# Score-based Outlier Generation via Controlling the Radon-Nikodym Derivative

arXiv: https://arxiv.org/abs/2609.12113 · HTML: https://mlsanity.com/paper/2609.12113/

## 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 14 of its 28 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Placed in

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

## Builds on (most cited first, 14 of 14)

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasets](https://mlsanity.com/paper/2311.15127/) (2023)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Fast High-Resolution Image Synthesis with Latent Adversarial Diffusion Distillation](https://mlsanity.com/paper/2403.12015/) (2024)
- [Non-Parametric Outlier Synthesis](https://mlsanity.com/paper/2303.02966/) (2023)
- [Dream the Impossible: Outlier Imagination with Diffusion Models](https://mlsanity.com/paper/2309.13415/) (2023)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation](https://mlsanity.com/paper/2210.04296/) (2022)
- [Density control of large-scale particles swarm through PDE-constrained optimization](https://mlsanity.com/paper/2104.06373/) (2021)
- [Beyond the Norm: A Survey of Synthetic Data Generation for Rare Events](https://mlsanity.com/paper/2506.06380/) (2025)
- [Synthetic Data from Diffusion Models Improve Drug Discovery Prediction](https://mlsanity.com/paper/2405.03799/) (2024)

## Abstract (arXiv)

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.