Research map / Learning theory and algorithms / Optimal transport
Fourier Sliced-Wasserstein Embedding for Multisets and Measures
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Problem
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Approach
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Key results
- On 1-Wasserstein approximation, the FSW+MLP architecture achieves the lowest mean relative error on most of six datasets (e.g., 1.1-1.4% on noisy-sphere-3 vs 2.2-4.6% for next-best methods on some sets).
- On ModelNet40, FSW+MLP reaches 83.5% accuracy with 31k parameters while PointNet achieves 54.1% at 35k parameters.
- FSW with m=1 already distinguishes a multiset pair that PSWE identifies as identical, and embedding distances converge to SW as m grows.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FSW + MLP instead of NProductNet | 1-Wasserstein distance approximation · noisy-sphere-3 | relative error: 2.2 % -> 1.4 % | Helped |
| Use FSW + MLP instead of NProductNet | 1-Wasserstein distance approximation · noisy-sphere-6 | relative error: 1.4 % -> 1.3 % | Helped |
| Use FSW + MLP instead of PSWE + MLP | 1-Wasserstein distance approximation · uniform | relative error: 2.1 % -> 2.4 % | Hurt |
| Use FSW + MLP instead of NProductNet | 1-Wasserstein distance approximation · ModelNet-small | relative error: 5.7 % -> 2.9 % | Helped |
| Use FSW + MLP instead of PSWE + MLP | 1-Wasserstein distance approximation · ModelNet-large | relative error: 2.4 % -> 2.6 % | Hurt |
| Use FSW + MLP instead of PSWE + MLP | 1-Wasserstein distance approximation · RNAseq | relative error: 1.2 % -> 1.1 % | Helped |
| Replace PointNet max-pooling with FSW embedding | PointNet-style ModelNet40 classifier · ModelNet40 (low-parameter regime) | accuracy: 54.1 % -> 83.5 % | Helped |
Surprising findings
- Embedding distances of PSWE are numerically zero (about 1e-7) for a multiset pair that FSW separates, showing PSWE loses injectivity on finite multisets.
- Proves that no bi-Lipschitz Euclidean embedding exists for distributions over R^d even with bounded support size and compact domain.
Limitations
- Bi-Lipschitz guarantee holds only for multisets; distributions admit only injectivity, and bi-Lipschitz embedding is impossible for them.
- Simple PointNet-style architecture does not reach best ModelNet40 results; the authors note more complex methods score higher.
- Experiments did not use fixed random seeds.
Experimental setup
Datasets: ModelNet40 noisy-sphere-3 noisy-sphere-6 uniform ModelNet-small ModelNet-large RNAseq
Benchmarks: ModelNet40 noisy-sphere-3 noisy-sphere-6 uniform ModelNet-small ModelNet-large RNAseq
Metrics: mean relative error classification accuracy training time embedding distance
Baselines: PSWE NProductNet NSDeepSets WPCE Sinkhorn PointNet
Builds on: sort embedding sliced Wasserstein distance PointNet Wasserstein Wormhole
Closest papers by meaning
- Slicing Wasserstein over Wasserstein via Functional Optimal Transport (ICLR 2026 · similarity 0.90)
- Learning Embeddings into Entropic Wasserstein Spaces (ICLR 2019 · similarity 0.89)
- Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions (NeurIPS 2018 · similarity 0.89)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.87)
- Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances (ICML 2025 · similarity 0.87)
- Kernel Quantile Embeddings and Associated Probability Metrics (ICML 2025 · similarity 0.86)
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