# Geometry-Aware Discretization Error of Diffusion Models

arXiv: https://arxiv.org/abs/2605.08392 · HTML: https://mlsanity.com/paper/2605.08392/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 8 May 2026.

## Placed in

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

## Builds on (most cited first, 13 of 13)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [DeepInverse: A Python package for solving imaging inverse problems with deep learning](https://mlsanity.com/paper/2505.20160/) (2025)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors](https://mlsanity.com/paper/2405.14250/) (2024)
- [The Hidden Linear Structure in Score-Based Models and its Application](https://mlsanity.com/paper/2311.10892/) (2023)
- [Towards More Accurate Diffusion Model Acceleration with A Timestep Tuner](https://mlsanity.com/paper/2310.09469/) (2023)
- [From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting](https://mlsanity.com/paper/2503.11615/) (2025)
- [Fast Diffusion Probabilistic Model Sampling through the lens of Backward Error Analysis](https://mlsanity.com/paper/2304.11446/) (2023)

## Built on by (most cited first, 3 of 3)

- [Gaussian Flow-Matching Schedules: Implications for Sampling and Training](https://mlsanity.com/paper/2609.25839/) (2026)
- [Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching](https://mlsanity.com/paper/2609.39488/) (2026)
- [Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis](https://mlsanity.com/paper/2610.12052/) (2026)

## Abstract (arXiv)

Practical diffusion sampling requires simulating a reverse-time ODE or SDE with a limited number of denoising steps, making the choice of sampling parameters crucial for minimizing discretization error. Non-asymptotic convergence bounds characterize sampling complexity, but their worst-case constants can obscure target geometry and thereby limit guidance on parameter optimization. Rather than bounding the error, we derive asymptotically exact small-stepsize expansions of Euler-Maruyama weak and Frechet errors for general smooth reverse diffusions, with explicit formulas for Gaussian data. These formulas provide tractable objectives for optimizing diffusion parameters, including the noise and rescaling schedules and the stochasticity coefficient, according to the target's covariance spectrum. In particular, our theory predicts lower optimal stochasticity at smaller step budgets, shows how to adapt the rescaling coefficient to the data power spectrum, and motivates a new family of effective noise schedules. A perturbative extension to Gaussian scale mixtures (GSMs) quantifies how departures from Gaussianity shift the optimal parameters. Finally, experiments on different real image datasets show that FID-optimal parameters agree with the qualitative theoretical predictions.