# Distributional Sliced-Wasserstein and Applications to Generative Modeling

arXiv: https://arxiv.org/abs/2002.07367 · HTML: https://mlsanity.com/paper/2002.07367/

## Key facts

- Cited by 69 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2020.

## Built on by (most cited first, 30 of 69)

- [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)
- [Energy-Based Sliced Wasserstein Distance](https://mlsanity.com/paper/2304.13586/) (2023)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [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)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Differentially Private Sliced Wasserstein Distance](https://mlsanity.com/paper/2107.01848/) (2021)
- [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)
- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Sliced Wasserstein Estimation with Control Variates](https://mlsanity.com/paper/2305.00402/) (2023)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Sliced Wasserstein with Random-Path Projecting Directions](https://mlsanity.com/paper/2401.15889/) (2024)
- [SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer](https://mlsanity.com/paper/2301.12811/) (2023)
- [PAC-Bayesian Generalization Bounds for Adversarial Generative Models](https://mlsanity.com/paper/2302.08942/) (2023)
- [Entropic Gromov-Wasserstein between Gaussian Distributions](https://mlsanity.com/paper/2108.10961/) (2021)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Model Selection for Bayesian Autoencoders](https://mlsanity.com/paper/2106.06245/) (2021)
- [Color Conditional Generation with Sliced Wasserstein Guidance](https://mlsanity.com/paper/2503.19034/) (2025)
- [Hilbert Curve Projection Distance for Distribution Comparison](https://mlsanity.com/paper/2205.15059/) (2022)
- [One Loss for Quantization: Deep Hashing with Discrete Wasserstein Distributional Matching](https://mlsanity.com/paper/2205.15721/) (2022)
- [Semi-Autoregressive Energy Flows: Exploring Likelihood-Free Training of Normalizing Flows](https://mlsanity.com/paper/2206.06672/) (2022)
- [Lidar Upsampling with Sliced Wasserstein Distance](https://mlsanity.com/paper/2301.13558/) (2023)
- [Statistical and Topological Properties of Gaussian Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2110.10524/) (2021)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)

## Abstract (arXiv)

Sliced-Wasserstein distance (SW) and its variant, Max Sliced-Wasserstein distance (Max-SW), have been used widely in the recent years due to their fast computation and scalability even when the probability measures lie in a very high dimensional space. However, SW requires many unnecessary projection samples to approximate its value while Max-SW only uses the most important projection, which ignores the information of other useful directions. In order to account for these weaknesses, we propose a novel distance, named Distributional Sliced-Wasserstein distance (DSW), that finds an optimal distribution over projections that can balance between exploring distinctive projecting directions and the informativeness of projections themselves. We show that the DSW is a generalization of Max-SW, and it can be computed efficiently by searching for the optimal push-forward measure over a set of probability measures over the unit sphere satisfying certain regularizing constraints that favor distinct directions. Finally, we conduct extensive experiments with large-scale datasets to demonstrate the favorable performances of the proposed distances over the previous sliced-based distances in generative modeling applications.