Research map / Learning theory and algorithms / Optimal transport
Spherical Tree-Sliced Wasserstein Distance
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Proposes Spherical Tree-Sliced Wasserstein (STSW), a metric for measures on the hypersphere built from a spherical Radon transform onto random spherical trees with O(d+1)-invariant splitting maps, yielding closed-form tree OT.
Problem
Comparing probability measures on the hypersphere with optimal transport is computationally expensive, and existing spherical sliced variants (projections onto lines/circles or great semicircles) lose topological information.
Approach
Replaces spherical slices with spherical trees (gluings of spherical rays rooted at a point), introduces a spherical Radon transform variant that splits each point's mass across tree edges via an O(d+1)-invariant softmax splitting map, and uses the closed-form tree Wasserstein distance, giving an orthogonally invariant metric.
Key results
- On the 12-vMF gradient flow task, STSW reaches log W2 of -4.69 and NLL of -5041.13 with runtime 1.89s, beating SSW (-3.36, 55.31s) and S3W variants.
- On CIFAR-10 linear evaluation over S^9, STSW achieves 80.53% accuracy on encoded features and 76.78% on projected features, the best among listed methods.
- In SWAE on CIFAR-10, STSW achieves the best log W2 (-3.4191) and NLL (-0.0051), though its BCE (0.6341) is slightly worse than SSW's 0.6309.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use STSW instead of ARI-S3W | Gradient flow toward a 12-vMF target · 12 vMF gradient flow | log W2: -4.39 ± 0.19 -> -4.69 ± 0.01 | Helped |
| Use STSW instead of ARI-S3W | Gradient flow toward a 12-vMF target · 12 vMF gradient flow | Runtime (s): 20.25 ± 0.15 -> 1.89 ± 0.05 | Helped |
| Replace Gaussian-kernel uniformity loss with STSW | ResNet18 contrastive learning · CIFAR-10 linear evaluation, encoded features | Accuracy (%): 80.08 -> 80.53 | Helped |
| Replace Gaussian-kernel uniformity loss with STSW | ResNet18 contrastive learning · CIFAR-10 linear evaluation, projected features | Accuracy (%): 75.12 -> 76.78 | Helped |
| Use STSW instead of ARI-S3W | Spherical density estimation · Quake | NLL: 0.77 ± 0.06 -> 0.68 ± 0.04 | Helped |
| Use STSW instead of ARI-S3W | Spherical density estimation · Flood | NLL: 1.24 ± 0.03 -> 1.23 ± 0.03 | Helped |
| Use STSW instead of ARI-S3W | Spherical density estimation · Fire | NLL: 0.08 ± 0.05 -> -0.07 ± 0.05 | Helped |
| Use STSW instead of ARI-S3W | Sliced-Wasserstein auto-encoder · CIFAR-10 test latent regularization | log W2: -3.3935 -> -3.4191 | Helped |
| Use STSW instead of ARI-S3W | Sliced-Wasserstein auto-encoder · CIFAR-10 test latent regularization | BCE: 0.6330 -> 0.6341 | Hurt |
| Use STSW instead of SW | Sliced-Wasserstein auto-encoder · CIFAR-10 SWAE | Time (s/ep.): 3.4060 -> 3.5460 | Hurt |
Surprising findings
- STSW's BCE in the SWAE experiment slightly underperforms several baselines despite better latent distribution metrics.
- In the density estimation table, STSW reports a negative NLL on Fire (-0.07), while the baselines are positive.
Limitations
- Sampling spherical trees and the splitting map rely on a tuning parameter zeta whose effect is only described qualitatively.
- Results on SWAE show slightly worse BCE than some baselines.
- The paper notes future work needed on sampling processes for spherical trees and generalized Radon transforms.
Experimental setup
Datasets: CIFAR-10 Fire, Earthquake (Quake), Flood earth event datasets
Benchmarks: CIFAR-10 linear evaluation Earth event density estimation (Quake, Flood, Fire) vMF mixture gradient flow
Metrics: log W2 negative log-likelihood linear evaluation accuracy runtime BCE
Baselines: SW SSW S3W RI-S3W ARI-S3W SimCLR Hypersphere
Builds on: Sliced Wasserstein distance Tree-Sliced Wasserstein on systems of lines Spherical Sliced Wasserstein (SSW) Stereographic Sliced Wasserstein (S3W) Sliced-Wasserstein Auto-Encoder
Closest papers by meaning
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.94)
- Tree-Sliced Wasserstein Distance: A Geometric Perspective (ICML 2025 · similarity 0.93)
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.92)
- Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data (ICLR 2025 · similarity 0.89)
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.88)
- Differentiable Generalized Sliced Wasserstein Plans (NeurIPS 2025 · similarity 0.87)
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