这篇论文把黎曼深度学习做成了统一理论，把批量归一化、逻辑回归推广到各种流形，还有新的双曲模型和高效几何，搞几何深度学习的别错过。
这篇论文提出了黎曼深度学习的统一框架，涵盖可复用模块、流形特定网络和底层几何设计。它将批量归一化从欧氏空间和单一流形推广到李群和陀螺群等广泛类别，将多项逻辑回归从欧氏空间扩展到SPD流形再推广到一般黎曼流形。论文还开发了无约束双曲空间模型、Busemann双曲学习以及全秩相关矩阵的神经网络。此外，引入了SPD流形上可学习的Log-Euclidean几何和快速的Cholesky几何。方法在视觉、信号处理、图学习和基因组学等应用中通过实验验证。
Riemannian Deep Learning:Modules, Networks, and Geometries
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.