ConceptGA: Geometric Algebra for Concept Representation

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:

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·b + a∧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 ∧ v_2 ∧ ... ∧ 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

sim(A, B) = ⟨A B̃⟩₀  (scalar part of geometric product)

3.2 Meet (Intersection)

A ∨ B = ⟨A* · B⟩  (dual of A contracted with B)

3.3 Join (Union)

A ∧ B  (outer product)

3.4 Dual (Negation)

¬A = A* = A I⁻¹  (dual w.r.t. pseudoscalar)

4. Reasoning Operations

4.1 Analogy

R_{A→B} = B A⁻¹  (versor transformation)
D = R_{A→B} C = B A⁻¹ C

4.2 Abstraction/Specification

abstract_j(A) = ⟨A⟩_j     (drop features)
specify(A, F) = A ∧ F      (add features)

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:

surprise = 1 - sim_w(P, O)

Counterfactuals:

CF_¬F(B) = B ∧ ¬F

6. Evaluation

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).