Identifiability in Finite Views for Spherical Sliced Optimal Transport: Geometry, Couplings, and the Critical Regime

Spherical Sliced-Wasserstein (SSW) distances compare distributions supported on the hypersphere $\mathbb{S}^{d-1}$ through projections onto great circles, where circular optimal transport is computable in closed form. Empirically, $\widehat{\mathrm{SSW}}_p$ losses are optimized over a finite number of views $L<\infty$. Here, we develop the identifiability theory of spherical sliced OT in finite views. We begin with the identification of the critical two-view regime on $\mathbb{S}^2$ ($L=2$) for general $n$: candidate ghost atoms may arise if and only if atoms of the target distribution lie in the same random sign hemisphere, thus allowing us to split competing measures into independent transportation polytopes. As a result, $n=1$ is identifiable almost surely; $n=2$ is identifiable with precise probability $d_{\mathbb{S}^2}(x_1,x_2)/\pi\in(0,1]$; and, for all $n\ge3$, non-identifiability is almost surely established using the Pigeonhole Principle. Given the presence of ghosts, we further find the minimum possible true support mass for arbitrary weights and show that 100% false support is achievable if and only if no atom has more than half of the weight in its sign block. Moreover, whereas in Euclidean space $\mathbb{R}^d$ a minimum of $d+1$ views is needed to reconstruct discrete targets through linear projections, we show that on $\mathbb{S}^{d-1}$ ($d\ge3$), $L\ge d$ generic views guarantee almost-sure identifiability of any discrete target distribution versus arbitrary admissible probability measures, and construct an open set embedding which establishes that $L=d$ is the sharp minimal universal guarantee for all $d\ge3$. Matched particle optimization experiments (200 runs) illustrate that, although $L\ge d$ precludes any spurious global zero-loss solutions, non-convex optimizer traps persist.