Research map / Learning theory and algorithms / Optimal transport
Tree-Sliced Wasserstein Distance: A Geometric Perspective
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Proposes Tree-Sliced Wasserstein on Systems of Lines (TSW-SL), replacing SW's 1D lines with tree systems of connected lines that admit closed-form tree-Wasserstein OT, with a new Radon transform variant shown injective.
Problem
Sliced Wasserstein projects measures onto one-dimensional lines, losing topological information in high-dimensional settings, while OT is computationally expensive.
Approach
Replace one-dimensional projection lines with tree systems (connected systems of lines with a spanning-tree structure) that are metrizable by a tree metric, use splitting maps to distribute mass across lines, and define a Radon transform on systems of lines whose tree-Wasserstein distance has a closed form.
Key results
- On Swiss Roll at iteration 2000, TSW-SL reaches 4.44e-8 vs SW 1.11e-3 Wasserstein distance (same 100 directions for SW, L=25,k=4 for TSW-SL).
- On STL-10 with 500 projections, TSW-SL (L=100,k=5) achieves FID 51.81 vs SW 53.52 (SN-GAN).
- On CIFAR-10 denoising diffusion, TSW-SL-DD reaches FID 2.83 vs SW-DD 2.90 and DDGAN 3.64.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace SW projections with tree systems | Gradient flows using TSW-SL · Swiss Roll, iteration 2500 | Wasserstein distance: 1.05e-3 -> 3.65e-8 | Helped |
| Use maximized tree-sliced directions instead of SW directions | Gradient flows · Swiss Roll, iteration 2500 | Wasserstein distance: 2.47e-2 -> 5.00e-7 | Helped |
| Replace SW projections with tree systems | Gradient flows using TSW-SL · 25 Gaussians, iteration 2500 | Wasserstein distance: 2.20e-2 -> 3.03e-7 | Helped |
| Use maximized tree-sliced directions instead of SW directions | Gradient flows · 25 Gaussians, iteration 2500 | Wasserstein distance: 8.48e-2 -> 5.19e-7 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 10-dimensional Gaussian, iteration 2500 | Wasserstein distance: 2.28e-3 -> 1.75e-3 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 50-dimensional Gaussian, iteration 2500 | Wasserstein distance: 35.75 -> 1.72 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 75-dimensional Gaussian, iteration 2500 | Wasserstein distance: 90.31 -> 31.61 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 100-dimensional Gaussian, iteration 2500 | Wasserstein distance: 128.29 -> 75.28 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 150-dimensional Gaussian, iteration 2500 | Wasserstein distance: 216.32 -> 182.63 | Helped |
| Replace SW projections with tree systems | High-dimensional Gaussian gradient flows · 200-dimensional Gaussian, iteration 2500 | Wasserstein distance: 301.51 -> 279.08 | Helped |
| Replace SW projections with tree systems | SN-GAN · STL-10 (96x96), 50 projecting directions | FID: 69.46 ± 0.21 -> 61.15 ± 0.37 | Helped |
| Replace SW projections with tree systems | SN-GAN · STL-10 (96x96), 50 projecting directions | IS: 9.08 ± 0.06 -> 10.00 ± 0.03 | Helped |
Surprising findings
- On 25 Gaussians, TSW-SL is worse than SW at iteration 500 (3.49e-1 vs 1.61e-1), and at higher-dimensional gradient flows the advantage is limited to the reported settings.
Limitations
- Authors do not expect TSW-SL to outperform recent SW variants; no comparison with advanced SW techniques.
- Time per iteration is somewhat higher than SW in several settings (e.g., DDGAN experiments 163s vs 140s per epoch).
Experimental setup
Datasets: Swiss Roll 25 Gaussians CelebA STL-10 CIFAR-10
Benchmarks: Swiss Roll 25 Gaussians CelebA STL-10 CIFAR-10
Metrics: Wasserstein distance FID IS time per iteration
Baselines: SW MaxSW SWGG LCVSW SW-DD DDGAN
Builds on: Sliced Wasserstein Max Sliced Wasserstein tree-Wasserstein distance SNGAN DDGAN
Closest papers by meaning
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.96)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.93)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.93)
- Differentiable Generalized Sliced Wasserstein Plans (NeurIPS 2025 · similarity 0.90)
- Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis (ICML 2025 · similarity 0.88)
- Learning Ultrametric Trees for Optimal Transport Regression (AAAI 2024 · similarity 0.88)
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