Shape Dissimilarity for Product-Spherically Embedded Trajectories

Traditional shape analysis in Euclidean space involves constructing pre-shapes by quotienting out Euclidean uniform scaling and translation. However, these operations are fundamentally undefined on spheres or product spheres. To overcome this geometric incompatibility, we introduce a novel framework for the shape analysis of product-spherically embedded trajectories that reside on or can be equivalently mapped to a product sphere. We propose the $\text{P}$roduct-$\text{S}$pherical $\text{S}$hape $\text{D}$issimilarity (PSSD) measure to rigorously quantify the shape differences between two such trajectories. Theoretically, we establish that the PSSD measure evaluates to zero if and only if two trajectories are equivalent up to spherical uniform scaling and reference-based spherical translation. Furthermore, we develop two algorithms: one extracts intrinsic spherical pre-shapes that remain embedded in the product sphere, and the other aligns the rotation of these pre-shapes. We also propose a rigorous smoothing estimator that recovers trajectories from discrete observations and guarantees uniform convergence. Extensive empirical evaluations across various synthetic and real-world datasets demonstrate that the PSSD measure achieves superior discriminative power in complex non-Euclidean classification tasks.