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Non-asymptotic convergence bounds for Wasserstein approximation using point clouds
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply one Lloyd step using optimal Power-cell barycenters to a sufficiently separated point cloud | Uniform quantization of a probability density · Uniform quantization | squared Wasserstein distance: C_{c,d,\Omega} N^{-1/d} | Helped |
| Apply one Lloyd step to a cloud whose minimum separation scales as N^{-1/β}, with β>d−1 | Uniform quantization · Uniform quantization | squared Wasserstein distance: \frac{C_{d,\Omega}}{C^{d-1}}N^{-\alpha} | Helped |
| Sample the initial cloud i.i.d. from a bounded density and apply one Lloyd step | Uniform quantization · Uniform quantization, probabilistic case | squared Wasserstein distance: N^{-\frac{1}{2d-1}} | Helped |
| Initialize with a random uniform cloud rather than an axis-aligned grid | One-step Lloyd quantization of a Gaussian density · Gaussian density on the unit square | quantization error F_N(B_N(Y_N)): N^{-0.8} -> N^{-0.95} | Helped |
| Apply one Lloyd step from a regular grid | Uniform quantization of a gray-scale image · Gray-scale image | squared Wasserstein distance: N^{-1.00} | Helped |
| Use an initial cloud concentrated on a (d−1)-dimensional set | One-step Power-cell barycenters · Uniform quantization | squared Wasserstein distance: \frac{1}{12} | Hurt |
| Use discrete gradient descent from a separated initial cloud with timestep tending to zero | Uniform quantization · Uniform quantization | squared Wasserstein distance: O_{N\to\infty} \left( \Wass_2^2\left(\rho,\delta_{Y_N^0}\right)^{1-\frac{1}{d}} .N^{\frac{-1}{d^2}+\alpha\left(1-\frac 1 | Helped |
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