# Bending Graphs: Hierarchical Shape Matching using Gated Optimal Transport

arXiv: https://arxiv.org/abs/2202.01537 · HTML: https://mlsanity.com/paper/2202.01537/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 79 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 30 of 48)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (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)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [SuperGlue: Learning Feature Matching with Graph Neural Networks](https://mlsanity.com/paper/1911.11763/) (2019)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [KPConv: Flexible and Deformable Convolution for Point Clouds](https://mlsanity.com/paper/1904.08889/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Scene Graph Generation by Iterative Message Passing](https://mlsanity.com/paper/1701.02426/) (2017)
- [Learning from Synthetic Humans](https://mlsanity.com/paper/1701.01370/) (2017)
- [3DMatch: Learning Local Geometric Descriptors from RGB-D Reconstructions](https://mlsanity.com/paper/1603.08182/) (2016)
- [Deep Closest Point: Learning Representations for Point Cloud Registration](https://mlsanity.com/paper/1905.03304/) (2019)
- [3D Menagerie: Modeling the 3D shape and pose of animals](https://mlsanity.com/paper/1611.07700/) (2016)
- [PointNetLK: Robust & Efficient Point Cloud Registration using PointNet](https://mlsanity.com/paper/1903.05711/) (2019)
- [Conditional Random Fields as Recurrent Neural Networks](https://mlsanity.com/paper/1502.03240/) (2015)
- [Learning 3D Semantic Scene Graphs from 3D Indoor Reconstructions](https://mlsanity.com/paper/2004.03967/) (2020)
- [PPFNet: Global Context Aware Local Features for Robust 3D Point Matching](https://mlsanity.com/paper/1802.02669/) (2018)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [SceneGraphFusion: Incremental 3D Scene Graph Prediction from RGB-D Sequences](https://mlsanity.com/paper/2103.14898/) (2021)
- [3D-CODED : 3D Correspondences by Deep Deformation](https://mlsanity.com/paper/1806.05228/) (2018)
- [CoFiNet: Reliable Coarse-to-fine Correspondences for Robust Point Cloud Registration](https://mlsanity.com/paper/2110.14076/) (2021)
- [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017)
- [FLOT: Scene Flow on Point Clouds Guided by Optimal Transport](https://mlsanity.com/paper/2007.11142/) (2020)
- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [Topology Adaptive Graph Convolutional Networks](https://mlsanity.com/paper/1710.10370/) (2017)
- [Three-D Safari: Learning to Estimate Zebra Pose, Shape, and Texture from Images "In the Wild"](https://mlsanity.com/paper/1908.07201/) (2019)

## Built on by (most cited first, 24 of 24)

- [Rotation-Invariant Transformer for Point Cloud Matching](https://mlsanity.com/paper/2303.08231/) (2023)
- [Deformable 3D Gaussian Splatting for Animatable Human Avatars](https://mlsanity.com/paper/2312.15059/) (2023)
- [MeshMAE: Masked Autoencoders for 3D Mesh Data Analysis](https://mlsanity.com/paper/2207.10228/) (2022)
- [RIGA: Rotation-Invariant and Globally-Aware Descriptors for Point Cloud Registration](https://mlsanity.com/paper/2209.13252/) (2022)
- [Self-Supervised Learning for Multimodal Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2303.10971/) (2023)
- [Deep Graph-based Spatial Consistency for Robust Non-rigid Point Cloud Registration](https://mlsanity.com/paper/2303.09950/) (2023)
- [CheckerPose: Progressive Dense Keypoint Localization for Object Pose Estimation with Graph Neural Network](https://mlsanity.com/paper/2303.16874/) (2023)
- [S3M: Scalable Statistical Shape Modeling through Unsupervised Correspondences](https://mlsanity.com/paper/2304.07515/) (2023)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Unsupervised Cross-Domain Image Retrieval via Prototypical Optimal Transport](https://mlsanity.com/paper/2402.18411/) (2024)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Physics-Encoded Graph Neural Networks for Deformation Prediction under Contact](https://mlsanity.com/paper/2402.03466/) (2024)
- [ESCAPE: Equivariant Shape Completion via Anchor Point Encoding](https://mlsanity.com/paper/2412.00952/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [Few-Shot Object Detection via Synthetic Features with Optimal Transport](https://mlsanity.com/paper/2308.15005/) (2023)
- [Tree-Sliced Wasserstein Distance with Nonlinear Projection](https://mlsanity.com/paper/2505.00968/) (2025)
- [FUSE: A Flow-based Mapping Between Shapes](https://mlsanity.com/paper/2511.13431/) (2025)
- [CloudAttention: Efficient Multi-Scale Attention Scheme For 3D Point Cloud Learning](https://mlsanity.com/paper/2208.00524/) (2022)
- [Shape-Graph Matching Network (SGM-net): Registration for Statistical Shape Analysis](https://mlsanity.com/paper/2308.06869/) (2023)
- [SelfGeo: Self-supervised and Geodesic-consistent Estimation of Keypoints on Deformable Shapes](https://mlsanity.com/paper/2408.02291/) (2024)
- [A Recipe for Geometry-Aware 3D Mesh Transformers](https://mlsanity.com/paper/2411.00164/) (2024)
- [Multiview Point Cloud Registration via Optimization in an Autoencoder Latent Space](https://mlsanity.com/paper/2504.21467/) (2025)

## Abstract (arXiv)

Shape matching has been a long-studied problem for the computer graphics and vision community. The objective is to predict a dense correspondence between meshes that have a certain degree of deformation. Existing methods either consider the local description of sampled points or discover correspondences based on global shape information. In this work, we investigate a hierarchical learning design, to which we incorporate local patch-level information and global shape-level structures. This flexible representation enables correspondence prediction and provides rich features for the matching stage. Finally, we propose a novel optimal transport solver by recurrently updating features on non-confident nodes to learn globally consistent correspondences between the shapes. Our results on publicly available datasets suggest robust performance in presence of severe deformations without the need for extensive training or refinement.