A Unified View of Score-Based and Drifting Models

Chieh-Hsin Lai; Bac Nguyen; Naoki Murata; Yuhta Takida; Toshimitsu Uesaka; Yuki Mitsufuji; Stefano Ermon; Molei Tao

cs.LGMar 8, 2026

A Unified View of Score-Based and Drifting Models

Authors: Chieh-Hsin Lai, Bac Nguyen, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yuki Mitsufuji, Stefano Ermon, Molei Tao

Organizations: Sony AI · Sony Group Corporation · Stanford University · Georgia Tech

Abstract

Drifting models train one-step generators by optimizing a kernel-induced mean-shift discrepancy between the data and model distributions, with Laplace kernels used by default in practice. At each point, this discrepancy compares the kernel-weighted displacement toward nearby data samples with the corresponding displacement toward nearby model samples, thereby defining a transport direction for generated samples. In this paper, we show that drifting is more closely connected to score-based generative modeling than it may first appear, establishing a precise link to the score-matching principle underlying diffusion models. For Gaussian kernels, the population mean-shift field exactly equals the difference between the scores (i.e., the gradient-log-densities) of the Gaussian-smoothed data and model distributions. This identity follows from Tweedie's formula, which links the score of a Gaussian-smoothed density to its conditional mean, and implies that Gaussian-kernel drifting is exactly a score-matching objective on smoothed distributions. More generally, we derive an exact decomposition for radial kernels in which mean shift equals a score-based field plus a residual term. For the practical Laplace kernel, we further show theoretically and empirically that this residual is negligible in high dimension, implying that the transport field used in practice is nearly score-based. Our results reveal a structural connection to diffusion models: both methods use score-mismatch transport directions, but drifting realizes the score nonparametrically through kernel-based estimates, whereas diffusion models learn it parametrically with neural networks.

Figures & tables

Figure 1: 2D visualization: Gaussian drifting exactly matches the score-mismatch direction. Under Gaussian smoothing, the mean-shift drifting field $\Delta_{p,q}$ in (a) is exactly parallel to the score-mismatch field $\Delta{\mathbf{s}}_{p,q}$ in (b), as shown in Theorem 1 ; see Section 2.3 for the definitions. Panels (c,d) visualize this exact alignment. Both fields are estimated from finite samples using the same kernel-based Monte Carlo procedure. Here $p$ is a three-mode distribution on the unit circle, and $q$ is a fixed model distribution obtained by rotating the mode centers of $p$ by a fixed angle.

Figure 2: Illustration of

\bm{\delta}_{\perp}({\mathbf{x}})

for the Laplace case. Here

\bm{\delta}_{\perp}({\mathbf{x}})\approx\bm{0}

because tangential contributions from different directions nearly cancel.

Figure 3: Empirical validity of drifting–score alignment as dimension grows. Field alignment between the drifting discrepancy

\Delta_{p,q}(\mathbf{x})

and the score discrepancy

\Delta\mathbf{s}_{p,q}(\mathbf{x})

as the dimension

D

increases, evaluated on Ring MoG and Raw MoG. (a) Absolute alignment error

\mathbb{E}_{q}\|\Delta_{p,q}(\mathbf{x})-C^{*}\Delta\mathbf{s}_{p,q}(\mathbf{x})\|_{2}^{2}

, where

C^{*}

is the best global least-squares scaling defined in Section C.2 . (b) Relative error normalized by the field energy

\mathbb{E}_{q}\|\Delta_{p,q}(\mathbf{x})\|^{2}

. (c) Directional misalignment

1-\cos\angle(\Delta_{p,q},\Delta\mathbf{s}_{p,q})

on a log scale. For both datasets, alignment improves monotonically with dimension: the errors decay close to the predicted

1/D

rate, while the cosine similarity approaches

1

, supporting the theory in Section 3.3 .

Figure 4: Illustration of the 2D datasets. Top row shows Ring MoG; bottom row shows Raw MoG.

Figure 5: Empirical validity of drifting–score alignment on CIFAR-10 under different pre-trained feature embeddings. Starting from Laplace-drifting checkpoints, we evaluate the decomposition-related quantities in the same frozen feature spaces used by training: the ConvNeXt and the MAE embeddings. (a) Relative error

\mathbb{E}_{q}\|\Delta_{p,q}-C^{*}\Delta{\mathbf{s}}_{p,q}\|_{2}^{2}/\mathbb{E}_{q}\|\Delta_{p,q}\|_{2}^{2}

. (b) Cosine similarity

\cos\angle(\Delta_{p,q},\Delta{\mathbf{s}}_{p,q})

. (c) pre-conditioner matching

\bar{\alpha}_{p}/\bar{\alpha}_{q}

. (d) Averaged residual gap

\mathbb{E}_{q}\|\bm{\delta}_{p}-\bm{\delta}_{q}\|_{2}^{2}

. Across training, the approximation error decreases, the drifting and score discrepancies become more aligned, the pre-conditioners become better matched, and the residual gap remains small, again consistent with Theorems 2 and 3.3 .

Method	Field Type	Update Field	Normalization?	Main Interpretation
Drifting Model	Mean-shift	$\displaystyle\Delta_{p,q}({\mathbf{x}})=\eta\big(\mathbf{V}_{p,k}({\mathbf{x}})-\mathbf{V}_{q,k}({\mathbf{x}})\big)$	$\checkmark$	Attractive–repulsive
Score-based Model	Score-mismatch	$\displaystyle\Delta{\mathbf{s}}_{p,q}({\mathbf{x}})=\mathbf{s}_{p,k}({\mathbf{x}})-\mathbf{s}_{q,k}({\mathbf{x}})$	$\checkmark$ (automatic via score)	Attractive–repulsive
GAN	kernel-potential/ MMD-type kernel	$\displaystyle\mathbf{F}_{p,q}({\mathbf{x}})=\nabla_{{\mathbf{x}}}p_{k}({\mathbf{x}})-\nabla_{{\mathbf{x}}}q_{k}({\mathbf{x}})$	$\times$	Attractive–repulsive
IGN	Self-map residual	$\displaystyle\Delta^{\mathrm{IGN}}_{{\bm{\theta}}}({\mathbf{u}})={\mathbf{f}}_{{\bm{\theta}}}({\mathbf{u}})-{\mathbf{u}}$	$\times$	Data-anchored manifold-restoring

Table 1: Comparison of four update-field realizations for one-step generative modeling. Drifting, score-mismatch, and Coulomb/MMD-style methods are all attractive–repulsive fields, while IGN is a self-map residual that enforces equilibrium through fixed points.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Illustration of

\bm{\delta}_{\perp}({\mathbf{x}})

in three illustrative examples. (a)

\bm{\delta}_{\perp}\gg\bm{0}

because most of the mass of

\uppi_{\tau}(\cdot|{\mathbf{x}})

lies far away (large

r

) and in directions perpendicular to

\hat{\mathbf{s}}

; (b)

\bm{\delta}_{\perp}\approx\bm{0}

because most of the mass of

\uppi_{\tau}(\cdot|{\mathbf{x}})

lies closer (small

r

) and in directions parallel to

\hat{\mathbf{s}}

; (c)

\bm{\delta}_{\perp}\approx\bm{0}

because the contributions from different directions nearly cancel out.

Figure 7: Empirical diagnostics for the Laplace-kernel mechanism. (a) The kernel-reweighted pre-conditioners concentrate and become indistinguishable, with

\bar{\alpha}_{p}/\bar{\alpha}_{q}\to 1

. (b) The residual-gap energy

\mathbb{E}_{{\mathbf{x}}\sim q}\|\bm{\delta}_{p}(\mathbf{x})-\bm{\delta}_{q}(\mathbf{x})\|_{2}^{2}

decays with dimension, indicating a vanishing covariance residual. (c) The theory-predicted scale

C_{\mathrm{theory}}=\rho\tau

matches the oracle least-squares scale

C^{*}

, with

C^{*}/C_{\mathrm{theory}}\to 1

. Together, these diagnostics support the mechanism behind Theorems 2 and 3 and show that the Laplace mean-shift discrepancy is not only directionally aligned with the score discrepancy, but also correctly calibrated in magnitude.

Figure 8: Illustration of 2D generation across different synthetic datasets. We compare the generation quality of drifting models using Laplace and Gaussian kernels, and evaluate them using Sliced Wasserstein Distance (SWD) and MMD. The two kernels achieve nearly identical performance on both metrics across the four datasets. This suggests that, even in low dimension, the pre-conditioning and covariance-residual terms in Theorem 2 have little practical effect on generation quality.

Panel of Figure 9 — Figure 9: Comparison of generation on CIFAR-10 under different feature embeddings and kernels. Single-step unconditional generation on CIFAR-10 at $32\times 32$ resolution using Laplace and Gaussian kernels, with the drifting loss evaluated in either ConvNeXt or MAE feature space. The first two panels show the ConvNeXt-based models, where Gaussian achieves FID 8.38 and Laplace achieves FID 8.66. The last two panels show the MAE-based models, where Gaussian achieves FID 4.72 and Laplace achieves FID 4.79. Across both feature spaces, the two kernels achieve near-parity under an otherwise identical training pipeline, suggesting that the Laplace-specific pre-conditioning and residual terms have limited effect on end-to-end sample quality in these settings. This observation is also consistent with concurrent findings on CelebA-HQ reported by [ 23 ] .

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

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LIX, Ecole Polytechnique, IP Paris · Kyutai

Drifting Models Score Matching

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Generative Modeling via Drifting~\citep{deng2026drifting} has recently achieved state-of-the-art one-step image generation through a kernel-based drift operator, yet its success is largely empirical and its theoretical foundations remain poorly understood. We observe that under a Gaussian kernel, the drift operator is exactly a score difference on smoothed distributions. This answers three 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 score-matching family. 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, potentially explaining the empirical preference for the Laplacian kernel. This suggests a fix: an exponential bandwidth annealing schedule

σ(t)=σ_0 e^{-rt}

that reduces convergence time from

\exp(O(K_{\max}^2))

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 not a heuristic but is derived from the frozen-field discretization mandated by the Jordan-Kinderlehrer-Otto (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, which we demonstrate with a Sinkhorn divergence drift. We validate our analysis on toy datasets and scale it up to ImageNet.

Abstract

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