The sliced Wasserstein flow (SWF), a nonparametric and implicit generative gradient flow, is transformed to a Liouville partial differential equation (PDE)-based formalism. First, the stochastic diffusive term from the Fokker-Planck equation-based Monte Carlo is reformulated to Liouville PDE-based transport without the diffusive term, and the involved density estimation is handled by normalizing flows of neural ODE. 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 SWF barycenter for fair regression demonstrates competent profiles in the accuracy-fairness Pareto curves.

翻译：切片Wasserstein流（SWF）作为一种非参数隐式生成梯度流，被转化为基于Liouville偏微分方程（PDE）的形式体系。首先，将基于Fokker-Planck方程的蒙特卡洛方法中的随机扩散项重新表述为基于Liouville PDE的无扩散项输运形式，并通过神经常微分方程的正则化流处理其中涉及的密度估计问题。其次，通过基于Liouville PDE的SWF重心来近似计算Wasserstein重心，并借助Kantorovich势的设定来引导生成该梯度流的样本。这两项工作表明，基于Liouville PDE的SWF及SWF重心在训练和测试中呈现出更优的收敛性能，同时降低了方差。将生成式SWF重心应用于公平回归任务，在准确率-公平性帕累托曲线中展现出具有竞争力的表现。