Finite-particles rates for drifting models

We view drifting from a Wasserstein gradient flow (WGF) perspective and propose a conservative method for one-step generative modeling, replacing the original displacement-based velocity with a KDE-gradient field given by the difference between kernel-smoothed data and model scores. Using a joint-entropy identity, we establish continuous-time finite-particle bounds on R^d, including a root residual-velocity rate of N^{-1/(d+4)} under bandwidth-uniform quadrature regularity. We also analyze the original non-conservative Laplace-kernel method through a sharp companion-kernel decomposition and show how the resulting residual-velocity bounds yield explicit one-step generation guarantees in terms of the drift size.