# Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions

arXiv: https://arxiv.org/abs/2606.30310 · HTML: https://mlsanity.com/paper/2606.30310/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 47 references that are in the map.
- First posted on arXiv 29 Jun 2026.

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

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Federated Learning with Personalization Layers](https://mlsanity.com/paper/1912.00818/) (2019)
- [Exploiting Shared Representations for Personalized Federated Learning](https://mlsanity.com/paper/2102.07078/) (2021)
- [Sliced Wasserstein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1903.04064/) (2019)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Practical Secure Aggregation for Federated Learning on User-Held Data](https://mlsanity.com/paper/1611.04482/) (2016)
- [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)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [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)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019)
- [Hierarchical Sliced Wasserstein Distance](https://mlsanity.com/paper/2209.13570/) (2022)
- [Approximate Bayesian Computation with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1910.12815/) (2019)
- [Dataset Dynamics via Gradient Flows in Probability Space](https://mlsanity.com/paper/2010.12760/) (2020)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [Amortized Projection Optimization for Sliced Wasserstein Generative Models](https://mlsanity.com/paper/2203.13417/) (2022)
- [Markovian Sliced Wasserstein Distances: Beyond Independent Projections](https://mlsanity.com/paper/2301.03749/) (2023)
- [Flowing Datasets with Wasserstein over Wasserstein Gradient Flows](https://mlsanity.com/paper/2506.07534/) (2025)
- [Slicing Unbalanced Optimal Transport](https://mlsanity.com/paper/2306.07176/) (2023)
- [Federated Wasserstein Distance](https://mlsanity.com/paper/2310.01973/) (2023)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025)

## Abstract (arXiv)

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.