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

Khai Nguyen, Nhat Ho

TL;DR

This work introduces Hierarchical Hybrid Sliced Wasserstein (H2SW), a scalable metric for comparing heterogeneous joint distributions whose marginals lie on different domains. It builds Hierarchical Hybrid Radon Transform (HHRT) by composing Partial Generalized Radon Transforms (PGRT) on marginals with a final Partial Radon Transform, yielding an injective transformation and enabling a Wasserstein-based distance across heterogeneous supports. The paper establishes the metric properties, analyzes sample complexity and Monte Carlo estimation, and demonstrates practical advantages over SW/GSW in 3D mesh deformation, deep mesh autoencoding, and Hadamard-manifold dataset comparisons. The results suggest H2SW as a robust tool for scalable, topology-aware distribution comparison across mixed domains, with potential extensions to more marginals and domain types.

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.

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

TL;DR

This work introduces Hierarchical Hybrid Sliced Wasserstein (H2SW), a scalable metric for comparing heterogeneous joint distributions whose marginals lie on different domains. It builds Hierarchical Hybrid Radon Transform (HHRT) by composing Partial Generalized Radon Transforms (PGRT) on marginals with a final Partial Radon Transform, yielding an injective transformation and enabling a Wasserstein-based distance across heterogeneous supports. The paper establishes the metric properties, analyzes sample complexity and Monte Carlo estimation, and demonstrates practical advantages over SW/GSW in 3D mesh deformation, deep mesh autoencoding, and Hadamard-manifold dataset comparisons. The results suggest H2SW as a robust tool for scalable, topology-aware distribution comparison across mixed domains, with potential extensions to more marginals and domain types.

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.
Paper Structure (21 sections, 6 theorems, 41 equations, 11 figures, 4 tables)

This paper contains 21 sections, 6 theorems, 41 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hierarchical Hybrid Sliced Wasserstein Distance
  4. Hierarchical Hybrid Radon Transform
  5. Hierarchical Hybrid Sliced Wasserstein Distance
  6. Experiments
  7. 3D Mesh Deformation
  8. Training deep 3D mesh autoencoder
  9. Comparing Datasets on The Product of Hadamard Manifolds
  10. Conclusion
  11. Proofs
  12. Proof of Proposition \ref{['prop:PGRT_injectivity']}
  13. Proof of Proposition \ref{['prop:HHRT_injectivity']}
  14. Proof of Theorem \ref{['theo:metricity']}
  15. Proof of Proposition \ref{['prop:connection_sliced']}
  16. ...and 6 more sections

Key Result

Proposition 1

For some defining function $g$ such as linear, circular, and homogeneous polynomials with an odd degree; the Partial Generalized Radon Transform is injective, i.e., for any functions $f_1,f_2 \in \mathbb{L}^1(\mathbb{R}^d)$, $(\mathcal{PGR}f_1)(t,\theta,y) = (\mathcal{PGR}f_2)(t,\theta,y)$$\forall t

Figures (11)

  • Figure 1: Generalized Radon Transform and Hierarchical Hybrid Radon Transform on a discrete distribution.
  • Figure 2: Visualization of deformation from the sphere mesh to the Armadillo mesh with $L=10$.
  • Figure 3: Visualization of deformation from the sphere mesh to the Stanford Bunny mesh with $L=10$.
  • Figure 4: Visualization of some randomly selected reconstruction meshes from autoencoders trained by SW, GSW, and H2SW in turn with the number of projections $L=100$ at epoch 2000.
  • Figure 5: Cost matrices between datasets from SW, CHSW, and H2SW with $L=2000$.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 1: Partial Generalized Radon Transform
  • Proposition 1
  • Definition 2: Hierarchical Hybrid Radon Transform
  • Proposition 2
  • Example 1
  • Definition 3
  • Theorem 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 2 more