RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization
Authors: Alonso Urbano
David W. Romero
Max Zimmer
Sebastian Pokutta
Venue: ICLR 2026 Poster (2026)
Keywords: symmetry discovery
canonicalization
equivariance
Primary area: learning on graphs and other geometries & topologies
TL;DR: An architecture-agnostic framework to align arbitrary canonical representations in class–pose methods with the data symmetries, enabling symmetry discovery and downstream practical applications

Abstract: Real world data often exhibits unknown, instance-specific symmetries that rarely exactly match a transformation group $G$ fixed a priori. Class-pose decompositions aim to create disentangled representations by factoring inputs into invariant features and a pose $g\in G$ defined relative to a training-dependent, \emph{arbitrary} canonical representation. We introduce RECON, a class-pose agnostic \emph{canonical orientation normalization} that corrects arbitrary canonicals via a simple right translation, yielding \emph{natural}, data-aligned canonicalizations. This enables (i) unsupervised discovery of instance-specific pose distributions, (ii) detection of out-of-distribution poses and (iii) a plug-and-play \emph{test-time canonicalization layer}. This layer can be attached on top of any pre-trained model to infuse group invariance, improving its performance without retraining.  We validate on 2D (images) and 3D (molecular ensembles), demonstrating fine-grained, accurate pose discovery, and matching or outperforming label-supervised
canonicalizations in downstream classification.

Published as a conference paper at ICLR 2026
RECON: ROBUST SYMMETRY DISCOVERY VIA
EXPLICIT CANONICAL ORIENTATION NORMALIZATION
Alonso Urbano David W. R
Max Zimmer'
Sebastian Pokutta
Department for AI in Society, Science, and Technology, Zuse Institute Berlin (ZIB), Germany
2 Cartesia AI. San Francisco, CA, USA
3 Institute of Mathematics, Technische Universität Berlin, Germany
(urbano, zimmer,
de dwromerog@@mail. com
ABSTRACT
Real world data often exhibits unknown, instance-specific symmetries that rarely exactly match transformation group G fixed priori. Class-pose decompositions aim to create disentangled representations by factoring inputs into invariant features and pose g G defined relative to a wrining-dependens arbitrary canonical representation. We introduce RECON, class-pose agnostic canonical orientation normalization that corrects arbitrary canonicals via a simple right translation, yielding natural, data-aligned canonicalizations. This enables (i) unsupervised discovery of instance-specific pose distributions, (ii) detection of out-of-diitributiin poses and (iii) plug-and-play test-time canonicalization layer. This layer can be attached on top of any pre-trained model to infuse group invariance, improving its performance without retraining. We validate on images and molecular ensembles, demonstrating accurate symmetry discovery, and matching or outperforming other canonicalizations in downstream classification.
Piest
Class- pose
RECON
Pirain
Data
Canon. Distrib.
Canon.
Distrib.
7
[1. 7... 7)
L
P/7]
7
Hm
15, 5,.. 51
5
Pi
Canon. (RECIO)
(a) Inputs with identical +300 symmetries.
Mi
high acc.
(arbitrary)
85
M5
A|S
t
low
(b) Relative distributions Ma] (M) vs normalized dis-
(c) Distribution shift (Priin VS Piea) induced by unseen symmetries at test time. corrected by RECON.
tributions M1x] (O) via RECON.
Figure 1: (a) Class-pose methods assign arbitrary (often distrribbuu)))))) canonicals per class. (b) This leads to distinct relative- pose distributions V(z). obscuring the shared =30 symmetries. RECON corrects these offsets, mapping inputs under same symmetries to the same distribution [*[x] and extracting their natural pose 7. (c) Our data-aligned canonicalization removes symmetry-induced distribution shifts, improving downstream performance of pre-trained backbones without architectural restrictions or retraining.
INTRODUCTION
Symmetry transformations like rotations arise naturally in many domains (Cohen Welling 2016 Higgins et al., 2018 Bronstein et al.) (021). with objects appearing in poses related by group transformations g E g (e.g., molecules different orientations) (Weiler et al. 2018: Brandstetter
Corresponding author: urbanodzii

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(a) SGM
(b) IE-AE
(c) RECON (Ours)
Figure 2: Comparison of canonical representations. Top row: Input MNIST digits under varying
rotations. Bottom row: canonicals generated by (a) SGM(Alliggham et al. 2024) and (b) IE-
AE (Winter et al. 2022 which yield arbitrary orientations, while RECON consistently produces data-aligned canonical poses (upright digits) (c).
et al.
2022). While G-equivariant neural networks exploit such structure (Cohen et al. 2018 2019
Romero al. 2020, Romero & Cordonnier 2020] Wang et al. | (2020), they can be overly constraining
when there is mismatch between the pre-fixed group G and the symmetries in the data (Weiler &
Cesa 2019 Romero Lohit 2022). effect, real-world symmetries are often (i) unknown priori,
(ii) partial (covering only part of the group). (iii) instance-dependent. This motivates methods that
discover symmetries from data (Benton et al. 2020 Forestano et al. 2023 van der Linden et al.
2024
Allingham et al. 2024).
However, existing approaches often require supervision or learn only dataset-level patterns (Sec. Our goal is to discover, in unsupervised manner, the group transformations inherent to each instance; in particular, we aim to learn probability distributions on G that describe the poses in which each instance appears in the data We argue that promising foundations lies class-
pose decomposition methods (Winter et al. 2022 Yokota & Hontani 2022] Marchetti et al. 2023
Allingham al., 2024). which disentangle imputs into invariant class features and a pose g 9 relative to learned canonical representation. The canonical's 's orientation is generally arbitrary (Winter et al. 2022, Allingham et al. 2024), which results in an disttriiiitiiitoo (OOD) canonicalization and in arbitrarily shifted relative-por pose distributions (Fig. TT.
We address arbitrary canonicals and propose a framework for Robust unsupervised discovery of intrinsic symmetry distributions via Explicit Canonical Orientation Normalization (RECON). We prove (Proposition 3.T) that by estimating the centroid (Fréchet mean) of the observed relative poses, which captures the offset induced by the arbitrary canonical, we can approximate symmetry distributions centered at the input's natural pose via a simple right translation. This yields instance- specific symmetry descriptions that are robust (independent of the arbitrary canonical), interpretable (centered at e e representing the input's 's natural pose), and comparable across classes (Fig. (Fig.[b). 2D image benchmarks and 3D molecular conformations beyond typical 2D-only settings of prior work (Benton et al. 2020 Romero & Lohit [2022] van der Linden et al., 2024 Allingham et al. 2024 Kim al. 2024). Lastly, provide practical applications in (i) OOD
RECON is validated
pose detection and (11) test-time canonical liiztii,,,,,, a drop-in method to grant group invariance to
frozen pre-trained models, improving downstream performance (Sec. 5.2)
Contributions
We propose RECON, method for discovery of instance-specific pose distributions from unlabeled data leveraging class-pose representation learning methods.
2. We achieve this through canonical orientation normalization (Proposition 3.1 an architecture- agnostic correction of arbitrary canonicals, yielding data-aligned natural canonicalizations and well-behaved symmetry distributions.
empirically validate on images and large-scale, real-world 3D data. We offer applications in OOD pose detection and test-time canonicalization, offering performance improvements to pre-trained backbones via simple plug-in canonicalization step.
Our code is publicly available at https: /ttpt/gggtttttt/////// com/2IB-IOO/recoom
PRELIMINARIES
Our approach leverages class- decomposition methods, a class of neural networks designed to disentangle input data into invariant (class) component and equivariant (pose) component based on a given transformation group G. In particular, we build upon Dnvariant-Qquvaari Autoencoders
Note how this differs from estimating the stahilizer S, describing the (self) symmetries of (ef. Appx D.1
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(IE-AEs) (Winter et al. 2022). We briefly review the concepts here; formal definitions regarding group theory, representations and G-equivariance are deferred to Appendix A
IE-AEs aim to learn group invariant and equivariant representations for an input I E X of a vector space, e.g., an image or a 3D structure. For a chosen group G. an IE-AE maps x to a G-invariant component E R" and G-equivariant group element (pose) g E g G
Invariant component and canonical representation a First, a G-invariant encoder vj X 2 C IR learns an invariant representation = 7/()). This embedding captures features of the input that are independent of its pose under G. meaning 2 remains unchanged if is transformed by any element of the group. A corresponding decoder å Z + X' then reconstructs the input ê = d(2) from this invariant representation. Note that since the decoder only sees the invariant embedding
2 17(2)) it will produce the same reconstruction å 6(n(z)) for all group transformations of the called the canonical representation. Notably, the specific pose orientation of ă is arbitrury, influenced by initialization or training dynamics (Winter et al.
input. This common reconstruction
2022). This behaviour is common to other class-pose decomposition methods (Yokota & Hontani
2022
Allingham et al. 2024).
Pose component g The canonical è has some fixed, arbitrary pose. To recover the original input 2, we have to determine the specific group transformation g E G that maps & back to x. This is the role of the group function 1b X + G, which predicts this transformation g = (x)) All the IE-AE components (17, 5, v/) are then trained jointly by minimizing the reconstruction loss d(px (z)) 8(n(=))) x>) where px is the group action of G on X.
In summary, IE-AE provides an invariant latent vector x=7(2), an arbitrary canonical (=) and the relative group transformation g=vh(x) that maps the canonical & back to z. Our method leverages these components, specifically the distribution of relative transformations g=v(2z) and the invariant latent space Z. to discover the specific distribution of transformations that appear in the data.
3
METHOD
3.1 PROBLEM STATEMENT
Our goal is to discover the characteristic distribu- tion of symmetry transformations associated with different classes of objects within unlabeled data X. Consider datasets such as GEOM (Axclrod & Gómez Bombarelli 2022), where each molecule class is rep- resented as an ensemble of multiple 3D molecular conformations; MNIST, where same digits share an underlying shape under varying handwriting styles. Generally, variations within these classes a com- bination of (a) non non-group structural distortions (e.g.. bond rotations or ring puckering in molecules; style variations in digits) and (b) transformations from a symmetry group & (like SO(3) rotations of molecu- lar conformations). We specifically aim to model the distribution of the underlying group transformations.
Z
[x]
Piel
Figure 3: Problem setup. Left: A class [a] is defined by inputs clustering together in the invariant space Z. Right: We model instances
[æ]] as (x(())))=) where "[x)] is refer- ence frame and is drawn from a distribution over rotation angles M[n]] The objective is to recover [[x] from the unlabeled data.
Therefore, to achieve this goal, we first need to (i) define way to group objects based on their intrinsic shape or similarity independently of their pose or minor deformations and under the absence of labels. and (ii) model their pose variations.
(i Modeling pose-invariant similarity We model pose invariant similarity through equivalence classes. The G-invariant latent space typical of class-pose decomposition methods is promising candidate for this purpose. In effect, we rely on the principle that structurally similar objects generally map to nearby points in Z. This assumption is empirically supported by previous work in class-pore pose decompositions methods showing that Z maps different classes into well-separated connected clusters (see t-SNE visualizations of Z in IE-AEs (Winter et al. 2022 ). Additionally, it aligns with broader
2 Figure 7 in the Appendix, adapted from Winter et al. (2022). visualizes this architecture.
Any estimator satisfying the proposed reconstruction loss is G-equivariant as proved in Winter et al.
2022
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findings indicating that deep features correctly capture perceptual and semantic similarity, even when learned without supervision Zhang et al., 2018
Formally, let 17 A'' - Z be such a G-invariant encoder, we then define equivalence relation ~ in X' based on connected proximity in Z:
=2 C X' for some N E N s.t. To = and dz( (aaa)ra)))((x))))))) 7(x=-1)) < Vi
where dz is a norm-induced distance in 2 and > O is a small threshold. Intuitively, this equivalence relation groups objects whose invariant features form connected component (Fig. [] left). We denote the resulting equivalence class containing x as [ar] 5 We empirically validate this equivalence class confirming effective capture of pose-invariant similarity, and analyze its
definition in Appendix D.3
limitations in detail.
(ii) Modeling pose variations Having established the classes [x]] we now focus on modeling the distribution of group transformations responsible for pose variations within each class. We model this as a probabilistic process.
Conceptually, there exists some true underlying probability distribution [[u] that governs how in- stances E [ar] are generated by transforming some reference pose Yiu X plus residual non-group variations. Formally, let 9 be a Lie group and consider Hlu]] probability distribution over G. We assume that instances a E [xr] can be generated by sampling a transformation g ~ M[x] and applying it to reference pose "[x)+ plus deviation term E, Per:
px(g) + with g ~ M[u) E, Pa.
Here, [a]]] represents the true probability distribution relative to the reference pose Mx)- The distribu- tion Per models variations that are not explained by the group transformation (e.g.. style variations in digits, internal conformational changes in ensembles of molecules) and is assumed to have zero mean and small variance (we assume ||>,|| < almost surely for simplicity). Fig. 3 (right) illustrates this model.
Note that under this model, there are several ways of representing the symmetries in the data, depending the reference pose Tz Consider dataset of handwritten '7's exhibiting rotational symmetries uniformly between +:30%: ......7. 2}- Using this model, we can describe the
symmetries in this dataset as M[7] Zd[[550) 2210s) for reference pose Y/ ='L', or as PM7] 4([-30" 30%)) for reference pose ?[7] ='7'. While both descriptions
mathematically valid, the latter, identity-centered representation is far more desirable. It aligns with our intuition of a naturally occurring pose and directly reflects the symmetries as deviations from a neutral reference frame (e. no transformation), which offers several advantages (as demonstrated in Sec.
Defining a natural pose Ideally, we aim to obtain such identity- centered descriptions of the data symmetries. As have just ex- emplified, this boils down to obtaining the symmetry distribution
Probabiiity censity Highest probabitty
Tukey-Fréchet
Min whose Yij is the "natural" pose in the data - like the upright '7' - centered at the group identity. But without labels defining what reference or true canonical pose is, how can we define natural pose? Specifically, how can we obtain canonical pose that is ge- ometrically grounded the symmetries in the data rather than just an arbitrary pose?
Probobility
While the most likely pose (i.e., with maximum probability den- sity) may seem like good candidate - for instance in the case of Gaussian or other unimodal distributions - it can be ill-defined (e.g..
for uniform distributions, see Fig. 4 top) or not representative (e.g., in the case of a distribution with an anomalous sharp spike its density, Fig. 4 middle). To establish a general definition, the Fréchet mean (Pennec (2066) or Riemannian center of mass, a measure of the central tendency of the distribution on the group
m/2
T
3m/2
2m.
Angle (rad)
Figure 4: Estimator compari- son for different distributions over 50(2).
'Structured per-class manifolds in latent spaces have also been shown algebraically PPellriiii et al. 2025) 'While we use this definition throughout our theoretical derivations (see Proofs practice, approxi- the classes (z) by simply computing the k-nearest neighbors of 17(z) in Z for efficiency (sce Algorithm
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manifold. Given any distribution i over G, its Fréchet mean F(u) is the unique transformation minimizing the expected squared Riemannian distance dR to samples from pc:
F(n) = argninneeg Ee-uldR(ys (v. 9)
Under mild conditions, the Fréchet mean provides a unique centroid (Afsari 2011 that aligns with intuition for common cases: it corresponds to the midpoint for symmetric uniform distributions and the peak for strongly unimodal ones (e.g. Gaussian). However, certain multimodal cases, e.g., a bimodal distribution on SO(2) with opposing identical peaks (Fig. 4 bottom), the Fréchet mean may fall outside the support of j.. This would lead to an out-of-distribution canonicalization, affecting downstream tasks (Sec. Therefore, we propose robust Fréchet mean extension based on robust M-estimators (Shevlyakov et al. 2008) that we coin the Tukey- Frechht mean Fr)))::
Fx(1) arg min Eg- (ml (dx (y. g):g: c)l,
'EG
where m(u; c) is the Tukey biweight loss (Rousseeuw & Hubert 2011 Shin & Oh 2022). This variation extends the Fréchet mean and converges consistently to the same mode in multimodal distributions (cf. Appendix (C. For simplicity, we derive our theoretical framework with the classical Fréchet mean, whose properties are well established, but in practice, the robust TTukey-Fréche can be used as drop-in replacement.
We thus define our target distribution M[a] in equation D as the probability distribution over 9 that is centered the group identity e the Fréchet sense, i.e., satisfying F(((z)) e. Consequently, the reference pose ?an (the natural pose) corresponds implicitly to the Fréchet mean (i.c., the upright '7' in the previous example). Our objective is to estimate this well-behaved [4s]] from unlabeled data. The subsequent section details our method for achieving this by normalizing the arbitrarily offset outputs of IE-AEs.
3.2 RECOVERING SYMMETRIES VIA CANONICAL ORIENTATION NORMALIZATION
The core idea is to leverage the disentangled representation (z, g) E Z x G of the IE-AE extract the symmetry distribution of an input, and correct for the arbitrary canonical pose to yield a canonical- pose independent distribution. Our main result provides way to recover the true symmetries M[=] from the relative transformations (z).. We provide proof in Appendix B
Proposition 3.1 (Approximation of pla) via Normalization). Let X be a metric space, G a Lie group and n, 8, s IE-AE where &b is continuous on compact domain X. Suppose that X' exhibits symmetries characterized by H(a) where F(H(z) = e as described above. Consider a random
sample { of [a:] and denote their images by sb as ([[==) [v(s:))M Let [(n) G be the empirical Fréchet mean of <(xx)). Then, the empirical distribution [Mt] corresponding to the normalized samples ((x])]
[x] 1-1 approximates the target distribution [lu]. Specifically, Pml) converges
in Wasserstein distance to P(x) as g 0 and N - xc.
Interpretation. Proposition 3.1 provides
Algorithm Canonical orientation normalization
a practical method for consistently retriev-
ing M4s)] through canonical orientation normalization, outlined in practice AI- gorithm[ [ The process involves identify-
Require:
Trained IE-AE (n, 6, v). input æ, number of neighbors k
1: Compute invariant embedding: = +- -=(())
ing the class [xr] and collecting the rela- tive transformations ((=[)) (representing poses relative to the arbitrary canonical i). The distribution of transformation in this set is offset w.r.t. M[x] by a translation in- duced by the canonical pose, which can be estimated through the empirical Fréchet mean i of the observed transformations ([==)) Then, right-multiplying by the in- verse [x] corrects this offset and centers
Find k-nearest neighbors 8, of based
dz (π{s,), 2) to approximate class [x]]
3: Collect relative transformations:
([x)) + (v(s,) 85 E k-NN(x))
Estimate offset via Fréchet mean:
[c]] argmin yeg -gre(((>)) dg(v.
Compute inverse offset: r [o]
Compute normalized transformations:
('([z)) + {
9i E ((x)))
[o]
the Fréchet mean at the identity consis-
7: return (xx(() b Samples approximating [[*]
tently for all classes. This removes the influence of the arbitrary choice of the canonical pose â. providing geometrically meaningful
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canonical (centered at the Fréchet mean) and enabling the retrieval of symmetry distributions with useful properties, as we show in our experiments (Sec. 5
3.3 INFERRING SYMMETRIES VIA LEARNED MAPPINGS
Algorithm provides a way to estimate the symmetry distribution and the centering transformation for any class in the training data. To enable efficient inference for unseen inputs without explicit class computations at test time, we can train learnable mappings. If M|s] has a known parametric form (or we approximate it parametrically), its parameters ((x)] can be estimated as
|00 - 4
(v ([x)) (++[) [M])
where yp is an appropriate estimator for the parameters of the distribution, e.g. maximum likelihood. Consider the estimates 0,n] and I [s)- We can learn two maps using these estimates as pseudo-labels:
A map θ predicting the parameters of the symmetry distribution of an input by minimizing
pp (z)) (/])).
A map A X - G. predicting the centering transformation by minimizing L. do (o)))
where and dg, dg are appropriate distances. The first mapping generalizes the estimation process: at test time, given I, we can predict its symmetry parameters as = ((x) without requiring class computations. The second mapping allows us obtain our RECON canonicalizations during inference as C(z) = Px (Ax)) .x(z) æ. Additionally, we can use these functionn in combination to detect out-of -itt----iitriuttion symmetries Section 5.2] We model and A as G-invariant networks, ensuring that predictions depend only on the object's s class [z)), not its specific input pose.
4 RELATED WORK
Class-pose decomposition methods Unsupervised learning of disentangled invariant and equiv- ariant representations via autoencoders or other learning paradigms has seen various propositions
(Shu et al., 2018: Guo et al. 2019 Feige 2019 Kosiorek et al. 2019 Koneripalli et al. 2020
Winter al. 2021 2022 Yokota & Hontani 2022). In this work, we build top of the IE-AE
framework (Winter et al. (022). Our main contribution is the discovery of the transformations in the data via invariant latent space search coupled with explicit canonical orientation normalization step (Proposition J. which corrects the pose offset introduced by arbitrary canonicals typical of class-pose decomposition methods like the IE-AE. In principle, however, this approach can be applied to any method which factors inputs into an invariant component and a symmetry component. For instance, Quotient Autoencoders (Yokota & Hontani 2022) also learn canonical representations in a similar spirit; Marchetti et al. (Marchetti et al. 2023) proposes class pose decomposition network akin to IE-AEs, albeit with different learning paradigm and advantages: SGM (Allingham et al.) 2024) also feature canonical representation or prototype and a relative transformation component. Our approach remains compatible with such backbones.
Learning symmetries from data While standard group equivariant networks impose fixed symme-
tries (Cohen & Welling 2016 Cohen et al. [2018; WWiler et al. 2018 Weiler & Cesa 2019 Cohen
ct al. 2019: Romero et al.) 2020 Romero & Cordonnier, 2020 Wang al. 2020), recent effort
have focused on learning symmetries from data. Approaches like Augerino (Benton et al. 2020)
learn data augmentations for non-equivariant models, while others implement relaxed equivariance constraints. Partial G-CNNs modulate the equivariance per-layer by learning a distribution over the group (Romero & Lohit 2022); ; Residual Pathway Priors (Finzi et al., (202T) propose handling partial equivariances through a combination of equivariant and non-equivariant models. These methods require supervision or learn dataset-level transformations rather than instance-specific distributions. Recently. Variational Partial Group Convolutions (VP (-C et al.) (2024) extended Partial G-CNNs to adapt to instance-level symmetries with a variational inference approach. While they can compute class-dependent "equivariance error", this method does not expose a clear symmetry distribution for each input. Equivariance via weight-sharing patterns (Ravanbakhsh et al. 2017 Zhou ct al. 2021 Yeh et al. 2022) can also be leveraged to adapt to partial or dataset-level symmetries. Akin to Partial G-CNNs, WSCNNs (van der Linden et al. 2024) introduce layers that can adjust their equivariance based on the data by modulating the Briigh-sharing pattern. This approach learns sin- gle set of transformations per layer, while our method discovers instance- symmetry distributions instead.

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Another line of work aims to learn dataset-level symmetry (sub)group H < 9 (or generators) of a prescribed ambience group G from data. LieGG (Moskalev et al. 2023) extracts infinitesimal generators from a trained model to reveal learned invariances; LieGAN Yang et al. 2023) adver- sarially learns Lie algebra generators and a dataset-level distribution over coefficients on the group, discovering subgroups of a prescribed G without supervision, well as subsets of the group through regularization strategies; LaLiGAN (Yang et al., (2024) lifts to a latent space to capture non-linear actions. Other methods reason through Lie derivatives (Otto et al., (025) or extend to broader types of symmetries (Forestano et al. 2023] Shaw et al.) 2024 Ko et al. (2024). In contrast to these methods, our approach discovers instance- -peancce distributions of symmetries over a group 9 and a data-aligned canonicalization function without supervision, rather than discovering dataset-level (sub)group G itself. We emphasize that our method is complementary and continuation to this line of work: these approaches can be used to infer suitable G for all the data, while RECON aims to infer instance-level distributions and a test-time canonicalization operator.
Symmetry-awarr Generative Model (SGM) (Allingham et al. 2024) is the most closely related method in terms of instance-level granularity. SGM uses a flow- based model to learn the relative distribution of transformations. However, this distribution is relative to arbitrary canonical, which presents several disadvantages that RECON addresses (Figures 5b). Note that SGM is a class- pose decomposition method, and therefore is compatible with RECON. Preliminary work (Urbano & Romero 2024) on class-pose methods proposed enforcing a constraint on the group action estimator via regularization term, which can be leveraged to learn symmetries, but this often leads to degenerate solutions where (z) R e for all x, limiting symmetry learning. Other methods model probability distributions over the group. Implicit-PDF (Murphy et al. 2022) models instance-level distributions over 50(3) with an implicit density and pose supervision; Alignist (Vutukur et al. 2024) estimates orientation distributions using CAD shape priors and correspondence distributions. In contrast, RECON requires neither pose labels nor CAD models and operates on general groups supported by the class-pose backbone.
5b).
Test-time canonicalization. Other works can grant model-agnostic group invariance to pre-trained models by inserting a small canonicalization in front of them. Spatial Transformer Networks (STN) (Jaderberg et al. (015) learn an input-dependent geometric warp via small network which can be seen as a group-free form of canonicalization, but it is typically trained end-to-end with discriminative objective and requires joint training with the predictor. Equivariant Adaptation (EquiAdapt) (Mondal et al. 2023) places an equivariant canonicalizer before the frozen predictor and trains only the canonicalizer using a canonicalization prior; for smaller classifiers they also propose joint fine-tuning of the predictor. In both cases, training the canonicalization is tied to a specific pretrained model and its size. In contrast, our canonicalization is trained independently of the downstream model and can be plugged in front of any model that operates on the same input domain. In a similar fashion, Affine Steerable EquivarLayer (Li al., 2025) learns a canonicalizer trained independently of the pre-trained model, but we note that it is trained using known random transforms, i.e. pose labels. Our canonicalization, however, is obtained without any supervision.
EXPERIMENTS
We empirically validate RECON on diverse datasets and rotational symmetry distributions. In par- ticular, we use benchmark image datasets and a large-scale, real-world geometric graphs dataset exhibiting known ground truth S2()) and SO(3) rotational symmetries respectively. We focus on rotation groups since rotations have been recognized as a significant challenge (specially in 3D) by several studies (Zhao et al., 2020 Shen et al.) 2021) even for small (<15%) angles (Sun et al. 2022), but our framework remains general. All our results are on test data, using the learned e and A predictors trained with the pseudo-labels obtained through Algorithm [ (Sec. 3.3 Further experimental details are provided Appx.
5.1 RECOVERING SYMMETRIES
Imaging We first validate RECON on rotated MNIST and FashionMNIST Axelrod & Gómez Bombarelli 2022) datasets where different classes exhibit distinct rotational symmetry patterns. For MNIST, we apply random rotations drawn uniformly from +60 for digits 0-4 and +900 for digits 5-9. For FashionMNIST, we apply rotations drawn from Gaussian (N(0, c) with o=0 for classes 0-2, o=32 for classes 3-5 and o-64 for classes 6-9. We aim to discover these input-dependent symmetries from the unlabeled datasets. Canonicalization plots (Fig. 2 show RECON'S consistent,
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329
True Predicted
Input
Canonical (TE-AR)
Canonical (RECON)
9
Symmetry parameterr
9
Dee Rutau MSE aorio ridecko
(b)
Predicted
AIC-ROC (*)
Experiment
IE-AE
RECON
MNIST
0.79
0.91
(a5e-4)
FashionMNIST
(::0-4))
GEOM-QM9
0.76
0.82
Class
4CD
300
1059
1208
(a)
(c)
(d)
predicted parameter of the label-dependent symmetry distribution in MNIST (top) and FashionMNIST (bottom) experiment. (b) Canonicals for digits of the class 6 (first row) and 9 (second row) obtained with IE-AE and RECON. (c) Out-of-distribution detection performance with IE-AE and RECON. (d) Histogram of average prediction error across molecules and scatter plot of their standard deviations for GEOM.
Figure 5: (a) Truc
upright canonicals across classes, compared other class-pose backbones. In Figure 9 Col.3, we visually confirm that our normalized distributions match the true sampling regime applied to each class in the MNIST experiment. Note how this clear distinction and direct interpretability lost when analyzing the unnormalized, relative distributions from the class-pose backbone (Figure Dareeoooo Col.2). Figure 5a shows quantitative results for MNIST (top) and FashionMNIST (bottom), confirming that the predicted distribution parameters (given by (xx)) align closely with the ground truth parameters, even when these symmetries stem from distributions with different shapes and scales per-class. We obtain these results using k= 10 neighbors for the class computation [ar]. which we found to give the best performance (details in App. D.2).
A notable strength of RECON is its ability to pick up partial symmetries.. RECON canonicalizations can distinguish between distinct classes that are related by a group transformation such as digits '6' and '9', which related by 180" rotation as opposed the IE-AE canonicalizations, which collapse both classes into the same canonica 2 (Fig. 5p)) This sensitivity to the data's 's contextual orientation allows RECON to handle cases beyond perfect group orbits where full equivariance is inappropriate. We discuss this in detail in Appendix D.4
Geometric graphs To demonstrate RECON's ability to discover symmetries higher-dimensional groups and complex data, we apply it to the Geometric Ensemble of Molecules (GEOM) dataset (Ax elrod & Gömez-Bombarell 2022), which provides multiple molecular conformations (i.e., unique 3D geometries or conformers) for several classes of molecules. We focus on the the QM9 (Ramakrishnan al. 2014) subset of GEOM, and to ensure sufficient pose variation each molecule class, select only molecules with at least 64 distinct low-energy conformers. This results in approximately 175k samples for training, 21 for validation, and 24k for testing across 2, 211 distinct classes of molecules. which can be identified by their SMILES (Anderson et al., 1987) string (more details in Appx. E We then apply random SÔ(3) rotations to each molecule class to create ground truth pose variations. Specifically, subsets of molecules are rotated using three distinct Matrix-Fisher distributions (Mar dia & Jupp 2009). each centered on a different axis C1, 622 63 to simulate rotations around a fixed direction. These parameterized distributions (M(F) M(F M(F are visualized on the sphere following Mohlin et al. (2020), where brighter regions indicate preferred orientations. For instance, in Fig. rotations of [Ce@] (0) [CêH] 1 [ @H) (0) conformers are sampled from M(F) ). which perturbs the ej axis locally while the orthogonal axes rotate around it freely forming ring.
Figure 6 evaluates how accurately RECON recovers each molecule's SO(3) symmetry distribution by measuring per-molecule average MSE between the true and predicted matrix-Fiste parameters. Panels (a-c) visualize randomly sampled molecules from three error regimes low, average, and high
TThis limitation also be observed in SGM Allingham
2024
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Input
Canon.
Reconst.
True sym.
Pred.
Input
Canon.
Reconst.
True sym.
Pred.
(a) Low-error sample, 18E-173.71.
(b) Average-error sample, MSE-368.91,
Input
Canon.
Reconst.
True sym.
Pred. sym.
(Trina/ees
Backbone
Canon.
Canon. Supervision
Test
(f)
MNIST (Orig Rot)
IE-AE
13.60 1.2%
ResNet18
EquiAdapt
94.52
RECON
90.96 X
0.5%
57.14 0.1%%
F-MNIST (Orig / Rot)
ResNet18
IE-AE
11.55
1.2%
EquiAdap
79.44
RECON
0.3% x
52.40 0.0%
GROM-QM9 (Orig / Rot)
GCN
X
EquiAdapi
/
52,96 1. 0..9%
(d) Test accuracies obtained using different test-time canonicalizations in pre-trained classifiers.
(c) High-erm error sample, MSE=1074.76.
Figure 6: (a-c) Test-time comparison for molecules randomly sampled from low error ( P1o tenth percentile average per-molecule MSE), average error (P4s Pss), and high error (z Poo) regimes. For each selected molecule, four sample conformers are shown using ball-and-stick model, displaying: (column I) input conformer, (2) our canonical reconstruction, (3) final reconstruction, (4) true symmetry distribution (constant for the molecule), and (5) per-conformer predicted symmetry distribution. (d) Test-time canonicalization experiment on the per-class rotated dataset variants.
MSE based on percentiles. The panel visualizations show close alignment between the predicted and the true distributions for molecules in the low and average error regime. Figure 5 provides global view, showing the histogram 80% of molecules fall within a region of accurate predictions (approx. MSE < 800 based on visualizations). demonstrating RECON'S ' ability to capture diverse SO(3) symmetry patterns from large-scale, unlabeled, realistic 3D data. Interestingly, we observe no clear correlation between MSE and (i) molecular size, (ii) flexibility in conformational variation or (iii) reconstruction error (Figure 14 reconstruction losses defined in Appendix E.2.4). This indicates that our symmetry estimates are robust to conformational complexity, and that the remaining error stems from other sources (see limitations in Sec. (). RECON also demonstrates remarkable data efficiency, providing accurate symmetry distributions inferences a dataset where most molecule classes have just few (<100) number of conformers per class (Fig 13b), highlighting its ability to work with limited per-class data.
MSE and their standard deviations. In general, about
5.2 DOWNSTREAM APPLICATIONS
OOD pose detection The centered distributions recovered by RECON yield a natural anomaly score. Given predictors (z)) and A(r) (Sec. 3.3), consider the absolute pose Jabs (<)) Then, the likelihood gade under the distribution parameterized by (xx) serves as an anomaly score:
s(x) log (gats)
(5)
Low s(x) indicates in-distribution, while high s(x) flags OOD. We empirically validate this by classifying randomly oriented S0(2) (images) and SO(3) (GEOM) test instances, showing strong OOD detection (AUC-ROC in Table 5 measuring separability between in and OOD predictions).

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While this score is efference-framt invariant in theory (we provide proof in Prop. B.2 centering via RECON improves optimization, resulting in consistently higher AUC-ROC scores against IE-AE (full ablation details in Appendix F.I ROC curves in Fig. [5].
Test-time canonicalization RECON's data-aligned canonicalizations can be used to create drop- in layer that grants group invariance to frozen backbones (e.g., classifiers, foundation models) at inference, with no retraining. Concretely, we canonicalize inputs at test time via canonicalization layer defined as C(z) = p.x (A(z) (xxx)) z, and feed the canonicalized sample to the pre-trained model: because RECON aligns imputs the training data distribution of poses (visualized in Dt), it recovers a large fraction of the accuracy lost to test-time transformations.
Figure Gdd shows considerable performance gains in classifiers. We improve consistently over the base- line and offer competitive (MNIST) or better performance (FashionMNIST. GEOM-QM9) against other test-time canonicalization methods such as EquiAdapt (Mondal et al. 2023), while noting that our canonicalization is trained without supervision, whereas EquiAdapt's 's canonicalization is trained end-to-end using class labels. Notably, in the imaging domain, our method consistently outperforms cetiivarinncelearniin architectures (SE(2) Partial G-CNNs (Romero & Lohit )022)) sometimes by large margin (67.72% 81.96% FashionMNIST), and stay competitive (MNIST) or even surpass (FASh)onMNISTT) fully equivariant architectures (SF(2) -equivarraa Steerable CNNs (Cesa et al. 2022)) (ef. Table B). Note that arbitrary canonicalizations from class-pose methods (e.g., IE-AE, SGM) map inputs to OOD poses and therefore degrade performance; benefits from these canonical- izations arise only when they are also applied at training time, thus requiring backbone retraining. Frame-averaging approaches face the same practical barrier, typically requiring incorporating the Trame-averaging into training to grant invariance (Puny et al. 2022 Kaba et al. 2023), which limits adoption due to the added training cost. Our canonicalizations work as plug-in -i solution during inference, facilitating its broader adoption. For further details, refer to Appendix F.2
6 LIMITATIONS AND FUTURE WORK
We addressed the challenge of learning instance-specific symmetries from unlabeled data by lever- aging class-pose decompositions and solving the problem of arbitrary canonicals. In our testing, RECON proved to be successful in recovering pose distributions at the instance-level including demonstrations in 3D data. We also showed how RECON enables downstream practical applications, notably in test-time canonicalization, which yields clear performance gains and has potential for broader adoption due to its plug-and-play, architecture agnostic, retrain-free nature.
Limitations A crucial limitation is that RECON relies on class-pose decompositions, where the relative transformation is defined with respect to the entire input X. This is effective in certain domains, but when we use class-pose methods as backbones for symmetry discovery e.g. in natural images, it exposes key mismatch: the group action p(g) moves the whole scene (input), whereas the object of interest occupies only a subset (e.g., a car within a street scene). Background and multi-object context make unsupervised association across instances difficult, hindering recovery of object-level symmetry distributions. A natural next step is moving from instance-level decomposition to an object-centric variant that predicts pose from token or mask-based features from e.g. strong self-supervised backbones. This preserves our pipeline's 's simplicity while targeting in-the-win settings where symmetries are local to objects rather than inputs. Complementary improvements target sampling and modeling; for instance, replacing nearest neighbors with more sophisticated sampling strategies. On the density side, more flexible estimators for the symmetry distribution - like flow-based models in the spirit of Allingham et al. (2024). can better capture complex symmetry patterns than a parametric family. Finally, relaxing the need to predefine & by inferring group structure or group-agnen aoou--ggoottcc transformations (van der Linden et al. (024) is promising direction. These extensions preserve RECON's ss simplicity while pushing toward more robust performance.
ACKNOWLEDGMENTY
We thank Moritz Wagner for valuable discussions and feedback on earlier drafts of the manuscript, and Carlos Saavedra Luque for his contributions to the design of the main figure. This research was partially supported by the Deutsche Forschungsgemeinschatt (DFG) through the DFG Cluster of Excellence MATH+ (EXC-2046/1, EXC-2046/2. project id 390685689), as well as by the German Fed- eral Ministry of Research, Technology and Space (research campus Modal, fund number 4ZMM, 05M20ZBM) and the VDI/VDE Innovation Technik GmbH (fund number 161823025B).
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REFERENCES
Bijan Afsari. Riemannian 1e center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139(02):655-653) 655, February 2011. ISSN 0002-9939, doi:
James U. Allingham, Bruno K. Mlodozenice, Shreyas Padhy, Javier Antorán, David Krueger, Richard E. Turner, Eric Nalisnick, and José M. Hernández-Lobato. A generative model of symmetry transformations. Advances in Neural Information Processing Systems, 37:11091-911330 December 2024.
E. Anderson, Gd Veith, and David Weininger. Smiles: a line notation and computerized interpreter for chemical structures., 1987. URL htt.httsss//ppp..eaaiiicaaaaaa wtw.cchhoh..aaaooooorossaa. 64884759
Andrea Aveni and Sayan Mukherjee. Uniform consistency of generalized fréchet means, August 2024.
0tt1//040880333444 arXiv:2408.0753. [math].
Simon Axelrod and Rafael Bonnzz--ombarelli... GEOM, for property prediction and molecular generation. Sci. Data, 9(1):185, April 2022.
molecular conformations
Gregory W. Benton, Marc Finzi, Pavel Izmailov, and Andrew Gordon Wilson. Learning invariances in neural networks. CoRR, abs/2010. 11882, 2020.
Lukas Biewald. Experiment tracking with weights and biases, 2020. URL https://www/ wandb com/ Software available from wandb.com.
Johannes Brandstetter, Rob Hesselink, Elise van der Pol, Erik J. Bekkers, and Max Welling. Geometric and physical quantities improve e(3) equivariant message passing, March 2022. URL http 02905 trXiv:2110.02900 [cs].
//arxiv. ortgabs///1aba000
Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges, May 2021. URL http::/aazx.v..r///////
2104.13478 arXiv:2104.13477 [es].
Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers, May 2021. URL
httt:://axiv..e wttpor//ab.///000
14294 arXiv:2104. 104.229004.1224 [es].
Gabriele Cesa, Leon Lang, and Maurice Weiler. A program to build E(N) ((N)varuivari steerable CNNs. In International Conference on Learning Representations, 2022. URL /tpp:tttp///oeeeeeeeeee
net fost// wid-wE nnw
Taco Cohen and Max Welling. Group equivariant convolutional networks. International conference machine learning, pp.
PMLR, 2016.
Taco Cohen, Mario Geiger, and Maurice Weiler. A general theory of equivariant cnns on homogeneous spaces. CoRR, abs/1811 118811100 1.02017, 2018.
Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling, Gauge equivariant convolu- tional networks and the icosahedral CNN. CoRR, abs/1902.04615, 2019.
Matthijs Douze, Alexandr Guzhva, Chenggi Deng. Jeff Johnson, Gergely Szilvasy, Mazaré, Maria Lomeli, Lucas Hosseini, and Hervé Jégou. The faiss library, 2024.
Ilya Feige. Invariant-equivarian representation learning for multi-class data, 2019.
Matthias Fey and Jan Lenssen. Fast graph representation learning with PyTorch Geometrie, In ICLR Workshop Representation Learning Graphs and Manifolds, 2019.
Marc Finzi, Gregory W. Benton, and Andrew Gordon Wilson. Residual pathway priors for soft equivariance constraints. CoRR, abs/2112.01333 2021.

---

Published as a conference paper at ICLR 2026
Roy T Forestano, Konstantin T Matchev, Katia Matcheva, Alexander Roman, Eyup B Unlu, and Sarunas Verner. Deep learning symmetries and their lie groups, algebras, and subalgebras from first principles. Machine Learning: Science and Technology, 4(2):025027, June 2023. ISSN 2632-2153. doi: 10. 100882233-2113//aae
Jean-Bastir Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent-a approach self-supervised learning. Advances in neural information processing systems, 33:21271-21288 2020.
Xifeng Guo, En Zhu, Xinwang Liu, and Jianping Yin. Affine equivariant autoencoder. In Proceedings International Joint Conference on Artificial Intelligence, IJCAI-19, pp. of the International Joint Conferences Artificial Intelligence Organization, 2019. doi: 2413-2419.
10.24963/ijcai.011999
Irina Higgins, David Amos, David Pfau, Sébastien Racanière, Loic Matthey, Danilo J. Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. CoRR, ab2/120..,2220, 2018.
Max Jaderberg, Karen Simonyan, Andrew Zisserman, and Koray Kavukcuoglu. Spatial transformer networks. CoRR, abs/1 2015.
Sékou-Oumai Kaba, Arnab Kumar Mondal, Yan Zhang, Yoshua Bengio, and Siamak Ravanbakhsh. Equivariance with learned canonicalization functions. In International Conference Machine Learning, pp. 15546-15566. PMLR, 2023.
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30(5):509-5444 1977. ISSN 1097-0312. doi: 10.1002/cpa.3110000502..
Hyunsu Kim, Yegon Kim, Hongscok Yang, and Juho Lee. Variational partial group convolutions for awpue partial equivariance of rotations and color-shifts, July 2024. URL httt:://aa/
htg/abs//ee /tbb///40.2 04271 arXiv:2407. .04271 [cs].
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017.
Gyeonghoon Ko, Hyunsu Kim, and Juho Lee. Learning infinitesimal generators of continuous symmetries from data, 2024.
Risi Kondor, Zhen Lin, and Shubhendu Trivedi. Clebsch- gordan nets: a fully fourier space spherical convolutional neural network. In Advances in Neural In- formation Processing Systems, volume 31. Curran Associates, Inc., 2018. URL
https://prrcceeiing...ss B3f0081.445511322066611111133133 htt.://ce....aatt.te..eeeeeeeaar
neur neurips cc Abstract.
Kaushik Koneripalli, Suhas Lohit, Rushil Anirudh, and Pavan Turaga. Rate-invariant autoencoding of time-series. In ICASSP 2020 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3732-3736 2020. doi: 10. 0..100//CASS444766220000333333
Adam R. Kosiorek, Sara Sabour, Yee Whye Teh, and Geoffrey E. Hinton. Stacked capsule autoen- coders, 2019.
Greg Landrum, Paolo Tosco, Brian Kelley, Ricardo Rodriguez, David Cosgrove, Riccardo Vianello, sriniker, Peter Gedeck, Gareth Jones, NadineSchneider, Eisuke Kawashima, Dan Nealschneider, Andrew Dalke, Matt Swain, Brian Cole, Samo Turk, Aleksandr Savelev, tadhurst-edd, Alain Vaucher, Maciej Wójeikowski, Ichiru Take, Rachel Walker, Vincent F Scalfani, Kazuya Ujihara, Daniel Probst, Juuso Lehtivarjo, Hussein Faara, Guillaume Godin, Axel Pahl, and Jeremy Monat. rdkit/rdkit: 2025_03 0025_03_1 (q) 2025) beta release, 2025.
Yikang Li, Yeqing Qiu, Yuxuan Chen, and Zhouchen Lin. Affine steerable equivariant layer for canonicalization of neural networks. In The Thirteenth International Conference Learning Representations, ICLR 2025, Singapore. April 24-28, 2025. OpenReview.net, 2025, [RLhttss //openreview. net. forum 5:i
12

---

Published as a conference paper at ICLR 2026
Giovanni Luca Marchetti, Gustaf Tegnér, Anastasiia Varava, and Danica Kragic. Equivariant rep- resentation learning via class-pose decomposition. In Proceedings of The 26th International Conference Artificial Intelligence and Statisties, pp. 4745-4756 PMLR, April 2023. URL
ngg ngs. ml press v206/ rewettceeittt 23b.. html
K.V. Mardia and P.E. Jupp. Directional Statistics. Wiley Series in Probability and Statistics. Wiley, 2009. ISBN 978-0-470-31781-55 URL
de/bokks
PINiCm40-MOC
David Mohlin, Josephine Sullivan, and Gérald Bianchi. Probabilistic orientation estimation with matrix fisher distributions. In Advances in Neural Information Processing Systems, volume 33, pp. 1884-4893. Curran Associates, Inc., 2020.
Arnab Kumar Mondal, Siba Smarak Panigrahi, Sekouba Kaba, Sai Rajeswar, and Siamak Ravan- bakhsh. Equivariant adaptation of large pretrained models. ArXiv, abs/2310.016477 2023.
Artem Moskalev, Anna Sepliarskaia, Ivan Sosnovik, and Aroold Smeulders. Liegg: Study- ing Icarned lie group generators, 2023. URL http://aawww...cr//aa////1...... arXiv:2210.04345 [cs].
Kieran Murphy, Carlos Esteves, Varun Jampani, Srikumar Ramalingam, and Ameesh Makadia. Implicit-pdff Non-parametriic representation of probability distributions the rotation manifold,
2022. URL httl:tttt///axiv..oom aaag///b///aee /2106 C5965 arXiv:2106.05%65 [cs].
Samuel E. Otto, Nicholas Zolman, J. Nathan Kutz, and Steven L. Brunton. A unified framework to enforce, discover, and promote symmetry in machine learning, 2025. URL http:
org//bs/2311.000 00212
arXiv:2311 1,00212 [cs].
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang. Zach De Vito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library, 2019.
Nico Pelleriti, Max Zimmer, Elias Wirth, and Sebastian Pokutta. Approximating latent manifolds in neural networks via vanishing ideals. In International Conference on Machine Learning (ICML), 2025.
Xavier Pennec. Intrinsic statistics riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 25(1):127-154, July 2006. ISSN 0924-9907, 1573- 7683. doi: 1100007/11851-00000 -06--228------
Omri Puny, Matan Atzmon, Heli Bam-Hamu, Ishan Misra, Aditya Grover, Edward J. Smith, and Yaron Lipman. Frame averaging for invariant and equivariant network design, March 2022. URL httt:://axxis org. www.absslli. 03336 arXiv:21 10.03336 [es].
Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and Anatole von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Sei. Data, 1(1):140022, August 2014.
Siamak Ravanbakhsh, Jeff Schneider. and Barabass Poczos. Equivariance through parameter-sharing, June 2017. URL http:://arxiv..or//aaaaaa/ 01t.//81912208989/// arXiv: 1702.08388 [stat].
David W. Romero and Jean- -aatistt Cordonnier. Group equivariant stand-alone self-attention for vision. CoRR, abs/2010.00977. 2020.
David W Romero and Suhas Lohit. Learning partial equivariances from data. Advances in Neural Information Processing Systems, 35:36466-36478, 2022.
David W. Romero, Erik J. Bekkers, Jakub M. Tomczak, and Mark Hoogendoorn. Attentive group equivariant convolutional networks. CoRR, abs/2002.03830, 2020.
Peter J. Roussceuw and Mia Hubert. Robust statistics for outlier detection. WIREs Data Mining and Knowledge Discovery, 1(1):33-79, January 2011. ISSN 1942-4787, 1942-4795. doi: 10.1002/ widm.2.
13

---

Published as a conference paper at ICLR 2026
Ben Shaw, Abram Magner, and Kevin R Moon. Symmetry discovery beyond affine transformations, 2024.
Wen Shen, Qihan Ren, Dongrui Liu, and Quanshi Zhang. Interpreting representa- tion quality of dnns for 3d point cloud processing. In Advances in Neural Infor- mation Processing Systems, volume 34, pp. 8857-8870. Curran Associates, Inc., 2021.
neurips wwtprrippppccpppppeeeeee ccc/paaper/2222/haslll 4330005188104411100000022------3
URL https://proceeiingg neeriiinnnnn.nus..aeeeee.m...
html
Georgy L. Shevlyakov, Stephan Morgenthaler, and Alexander M. Shurygin. Redescending m-
estimators. Journal of Statistical Planning and Inference, 138:2906 22127699-917,,, 2008. URL https:
7/api. Str.llseeentticcsc www.hntccccchhho
12:1111141
Ha- Young Shin and Hee-Seok Oh. Robust geodesic regression, January 2022. URL http://
04518 arXiv:2007.044 000510 [stat].
Zhixin Shu, Mihir Sahasrabudhe, Riza Alp Güler, Dimitris Samaras, Nikos Paragios, and lasonas Kokkinos. Deforming autoencoders: Unsupervised disentangling of shape and appearance. CoRR, abs/1806.06503, 2018.
Jiachen Sun, Qingzhao Zhang, Bhavya Kailkhura, Zhiding Yu, Chaowei Xiao, and Z. Morley Mao. Benchmarking robustness of 3d point cloud recognition against common corruptions, January 2022. URL /ttpttttt///iiii...tt org/ 2201. 12296 arXiv:2201.12299 [cs].
Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang. Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation- and translation-equuiariann neural networks for 3d point clouds, May 2018.
1rXiv:1802.08219 [cs].
Alonso Urbano and David W. Romero. Self-supervised detection of perfect and partial input- dependent symmetries. In Proceedings of the Geometry groounddd Representation Learning and Generative Modeling Workshop (GRaM), volume 251 of Proceedings of Machine Learning Research, pp. 113-131, PMLR, 29 Jul 2024. URL httts:://proseee ngs. . ml r.prrss// w251/urbano244 htmm
Putri A. van der Linden, Alejandro
Sharvaree Vadgama, Thijs P. Kuipers, and Brik J. Bekkers. Learning symmetries via weight-sharing with doubly stochastic tensors. Advances December 2024.
in Neural Information Processing Systems, 17:42738-42777
Shishir Reddy Vutukur, Rasmus Laurvig Haugaard, Junwen Huang, Benjamin Busam, and Tolga Birdal. Alignist: Cad-informed orientation distribution estimation by fusing shape and correspon- dences, 2024. /
http: wttpvaiiiivv.. wttp:/iww..rg/b////20000 06683 2XXi:22409006683 [es].
Yude Wang, Jie Zhang, Meina Kan, Shiguang Shan, and Xilin Chen. Self-supervised equivariant attention mechanism for weakly supervised semantic segmentation. CoRR, Abs/2004.04581, 2020.
Maurice Weiler and Gabriele Cesa. General e(2)-equivariant steerable cnns. CoRR, https/9911,082555 2019.
Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco Cohen. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. CoRR, abs/1807.0255 2018.
Robin Winter, Frank Noé, and Djork-Aréé Clevert. Permutation-in iiniuuattiiooiiniiannt invariant variational autoencoder for geaph-level representation learning. CoRR, abs/2104.09856 2021.
Robin Winter, Marco Bertolini, Tuan Le, Frank Noé, and Djork-Aré Clevert. Unsupervised learning of group invariant and equivariant representations, Advances in Neural Information Processing Systems, 35:31942-31956, 2022.
Andrew Wood. Estimation of the concentration parameters of the fisher matrix distribution on 50(3) and the bingham distribution sq. q >= 2. Australian Journal of Statistics, 35:69 - 79, 06 2008. doi: 10. 11j. 011777842X.199.1011111
Jianke Yang, Robin Walters, Nima Dehmamy, and Rose Yu. Generative adversarial symmetry discovery, 2023. URL URL httttt/pp//:xiiii org/ abs /2002. 00236 arXiv:2302.00236 [cs).
14

---

Published as a conference paper at ICLR 2026
Jianke Yang, Nima Dehmamy, Robin Walters, and Rose Yu. Latent space symmetry discovery, August 2024. /ALLAttp:///ax.v.ooz////////2/// c0105 arXiv:2310.00105 [cs].
Raymond A. Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing. In Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera (eds.), Proceedings of The 25th International Conference on Artificial Intelligence and Statisties, volume 151 of Proceedings of Machine Learning Research, pp. 1527-1545 PMLR. 28-30 Mar 2022. URL https: int
mir pte..m/.....ees./............. html
Tatsuya Yokota and Hidekata Hontani. Manifold modeling in quotient space: Learning invariant mapping with decodability image patches, 2022.
Richard Zhang, Phillip Isola, Alexei A. Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features perceptual metric, April 2018. URL http:://xxx....rrg/ arXiv: 100.088013394 [cs].
abs/1801. 03924
Yue Zhao, Yuwei Wu, Caihua Chen, and Andrew Lim. On isometry robustness of deep 3d point cloud models under adversarial attacks. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1207, Seattle, WA, USA, June 2020. IEEE. ISBN
978-1-7281-7168-5. doi: 10.1 1099CCVP2220002200000022 URL htttps:
ieee wttp://g//dootto//eo/eeeee http:/mmmmnn99555909.0
Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-leaming symmetries by reparameterization, 2021. /
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A BACKGROUND
Groups and group actions. A group 9 is a set equipped with a closed, associative binary operation such that contains an identity element a given set X' and group G, the (left) group action of G on X' is map p G x X + X' that preserves the group structure. Intuitively, it describes how set elements transform by group elements.
G and every element 9 E & has an inverse g G. For
Group representations. In this work, we focus on cases where X' is vector space. In such scenarios, the group acts on by means of group representations. A representation of a group 9 on a vector space A' is homomorphism px 9 - GT.(X) mapping each g E G to invertible linear operator on X. We consider our dataset be of the form (= f V W}. where is a set on which g acts, and W is a vector space on which G may also act. As an example, molecular conformations can be interpreted as functions f V + R that map atoms to their spatial coordinates. Following this definition, a group element acts in data sample as
(xx(g))((z) = pw(g)f (pv(g~")x)
(6)
For instance, rigid transformation 9 might shift or rotate node coordinates via (W( while leaving the nodes unchanged or transforming them via py()). Whenever we speak representation px on X, it is understood that we are implicitly referring to the previous equation to understand the transformation each component.
Orbits. The orbit of x, (x=(px(()=)))pcon the action of all elements of G.
all possible transformations of resulting from
Equivalence classes and quotient sets. Our analysis relies on the definition of equivalence classes and their quotient sets. Let ~ be an equivalence relation on X and consider the equivalence classes [x]=[y E X, s.t. y) of X. The quotient set //~ ~~ is defined the collection of all equivalent classes in X' under the relation ~..
Group equivariance and group invariance. A map V W is G-equivariant with respect to the representations pv: Pw if h(pr(g)z) = Pw(g) Vg E G, V. E X. In the context of neural networks, G-CNNs (Cohen & Welling, 2016) designed to be G-equivariant by using only G-equivariant layers in their constructions. This ensures that applying a transformation E G before layer yields an equivalently transformed output. Analogously, a map h is G-invariant with respect to if ()))())))==((((((())))))))) Vg E g. Vz E X. That is, if G-transformations of the input yield the same result.
B PROOFS
Proposition B.1 (Approximation of Pin) via Normalization). Let X' be metric space, Lie group and 5, 2b an IE-AE where vp is continuous on a compact domain X. Suppose that X exhibits symmetries characterized by Hla) where F(H(z) = as described above. Consider a random sample {s:),N1 of [x]] and denote their images by vo as v((x)) [v(x))] Let [w] G be the empirical Fréchet mean of ((xx)). Then, the empirical distribution Man corresponding to the normalized samples ((x)))]]]] [z] approximates the target distribution Hlz]] Specifically, Pax) converges
in Wasserstein distance to P[x] as s' - 0 and N - Jc.
Proof. Consider N independent samples N drawn from class [z] according the model $; = px (x) [=)] + where gi ~ Hlx i.i.d. and E, ~ P. iidd, with ||=/|| < e' almost surely. Let ([=])) = ( be the set of observed relative transformations and Mul] be the distribution over g from which (ss;;) are sampled. Our first goal is to show that 17)] and [1xx] differ by some translation
[ E G.
Let s
px (x(g)) Mao be the noise-free (i.e., e; = 0) counterparts of s;, and consider their images (s')). Denote V/x) as the corresponding distribution over & from which (s({) are sampled. Then,
(5,)) = )((("))) = 9.L [m]
1
G-09
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1
6
Figure IE-AE architecture visualization. Image taken from Winter et al. 2022
for [|2] == (1a1)) E G. Since 9: ~ H[a] the distribution 1/a) is exactly the right-translate Hla Tini[pp [x]
That is, (Mn1 M[x) This relates the noise-free free v/x; and M[a]- Now, we will show that this relation also holds approximately for M[s]-
By the uniform continuity of & (which holds since X' is compact and e is continuous), there cxists a modulus of continuity such that lims- (6) and for 8, s X, it holds:
(s()) vs())) I w(dx(s;, x()) ((|e<]l)) < w('') K,
(8)
for some small K >> O, since g is presumably small and ( = 0. This establishes pointwise closeness almost surely.
Consider the Wasserstein distance between both probability distributions, defined as 1/P
W (y ()) Ma)) = inf (() ( (2), der
(9)
where [[]] (aj) is the of all possible couplings (joint probability measures) on with marginals Mis] and '/ar Consider the natural coupling That determined by the joint random variable pair (1b(s;), ((())) Tnat is valid coupling in [[ (M))) since its marginals are Mss] and [/x1] respectively. Since the Wasserstein metric is defined as the minimum over all possible couplings, must hold that
W Wo(m (Jeve dg(و1, (2) 92 dinat(91.92)
1/P
(10)
- -Vose do(((si), (($))) d
1/P
( w(E')p dnnat (91
1/v
= w(E").
(11)
This inequality shows that the distribution M|s] generating our samples (<xx) is close to the distribu- tion '/x] = Mi [x] in Wasserstein distance, with proximity controlled by the noise bound e via the modulus of continuity W. Let's show now how to estimate the unknown I Jaj-
First, note that we can calculate the Fréchet mean of 1/11 as follows:
F(v(4)) = F(4)))
II' inj el'(z] [u)
(12)
where used the property that the Fréchet mean is equivariant under isometries (such right- translation by [[x] when using right- ingarianarriant metric), F(uI) = F(H) (Karcher) 1977). This means that the unknown [(z] the Fréchet mean of (()- However, we just proved that ((a) and [[x] We want to prove now that their Fréchet means are also
are close, bounded by w(E') (equation 10 close.
Consider the Fréchet mean F()(z) of the actual data distribution. The Fréchet functional is known to be strictly convex within geodesic balls of certain radius ro determined by the manifold's geometry (Karcher, 1977). Within such regions, the Fréchet mean map μ + F(u) is Lipschitz continuous with respect to the Wasserstein distance (Afsari 2011). Therefore, exists a constant Cr, 0 such that
dg dg(FFyia) S CLWp(Mia) 1/4) < Czw(s').
(13)
Substituting F(v()) [2] we have (F(())))) (1)) < Czw(s'). This shows the true Fréchet mean of the distribution we sample from is close to I [m Therefore, we can estimate I [x] by obtaining the population Fréchet mean of Uxx]
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Let's show that using the sample Fréchet mean of (xxx) estimates the true Fréchet mean. In effect, the sample Fréchet mean is a statistically consistent estimator of the population Fréchet mean F(M))) (Aveni & Mukherjee 2024):
E, F(Ma) as N oc.
[o]
(14)
Combining all the above, we see that for large N and small e', [s)] provides a good approximation of [s]]-
Finally, consider the Fréchet-normalle empirical measure P = Σ f: By the continu-
ous mapping theorem, and because group inversion and multiplication are continuous operations in the Lie group, it follows that ji converges to [[a] in Wp distance.
Proposition B.2 (Right-translation invariance the log density). Let 9 be a Lie group with a right Haar measure X. Let p be probability measure on absolutely continuous w.r.t. X, with density du (A). Fix y 9 and let rr(9) 97. If (r7)-14, then
Pp = 2x
Pu(h) - pn(hy=1) for almost every h E G.
(15)
Consequently,
Pogpo(h) log po for almost every h 9, interpreted in the extended reals with the convention - log 0 = +oc.
(16)
Proof. For any bounded measurable f R.
f / ((h)av()) = / frr(())) du(g) = -((()(1))))) f(r-(0)) dX(g).
(17)
Set h gy. Right invariance of x gives dd(h) dD((g), hence
/ f(N) =| / f(h) dA(h).
(18)
By uniqueness of GadonnNikodyy derivatives, (h) p1(h)=1) for almost every h. Taking log yields equation 16 (with log 0 +00).
C ROBUST FRÉCHET MEAN EXTENSION FOR MULTIMODAL DISTRIBUTIONS
While the Fréchet mean provides a robust and well-defined centroid for many distributions, as discussed above, it may fall outside the support in certain multimodal cases, leading to non-natural canonical poses that do not align with the data's intrinsic symmetries. For instance, consider bimodal distribution on 50(2) with identical modes at 0 and æ radians. Here, the Fréchet mean minimizes the sum of squared geodesic distances and, due to the symmetry, converges to point midway between the modes (H 12), which lies in region of near-zero probability density, outside the effective support of the distribution. This violates the desired natural (in-distribution) reference pose, as may orientation not observed in the data, potentially degrading downstream applications like invariance granting or interpretability.
correspond
To address this limitation while preserving the generality of the Fréchet mean, propose extension using redescending M-estimators (Shevlyakov et al., 2008 Rousseeuw & Hubert 2011). These estimators replace the squared loss with a bounded, redescending function that caps the influence of distant points, providing robustness to outliers or separated modes. We specifically propose using the Tukey biweight loss (Rousseeuw & Hubert
2011; (Shin Oh 2022):
- (1-1-(65 (2) ))
if
m(w;c) c)
(19)
if
In the standard Fréchet mean, minimize E, dly. 9.) which is unbounded quadratic in distances; large distances (outliers or far modes) have disproportionate pull because their squared penalty grows without limit. However, we can provide robust version by minimizing Σ; m(d(y. g:)s instead.
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Angareet
Aeprose
(a) Unimodal distribution (von Mises with concen- tration & 20, mode at 7/2 rad).
(b) Bimodal distribution (two Mises with modes and rad, 20).
(c) 5-modal distri-
(d) Uniform distribution
bution (five Mises with
in interval in SO(2).
modes
at 0, 2m 27/5, ..... ......... 8T /5 rad,
K = 20).
Fréchet mean
Tukey-Frécher Fréchet mean
Figure 8: Comparison of Fréchet mean and robust Tukey-Frécchee mean variation SO(2) distribu- tions. Plots generated by sampling = 2000 points from each distribution, computing the means, and overlaying the theoretical density. The robust estimator localizes high-density regions in multimodal cases, while converging to the standard Fréchet mean in unimodal/uniform scenarios.
In effect, the robustness arises from the bounded influence the Tukey biweight loss; m is such that for small u, it holds that m(u) Rs u² behaving like the Fréchet mean locally (quadratic). However, for large distances u >> c, m is constant at e² e²/6, assigning zero gradient to distant points:
dm (w: du
u (²)))²
[ S
(20)
0
| >
The function dm/du determines how much each sample affects the final estimate during optimization (we gradient descent to solve the minimization problem in Shin & Oh (2022 Therefore, a multimodal distribution with well-separated clusters, large distances ([u) >> c) have zero influence on the gradient/update, and the objective effectively ignores points from secondary modes when evaluated near primary mode, minimizing only over the local cluster. To mitigate ambiguity, initialization common starting point can guide convergence contrast, the Fréchet mean's 's quadratic penalty (u²) grows unbounded, pulling the minimizer toward global compromise, often outside any cluster. We provide some simulations in SO(2) in Fig.
"principal" consistent mode. In
The parameter > 0 controls the robustness threshold (e.g., c = π for SO(2), half the maximum geodesic distance n; more generally, e can be set adaptively, but intuitively, smaller values of induce a "shorter-sipee siihht" on the estimator).
Formally, given a distribution μ over the Lie group & with geodesic distance dR, the robust centroid or FrécheFrrécht mean Fr) is defined as:
Fr(m) = arg min Eg g): c)],
(21)
where m(u c) is the Tukey biweight loss. For empirical samples {g ) the estimator becomes:
= arg min
Σ mldx(y.9):))
(22)
Peg
C.I EXAMPLES
We show comparison via simulations between the Fréchet mean and the robust variation for common distributions (Fig. 8) In SO(2), the Fréchet mean has closed-form expression (circular mean). The robust variation lacks a closed form and requires optimization, e.g., scalar minimization over [0, 2x) but as one-time computation for pseudo-labels (Algorithm[1 []. this is feasible (e.g., <Is for n = 2000 on standard hardware). In multimodal cases, converges to a mode (e.g., ा for bimodal, shown in 8bb while Fréchet converges to f In SO(3), there is closed form for either Fréchet mean or the robust variation. In this case, the Fréchet mean can be found via singular value decomposition Riemannian gradient descent.
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D ADDITIONAL INSIGHTS AND EXPERIMENTS DETAILS
D.1 ORBITS, STABILIZERS, AND RELATION TO INSTANCE-LEVEL POSE DISTRIBUTIONS
We clarify how our notion of instance-level pose/orbit distributions relates to the group-theoreti notions of orbits and stabilizers. Given x E X' and a group g acting on X' the stabilizer S₂ {gg E 9 gø =) collects the (self) symmetries of x. For example, on planar rotations, perfect circle has cincks
Digit
Class-pose (v)
RECON (μ)
0
1
(2),), whereas a perfect square has Sqquare equal to the group of 90" rotations.
2
In contrast, our instance-level distributions Mis] do not model stabilizers explicitly; instead, they approximate the part of the orbit O₂ that is actually observed in the data, When x has a non-trivia stabilizer Sz. many group ele- ments act identically on 2;, so the pose distribution only identifiable modulo S₂ and becomes broad or multimodal along the corresponding symmetry directions. This makes canonical poses inherently ambiguous for highly symmet- ric inputs, but our Tukey-Frécht estimator explicitly designed to remain robust in this regime.
3
4
5
6
7
This is visible for the digit "8" in the per-class rotated MNIST experiment: for small fraction of very sym- metric handwritten 8s, upright and upside down config- urations are effectively indistinguishable, so the learned pose distribution shows non-negligible mass near 180 (pose ambiguity), unlike digits such as 3, 4, 7 whose histograms are sharply concentrated. In such cases, our Mulkey-Fréchen objective does not average the modes into an unstable in-between pose; it admits minimizers aligned with one of the symmetric modes, yielding stable in-distribution canonicalizations (ef. Appendix and enabling test-time canonicalization.
8
9
Figure 9: Probability density of recov-
ered SO(2) distributions on MNIST ex- periment: IE-AE, vs RECON.
D.2 IMPACT OF NEIGHBORHOOD SIZE
Our symmetry estimation relies approximating the class [x]] via k-nearest neighbors in the invariant space Z. Figure 10 shows the impact of i on the MAE of the predicted pea-class symmetry parameter 0 for the MNIST experiment. While Proposition] 3.1 suggests convergence as N - co (larger k), practical datasets have finite class separation and noisy samples. Large can include samples from different underlying classes in the k-NN approximation, increasing noise and impacting performance. Conversely, very small & may not provide a representative sample. Overall, k = 25 offers the best balance for GEOM-QM9 and k 10 for MNIST and FashionMNIST. and the performance on the symmetry discovery task is not overly sensitive to the neighborhood size within a reasonable range of neighbors, indicating that our method is stable w.r.t. the choice of k.
D.3 ANALYSIS OF EQUIVALENCE CLASS DEFINITION
To quantitatively assess the quality of the computed equivalence classes [xi], we introduce hit rate metric that, for each E X. measures the proportion of k-nearest neighbors in the computation of [ar] that belong to the same molecular class (SMILES) as æ. high hit rate indicates that our definition of equivalence class captures iosc-invariant similarity successfully.
Figure| 13a shows the distribution of hit rates across all 174, 481 conformers in our GEOM-QM9 dataset. The average hit rate is 0.95, with 76.8% of distinct molecules achieving an average hit rate above 0.9. This confirms that the proposed equivalence class definition behaves as expected, abstracting molecular identity across conformational variations and group transformations, while maintaining discriminative power between different molecular structures. Figure 13b shows that molecules with more conformers in the dataset tend to achieve slightly higher hit rates (correlation
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MNIST
Fashion-MNIST
Prediction errer
Prediction error (MAE) (9)
Highest per-molecule MSE
Error (*)
2000
1000
10 15 20
25
30
35
10
20
30 40
Number of nearest neighbors
Number of nearest neighbors
Figure 10: Prediction error in symmetry discovery task vs number of neighbors in class computation for MNIST and FashionMNIST experiments (left) and GEOM-QM9 (right). On the left, plot average prediction error, which is defined as the MAF (in degrees) between the ground truth symmetry parameter and the per-class average predicted parameter, averaged across classes; shaded bands show the corresponding standard deviation across multiple runs with different seeds. On the right, we plot average and standard deviation of the prediction error, which is defined by the MSE between predicted and true the highest per-molecule mean MSE across conformers, illustrating the trade-off between average accuracy and worst case error as the number of neighbors varies.
parameters, averaged over all conformers. The orange curve reports
coefficient 0.149). This is expected; molecules with small number of conformers are more likely to include neighbors that belong to other molecules. This supports the insights about how the number of neighbors affects the quality of the symmetry discovery (Fig. [O), since greater number of neighbors increases the probability of a reduced hit rate on low-conformer molecules.
We also report hit rate metrics for varying number of neighbors on the imaging experiments in Fig. 13c Note that configurations with the highest hit rate (k = 5) do not necessarily correspond to configurations with lowest prediction error (k- = 10) in the symmetry discovery task (Fig. I0] left). In effect, lower & generally yields higher hit rates, but using too few neighbors provides insufficient samples for accurate estimation. This explains the observed trade-off Fig. 10, where i 10 achieves the lowest prediction error despite h = 5 having the highest hit rate.
Limitations of the proposed equivalence class We discuss the most prominent failure modes of our equivalence class definition by examining lowest-hit hit- rate cases in FashionMNIST, the dataset which exhib- ited the lowest hit rate score in our experiments (cf. Figure 13c). (3c). The heatmap in Figure shows that most inputs with equivalence class construction imprecisions concentrate few semantically similar labels, partic- ularly the footwear block (Sneaker ++ Ankle boot) and
3.6
Tishirotop
0222 0.00 0.39
5.0 0.30 026 0.33
0.28 0.14 0.1 0.26 3.07 0.52 0.00 0.03 0.00
0.8
Pullover
0.23 0.00 0.3 0.01 8 -0.0 119-3.1..000
Weighood fraction
0.24 0.52 033 027 0.02 926
Input flowest
Coat 0 -0.00 0.4 818
0.00
Sandal 0.63 10. to 0.04
0.62
0.62
Shirt 0.00
-1.10.1
Sneeker no 10.33-0 02 0.00 <1.1
03.033 0.02
0.2
Bag 0.00 0.00 0.55 034 337
020 0.10
tops (Coat ++ Shirt). For qualitative analysis, we visu-
s0 83-00 di-a Bi-a t3-0 0:1 0:0
alize at random some equivalence classes with low and zero hit rate in Figure 12 We observe that neighbors of each input are visually nearly indistinguishable from the input, up to small cues (e.g., ankle height), partly due to small resolution of the data. As a result, the encoder maps them together the latent space, despite having different semantic labels, which generates the feature overlap.
Tasethop
Phouter
Shoster
Heg
Hokel Boot
0.5
Nelahbors class
Figure 11: Neighbor confusion matrix (FashionMNISTT) inputs with the lowest neighbor label hit rate (worst 5k samples). Each row corresponds to an input class and each column to a neighbor class; entry (t.j) gives, averaged over low-hit hit inputs of class the fraction of their k-NN that belong to class j.
Overall, these figures indicate that the most challenging scenarios for our method involve inputs with high inter- class overlap in the latent space (and not large intra-class deformation). In effect, our method thrives in latent spaces with clear separation between classes, and is not required have small within-class variations in the data.
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Input
Nearest neighbors ls k-NN(a) (k 10)
Ankle boot
Sneaker
Cost
Shirt
Shirt
Coai
Sneaker
Ankle boot
T-Shirt
Tronser)
Figure 12: Qualitative analysis of lowest hit equivalence classes in FashionMNIST. Each row shows a training sample (left) with low hit rate (in particular, zero for rows 1-4) and its 10 nearest neighbors in the learned latent space (right), ordered by cosine similarity. Text underneath denotes labels. This illustrates inter-class feature overlap in the latent space: some inputs from different semantic classes exhibit very similar features (e.g., some Sneaker models look very similar some Ankle boot models), therefore they are clustered together despite having different labels. This overlap the primary cause of reduced hit rates in FashionMNIST for certain inputs, and highlights the regime where our equivalence class definition is most challenged.
As an empirical example, GEOM-QM9 exhibits non-trivial intra-class conformational variations (RMSD <. 1.5 Angstrom) yet achieves 0.95 hit rate (cf. Figure 13a). This happens because molecular identity has very strong and clear signal from compositional and topological invariants, e.g. number atoms of given element, that remain perfectly constant across conformers, which allows the encoder to more easily separate conformers of different molecules. Our equivalence class definition behaves well as result.
Mitigation strategies for inter-class overlap Mitigation strategies for these cases should therefore target inter-class overlap, i.c., increasing class separation in the invariant latent space. This challenge well-studied, and established approaches with contrastive or self-supervised objectives (e.g., DINO (Caron et al. 2021), BYOL. (Grill et al.) (020) are known to sharpen class boundaries without supervision. Such objectives are perfectly compatible with our framework and directly address its main failure mode. Additional mitigation strategies, discussed in limitations section, include richer and more robust neighborhood sampling to further stabilize equivalence class construction.
D.4 PICKING UP ON PARTIAL SYMMETRIES
notable strength of RECON its ability to distinguish between distinct classes that are related by group transformation such digits "6" and '9', which are related by 180" rotation. Fully SO(2)- equivariant methods map these inputs to an equivalent representation, and therefore, downstream tasks struggle distinguishing between them. This is a well-known example that has motivated partially equivariant methods in the past (Romero & Lohit 2022). RECON leverages the clustering of the input's invariant features and normalizes their pose distributions separately, which addresses this problem. Other class class-pose pose decomposition methods, much like classical equivariant networks, can not pick up partial symmetries, and therefore collapse both '6's and '9's into the same canonical
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Table I: RECON pseudo-label generation (Algorithm wall-clock runtime for different datasets
(naive implementation).
Dataset
Number of samples
Nearest neighbors
Runtime (seconds)
MNIST
12,000
10
15
Fashion-MNIST
60,000
10
173
GEOM-QM9
174,481
25
1033
reconstruction (Figure 5p Col. 2) Consequently, the distributions M[] and M[9] of relative poses are different with opposing peaks (Figure 2 Col. 2, Digits 6 and 9). On the contrary, RECON estimates distinct offsets (T'6) and I (1g)) based how each digit class typically appears relative to the arbitrary canonical. As a result, normalization yields not only a shared symmetry pattern (Hi R [lo] Figure Col. 3) but also correctly associates them with distinct natural poses (Figure 5p Col. 3). This sensitivity to the data's contextual orientation allows RECON to handle cases beyond perfect group orbits where full equivariance is inappropriate.
D.5 COMPUTATIONAL ANALYSIS
Our method introduces three computational components on top of the class-pose backhone: (i) (one time) pseudo-label generation step based on k-nearest neighbors (Algorithm [T]. (ii) training of & and A (computation and implementation details in Appendix [E) and (iii) the downstream symmetty.00D/eanooiilaaattt
inferences with the learned mappings + and A. We analyze and (iii) in terms of complexity and runtime in practice.
Pseudo-label generation Let be the number of training examples, d the dimensionality of the invariant embedding 2, and i the number of neighbors for Algorithm 1 The dominant cost of pseudo-label generation is the k-NN computation, with complexity O(N2d). This step is ran once per dataset, and the resulting pseudo-labeled dataset is saved to disk and subsequently used to train @ and A. We emphasize that the pseudo-label generation is one-time computation that does not happen at training inference time. On our hardware (see Appendix E) a naive, non-optimized implementation of the pseudo-labe generation process completes in the order of few seconds minutes per dataset; wall-clock times are reported in Table For scaling to substantially larger N. once can use batch processing to reduce memory and replace the k-NN search by highly optimized neighbor search implementations (e.g.. FAISS (Douze et al. 2024
which greatly reduces the effective cost.
RECON canonicalization overhead Canonicalization at test time makes of two forward passes: the IE-AE encoder on the input to obtain the relative pose (xx) (which yields the IE-AE canonical- ization), and the forward pass A(x) to obtain the centering transformation (which alongside (xx) yields the RECON canonicalization, C(x)= = px (x) vbxx)~1) -1) x)) We quantify RECON's 's canonical- ization overhead by measuring the average wall-clock inference time (in ms/sample) of the centering transformation computation. The computational overhead is of 0.351 ms/sample for MNIST, 0. 113 ms/sample for FashionMNIST and 0.068 ms/sample for GEOM-QU QM9, Our canonicalization adds only small overhead, and the performance gains outweigh this cost.
E IMPLEMENTATION DETAILS OF RECON AND IE-AE BACKBONE
Our experiments are implemented using Python 3.11, primarily with Pytorch (Paszke et al. 2019) and Pytorch Geometric (Fey & Lenssen 2019) for neural network implementation and training. All experiments were seeded with a fixed random seed for reproducibility and logged using Weights & Biases (Biewald] 2020).
This limitation also be observed in SGM Allingham
2024
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Near:
0.1
0.7
133
20
HD 1200
neightomm
(a) Per-input hit rate in GEOM- (b) Average hit rate per-molecule vs (c) Average and standard deviation number of conformers. Molecules with of hit rate across neighbors imag-
QM9, measuring quality of the computed equivalence classes less conformers present noisier equiva- ing experiments. [x)].
lence classes.
Figure 13: Quantitative analysis of the proposed equivalence class definition for the GEOM-QM9 experiment.
E.1 IMAGES
E.1.1 DATA PREPARATION
We use augmented versions of the MNIST/FashionMIIIST datasets with different true symmetry distributions for each class. We load the original training and testing sets, split images by their class label (0-9), and apply rotation to each image sampled from label-dependent distribution (c.g. uniformly from (-600, ,60%) for labels 0-4, and [-90", -00) for labels 5-9 MNIST).
E.1.2 MODEL IMPLEMENTATIONS
For the equivariant architectures, we use the escnn library (Weiler & Cesa 2019 Cesa et 2022).
The encoder processes the input image through series of SE(2)-cqqivariant blocks (escnn. nn. R2Conv layers with kernel sizes 7, 5, 5, 5, 5, 3, 1, using gapaces. with additional batch normalization and non-linearities (ReLU, NorNNonLi nearittt).. The final layer outputs features decomposed into invariant scalar fields (128 channels, representing the invariant component n(x)), and equivariant vector fields from which the relative pose (bxx) parameterized by an angle - is derived.
Encoder.
Decoder. A standard CNN takes the 128-dimensional invariant latent vector as input. uses sequence of torch. nn. Conv2d, Ba catchNorme Dropout2d (p=0.2), and ReLU layers, along with bilinear interpolation for upsampling, reconstruct the input image. The output passes through Sigmoid activation.
Learnable mappings and A. First, θ computes an invariant embedding from the input using escnn. nn. R2Conv layers as before. Then, this embedding is passed through small MLP consisting of two torch. nn. Li Lian layers with ReLU activations, predicting a single scalar, parameter of the target distribution (an angle for MNIST experiment and a standard deviation angle for FashionMNIST experiment). built equivalently the @ predictor, but outputs a single scalar value representing the predicted transformation offset î parameterized as an angle.
E. TRAINING
Best hyperparameter configuration (as in configuration that yields the lowest reconstruction error during validation) was found by hyperparameter tuning. The IE-AE components were trained for 700 epochs using the Adam optimizer (Kingma & Ba 2017) with a learning rate s 8 x 10-4 and a batch size of 128. We save model weights corresponding the lowest validation reconstruction loss.
Training the IE-AE:
Canonical orientation normalization: Using the frozen pre-trained encoder, compute the invariant embedding and the relative rotation angle degrees for each training sample. We then compute the 10 nearest neighbors (for each sample) in the 7 space based on cosine
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similarity Finally, for each sample x, compute two pseudo-labels for training A and Q respectively:
in]: the Fréchet mean (an angle in degrees) which is equivalent the circular mean in SO(2) of the set of neighbor angles (v(x)) IxEN())))
]m]] the set of neighbor angles (1(=)) is then normalized using the previous Fréchet mean: {+)()) [x] (a)) From this set of normalized angles, the pseudo-
label [1]] (estimate of the parameter of the symmetry distribution) calculated using standard parameter estimation methods. In our case, we use methods based moments robust to outliers.
Training the learnable mappings @ and A: The @ predictor and predictor were trained jointly for 600 epochs by minimizing MSE between the network outputs and the pseudo- labels. We used combined Adam optimizer targeting the parameters of both predictors, with a learning rate of approximately 1.35 x 10 10- We use batch size of 128. We weight both losses by a weighting factor of 0.25 applied the loss of A. We save model weights corresponding to the best validation loss.
For all our trainings, we employ a cosine annealing learning rate scheduler with a warm-up phase of epochs and restarts. Trainings were performed using NVIDIA running for approximately + 1 hours for MNIST and 20 + 3 hours for FashionMNIST (IE-AE + learnalle mappings phase).
graphics cards,
E.2
MOLECULAR CONFORMATIONS
E.2. DATA PREPARATION AND SELECTION
Our starting point is the GEOM dataset (Axelrod & Gómez Bombazel 2022), focusing on its QM9 subset (Ramakrishnan et al.] 2014). From this, we select molecules that possess at least 64 distinet conformers. We focus on low-energy states and only retain conformers with Root Mean Square Deviation (RMSD) minimum-enerry conformer. This threshold is applied using RDKit's (Landrum et al. (025) rdkit. rdMo rdMo Wwl.inni.nn....... GetBeet RMS function, which also aligns each qualifying con- former its corresponding minimum-ene minimum-enerry reference structure. This alignment step provides orientation-neutral base for each set of conformers.
less than 1.5 Angstrom from their respective molecule's
The conformers for each selected molecule then randomly split into training, validation, and test sets with an 0.8, 0.1, 0.1 ratio (per molecule) respectively. This results in 174, 481 training samples, 20, 902 validation samples and 23, 805 test samples across 221 distinct classes of molecules.
To introduce controlled and diverse controlled global orientations to serve as ground truths, we augment the data by applying random rotations the aligned conformers. These rotations sampled from matrix-Fisher distributions (Mardia & Jupp/ 2009), a unimodal distribution on SO(3) suitable for modeling varied directional concentrations.
Specifically, we utilize three distinct matrix-Fisher parameter matrices Firue that simulate rotations around the standard e1, E2, 63 axes in the 3D space:
Firue diag( (100, 0.001, 0.001)
F2 Fru (0.000 100, 0.001)
F Prue (iagg(0001, 0.001,
Each molecule, along with its conformers, randomly assigned one of these Firue matrices. Rotations are then sampled for each conformer from its assigned Firue matrix. This process allows us to simulate distinct, realistic and parametrically defined orientation preferences across different molecules in the dataset.
F.2.2 DATA PRE-PROCESSING
In our graph-based framework, molecules are represented as graphs 9 = (V,E), where nodes V correspond to atoms and edges represent connections between them. The Steerable F(3)--qqivariant
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Graph Neural Network (SEGNN) (Brandstetter et al. 2022 processes these graphs, leveraging their geometric and chemical information.
Initial node features. Each atom i E V is ini- tially characterized by feature vector x,. This vector is a one-hot encoding of the atom type (e.g., Hydrogen, Carbon, Nitrogen, Oxygen, Flu- orine). For our SEGNN model, this corresponds to an input irreducible representation (irrep) of 5 x Oe, representing five distinct scalar atom types.
Mo. attribua count
0.42
Rocanct. loss
0.25
Graph connectivity and edge definition. The graph's 's edges are determined using a radius graph approach. An edge (i.5) exists between atom i and atom j if their Euclidean distance within a predefined cutoff radius Teur- In our experiments,
0.002
Aram sype
200
400
500
800
1000
1200
Average MSE per molecule
Figure 14: MSE vs reconstruction error and molecule attributes.
Teut 2.0. The connec-
tivity is stored in edge_ index as per Py Torch Geometric conventions.
Edge attributes. To incorporate 3D geometry in equivariant manner, edges are augmented with attributes derived from the relative positions of the connected atoms (Brandstetter et al. 2022). Specifically, for edge (i.j) connecting atom at position p, E R to atom j at position Pj R
we compute the relative position vector rij P, Рş- These relative position vectors are then
transformed into spherical harmonics up to a maximum degree /edge (we use /edge = 3). We use the mmuz e3nn (Weiler et al. 2018: Thomas al. 2018 Kondor et al.) 2018) library to compute spherical harmonics Y (r.;)). which serve as edge attributes for the SEGNN. The raw relative squared Euclidean distance de, = ||r is also added and passed to the SEGNN as an additional scalar 1 x Oe type feature for each edge.
Node attributes. In addition the initial atom type features, nodes also assigned geometric attributes based on the mean of the spherical harmonic attributes of their incoming edges. These node attributes are processed by the SEGNN.
E.2.3 MODEL IMPLEMENTATION
SEGNN-based encoder. The SEGNN architecture learns equivariant node representations through message passing. Our implementation consists of message passing blocks. Each layer operates on node features represented by a combination of irreducible representations (irreps): we use 100 scalar channels (type Oe irreps). 64 vector channels (type 1o irreps), 16 type 2e tensor channels, and 8 type 30 tensor channels. Equivariant tensor products and gated Sigmoid I. Linear Units (SiLU) non-linearities are used throughout these layers to ensure equivariance. Instance normalization is applied to the features within the SEGNN layers.
After the main message passing sequence, final equivariant projection layer maps the processed node features to target set of di = 512 scalar channels and deg = 512 vector (type lo) channels per node. These structured nove-level features are then processed as:
1. Pooling:
The scalar (0e) and vector (Io) components of the node features are independently pooled across all nodes to obtain graph-level invariant and equivariant features (per-graph: an invariant embedding Spoot E RR52 and an equivariant feature set Vpool consisting of 512 3D vectors). Additionally, the invariant embedding is passed through a Multi-Layer Perceptron (linear layers with ReLU activations) produce a final graph-level invariant latent encoding Zino E 5R512
2. SE(() pose prediction: The equivariant embedding must be processed to obtain group transformations (R, t) E SE).). We pass the pooled equivariant features v poot through an equivariant MI.P to obtain three equívariant 3D output vectors: y1, Y2 and yt. A rotation
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matrix R SO(3) is then derived from y1 and y2 using procedure. The translation vector E R is simply given by yt-
orthogonalization
This construction leads to an embedding Zine invariant to SE(3) transformations of the input graph, and to a predicted pose (R, t) S3) that transforms in equivariant manner.
The decoder reconstructs the molecule from the invariant latent code Zine. It comprises two main components: a position decoder and an atom-type decoder. Both are implemented as MLPs with residual blocks and ReLU activations.
Decoder.
1. The Position decoder takes Xine as input and outputs set of 3D coordinates for a maximum number of 29 atoms in our dataset. This set of 3D coordinates represents the molecule in learned canonical orientation.
2. The Atom-type decoder also takes Zine and predicts the logits for atom types for each of the 29 positions.
The final reconstructed atom positions are obtained by applying the predicted equivariant transforma- tion (R, t) to the canonical positions obtained by the decoder.
Learnable mappings θ and A. During this self self-suuer learning phase we train two separate networks using the computed pseudo-labels as outlined Section 3.3 Both networks use the same architectural pattern as the encoder: an SEGNN backbone to generate invariant graph-level embedding, followed by an MLP head.
The θ network outputs parameters to form the 3 x 3 matrix-Fishen parameter matrix Fpred- The A
network outputs a rotation matrix representing the predicted offset I'.
E.2.4
TRAINING
Training the IE-AE: The encoder and a decoder based on Squuieaquivarian graph neural networks as defined previously were trained for 600 epochs using the Adam optimizer with a learning rate of x 8.89 x 10-5 and a batch size of 128. We compute two loss functions for each of the decoder outputs molecule's 's positions and molecule's 'ss atom types respectively.
positional reconstruction loss, computed as an LI loss (mean absolute error) between the true node coordinates and the coordinates obtained by applying the predicted transformation (R, t) to the decoder's output positions.
An atom-type reconstruction loss, which is cross-entropy loss between the true atom types and the atom types predicted by the decoder.
The loss contribution by the atom-type loss is weighted by factor of R 3.614. We save model weights (both encoder and decoder) corresponding to the lowest validation positional reconstruction loss.
Canonical orientation normalization: Using the frozen pre-trained encoder, we compute the invariant embedding and the observed relative rotation matrix for each training sample (molecule). We then compute the k 25 nearest neighbors for each sample in the n) space based on cosine similarity. Finally, for each sample æ we compute two pseudo-labels for training the A and θ predictors respectively:
0: The set of neighbor rotation matrices (+(x)) (e)))(() is then normalized using the inverse of the centering rotation as (v(x)) [e] From this set of normalized
rotation matrices, we estimate the parameters of the matrix-Fisher distribution (Ø Ê) on these aligned rotations via the moment-matching approach (inverting A(s)
coth(s) - 1/8 by Newton's method and reconstructing via SVD) Wood 2008
Mardia & Jupp 2009). This Ô is the pseudo-label for the @ network.
ı=]; This is the Fréchet mean on 50(3), estimated as the mode of the matrix-Fisher distribution fitted to the set of observed neighbor rotations [v(=)))eNNW) We compute it using the SVD-based moment matching estimator for the mode standard
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in literature (Wood 2008: Mardia & Jupp 2009). This mode [sx] then serves as the pseudo-label for the A network.
Training the learnable mappings θ and A: The θ and A predictor were trained jointly for 150 epochs. The SEGNNs for these predictors use 4 layers, with x0 + 32 1o 8x2e 4x30 hidden irreps, and instance normalization. We minimized the MSE between the network outputs and their respective pseudo-labels. A combined Adam optimizer was used for the parameters of both predictors, with learning rate of approximately 4.83 x 10-1 and a batch size of 128. The loss contribution from the A predictor weighted by factor 500. We save model weights to best validation loss.
For all our trainings, we employ a cosine annealing learning rate scheduler with a warm-up phase of 5 epochs. Gradient clipping with maximum norm of 1.0 is applied during the training of the learnable mappings phase. Training was performed using an NVIDIA A100-SXM4-80GB graphics cards, running for approximately 6 + 1.5 days (IE-AE + learnable mappings phase).
F IMPLEMENTATION DETAILS OF DOWNSTREAM APPLICATIONS
F.1 OOD DETECTION
The distributions recovered by RECON can used to identify objects in unnatural (out-of-disrribution) poses relative to their learned symmetry profile. Given predictors (xx) and Mz) (Sec. 3.3), consider
the absolute pose as Jabs = (bxx) A(x) (normalized relative pose). The likelihood Jads under the distribution parameterized by O serves anomaly score,
8 s(x) == log (((u))
(23)
Low s(z) indicates in-distribution, while high s(x) indicates OOD, We empirically validate this by classifying randomly oriented SO(2) (images) and SO(3) (GEOM) test instances.
Theoretically, one could skip RECON centering and score with an uncentered class-pose density using arbitrary canonicals, i.e., using the score (ze)(z) == log Pog where ((x) trained on pseudo-labels derived from raw relatives (x)) In effect, we show (Proposition B.2 that the log density (and therefore the proposed score s(x)) is invariant the choice of reference frame. However, in practice, centered distributions (RECON) perform better. Centered distributions facilitate optimization: they concentrate probability mass near the identity (reducing variance and dynamic range of the loss), decouple the center I' from the shape parameters (which are learned by +), and overall result in more stable training. On the contrary, uncentered distributions introduce class- specific, arbitrary shifts f [mn] that must be jointly learned by θ, increasing complexity. Empirically this results in consistently higher AUC-ROC across datasets for RECON (Table 5 cf. IE-AE vs. RECON, ROC curve plots in Fig. 15).
We now offer detailed implementation and discussion of the OOD experiment, including dataset. preparation, metrics and uncentered (IE-AE) ablation.
F.1.1 IMAGING
Pre-processing and metries. Each MNIST test image is rotated by random angle 0 ~ u(-1800, 180%.. An input is labeled in-distribution (ID) if its applied rotation lies within the class-specife training symmetry range (digits 0-4: = digits 5-9: =90%), and out-of-distribure (OOD) otherwise. All angular differences are computed on (-180, 180%] we write angdiff (a, B) wrap(a ().
We aim to verify that our anomaly score s(x) assigns larger values to OOD inputs than ID. For this, we report AUC-ROC, Area Under the Receiver Operating Characteristic Curve (using sk sklearn's sklearn. cetrarnmmericcccc
roc_auc scorsc which summarizes performance across all decision thresholds 7 (if we classify as OOD when s(x) r). We use AUC-ROC since it is a threshold-free measure and works well in class imbalance scenarios, a property that other metrics (e.g., accuracy) lacks (they require fixed threshold, whose choice depends the use for deployment, e.g. if one wants to optimize threshold for minimizing false positives). We also provide AUC-ROC curve plots in Fig 15
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Thue Postbvv Rote
Thue Poottive Rote:
2.
2.2
2.4
Thus
RECON IF-AE
RECON
D'
0.2
6-6
D'
Positire
(a) MNIST
(b) FashionMNIST
(c) GEOM-QM9 MMM
Figure 15: ROC curves for IE-AE and ReCON-basel anomaly scores for (a) MNIST, (b) FashionM- NIST, and (c) GEOM-QM9 experiment.
Given z, the model predicts uncentered uniform support ((x) (az, bu) (in degrees). Let the input's relative pose be Irel wrap(bbx))) and the
Uncentered distributions (baseline, IE-AE)
support midpoint Pa = wrap Buthh. ). We use the absolute deviation
(z)) |angdiff (grel, px)
(24)
(Equivalently, one may use distance the predicted support,
(re) (z) max (0, langdiff Mx)] Wex Ws = wrap(b, - az).
(25)
Both are surrogates of the uniform negative log likelihood, differing only by within-support constant.)
Centered distributions (RECON) Let be the predicted centering transformation (in degrees) from Azz) and define the absolute (centered) pose Jabs = wrap gree - -rx). For identity-centered supports, the anomaly score reduces to
(r) = ggabsl-
(26)
For FashionMNIST, mirror the MNIST setup, but assign ID/OOD labels using the FashionMNIST class-specific symmetry ranges specified in Section S
No other changes are made.
F.1.2 GEOMETRIC GRAPHS
Pre-processing and metrics. We start from the (augmented) GEOM-QM9 test set described in Section 5 and group conformers by molecule (SMILES), For each molecule, we randomly split its conformers into two halves: one half kept as in-distribution (ID) and left unchanged; for the other half we generate OOD candidates. This split avoids trivial task: because SO(3) is large, naively applying random rotation to every conformer would produce an overwhelming fraction of OOD examples.
For OOD candidates, we first canonicalize by undoing the augmentation used to create the conformer (apply the rotation used to create the augmented sample Raug. to bring coordinates to the original canonical frame), and then apply a fresh random rotation Reandidate (aar(SO(3)) to obtain the new coordinates. ID examples are those left unmodified.
Ground-truth labels for the candidate half are decided through likelihood: if the sample rotation Reandidate has high likelihood of coming from F (the molecule's matrix-Fisher parameter that was assigned during the creation of the augmented dataset), then that sample labeled as ID. Otherwise, the sample is labeled OOD.
Note that since the matrix-Fisse Fisher density on SO(3) is proportional to exp(tr(F" RR)). then the unnormalized likelioid of Reandidate tr FT Reandidate) up to an additive constant. Therefore, we mark a candidate as OOD if tr (F Reandidate) < r (low likelihood), where 7 is a fixed threshold controlling class balance (we use 96).
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Uncentered distributions (baseline, IE-AE) Let (xx) SO(3) be the predicted pose and let F(x) E R3x3 be the matrix-Fishhe parameter predicted from x using the learned mapping. We score anomalies with the negative unnormalized log likelihood
(x) = - ((F(2), v(x)) (F(((()) ((x)),
where (= B). When F(x) is uxis-symmetrie of the form F(z) = <(x) pe with mode [4x ESO(3), (x) a monotone surrogate of the geodesic deviation
Centered distributions (RECON) define the absolute (centered) pose
Let A(x) E SO(3) be the predicted centering transform and
Jabs = A(x))
We then score
s(x) (F(),, Jabs) = tr((((())) gabs): a monotone surrogate of the geodesic angle @ (gabs).
F.2
GRANTING GROUP INVARIANCE TO PRE-TRAINED BACKBONES
F.2.1 IMAGING
We evaluate inference-nnly canonicalization using classifier (ResNet18 backbone) trained on the raw (no augmentations) MNIST and FashionMNIST datasets. We then evaluate the performance of this pre-trained model on the rotated dataset variations created for the symmetry discovery experiment outlined in Section 5ection55ttha (that is, + 100 rotations for MNIST, etc). For both MNIST and FashionMNIST, we train a ResNet-18 with cross-entropy and Adam optimizer (Ir=le 3, batch size 128, 100 epochs) and save the best classifier checkpoint based on best validation accuracy. At test time, we compare three input pre- processesssngg modes before feeding images into the classifier: (i) no canonicalization (using the pre-trained classifier), (ii) arbitrary canonicalization (taken from the IE-AE) and (iii) RECON canonicalization. As a reference value, we additionally report accuracy on the original (un-augmented) test set (Table 2). which matches the training distribution by construction. This corresponds to a perfect canonicalization (reversing the augmentations), and serves as an upper bound of the best accuracy that can be obtained if we provide perfect canonicalization function during inference.
Comparisons We compare with EquiAdapt (Mondal et al. 2023), another test-time canonical- ization method. We reproduce their zero-shot setup, that is, only the canonicalization function is trained, and the pre-trained model weights are kept frozen (same setting as RECON). We use the SO(2) canonicalization prior and steerable network from their public equiadapt package for the canonicalization function. The canonicalization function is attached on top of the pre-trained ResNet18 classifier, and is trained for 50 epochs with batch size of 128. We choose the best hyperparameter configuration based on best test accuracy (see hyperparameter search in Figure D.
For reference, Table reports results for specialized group equivariant (and partially equivariant) architectures trained from scratch on the same non-rotated datasets. Equivariant models are by construction designed be robust to symmetry-inducee our canonicalization (RECON, 90 .6%%) shows a moderately small gap w.r.t. the -qquiqaauia ESCNN classifier (94.72%), while keeping the benefits of operating purely as unsupervised, test- time, plug-and performs better than the fully equivariant ESCNN. Lastly, we consistently outperform Partial G-CNNs in both datasets, sometimes by large margin (67.72% vs 81.96% in FashionMNIST).
test-time distribution shifts. On MNIST,
module for arbitrary classifiers. On FashionMNIST however, RECON actually
The ESCNN classifier built using the same architecture as the ESCNN backbone for the IE- AE described in Appendix E. but with a 2-layer MLP classification head attached at the end. The invariant embedding dimension before the classification head of size 1024 and the ESCNN backbone has hidden dimension of 128, totaling 3, 384, 463 trainable parameters in this case. We train for 50 epochs with batch size 128 and learning rate 1e 3. Hyperparameter configuration was chosen based on best test accuracy across different learning rates, hidden dimensions and embedding dimensions (Table The Partial G-CNNs were configured and trained similarly, aiming to keep an approximate same number of parameters as the ESCNN architectures.
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Table 2: Test accuracy on the in-distribution test (i.e., non-augmented) obtained with the pre- trained classifier trained on non-rotated, vanilla datasets: reference canonicalization upper bound representing a perfect canonicalization function.
Dataset
Augmentation--eevensee test set acc. (upper bound, perfect canon.)
MNIST
98. 14 = 0.02%
FashionMNIST GEOM-QM9
91.11 0.1%
96.40 + 0.0%
Table 3: Test accuracy on the per-class rotated datasets obtained with fully and partially equivari- classifiers trained on non-rotated, vanilla datasets; auxiliary reference metric representing the performance obtained with specialized equivariant architectures that are, by construction, robust to symmetry-induced distribution shifts during inference.
Dataset (Train Test)
Group
Equivariance
Backbone
Test set acc.
MNIST
Full Learned
ESCNN
94.59%
SE(2)
(Orig / Rot)
Partial G-CNN
90.20%
FaSHi
Full
ESCNN
77.94%
SE(2)
(Orig / Rot)
Learned
Partial G-CNN
67.72%
GEOM-QM9
SE(3)
Full
SEGNN
98.55%
(Orig Rot)
F.2.2 GEOMETRIC GRAPHS
We evaluate inference-only canonicalization GEOM-QM9 molecular graphs. We first train graph CNN on aligned conformers only: a layer GCN (using Pytorch Geometric's 's GCNConv layer with 128 hidden size) followed by two layer MI.P head (64 hidden units, dropout 0. trained with cross-entropy and Adam (batch size 128 for 50 epochs), saving the best checkpoint by validation accuracy. Node inputs to the classifier spherical harmonic attributes their incoming edges (see details in data pre-pre pre-prceessing section of GEOM-QMS dataset in Appendix E.2.2p. At test time, evaluate the rotated conformer test set described in Appendix 1 and compare three input pre-processing modes applied per graph before classification: (i) none (no canonicalization, using the pre-trained classifier), (ii) arbitrary canonicalization (taken from the IE-AE) and (iii) RECON canonicalization. Note that after rotation by either canonicalization method, we have recompute the geometric features (edge/node attributes) from the updated coordinates, since those are the input to the pre-trained GCN, For reference on the upper bound of perfect canonicalization, we also report accuracy on the test dataset obtained after reversing the test-set augmentations (Table 2 All runs use a fixed global seed with deterministic settings.
geometric attributes based on the mean of the
Comparisons For comparison against EquiAdapt (Mondal et al. 2023 we attach and train their continuous SO(3) point-cloud canonicalizer from the public equ i aadapt package on top of the pre-trained GCN. For each input molecular conformation, we canonicalize its atomic coordinates with the canonicalizer, and then compute the resulting conformation's geometric features (as in Appx. (.2.2) for input into the frozen GCN (same approach as in our test-time canonicalization). We use Adam with learning rate Te-3, batch size 128, and train for epochs. During training, the GCN weights are frozen and only the canonicalizer is optimized (coined the zero-shot setup in EquiAdapt (Mondal et al.) 2023), equivalent to our test-time canonicalization setup), as opposed to the alternative fine-tuning setup in which the pre-trained model is retrained.
On GEOM-QM9, the SO(3) (he)sSO33-euuaavaii equivvarat SEGNN reaches 98.55% accuracy (Table 3). far above the pre-trained non-equivariant baseline (with or without canonicalization). This evidences that test-time canonicalization significantly more challenging in the 3D domain than in images (at least for molecular conformations with varying per-molecule rotations). Nevertheless, RECON is the
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Table EquiAdapt hyperparameter search and final accuracy on the per-class rotated MNIST, FashionMNIST and GEOM-QM9 dataset variations.
Test acc. (MNIST)
Test acc. (FashionMNISTT
Test acc. (GEOM-QM9)
Learning rate
Prior weight
le-4
0.5
89.05%
62.22%
40.12%
le-4
1.0
91.60%
54.85%
42.63%
le-4
10.0
93.83%
70.30%
49.41%
le-4
50.0
94.05%
78.28%
39.15%
le-4
100.0
92.97%
79.00%
46.11%
3e-4
0.5
92.39%
59.37%
37.54%
3e-4
1.0
92.41%
65.35%
43.82%
3e-4
10.0
94.09%
71.76%
44.81%
3e-4
50.0
94.67%
75.96%
46.75%
3e-4
100.0
94.86%
78.50%
40.64%
le-3
0.5
92.17%
54.75%
43.22%
le-3
1.0
92.02%
63.86%
42.57%
le-3
10.0
94.76%
71.92%
48.17%
le-3
50.0
94.48%
77.82%
46.78%
le-3
100.0
95.00%
78.13%
47.28%
5e-3
0.5
92.47%
54.61%
45.33%
5e-3
1.0
92.77%
64.66%
52.19%
5e-3
10.0
93.82%
72.13%
47.50%
5e-3
50.0
94.74%
78.19%
53.81
5e-3
100.0
94.70%
79.13%
48.72%
Table 5: ESCNN hyperparameter search on MNIST and FashionMNIST. Test accuracy on the per-
class rotated test sets.
Emb. dim
Hidden dim
Learning rate
Test acc. (MNIST)
Test acc. (FashionMNIST)
512
128
le 3
91.76%
512
128
3e
93.1 1..1%% 1%
77. 77.13%
512
192
le-3
92.87%
75.61 %
512
192
3
92.11 %
76.06%
1024
128
1e-3
94.59%
77.94%
1024
128
3e 4
94.02%
76.25%
1024
192
le 3
93.99%
73.52%
1024
192
3e 4
91.93%
75.94%
only test-time canonicalization that offers improvements over the baseline, with both EquiAdapt and IE-AE reducing performance over the pre-trained GCN. The SEGNN classifier is built using the SEGNN backbone as described in Appendix E.2.3 and was trained for 50 epochs with batch size 128 and learning rate le 3.
G STATEMENTS
G.1 USE OF LARGE LANGUAGE MODELS
Large language models were used aid in writing (polishing text), retrieval of related work, generat- ing code for plots, and implementing standard components.
G.2 REPRODUCIBILITY STATEMENT
The experiments in this paper can be reproduced using the code provided in the repository at https: htt..//ttttttttceee
schub.com/ Cwt/////-TTt//eeoo.
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G.3 ETHICAL CONCERNS
Because our method relies on a data-driven approach to identify natural poses, it may be susceptible to dataset bias. Beyond this, do not anticipate significant ethical concerns negative societal impacts.
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Reviews:
[{"review_id": "XNwg3bHk3l", "reviewer": "Reviewer_jX6K", "summary": "The authors introduce a method for identifying the natural pose of a class in an unsupervised way. In contrast to class-pose methods such as IE-AE, the proposed approach learns a data-aligned canonical. This is achieved using the machinery of IE-AE. Embeddings of IE-AE suggest transformation distributions for dataset instances in connected components. The natural pose is defined as the center of this distribution and obtained using the Frechet mean. The authors argue that a data-aligned canonical is important for various downstream tasks and demonstrate its utility for OOD detection and test time canonicalization.", "questions": "At this phase of the review process I do not have clarifying questions.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "**Originality.** The work appears original. \n \n**Quality.** I rate the quality of this work as good. The work is well motivated, the proposed approach addresses the gap in the literature, and the empirical analysis support the paper claims.\n \n**Clarity.** The organization is good, and the writing is very good. \n \n**Significance.** I rate the significance of this work as good. The work has the potential for broad applicability.", "weaknesses": "No notable weaknesses", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SdyEVCX8JB", "reviewer": "Reviewer_ff67", "summary": "The paper introduces RECON, an unsupervised framework for discovering instance-specific symmetries from data.\nRECON builds upon class-pose decomposition methods (e.g., IE-AE) and addresses the key limitation of arbitrary canonical poses by normalizing them into natural orientations through an estimated Tukey-Fréchet mean of relative transformations.\nThis canonical orientation normalization yields interpretable, data-aligned symmetry distributions and enables downstream applications such as (i) OOD pose detection and (ii) test-time canonicalization that grants invariance to pretrained models without retraining.\nExperiments on 2D image datasets (MNIST, FashionMNIST) and 3D molecular data (GEOM-QM9) demonstrate the effectiveness of the proposed symmetry discovery method itself as well as its benefits in downstream tasks.", "questions": "- The equivalence class construction plays a crucial role in the proposed pipeline. Could the authors also provide hit rate results in the image domain?\n\n- Could the authors further discuss the practical utility of discovering symmetry distributions?\nIt seems that the most tangible application demonstrated is test-time canonicalization, yet other works such as [3] and [4] can achieve a similar effect given a known group structure, without explicitly estimating instance-level distributions.\n\n- In the implementation, does each equivalence class correspond to the entire semantic class, or is it constructed per-sample with a fixed number of neighbors (e.g., 25 as mentioned)?\n\n- How would the method perform on real-world datasets where each object appears at a unique orientation and images are not repeated across poses, unlike the synthetic settings used in this paper?\n\nReference:\n\n[1] Generative Adversarial Symmetry Discovery.\n\n[2] Latent Space Symmetry Discovery.\n\n[3] Affine Steerable Equivariant Layer for Canonicalization of Neural Networks.\n\n[4] Equivariant Adaptation of Large Pretrained Models.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The approach discovers symmetry distributions from unlabeled data and provides meaningful, identity-centered canonicalizations that align with intuitive notions of natural pose.\n\n- Interesting and elegant idea: The proposed Tukey-Fréchet-mean–based canonical orientation normalization provides a simple yet effective solution to the issue of arbitrary canonicalization in class-pose methods. The modeling is conceptually clear, mathematically grounded, and intuitively appealing.\n\n- RECON can grant group invariance to pretrained models without retraining, functioning as a plug-in canonicalization step that applies irrespective of model architecture. This design significantly enhances the method’s usability and potential impact.\n\n- The paper is clearly structured and communicates its ideas effectively, making both the theoretical formulation and practical implications easy to understand.", "weaknesses": "- Inaccurate contribution claim. The paper claims to “-for the first time– extend symmetry discovery to 3D groups,” but prior works such as [1] and [2] have already explored 3D group symmetries.\n\n- The current method is restricted to known transformation groups, primarily rotations (SO(2) and SO(3)), limiting its applicability to more general groups or unknown symmetry settings where the underlying group structure must be inferred.\n\n- The method assumes that variations within each equivalence class are small and can be cleanly separated from group-induced transformations. It remains unclear how reasonable this assumption is in more complex, natural data.\n\n- Computational cost and scalability: The reliance on k-nearest-neighbor search in the invariant space for each input (Algorithm 1) may be computationally expensive for large-scale or high-dimensional datasets.\nThe paper does not discuss computational cost or whether neighbor computations can be cached or reused during training and inference.\n\n- Some key quantitative results (e.g., Figure 6) appear to be based on single runs without repeated trials or statistical analysis, making it difficult to assess robustness and reliability.\nMoreover, the results in Figure 6(d) show a noticeable gap from the so-called upper bound in Table 1.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YKgaL02Vzq", "reviewer": "Reviewer_BHNi", "summary": "The paper introduces a method for unsupervised discovery of instance-specific symmetry distributions in data. Building upon class–pose decomposition frameworks (like Invariant-Equivariant Autoencoders), it corrects arbitrary canonical poses by estimating a Fréchet mean offset and applying a canonical orientation normalization. This yields consistent, data-aligned canonical representations and enables applications such as OOD pose detection and test-time canonicalization. The method is validated on both 2D datasets (MNIST, FashionMNIST) and 3D molecular datasets (GEOM-QM9).", "questions": "- What is the training overhead and inference time for RECON’s symmetry detection/OOD pose detection/ test-time canonicalization/reconstruction?\n\n- There are many learning-based methods that take take 3D object model as input and model symmetry pattern/distribution in SO(3), e.g., Implicit-PDF[1], Alignist[2]. Could you explain if RECON can be extended to a more generalized setup like the above-mentioned methods, and what would be the main advantage of RECON compared to those methods if the same datasets apply (for example, Symsol)? \n\nReferences:\n\n[1] Murphy, K.A., Esteves, C., Jampani, V., Ramalingam, S., Makadia, A.: Implicitpdf: Non-parametric representation of probability distributions on the rotation manifold. In: Proceedings of the 38th International Conference on Machine Learning. pp. 7882–7893 (2021).\n\n[2] Vutukur, Shishir Reddy, Rasmus Laurvig Haugaard, Junwen Huang, Benjamin Busam, and Tolga Birdal. \"Alignist: CAD-Informed Orientation Distribution Estimation by Fusing Shape and Correspondences.\" In European Conference on Computer Vision, pp. 351-369. Cham: Springer Nature Switzerland, 2024.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The canonical orientation normalization via the Fréchet mean is simple yet theoretically grounded and effective for unsupervised symmetry discovery.\n\n- The paper provides rigorous proofs and clear geometric intuition, which is scalable to both SE(2) for imaging and SO(3) for geometric graphs.\n\n- This method can be applied to any invariant-equivariant backbone, making it broadly useful.\n\n- The test-time canonicalization and OOD detection tasks are compelling, showing both research and applied potential.", "weaknesses": "- Lack of real-world validation in the experiments. Most experiments involve synthetic rotations or clean 3D datasets; robustness under complex, cluttered scenes for real-world objects, for example, objects in 2D images in the wild, like COCO/ImageNet, or scanned 3D objects in OmniObject3D/GSO. \n\n- The sensitivity to hyperparameters (e.g., k-nearest neighbors in class construction) could be explored more systematically.\n\n- Lacks direct quantitative comparisons with very recent equivariance-learning approaches (e.g., Partial G-CNNs, VP-GCNNs) beyond conceptual discussion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tkwG7d2rkK", "reviewer": "Reviewer_XXPp", "summary": "This paper seeks to build on IE-AEs by applying canonical orientation normalization to obtain a different equivalence class than with a typical IE-AE. The authors address instance-specific symmetries, such as rotations in images. The authors also present test-time canonicalization, which allows pre-trained models to be granted invariance with respect to a group action. The authors experiment with a couple of image benchmark datasets, as well as geometric graphs, with SO(2) and SO(3) symmetries.", "questions": "1. Near line 150, the authors write: \"In effect, we rely on the principle that structurally similar objects generally map to nearby points in Z.\" Is this really true in general? The authors mention empirical evidence, but what symmetries have been considered in these experiments? What does \"nearby\" mean in the G-invariant latent space? I think we need to be more precise about what this d_Z norm-induced distance is, or perhaps prove that *any* norm-induced distance suffices.\n\n2. What kind of symmetries can you learn? (It seems like the bottle-neck could be the G-invariant encoder, which may be limited in the types of symmetries it can discover.) It seems only rotations are experimented with. The claim is that the method is quite general, but a claim that *any* symmetry can be discovered is quite doubtful, particularly in light of the lack of experimental evidence.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors demonstrate the potential advantages of their approach over the most comparable baselines.\n\n2. The presented method appears to be novel.", "weaknesses": "1. *The symmetries considered are limited.* The groups SO(2) and SO(3) do not constitute a diverse collection of symmetries, as lines 348-354 suggest. \n\n2. *This is not the first work to consider 3-d symmetry,* contrary to the claim made in the abstract. In fact, this statement can mean one of two things: (1) the symmetry group is 3-dimensional (perhaps it is a Lie group); (2) the group action acts on a three-dimensional space. In either case, the claim made by the authors would be false. In fact, I do not believe that this work is the first to consider SO(3) symmetry: for example, see \"Image to Icosahedral Projection for SO(3) Object Reasoning from Single-View Images\" (Klee et al., 2022).\n\n3. *There are a few key missing references.* In particular, the idea of using the Lie derivative to discover symmetry (\"A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning\" by Otto et al., 2023) and to subsequently use the discovered symmetries to construct an invariant feature space (\"Symmetry Discovery Beyond Affine Transformations by Shaw et al., 2024) seems closely related to the present method and should be discussed, if not experimentally compared with. Another key reference in symmetry discovery is \"Learning Infinitesimal Generators of Continuous Symmetries from Data\" by Ko et al., 2924.\n\n4. *More experimental comparison would be helpful.* I appreciate the experimental comparison which has been made, but it would be nice to see a comparison against a representative sample of other methods seeking to enforce invariance, such as an equivariant NN or else a method or two mentioned in weakness 3.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)