IMPACT OF REGULARIZATION ON SPECTRAL CLUSTERING
成果类型：
Article
署名作者：
Joseph, Antony; Yu, Bin
署名单位：
University of California System; University of California Berkeley
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/16-AOS1447
发表日期：
2016
页码：
1765-1791
关键词：
摘要：
The performance of spectral clustering can be considerably improved via regularization, as demonstrated empirically in Amini et al. [Ann. Statist. 41 (2013) 2097-2122]. Here, we provide an attempt at quantifying this improvement through theoretical analysis. Under the stochastic block model (SBM), and its extensions, previous results on spectral clustering relied on the minimum degree of the graph being sufficiently large for its good performance. By examining the scenario where the regularization parameter tau is large, we show that the minimum degree assumption can potentially be removed. As a special case, for an SBM with two blocks, the results require the maximum degree to be large (grow faster than logn) as opposed to the minimum degree. More importantly, we show the usefulness of regularization in situations where not all nodes belong to well-defined clusters. Our results rely on a 'bias-variance'-like trade-off that arises from understanding the concentration of the sample Laplacian and the eigengap as a function of the regularization parameter. As a byproduct of our bounds, we propose a data-driven technique DKest (standing for estimated Davis-Kahan bounds) for choosing the regularization parameter. This technique is shown to work well through simulations and on a real data set.