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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

Indonesian

Geometric algebra (GA) menyatukan primitif geometris dan transformasi dalam kerangka algebra tunggal. Jaringan saraf yang dirancang di sekitar kerangka ini sering mewakili fitur seperti multivektor dan operasi lapisan membatasi untuk menghormati struktur algebra, memungkinkan perawatan prinsip dari transformasi geometris dan, dalam banyak kasus, built-in equivariance. Survei ini meninjau jaringan saraf algebra geometris, termasuk blok bangunan mereka, sifat equivariant, domain aplikasi, kinerja empiris, dan dukungan perangkat lunak. Kami menyediakan panduan pemilihan algebra, perbandingan empiris dari transformasi geometris berdasarkan GA dan arsitektur ekuivariant lainnya, dan analisis mode kegagalan yang mencakup patologi pelatihan, pembatasan implementasi, dan batasan kinerja. Di seluruh literatur yang diteliti, bukti terku

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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