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Smooth $p$-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply Gaussian smoothing before computing the Wasserstein distance | Empirical p-Wasserstein estimation · Empirical approximation, d ≥ 3 | expected empirical convergence rate: n^{-1/d} -> n^{-1/2} | Helped |
| Use the elementary maximal-coupling bound instead of the Sobolev comparison argument | Smooth p-Wasserstein empirical estimation · Empirical approximation | expected empirical convergence rate: O(n^{-1/(2p)}) -> O(n^{-1/2}) | Hurt |
| Compare the smooth Wasserstein distance to a smooth Sobolev IPM and use its Donsker limit theory | Smooth p-Wasserstein empirical estimation · Empirical approximation under the stated tail condition | expected empirical convergence rate: O(n^{-1/(2p)}) -> O(n^{-1/2}) | Helped |
| Use the MMD formulation for d_2^(σ) | Smooth Sobolev IPM computation · Arbitrary dimension d | kernel evaluation complexity for expected error ε: O(ε^{-4}) | Helped |
| Increase the smoothing parameter σ | Smooth p-Wasserstein empirical estimation · Uniform distributions on [-1,1]^d | empirical convergence | Helped |
| Use the Sobolev IPM upper bound as a practical approximation to smooth W_2 | Smooth p-Wasserstein estimation · Uniform distributions on [-1,1]^d | approximation tightness | Hurt |
| Increase σ beyond the range compatible with the stated sub-Gaussian condition | Empirical smooth Sobolev IPM d_2^(1) · Gaussian distributions, d = 5 | convergence to the limit distribution | Hurt |
| Use bootstrap-calibrated smooth Wasserstein test statistics | Two-sample testing · H0: μ = ν | false-alarm probability | No clear effect |
| Fit the model by minimum smooth Wasserstein estimation | Generative modeling · One-dimensional Gaussian-mixture fitting | estimator convergence rate: n^{-1/2} | Helped |
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