Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivity

TL;DR

This paper simplifies the theoretical analysis of diffusion models, showing that alternative discrete noise can replace Gaussian noise without losing convergence, and validates these findings through experiments.

Contribution

It provides a simplified convergence analysis for diffusion models and demonstrates the viability of using discrete noise instead of Gaussian noise.

Findings

Discrete noise achieves comparable sample quality to Gaussian noise when variance is matched.

Performance drops if the noise variance is scaled incorrectly.

The convergence rate of the discretized SDE is $O(T^{-1/2})$ under standard assumptions.

Abstract

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 Gr\"onwall'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.