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

Amartya Mukherjee, Tristan Milne, Kry Yik-Chau Lui, Stephanie Hazlewood, Jun Liu

arXiv:2609.12113v1cs.LGmath.APmath.OCmath.PRstat.ML

TL;DR

The paper addresses the lack of explicit likelihood control in outlier generation by defining outliers through distributions of log-likelihood values. It derives an RN-based modification of the diffusion score and uses it for controlled sampling; experiments report low-likelihood generation that remains consistent with data geometry.

Abstract

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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.

I. BACKGROUND

Diffusion models learn a score to reverse noise into data, while their distributions evolve under the Fokker-Planck-Kolmogorov equation. The Ornstein-Uhlenbeck dynamics and operator provide the diffusion framework used in the paper’s analysis.

A. Score-Based Diffusion Models

  • Diffusion models generate data by reversing a stochastic process from data to noise, using a learned score that approximates the log-density gradient.Sampling can use the reverse-time SDE or the probability-flow ODE.Source: A. Score-Based Diffusion Models · p. 2

B. Fokker-Planck-Kolmogorov Equation

C. Ornstein-Uhlenbeck Operator

  • The forward diffusion is an Ornstein-Uhlenbeck process, whose generator is self-adjoint with non-positive integer spectrum and Hermite-polynomial eigenfunctions.Source: C. Ornstein-Uhlenbeck Operator · p. 2

II. CONTROLLED LIKELIHOOD GENERATION

The paper frames outliers as a distribution-level likelihood-control problem, rather than as a search for individual unusual samples. It formalizes the target through stochastic ordering and derives a score-based control for diffusion sampling.

  • Because diffusion models learn probability measures rather than individual samples, principled outlier generation requires defining outliers at the distribution level.Source: §II · p. 2
  • The framework steers the distribution of log-likelihood values toward lower values, using stochastic ordering to express greater target mass in low-likelihood regions.Source: §II · p. 2
  • Likelihood reweighting induces an RN-derivative control that multiplicatively modifies the score and motivates a controlled probability-flow ODE.Source: §II · p. 2

A. Problem Formulation and Definition of Outlier

The paper represents a data distribution by the induced distribution of its log-likelihood values, then defines structured outliers by reweighting that distribution. The outlier condition requires a downward likelihood shift while retaining the data’s conditional structure at each likelihood level.

B. Radon-Nikodym Structure of Likelihood Reweighting

Likelihood reweighting yields a pointwise score relation in which the reference diffusion score is modified by a likelihood-dependent term derived from the Radon-Nikodym structure. This lets the method control likelihood without training a separate score function.

  • The target likelihood family may interpolate toward the reference likelihood distribution so that the reweighted measure at terminal time matches the reference measure.The passage notes that this terminal reference measure approaches the standard Gaussian for sufficiently large time.Source: B. Radon-Nikodym Structure of… · p. 4
  • The score identity holds pointwise in time under the stated smoothness assumptions on the reference density and likelihood-ratio function.Source: B. Radon-Nikodym Structure of… · p. 4
  • Likelihood reweighting induces a multiplicative modification of the diffusion score, with the correction determined by the likelihood distributions’ Radon-Nikodym structure.Source: B. Radon-Nikodym Structure of… · pp. 3–4
  • The modified score is expressible using the original diffusion score and a scalar likelihood-dependent correction, so likelihood control requires no new score-function training.Source: B. Radon-Nikodym Structure of… · p. 4

III. IMPLEMENTATION

The implementation motivates a controlled reverse-time ODE from the likelihood-reweighted score relation. Its OU analysis links exponential convergence of the likelihood distribution to an exponentially decaying controller.

III. IMPLEMENTATION

  • The implementation introduces a controlled reverse-time ODE ansatz motivated by the score relation.Source: §III · p. 4
  • The assumptions require μ_t ≪ γ and define h_t(x) as the Radon–Nikodym derivative; they also require some m > 0 with h_t(x) ≥ m for μ_t-almost all x and t.Source: §III · p. 4
  • The OU semigroup represents g_t as P_tg_0, enabling analysis through its action on the Hermite basis.The section identifies g_t with the unique semigroup solution and notes that P_t acts diagonally on that basis.Source: §III · p. 4
  • The likelihood distribution L_t converges exponentially toward the Gaussian likelihood equilibrium, and the slowest nonconstant OU rate e^-t motivates the controller.The section chooses a target family that vanishes toward the same equilibrium.Source: §III · p. 5

A. Approximating a control function

The section approximates the time-dependent control with an exponentially interpolated controller, motivated by OU convergence. Under stated assumptions, it establishes endpoint conditions and bounds on the approximation error.

A. Approximating a control function

  • The true control is computationally intractable, so the section uses an exponentially interpolated controller motivated by the OU semigroup's slowest nonconstant decay rate e^-t.The approximation is stated to have a reasonable error bound over small and large t and to respect boundary conditions.Source: A. Approximating a control function · p. 5
  • The controller's guarantees assume absolutely continuous likelihood measures converging exponentially to Gaussian likelihood equilibrium, along with the listed ordering and regularity conditions.The stated conditions include η0 ≪ L0, η0 ≤st L0, W1(η0,L0) ≥ ρ, and assumptions A1 and A2.Source: A. Approximating a control function · p. 5
  • The exponentially interpolated controller satisfies the boundary condition at t=0 and its error converges exponentially to zero in L2(γ).Source: A. Approximating a control function · p. 5
  • The true-control and interpolated-control terms are bounded in error for small and large t under the assumptions of Theorems 4 and 5.Source: A. Approximating a control function · p. 6

B. Approximating the target Radon-Nikodym derivative

The section uses a chi-square approximation for L0 to make controller coefficients computable. It introduces a linear map and describes constant and exponentially interpolated controls with a normalized proxy log-RN derivative.

B. Approximating the target Radon-Nikodym derivative

  • A chi-square approximation for L0 yields closed-form control coefficients, and the section states that the approximation is verified in numerical experiments.The linear map is defined as La,C(x) = -ax+C.Source: B. Approximating the target… · p. 6
  • The chi-square construction gives a constant control and a proxy log-RN derivative under the exponentially decaying control c~_t(u) = c0e^-t.Source: B. Approximating the target… · p. 6
  • The normalization condition on β(t) ensures smooth interpolation between z_T ≈ 1 and z_0, while linear dependence on u simplifies implementation and analysis.Source: B. Approximating the target… · p. 6

A. Mixture of Two Gaussians

The Gaussian-mixture experiments assess a chi-square approximation to the likelihood distribution and test whether controlled PF-ODE sampling can target lower-likelihood regions. Generated likelihood shifts closely match prescribed ρ values while satisfying stochastic dominance.

A. Mixture of Two Gaussians

  • The closed-form mixture score permits direct solution of the PF-ODE and log-FPK equation without training a neural network.The experiments use m = ˆm1_n and aim to generate ρ-outliers.Source: A. Mixture of Two Gaussians · p. 6
  • Across several dimensions and values of ˆm, the chi-square model provides a good approximation of the log-likelihood distribution, assessed using empirical W1 distances from 10,000 sampled points.An example empirical CDF is shown in Figure 2.Source: A. Mixture of Two Gaussians · p. 6
  • Controlled PF-ODE samples concentrate in low-likelihood tails and the low-density region between mixture modes.Figure 1 compares controlled and uncontrolled samples in the one-dimensional example at ρ = 0.5.Source: A. Mixture of Two Gaussians · pp. 6–7, Figure 1 · p. 7
  • Generated and reference likelihood distributions have empirical W1 distances that closely match prescribed ρ values, with stochastic dominance satisfied in every experiment.The table varies dimension n across {1,4,16,64,256} and target ρ across {0.2,0.3,0.4,0.5}.Source: Table I · p. 6, Table II · p. 7

B. Image Data: CIFAR-10

On CIFAR-10, the controller shifts generated likelihood distributions progressively lower as ρ increases, although realized shifts do not exactly match prescribed values. At moderate strengths, samples remain visually consistent with CIFAR-10.

B. Image Data: CIFAR-10

  • The evaluation uses the pretrained google/ddpm-cifar10-32 diffusion model and approximates L0 from reverse PF-ODE samples with the log-FPK equation.Source: B. Image Data: CIFAR-10 · p. 7
  • Realized W1 shifts do not exactly match prescribed ρ values, but increase monotonically with ρ on CIFAR-10.Table III reports empirical W1 distances between generated and reference likelihood distributions.Source: B. Image Data: CIFAR-10 · p. 7, Table III · p. 7
  • Empirical CDFs show that increasing ρ progressively shifts the generated likelihood distribution toward lower values.The figure compares likelihood distributions for different ρ values.Source: B. Image Data: CIFAR-10 · p. 7, Figure 3 · p. 7
  • At moderate control strengths, generated samples remain visually consistent with CIFAR-10.The reported visual assessment concerns samples in Figures 4a and 4b.Source: B. Image Data: CIFAR-10 · p. 7

V. CONCLUSION

The conclusion frames outliers through distributions of log-likelihood values and uses likelihood reweighting to motivate controlled reverse-time diffusion. Experiments support steering toward lower-probability regions while preserving data structure.

V. CONCLUSION

  • The paper formalizes ρ-outliers through discrepancies between likelihood distributions and modifies reverse-time diffusion using a score adjustment induced by the Radon–Nikodym derivative.The method uses this relationship to motivate a controlled PF-ODE.Source: §V · p. 7
  • Numerical experiments on synthetic and image datasets support the theoretical predictions of the framework.Source: §V · p. 7
  • Generated CIFAR-10 ρ-outliers have lower likelihood values while remaining visually consistent with the underlying data distribution.Source: Figure 4 · p. 8

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