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Quasi-Monte Carlo for 3D Sliced Wasserstein
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Proposes quasi-sliced Wasserstein (QSW) and randomized QSW (RQSW) approximations of the sliced Wasserstein distance using QMC point sets on the 3D unit hypersphere, with proofs of asymptotic convergence and unbiasedness and experiments on 3D point-cloud tasks.
Problem
Monte Carlo estimation of the Sliced Wasserstein distance has suboptimal approximation error because i.i.d. uniform directions on the hypersphere are not well spread; stochastic optimization with SW also needs unbiased gradient estimates.
Approach
Replace random projecting directions with quasi-Monte Carlo point sets on the 3D unit sphere (Gaussian-based mapping, equal area mapping, generalized spiral, distance maximization, Coulomb energy minimization), assess them with spherical cap discrepancy, and randomize them (scrambling pushforward or random rotation) to obtain unbiased RQSW estimators.
Key results
- QSW variants (notably CQSW and DQSW) give lower absolute approximation error of SW_2^2 than Monte Carlo SW across number of projections L from 10 to 10000 on 3D point clouds.
- In point-cloud interpolation with L=100, RQSW variants such as RSQSW, RDQSW, and RCQSW give lower W_2 than SW at steps 100-500, while deterministic QSW variants fail to produce terminating curves.
- In deep point-cloud autoencoder training with L=100, CQSW performs best at the last epoch, and random rotation RQSW variants perform consistently well; the authors recommend RCQSW as a safe choice.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace conventional Monte Carlo directions with QMC directions | Sliced Wasserstein approximation · ShapeNet point-cloud comparison, approximation error | absolute approximation error | Helped |
| Use randomized QMC directions instead of fixed deterministic QMC directions | Point-cloud interpolation · ShapeNet points 1–3, L=100 | W2: $0.004\pm0.001$ -> $0.002\pm0.0$ | Helped |
| Use deterministic QMC directions instead of conventional Monte Carlo directions | Point-cloud interpolation · ShapeNet points 1–3, L=100 | W2: $0.004\pm0.001$ -> $0.065\pm0.0$ | Hurt |
| Use Gaussian-based mapping points rather than conventional Monte Carlo directions | Image style transfer · Epoch 400, L=100 | W2: $9.21\pm0.06$ -> $37.99 \pm 0.05$ | Hurt |
| Use QSW or RQSW directions rather than conventional Monte Carlo directions | Image style transfer · Epoch 400, L=100 | W2: $9.21\pm0.06$ -> $37.99 \pm 0.05$; $9.21\pm0.06$ -> $9.06 \pm 0.02$ | Mixed |
Surprising findings
- Deterministic QSW variants do not converge to smooth shapes in point-cloud interpolation because a fixed set of directions is reused across all steps, making them worse than MC SW there.
- Gaussian-based mapping QMC point sets perform worst among QMC constructions and GQSW suffers numerical issues in autoencoder training.
Limitations
- Only the 3D setting (d=3) is studied; extension to higher dimensions is left to future work.
- Deterministic QSW can be worse than MC for optimization-based applications with small L, and results vary across variants without a single dominant choice.
- Gaussian-based and equal-area constructions have weaker discrepancy properties than spiral or energy-based point sets.
Experimental setup
Datasets: ShapeNet Core-55 ModelNet40
Benchmarks: ShapeNet Core-55 point cloud approximation point-cloud interpolation Wasserstein-2 image style transfer Wasserstein-2 ModelNet40 autoencoder reconstruction
Metrics: absolute approximation error of SW_2^2 W_2 distance SW_2 distance spherical cap discrepancy runtime
Baselines: SW (Monte Carlo sliced Wasserstein) near-orthogonal Monte Carlo GQSW EQSW SQSW CQSW DQSW
Builds on: sliced Wasserstein distance Monte Carlo integration sobol sequence orthogonal Monte Carlo
Closest papers by meaning
- Sliced Wasserstein Estimation with Control Variates (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 0.87)
- Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates (ICML 2024 · similarity 0.87)
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.85)
- Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis (ICML 2025 · similarity 0.85)
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