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Towards Marginal Fairness Sliced Wasserstein Barycenter
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Defines a marginal-fairness sliced Wasserstein barycenter (MFSWB) as a constrained SWB problem, proposes three hyperparameter-free tractable surrogates (max-SW, expected max-1D-W2, and energy-based slicing), and tests them on Gaussians, point-cloud averaging, color harmonization, and fair SWAE training.
Problem
Sliced Wasserstein barycenters with uniform weights do not yield barycenters that are approximately equidistant from all marginals (marginal fairness); the formal constrained definition is computationally hard and lacks hyperparameter-free tractable alternatives.
Approach
Replace the constrained fairness objective with three surrogates: minimizing the maximum SW distance to marginals, minimizing the expected maximum one-dimensional W_p^p over projections (unbiased gradient), and reweighting projection directions with an energy-based slicing distribution that favors marginally unfair directions.
Key results
- On point-cloud averaging at iteration 10000, es-MFSWB reaches F=0.84 vs 1.55 for USWB, with the lowest W among methods (39.18 vs 39.81).
- In the SWAE experiment, s-MFSWB and es-MFSWB reduce the latent fairness F to 2.037 and 1.760 (x10^-2) versus 5.190 for USWB, though us-MFSWB and es-MFSWB do not uniformly win on the other metrics.
- On Gaussian barycenters, the formal MFSWB with lambda=1 gives the fairest result but is sensitive to lambda, while es-MFSWB gives the most fair surrogate result.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the es-MFSWB surrogate instead of USWB | barycenter optimization · ShapeNet Core-55 car point-cloud averaging, iteration 10000 | F: 1.55±0.48 -> 0.84±0.22 | Helped |
| Use the es-MFSWB surrogate instead of USWB | barycenter optimization · ShapeNet Core-55 car point-cloud averaging, iteration 10000 | W: 39.81±0.18 -> 39.18±0.08 | Helped |
| Use the es-MFSWB surrogate instead of USWB | SWAE class-fair representation · MNIST test | F × 10^2: 5.190 -> 1.760 | Helped |
| Use the us-MFSWB surrogate instead of USWB | SWAE class-fair representation · MNIST test | F_images: 7.140 -> 6.898 | Helped |
| Add a barycenter loss to conventional SWAE | SWAE class-fair representation · MNIST test | RL: 3.002 -> 3.195 | Hurt |
| Add a barycenter loss to conventional SWAE | SWAE class-fair representation · MNIST test | F × 10^2: 17.661 -> 5.190 | Helped |
| Use the es-MFSWB surrogate instead of USWB | SWAE class-fair representation · MNIST test | W₂,image² × 10²: 27.446 -> 28.012 | Hurt |
| Use fairness-energy slicing instead of uniform slicing | MFSWB surrogate barycenter optimization · 3D point-cloud averaging | F | Helped |
| Use fairness-energy slicing instead of uniform slicing | MFSWB surrogate barycenter optimization · color harmonization | F | Helped |
| Choose MFSWB λ=1 rather than λ=10 | formal MFSWB · ShapeNet Core-55 car point-cloud averaging, iteration 10000 | F: 4.13±2.48 -> 1.03±0.06 | Helped |
| Choose MFSWB λ=10 rather than λ=1 | formal MFSWB · ShapeNet Core-55 car point-cloud averaging, iteration 1000 | W: 79.08±0.15 -> 71.24±0.9 | Helped |
Surprising findings
- The formal constrained MFSWB performs poorly or inconsistently for some lambda values, e.g. lambda=10 gives high F and W in point-cloud averaging, so the surrogates are more practical.
- Surrogate MFSWB methods trade off image reconstruction and generative quality (higher W2 image) for latent class fairness in SWAE.
Limitations
- Marginal fairness barycenter may not exist for small epsilon, and the formal constrained problem is sensitive to the Lagrange multiplier lambda.
- The es-MFSWB gradient is only asymptotically unbiased due to importance sampling.
- The SMW with maximal ground metric is left for future investigation.
Experimental setup
Datasets: ShapeNet Core-55 MNIST
Benchmarks: ShapeNet Core-55 point-cloud averaging color harmonization images MNIST SWAE
Metrics: F-metric (marginal fairness) W-metric (centrality) reconstruction loss W2 latent W2 image F_images
Baselines: USWB formal MFSWB (dual form with lambda in {0.1, 1, 10}) SWAE
Builds on: sliced Wasserstein distance Fair PCA Danskin's envelope theorem sliced Wasserstein autoencoder sliced multi-marginal Wasserstein distance
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