Improved analysis of spectral algorithm for clustering

Spectral algorithms are graph partitioning algorithms that partition a node set of a graph into groups by using a spectral embedding map. Clustering techniques based on the algorithms are referred to as spectral clustering and are widely used in data analysis. To gain a better understanding of why spectral clustering is successful, Peng et al. (In: Proceedings of the 28th conference on learning theory (COLT), vol 40, pp 1423-1455, 2015) and Kolev and Mehlhorn (In: 24th annual European symposium on algorithms (ESA 2016), vol 57, pp 57:1-57:14, 2016) studied the behavior of a certain type of spectral algorithm for a class of graphs, called well-clustered graphs. Specifically, they put an assumption on graphs and showed the performance guarantee of the spectral algorithm under it. The algorithm they studied used the spectral embedding map developed by Shi and Malik (IEEE Trans Pattern Anal Mach Intell 22(8):888-905, 2000). In this paper, we improve on their results, giving a better performance guarantee under a weaker assumption. We also evaluate the performance of the spectral algorithm with the spectral embedding map developed by Ng et al. (In: Advances in neural information processing systems 14 (NIPS), pp 849-856, 2001).