Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data

Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into onedimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into L 2 spaces, where the L 2 distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into L 2 spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications. * Data used in preparation of this article were partially obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this paper. A complete listing of ADNI investigators can be found at: https://adni.loni.usc.edu/ wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf . Contributions. In summary, our contributions are as follows: 1. We propose Linear Spherical Sliced Optimal Transport (LSSOT) to embed spherical distributions into L 2 space while preserving their intrinsic spherical geometry. 2. We prove that LSSOT defines a metric, and demonstrate the superior computation efficiency over other baseline metrics. 3. We conduct a comprehensive set of experiments to show the effectiveness and efficiency of LSSOT in diverse applications, from point cloud analysis to cortical surface registration. BACKGROUND 2.1 CIRCULAR OPTIMAL TRANSPORT AND LINEAR CIRCULAR OPTIMAL TRANSPORT Consider two circular probability measures µ, ν ∈ P(S 1 ), where S 1 denotes the unit circle in R 2 . Let us parametrize S 1 with the angles in between 0 and 1 and consider the cost function c(x, y) := h(|x -y| S 1 ), where h : R → R + is a convex increasing function and |x -y| S 1 := min|x -y|, 1 -|x -y| for x, y ∈ [0, 1). The Circular Optimal Transport (COT) problem between µ and ν is defined by the following two equivalent minimization problems: where in the first expression Γ(µ, ν) is the set of all couplings between µ and ν, and in the second expression F µ (respectively, F ν ) is the cumulative distribution function of µ on S 1 (i.e., F µ (y) := µ([0, y)) = y 0 dµ, ∀y ∈ [0, 1)) extended on R by F µ (y + n) = F µ (y) + n, ∀y ∈ [0, 1), n ∈ Z, and its inverse (or quantile function) is defined by F -1 µ (y) = infx : F µ (x) > y. When h(x) = |x| p for 1 ≤ p < ∞, we denote COT h (•, •) as COT p (•, •), and COT p (•, •) 1/p defines a metric on P(S 1 ). Moreover, if µ = Unif(S 1 ) and h(x) = |x| 2 , the minimizer α µ,ν of (1) is the antipodal of E(ν), i.e.,