Semantic Level of Detail (SLoD)
Multi-scale knowledge representation via heat kernel diffusion on hyperbolic manifolds.
Edward Izgorodin — arXiv:2603.08965
TL;DR
- SLoD brings Level of Detail from computer graphics to semantic data — a continuous zoom parameterized by scale
sigma, from individual facts to global themes.- The substrate is hyperbolic space: the Poincaré ball embeds tree-structured hierarchies with
O(log n)distortion, and heat-kernel diffusion provides the zoom.- A boundary scanner detects natural abstraction levels from spectral gaps — no manual tuning required.
- Backed by theoretical guarantees (hierarchical coherence,
O(sigma)approximation error) and validation on HSBM and WordNet; production does not yet run native hyperbolic geometry.
The problem
AI memory systems store knowledge as flat vector collections, discarding the hierarchical structure inherent in semantic information. A software project has architecture-level concepts, module-level patterns, and line-level details — but a vector database treats them all the same.
The idea
SLoD borrows Level of Detail from computer graphics — where 3D engines render geometry at variable resolution depending on distance — and builds an analogous operator for semantic data.
The key insight: hyperbolic space is the natural substrate. The Poincare ball has exponential volume growth, embedding tree-structured hierarchies with distortion O(log n) — provably optimal. Heat kernel diffusion on this space provides a continuous zoom parameterized by scale sigma:
sigma -> 0 Fine detail (individual facts, specific memories)
sigma -> inf Global summary (high-level themes, abstract concepts)Components
| Component | What it does | Paper reference |
|---|---|---|
| SLoD Operator | Heat kernel weights + Frechet mean on Poincare ball | Algorithm 1, Def. 2 |
| Hierarchical Coherence | Bounded error O(sigma), distortion O(log n) | Theorems 1-2 |
| Boundary Scanner | Detects natural abstraction levels from spectral gaps | Algorithm 2, Prop. 3 |
| Multi-Center Extension | Mixture representation when single summary is lossy | Def. 3, Corollary |
Boundary detection
Three complementary signals detect where the representation undergoes qualitative transitions:
- V(sigma) — representation velocity: how fast does the summary move in hyperbolic space?
- D_w(sigma) — weight divergence: Jensen-Shannon between consecutive weight distributions
- C_k(sigma) — neighborhood churn: do the nearest neighbors change?
Peaks in the composite score reveal natural abstraction boundaries — no manual tuning required.
Theoretical guarantees
Theorem 1 (Hierarchical Coherence): For a tree with n nodes embedded in the Poincare ball:
d_H(Phi_s1, Phi_s2) <= C * |s2 - s1| * log(n)Nearby scales produce semantically related representations.
Theorem 2 (Approximation Error): Memories within cognitive distance R can be approximated with error O(sigma).
Proposition 3 (Spectral Boundaries): JSD peaks near sigma* ~ 1/lambda_k when spectral gap ratio exceeds threshold R.
Experimental validation
| Experiment | Data | Key result | Status |
|---|---|---|---|
| Boundary Recovery | HSBM (1024 nodes, 3 levels) | ARI macro=1.00, meso=0.89 [0.86, 0.92] at r=200 | Complete (8/8 pass) |
| WordNet Consistency | WordNet 3.0 (82K synsets, depth 19) | Kendall tau=0.79, Recall@2=0.75 | Complete |
Citation
@article{izgorodin2026slod,
title = {Semantic Level of Detail for Knowledge Graphs: Discovering
Abstraction Boundaries via Spectral Heat Diffusion},
author = {Izgorodin, Edward},
year = {2026},
eprint = {2603.08965},
archivePrefix = {arXiv}
}Paper
- arXiv: 2603.08965
- Format: 21 pages (arXiv v2, revised May 2026)
Common questions
What is Semantic Level of Detail (SLoD)?
SLoD borrows Level of Detail from computer graphics — where 3D engines render geometry at variable resolution depending on distance — and builds an analogous operator for semantic data, so memory can be read at multiple scales from individual facts to high-level themes.
Why does SLoD use hyperbolic space?
Hyperbolic space is the natural substrate: the Poincaré ball has exponential volume growth, embedding tree-structured hierarchies with distortion O(log n) — provably optimal — which matches the hierarchical structure inherent in semantic information.
What does the scale parameter sigma control in SLoD?
sigma is the scale of a heat-kernel diffusion: as sigma → 0 the representation is fine detail (individual facts, specific memories); as sigma → ∞ it becomes a global summary (high-level themes, abstract concepts).
How does SLoD detect abstraction boundaries?
A boundary scanner combines three signals — representation velocity V(sigma), weight divergence D_w(sigma), and neighborhood churn C_k(sigma) — whose composite peaks reveal natural abstraction boundaries, with no manual tuning required.
Is SLoD running in the production memory engine?
No. Production does not yet run native hyperbolic geometry — SLoD is the research operator, and its geometry is implemented in the Tensor-Hyperbolic Graphs library.
Related
- Tensor-Hyperbolic Graphs — the library implementing SLoD geometry
- Building Memory That Scales — why production does not yet run native hyperbolic geometry
- Manifesto — why hyperbolic geometry matters for memory
- Scientific sources — bibliography underpinning the approach
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