Research map / Learning theory and algorithms / Optimal transport
Distance-Based Tree-Sliced Wasserstein Distance
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Proposes Db-TSW, a tree-sliced Wasserstein metric on systems of lines using Euclidean-invariant distance-based splitting maps and root-concurrent tree sampling, which is Euclidean invariant and GPU-friendly.
Problem
Tree-Sliced Wasserstein on Systems of Lines (TSW-SL) uses splitting maps that ignore tree/line positions and are not Euclidean invariant, limiting accuracy of sliced-type OT distances on Euclidean space.
Approach
Generalize splitting maps to depend on both points and the tree system, using softmax over point-to-line distances (which are Euclidean invariant), and sample tree systems as k concurrent lines sharing a root so the computation is parallel and as fast as SW.
Key results
- Db-TSW-DD$^\perp$ achieves FID 2.525 on CIFAR-10 vs 2.70 for IWRPSW-DD and 3.64 for DDGAN, with ~162 s/epoch.
- On Gaussian 20d gradient flow at iteration 2500, Db-TSW reaches W2 4.60e-3 versus 10.30 for vanilla SW and 7.34 for SWGG.
- Color transfer: Db-TSW and Db-TSW-perp reach 0.21 and 0.12 vs 9.58 for SW and MaxSW.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace IWRPSW with Db-TSW in DDGAN | DDGAN on CIFAR-10 · CIFAR-10 | FID: 2.70 -> 2.60 | Helped |
| use orthogonal directions instead of unrestricted directions (ablation) | Db-TSW-DD · CIFAR-10 | FID: 2.60 -> 2.525 | Helped |
| replace IWRPSW with Db-TSW | DDGAN on CIFAR-10 · CIFAR-10 | Time/Epoch(s): 152 -> 160 | Hurt |
| use Db-TSW instead of TSW-SL | gradient flow · Swiss Roll, iteration 2500 | Wasserstein distance: 3.65e-8 -> 3.01e-8 | Helped |
| use orthogonal directions instead of unrestricted directions (ablation) | Db-TSW · Swiss Roll, iteration 2500 | Wasserstein distance: 3.01e-8 -> 3.78e-8 | Hurt |
| use Db-TSW instead of TSW-SL | gradient flow · Gaussian 20d, iteration 2500 | Wasserstein distance: 5.71e-3 -> 4.60e-3 | Helped |
| use orthogonal directions instead of unrestricted directions (ablation) | Db-TSW · Gaussian 20d, iteration 2500 | Wasserstein distance: 4.60e-3 -> 5.03e-3 | Hurt |
| use Db-TSW instead of LCVSW | gradient flow · Swiss Roll | Time/Iter(s): 0.010 -> 0.006 | Helped |
| use Db-TSW⊥ instead of RRQDSW (ablation) | color transfer · Color transfer | Wasserstein distance: 0.08 -> 0.12 | Hurt |
| use Db-TSW instead of SW | color transfer · Color transfer | Wasserstein distance: 9.58 -> 0.21 | Helped |
| use orthogonal directions instead of unrestricted directions (ablation) | Db-TSW · Color transfer | Wasserstein distance: 0.21 -> 0.12 | Helped |
Surprising findings
- The Euclidean-invariant splitting map forces the original constant-type splitting class to be trivial, so invariance must be imposed on the richer class.
Limitations
- Db-TSW-DD and variants are only modestly better than IWRPSW-DD in FID (gaps of 0.1 to 0.175)
- Injectivity and invariance proofs rely on a simplified argument (remark notes formal proof would need Haar measure theory)
- Orthogonal tree-system variant limited to k <= d
Experimental setup
Datasets: CIFAR-10 Swiss Roll Gaussian 20d
Benchmarks: CIFAR-10
Metrics: FID Wasserstein distance time per epoch time per iteration
Baselines: SW MaxSW SWGG LCVSW TSW-SL DSW EBSW RPSW IWRPSW QSW variants
Builds on: Sliced Wasserstein TSW-SL DDGAN IWRPSW
Closest papers by meaning
- Tree-Sliced Wasserstein Distance: A Geometric Perspective (ICML 2025 · similarity 0.96)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.95)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.92)
- Differentiable Generalized Sliced Wasserstein Plans (NeurIPS 2025 · similarity 0.89)
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.87)
- Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis (ICML 2025 · similarity 0.87)
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