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Sliced Optimal Transport on the Sphere

Michael Quellmalz, Robert Beinert, Gabriele Steidl

TL;DR

This work extends sliced optimal transport to the sphere by introducing two spherical Radon-type transforms: the vertical slice transform $\mathcal{V}$ and the normalized semicircle transform $\mathcal{W}$. It develops rigorous definitions for both transforms on measures, derives their singular value decompositions, proves injectivity results, and defines spherical sliced Wasserstein distances $\mathrm{VSW}_p$ and $\mathrm{SSW}_p$. The authors implement discretizations via specialized quadratures and fast transform algorithms, and address inversion through Moore–Penrose pseudoinverses and a variational entropy-regularized approach to ensure probability densities. Numerical experiments demonstrate interpolation (barycenters) and classification of spherical measures, highlighting the potential of spherical CDT/CDT-like methods for shape and distribution analysis on the sphere. The proposed framework provides a principled, computationally tractable approach to spherical OT with strong rotational invariance properties and practical applicability to spherical data analysis.

Abstract

Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback--Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.

Sliced Optimal Transport on the Sphere

TL;DR

This work extends sliced optimal transport to the sphere by introducing two spherical Radon-type transforms: the vertical slice transform and the normalized semicircle transform . It develops rigorous definitions for both transforms on measures, derives their singular value decompositions, proves injectivity results, and defines spherical sliced Wasserstein distances and . The authors implement discretizations via specialized quadratures and fast transform algorithms, and address inversion through Moore–Penrose pseudoinverses and a variational entropy-regularized approach to ensure probability densities. Numerical experiments demonstrate interpolation (barycenters) and classification of spherical measures, highlighting the potential of spherical CDT/CDT-like methods for shape and distribution analysis on the sphere. The proposed framework provides a principled, computationally tractable approach to spherical OT with strong rotational invariance properties and practical applicability to spherical data analysis.

Abstract

Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback--Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.
Paper Structure (30 sections, 20 theorems, 144 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 20 theorems, 144 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Outline of the Paper
  4. Preliminaries
  5. Measures and Optimal Transport
  6. Optimal Transport on the Interval
  7. Optimal Transport on the Circle
  8. Sphere and Rotation Group
  9. Unit Sphere
  10. Rotation Group
  11. Vertical Slice Transform
  12. Vertical Slice Transform of Functions
  13. Vertical Slice Transform of Measures
  14. Normalized Semicircle Transform
  15. Normalized Semicircle Transform of Functions
  16. ...and 15 more sections

Key Result

Proposition 1

Let $1 \le p \le \infty$. For every $f \in L^p(\mathbb{S}^2)$, it holds Let $\psi \in \mathbb{T}$. The operators $\mathcal{V}_\psi \colon L^p(\mathbb{S}^2) \to L^p(\mathbb{I})$ and $\mathcal{V} \colon L^p(\mathbb{S}^2) \to L^p(\mathbb{T} \times \mathbb{I})$ are bounded with Moreover, it holds $\mathcal{V}_\psi \colon C(\mathbb{S}^2) \to C(\mathbb{I})$ and $\mathcal{V} \colon C(\mathbb{S}^2) \to

Figures (6)

  • Figure 1: Areas of integration of the spherical transforms.
  • Figure 2: Semicircles $M_{\alpha,\beta}^\gamma$ (red) starting at a fixed point $\Phi(\alpha,\beta)$ and with varying $\gamma \in\mathbb{T}$. Here $\beta$ is the angle of $\Phi(\alpha,\beta)$ to the north pole and $\alpha$ the angle of its projection in the $\xi_1$-$\xi_2$ plane to the $\xi_1$ axis. The blue circle is orthogonal to the semicircles.
  • Figure 3: Density functions of two symmetrized vMF distributions \ref{['eq:vmf']}.
  • Figure 4: CDT interpolation with $\delta=0.5$ of the vMF distributions from \ref{['fig:vmf']}.
  • Figure 5: CDT interpolation of density functions with $\delta=0.5$.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Proposition 1
  • proof
  • Theorem 3.1: HiQu15circav
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 26 more