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A Probabilistic Perspective on Wasserstein-Based Evidential Uncertainty for Out-of-Distribution Segmentation

Arnold Brosch, Abdelrahman Eldesokey, Michael Felsberg, Kira Maag

arXiv:2610.10116v1cs.CV

TL;DR

OOD segmentation needs reliable identification of unfamiliar regions, yet softmax confidence can remain overconfident and Bayesian alternatives require repeated passes. This paper uses Wasserstein-based evidential learning to study objective orders across convolutional and transformer architectures; the preferred order differs by backbone, and the method outperforms evaluated baselines on most comparisons.

  • Problem

    OOD segmentation needs reliable identification of regions outside the training distribution, but softmax uncertainty can remain overconfident and Bayesian approaches require costly repeated forward passes.

    Source: §1 · pp. 2–3

  • Method

    The paper studies W1, W2, and W3 objectives within evidential learning for semantic segmentation across convolutional and transformer architectures.

    Source: §1 · p. 3

  • Results

    W2 achieves the strongest overall detection on three of four benchmarks for DeepLabV3+, while adapted SegFormer leads on RoadObstacle21, RoadAnomaly21, and Fishyscapes; DeepLabV3+ remains stronger on LostAndFound.

    Source: §5 · p. 12

  • Takeaways & Limitations

    Evidence activation and transport order should be selected jointly with the architecture.

    Source: §5 · p. 12

    The analysis uses a Dirac transport cost that assigns equal cost to all class confusions and considers only integer Wasserstein orders.

    Source: §5 · p. 12

Abstract

from arXiv · show

Semantic segmentation networks operate on a fixed set of classes and therefore fail when out-of-distribution (OOD) objects appear during deployment, a critical limitation for safety-critical applications such as autonomous driving. Reliably identifying OOD objects requires well-calibrated epistemic uncertainty, yet common softmax-based confidence scores remain overconfident, while Bayesian alternatives such as Monte Carlo dropout or deep ensembles require costly repeated forward passes. Evidential Deep Learning (EDL) offers an efficient alternative by modeling class probabilities as a Dirichlet distribution learned from a single deterministic forward pass. Existing EDL formulations rely on Euclidean objectives that push predictions towards the simplex vertices, encouraging overconfidence rather than preserving uncertainty for unfamiliar inputs. We instead employ Wasserstein-based objectives, which respect the geometry of the probability simplex, and study the influence of the Wasserstein order on segmentation accuracy and OOD detection within a unified evidential framework. We evaluate this framework on a convolutional (DeepLabV3+) and a transformer-based (SegFormer) architecture on the SegmentMeIfYouCan benchmark, including LostAndFound, RoadObstacle21, RoadAnomaly21, and Fishyscapes. Our results show the optimal Wasserstein order is architecture-dependent: second-order objectives dominate on the convolutional backbone, third-order objectives on the transformer backbone, and our framework surpasses comparable baselines on most metrics, with a single deterministic forward pass.

1. Introduction

Semantic segmentation must identify regions outside its training distribution, but common confidence estimates can remain overconfident on unfamiliar inputs, while Bayesian alternatives require repeated forward passes. This work studies Wasserstein-based evidential learning across objective orders, architectures, and OOD benchmarks.

1. Introduction

  • OOD segmentation must identify regions outside the training distribution, where softmax-derived uncertainty can remain overconfident and Bayesian estimates can require costly repeated passes.This matters in safety-critical settings such as autonomous driving, where unknown objects may be misclassified and affect downstream decisions.Source: §1 · pp. 2–3
  • EDL represents class probabilities with a Dirichlet distribution and derives uncertainty from accumulated evidence in a single deterministic forward pass.The paper contrasts this with Euclidean objectives that may encourage overconfident predictions and investigates Wasserstein objectives that respect probability-simplex geometry.Source: §1 · p. 3
  • The study compares first-, second-, and third-order Wasserstein objectives for segmentation performance, uncertainty calibration, and OOD segmentation capability.It evaluates both DeepLabV3+ and SegFormer to examine how the objective interacts with convolutional and transformer architectures.Source: §1 · p. 3
  • Experiments cover LostAndFound, RoadObstacle21, RoadAnomaly21, and Fishyscapes, analyzing trade-offs between in-distribution accuracy and OOD awareness.The authors describe the benchmarks as the complete SegmentMeIfYouCan benchmark.Source: §1 · p. 3

2. Related Work

Prior OOD-segmentation work includes uncertainty-, feature-space-, auxiliary-model-, and additional-data-based approaches. This work positions evidential Wasserstein segmentation as a deterministic uncertainty-based method and extends an earlier first-order study to multiple orders, architectures, and benchmarks.

2. Related Work

  • OOD-segmentation methods include uncertainty-based, feature-space, auxiliary-model, and additional-training-data approaches.Examples of auxiliary-data methods use real OOD samples or synthetic negatives during training.Source: §2 · pp. 3–4
  • Uncertainty-based alternatives range from softmax scores, which can remain overconfident on unfamiliar inputs, to sampling methods that improve calibration at higher inference cost.EDL instead parameterizes a Dirichlet distribution and estimates epistemic uncertainty from evidence in one deterministic forward pass.Source: §2 · p. 4
  • The proposed method derives uncertainty from the segmentation network’s evidential output without auxiliary architectures, extra OOD training data, feature-space postprocessing, backward passes, or input perturbations.Its inference is computationally closest to output-based uncertainty baselines such as maximum softmax probability.Source: §2 · p. 4
  • Compared with the prior fixed first-order study, this work varies W1, W2, and W3, evaluates DeepLabV3+ and SegFormer, and adds Fishyscapes to the benchmark evaluation.The broader evaluation examines objective-order effects on the accuracy–OOD-awareness trade-off across architectures.Source: §2 · pp. 4–5

3. Method

The method combines evidential prediction with Wasserstein objectives to represent uncertainty for OOD segmentation. It varies the Wasserstein order and adds regularizers to balance uncertainty calibration, segmentation structure, and optimization stability.

Evidential Deep Learning

  • EDL represents class beliefs with a Dirichlet distribution, using total evidence to distinguish confident predictions from uncertainty due to insufficient evidence.The method obtains non-negative evidence from logits and uses it to parameterize the Dirichlet distribution.Source: §3 · p. 6

Wasserstein-based Loss

  • Unlike MSE objectives that push predictions toward simplex vertices, Wasserstein distances account for distribution geometry and can preserve uncertainty across classes when evidence is inconclusive.The paper motivates this choice for OOD detection, where preserving uncertainty is preferable to unsupported confident assignments.Source: §3 · pp. 6–7
  • Wasserstein distances frame OOD prediction as mass redistribution when test inputs extend beyond the training support, allowing insufficient evidence to remain uncertain rather than forcing assignment to a known class.The approach uses Wasserstein distance as a geometry-aware alternative to Euclidean objectives within evidential learning.Source: §3 · pp. 5–7

Wasserstein Order

  • The Wasserstein order controls the relative influence of transport costs: W1 weights discrepancies linearly, while W2 and W3 increasingly emphasize larger deviations.The study frames order selection as a trade-off between segmentation accuracy and OOD awareness.Source: §3 · p. 7

Regularization Terms; Loss Construction

  • Dice loss promotes spatial consistency and coherent object regions, while KL regularization encourages a uniform Dirichlet prior when evidence is insufficient.A small MSE contribution is retained to stabilize convergence and preserve sharp semantic boundaries; the combined objective includes Wasserstein, Dice, KL, and MSE terms.Source: §3 · p. 8

Loss Construction

  • The framework is evaluated on convolutional and transformer-based segmentation architectures to assess whether Wasserstein-order behavior generalizes across representation-learning paradigms.The architectures are convolutional and transformer-based, respectively.Source: §3 · p. 8

4. Experiments

Experiments across convolutional and transformer segmentation models show that the preferred Wasserstein order depends on architecture, and the selected method outperforms most uncertainty-based baselines on the benchmark. The results also reveal architecture-dependent trade-offs between OOD detection and in-distribution segmentation quality.

4.1. Experimental Setting

  • The evaluation trains on Cityscapes and tests on the four SegmentMeIfYouCan datasets, comparing DeepLabV3+ with SegFormer across pixel-level, segment-level, and in-distribution metrics.SegFormer results use Softplus instead of ReLU to produce non-zero evidence for pixels with negative logits.Source: §4.1 · pp. 8–9
  • KL regularization is annealed from zero to its target weight between iterations 40K and 48K to avoid suppressing evidence generation too early.The schedule lets the model first learn discriminative semantic features before shifting optimization toward uncertainty calibration and regularization.Source: §4.1 · p. 9
  • For SegFormer, Softplus replaces ReLU because approximately 12% of pixels have only negative logits and would otherwise receive zero evidence.The replacement ensures non-zero evidence while preserving the ordering of logits.Source: §4.1 · p. 9

4.2. Numerical Results

  • Combining Wasserstein and residual MSE generally outperforms either objective alone; intermediate Dice weighting helps W1, while annealed KL has a small, dataset-dependent OOD effect.For DeepLabV3+ W1 on LostAndFound, MSE weight 0.45 gives 53.2 AuPRC versus 47.5 for pure Wasserstein and 25.7 for pure MSE; Dice weight 0.75 reduces FPR95 from 23.2 to 17.6.Source: §4.2 · pp. 9–10
  • W2 is generally preferred for DeepLabV3+, while SegFormer favors W3, which achieves its best AuPRC on all four benchmarks and reaches 64.5 on RoadAnomaly21.DeepLabV3+ W2 reaches 57.9 AuPRC on RoadAnomaly21 versus 43.7 for its best W3 configuration; Fishyscapes is the exception to W2's lead on DeepLabV3+.Source: §4.2 · p. 10
  • On the benchmark tables, the fixed DeepLabV3+ W2 model beats most uncertainty baselines, including PGN on RoadAnomaly21 AuPRC (57.9 vs. 42.8), but trails gradient- and adversarial-based methods on RoadObstacle21 and Fishyscapes.The baseline comparison uses a fixed DeepLabV3+ configuration rather than a per-metric optimum.Source: Table 1 · p. 11, Table 2 · p. 11, §4.2 · p. 11
  • SegFormer mIoU stays within 80.54%–80.61% across orders, whereas DeepLabV3+ averages 56.14% for W2 and 67.67% for W1, with configuration spreads reaching 24.98 points.The DeepLabV3+ W2 order favored for OOD detection is accompanied by an 11 percentage point mIoU drop; SegFormer remains above 80.09% across configurations.Source: Table 3 · p. 12, §4.2 · p. 12

5. Conclusion

The paper concludes that Wasserstein-based evidential OOD segmentation benefits from architecture-specific choices of transport order and evidence activation. It reports stronger baseline comparisons on most obstacle-dataset comparisons and identifies limits of its transport cost and order analysis.

5. Conclusion

  • Wasserstein, Dice, and annealed KL regularization are complementary, and Dice provides the clearest reduction in FPR95.The framework uses a geometry-aware loss to address overconfidence encouraged by Euclidean MSE.Source: §5 · p. 12
  • DeepLabV3+ achieves its strongest overall detection with W2 on three of four benchmarks, while SegFormer leads on RoadObstacle21, RoadAnomaly21, and Fishyscapes with softplus evidence activation and W3.DeepLabV3+ remains stronger on LostAndFound.Source: §5 · p. 12
  • Across LostAndFound and RoadObstacle21, the method outperforms evaluated deterministic and sampling-based baselines on most comparisons, with particularly clear gains on RoadObstacle21.It uses a single deterministic forward pass, with no auxiliary models, gradient access, or additional OOD training data.Source: §5 · p. 12
  • The analysis uses integer Wasserstein orders and a Dirac transport cost that assigns equal cost to all class confusions.The authors identify learned semantic cost matrices and fractional orders as future directions.Source: §5 · p. 12

Funding

The funding statement names Swedish Research Council and Linköping University support and says funders had no role in the study.

Funding

  • M. Felsberg received partial support from the Swedish Research Council and Linköping University; the funders had no role in study design, data collection and analysis, manuscript preparation, or submission decisions.The support was through project 2022-04266 and the Center of Excellence AI4x, respectively.Source: Funding · p. 13

A. More Numerical Results

The appendix presents numerical results for parameter configurations in tables covering both DeepLabV3+ and SegFormer across the named OOD datasets.

A. More Numerical Results

  • The section states that its numerical results are presented in tabular form for various parameter configurations.Source: A. More Numerical Results · p. 13
  • Tables A.1–A.6 report DeepLabV3+ OOD segmentation results for LostAndFound, RoadObstacle21, Fishyscapes, and RoadAnomaly21, separated by Wasserstein order.Tables A.1–A.3 cover orders W1, W2, and W3 on LostAndFound and RoadObstacle21; Tables A.4–A.6 cover those orders on Fishyscapes and RoadAnomaly21.Source: Table A.1 · p. 14, Table A.2 · p. 14, Table A.3 · p. 14
  • Tables A.7–A.12 report SegFormer OOD segmentation results for LostAndFound, RoadObstacle21, Fishyscapes, and RoadAnomaly21, separated by Wasserstein order.Tables A.7–A.9 cover orders W1, W2, and W3 on LostAndFound and RoadObstacle21; Tables A.10–A.12 cover those orders on Fishyscapes and RoadAnomaly21.Source: Table A.7 · p. 16, Table A.8 · p. 16, Table A.9 · p. 16

CRediT authorship contribution statement

The statement assigns conceptualization across the listed authors, with other contributions spanning methods, implementation, analysis, writing, and supervision.

CRediT authorship contribution statement

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