  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.
