Liouville PDE-based sliced-Wasserstein flow

arXiv:2505.17204v2 Announce Type: replace
Abstract: 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.