A Tighter Analysis of Spectral Clustering, and Beyond
This work offers theoretical insights and practical improvements for spectral clustering, an incremental advancement in graph-based machine learning.
The authors provided a tighter analysis of spectral clustering, showing it works under weaker conditions than previously known, and demonstrated that using fewer eigenvectors can improve performance in practice, with empirical validation on synthetic and real-world datasets.
This work studies the classical spectral clustering algorithm which embeds the vertices of some graph $G=(V_G, E_G)$ into $\mathbb{R}^k$ using $k$ eigenvectors of some matrix of $G$, and applies $k$-means to partition $V_G$ into $k$ clusters. Our first result is a tighter analysis on the performance of spectral clustering, and explains why it works under some much weaker condition than the ones studied in the literature. For the second result, we show that, by applying fewer than $k$ eigenvectors to construct the embedding, spectral clustering is able to produce better output for many practical instances; this result is the first of its kind in spectral clustering. Besides its conceptual and theoretical significance, the practical impact of our work is demonstrated by the empirical analysis on both synthetic and real-world datasets, in which spectral clustering produces comparable or better results with fewer than $k$ eigenvectors.