# Generalized Sliced Wasserstein Distances

arXiv: https://arxiv.org/abs/1902.00434 · HTML: https://mlsanity.com/paper/1902.00434/

## Key facts

- Cited by 208 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 30 of 208)

- [No Fear of Heterogeneity: Classifier Calibration for Federated Learning with Non-IID Data](https://mlsanity.com/paper/2106.05001/) (2021)
- [3D Gaussian Splatting as Markov Chain Monte Carlo](https://mlsanity.com/paper/2404.09591/) (2024)
- [Lossy Compression with Gaussian Diffusion](https://mlsanity.com/paper/2206.08889/) (2022)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Being-H0.5: Scaling Human-Centric Robot Learning for Cross-Embodiment Generalization](https://mlsanity.com/paper/2601.12993/) (2026)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Improving Mini-batch Optimal Transport via Partial Transportation](https://mlsanity.com/paper/2108.09645/) (2021)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Universal Approximation Under Constraints is Possible with Transformers](https://mlsanity.com/paper/2110.03303/) (2021)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution](https://mlsanity.com/paper/2204.01188/) (2022)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Fairness and Explainability: Bridging the Gap Towards Fair Model Explanations](https://mlsanity.com/paper/2212.03840/) (2022)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Optimal transport for causal discovery](https://mlsanity.com/paper/2201.09366/) (2022)
- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [On the Inherent Regularization Effects of Noise Injection During Training](https://mlsanity.com/paper/2102.07379/) (2021)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Amortized Projection Optimization for Sliced Wasserstein Generative Models](https://mlsanity.com/paper/2203.13417/) (2022)
- [On Transportation of Mini-batches: A Hierarchical Approach](https://mlsanity.com/paper/2102.05912/) (2021)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)

## Abstract (arXiv)

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We also show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed distances. Finally, we compare the numerical performance of the proposed distances on several generative modeling tasks, including SW flows and SW auto-encoders.