arXiv cs.AI· Ziheng Chen·· 14 小时前AI 评分26
黎曼深度学习：模块、网络与几何
Riemannian Deep Learning: Modules, Networks, and Geometries
AI 导读
一篇博士论文提出黎曼深度学习统一框架，从可复用神经模块、流形专用网络架构与底层几何设计三个层面展开。工作将批归一化从欧氏空间推广到李群与陀螺群，并把多项逻辑回归扩展到 SPD 流形及一般黎曼流形。论文还给出双曲空间无约束模型、基于 Busemann 的双曲学习、满秩相关矩阵网络，以及 SPD 流形上可学习的 Log-Euclidean 几何与基于 Cholesky 的快速稳定几何。
正文 · 原文
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.
| Comments: | PhD thesis manuscript, University of Trento; defense pending |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Differential Geometry (math.DG) |
| Cite as: | arXiv:2607.19305 [cs.LG] |
| (or arXiv:2607.19305v5 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2607.19305 arXiv-issued DOI via DataCite |
Submission history
From: Ziheng Chen [view email]
[v1]
Tue, 21 Jul 2026 17:17:35 UTC (23,055 KB)
[v2]
Wed, 22 Jul 2026 21:13:10 UTC (23,055 KB)
[v3]
Sun, 9 Aug 2026 18:07:22 UTC (23,055 KB)
[v4]
Tue, 15 Sep 2026 21:42:44 UTC (31,050 KB)
[v5]
Thu, 29 Oct 2026 08:57:44 UTC (23,055 KB)
来源：arXiv cs.AI · arxiv.org