Understanding the Generalization Performance of Spectral Clustering Algorithms

The theoretical analysis of spectral clustering is mainly devoted to consistency, while there is little research on its generalization performance. In this paper, we study the excess risk bounds of the popular spectral clustering algorithms: relaxed RatioCut and relaxed NCut. Our analysis follows the two practical steps of spectral clustering algorithms: continuous solution and discrete solution. Firstly, we provide the convergence rate of the excess risk bounds between the empirical continuous optimal solution and the population-level continuous optimal solution. Secondly, we show the fundamental quantity influencing the excess risk between the empirical discrete optimal solution and the population-level discrete optimal solution. At the empirical level, algorithms can be designed to reduce this quantity. Based on our theoretical analysis, we propose two novel algorithms that can penalize this quantity and, additionally, can cluster the out-of-sample data without re-eigendecomposition on the overall samples. Numerical experiments on toy and real datasets confirm the effectiveness of our proposed algorithms.