Geometric Algebra for NLP Semantics

Current natural language processing, dominated by distributional and neural methods, relies heavily on conventional linear algebra. While empirically successful, these approaches exhibit inherent limitations in capturing the nuances of compositional semantics, type sensitivity, and true interpretability. This paper introduces geometric algebra (GA), specifically Clifford algebras, as a mathematically superior foundation for semantic representation, proposing a Functional Geometric Algebra (FGA) framework.

Beyond Vector Spaces: The Power of Multivectors

The core argument posits that GA offers a qualitatively different approach to representing meaning. Instead of merely increasing dimensionality as in current vector-based models, GA expands an $n$-dimensional embedding space into a $2^n$ dimensional multivector algebra. This expansion is not arbitrary; it provides a single, principled algebraic framework where base semantic concepts and their higher-order interactions are naturally organized. This structural richness is GA's primary advantage over traditional linear algebra for advancing geometric algebra natural language semantics.

Enhanced Compositionality and Type Sensitivity

GA's graded type system and multivector representations are designed to address key shortcomings in current semantic models. The framework inherently supports inference and transformation through its algebraic operations, enabling a more robust handling of compositional semantics and type coercion. This is a critical step towards models that not only process language but also understand its underlying structure and meaning with greater fidelity. The paper details three core capabilities that GA provides and linear algebra lacks, illustrating these with operator-level semantic contrasts.

Bridging Theory and Practice in Neural Architectures

The research demonstrates that operations implicit in current transformer architectures can be made explicit and extended using GA. This suggests a path for integrating this more powerful algebraic foundation into existing neural language models without discarding current successes. The ultimate promise is a significant leap in the structural organization and interpretability of natural language semantics, moving beyond empirical performance to a deeper, principled understanding of language.