Learning Molecular Chirality via Chiral Determinant Kernels
Authors: Runhan Shi
Zhicheng Zhang
Letian Chen
Gufeng Yu
Yang Yang
Venue: ICLR 2026 Poster (2026)
Keywords: chirality
molecular representation learning
axial chirality
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Chirality is a fundamental molecular property that governs stereospecific behavior in chemistry and biology. Capturing chirality in machine learning models remains challenging due to the geometric complexity of stereochemical relationships and the limitations of traditional molecular representations that often lack explicit stereochemical encoding. Existing approaches to chiral molecular representation primarily focus on central chirality, relying on handcrafted stereochemical tags or limited 3D encodings, and thus fail to generalize to more complex forms, such as axial chirality. In this work, we introduce \textbf{ChiDeK} (\textbf{Chi}ral \textbf{De}terminant \textbf{K}ernels), a framework that systematically integrates stereogenic information into molecular representation learning. We propose the chiral determinant kernel to encode the SE(3)-invariant chirality matrix and employ cross-attention to integrate stereochemical information from local chiral centers into the global molecular representation. This design enables explicit modeling of chiral-related features within a unified architecture, capable of jointly encoding central and axial chirality. To support the evaluation of axial chirality, we construct a new benchmark for electronic circular dichroism (ECD) and optical rotation (OR) prediction. Across four tasks, including R/S configuration classification, enantiomer ranking, ECD spectrum prediction, and OR prediction, ChiDeK achieves substantial improvements over state-of-the-art baselines, most notably yielding over 7\% higher accuracy on axially chiral tasks on average.

Published as a conference paper at ICLR 2026
LEARNING MOLECULAR CHIRALITY VIA CHIRAL DE- TERMINANT KERNELS
Runhan Shi', Zhicheng Zhang', Letian Chen' Gufeng Yu', Yang Yang AGI Institute, School of Computer Science, Shanghai Jiao Tong University
Shanghai Innovation Institute
{han. run. jiangming, zhi zhicheng. zhang, clt2001, jm5820zz)@sjtu. yangyangecs. jjtu. edu.
ABSTRACT
Chirality is a fundamental molecular property that governs stereospecific behavior in chemistry and biology. Capturing chirality in machine learning models remains challenging due to the geometric complexity of stereochemical relationships and the limitations of traditional molecular representations that often lack explicit stereochemical encoding. Existing approaches to chiral molecular representation primarily focus on central chirality, relying on handcrafted stereochemical tags or limited 3D encodings, and thus fail to generalize to more complex forms such as axial chirality. this work, we introduce ChiDeK (Chiral Determinant Kernels), a framework that systematically integrates stereogenic information into molecu- lar representation learning. We propose the chiral determinant kernel to encode the SE(3)-invariant chirality matrix and employ coss-altention integrate stere- ochemical information from local chiral centers into the global molecular repre- sentation. This design enables explicit modeling of chiral-related features within a unified architecture, capable of jointly encoding central and axial chirality. To support the evaluation of axial chirality, construct new benchmark for elec- tronic circular dichroism (ECD) and optical rotation (OR) prediction. Across four tasks, including R/S configuration classification, enantiomer ranking, ECD spec- trum prediction, and OR prediction, ChiDeK achieves substantial improvements over state-of thettr- baselines, most notably yielding over 7% higher accuracy on axially chiral tasks on average.
INTRODUCTION
Chirality, the property of molecules that are non-superimposable mirror images, fundamental concept in chemistry, biology, and materials science (Moss) 1996). It plays central role in shap- ing molecular behavior across diverse applications, including biomolecular recognition (Ceramella
et al. 2022), stereoselective synthesis (Gaucherand et al. 2024), and enantioselective catalysis (He ct al. 2023). Chiral molecules, often existing as pairs enantiomers, can exhibit drastically differ- chemical and biological properties despite sharing identical compositions (Peluso & Chankve- tadze 2022). Such differences are especially critical drug efficacy (Liu et al. (003b)). toxic- ity (MeVicker & O' O'Boyle 2024), and protein-binding affinity (Schneider al.] 2018]] Shen al.)
2013). underscoring the paramount importance of accurately modeling stereochemical information molecular representation learning.
Despite this significance, capturing chirality in machine learning models remains challenging. Ex- isting molecular representation learning (MRL) approaches predominantly focus central chirality and often rely on implicit encoding strategies that fail to extract explicit stereochemical features. While recent advances in SE(3)-invariann architectures have shown promise through the incorpora- tion of torsion angles (Klicpera et al. 2021 Liu et al. (222) and explicitly chirality-wwar designs (Adams et al.) [0222] Yan et al.) (025). research on comprehensive chirality modeling within MRL. remains limited. More complex stereogenic forms, such axial, planar, and helical chirality, are largely unaddressed, representing a significant gap current methodologies.
Corrreponniigg author.

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Published as a conference paper at ICLR 2026
(a)
(b)
HO
HO
Chiral atons
Chiral-rolated atoms
Enantioners (nirror images)
Examples of central chirality (a) and axial chirality (b). The configuration of the chiral atoms (in green) is determined by the spatial arrangement of chiral-related atoms (in orange).
Figure 1:
Among the various forms of molecular chirality, central chirality and axial chirality represent the two most prevalent categories. Central chirality arises when a tetrahedral atom (typically carbon) is bonded to four distinct substituents. Such an atom constitutes a chiral center, and its stereo- chemical configuration (R or S) (Cahn et al., 1966. Favre Powelll 2013) determined by the spatial arrangement of its substituents, as illustrated in Figure aa). Axial chirality typically results from restricted rotation around a chemical axis, often bond connecting two aromatic or aliphatic groups. Here, chirality (Ra/M Sa/P) is determined by the relative arrangement of substituent groups around the stereogenic axis, as shown Figure[[b)
Existing computational approaches encounter fundamental limitations in capturing chirality. H(3)- invariant architectures that rely exclusively on pairwise distances or bond angles, such as SchNet (Schütt et al.) 2017) and DimeNet (Gasteiger al. 2020b a). are inherently incapable of distin- guishing enantiomers since these geometric quantities remain unchanged under reflection. While SE(3)-invariant models, including SphereNet (Liu et al.] (2022) ChiRo (Adams et al. (022). and ChiGNN (Yan et al.] 2025), address this limitation through directional information and equivariant features, they face critical challenges. Most notably, molecular graphs typically treated as homo- geneous structures where chiral atoms are not explicitly differentiated, causing stereochemical sig- nals be diluted during global aggregation. Recent approaches like ChiGNN attempt to incorporate chirality through permutation-resot resolution strategies, but they still fail to extract expressive, localized descriptors that faithfully characterize chiral centers or stereogenic axes. These constraints highlight the urgent need for unified framework capable of systematically modeling diverse chirality types while explicitly encoding stereochemical features that govern molecular behavior.
To address the above limitations, we propose the ChiDeK model (Chiral Determinant Kernels), which learns molecular chirality based on the determinant the chirality matrix and applies cross- attention between chiral and non-chiral atoms to explicitly capture stereogenic information. The introduced chiral determinant kernels embed the SE(3)-invarizn chirality matrix (Shi et al. (026) for each chiral atom, capturing stereogenic information for both central and axial configurations. A cross-attention mechanism enhanced by pair representations then propagates the influence of chiral atoms (queries) with chiral-related and non-chiral atoms (keys and values) into the global molecular representation, enhancing stereochemical sensitivity. Our main contributions are:
We introduce ChiDeK, unified architecture that systematically encodes both central and axial chirality, and evaluate it comprehensively on multiple chirality-aware prediction tasks.
We design the chiral determinant kernel for chiral atoms that embeds the SE(3)-invariant chirality matrix, effectively capturing stereogenic features for chirality.
We construct and release dataset of axially chiral molecules for the prediction of elec- tronic circular dichroism (ECD) and optical rotation (OR), providing benchmark for this underexplored stereogenic type.
Across multiple tasks, including R/S classification, enantiomer ranking, ECD prediction, and OR prediction, ChiDeK demonstrates substantial improvements over state-of-the. the-art art baselines, particularly for the axial chirality task.
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2 RELATED WORK
Learning 3D Representations of Molecules. Recent advances in molecular machine learning have heavily leveraged 3D geometric message passing and attention architectures, where interatomic in- teractions are modeled via internal coordinates such as distances, angles, and torsions. SchNet (Schütt et al.) 2017). pioneering architecture for predicting quantum mechanical properties, param- eterizes messages using radial basis expansions of pairwise distances. DimeNet (Gasteiger et al.) 2020b) and DimeNet++ (Gasteiger al., (2000a) extend this framework by incorporating angular features through spherical Bessel functions, while DeepH-E3 (Gong et al. (2023) learns Hamiltoni- from density functional theory (DFT) calculations. BOTNet (Batatia al. 2025) further sim- plifies neural equivariant interatomic potentials (NequlP) (Batzner et al. 2022) a high-performing equivariant message passing framework. These models are designed be E(3) iivariat,,,,,, which are robust to translations, rotations, and reflections. While this invariance is desirable for many prop- erty prediction tasks, enforces reflection -ymmmmttt--ymmttt.......... rendering such models unable to distinguish enantiomers that differ only by mirror inversion (Satorras et al. [021] Dumitrescu et al. (025).
To remedy this shortcoming, SEv3)-invariant architectures have been developed. which maintain sensitivity to reflections. Tensor Field Networks (Thomas et al. (018)) and
(Fuchs al. 2020) achieve equivariance through the use of spherical harmonics and irreducible representations. Path-MPNN (Flam-Shephen) et al. 2021), GemNet (Klicpera et al.] (202T), and SphereNet (Liu et al. 2022) further enhance 3D GNNs by embedding angular and torsional infor- mation. More recently, MoleculeSED (Liu et al. 2023a) introduces reflection-sensitiee stochastic differential equations for molecular modeling.
Explicit Representation Learning of Molecular Chirality. A number of studies explicitly in- corporate stereochemical information into molecular representations. Early efforts introduce hand- crafted descriptors such the chirality code (Aires-de Sousa al.. 20004), physicochemical atomic stereo (PAS) descriptors (Zhang & Aires-de Sousa (206)) relative chirality indices (RCI) (Natarajan) al. 2007). and chiral cliffs (Schneider al.J 2018). More recently, MAP4C (Orsi & Reymond
2024 extends molecular fingerprints to capture stereochemical tags directly. Linear notations such
as SMILES (Weininger 1988), SELFIES (Krenn et al., 2022). and fragSMILES (Mastrolorito al.
2025) encode molecules as sequences that can include chirality tags, enabling models like Trans- formers (Vaswani al.) (017)) to process stereochemical information (Yoshikai et al.. [0224] Yang
et al. (005)). However, these tags serve only as symbolic indicators without explicit 3D structural context, making the resulting representations less informative for capturing stereogenic geometry. Graph neural networks have also been adapted to represent chirality more explicitly. Tetra-DMPNN (Pattanaik et al. 2020) modifies message passing in 2D GNNs with asymmetric updates to en- code tetrahedral chirality. SPMS (Xu et al. 2021 introduces spherical projection descriptors with convolutional architectures for stereostructure modeling. ChiRo (Adams et al.. (022) learns 3D rep- resentations invariant to bond rotations yet sensitive to stereoisomers. MolKGNN (Liu al. (023c) leverages the tetrahedron volume calculation to capture the neighbor ordering for chirality. CFFN (Du et al. 2023) combines 2D graph topology and 3D geometry to build strreochemistty-awware
molecular representations. GCPNet (Morehead & Cheng 2024) develops geometric representations tailored for biomolecular graphs. ChiENN (Gainski et al., (023) and ChiGNN (Yan al.) 2025) explicitly consider atom permutations related to chirality to improve stereochemical discrimination. ChiralCat (Peng et al., 2025) integrates 3D) molecular structures with A...--gnnrraee textual de- scriptions to perform chiral type classification.
Despite these advances, the vast majority prior work focuses primarily on central chirality. Other stereogenic forms, such as axial chirality, remain largely underexplored, limiting current models' ability to generalize across the full spectrum of stereochemistry. Although methods such as SPMS theoretically capable of representing additional forms of chirality, they have not been systemati- cally evaluated in these settings, leaving their practical effectiveness uncertain.
PRELIMINARIES
3.1 CHIRAL AND CHIRAL-RELATED ATOMS
To construct a meaningful molecular representation that explicitly encodes chirality, we distinguish between chiral atoms and their corresponding chiral-related atoms (Figure T] Given molecule
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with M atoms, where the index set of atoms is == (I....,Mf} define three disjoint subsets
Ics I. = Ub:JEN()) Voo
Iz: In - I (Io UI,).
(I)
iEI.
where Iz. Iy,, and In c are the index sets of chiral, chiral-related, and non-chiral atoms, respec- tively, and (i) denotes the index set of substituents around chiral atom
For central chirality, chiral atom E Ie typically has four different substituent groups. The corresponding four neighboring atoms define its chiral-related set (i)). For axial chirality, chirality arises along a stereogenic axis (e.g., in biaryls), where steric hindrance prevents free rotation. In this case, the substituent groups are partitioned into four distinet sets around the axis, and select the nearest atom from each group as the chiral-related atoms.
3.2 CHIRALITY MATRIX
To explicitly encode stereogenic information, we adopt the chirality matrix introduced in Chi- ralFinder (Shi et al., 2026). Formally, for chiral atom i E Ie with ordered substituents ((i) (ri,1, F2,,, f1,3, T2,4), the chirality matrix is defined as
Mc(1) ((0r,,1 -x;) x.) (ær,,
vi)
(Xris Fria))
(2)
where x; R denotes the 3D position of the chiral atom and ær,, denotes the position of its j-th substituent. The corresponding chirality product is then defined as
Po(i)= ((#r.. x.) (ærr,a -x:)) (IFr.. - Fr..)) det (Mc(i)),
(3)
where denotes the vector product, denotes the dot product, and det(-) is the determinant of a given matrix. Appendix DrpeendixD.IIppooee provides details on how (()) is determined.
Proposition 3.1. Given chiral atom i, the chirality product Pc(i) is invariant under rigid-body translation and rotation, and changes sign under reflection:
det (Mc(1)) = det (R1Mc(i)), det (R2Mc(1) det (Mc(i)).
(4)
where R1 E SS((3) denotes any rigid-body motion with rotation r E SO(3) and translation E TR3,
and R2 E E0-(3) (3) denotes any reflection (orthogonal transformation with determinant -1).
Lemma 3.1. Let (2) be a pair of enantiomers differing only in the stereochemical configuration of atom i. Then
R.
det(Mc > 0,
configuration())
(5)
S,
det dec(M(()) <
Proof of Lemma 3.1 provided in Appendix
F.1
METHOD
4.1 CHIDEK FRAMEWORK
The proposed ChiDeK framework, illustrated in Figure 2aa), , is designed to capture stereochemi- cal information by explicitly modeling chiral atoms through chiral determinant kernels. Given molecule = (X, H), where X (æI..... ...) E R3M specifies the 3D coordinates of the (hi..... har) RdM encodes atom-level features, we construct a molecular
M atoms, and graph G (V, E). where the node set V is partitioned into chiral atoms, chiral-related atoms, and non-chiral atoms, and the edge set & incorporates geometric distances between atoms.
The forward pass of ChiDeK proceeds in three stages: (1) a chiral encoder computes chiral-sensitive embeddings for chiral atoms via chiral determinant kernels, while three separate multi-layer projec- tors compute chiral-invarios multi-layer chiral pross-attenticn mechanism refines embeddings chiral atoms (queries) by at- tending to heterogeneous neighbors (keys/values), guided by distance-awar Gaussian Kernel with Pair Type (GKPT) biases; (3) the resulting chiral-aware representations are aggregated and passed through task-specific heads (e.g., chirality classification, spectrum regression).
embeddings for chiral, chiral-related, and non-chiral atoms; (2) the
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Predictor
Mc(c) (g33
Longurs
Lómear-K
Limuur-Q
Transformer
Lincar & Normalize
MatMul & Mask
Feed forward
B (((xxxx3)
p
Scale
Layer
QR-cocomposition
MatMal
Cross-atteziion
X
Kemels
Finheddings
Eibbodding Layer
Chiral Encoder
R E ((x3x3)
Chinal
Determinant
Input
DeD R
Non-chtral
(a) ChiDeK architccture
(b) Chiral encoder
(c) Chiral cross-attention
Figure 2: Overview of the ChiDeK architecture (a). consists of a chiral encoder (b), chiral trans- former incorporating T. cross attention layers (c), and predictor for predicting chiral properties.
4.2 CHIRAL ENCODER
key requirement for learning molecular chirality is to design features that are sensitive to reflec- tions but invariant global translations and rotations. The classical determinant matrix Mc(i) provides exactly such a property, as it encodes the signed volume spanned by three bond vectors around a chiral atom. To generalize this construction into a high-dimensional latent space, we introduce the chiral encoder based on chiral determinant kernels.
3 chirality
Chiral Determinant Kernel. To explicitly represent chiral atoms with stereogenic information, introduce the chiral determinant kernel, a determiinnt--baee -basd kernel that encodes the oriented volume spanned by triplets of projected atomic vectors with QR decomposition.
Let Mc E IR 3x333 denote the batch of chirality matrices and W E R the learnable weights, where k is the number of kernels and d, is the projection dimension. After broadcasting M6 R xxx33x and W BBBxxxd x3
we compute
Ob
W O Bxxxxxx3
where M E 3xx3 the b-th chirality matrix, W E R4 the k-th kernel, and multiplication standard matrix multiplication along the last two dimensions. Then, we reshape the output O from Bxkxd, O' (d, x3, Applying layer normalization along the dp dimension yields
Ô LayerNorm (O') (BK)xxxxx
Next, we perform QR decomposition for each slice:
Ô ((kk) = ((x)) R(RR))
Q (ha)) (bR) = Is. Q(bR) R4
R(ba))
3x3
(8)
for all b (1...., B} and & E {1,..., k... We then compute the determinant of each factor:
Deb. det (R(n)), D R Rxn
(9)
Thus, each chiral atom is embedded into a k-dimensional representation that generalizes the determi- nant, provides differentiability (Roberts & Roberts (020), and explicitly encodes stereochemistry. Then the initial hidden representations of chiral, chiral-related, and non-chiral atoms are
H. D. fa (HI.), H. fr (Hz.), H (Hz.),
(10)
where fer fr. fn are embedding layers for atomic features. We finally prepend a learnable global chiral token before chiral atoms enhance chiral representation.

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4.3 CHIRAL TRANSFORMER
Gaussian kernel with pair type (GKPT). Distances between atoms provide crucial stereochemical cues. To encode them, we adopt GKPT (Shuaibi al. 2021 Zhou et al. 2023). Formally, for input distance x and pair type e, the GKPT is defined as
GKPT ((2,, c), 14,0 = D (c) . x H.)))
(11)
1 / exp (+=") E1, E2 E N. are learnable embeddings, No
where 9 0))
the number of pair types, and μ, R parameterize the Gaussian kemels. Here, we distinguish between two types of edges: chiral atoms with chiral-related atoms and with non-chiral atoms. The GKPT module applied geometric distance Tay with edge type Cijs between atom Ie and j E I, UI,, producing the bias term
Pg = GKPT ((11)) Cig), 14, σ Upp
(12)
with learnable projection wp. The resulting pairwise bias matrix is
p0)
Pij
E IR (Z1
(13)
This pairwise representation encodes chemically relevant interactions, informing the subsequent attention computation and enabling the model to learn complex spatial dependencies.
Chiral Cross-attention. We introduce a cross-attention mechanism enhanced by pairwise repre- sentations, where chiral atoms serve queries, while chiral-related and non-chiral atoms provide keys and values. Let H., H,, and H denote the hidden representations chiral atoms Iz, chiral- related atoms I, and non-chiral atoms In, respectively. At each layer l (totally L. layers), the linear projections for keys and values are separated according to atom types and defined as
(2) = -H.W K,
V(0) H.,W H.W
K(O = H,, W (E)T K.
( = H WOT Va
(14)
where W K., w:( Who K., and W E k are learnable projection matrices. Let H the output at layer E, the resulting queries, keys, and values are
denotes
( = W!O K (
K V [V70) V
(15)
where w E Rick is the learnable query projection, H (0) H and [ denotes concatenation.
At each layer f, we maintain a learnable pairwise bias Pij between atom i (chiral) and atom (chiral-related or non-chiral), updated via the query-key interaction. See Appendix C foor details.
(
Predictor. We utilize lightweight predictor to predict chirality properties. It consists of two linear layers with GELU activation Hendrycks Gimpel (2016). This simple yet effective architecture maps the final global representation to the target prediction space.
4.4 THEORETICAL ANALYSIS FOR CHIRAL ENCODER
The chirality product obtained from the chirality matrix (Shi et al. 2026) exhibits inherent sensitivity stereochemical variation. We demonstrate that a chiral determinant kernel maintains this desirable characteristic.
Let Mc(i) E 3333 be the chirality matrix of atom i, and let W E 124,33 be a full column-rant projection. Define QR decomposition of 0 W Mc(i) E R4,x3 as
Lemma 4.1.
0 O=QR,
Q===1 1s, Q xd X3
R 3x3
(16)
Then the generalized chirality product Pw (i) = del R satisfies
Pw(1) = a: W) Pe(i).
(17)
where Pi(i) is the chirality product, and a (W) > is a scaling factor depending only on W but not on the atom coordinates. Pw (i) satisfies Proposition and Lemma 3.1

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The proof of Lemma 4.1 is provided in Appendix F2
Discussion on rank deficiency. If W is not full column-rank, then becomes rank-deficient and the determinant det(R) [)) degenerates to 0, resulting in the loss of chirality information. While the magnitude (et((R) depends on of (.), the sign remains a reliable indicator chiral configura- tions long W has full column rank. Hence, ensuring rank sufficiency of W is critical for stability and discriminative power. To guarantee full column rank of W, we adopt two strategies in practice:
Weight regularization. We introduce a penalty term on the weight as part of the loss function:
Leg | W -3
(18)
encouraging W to preserve rank during training.
Auxiliary QR on weights. Alternatively, we apply QR decomposition directly on W before pro- jection, ensuring that projected neighbor vectors remain linearly independent.
5
EXPERIMENTS
5.1 MAIN EXPERIMENTS
Datasets. We evaluate our method on both centrally and axially chiral molecules using multiple benchmarks, shown in Appendix Table 4 R/S classification and enantiomer ranking datasets for central chirality are from ChIRo (Adams al., 2022). ECD prediction for central chirality is from CMCDS Li al. 2025). ECD and OR prediction for axial chirality are constructed by ourselves, named ACMP (axial chiral molecular properties). See Appendix A for details about dataset processing, construction, and splits.
Baselines. We benchmark ChiDeK against both an E(3)-invariant baseline, DimeNett+ Gasteiger al.) (2020a). and several SE(3)-invariant baselines, including SphereNet (Liu et al. 2022), ChiRo [Adams et al. 2022), ECDFormer (Li et al. 2025), and ChiGNN (Yan et al. 2025). We also consider
the 2D chiral GNN Tetra-DMPNN with permutation (p) and concatenation (c) variants (Pattanaik
et al. 2020), and the stereostructure descriptor- SPMS Xu et al. (0221.
Implementation Details. All experiments are implemented PyTorch (Paszke et al. 2019 and conducted on NVIDIA RTX 3090 GPUs. See Appendix for more training details.
5.1.1 EXPERIMENTAL RESULTS FOR CENTRAL CHIRALITY
Table I:
R/S classification, enantiomer ranking, and ECD prediction results for central chirality. Best and second-bes best are marked.
R/S (%)
Ranking (%)
Position
Number
Symbol (%
Method
Ace f
Acc 1
RMSE +
RMSE +
Acc t
DimeNet++
65.7 + 2.9
58.4 + 0.2
2.12 + 0.19
1.04 + 0.15
50.8 + 0.1
Detra-DMPNN (c)
99.7 + 0.1 0.1
70.1
2.38 + 0.12
1.25 + 0.09
50.0 0.1
Satra-MMPN (p)
0.1
6.66
2.36
0.16
1.28 + 0.08
50.0 + 0.1
SphereNet
98.2
6.3
0.15
1.02 + 0.11
51.9 + 0.3
ChiRo
0.2
7.5
0.13
1.22 0.13
51.0 + 0.4
ECDFormer
9.2
5..
0.10
1.01 + 0.09
50.3 0.3
ChiGNN
0.9
59.
1.24 2.19
49.9 0.5
SPMS
81.4
60.4
2.58 0.17
1.22 0.12
50.9 0.3
ChiDeK (Ours)
99.8 + 0.1
72.8 0.2
0.14
1.18 + 0.09
53.3 + 0.6
R/S Classification. Table I reports the results. ChiDeK achieves the best performance close to 100%. This clearly demonstrates the effectiveness of our chirality- design. In contrast, the E(3)-invariant DimeNet++ completely fails on this task, which is as expected. We present a visual- ization of cross attention weights in Appendix G Note that, emphasized in prior work (Adams)

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et al. 2022), this R/S classification task is necessary but not sufficient for meaningful chiral learn- ing. Nonetheless, strong performance on this benchmark is an essential prerequisite for chirality- sensitive applications.
Enantiomer Ranking. Table T demonstrates that ChiDeK achieves optimal performance, surpass- ing the second-best method ChiRo by 0.8%. The modest performance gap between ChiDeK and ChiRo can be attributed to the limited structural diversity the dataset, as all molecules contain only single chiral center.
ECD Spectrum Prediction. Since the number and positions of peaks remain invariant across enan- tiomers while the symbols of peak intensities are inverted, accurately predicting peak heights repre- sents the most critical aspect of ECD spectrum modeling. A model that successfully recovers peak counts and positions but fails distinguish peak height symbols is ineffective and impractical. Con- sequently, we require a model that jointly predicts all three components, providing comprehensive ECD spectrum prediction. As shown in T ChiDeK achieves optimal performance in predicting peak height symbols while maintaining comparable accuracy for peak positions and counts.
However, the sign accuracy remains only slightly above 50% across all models. We hypothesize that this is largely due to datasel inconsistenciex: the original CMCDS dataset provides RDKit-generatee (Landrum) 2025) conformers, whereas the ECD labels are computed from optimized geometries. This mismatch causes poor performance. This underscores the inherent difficulty of reliable ECD spectrum modeling and suggests that consistent datasets are essential for more robust evaluation.
5.1.2 EXPERIMENTAL RESULTS FOR AXIAL CHIRALITY
Table 2: OR and ECD prediction results for axial chirality. Best and second-best are marked.
Rotation (%)
Position
Number
Symbol ( (%)
Method
Ace f
RMSE +
RMSE +
Acc 1
DimeNet++ Retra-DMPNN (c)
50.0 0.0
3.62 0.17
50.0 + 0.1
50.0 0.1
3.15 + 0.12
1.20
52.8 + 0.2
Tetra-DMPNT (p)
50.0 0.1
0.10
1.05 +
52.8 0.2
SphereNet
52.5
3.36
1.08 +.099
5..4
ChiRo
5..0
1.11 0.18
0.3
ECDFormer
5.3
+.25
1.16
0.5
ChiGNN
0.1
2.89 + 0.12
1.06 0.133
0.8
SPMS
0.6
3.69
0.21
1.15
60.4
ChiDeK (Ours)
69.2 + 0.5
3.24 0.14
1.05 + 0.12
71.2 = 0.6
OR Prediction. Table 2 shows that ChiDeK delivers the strongest performance, exceeding the second-best method by noticeable 4.2% margin. Notably, all baseline models, except SPMS, achieve accuracy below 55%, indicating their inability to capture axial chirality due to their neglect of stereogenic axes. SPMS, which incorporates spherical stereostructural projections, can partially encode such information. These findings underscore the inherent complexity of the task and high- light the limitations of previous models in accurately representing complex chirality.
ECD Spectrum Prediction. Table 2 reports the results on the axial chirality ECD prediction task. ChiDeK achieves the highest accuracy in predicting the symbols of peak heights, exceeding the second-best model by substantial margin of 10.8%. It also delivers competitive performance in pre- dicting peak positions and numbers. Similar to OR prediction, SPMS exhibits limited performance, while all other baseline models achieve less than 55% accuracy in symbol prediction. For fine- grained analysis, we present performance breakdown across seven -xial-chirality subtypes in Ap- pendix B.I Additionally, we evaluate general molecular property prediction tasks in Appendix B.2 to confirm that our model maintains full expressive power, where chirality is less critical.
Figure B presents representative prediction comparison. ChiDeK successfully assigns opposite peak-heigg height symbols to opposite stereochemical configurations, thereby correctly distinguishing ax- ial enantiomers. In contrast, ChiGNN and Tetra-DMPNN (permute) produce identical peak-height
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Chille
CRIGNN
Tetra-DMPPN (pormute))
Wavelength (mm)
Navelenggt
Wawelength
Dotstory Strenggh oarb. unttl
Wawelength
Wavdlongth
Wervelentth (1mo
Figure 3: An axial ECD prediction example of pair of enantiomers. Each shows the predictions for one configuration across models, with the two rows corresponding to opposite configurations.
symbols across enantiomers, failing to capture stereochemical differences. These results highlight the robustness and stereochemical sensitivity of ChiDeK handling axial chirality. See Appendix for more explanations and prediction examples.
investigate how molecular represen- tations generated by ChiDeK vary under rotation along the chiral axis, systematically vary the torsion angle, with details provided in Appendix E.3 Figure illustrates two representative ex- amples. In both cases, the 18 conformers comprising 9 conformers), consistent with the expected stereochemical symmetry. The trajectory reflects transition from one configuration its opposite and then back the original, while the cosine similarity plots show high similarity within the same configuration and vice versa.
Rotation Analysis of Axial Chirality Representations.
partitioned into two opposite configurations (each
Similarity vs rotation 90*
(b)
Similarity rotation 90
Wajectory
0.0
Trjectory (2D)
0.5
180
-1.0
0
180
-1.0
-5
225
315
10-
225
315"
270
270*
-10
200
Figure 4: Visualization of how ChiDeK representations change under rotations along the chiral axis for two molecules. Each point in the trajectory plot obtained by UMAP (McInnes et al.) 2018) represents the learned embedding at given rotation angle, while each point in the polar plot denotes the cosine similarity between the embedding and the reference at degree O.
5.2 ABLATION STUDIES
We perform ablation studies on the axial ECD prediction task, using the accuracy of peak height symbols the primary evaluation metric. We examine strategies to address rank deficiency (QR and Reg denote QR decomposition and regularization, respectively), types of chiral encoders, and atom separation strategies (Sep.), as shown in Table The ablation of missing mislabeled chiral atoms

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Published as a conference paper at ICLR 2026
is presented in Appendix B.3 Appendix B.4
The comparison with ChiralFinnder-enhannedd model is presented in
Table 3: Ablation model components.
Model Components
Acc (%) 1
Chiral Encoder
Rank strategy
Separate
Linear w/o. Mc
None
51.2 + 1.2
Linear w. Mc
None
/
68.7 0.6
Kernel-based
QR
X
69.8 = 0.4
Kernel-based
Reg
71.0 0.5
Kernel-based None
68.3 + 0.6
Kernel-based
QR
71.2 0.6
(i) Strategies Address Rank Deficiency. Table compares three approaches for handling rank deficiency; regularization (Reg), QR decomposition (QR), and no intervention (None). When han- dling rank deficiency, both Reg and QR outperform None, achieving similar performance.
(ii) Choice of Chirality Encoder. We examine alternative chiral encoder designs in Table A
linear encoder without access Mc performs poorly (close random guessing), indicating that cannot capture chirality. In contrast, our kemel-based encoder yields higher accuracy, confirming its ability to encode stereogenic features effectively.
(iii) Separating Chiral Atom Types. We test the effect of explicitly distinguishing chiral-related atoms (I,) from non-chiral atoms (I,). Models that separate chiral-related atoms and non-chiral atoms outperform those that treat them jointly, highlighting the importance of modeling their distinct roles within stereochemical environments.
CONCLUSION
We present ChiDeK (Chiral Determinant Kernels), unified framework for learning chiral molec- ular representations. By embedding the SE(3)-invaraant chirality matrix through chiral determinant kernels and employing cross-attention between chiral and non-chiral atoms, ChiDeK explicitly cap- tures both central and axial stereogenic features. Across multiple benchmarks, ChiDeK substantially outperforms baselines, particularly for axially chiral molecules. We further contribute benchmark dataset for axial chirality, encompassing both ECD and OR prediction tasks. This dataset provides a foundation for future research in stereochemistry-wwaee aware machine learning. future work, we plan to extend our unified stereochemical representation to additional forms of chirality, such as planar and helical chirality, and evaluate its utility in broader downstream applications, including docking-score prediction for chiral ligand-protein interactions and enantioselectivity prediction in asymmetric catalysis.
ACKNOWLEDGMENTT
This work was supported by the National Key R&D Program of China (No. 2023YFC2811500) and the National Natural Science Foundation of China (No. 62272300).
REPRODUCIBILLTTY STATEMENT
Code and data are available httpppss//gttuub...o////tterrrrr
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A DATASETS
Table 4: Summary of downstream datasets.
Dataset
Chirality
Task
Size
Split
ChIRo (R/S)
Central
R/S classification
466K conformers/ / 78K mols
70/15/15
ChiRo (Ranking)
Central
Enantiomer ranking
335K conformers 69K mols
70/15/15
CMCDS
Central
ECD prediction
22,182 conformers mols
80/10/10
ACMP
Axial
ECD and OR prediction
1.192 conformers mols
80/10/10
A.1 CENTRAL CHIRALITY
We adopt datasets from ChIRo (Adams et al. 2022): (i) R/S classification: the task is to classify the configuration (R S) of tetrahedral chiral center. The dataset is subset of PubChem3D, comprising about 466K conformers of 78K enantiomers, each with exactly one tetrahedral chiral center. The same 70/15/15 splits are used. (ii) Enantiomer ranking: the task is to predict docking score rankings, evaluating whether the predicted relative ordering between enantiomers matches the ground truth. This dataset includes 335K conformers of 69K enantiomers (34.5K pairs), with enantiomers kept in the same data partition. We sample one conformer for each enantiomer in the batch during training. The same 70/15/15 splits are used.
In addition, we employ the CMCDS dataset for ECD prediction (Li et al. 2025). formulated
multi-task, multi-label problem with three sub-tasks: predicting (1) the number of spectral peaks, (2) their positions, and (3) the sign of peak heights. CMCDS consists of ECD spectra for 22,190 chiral molecules (11,095 pairs), generated by large-scale theoretical calculations using Gaussian16 B.01 (Frisch et al., 2016b). After excluding four molecules without chiral centers, the dataset contains 28,182 conformers (11,091 pairs). We use random 80/10/10 splits while ensuring enantiomers are within the same partition.
A.2 AXIAL CHIRALITY
We introduce a new benchmark, ACMP, for electronic circular dichroism (ECD) and optical rotation (OR) prediction. Specifically, we curate 650 axially chiral molecules from ChiralFinder (Shi al.] 2026) and generate initial 3D conformations using RDKit (Landrum) 2025). To refine these struc- tures, preliminary conformational optimization is first carried out using GFN2-xTB (Bannwarth) al. 2019), which is semiempirical quantum mechanical method. Following this, the molecular geometries are fully optimized using the Gaussian 09 package (Frisch et al. 2016a), employing the B3LYP functional and the 6-31G(d) basis set.
For ECD, to obtain the spectral data, time-dependent DFT (TD-DFT) calculations are performed to compute the first 30 electronic excited states. These calculations use the long-range corrected CAM-B3LYP functional and the 6-31)+G(d.p)) basis set, yielding the excitation wavelengths and their corresponding rotatory strengths. ECD spectra curves are generated with Gaussian broaden- ing (FWHM = 2/3), and an example is presented in Figure [S] We uniformly sample spectra at nm intervals produce sequence-form ECD representations. The labeling process follows the same procedure as CMCDS: peak positions are uniformly discretized into 20 classes, while peak heights are encoded using their sign (positive or negative). The number of peaks ranges from 0 to 6, and its distribution is illustrated in Figure [6] For OR, we compute the optical rotation at 589.3 nm using CAM-B3LYP/6331C** The resulting scalar rotation value is then mapped to a binary classification label, forming two-class prediction task. All Gaussian calculations are conducted using 400 GB of memory with 192 CPU cores. To obtain enantiomeric counterparts, we generate opposite configurations by reflecting atomic coordinates across the z-axis. We finally get dataset of 1,192 enantiomers (596 pairs). For evaluation, we adopt random 80/10/10 vain/valiattin/testt splits, ensuring that enantiomer pairs kept within the same partition.
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ECD Spectrum
10
ECD
Rotatory Strength (arb. units)
-4
100
150
200 Wavelength (nm)
250
300
350
Figure 5: The ECD curve example for an axial chiral molecule.
Peak Count Distribution
600
400
Frequency
200
Number Peaks
Figure 6: Distribution of ECD peak numbers in ACMP dataset.
B ADDITIONAL EXPERIMENTS
B. FINE-GRAINED PERFORMANCE AXIAL-CHRAAITY SUBTYPES
To further investigate ChiDeK's performance across structural diversity, provide subtype-level breakdown on the test set, covering seven axial-chirality subtypes. Tebles detailed results for axial ECD-height prediction.
We observe that ChiDeK achieves strong performance on most subtypes (2 70%), while Heter- obiaryl (C-N) and Heterobiaryl (C-B) show lower performance (< 60%), highlighting potential directions for future improvement.
B.2 EVALUATION ON GENERAL MOLECULAR PROPERTY PREDICTION
To assess the broader applicability ChiDeK beyond chirality tasks, we evaluate the model several moderate-sized MoleculeNet datasets Wu et al. (2018) where chirality is less
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Table 5: Performance breakdown of seven subtypes for the axial ECID-height task.
Subtype
# Enantiomers
Height symbols
Acc (%) f
Chiral atom pair
10
26
71.5 + 1.8
Allene-like
2
80.0 + 9.9
Spiral atom and chain
4
8
90.0 = 5.0
Biaryl
70
194
72.9 0.5
Heterobiaryl (C-N)
12
20
57.0 4.0
Heterobiaryl (C-B)
16
60.0 3.1
Nonbiaryl
16
46
70.0 = 1.6
Total/Average
120
314
71.2 = 0.6
critical, including FreeSolv, BACE, BBBP, ClinTox, and SIDER. These benchmarks test general molecular property prediction rather than chiral discrimination.
As summarized in Table C ChiDeK achieves performance comparable to standard GNNs without pretraining (D-MPNN, Attentive FP). While does not surpass highly optimized SOTA models on these general tasks, the results indicate that ChiDeK introduces no harmful inductive bias and retains broad representational capacity for generic molecular property prediction.
Table 6: Results less chiral-related MoleculeNet datasets.
Metric
ROC-AUC (%)
RMSE
Datasets
BBBP 2039 1
BACE 1513
ClinTox 1478 2
SIDER 1427
FreeSoly 642
Molecules
# Tasks
27
Woo. pre-training
D-MPNN
71.0 0.3
90.6 = 0.6
57.0 0.7
2.082 + 0.082
Attentive FP
64.3 = 1.8
0.0
84. 0.3
60.6 3.2
2.073 + 0.183
ChiDeK (Ours)
68.9 0.8
7.9
7922=111
58.3 0.8
2,412 + 0.109
W. pre-training
PretrainGNN
68.7 = 1.3
84.5 1 0.7
72.6 = 1.5
62.7 0.8
2.764 + 0.002
Uni-Mol
72.9 = 0.6
85.7 + 0.2
91.9 1.8
65.9 - 1.3
1.480 + 0.048
B.3 ROBUSTNESS NOISY OR INCOMPLETE CHIRALITY ANNOTATIONS
To evaluate ChiDeK's robustness imperfect chirality annotations, we conduct experiments on both central and axial chirality.
Central chirality (R/S classification). We systematically introduce noise removing or mislabel- ing (1:1) chiral atoms in 2.5%, 5%, 7.5%, and 10% of molecules. For mislabeling, chiral centers are replaced with non-chiral atoms; for missing centers, the labels are simply removed. Table shows smooth and gradual degradation rather than sharp collapse. With moderate levels of noise, the model maintains strong accuracy, demonstrating that the cross-attention architecture pro- vides meaningful tolerance incomplete partially incorrect stereochemical inputs. These results confirm that ChiDeK is robust to moderate annotation errors, while also showing that reliable chiral identification naturally leads to stronger performance.
Axial chirality (ECD-height prediction). We evaluate the effect of removing ground-truth axial labels (with noise) ECD-hen CeDhheight prediction performance. In this setting, the average coverage (percentage of identified chiral atoms covering true chiral atoms) 0.924, and the average loU (intersection over union) between identified and true chiral atoms is 0.645. Similar central chiral-
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Table 7: Results of central R/S classification with different ratios of noise.
Ratio
W/o. noise
2.5%
5%
7.5%
10%
ChiDeK
99.8 + 0.1
99.1 + 0.0
98.2 + 0.0
96.9 + 0.1
96.1 1 + 0.2
ity, removing ground-trune guidance reduces performance as shown Table reliable chiral identification is essential for optimal model performance.
demonstrating that
Table 8: Results of axial ECD-height prediction with without noise.
Ratio
W/o. noise
W. noise
ChiDeK
71.2 + 0.6
65.8 0.4
B.4 COMPARISON WITH CHIRALFINDER-ENTIANN MODEL
We integrate the ChiralFFnder-derived chirality matrix into ChiRo by concatenating the matrix fea- tures with the atomic embeddings (using zero vectors for non-chiral atoms) and evaluate this aug- mented model on the same axial-chirality benchmarks.
As shown in Table 9 the inclusion of the chirality matrix yields clear improvement over the original ChiRo. Nevertheless, ChiDeK continues to outperform both variants. This result indicates that while the chirality matrix is beneficial, ChiDeK provides stronger representational capacity for modeling axial chirality.
Table 9: Comparison with ChiRo enhanced by the chirality matrix from ChiralFinder.
Method
Rotation (%)
Symbol (%)
ChIRo
50.0 + 0.1
51.1 = 0.3
ChIRo Chirality matrix
62.1 + 0.3
61.8 = 1.0
ChiDeK
69.2 + 0.5
71.2 + 0.6
C CHIRAL CROSS-ATTENTION
(4) The learnable pairwise bias Pij is updated by:
Q.0 (K)²)
P5
(8+1)
Pij
(8)
(19)
V
where ls is the hidden dimension. The attention outputs then computed
@ (K())T +
Attention (o! (0) v(0)=
softmax
(e-1)
v(0)
(20)
Vk
Finally, the outputs of L chiral cross-attention layers are aggregated by average pooling to get
Hglobal
(21)
I
LET.
This global representation serves as the input for downstream chirality property prediction.
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D IMPLEMENTATION DETAILS
Whenever possible, baseline results are taken directly from the original publications; otherwise, we reproduce them using publicly available code. All results are reported over five test folds or five random runs.
D.1 INITIALIZATION ATOMS
Identification of Chiral and Chiral-related Atoms. For centrally chiral molecules, we use RDKit (Landrum 2025) to detect chiral cen- ters and their corresponding neighbors as chiral-related atoms. For axially chiral molecules, the chiral axis is labeled by chemists, while chiral-related atoms are determined using ChiralFinder (Shi et al. 2026). If ChiralFinder fails, we fall back to selecting up to four near- neighbors chiral-related atoms.
D:5
C:2
NH
HO
A:1
B:
An example for ordering chiral-related atoms (i) for atom i is pre- sented in Figure 7 After identifying the chiral atom (in green), sup- pose the indices of its chiral-related atoms (in orange) are A, B, C, and D, respectively. We RDKit to calculate the CIP value for each atom (1, 4, 2, 5, respectively) and sort the indices in ascending order of their CIP priorities.
(,0=AA,,,B,D)
Figure Ordering chiral- related atoms.
Initial Atom Features. Each atom is initialized with 52-dimensional feature vector, consistent with ChiRo (Adams et al. 2022). including atom type, degree, formal charge, number of hydro- gens, and hybridization state. Chiral atoms are additionally associated with their chirality matrix, while non-chiral and chiral cerrae-eelatd atoms share the same feature design. The embedding layers are implemented multi-layer perceptrons.
D.2 HYPERPARAMETER OPTIMIZATION
We tune hyperparameters via grid search (Table 10). Training epochs are 10 for R/S configu- ration classification, 200 for enantiomer ranking, and 50 for both central and axial ECD prediction. We use Adam optimizer (Kingma Ba (0044) and adopt cosine learning rate schedule with a min- imum learning rate of 0.1x the intiil value. As for the enantiomer ranking task, we follow ChiRo Adams al. 2022) and add the margin ranking loss.
Table 10: Hyperparameter search space for ChiDeK.
Hyperparameter
Search Space
# Chiral Transformer Layers
{4, 8, 12, 16} {2, 4, 8]
# Chiral Transformer Heads
Hidden Dimension
{64, 128, 256, 512]
Projection Dimension
{32, 64, 128, 256, 512]
Rank Strategy
(QR, Reg) (1e-4,5e-4)
Learning Rate
Batch Size
{16,32, 64, 128, 256}
Weight of MarginRankingLoss
{0.1, 0.5, 1.0]
E ANALYSIS OF AXIAL ECD PREDICTION
E. BASELINES FAIL TO LEARNING AXTAL CHIRALITY
Axial chirality presents unique challenges for ECD prediction. Our results demonstrate that ChiDeK achieves strong performance in this setting, primarily due its explicit encoding of chirality through
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the proposed chirality determinant kernlls. By directly modeling stereochemically relevant features, ChiDeK is capable of distinguishing axial enantiomers and accurately predicting peak height sym- bols, a property that is essential for practical chirality spectrum prediction.
By contrast, all baseline methods except SPMS perform poorly on axially chiral molecules, with each exhibiting specific limitations for distinct reasons, analyzed below:
DimeNet++: As E(3)-invariant model that relies only distances and angles, DimeNet++ is fundamentally incapable of distinguishing mirror-related structures. Its poor performance is therefore expected.
Tetra-DMPNN: This model augments 2D GNNs with central-chirality readout, but has no mechanism to capture axial chirality. As result, it fails on this task.
SphereNet and ChiRo: In principle, these models incorporate torsional information and should have some sensitivity to stereochemistry. However, axial chirality involves subtle global spatial arrangements that not well captured by their architectures, leading to limited performance.
While this model introduces additional chiral encoding, it is designed for central chirality and lacks the necessary generalization axial cases.
ECDFormer:
This method explicitly resolves atom permutations around tetrahedral centers, making it effective for central chirality. However, it does not generalize axial stereogenic axes and thus completely fails in this setting.
ChiGNN:
E.2 ECD PREDICTION EXAMPLES
Figure 8 presents two additional representative prediction comparisons. ChiDeK successfully as- signs opposite peak-height symbols to opposite stercochemical configurations, while ChiGNN and Tetra-DMPNN (permutation variant) produce identical peak-height symbols across enantiomers, failing to capture stereochemical differences.
E.3 DETAILS FOR ROTATING CHIRAL AXIS
We generate 18 conformers by rotating the torsion angle in 20-degree increments over 360 degrees. Each conformer corresponds to distinct coordinates, yielding different chirality matrices and thus different representations. We extract the embeddings before the predictor, apply UMAP (McInnes) et al. 2018) for visualization, and record the trajectory as the torsion angle progresses from 340 degrees. Additionally, calculate the cosine similarity between each embedding and the reference conformer at degree 0, and visualize the results using polar plots.
F PROOF DETAILS
F.I CHIRALITY MATRIX
Proof of Lemma 3.7 Let (21, be pair of enantiomers that differ only in the stereochemical configuration of atom i. By definition of enantiomers there exists an isometry T of R that maps the atomic coordinates of a reflection; equivalently, one may write T = Rg o R2 with RS E SE(3) and R2 E O- (3) (an orthogonal transformation with determinant -1). In particular, the local coordinate frame used to form the chirality matrix Mc(i) for 22 is obtained from that for z by applying the linear part of T, which has determinant 1.
those of and which is a composition of rigid-body motion and
Since is chiral atom, its substituent vectors (used to build Mc(i)) are non-coplanar, hence det (Mc(i) # 0. The sign of the determinant is well-defined.
Applying Proposition 3.1 first note that the rigid-body component R1 E SE(3) leaves the determi-
nant invariant:
det (M) det (R, (())(())
(22)
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(permate)
Saletory Snoweghh (eb. unbt)
Talattry
Werelength
Wereelength
Wavelength
Salatory Stoeoggh Each urbs)
Salatory Soosoghh twth
Mezelerngth brit
Wareherggh (rn)
Wandlergth
Redl
Spectiun
Reel
Dabattrr Strenggt unite)
Przz
Bobetory
Wioherggh (mm)
Wacowwngth (1])
Weoelentth (1m))
teto (nnrr))
Peal
Rabeary Stwweghh tafto unto)
Vauerength
ength
Whwelength
Figure 8: Axial ECD prediction examples of two pairs of enantiomers.
Then the reflection R2 (3) flips the sign of the determinant:
det M(
det R M ( = det (M
(23)
R2R,M is exactly the chirality matrix for atom in the enantiomer 22, i...
M (z2) (i) RgR M
(x) (i).
(24)
Therefore,
det (M det (M (1).
(25)
follows that the determinants for the two enantiomers have opposite signs. By the sign conven- tion in the lemma (positive determinant 'R', negative determinant 'S'), the stereochemical configuration of is RR' when (M(( > 0 and 'S" when det(Mc(i)) < O.
Figure illustrates how reflection flips the sign of the volume (determinant), yielding an opposite configuration.
F.2 CHIRAL DETERMINANT KERNEL
Proof of Lemma 4.1 Let M Mc(i) R3x3 and W E Rd,x33 be full column rank (so the rank
3). Define O WM E [pdx and perform QR decomposition:
0 QR, Q Is. Q x4x3
R R 3x3 upper-triangular.
(26)
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Positive volume (det=0.65>0)
Negative (det=-0.65-0)
10
0.0
3.0
-02
0.6
0.4
0.0
1.0
1.0
0.8
0.5
0.0
0.6
0.2
0.4
3.4
0.6
0.2
0.4
0.2
0.4
0.0
0.2
1.0
1.2
0.0
0.8
10
Figure 9: An example of reflection that flips the sign of volume.
Unlike the conventional QR convention, here allow the diagonal entries of R be negative, as is the case in numerical packages such as PyTorch. Consequently, the sign det(()) (t) is not predetermined but remains well defined.
Consider the Gram matrix
G W W.
(27)
Since has full column rank, G and det(G) > We have
R R = - = (WM) (WM) M w WM M GM.
(28)
Taking determinants gives
det de) = del(M GM) (M) del det())
(29)
Hence
[det(H) det (M) det
(30)
To recover the sign, observe that det (0) = dct(Q) (R). Since 0 = W M, this implies
det [ det(WW M)
(31)
det(Q)
Here det(Q) E {+1} because Q has orthonormal columns. Substituting gives
()) det W) det(M)
(32)
det(() (Q)
Combining with equation 30 the factor depending on W is
a (W) = det det(W))
[e) (dw W)((WT))).
(33)
Since det Q) = =1, this ambiguity only flips the sign in way consistent with (M). Therefore, we can absorb into the proportionality constant and write
det R) = (d) det a aWW)
(34)
with (W) = /det W W) > independent of M.
Finally, the invariance properties follow directly: if M + rM for SO(3), then det(M) is unchanged, hence Pw (7) invariant. M + sM for s E 0 with det 1, then det M
flips sign, hence (i) flips sign as well. Thus Pw (1) inherits Proposition 3.1 and Lemma 3.1
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Chiml-rrlated atoms
Chiral atems
Chiral-rolated atoms
Chiral atoms
Figure 10: Two examples of cross attention weights. The chiral axis for each molecule is highlighted in green.
G VISUALIZATION OF THE CHIRAL TRANSFORMER
To better understand how ChiDeK leverages chiral information, we visualize the cross-attention weights between chiral atoms (used queries) and all other atoms (used as keys and values) on the R/S classification task. Figure TO illustrates two representative test examples from the axial ECD prediction task, depicting the head-averaged cross-attention weights in the final layer of the chiral transformer. In both cases, we observe that attention weights assigned to chiral-related atoms are higher than those assigned non-chiral atoms. This demonstrates that ChiDeK successfully pri- oritizes stereochemically relevant atoms when integrating local chiral information into the global molccular representation. Conversely, non-chiral atoms receive comparatively lower weights, sug- gesting that the model effectively distinguishes their less critical role in stereospecific interactions.
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Reviews:
[{"review_id": "BqmDLocxKF", "reviewer": "Reviewer_VtGG", "summary": "This paper proposes ChiDeK (Chiral Determinant Kernels), a new architecture for learning stereochemistry-aware molecular representations. The key goal is to encode both central chirality (classical R/S stereocenters) and axial chirality (stereogenic axes, e.g. biaryls with restricted rotation) in a unified framework.", "questions": "1. The method assumes a correct partition of atoms (and axes) into chiral / chiral-related / non-chiral, in some cases with manual labeling by chemists. The model does not learn these on its own. Robustness to mistakes in this preprocessing step is not evaluated, so it’s unclear how well the approach holds up under noisy or incomplete chirality annotations. For example, if the stereocenter or chiral axis is mislabeled or partially missed, does performance degrade sharply, or is the model tolerant to some noise in these assignments?\n\n2. The strongest axial chirality results are on ECD peak-sign prediction. It is not shown that the same architecture improves other chirality-sensitive endpoints (e.g., enantioselective binding, optical rotation). This leaves open whether the method is generally useful for axial chirality in drug discovery, or mainly tuned to ECD-style tasks. \n\n3. For central chirality ECD prediction, the model’s sign accuracy is near random (~53%), and the explanation is attributed to noisy conformers. The paper does not test whether using higher-quality conformers fixes this. So it’s not clear whether the limitation is data quality or a weakness of the architecture on central chirality.\n\n4. Most of the axial chirality evaluation is framed around ECD spectrum prediction. Do you expect the same architecture to transfer to other chirality-sensitive endpoints (e.g., enantioselective binding affinity, optical rotation, chiral toxicity)?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. A major strength of the paper is that it propose an approach to build a representation that can capture both central and axial chirality, which has practical important in drug discovery.\n\n2. The paper contributes the AxialECD dataset , providing a good benchmark for the analysis in the area.", "weaknesses": "1. The method assumes a correct partition of atoms (and axes) into chiral / chiral-related / non-chiral, in some cases with manual labeling by chemists. The model does not learn these on its own. Robustness to mistakes in this preprocessing step is not evaluated, so it’s unclear how well the approach holds up under noisy or incomplete chirality annotations. For example, if the stereocenter or chiral axis is mislabeled or partially missed, does performance degrade sharply, or is the model tolerant to some noise in these assignments?\n\n2. The strongest axial chirality results are on ECD peak-sign prediction. It is not shown that the same architecture improves other chirality-sensitive endpoints (e.g., enantioselective binding, optical rotation). This leaves open whether the method is generally useful for axial chirality in drug discovery, or mainly tuned to ECD-style tasks. \n\n3. For central chirality ECD prediction, the model’s sign accuracy is near random (~53%), and the explanation is attributed to noisy conformers. The paper does not test whether using higher-quality conformers fixes this. So it’s not clear whether the limitation is data quality or a weakness of the architecture on central chirality.\n\n4. Most of the axial chirality evaluation is framed around ECD spectrum prediction. Do you expect the same architecture to transfer to other chirality-sensitive endpoints (e.g., enantioselective binding affinity, optical rotation, chiral toxicity)?", "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": "QLgk08O7QQ", "reviewer": "Reviewer_pVhT", "summary": "This paper proposes ChiDeK, a unified framework for learning molecular chirality representations. The method introduces a chiral determinant kernel to encode stereochemical information derived from the chirality matrix, and integrates it into a cross-attention architecture to propagate chiral information across the molecular graph. The model is able to jointly handle central and axial chirality, and a new benchmark dataset for axial-chiral ECD prediction is constructed. Experiments show improvements across R/S classification, enantiomer ranking, and ECD spectrum prediction tasks.", "questions": "1. How does the model’s performance vary when using different conformer generation pipelines? For example, RDKit vs. xTB-relaxed vs. DFT-optimized conformers.\n\n2. Could the authors provide runtime and memory usage comparisons, particularly regarding QR decomposition, compared to SE(3)-equivariant baselines?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Provides a unified representation for both central and axial chirality, which is not addressed by prior models.\n\n2. The chiral determinant kernel is mathematically well-motivated and reflection-sensitive.\n\n3. The newly constructed AxialECD dataset fills a gap in evaluating axial chirality.\n\n4. Demonstrates clear improvements in ECD prediction, especially peak sign prediction for axial chirality.", "weaknesses": "1. Limited diversity and size of the AxialECD dataset. \n\nThe proposed AxialECD benchmark includes ~600 axial chiral molecules, which represents a relatively narrow stereochemical space and may not generalize to other classes of axial chirality such as atropisomeric biaryls with flexible steric barriers or complexes with metal-coordinated axes. I recommend that authors report performance breakdown by molecular subtypes, or perform zero-shot or cross-dataset evaluation if another axial-chirality dataset (or cases from published articles) becomes available.\n\n2. Scalability and efficiency of the chiral determinant kernel.\n\nThe chiral determinant kernel relies on QR decomposition for each chiral center, which may incur non-trivial computational overhead when scaling to large molecules (e.g., natural products, macrocycles) or conformer ensembles. Provide runtime comparisons with SE(3)-equivariant baselines and discuss strategies for batch-efficient QR computation.\n\n3. Lack of evaluation beyond chirality-sensitive tasks.\n\nWhile the model is clearly designed for chirality-aware tasks, all experiments are conducted on datasets where chirality is explicitly involved in the label. It is unclear whether ChiDeK introduces unnecessary inductive bias or reduced performance in general molecular property prediction tasks where chirality is less relevant. Please include control experiments on standard property prediction datasets (e.g., QM9, PCBA, MoleculeNet) to confirm no loss of expressivity.", "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": "py1WJbFlqf", "reviewer": "Reviewer_97qz", "summary": "The authors developed a deep learning-based tool named ChiDeK for distinguishing point chirality and axial chirality, and applied it to tasks including R/S label classification, enantiomer ranking, and prediction of the sign and position of ECD.", "questions": "See weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Chirality recognition is a central issue in chemical research, and accurately capturing the stereochemical environment of molecules is key to distinguishing enantiomers. The proposed ChiDeK architecture leverages an SE(3)-invariant chirality matrix and cross-attention mechanisms to effectively extract molecular stereochemical information, enabling the simultaneous identification of both point and axial chirality. The model demonstrates strong performance in R/S label classification, enantiomer ranking, and ECD sign and position prediction for chiral molecules, achieving particularly notable accuracy in the binary classification of ECD signs for axially chiral molecules.", "weaknesses": "1. Although acquiring chiral information of molecules is important, using a model to distinguish R/S configurations of point and axial chirality appears to be of limited practical significance. Well-established chemical rules, such as those implemented in RDKit, already exist for such differentiation. While the authors generate discriminative features for enantiomers via the chirality matrix and deep learning—which can indeed distinguish R/S configurations—this should not be the ultimate goal. The application of these chirality-related features should target more valuable downstream tasks, such as predicting enantioselectivity in asymmetric catalytic reactions or binding affinity between chiral drug molecules and protein targets.\n2. This work can largely be viewed as an extension of the ChIRo study, expanding its scope from point chirality to axial chirality. However, after demonstrating that ChIRo could generate discriminative features for enantiomers, its authors further conducted a practically meaningful downstream application: ranking enantiomers by docking scores in an enantiosensitive protein pocket. In contrast, the present authors only replicated similar benchmark tests on point chiral data and did not perform valuable downstream application experiments on the axially chiral molecules emphasized in this work.\n3. The authors claim that this is the first architecture capable of jointly encoding central and axial chirality. In reality, existing methods such as ChiralFinder and SPMS, which are mentioned in the paper, can also extract features that distinguish point and axial chirality. Moreover, the authors' model itself utilizes the chirality matrix generated by ChiralFinder. To ensure a rigorous performance comparison, the authors should benchmark their model against features derived from ChiralFinder and SPMS on the same set of tasks.\n4. The core chiral information in this work is derived from ChiralFinder, which appears crucial to ChiDeK's ability to distinguish chirality. As a result, the unique contribution of ChiDeK itself to the task of axial chirality discrimination remains unclear.\n5. According to the results in Table 1 for point chiral molecules across R/S classification, enantiomer ranking, and ECD prediction tasks, the model does not show a significant advantage in point chirality-related predictions. In particular, the accuracy for ECD sign prediction is only 53.3%, which—assuming a 1:1 positive-to-negative sample ratio—is barely better than random guessing (50%).\n6. The authors do not appear to have provided the complete data used in the study. The data folder supplied is empty, and the accompanying README file is too brief, offering no guidance on how to obtain the necessary dependencies or datasets.", "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": "IrgTfq2JzL", "reviewer": "Reviewer_QnyP", "summary": "The paper proposes ChiDeK, a novel framework that explicitly models molecular chirality using chiral determinant kernels and a chiral-aware cross-attention transformer to capture stereochemical interactions. It achieves chiral-sensitive, rotation- and translation-invariant molecular representations for tasks like chirality classification and spectrum prediction.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Paper is well written\n- Introduces differentiable chiral determinant kernels to encode stereochemistry.\n- Uses chiral-aware cross-attention to capture interactions between atom types.", "weaknesses": "- How does the chiral matrix characterize the axial chirality? It seems it can only characterize central chirality. \n- What is the computational complexity of the method, since it computes the determinant, which is expensive in higher dimensions?\n- Can this method be generalizable to the datasets where we have a combination of molecules, of which some are chiral and some are not?\n- Can the method characterize Diastereomers?", "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)