Riemannian Deep Learning: Modules, Networks, and Geometries

What the math gives to ML

The paper provides reusable intrinsic neural modules for non-Euclidean representations, with the clearest transfer opportunities being its unconstrained proper-velocity parameterization of hyperbolic space and its Cholesky-factorized SPD geometry. Proper velocity removes the Lorentz hyperboloid constraint from the learnable spatial tensor, allowing ordinary affine maps and activations while reconstructing the time coordinate analytically. The SPD construction replaces expensive eigendecomposition-based operators with lower-triangular and diagonal operations, making covariance-aware residual blocks and classifiers more practical. These ideas are most promising as drop-in replacements for constrained hyperbolic layers and as efficient covariance-aware heads for vision, graph, and sequence models.