SIEVE: Overcoming Topological Obstruction in Equivariant Self-Supervised Learning

Jongann Lee; Sun Woo Park; Yun Young Choi

cs.LGOct 15, 2026

SIEVE: Overcoming Topological Obstruction in Equivariant Self-Supervised Learning

Authors: Jongann Lee, Sun Woo Park, Yun Young Choi

Organizations: Max Planck Institute of Mathematics Bonn, Germany · SolverX Seoul, Republic of Korea

Abstract

We identify a theoretical incompatibility between 3D point cloud masked autoencoding and SE(3)-equivariant networks. We prove that standard masking triggers a topological collapse, effectively forcing the model to hallucinate orientation without a reference frame. To resolve this, we introduce SIEVE (Scalar Invariant Extraction Via Equivariance), an SE(3)-equivariant self-supervised framework that compresses point cloud geometry into invariant scalar embeddings. At its core is the spherical projection mask, which preserves global pose while obfuscating local semantics, sidestepping the obstruction by construction. Experiments confirm that SIEVE circumvents the predicted collapse where constant and noise masks fail. On downstream point cloud tasks, the resulting embeddings improve over coordinate-based baselines and hand-crafted geometric descriptors, with the largest gains on generalization to unseen shape categories.

Figures & tables

Figure 1: Random masking causes representational collapse; spherical projection masking does not. Random masking functions ignore subtle geometric data inherent in point clouds. This imposes topological obstructions in reconstructing the desired non-trivial vector bundles characterizing the underlying geometry of point clouds, as demonstrated in Theorem 2.6 . As a result, representational collapse can occur as shown in Section 4.2 . SIEVE utilizes a spherical projection mask, thereby possessing capabilities in reconstructing a class of pullbacks of vector bundles defined over a wide variety of 2 and 3-dimensional manifolds as demonstrated in Theorem 3.1 .

Figure 2: SIEVE Framework. SIEVE extracts SE(3)-invariant embeddings using self-supervised reconstruction on an equivariant bottleneck. The equivariant encoder compresses raw point coordinates into an invariant scalar encoding, while the vector pipeline is blocked. The spherical projection mask is applied to the input point cloud and supplied as the new vector embedding. It and the semantic details from the scalar embedding are used by the equivariant decoder to reconstruct the point cloud on a node-level.

Figure 3: Pre-trained SIEVE Encoder Embeddings. We visualize the PCA components of the pre-trained SIEVE embeddings. These embedding are invariant to point cloud rotation and translation.

Figure 4: Validation Loss of SIEVE on two baselines: GATr and SE(3)-Transformer. Trained on the synthetic dataset.

Figure 5: Reconstruction Comparison and Representational Collapse. We evaluate the reconstruction (blue) of the input point cloud (red) under different masking strategies. Using our spherical projection mask, the model achieves near-perfect reconstruction. Reconstructing with a constant vector mask or the ellipsoid noise mask results in a catastrophic representational collapse. We refer to Appendix A.3 for further discussions.

Model	$xyz$	$r$	$\kappa_{1},\kappa_{2}$	SIEVE
PointNet++	71.55	75.54	72.90	86.28
DeltaNet	87.21	88.07	86.86	89.58
DiffusionNet	81.20	85.89	85.97	89.30

Table 1: Test accuracies(%) on the RNA Segmentation Task

Seen Categories	PCN	PCN + SIEVE	Unseen Categories	PCN	PCN + SIEVE
Airplane	1.6508	1.8674	Bus	4.2137	4.2908
Cabinet	4.8287	4.3237	Bed	33.1502	29.4725
Car	2.6543	2.7663	Bookshelf	13.6558	12.5375
Chair	5.7248	5.6236	Bench	7.4325	6.3300
Lamp	7.4924	7.2977	Guitar	7.4999	5.2607
Sofa	5.7775	5.6728	Motorbike	9.4908	8.9940

Table 2: Reconstruction

\mathcal{L}2

CD Loss

(\times 10^{4})

on the 3D-EPN Dataset

Model	MSE( $R$ )	RMSE( $R$ )	MAE( $R$ )	MSE( $t$ )	RMSE( $t$ )	MAE( $t$ )
Shared Categories
DCP	1.475760	1.214808	0.826500	0.000004	0.002040	0.001239
DCP + SIEVE	0.703465	0.838728	0.479402	0.000002	0.001539	0.000774
Separate Categories
DCP	6.948212	2.635946	1.782947	0.000025	0.005048	0.003669
DCP + SIEVE	0.874955	0.935390	0.511852	0.000002	0.001509	0.000813

Table 3: Point Cloud Registration Results on the ModelNet40 Dataset

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Module	Hyperparameter	GATr	SE(3)-Transformer
Encoder	Input Channel (s,v)	(1,1)	(1,1)
	Hidden Channel (s,v)	(32,8)	(32,8)
	Output Channel (s,v)	(16,1)	(16,1)
	Number of Attention Heads	4	1
	Number of Layers	6	6
Decoder	Hidden Channel (s,v)	(32,8)	(32,8)

Appendix

Table 4: Model Hyperparameters for the Mask Ablation Study

Panel of Figure 6 — Figure 6: SIEVE on DeepJEB. DeepJEB is a point cloud dataset that contains objects with identical topology. Therefore, both the constant mask SIEVE and the spherical mask SIEVE is able to do near-perfect reconstruction.

Figure 7: Input Noise Ablation. We train SIEVE on ModelNet40 with three different input: local point cloud feature sv, a constant array of ones, and noise. Either noise or sv input enables SIEVE to reconstruct the input point cloud. SIEVE with constant input shows significantly worse performance when reconstructing the input point cloud.

Figure 8: Noise Input Trained SIEVE with SV as Input. The above figure demonstrates the result (blue, right) of reconstructing the input point cloud (red, left) using a noise-input pre-trained SIEVE with SV as scalar input. The distorted reconstruction implies that the encoder does not simply ignore the scalar noise input.

	seed 42	seed 67	seed 721
Validation Loss	$1.004\times 10^{-2}$	$1.002\times 10^{-2}$	$0.9981\times 10^{-2}$

Appendix

Table 5: Reconstruction Validation Loss

(\mathcal{L}1)

of SIEVE with different random seeds on the ModelNet40 dataset

Figure 9: SIEVE Embedding PCA Analysis. We perform PCA on the SIEVE embedding $s_{code}$ and plot the percentage of each eigenvalue $\lambda_{i}$ over the sum of all the eigenvalues. In addition, the cumulative percentage is also plotted. From the top left, we plot the analysis results for SIEVE with no noise, ‘simple noise’ strategy, ‘curv and norm’ strategy, and ‘nearest neighbor attention mask’ strategy.

Figure 10: Embedding Radius Correlation. This plot illustrates the correlation between each channel of the SIEVE embedding and the radius $r=\|\mathbf{x}_{i}-\overline{\mathbf{x}}\|$ .

$xyz$ only	radius $r$	No Noise	Simple Noise	Curv and Norm	NN Attention Mask
71.55	75.54	76.08	75.94	78.05	84.99

Appendix

Table 6: Test accuracies(%) on the RNA segmentation task with different SIEVE strategies

	PointNet++	DeltaNet	DiffusionNet
$xyz$ as feature	71.55	87.21	81.20
random init SIEVE	67.47	77.59	82.52
pre-trained SIEVE	86.28	89.58	89.30

Appendix

Table 7: Comparison of untrained SIEVE and pre-trained SIEVE on the RNA segmentation task (%)

$s_{code}$ injection location	MSE( $R$ )	RMSE( $R$ )	MAE( $R$ )	MSE( $t$ )	RMSE( $t$ )	MAE( $t$ )
Before DGCNN (chosen)	7.05	2.65	1.39	0.000004	0.0020	0.00092
DGCNN Intermediate	7.34	2.71	1.55	0.000006	0.0025	0.0015
Before Transformer	12.43	3.53	2.01	0.000029	0.0054	0.0036
Before SVD	24.59	4.96	3.12	0.000452	0.021	0.015

Appendix

Table 8:

s_{code}

injection location comparison on the geometric registration task

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Abstract

Self-Supervised Point Cloud Learning

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