Riemannian Sliced-Wasserstein Distances on Correlation Matrices

Covariance matrices are widely used in Riemannian learning, but they change with marginal signal amplitudes even when the underlying dependence structure remains similar. This paper studies sliced-Wasserstein regularization on full-rank correlation matrices. We consider two geometric branches. The first uses four zero-curvature pullback metrics ECM, LECM, OLM, and LSM, for which the geodesic and Busemann coordinates differ only by sign and CorSW is equivalent to Euclidean sliced-Wasserstein in the corresponding coordinate space. The second branch, PHCML, lifts normalized Cholesky coordinates to the Lorentz model and gives two distinct discrepancies based on Busemann and geodesic projections. We use these discrepancies either alone or together with covariance-based SPD sliced-Wasserstein regularization. Experiments across multiple datasets and network architectures show that CorSW consistently improves downstream performance. For the four flat metrics, caching the embeddings reduces subsequent projections to $\mathcal{O}(d^2)$ inner-product operations per sample