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: 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 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 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)=σ_0 e^-rt that reduces convergence time from (O(K_^2)) to O( K_). 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.

Reproductions