Structure-Aware Spectral Sparsification via Uniform Edge Sampling

2025-10-14

AI Summary

Scalability of traditional spectral clustering is hindered by its reliance on full-graph eigendecomposition. While existing sparsification methods preserve spectral properties, they incur prohibitive preprocessing costs due to expensive effective resistance computation. This paper establishes, for the first time, that uniform edge sampling—agnostic to graph structure—suffices to preserve the principal eigenspace required for spectral clustering under a strong separability assumption. Our key innovations include: (i) an upper bound on intra-cluster edge resistances; (ii) a novel characterization of rank-$(n-k)$ effective resistance; and (iii) a matrix Chernoff inequality tailored to the principal eigenspace. Theoretically, we prove that sampling only $O(gamma^2 n log n / varepsilon^2)$ edges guarantees $varepsilon$-accurate clustering. Empirically, our method efficiently constructs spectrally faithful sparse graphs on large-scale benchmarks, drastically reducing preprocessing overhead.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Spectral clustering is a fundamental method for graph partitioning, but its reliance on eigenvector computation limits scalability to massive graphs. Classical sparsification methods preserve spectral properties by sampling edges proportionally to their effective resistances, but require expensive preprocessing to estimate these resistances. We study whether uniform edge sampling-a simple, structure-agnostic strategy-can suffice for spectral clustering. Our main result shows that for graphs admitting a well-separated $k$-clustering, characterized by a large structure ratio $Upsilon(k) = lambda_{k+1} / ho_G(k)$, uniform sampling preserves the spectral subspace used for clustering. Specifically, we prove that uniformly sampling $O(gamma^2 n log n / epsilon^2)$ edges, where $gamma$ is the Laplacian condition number, yields a sparsifier whose top $(n-k)$-dimensional eigenspace is approximately orthogonal to the cluster indicators. This ensures that the spectral embedding remains faithful, and clustering quality is preserved. Our analysis introduces new resistance bounds for intra-cluster edges, a rank-$(n-k)$ effective resistance formulation, and a matrix Chernoff bound adapted to the dominant eigenspace. These tools allow us to bypass importance sampling entirely. Conceptually, our result connects recent coreset-based clustering theory to spectral sparsification, showing that under strong clusterability, even uniform sampling is structure-aware. This provides the first provable guarantee that uniform edge sampling suffices for structure-preserving spectral clustering.

Problem

• Investigating uniform edge sampling for spectral clustering scalability
• Preserving spectral subspace structure without importance sampling
• Providing theoretical guarantees for structure-aware graph sparsification

Innovation

• Uniform edge sampling for spectral sparsification
• Bypassing importance sampling with resistance bounds
• Connecting coreset theory to spectral clustering