Log-Euclidean geometry

Unified framework for Riemannian deep learning: modules, architectures, and geometries

A doctoral thesis proposes a unified framework for deep learning on manifold-valued representations, addressing limitations of existing approaches tied to specific manifolds or relying on Euclidean approximations. The work generalizes batch normalization to Lie groups and gyrogroups, extends multinomial logistic regression to SPD and general Riemannian manifolds, and introduces new architectures for hyperbolic space and correlation matrices. It also contributes adaptive, computationally efficient Riemannian metrics including learnable Log-Euclidean and Cholesky-based geometries, validated across vision, signal processing, graph learning, and genomics tasks.