Ornstein-Uhlenbeck Is Hard to Beat, Yet Superlinear Drift Ships Lower Transport Costs

Attila Lovas; Lóránt Nagy

stat.MLSep 29, 2026

Ornstein-Uhlenbeck Is Hard to Beat, Yet Superlinear Drift Ships Lower Transport Costs

Authors: Attila Lovas, Lóránt Nagy

Organizations: HUN-REN Alfréd Rényi Institute of Mathematics Budapest, Hungary

Abstract

Brešar and Mijatović \cite{bresar2025} show that Ornstein--Uhlenbeck diffusion is hard to beat in forward convergence under assumptions that exclude superlinear drift. We instead test superlinear Langevin diffusions for score-based image generation, computing their conditional scores numerically from a Fokker--Planck equation. In our experiments, the superlinear models beat the Ornstein--Uhlenbeck baseline on empirical Wasserstein distance across nearly the entire tested grid and show less variation across diffusion horizons. The ``hard to beat'' verdict of \cite{bresar2025} thus fails to be universal.

Figures & tables

	$T$
$\alpha$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$3$	7.4895	7.1817	7.2128	7.3525	7.5041	7.6127	7.6784	7.7590
$2$	8.0436	7.1735	7.1995	7.2805	7.4185	7.5190	7.7112	7.6554
$1$	10.7980	8.1018	9.5526	7.4124	7.4386	7.4421	7.7372	7.7217

Table 1:

\mathcal{W}_{1}

for

N=100

. Lower is better; darker shading indicates a smaller value.

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Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis

Adam Perbost, Francis Bach, Pierre Marion

Inria, École normale supérieure – PSL Research University Paris, France

Diffusion Sampling Score-Based Generative Modeling Langevin Dynamics

Open paper AskPDF arXiv

Despite their empirical success, why diffusion models overcome the bottlenecks of classical score-based samplers remains unclear. In this work, we leverage Gaussian distributions to isolate this phenomenon. We establish 2-Wasserstein convergence bounds for optimized hyperparameters, showing that diffusion processes achieve a sampling error of

O(\sqrt{dλ_{\max}}\log N/N)

, where

d

is the dimension,

N

the number of sampling steps, and

λ_{\max}

the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional

\sqrtκ

factor, where

κ

is the condition number. These rates follow from spectral bounds which are sharp: we confirm them via matching first-order asymptotics as

N\rightarrow\infty

. Our analysis provides a rigorous characterization, in the Gaussian setting, of how time-dependent score trajectories remove condition-number dependence during sampling. By contrast, in the learning phase, we show that estimating the unnoised score by gradient descent leads to essentially the same estimator as estimating a noisy score, which suggests that the benefits of noising do not come from the learning phase.

Abstract

Explore similar work

Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis

On the Robustness of Langevin Dynamics to Score Function Error

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