# Combinative Matching for Geometric Shape Assembly

arXiv: https://arxiv.org/abs/2508.09780 · HTML: https://mlsanity.com/paper/2508.09780/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 58 references that are in the map.
- First posted on arXiv 13 Aug 2025.

## Builds on (most cited first, 30 of 34)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [SuperGlue: Learning Feature Matching with Graph Neural Networks](https://mlsanity.com/paper/1911.11763/) (2019)
- [Circle Loss: A Unified Perspective of Pair Similarity Optimization](https://mlsanity.com/paper/2002.10857/) (2020)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [PREDATOR: Registration of 3D Point Clouds with Low Overlap](https://mlsanity.com/paper/2011.13005/) (2020)
- [Deep Global Registration](https://mlsanity.com/paper/2004.11540/) (2020)
- [Geometric Transformer for Fast and Robust Point Cloud Registration](https://mlsanity.com/paper/2202.06688/) (2022)
- [CoFiNet: Reliable Coarse-to-fine Correspondences for Robust Point Cloud Registration](https://mlsanity.com/paper/2110.14076/) (2021)
- [PQ-NET: A Generative Part Seq2Seq Network for 3D Shapes](https://mlsanity.com/paper/1911.10949/) (2019)
- [Rotation-Invariant Transformer for Point Cloud Matching](https://mlsanity.com/paper/2303.08231/) (2023)
- [You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptors](https://mlsanity.com/paper/2109.00182/) (2021)
- [Generative 3D Part Assembly via Dynamic Graph Learning](https://mlsanity.com/paper/2006.07793/) (2020)
- [Learning 3D Part Assembly from a Single Image](https://mlsanity.com/paper/2003.09754/) (2020)
- [Form2Fit: Learning Shape Priors for Generalizable Assembly from Disassembly](https://mlsanity.com/paper/1910.13675/) (2019)
- [Shonan Rotation Averaging: Global Optimality by Surfing SO(p)n](https://mlsanity.com/paper/2008.02737/) (2020)
- [Neural Shape Mating: Self-Supervised Object Assembly with Adversarial Shape Priors](https://mlsanity.com/paper/2205.14886/) (2022)
- [Breaking Bad: A Dataset for Geometric Fracture and Reassembly](https://mlsanity.com/paper/2210.11463/) (2022)
- [Learning Part Generation and Assembly for Structure-aware Shape Synthesis](https://mlsanity.com/paper/1906.06693/) (2019)
- [DiffAssemble: A Unified Graph-Diffusion Model for 2D and 3D Reassembly](https://mlsanity.com/paper/2402.19302/) (2024)
- [Equivariant Point Cloud Analysis via Learning Orientations for Message Passing](https://mlsanity.com/paper/2203.14486/) (2022)
- [Jigsaw: Learning to Assemble Multiple Fractured Objects](https://mlsanity.com/paper/2305.17975/) (2023)
- [Leveraging SE(3) Equivariance for Learning 3D Geometric Shape Assembly](https://mlsanity.com/paper/2309.06810/) (2023)
- [Category-Level Multi-Part Multi-Joint 3D Shape Assembly](https://mlsanity.com/paper/2303.06163/) (2023)
- [Learning Robotic Assembly from CAD](https://mlsanity.com/paper/1803.07635/) (2018)
- [VectorAdam for Rotation Equivariant Geometry Optimization](https://mlsanity.com/paper/2205.13599/) (2022)
- [PARE-Net: Position-Aware Rotation-Equivariant Networks for Robust Point Cloud Registration](https://mlsanity.com/paper/2407.10142/) (2024)
- [PuzzleFusion++: Auto-agglomerative 3D Fracture Assembly by Denoise and Verify](https://mlsanity.com/paper/2406.00259/) (2024)

## Built on by (most cited first, 3 of 3)

- [TORA: Topological Representation Alignment for 3D Shape Assembly](https://mlsanity.com/paper/2604.04050/) (2026)
- [CRAG: Can 3D Generative Models Help 3D Assembly?](https://mlsanity.com/paper/2602.22629/) (2026)
- [LLM2R: From Large Language Models to 3D Reassemblers](https://mlsanity.com/paper/2610.18604/) (2026)

## Abstract (arXiv)

This paper introduces a new shape-matching methodology, combinative matching, to combine interlocking parts for geometric shape assembly. Previous methods for geometric assembly typically rely on aligning parts by finding identical surfaces between the parts as in conventional shape matching and registration. In contrast, we explicitly model two distinct properties of interlocking shapes: 'identical surface shape' and 'opposite volume occupancy.' Our method thus learns to establish correspondences across regions where their surface shapes appear identical but their volumes occupy the inverted space to each other. To facilitate this process, we also learn to align regions in rotation by estimating their shape orientations via equivariant neural networks. The proposed approach significantly reduces local ambiguities in matching and allows a robust combination of parts in assembly. Experimental results on geometric assembly benchmarks demonstrate the efficacy of our method, consistently outperforming the state of the art. Project page: https://nahyuklee.github.io/cmnet.