Title: Rigid invariant sliced Wasserstein via independent embeddings
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: optimal transport, Wasserstein, geometric embedding, pseudometric, PCA

Abstract: 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 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 the Gaussian case 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.

TL;DR: a rigid invariant distance for shape registration via independent sliced Wasserstein, with promising empirical performance and scalable computational cost