From Embedding Geometry to Spectral Search: Energy Dispersion Networks for Vector Retrieval

Jul 16, 2026

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High-dimensional embedding spaces are commonly used through geometric relationships, yet their dense semantic structure also induces a spectral energy-network view. The work formalizes embeddings as spectral energy networks derived from the topology of the underlying feature-space manifold, enabling improvements for downstream vector retrieval. It introduces Graph Wiring and its search instantiation Spectral Indexing, coupling geometric similarity with spectral signals to enhance head–tail coherence, semantic alignment, and adaptive τ-modulated search in RAG pipelines. The paper provides a complete algorithm, theoretical support via epiplexity, and evaluations on benchmarks and industrial settings using the arrowspace library.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages24

Reading Time1 h 0 min

Introduction

Semantic gap and limitations of geometric scoring

Proposed approach: Graph Wiring and Spectral Indexing (SPIN)

Theoretical foundation via epiplexity

Evaluation setup and datasets

Experimental results

FROM EMBEDDING GEOMETRY TO SPECTRAL SEARCH: ENERGY DISPERSION NETWORKS FOR VECTOR RETRIEVAL Lorenzo Moriondo [tuned.org.uk-Genefold AI](tuned.org.uk-Genefold AI)[ ](tuned.org.uk-Genefold AI)[tunedconsulting@gmail.com](tunedconsulting@gmail.com)[ ](tunedconsulting@gmail.com)[lorenzo@genefold.ai](lorenzo@genefold.ai) Ilias Azizi LIPADE, Université Paris Cité France [ilias.azizi@u-pariscite.fr](ilias.azizi@u-pariscite.fr) arXiv :2606 .2 1535v2 [ cs .IR] 13 Jul 2026 ABSTRACT High-dimensional vector spaces, particularly embedding spaces with dense semantic structure, are often interpreted primarily leveraging solely geometric relationships. In this work, we show that they can also be viewed as spectral energy networks induced by the topology of their underlying featurespace manifold with relevant improvements for downstream tasks. Building on this perspective, we introduce Graph Wiring, a general framework for exploiting feature-space spectral structure, together with Spectral Indexing, its task-specific instantiation for vector search. By coupling geometric similarity with spectral information, the proposed method improves head–tail coherence and semantic alignment relative to purely geometric retrieval methods. It further supports adaptive search behavior through τ-modulation, providing the flexibility increasingly required by modern Retrieval-Augmented Generation (RAG) pipelines. We present the complete algorithmic pipeline, establish its theoretical foundation through epiplexity, and evaluate the approach across benchmark and industrial settings using the open-source arrowspace library. Keywords physical networks, spectral graph theory, search, vector search, vector similarity, ranking, RAG, OOD, epiplexity, structural information 1 Introduction Vector retrieval systems overwhelmingly rely on cosine similarity, inner product, or related vector-matrix product scores as their primary ranking signal [1, 2] . Given a query and a corpus item, cosine similarity measures their angular proximity in the ambient embedding space, but it does not explicitly account for the statistical and topological structure of the corpus from which those embeddings are drawn [3, 4] . This creates a descriptive limitation: items may be geometrically close while belonging to different regions of the feature manifold, whereas geometrically distant items may still share structural affinity through the corpus topology. This limitation is especially visible in the tail of ranked retrieval results, where weakly grounded or noisy evidence can affect downstream applications (i.e Retrieval-Augmented Generation systems) [5, 6] . A result set with high average cosine similarity may still be topologically scattered: its items can originate from or induce poorly connected corpus communities; therefore providing the language model with a cluttered context despite strong local geometric scores. Similarly, out-of-distribution (OOD) queries receive no explicit warning signal from cosine scoring alone: a query that falls in a spectrally irregular or high-energy region of the feature space can be ranked in the same way as a well-grounded in-distribution query [7] . We refer to this missing structural component as a semantic gap: a corpus-level signal that isnot directly encoded by pairwise geometric similarity but is relevant for semantic retrieval, this gap can be addressed by semantic search leveraging spectral information [8, 9] . This paper introduces SPectral INdexing (SPIN), a search-specific procedure that leverages the feature-space graph Laplacian LF for retrieval and ranking that addresses the semantic gap, providing fully-featured semantic search. SPIN uses the spectral signal induced by Graph Wiring (GW), a physical-network-inspired framework for topological search, to capture corpus-level structure that is not visible to geometric search. SPIN constructs a feature-space graph (as computed by GW) to expose spectral structure from the e

FAQs

Why can cosine similarity be insufficient for vector retrieval?

Cosine scoring measures angular proximity in the ambient embedding space but does not explicitly reflect the statistical and topological structure of the corpus manifold. This can lead to geometrically close items being topologically scattered, harming semantic retrieval and failing to flag out-of-distribution queries.

What are Graph Wiring (GW) and Spectral Indexing (SPIN)?

Graph Wiring is a general framework for exploiting feature-space spectral structure via a feature-space graph induced by a topology-oriented construction. Spectral Indexing is the retrieval- and ranking-specific instantiation that augments geometric similarity with discrete spectral signals derived from the graph Laplacian.

How does SPIN improve retrieval quality and RAG behavior?

By coupling geometric similarity with spectral information, SPIN improves head–tail coherence and semantic alignment compared to purely geometric retrieval. It also supports adaptive search behavior via τ-modulation, offering flexibility needed by modern Retrieval-Augmented Generation pipelines.

Jul 16, 2026

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