Spectral clustering is one of the most popular algorithms to group high- dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been fully understood. In this paper, we show that spectral clustering is minimax optimal in the Gaussian mixture model with isotropic covariance matrix, when the number of clusters is fixed and the signal-to-noise ratio is large enough. Spectral gap conditions are widely assumed in the literature to analyze spectral clustering. On the contrary, these conditions are not needed to establish optimality of spectral clustering in this paper.
M. Löffler gratefully acknowledges financial support of ERC grant UQMSI/ 647812 and EPSRC grant EP/L016516/1, which funded a research visit to Yale University, where parts of this work were completed. These grants also funded M. Löffler during his PhD studies at the University of Cambridge.
The authors would like to thank Zhou Fan from Yale University for pointing out the references [52, 32]. The authors are further grateful to the Co-Editor, Ming Yuan, an anonymous Associate Editor and three anonymous referees for careful reading of the manuscript and their valuable remarks and suggestions.
"Optimality of spectral clustering in the Gaussian mixture model." Ann. Statist. 49 (5) 2506 - 2530, October 2021. https://doi.org/10.1214/20-AOS2044