统一大规模流形上的深度学习框架，提升模型通用性与计算效率。
Riemannian Deep Learning: Modules, Networks, and Geometries
- 提出可复用的流形神经模块，支持广义李群与旋群的统一建模
- 实现流形上批归一化与逻辑回归扩展，支持双曲空间与协方差矩阵表示
- 设计高效稳定的流形度量，适合视觉、信号、基因组等多领域应用
流形上的深度神经网络受到越来越多关注，但许多基础组件仍局限于特定流形、依赖欧氏近似或需高成本且数值不稳定的几何运算。本文从三个互补视角构建统一的黎曼深度学习框架：可复用的神经模块、流形特异的网络架构，以及底层几何结构的设计。将批归一化从欧氏空间和单一流形推广至广义李群与旋群，将多项式逻辑回归从欧氏空间扩展至对称正定（SPD）流形，并进一步推广至一般黎曼流形。开发了多种重要几何表示的神经网络，包括无约束双曲空间模型、基于Busemann函数的双曲学习，以及满秩相关矩阵表示。最后，提出自适应且计算高效的SPD流形度量，包含可学习的Log-Euclidean几何与快速稳定的基于Cholesky分解的几何。所提方法经理论分析与数值实验验证，在视觉、信号处理、图学习和基因组学等领域取得应用成效。
原文摘要 · Abstract (English)
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.
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