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

German

Geometrische Algebra (GA) vereint geometrische Primitiven und Transformationen in einem einzigen algebraischen Rahmen. Neuralnetze, die um diesen Rahmen entworfen wurden, repräsentieren häufig Funktionen wie Multivektoren und Einschränkungen der Schichtoperationen, um die algebraische Struktur zu respektieren, wodurch eine Prinzipienbehandlung von geometrischen Transformationen und in vielen Fällen eingebaute Äquivalenz ermöglicht wird. Diese Umfrage untersucht geometrische Algebra-Nervennetzwerke, einschließlich ihrer Bausteine, Äquivalent-Eigenschaften, Anwendungsdomänen, empirische Leistung und Software-Unterstützung. Wir bieten eine Algebra-Auswahl-Anleitung, einen empirischen Vergleich von GA-basierten und anderen äquivalenten Architekturen

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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