Paper Overview
- Field: NLP
- Author: James Pustejovsky
- Published: 2026-04-29
- arXiv: 2504.21168
- Develops the formal foundations of FGA for typed, compositional semantics.
- Identifies three core capabilities that GA offers beyond conventional linear algebra.
- Provides a detailed worked example demonstrating operator-level semantic contrast.
- Shows how GA-based operations implicitly present in current Transformer architectures can be made explicit and extended.
Summary
Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable empirical success, yet they face persistent structural limitations in compositional semantics, type sensitivity, and interpretability. The paper argues that geometric algebra (GA) -- specifically, Clifford algebras -- provides a mathematically superior foundation for semantic representation, and that a Functional Geometric Algebra (FGA) framework extends GA toward a typed, compositional semantics capable of supporting inference, transformation, and interpretability while retaining full compatibility with distributional learning and modern neural architectures.
Key contributions
Core claim
The proposal is not about adding dimensions but about adding structural organization: GA extends an n-dimensional embedding space into a 2^n multivector algebra, where base semantic concepts and their higher-order interactions are represented within a single, principled algebraic framework.
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*Auto-collected on 2026-04-30*