# PolyNet: Polynomial Neural Network for 3D Shape Recognition with PolyShape Representation

arXiv: https://arxiv.org/abs/2110.07882 · HTML: https://mlsanity.com/paper/2110.07882/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 62 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 30 of 38)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [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)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [KPConv: Flexible and Deformable Convolution for Point Clouds](https://mlsanity.com/paper/1904.08889/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Relation-Shape Convolutional Neural Network for Point Cloud Analysis](https://mlsanity.com/paper/1904.07601/) (2019)
- [OctNet: Learning Deep 3D Representations at High Resolutions](https://mlsanity.com/paper/1611.05009/) (2016)
- [Learning Convolutional Neural Networks for Graphs](https://mlsanity.com/paper/1605.05273/) (2016)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [SpiderCNN: Deep Learning on Point Sets with Parameterized Convolutional Filters](https://mlsanity.com/paper/1803.11527/) (2018)
- [Deep Convolutional Networks on Graph-Structured Data](https://mlsanity.com/paper/1506.05163/) (2015)
- [SO-Net: Self-Organizing Network for Point Cloud Analysis](https://mlsanity.com/paper/1803.04249/) (2018)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [Point Convolutional Neural Networks by Extension Operators](https://mlsanity.com/paper/1803.10091/) (2018)
- [Generative and Discriminative Voxel Modeling with Convolutional Neural Networks](https://mlsanity.com/paper/1608.04236/) (2016)
- [MeshNet: Mesh Neural Network for 3D Shape Representation](https://mlsanity.com/paper/1811.11424/) (2018)
- [Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation](https://mlsanity.com/paper/1905.02876/) (2019)
- [Local Spectral Graph Convolution for Point Set Feature Learning](https://mlsanity.com/paper/1803.05827/) (2018)
- [GNN-FiLM: Graph Neural Networks with Feature-wise Linear Modulation](https://mlsanity.com/paper/1906.12192/) (2019)
- [RGCNN: Regularized Graph CNN for Point Cloud Segmentation](https://mlsanity.com/paper/1806.02952/) (2018)
- [O-CNN: Octree-based Convolutional Neural Networks for 3D Shape Analysis](https://mlsanity.com/paper/1712.01537/) (2017)
- [TextureNet: Consistent Local Parametrizations for Learning from High-Resolution Signals on Meshes](https://mlsanity.com/paper/1812.00020/) (2018)
- [Learning 3D Shape Completion under Weak Supervision](https://mlsanity.com/paper/1805.07290/) (2018)

## Built on by (most cited first, 9 of 9)

- [A Survey on Graph Neural Networks and Graph Transformers in Computer Vision: A Task-Oriented Perspective](https://mlsanity.com/paper/2209.13232/) (2022)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Equivariant Hypergraph Neural Networks](https://mlsanity.com/paper/2208.10428/) (2022)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [ACL-SPC: Adaptive Closed-Loop system for Self-Supervised Point Cloud Completion](https://mlsanity.com/paper/2303.01979/) (2023)
- [The Alexander-Hirschowitz theorem for neurovarieties](https://mlsanity.com/paper/2511.19703/) (2025)
- [Self-supervised Learning for Enhancing Geometrical Modeling in 3D-Aware Generative Adversarial Network](https://mlsanity.com/paper/2312.11856/) (2023)
- [Transferable and Undefendable Point Cloud Attacks via Medial Axis Transform](https://mlsanity.com/paper/2507.18870/) (2025)

## Abstract (arXiv)

3D shape representation and its processing have substantial effects on 3D shape recognition. The polygon mesh as a 3D shape representation has many advantages in computer graphics and geometry processing. However, there are still some challenges for the existing deep neural network (DNN)-based methods on polygon mesh representation, such as handling the variations in the degree and permutations of the vertices and their pairwise distances. To overcome these challenges, we propose a DNN-based method (PolyNet) and a specific polygon mesh representation (PolyShape) with a multi-resolution structure. PolyNet contains two operations; (1) a polynomial convolution (PolyConv) operation with learnable coefficients, which learns continuous distributions as the convolutional filters to share the weights across different vertices, and (2) a polygonal pooling (PolyPool) procedure by utilizing the multi-resolution structure of PolyShape to aggregate the features in a much lower dimension. Our experiments demonstrate the strength and the advantages of PolyNet on both 3D shape classification and retrieval tasks compared to existing polygon mesh-based methods and its superiority in classifying graph representations of images. The code is publicly available from https://myavartanoo.github.io/polynet/.