GLASS: A Differentiable Geometric Alignment Layer on Manifolds and Graphs via Learned Slices and Soft Optimal Transport

We introduce GLASS (Geometric Learned Alignment via Soft Slices), a lightweight, fully differentiable layer for aligning non-Euclidean representations on compact manifolds and graphs. GLASS learns intrinsic one-dimensional projections, performs entropic one-dimensional optimal transport (a smooth soft sorting), and lifts the soft plan back to the ambient geometry to produce a geometry-aware loss and a reusable coupling. We prove equivariance to isometries and automorphisms, approximation guarantees to manifold $W_2$ and a diffusion-feature surrogate on graphs up to curvature, truncation, and entropy terms, uniform generalization bounds, and end-to-end differentiability. On synthetic spheres and stochastic block model graphs, GLASS achieves compelling accuracy and efficiency with near-linear runtime via banded Sinkhorn, and outperforms naive baselines when gate widths are tied or learned relative to capacity. The layer is plug-and-play and directly applicable as a geometry-aware alignment head inside foundation-model pipelines operating over manifolds and graphs.