Sliced Optimal Transport (OT) simplifies the OT problem in high-dimensional spaces by projecting supports of input measures onto one-dimensional lines and then exploiting the closed-form expression of the univariate OT to reduce the computational burden of OT. Recently, the Tree-Sliced method has been introduced to replace these lines with more intricate structures, known as tree systems. This approach enhances the ability to capture topological information of integration domains in Sliced OT while maintaining low computational cost. Inspired by this approach, in this paper, we present an adaptation of tree systems on OT problems for measures supported on a sphere. As a counterpart to the Radon transform variant on tree systems, we propose a novel spherical Radon transform with a new integration domain called spherical trees. By leveraging this transform and exploiting the spherical tree structures, we derive closed-form expressions for OT problems on the sphere. Consequently, we obtain an efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) distance. We provide an extensive theoretical analysis to demonstrate the topology of spherical trees and the well-definedness and injectivity of our Radon transform variant, which leads to an orthogonally invariant distance between spherical measures. Finally, we conduct a wide range of numerical experiments, including gradient flows and self-supervised learning, to assess the performance of our proposed metric, comparing it to recent benchmarks.

Spherical Sliced-Wasserstein (SSW) has recently been proposed to measure the discrepancy between spherical data distributions in various fields, such as geology, medical domains, computer vision, and deep representation learning. However, in the original SSW, all projection directions are treated equally, which is too idealistic and cannot accurately reflect the importance of different projection directions for various data distributions. To address this issue, we propose a novel data-adaptive Discriminative Spherical Sliced-Wasserstein (DSSW) distance, which utilizes a projected energy function to determine the discriminative projection direction for SSW. In our new DSSW, we introduce two types of projected energy functions to generate the weights for projection directions with complete theoretical guarantees. The first type employs a non-parametric deterministic function that transforms the projected Wasserstein distance into its corresponding weight in each projection direction. This improves the performance of the original SSW distance with negligible additional computational overhead. The second type utilizes a neural network-induced function that learns the projection direction weight through a parameterized neural network based on data projections. This further enhances the performance of the original SSW distance with less extra computational overhead. Finally, we evaluate the performance of our proposed DSSW by comparing it with several state-of-the-art methods across a variety of machine learning tasks, including gradient flows, density estimation on real earth data, and self-supervised learning.

Applications involving distributions defined on a hypersphere are remarkably diverse, highlighting the importance of spherical geometries across various disciplines. These applications include: 1) mapping the distribution of geographic or geological features on celestial bodies, such as stars and planets [39, 8, 60], 2) magnetoencephalography (MEG) imaging [75] in medical domains, 3) spherical image representations and 360 images [13, 38], such as omnidirectional images in computer vision [40], 4) texture mapping in computer graphics [24, 21], and more recently, 5) deep representation learning, where the latent representation is often mapped to a bounded space, commonly a sphere, where cosine similarity is utilized for effective representation learning [11, 76]. The analysis of distributions on hyperspheres is traditionally approached through directional statistics, also referred to as circular/spherical statistics [37, 52, 50, 61]. This specialized field is dedicated to the statistical analysis of directions, orientations, and rotations. More recently, with the growing application of optimal transport theory [74, 62] in machine learning, due in part to its favorable statistical, geometrical, and topological properties, there has been an increasing interest in using optimal transport to compare spherical probability measures [14, 32]. One of the main bottlenecks in optimal transport theory is its high computational cost, generally of cubic complexity.