特伦托大学博士论文：黎曼深度学习的模块、网络与几何统一框架
Riemannian Deep Learning: Modules, Networks, and Geometries
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特伦托大学博士论文提出黎曼深度学习统一框架，从可复用神经模块、面向流形的网络架构和底层几何设计三方面展开。论文将批归一化推广到李群与 gyrogroups，把多项逻辑回归扩展到 SPD 流形及一般黎曼流形，并开发了双曲空间无约束模型、基于 Busemann 的双曲学习和满秩相关矩阵网络，还提出可学习 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