Transferable Sliced Transport Plans on Riemannian Manifolds

Sliced optimal transport plans are computationally efficient alternatives to optimal transport, producing couplings between measures by lifting 1-D optimal plans to the ambient space. Recently introduced, min-STP optimizes the slicer and is transferable; however, it is defined on compact subsets of $\mathbb R^d$ and utilizes a Laplace perturbation. We analyze what the perturbation must accomplish, reduce those requirements to a single property, and show that they hold exactly when the pairwise increments of the perturbation are atomless. We then find transferability extends to any closed, connected Riemannian manifold with geodesic cost. From this, we establish feature-map perturbations with quantitative bounds and an intrinsic instantiation from the Laplace--Beltrami spectrum. Experimentation on the sphere $\mathbb S^2$ and on a Gaussian random field-perturbed sphere, where the drift is not an isometry and the feature map is estimated from samples, shows that transferred slicers start near the converged optimum, and measures the bi-Lipschitz constants that enter the bounds.