On the Power of SVD in the Stochastic Block Model

TL;DR

This paper investigates the effectiveness of vanilla SVD in accurately recovering clusters within the symmetric stochastic block model, confirming its power as a clustering tool.

Contribution

It proves that vanilla SVD can perfectly recover clusters in the symmetric SBM, answering an open question in the field.

Findings

Vanilla SVD recovers all clusters correctly in symmetric SBM.

Spectral methods enhance clustering performance.

Theoretical validation of SVD's power in community detection.

Abstract

A popular heuristic method for improving clustering results is to apply dimensionality reduction before running clustering algorithms. It has been observed that spectral-based dimensionality reduction tools, such as PCA or SVD, improve the performance of clustering algorithms in many applications. This phenomenon indicates that spectral method not only serves as a dimensionality reduction tool, but also contributes to the clustering procedure in some sense. It is an interesting question to understand the behavior of spectral steps in clustering problems. As an initial step in this direction, this paper studies the power of vanilla-SVD algorithm in the stochastic block model (SBM). We show that, in the symmetric setting, vanilla-SVD algorithm recovers all clusters correctly. This result answers an open question posed by Van Vu (Combinatorics Probability and Computing, 2018) in the symmetric setting.