Graph Laplacian retrieval
The mathematics behind spectral retrieval: how a graph over embeddings yields smoothness energies, and how those energies become bounded ranking scores.
Graph Laplacian retrieval builds a graph over feature-space vectors, forms its graph Laplacian L = D − A, and uses the Rayleigh quotient R(x) = xᵀLx / xᵀx as a smoothness energy of each item relative to the graph. A bounded ratio turns this energy into a comparable score used for retrieval and diagnostics.
Setup and notation
Let X = {x₁, …, xₙ} be a set of embedding vectors. Build a weighted, undirected graph G = (V, E, A) whose vertices are the items and whose adjacency matrix A encodes local relationships, typically k-nearest neighbours with similarity-based weights. D is the degree matrix of A, and the graph Laplacian is
This Laplacian is computed on the graph of data items in feature space. It is distinct from the Laplacians used in graph neural networks over message-passing graphs; here the graph is built from the embedding geometry itself.
Smoothness energy
For a vector x with unit-scale energy, the Rayleigh quotient
measures how strongly x is connected relative to its neighbourhood: low values mean the item sits in a smooth, well-connected region; high values mean it is inconsistent with its neighbourhood or weakly connected.
ArrowSpace converts this unbounded energy into a bounded, comparable score by normalising with a small constant ε (Engineering 001):
The bound holds regardless of the corpus energy scale, which is what makes scores comparable across collections, time windows, and embedding model updates. Edge-wise dispersion statistics from the graph complement the Rayleigh quotient in the full taumode score described in the ArrowSpace paper.
Graph wiring
The results depend on how edges are chosen and weighted, which the literature calls graph construction and Genefold calls graph wiring. Wiring parameters (neighbourhood size, weighting scheme, and dispersion-based edge selection) control which structure the Laplacian sees. The empirical study Epiplexity and Graph Wiring evaluates how construction choices affect retrieval behaviour, and Engineering 001 documents the practical pipeline.
From energies to retrieval
The bounded scores enter retrieval as a second signal: the spectral difference between query and item is combined with cosine similarity through a runtime blend parameter. The full formulation and its evaluation are in spectral vector search.
Assumptions and limitations
- The graph must reflect real neighbourhood structure; poor wiring produces misleading spectral signals.
- Building and storing the graph costs compute and memory proportional to corpus size and neighbourhood parameters.
- Smoothness is a property of the chosen graph and weights, not of the embedding model alone; results are dataset-specific.
- The method complements geometric similarity; it does not replace it.