Rigid-Invariant Sliced Wasserstein via Independent Embeddings

Comparing probability measures when their supports are related by an unknown rigid transformation is an important challenge in geometric data analysis, arising in shape matching and machine learning. Classical optimal transport (OT) distances, including Wasserstein and sliced Wasserstein, are sensitive to rotations and reflections, while Gromov-Wasserstein (GW) is invariant to isometries but computationally prohibitive for large datasets. We introduce \emph{Rigid-Invariant Sliced Wasserstein via Independent Embeddings} (RISWIE), a scalable pseudometric that combines the invariance of NP-hard approaches with the efficiency of projection-based OT. RISWIE utilizes data-adaptive bases and matches optimal signed permutations along axes according to distributional similarity to achieve rigid invariance with near-linear complexity in the sample size. We prove bounds relating RISWIE to GW in special cases and empirically demonstrate dimension-independent statistical stability. Our experiments on cellular imaging and 3D human meshes demonstrate that RISWIE outperforms GW in clustering tasks and discriminative capability while significantly reducing runtime.