# Liouville PDE-based sliced-Wasserstein flow

arXiv: https://arxiv.org/abs/2505.17204 · HTML: https://mlsanity.com/paper/2505.17204/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 44 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 26 of 26)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Fair Regression: Quantitative Definitions and Reduction-based Algorithms](https://mlsanity.com/paper/1905.12843/) (2019)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [Generative Modeling using the Sliced Wasserstein Distance](https://mlsanity.com/paper/1803.11188/) (2018)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [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)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Convergence of flow-based generative models via proximal gradient descent in Wasserstein space](https://mlsanity.com/paper/2310.17582/) (2023)
- [Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm](https://mlsanity.com/paper/2002.04783/) (2020)
- [Projected Wasserstein gradient descent for high-dimensional Bayesian inference](https://mlsanity.com/paper/2102.06350/) (2021)
- [Self-Consistency of the Fokker-Planck Equation](https://mlsanity.com/paper/2206.00860/) (2022)
- [Equal Opportunity of Coverage in Fair Regression](https://mlsanity.com/paper/2311.02243/) (2023)
- [Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flows](https://mlsanity.com/paper/2305.02164/) (2023)
- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Fairness Measures for Regression via Probabilistic Classification](https://mlsanity.com/paper/2001.06089/) (2020)
- [Fair Regression for Health Care Spending](https://mlsanity.com/paper/1901.10566/) (2019)
- [Relative Entropy Gradient Sampler for Unnormalized Distributions](https://mlsanity.com/paper/2110.02787/) (2021)
- [Differentially Private Gradient Flow based on the Sliced Wasserstein Distance](https://mlsanity.com/paper/2312.08227/) (2023)

## Abstract (arXiv)

The sliced Wasserstein flow (SWF), a nonparametric and implicit generative gradient flow, is transformed into a Liouville partial differential equation (PDE)-based formalism. First, the stochastic diffusive term from the Fokker-Planck equation-based Monte Carlo is reformulated as a Liouville PDE-based transport without the diffusive term, essentially reflecting the probability flow ODE. The involved density estimation is handled by normalizing flows of neural ODE without an explicitly defined score function. Next, the computation of the Wasserstein barycenter is approximated by the Liouville PDE-based SWF barycenter with the prescription of Kantorovich potentials for the induced gradient flow to generate its samples. These two efforts show outperforming convergence in training and testing Liouville PDE-based SWF and SWF barycenters with reduced variance. Applying the generative Liouville PDE-based SWF barycenter for fair regression demonstrates competent profiles in the accuracy-fairness Pareto curves, with comparable and alternative choices against the standard SWF, and significant benefit in improving fairness with scalability in comparison to the exact Wasserstein barycenter.