ConceptGA: Geometric Algebra for Concept Representation and Reasoning
Abstract
We present ConceptGA, a novel framework for representing and manipulating concepts using k-blades from Geometric Algebra instead of traditional vector embeddings. Unlike standard embeddings where concepts are points on a hypersphere, ConceptGA represents concepts as directed volumes (k-blades) where grade corresponds to abstraction level and orientation encodes asymmetric relations.
1. Introduction
Standard embeddings map concepts to vectors on a hypersphere — similarity is cosine angle. This has limitations:
| Limitation | Vector Embeddings | ConceptGA (k-blades) |
|---|---|---|
| Compositionality | Linear superposition only | Geometric product encodes relations (meet/join) |
| Hierarchy | Flat | Grade = conceptual abstraction level |
| Directionality | Single orientation | Edge directions = asymmetric relations (IS-A, PART-OF, CAUSES) |
| Negation | -v = opposite point | Dual = orthogonal complement = "not this concept" |
| Intersection | Not defined | Meet (\(\wedge\)) = shared subspace = common substructure |
2. Algebraic Foundations
2.1 Geometric Algebra G(V, Q)
Let V be an n-dimensional vector space. The geometric product of vectors a, b:
\[ ab = a \cdot b + a \wedge b \]
where a·b is the symmetric inner product and a∧b is the antisymmetric outer product.
2.2 k-Blades as Concepts
A k-blade is the outer product of k linearly independent vectors:
\[ B_k = v_1 \wedge v_2 \wedge \dots \wedge v_k \]
- Grade k = abstraction level (1 = atomic, higher = composite)
- Magnitude = salience/centrality of concept
- Basis indices = defining features
- Orientation = asymmetric relations (IS-A, PART-OF, CAUSES)
3. Core Operations
3.1 Similarity
\[ \operatorname{sim}(A, B) = \langle A \widetilde{B} \rangle_0 \quad \text{(scalar part of geometric product)} \]
3.2 Meet (Intersection)
\[ A \vee B = \langle A^{*} \cdot B \rangle \quad \text{(dual of } A \text{ contracted with } B\text{)} \]
3.3 Join (Union)
\[ A \wedge B \quad \text{(outer product)} \]
3.4 Dual (Negation)
\[ \neg A = A^{*} = A I^{-1} \quad \text{(dual w.r.t. pseudoscalar)} \]
4. Reasoning Operations
4.1 Analogy
\[ \begin{aligned} R_{A \to B} = B A^{-1} \quad \text{(versor transformation)} \\[2pt] D = R_{A \to B}\, C = B A^{-1} C \end{aligned} \]
4.2 Abstraction/Specification
\[ \begin{aligned} \operatorname{abstract}_j(A) = \langle A \rangle_j \quad \text{(drop features)} \\[2pt] \operatorname{specify}(A, F) = A \wedge F \quad \text{(add features)} \end{aligned} \]
5. CKB Integration
Each CKB belief becomes a concept blade:
"Vaccines reduce severe disease"
→ e_vaccine ∧ e_reduce ∧ e_severe ∧ e_disease ∧ r_CAUSES
Surprise as prediction error:
\[ \operatorname{surprise} = 1 - \operatorname{sim}_w(P, O) \]
Counterfactuals:
\[ \mathrm{CF}_{\neg F}(B) = B \wedge \neg F \]
6. Evaluation
| Metric | Target |
|---|---|
| Belief similarity nDCG@10 | > Vector baseline |
| Contradiction detection F1 | > Textual entailment |
| Analogy completion accuracy | > Word2vec/GPT |
| Surprise prediction AUC | > Heuristic |
7. Conclusion
ConceptGA provides a mathematically rigorous, compositional framework for concept representation that naturally supports hierarchy, relations, negation, and reasoning operations — all grounded in the single algebra that also powers spatial reasoning (SGA engine).