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Semantic Level of Detail: Multi-Scale Knowledge Representation via Heat Kernel Diffusion on Hyperbolic Manifolds

Edward Izgorodin

TL;DR

It is shown that spectral gaps in the graph Laplacian induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- which can be detected automatically without manual resolution parameters, and shown that spectral gaps in the graph Laplacian induce emergent scale boundaries which can be detected automatically without manual resolution parameters.

Abstract

AI memory systems increasingly organize knowledge into graph structures -- knowledge graphs, entity relations, community hierarchies -- yet lack a principled mechanism for continuous resolution control: where do the qualitative boundaries between abstraction levels lie, and how should an agent navigate them? We introduce Semantic Level of Detail (SLoD), a framework that answers both questions by defining a continuous zoom operator via heat kernel diffusion on the Poincaré ball $\mathbb{B}^d$. At coarse scales ($σ\to \infty$), diffusion aggregates embeddings into high-level summaries; at fine scales ($σ\to 0$), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error $O(σ)$ and $(1+\varepsilon)$ distortion for tree-structured hierarchies under Sarkar embedding. Crucially, we show that spectral gaps in the graph Laplacian induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- which can be detected automatically without manual resolution parameters. On synthetic hierarchies (HSBM), our boundary scanner recovers planted levels with ARI up to 1.00, with detection degrading gracefully near the information-theoretic Kesten-Stigum threshold. On the full WordNet noun hierarchy (82K synsets), detected boundaries align with true taxonomic depth ($τ= 0.79$), demonstrating that the method discovers meaningful abstraction levels in real-world knowledge graphs without supervision.

Semantic Level of Detail: Multi-Scale Knowledge Representation via Heat Kernel Diffusion on Hyperbolic Manifolds

TL;DR

It is shown that spectral gaps in the graph Laplacian induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- which can be detected automatically without manual resolution parameters, and shown that spectral gaps in the graph Laplacian induce emergent scale boundaries which can be detected automatically without manual resolution parameters.

Abstract

AI memory systems increasingly organize knowledge into graph structures -- knowledge graphs, entity relations, community hierarchies -- yet lack a principled mechanism for continuous resolution control: where do the qualitative boundaries between abstraction levels lie, and how should an agent navigate them? We introduce Semantic Level of Detail (SLoD), a framework that answers both questions by defining a continuous zoom operator via heat kernel diffusion on the Poincaré ball . At coarse scales (), diffusion aggregates embeddings into high-level summaries; at fine scales (), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error and distortion for tree-structured hierarchies under Sarkar embedding. Crucially, we show that spectral gaps in the graph Laplacian induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- which can be detected automatically without manual resolution parameters. On synthetic hierarchies (HSBM), our boundary scanner recovers planted levels with ARI up to 1.00, with detection degrading gracefully near the information-theoretic Kesten-Stigum threshold. On the full WordNet noun hierarchy (82K synsets), detected boundaries align with true taxonomic depth (), demonstrating that the method discovers meaningful abstraction levels in real-world knowledge graphs without supervision.
Paper Structure (42 sections, 5 theorems, 13 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 42 sections, 5 theorems, 13 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. The Poincaré Ball Model
  5. Exponential and Logarithmic Maps.
  6. Heat Kernel on Hyperbolic Space
  7. Fréchet Mean on Hadamard Manifolds
  8. Semantic Level of Detail
  9. Problem Formulation
  10. Diffusion-Based Aggregation
  11. Practical Approximation.
  12. Theoretical Analysis
  13. Scalability.
  14. Why Hyperbolic?
  15. Emergent Scale Selection
  16. ...and 27 more sections

Key Result

Theorem 1

Let $\mathcal{T}$ be a weighted tree with $n$ nodes embedded in $\mathbb{B}^d$ via a Sarkar-type embedding with distortion $\delta = (1+\varepsilon)$ for $\varepsilon > 0$. For any node $v$ at depth $d_v$ and $\sigma_1 < \sigma_2$: where $\mathcal{V}_v$ is the set of descendants of $v$ and $C$ depends only on the curvature.

Figures (3)

  • Figure 1: Effective dimensionality $K^*(\sigma)$ as a function of diffusion scale for varying signal strength $r$. Horizontal dashed lines mark planted hierarchy levels ($K=2$ macro, $K=8$ meso, $K=64$ micro). At high $r$, $K^*$ exhibits clear transitions at the planted levels; at low $r$ (below the Kesten--Stigum threshold), transitions blur.
  • Figure 2: Phase transition in boundary recovery. (a) ARI at macro ($K^*=2$) and meso ($K^*=8$) scales vs. hierarchy strength $r$; shaded region marks the Kesten--Stigum threshold. (b) Spectral gap $\lambda_3/\lambda_2$ grows with $r$, confirming increasing signal separability. (c) Signal-to-noise ratio at macro and meso levels; the dashed line marks SNR$\,=1$.
  • Figure 3: Detected boundary scale $\sigma^*$ vs. true ancestor depth for 100 stratified leaf nodes in WordNet ($N=82{,}115$). Strong positive correlation (Kendall $\tau = 0.79$) confirms that larger diffusion scales correspond to shallower (more abstract) ancestors.

Theorems & Definitions (14)

  • Remark 1: No bifurcation on $\mathbb{B}^d$
  • Definition 1: Heat Kernel Weights
  • Definition 2: Semantic LOD Operator
  • Remark 2: Practical computation
  • Theorem 1: Hierarchical Coherence
  • proof : Proof sketch
  • Theorem 2: Scale-Dependent Approximation
  • proof : Proof sketch
  • Proposition 1: Computational Complexity
  • Proposition 2: Spectral Scale Boundaries
  • ...and 4 more