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Geometric Algebra for Neural Networks: A Survey

Tao Zhong, Kaison Fong, Keqin Wang, Christine Allen-Blanchette · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.30121 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

Geometric algebra (GA) 在一个单一的 algebraic 框架中统一了几何原理和转变。围绕这个框架设计的神经网络通常代表了多重引擎和限制层操作等特征,以尊重 algebraic 结构,允许对基于 GA 和其他等级架构的原理处理,并在许多情况下嵌入到等级中。本调查审查了几何原理神经网络,包括它们的构建块,等级属性,应用领域,经验性能和软件支持。我们提供了一个 algebra-selection 指南,对基于 GA 和其他等级架构的经验比较,以及覆盖培训病理学,部署限制和性能限制的故障模式分析。在调查的文献中,基于 GA 基础架构的最强有力的证据出现在明确的

Machine translation (Meta M2M100); the English original follows.

Abstract

Geometric algebra (GA) unifies geometric primitives and transformations in a single algebraic framework. Neural networks designed around this framework often represent features as multivectors and constrain layer operations to respect the algebraic structure, enabling principled treatment of geometric transformations and, in many cases, built-in equivariance. This survey reviews geometric algebra neural networks, including their building blocks, equivariant properties, application domains, empirical performance, and software support. We provide an algebra-selection guide, an empirical comparison of GA-based and other equivariant architectures, and a failure-mode analysis covering training pathologies, deployment constraints, and performance limitations. Across the surveyed literature, the strongest evidence for GA-based architectures appears in settings with explicit geometric symmetries and heterogeneous geometric entities; evidence for broader advantages in parameter efficiency, interpretability, and dense prediction is comparatively limited and uneven. We therefore conclude that geometric algebra neural networks are not universally superior to well-tuned non-GA equivariant baselines, and outline open problems in algebra selection, numerical stability, scalable implementation, and reproducible evaluation.

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Topics: Equivariant map, Geometric algebra, Conformal geometric algebra, Algebra over a field, Artificial neural network, Geometric networks

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