Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions

Abstract: Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability. However, the SW and the GSW are only defined between distributions supported on a homogeneous domain. This limitation prevents their usage in applications with heterogeneous joint distributions with marginal distributions supported on multiple different domains. Using SW and GSW directly on the joint domains cannot make a meaningful comparison since their homogeneous slicing operator i.e., Radon Transform (RT) and Generalized Radon Transform (GRT) are not expressive enough to capture the structure of the joint supports set. To address the issue, we propose two new slicing operators i.e., Partial Generalized Radon Transform (PGRT) and Hierarchical Hybrid Radon Transform (HHRT). In greater detail, PGRT is the generalization of Partial Radon Transform (PRT), which transforms a subset of function arguments non-linearly while HHRT is the composition of PRT and multiple domain-specific PGRT on marginal domain arguments. By using HHRT, we extend the SW into Hierarchical Hybrid Sliced Wasserstein (H2SW) distance which is designed specifically for comparing heterogeneous joint distributions. We then discuss the topological, statistical, and computational properties of H2SW. Finally, we demonstrate the favorable performance of H2SW in 3D mesh deformation, deep 3D mesh autoencoders, and datasets comparison.