Riemannian Deep Learning:Modules, Networks, and Geometries
2026-07-21 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors created a new general approach to help deep learning work better with data that lives on curved spaces, called manifolds. They improved common techniques like batch normalization and logistic regression to work on many types of special geometric spaces, not just flat ones. They also built new neural network designs for tricky spaces like hyperbolic space and correlation matrices, making the computations faster and more stable. Their work was tested theoretically and on practical problems in areas like vision and genomics.
Riemannian manifoldbatch normalizationlogistic regressionLie groupsgyrogroupshyperbolic spaceSPD manifoldsCholesky decompositiondeep neural networks
Authors
Chen Ziheng
Abstract
Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.