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

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 develop a method for generating outliers by modifying the reverse-time dynamics of diffusion models through likelihood reweighting. The resulting distributional change can be implemented by scaling the score function with a control term derived from the Radon-Nikodym derivative, requiring 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.