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.

翻译：切片Wasserstein流（SWF）——一种非参数化隐式生成梯度流——被转化为基于Liouville偏微分方程（PDE）的形式体系。首先，基于Fokker-Planck方程的蒙特卡罗方法中的随机扩散项被重构为不含扩散项的Liouville PDE输运过程，本质上反映了概率流常微分方程。其中涉及的密度估计通过神经ODE的正则化流处理，无需显式定义的得分函数。其次，Wasserstein重心的计算被近似为基于Liouville PDE的SWF重心，通过指定Kantorovich势能诱导梯度流生成样本。这两项工作在训练和测试中展现出更优的收敛性，降低了基于Liouville PDE的SWF和SWF重心的方差。将生成式Liouville PDE-SWF重心应用于公平回归，在准确率-公平性帕累托曲线上展现出与标准SWF相当且可替代的性能特征，相较于精确Wasserstein重心，在提升公平性方面具有显著优势及可扩展性。