Learning Generalizable Shape Completion with SIM(3) Equivariance
Authors: Yuqing Wang
Zhaiyu Chen
Xiao Xiang Zhu
Venue: NeurIPS 2025 poster ()
Keywords: Shape completion
Point cloud completion
Equivariance
Generalizability
3D representation
Primary area: deep_learning
TL;DR: We present the first SIM(3)-equivariant model for generalizable shape completion, achieving state‑of‑the‑art results on synthetic and real scans.

Abstract: 3D shape completion methods typically assume scans are pre-aligned to a canonical frame. This leaks pose and scale cues that networks may exploit to memorize absolute positions rather than inferring intrinsic geometry. When such alignment is absent in real data, performance collapses. We argue that robust generalization demands architectural equivariance to the similarity group, SIM(3), so the model remains agnostic to pose and scale. Following this principle, we introduce the first SIM(3)-equivariant shape completion network, whose modular layers successively canonicalize features, reason over similarity-invariant geometry, and restore the original frame. Under a de-biased evaluation protocol that removes the hidden cues, our model outperforms both equivariant and augmentation baselines on the PCN benchmark. It also sets new cross-domain records on real driving and indoor scans, lowering minimal matching distance on KITTI by 17\% and Chamfer distance $\ell1$ on OmniObject3D by 14\%. Perhaps surprisingly, ours under the stricter protocol still outperforms competitors under their biased settings. These results establish full SIM(3) equivariance as an effective route to truly generalizable shape completion.

Learning Generalizable Shape Completion with SIM(3) Equivariance
Yuqing Wang""
Zhaiyu Ch
Xiao Xiang Zhu
TTechncaal University of Munich
Munich Center for Machine Learning
Abstract
3D shape completion methods typically assume scans are pre-aligned canonical frame. This leaks pose and scale cues that networks may exploit to memorize absolute positions rather than inferring intrinsic geometry. When such alignment is absent in real data, performance collapses. We argue that robust generalization demands architectural equivariance the similarity group, SIM(3), so the model remains agnostic to pose and scale. Following this principle, we introduce the first (LM(3))-qqivvriia shape completion network, whose modular layers successively canonicalize features, reason over similarity- invariaat geometry, and restore the original frame. Under a de-biased evaluation protocol that removes the hidden cues, our model outperforms both equivariant and augmentation baselines on the PCN benchmark. It also sets new cross- doms-n records on real driving and indoor seans, lowering minimal matching distance on KITTI by 17% and Chamfer distance él on OmniObject3D by 14%. Perhaps surprisingly, ours under the stricter protocol still outperforms competitors under their biased settings. These results establish full SIM(3) equivariance as an effective route to truly generalizable shape completion. Project page: https: / //t:://imm /simemmmma////////// completion ceciiinniittutt github.ic io.
Introduction
3D scans are often riddled with gaps due occlusions and limited sensor coverage. Completing the missing geometry lets robots plan stable grasps, autonomous vehicles reason about hidden traffic, and curators digitize heritage artifacts without repeated scanning [1, 2, 3, 4J. However, most shape curated benchmarks where every scan is pre-
completion methods [5, 6, 7, 8] are developed aligned to a canonical frame with a fixed pose and scale relative to ground truth. These leaked cues inadvertently bias learning: instead of inferring intrinsic geometry, neural networks tend to memorize where shapes reside that frame, leading to inflated performance that collapses once the alignment is removed in practice [9, 10]. The resulting gap between benchmark success and real-world reliability highlights the challenge of exploiting geometry without inheriting extrinsic transforms that convey
Data augmentation mitigates this alignment bias by randomizing transforms during training to approximate inference-time invariance, but it entangles those transforms with underlying geometry and leaves the core ambiguity unresolved. Architectural equivariance, by contrast, aims to ensure that applying a transform to the input induces the same transform in the prediction, thereby isolating geometry from transforms and sharpening learned representations [11, 12]. However, existing equivariant methods still struggle to enforce this separation. -quuiiraana shape completion [13, 14] typically normalizes inputs using ground-truth centroids and scales, while (B(3)-cquivart) variants [10, 15, 16] still rely on ground-truth scale to canonicalize scans. Relying on such privileged information effectively reduces these models to explicit canonicalization (Fig. 1), undermining the true purpose of equivariance. To our knowledge, no existing architecture fully eliminates alignment bias, all still require some ground-truth alignment that is unavailable in practice.
Equal contribution. Corresponding author: zhaiyu. chen@tum. de.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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We argue that true generalization hinges on handling arbitrary similar- ity transforms, SIM(3), including ro- tation, translation, and scaling. To this end, we present the first shape completion architecture whose mod- ules are sign. During training, the network learns representations agnostic to pose and scale; at test time, any similarity transform applied to the input induces identical changes in the prediction (Fig. 1). To recover the completed shape in the sensor frame, we intro- duce lightweight restoration path that re-injects the transform informa- tion progressively. By disentangling intrinsic geometry from extrinsic transforms, our model trained on synthetic data transfers directly to real scans under a fair, de-biased evaluation protocol. In summary, our contributions include:
Canonicalization
Augmentation
SIM(3) equivariance
Ammbiguity
Agnastic to arbitrary similarity transfom
Cue leaked
by de-
Figure 1: Three paradigms for shape completion. Explicit canonicalization, including SO(3)- and -qqiivarva variants, leak pose and scale cues and fail on non-canonical inputs. Data augmentation mitigates the alignment bias but incurs ambiguity. We present (MM(())-quivarian approach that generalizes to arbitrary similarity transforms.
Problem identification. We reveal pose and scale bias in existing shape completion methods, and identify SIM(3) equivariance as prerequisite for reliable, in-the-wild generalization.
2. Generalizable framework. We develop the first fully completion. It integrates feature canonicalization, similarity-invariant geometric reasoning. and a transform restoration path into a modular design, generalizing from synthetic to real scans.
network for shape
3. Protocol and resources. We establish a rigorous evaluation protocol that eliminates hidden pose and scale bias, release code for reproducibility, and provide thorough analyses that pinpoint where equivariance delivers its gains. Under this protocol our method sets a new state of the art.
2 Related Work
Equivariant 3D representations. Equivariant neural networks learn features that transform con- sistently under input symmetry operations. Grounded in group theory and representation learn- ing [11, 17, 18. 19, 20. 21, 22. 23, 24, 25, 26, 27], 3D equivariant models have been developed to handle variability in data transforms. One approach employs group convolution [28, 29, 30] to encode symmetry, but these methods often remain bound to specific architectures and lack gener- ality. Another leverages tensor algebra with spherical harmonics as irreducible representations to achieve equivariance [12, 31J. Vector neurons (VN) [32] replaced high-order tensors with structured alternative. Subsequent extensions integrated attention mechanisms [33] and translation equivariance [34], yet most VN-based networks remain relatively shallow, which limits their applicability to complex 3D tasks. Nonetheless, both paradigms have driven advances in 6- DoF pose estimation [35, 36, 37], point cloud registration [38, 39], robotic
3D vectors, providing modular
manipulation [40, 41]. and 3D reconstruction [42, 43, 44], among others [45, 46]. However, most methods are confined to (3( or SE(3) equivariance and require centering or scale normalization. Both assumptions break down in real-world scenarios without ground truths. Although SIM(3)- equivariance can overcome these limitations, existing efforts [40, 41, 46) remain sparse, depend on near-complete inputs, and struggle on partial observations. We close this gap with fully SIM(3)- cquivariant Transformer architecture.
3D shape completion. Early methods represented geometry in voxels and applied 3D CNNs [5, 47, 48, 49], but cubic complexity limited resolution. The use of symmetrie functions for permuta- tion invariance [50] led to the development of shape completion networks that directly consume 3D points [51, 52]. More recently, Transformers [53] have recast shape completion as set-to-set translation [6], and now lead the field [7. 8, 54, 55, 56, 57]. Almost all prior work, however, presumes that inputs are pre-aligned to the training frame, letting pose and scale cues leak into models and collapse performance on raw scans without special adaptation [9, 58). Existing remedies follow two paradigms. Data augmentation randomizes transforms during training, but entangles extrinsic transforms with intrinsic geometry and incurs ambiguity at test time. Equivariance- based methods
2

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either estimate a canonical pose prior to completion [15, 16] or replace standard layers with equiv- ariant variants [10, 13]. The former relies on fragile pose estimator that misaligns under partial observations and propagates errors for the downstream completion. The latter often loses fine-grained details and underperforms Transformer models with data augmentation [13]. recent anchor-point scheme extends equivariance to SE(3) [10], but its dependence on brittle anchor selection hampers performance and still falls behind augmented baselines. Moreover, all these methods still rely on ground-truth bounding boxes to cancel scale variance, re-introducing the very cues they aim to discard. In contrast, we integrate full (3) equivariance into every layer, inherently agnostic to arbitrary pose and scale, delivering the first shape completion method that truly generalizes to completely unaligned real-world scans.
Method
3.1 Preliminaries
Formulation.
Shape completion, fo: x -> ý, takes a partial observation x {x; E R3 (e.g.. a point set) and aims to reconstruct a set ÿ = {ūr E R expressed in the original sensor frame. Due to varying capture conditions, and its ground truth y may undergo a shared unknown similarity transform = (s, R) E SIM(3):
Ji=l representing the completed shape, both
x = x sRx + + y' g- sRy + t,
E EE SO(3). t tE RR, (1)
where x' and y' are transformed representations of the same object. Although the transform g alters coordinates, the intrinsic geometry remains invariant. To guarantee consistent predictions under any similarity transform, we enforce SIM(3) equivariance in fo. Paired with permutation invariant loss (x)(xx),)) (e.g., Chamfer distance), we solve the constrained optimization problem:
min ((fo(x),)) s.t. folg x) = g- fo(x) Vg E SIM(3).
(2)
Vector neurons. To enforce the equivariance constraint in Eq. (2), we build on the vector neuron (VN) framework [32]. which replaces scalar neurons with 3D vector ones. At layer I, organize D vector channels into M vector features:
V (V V E D'33
The VN framework defines linear, nonlinear, and pooling operations on these vector neurons to preserve symmetry under the SO(3) group action. We adapt VN representations and integrate these operations as building blocks into our -qquivreuivaariant architecture. For simplicity, we omit the layer index / in the following unless cross-layer operations are involved. See Appendix C for details.
Challenges. We decompose the overall objective in Eq. (2) into three complementary requirements, separating geometric reasoning from transform alignment:
f(g fo(x), y) Req. (1) geometric reasoning
min
s.t. folg ()0xx)=0 9 ff((x)
(4)
11g* - I|
Req. (3) transform alignment
Req. (2) equivariance
Here 9 denotes the optimal alignment between the prediction and the ground truth, such that A(gg (x), y) measures geometric discrepancy independent of pose and scale. In this formulation:
Req. (1) Geometric reasoning. The network must infer the complete geometry of missing regions, even when x are sparsely and heterogeneously sampled. This demands strong structural priors and fine-grained feature extraction that shallow architectures cannot achieve.
Req. (2) Equivariance. The model must respect SIM(3) symmetry, which needs to be enforced throughout, because any single layer that is not equivariant will break global equivariance. Thus, each operator must be redesigned to commute with the group actions.
Req. Transform alignment. The completed shape fo(x) must be presented in the sensor frame that no further alignment is required for downstream tasks. Equivalently, g should remain close to the identity transform I under any reasonable metric || ||. This requires propagating pose and scale information throughout the network to preserve the original frame
The next section details how each component of our architecture satisfies these requirements.
3

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Query generator
Encoder
Decoder
JR Mr
R
DOUNN
Input
Output
(ins]
I thx
R Mxx3
(x) Canonicalization
(ix) Shape reasoning
Input feature
Output featire
Equivariance
Rotation
Translation
Scale
Queries
Invariance
Figure 2: Overview of our (1M(()-equivarian shape completion pipeline. We extract point patch features with VN-DGCNN [32] and feed them into Transformer encoder-decoder. Within each layer module, we (i) canonicalize features to be translation- and scale-invariant, (ii) reason intrinsic geometry via (MMM(33-invariia)) attention, and (éit) restore the original transform. This guarantees that both intermediate features and the reconstructed shape adhere to SIM(3) transforms.
3.2
shape completion
We progressively address the three challenges in Eq. (4) via -quuiaaeeuivvr blocks, which enforce the equivariance constraint in Req. (2) by design:
B' R' A' c'o fa(z) = B²
B¹ (z).
(5)
Each block B' comprises three sequential stages (Fig. 2): (i) feature canonicalization C¹ produces translation- and scale-invariant feature vectors; (ii) iiiilarity-ivvrriane shape reasoning .'' optimizes Req. (1); and (iii) pose and scale restored via R' to satisfy the hansform-alignnenn
objective in Req. (3). By stacking these blocks and end-to-end optimizing Eq. (4), the network iteratively refines its prediction and converges the completed shape expressed in the input frame.
Canonicalization. Robust geometric reasoning requires removing transform variance embedded in the feature repre- sentation. As shown in Fig. 3, we ex- tend layer normalization [33, 59, 60] explicitly factor out global translation and scale from shape features. Specif- ically, let V, E R3 denote the channel- wise mean of the latent vectors. We first subtract this mean from each V; eliminate translation, then normal- ize the centered vector to remove scale variation. followed by vanilla layer normalization applied to the row-wise norm V Dx1 stabilize training without altering the direction of the vector neurons, thus preserving rotational consistency. Formally, the ex-
Chernel-oris mean
Rew-sist rorm
Layer
Teput
Row-wise divicon
Fguivarincc
Incerience
Multiplicatinn
Figure 3: Layer normalization. We extend standard layer normalization [59] to producing translation- and scale- invariant features and preserving rotation equivariance.

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tended layer normalization is defined as:
C² V = layernorm
Vill2)
(6)
V
This procedure canonicalizes features to implicitly defined canonical feature frame.
Shape reasoning. In the scale- and translationiivvaiiai canonical feature frame after C² perform similarity-invariant shape reasoning via the rotation-invariant attention weights from VN-Transformer [33], ensuring no residual rotational bias. Since attention is invariant under any E SIM(3), the local form of the objective in Req. (1), for each shape reasoning layer, reduces to:
A': min ((fe))) y),
where B:j = softmax,
V3D (WoV', WxV'))) 0i E Ax).
(8)
Here, the query and key projections We, W E RXD, included the model parameters define how shape features interact. The Frobenius inner product (+)p is invariant to joint rotation of V', and V',, making the attention weights depend solely on their relative geometry. The attention weights satisfy Agg x) (x) for any g SIM(3), which decouples intrinsic shape features from transforms.
Transform restoration. SIM(3) equivariance alone does guarantee that the shape reasoning output is aligned with the original sensor frame, as it preserves only relative pose and scale. The final challenge is to recover this absolute alignment, as required Req. (3). To achieve this, we introduce transform restoration path to propagate input pose and scale via residual connections (Fig. 2). After each -invaainn shape reasoning step, the restoration path reinjects translation and scale to recover spatial grounding. Rotation is implicitly preserved through the attention output Z. = >, a,, Wv V where Wv is the value projection weight. Translation and scale are injected in accordance with their group actions via addition and multiplication, respectively:
R':
V v (2'Z),
where Ep [V V!ll,, is a global scale statistic computed from the average norm of centered input features, and Φ is VN linear layer that fuses spatial and geometric features to guide alignment. By restoring translation and scale at each stage, we reestablish full SIM(3) equivariance the module output (see Appendix C for the proof), ensuring consistent spatial grounding across layers, as shown in Fig. 4.
3.3 Network architecture
We build on the AdaPoinTr [7] backbone, which features a coarse-to-fin shape comple- tion scheme. We replace the original DGCNN with VN-DGCNN [32] for local geometrie fea- ture extraction while retaining SIM(3) equiv- ariance, and replace every Transformer layer with our HMM33-equivartt duced in Sec. 3.2. Key components, such as the query generator and the reconstruction head, are likewise implemented to be equiv- ariant or invariant when appropriate. The net- work takes a partial input point cloud with 2,048 (N,,.) points and predicts complete shape of 16,384 (Nout) points. Aside from these changes, we retain AdaPoinTr's network depth, loss function, and training settings to ensure fair comparison. Fig. illustrates the architecture. Further architecture and implementation details are provided in Appendix C and D.
None
Rotated
Scaled
Translated
Jopus
module intro-
Ourpet
Figure 4: SIM(3) equivariance. Our outputs follow arbitrary similarity transforms applied to inputs.
5

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Experiments
4.1 Experimental setup
Datasets and baselines. We first evaluate on the PCN benchmark [51], which comprises eight categories from ShapeNet [61] with paired partial and complete point clouds. To assess cross-domain transferability, we directly apply PCN-trained models, without further normalization, to world scans from KITTI [62] and OmniObject3D [63]. We compare against leading non-equivariant shape completion methods, namely PoinTr [6], SeedFormer [54], SnowflakeNet [64]. AnchorFormer [8], and AdaPoin'T Tr [7]. each trained with SIM(3) augmentations for fairness. Because no prior model offers full (3) equivariance, we resort including the -equivaraaa EquivPCN [13] and the -eeuivreuui SCARP [15] and ESCAPE [10] as baselines.
Evaluation protocol. For our model, which requires no training time augmentation, we adopt the train/test setting of I/SIM((3 where I denotes the identity transform. Each baseline is first evaluated under the group it was designed for (i.e., 1/SO(3) for EquivPCN [13]. 1/SE(3) for ESCAPE [10]. and SE(3) for SCARP [15]) reveal their upper-bound performance when pose/scale cues are still partly available. We then report their performance under SIM(3) /SIM(3) with additional data augmentation. On PCN, rotations are sampled uniformly from SO(3), while each partial input is itself centered and scaled to the unit sphere. This prevents cue leakage from ground truths and realistically simulates real-world inputs that models actually have access to. On KITTI and OmniObject3D, we transfer our model without any cue leakage, while competing equivariant methods receive canonicalized inputs via ground-truth alignment; otherwise, they fail completely. For PCN and OmniObjeet3D, report Chamfer distance l (CD-&1, scaled by 10a) and F-score @ 1% (FI). For experiments on KITTI, we follow prior practices [6, 7, 8, 54] and report the Fidelity and Minimal Matching Distance (MMD) metrics. All metrics are computed in the common canonical frame with unit scale for direct comparison. We refer to our method as SIMECO in all comparisons.
4.2 De-biased benchmark evaluation
Against data augmentation. Table 1 compares our SIM(3) -equivvaqqva model against leading non-equivariant networks trained with augmentation. Our method achieves the lowest average CD-L1 and the highest F1 score, outperforming AdaPoinTr by 10% and 8%, respectively. It yields the best score in every category, with consistent error reductions across the board. Qualitative results in Fig. show that our completions faithfully recover fine geometric details such as sharp airplane wings, slender lamp stems, and thin table legs, whereas the Iugmentation-bass baseline produces blurrier or distorted shapes. Notably, AdaPoin'rr without augmentation collapses under the de-biased protocol. These results confirm the superiority of our architectural equivariance over heavy data augmentation.
Table 1: Evaluation on PCN. We compare methods supporting only SO(3) (top) and SE(3) (middle), and those with S3M(3) augmentation (bottom). "Transform" indicate train/test settings. Our model outperforms competitors limited to partial transform groups and those with data augmentation. CD-( values are scaled by factor of 1000. Bold numbers indicate the best STM(3) results.
Method
Transform
Airpl.
Cab.
Car
Chair
Lamp
Sofa
Table
Wat.
CD-1 1
F1 t
EquivPCN [13]
1/SO(3)
8.38
13.74
11.81
14.31
12.50
15.68
12,86
11.02
12.54
0.569
AdaPcinTr
80(3)80(3)
5.76
11.27
9.63
9.61
6.71
11.53
8.15
7.81
8.81
0.693
SCARP
8B(3)/SE(3)
10.05
40.82
23.02
22.92
29.17
62.51
$7.82
37.59
35.49
0.223
ESCAPE [10]
(SS((3)
8.13
13.18
10.43
10.62
8.07
13.74
9.35
9.81
10.41
0.650
AdaPoinTr
SE(3)/S((3)
6.16
12.56
10.48
9.77
7.10
12.36
8.34
8.57
9.42
0.685
Pvaluation under de-hiosed protocol
AdaPoinTT
ISIM(3)
31.93
78.90
63.51
60..5
61.47
70.77
71.94
42.74
60.23
0.206
EquivPCN [13]
SIM(3)/LM)3)
9.10
14.60
13.09
15.74
13.74
16.42
14.74
11.74
13.65
0.523
ESCAPE [10]
(3/SS)))
12.59
22.54
18.63
15.86
12.68
24.38
14.17
14.48
16.88
0.515
PoinT:]6)
(IM(3)/LMM3)
10.18
17.97
15.61
16.94
13.39
16.80
17.75
12.18
15.10
0.434
SeedFoumer [59]
SIM(3)/MM
8.42
17.32
15.08
12.10
8.25
17.19
11.38
9.62
12,42
0.616
Snowflake [64]
(TM(3)/SMM(3
7.99
15.59
13.81
8.58
15.66
10.59
9.72
11.23
AnchoeFoomer [8]
(3)SSMM)
7.77
13.61
12.13
12.71
9.16
14,26
10.95
9.35
11.24
0.599
ODGNet [56]
(IM(
6.16
11.15
AdaPoiaTs [7]
(3/SL)
6.46
12.17
10.51
10.29
7.59
12.26
8.90
8.14
9.54
0.661
SIMECO (ours)
(3SM(33
6.02
9.25
7.77
0.714

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Against equivariant networks. Table benchmarks our method against other (3 and SE(3)- equivariant networks, each evaluated under its native transform group. In contrast, by tackling the full STM(3) group, we address substantially harder setting. Despite this, our model reduces average CD-& from 10.41 to 8.59 (-17%) and raises from 0.650 0.714 (+ 10%) relative to ESCAPE, which uses the same AdaPoinTr backbone. EquivPCN (SO(3)) and SCARP (SE(3)) lag even further, confirming that full (3)(3) equivariance enables learning more intrinsic shape representations. Moreover, neither EquivPCN nor ESCAPE can outperform ingmentation-asee baselines in their respective groups. And training ESCAPE and EquivPCN with SIM(3) augmentation degrades their performance, highlighting that equivariance not built into the architecture hard to acquire through augmentation alone. Figs and respectively demonstrate that our model preserves fine details and delivers consistent outputs under various pose and scale perturbations.
Input
GT
SIMECO (ours)
AdaPoinTr STM(3//SM((3)
ESCAPE
EquivPCN 1/S0(3)
SCARP SECI/SB(
T/SIM(3)
1/SB(8)
Lamp
Table
Figure 5: Comparison on PCN. Our (MM((3)-qquvarrii)) model outperforms other equivariant meth- ods restricted to S3(3) and SE(3) and non non-cquivariant baseline trained with SIM(3) augmentation.
4.3 Cross-domain generalization
Unseen driving scans (KITTI). Table 2 presents cross-domain performance on KITTI using models trained solely synthetic, canonicalized PCN data. Even under full STM(3) variation, model reduces MMD from 6.47 to 5.35 (- (-1%%) compared to the strongest non-equivariant baseline and ESCAPE'S Fidelity error from 1.81 0.56 (-69%), EquivPCN achieves even lower MMD, but only under its native SO(3) setting. Crucially, all competing methods, apart from those using data augmentation, rely on ground-truth bounding boxes normalize KITTI inputs. SO(3) models use them for translation and scale normalization; SE(3) models use them for scale. This requirement leaks information and impractical in real deployments. By contrast, our fully SIM(3) -qquivaaquva architecture requires no external normalization. In Fig. 7, our model recovers car wheels and indoor details more faithfully, while AdaPoinTr with augmentation produces oversmoothed outputs.

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Input
GT
SIMECO (ours)
AdaPoinTr SIMC3//SMM())
ESCAPE (1M(9)
EquivPCN SIM(3//SIM(3)
(SSI((3)
(3)
Figure 6: Robustness to pose and scale perturbations. Under larger pose and scale changes. our (iMu3)--quivarrian model maintains completion quality, whereas competing methods degrade.
Irput
GT
Ours
AdaPoinTr
N.A.
Chaic
Sola
Toy
Figure 7: Cross-domain generalization to real scans. Our PCN-trained model completes driv- ing (KITTI) and indoor (OmniObject3D) scans, with more details than the augmented baseline.
Figure 8: Feature consistency. Despite sig- nificant SIM(3) variations, feature maps from the PCN sample (outlined) and OmniObject3D scans exhibit matching structural patterns.
Unseen indoor scans (OmniObject3D). Table 3 presents cross-domain results on the diverse OmniOb- ject3D benchmark. Our model achieves the lowest average CD-4, and highest Fl. Compared top non-equivariant baseline, we reduce CD- é by 14% and increase F1 by 5%. Relative the SE(3)- equivariant ESCAPE, we achieve a 17% reduction in CD-I, These gains hold across seven categories, with particularly notable improvements on Cabinet (14.69 vs. 17.15) and Lamp (11.07 vs. 14.03). Aside from augmentation-bent based methods, ours is the only
Table 2: Cross-domain performance
KITTI. All methods are trained on PCN Cars.
the
Method
Transform Fidelity J MMD
EquivPCN [13]
I/SO
0.413
3.293
SCARP [15] ESCAPE [10]
SE(3)/S((3)
2.733
10.020
ISE(3)
1.810
5.930
PoinTr
SIM(3)SSM(()
AdaPoin [7]
(3M(S)S
0.537
6.468
(ouns)
ISIM(3)

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model that generalizes without bounding-bor box normalization. Fig. shows that our completions better preserve intricate geometric details.
Table 3: Cross-domain performance on OmniObject3D. Our model outperforms S0(3)- and -equivvrria methods and non-equivariant baselines trained with SIM(3) augmentation.
Method
Transform
Aipl.
Chair
Lamp
CD-41
EquivPCN [13]
VSO(()
12.05
16.06
13.91
13.68
16.84
13.47
12.67
14.10
0.543
SCARP [15]
(B(3)/
37.40
56.64
44.79
45.99
70.24
38.66
59.50
50.46
0.106
ESCAPE (10)
USE(3)
9.89
15.79
11.87
7.78
19:49
11.12
10.41
12.34
0.679
PoinTT [6]
(IM(3)/IM(3)
12.11
26.40
18.98
12.48
25.38
16.83
14.81
18.14
0.515
AdaPoinTT [7]
(IM(3)
11.48
17.15
12.10
7.44
14.03
10.73
10.38
1190
0.664
SIMECO tours)
USIM(
11.20
14.69
10.10
6.44
11.07
9.11
9.12
10.25
0.698
Feature visualization. Fig. 8 compares feature maps for PCN sample alongside those from
several OmniObject3D scans under different SIM(3) transforms. Despite large variations in pose and scale, the feature maps share strikingly similar structures, demonstrating that our network learns pose- and scalc-invarian features that generalize effectively to real-world data.
4.4 Ablations and analyses
Can pose estimation replace equivariance? To assess whether explicit pose estimator can substitute for built-in equivariance, we prepend ConDor [36]. a state-of-the-art self-supervised SE(3) pose canonicalizer (no equivalent exists for STM(3) our knowledge), baselines (PoinTr and AdaPoinTr) on the PCN dataset. As shown in Table 4, ConDor AdaPoin still fails match the augmentation--asee baseline (Table 1). and yields CD-(, 15% higher and an Fl score 3% lower than our SIM(3) -equivarii model; ConDor PoinTr performs even worse. In contrast, our approach achieves superior accuracy without any explicit pose estimation, demonstrating that architectural equivariance is more effective design choice.
two non-equivariant
Table 4: Pose estimation equivariance. Our model ourperforms baselines with pose estimator on PCN.
Table 5: Sensitivity to training-time transforms on PCN Car.
Method
Transform
CD-61
Fl
Tranffrmm
CD-f1
Fl
ConDor PoinTr [6]
ConDor [36] AdaPoinTr [7]
SE(3/SE((3
(3(SS)))3)
18.56
9.92
0.692
0.408
SF(3)SSM(3
50(3)SIM
(3)
8.78
8.88
0.705
0.708
0.707
SIMECO (ours)
I/SI((3)
8.59
0.714
1/SIM(3)
8.76
0.712
How much equivariance is necessary? Figure 9 plots performance on the PCN Car subset as we progressively swap non-equivariant for (IM(3)-equivarin) the encoder/decooder. The non-equivariant baseline [0/0] yields the worst CD-6, and lowest F1. Equipping the coder with six equivariant layers [6/0] reduces CD-&, and boosts FI. Further introducing four equivariant decoder layers [6/4] yields a slight performance drop, likely due to the overhead from mixing equivariant and non-equivariant modules. Nevertheless, the fully equivariant setup [6/8] attains the best CD-4 and F1, confirming that preserv- ing end-to-end SIM(3) symmetry is essential for optimal shape completion.
0.8
15.0
layers in
12.5
0.7
10.0
5.0
0.5
0.0
0.4
[Encoddr/Deunder equivariant layers
Which equivariance group matters most? We ablate training-tinn time equivariance on the PCN Car subset across four operational design domains (ODD): rotation (R), ro- tation + translation (R T), rotation + scale (R + S), and full SIM(3) (R + S + T). When directly transferred to KITTI seans (Fig. 10), only the STM(3) model attains the best Fidelity and MMD. Omitting scale (R + T) translation
Figure 9: Equivariant layers ablation. On PCN Car, performance increases as non-equivariant layers are progressively replaced with (EM((3)-euivarr)) the encoder/decoder. The fully equiv- ariant setup delivers the best results.
ones
(R + S) equivariance increases errors

---

on both metrics. Notably, the gap between SIM(3) and its subgroups highlights how much more challenging full (3) equivariance compared the subgroups. This synthetic-to- analysis confirms that S3M(3) equivariance is essential for robust, in-the- hiil shape completion and further validates our model's 's advantage over methods limited to SO(3) or SE(3).
Must we canonicalize training data? Real-world data seldom provide objects in a common reference frame, practical completion model should tolerate arbitrary pose and scale training time as well. We therefore Irain our network on the PCN Car subset under four configurations, I, SO(3), SE(3), and SIM(3), and report test performance in Table 5. Across these settings, the average CD-&, varies by less than 0.15 and the F-score by at most 0.007. The negligible difference demonstrates that our STM(3) -3eeivivrrri architecture learns shape priors that robust to the transform of training data. Thus, explicit canonicalization of the training data is not required.
Robustness to input noise and point dropout. Our model, trained exclusively clean PCN data, degrades gracefully under Gaussian noise up to 0.5% of the object scale, with average CD-& rising modestly from 8.59 to 9.34 (Fig. 11). It also tolerates substantial point dropout: with 25% additional dropout rate, Fl remains above 0.69 while CD-4 stays near its drop-free level. These results demonstrate the robustness of our architecture real-world scans subject to noise and sparsity. We provide additional analyses in Appendix B.
Ourn R.
Our RAS
Dure R-T
Ours Rt R
20c
0.75
Fildity
0.50
0.25
01P4
0,8%
0.00
Noise
1.00
0.75
1010
0,50
10
0.25
R13-T
8'
0DD
Real-word
Dropoul
Figure 10: Equivariance group ablation. PCN- trained models evaluated directly on KITTI are endowed with equivariance to rotation (R), trans- lation (T), and scale (S). Each added symmetry group improves performance, with the full SIM(3) model performing best in real-world ODD.
Figure 1I: Robustness to noise and dropout. Shaded regions show category-wise min-max. Trained solely on clean PCN data, our model remains robust under increasing Gaussian noise and to 25% additional point dropout.
5
Conclusion
We identified SIM(3) equivariance as essential for tackling the persistent pose and scale bias in shape completion and achieving robust generalization. To this end, we introduced the first shape completion architecture composed of inherently (ieu3)--euivaariant modules, which effectively disentangle intrinsic geometry from extrinsic transforms. Under strict, unbiased evaluation protocol that removes all alignment cues, our method sets new state of the art both on synthetic benchmarks and in direct transfer to unconstrained real scans. These results confirm architectural SIM(3) equivariance as principled remedy for truly generalizable shape completion. While our current implementation is limited to single-shape completion, extending this framework multi-object and large-scale scene modeling opens compelling avenues for future work.
Acknowledgment
This work was supported by TUM Georg Nemetschek Institute under the AI4TWINNING project. We thank Burak Bekei for sharing pretrained ESCAPE weights, Miaowen Dong for helpful discussions, and anonymous reviewers for their constructive comments.
10

---

A
Reproducibility
The code repository and demo are publicly accessible via the project page'. Detailed instructions for setup and running the code are described in the repository's README, md file.
B Additional Analysis
B.1 Input normalization
Table 6 compares two common scale normalization schemes: per-scan bounding-bor box extents and the global category maximum. Both schemes are consistently applied training and testing. We observe that the non-equivariant AdaPoin'r [7] is sensitive to the choice and performs better with bounding-bol box normalization, which we adopt for all competing methods in Sec. 4.2. Our model outperforms AdaPoin'Ti Ir by substantial margin, with only marginal improvements when ground-truth extents are available.
Table 6: Effect of input scale normalization. "B. box" scales each scan by its bounding box extent, while "max" uses the category's ' global maximum extent.
Methed
Source
Exteat
CD-E; 4.
F1 f
AdaPoinTTs [7]
Input
B. box
9.97
0.629
AraPoinTr [7]
Input
Max
12.46
0.557
SIMFCO (ours)
Input
B. box
8.88
0.705
SIMECO (ours)
GT
B. box
8.76
0,712
B.2 VN-SPD constraint
Imposing the VN-SPD constraint [34] on linear weights yields a more principled optimization than centering- based VN networks, since the center is learned rather than fixed. We validate this with an ablation that replaces VN-SPD with standard VN layers plus centering and normalization on partial inputs. As shown in Table 7, our full model clearly outperforms the centering variant.
Table 7: Effect of VN-SPD constraint [34]. The constraint yields more principled optimization
than the centering- based variant.
Mothod
Transform
Airpl.
Cab.
Car
Chair
Lamp
Sofa
Table
Wat.
CD-1; 1
F1 1
AdaPcieT: (71
SIM (3)
6.46
12.17
10.51
10.29
7.59
12.26
8.90
8.14
9.54
0.561
SIMECO (ezntered)
1/SMM(3)
7,50
SIMECD (ours)
(SSM(33)
6.02
10.75
9.27
9.25
6.66
11.16
7.82
7.77
8.59
0.714
B.3
Computational efficiency
With batch size of 40, SIMECO trains at about hour per epoch on two NVIDIA A40 GPUs. Fig. 12 reports training losses and validation metrics of SIMECO and AdaPoinTr [7]. SIMECO converges much faster in terms of epochs. AdaPoin'Tr needs about 140 epochs to reduce CD-4 below 10, whereas model does in only 50 epochs.
Table 8 reports the per-scan latency on PCN. Our model processes a scan in 76 ms end-to-end, about twice fast as the next-quickest equivariant competitor, ESCAPE [10] (148 ms), and more than twice as fast as EquivPCN [13] (172 ms) and SCARP [15] (172 ms). ESCAPE and SCARP spend extra time in AdaPoinTr remains faster at ms but achieves this speed without any built-in equivariance. Overall, method offers the best combination of speed and high-level SIM(3) symmetry preservation.
alignment steps. which inflate total latency beyond the raw inference cost.
To isolate the effect of built-in S3M(3) equivariance from model capacity, we compare methods under similar parameter budget on PCN (Table 9). When scaled to comparable parameter count, AdaPoin'T Tr improves over its smaller variant yet still trails our model by 0.34 in CD-&1, suggesting that the advantage of equivariance persists after controlling for parameter count.
httpp:: htapp:://smmmme completion. github.io

---

Adohanti
140
SIMBCO
SIMECO
Eocsu (x10")
(x10")
100
10
30
8
60
40
20-
50
100
150
200
50
100
150
200
Epoch
Epoch
AdaPuinTs SIMPCC
25
AdhPvinTt
SIMICO
0.65
20
0.60
0.55
15
epochs faster
12
0.40
50
100
150
200
0
50
100
150
200
Epoch
Epoch
Figure 12: Training losses and validation metrics. Our model converges faster than the baseline.
Table 8: Average per-scan latency on PCN. "Inference" " denotes the network forward time, measured NVIDIA A40 GPU; "Total" adds any pest-processing overhead.
Latency (ms)
Method
Equivariance
Inference
Total
AdaPoin'I [7]
I
16.4
16.4
EquisPCN [13]
80(3)
171.6
SCARP [15]
SE(3)
159.2
172.0
ESCAPE [10]
SE(3)
18.1
148.4
SIMECO (ours)
STM(3)
76.4
76.4
Table PCN results by parameter count. "#Param. (M)" is the number of parameters (millions).
Method
Transform
#Param. (M)
CD-61
FI
AdaPoinTr STM(3)SMM(3)
32.49
9.54
0.661
AdaPoinTr [?]
(IM(3)SIMM(3
$7.12
8.93
0.693
(murs)
TIST((3)
8.59
B.4 Performance on thin structures
To evaluate performance shapes with pronounced thin structures, compute local PCA-based anisotropy score: I, = d1-d2 on each point's k-nearest neighbors (k- = 30), where σi 2 02 2 d3 are the singular values of the local covariance matrix. high L indicates a spindly, edge-like neighborhood. We then select shapes in which more than 0.5% of points satisfy L > 0.8, yielding 50 thin-structure cases out of 1200 total. Quantitatively, on this subset, our method achieves an average CD-(1 of 6.83, compared 8.59 over the entire test set, demonstrating performance on such samples with thin structures. Moreover, qualitative examples such as chair and table legs in Fig. and Fig. 7 further confirm that method preserves fine details effectively.
better overall
B.5 Limitations
Despite its strengths, approach has several limitations:
12

---

1. Pose- and scale-dependent features. By construction, we remove any dependence on absolute pose or scale. While this makes the model robust arbitrary similarity transforms, it can also discard helpful cues when objects always appear in canonical frame. For instance, chair back with no visible legs might be mistaken for sofa because the two shapes coincide under similarity transform (see Fig. 13). Nevertheless, in realistic settings our method consistently outperforms non-equi non-equivariant baselines.
2. Symmetries across partial observations. The equivariance property in our framework is defined with respect single partial sean. For different partial observations of the same object, initialization variability cannot be fully eliminated, thus cross-view symmetries cannot be explicitly enforced and must be learned implicitly from data.
3. Articulated complex scenes, Our method excels at completing shapes under arbitrary similarity transforms, but does not explicitly account for independently moving sub-parts (e.g. human joints, robotic arms, or scenes with multiple objects). Incorporating category aetegor--specific shape priors or allowing multiple local transforms would be natural extensions to address these more complex scenarios.
Computational overhead. Vector-valued features and fully equivariant modules incur substan- tial computation by factor of three compared to scalar-valued layers. As result, runtime latency is higher than that of non-equivariant baselines (see Table 8), which may limit real-time resource-constrained deployments.
B.6 Failure cases
Fig. 13 shows two failure cases that rarely occurred in experiments. Ambiguous partial geometry can entice the network to produce a completion that is plausible yet incorrect. Severe sparsity or noise may disrupt the transform restoration and cause the completed shape to drift from the input frame.
Input
SIMECO (ours)
Inpul
SIMECO (ows)
N.A.
Figure 13: Failure cases. Top: ambiguous partial scans leave the network unsure how to complete the shape. Bottom: poor input quality disrupts the transform restoration module and yields misalignment.
C Proof of SIM(3) Equivariance
In this section, for convenience, we represent 3D vectors as row vectors and stack them into matrices. Specifically, let V Dx3 denote vector feature, where Vid] E R is its d-th channel, and collect M such features in the set V [V: -=1 Broadcast vectors (e.g., 1n [1, 1..., 1]T RDx11 column vectors.
C.1 Definitions
Definition 1 (Group invariance). A mapping f is G-invariant (e.g., [[g x) = f(x) for all g E G and admissible inputs x.
if it satisfies
Definition 2 (Group equivariance). A mapping f is G-equivariant (e.g., (equivvaauaa)) if satisfies f(g x) = f(xx for all g G and admissible inputs x.
13

---

C.2 -IM(3)- vector neurons
We assume the input is transformed by an arbitrary g = (s, R. t) SIM(3), acting on each 3D vector in the VN architecture [32] and the associated matrix as follows:
g V[d] -RV [d] + g V
VVR 1pt,
S R E SO(3), e R
(10)
VN-Linear. The VN- layer is defined as linear transformation shared across the three columns of V:
VN--Lnear(V) WV, WERD For translation equivariance, we constrain each row of the weight matrix W sum to one [34]:
(11)
wij Wij W e (L....,D',, = Wip 1p
(12)
Proposition 1. VN-Tinear()) is (3)
Proof. For all 9 E SIM(3),
VN-Linear(g- V) = W(sVR + 1
(13)
&WVR W 1nb
(14)
+sWVV)R 1n
(15)
VN-Linear (V) R +1pp
(16)
g (X-Linear(V),
(17)
The VN-ReLU layer is constructed via three VN-Linear layers that produce a feature F, direction B, and origin O, followed by centering with respect to O:
VN-ReLU.
(F, B, O) = VN--incarrV),, Fo = O, Bo = O.
(18)
The nonlinearity removes the negative projection of Fo onto the normal Bo of the plane through O:
Fo
(Fo. Bo) F 2
VN-ReLU(V)
Bo
Bo
0 Fo
Fo,
O.W.
Bo 2
F Boll
F
if (Fo. Bo) 20
Bo
(19)
F (Fo, Bo) |Bol
o.W.
Proposition 2. VN-ReLU() SIM(3) eqqivvarian.
Proof. For all g E SIM(3),
9 9.F
if (sFo,, sBol) 2
VN-ReLU(g v)
sBo
(20)
sFR + 1pt (sFoR, R)oR)
BBoR
F.F F
if & Fo. Bo) 20
sBoR
(21)
sFR + 1 - (Fo. Bo)
O.W.
3
BB
g. F.
if Fo. Bo) 20
Bo
(22)
sFR + 1n =F Bolr
o.w.
Bo
g F
Fo. Bo) 2
Bo
(23)
(F (Fo. Bo)
R
1pt
Bo
g F
if (Fo.
Bo) 2 0
Bo
(24)
(F (Fo. Bo)s
Bo
O.W.
VN-ReLU(V).
(25)
14

---

Here, (+) holds because F, B.O are
(Prop. 1), and translation cancels in Fo
and Bo (**) holds as the Frobenius inner product and the -orm are
Re-L.. The VN--LakkReLLU layer is minor variant of VN-ReLU:
VN-Leaky<RLLU(V) aV (1 a) VN--RLU((V), (0,
(26)
This operation is trivially -iqui)-eetiivarraat.
VN-Max. The VN-Max layer is defined on a set of vector features V by applying two shared
VN-Linear layers each V, E producing a direction B; and an origin O,, followed by centering with respect to O
(B O,) = (V-L.near(V.). Bo.: = B; - -,, Vo.
V, O
(27)
VN-Max selects, for each channel d, the feature whose centered representation Vo,id] is most aligned with its corresponding centered direction Bo.:[d]:
VN-Max(V)[d = V.- [4] with 8" = arg max Bo.; d))r.
(28)
Proposition VN-Max(-) - equivariant.
Proof. For all g E SIM(3),
VN-Max(g Vi[d] g. V.-(d), with i arg argmax(sVo..dd RBo.ddR)
(29)
arg max (o.. (d)
(30)
arg memax(Vo.. [4]. Bo.(dd)r
(31)
VN-Max(g [d g VN-Max(V)ld]
VN-Max(g - V) = VN-Max(VV.
(32)
Here, (+) holds since the Frobenius inner product is scaling factor x2 preserves the ordering, so the index i" remains unchanged.
(**) holds as the positive
C.3 SIM{3)-cquivariint Transformer
Canonicalization.
VN-LayerNorm follows the definition in Sec. 3.2:
V V
D
V = VN-LayerNorm VV = layernorm (/|v - 1 |
with
Σ vid.
[V-V
D
(33)
Proposition 4. VN-LayerNorm()) is invariant to scaling and translation, and equivariant
tion.
Proof. For all g E SIM(3),
(sVR - -VR)
VN-LayerNorm(g- V)
layernorm -VVVRssVRR
(34)
SR SVR]
V
layernorm (s | -V(I)
R
(35)
| V
(w+*) layernorm (/V Vll2)
V V
(36)
|| V
VN-LayerNorm (
(37)
(+) holds since Vis is (IM(3))equivvarian, and translation cancels differences. (**) holds because the -orm is rotation invioonnttiaiitn...mm (* *) holds as layer normalization is invariant to positive scaling. D
15

---

Shape reasoning.
VN-Attn follows the definition in Sec. 3.2. For self- the input features satisfy V & = =V V": for cross-attention, V, and Vi may differ. These features are the outputs of the canonicalization step, which removes the effects of translation and scale. The query and key are computed via shared VN-Linear layers:
Q3 VN-Linear (V Kj VN-Linear(VV...
(38)
The attention weight and output are then computed following IN-Transformer [33]:
Ci.j = VVN-AtttQ.... = softmax,
(Q Kj)p)
(39)
V3D
Z: Σ aig VN-LLnear (VV.))
(40)
Proposition 5. VN-Attu(;s)) is invariant to rotation, and Z, is equivariant rotation.
Proof. Rotation invariance of VN-Attn(-,-) follows immediately from the fact that the Frobenius inner product is notation-invarian Because Oi.j is rotation- -nviarnnn and, by Prop. 1, VN-Linear (-) rotation-equiis -qutivrneuu follows that Z; is rotation-equivariant.
Transform restoration. Transform restoration follows the definition in Sec. 3.2. Given the module input V and the attention output Z, the restored output is then computed
D
TR(A, V, Z) = VN-Linear(uZ), with = ED B.V, Vollg V = D
Σ V Vi[d]. d=1
(41)
Proposition 6. :R(: can recover SIM(3) equivariance.
Proof. For all 9 E SIM(3)
TR(g (H, V. Z))
sVR + 1pt VN-Lincar(sy(ZR)
(42)
(xx)
sVR + 1pt S/VN-Liner)))) R
(43)
sVV + */VN-Linear(Z))RR 1pt
(44)
= g [R, Z))
(45)
Here, (+) holds because the attention output Z encodes only the effect of rotation (Prop. and Prop. 5). The scalar μ scales with s, as translation is eliminated by differencing, and the f--n is intation-ivvarian Hence,
ED (g (V, - V.) = & ED E.(V -V::-V))l>
(46)
(**) holds because VN-Linear(.) is
(Prop. 1).
C.4 Other modules
VN-DGCNN. VN-DGCNN performs edge feature extraction and aggregation across layers [7, 32]:
1 M
V-- = VN-MaxjeN, (VNLA ((V) v' v!) V!)), with V
(47)
where M2 is the KNN neighborhood of point i, and @ denotes feature concatenation. VNLA(.) applics VN-Linearr.) followed by VN-LeakyRoLU()) Because each edge feature (V), V VV) V preserves S3M(3) equivariance, and both VNLA(.) and VN-Max(-) are (3e(a)--euivaaiann (Prop. I, Prop. 2, and Prop. 3), each layer output remains equivariant. By layer-wise induction, the entire VN-DGCNN is
We initialize all vector features V with the 3D coordinates of
the input points.
16

---

Query generator. The query generator produces fused query set Q [Q, QGl [7], where Q sampled from the partial input and Qa LVN-eer(V--Max((( with denoting the output of the final encoder layer. Q is (3M(3) eequivarra,,,,, as Q follows the transformed input, and QG inherits equivariance from the encoder through -qquivaal operations (Prop. 1 and Prop. 3).
Reconstruction head.
The reconstruction head produces the final output point set y as:
1 - E Vid. D d=1
y VN--inear(V V) Q with V
(48)
where V is the decoder output. is accumulation from V and Q, with both the VN-Linear(-) and Q preserving equivariance (Prop. 1).
as centering V prevents translation
C.5 Summary and approximate equivariance bound
The entire network architecture is from the above-mentioned -eeuui-euuaaarian modules. To stabilize training, we follow the practice of Assaad et al. [33] and introduce small norm-controlled bias to VN-Lincar layers. Although this modification introduces a minor deviation from exact equivariance, its effect in each layer is bounded by constant et, and remains insignificant across layers as proved in VN-Transformer [33]. As result, the overall network is effectively appro--apro equivariant.
by construction, since it is built exclusively
D Implementation Details
SIMECO is implemented in PyTorch and optimized using the Adam optimizer with an initial learning rate of 10-1, a weight decay of 5 x 10-1 and learning-rate decay factor of 0.9 every 15 epochs. We adopt the same architectural depth and hyperparameters as AdaPoinTr [7]. The models, including baselines, were trained for 200 epochs on two NVIDIA A40 GPUs. All other completion methods [6, 7, 8, 10, 13, 15, 54, 64] were used with their default settings.
E
More Visualizations
Fig. 14 expands the PCN comparison with more methods. Fig. 15 presents further qualitative results on KITTI and OmniObject3D scans. Fig. 16 shows how the methods respond to controlled pose and scale perturbations.
17

---

SIMECO
AdaPeinTr (SSM(())
AnchorFormer
Snowilaeeeet SSM(A)SSIM())
SeedForer
PoinTr
ESCAPE
EquxPCN
1/SIM(3)
SIMMA)/SLM(()
SDM(3)SSMM(3)
SDM(3)/SIM(3) SIMMAaSSIM()
lang
TaHo
Figure 14: Extended comparison PCN. Our model outperforms other equivariant methods and
non-equivariant baselines trained with SIM(3) augmentation. Complements Fig. 5.
Imput
Ours
ESCAPE
EquivPCN
KUm
N.A
Chair
3ofa
Toy
Figure Extended cross-domain comparison. Our PCN-trained model completes driving (KITTI)
and indoor (OmniObject31) scans more accurately than other methods with SIM(3) augmentation. Complements Fig. 7.
18

---

Input
GT
SIMECO (ours)
AdaPoinTr SIM(33/SIM(8)
ESCA PE (IM(3/SSMM()
EquivPCN
1/SIM(3)
SIM(3)/SD((3)
S3M(3)
Figure 16: Extended comparison of robustness to pose and scale perturbations. Under larger pose and scale changes, our -qquiiaraaa model maintains completion quality. whereas competing methods degrade. Complements Fig. 6.
19

---

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Reviews:
[{"review_id": "33LOGWc89s", "reviewer": "Reviewer_AFMZ", "summary": "This paper claims that they developed a SIM(3)-equivariant point cloud completion network. It aims to address the problem of shape completion in any reference frame, which is a very important problem, in my opinion. The method is through translation and scale normalization, building on top of the Vector Neurons feature representation. Experiments are conducted under a \"debiased protocol\", meaning that the bounding box and center information are not given during inference, which aligns much better with the practical use case.", "questions": "Explain how the vector features are initialized for each point, and discuss the equivariance property at this step. \n\nThe layer normalization seems to be done per point, but why is the scale restoration common across all points? Does that create a discrepancy? \n\nClarify the dimensions in the canonicalization step. \n\nExplain equation 47 in the appendix. Why is it designed this way? It seems to be different from typical VN-DGCNNs. What is the dimension of $\\bar{V}_i^l$, and what's its meaning? The per-point mean across the feature dimension? \n\nIs the definition of $mu^l$ wrong? The $E_i$ should be outside of the norm? Is $E$ expectation?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I'm glad that this paper aimed to address the issue of shape completion from arbitrary reference frames without privileged ground-truth information in terms of the center or scale. This is the real challenge in practice, and solving it adds real value. \n\nIn 3D shape completion, I think we need to be careful about what equivariance we are talking about. The challenge of applying equivariance in 3D point clouds mainly comes from the partial observation. It means that the observations from different perspectives are no longer simply connected by a transformation, and it also brings ambiguities in terms of the reference frame. In this context, discussing equivariance to the transformation of a single input partial-view point cloud itself is not very useful, because you can always centralize and scale-normalize the point cloud, leaving only rotation variations to be concerned with, in which case, only rotation equivariance is needed. However, what's more important is finding the internal symmetries among different partial observations. While they can be associated using a common center, the underlying center is not in the data, and centering each input individually breaks the association across observations. Furthermore, the scale-normalization step, which is common in point cloud learning, leads to further discrepancy across observations, because different parts have different scales. The above calls for translation and scaling equivariance for the shape completion task. \n\nI don't think any work, including this work, has a perfect solution to the equivariance discussed above, because the centering and scaling information are not directly observable in a single input point cloud. However, most previous work (cited in this paper) only attempted to approximate translation equivariance (for example, using anchor points as in ESCAPE [10]), but left scaling out. This paper does not do any better in dealing with translation equivariance (it still relies on centering the input point cloud), but the scale normalization in the layer-normalization step, done for the features of each point, addressed the scaling variations in the partial view point clouds after the unit-ball normalization. I think this is the key contribution and differentiator of this work. \n\nThe proposed strategy showed its practical value in the experiments under the realistic evaluation protocol, which I appreciate. \n\nWeaknesses: \nAs in the previous discussion, this paper also followed the notation of equivariance of a single point cloud, and thus, the proof of equivariance is equally trivial and impractical as previous work. Specifically, the problem still lies in the translation. For different partial observations, the centering step subtracts a different offset, resulting in different outcomes, thus breaking the symmetry. \n\nYou may argue that the per-point feature canonicalization removed the dependency of perfect centralization. However, if I understand correctly, the vector features for each point in Vector Neurons are initialized with the absolute coordinate of each point and the local differences with other points. Global translation will change the global component but not the local component, resulting in different overall vector features, which cannot be cancelled by point-wise canonicalization. \n\nThe notations in the method explanation are unclear, especially in terms of dimensions. For example, the canonicalization step is written with a lot of ambiguities. What are \"channel-wise mean\", \"row-wise norm\", and \"modified layer normalization\"? I need to think very carefully to make sense of these descriptions. Use letters to disambiguate the dimensions. Figure 3 also didn't help too much.", "was_revised": true, "final_justification": "The author addressed my concerns. I recommend accepting this paper.", "rebuttal": "Dear Reviewer AFMZ,\n\nThank you for your comments. We address your concerns and questions below.\n\nThroughout the discussion, we denote shape features as $\\mathcal{V} = \\\\{\\mathbf{V}\\_i\\\\}\\_{i=1}^M$, where $\\mathbf{V}\\_i \\in \\mathbb{R}^{D \\times 3}$. Here, $M$ denotes the number of vector features, $D$ the feature dimension, and $3$ the vector dimension.\n\n> Specifically, the problem still lies in the translation. For different partial observations, the centering step subtracts a different offset, resulting in different outcomes, thus breaking the symmetry. You may argue that the per-point feature canonicalization removed the dependency of perfect centralization. However, if I understand correctly, the vector features for each point in Vector Neurons are initialized with the absolute coordinate of each point and the local differences with other points. Global translation will change the global component but not the local component, resulting in different overall vector features, which cannot be cancelled by point-wise canonicalization.\n\n> This paper does not do any better in dealing with translation equivariance (it still relies on centering the input point cloud)\n\nWe believe this impression may stem from a misunderstanding of our method design. Unlike earlier vector neuron-based methods [1, 2, 3] which **explicitly** subtract the input centroid and thus result in reference frame ambiguities, our method does not rely on any centering of the input points' coordinates. Instead, we follow the practice of VN-SPD [4] by constraining $\\mathbf{W} \\mathbf{1}\\_D =\\mathbf{1}\\_{D'}$, where  $\\mathbf{W} \\in \\mathbb{R}^{D'\\times D}$ is the weight matrix for linear layers, and design each SIM(3)-equivariant block to preserve strict equivariance. This way, no ambiguity exists because the network can **inherently** distinguish changes in input arising from translation or from intrinsic geometry. For a full description of our SIM(3)-equivariant architecture, please see Suppl. Mat. C.2 (Propositions 1-3).\n\nWe then further clarify below two pieces of detail that might have caused the misunderstanding.\n\n> Explain how the vector features are initialized for each point, and discuss the equivariance property at this step.\n\nEach point's vector features are initialized as in Eq. (47):\n\n$$\n \\mathrm{VN\\text{-}Max}\\_{j \\in \\mathcal{N}\\_i} \\left( \\mathrm{VNLA} \n\\left((\\mathbf{x}\\_j-\\mathbf{x}\\_i+\\bar{\\mathbf{x}}\\_i)\\oplus \\mathbf{x}\\_i \\right)\\right),\n$$\n\nwhere $\\mathcal{N}\\_i$ is the $k$-nearest neighbors of point $i$, and $\\oplus$ denotes feature concatenation. $\\mathrm{VNLA}(\\cdot)$ applies $\\mathrm{VN\\text{-}Linear}(\\cdot)$ followed by $\\mathrm{VN\\text{-}LeakyReLU}(\\cdot)$. $\\bar{\\mathbf{x}}\\_i \\in \\mathbb{R}^{1\\times 3}$ is the mean coordinate of all input points  $ \\mathbf{x} =\\\\{x\\_i \\in \\mathbb{R}^3\\\\}\\_{i=1}^{N\\_\\mathrm{in}}$,\nwhich preserves translation offset for the local difference $(\\mathbf{x}\\_j-\\mathbf{x}\\_i+ \\bar{\\mathbf{x}}\\_i)$.\n\nFor all $g \\in \\mathrm{SIM}(3)$,\n$$\ng \\cdot \\left(( \\mathbf{x}_j-\\mathbf{x}_i + \\bar{\\mathbf{x}}_i) \\oplus \\mathbf{x}_i \\right) \n= \\left( ((s R \\mathbf{x}_j+t)-(s R\\mathbf{x}_i+t)+(s R \\bar{\\mathbf{x}}_i+t))\\oplus (sR\\mathbf{x}_i +t) \\right) \\notag \n= sR\\left((\\mathbf{x}_j-\\mathbf{x}_i+\\bar{\\mathbf{x}}_i )\\oplus \\mathbf{x}_i \\right)+t\n$$\n\nThus, the initialized vector features remain SIM(3)-equivariant.\n\n> Explain equation 47 in the appendix. Why is it designed this way? It seems to be different from typical VN-DGCNNs. What is the dimension of $\\bar{\\mathbf{V}}\\_i^l$, and what's its meaning? The per-point mean across the feature dimension?\n\nAs you correctly noted, typical VN-DGCNN initializes vector features with $\\left( (\\mathbf{x}\\_j-\\mathbf{x}\\_i) \\oplus \\mathbf{x}\\_i \\right)$ and constructs vector features per layer in the same way $\\left( (\\mathbf{V}\\_j^l-\\mathbf{V}\\_i^l) \\oplus \\mathbf{V}\\_i^l \\right)$. While $\\mathbf{V}\\_i^l $ is translation-equivariant, the subtraction $(\\mathbf{V}\\_j^l-\\mathbf{V}\\_i^l)$ cancels the translation term $t$, which would break the equivariance. To address this, we follow the practice from VN-SPD [4] (Eq. (47)), \n$$\\mathrm{VN\\text{-}Max}_{j \\in \\mathcal{N}_i} \\left( \\mathrm{VNLA} \\left( (\\mathbf{V}_j^l - \\mathbf{V}_i^l+\\bar{\\mathbf{V}}_i^l) \\oplus \\mathbf{V}_i^l \\right) \\right),$$\nwhere  $\\bar{\\mathbf{V}}\\_i^l \\in \\mathbb{R}^{D^l \\times 3}$ is the mean across $M$ features and is added back to the local difference $(\\mathbf{V}\\_j^l-\\mathbf{V}\\_i^l+\\bar{\\mathbf{V}}\\_i^l)$ to restore $t$.\n\n> As in the previous discussion, this paper also followed the notation of equivariance of a single point cloud, and thus, the proof of equivariance is equally trivial and impractical as previous work.\n\nWe appreciate the opportunity to clarify our perspective. As previously explained, our strict SIM(3) equivariance ensures that we never need to centralize input points as prior works typically do. For the same object, all partial observations, their learned equivariant features, and the outputs, naturally reside in the original object frame. Importantly, any higher-level symmetry across arbitrary observations cannot be enforced explicitly and must instead be learned, as equivariance can only be defined per observation and can only help us this far. Our approach embraces this principle by letting the network learn truly intrinsic geometric features that are inherently invariant to pose and scale, rather than forcing all inputs into a common reference frame.\n\n> The notations in the method explanation are unclear, especially in terms of dimensions. For example, the canonicalization step is written with a lot of ambiguities. What are \"channel-wise mean\", \"row-wise norm\", and \"modified layer normalization\"? Use letters to disambiguate the dimensions.\n\n> Clarify the dimensions in the canonicalization step.\n\nThank you for pointing out the ambiguity and your suggestion. \n\nThe \"channel-wise mean\" $\\bar{\\mathbf{V}}\\_{i} $ is defined as\n\n$$\n\\bar{\\mathbf{V}}\\_{i} = \\frac{1}{D} \\sum\\_{d=1}^{D} \\mathbf{V}\\_{i,d},\\quad\\bar{\\mathbf{V}}\\_{i}\\in\\mathbb{R}^{1\\times3}.\n$$\n\nThe \"row-wise norm\" $\n\\big\\Vert\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_{2}\n$  is defined as the $\\ell\\_2$-norm across 3 vector dimensions:\n\n$$\n\\Vert\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_{2} = \\sqrt{\\sum\\_{k=1}^{3} \\left( \\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i} \\right)^{2}\\_{:,k}}\n, \\quad\n\\big\\Vert\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_{2} \\in \\mathbb{R}^{D \\times 1}.\n$$\n\nThe \"modified layer normalization\" corresponds to Eq. (6), ${\\mathbf{V}^{\\prime}}\\_{i}\n= \\mathrm{layernorm}(\\big\\Vert\\mathbf{V}\\_{i} -\\bar{\\mathbf{V}}\\_{i}\\big\\Vert_2)  \\frac{\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}}{\\big\\Vert\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_2}$,\nwhere the $\\mathrm{layernorm}$ operation is applied to the scalar-valued \"row-wise norm\"\n$\\big\\Vert\\mathbf{V}\\_{i}-\\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_{2}$. $\\big\\Vert\\mathbf{V}\\_{i} - \\bar{\\mathbf{V}}\\_{i}\\big\\Vert\\_{2}$ produces scalar neurons for standard layer normalization. We have revised the manuscript to explicitly use letters for solving the ambiguity over dimensions.\n\n> The layer normalization seems to be done per point, but why is the scale restoration common across all points? Does that create a discrepancy?\n\nThe layer normalization (canonicalization) serves two purposes: \n(1) For the same partial scan under SIM(3) transformation, the canonicalization step can factor out the global translation and scale from the shape features (Proposition 4 in the Suppl. Mat.):\n$\n\\mathrm{VN\\text{-}LayerNorm}(s \\mathbf{V}R + t)=\\mathrm{VN\\text{-}LayerNorm}(\\mathbf{V}) R.$\n(2) It performs the mean-centering $(\\mathbf{V}-\\bar{\\mathbf{V}})$ and variance normalization of standard layer normalization, extended to vector features, which stabilizes the attention mechanism during training.  \n\nThe scale restoration is used to align the completion result with the original sensor frame, that is, to recover $s$. We apply this single, common scale only after the attention-based shape reasoning, since the relative magnitudes of feature vectors encode intrinsic geometric information. Introducing per-feature scaling would re-entangle the intrinsic geometry with the external transform and potentially disrupt the attention. By using one global scale, we preserve the relative magnitudes while correctly restoring the output to the sensor frame.\n\n> Is the definition of $\\mu\\_l$ wrong? The $E\\_i$ should be outside of the norm? Is $E$ expectation?\n\n$E$ is indeed empirical expectation. The definition of $\\mu\\_l$ is correct as we intentionally place the point-wise expectation $E\\_i$ inside the norm.\n$E\\_i(\\mathbf{V}^{l}\\_{i}-\\bar{\\mathbf{V}}^{l}\\_{i}) \\in \\mathbb{R}^{D \\times 3}$ is the  mean across $M$ features and $E\\_{D^l}$ is the mean across $D$ dimensions for $\\big\\Vert E\\_i(\\mathbf{V}^{l}\\_{i}-\\bar{\\mathbf{V}}^{l}\\_{i})\\big\\Vert\\_2 \\in \\mathbb{R}^{D}$. Taking the mean across $M$ features first provides a stable global representation, and applying the $\\ell\\_2$-norm across the vector dimension before averaging over channels ensures that $\\mu\\_l$ purely reflects the magnitude and ensures stability. We have revised the manuscript to use $\\mathbb{E}[\\cdot]$ more consistently.\n\nThank you again for your insightful feedback. We hope our response has resolved your concerns and are happy to clarify further if needed.\n\nReferences:\n\n[1] Lei et al. EFEM: Equivariant Neural Field Expectation Maximization for 3D Object Segmentation without Scene Supervision. CVPR 2023.\n\n[2] Yang et al. EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning. CoRL 2024.\n\n[3] Yang et al. EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulation. ICRA 2024.\n\n[4] Katzir et al. Shape-Pose Disentanglement using SE(3)-Equivariant Vector Neurons. ECCV 2022."}, {"review_id": "uADlyQ6R8m", "reviewer": "Reviewer_eyhY", "summary": "This paper aims to address the issues that many point cloud completion methods leak pose and scale cues to the model so that networks may exploit to memorize absolute positions rather than inferring intrinsic geometry. The authors proposed the first shape completion architecture whose modules are SIM(3)-equivariant by design. The experiment shows the good performance of this method.", "questions": "1. what is the computation cost of the proposed method", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The problem setting is very interesting.\n2. The formulation is clear and intetersting.\n\nWeaknesses:\n\n1. SIM(3)-equivalence does not convey any meaning if the reader does not know the concept. The author may need to give formal definition, clear introduction earlier or reference for readers to understand what the method aims to do and what was proposed. Similar to SO(3) and SE(3)\n\n2. The compared baseline are out-of-date. The authors need to compare with more recent baselines.\n\n3. Insufficient experiment: For De-biased benchmark evaluation, this paper only provided results on the PCN dataset, which is insufficient as most recent point cloud completion paper will also present results on other ShapeNet-derived datasets to comprehensively evaluate their method.\n\n4. Concern about zero-shot concern: It is invalid to claim it works on zero-shot case. It was tested on KITTI cars, while cars are included in the PCN dataset, therefore it is totally invalid to claim it is zero-shot. Also, many previous methods were also tested on KITTI, but, to my best knowledge, no one claimed that this is zero-shot.", "was_revised": true, "final_justification": "My final justification is maintaining my positive rating.", "rebuttal": "Dear Reviewer eyhY,\n\nThank you for your comments. We address your concerns and questions below.\n\n> SIM(3)-equivalence does not convey any meaning if the reader does not know the concept. The author may need to give formal definition, clear introduction earlier or reference for readers to understand what the method aims to do and what was proposed. Similar to SO(3) and SE(3).\n\nThank you for this suggestion. In the revised manuscript, we have added an introductory explanation of SIM(3) equivariance, explicitly contrasting it with SO(3) and SE(3), before presenting the formal definition in Sec. 3.1.\n\n\n> The compared baseline are out-of-date. The authors need to compare with more recent baselines.\n\nWe focus our comparison on equivariant methods, as they provide the fairest baseline for assessing the impact of equivariance. The most recent equivariant method to our knowledge, ESCAPE [1], supports only SE(3) equivariance and underperforms AdaPoinTr [2] when the latter is trained with data augmentation. In contrast, our method guarantees full SIM(3) equivariance by design and outperforms both ESCAPE (even under its SE(3) setting) and the augmented AdaPoinTr. Since both our method and ESCAPE use the same AdaPoinTr backbone, these comparisons are directly equitable. While more recent non-equivariant methods may show stronger performance under biased settings, they do not address the core issue of pose and scale bias. Our equivariance mechanism is backbone-agnostic and directly remedies these biases, making our method a principled solution to fundamental limitations in existing shape completion pipelines, rather than merely another state-of-the-art variant.\n\nNevertheless, to provide additional context, we also evaluated ODGNet [3], a recent non-equivariant method. As shown in the table below, under the same debiased setting, our method consistently outperforms ODGNet across all categories and metrics, further highlighting the advantage of built-in SIM(3)-equivariance.\n\n|Method|Transform|Airpl.|Cab.|Car|Chair|Lamp|Sofa|Table|Wat.|CD-$\\ell_1$↓|F1↑|\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|AdaPoinTr [2]|I/I|3.68|8.82|7.47|6.85|5.47|8.35|5.80|5.76|6.53|0.845|\n|ODGNet [3]|I/I|3.77|8.77|7.56|6.84|5.09|8.47|5.84|5.66|6.50|0.833|\n|AdaPoinTr [2]|I/SIM(3)|31.93|78.90|63.51|60.56|61.47|70.77|71.94|42.74|60.23|0.206|\n|ODGNet [3]|I/SIM(3)|15.03|52.36|41.04|39.86|30.22|50.01|46.73|24.45|37.46|0.296|\n|AdaPoinTr [2]|SIM(3)/SIM(3)|6.46|12.17|10.51|10.29|7.59|12.26|8.90|8.14|9.54|0.661|\n|ODGNet [3]|SIM(3)/SIM(3)|6.16|11.60|11.15|10.13|6.81|13.12|9.48|8.19|9.58|0.659|\n|**SIMECO (ours)**|I/SIM(3)|**6.02**|**10.75**|**9.27**|**9.25**|**6.66**|**11.16**|**7.82**|**7.77**|**8.59**|**0.714**|\n\n> Insufficient experiment: For De-biased benchmark evaluation, this paper only provided results on the PCN dataset, which is insufficient as most recent point cloud completion paper will also present results on other ShapeNet-derived datasets to comprehensively evaluate their method.\n\nWe agree that evaluating on other ShapeNet-derived benchmarks can be helpful. However, our primary goal is to identify and understand the theoretical benefits of equivariance itself, rather than to chase state-of-the-art scores across all datasets. To this end, we matched exactly the three benchmarks used by the recent method ESCAPE [1] to enable a direct, apples-to-apples comparison, but also focused our compute budget on a detailed analysis of when and why equivariance delivers gains (Tabs. 4-5, Figs. 8-9). We believe this targeted evaluation on PCN [4], KITTI [5], and OmniObject3D [6], together with dense ablations and analyses, provides meaningful insight into the fundamental behavior of equivariance in shape completion.\n\n\n> Concern about zero-shot concern: It is invalid to claim it works on zero-shot case. It was tested on KITTI cars, while cars are included in the PCN dataset, therefore it is totally invalid to claim it is zero-shot. Also, many previous methods were also tested on KITTI, but, to my best knowledge, no one claimed that this is zero-shot.\n\nThank you for pointing this out. You are correct that \"zero-shot\" may suggest unintended implications. To align with prior works, we have removed all instances of \"zero-shot\" throughout the manuscript.\n\n> What is the computation cost of the proposed method?\n\nWe report computational efficiency in Suppl. Mat. B.2, with average per-scan latencies in Table 7. Despite offering full SIM(3)-equivariance, our method processes a scan end-to-end in just 76 ms—roughly half the runtime of the next-fastest equivariant method, ESCAPE [1] (148 ms), and even faster than that of EquivPCN [7] and SCARP [8] (172 ms each). ESCAPE and SCARP spend extra time in post-processing steps, which inflate total latency. Although AdaPoinTr runs faster at 16 ms, it does so without any built-in equivariance. Overall, our method achieves the best balance of speed and rigorous SIM(3) symmetry preservation.\n\nThank you again for your insightful feedback. We hope our response has resolved your concerns and are happy to clarify further if needed.\n\nReferences:\n\n[1] Bekci et al. ESCAPE: Equivariant Shape Completion via Anchor Point Encoding. CVPR 2025.\n\n[2] Yu et al. AdaPoinTr: Diverse Point Cloud Completion with Adaptive Geometry-aware Transformers. TPAMI 2023.\n\n[3] Cai et al. Orthogonal Dictionary Guided Shape Completion Network for Point Cloud. AAAI 2024.\n\n[4] Yuan et al. PCN: Point Completion Network. 3DV 2018.\n\n[5] Geiger et al. Vision Meets Robotics: The KITTI Dataset. IJRR 2013.\n\n[6] Wu et al. OmniObject3D: Large-Vocabulary 3D Object Dataset for Realistic Perception, Reconstruction and Generation. CVPR 2023.\n\n[7] Wu and Miao. SO(3) Rotation Equivariant Point Cloud Completion Using Attention-Based Vector Neurons. 3DV 2022.\n\n[8] Sen et al. SCARP: 3D Shape Completion in Arbitrary Poses for Improved Grasping. ICRA 2023."}, {"review_id": "4tzpXNHChu", "reviewer": "Reviewer_ztk1", "summary": "In this paper, to address the bias learning caused by leaked pose and scale cues in most 3D shape completion methods, the authors proposed a more robust shape completion network that ensures SIM(3)-equivariant. With this novel shape completion network, a completed 3D shape can be inferred without the need for any ground-truth alignment, which makes the proposed method more available in practice use. The main contribution of the paper is stating the bias learning caused by leaked pose and scale cues in most traditional 3D shape completion methods; meanwhile, to address this issue, the authors decompose the optimizing objective into three parts: geometric reasoning, equivariance, and transform alignment. With this designed fully SIM(3)-equivariant network for shape completion, the inference 3D shapes follow arbitrary similarity transforms applied to inputs. The experiments show the effectiveness of the proposed method.", "questions": "Please refer to the weakness part.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The motivation and idea of this paper are reasonable, and the problem that poses and scale bias existing in shape completion methods is clearly stated, which shows the importance of the proposed method.\n2. A fully SIM(3)-equivariant network for shape completion is designed to address the identified problem, which includes feature canonicalization to remove translation and scale variation, shape reasoning to eliminate rotation transformations, and transform restoration to recover the absolute alignment. With all of these designs, the proposed network can learn representations agnostic to pose and scale in the training stage, and any transformations to inputs can be applied to the completed outputs.\n3. The visualizations in Fig. 5 and Fig.6 show the effectiveness of the proposed network, which can generate completed 3D shapes with more details.\n\n\nWeaknesses:\n1. The training time and datasets used for training are not clearly stated in the paper.\n2. It would be better if the authors could show the outputs given the same input partial shape with different poses and scales, which could better illustrate the proposed network is agnostic with pose and scale.\n3. I am curious about the performance of the proposed network if the GT completed shape with many thin structures.", "was_revised": true, "final_justification": "I thank the authors for their comprehensive responses, which have satisfactorily addressed all of my initial comments. I suggest including these discussions in the final version for completeness. I will therefore maintain my original positive rating.", "rebuttal": "Dear Reviewer ztk1, \n\nThank you for your comments. We address your questions below.\n\n> The training time and datasets used for training are not clearly stated in the paper.\n\nThank you for pointing out this omission. The datasets are specified in Sec. 4.1 under \"Datasets and baselines\" and include PCN [1], KITTI [2], and OmniObject3D [3]. As detailed in Suppl. Mat. D (\"Implementation Details\"), all methods were trained for 200 epochs on the PCN dataset using two NVIDIA A40 GPUs. With a batch size of 40, SIMECO's average training time is approximately 63 min/epoch. We have added these details in the revised manuscript.\n\n> It would be better if the authors could show the outputs given the same input partial shape with different poses and scales, which could better illustrate the proposed network is agnostic with pose and scale.\n\nIn Fig. 4, we demonstrate SIM(3) equivariance by applying diverse similarity transforms (rotations, translations, and scalings) to the same partial input and showing that the completed outputs follow these transforms exactly. For more extreme pose and scale variations, Fig. 13 in the Suppl. Mat. spans a full page with extensive examples, further confirming that our method consistently maintains high reconstruction quality under large input perturbations. We have also provided an interactive demo in the Suppl. Mat. A (L823) where readers can explore completion results under arbitrary SIM(3) transformations, and we warmly invite you to try it out.\n\n> I am curious about the performance of the proposed network if the GT completed shape with many thin structures.\n\nTo evaluate performance on shapes with pronounced thin structures, we compute a local PCA-based anisotropy score: \n$$\nL = \\frac{\\sigma_1 - \\sigma_2}{\\sigma_1}\n$$\non each point's $k$-nearest neighbors ($k=30$), where \n$\\sigma_1 \\geq \\sigma_2 \\geq \\sigma_3$ are the singular values of the local covariance matrix. A high $L$ indicates a spindly, edge-like neighborhood. We then select shapes in which more than 0.5\\% of points satisfy $L > 0.8$, yielding 50 thin-structure cases out of 1200 total. Quantitatively, on this subset, our method achieves an average CD-$\\ell1$ of 6.83, compared to 8.59 over the entire test set, demonstrating even better overall performance on such samples with thin structures. Moreover, qualitative examples such as chair and table legs in Fig. 5 and Fig. 6 further confirm that our method preserves fine details effectively.\n\nThank you again for your insightful feedback. We hope our response has resolved your concerns and are happy to clarify further if needed.\n\nReferences:\n\n[1] Yuan et al. PCN: Point Completion Network. 3DV 2018.\n\n[2] Geiger et al. Vision Meets Robotics: The KITTI Dataset. IJRR 2013.\n\n[3] Wu et al. OmniObject3D: Large-Vocabulary 3D Object Dataset for Realistic Perception, Reconstruction and Generation. CVPR 2023."}, {"review_id": "RWxiWTbCYJ", "reviewer": "Reviewer_1dGn", "summary": "The paper proposes a SIM(3)-equivariant shape completion network architecture based on an adaptation of the AdaPoinTr backbone. The original Transformer layers are replaced with novel SIM(3)-equivariant modules.\n\nThe proposed SIM(3)-equivariant module consists of three subcomponents:\n1. a canonicalization  module (layer norm extension) that normalizes scale and translation, while being rotation equivariant,\n2. a shape reasoning module (VN-transformer) that is rotation invariant, to obtain a fully SIM(3)-invariant representation,\n3. and a transform restoration module to restore scale and translation, to obtain a SIM(3)-equivariant output.\n\nThe resulting model, SIMECO, achieves significant improvements over state-of-the-art methods trained with SIM(3) data augmentation. On the PCN benchmark, it outperforms AdaPoinTr by 10% in CD-L1 and 8% in F1-score. However, SIMECO’s forward pass latency is higher (76 ms vs. 16 ms for AdaPoinTr).", "questions": "1. **Training Convergence** - All methods are reported to be trained for 200 epochs. However, did all of them fully converge? It would be helpful to include training and validation loss curves—especially for AdaPoinTr vs. SIMECO. I suspect that SIMECO may converge faster since it does not rely on SIM(3) data augmentation, but it's not entirely clear from the current presentation.\n\n2. **Table 2 – Possible Typo** - In Table 2, PoinTR is shown with a Fidelity score of 0.000. Is this a typo, or is there an explanation?\n\n3. **Clarification on Figure 10** - If I understood Figure 10 correctly, the model is trained with $N_{in}=2018$ input points, but evaluated at test time with fewer points. Is that interpretation correct?", "limitations": "The limitations are discussed in the supp. mat.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n1. The paper proposes the first SIM(3)-equivariant network architecture for shape completion.\n2. The proposed SIM(3)-equivariant module is intuitive and simple.\n3. The paper is well illustrated, especially with the use of color-coded circles and triangles to clearly highlight different types of equivariances and invariances.\n4. An extensive ablation study supports the architectural and design choices.\n\n\n## Weaknesses\n1. The motivation for introducing the relaxed problem in Equation (4) is unclear. Since Equation (2) appears sufficient, the use of a relaxed version where $g^* \\neq I$ seems to unnecessarily complicate the formulation. Clarification or justification would strengthen this point.\n\n2. References to the supplementary material are missing in several places. Explicitly pointing to the supplement would help readers, especially since important derivations and explanations are located there.\n\n3. Inconsistencies in notation: below Equation (9), the terms $E_{D^l}$​ and $E_i$​ appear to denote expectations, yet the standard expectation notation $\\mathbb{E}$ is not used (even though it is sometimes used in the supp. mat., albeit inconsistently).\n\n4. SIMECO is much slower (76ms vs 16ms) than AdaPoinTr and only increases the performance by 10%. The trade-off between accuracy and efficiency could be more thoroughly discussed.", "was_revised": true, "final_justification": "The author addressed my concerns, as a consequence, I recommend accepting the paper.", "rebuttal": "Dear Reviewer 1dGn,\n\nThank you for your comments. We address your concerns and questions below.\n\n> The motivation for introducing the relaxed problem in Equation (4) is unclear. Since Equation (2) appears sufficient, the use of a relaxed version where seems to unnecessarily complicate the formulation. Clarification or justification would strengthen this point.\n\nWe appreciate your suggestion to improve the exposition. While Eq. (2) might seem sufficient from a theoretical standpoint, it entangles geometric inference and transform alignment. To remove the bias embedded in the transform, we need to disentangle the geometry from the transform and let the network learn the former. Once we do so, the immediately inferred geometry (in a canonical frame) is invariant to the transform. Thus, to re-align the geometry to the input frame, the transform must then be restored. Accordingly, $\\min_{\\theta} \\mathcal L\\bigl(f_{\\theta}(\\mathbf{x}), \\mathbf{y}\\bigr)$ in Eq. (2) needs to be decomposed into two components: Req. (1) and Req. (3) in Eq. (4), for geometric reasoning and transform alignment, respectively. We have made the motivation behind Eq. (4) more explicit in the revised manuscript.\n\n> References to the supplementary material are missing in several places. Explicitly pointing to the supplement would help readers, especially since important derivations and explanations are located there.\n\nThank you for the suggestion. We have added explicit references to the Suppl. Mat. at all relevant points, directing readers to the detailed derivations and explanations.\n\n> Inconsistencies in notation: below Eq. (9), the terms and appear to denote expectations, yet the standard expectation notation is not used (even though it is sometimes used in the supp. mat., albeit inconsistently).\n\nThank you for pointing out this issue. We have revised Eq. (9) and Eq. (46) (and all related occurrences) to use the standard expectation notation $\\mathbb{E}[\\cdot]$ consistently throughout both the main text and the Suppl. Mat.\n\n> SIMECO is much slower (76ms vs 16ms) than AdaPoinTr and only increases the performance by 10\\%. The trade-off between accuracy and efficiency could be more thoroughly discussed.\n\nWhile SIMECO incurs higher latency, it delivers substantial quality gains, reducing MMD on KITTI by 17\\% and CD-$\\ell1$ on OmniObject3D by 14\\% without any data augmentation. For a fair comparison among equivariant completion methods, SIMECO is not only the fastest but also uniquely outperforms an augmented AdaPoinTr within its target equivariance group. Notably, Fig. 13 in the Suppl. Mat. demonstrates that SIMECO maintains performance under large pose and scale perturbations, whereas AdaPoinTr degrades. In real-world settings where object scales vary significantly and cannot always be normalized (e.g., due to outliers), SIMECO’s equivariance ensures more reliable and robust completions.\n\n> Training Convergence - All methods are reported to be trained for 200 epochs. However, did all of them fully converge? It would be helpful to include training and validation loss curves—especially for AdaPoinTr vs. SIMECO. I suspect that SIMECO may converge faster since it does not rely on SIM(3) data augmentation, but it's not entirely clear from the current presentation.\n\nThank you for the suggestion. All models were trained for 200 epochs and fully converged. While the rebuttal format precludes the inclusion of rich media, we have tabulated training losses and validation metrics at 10-epoch intervals (see table below) and added the full curves to the revised manuscript. These show that SIMECO's (S) losses and metrics improve far more effectively than AdaPoinTr’s (A). For instance, AdaPoinTr requires roughly 140 epochs to drive CD-$\\ell_1$ below 10, whereas SIMECO reaches the same threshold in only about 50 epochs.\n\n|Epoch|$\\mathcal{L}_{\\mathrm{denoise}}$↓ (A)|$\\mathcal{L}_{\\mathrm{denoise}}$↓ (S)|$\\mathcal{L}_{\\mathrm{distance}}$↓ (A)|$\\mathcal{L}_{\\mathrm{distance}}$↓ (S)|CD-$\\ell_1$↓ (A)|CD-$\\ell_1$↓ (S)|F1↑ (A)|F1↑ (S)|\n|---|---|---|---|---|---|---|---|---|\n|0|13.86|6.26|139.44|58.17|24.76|16.78|0.406|0.487|\n|10|7.90|3.67|84.64|39.84|13.27|12.26|0.552|0.581|\n|20|7.52|3.40|80.47|36.78|12.39|11.11|0.578|0.608|\n|30|7.36|3.30|79.91|35.60|12.29|10.60|0.581|0.629|\n|40|7.17|3.21|76.58|34.13|11.59|10.29|0.595|0.644|\n|50|7.07|3.16|75.67|33.10|11.35|10.09|0.603|0.648|\n|60|6.98|3.12|73.90|32.69|10.84|10.03|0.613|0.647|\n|70|6.91|3.06|73.42|31.99|11.03|9.64|0.611|0.666|\n|80|6.85|3.00|72.57|30.97|10.61|9.43|0.624|0.672|\n|90|6.80|2.97|71.86|30.34|10.47|9.20|0.625|0.685|\n|100|6.77|2.92|71.62|29.89|10.29|9.17|0.637|0.686|\n|110|6.74|2.89|71.09|29.47|10.11|8.98|0.643|0.693|\n|120|6.68|2.88|69.89|29.29|9.97|8.98|0.648|0.695|\n|130|6.65|2.87|69.44|29.06|9.93|8.95|0.649|0.697|\n|140|6.62|2.86|69.66|28.86|9.99|8.91|0.650|0.696|\n|150|6.58|2.84|68.39|28.66|9.78|8.83|0.655|0.702|\n|160|6.54|2.82|67.83|28.39|9.65|8.80|0.658|0.702|\n|170|6.48|2.81|67.30|28.28|9.70|8.74|0.657|0.707|\n|180|6.44|2.80|67.13|27.92|9.72|8.70|0.655|0.707|\n|190|6.42|2.79|67.12|27.88|9.71|8.59|0.654|0.714|\n|200|6.37|2.79|66.23|28.00|9.64|8.71|0.658|0.709|\n\n> Table 2 – Possible Typo - In Table 2, PoinTR is shown with a Fidelity score of 0.000. Is this a typo, or is there an explanation?\n\nThis is not a typo—PoinTr’s fidelity score is indeed 0.000 by design. Fidelity is defined as the average nearest-neighbor distance from each input point to the output shape, so a score of zero means the model leaves every input point exactly in place. PoinTr trades off this perfect input preservation against higher generative capacity, which is why it reports zero fidelity in both the original PoinTr [1] and AdaPoinTr [2] papers. We believe that Minimal Matching Distance (MMD) is often a more intuitive metric for overall completion quality. On MMD (i.e. Chamfer distance to the closest PCN car), SIMECO outperforms PoinTr by 23\\% (5.353 vs. 6.929 in Table 2), demonstrating stronger completion fidelity to typical car geometries.\n\n> Clarification on Figure 10 - If I understood Figure 10 correctly, the model is trained with N=2018 input points, but evaluated at test time with fewer points. Is that interpretation correct?\n\nYes, your interpretation is correct, except that we train with $N_\\mathrm{in}=2048$ (not 2018) input points. Our encoder (following the original DGCNN design [3], with VN modifications to ensure SIM(3) equivariance [4, 5]) aggregates point features onto a fixed set of \"point proxies\" via $k$-nearest neighbors. As a result, at test time the network can gracefully handle any reasonable number of input points without retraining, making it well suited for real-world scans.\n\nThank you again for your insightful feedback. We hope our response has resolved your concerns and are happy to clarify further if needed.\n\nReferences:\n\n[1] Yu et al. Diverse Point Cloud Completion with Geometry-aware Transformers. ICCV 2021.\n\n[2] Yu et al. Diverse Point Cloud Completion with Adaptive Geometry-aware Transformers. TPAMI 2023.\n\n[3] Wang et al. Dynamic Graph CNN for Learning on Point Clouds. TOG 2019.\n\n[4] Deng et al. Vector Neurons: A General Framework for SO(3)-Equivariant Networks. ICCV 2021.\n\n[5] Katzir et al. Shape-Pose Disentanglement using SE(3)-Equivariant Vector Neurons. ECCV 2022."}]

Author rebuttal:

Meta-review: The paper proposes a shape completion method which is equivariant to similarity transforms. 
All reviewers are in favor of acceptance and note the paper is sound and well evaluated. The AC panel recommends acceptance.
Decision: Accept (poster)