# Point-set Distances for Learning Representations of 3D Point Clouds

arXiv: https://arxiv.org/abs/2102.04014 · HTML: https://mlsanity.com/paper/2102.04014/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 57 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 30 of 42)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Deep Hough Voting for 3D Object Detection in Point Clouds](https://mlsanity.com/paper/1904.09664/) (2019)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [3DMatch: Learning Local Geometric Descriptors from RGB-D Reconstructions](https://mlsanity.com/paper/1603.08182/) (2016)
- [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019)
- [PU-Net: Point Cloud Upsampling Network](https://mlsanity.com/paper/1801.06761/) (2018)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [PPFNet: Global Context Aware Local Features for Robust 3D Point Matching](https://mlsanity.com/paper/1802.02669/) (2018)
- [3D Point Cloud Generative Adversarial Network Based on Tree Structured Graph Convolutions](https://mlsanity.com/paper/1905.06292/) (2019)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [Learning Efficient Point Cloud Generation for Dense 3D Object Reconstruction](https://mlsanity.com/paper/1706.07036/) (2017)
- [Pointwise Convolutional Neural Networks](https://mlsanity.com/paper/1712.05245/) (2017)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [3D Point Capsule Networks](https://mlsanity.com/paper/1812.10775/) (2018)
- [PointGrow: Autoregressively Learned Point Cloud Generation with Self-Attention](https://mlsanity.com/paper/1810.05591/) (2018)
- [Point Cloud GAN](https://mlsanity.com/paper/1810.05795/) (2018)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [JSIS3D: Joint Semantic-Instance Segmentation of 3D Point Clouds with Multi-Task Pointwise Networks and Multi-Value Conditional Random Fields](https://mlsanity.com/paper/1904.00699/) (2019)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)

## Built on by (most cited first, 30 of 48)

- [Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion](https://mlsanity.com/paper/2111.12702/) (2021)
- [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (2021)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Topologically-Aware Deformation Fields for Single-View 3D Reconstruction](https://mlsanity.com/paper/2205.06267/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Beyond Statistical Similarity: Rethinking Metrics for Deep Generative Models in Engineering Design](https://mlsanity.com/paper/2302.02913/) (2023)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity](https://mlsanity.com/paper/2108.07992/) (2021)
- [Lidar Upsampling with Sliced Wasserstein Distance](https://mlsanity.com/paper/2301.13558/) (2023)
- [3DMambaComplete: Exploring Structured State Space Model for Point Cloud Completion](https://mlsanity.com/paper/2404.07106/) (2024)
- [Self-Attention Amortized Distributional Projection Optimization for Sliced Wasserstein Point-Cloud Reconstruction](https://mlsanity.com/paper/2301.04791/) (2023)
- [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Dynamic 3D Point Cloud Sequences as 2D Videos](https://mlsanity.com/paper/2403.01129/) (2024)
- [Tree-Sliced Wasserstein Distance: A Geometric Perspective](https://mlsanity.com/paper/2406.13725/) (2024)
- [Scalable Counterfactual Distribution Estimation in Multivariate Causal Models](https://mlsanity.com/paper/2311.00927/) (2023)
- [Adversarial Support Alignment](https://mlsanity.com/paper/2203.08908/) (2022)
- [DDM: A Metric for Comparing 3D Shapes Using Directional Distance Fields](https://mlsanity.com/paper/2401.09736/) (2024)
- [Point2SSM++: Self-Supervised Learning of Anatomical Shape Models from Point Clouds](https://mlsanity.com/paper/2405.09707/) (2024)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Hybrid-CSR: Coupling Explicit and Implicit Shape Representation for Cortical Surface Reconstruction](https://mlsanity.com/paper/2307.12299/) (2023)
- [iBA: Backdoor Attack on 3D Point Cloud via Reconstructing Itself](https://mlsanity.com/paper/2403.05847/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)
- [Improving Generative Flow Networks with Path Regularization](https://mlsanity.com/paper/2209.15092/) (2022)
- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)

## Abstract (arXiv)

Learning an effective representation of 3D point clouds requires a good metric to measure the discrepancy between two 3D point sets, which is non-trivial due to their irregularity. Most of the previous works resort to using the Chamfer discrepancy or Earth Mover's distance, but those metrics are either ineffective in measuring the differences between point clouds or computationally expensive. In this paper, we conduct a systematic study with extensive experiments on distance metrics for 3D point clouds. From this study, we propose to use sliced Wasserstein distance and its variants for learning representations of 3D point clouds. In addition, we introduce a new algorithm to estimate sliced Wasserstein distance that guarantees that the estimated value is close enough to the true one. Experiments show that the sliced Wasserstein distance and its variants allow the neural network to learn a more efficient representation compared to the Chamfer discrepancy. We demonstrate the efficiency of the sliced Wasserstein metric and its variants on several tasks in 3D computer vision including training a point cloud autoencoder, generative modeling, transfer learning, and point cloud registration.