Search embeddings with spectral graph awareness

ArrowSpace augments nearest-neighbor vector search with spectral graph features: it builds a graph Laplacian over the item embeddings and uses the Rayleigh quotient to compute a lambda-tau (λτ) score per item, so retrieval accounts for both semantic similarity and each item's structural role in the embedding graph - something plain cosine or L2 similarity can miss. Setup is:

pip install arrowspace

Usage takes an (N, d) float64 NumPy array of embeddings, builds an ArrowSpaceBuilder with graph parameters (eps for the connectivity radius, k for neighbor count, topk, p for the distance norm, sigma for kernel width), and calls .build() to get an ArrowSpace object exposing .lambdas() (scores indexed by insertion order) and .lambdas_sorted() (score, index pairs in ascending order). Higher λτ values mark items that are both semantically close to their neighbors and structurally central in the graph. The skill documents concrete tuning guidance: normalize embeddings to unit norm first, start eps proportional to 1/sqrt(embedding dimension), keep k between 3 and 25 (roughly N/50), and set sigma=None to auto-select kernel width from the observed distance distribution.

Use it when cosine or L2 similarity is missing latent structure in an embedding space, when graph-based retrieval with spectral awareness is wanted, when characterizing the spectral properties of an embedding space, or when building RAG pipelines where an item's contextual role matters alongside its raw semantic content. Do NOT use it with fewer than 10 items - the skill states graph structure isn't meaningful at that scale - and do NOT use it for real-time streaming data, since ArrowSpace is explicitly batch-oriented, not an incremental index. An eps set too small produces a disconnected graph; too large a k produces a dense graph with washed-out spectral signal, so both need tuning per dataset.

Input is an (N, d) float64 NumPy array of embedding vectors plus a graph_params dict (eps, k, topk, p, sigma). Output is an ArrowSpace object whose .lambdas() and .lambdas_sorted() methods return per-item spectral scores that can be compared directly against a cosine-similarity ranking, as shown in the skill's worked example contrasting cosine_order and spectral_order on the same item set.

A Python library (pip install arrowspace) that composes with NumPy and, for comparison purposes, sklearn.metrics.pairwise.cosine_similarity; runs as a skill inside Claude, Cursor, Codex, Gemini, and OpenCode. Related skills referenced for adjacent work are vector-database-engineer, embedding-strategies, similarity-search-patterns, and hybrid-search-implementation.

Engineers building semantic search or RAG retrieval who want structural, graph-aware ranking alongside plain similarity search, on batch (not streaming) embedding sets of at least a few dozen items.