Error estimates of a training-free diffusion model for high-dimensional sampling

TL;DR

This paper provides rigorous, numerically verifiable error estimates for training-free diffusion models used in high-dimensional data sampling, demonstrating classical convergence rates and favorable dimension dependence.

Contribution

It develops a comprehensive error analysis for training-free diffusion models, avoiding score approximation errors and providing verifiable bounds that align with observed numerical behavior.

Findings

Error bounds scale as O(d) in ℓ₂ norm and O(log d) in ℓ_∞ norm.

Classical convergence rates like first-order for Euler method are recovered.

Error estimates are fully numerically verifiable.

Abstract

Score-based diffusion models are a powerful class of generative models, but their practical use often depends on training neural networks to approximate the score function. Training-free diffusion models provide an attractive alternative by exploiting analytically tractable score functions, and have recently enabled supervised learning of efficient end-to-end generative samplers. Despite their empirical success, the training-free diffusion models lack rigorous and numerically verifiable error estimates. In this work, we develop a comprehensive error analysis for a class of training-free diffusion models used to generate labeled data for supervised learning of generative samplers. By exploiting the availability of the exact score function for Gaussian mixture models, our analysis avoids propagating score-function approximation errors through the reverse-time diffusion process and recovers classical convergence rates for ODE discretization schemes, such as first-order convergence for the Euler method. Moreover, the resulting error bounds exhibit favorable dimension dependence, scaling as $O(d)$ in the $\ell_2$ norm and $O(\log d)$ in the $\ell_\infty$ norm. Importantly, the proposed error estimates are fully numerically verifiable with respect to both time-step size and dimensionality, thereby bridging the gap between theoretical analysis and observed numerical behavior.