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

TL;DR

This paper introduces Local Adjacency Spectral Embedding (LASE), a method that captures local low-dimensional structures in networks, improving over traditional global spectral embedding for better local reconstruction and visualization.

Contribution

The paper proposes LASE, a novel spectral embedding technique that uncovers local low-dimensional structures in networks, with theoretical guarantees and practical improvements.

Findings

LASE improves local reconstruction accuracy.

LASE reveals spectral gaps indicating local low-dimensionality.

Experiments demonstrate enhanced visualization of networks.

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.