Antithetic Noise in Diffusion Models

TL;DR

This work uncovers a universal antithetic noise principle in diffusion models: pairing each initial Gaussian noise with its negation induces strong negative output correlation across architectures, datasets, and conditioning schemes, enabling substantial variance reduction in uncertainty estimates without affecting generation quality. The authors propose a symmetry conjecture that the learned score function is approximately affine antisymmetric at each diffusion time, supported by both theory and extensive empirical evidence. They formalize uncertainty quantification improvements via Antithetic Monte Carlo and extend to Quasi-Monte Carlo, achieving up to 90% tighter confidence intervals and major cost savings. Beyond UQ, the antithetic design increases output diversity and provides a plug-in tool for image editing and benchmarking diffusion-based methods, with evidence across pixel statistics, diffusion inverse solvers, and editing tasks, all without training or runtime overhead.

Abstract

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a \textit{symmetry conjecture} that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to $90\%$ narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Figures (28)

• Figure 1: Use antithetic noise $-z$ and $z$ (with condition $c$) to generate visually “opposite” images.
• Figure 2: Histograms of standard and centralized Pearson correlation coefficients for CelebA-HQ, and DiT class and Glow in single class. Dashed lines indicate the average.
• Figure 3: Correlation of $x_t$ (solid) and $\epsilon_\theta^{(t)}$ (dash) between antithetic (PN) pairs. Step 50 is the initial noise and Step 0 is the generated image. Shaded bands show $\pm1$ std. dev.
• Figure 4: First-coordinate output of the pretrained score network on CIFAR10 as a function of the interpolation scalar $c$ for a 50-step DDIM at $t=1, 3,\dots,19$.
• Figure 5: Pearson Correlation histograms for PN and RR pairs across three datasets using DDPM. Dashed lines indicate the mean Pearson correlation for each group.

Theorems & Definitions (28)

• Lemma 1
• proof : Proof of Lemma \ref{['lemma:central symmetry']}
• Theorem 1: Score converges to the Gaussian score
• proof
• Corollary 1: Correlation of 1-step DDIM
• proof
• Proposition 1
• Proposition 2
• proof
• Proposition 3