Convergence Rates for Distribution Matching with Sliced Optimal Transport
This work addresses distribution matching for computational statistics, but it is incremental as it focuses on Gaussian cases and builds on existing sliced optimal transport methods.
The paper tackles the problem of distribution matching using sliced optimal transport, deriving non-asymptotic convergence rates for Gaussian distributions and showing that eigenvalues can be controlled with random orthonormal bases.
We study the slice-matching scheme, an efficient iterative method for distribution matching based on sliced optimal transport. We investigate convergence to the target distribution and derive quantitative non-asymptotic rates. To this end, we establish __ojasiewicz-type inequalities for the Sliced-Wasserstein objective. A key challenge is to control along the trajectory the constants in these inequalities. We show that this becomes tractable for Gaussian distributions. Specifically, eigenvalues are controlled when matching along random orthonormal bases at each iteration. We complement our theory with numerical experiments and illustrate the predicted dependence on dimension and step-size, as well as the stabilizing effect of orthonormal-basis sampling.