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Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions
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Proposes Hierarchical Hybrid Sliced Wasserstein (H2SW), built on a new Hierarchical Hybrid Radon Transform, to compare heterogeneous joint distributions with marginals on different domains, and shows gains on 3D mesh deformation, mesh autoencoder training, and dataset comparison.
Problem
Sliced Wasserstein and generalized sliced Wasserstein are defined only for distributions on a homogeneous domain, so they cannot directly compare joint distributions whose marginals live on different domains (e.g., points in R^3 with normals on S^2).
Approach
Introduce Partial Generalized Radon Transform (PGRT) to apply non-linear defining functions to one marginal, and compose per-marginal PGRTs with a final partial Radon Transform (HHRT); using HHRT as the slicing operator yields H2SW, proven to be a valid metric with SW-like complexity.
Key results
- On Armadillo deformation with L=10 at step 5000, H2SW reaches 2.602 joint Wasserstein vs 2.726 for SW and 4.743 for GSW.
- On ShapeNet autoencoder at epoch 2000 with L=1000, H2SW gets 129.87 vs 130.93 (SW) and 130.76 (GSW) joint Wasserstein reconstruction error.
- On dataset comparison with L=2000, H2SW relative error is 3.998 vs 4.035 for CHSW and 4.198 for SW (100 runs).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use H2SW instead of SW | 3D mesh deformation from sphere to Armadillo · 3D mesh deformation, step 100, L=10 | joint Wasserstein distance: 1852.519$\pm$3.236 -> 1840.73$\pm$1.282 | Helped |
| Use H2SW instead of SW | 3D mesh deformation from sphere to Armadillo · 3D mesh deformation, step 5000, L=10 | joint Wasserstein distance: 2.726$\pm$0.305 -> 2.602$\pm$0.201 | Helped |
| Use H2SW instead of SW | 3D mesh deformation from sphere to Armadillo · 3D mesh deformation, L=100 | joint Wasserstein distance: 1847.572$\pm$0.303 -> 1839.347$\pm$1.986 (step 100); 1.171$\pm$0.056 -> 1.086$\pm$0.177 (step 5000) | Helped |
| Use H2SW instead of GSW | 3D mesh autoencoder · ShapeNet test set, epoch 2000, L=100 | joint Wasserstein reconstruction error: $130.80\pm0.46$ -> $\mathbf{130.24\pm 0.47}$ | Helped |
| Use H2SW instead of GSW | 3D mesh autoencoder · ShapeNet test set, epoch 2000, L=1000 | joint Wasserstein reconstruction error: $130.76\pm0.26$ -> $\mathbf{129.87\pm 0.44}$ | Helped |
| Increase projections from L=100 to L=1000 | SW variants in 3D mesh autoencoder · ShapeNet test set, epoch 2000 | joint Wasserstein reconstruction error: $131.10\pm 0.34$ -> $130.93\pm 0.11$ (SW); $130.80\pm0.46$ -> $130.76\pm0.26$ (GSW); $\mathbf{130.24\pm 0.47}$ -> $\math | Mixed |
| Use H2SW instead of CHSW | Hadamard manifold dataset comparison · Dataset comparison, L=100 | relative error: $4.449\pm 0.497$ -> $\mathbf{4.381\pm 0.695}$ | Helped |
| Increase projections from L=100 to L=2000 | SW variants in Hadamard manifold dataset comparison · Dataset comparison | relative error: $4.618\pm0.744$ -> $4.198\pm0.238$ (SW); $4.449\pm 0.497$ -> $4.035\pm0.145$ (CHSW); $\mathbf{4.381\pm 0.695}$ -> $\math | Helped |
Surprising findings
- H2SW is not uniformly best: GSW and SW outperform H2SW at some later steps of Armadillo deformation (e.g., step 4000 with L=100 and L=10 SW is best at step 4000).
Limitations
- HHRT costs slightly more computation than SW.
- Injectivity of HHRT depends on the injectivity of the component PGRTs, which is only shown for some defining functions.
- Experiments are limited to 3D meshes and a small set of dataset-comparison benchmarks with modest neural architectures.
Experimental setup
Datasets: Armadillo mesh Stanford Bunny mesh ShapeNet MNIST EMNIST Fashion MNIST KMNIST USPS
Benchmarks: Armadillo mesh deformation Stanford Bunny mesh deformation ShapeNet autoencoder reconstruction MNIST-family dataset comparison
Metrics: joint Wasserstein distance relative error of cost matrix
Baselines: SW GSW CHSW
Builds on: sliced Wasserstein generalized sliced Wasserstein partial Radon transform Cartan-Hadamard sliced Wasserstein
Closest papers by meaning
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.87)
- Sliced Wasserstein with Random-Path Projecting Directions (ICML 2024 · similarity 0.86)
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.85)
- Quasi-Monte Carlo for 3D Sliced Wasserstein (ICLR 2024 · similarity 0.85)
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.85)
- Tree-Sliced Wasserstein Distance: A Geometric Perspective (ICML 2025 · similarity 0.84)
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