Research map / Learning theory and algorithms / Optimal transport
Stereographic Spherical Sliced Wasserstein Distances
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Proposes Stereographic Spherical Sliced Wasserstein (S3W) distances that compose stereographic projection with an injective map and the generalized Radon transform, plus a rotationally invariant variant (RI-S3W), for comparing spherical probability measures.
Problem
Computing optimal transport distances between probability measures on hyperspheres is expensive; existing spherical sliced Wasserstein methods require circular OT, which is slower than real-line OT, and stereographic projection distorts distances.
Approach
Map the sphere to Euclidean space via stereographic projection, then apply an injective nonlinear map h (analytic h_1 or a learned h_NN) approximating arclength, and slice with the classic Radon transform so that 1D Wasserstein on the real line (sorting) can be used; add a Haar-averaged rotation-invariant version.
Key results
- On CIFAR-10 SSL (d=9), RI-S3W(5) reaches 79.97% encoded accuracy vs 70.46% for SSW, while being roughly 1.5x faster per epoch (~21.6 s vs 33.1 s).
- In gradient flow on the sphere, S3W variants are on par or better than SSW while up to 20x faster.
- On Earth density estimation, ARI-S3W(50) achieves the lowest NLL-type scores (Quake 0.78, Flood 1.24, Fire 0.10) versus SSW (0.84, 1.26, 0.24).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the Gaussian-kernel uniformity loss with S3W | ResNet18 contrastive SSL · CIFAR-10 linear evaluation, encoded features | accuracy (%): 70.46 -> 78.54 | Helped |
| Use rotationally invariant S3W with 5 rotations instead of S3W | ResNet18 contrastive SSL · CIFAR-10 linear evaluation, encoded features | accuracy (%): 78.54 -> 79.97 | Helped |
| Use amortized rotationally invariant S3W with 5 rotations instead of S3W | ResNet18 contrastive SSL · CIFAR-10 linear evaluation, projected features | accuracy (%): 73.84 -> 75.07 | Helped |
| Use rotationally invariant S3W with 5 rotations instead of S3W | SWAE · CIFAR-10 | log W2: -3.3381 -> -3.1424 | Hurt |
| Use amortized rotationally invariant S3W with 5 rotations instead of S3W | SWAE · CIFAR-10 | log W2: -3.3381 -> -3.3853 | Helped |
| Use rotationally invariant S3W instead of S3W | Spherical normalizing-flow density estimation · Earthquake | score: 0.88 ± 0.09 -> 0.79 ± 0.07 | Helped |
| Use amortized rotationally invariant S3W instead of S3W | Spherical normalizing-flow density estimation · Fire | score: 0.36 ± 0.04 -> 0.10 ± 0.04 | Helped |
Surprising findings
- The naive stereographic projection distorts spherical distances severely: nearby points on the sphere can map to arbitrarily far points in R^d, making naive SP-plus-OT unsuitable.
Limitations
- S3W is only a pseudo-metric in general; metric property requires injective h (RI-S3W variant is a metric only with injective h)
- Rotation-invariant variants add cost (N_R rotations) and ARI-S3W introduces bias from pool subsampling
- Experiments are on low-dimensional spheres (mostly S^2 and S^9) and small benchmarks
Experimental setup
Datasets: CIFAR-10 MNIST Earthquake Flood Fire von Mises-Fisher mixtures (synthetic)
Benchmarks: CIFAR-10 linear evaluation CIFAR-10 SWAE reconstruction/latent loss Earth density estimation (Quake, Flood, Fire) vMF gradient flow
Metrics: accuracy runtime per epoch log W2 NLL BCE negative log-likelihood runtime
Baselines: Wasserstein Sinkhorn SW SSW Hypersphere SimCLR
Builds on: sliced Wasserstein distance spherical sliced Wasserstein (SSW) generalized Radon transform SWAE GeoTorch
Closest papers by meaning
- Towards Better Spherical Sliced-Wasserstein Distance Learning with Data-Adaptive Discriminative Projection Direction (AAAI 2025 · similarity 0.91)
- Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data (ICLR 2025 · similarity 0.90)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.89)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.88)
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.87)
- An in depth look at the Procrustes-Wasserstein distance: properties and barycenters (ICML 2025 · similarity 0.87)
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