Slicing Wasserstein over Wasserstein via Functional Optimal Transport

Core task: comparing distributions over non-Euclidean objects using optimal transport. The field has evolved into a rich taxonomy spanning seven major branches that address distinct but interconnected challenges. Computational Methods and Algorithmic Frameworks focus on efficient solvers and projection-based techniques such as sliced approaches, which reduce high-dimensional transport problems to tractable one-dimensional projections. Non-Euclidean Geometries and Manifold Structures tackle transport on curved spaces—ranging from hyperbolic geometries to Riemannian manifolds—where classical Euclidean assumptions break down. Reproducing Kernel Hilbert Spaces and Feature Embeddings leverage kernel methods to handle complex data types, while Theoretical Foundations and Mathematical Analysis provide rigorous convergence guarantees and geometric insights. Regression, Inference, and Statistical Modeling apply transport-based distances to prediction and uncertainty quantification, Applications and Domain-Specific Methods demonstrate utility in biology, recommendation systems, and medical imaging, and Specialized Formulations and Extensions explore variants like unbalanced or Gromov-Wasserstein distances that relax standard constraints. Within the computational branch, sliced and projection-based methods have emerged as a particularly active line of work, balancing scalability with approximation quality. Sliced Optimal Transport Introduction[1] and Non-Euclidean Sampling Software[47] illustrate how projections can be extended beyond Euclidean settings, while Sliced Fused Gromov-Wasserstein[22] combines slicing with structure-aware matching. The original paper, Slicing Wasserstein Functional[0], sits squarely in this projection-based cluster, contributing new theoretical or algorithmic insights into how slicing strategies generalize to non-Euclidean domains. Compared to Non-Euclidean Sliced Sampling[8], which emphasizes sampling procedures on manifolds, Slicing Wasserstein Functional[0] appears to focus more directly on the functional analytic properties of sliced transport. Meanwhile, works like Gaussian Process Wasserstein Kernels[2] and RKHS Optimal Transport[14] pursue complementary kernel-based embeddings, highlighting an ongoing trade-off between projection simplicity and the expressiveness of feature-space methods.