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Differentiable Generalized Sliced Wasserstein Plans
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Reformulates min-SWGG as a bilevel problem and proposes DGSWP, a differentiable, Gaussian-smoothed generalized sliced Wasserstein plan with nonlinear projections, applied to OT plan approximation, gradient flows, and sliced-OT conditional flow matching.
Problem
Exact OT plans are expensive to compute at scale; min-SWGG sliced plans are limited to linear projections, need exponentially many slices in high dimension, and their bilevel objective is discontinuous and hard to optimize with gradients.
Approach
Generalize min-SWGG to nonlinear scalar-field projections parameterized by theta, and make the bilevel value function differentiable by Gaussian smoothing over theta, with a Stein-lemma gradient estimator using a control variate and 1D OT solves.
Key results
- On CIFAR-10 with 100-step Euler sampling, DGSWP-CFM (NN) achieves FID 3.62 versus 4.61 for I-CFM and 4.75 for OT-CFM.
- With DoPri5, DGSWP-CFM (NN) reaches FID 3.95 with 120.04 NFE, comparable to OT-CFM (3.81 FID, 132.56 NFE).
- In the 8Gaussians to Two Moons plan experiment, DGSWP (NN) attains transport cost 13.80 versus 22.22 for min-SWGG and exact OT 13.50.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use a neural-network projector instead of a linear projector (ablation) | DGSWP-CFM · CIFAR-10, Euler | FID: 4.16 $\pm$ 0.09 -> \textbf{3.62} $\pm$ 0.05 | Helped |
| use a neural-network projector instead of a linear projector (ablation) | DGSWP-CFM · CIFAR-10, DoPri5 | FID: 4.45 $\pm$ 0.08 -> 3.95 $\pm$ 0.09 | Helped |
| use DGSWP-CFM (NN) instead of OT-CFM | CIFAR-10, Euler · CIFAR-10, Euler | FID: 4.75 $\pm$ 0.05 -> \textbf{3.62} $\pm$ 0.05 | Helped |
| use DGSWP-CFM (NN) instead of OT-CFM | CIFAR-10, DoPri5 · CIFAR-10, DoPri5 | FID: 3.81 $\pm$ 0.04 -> 3.95 $\pm$ 0.09 | Mixed |
| use the DGSWP variance-reduction strategy (ablation) | DGSWP · 8 Gaussians to Two Moons | transport cost and learning stability | Helped |
| use DGSWP (linear) instead of min-SWGG (optim) at the same time budget | Swiss Roll gradient flow · Swiss Roll gradient flow, 500 iterations | log10 W2^2: \textbf{-0.81} -> -0.49 | Hurt |
| use DGSWP (linear) instead of min-SWGG (optim) at the same time budget | Swiss Roll gradient flow · Swiss Roll gradient flow, 1000 iterations | log10 W2^2: \textbf{-0.89} -> -0.84 | Hurt |
| use DGSWP (linear) instead of min-SWGG (optim) at the same time budget | Swiss Roll gradient flow · Swiss Roll gradient flow, 1500 iterations | log10 W2^2: -0.80 -> \textbf{-1.07} | Helped |
| use DGSWP (linear) instead of min-SWGG (optim) at the same time budget | Swiss Roll gradient flow · Swiss Roll gradient flow, 2000 iterations | log10 W2^2: -0.79 -> \textbf{-1.28} | Helped |
| use DGSWP (NN) instead of Expected Sliced Transport Plans | 8 Gaussians to Two Moons transport · 8 Gaussians to Two Moons | transport cost: 32.28 -> 13.80; 21.30 -> 13.80; 20.09 -> 13.80 | Helped |
Surprising findings
- Linear DGSWP underperforms NN-based DGSWP in CFM, and the variance-reduced gradient estimator is needed for convergence at the same learning rate.
Limitations
- Injectivity of the projection is required for a proper distance and is not guaranteed for neural projections
- Gradient estimation uses N+1 1D OT solves per step, and the value function is discontinuous, so theoretical guarantees are limited to smoothing consistency
- No statistical guarantees for general nonlinear projections are established yet
Experimental setup
Datasets: CIFAR-10 Two Moons 8Gaussians Swiss Roll
Benchmarks: CIFAR-10
Metrics: FID NFE transport cost log10 Wasserstein distance
Baselines: min-SWGG SWD ASWD HHSW Expected Sliced Transport Plans I-CFM OT-CFM
Builds on: min-SWGG sliced Wasserstein distance generalized sliced Wasserstein OT-CFM augmented sliced Wasserstein
Closest papers by meaning
- Expected Sliced Transport Plans (ICLR 2025 · similarity 0.93)
- Efficient Transferable Optimal Transport via Min-Sliced Transport Plans (NeurIPS 2026 · similarity 0.90)
- Tree-Sliced Wasserstein Distance: A Geometric Perspective (ICML 2025 · similarity 0.90)
- Distance-Based Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.89)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.88)
- Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows (ICML 2025 · similarity 0.88)
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