Score-based Outlier Generation via Controlling the Radon-Nikodym Derivative
Abstract
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.
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BibTeX
@misc{mukherjee2026scorebased,
title = {{Score-based Outlier Generation via Controlling the Radon-Nikodym Derivative}},
author = {Amartya Mukherjee and Tristan Milne and Kry Yik-Chau Lui and Stephanie Hazlewood and Jun Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12113},
eprint = {2609.12113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12113},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Score-based Outlier Generation via Controlling the Radon-Nikodym Derivative
AU - Amartya Mukherjee
AU - Tristan Milne
AU - Kry Yik-Chau Lui
AU - Stephanie Hazlewood
AU - Jun Liu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12113
UR - https://arxiv.org/abs/2609.12113
AB - 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.
ER -