Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

TL;DR

This work introduces parallelly sliced OT on the sphere ${\mathbb S}^{d-1}$ and the rotation group ${\rm SO}(3)$, reducing multivariate OT to 1D problems via a novel parallel slice transform that yields a rotationally invariant metric on spherical measures. It develops a two-dimensional Radon transform on ${\rm SO}(3)$ with a complete singular value decomposition, enabling sliced OT on rotations and enabling efficient barycenter computations. The paper provides both free- and fixed-support barycenter algorithms and Radon-based barycenters, with extensive synthetic experiments on the 2-sphere demonstrating speedups (40–100x) over semicircular slicing while maintaining accuracy in most settings. The approach offers fast, scalable tools for barycenter computations in spherical and rotational domains, with potential extensions to higher-dimensional spheres and gradient-flow applications on manifolds.

Abstract

Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere $\mathbb{S}^{d-1}$ and on the rotation group SO(3). We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For SO(3), we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on SO(3). As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free and fixed support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

Figures (11)

• Figure 1: Our proposed parallel slicing (left): each red circle is projected to a single point on the blue line segment. The vertical slicing QueBeiSte23 can be regarded as special case that keeps the blue line direction fixed. Semicircular slicing from Bon23 (right): each red semicircle is projected to one point on the blue circle.
• Figure 2: PSB (left) and SSB (right) of two vMF distributions. The green and orange points represent the two input measures, and the blue points the barycenter.
• Figure 3: PSB (left) and SSB (right) of highly concentrated measures. The green and orange points represent the two input measures, and the blue points the barycenter.
• Figure 4: PSB (left) and SSB (right) for 2 "croissants" measures. The green and orange points represent the two input measures, and the blue points represent the barycenter.
• Figure 5: Loss function (left) and step norm (right) on $\mathbb{S}^{d-1}$ for $d=3$ (top row) and $d=10$ (bottom row) depending on the number of iterations. Reported are the mean and standard deviation (vertical lines) over 100 runs.

Theorems & Definitions (46)

• Proposition 1: Integration in $t$
• Theorem 3.1: Positivity
• proof
• Theorem 3.2: Singular value decomposition
• proof
• Theorem 3.3: Adjoint
• proof
• Proposition 2: Connection with adjoint
• proof
• Proposition 3: Absolutely continuous measures