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：切片沃瑟斯坦距离（SW）和广义切片沃瑟斯坦距离（GSW）因其计算与统计可扩展性而被广泛应用于实际场景。然而，SW与GSW仅适用于定义在齐次域上的分布之间的比较。这一限制使其无法应用于边缘分布支持多个不同域的异质联合分布场景。若直接对联合域使用SW或GSW，其齐次切片算子（即拉东变换（RT）与广义拉东变换（GRT））表达能力不足，无法捕捉联合支持集的结构，因此无法进行有意义的比较。为解决该问题，我们提出两种新型切片算子：部分广义拉东变换（PGRT）与分层混合拉东变换（HHRT）。具体而言，PGRT是部分拉东变换（PRT）的推广，可对函数的部分参数进行非线性变换；而HHRT则是由PRT与多个域特定PGRT作用于边缘域参数后组成的复合变换。通过引入HHRT，我们将SW扩展为专门用于比较异质联合分布的分层混合切片沃瑟斯坦距离（H2SW）。随后，我们讨论了H2SW的拓扑性质、统计性质及计算性质。最后，我们在三维网格形变、深度三维网格自编码器及数据集比较任务中验证了H2SW的优越性能。