# From Modes to Memories: Characterizing the Scale-Space Dynamics of Diffusion Models

arXiv: https://arxiv.org/abs/2609.39648 · HTML: https://mlsanity.com/paper/2609.39648/

## 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 31 of its 42 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Placed in

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

## Builds on (most cited first, 30 of 31)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Are Diffusion Models Vulnerable to Membership Inference Attacks?](https://mlsanity.com/paper/2302.01316/) (2023)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Detecting, Explaining, and Mitigating Memorization in Diffusion Models](https://mlsanity.com/paper/2407.21720/) (2024)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [Membership Inference Attacks against Diffusion Models](https://mlsanity.com/paper/2302.03262/) (2023)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [A Reproducible Extraction of Training Images from Diffusion Models](https://mlsanity.com/paper/2305.08694/) (2023)
- [Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later](https://mlsanity.com/paper/2303.02490/) (2023)
- [Towards More Realistic Membership Inference Attacks on Large Diffusion Models](https://mlsanity.com/paper/2306.12983/) (2023)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [Membership Inference on Text-to-Image Diffusion Models via Conditional Likelihood Discrepancy](https://mlsanity.com/paper/2405.14800/) (2024)
- [Finding NeMo: Localizing Neurons Responsible For Memorization in Diffusion Models](https://mlsanity.com/paper/2406.02366/) (2024)
- [Memorization to Generalization: Emergence of Diffusion Models from Associative Memory](https://mlsanity.com/paper/2505.21777/) (2025)
- [Classifier-Free Guidance inside the Attraction Basin May Cause Memorization](https://mlsanity.com/paper/2411.16738/) (2024)
- [Understanding and Mitigating Memorization in Generative Models via Sharpness of Probability Landscapes](https://mlsanity.com/paper/2412.04140/) (2024)
- [Exploring Local Memorization in Diffusion Models via Bright Ending Attention](https://mlsanity.com/paper/2410.21665/) (2024)
- [Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusion](https://mlsanity.com/paper/2502.21278/) (2025)

## Abstract (arXiv)

Diffusion models are typically viewed as stochastic processes that transform noise into data. We take a complementary perspective: a diffusion model defines a family of deterministic dynamical systems indexed by noise scale. At each fixed scale σ, we treat the denoiser as a self-map and study its dynamics. For an exact denoiser, fixed points correspond to critical points of the smoothed data density, while attractors correspond to its modes; as σ increases, sample-level modes merge into progressively coarser ones. This suggests a geometric view of memorization: examples that receive excess probability mass due to duplication or overfitting, as well as outliers, should remain distinguishable under stronger smoothing than ordinary examples. We quantify this persistence by the critical scale σc, the largest noise scale at which an example is retained by the fixed-scale dynamics. In conditional models, the same construction extends naturally to image--caption pairs. Experiments in controlled settings and on large-scale models show that σc tracks memorization arising from duplication, overfitting, and outliers, and identifies both memorized and partially memorized examples in Stable Diffusion. Moreover, σc yields interpretable measures of the image spatial distribution and caption dependence of memorization.