Generative Drifting is Secretly Score Matching: a Spectral and Variational Perspective

Abstract: Generative Modeling via Drifting has recently achieved state-of-the-art one-step image generation through a kernel-based drift operator, yet the success is largely empirical and its theoretical foundations remain poorly understood. In this paper, we make the following observation: \emph{under a Gaussian kernel, the drift operator is exactly a score difference on smoothed distributions}. This insight allows us to answer all three key questions left open in the original work: (1) whether a vanishing drift guarantees equality of distributions ($V_{p,q}=0\Rightarrow p=q$), (2) how to choose between kernels, and (3) why the stop-gradient operator is indispensable for stable training. Our observations position drifting within the well-studied score-matching family and enable a rich theoretical perspective. By linearizing the McKean-Vlasov dynamics and probing them in Fourier space, we reveal frequency-dependent convergence timescales comparable to \emph{Landau damping} in plasma kinetic theory: the Gaussian kernel suffers an exponential high-frequency bottleneck, explaining the empirical preference for the Laplacian kernel. We also propose an exponential bandwidth annealing schedule $σ(t)=σ0 e^{-rt}$ that reduces convergence time from $\exp(O(K{\max}^2))$ to $O(\log K_{\max})$. Finally, by formalizing drifting as a Wasserstein gradient flow of the smoothed KL divergence, we prove that the stop-gradient operator is derived directly from the frozen-field discretization mandated by the JKO scheme, and removing it severs training from any gradient-flow guarantee. This variational perspective further provides a general template for constructing novel drift operators, demonstrated with a Sinkhorn divergence drift.