Rigid Invariant Sliced Wasserstein via Independent Embeddings

Zakk Heile, Peilin He, Jayson Tran, Ruiling Wang, Shrikant Chand

Track: long paper (up to 8 pages)

Keywords: embeddings, rigid-invariance, distance, optimal transport, wasserstein, geometric data analysis, shape matching, alignment, registration

TL;DR: We introduce a polynomial-time computable rigid-invariant distance for comparing point clouds or probability distributions that performs at least as good as existing rigid-invariant optimal transport distances while being orders of magnitudes faster.

Abstract: Comparing probability measures modulo unknown rigid transformations is a central challenge in geometric data analysis. Classical optimal transport (OT) distances, including Wasserstein and sliced Wasserstein, are sensitive to rotations and reflections, whereas Gromov-Wasserstein (GW) distances are invariant to isometries but computationally prohibitive for large datasets. We introduce Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE), a scalable distance 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 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.

Anonymization: This submission has been anonymized for double-blind review via the removal of identifying information such as names, affiliations, and identifying URLs.

Submission Number: 32