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Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use projection-averaging instead of classic Wasserstein distance for robust estimation | Sliced Wasserstein estimation under TV contamination · Distributions with bounded projected qth moments | minimax robust estimation risk: classic \Wp acquires an extra \sqrt{d} factor | Helped |
| Use average-sliced rather than max-sliced distance for robust estimation | Sliced Wasserstein estimation under TV contamination · Distributions in \cG_q(\sigma) | minimax robust estimation risk: \sigma \sqrt{(1 \lor d/q)(1 \land p/d)} \,\eps^{1/p-1/q} vs. \sigma \eps^{1/p-1/q} | Helped |
| Increase dimension under bounded covariance and bounded mean difference | Monte Carlo estimator for average-sliced Wasserstein · Log-concave distributions | Monte Carlo integration error: 1/\sqrt{md} | Helped |
| Use subgradient ascent instead of Riemannian optimization | Max-sliced Wasserstein computation · Fragmented hypercube, across tested dimensions | iteration count to convergence | Helped |
| Use subgradient ascent instead of Riemannian optimization | Max-sliced Wasserstein computation · Fragmented hypercube, across tested dimensions | computation time | Helped |
| Use LIPO for global max-sliced optimization rather than local subgradient optimization | Max-sliced Wasserstein computation · High-dimensional settings | number of function evaluations: exponentially with dimension | Hurt |
| Estimate \MSWone with iterative filtering and sliced distance rather than classic \Wone | Robust estimation · Synthetic contamination experiment | dimension dependence of estimation error: \sqrt{d} | Helped |
| Use the subgradient-based method without regularization | Max-sliced Wasserstein computation for one-dimensional projections · Empirical max-sliced Wasserstein with p=2 | computational complexity: O(\epsilon^{-4}) | No clear effect |
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