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

arXiv:2603.08965v1 Announce Type: new 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 Poincar\'e ball $\mathbb{B}^d$. At coarse scales ($\sigma \to \infty$), diffusion aggregates embeddings into high-level summaries; at fine scales ($\sigma \to 0$), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error $O(\sigma)$ 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 rep

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Edward Izgorodin · · 1 min read · 7 views
#cs.LG #cs.AI

arXiv:2603.08965v1 Announce Type: new 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 Poincar\'e ball $\mathbb{B}^d$. At coarse scales ($\sigma \to \infty$), diffusion aggregates embeddings into high-level summaries; at fine scales ($\sigma \to 0$), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error $O(\sigma)$ 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 ($\tau = 0.79$), demonstrating that the method discovers meaningful abstraction levels in real-world knowledge graphs without supervision.

Executive Summary

The article introduces Semantic Level of Detail (SLoD), a novel framework for multi-scale knowledge representation using heat kernel diffusion on hyperbolic manifolds. SLoD addresses the critical gap in AI memory systems by enabling continuous zoom between abstraction levels via a mathematically grounded mechanism. The framework leverages heat kernel diffusion on the Poincaré ball to manage resolution control, aggregating embeddings at coarse scales and preserving detail at fine scales. The authors provide theoretical guarantees on hierarchical coherence and distortion bounds, and identify spectral gaps as automatic indicators of scale transitions. Empirical validation on synthetic and real-world datasets—HSBM and WordNet—demonstrates robust boundary detection with high accuracy and alignment with taxonomic depth. This represents a significant advancement in scalable, adaptive knowledge modeling.

Key Points

  • ▸ Introduction of SLoD as a continuous zoom operator via heat kernel diffusion
  • ▸ Theoretical bounds on approximation error and distortion for tree-structured hierarchies
  • ▸ Automatic detection of scale boundaries via spectral gaps without manual parameters

Merits

Theoretical Rigor

The authors provide rigorous mathematical proofs for hierarchical coherence and distortion bounds, enhancing credibility and applicability.

Empirical Validation

Application to both synthetic and real-world knowledge graphs (WordNet) confirms the practical relevance and effectiveness of the framework.

Demerits

Assumption Dependency

The framework relies on specific embedding assumptions (e.g., Sarkar embedding), which may limit applicability to non-tree or heterogeneous graph structures.

Scalability Constraints

Performance implications of diffusion-based diffusion on very large graphs remain unaddressed, potentially affecting applicability in massive-scale AI systems.

Expert Commentary

This work represents a pivotal shift in the paradigm of knowledge representation within AI. By formalizing abstraction control via diffusion on hyperbolic manifolds, the authors transcend conventional static graph-level abstractions and introduce a dynamic, mathematical mechanism for adaptive resolution. The identification of spectral gaps as indicators of emergent scale transitions is particularly innovative—it transforms an opaque, parameter-dependent process into an observable, algorithmic phenomenon. While the Sarkar embedding dependency introduces a limitation, the framework’s ability to detect meaningful abstraction levels without supervision on real-world data (e.g., WordNet) signals a major leap toward intelligent, self-organizing knowledge architectures. The implications extend beyond AI memory systems into semantic web, ontology design, and cognitive modeling domains. The paper bridges theoretical mathematics and practical AI in a manner rarely achieved, offering a blueprint for future adaptive systems.

Recommendations

  • ✓ 1. Extend SLoD to heterogeneous and directed graphs beyond tree structures via adapted diffusion variants.
  • ✓ 2. Integrate SLoD into open-source knowledge graph platforms (e.g., Wikidata, RDF) to enable adaptive resolution for downstream NLP and AI applications.

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