# A Theoretical Analysis of Memory and Overfitting Phenomena in Stochastic Interpolation Models

arXiv: https://arxiv.org/abs/2606.08554 · HTML: https://mlsanity.com/paper/2606.08554/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 28 references that are in the map.
- First posted on arXiv 7 Jun 2026.

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scaling Diffusion Language Models via Adaptation from Autoregressive Models](https://mlsanity.com/paper/2410.17891/) (2024)
- [Understanding and Mitigating Copying in Diffusion Models](https://mlsanity.com/paper/2305.20086/) (2023)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [On the Edge of Memorization in Diffusion Models](https://mlsanity.com/paper/2508.17689/) (2025)
- [How Do Flow Matching Models Memorize and Generalize in Sample Data Subspaces?](https://mlsanity.com/paper/2410.23594/) (2024)
- [A Closer Look at Model Collapse: From a Generalization-to-Memorization Perspective](https://mlsanity.com/paper/2509.16499/) (2025)
- [Provable Separations between Memorization and Generalization in Diffusion Models](https://mlsanity.com/paper/2511.03202/) (2025)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [Smoothing the Score Function to Enhance Generalization in Diffusion Models](https://mlsanity.com/paper/2601.19285/) (2026)
- [Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models](https://mlsanity.com/paper/2602.17846/) (2026)
- [A Novel Metric for Detecting Memorization in Generative Models for Brain MRI Synthesis](https://mlsanity.com/paper/2509.16582/) (2025)
- [Diffusion Models Memorize in Training -- and Generalize in Inference](https://mlsanity.com/paper/2603.13419/) (2026)
- [Manifold Generalization Provably Proceeds Memorization in Diffusion Models](https://mlsanity.com/paper/2603.23792/) (2026)
- [Memorization Control in Diffusion Models from Denoising-centric Perspective](https://mlsanity.com/paper/2601.21348/) (2026)

## Abstract (arXiv)

This paper provides a theoretical account of memorization in stochastic interpolation models. By leveraging closed-form expressions for the optimal velocity field and the associated score function, we show that, in the continuous-time oracle setting, both deterministic and stochastic generation processes recover training samples. Under Euler discretization, generated samples remain centered around training samples, with deviations controlled by the step size. We further analyze generation in the presence of estimation errors and show that accumulated estimation errors control the endpoint deviation from the training set. These results imply that the generated sample admits a representation as a training sample perturbed by three controlled terms: a discretization-induced bound, an estimation-error-induced bound, and stochastic Gaussian noise. Based on this characterization, we provide theoretical definitions of overfitting and underfitting in generative models. Synthetic simulations support our theoretical findings.