Research map / Learning theory and algorithms / Optimal transport
Tree-Sliced Wasserstein Distance with Nonlinear Projection
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Proposes nonlinear-projection tree-sliced Wasserstein distances (CircularTSW and SpatialTSW, plus a spherical variant) by generalizing the Radon transform on systems of lines with circular and spatial defining functions, with injectivity and metric proofs and experiments on gradient flows, diffusion GANs, SSL and autoencoders.
Problem
Tree-sliced Wasserstein distances are restricted to linear projections onto tree systems of lines, limiting their ability to capture topological structure; the paper asks whether nonlinear projections can be incorporated while keeping injectivity and metric properties.
Approach
Replace linear projections in the Radon Transform on Systems of Lines with nonlinear ones (circular defining function and spatial map h with E(d)-invariant softmax splitting), prove injectivity and metric properties, and extend to spherical trees (SpatialSTSW).
Key results
- CircularTSW-DD achieves FID 2.33 on CIFAR-10 versus 2.53 for Db-TSW-DD-perp, the best prior tree-sliced DDGAN baseline.
- SpatialTSW reaches final W2 of 1.17e-7 on 25 Gaussians gradient flow versus 1.30e-7 for Db-TSW and 3.59e-2 for vanilla SW.
- SpatialSTSW gets 77.31% projected-feature linear probe accuracy on S9 versus 76.79% for STSW.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CircularTSW instead of Db-TSW-DD⊥ in DDGAN | DDGAN · CIFAR-10 unconditional generation | FID: 2.53 -> 2.33 | Helped |
| Use SpatialTSW instead of Db-TSW in gradient flow | Gradient flow · 25 Gaussians, iteration 2500 | W₂ distance: 1.30e-7 -> 1.17e-7 | Mixed |
| Use CircularTSW rather than vanilla SW | Gradient flow · 25 Gaussians, iteration 2500 | W₂ distance: 3.59e-2 -> 7.86e-3 | Helped |
| Use SpatialSTSW instead of STSW | Spherical gradient flow · 12-vMF mixture, epoch 250 | log W₂ distance: -4.6486 -> -4.6859 | Helped |
| Replace the Gaussian-kernel uniformity loss with SpatialSTSW | ResNet18 self-supervised learning · CIFAR-10, encoded features | accuracy: 80.51 -> 80.68 | Helped |
| Use SpatialSTSW instead of STSW for latent regularization | SWAE · CIFAR-10 | log W₂: -3.4098 -> -3.4254 | Helped |
| Set the CircularTSW radius to r=0 (ablation) | CircularTSW gradient flow · 25 Gaussians, iteration 2500 | W₂ distance: 7.86e-3 -> 8.94e-3 | Hurt |
Surprising findings
- CircularTSW with r=0 performs well in tree settings but poorly in the original sliced setting, indicating tree structure and distance-based splitting are necessary.
Limitations
- Injectivity and metric guarantees depend on E(d)-invariant splitting maps and injective h; the general nonlinear case is not fully characterized.
- Gains on some tasks are small (e.g., FID differences of 0.01 to 0.2), and CircularTSW underperforms SpatialTSW in gradient flow.
Experimental setup
Datasets: CIFAR-10 25 Gaussians mixture of 12 von Mises-Fisher distributions
Benchmarks: CIFAR-10 FID 25 Gaussians W2 gradient flow vMF mixture log W2 CIFAR-10 linear probe accuracy on S9 CIFAR-10 SWAE log W2/NLL
Metrics: FID Wasserstein distance linear classifier accuracy NLL BCE time per epoch
Baselines: SW DSW EBSW RPSW IWRPSW TSW-SL Db-TSW Db-TSW-perp MaxSW LCVSW SWGG SSW
Builds on: Tree-Sliced Wasserstein on systems of lines (TSW-SL) Db-TSW Generalized Radon Transform Circular Radon Transform Spatial Radon Transform
Closest papers by meaning
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.95)
- Revisiting Tree-Sliced Wasserstein Distance Through the Lens of the Fermat–Weber Problem (ICLR 2026 · similarity 0.94)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.94)
- Tree-Sliced Wasserstein Distance: A Geometric Perspective (ICML 2025 · similarity 0.93)
- Slicing Wasserstein over Wasserstein via Functional Optimal Transport (ICLR 2026 · similarity 0.92)
- Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis (ICML 2025 · similarity 0.89)
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