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Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data
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Proposes Linear Spherical Sliced Optimal Transport (LSSOT), which slices the sphere into circles and applies linear circular OT embeddings to get an L2 metric for spherical distributions, and shows it is metric-valid and faster on cortical registration, VMF geometry, and point cloud interpolation.
Problem
Comparing spherical probability distributions with optimal transport is computationally expensive (cubic cost), and pairwise comparisons across many distributions are costly.
Approach
Extends linear circular OT (LCOT) to hyperspheres by averaging LCOT embeddings over slices obtained from Stiefel-manifold projections onto great circles, yielding an L2-embedding whose distance is a metric on spherical measures (proved via a Semicircle/Spherical Radon transform injectivity argument).
Key results
- On NKI left hemisphere, LSSOT achieves the lowest LSSOT distance (0.1890) and highest Dice (0.8462) among methods, with training time of 101 s versus 1351 s for SSW.
- On ADNI left hemisphere, LSSOT reaches Dice 0.8119 and lowest edge distortion 0.0979, with training time 84.5 s versus 1094 s for SSW.
- Pairwise distance computation of LSSOT is reported as the fastest among compared methods across sample sizes up to 15000 points.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LSSOT instead of SSW as the registration similarity loss | S3Reg · NKI and ADNI cortical surface registration, all four hemispheres | LSSOT: 0.2079±0.0369 -> 0.1890±0.0361; 0.1341±0.0209 -> 0.1121±0.0253; 0.1262±0.0261 -> 0.1041±0.0277; 0.1349±0.0541 -> 0.1295± | Helped |
| Use LSSOT instead of SSW as the registration similarity loss | S3Reg · NKI and ADNI cortical surface registration, all four hemispheres | Dice: 0.8429±0.0692 -> 0.8462±0.0548; 0.7952±0.0751 -> 0.7996±0.0706; 0.7987±0.0775 -> 0.8119±0.0940; 0.8622±0.0915 -> 0.8518± | Mixed |
| Use LSSOT instead of SSW as the registration similarity loss | S3Reg · NKI and ADNI cortical surface registration, all four hemispheres | Time (seconds) per epoch: 1350.96 -> 101.01; 1362.16 -> 100.68; 1094.06 -> 84.51; 1114.83 -> 85.37 | Helped |
Surprising findings
- LSSOT is slower than SWD in some cases only because SWD recomputes both surfaces each iteration; LSSOT pre-embeds the fixed template.
- On ADNI right hemisphere, LSSOT's CC (0.8013) is notably lower than competing methods, and several metrics are not best.
Limitations
- LSSOT is defined only for probability measures, so unnormalized or signed spherical signals are not handled.
- Polar points handled via epsilon-cap approximation in numerical implementation.
- Registration results are mixed; SSW and S3W are competitive on several metrics.
Experimental setup
Datasets: NKI ADNI ModelNet DKT Atlas Freesurfer fsaverage
Benchmarks: NKI ADNI
Metrics: LSSOT SWD MAE Pearson CC Dice edge distortion area distortion training time pairwise distance runtime
Baselines: MSE S3W SWD SSW Spherical OT Sinkhorn divergence
Builds on: Linear Circular Optimal Transport (LCOT) Spherical Sliced Wasserstein (SSW) S3Reg Linear optimal transport Set Transformer
Closest papers by meaning
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.90)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.89)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.87)
- Differentiable Generalized Sliced Wasserstein Plans (NeurIPS 2025 · similarity 0.87)
- Efficient Transferable Optimal Transport via Min-Sliced Transport Plans (NeurIPS 2026 · similarity 0.87)
- Towards Better Spherical Sliced-Wasserstein Distance Learning with Data-Adaptive Discriminative Projection Direction (AAAI 2025 · similarity 0.86)
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