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).

频谱算法是图划分算法，它通过使用频谱嵌入图将图的节点集划分为组。基于算法的聚类技术被称为频谱聚类，并广泛用于数据分析中。为了更好地理解为什么频谱聚类成功的原因，Peng等。（在：第28届学习理论会议论文集（COLT），第40卷，第1423–1455页，2015年）和Kolev和Mehlhorn（在：第24届欧洲算法年会上（ESA 2016），第57卷，第57：1页） –57：14，2016）研究了一类图的某种光谱算法的行为，该图称为聚类图。具体来说，他们在图形上进行了假设，并显示了其下频谱算法的性能保证。他们研究的算法使用了Shi和Malik（IEEE Trans Pattern Anal Mach Intell 22（8）：888-905，2000）开发的频谱嵌入图。在本文中，我们改进了它们的结果，在较弱的假设下提供了更好的性能保证。我们还利用Ng等人开发的光谱嵌入图评估了光谱算法的性能。（在：神经信息处理系统的进展14（NIPS），第849–856页，2001年）。