Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models
Abstract
We analyze finite-particle drifting models for one-step generative modeling. For a conservative velocity given by the difference of the kernel-smoothed data and model scores, a joint-entropy identity yields continuous-time bounds for the smoothed Fisher discrepancy and the squared particle velocity. The finite-particle correction involves reciprocal kernel density estimates. We give local-occupancy conditions and exact expectation bounds, and show that these expectations diverge for full-support initial densities in a shrinking-bandwidth regime. Keeping the bandwidth dependence of the quadrature constants explicit yields a conditional root residual-velocity rate of $N^{-1/(d+4)}$ under uniform regularity, with a corresponding rate under weaker quadrature growth conditions. We also analyze the original displacement field with the exact Laplace kernel. A companion kernel gives a weighted coercivity estimate that accommodates the unbounded empirical scale and the kernel's nondifferentiability. Localization controls the particle velocity, while relative scale alignment absorbs the mismatch into dissipation. Both analyses quantify the size of an additional drift correction under explicit regularity and stability assumptions.
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