Research map / Probabilistic and scientific modeling / Uncertainty and Gaussian processes
Combining Statistical Depth and Fermat Distance for Uncertainty Quantification
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Proposes a non-intrusive, parameter-free OOD uncertainty score that applies Lens Depth with a modified sample Fermat distance directly in a trained network's feature space.
Problem
Out-of-distribution uncertainty quantification for neural network classifiers without retraining, without distributional assumptions (Gaussian fitting fails on non-Gaussian clusters), and without impacting original model accuracy.
Approach
Compute statistical Lens Depth in the penultimate feature space using a modified sample Fermat distance (adds distance from test point to a free starting point in the dataset so depth vanishes far from data), and score each point by max LD over per-class clusters.
Key results
- FashionMNIST vs MNIST: LD achieves AUROC 0.971 ± 0.001, above Euclidean (0.943), Mahalanobis (0.942), DUQ (0.955), and Deep Ensembles (0.861), but below LL ratio (0.994).
- CIFAR10 vs SVHN: LD achieves AUROC 0.936 ± 0.006, above DUQ (0.927), Deep Ensembles 3/5 models (0.926/0.933), LL ratio (0.930), Energy (0.912), and KNN (0.926).
- Consistency curves show accuracy on retained samples increases with rejection percentage across 5 runs on both benchmark pairs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lens Depth instead of Euclidean Distance | OOD detection · FashionMNIST vs MNIST | AUROC: 0.943 \scriptstyle \pm 0.009 -> 0.971 \scriptstyle \pm 0.001 | Helped |
| Use Lens Depth instead of Mahalanobis Distance | OOD detection · FashionMNIST vs MNIST | AUROC: 0.942 -> 0.971 \scriptstyle \pm 0.001 | Helped |
| Use Lens Depth instead of DUQ | OOD detection · FashionMNIST vs MNIST | AUROC: 0.955 -> 0.971 \scriptstyle \pm 0.001 | Helped |
| Use Lens Depth instead of Deep Ensembles (5 models) | OOD detection · FashionMNIST vs MNIST | AUROC: 0.861 -> 0.971 \scriptstyle \pm 0.001 | Helped |
| Use LL ratio instead of Lens Depth | OOD detection · FashionMNIST vs MNIST | AUROC: 0.971 \scriptstyle \pm 0.001 -> 0.994 | Hurt |
| Use Lens Depth instead of DUQ | OOD detection · CIFAR10 vs SVHN | AUROC: 0.927 \scriptstyle \pm 0.013 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Use Lens Depth instead of Deep Ensembles (3 models) | OOD detection · CIFAR10 vs SVHN | AUROC: 0.926 \scriptstyle \pm 0.010 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Use Lens Depth instead of Deep Ensembles (5 models) | OOD detection · CIFAR10 vs SVHN | AUROC: 0.933 \scriptstyle \pm 0.008 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Use Lens Depth instead of LL ratio | OOD detection · CIFAR10 vs SVHN | AUROC: 0.930 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Use Lens Depth instead of Energy | OOD detection · CIFAR10 vs SVHN | AUROC: 0.912 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Use Lens Depth instead of KNN | OOD detection · CIFAR10 vs SVHN | AUROC: 0.926 \scriptstyle \pm 0.002 -> \mathbf{0.936} \scriptstyle \pm 0.006 | Helped |
| Select inner points using K-means centroids rather than random sampling (ablation) | Reduced Lens Depth · FashionMNIST vs MNIST, 500 examples/class | AUROC: 0.9368 -> 0.9543 | Helped |
Surprising findings
- Euclidean distance to class centroids captured the two-moons feature distribution better than Mahalanobis distance, suggesting clusters in feature space are non-Gaussian.
- On FashionMNIST vs MNIST, the LL ratio baseline (AUROC 0.994) outperforms the proposed method, which the authors attribute to its extra generative models.
Limitations
- Computational cost of LD scales with O(C N^2) per point, requiring subsampling of training points.
- Hyperparameter alpha must be large enough; stability only holds in a proper range.
- FashionMNIST vs MNIST result trails LL ratio baseline.
Experimental setup
Datasets: FashionMNIST MNIST CIFAR10 SVHN two-moons spiral (simulated)
Benchmarks: FashionMNIST vs MNIST CIFAR10 vs SVHN two-moons qualitative
Metrics: AUROC consistency curve accuracy vs rejection rate
Baselines: Euclidean distance Mahalanobis distance DUQ Deep Ensembles LL ratio Energy KNN
Builds on: lens depth sample Fermat distance Deep Ensembles LL ratio Energy-based OOD deep k-nearest neighbors
Closest papers by meaning
- Depth Uncertainty in Neural Networks (NeurIPS 2020 · similarity 0.90)
- Uncertainty-Aware CNNs for Depth Completion: Uncertainty from Beginning to End (CVPR 2020 · similarity 0.88)
- Uncertainty Quantification with the Empirical Neural Tangent Kernel (NeurIPS 2025 · similarity 0.85)
- SURE: SUrvey REcipes for building reliable and robust deep networks (CVPR 2024 · similarity 0.85)
- One Step Closer to Unbiased Aleatoric Uncertainty Estimation (AAAI 2024 · similarity 0.85)
- A Rate-Distortion View of Uncertainty Quantification (ICML 2024 · similarity 0.85)
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