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

2026-03-12

AI Summary

This work addresses the limitations of traditional adjacency spectral embeddings, which rely on a global low-rank assumption and fail to capture the locally sparse, transitive geometric structures prevalent in real-world networks, thereby blurring local features. To overcome this, the authors propose Local Adjacency Spectral Embedding (LASE), which leverages weighted spectral decomposition to uncover latent low-dimensional local structure under a latent position model and employs kernel feature maps to embed nodes into locally low-dimensional subsets of an infinite-dimensional space. The study establishes, for the first time, a finite-sample theoretical trade-off between localization strength and truncation error, proving that sufficient localization induces rapid spectral decay and a pronounced spectral gap, thus providing theoretical guarantees for local low-dimensional embeddings. Experiments demonstrate that LASE outperforms existing global and subgraph-based methods in local reconstruction and visualization, and that the UMAP-LASE fusion algorithm efficiently integrates overlapping local embeddings to produce high-fidelity global visualizations.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• spectral embedding
• local low-dimensional structure
• network geometry
• latent position model
• sparsity

Innovation

• Local Adjacency Spectral Embedding
• locally low-dimensional structure
• spectral gap
• latent position model
• UMAP-LASE