Riemannian Geometry in Cognitive Memory

[sbharatjoshi]

🗣️ Discussion: Riemannian Geometry in Cognitive Memory

Research discussion about applying differential geometry to Spector's memory scoring and entity representation.

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Research Directions

1. Hyperbolic Embeddings (Poincaré Ball)

• Hierarchical entity relationships (is-a taxonomies) in hyperbolic space
• Exponentially more space near boundary — ideal for tree structures
• d(u,v) = arccosh(1 + 2||u-v||² / ((1-||u||²)(1-||v||²)))

2. Spherical Affect Space

• Valence-arousal emotions mapped to unit sphere
• Great-circle distance for emotion-aware recall
• Wraps around smoothly — no edge effects

3. Tangent Space Projection for HNSW

• Project manifold points to tangent space for Euclidean HNSW
• Preserves local geometry while enabling standard ANN search

4. Product Manifold Scoring

• Combine R^n (semantic) × B^n (hierarchical) × S² (affect) × R+ (temporal)
• Each factor has its own geometry, fused via weighted product distance

Why Riemannian Geometry?

Standard Euclidean spaces distort hierarchies and circular structures. Riemannian manifolds respect the intrinsic geometry of different data types:

• Trees → Hyperbolic space
• Emotions → Spherical space
• Semantics → Euclidean space
• Time → Half-line

Questions for Discussion

• Is the mathematical complexity justified for the recall quality improvement?
• Should this be a separate research module or integrated into the core scorer?
• How do we benchmark geometry-aware recall vs standard cosine similarity?
• Should we start with just hyperbolic (simplest non-Euclidean) and validate?

Related Issues

#79 Hyperbolic Embeddings, #80 Tangent Space Projection, #82 Spherical Affect, #83 Product Manifold

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This is bleeding-edge research. We'd love input from mathematicians and ML researchers!