Spectral embedding is a popular technique for the representation of graph data. Several regularization techniques have been proposed to improve the quality of the embedding with respect to downstream tasks like clustering. In this paper, we explain on a simple block model the impact of the complete graph regularization, whereby a constant is added to all entries of the adjacency matrix. Specifically, we show that the regularization forces the spectral embedding to focus on the largest blocks, making the representation less sensitive to noise or outliers. We illustrate these results on both on both synthetic and real data, showing how regularization improves standard clustering scores.
This paper analyzes the effect of regularization on spectral embeddings in a deterministic block model and explicitly characterizes the spectra of the Laplacian of the regularized graph in terms of the regularization parameter and block sizes. To my knowledge, this has not been done before. Prior work either derives sufficient conditions for the recovery of all blocks in the asymptotic limit of an infinite number of nodes in the case of (Joseph & Yu, 2016), or lower bounds the number of small eigenvalues of the Laplacian of the unregularized graph on random graphs in expectation (therefore arguing in favor of regularization) in the case of (Zhang & Rohe, 2018). This paper, on the other hand, gives a precise characterization of the eigenvalues and eigenvectors (albeit in the case of a deterministic graph); the results are elegant and the analysis uses simple elementary techniques, which is very satisfying and seems to be easy to build on. The authors mention that they would like to extend this analysis to stochastic block models, which would indeed be interesting. The paper is also well written and the results are clearly presented. Overall, this is a nice contribution to spectral graph theory and so I recommend acceptance.