# Approximately Piecewise E(3) Equivariant Point Networks

arXiv: https://arxiv.org/abs/2402.08529 · HTML: https://mlsanity.com/paper/2402.08529/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 56 references that are in the map.
- First posted on arXiv 13 Feb 2024.

## Builds on (most cited first, 30 of 45)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [4D Spatio-Temporal ConvNets: Minkowski Convolutional Neural Networks](https://mlsanity.com/paper/1904.08755/) (2019)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [Rotation Invariant Convolutions for 3D Point Clouds Deep Learning](https://mlsanity.com/paper/1908.06297/) (2019)
- [Controlling Neural Level Sets](https://mlsanity.com/paper/1905.11911/) (2019)
- [MultiBodySync: Multi-Body Segmentation and Motion Estimation via 3D Scan Synchronization](https://mlsanity.com/paper/2101.06605/) (2021)
- [Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimation](https://mlsanity.com/paper/2111.00190/) (2021)
- [ARAPReg: An As-Rigid-As Possible Regularization Loss for Learning Deformable Shape Generators](https://mlsanity.com/paper/2108.09432/) (2021)

## Built on by (most cited first, 4 of 4)

- [Rectified Point Flow: Generic Point Cloud Pose Estimation](https://mlsanity.com/paper/2506.05282/) (2025)
- [ETCH: Generalizing Body Fitting to Clothed Humans via Equivariant Tightness](https://mlsanity.com/paper/2503.10624/) (2025)
- [Efficient Continuous Group Convolutions for Local SE(3) Equivariance in 3D Point Clouds](https://mlsanity.com/paper/2502.07505/) (2025)
- [ARAPDiffusion: ARAP Regularization for Diffusion-Based Deformable Shape Space Learning](https://mlsanity.com/paper/2606.06887/) (2026)

## Abstract (arXiv)

Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are E(3) equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are E(3) equivariant, to accommodate inputs made of multiple parts, each of which exhibits local E(3) symmetry. In practical settings, however, the partitioning into individually transforming regions is unknown a priori. Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-E(3) equivariant point networks. Our primary insight is that functions that are equivariant with respect to a finer partition will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a design where the equivariance approximation error at each layers can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one. We demonstrate the effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise E(3) symmetry into network design, showing a distinct improvement in generalization compared to prior works for both classification and segmentation tasks.