# Banana: Banach Fixed-Point Network for Pointcloud Segmentation with Inter-Part Equivariance

arXiv: https://arxiv.org/abs/2305.16314 · HTML: https://mlsanity.com/paper/2305.16314/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 77 of its 89 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 30 of 77)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [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)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [PartNet: A Large-scale Benchmark for Fine-grained and Hierarchical Part-level 3D Object Understanding](https://mlsanity.com/paper/1812.02713/) (2018)
- [3D Shape Generation and Completion through Point-Voxel Diffusion](https://mlsanity.com/paper/2104.03670/) (2021)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [PointGroup: Dual-Set Point Grouping for 3D Instance Segmentation](https://mlsanity.com/paper/2004.01658/) (2020)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [FlowNet3D: Learning Scene Flow in 3D Point Clouds](https://mlsanity.com/paper/1806.01411/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [CNN-RNN: A Unified Framework for Multi-label Image Classification](https://mlsanity.com/paper/1604.04573/) (2016)

## Built on by (most cited first, 19 of 19)

- [ManipLLM: Embodied Multimodal Large Language Model for Object-Centric Robotic Manipulation](https://mlsanity.com/paper/2312.16217/) (2023)
- [Robot See Robot Do: Imitating Articulated Object Manipulation with Monocular 4D Reconstruction](https://mlsanity.com/paper/2409.18121/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulation](https://mlsanity.com/paper/2310.16050/) (2023)
- [PhysPart: Physically Plausible Part Completion for Interactable Objects](https://mlsanity.com/paper/2408.13724/) (2024)
- [A Self-Correcting Vision-Language-Action Model for Fast and Slow System Manipulation](https://mlsanity.com/paper/2405.17418/) (2024)
- [Living Scenes: Multi-object Relocalization and Reconstruction in Changing 3D Environments](https://mlsanity.com/paper/2312.09138/) (2023)
- [Survey on Modeling of Human-made Articulated Objects](https://mlsanity.com/paper/2403.14937/) (2024)
- [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)
- [MonoMobility: Zero-Shot 3D Mobility Analysis from Monocular Videos](https://mlsanity.com/paper/2505.11868/) (2025)
- [GAPartManip: A Large-scale Part-centric Dataset for Material-Agnostic Articulated Object Manipulation](https://mlsanity.com/paper/2411.18276/) (2024)
- [Part Segmentation and Motion Estimation for Articulated Objects with Dynamic 3D Gaussians](https://mlsanity.com/paper/2506.22718/) (2025)
- [Approximately Piecewise E(3) Equivariant Point Networks](https://mlsanity.com/paper/2402.08529/) (2024)
- [Subequivariant Reinforcement Learning in 3D Multi-Entity Physical Environments](https://mlsanity.com/paper/2407.12505/) (2024)
- [PAWS: Perception of Articulation in the Wild at Scale from Egocentric Videos](https://mlsanity.com/paper/2603.25539/) (2026)
- [BiEquiFormer: Bi-Equivariant Representations for Global Point Cloud Registration](https://mlsanity.com/paper/2407.08729/) (2024)
- [Physics-Driven Local-Whole Elastic Deformation Modeling for Point Cloud Representation Learning](https://mlsanity.com/paper/2505.13812/) (2025)
- [Part2GS: Part-aware Modeling of Articulated Objects using 3D Gaussian Splatting](https://mlsanity.com/paper/2506.17212/) (2025)

## Abstract (arXiv)

Equivariance has gained strong interest as a desirable network property that inherently ensures robust generalization. However, when dealing with complex systems such as articulated objects or multi-object scenes, effectively capturing inter-part transformations poses a challenge, as it becomes entangled with the overall structure and local transformations. The interdependence of part assignment and per-part group action necessitates a novel equivariance formulation that allows for their co-evolution. In this paper, we present Banana, a Banach fixed-point network for equivariant segmentation with inter-part equivariance by construction. Our key insight is to iteratively solve a fixed-point problem, where point-part assignment labels and per-part SE(3)-equivariance co-evolve simultaneously. We provide theoretical derivations of both per-step equivariance and global convergence, which induces an equivariant final convergent state. Our formulation naturally provides a strict definition of inter-part equivariance that generalizes to unseen inter-part configurations. Through experiments conducted on both articulated objects and multi-object scans, we demonstrate the efficacy of our approach in achieving strong generalization under inter-part transformations, even when confronted with substantial changes in pointcloud geometry and topology.