## The Annals of Statistics

### Impact of regularization on spectral clustering

#### Abstract

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

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

Dates
Revised: January 2016
First available in Project Euclid: 7 July 2016

https://projecteuclid.org/euclid.aos/1467894715

Digital Object Identifier
doi:10.1214/16-AOS1447

Mathematical Reviews number (MathSciNet)
MR3519940

Zentralblatt MATH identifier
1357.62229

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

#### Citation

Joseph, Antony; Yu, Bin. Impact of regularization on spectral clustering. Ann. Statist. 44 (2016), no. 4, 1765--1791. doi:10.1214/16-AOS1447. https://projecteuclid.org/euclid.aos/1467894715

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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.