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.

翻译：切片Wasserstein距离（SW）和广义切片Wasserstein距离（GSW）因其计算与统计可扩展性在各类应用中广泛使用。然而，SW和GSW仅适用于定义在同质域上的分布之间，这一局限阻碍了它们在边际分布支撑于多个不同域的异构联合分布中的应用。直接在联合域上使用SW和GSW无法进行有意义的比较，因为其同质切片算子——即Radon变换（RT）和广义Radon变换（GRT）——的表达能力不足以捕捉联合支撑集的结构。为解决该问题，我们提出两种新型切片算子：偏广义Radon变换（PGRT）与分层混合Radon变换（HHRT）。具体而言，PGRT是偏Radon变换（PRT）的推广，可对函数参数子集进行非线性变换；而HHRT由PRT与多个领域特定PGRT在边际域参数上的复合构成。通过引入HHRT，我们将SW扩展为专为比较异构联合分布设计的分层混合切片Wasserstein距离（H2SW）。随后，我们讨论了H2SW的拓扑性质、统计特性及计算性能。最后，我们在三维网格形变、深度三维网格自编码器以及数据集比较中展示了H2SW的优越表现。