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.

翻译：一种广受欢迎的聚类改进启发式方法是在运行聚类算法前先进行降维。已有研究表明，以PCA或SVD为代表的谱降维工具能在许多应用中提升聚类算法的性能。这一现象表明，谱方法不仅充当降维工具，还在某种意义上对聚类过程本身有所贡献。理解谱步骤在聚类问题中的行为是一个有趣的问题。作为该方向的初步探索，本文研究了朴素SVD算法在随机块模型（SBM）中的表现。我们证明，在对称设定下，朴素SVD算法能够正确恢复所有聚类。该结果回答了Van Vu（《组合学概率与计算》，2018）在对称设定下提出的一个开放性问题。