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

arXiv: https://arxiv.org/abs/2610.11380 · HTML: https://mlsanity.com/paper/2610.11380/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 8 of its 32 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 8 of 8)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Input Perturbation Reduces Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2301.11706/) (2023)
- [SenCache: Accelerating Diffusion Model Inference via Sensitivity-Aware Caching](https://mlsanity.com/paper/2602.24208/) (2026)
- [Quantifying Error Propagation and Model Collapse in Diffusion Models](https://mlsanity.com/paper/2602.16601/) (2026)
- [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025)
- [EpaCache: Error-Propagation-Aware Caching for Accelerating Diffusion-Based Visual Generation](https://mlsanity.com/paper/2608.29264/) (2026)

## Abstract (arXiv)

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.