Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis
Abstract
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(√(dλmax)log N/N), where d is the dimension, N the number of sampling steps, and λmax the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional √κ 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→∞. 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.
Related
- Persistence Pays: Square-root condition-number dependence for Metropolis-adjusted kinetic Langevin Monte Carlo with a non-reversibly updated accept/reject uniform
- Geometric Functional Inequalities for Markov Processes: An Invitation via the Langevin Diffusion
- Conditioning Gaussian Processes on Almost Anything
- Scalable Logistic Gaussian Process Density Regression with Kinetic Langevin Sampling
- Kinetic Langevin Meets Split Gibbs: Accelerated Posterior Sampling for Imaging Inverse Problems with Diffusion Priors
BibTeX
@misc{perbost2026diffusion,
title = {{Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis}},
author = {Adam Perbost and Francis Bach and Pierre Marion},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12052},
eprint = {2610.12052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12052},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis
AU - Adam Perbost
AU - Francis Bach
AU - Pierre Marion
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12052
UR - https://arxiv.org/abs/2610.12052
AB - 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(√(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 √κ 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→∞. 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.
ER -