Neural Projection Fusion for Sliced Wasserstein on the Hypersphere

Abstract

The Spherical Sliced Wasserstein (SSW) distance has emerged as a fundamental tool for comparing probability measures on the hypersphere, with broad applications spanning geology, medical imaging, computer vision, and representation learning. However, its reliance on uniformly sampling a large number of projection directions from the unit hypersphere can lead to suboptimal performance, as many directions are weakly informative and fail to capture salient differences between distributions. We address this limitation through Fusion Stereographic Spherical Sliced Wasserstein (FS3W), a data-adaptive divergence that incorporates distribution-dependent information into projection selection. FS3W uses a neural slicing fusion mechanism to refine prior directions, steering them toward projections aligned with discriminative geometric structures in the data. This adaptive approach produces more expressive and accurate characterizations of distributional discrepancies on the hypersphere. We evaluate FS3W across five diverse tasks, including gradient flows, Earth density estimation, and self-supervised representation learning. Empirical results consistently demonstrate that FS3W outperforms existing spherical optimal transport–based methods.