Newest papers

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

Japanese

ジオメトリックアルジェベラ(GA)は、ジオメトリック原理と変換を単一のアルジェベラフレームに統合します。このフレームワークの周りに設計されたニューラルネットワークは、多くの場合、アルジェベラ構造を尊重するためのマルチベクターおよび制限層操作などの機能を表します。この調査では、ジオメトリックアルジェベラのニューラルネットワークを調査し、その構造ブロック、エクイヴァリアント属性、アプリケーションドメイン、実験的パフォーマンス、およびソフトウェアサポートを含みます。我々は、アルジェベラ選択ガイド、GAベースと他のエクイヴァリアントアーキテクチャーの実験的比較、および訓練病理

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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