot the output of chart predictor after applying vari- ous linear actions in Figure 12. The figure highlights that the predictor is mildly equivariant to a wide range of actions. In fact, Figure demonstrates comparable magnitudes of the generators. All of these suggest that the atlas equivariance group is GL+ (2).
MODEL
PARAM 1
BG f
TC1
AR f
MEAN
SO(2) GL (2)
766K
0.911
0.174
0.384
0.490
111K
0.909
0.172
0.379
0.487
HUMAN
0.914
0.248
0.347
0.503
We compare using the discovered atlas equivariance group to the structure group in a downstream gauge equivariant CNN. Specifically, we use an icoCNN architecture in two different settings (Diaz-Guerra et al., 2023). For the base- line, we set the gauge group of the icoCNN to be SO(2) (the structure group). While is not easy to construct gauge equivariant CNN using steerable kernels (Weiler & Cesa, 2019) for non-compact group such GT. (2), the closest approximation is to have the kernel be spatially uniform. That is, all values for a given outut- channel pair are the same for a particular filter. In Table 2, we show that the "flat" kernel CNN is able to match the baseline performance despite having times fewer parameters. This highlights benefit using the discovered group as the gauge group versus choosing the structure group of the manifold.
finding is that global symmetries are not enough to describe all useful symmetries of a system. We demonstrate the abil- ity to discover infinitesimal generators and cosets of the atlas equivariance group from dataset. Moreover, the re- sults prove the atlas equivariance group also serves as an inductive bias in downstream gauge equivariant networks.
While our method effectively discovers atlas equivariances, atlas equivariances only describe a subset of all diffeomor- phisms manifold. We set the stage for future work to explore the discovery larger groups. In addition, while lasD proved resilient modifications of the given atlas, priori knowledge of suitable atlas is still important and may not always be available. An extension to our work can develop a method that discovers the atlas in tandem to the atlas equivariance group. Finally, one may consider symme- tries that act on both the manifold and the features, such as point symmetries in partial differential equation systems.
6. Conclusion
In this paper, we introduce atlas equivariance and propose automatic local symmetry discovery (AtlasD) as an ar- chitecture capable of learning local symmetries. Our key

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AtlasD: Automatic Local Symmetry Discovery
Impact Statement
De Haan, P., Weiler, M., Cohen, T., and Welling, M. Gauge equivariant mesh cons: Anisotropic convolutions on geo- metric graphs. arXiv preprint arXiv:2003.05425 2020.
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which feel must be specifically highlighted here.
Decelle, A., Mayor, V., and Scoane, B. Learning
local symmetry with neural networks. Phys. Rev. 100:050102, Nov 2019. doi: 10.1 10.1105/PhysRevE.100 050102. URL https: //tp:////iii//e///eeeeeeeeeee.. aps wtt..aas..rg/dd//110... 1103/PhysRevE. 100. 100.00000.. 050102.
Acknowledgement
This work was supported in part by NSF Grants #2205093, #2146343, #2134274, CDC-RRAA-FT-23-006) the U.S. Army Research Office under Army-ECASE award W91 the U.S. Department Of Energy, Office of Science, IARPA HAYSTAC Program, DARPA AIE FoundSci and DARPA YFA.
Dehmamy, N., Walters, R., Liu, Y., Wang, D., and Yu, R. Automatic symmetry discovery with lie algebra con- volutional network. In Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Ad- vances in Neural Information Processing Systems, vol- ume 34, pp. 2503-2515
Curran Associates, Inc., 2021.
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AtlasD: Automatic Local Symmetry Discovery
A. Math
A.1. Connection between Sheaf Morphism and Atlas Locality
Sheaves provide a framework for relating local data across overlapping regions and are relevant in the context of local functions. particular, we show that being a sheaf morphism a strictly stronger condtion than being atlas local. We first summarize the definitions related sheaves and sheaf morphisms, and refer Vakil (2022) for more detailed introduction.
Suppose that X is topological space. F is said to be presheaf (of sets) on X if the following conditions are met:
1. For any open set U S X, we have set F(U) containing the sections of F over U.
2. If U and V are open sets such that U S V, we have a restriction function resv,u F(V) F(U).
3. resu,u must be the identity.
4. For any triplet of open sets such that U C V C W, we have resw,u resy,u resw.V.
Then, any presheaf F is said to be a sheaf (of sets) if the additional two conditions are satisfied:
1. Let / be an indexing set and suppose we have an open cover {Uihier of where U₁ CU. For any f1, E F(U), if
resu,u, fi = resu,u, f2 for all E I, we must have fi = f2
2. For any open cover {Uther of U and family of sections {ff E U, i 1}, if resu,,unu,, ft = resu, fs for all j E I, then there must exist f F(U) so that resu,u, f fi for all
Finally, a sheaf morphism between the sheaves F and G is morphism & :: F G specified by map ou F(U) + G G(U) for each open set U S X so that o commutes with restriction, i.e., if V C U we have rest,v ou (F(U)) σv (res F(U))
Now, suppose we have a manifold M and atlas specified by collection of charts {U.)= We may consider M as a topological space whose subbasis is given by the collection of charts. Note that for any fixed d, the set of feature fields specified by functions F : .M Rd forms a sheaf. In particular, let & be a sheaf morphism between feature fields of dimension din to those of dimension dour- Then, for each Ue and arbitrary F M + Rdd we have that M, (F((F)= = óu, (res)M, F) for all Uc. To finally prove = Φ. is atlas local, we can construct each &. using du.. This is because for any chart Uc, the sheaf morphism condition guarantees that the local behavior of ΦM fully captured by ΦU.-
However, unlike sheaf morphisms, atlas locality does not require localized map on all open subsets, e.g., the intersection of charts, and is thus weaker condition.
A.2. Atlas equivariance of gauge equivariant CNN
Theorem 4.3. Let M be gauge equivariant CNN that (a) has a linear gauge group G. (b) is A atlas local for some atlas A with trivial charts, and (c) operates on Euclidean spuce. Then, M is A atlas equivariant to G.
Proof. We first clarify a few definitions. By trivial charts, mean that Pe is the inclusion map. We model a gauge equivariant CNN as a series of convolutional layers and pointwise nonlinearities (Cohen et al., 2019). To enforce gauge equivariance, we will require all kernels K Rdd -> TR. CCa in the network to satisfy K(g-lo) = K(v) for E G.
To show that such a network M G atlas equivariant, we must prove that there exists G-equivariant localized functions M.. Recall that in the definition of A atlas local, Mc has no restriction on its output field outside (U.)). Consequently, since all charts are trivial and M is assumed to be A atlas local to begin with, M itself is suitable for each Mc.
It remains to show that M is G equivariant. Indeed, a G gauge equivariant convolutional layer Euclidean space as presented above is equivalent to G-steerable convolution (Weiler & Cesa, 2019). Moreover, G-steerable convolutions are globally equivariant. As we assume feature vectors transform trivially in response to a group action, all pointwise nonlinearities are automatically G equivariant. M, the composition of G equivariant layers, is then G equivariant.
Thus, M is A atlas equivariant with respect to G.
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AtlasD: Automatic Local Symmetry Discovery
A.3. Argmin of Standard Basis Regularization
Let V be a k-dimensional subspace of IR... We call a basis {bi of V disjoint if the of indices of all non-zero elements of b, and the similar set for by are disjoint whenever i + j. The following theorem gives result about the arguments of the minima of Labr.
Theorem A.1. Let V be a k-dimensional subspace of tn for which there exists a disjoint basis. Then, among all possible bases (b, 1= of V. (bbr(b) is minimal if and only if b disjoint.
Proof. Suppose } is a basis of V. If b is disjoint then for all i * j we have [6x| |by] = O since for each index é, at least one of & or by. will be zero. Conversely, if is not disjoint, then there exist some * j such that [b,] [6y| > To see this, note that we must have some i * where b; and by share a non-zero element at index &. [b:] -byll the sum of non-negative numbers, is then greater than or equal to [br.ellb [bb.e O.
In particular, this implies that for disjoint basis b, we have (b) 0, but otherwise Cabr(b) >> 0. By assumption, there exists at least one disjoint basis so then the minimum of Lytr over all possible bases of V is 0. This minimum is attained exactly when the input basis is disjoint.
B. Additional Implementation Details
common degenerate solution in discovering the Lie algebra basis when all basis vectors tend towards 0, corresponding to the identity transformation. To prevent this, we add the following growth regularization, where and are hyperparameters.
Lgr
mi:(|
The min term ensures that the model does not produce arbitrarily large generators. In the experiment details, we refer to as the growth factor and 3 as the growth limit.
An important hyperparameter in the discovery of infinitesimal generators is k the dimension of the basis. Forestano et al. (2023) suggest setting the dimension as the highest number that results in vanishing loss. However, we find that the threshold for what constitutes "vanishing" can become ambiguous in real-world datasets. Therefore, to determine the final value of k we first run the model repeatedly, varying the basis dimension in different runs. We initially set as the minimum value such that a model with k generators always converges to the same algebra, irrespective of the starting conditions. Then, we increment & one by one until the norm of the weakest generator drops below a threshold.
C. Experiment Details
In this section, we include some additional details for the performed experiments, including the hyperparameters and synthetic dataset configurations.
C.1. Global Symmetry Discovery
The predictor for this task is 3-layer MLP that takes the input of 30-leading constituents for each sample, constructed of 4-momenta (E/c, Pa, Py. Pz). This results in an input dimension of 120. The predictor is trained for 10 epochs with a learning rate of 0.001 prior to the discovery process. We use cross-entropy loss for training. We find that the predictor can be relatively naive and still be suitable for symmetry discovery.
In the infinitesimal generator discovery, we seed the basis with elements by our criteria for choosing the dimension of the basis. Although the Lorentz group is 6 dimensional, Interestingly, Yang et al. (2023) find similar generator using their methodology. We run the model for 10 epochs using cross-entropy loss. We set the coefficient of standard basis regularization to be 0.1 and the growth factor of the generators to be 1. We do note set a growth limit. The learning rate is 0.001.
model occasionally finds additional scaling generator.
For coset discovery, we seed the model with K = 64 basis elements and run 3 epochs with cross-entropy loss. To filter out the final representatives, we find all the unique cosets in the top q = 16 matrices. The learning rate is 0.001.
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AtlasD: Automatic Local Symmetry Discovery
In the downstream task, we replace all Minkowski norms and Minkowski inner products of NorentzNet (Gong al., 2022) with those appropriate our discovered metric. We construct the model with 6 group equivariant blocks with 72 hidden dimensions and train it with a batch size of 32 for 35 epochs with dropout rate of 0.2, weight decay rate of 0.01, and learning rate of 0.0003. Note that while our predictor used for symmetry discovery was limited to the 30 leading components, the downstream model does not face the same restriction. We run our model and the baselines 3 times and record the average and standard deviation in Table 1.
C.2. Partial Differential Equation
For this experiment, we create a dataset of 10000 samples, each of size 128x128. The exclusion region spans from (0.1, 0.2) to (0.3, 0.5). where (o, 0) is the top left, and (1, 1) is the bottom right. The initial condition is given by creating a purely vertical sinusoid with random parameters and adding it to a purely horizontal sinusoid with random parameters. To construct the output for each input, we approximate the heat equation using a finite difference method. We use = 1. In particular, we numerically integrate 50 times with di- 0.01. We use the Dirichlet boundary condition, where all boundary values (including those on the excluded region) take the value /2.
The charts in the first atlas have an in-radius of 14 pixels (full dimension 29x29) and an out-radius of 10 (full dimension 21x21). There are a total of 19 charts centered at the following locations specified in the previously defined coordinate space: (0.5, 0.15), (0.675, 0.15), (0.85, 0.15), (0.5, 0. 025),, (0.675 0. 0.325),
(0. 80,85,
0.325), (0.5, 0.5), (0.675, 0.5), (0.85, 0.5),
15, 675),, 325, 0. 675),
(0.5, 0,57 67.675, 0. 0.675), (0.85, 0. 0.675), , 15, 0. 0.85), (0. 325, 0. 0.85), (0.5, 0.85), , (0. 675, 0. 85), , (0.85, 0. 0.) The pe do not perform any distortion, but do recenter each chart.
The charts in the second atlas have in-radius 26 (full dimension 53x53) and out-radius 20 (full dimension 41x41). They are centered at the following locations: (0.65, 0.3), , (7.675, 0. 625), , (0.35, 0. 75). The Pe act the same way as in the first chart.
The predictors are simple 4-layer CNNs. They are trained for 10 epochs in tandem with the discovery process. In the discovery process we seed the model with a single infinitesimal generator. The growth factor is set to 0.1 and growth limit is 1. We use mean absolute error as the loss. We seed the model with K 16 cosets and took the top q = 8 matrices before filtering duplicates. We run our model for 10 epochs. The learning rate is 0.001.
C.3. MNIST on Sphere
The dataset is constructed by creating 10000 spheres. Each sphere has 3 randomly selected digits from the MNIST dataset projected onto its equator at fixed positions. In particular, we first rotate each of the three digits +60 degrees. Then, all three digits of size 28x28 are placed onto a cylinder of dimensions 120x60 at equal intervals. Finally, they are projected onto a sphere using an equirectangular projection. To compute the output sphere, we label all pixels that are fully black as background. The pixels that have non-zero color are labeled with their numeric value. Consequently, there are a total of 11 classes.
The chosen atlas uses 3 charts located the locations of each of the three digits. In particular, the in- and out-radius of each chart is 14 (full dimension 29x29). The predictors for each chart are CNNs that identical in architecture but independently trained. When training the predictors, use cross-entropy loss and weigh background pixels 10 times less than numeric pixels. The growth factor of the generator is set to 0.35 and growth limit is 1. The predictors are trained in tandem with the discovery process. In particular, we run the discovery process for 20 epochs with a learning rate of 0.001. As a baseline, we compare to a modified LieGAN that can discover subgroups of SO(3). LieGAN is given a single continuous generator as well. LieGAN is run for 20 epochs with learning rate of 0.0002 for the discriminator and 0.001 for the generator.
In the downstream task, train two CNNs that are identical in design, except that one has Zu stecrable kernels. The model that has Z4 steerable kernels has less than third of the parameters of the unmodified CNN. Both models are trained for 100 epochs on the dataset. During training, to compensate for the abundance of background pixels, we weigh the background pixels 0.005 times as much as the numeric counterparts. In reporting accuracy, we fully ignore the background pixels and focus only on the numeric pixels.
15

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AtlasD: Automatic Local Symmetry Discovery
C.4. ClimateNet
We use ClimateNet dataset, which is an expert labeled open dataset provided by Prabhat al. (2021). There are roughly 200 input images in the training set, with some images having multiple human expert labelers.
In the symmetry discovery process, we use an atlas of 4 partially overlapping charts that are scattered across the equator. The in-radius is set to 200 (full dimension 401x401) whereas the out-radius is 150 (full dimension 301x301). The individual Pe do not do any additional projection, i.e. they keep the projection that the dataset used to parameterize the sphere as a rectangle. We use a modified CGNet (Wu et al., 2018; Prabhat et al., 2021) as the predictor for each chart. In particular, it is given four atmospheric variables as input: TMQ. U850, V850, PSL. The predictors are trained in tandem with the discovery process. The model is seeded with 4 generators, and use batch size of 16 and for 30 epochs using cross-entropy loss. The coefficient of the standard basis regularization is set to 0.05, the growth factor is 5.0, and the growth limit is 1.0. The learning rate is 0.001.
The downstream models are implemented using U-net (Ronneberger et al., 2015) version of icoCNN Diaz-Guerra et al., 2023). We also add strided convolutions and replace layer norm with batch norm. Since batch norm is typically not equivariant, we perform average pooling beforehand when necessary. We set the resolution of the icosahedron to r = 6. We train each model for 20 epochs with batch size of with learning rate of 0.001. Note that the downstream models, we give them all 16 atmospheric variables.
Table 3. ClimateNet dataset accuracy full results.
MODEL
PARAM 4
BG
TC 1
AR f
MEAN 1
PRECISION t
RECALL 1
SO(2) MODEL
766K
0.9107
0.1744
0.3839
0.4896
0.5983
0.6274
GL (2) MODEL
111k
0.9086
0.1720
0.3790
0.4865
0.5846
0.6344
HUMAN
0.9137
0.2475
0.3467
0.5026
We elaborate on the results of table 3. For the first two rows, the mean IoU, precision, and recall are calculated between the model predictions and every human expert label that exists for that image and then averaged. Then, these results themselves averaged across all input images in the test dataset Prabhat al. (2021). We run each model 10 times and include the run with the highest mean IoU. In the third row, we compute the mean IoU between human labels for the same input image in the training set. All scores are computed after projection onto an icosahedron.
D. Additional Experiments
D.1. Coset Discovery on Imperfect Data
We want to see if Atlash can discover multiple cosets in situations where the loss of a one coset (typically the identity) lower than the loss of another coset in the symmetry group. This could happen when the trained predictor not super accurate or if the symmetry of the true function itself is not perfect. To do so, experiment on the global discrete symmetries of the function fz, y) arctan 2+00.11 The identity transformation is a perfect symmetry of f and rotation by 180 degrees is an approximate symmetry. Although there is some variability between runs, Figure 13 highlights we are able to consistently discover rotation coset representatives among the top 24 cosets.
D.2. Top-Tagging Experiment without Standard Basis Regularization
To test how effective Lab is at regularizing basis towards standard form, we retry the the infinitesimal generator discovery of the top-tagging experiment. This time, we replace Cat with cosine similarity and plot the resultant basis in Figure 14.
In this basis, all 21 pairs generators share at least one non-zero term. On the other hand, the one produced by Lat (Figure 4) has a single such pair. We conclude that Lav is effective at forming a standard basis.
D.3. Top- Tagging and PDE Experiment without Coset Normalization
We want to see the usefulness of the normalization step during coset discovery. To do so, we repeat the discrete discovery step of both the top-tagging and PDE experiments without normalization.
16

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AtlasD: Automatic Local Symmetry Discovery
Descoeecc
Figure 133. Discovered cosets of aretan symmetry group. In each column, plot the distribution of the top 24 cosets for given Red denotes the identity component and blue the rotation component.
1280
415
14
Figure 14. Top-tagging basis using cosine similarity.
In the top-tagging experiments, we discover two cosets (Figure 15). There is more noise and a slight scaling factor as compared to the result with normalization (Figure 4), but we are still able to discover the parity and identity components. In the PDE experiment, we are usually able to discover the identity and reflection components (Figure 16), but the filtration process fails and extraneous cosets are also included.
1.0
0.5
0.0
0.0
-0.
Figure 15, Top-tagging cosets without normalization.
In both cases, we manage to find the ground truth cosets but there is more noise overall. We conclude that normalization is helpful step of the discrete discovery process.
D.4. Comparison with LieGG in PDE Experiment
To further highlight the necessity of considering local symmetry, we compare our results from the PDE experiment to those of LieGG (Moskalev et al., 2022). We generalize LieGG to be able to discover global equivariances when the group acts on R Specifically, we treat our dataset as modelling a collection of input and output feature fields (Fe, Gt) where Fz, G. R22 + IR. This given us in discretized form so that we only know Fş and Ge along some sampling
17

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AtlasD: Automatic Local Symmetry Discovery
0.5
0.5
0.5
0.5
0.5
-0.5
-0.5
-0.5
-0.5
-0.5
Figure PDE cosets without normalization.
(x), (12, x2) (1 (?) of RR2. To construct the polarization matrix, for every single output location z; in every single datapoint (P', Ge), we add
given by the following linear equation equation 4. The equation is modified from the original equation (3) from Moskalev et al. (2022) to learn equivariance instead of invariance. We keep the notations consistent with those in Moskalev et al. (2022). We note that this procedure is rather space-intensive as the number of rows proportional to the number of total output pixels.
@Gr(z.)
0F((Ep) 0Ge(2z)
Σ j.AE(1,2)
heu
Σ
op Oxk 0F((Tp)
(4)
dx
For simplicity, we consider the setting where LieGG is given access the ground truth partial differential equation instead of a predictor network. We also do not perform time stepping and focus solely on the global symmetries of du ät = a 0²
dzd + 0² ).
This setup, while slightly different from the experiments for our method, only makes easier for LieGG learn the symmetry. The singular values of the discovered generators were shown in Figure 8.
When remove the heat source and there is true global symmetry, the smallest singular value is 1.137 10-6 and is associated with the SO(2) generator. When there is a heat source like the one in the PDE experiment, all singular values become much higher. We conclude that global symmetry discovery methods such as LieGG are unable to discover meaningful symmetries in systems that only have local symmetries.
D.5. Coset Discovery of Complex Component Groups
Next, we verify that the coset discovery algorithm of At lasD can discover more complex component groups. In particular, we search for the symmetries the function f(r y) = | + |y|- The ground truth symmetry group is D4, which contains 8 elements.
We seed the discovery algorithm with K 256 representatives and report the unique cosets among the top q 128. In Figure 17, show that Aasa finds exactly the 8 elements of D4.
D.6. Discovery under Sheared Charts
To further verify the resilience of At AtlasD under the choice of atlas, we repeat the PDE experiment under third atlas where one of the coordinate charts is partially sheared (Figure 18).
0.368
-1.035
We discover a single generator
and both cosets. While the noisy chart does worsen the learned generator,
1.101
0.386
remains recognizable rotation.
18

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AtlasD: Automatic Local Symmetry Discovery
10
ka
-35 -as
Figure 17. Discovered symmetry group for [a]] + [yv
Figure 18. Atlas with sheared chart
E. Algorithm Analysis
We analyze the space and time complexity the different parts of our algorithm. our analysis, we assume that dim M fixed as constant. We use the notation that 7 the number of training iterations for given run, k the dimension the ground truth symmetry, P is the number of localized predictors, K is the total number of cosets used during training, and q is the max number of cosets reported.
E.1. Infinitesimal Generators
Our algorithm requires storing the discovered Lie algebra basis, which takes space proportional to k. We also need to store each of the P localized predictors, giving us space complexity of O(k P).
We first consider the amount of time a single training step takes. Calculating the standard basis regularization takes O(k2) time. For a given predictor, computing the main loss takes time proportional to k, which is needed for the sampling of the group element. In our implementation, we evaluate all P predictors training step. There are T total training steps in given run. Then, we require most total runs determine the optimal dimension of the basis. This gives the total time complexity as (K? + kP)) (k2 P))
The above result is somewhat misleading as it hides the high constant factor that the predictor evaluation entails. If we ignore the regularizations and only consider the predictor evaluation, the time complexity becomes OKT
E.2. Discrete Symmetries
We require storing the K cosets well as the P predictors. The space complexity is then O(K + P).
In each training step, we evaluate each of the /' predictors on K transformed inputs, corresponding to the K cosets. This is repeated for all T training iterations. After the training process, we must filter the duplicate cosets. the worst case, we report & cosets in which case we need to do (KK)) comparisons. The total time then OK +
19


Reviews:
[{"review_id": "BgQW1yMMzn", "reviewer": "Reviewer_9veb", "summary": "This paper introduces AtlasD, a framework for discovering local symmetries, specifically, atlas equivariance, within datasets. Atlas equivariance is a kind of gauge equivariance, where a global symmetry group $G$ acts differently in each local coordinate system. To identify such a $G$, AtlasD assumes a predefined atlas, trains local predictor networks  (which learn $G$-equivariance implicitly from data) for each coordinate chart, and extracts the Lie algebra of $G$ that is commonly present across the local predictors. The proposed method also extends to discrete symmetries by identifying cosets of the identity group action. Furthermore, the authors establish a theoretical connection between atlas equivariance and gauge symmetry, ultimately constructing a pipeline to extract local symmetries from data and integrate them into gauge-equivariant CNNs. The approach is validated across four different datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> A major concern with this method is the assumption that a suitable atlas for the dataset is known. \n\nWhile this assumption may initially appear overly ideal, in practice it is achievable. The primary requirement for an atlas is that the charts are large enough for the function to truly be atlas local, but small enough that each region is approximately flat. In all of our experiments, we only needed minor additional tuning once this relatively weak condition was met.\n\nWe showed in section 5.2 that AtlasD is successful under two diverse atlases, implying one has large freedom in the exact atlas they choose. To provide more evidence, we have created an additional atlas for the PDE experiment with heavy overlap, missing regions, and a sheared/noisy chart. \nhttps://i.ibb.co/xtgDr7ZF/Figure-1.png\n\nWe discover a single generator\n$\n\\\\begin{vmatrix} \n-0.368 & -1.035 \\\\\\\\\n 1.101 & 0.386\n\\\\end{vmatrix}\n$  and both cosets. The noisy chart does worsen the discovered generator, but it remains recognizable as a rotation.\n\n> Atlases constructed using data-driven approaches\n\nWe believe the referenced works slightly differ from our setup. In particular, these works model an unknown data manifold embedded in higher-dimensional Euclidean spaces. On the other hand, we deal with feature fields on a simple, explicitly known manifold, such as a sphere or 2D region. \n\n> Computational complexity w.r.t hyperparameters $k$ and $K$.\n\n$k$ is simple to tune in practice since we are dealing with low-dimensional spaces and only need several reruns during training. Moreover, only $k$ is chosen in the high-to-low process. $K$, the number of cosets, is fixed once. \n\n> It would be beneficial to include Figure 15 from the Supplementary Material in the main manuscript.\n\nWe agree that Appendix D.4 is important and will be sure to include it and Figure 15 in the main manuscript.\n\n> The proposed method identifies the global symmetry group by aggregating multiple local predictor models. Does this approach enhance the robustness of symmetry discovery against noise?\n\nTo be specific, we identify the local symmetry group of $\\Phi$ by finding the common global symmetries of all $\\Phi_c$. We argue that performing symmetry discovery across multiple predictors effectively multiplies the amount of data elements we have available, making our method resilient to noise. \n\n> If the local predictors are not well optimized, the discovered symmetry group might be unreliable.\n\nYes, this is true. However, the predictors generally have easy tasks in our setup, since they only need to predict locally. This gives us an advantage over global predictors as it is inherently more difficult to predict on a global scale. Also, compared to existing works that use a GAN discriminator, our predictor-based approach does not suffer from the training instability issue of GAN.\n\n> The authors mention that standard basis regularization provides more interpretable results. What does \"interpretable results\" mean in this context? Additionally, could you conduct an ablation study to compare the effectiveness of standard basis regularization and cosine regularization?\n\nWe provide an ablation in Appendix D.2 to compare standard basis regularization against cosine similarity. \n\nWhen cosine similarity is used, we are still able to find an orthogonal basis, but each generator has many non-zero elements (Figure 12). This makes it difficult to understand what physical action each generator corresponds to. In contrast, in the basis discovered using standard basis regularization (Figure 4), each generator has only a few non-zero elements which allows it to easily be classified as a boost or rotation. Hence, “interpretability” in this context means sparsity of the generators. \n\n> Algorithm 1, though providing a compact summary, is too vague.\n\nWe will be sure to include subroutines for each step in Algorithm 1. If there are any other specific points the reviewer feels should be included in the overview itself, we would be happy to add them.", "claims_and_evidence": "The proposed method is based on the strong assumption that the presence of known atlas. Building on this assumption, the authors’ claims are empirically supported as follows:\n\n- AtlasD detects local symmetries, defined as atlas equivariance (Definition 4.2), where global methods fail. This is empirically supported by the PDE experiment (Figure 15) and the MNIST-on-sphere experiment (Figure 8).\n\n- AtlasD identifies discrete symmetries (more specifically, disconnected Lie group), including both positive and negative determinant cases. This is demonstrated in the top quark tagging experiment (Section 5.1), where AtlasD finds Section 5.1 ($O^+(1, 3)$ from AtlasD vs. $SO^+(1,3)$ from LieGAN).\n\n- The extracted local symmetry provides a useful inductive bias that compatible with gauge-equivariant neural networks: this claim is supported by several downstream tasks.", "methods_and_evaluation_criteria": "- A major concern with this method is the assumption that a suitable atlas for the dataset is known. It is unclear whether this assumption is reasonable, and the authors should provide more empirical evidence on how different atlas choices impact the discovered symmetry. While this is partially addressed in the PDE example (19 charts vs. 3 charts), additional ablation studies would strengthen the validation, including:\n\n1. The effects of overly sparse or excessively dense (overlapping) charts.\n\n2. Performance under missing or incorrect (noisy) charts.\n\n3. Cases where local coordinates are embedded in higher-dimensional Euclidean spaces.\n\n4. Atlases constructed using data-driven approaches, such as chart auto-encoders [1,2].\n\n[1] Schonsheck, S., Chen, J., & Lai, R. (2019). Chart auto-encoders for manifold structured data. arXiv preprint arXiv:1912.10094.\n\n[2] Floryan, D., & Graham, M. D. (2022). Data-driven discovery of intrinsic dynamics. Nature Machine Intelligence, 4(12), 1113-1120.\n\n- Another concern is the computational complexity of the proposed method. Several crucial hyperparameters, such as the number of basis functions k and the number of cosets for disconnected symmetries K, are determined heuristically by progressively reducing their values from high to low. This approach appears inefficient and may benefit from a more systematic or scalable selection strategy.", "theoretical_claims": "This paper primarily explores applications of symmetry discovery but also presents an interesting theoretical connection between atlas equivariance and gauge-equivariant CNNs. I briefly reviewed the theorem and its proof and did not find any significant flaws.", "experimental_designs_or_analyses": "As mentioned in the Methods and Evaluation Criteria section, I believe the authors should provide a more rigorous ablation study regarding the selection of atlases. Please refer to this section for details. Apart from that, I am satisfied with the experimental procedure and benchmarks.", "relation_to_broader_scientific_literature": "This paper is relevant to gauge theory in theoretical and high-energy physics. While the authors establish a connection between atlas equivariance and gauge-equivariant CNNs and present experimental results using a quark dataset, its focus remains within the realm of machine learning. If the authors illustrate the practical applications of gauge symmetry and local symmetry through examples from physics or other broader scientific fields in the introduction, the practical usefulness of this paper will be further emphasized.", "essential_references_not_discussed": "This paper provides a structured overview of relevant works on equivariant neural networks and automatic symmetry discovery. However, I believe the authors would benefit from discussing representation learning literatures, for example, [1,2], which can be seen as the automatic construction of charts and embeddings for manifold data.\n\n[1] Schonsheck, S., Chen, J., & Lai, R. (2019). Chart auto-encoders for manifold structured data. arXiv preprint arXiv:1912.10094.\n\n[2] Floryan, D., & Graham, M. D. (2022). Data-driven discovery of intrinsic dynamics. Nature Machine Intelligence, 4(12), 1113-1120.", "other_strengths_and_weaknesses": "Overall, the motivation is clear, the proposed method is well-founded, and the evaluation is reasonable. However, to make the work publication-worthy, the authors should provide additional theoretical or empirical validation to better clarify its sensitivity to the predefined local charts.", "other_comments_or_suggestions": "To highlight the importance of local symmetry detection, it would be beneficial to include Figure 15 from the Supplementary Material in the main manuscript.", "questions_for_authors": "Please see the Method and Evaluation Criteria Section for major questions and concerns. Some additional questions are:\n\n- The proposed method identifies the global symmetry group by aggregating multiple local predictor models. I am curious whether this approach enhances the robustness of symmetry discovery or makes it more susceptible to noise. From an ensemble perspective, it seems to improve robustness of discovered symmetries; however, if the local predictors are not well optimized, the discovered symmetry group might be unreliable. I would like to hear the authors' thoughts on this aspect.\n\n- The authors mention that standard basis regularization provides more interpretable results, albeit at the cost of a higher rate of duplicate generators, possibly in comparison to cosine similarity regularization. What does \"interpretable results\" mean in this context? Does it simply imply that the discovered symmetries are more consistent across different runs? Additionally, could you conduct an ablation study to compare the effectiveness of standard basis regularization and cosine regularization?\n\n- Algorithm 1, though provides a compact summary of the proposed framework, is too vague. Providing more details on practical computational methods would enhance its comprehensibility.", "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": "", "extra_scores": {}}, {"review_id": "Xo1Vl4Vy5p", "reviewer": "Reviewer_YBVF", "summary": "The paper introduces the concept of atlas equivariance, which formalizes the notion of local symmetry in contrast to traditional global symmetry approaches. The proposed method discovers local symmetries by learning a Lie group basis for each chart (a local region of the input manifold). This is achieved by training local predictor networks and optimizing the equivariance loss with respect to learnable group generators. These generators are defined in the Lie algebra of the target symmetry group, which acts on each local chart.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The reasoning for the necessity of local symmetry is not convincing.\n\nThe motivation for local symmetry is that arbitrary manifolds (such as a Möbius strip) do not have global symmetries. In such cases, there is nothing for global discovery methods to learn. However, all manifolds do have local symmetries, making them broadly applicable. Such local symmetries are interesting because they can be used as an inductive bias in gauge equivariant neural networks to improve performance in computer vision, climate segmentation, and other real world tasks [1]. On the other hand, the global symmetries that prior works discover are incompatible with gauge equivariant networks.\n\n>  The experiments focus only on locally varying symmetry. It is also important to demonstrate that the method consistently discovers the same symmetry for every chart when only global symmetry is present, which prior methods can easily achieve.\n\nWe clarify that the atlas equivariance group describes the global symmetries of the local predictors, where each local predictor is the task function restricted to a particular chart. Crucially, it is the *common* symmetry group for these predictors, rather than varying with each chart. This means that in our experiments, we have in fact been discovering the symmetry group that is the same for all charts.\n\n> I also wonder about the memory and time complexity required for learning the group generators.\n\nWe include an analysis of space and runtime in Appendix E. In summary, the time and space complexity scale linearly with the number of charts. \n\n> Additionally, the paper should provide guidelines on how many neighborhoods are needed\n\nThe primary requirement for an atlas is that the charts are large enough for the function to truly be atlas local, but small enough that each region is approximately flat. The exact number of charts is a hyperparameter that may depend on domain-specific factors (e.g. geometry, boundary, data coverage). However, Sec 5.2 demonstrates that our algorithm works under a diverse set of atlases, indicating its robustness to the chart size and count in practice.\n\n> I do not see how the proposed loss functions or parameterization are specifically designed for discovering local symmetry.\n\nCompared to those in LieGG or LieGAN, the loss function and parameterization used by AtlasD are specifically tailored to local symmetry discovery. LieGG requires one row in the polarization matrix for every single output pixel in the dataset (Appendix D.4). However, local symmetry datasets are detailed feature fields, making this method impractical due to the memory required. On the other hand, an approach based on LieGAN requires adversarial training, which is often unstable. Moreover, a discriminator-based setup cannot be used in the discrete discovery algorithm because losses for different cosets are no longer comparable. Hence, the predictor loss method used by AtlasD is most applicable to discovering the full local symmetry group.\n\nThe reviewer is correct that the discrete discovery is somewhat independent of local symmetry, but we argue that this only boosts our contribution as our method can then be applied to broader domains.\n\n**References**\n\n[1] Gauge equivariant convolutional networks and the icosahedral cnn, Taco Cohen et al. 2019", "claims_and_evidence": "The reasoning for the necessity of local symmetry in general is not convincing. The paper states that “local symmetries are more generalized,” but it does not strongly justify why local symmetry is essential for either the machine learning or natural science communities. Although it also mentions that “it generalizes symmetry discovery to arbitrary manifolds and allows for downstream use in gauge equivariant networks,” this is also feasible with prior methods when data is defined as a manifold. Moreover, the performance gains reported in the experiments from utilizing discovered local symmetry are only marginal.", "methods_and_evaluation_criteria": "The suggested approach partially makes sense. However, I do not understand why parameterizing only the Lie algebra and minimizing the equivariance loss with a pretrained predictor is sufficient for learning the symmetry group. This approach may lead to unstable symmetry discovery depending on the initialization of the Lie algebra and the accuracy of the pretrained predictor, especially for complex data like PDE solutions. This is why prior methods choose to provide some group generators and train their coefficients [1,2] or use well-designed loss functions, such as Jacobian-based losses [3] or cosine similarity of output features [4]. The paper should discuss these potential issues or provide an empirical study to address them.\n\n[1] Learning Invariances in Neural Networks, Benton et al. 2020.\n\n[2] Generative Adversarial Symmetry Discovery, Yang et al. 2023.\n\n[3] LieGG: Studying Learned Lie Group Generators, Moskalev et al. 2023.\n\n[4] Learning Infinitesimal Generators of Continuous Symmetries from Data, Ko et al. 2024.", "theoretical_claims": "I did not check the proof of Theorem 4.3. but statement at least makes sense.", "experimental_designs_or_analyses": "1. The experiment excluding a certain region in the PDE setting is sound and well-designed.\n2. However, the experiments focus only on locally varying symmetry. It is also important to demonstrate that the method consistently discovers the same symmetry for every chart when only global symmetry is present, which prior methods can easily achieve.\n3. I also wonder about the memory and time complexity required for learning the group generators. I guess the learning time is proportional to the number of charts available\n4. Additionally, the paper should provide guidelines on how many neighborhoods are needed to form a chart to obtain reasonable local symmetry for different types of data. The choice of the number of neighborhoods may significantly impact the discovered symmetry.", "relation_to_broader_scientific_literature": "The main concern is the contribution of the paper. Local symmetry can also be found using baselines like LieGG and LieGAN by defining charts and treating the data as a manifold. The main novelty appears to be the new loss function for discovering discrete symmetry, which was limited in prior works. However, this contribution is independent of finding local symmetry. I do not see how the proposed loss functions or parameterization are specifically designed for discovering local symmetry.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "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": "", "extra_scores": {}}, {"review_id": "SdmziJSvuY", "reviewer": "Reviewer_hC8P", "summary": "The paper proposes a novel pipeline to discovery symmetries in a dataset and, then, employ them to enforce the correct inductive bias in a machine learning model.\nIn particular, the proposed method can discover not only global symmetries but also local symmetries by restricting the attention to local patches of the input data via the charts in a user-defined atlas.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The manuscript is missing a few relevant citations\n\nWe appreciate the suggestion for additional citations. The mentioned works on approximate equivariance modify equivariant architectures to handle situations where no perfect global symmetry is present. These works are indeed relevant and we will be sure to include them in our revised manuscript.\n\nThe fluid simulation benchmark also appears relevant. We have started work towards an experiment, but are currently addressing challenges related to the chaotic and non-local nature of the dataset.\n\n>  Eq. 3 Why not train the basis using SVD?\n\nSVD achieves orthogonality but is not suitable for encouraging sparsity. Our standard basis regularization explicitly promotes sparsity by penalizing shared non-zero terms among the generators.\n\n>  “This implies we only need to consider transformations whose determinant has absolute value 1.” Why?\n\nWhen the component group of a matrix Lie group $G$ is finite, we may find a finite subgroup $H$ that contains at least one element from each connected component [1]. As $H$ is a finite subgroup of $GL(n)$, for each $h\\\\in H$, $h^{|H|}=I_n$ so $(\\\\det h)^{|H|}=1$ and $|\\\\det h|=1$. Hence, each coset has at least one representative whose determinant has absolute value $1$.\n\n> Can you make Fig 4 clearer?\n\nCurrently, the subfigures are in row-major order. The pink and green heatmap is the computed invariant metric tensor, using the methodology by [2]. We will make the numbering clearer and split this figure into multiple subfigures.\n\n> What are the $\\Phi_c$ functions in sec. 4.2.1/5.1?\n\nThe description you provided is correct. We will be sure to include more details and examples in the main paper.\n\n> 5.4 Why so few charts?\n\nThe chart size was primarily constrained by the fact that when small charts were used, the atmospheric river class would take up the entirety of the chart. So we instead chose to use a smaller number of larger charts.\n\n> If a network is trained on a real dataset, it is typically not equivariant\n\nThis is a problem with global symmetry discovery when datasets are canonicalized (e.g. rotation of images removed). It is uncommon to perform such canonicalization at the local scale. Thus, our predictor networks will not face this issue and remain equivariant. \n\n> 4.2.1 Sampling $\\eta \\sim N(0,I)$ will only enforce stability to small transformations\n\nAlthough there are many distributions to try, it’s difficult to sample uniformly from the group as the hypothesis space $\\mathrm{GL}(n)$ is not compact. To sample more extreme transformations, one can increase the standard deviation. However, due to the unboundedness of the search space, the distribution must necessarily be biased towards the origin.\n\n>  Why does the discrete discovery algorithm work?\n\nThe search space is relatively low-dimensional and is further reduced in dimensionality by our assumption that the component group is finite. Moreover, by setting $K$ to be significantly larger than the expected number of cosets, at least one representative likely converges toward each ground truth coset. This is confirmed experimentally in 5.1 and 5.2. \n\nTo verify if our discrete discovery pipeline is able to discover complex groups, we ask it to find the global symmetries of the function $f(x,y)=|x|+|y|$. The ground truth symmetry group is $D_4$ and our algorithm can recover all $8$ elements. \nhttps://i.ibb.co/CKy1wvLc/output.png  \n\n> 4.2.2 Even if $C_i C_j^{-1}$ is not in the identity component, they might be redundant\n\nNote that we claim to discover all the cosets, not just the generators of the component group. In practice, once one has all the elements of the component group, it’s easy for a human to identify the generators.\n\n> 4.2.2 How is the minimization performed? Is the function convex?\n\nThe function is not globally convex (especially since in certain directions $\\\\exp$ is periodic). Nevertheless, we have found gradient descent to be successful empirically and are yet to run into any issues. In case gradient descent fails, one can turn to higher-order methods like L-BFGS.\n\n> 5.2 What happens if you seed with more than 1 generator?\n\nWhen we use multiple generators, the algorithm produces one rotational generator and one that corresponds to a weak scale. The rotational generator is still recognizable, though admittedly less accurate than when the algorithm is seeded with 1 generator.\n\n> 5.3 Assigning a chart to the region of each digit seems unfair\n\nWe acknowledge the MNIST experiment has an idealized setting. Its purpose is to demonstrate the viability of the full pipeline and highlight the difference between local and global transformations. In more realistic scenarios (like our PDE or climate examples), one does not know a priori which charts exhibit local symmetries. Despite having less knowledge, our method still succeeds.\n\n**References**\n\n[1] A note on free subgroups in linear groups, Wang 1981.\n\n[2] Generative Adversarial Symmetry Discovery, Yang et al. 2023.", "claims_and_evidence": "See comments below", "methods_and_evaluation_criteria": "See Experimental Designs or Analysis.", "theoretical_claims": "I didn't check the theoretical proofs in detail.", "experimental_designs_or_analyses": "The evaluation criteria seems suitable for evaluating the proposed idea, although I think there is space of improvment in some experiemnts.\n\n\n- Sec. 5.2: what happens if you seed with more than 1 Lie Algebra generator? Do you still find a single relevant generator? \n\n\n- Sec 5.4, isn't the choice of isotropic filters making the model equivariant to GL(2)? If so, this seems to imply you ignored the discovered GL+(2) symmetry and, instead, implemented a model equivariant to all possible transformations considered in the initial search space. Then, I am not sure this experiment supports the benefit of the symmetry discovery method. That being said, I agree on the difficulty of implementing a GL+(2) equivariant model, but I think this suggests this is not the best task to evaluate the proposed method.\n\n\n- Why not consider the benchmark of fluid simulation from (Wang et al., 2022), which already compares many learnable equivariance methods in the literature? This could provide a simple and effective way to compare with most of the previous literature. This dataset seems to also feature local rotational symmetries (similar to the PDE dataset in this manuscript) and broken global symmetries due to boundary conditions.\n\n\n- All experiments consider at most order-2 discrete groups (such as the flip group and the parity group). I suspect this significantly limits the complexity of discovering discrete symmetries (see Questions for the Authors below); it would be interesting to experiemnt with tasks featuring bigger discrete symmetries e.g. C_4 or D_4 rotations. \n\n\n\nWang et al., 2022, Approximately equivariant networks for imperfectly symmetric dynamics", "relation_to_broader_scientific_literature": "I think this is a relevant novel work.\n\nThe paper claims previous works on symmetry discovery mostly focused on global symmetries.\nHowever, there's a few previous works which considered learning local symmetries by leveraging the idea of symmetry breaking at different scales in CNNs with layer-wise learnable equivariance.\nSee for example:\n\nRomero & Lohit, 2022, Learning partial equivariances from data\nVeefkind & Cesa, 2024, A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs", "essential_references_not_discussed": "I think the manuscript is missing a few relevant citations from the literature on learnable equivariance and approximate equivariance.\nBelow are a few examples:\n\n\nFinzi, M., Benton, G., and Wilson, A. G. Residual pathway priors for soft equivariance constraints.\n\nWang, R., Walters, R., and Yu, R. Approximately equivariant networks for imperfectly symmetric dynamics\n\nWang, D., Zhu, X., Park, J. Y., Platt, R., and Walters, R. A general theory of correct, incorrect, and extrinsic equivariance\n\nvan der Ouderaa, T., Romero, D. W., and van der Wilk, M. Relaxing equivariance constraints with non-stationary continuous filters.\n\nvan der Ouderaa, T. F., Immer, A., and van der Wilk, M. Learning layer-wise equivariances automatically using gradient\n\nPetrache, M. and Trivedi, S. Approximation-generalization trade-offs under (approximate) group equivariance\n\nVeefkind, L., and Cesa, G. A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs\n\nvan der Ouderaa et al., Noether's Razor: Learning Conserved Quantities", "other_strengths_and_weaknesses": "Strengths.\n\nThe atlas approach to local symmetries is novel and the idea proposed is interesting. \nThe experimental analysis shows the potential benefits of the proposed method in different settings.\n\nWeaknesses.\n\nThe paper misses the comparison with some previous works and the experimental analysis has a few issues.\nI am happy to raise my score if the authors address these concerns.", "other_comments_or_suggestions": "- in Sec. 4.2.1, it is not very clear to the reader what kind of functions the \\Phi_c should be at this point of the manuscript. Might be worth adding some concrete examples earlier in the manuscript.\n\n- Eq. 3 why not just training the basis while enforcing its elements to be orthogonal to each other? One would only need to enforce the matrix B (containing the vecorized B_i in its columns) to be an orthogonal matrix, which you can easily do via SVD, no?\n\n- Sec. 4.2.2: page 5, second column, \"This implies we only need to consider transformations whose determinant has absolute value 1.\" Why is that the case?\n\n\n- can you make Fig 4 clearer? the colorbars are hardly readable. Also, the numbering of the generators in the caption are ambiguous. Maybe add the numbers in the figure too. Also, what is the pink and green heatmap exaclty? It seems that Fig 4 includes different types of matrices (continuous generators, some metric and the discrete generators) but puts them all together with no explanation in the image.\n\n- what are the $\\Psi_c$ functions used for the symmetry discovery algorithm in sec. 5.1? Do you just train a normal neural network first, discover the symmetry and, then, train a new network equivariant to the new symmetries? I think more details about this could be included in the main paper rather than just the appendix.\n\n- 5.4 why so few charts? Also, the width of the charts seem a very important parameter here to manage the scale of the local symmetries. How is it chosen?", "questions_for_authors": "- if a neural network is trained on a real dataset, it is typically never numerically equivariant unless the data contains sufficient data augmentation. At that point, however, it is assumed the user already knows the underlying symmetry. I feel this aspect could get some more discussion in the main paper.\n\n\n- Sec. 4.2.1 sampling $\\eta \\sim N(0, I)$ will only enforce stability to (relatively) small transformatoins close to the identity, isn't it? Does't it make more sense to sample uniformly from the group? \n\n\n- Sec 4.2.2: if I understand correctly, the method uses a simple gradient descent strategy in a rather low-dimensional space (the parameter space of the $C_l$ matrices) to find all minima of a probably very non-convex function. Why do you expect this solution to work well to find the discrete symmetries?\n\n\n- Sec 4.2.2 even if $C_i C_j^{-1}$ is not in the identity component, they might still be redundant, e.g. if $C_i$ is a power of $C_j$. Also, how is the minimization perfomed? Is the objective convex such that a simple gradient descent can be expected to converge? I am not sure it is possible to claim that the filtration process produces a list of unique representatives of the cosets. Note also that this is probably not a problem in these experiments since only order-2 discrete groups have been considered (such as the flip group and the parity group). You should experiment with other discrete symmetries, e.g. C_4 or D_4 rotations.\n\n\n- 5.2: what happens if you seed with more than 1 Lie Algebra generator? Do you still find a single relevant generator? \n\n\n- 5.3: assigning a chart to the region of each digit seems a bit like cheating, since the users are implicitly pointing to the regions (and restricting the attention to only those regions) where they know there are exact local symmetries. Did you consider other choices of charts?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper presents AtlasD, a framework for discovering local symmetries by formalizing the natural concept of atlas equivariance. Unlike existing methods that focus on global symmetries, AtlasD trains local predictor networks on coordinate charts and extracts a Lie group that captures equivariance across these charts. The method accommodates both continuous and discrete symmetries and demonstrates effectiveness across a range of relevant tasks.

Prior to the rebuttal, the paper was assessed as borderline. Although reviewers recognized the conceptual novelty of introducing atlas equivariance for local symmetry discovery, they also identified several significant concerns—chief among them the flexibility in atlas selection, the limited scope of empirical evaluation, and the unclear comparative advantages over existing methods. The discussion phase was constructive, with the authors presenting additional experiments demonstrating robustness to atlas choice, clarifying aspects of the methodological framework, and providing more meaningful comparisons to prior work. Reviewers YBVF and 9veb subsequently revised their scores upward, with the latter indicating that major concerns had been addressed to a satisfactory degree. Nevertheless, some reservations remained, particularly regarding the stability of the approach and the need for further comparison to existing methods. Overall, however, the consensus shifted in a more favorable direction. In light of the clarified contributions, the paper makes a reasonable case for acceptance, primarily on the strength of its conceptual novelty, provided there is room.

The authors are encouraged to incorporate the valuable feedback provided by the knowledgeable reviewers.
Decision: Accept (poster)