# Properties of Discrete Sliced Wasserstein Losses

arXiv: https://arxiv.org/abs/2307.10352 · HTML: https://mlsanity.com/paper/2307.10352/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 19 Jul 2023.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [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)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Sliced Wasserstein Generative Models](https://mlsanity.com/paper/1904.05408/) (2019)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Convergence of constant step stochastic gradient descent for non-smooth non-convex functions](https://mlsanity.com/paper/2005.08513/) (2020)
- [Analysis of nonsmooth stochastic approximation: the differential inclusion approach](https://mlsanity.com/paper/1805.01916/) (2018)
- [Approximate Bayesian Computation with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1910.12815/) (2019)
- [Non-asymptotic convergence bounds for Wasserstein approximation using point clouds](https://mlsanity.com/paper/2106.07911/) (2021)
- [Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation](https://mlsanity.com/paper/1603.05031/) (2016)

## Built on by (most cited first, 12 of 12)

- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Convergence of SGD for Training Neural Networks with Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.11714/) (2023)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Learning with Differentially Private (Sliced) Wasserstein Gradients](https://mlsanity.com/paper/2502.01701/) (2025)
- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [LaCoOT: Layer Collapse through Optimal Transport](https://mlsanity.com/paper/2406.08933/) (2024)
- [Canonical Variates in Wasserstein Metric Space](https://mlsanity.com/paper/2405.15768/) (2024)
- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)
- [Sliced-Wasserstein Distribution Alignment Loss Improves the Ultra-Low-Bit Quantization of Large Language Models](https://mlsanity.com/paper/2601.07878/) (2026)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)

## Abstract (arXiv)

The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is common to optimise some parameters in order to minimise SW, which serves as a loss function between discrete probability measures (since measures admitting densities are numerically unattainable). All these optimisation problems bear the same sub-problem, which is minimising the Sliced Wasserstein energy. In this paper we study the properties of E: Y longmapsto SW22(γY, γZ), i.e. the SW distance between two uniform discrete measures with the same amount of points as a function of the support Y ∈ ℝn × d of one of the measures. We investigate the regularity and optimisation properties of this energy, as well as its Monte-Carlo approximation Ep (estimating the expectation in SW using only p samples) and show convergence results on the critical points of Ep to those of E, as well as an almost-sure uniform convergence and a uniform Central Limit result on the process Ep(Y). Finally, we show that in a certain sense, Stochastic Gradient Descent methods minimising E and Ep converge towards (Clarke) critical points of these energies.