Diffusion removes condition number dependency: a sharp comparison with Langevin on Gaussian data

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 rate of $\mathcal{O}(\sqrt{d\lambda_{\max}}\log N/N)$, where $d$ is the dimension, $N$ the number of discretization steps, and $\lambda_{\max}$ the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional $\sqrt\kappa$ 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 $N\rightarrow\infty$. Our analysis provides a rigorous characterization, in the Gaussian setting, of how time-dependent score trajectories removes condition-number dependence.