Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivity

Diffusion models, typically formulated as discretizations of stochastic differential equations (SDEs), have achieved state-of-the-art performance in generative tasks. However, their theoretical analysis often involves complex proofs. In this work, we present a simplified framework for analyzing the Euler–Maruyama discretization of variance-preserving SDEs (VP-SDEs). Using Grönwall’s inequality, we derive a convergence rate of $O(T^{-1/2})$ under standard Lipschitz assumptions, streamlining prior analyses. We then demonstrate that the standard Gaussian noise can be replaced by computationally cheaper discrete random variables (e.g., Rademacher) without sacrificing this convergence guarantee, provided the mean and variance are matched. Our experiments validate this theory, showing that (i) discrete noise achieves sample quality comparable to Gaussian noise provided the variance is matched correctly, and (ii) performance degrades if the noise variance is scaled incorrectly.

💡 Research Summary

This paper revisits the theoretical foundations of diffusion models, focusing on the variance‑preserving stochastic differential equation (VP‑SDE) that underlies most modern generative pipelines. The authors first present a streamlined convergence analysis for the Euler‑Maruyama discretization of the reverse‑time VP‑SDE. By rescaling time (τ = t/T) and applying the continuous and discrete versions of Grönwall’s inequality, they show that under standard Lipschitz conditions on the drift b(t,x) and diffusion coefficient σ(t), the strong error satisfies
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