# Local Diffusion Models and Phases of Data Distributions

arXiv: https://arxiv.org/abs/2508.06614 · HTML: https://mlsanity.com/paper/2508.06614/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 31 of its 81 references that are in the map.
- First posted on arXiv 8 Aug 2025.

## Placed in

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

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Patch Diffusion: Faster and More Data-Efficient Training of Diffusion Models](https://mlsanity.com/paper/2304.12526/) (2023)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [Understanding Hallucinations in Diffusion Models through Mode Interpolation](https://mlsanity.com/paper/2406.09358/) (2024)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Critical windows: non-asymptotic theory for feature emergence in diffusion models](https://mlsanity.com/paper/2403.01633/) (2024)
- [Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model](https://mlsanity.com/paper/2305.03486/) (2023)
- [Patched Denoising Diffusion Models For High-Resolution Image Synthesis](https://mlsanity.com/paper/2308.01316/) (2023)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [Towards a Mechanistic Explanation of Diffusion Model Generalization](https://mlsanity.com/paper/2411.19339/) (2024)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)
- [Probing the Latent Hierarchical Structure of Data via Diffusion Models](https://mlsanity.com/paper/2410.13770/) (2024)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Cascade of phase transitions in the training of Energy-based models](https://mlsanity.com/paper/2405.14689/) (2024)
- [Blink of an eye: a simple theory for feature localization in generative models](https://mlsanity.com/paper/2502.00921/) (2025)
- [A Benchmark Suite for Evaluating Neural Mutual Information Estimators on Unstructured Datasets](https://mlsanity.com/paper/2410.10924/) (2024)
- [Computational Implications of Reducing Data to Sufficient Statistics](https://mlsanity.com/paper/1409.3821/) (2014)

## Built on by (most cited first, 8 of 8)

- [Quantum Flow Matching](https://mlsanity.com/paper/2508.12413/) (2025)
- [An exact information theory of generalization phase transitions in Bayesian diffusion models](https://mlsanity.com/paper/2607.08041/) (2026)
- [Unlearnable phases of matter](https://mlsanity.com/paper/2602.11262/) (2026)
- [Learning and Generating Mixed States Prepared by Shallow Channel Circuits](https://mlsanity.com/paper/2604.01197/) (2026)
- [Concurrence of Symmetry Breaking and Nonlocality Phase Transitions in Diffusion Models](https://mlsanity.com/paper/2605.04830/) (2026)
- [Stability of mixed-state phases under weak decoherence](https://mlsanity.com/paper/2511.01976/) (2025)
- [Renormalization Group Flow Matching for Scalable Local Generative Modeling](https://mlsanity.com/paper/2608.23696/) (2026)
- [Breakdown of Local Denoising as Semantic Speciation](https://mlsanity.com/paper/2609.38176/) (2026)

## Abstract (arXiv)

As a class of generative artificial intelligence frameworks inspired by statistical physics, diffusion models have shown extraordinary performance in synthesizing complicated data distributions through a denoising process gradually guided by score functions. Real-life data, like images, is often spatially structured in low-dimensional spaces. However, ordinary diffusion models ignore this local structure and learn spatially global score functions, which are often computationally expensive. In this work, motivated by recent advances in non-equilibrium statistical physics, we develop a generic framework for defining phases of data distributions and use it to analyze the locality requirements of denoisers in diffusion models. We define two distributions as belonging to the same data distribution phase if they can be mutually connected via spatially local operations such as local denoisers, along the same evolution path as the diffusion. We demonstrate that the reverse denoising process consists of an early trivial phase and a late data phase, sandwiching a rapid phase transition where local denoisers must fail. We further demonstrate that the performance of local denoisers is closely tied to spatial Markovianity, which provides an operational criterion for diagnosing such phase transitions. We validate this criterion through numerical experiments on real-world datasets. Our work suggests guidance for simpler and more efficient architectures of diffusion models: far from the phase transition point, we can use small local neural networks to compute the score function; global neural networks are only necessary around the narrow time interval of phase transitions. This result also opens up new directions for studying phases of data distributions, the broader science of generative artificial intelligence, and guiding the design of neural networks inspired by physics concepts.