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Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport
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Proposes U-NOTB, a continuous semi-unbalanced OT barycenter solver built on a semi-dual min-max formulation with neural maps and potentials, shown robust to outliers and class imbalance on toy and MNIST-style data.
Problem
Estimating OT barycenters robust to outliers and class imbalance in reference measures, in the continuous (out-of-sample) setting where barycenter is learned from samples.
Approach
Derives a semi-dual maximin form of the semi-unbalanced OT barycenter problem (softening only the source marginals with a psi-divergence), with m-congruent neural potentials and stochastic neural plans; new inputs are handled by rejection sampling using the learned density ratio, enabling robust continuous barycenters with general costs.
Key results
- On Spiral-to-Gaussian Mixture, U-NOTB achieves L2 0.25 and W2 0.12, better than UOTM (0.65, 0.29) and Mini-Batch UOT (0.31, 0.15).
- On Moon-to-Spiral, U-NOTB is best on L2 (0.52) but UOTM is best on W2 (0.59 vs 0.81 for U-NOTB).
- In the shape-color experiment, accepted probability for outlier digit '7' is 0.19 vs 0.63 for in-distribution digits '2' and '3'.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use U-NOTB instead of the strongest baseline, Mini-Batch UOT | Synthetic barycenter estimation · Spiral → Gaussian Mixture | L2: 0.31 -> 0.25 | Helped |
| Use U-NOTB instead of the strongest baseline, Mini-Batch UOT | Synthetic barycenter estimation · Spiral → Gaussian Mixture | W2: 0.15 -> 0.12 | Helped |
| Use U-NOTB instead of the strongest baseline, Mini-Batch UOT | Synthetic barycenter estimation · Moon → Spiral | L2: 0.64 -> 0.52 | Helped |
| Use U-NOTB instead of the strongest baseline, UOTM | Synthetic barycenter estimation · Moon → Spiral | W2: 0.59 -> 0.81 | Hurt |
| Lower the unbalancedness parameter τ | U-NOTB · Gaussian Mixture barycenter with class imbalance | Robustness to class imbalance | Helped |
| Increase the unbalancedness parameter τ | U-NOTB · Gaussian Mixture barycenter with class imbalance | Similarity to OT barycenter | Helped |
| Lower the unbalancedness parameter τ | U-NOTB · Gaussian Mixture barycenter with 5% outliers | Outlier robustness | Helped |
| Increase the unbalancedness parameter τ | U-NOTB · Gaussian Mixture barycenter with outliers | Influence of outliers on the barycenter | Hurt |
| Use shape-color costs with U-NOTB | U-NOTB · Shape-color experiment | Acceptance rate: 0.63 -> 0.19 | Hurt |
Surprising findings
- On Moon-to-Spiral, U-NOTB does not win on W2, where UOTM is best (0.59 vs 0.81).
Limitations
- Strong convexity assumptions used in quality bounds are not guaranteed in practice.
- No closed-form ground truth; discrete POT solution used as reference for toy experiments.
- Outlier mappings have no reasonable interpretation.
Experimental setup
Datasets: MNIST synthetic Spiral-to-Gaussian Mixture Moon-to-Spiral colored MNIST (digits 2 and 3) for StyleGAN pretraining
Benchmarks: Spiral-to-Gaussian Mixture Moon-to-Spiral Gaussian Mixture with outliers shape-color MNIST experiment
Metrics: W2 distance L2 transport map error acceptance rate
Baselines: UOTM Mini-Batch UOT NOTB
Builds on: Neural OT Energy-guided Neural OT unbalanced OT semi-dual OT barycenter congruence condition
Closest papers by meaning
- Estimating Barycenters of Distributions with Neural Optimal Transport (ICML 2024 · similarity 0.89)
- Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections (NeurIPS 2021 · similarity 0.88)
- Energy-Guided Continuous Entropic Barycenter Estimation for General Costs (NeurIPS 2024 · similarity 0.87)
- Outlier-Robust Distributionally Robust Optimization via Unbalanced Optimal Transport (NeurIPS 2024 · similarity 0.86)
- Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization (ICML 2025 · similarity 0.86)
- Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows (ICML 2025 · similarity 0.85)
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