Optimality of Spectral Clustering in the Gaussian Mixture Model
This provides theoretical justification for spectral clustering in high-dimensional data analysis, though it is incremental as it builds on existing models.
The paper demonstrates that spectral clustering achieves minimax optimality in the Gaussian Mixture Model with isotropic covariance, under conditions of a fixed number of clusters and sufficiently high signal-to-noise ratio, without requiring spectral gap assumptions.
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.