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A Geometric Explanation of the Likelihood OOD Detection Paradox
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Explains the likelihood OOD detection paradox geometrically via local intrinsic dimension (LID) and proposes a dual-threshold detector that pairs likelihood with LID estimated from a single pre-trained normalizing flow or diffusion model.
Problem
Likelihood-based deep generative models assign higher likelihood to OOD data from simpler datasets yet never generate such samples, making likelihood-based OOD detection unreliable.
Approach
High likelihood on low-dimensional OOD manifolds can coexist with negligible probability mass because the model assigns small local volume there; estimating LID from a single pre-trained DGM and thresholding both likelihood and LID (dual threshold) identifies such regions.
Key results
- NF dual threshold achieves AUC-ROC 0.951 on FMNIST vs MNIST and 0.936 on CIFAR10 vs SVHN, versus 0.073 and 0.063 for NF likelihood alone.
- NF dual threshold outperforms all NF baselines on most pathological tasks, e.g. CIFAR10 vs CelebA 0.655 vs 0.479 for complexity correction.
- DM dual threshold matches the strongest DM baseline (reconstruction) and beats it on five of eight tasks, while being fully unsupervised; it reaches 0.944 on CIFAR10 vs SVHN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Combine likelihood and LID thresholds instead of using likelihood alone | Normalizing flow · FMNIST-trained: MNIST OOD | AUC-ROC: 0.073 -> 0.951 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Normalizing flow · CIFAR10-trained: SVHN OOD | AUC-ROC: 0.063 -> 0.936 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Normalizing flow · CIFAR10-trained: CelebA OOD | AUC-ROC: 0.391 -> 0.655 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Normalizing flow · MNIST-trained: Omniglot OOD | AUC-ROC: 0.796 -> 0.855 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Diffusion model · FMNIST-trained: MNIST OOD | AUC-ROC: 0.240 -> 0.912 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Diffusion model · CIFAR10-trained: SVHN OOD | AUC-ROC: 0.064 -> 0.944 | Helped |
| Combine likelihood and LID thresholds instead of using likelihood alone | Diffusion model · CIFAR10-trained: CelebA OOD | AUC-ROC: 0.360 -> 0.648 | Helped |
| Use LPCA-calibrated dataset-specific threshold instead of fixed τ (ablation) | Normalizing flow dual-threshold detector · CIFAR10 vs. SVHN | AUC-ROC: 0.737 -> 0.936 | Helped |
| Use LPCA-calibrated dataset-specific threshold instead of fixed τ (ablation) | Normalizing flow dual-threshold detector · FMNIST vs. MNIST | AUC-ROC: 0.961 -> 0.951 | Hurt |
Surprising findings
- Likelihoods of generated samples are often lower than those of OOD data, a new pathology not previously reported.
- Complexity correction and likelihood-ratio baselines degrade on non-pathological tasks (e.g. MNIST and SVHN trained models), suggesting they overfit to pathological tasks.
Limitations
- Method only applies to normalizing flows and diffusion models because LID is tractable only for these models.
- LID estimator for diffusion models yields very different estimates on image datasets than established estimators; performance depends on threshold tau set via LPCA calibration.
- Dual threshold requires choosing thresholds; no single fixed tau works across all datasets.
Experimental setup
Datasets: FMNIST MNIST Omniglot EMNIST SVHN CIFAR10 CIFAR100 Tiny ImageNet CelebA
Benchmarks: FMNIST vs MNIST FMNIST vs Omniglot CIFAR10 vs SVHN CIFAR10 vs CelebA MNIST vs FMNIST MNIST vs Omniglot SVHN vs CIFAR10 SVHN vs CelebA
Metrics: AUC-ROC
Baselines: NF Likelihood Complexity Correction NF Likelihood Ratios DM Likelihood DM Reconstruction DM Likelihood Ratios Normalized autoencoders (NAE)
Builds on: Glow RQ-NSF Denoising score matching Score-based SDE Local PCA intrinsic dimension estimation
Closest papers by meaning
- Understanding Likelihood of Normalizing Flow and Image Complexity through the Lens of Out-of-Distribution Detection (AAAI 2024 · similarity 0.90)
- Forte : Finding Outliers with Representation Typicality Estimation (ICLR 2025 · similarity 0.89)
- Detecting Out-of-Distribution Through the Lens of Neural Collapse (CVPR 2025 · similarity 0.87)
- Double Descent Meets Out-of-Distribution Detection: Theoretical Insights and Empirical Analysis on the Role of Model Complexity (NeurIPS 2025 · similarity 0.87)
- The Best of Both Worlds: On the Dilemma of Out-of-distribution Detection (NeurIPS 2024 · similarity 0.87)
- Out-of-Distribution Detection with Prototypical Outlier Proxy (AAAI 2025 · similarity 0.86)
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