Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding

• URL: http://arxiv.org/abs/2603.11965v1
• Date: Thu, 12 Mar 2026 14:10:55 GMT
• Title: Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding
• Abstract summary: Local Adjacency Spectral Embedding (LASE)<n>LASE uncovers locally low-dimensional structure via weighted spectral decomposition.<n>Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines.
• Score: 7.075759309995285
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Standard Adjacency Spectral Embedding (ASE) relies on a global low-rank assumption often incompatible with the sparse, transitive structure of real-world networks, causing local geometric features to be 'smeared'. To address this, we introduce Local Adjacency Spectral Embedding (LASE), which uncovers locally low-dimensional structure via weighted spectral decomposition. Under a latent position model with a kernel feature map, we treat the image of latent positions as a locally low-dimensional set in infinite-dimensional feature space. We establish finite-sample bounds quantifying the trade-off between the statistical cost of localisation and the reduced truncation error achieved by targeting a locally low-dimensional region of the embedding. Furthermore, we prove that sufficient localisation induces rapid spectral decay and the emergence of a distinct spectral gap, theoretically justifying low-dimensional local embeddings. Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines, and we introduce UMAP-LASE for assembling overlapping local embeddings into high-fidelity global visualisations.

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