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Sliced Wasserstein Estimation with Control Variates
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Proposes lower- and upper-bound control variates built from Gaussian approximations of projected measures to reduce the variance of Monte Carlo sliced Wasserstein estimation at no extra asymptotic cost.
Problem
Monte Carlo estimation of the sliced Wasserstein distance has high variance for a finite number of projections, and prior work has not improved the Monte Carlo scheme itself.
Approach
Fit 1D Gaussians to each projected measure (moment matching / KL), use the closed-form W2 between these Gaussians as a control variate, and use a lower bound (mean difference squared) and an upper bound (adds both variances) that have closed-form expectations, yielding linear-time control variate SW estimators.
Key results
- On CIFAR10 with L=10, LCV-SW reduces FID from 21.72 to 18.67 (about 14%) relative to SW with similar runtime.
- On MNIST 0-1 comparison, LCV-SW variance drops from 4700.87 to 0.045 versus conventional SW estimator at the same number of projections.
- In point-cloud gradient flows with L=100, LCV-SW and UCV-SW reach nearly identical W2 to SW's trajectory with comparable time, converging faster at intermediate steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=10 | FID: 21.72±0.37 -> 18.67±0.39 | Helped |
| Use LCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=10 | IS: 7.54± 0.06 -> 7.74± 0.04 | Helped |
| Use LCV-SW instead of SW | SNGAN · CelebA (64x64), L=10 | FID: 11.14 ± 1.12 -> 10.17± 0.73 | Helped |
| Use UCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=10 | FID: 21.72±0.37 -> 21.71±0.36 | No clear effect |
| Use UCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=10 | IS: 7.54± 0.06 -> 7.57± 0.01 | No clear effect |
| Use UCV-SW instead of SW | SNGAN · CelebA (64x64), L=10 | FID: 11.14 ± 1.12 -> 11.00± 0.04 | No clear effect |
| Use LCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=1000 | FID: 14.07±0.97 -> 13.58±0.12 | Helped |
| Use UCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=1000 | FID: 14.07±0.97 -> 13.21±0.49 | Helped |
| Use LCV-SW instead of SW | SNGAN · CIFAR10 (32x32), L=1000 | IS: 8.07± 0.06 -> 8.23± 0.02 | Helped |
| Use UCV-SW instead of SW | SNGAN · CelebA (64x64), L=1000 | FID: 10.05± 0.40 -> 9.51 ± 0.30 | Helped |
| Use LCV-SW instead of SW | SW point-cloud gradient flow · ShapeNet Core-55, step 8000, L=10 | W2: 2.1e-5±1.0e-5 -> 1.5e-5±0.2e-5 | Helped |
| Use LCV-SW instead of SW | SW projected-distance estimator · MNIST 0-1 | Variance: 4700.87± 59.12 -> 0.045±0.008 | Helped |
Surprising findings
- The upper-bound control variate UCV-SW needs more projections than LCV-SW to approximate the optimal coefficient well, so it is sometimes worse at L=10 on generative modeling.
Limitations
- Control variate estimator is only asymptotically unbiased after p-th root with finite L.
- Upper-bound variant requires more projections to estimate the optimal coefficient well.
- Gaussian approximation of projected measures may be inaccurate for non-Gaussian projections.
Experimental setup
Datasets: MNIST ShapeNet Core-55 CIFAR10 CelebA
Benchmarks: MNIST ShapeNet Core-55 CIFAR10 CelebA
Metrics: estimation error variance wasserstein-2 distance FID inception score computational time
Baselines: SW (conventional Monte Carlo estimator) sliced Wasserstein generator
Builds on: sliced Wasserstein distance distributional sliced Wasserstein sliced Wasserstein generator SNGAN
Closest papers by meaning
- Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates (ICML 2024 · similarity 0.93)
- Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances (NeurIPS 2022 · similarity 0.92)
- Slicing Wasserstein over Wasserstein via Functional Optimal Transport (ICLR 2026 · similarity 0.92)
- Quasi-Monte Carlo for 3D Sliced Wasserstein (ICLR 2024 · similarity 0.89)
- Sliced Wasserstein with Random-Path Projecting Directions (ICML 2024 · similarity 0.89)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.89)
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