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

arXiv:2610.12052v1 Announce Type: cross 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(sqrt{dlambda_{max}}log N/N), where d is the dimension, N the number of sampling steps, and lambda_{max} the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional sqrtkappa factor, where kappa is the condition number. These rates follow from spectral bounds which are sharp: we confirm them via matching first-order asymptotics as Nrightarrowinfty. 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.

Source: arXiv cs.LG | 2026-10-09