Take Note: Your Molecular Dataset Is Probably Aligned
Authors: Peter Lippmann
Roman Remme
Manuel V. Klockow
Fred A. Hamprecht
Venue: ICLR 2026 Poster (2026)
Keywords: molecular machine learning
datasets
orientation bias
equivariance
3D orientation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Massive training datasets are fueling the
astounding progress in molecular machine learning.
Since these datasets are typically generated with computational chemistry codes which do not randomize pose, the resulting molecular geometries are usually not randomly oriented. While cheminformaticians are well aware of this fact, it can be a real pitfall for machine learners entering the burgeoning field of molecular machine learning.
We demonstrate that molecular poses in the popular datasets QM9, QMugs, and OMol25 are indeed biased. While the fact can easily be overlooked by visual inspection alone, we show that a simple classifier can separate original data samples from randomly rotated ones with high accuracy. Second, we empirically validate that neural networks can and do exploit the orientation bias in these datasets by successfully training a model on chemical property prediction using molecular orientation as _sole_ input.
Third, we present visualizations of all molecular orientations and confirm that chemically similar molecules tend to have similar canonical poses. 
In summary, we recall and document orientation bias in the prevalent datasets that machine learners should be aware of.

Published as a conference paper at ICLR 2026
TAKE NOTE: YOUR MOLECULAR DATASET Is PROBA- BLY ALIGNED
Peter Lippmann, Roman Remme, Manuel V. Klockow & Fred A. Hamprecht Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 69120 Heidelberg. Germany {peter. 1i pippann,, roman. remme, manuel, k1 klockow fred. hamprecht) @iwr. www-uniidedeeerreeeeeee de
ABSTRACT
Massive training datasets are fueling the astounding progress in molecular ma- chine learning. Since these datasets typically generated with computational chemistry codes which do not randomize pose, the resulting molecular geome- tries usually not randomly oriented. While cheminformaticcans this fact, it can be a real pitfall for machine learners entering the burgeoning field of molecular machine learning. We demonstrate that molecular poses in the popular datasets QM9, QMugs, and OMol25 are indeed biased. While the fact can easily be overlooked by visual inspection alone, we show that simple clas- sifier can separate original data samples from randomly rotated ones with high accuracy, Second, we empirically validate that neural networks can and do exploit the orientation bias in these datasets by successfully training a model on chemical property prediction using molecular orientation as sole input. Third, we present visualizations of all molecular orientations and confirm that chemically similar molecules tend to have similar canonical poses. In summary, we recall and doc- ument orientation bias in the prevalent datasets that machine learners should be aware of.
well aware
INTRODUCTION
Machine learning has become a well-established tool for designing, discovering, and studying molecular systems, for instance in drug discovery, materials science and physical chemistry. Much of the progress in the field is enabled by the curation of large-scale datasets that provide accurate molecular properties. Computational chemistry codes used in the underlying data-gen nata-generating processes usually do not generate molecular geometries in random orientations. At the same time, handling the arbitrariness of coordinate systems by incorporating symmetries into machine learning models has become central theme in geometric deep learning. S0(3)-equivariant neural networks guarantee well-defined transformation behavior under rotations. In other words, equivariant models produce consistent predictions for inputs that differ only by rotation (or equivalently, by the choice of reference frame). Consequently, equivariant architectures are agnostic to the orientation of molecular geometries in ML datasets. However, most existing equivariant architectures rely non-standard building blocks, such as specialized normalization layers, nonlincarities, and tensor operations, which often computationally demanding (Passaro & Zitnick] 2023) and can be challenging to tune in practice (Abramson et al. 2024] Pertigkiozoglon et al. 2024 Elhag et al., 2024]. As a consequence, while exact equivariance desirable in principle, softening the constraint by learning approximate symmetries or breaking built-in equivariance is (re- mee)eemegging trend in molecular machine learning (Langer al.
2024 Fissler et al., (202).. fueled by prominent
examples such as Alphafold 3 Abramson et al. 2024). Other recent examples include Wang et al.
(2023b 2025); Zhang et al. (2025): Joshi et al. (2025). In that case, possible orientation bias in molecular datasets might affect machine learning workflows. Building on and extending the recent work by Lawrence et al. (2025a), we show that molecules in some of the most popular molecular datasets, including QM9, QMugs and OMol25, are by default not presented in random orientations, and we discuss implications for machine learning practitioners.

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Figure 1: Molecules in many popular molecular datasets (here: QM9) not randomly ori- ented. Structurally similar molecules are oriented similarly. 2D visualization of normalized principal components (PCs) of all QM9 molecules reveals clear non-uniform structure. Orientation reference frames correspond to triplets of points in the 2D area-preserving (Mollweide) projection (blue PCI, yellow PC2, magenta PC3). Left and right: two groups structurally similar molecules that share nearly the same PC orientation (left orientation O right orientation A).
To date, many architectures for molecular machine learning are benchmarked on the widely used QM9 dataset (Ruddigkeit al., 2012] Ramakrishnan et al., 2014), a collection of around 134,000 small organic molecules with up nine heavy atoms. While QM9 is the gold standard for property prediction on smaller molecules, the QMugs collection (Isert et al.] 2022) comprises quantum mechanical properties for nearly million larger drug-like like molecules with up to 100 heavy atoms extracted from the ChEMBL database (Mendez et al., 2019). The OMol25 dataset (Levine et al.) 2025) combines broad chemical diversity with a high level of accuracy at unprecedented scale (100M systems), comprising not only neutral organics but also biomolecules, metal complexes, and electrolytes, spanning 83 different elements.
We want raise awareness of the fact that molecules in ML datasets are oriented randomly for two crucial reasons. First, non-equivariant architectures trained on these datasets must employ explicit data augmentation; otherwise, their test performance will degrade significantly when evaluated on randomly oriented molecules. Second, orientation bias may have an undesired effect on the performance of architectures that introduce symmetry breaking or are only approximately equivariant (Wang et al., 2023b) Pertigkiozzolou et al. 2024, Kaba & Ravanbakhsh [024] Elhag al.) 2024 2025] Lawrence al. [2025b]: for instance, exploiting spurious information contained in biased orientations of molecules may artificially inflate the performance of non-equivariant models. addition, when molecular properties such as the ground state electron density are evaluated grid that is not spherically symmetric (Brockherde 2020 Jørgensen & Bhowmik 2022] Li et al.) 2025), the canonical orientation may introduce systematic bias in the grid orientation, even when equivariant architectures used later in the prediction.
al. 2017; Bogojeski et al.,
In this paper, we make the following contributions: we demonstrate, using QM9, QMugs, and OMol25 prominent examples, that molecules in many popular ML datasets are not randomly ori- ented by training a simple classifier that distinguishes between rotated and unrotated samples with very high accuracy. We show that the accuracy remains high even when the default atom positions are perturbed with substantial noise and random rotations of up to 90". Further, we demonstrate that neural networks can leverage the canonical orientation to achieve artificially high accuracy in extreme scenario: using only the normalized principal components of atom positions in- put, we regress molecular properties and observe performance on the three standard datasets that exceeds the best possible accuracy expected for randomly oriented data. Lastly, we visualize the orientations of all molecules in these datasets and show that chemically similar molecules tend to be oriented similarly (see Fig. []. We make code publicly available toolbox to visualize and quantify orientation bias in molecular datasets at https: gig thub. con/ sciai-iiiiii aacciillabbl lab/ are-my-mo lecu les les-aligned igned
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2 BACKGROUND AND RELATED WORK
Formally, a function o V + W between vector spaces and W, is equivariant under a group G if Pout (g) vp(x) (P (g) x) for all G and E V. Here, Pln Pout group representations on V and W, which define how elements of G act on V and W, respectively. In diagrammatic form, equivariance means that the following commutes:
(xx)
(a)
(oma(o)
(0xx')
Several approaches to incorporating exact SO(3) eequivariie into neural networks exist. Most famously, tensor field networks tensorial features in all hidden layers to maintain well-defined
transformation behavior throughout the network (Thomas et 2018 Geiger & Smidt 2022 Batatia
ct al. 2022: Liao al., 2024; Aykent & Xia 2025). Similarly, specialized architectures exist to
use elements of the projective geometric (or Clifford) algebra to achieve Euclidean rotation and translation equivariance (Brehmer al., [2023] Ruhe et al. (023). In contrast, canonicalization
approaches avoid the need for specialized architectural building blocks and use canonical reference frames to guarantee exact equivariance (Puny et al. [022 Pozdnyakov & Ceriotti 2024] Spinner et al., 2025; Lippmann et al.) 2025). Data augmentation offers simple and practical alternative where equivariance is learned approximately by presenting the network with randomly rotated inputs (and targets). It is an open research question whether built-in equivariance is favorable over data augmentation, and a fair comparison is often non-trivial (Brehmer et al. 2024). Some evidence points to the superiority of non-equivariant architectures (Langer al. 2024 Lippmann et al. 2025
possibly due to their less constrained design space.
In their recent work, Lawrence et al. 2025a) introduced the idea that orientation biases exist in
molecular datasets. While the focus of their analysis lies on the detection of orientation bias, we here introduce a set of alternative and complementary methods systematically study the distribution of molecular orientations.
Related broken symmetries in geometric datasets, there are several systems of interest in which symmetries are spontaneously broken, such as dynamical phase transitions (Back et al. (017] or po-
lar fluids (Gib et al. 2024). For these tasks, networks with exact, built-in equivariance might be too constrained and functional symmetry breaking (Wang et al., (223a) offers a possible solution (Wang) jet al. 2022). The authors of (Wang et al. 2023a) further introduced the notion of distributional symmetry breaking. The term applies to data distributions under which a data sample and its trans- formed version are not equally likely, such as particle collisions recorded at particle colliders, where the beam axis singles out a preferred direction (Favaro et al. (2025). Orientation bias in molecular datasets can be seen as another instance of distributional symmetry breaking w.r.t. the symmetry group of rotations.
2.1 RELATED WORKS THAT MAY BE AFFECTED BY ORIENTATION BIAS
As pointed out above, one subtle example where an orientation bias may easily be overlooked is charge density prediction. In the field, several standard benchmarks exist that store DFT ground state electron densities on regular Cartesian (axis-aligned) grids as training targets (Brockherde et al.,
2017 Bogojeski al. 2020 Jørgensen & Bhowmik 2022 Li al. 2025
We have investigated all four datasets and found that they are indeed strongly canonicalized (see App. E.Ip. The combined datasets (Brockherde et al. 2017 et al. (202)) contain molec- ular dynamics trajectories for six different molecules (around 1000 data samples each). The datasets (Jörgensen & Bhowmik 2022 Li et al. (025) both contain electron densities evaluated on regular Cartesian grids for all QM9 molecules. The two datasets differ in the DFT software used to calculate the reference electron density: VASP in (Joggesen & Bhowmik (022) and PySCF in (Li et al. 2025). However, both datasets use the standard QM9 dataset which exhibits orientation bias (cf. Fig. D. Since the volumetric grids are not spherically symmetric, the evaluation the loss function is not ontationally-invarian
which may introduce systematic bias even when equivariant architectures are used later in the prediction. We can think of two scenarios in which such biases may lead problems: (a) when practitioners deploy trained models, e.g. in MD simulations, they
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(a) QM9
(b) QMugs
(c) OMol25 (centered)
Figure 2: Overlay of 100 random molecular geometries from QM9, QMugs, and OMol25 re- spectively. QM9 shows strong alignment of the bond adjacent the origin with the Cartesian y-axis (yellow). Bulges of QMugs and OMol25 molecules exhibit less structure, but are not spheri- cally symmetric.
will. most likely, rotate the simulated molecules into preferred orientation, which may decrease the performance of the model; and (b) when studying extrapolation capabilities of trained models, e.g. for larger molecules, the generalization to new datasets may be hindered by orientation biases.
Furthermore, mentioned previously, using non-equivariant architectures for (equivariant) molec- ular tasks has emerged as recent but growing trend. Some of the proposed architectures do not use explicit data augmentation, making them susceptible to orientation biases in molecular datasets; for instance, the authors of (Pertigkiozoglou et al. 2024) relax the equivariance constraint the inter- mediate layers of networks during training by introducing progressively constrained until one arrives at a fully equivariant solution. Therefore, orientation bias in the datasets used (e.g. in MD17, trajectory of such networks. The authors of (Elhag et al.] 2024) 2025) present machine learned in- teratomic potentials (MLIPs) trained to learn rotational equivariance approximately by minimizing an additional loss. The MLIPs are trained on OMol25 and the MD17 dataset, which can both be shown to exhibit strong orientation bias (cf. Fig. 5p and Fig. IT). which may influence the train- ing and evaluation of these models. Motivated by the "bitter lesson" SSutton (2019), the conformer generation model presented in (Wang al.) 2023b) based on an efficient and scalable di diffusion model that operates directly on 3D atomic positions without enforcing rotational equivariance. The authors conduct experiments on QM9 and the strongly aligned GEOM dataset (cf. Fig. 7 Notably, the authors observe that randomly rotating their training prior to training negatively impacts their performance and hypothesize that the reason may be that "DFT simulations used to generate the data might be implicitly encoding canonical coordinate system, which affects generalization if broken" (Wang et al.) (023b) (p. 8).
additional non-equivariaat term that is
Fig. 11 could have undesired effect the optimization
INVESTIGATING ORIENTATIONS IN MOLECULAR DATASETS: METHODS, RESULTS, AND IMPLICATIONS
Clearly, the first step that comes to mind when investigating the orientations of molecular geometries visually inspect the 3D geometries for obvious alignment. Figure shows 100 randomly sam- pled molecular geometries from each dataset. For QM9, clear structure is visible. Most strikingly, the first bond (adjacent to the origin) almost perfectly aligns with the Cartesian y-axis. The original QM9 paper (Ramakrishnan et al. 2014) invoked the cheminformatics tool Corina (version 3.491, 2013) (Sadowski Gasteiger] 1993) to generate 3D structures from SMILES strings. The geome- tries were then relaxed using Kohn-Sham DFT caleulations at the B3LYP/6-3100 G(((ff..)) level. The Corina algorithm detosed-suuree is likely responsible for the alignment with the y-axis, while the subsequent geometry relaxation softens the strict alignment. For QMugs and OMol25, similarly
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Figure 3: Canonical orientations in QM9, QMugs, and OMol25 are highly consistent and de- tectable. A simple geometric message passing network accurately discerns canonical samples from randomly rotated ones, even if the atom positions are previously perturbed and the molecule ran- domly rotated by an angle up to a.
distinct structure is visible regarding the orientation, but the central bulge of molecules is clearly not spherically symmetric, hinting at systematic orientation bias.
3.1 LEARNED DETECTION OF DEFAULT ORIENTATIONS
To empirically validate that molecules in ML datasets exhibit orientation bias, we have trained simple message passing network to distinguish between molecules in their original ("canonical") orientation and ones that have been randomly rotated. If the dataset were truly orientation invariant, such a classification task would be impossible beyond random guessing. This test is known in the literature classifier two-sample test (Lopez-Paz & Oquab 2017) and has previously been applied to molecular datasets by Lawrence et al. (2025a).
For each sample in the datase we randomly decide whether to apply a global rotation, sampling a rotation matrix uniformly from SO(3). To ensure that the learned detection is not just based on simple geometric pattern (such particular edge being aligned with coordinate axis, cf. Fig. 2)
we apply Gaussian noise (with standard deviation ó up to Á) to the default atomic positions as well as random rotations up to a maximum angle a (before applying the uniformly sampled rotation that should be detected by the network). A noise level of 5 = 1 Áis quite substantial, considering that the length of carbon-carbon single bond is 1.5 Å, and shorter for a double or triple bond. The network is then trained to minimize binary cross-entropy loss, predicting whether the input molecule has been randomly rotated or is in its (perturbed) default pose. We employ a straightforward point cloud architecture, consisting of three layers of message passing:
(R++11)
MLP(F emb(z; x2)). JEN(()
where ((()) denotes the feature vector of atom i in layer k; 4; its position, an) its neighborhood (defined by radial cutoff of 10 Å). This network is neither rotation-equivariant. nor invariant, but petttion-dependent by design. During message passing, the angular and radial part of the relative distance vectors I; I are embedded using Gaussian basis functions. Prior to these message passing layers, we combine learned embedding of each atom's neighborhood geometry with an
embedding of the atom type to initialize the node features p!()) We use the same architecture for all three datasets, see App. for details.
The results, summarized in Fig. 3 reveal that even simple classifier can discern randomly rotated samples from canonical ones with very high test accuracy, even when the default atom positions are substantially perturbed. This provides clear evidence that molecules in the considered datasets are not randomly oriented, and that the canonical poses are highly consistent and detectable.
For all our experiments, we use the QM9 dataset readily available in PyTorch Geometric Fey & [ Lenssen 2019), QMugs from httph::t able through the fai rchem http://ayyyooo///gggoooooooooooe
and OMol25 avail-
rccher
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3.2 QUANTITATIVE ORIENTATION ANALYSIS
To systematically study the orientation of molecules in the respective datasets, we devise a map- ping Ωt from the set of molecular geometries M =f) to the set of orientations SO(3). For each molecule, consisting of atoms A with charges 2u and positions Ta it outputs an orienta- tion. Clearly, for general molecules, no canonical orientation function exists. However, it very reasonable to require that orientations predicted for two molecules that differ just by a rotation must be consistent. Intuitively, the orientation (M) can be thought of as a coordinate frame attached to the molecule that should rotate along when the molecule is rotated. Formally, we thus demand that the mapping Q: M - SO(3) is equivariant under rotations It applied to the molecular geometry:
(2) To understand the constraint imposed by Eq. ( we view ((M) E SO(3) as a collection of three row vectors et E R² i 2, 3, i.c. ((M) (E1, e2, 63) The e, are precisely the basis vectors
S(RM) := (z(: R
(a))aa) ))T = =QM
of the reference frame that rotates along with the molecular geometry M.
M' RMM
e = Re; equivalently:
(2RM) (Rei, Rez, Reg) ) (e1, e3) "T'TR" RT = ((M)RT
as demanded per Eq. 2 A simple such orientation function is obtained by choosing the basis vectors e; to be the normalized principal components of the centered atom positions. To account for the fact that the sign of eigenvectors is ambiguous, we choose the sign of the first two principal components such that they point in the direction of the largest absolute projection, that is, we orient them to satisfy
max Ir - es| max Ia EG for == 1,2. aeA
aeA
The orientation of the third principal component C3 is fixed by the constraint that det((e), 2,, T)) 1. While one may also use weighted covariance matrix based on atomic masses (yielding the same eigenvectors as the moment of inertia tensor), we opted for the simpler unweighted version which is robust against exchange of atom types.
To compare the orientations of different molecules, we define distance measure based on the following fact: Any rotation can be described by rotation axis (vectors pointing along this axis are left invariant) and the rotation angle, specifving how much to rotate around that axis. This angle is given by acos((t((R) 1)/2), see App. A Then, for rotations R1, R2 SO(3), we use the rotation angle of the relative rotation matrix R. R2 as a distance measure:
[tr[R] R2)
0(R1, R2)
arccos
(4)
2
Intuitively, ((R,, R2) is the angle of the rotation that maps from reference frame R, to R₂ and vice versa. The properties of the trace imply that 0(R,, R2) = 0 R1) 0 125). If the orientations of molecules in dataset D [M;) were truly random, the set of all orientations would be uniformly distributed on SO(3). In that case, for any given reference orientation R the empirical distribution of distances {O, ((M))) | M; E D) should approximate the following
distribution (see App. B):
2 π
p(0) (0/())))
8 10.
(5)
To characterize deviations from this distribution, we proceed as follows. For each dataset, we first compute the distance matrix O2 = (()(M)) (,))),,,MMM)) (M,))) for all M,, M, E D (very large datasets subsampled randomly). Afterwards, each entry in θ and sum over the rows of the resulting matrix to obtain the following kernel density estimate at 0 = 0:
apply 1D Gaussian kemne (00pp, o²) centered at 0 to
k
N
N
1
KDE(#- 0)M,)
Σ N
k(010ij,02)
02,02)
(6)
N
j=1
1
(0 - 12)
with kernel k(0lu, o²)
exp
2
20
"Given that we work with small a, approximate the von Mises distribution the circle with Gaussian kernel.

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QM9
(b) QMugs
(c) OMo125
Figure 4: Quantitative comparison of canonical orientations vs. uniform random orientations. The empirical distribution of relative angles between each orientation in the dataset and the most common reference orientation differs significantly from the same distribution for randomly ro- tated dataset and from the thcoretical expectation for uniform random orientations. Histograms for QMugs and OMol25 are based on 130,000 randomly sampled molecules.
The most prominent orientation in the dataset is then given by
(M;-)) with i = arg max KDE(0- = O).
(8)
ED
Empirically, we find that ((M+-) is fairly robust against the choice of a. Using (2 as refer- ence rotation, the distribution of relative rotation angles in row +,, differs strongly from the theo- retically expected distribution for uniformly sampled orientations given by Eq. (5), see Fig. 4 For both QMugs and OMol25, the most common principal component directions (using
0.5) in-
deed align closely with the standard Cartesian coordinate frame (el = 01, E2 = 1,
63 (0.0, )) see also Fig. 5 For QM9, the most common principal component direction is more ambiguous (compare Figs. and 55. Our analysis has identified the most common prin- cipal component orientation
QM9 to to be approximately é (-0.34, 0..4, 0) 62 (0.94, 00.34, 0) 63 (0, 0, 1)T All three datasets contain significantly more molecules in the most common orientation than expected for a uniform distribution of orientations as well as signifi- cantly more orientations that differ by a rotation of 180 forming the peak at 0 π. The latter likely correspond to the same principal components, but with orientations flipped relative to the reference.
Measuring the non-uniformity of orientations in a single number. To quantify the deviation of the empirical distribution of orientations (described by rotation matrices) from the uniform distri- bution (Haar measure) on SO(3), we estimate the Kullback-Leibler (KL) divergence between the empirical sample distribution and the uniform reference distribution. As an empirical estimator, we employ a version of the classical Kozachenko--Lennekko estimator (Kozachenko) 1987) adapted to the curved domain of SO(3) (see App. C.I for details). Our estimated KL divergences align very well with the visual impressions from the Mollweide plots. We obtain estimates of 0.90 for the QM9 dataset, 1.76 for QMugs (from 130,000 samples), and 1.04 for OMol25 (from 130k samples). A higher number indicates a stronger non non-uniformity of orientations in the dataset. Furthermore, sorting the OMol25 subsets by this non non-uniformity measure yields visually convincing ordering (see Fig. 77 In addition, we have devised a Kolmogorov-Smimin distance distribution between orientations and the expected reference given by Eq. (5) and have studied the robustness of both (non- noonnJuniformity tests (see App. C.2 [.22[[nddd and C.3). Alternative statistical tests for quantifying distributional symmetry are based on kernel methods, such as maximum mean discrepancy (MMD), that base their estimates on empirical means over distances between individual
test to compare the empirical
samples Chiu & Bloem Meddm-eedyyla (2023); (Soleymani et al. (2025) Lawrence et al. 2025a).
3.3 EXPLOITING THE CANONICAL ORIENTATION FOR PROPERTY PREDICTION
While our previous investigations demonstrate that molecules in QM9, QMugs, and OMol25 are not randomly oriented, they do not yet address the possible impact on the performance of machine learning models. In the following, "canonical" orientation of molecules in typical machine learning tasks such as molecular property prediction.
will investigate whether neural networks can exploit the
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Table I: Molecular property prediction from a molecule's orientation alone. A simple MLP
trained on the canonical datasets to regress molecular properties using only the normalized principal components of atomic positions as imput significantly outperforms the test performance achievable with uninformative input features (mean + std over 5 runs).
Dataset
Property
Random orientation
no MSE of mean (test)
MSE of MLP (test)
1.6355
1,4237 0.0048
QM9
CLUM [cV]
yes
1.6355
1.6367 + 0.0001
QM9
ZPVE [eV]
no
0.8107
0.6204 0.0011
yes
0.8107
0.8111 + 0,0001
QM9
cal mol
no
16.169
13.814 +: 0.083
yes
16.169
16.173 = 0.001
Mugs
URT [Fm)]
no
890.54
843.48 + 0.09
yes
890.54
890.55 + 0.02
QMugs
Vee [En]
no
4,894.0 x 10
(4,596.6 + 0.7) x 10
yes
4,894.0 x 10
(4,894.6 + 0.2) x 10a
OMol25
Evot [eV]
no
14,394.3 x 106
(13,689.1 + 1.7) x 106
yes
14,394.3 X 10
(14,398.7 + 0.3) x 10
Here, we consider an extreme scenario: Can network learn anything about molecular proper- ties when presented with the orientation of the molecule alone, without any information about the molecule's constitution or geometry? In the following, we investigate the regression performance of a model that receives only normalized principal components of the atom positions input fea- tures (cf. Sec. 3.2). once canonical pose and once after random rotation. For a fair comparison, deterministic rotation conditioned the molecule index that the same sample will be transformed with the same rotation when revisited during training. This is equivalent to using a version of the dataset in which each molecule has been rotated once prior training. For molecules in random orientations, the normalized principal components do not contain any chemically relevant information. Under an MSE (mean squared error) loss, the best model can do in this setting is to predict the mean of the target feature, since
mean((y:)) arg min Σ (x y2)
Therefore, if the trained model achieves significantly better MSE on the test set than the mean of the test targets does, the model has learned a non-trivial pattern from the normalized principal components. This indicates that chemically similar molecules, by default, tend to have similar orientations. The results, presented in Tab. [ reveal precisely that. Indeed, simple MLPs trained and tested on the canonical datasets significantly outperform the theoretically best possible results, while models trained on the transformed datasets do not, as expected. The chemical properties used for regression were chosen based on the amount of structure in the visualization of all molecular orientations when using the respective properties as a heat map (see Sec. 3.4).
This provides empirical evidence that neural networks may learn unphysical pattern by by map- ping canonical orientation alone to chemical properties. In that case, the spurious information con- tained the biased orientations could be exploited by non-equivariant models, potentially leading to artificially inflated performance metrics in the absence of randomly oriented test set. In an additional experiment, we have studied what extent equivariitt)) transformer architecture trained on property prediction can be influenced by the orientation bias in the selected datasets (see App. for details).
3.4 VISUALIZATION OF MOLECULAR ORIENTATIONS
To further investigate whether chemically similar molecules in molecular datasets tend to be sim- ilarly oriented, we have devised the following visualization strategy: for every molecule M,
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PC3
PC2 mort-commion
CO0
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99..0
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(a) QMugs
(h) OMol25
Figure 5: 2D visualization of normalized principal components of all molecules in QMugs and 10 random samples from OMol25. The equal-area Mollweide projection of the three principal axes reveals that the most common principal axes orientation aligns with the standard Cartesian coordinate system. Orientation frames consist of triplet of points in the 2D projection (blue PCI, yellow ~ PC2, magenta PC3).
PC
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PC1
PC1
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(b) QMugs
(c) OMol25
Figure 6: Default orientations of molecules are correlated with chemical properties. Plots are colored by chemical properties in in the 2D Mollweide projections of the three normalized principal component directions reveal substructure within the distributions. Non-equivariant architectures may exploit such correlation to artificially inflate performance.
compute the normalized principal components 61, C3 E 3 and combine them into a rotation matrix ((M) (c1,62, C3) as described in Sec. 3.2 The e1, 62, S are projected using the equal-area Mollweide projection, and are each visualized in a different color (blue ~ PCI, yellow PC2, magenta PC3), such that orientation frame P(M) consists of triplet of orthogonal points in the projection, see Figs. | form distribution would also be perceived as uniform distribution. show a clear pattern and illustrate the previous finding (Sec. 3.2) that the most common principal component orientation in the QMugs and OMol25 datasets aligns with the standard Cartesian co- ordinate system. Similarly, Fig. [ reveals the structure in all principal component orientations for QM9. Based on the distance measure defined Eq. (4), we have cherry-picked two groups QM9 molecules of practically identical orientation that arguably have very similar chemical constitution and geometry. These examples nicely illustrate that, as one of the signatures of the cheminformatics codes used in the data- gentraggnnating process, chemically similar molecules tend to have similar canon- ical orientations. In Fig. 6 we show one selected chemical property for each dataset as a heat map in the projections instead of using distinct colors to distinguish the principal components. The three
and 5 Notably, using an equal-area projection, a perfectly uni- contrast, the visualizations

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(a) GEOM (5.613)
(b) ANI-2X (4.328)
(c) Reactivity (1.992)
(d) Transition-IX (1.861)
Orbnet Denali (1.656)
(1) RGDI (1.046)
(g) Metal
(h) Electrolytes
(i) Biomolecules (0.070)
(j) SPICE2 (0.005)
Complexes (0.243)
(0.152)
Figure 7: Visualization of molecular orientations in OMol25 subsets. Molecules some subsets display strong alignment of principal components (PCs) with the standard Cartesian coordinate system (a, c, d, c, f). For others, orientations are more uniformly distributed (g. h, i, j). as indicated by the estimated KL-divergence to the uniform distribution over rotations (in brackets, App. PCs are projected using the equal-area Mollweide projection and colored as Fig. 5
visualizations visibly confirm the correlation between chemical properties of molecules and their default orientations.
Lastly, we show that the orientation distribution in the large collection OMol25 differs strongly between different subsets of the dataset (subsets are based on the "data id" field of OMol25 sam- ples), see Fig. 7 The plots demonstrate that while molecules in some subsets are visibly strongly aligned with the standard Cartesian axes, for other subsets the distribution of orientation is (almost) perceptually uniform (c.g. for the SPICE2 dataset (Eastman et al., 2023), Fig. 7i)). In particular small biases in the Biomolecules subset (Fig. (i)) may be easily overlooked, highlighting the need for random rotations even in the absence of an obvious alignment.
4 CONCLUSION
We demonstrate in various ways that the default orientations of molecules some the most pop- ular molecular datasets (QM9, QMugs, OMol25) are far from random, and that the alignment of chemically similar molecules can in principle be exploited by machine learning models. Extending the work by Lawrence al. (2025a), this paper presents systematic analysis of orientations in the selected datasets. We introduce an interpretable visualization method to jointly plot all molecular orientations. We summarize the distribution of orientations by pairwise distances to identify the most common poses and quantify the degree of orientation bias. In addition, we highlight differ- scenarios in which ML pipelines may be affected by orientation bias. Given that universally agreed-upon canonical orientations do not-and probably bias not a deficit of these important community resources. However, for researchers entering the field of molecular machine learning, the assumption that molecular poses are fully random can be a significant source of eиror. Based on our findings, we thus recommend the following best prac- tices for rigorous evaluation and development of molecular machine learning models: First, it is essential to report equivariance errors for equivariant models, and as a sanity check, evaluate equiv- ariant models randomly oriented test sets. This ensures that any claimed equivariant behavior is genuine and bug-free. Second, for non-equivariant or only approximately equivariant models, crucial use data augmentation during training prevent overfitting to any canonical orienta- tions and to provide a more realistic assessment of model generalization. It is quite likely that other Bata-generating processes may also introduce preferred orientations. Therefore, when in doubt, recommend following the same best practices for other geometric datasets as well.
the presence of orientation
At the same time, our results highlight the potential benefits of leveraging well-defined canonical orientation, as explored in recent work (e.g. by Baker et al. (2024)). In scenarios where meaningful canonicalization available and justified by the application, it can advantageous incorporate this information explicitly.
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ACKNOWLEDGMENTS
This study has received funding from the Klaus Tschira Stiftung gGmbH (Simplaix project) and the Wildcard program from the Carl Zeiss Stiftung. Further, this work is supported by Deutsche Torschungsgemeinschatt (DFG) under Germany's Excellence Strategy EXC-2181/1 - -390900948 (the Heidelberg STRUCTURES Excellence Cluster) well under project number 240245660 SFB I 1129. The authors acknowledge support by the state of Baden- Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST 35/1597-1 FUGG.
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A DERIVATION OF THE ROTATION ANGLE OF A GIVEN ROTATION MATRIX
Let R E (3)3) a rotation matrix. Since real orthogonal, all its eigenvalues lie on the unit circle and complex ones occur in conjugate pairs. With det R the three eigenvalues must be {1, elP, e } for some E []. Let u be unit eigenvector with Ru = (the rotation axis). Now, let extend this basis R leaves span (1, 62) invariant and acts on the corresponding subspace as a 2 x 2 planar rotation by Hence R is (by change of basis) orthogonally similar to
an orthonormal basis {ei, e2, J with appropriate basis vectors e and e2. In
cos sin 0 sin
0
cos 0
1
0
0
Since the trace is invariant under similarity, we have
tr( cos 0 + cos 1= 1 + cos 0,
(10)
which yields the rotation angle of R
0 = arccos
1r (r(R)
- 2
E [0, m].
(11)
Furthermore, the fact that the trace of R also given by the sum of its eigenvalues tr(R) = + e"P + + cps reveals that yp 0.
B DERIVATION OF ANGLE DISTRIBUTION FOR UNIFORMLY SAMPLED ROTATIONS
Let R SO(3) be Haar-uniform, i.e. drawn from the unique probability measure on the rotation group that is invariant under multiplying by any fixed rotation on the left right. Further, we identify each E S3(3) with unit quatermion
(w, i) E S IR modulo the antipodal map q -q. The rotation angle 0 e [0, π] of R is related to q by
0
2arcoos (w])..
(12)
Further, the rotation axis πt of R is related to e by π /// |0 Now, if we parametrize the 3-sphere by = (cos X, sin π with 7 E S2 and x [| the polar angle as measured from the "north pole" in the w-direction, the uniform surface element on S factorizes
dogs
sin" x d52,.
(13)
where d02 is the uniform measure on $2 Since w is directly related to the rotation angle 0 by Eq. 12). interested in the marginal distribution of To get the marginal of w. compute the area (hence the probability mass for the uniform measure) of the "spherical band" between and x + dx. This area is proportional to
1 dw sin x dw v1 - w2 dw, dx dw sin? x sin x
sin? x dx sin x
(14)
dw
where we have used that w = cos x and that du = sin dx. The marginal density of w is given by
(w) x V1- 11-w" w
w E [-111.
(15)
Normalizing with 1", vI 2 dw = TT/ gives
fu(w) = V1 - 22, π
w 1-11].
(16)
Now, since q and -q represent the same rotation, let us consider the density of [w]:
4 v1 u²
S (1) (n)
u E 10, 1].
(17)
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Using that |w| c/s( (ef. Eq. (12) and thus du/d0 sin(0/2), the density of 9 E [0. follows by change of variables:
0()) Lilll cos(0/22)
cos(0/2)
do
4
1
V (0/2)
sin(0/2)
π
2
2 sin
(0/2)
0 E [0,
(18)
Hence, the principal rotation angle of Haar-uniform It E SO(3) has density p(0) 2 sin on [0, T].
(0n2(
C MEASURING THE NON-UNIFORMITY OF ORIENTATIONS IN A SINGLE NUMBER
C.I. EMPIRICAL ESTIMATE OF THE KL DIVERGENCE TO THE UNIFORM DISTRIBUTION OF ROTATIONS
To quantify the deviation of the empirical distribution of orientations (described by rotation matrices) from the uniform distribution (Haar measure) on SO(3). we estimate the Kullback-Leibler (KL) divergence between the empirical sample distribution, denoted by P, and the uniform reference distribution U.
Given that (3)3) compact manifold with finite volume, the KL divergence can be expressed solely in terms of the differential entropy H(P): DKL(Pl log(8m²)) - HP)),
(19)
where 8 is the total volume of SO(3) under the standard bi-invariant metric, and HP)) is the geometric differential entropy. A KL divergence of indicates perfect uniformity, while higher values indicate clustering anisotropy in the rotational distribution.
To estimate H (P)) from a finite set of samples {R, J we employ a non-parametric k-Nearest Neighbor (k-NN) estimator. We follow the manifold-corrected approach described by [Singh & es-
Póczos (016)): Heinz & Grubmüller (2019). which extends the classical timator (ozachenko 1987) the curved geometry of the rotation group. Unlike standard Euclidean estimators for _ which assume that local volumes scale as rd, this method explicitly corrects for the manifold's curvature by using the exact volume of a geodesic ball on SO(3). The estimator is defined as:
= (kk) 1 EE log (V(+.))
(20)
where is the digamma function (defined as the logarithmic derivative of the gamma function). Further, N is the number of samples, k is the number of neighbors (set to k = 5 for all experiments), 8i is the geodesic distance from sample R, to its k-th nearest neighbor and V(Ox)) is the volume of geodesic ball with radius lx.-
The geodesic distance Ozj between two rotation matrices Rg and Ry defined as the angle of the relative rotation (as in Eq.
(r(( (g) Bgg E m0
02 = R, R2) arccos
(21)
Geodesic ball volume on SO(3). Crucially, the volume V0) of a geodesic ball with radius on SO(3) deviates from the Euclidean approximation ()) due to the positive curvature of the manifold. In the following, we sketch how the exact volume formula derived from the integration of the Haar measure. The Haar measure
compact Lie group provides an intrinsie notion of "volume": : it assigns the same measure to any set C SO(3) and its left- (or right-)translatee Itranslated copy, i.e. g8 [gs] E 8) or Sg {sgl E 5) for given g E SO(3). Intuitively, this is the
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0.4
Motric volue
0.
0.0
tranns
cranet, Menali
metal compresus
elytas
Boondeeces
spice
gromm oraat
metecieyy
Danzza
eranet Menair
rlrl
metal carrreee
green
Subset rolass laben
Subset (cless Iabe)
(a) Sullback-Leiblee divergence
(b) Kolmogorov Smirnov test statistic
Selmogorov-Smirnov
Figure 8: Comparison of estimated non-uniformity of orientation distributions of OMol25 sub- sets. Estimates (KL divergence and KS test statistic) are based on 19,000 samples from each subset. The estimates imply an ordering that is compatible with the visual impressions of the distributions in Fig.
unique way of measuring subsets of (3(3) that does depend on the chosen coordinate system or reference frame.
Let μ denote the (unnormalized) Haar volume measure on SO(3) and let Vol(SO(3) == ==((33)) We now fix reference rotation R (e.g. the identity) and define the geodesic ball
(R) = {R SO(3) d(R, R) < <)) V(6) == (B)(
A [0, n..
R is Haar-uniformly distributed, then the probability that d(fi, R) S 0 is given by
V(O)
P(d R) < 0)
Vol(SO(3))
Hence V(0) is proportional to the CDF of the random distance di f. R):
v(0) Vol(SO(3)) / p(t) dt,
(22)
where p is the distance (angle) density derived in App. B
2 sin 2 (0/2).
(())
0 E [0,
The integral can be easily computed as
L p(1) di -0 sin 0).
Finally, using the known result that the total Haar volume of SO(3) is Vol(SO(3)) e.g. Eq. (17) in Ecker & Kolev (205)), we obtain the exact geodesic ball volume
8T7 (see
V(0) 8T (0 sin 0).
0 E [0,
(23)
As sanity check, for small 0 we indeed recover the Euclidean approximation:
V(0) 8T (0 sin 0)
8T ( (Ø Ø² e + +((9))
4 TB3 + O
(24)
Substituting Eq. entropy.
23 into Eq. (20 yields a statistically consistent, coordinate free estimate of the
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C.2 KOLMOGOROV SMIRNOV TEST FOR NON-UNIFORMITY OF ORIENTATIONS
In Sec. 3.2, we describe how to characterize the distribution of all orientations in molecular dataset in terms of all pairwise distances between the associated rotation matrices. The quadratic distance matrix ④ E IR NxN (for N orientations) holds the geodesic distance between any two orientations R2 R E (3) given by the rotation angle of the relative rotation (cf. Eq. 4)::
(IE(R) -
ig = - R2) = arcoos
(25)
2
Each row (or column) of Gij may be interpreted as samples from the empirical marginal "radial" distribution of the full distribution of orientations relative to the respective reference orientation.
Furthermore, we show in App. that, given a sample from the Haar-uniform distribution on SO(3), the distribution of distances relative to any reference orientation should approximate
2 (()) sin()/2)² 0 0 10
(26)
π
We can therefore compare each empirical distribution of distances (given by a row of Q2y) the ex- pected marginal of the uniform distribution to quantify the deviation from (3nifoonifrrm Since the radial distance distributions are 1D, is straightforward to compute their empirical cumulative distribution functions (CDFs) and compare them in (one-sample) Kolmogorov-Sminne (KS) test against the known theoretical reference given by
8 F(O) = { si (0)
1 10 (0), 0 E -0
(27)
The empirical CDF F, of the i-th row of θ given by
(((0) number of entries in O with O; S
N
Σ
1 10.0(
(28)
N
where [0: ) is the indicator function, equal to I if O,j < 0 and equal to 0 otherwise.
The Dolmogorov-Smimee test statistic is then given by
D. = sup (Ā:(0) - F(O))
(29)
describe how to obtain estimate for the most common pose in dataset using kernel density estimate. We plot the distance distribution relative to the most common orientation and compare it against the reference distribution (Fig. 4). Here, we want leverage the D; from all empirical CDFs to obtain an estimate that is not sensitive to selected reference orientation. A natural way combine all D, is the average:
In Sec. 3.2
D
(30)
Higher values for D indicate a larger discrepancy from the uniform distribution. We have applied this KS test and the KL divergence estimation to the subsets of the OMol25 dataset (see Fig 8). The ranking obtained from both methods is consistent, except for one transposition in the ordering of the "Metal complexes" and the "Flectrolytes" subset, whose relative ordering is arguably also difficult to judge visually (Fig. Zp. h).
For the KS test, it is standard procedure to determine the significance of the test in terms of p- values, e.g. from s scipy. stats. kstest. In our case, the null hypothesis is that a sample is distributed according to the reference distribution, and hence for p-values below the threshold of 0.05 the null hypothesis can be rejected and we can say with statistical significance that the sample is not uniformly distributed. We have tested the significance of the non-uniformity in the OMol25 subsets in the following way: first, we split each subset (we use 19,000 samples) in two halves, (a) and (b). With split (a) we determine independent estimate of the most common orientation
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0.25
15.
0,20
Unternm QN
Uniterm
CHR
QM0-0S
0.05
320
3000
30350
33502
200
3000
3030
10320
20000
Sibset sicp
Subsect
(a) Kullback-Leibler divergence
(b) Kolmogorov-Smirnon test statistic
Figure 9: Robustness of KL divergence and Balnogorov-Smiman (KS) test as measure of (non-)unifoon Junffortif for the orientations in selected molecular datasets. Higher values indicate larger discrepancy the Haar-uniform distribution on SO(3).
in the subset (as described in Sec. 3.2 Second, compare the distribution of distances in split (b) relative to the determined reference orientation via a single (one-sample) KS test and determine the p-value. The independence of the reference orientation is important not artificially bias the KS-test towards a non-uniform distribution. The p-values and KS test statistics for each subset are shown in Tab. 2
In App- we test the robustness of the Kolmogorov-Sminne -Smirnnv test presented here and the empirical estimate of the KL divergence presented in App. C.I across different sample sizes.
Table Significance of non-uniformity in OMol25 subsets. Canonical orientations in all but the SPICE subset differ significantly from the uniform distribution of rotations.
Data Subset
KS Statistic
p-value
geom orcaó
0.5325
0.0000e+00
ani2x
0.4106
0.0000e+00
reactivity
0.3010
0.0000e+00
trans
0,2162
0.0000c+00
rgd
0.2158
0.0000e+00
orbnet.denal
0.1711
1.2694e-243
elytes
0.0641
2.1944e-34
metal complexes
0.0406
5.1420e-14
biomolecules
0.0183
3.4879e-03
spice
0.0078
6.0437e-01
C.3 ROBUSTNESS SO(3)-UNIFORMITY
In App. C.2 we describe a Kolmogorov-Smime (KS) test to quantify the (non-)uniformity of molecular orientations. The KS test provides alternative to the KI. divergence presented as (non-)uniforming measure in App. C.I To empirically study the robustness of both methods, we have subdivided ~ 130,000 samples from each dataset (QM9, QMugs, and OMol25) as well as 130,000 samples from reference set of uniformly sampled rotations into k (between 4 and 50) dis- joint sets for various sizes n and computed the KL divergences and KS test statistics on each subset. Box plots that illustrate the spread and biases for the different dataset sizes shown in Fig. 9
The KS test statistic seems to be bit more robust, i.c. converges faster to stable value with growing subset size. The reason that the QM9 dataset shows the strongest variation of the datasets is most likely due to the fact that the orientation distribution has substructure (non-uniformity) at a finer resolution which can only be resolved with sufficiently large sample size.
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Table 3: Hyperparameters for training our simple message passing network on default orien- tation detection.
Architecture hyperparameter
Bessel frequencies (radial)
32
Bessel frequencies (angular)
20
Num. message passing layers (Eq. 31))
3
Aggregation operation in message passing
max
Node feature dimension
512
Hidden layers in message MLP
[128]
Activation function
SiLU
Radial cutoff for message passing
10 Å
Hidden layers in readout MLP
[512, 128, 32]
Training hyperparameter
Optimizer
Adam W
Weight decay
5e-3
Learning rate
5e-4
Scheduler
Cosine-I.R
Epochs
200 for OMol25, 100 otherwise
Warm-up epochs
5
Gradient clip
0.5
Loss function
BCE-loss
D DETAILS REGARDING MODEL TRAINING AND ARCHITECTURES
Details on the message passing architecture used for the detection of default orientations. In demonstrate that simple geometric message passing network accurately discerns canonical samples from randomly rotated ones. For the model, we employ a straightforward point cloud architectrre, consisting of three layers of message passing:
Sec. 3.1
I (R+1)
θ JEA(()
(aaPs,))emmbb/ 1 (em((x=-;)))
(31)
where s!(x) denotes the feature vector of atom in layer k; x, its position and (i) its neighborhood (defined by radial cutoff of 10 A). As aggregation function D) we use the component-wist max operation. The input these message passing layers consists of learned embedding of the neighbor geometry combined with an embedding of the atom type. For the initial node embedding, we use Bessel functions of the first kind with 32 learnable frequencies to embed the relative distance Ti = ||=; x; l|2 Similarly, we use Bessel functions with 20 learnable frequencies to separately embed each component of the normalized relative distance vector as angular embedding. The aggre- gated (summed) angular and radial embeddings from the local neighborhood are combined with
one-hot embedding of the atom type to form the input node features f((), During message passing. the angular and radial part of the relative distance vectors 2; xj are embedded using 64 Gaus- sian radial basis functions (spaced equidistantly between 0 and 10 Å). The network is then trained to minimize a binary cross-entropy loss, predicting whether the input molecule has been randomly rotated or is in its (perturbed) default pose. All hyperparameters summarized in Tab. 3
MLP used for the molecular property prediction from molecular orientations alone. In demonstrate that simple MLP receiving as input only normalized principal com- ponents (PCs) of atom positions can successfully regress molccular propertics. The MLP receives the first normalized PCs as input, uses SiLU activations and four hidden channels with 256 fea- tures each. trained with MSE loss without weight decay. We have trained one separate model of the same architecture for each property from the different datasets reported in Tab.
Sec. 3 3
Dealing with the dataset size. For QM9 we train all models for 100 epochs on the full dataset using train-val-test split of (110,000, 10,000, 200,000). For the larger QMugs dataset
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Prai-val-tess split of (80%, 10%, 10%). However, in order to keep the training time and learning rate scheduling comparable to the one in QM9, we train all QMugs models for 100 epochs with different random training subsets of size 110000 for each epoch. For the massive OMol25 dataset we use train-val-test split of (99.8%, 0.1%, 0. ()) and train for total of 200 epochs again on different random subsets of the training set of size 110000.
E ADDITIONAL EXPERIMENTS AND ABLATION STUDIES
E.1 INVESTIGATING ORIENTATION BIAS ADDITIONAL DATASETS
As mentioned in Sec. 2.1 one subtle example where an orientation bias may easily be overlooked is the field of charge density prediction. In the field, several standard benchmarks exist that store DFT ground state electron densities regular Cartesian (axis-aligned) grids as training targets (Brock- herde et al. 2017 Bogojeski et al., 2020 Jørgensen Bhowmik 2022 Li al. 2025). For the combined MD dataset Brockherde et al. (0017); Bogojeski et al. (020). we visualize the distribu- tion of orientations (as given by the principal components, cf. Sec. 3.2) in a Mollweide projection (Sec. 3.4) in Fig. I0 Additionally, show the empirical distribution of relative angles between all orientations in the respective dataset and the most common orientation (described See. Scc.3.2).
The two other datasets also contain electron densities stored on regular Cartesian (axis-aligned) grids. They are both based on the standard QM9 dataset, which we have shown to exhibit strong orientation bias. Indeed, could confirm that the molecular orientations are exactly the same as for the QM9 dataset presented in the main text.
Since several of the recently proposed approaches for relaxed equivariance train models the molecular dynamics dataset rMD17 (Christensen Von Lilienfeld, 2020),, we have decided to in- corporate it into our analysis (complementing the prior work by Lawrence et al. (2025a)). Fig- ure II shows the plots equivalent to those described above for rMD17. We have used the rMD17 dataset available via https: /tttss://fighaae.e com com/articlest dataeetttsee/teeeeee /eeeeet/Reeiedd MD17 dataset rMD17 /2//1265 12672038 In particular, for the subsets of the rMD17 molecular dy- namics dataset, it is surprising that the orientation distributions are qualitatively so different, given that all subsets should be generated in a comparable way.
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Enaicee argedat:
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00
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(c) Ethane
argicca Rng Theiy vasse
Fis commron
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30 Rolidie ritaion angle
rtss
(f) Malonaldehyde
Figure 10: Mollweide projections of all molecular orientations (given by principal components) and distribution of relative rotation angles (relative to the most common orientation) for the MD electron density datasets of (Brockherde al. 2017 Bogojeski et al. 2020), containing between 1000 and 2000 conformations for each molecule.
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MA
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(3
Oriinial ang
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KC
Ling sigh
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PC UBLL
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(d) Malonaldehyde (rMD17)
Frig Theiry sance
3401
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13,0.1
(c) Toluene (rMD17)
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SAR
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PC3
und los ridue misat
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rszal conminnn ortanaaiinn
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Paracetamol (rMD17)
Figure 11: Mollweide projections of all molecular orientations (given by principal components) and distribution of relative rotation angles (relative to the most common orientation) for the rMD17 dataset (Christensen & Von Lilienfeld 2020), containing 100,000 conformations for each molecule.
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1.0
0.9
0.9
0.9
0.0
0.0
0.0
0.7
0.7
ayers message passing
ayers mezsage passing
ayar massage sossing
ayars message passing
ayys mezaage passing
0.5-
ayer massage so5sing timyl
massage sessing (Ery)
ayer massage sassing thry)
ayar massage sessing (nans)
massage passing (nano)
ayar massage sessing snans)
0.4
uyer massage sessing (pica
0.4
wyer massage pessing (pical
0.4-
uyer massage sessing (pical
10
30
50 50 70
00
90
10
25 30 40
50 50 degree
00
90
10
23 30
50 so 10 defiee ofrotacion [deg)
95
defiee ofrotation [deeg]
a
QM9
(b) QMugs
(c) OMo125
Figure 12: Much smaller versions of the simple geometric message passing network can still ac- curately discern canonical samples from randomly rotated ones, even when pre-rotated by up to a before applying the random rotation to be detected. For each angle we train a separate model. The I-layer "pico" models only have 238 trainable parameters.
E.2 DETECTING ORIENTATION BIAS WITH EVER SMALLER MESSAGE PASSING MODELS
The results presented in Fig. show that a simple geometric message passing network can accurately discern canonical samples from randomly rotated ones. is an interesting and relevant question to study the effect that shrinking the network size has on the classifier's accuracy. Reducing the number of trainable parameters lowers the training times and hardware requirements for probing a dataset for orientation bias. To that end, we have conducted ablation study on QM9 by training a 2-layer and 1 I-layer message passing model. Both models still accurately discern the canonical dataset from randomly rotated one (see Fig. [2}. Further, have drastically reduced the number of learnable parameters on the I-layer network from 642,000 to a "tiny" version with 25,000, to a "nano" version with 2350, and to a "pico" version with only 238 trainable parameters. Very surprisingly, even the smallest versions only sacrifice a bit of accuracy but still adequately detects the orientation biases in all three datasets (see Fig. [2). This result seems to illustrate the remarkable inductive bias in the geometric message passing network and suggests that in practice the classifier test in Sec. 3.1 may be performed with a much smaller network.
E.3 ADDITIONAL PROPERTY PREDICTION FROM A MOLECULE'S ORIENTATION ALONE
In Tab. have demonstrated, for selection of molecular properties, that simple MLP that is trained on property prediction on the canonical datasets and ingests only normalized principal components can significantly outperform the test performance achievable with uninformative in- put features. We have now additionally trained one separate MLP model to regress each (scalar) molecular property in the QM9 and QMugs datasets based solely on the principal components of the molecular geometries. In Tab. A and 5 we report the mean squared error (MSE) that the MLP achieves when trained on the standard datasets and on randomly rotated versions of them. We compare the MSE against the optimal test performance achievable for uninformative input features (as Tab. D. For (almost) all properties we find that the MLPs trained on the standard datasets achieve significantly lower MSE than the mean of the test set. For several properties the rela- tive improvements are comparable to those of the selected properties shown in Tab. [] The im- provement relative to the MSE of the mean of the test set is shown in percent and computed as (MSE( E mean)) ( //M (test mean). Since the standard deviations in Tab. [ are comparatively small, we have conducted only one training for each property in this case.
E.4 TRAINING AN MLP CLASSIFIER TO DISCERN ORIENTATIONS DIRECTLY
The results presented in Fig. 3 and 12 show that a simple geometric message passing network can accurately discern canonical samples from randomly rotated ones. The trained message passing networks receive as input the molecular geometry comprised of the atomic coordinates and the atom types. Since the distribution far from uniform and contain non-trivial patterns, it is an interesting question how well classifier can detect orientation bias from the principal eigenvectors directly. For that, we have trained vanilla MLPs ingesting only the molecule's 's orientation (in the form of the principal components) predict
orientations (given by the principal components) e.g. in Fig. [ is
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Table 4: Molecular property prediction on QM9 from molecule's orientation alone. sim- ple MLP trained on the canonical datasets to regress molecular properties given only normalized principal components of atom positions as input outperforms the test performance achievable with uninformative imput features on all properties. Shown in percent is the improvement of the MLP relative to the test mean.
Property
MSE MLP (w/ rot.)
MSE of mean (test)
MSE of MLP rot)
"
[Debye]
2.35 (0. (%)
2.35
2.24 (-4.848)
[Bohr"]
66,40 (0.02%)
66.38
63.54 (-4,29%)
<HoMo [eV]
0.37 (0.12%)
1.36
0.34 (-8.205))
ELGAMO [eV]
1.64 (0.08%)
1.64
1.42 (-12.88%)
Essp [eV]
1.66 (0.10%)
1.66
1.40 (-15.72%)
(P)) [Bohr"]
75589.49 49 (0 (0.02%))
75573.68
67985.68 (-10.0455)
Barva [eV]
0.81 (0.10%)
0.81
0.62 (-23.09%)
[cV]
1234359. (-0.01%)
1234518,12
1209677 25 (-2.01%)
U [eV]
1234698.00 00 (0.025)
1234504. 12
1209264, 88 (-2.04%)
[eV]
1234502. 12 (-0.00%)
1234504.38
1209401.00 (-2.03%)
G [eV]
1234789.88 (0.02%)
1234547.75
121048 6.22 (-1.95%)
C. [cal mol
K-'
16.17 (0.025))
16.17
13.79 (-14,7 44.7.199 (E)
Ug [cV]
106.79 (0.05%)
106.73
87.31 (-18. 20%)
gattom [eV]
108.68 (0.04%)
108.64
88.33 (-18 69%)
Hatom [eV]
110.23 (0.04%)
110.18
89.26 (-1..6-111999)1 (-18.99%)
Gutom [eV]
90.41 (0.03%)
90.39
73.87 (8%%
[GHz]
10.18 (0.02%)
10.18
9.81 (-3.58%)
B [GHz]
8.20 (0.01%) %)
8.20
8,16 (-.43%%
C' [GHz]
3.56 (0.01%)
3.56
3,54 (-0.46%)
Table 5: Molecular property prediction on QMugs from molecule's orientation alone. A simple MLP trained on the canonical datasets to regress molecular properties given only normalized principal components of atom positions input outperforms the test performance achievable with uninformative input features on most propertics. Shown in percent is the improvement of the MLP relative to the test mean.
Property
MSE of MLP (w/ rot.)
MSE of mean (test)
MSE of MLP (w/o rot.)
GEN2:TOTAL ENERGY
890.55 (0.01%)
890.48
843.62 (-5.26%)
GFN2-ATOMIC ENERGY
653.21 (0.01%)
653.17
618.43 (-5.32%)
GPN2/FORMATION ENERGY
19.69 (0.01 S)
19.69
18.76 (-4.72%)
GIN2:TOTALEETTIALPP
880.71 (0.01%)
880.66
834.10 (-5.29%)
GFN2/TOTAI_ BBAM___RF__EEE_E FREGGYNERGY
882.03 (0.01%)
881.91
835.28 (-5.29%)
GIN2:HOMO, ENERGY
0,0003285 (0.0031%)
0.0003285
0.0003285 (-0.0013%)
GFN2:LUMO ENERGY
0.0007593 (0.0068%)
0.0007593
0.0007592 (-0.00345)
GFN2:HOMOJ JUMOGAPP
0.00070088 (0.003895
0.0007007
0.0007006 (-0.0178%)
GIN2:FERMI LEVEL
0.0003687 (0.0099%)
0.0003687
0.0003687 (0.0004%)
GENE.DISPERSION WWW..DDSPERSION.OEEFFCIII .COEFFICIENT WWWFFFCCENTTOLBBLLLL..
1035579456 (0.02%)
1035421632
997401536 (-3.67%)
LIRIABLLARIZBBIIIYY MOLECULAR
8677.71 (0.01%)
8677.08
8224.04 (-5.22%)
TFTATL ENERGY
618583 (0.01%)
618508
603139 (-2.48%)
AFT.ATOMMC ENERGY
616063 (0.00%)
616037
600839 (-2.47%)
DIT:FORMATION, ENERGY
JENERYY
11.37 (0.01%)
11.37
10.85 (-4.59%)
EENEGY
2287.17 (0.01 %)
2287.04
2169.66 (-5.13%)
DFT. REPULSION ENERGY
_ION.ENERGY
3938205 (0.01 S+)
3937772
3695442 (-6.159) 15%)
DFT: ELECTRON ENERGY
22159716 16 (0.01%)
22157832
20835828 (-5.97%)
WC REERGY
4894430 (0.01 9-)
4894038
4595915 (-6.099%
DET.HMMO, JENERGY
0.0003445 (0.0074%)
0.0003445
0.0003438 (-0. (0.6)))))
DFT.LUMO ENERGY
0.0007075 (0.0054%))
0.0007075
0.0007056 (-0.2714%)
DOTO LUMOOOLLMM.CCO .GAP
0.0009014 (0.0055%)
0.0009013
0.0008966 (-0.5185%)
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whether the PCs come from a previously rotated molecular geometry not. Conceptually, this test is similar to our non-parametric uniformity test (discussed in Sec. 3.2 and App. C.T). Indeed, the MLP classifier can still distinguish well between samples from the two distributions (as shown in Fig. 13). However, since the principal components arguably contain fewer cues about a molecule's orientation than the full geometry. it seems plausible that the classifier is overall less accurate than the message passing network. For instance, in some cases the canonical orientation may also be detectable from the orientation of local substructure in the molecule information which is not contained in the PCs.
1.0
0.9
0.7
0.6
layers message passing
MLP using PCA input
10
20
30
40
60
70 [deg]
80
90
degree
Figure 13: A vanilla MLP ingesting only the molecule's orientation (in the form of the principal components) can still predict whether the PCs come from previously rotated molecular geome- try not (dataset: QM9). Similar to the experiments presented in Fig. and 12 the canonical orientations were previously randomly rotated by an angle up to a to examine the robustness of the detection. Overall, the MLP performs worse than geometric message passing network which receives the molecular geometry as input. The reason for that may simply be that the molecular geometry contains more cues about the exact orientation of molecule.
E.5 STUDYING THE EMPIRICAL EFFECT OF ORIENTATION BIAS ON NNS
Beyond the experiments reported in Tab. 1 5 it is an interesting but involved research question to study which type of architectures and on which tasks (c.g. prediction of scalar quantities vs. tenso- rial quantities) are how strongly affected by orientation bias. As a first step, we have implemented an adaptation of the Graphormer architecture (Ying et al. 2021) to study the effect of orientation bias on typical transformer model. Indeed, despite its simplicity, the Graphormer is still actively used in molecular Zhang et (004)) ding similar the used for the message passing network (ef. App.D), followed by 4 Graphormer layers (32 heads, 512 channels) in which a radial (and optionally an angular) embedding are used as and only uses
et al. (2025). Our adaptation uses a geometric embed-
offsets on the attention weights. By default, the Graphormer is potationally-invariie radial embeddings of relative positions in the attention mechanism. However, for comparison, we optionally modify the architecture to further ingest simple angular embedding of the normalized Importantly, this naive angular embedding breaks the exact in- relative distance vectors 12, -
variance of the architecture. We have trained the invariant and non-invariant version this model on molecular property prediction using a) the canonical QM9 and QMugs dataset and b) dataset versions in which orientations were previously randomized (see Tab. 66) All models were trained with Ll-loss for 100 epochs (learning rate: 5 10- and weight decay 5 x 10-3). As molecular
properties we have chosen the ZPVE for QM9 and the DFT:NUCLEAR REPULSION _NEERGY for QMugs, since these showed the strongest relative improvements Tab, and 5 respectively
As expected, in all cases, we find that the non-equivariant models trained on the canonical data strongly rely on the alignment of orientations. When evaluating these models on the datasets with randomized orientations the error metrics (mean absolute error and mean squared error) increase significantly. For the Quus property, it seems that the orientation-randomii version of the data help the model to learn more informative patterns (comparable to the effect of data augmentation). Here, the model trained on randomized data achieves a better performance than the model trained canonicalized data, even when evaluating canonicalized data. However, highlighted in bold (QM9, MSE) we see a scenario in which the opposite is the case and the model trained canonicalized data also outperforms the model trained on randomized orientations. This
and thus more varied
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is the critical case, in which non-equivariant model leverages the orientation bias in the dataset to artificially inflate its performance. Further, we find that for both properties (except in the MAE metric for the QM9 property) the fully-invariant model performs best. A toy example in which the canonicalization should show a strong effect is the regression of the first principal component, given the molecular geometry input (see Tab. (G)) Indeed, in this scenario, the non-equivariant Graphormer trained on the canonical dataset leverages the orientation bias in the canonical dataset outperform the same model trained randomized orientations. This experiment is intended to serve as proxy for the prediction of vectorial quantities (such as the dipole moment) which could be highly correlated with the orientation of molecule.
All in all, it is very reasonable to assume that the effect of the orientation bias on model performance strongly varies depending on the architecture, the dataset size and the nature of the molecular target.
Table 6: Invariant and non-invariant Graphormer adaptations trained and evaluated on molecular property prediction with and without canonicalized molecular orientations.
(a) Regression of ZPVE [eV] on QM9
test
MAE canonical
MAE randomized
MSE canonical
MSE randomized
train
canonical
0.0049
0.0051
0.000045
0.000048
randomized
0.0031
0.0031
0.000062
0.000062
invariant model
0.0036
0.0036
0.000027
0.000027
(h) Regression of DFT:NUCLEAR REPULSION ENERGY [Ea] QMugs
test
MAE canonical
MAE randomized
MSE canonical
MSE randomized
train
canonical
107.2
150.8
77127
randomized
85.7
85.8
66544
66800
invariant model
63.6
63.6
51237
51237
(c) Regression of PC1 on QMugs
test
MAE canonical
MAE randomized
MSE canonical
MSE randomized
train
canonical
0.3556
0.6296 0.3565
0.3442
0.6190
randomized
0.3569
0.3604
0.3597
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rC
PC
PC 0
zaro point vibritoral energy (e4]
Two electron energy Ra [F.]
PC1
PC11
5000
-50000
-000000
2000
(a) QM9
(b) QMugs
(c) OMol25
Figure 14: Combined Mollweide projections all molecular orientations (given by principal com- ponents) and molecular properties. Each orientation consists of a triplet of points in the 2D projec- tion. In the top row, the triplets are plotted jointly (blue PCI, yellow ~ PC2, magenta ~ PC3). The plots below show the distribution of each PC separately colored by chemical properties. The substructure reveals the correlation between canonical orientations and chemical properties.
E.6 COMBINED VISCALIZATION OF MOLECULAR ORIENTATIONS AND MOLECULAR PROPERTIES
For completeness, we present combined visualization of all molecular orientations and the corre- sponding property-colored visualizations reveal correlations between orientations and molecular properties (Fig. [4].
LLM usage. Large Language Models (LLMs) were used in the preparation of this submission polish the writing regarding formulations and wording. addition, we have used LLMdbbsed auto-completion in the development of our research code.
29


Reviews:
[{"review_id": "2cI9xAEEL2", "reviewer": "Reviewer_wjTX", "summary": "The paper demonstrates that the molecules in the most commonly used molecular benchmark datasets are not in random orientations. Worse, the orientation implicitly encodes information about some of the properties that algorithms are supposed to be predicting.", "questions": "see \"weaknesses\" for how the paper could be improved", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper makes an important observation about the most popular molecular benchmarks and demonstrates it quite convincingly. It is important to convey this message to the community because it might be biasing our understanding of the relative strengths of different architectures. It is remarkable that this issue persists across multiple datasets.", "weaknesses": "- The degree of deviation from uniformity of the orientations could be summarized in a single number, I don't see that in the paper.\n- It would be very interesting to see a putative explanation for why the way the datasets were generated lead to bias in orientation.\n- It is remarkable that a 3-layer message passing architecture is sufficient to decode the hidden signal in the orientations. I would be interested to see what the minimal architecture is that can do this. Relatedly, it would be interesting to see a how well this strategy works on all the target properties, not just the few cherry-picked ones reported in Table 1.\n- Section 3.2 suggests that to some extent the molecules are aligned by their principle components. A natural question then might be whether a very simple model trained on just the principle eigenvectors could do the same as the message passing network.\n- Of course it would be great to see how each of the models mentioned in the paper (both fully equivariant and non-equivariant) actually do on a randomly rotated version of these benchmarks, but I understand that running all the models is somewhat outside the scope of this paper.", "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": "gU6N3BBdEr", "reviewer": "Reviewer_YdZo", "summary": "The paper studies the prevalence of canonicalized poses in commonly used molecular datasets (QM9, QMugs, OMol25). The authors use a simple classification test to distinguish between molecules in their original orientation in the dataset and ones that are randomly rotated. They also present visualization of principal components of molecular orientations for each dataset and the distributions of relative rotation angles between molecules. Finally, they demonstrate that one can use the orientation alone to regress on molecular properties and that this achieves better performance than using randomly rotated features.", "questions": "In Table 1, were any other QM9 properties explored? It would be interesting to see if the performance varies per property (although I expect it would not).\n\nFor Figure 6, it would be helpful to label each plot in a larger font with the chemical property being plotted.\n\nIn the conclusion L438, the authors state “it is essential to evaluate equivariant models on a randomly oriented test set as a sanity check for true equivariance.” Would it not be more robust to test the equivariance error of the model? E.g. for some sample $x$ and a rotation $R$\n$$\n\\epsilon(x, R) = \\frac{\\| f(R \\cdot x) - D(R) \\cdot f(x) \\|}{\\| f(x) \\|}\n$$\nand then average over multiple rotations/the data distribution.\n\nCould the authors elaborate on/connect their work to prior work on functional vs. distributional symmetry breaking? E.g. see [5]. Functional symmetry breaking is where the mapping between inputs and outputs is not fully equivariant. Approximately equivariant models have mostly been applied in this setting. The setting studied in this paper is distributional symmetry breaking, where a datapoint $x$ and its transform $Rx$ are not equally likely under the data distribution. This is also stated in [1]. Given that approximately equivariant models are built for the setting of functional symmetry breaking, it is not clear that they “might artificially inflate their performance by exploiting extrinsic canonicalization” (L082).\nIn general, I would like the authors to relate their work to the classifier test presented in [1] and explain the novelty of their work.\n\n[1] Lawrence et. al, Detecting Symmetry-Breaking in Molecular Data Distributions, https://openreview.net/forum?id=yEvdOXW5iY#discussion\n\n[2] David Lopez-Paz and Maxime Oquab. Revisiting classifier two-sample tests. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?\nid=SJkXfE5xx.\n\n[3] Kenny Chiu and Benjamin Bloem-Reddy. Non-parametric hypothesis tests for distributional group symmetry. In NeurIPS AI for Science Workshop, 2023.\n\n[4] Ashkan Soleymani, Behrooz Tahmasebi, Stefanie Jegelka, and Patrick Jaillet. A robust kernel\nstatistical test of invariance: Detecting subtle asymmetries. In Yingzhen Li, Stephan Mandt,\nShipra Agrawal, and Emtiyaz Khan (eds.), Proceedings of The 28th International Conference\non Artificial Intelligence and Statistics, volume 258 of Proceedings of Machine Learning Research, pp. 4816–4824. PMLR, 03–05 May 2025. URL https://proceedings.mlr.press/v258/\nsoleymani25a.html.\n\n[5] Rui Wang, Elyssa Hofgard, Robin Walters, and Tess Smidt. Discovering symmetry breaking in physical systems with relaxed group convolution. arXiv preprint arXiv:2310.02299, 2024c.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper is well written, organized, and easy to follow.The experiment with property regression from molecular orientation is quite interesting and illuminates that one should be cautious when using non-equivariant models (the model could memorize spurious information contained in the molecular orientation). The visualizations of correlations of molecular orientation with chemical properties (Figure 6) are also quite interesting. In general, the paper makes a good point that one should be cautious about data pre-processing and pre-existing canonicalizations in the data.", "weaknesses": "A paper at the ICLR 2025 AI4Mat workshop (posted Mar 3, 2025) essentially presented the same idea [1] of a two-sample classifier test to detect whether a dataset is canonicalized or not, also applied to molecular datasets. [1] introduced the idea that these canonicalization biases may exist in molecular datasets. Furthermore, [1] showed that orientation of molecules in commonly used materials science benchmarking datasets is non-uniform (applied to QM9, MD17, OC20) and suggested that non-equivariant models may be strongly benefiting from canonicalization of the molecules’ orientations. This work seems to be quite similar, so I am not sure of the novelty ([1] also visualizes the distribution of principal moments of inertia for QM9 in the appendix). The framing and setup of [1] is very similar, but the authors of this work do not cite or mention this prior work. [1] also provided a more detailed treatment of the classifier test, showing how to compute p-values for a test of level $\\alpha$ and validation of classifier performance, which this paper does not discuss. The authors should clarify the novelty of their work relative to [1].\n\nAdditionally, in relation to section 3.1, two-sample classifier tests have already been presented in the literature, so I believe [2] should be cited. There also exists literature on non-parametric hypothesis tests for distributional group symmetry and statistical tests for invariance [3,4], so these works should be cited or discussed. \n\nWhile the visualization of molecular principal components are quite aesthetically pleasing, I am also not sure of their novelty. It is reasonable to expect that a machine learning practitioner would perform PCA on their data or visualize certain molecules prior to training a model, but much of the paper seems to rely on these visualizations. For example, take the observation that “chemically similar molecules tend to have similar canonical orientations.” For QM9, this most likely follows directly from the way conformers are generated (via CORINA from SMILES strings). Since CORINA uses deterministic geometry generation rules, this would be expected (the authors also state this). Therefore, I am not sure of the novelty of this finding/if it needs to be presented in the main part of the paper. For the other datasets (QMugs and OMol25) the authors should add further information as to why they think it has an orientational bias (L155). \n\nFrom my understanding, equivariant models generally perform best for molecular property regression on many of these datasets (outperforming non-equivariant models e.g. https://benchmarks.rowansci.com/). If there is useful information contained in the molecular orientations, shouldn’t we expect the non-invariant models to do better? There are no experiments comparing equivariant, non-invariant, or approximately equivariant models to support the claims such as “architectures that introduce symmetry breaking or are only approximately equivariant…might artificially inflate their performance by exploiting the extrinsic canonicalization” (L082). Thus, it is hard to draw meaningful conclusions. To strengthen the paper, it would be good to have more experiments comparing models on molecular property prediction.\n\nThe test in Section 3.3 is interesting, but the claim it is “empirical proof that canonical orientation alone holds information about a molecule’s properties” (L355-356) is misleading. Would it be possible for the model to effectively memorize the poses in association with properties, as the orientation correlates with the type of molecule? For example, $\\epsilon_{LUMO}$, ZVPE, and $c_V$ are all invariant properties, so the orientation physically should not hold information about the property. Rather, it should be made clear that this is a dataset level artifact. The model is essentially learning biases in the dataset’s conventions, rather than some physically meaningful relation between orientation and molecular property. I believe this may be what the authors meant but clarification would be helpful.\n\nI believe these weaknesses, particularly in terms of novelty, are somewhat substantial.", "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": "JUxLKGqLTp", "reviewer": "Reviewer_pzKZ", "summary": "This paper provides clear evidence of subtle leakage in major molecular conformation datasets used for public comparisons of machine learning applications to drug discovery. Notably, the leakage arises because molecules with similar properties have similar geometric orientations in these datasets. The authors demonstrate this defect with increasing levels of rigor: from simple visualization, to discriminating rotated vs unrotated molecules with a simple classifier (even in the presence of substantial noise), to comparing the densities of orientational distributions.  Importantly, the paper shows that simply reading a molecule's orientation enables property prediction with a substantial reduction in variance compared to the test set mean. Finally, the paper includes equal-area Mollweide projections of the overall orientational distributions in small molecule datasets that show the relative degree of the lack of true randomness across datasets.", "questions": "Due to the possibility of reflection symmetries in small molecules (think small symmetric fragments), exact degeneracies can arise from their geometries. How do the authors handle degeneracies in the ranking of the principal components or the max function in Eq 3?\n\nFig 4 convincing and clear. I'd recommend that future datasets randomize orientations and include this analysis and the Mollweide equal-area plots.  Did the authors perhaps also try to use a simple summary metric of Fig 4, e.g. the distance between the actual vs intended CDE of the distributions?  Would the ranking of such a summary metric agree with the visual ranking in Fig 7?\n\nDoes this work imply issues in any specific prior publications that used symmetry-breaking, approximate-equivariant, or non-equivariant methods that relied on experimental evaluations? Highlighting any suspicious references (even in supplements or the openreview discussions) could benefit the community, though I understand that it's not the paper's main focus and nobody likes to be that guy.", "limitations": "", "rating": 10, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "This paper is original and significant because it analyzes a subtle effect that might prematurely and incorrectly influence the selection of models based on their observed performance. It is particularly timely, as limitations imposed from forced SE3 equivariance in existing architectures could be replaced by learned equivariance with faster/stronger architectures and larger datasets. Unfortunately, careful preprocessing of the standardized datasets is often avoided, but the magnitude of the effects shown in this paper clarify that such practice is unacceptable going forward. The quality of the work is high; as a subtle example: the authors apply single fixed rotations to un-randomized molecules when demonstrating their orientation-based property predictors to minimize any accidental deviation in their data pipelines. The flow of the presentation is clear and the message is strong. Duly Noted!", "weaknesses": "The main weakness is the lack of accompanying code for reproducibility, which should be resolved prior to publication. The lack of code would unfairly reduce the applicability of this work in future dataset preparations and benchmarks (although the implementations appear simple, rotations warrant care.)  A minor weakness is the hard-to-discern densities in some combined Mollweide plots.  While the lack of uniformity across datasets makes them memorable, comparing Fig 5 to 6 I wonder if the order of plotting the dots in Fig 6 is deterministic, rather than random, and hides information. A suggestion: perhaps also try a more distinct/color-blind friendly color palette (#0072B2 #E69F00 #CC79A7) and add a combined QM9 plot in the supplement for completeness.", "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": "jgOEJglweP", "reviewer": "Reviewer_F9xk", "summary": "The paper describes the inherent bias present in the open-source molecular datasets (QM9, QMugs, and OMol25) in terms of molecular poses when applied to the task of predicting properties of these molecules.", "questions": "- Can the authors describe how the random rotation operation is being executed in Sec 3.3, as it's not fully clear how it is applied to the canonical molecular pose? Is it a global or local transformation applied to the canonical position?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The issue of dataset-induced bias in molecular ML is novel, especially as orientation-invariant architectures (e.g., SE(3)/E(3)-equivariant networks) are increasingly used.\n- The Mollweide projections and density analyses effectively highlight the non-uniformity of molecular orientations.", "weaknesses": "- I find the paper’s argument about “pose bias” somewhat unclear. The 3D molecular geometries in datasets like QM9, QMugs, and OMol25 are generated using physics-based computational chemistry software, and therefore correspond to energetically stable conformers of each molecule. These conformations are naturally determined by the molecule’s 2D topology, atom types, and underlying energy landscape. From a physical standpoint, certain configurations or poses are expected to be more stable and thus more frequently observed — similar to how structures in the Protein Data Bank (PDB) reflect biologically preferred conformations. Given this, it seems there may be a misinterpretation between “pose bias” (a coordinate or orientation artifact) and physical realism (the fact that molecules adopt specific stable conformations). Could the authors clarify whether the observed “pose bias” truly represents an artifact of how datasets are stored and oriented, or whether it simply reflects physically meaningful conformational preferences inherent to molecular systems?\n\n- I understand that, when developing methods for downstream applications on these molecular datasets, it is important to account for how global molecular poses relate to their representation in Cartesian coordinates. However, most state-of-the-art molecular machine learning architectures are E(3)-equivariant or invariant, meaning they operate on geometric quantities such as interatomic distances, angles, and relative orientations, rather than on absolute Cartesian coordinates. Consequently, these models are inherently insensitive to global rotations and translations of the molecule. This raises the question of how practically relevant the reported orientation bias is, given that such equivariant models should, by design, be unaffected by the dataset’s global coordinate frame.\n\n- I think it would be more beneficial to check if this so-called \"pose bias\" is demonstrated in the models that have been trained on these datasets such as EGNNs, etc, to see if they are also susceptible to the variations wrt random rotations.", "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)