用李维维尔方程重构切片沃瑟斯坦流，提升生成效率与公平性。
Liouville PDE-based sliced-Wasserstein flow
- 将切片沃瑟斯坦流转化为无扩散项的李维维尔PDE形式，简化概率流计算。
- 在训练和测试中收敛更快，方差更低，生成质量更优。
- 适合需要高公平性与可扩展性的生成建模任务，如公平回归。
切片沃瑟斯坦流（SWF）作为一种非参数隐式生成梯度流，被重新表述为基于李维维尔偏微分方程（PDE）的形式。首先，将基于福克-普朗克方程的蒙特卡洛随机扩散项重写为无扩散项的李维维尔PDE传输机制，本质反映概率流常微分方程。密度估计通过神经常微分方程的归一化流实现，无需显式定义得分函数。其次，通过施加坎托罗维奇势函数，用李维维尔PDE-based SWF巴氏中心近似计算沃瑟斯坦巴氏中心，以生成样本。两项改进均在训练与测试中表现出更优收敛性及更低方差。将该生成式李维维尔PDE-based SWF巴氏中心应用于公平回归，其准确率-公平性帕累托曲线表现优异，相较标准SWF具有竞争力，且在提升公平性方面显著优于精确沃瑟斯坦巴氏中心，并具备良好可扩展性。
原文摘要 · Abstract (English)
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.
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