Hierarchical Sliced-Wasserstein on Symmetric Positive Definite Matrices

Comparing distributions of symmetric positive definite (SPD) matrices is a recurring operation in covariance-based learning, but high-dimensional projections can make repeated sliced-Wasserstein evaluations costly. We present SPD hierarchical sliced-Wasserstein (SPDHSW), which computes a small bank of projections in symmetric log-space and mixes their coordinates into many one-dimensional comparisons. Resampling the projection family at each update avoids confining optimization to a fixed subspace. Under Frobenius-uniform sampling, SPDHSW preserves the squared SPDSW population objective. Because the mixed projections share one bank, they are not independent, which prior analysis of hierarchical slicing overlooks; we characterize this dependence through an exact variance decomposition and harmonic analysis, and extend the results to the gradients used in optimization. This establishes separate precision criteria for discrepancy values and training gradients. Experiments on synthetic distributions, hand-action covariances, and EEG adaptation demonstrate improved alignment at a fixed matrix-projection count per update and reduced projection-dominated loop cost relative to large-budget flat slicing under shared optimization settings. Together, the construction and analysis provide a principled approach to allocating projection computation in learning systems that repeatedly compare SPD-valued distributions.