Optimality of spectral clustering in the Gaussian mixture model
Matthias Löffler, Anderson Y. Zhang, Harrison H. Zhou
The Annals of Statistics · 2021 · Volume 49 · Issue 5
Abstract
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.
Keywords
62H30, Gaussian mixture model, K-means, spectral clustering, spectral perturbation
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