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~\cite{deng2026drifting}. 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：我们提出并分析一种用于单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，解决了基于位移的通用漂移场中已识别的非保守性问题。我们证明了在$\R^d$上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反KDE自相互作用项，并给出了控制该项的确定性和高概率局部占据条件。我们保持求积常数显式化，并追踪其可能的带宽依赖性：在附加$h$均匀求积正则条件下，残差速度的根本速率为$N^{-1/(d+4)}$；而更一般的增长条件则产生优化后的根本速率$N^{-(2-β)/(2(d+4-β))}$，其中$0\le β<2$。我们还分析了使用拉普拉斯核的非保守漂移方法，对应文献~\cite{deng2026drifting}中提出的原始基于位移的速度。对于该方法，一个锐利伴随核将速度分解为锐利得分不匹配的正标量预处理加上拉普拉斯尺度不匹配残差，从而产生类似的有限粒子速率，并带有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大小$η$将连续时间残差速度界转化为单步生成保证。