# When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets

arXiv: https://arxiv.org/abs/2506.19031 · HTML: https://mlsanity.com/paper/2506.19031/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 32 of its 44 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Placed in

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

## Builds on (most cited first, 30 of 32)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [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)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Neural Networks are Convex Regularizers: Exact Polynomial-time Convex Optimization Formulations for Two-layer Networks](https://mlsanity.com/paper/2002.10553/) (2020)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Overparameterized Neural Networks Implement Associative Memory](https://mlsanity.com/paper/1909.12362/) (2019)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions](https://mlsanity.com/paper/2209.15055/) (2022)
- [Memorization to Generalization: Emergence of Diffusion Models from Associative Memory](https://mlsanity.com/paper/2505.21777/) (2025)
- [Vector-output ReLU Neural Network Problems are Copositive Programs: Convex Analysis of Two Layer Networks and Polynomial-time Algorithms](https://mlsanity.com/paper/2012.13329/) (2020)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Feature Learning in L2-regularized DNNs: Attraction/Repulsion and Sparsity](https://mlsanity.com/paper/2205.15809/) (2022)
- [Bottleneck Structure in Learned Features: Low-Dimension vs Regularity Tradeoff](https://mlsanity.com/paper/2305.19008/) (2023)
- [On Investigating the Conservative Property of Score-Based Generative Models](https://mlsanity.com/paper/2209.12753/) (2022)
- [The Convex Geometry of Backpropagation: Neural Network Gradient Flows Converge to Extreme Points of the Dual Convex Program](https://mlsanity.com/paper/2110.06488/) (2021)
- [Associative Memory in Iterated Overparameterized Sigmoid Autoencoders](https://mlsanity.com/paper/2006.16540/) (2020)
- [How do Minimum-Norm Shallow Denoisers Look in Function Space?](https://mlsanity.com/paper/2311.06748/) (2023)

## Built on by (most cited first, 6 of 6)

- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [A Survey of Inductive Reasoning for Large Language Models](https://mlsanity.com/paper/2510.10182/) (2025)
- [Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data](https://mlsanity.com/paper/2603.22962/) (2026)
- [On the Memorization of Consistency Distillation for Diffusion Models](https://mlsanity.com/paper/2604.23552/) (2026)
- [Diffusion Models, Denoiser Architecture and Creativity](https://mlsanity.com/paper/2605.16415/) (2026)
- [Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine](https://mlsanity.com/paper/2605.20235/) (2026)

## Abstract (arXiv)

While diffusion models generate high-quality images via probability flow, the theoretical understanding of this process remains incomplete. A key question is when probability flow converges to training samples or more general points on the data manifold. We analyze this by studying the probability flow of shallow ReLU neural network denoisers trained with minimal ℓ2 norm. For intuition, we introduce a simpler score flow and show that for orthogonal datasets, both flows follow similar trajectories, converging to a training point or a sum of training points. However, early stopping by the diffusion time scheduler allows probability flow to reach more general manifold points. This reflects the tendency of diffusion models to both memorize training samples and generate novel points that combine aspects of multiple samples, motivating our study of such behavior in simplified settings. We extend these results to obtuse simplex data and, through simulations in the orthogonal case, confirm that probability flow converges to a training point, a sum of training points, or a manifold point. Moreover, memorization decreases when the number of training samples grows, as fewer samples accumulate near training points.