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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Mar 10, 2026

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

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.

Erkan Turan, Nicolas Dufour, Maks Ovsjanikov

LIX, Ecole Polytechnique, IP Paris · Kyutai

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May 11, 2026

Kernel-Gradient Drifting Models

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.

Maria Esteban-Casadevall, Jorge Carrasco-Pollo, Max Welling +3

AMLab · University of Amsterdam · CuspAI +1

Synthetic Data Generation One-Step Generative Modeling Generative Modeling Drifting Models

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May 7, 2026

A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models

We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models, score-based generative models, and flow matching are instances of learning a time-dependent vector field that induces a family of marginals

(ρ_t)_{t \in [0,1]}

governed by continuity and Fokker-Planck equations. Such a unified theory is timely because these methods are converging methodologically, yet fragmented notation and competing derivations continue to obscure their shared structure and the practical tradeoffs governing sampling, stability, and computation. Within this framework, we (i) derive reverse-time sampling for diffusion and score-based models as controlled stochastic dynamics, (ii) show that the probability flow ODE yields identical marginals and connects diffusion to likelihood-based normalizing flows, and (iii) interpret flow matching as direct regression of the velocity field under a chosen interpolation, clarifying when it coincides with or differs from score-based training. We compare objectives, sampling schemes, and discretization errors under unified notation, discuss connections to Schrodinger bridges and entropic optimal transport, and summarize theoretical guarantees and open problems on approximation, stability, and scalability.

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Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94551, USA · Department of Electrical Engineering University of California Merced, CA 95343, USA

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