Spectral symmetry fingerprints in equivariant representations

Jiacheng Cen; Anyi Li; Ning Lin; Zihe Wang; Ji-Rong Wen; Wenbing Huang

cs.LGNov 5, 2026

Spectral symmetry fingerprints in equivariant representations

Authors: Jiacheng Cen, Anyi Li, Ning Lin, Zihe Wang, Ji-Rong Wen, Wenbing Huang

Organizations: Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China · Beijing Key Laboratory of Research on Large Models and Intelligent Governance, Beijing, China · Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE, Beijing, China · Beijing Academy of Artificial Intelligence, Beijing, China

Abstract

Symmetry governs the organization of physical structures, while symmetry breaking reveals their transformations. In geometric deep learning, symmetry imposes rank constraints on equivariant representations, limiting the expressive capacity of equivariant graph neural networks (GNNs). Here we show that these constraints can instead be exploited: the singular-value spectra of equivariant representations provide a unified fingerprint of structural symmetry. We introduce spectral symmetry fingerprints (SymFP), which enable discrete point-group identification through rank-template matching and continuous symmetry analysis by quantifying spectral energy beyond target rank constraints. SymFP is orientation invariant and architecture agnostic, enabling a parameter-free GNN implementation. Its spectra can be reused across multiple target groups, and the resulting leakage scores satisfy subgroup monotonicity. Across synthetic structures, molecules, coordination complexes and large biological assemblies, SymFP accurately identifies point groups, tracks controlled symmetry breaking and reveals local and degree-resolved symmetry changes that can be obscured by global geometric measures. These results establish the spectral structure of equivariant representations as a general framework for characterizing symmetry across physical systems.

Figures & tables

Figure 1: Overview of SymFP. a , Pipeline for constructing SymFP from a geometric graph. The spectra support discrete point-group identification through rank-template matching and continuous symmetry analysis through leakage scores. b , Rank templates of the 32 crystallographic point groups. c , Subgroup relations and monotonicity of rank templates and leakage scores. In b,c , fill colour denotes the crystal system, with grey for blocks spanning multiple systems.

Figure 2: Discrete point-group identification across equivariant GNNs. a , Point-group identification accuracy and rank-template match rate across GNN architectures and training states. The $30/32$ reference line marks the identification accuracy limit at $l=0,\ldots,6$ on this balanced benchmark. Owing to computational constraints, learnable GNNs are evaluated only at random initialization for $l=0,\ldots,9$ . b , Sensitivity of point-group identification accuracy to the relative rank tolerance for randomly initialized models at $l=0,\ldots,6$ ; the shared tolerance $3\times 10^{-5}$ used across models is indicated. c , Ablations of the PF node and edge branches and radial-basis count and width at $l=0,\ldots,6$ . $N$ denotes the number of radial bases per branch, and $\Delta$ their centre spacing.

a Retrieval performance
		AUC			Recall at 25% budget (%)			pLS W/T/L
Group	Target symmetry	pLS	CSoM	Engel score	pLS	CSoM	Engel score	vs CSoM	vs Engel
Overall	—	$\mathbf{0.794}$	$0.747$	$0.787$	$\mathbf{68.8}$	$66.9$	$66.3$	41/36/9	23/53/10
All	w/	$\mathbf{0.927}$	$0.920$	$0.915$	$\mathbf{99.4}$	$98.3$	$95.3$	4/81/1	13/72/1
All	w/o	$0.593$	$0.475$	$\mathbf{0.597}$	$19.5$	$16.9$	$\mathbf{20.0}$	40/10/9	12/34/13
Maxima	w/	$\mathbf{0.916}$	$0.914$	$0.900$	$\mathbf{100.0}$	$99.0$	$94.4$	3/70/7	14/65/1

Table 3

a Energy extrema
GNN	Change in pLS			Improved molecules		$\Delta$ AUC
	Maxima	Minima	Gap	All ( $n=86$ )	Validation ( $n=17$ )
Total-energy pretraining
HEGNN	$-0.4718$	$-0.7146$	$+0.2428$	54/86 (62.8%)	10/17 (58.8%)	$+0.0061$
TFN	$-0.4048$	$-0.4192$	$+0.0143$	45/86 (52.3%)	9/17 (52.9%)	$-0.0088$
MACE	$-1.5250$	$-1.6448$	$+0.1198$	56/86 (65.1%)	9/17 (52.9%)	$+0.0006$

Table 4

a Dy-DOTA-water (30 heavy atoms): CPU total time (ms)
Multiple targets share computations within each call; baseline timings include orientation search.
Method	1 target	2 targets	4 targets	7 targets
	$\mathrm{C}_{2}$	$\mathrm{C}_{2},\mathrm{C}_{4}$	$\mathrm{C}_{2},\mathrm{C}_{4},$ $\mathrm{C}_{6},\mathrm{D}_{2}$	$\mathrm{C}_{2},\mathrm{C}_{4},\mathrm{C}_{6},$ $\mathrm{D}_{2},\mathrm{D}_{4},\mathrm{D}_{6},\mathrm{O}$
SymFP	6.46	6.59	6.87	7.21
CSoM	1575.40	5605.46	15901.44	96382.65
Engel, grid search	611.77	619.24	620.43	627.62

Table 5

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Schoenflies	Hermann–Mauguin	Fixed-subspace dimension $m_{l}$ (natural parity)												$n_{\mathrm{CF}}$
		$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\mathrm{C}_{1}$	$1$	1	3	5	7	9	11	13	15	17	19	21	23	27
$\mathrm{C_{i}}$	$-1$	1	0	5	0	9	0	13	0	17	0	21	0	27
$\mathrm{C}_{2}$	$2$	1	1	3	3	5	5	7	7	9	9	11	11	15
$\mathrm{C_{s}}$	$m$	1	2	3	4	5	6	7	8	9	10	11	12	15
$\mathrm{C_{2h}}$	$2/m$	1	0	3	0	5	0	7	0	9	0	11	0	15

Appendix

Supplementary Table A.1: Fixed-subspace dimensions of crystallographic point groups. Values use natural-parity representations with

l=0,\ldots,11

. The final column gives the crystal-field parameter count

n_{\mathrm{CF}}=\sum_{k\in\{2,4,6\}}m_{k}

( § 2.5 ).

Training regime	GNN	Identification accuracy (%)	Template match rate (%)
—	PF	93.75	97.19
Random init.	HEGNN 6	93.75	92.50
	TFN 7	93.75	100.00
	MACE 8	93.75	100.00
	Equiformer 9	93.75	100.00
Pretrained	HEGNN 6	93.75	91.88

Appendix

Supplementary Table C.2: Discrete point-group identification across training regimes. Evaluations use natural-parity degrees

l=0,\ldots,6

and relative rank tolerance

\tau_{\mathrm{rel}}=3\times 10^{-5}

GNN	Identification accuracy (%)	Template match rate (%)
PF	99.38	93.75
HEGNN 6	99.06	92.19
TFN 7	100.00	100.00
MACE 8	100.00	100.00
Equiformer 9	99.38	99.38

Appendix

Supplementary Table C.3: Discrete point-group identification using degrees

l=0,\ldots,9

. Evaluations use natural-parity representations and validation-selected relative rank tolerance

\tau_{\mathrm{rel}}=10^{-6}

( § C.2 ).

$L$	GNN	Identification accuracy (%)				Template match rate (%)
		Seed 42	Seed 43	Seed 44	Mean $\pm$ s.d.	Seed 42	Seed 43	Seed 44	Mean $\pm$ s.d.
6	HEGNN	93.75	93.44	93.13	$93.44\pm 0.31$	92.50	92.50	91.25	$92.08\pm 0.72$
	TFN	93.75	93.75	93.75	$93.75\pm 0.00$	100.00	100.00	100.00	$100.00\pm 0.00$
	MACE	93.75	93.75	93.75	$93.75\pm 0.00$	100.00	100.00	100.00	$100.00\pm 0.00$
	Equiformer	93.75	93.75	93.75	$93.75\pm 0.00$	100.00	100.00	100.00	$100.00\pm 0.00$
9	HEGNN	99.06	99.06	99.06	$99.06\pm 0.00$	92.19	93.13	92.81	$92.71\pm 0.48$

Appendix

Supplementary Table C.4: Discrete point-group identification across random initializations.

$\tau_{\mathrm{rel}}$	PF		HEGNN 6		TFN 7		MACE 8		Equiformer 9
	Group	Match	Group	Match	Group	Match	Group	Match	Group	Match
PF and random init.
$1\times 10^{-7}$	87.50	77.81	93.75	98.75	93.75	99.69	93.75	100.00	28.44	15.62
$3\times 10^{-7}$	93.75	93.44	93.75	99.06	93.75	100.00	93.75	100.00	86.56	84.38
$1\times 10^{-6}$	93.75	97.19	93.75	97.81	93.75	100.00	93.75	100.00	93.75	99.69
$3\times 10^{-6}$	93.75	97.19	93.75	96.88	93.75	100.00	93.75	100.00	93.75	99.69

Appendix

Supplementary Table C.5: Sensitivity to rank tolerance. Group and Match denote point-group identification accuracy and rank-template match rate (%). Bold marks the validation-selected tolerance.

Variant	Branch	$N$	$\sigma$	Identification accuracy (%)	Template match rate (%)
node default	node	16	$0.0625$	57.50	52.81
edge default	edge	16	$0.0625$	92.81	94.69
hybrid default	hybrid	16	$0.0625$	93.75	97.19
hybrid small $N$	hybrid	8	$0.1250$	93.75	96.25
hybrid large $N$	hybrid	32	$0.03125$	93.75	97.19
hybrid $\sigma=\Delta/2$	hybrid	16	$0.0333$	93.75	96.88

Appendix

Supplementary Table C.6: PF ablation on synthetic structures.

N

is the number of radial bases per branch,

\sigma

their Gaussian width, and

\Delta=1/(N-1)

their centre spacing on

[0,1]

. Bold marks the default hybrid configuration.

$\tau_{\mathrm{rel}}$	Worst Group (%)	Mean Group (%)	Mean Match (%)
$10^{-8}$	3.12	30.99	18.82
$3\times 10^{-8}$	3.12	44.52	37.14
$10^{-7}$	3.12	62.64	59.40
$3\times 10^{-7}$	4.06	84.47	87.40
$10^{-6}$	41.25	89.38	92.88
$3\times 10^{-6}$	90.00	93.27	98.08

Appendix

Supplementary Table C.7: Validation-based rank-tolerance selection. Group and Match denote point-group identification accuracy and rank-template match rate (%); Worst and Mean are taken across GNN and training-regime configurations.

Target	SymFP	CSoM	Engel
$\mathrm{C}_{4}$	$4\alpha$	$16\alpha^{*}$	$4\alpha$
$\mathrm{C_{4v}}$	$4\alpha$	$8\alpha$	$4\alpha$
$\mathrm{C}_{2}$	$4\alpha$	$21\alpha^{*}$	$4\alpha$
$\mathrm{C_{2v}}$	$4\alpha$	$4\alpha$	$4\alpha$
$\mathrm{C_{s}}$	$8\alpha$	$2\alpha$	—

Appendix

Supplementary Table C.8: Dominant harmonics of capping-water rotation. Entries give the largest-amplitude harmonic among

n=1,\ldots,24

after mean subtraction. Stars mark noise-sensitive components of near-constant CSoM profiles; a dash denotes an unavailable target.

GNN	Axial staircase ( $\|m\|=1$ )	In-plane stretch ( $\|m\|=2$ )	In-plane radial ( $\|m\|=3$ )	Axial pucker ( $\|m\|=3$ )
PF	1	2	3	3
HEGNN 6	1	2	3	$4^{\dagger}$
TFN 7	1	2	3	3
MACE 8	1	2	3	3
Equiformer 9	1	2	3	3

Appendix

Supplementary Table D.9: Onset degrees across deformation modes. Onsets exclude round-off blocks ( Eq. 82 ).

\dagger

The

l=3

block remains at round-off; the first energy-supported response occurs at

l=4

Domain	neutral pH	state I	state II	state III	state IV
HA1 head	$0.26$	$0.26$	$0.54$	$0.85$	$1.05$
HA2 stem	$0.23$	$0.22$	$0.54$	$0.51$	$0.49$

Appendix

Supplementary Table D.10: Domain-resolved self-match residuals in haemagglutinin. Residuals (Å) use a cyclic

120^{\circ}

rotation about the shared protomer-centroid axis.

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Molecular representations are essential for the evaluation of molecular similarity and the development of structure-property relationships. Despite the known importance of 3D structure to determine chemical and physical properties, the most widely used molecular fingerprints encode only two-dimensional connectivity. Such representations fail to distinguish similar but distinct stereoisomers and conformers. Alternative 3D methods are typically defined pairwise, making their application to large chemical spaces prohibitive, while deep learning embeddings are expressive but uninterpretable and limited by their training data diversity. Here, we introduce novel physics-inspired molecular fingerprints based on principles from spectral graph theory. We represent molecules as a complete graph in 3D space, with edge weights encoding heuristic physical interactions. Eigenvalue decomposition of the resulting graph Laplacian matrix results in a computationally efficient fixed-length chemical fingerprint that encodes 3D structure while obeying necessary physical symmetries of permutation and E(3) invariance. Spectral fingerprints differentiate between unique molecular structures with identical 2D connectivity, overcoming a limitation of 2D descriptors, while maintaining the low computational cost needed for efficient screening of vast chemical spaces. We evaluate our fingerprints with community detection algorithms and observe strong performance against representative baselines across datasets from organic, inorganic, biological, reticular, and reaction chemistry. Nearest-neighbor property estimation and applicability domain analyses reveal the utility of our molecular representation in machine learning and cheminformatics. We anticipate that spectral fingerprints will serve as generalizable, interpretable, and efficient measures of chemical similarity that incorporate 3D information at minimal cost.

Abstract

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