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

arXiv: https://arxiv.org/abs/2610.12052 · HTML: https://mlsanity.com/paper/2610.12052/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 3 of its 38 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 3 of 3)

- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting](https://mlsanity.com/paper/2503.11615/) (2025)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)

## Abstract (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(√dλmaxlog 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 √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 Nrightarrow∞. 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.