Equal Path Cost, Unequal Output Effects: Understanding Perturbation Propagation in Diffusion Models

2026-10-08

AI Summary

This study investigates how sampling perturbations in diffusion models affect generated outputs and why their effects remain uneven under identical path costs. To address this, we construct a theoretical framework integrating dynamical analysis with information theory to quantify perturbation propagation and derive response identities. We introduce the concept of "path cost," validated through KL divergence and intervention experiments on pretrained models. Our contributions reveal that while path cost bounds output variation, it does not strictly determine it, demonstrating that first-order responses may obscure underlying distributional discrepancies. Furthermore, this work elucidates sensitivity patterns across different sampling stages and spatial frequencies, and successfully diagnoses error sources in caching-based acceleration techniques.

Abstract

Diffusion models have achieved remarkable success in generative modeling, with their sampling procedures routinely modified to control generation and improve efficiency. These modifications introduce perturbations along the sampling trajectory, raising a central question: how do such perturbations affect generated output? To address this question, we develop a theoretical framework to investigate perturbation propagation, combining dynamical analysis of the sampling process with an information-theoretic characterization of output responses. Within this framework, we quantify perturbation strength using the Kullback--Leibler (KL) divergence between perturbed and reference trajectory distributions, termed as path cost, which is shown to bound, but do not determine, changes in the output distribution. Building on this analysis, we derive a response identity that connects the propagation and accumulation of local perturbations with the information captured by a selected feature mean, explaining why changes in the output distribution can remain undetected by its first-order response. We test our theoretical analysis through controlled interventions at equal path cost in pretrained diffusion models, revealing distinct patterns of output sensitivity across sampling stages and spatial frequencies. To assess whether our framework can diagnose perturbations arising from practical approximations, we apply it to cache-based acceleration and show that our propagation analysis reliably identifies sampling intervals where caching causes larger image errors.

Problem

• Diffusion Models
• Perturbation Propagation
• Path Cost
• Sampling Trajectory
• Output Sensitivity

Innovation

• Diffusion Models
• Perturbation Propagation
• Path Cost
• Response Identity
• Cache-based Acceleration