The Annals of Statistics

Impact of regularization on spectral clustering

Antony Joseph and Bin Yu

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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 $\log n$) 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.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1765-1791.

Received: July 2014
Revised: January 2016
First available in Project Euclid: 7 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 62H99: None of the above, but in this section

Spectral clustering regularization network analysis community detection stochastic block model


Joseph, Antony; Yu, Bin. Impact of regularization on spectral clustering. Ann. Statist. 44 (2016), no. 4, 1765--1791. doi:10.1214/16-AOS1447.

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Supplemental materials

  • Supplementary Material: Supplement to “Impact of regularization on spectral clustering”. The supplementary file contains the proof of the claims in the paper that were not included in the main body.