Understanding the Generalization Performance of Spectral Clustering Algorithms
This work addresses a theoretical gap in spectral clustering for machine learning researchers, offering incremental improvements with new algorithms and empirical validation.
The paper tackles the lack of theoretical analysis on the generalization performance of spectral clustering algorithms, deriving excess risk bounds with a O(1/√n) convergence rate and proposing two novel algorithms that penalize key quantities and handle out-of-sample data efficiently.
The theoretical analysis of spectral clustering mainly focuses on consistency, while there is relatively little research on its generalization performance. In this paper, we study the excess risk bounds of the popular spectral clustering algorithms: \emph{relaxed} RatioCut and \emph{relaxed} NCut. Firstly, we show that their excess risk bounds between the empirical continuous optimal solution and the population-level continuous optimal solution have a $\mathcal{O}(1/\sqrt{n})$ convergence rate, where $n$ is the sample size. Secondly, we show the fundamental quantity in 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 not only penalize this quantity, but also cluster the out-of-sample data without re-eigendecomposition on the overall sample. Experiments verify the effectiveness of the proposed algorithms.