# ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds

arXiv: https://arxiv.org/abs/2111.15341 · HTML: https://mlsanity.com/paper/2111.15341/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 56 references that are in the map.
- First posted on arXiv 30 Nov 2021.

## Builds on (most cited first, 30 of 35)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](https://mlsanity.com/paper/1502.03044/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (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)
- [YFCC100M: The New Data in Multimedia Research](https://mlsanity.com/paper/1503.01817/) (2015)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Learning to Find Good Correspondences](https://mlsanity.com/paper/1711.05971/) (2017)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Learning Two-View Correspondences and Geometry Using Order-Aware Network](https://mlsanity.com/paper/1908.04964/) (2019)
- [Set Transformer: A Framework for Attention-based Permutation-Invariant Neural Networks](https://mlsanity.com/paper/1810.00825/) (2018)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)

## Built on by (most cited first, 15 of 15)

- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [A case for using rotation invariant features in state of the art feature matchers](https://mlsanity.com/paper/2204.10144/) (2022)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [A Simple and Universal Rotation Equivariant Point-cloud Network](https://mlsanity.com/paper/2203.01216/) (2022)
- [Normalization-Equivariant Neural Networks with Application to Image Denoising](https://mlsanity.com/paper/2306.05037/) (2023)
- [Complete Neural Networks for Complete Euclidean Graphs](https://mlsanity.com/paper/2301.13821/) (2023)
- [Weisfeiler Leman for Euclidean Equivariant Machine Learning](https://mlsanity.com/paper/2402.02484/) (2024)
- [Investigating how ReLU-networks encode symmetries](https://mlsanity.com/paper/2305.17017/) (2023)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [Affine steerers for structured keypoint description](https://mlsanity.com/paper/2408.14186/) (2024)
- [Equivariant geometric convolutions for emulation of dynamical systems](https://mlsanity.com/paper/2305.12585/) (2023)
- [TetraSphere: A Neural Descriptor for O(3)-Invariant Point Cloud Analysis](https://mlsanity.com/paper/2211.14456/) (2022)
- [On normalization-equivariance properties of supervised and unsupervised denoising methods: a survey](https://mlsanity.com/paper/2402.15352/) (2024)
- [SIM2E: Benchmarking the Group Equivariant Capability of Correspondence Matching Algorithms](https://mlsanity.com/paper/2208.09896/) (2022)
- [In Search of Projectively Equivariant Networks](https://mlsanity.com/paper/2209.14719/) (2022)

## Abstract (arXiv)

In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries. We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision.