AI 中文总结
研究为黎曼深度学习构建统一框架，从可复用神经模块等三个角度出发，推广批量归一化等，开发相关神经网络及度量，经理论分析、数值实验和多领域应用验证方法的有效性。
AI 中文摘要
流形值表示上的深度神经网络引发了越来越多的关注，但许多基本组件仍与特定流形相关，依赖欧几里得近似，或需要昂贵且数值不稳定的几何运算。本文从三个互补的角度为黎曼深度学习开发了一个统一框架：可复用神经模块、特定流形的网络架构以及底层几何设计。它将批量归一化从欧几里得空间和单个流形推广到广泛的李群和回旋群，将多项式逻辑回归从欧几里得空间扩展到对称正定（SPD）流形，再到一般黎曼流形。还为几种重要的几何表示开发了神经网络。最后，在SPD流形上引入了自适应且计算高效的黎曼度量。所提方法得到理论分析支持，并通过数值实验及在视觉、信号处理、图学习和基因组学中的应用得到验证。
英文摘要
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.
CommentsPhD thesis manuscript, University of Trento; defense pending
