Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions
arxiv(2024)
摘要
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.
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