Asymmetry Tunneling Beyond Adaptive Equivariance Relaxation

Eunho Jeong; Joon Jang; Hyeonjin Kim

cs.LGOct 21, 2026

Asymmetry Tunneling Beyond Adaptive Equivariance Relaxation

Authors: Eunho Jeong, Joon Jang, Hyeonjin Kim

Organizations: Seoul National University.

Abstract

Equivariant neural networks encode prescribed symmetries as powerful inductive biases, yet real-world data may exhibit symmetry-breaking structure that these priors cannot fully capture. Prior work has largely addressed this challenge through model-side equivariance relaxation, seeking an appropriate level of equivariance for the data. This raises a complementary question: must equivariance relaxation be adaptively controlled? To investigate this question, we introduce a residual path alongside a strictly equivariant model and deliberately restrict their interface to examine the transmission of task-relevant structure. Across five benchmarks, we observe both residual collapse and survival despite these restrictions; we call the latter phenomenon asymmetry tunneling. We further characterize this phenomenon through four complementary diagnostics. Our results suggest that adaptive control of equivariance relaxation is not a prerequisite for capturing potential symmetry-breaking structure.

Figures & tables

Figure 1: Overview of the restricted residual architecture. The residual path reads detached anchor-layer representations, and its gated contribution is fused once after the final anchor layer.

Setting	Anchor	+ ours	$\kappa_{r}$
N-body	0.00625	0.00626	0.0080
ETH	0.40/0.61	0.40/0.58	0.0970
Hotel	0.12/0.18	0.16/0.23	0.0697
Univ	0.23/0.43	0.24/0.43	0.0644
Zara1	0.18/0.32	0.20/0.34	$7.9{\times}10^{-5}$
Zara2	0.13/0.23	0.12/0.21	0.0966

Table 1: Residual survival and task performance on Charged N-body and ETH/UCY. Charged N-body uses SEGNN ( Brandstetter et al., 2021 ) as the equivariant anchor and reports mean squared error (MSE). ETH/UCY uses EqMotion ( Xu et al., 2023 ) and reports average/final displacement error (ADE/FDE; lower is better).

\kappa_{r}

measures residual survival as defined in Equation ( 9 ).

Figure 2: CMU Run example. (a) The anchor predicts narrower leg separation than the ground truth. (b) The anchor shows little forward progression, whereas the fused prediction recovers the running trajectory.

Method / Metric	Sub #35 (Walk)	Sub #9 (Run)
TFN	32.0 $\pm$ 1.8	56.6 $\pm$ 1.7
SE(3)-Tr.	31.5 $\pm$ 2.1	61.2 $\pm$ 2.3
EGNN	28.7 $\pm$ 1.6	50.9 $\pm$ 0.9
EGNO	8.1 $\pm$ 1.6	33.9 $\pm$ 1.7
EGNO + ACE	7.4 $\pm$ 0.2	23.8 $\pm$ 1.5
EGNO + ours	6.97 $\pm$ 0.90	22.83 $\pm$ 2.04

Table 2: CMU motion results. Baseline results follow TFN ( Thomas et al., 2018 ) , SE(3)-Transformer ( Fuchs et al., 2020 ) , EGNN ( Satorras et al., 2021 ) , EGNO ( Xu et al., 2024 ) , and EGNO+ACE ( Manolache et al., 2025 ) ; EGNO is the equivariant anchor. MSE is reported in units of

10^{-2}

. Local-SVD uses the globally centered trajectory representation with

K=32

\delta=.95

, and

H=32

Figure 3: CMU physical tracing. (a,b) Joint-wise localization $L_{p}$ . (c–f) Motion-conditioned residual patterns. (g,h) Left–right residual-energy relations.

Figure 4: Residual Grad-CAM visualization on ScanObjectNN under

Z/Z

(left) and

\mathrm{SO}(3)/\mathrm{SO}(3)

(right). The label [table] denotes the object category. Red highlights the top

5\%

attributed points; circles and crosses denote object and background points.

Accuracy (%)
Method / Metric	Z/Z	SO(3)/SO(3)
Li et al.	74.6	74.9
Yu et al.	77.2	77.4
VN-DGCNN	77.9	78.5
TetraSphere ( $K=4$ )	79.2	79.0
VN-DGCNN + ours	80.7	79.5

Table 3: ScanObjectNN results. Accuracy baselines follow Li et al. (2021) , Yu et al. (2023) , VN-DGCNN ( Wang et al., 2019 ; Deng et al., 2021 ) , and TetraSphere ( Melnyk et al., 2024 ) ; VN-DGCNN is the equivariant anchor. Local-SVD uses

K=128

\delta=.95

, and

H=32

. Lift uses the top-

5\%

Grad-CAM support and is reported as Object/Background (Obj/Back), with

1

denoting chance.

Prediction performance
Method / Metric	GVP+Mixed	GVP+Had.	GVP
RMSE $\downarrow$	1.449	1.518	1.565
$R$ $\uparrow$	0.531	0.521	0.428
Residual diagnostics
$\kappa_{r}$	0.223	0.298	–
$\Delta\rho^{\perp}$ (%) $\uparrow$	44.1	12.9	–

Table 4: PDBbind results. GVP denotes the reproduced GVP-GNN baseline ( Jing et al., 2021 ) . Mixed and Had. denote the mixed and Hadamard operators used in the restricted residual path, respectively. Local-SVD uses

K=32

\delta=.95

, and

H=32

. Lift uses the top-

5\%

Grad-CAM support and is reported as Ligand/Protein (L/P), with

1

denoting chance.

Equivariant Neural Networks

Equivariant neural networks encode transformations such as rotations, translations, and permutations so model outputs change consistently with their inputs.

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Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration

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Open paper AskPDF arXiv

Image restoration is an inherently ill posed inverse problem. Equivariant networks that embed geometric symmetry priors can mitigate this ill posedness and improve performance. However, current understanding of the relationship between network equivariance and data symmetry remains largely heuristic. Particularly for real world data with imperfect symmetry, existing research lacks a systematic theoretical framework to quantify symmetry, select transformation groups, or evaluate model data alignment. To bridge this gap, we conduct an analysis from an optimization perspective and formalize the intrinsic relationship among data symmetry priors, model equivariance, and generalization capability. Specifically, we propose for the first time a quantifiable definition of non strict symmetry at the dataset level (rather than sample level) and use it as a constraint to formulate the restoration inverse problem. We then show that the equivariance for restoration models can be naturally derived from this inverse problems incorporated the proposed symmetry constraints, and that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size. Furthermore, by analyzing the network's empirical risk, we demonstrate that aligning equivariance with data symmetry optimizes the bias variance trade off, minimizing the total expected risk. Guided by these insights, we propose a Sample Adaptive Equivariant Network that uses a hypernetwork and transformation learnable equivariant convolutions to dynamically align with each sample's inherent symmetry. Extensive experiments on super resolution, denoising, and deraining validate our theoretical findings and show significant superiority over standard baselines and traditional equivariant models. Our code and supplementary material are available at https://github.com/tanfy929/SA-Conv.

Abstract

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