We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in Deng et al., 2026 (arxiv:2602.04770). For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

翻译：我们提出并分析了一种用于一步生成建模的保守漂移方法。该方法将原始的基于位移的漂移速度替换为核密度估计（KDE）梯度速度，即核平滑数据得分与核平滑模型得分的差值。该速度为梯度场，解决了一般基于位移的漂移场中存在的非保守性问题。我们在$\R^d$上证明了保守方法的连续时间有限粒子收敛界：联合熵恒等式给出了经验Stein漂移、KDE的平滑Fisher差异以及平方中心速度的界。主要的有限粒子修正项是互逆KDE自相互作用项，我们给出了该项受控的确定性和高概率局部占用条件。我们保持求积常数显式化，并追踪其可能的带宽依赖性：在额外的$h$一致求积正则条件下，根残差速度率为$N^{-1/(d+4)}$；而更一般的增长条件则产生优化根速率$N^{-(2-β)/(2(d+4-β))}$，其中$0\le β<2$。我们还分析了使用Laplace核的非保守漂移方法，对应于Deng等人于2026年提出的基于位移的原始速度（arXiv:2602.04770）。对于该方法，一个尖锐的伴随核将速度分解为尖锐得分失配的正标量预处理项与Laplace尺度失配残差项，从而产生类似的有限粒子速率，但包含一个不可避免的残差项。最后，我们解释了如何通过显式漂移步长$η$将连续时间残差速度界转化为一步生成保证。