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Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates
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Proposes SHCV, a Monte Carlo estimator of the sliced-Wasserstein distance that uses spherical harmonics as control variates fitted by ordinary least squares, giving exactness for some Gaussian cases and a faster-than-MC convergence rate.
Problem
Monte Carlo estimation of the sliced-Wasserstein distance over random projections has O(n^{-1/2}) integration error, which limits accuracy at a fixed number of projections.
Approach
Treat the SW integrand as a function on the sphere and use spherical harmonics (an orthonormal basis of L2 of the sphere, centered under the uniform measure) as many control variates, estimating the integral as the OLS intercept over projections.
Key results
- In the exact Gaussian case with proportional covariances, SHCV achieves zero error once degree-2 harmonics are included, while baselines do not.
- On sampled Gaussians in d=5 with n=500 projections, SHCV reaches MSE 2.68e-6 versus 1.45e-3 for MC and 5.80e-5 for RQMC.
- Kernel SVM on digits with gamma=1 improves test accuracy from 93.3% (MC) to 94.3% (SHCV) at n=75 and n=100 projections.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use spherical harmonics as control variates instead of RQMC | Sliced-Wasserstein estimation · Sampled Gaussian, d=5, n=500 | MSE: 5.80e-5 -> 2.68e-6 | Helped |
| Use spherical harmonics as control variates instead of QMC | Sliced-Wasserstein estimation · Sampled Gaussian, d=10, n=500 | MSE: 2.37e-4 -> 1.93e-4 | Helped |
| Use spherical harmonics as control variates instead of CV_low | Sliced-Wasserstein estimation · Sampled Gaussian, d=20, n=500 | MSE: 3.82e-4 -> 2.95e-4 | Helped |
| Use spherical harmonics as control variates instead of RQMC | Sliced-Wasserstein estimation · Exact Gaussian, d=5, n=500 | MSE: 7.5e-4 -> 4.8e-5 | Helped |
| Use spherical harmonics as control variates instead of QMC | Sliced-Wasserstein estimation · ShapeNetCore plane/lamp, n=1000 | MSE: 2.8e-8 -> 2.1e-8 | Helped |
| Use spherical harmonics as control variates instead of RQMC | Sliced-Wasserstein estimation · ShapeNetCore lamp/bed, n=1000 | MSE: 1.3e-7 -> 1.4e-8 | Helped |
| Use spherical harmonics as control variates instead of CVNN | Sliced-Wasserstein estimation · ShapeNetCore plane/bed, n=1000 | MSE: 3.3e-7 -> 3.6e-8 | Helped |
| Estimate the Sliced-Wasserstein kernel using SHCV instead of MC | Kernel SVM · Digits test, gamma=0.01, n=25 | accuracy: 86.0±0.6 -> 87.3±0.3 | Helped |
| Estimate the Sliced-Wasserstein kernel using SHCV instead of MC | Kernel SVM · Digits test, gamma=1, n=25 | accuracy: 92.7±0.6 -> 93.3±0.6 | Helped |
| Estimate the Sliced-Wasserstein kernel using SHCV instead of MC | Kernel SVM · Digits test, gamma=0.01, n=100 | accuracy: 86.9±0.3 -> 87.7±0.1 | Helped |
| Estimate the Sliced-Wasserstein kernel using SHCV instead of MC | Kernel SVM · Digits test, gamma=1, n=100 | accuracy: 93.3±0.3 -> 94.3±0.2 | Helped |
Surprising findings
- The SW integrand is even, so odd-degree spherical harmonics can be dropped from the control variate set without loss.
Limitations
- Convergence rate of O_P(L^{-1} n^{-1/2}) requires L = o(n^{1/(2(d-1))}), giving only about n^{-3/4} in d=3 and degrading with dimension.
- Method is developed for the uniform distribution on the sphere; extension to general projection distributions is left to importance sampling.
- Number of control variates is fixed in advance rather than selected by lasso with cross-validation.
Experimental setup
Datasets: ShapeNetCore digits (scikit-learn)
Benchmarks: synthetic Gaussian SW_2^2 estimation ShapeNetCore 3D point cloud SW estimation digits kernel SVM classification
Metrics: mean squared error computing time (ms) test accuracy
Baselines: Monte Carlo CV_low CV_up CVNN QMC RQMC
Builds on: OLS Monte Carlo control variates sliced-Wasserstein distance lower/upper CV Gaussian control variates (Leluc et al. / Nadjahi et al.) control neighbors randomized quasi-Monte Carlo sliced-Wasserstein
Closest papers by meaning
- Sliced Wasserstein Estimation with Control Variates (ICLR 2024 · similarity 0.93)
- Quasi-Monte Carlo for 3D Sliced Wasserstein (ICLR 2024 · similarity 0.87)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.86)
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.86)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.85)
- Scalable Approximation Algorithms for $p$-Wasserstein Distance and Its Variants (ICML 2025 · similarity 0.85)
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