Optimality of Spectral Clustering in the Gaussian Mixture Model

TL;DR

This paper proves that spectral clustering is minimax optimal for Gaussian Mixture Models with isotropic covariance, without relying on spectral gap conditions, when the number of clusters is fixed and the signal-to-noise ratio is high.

Contribution

It establishes the minimax optimality of spectral clustering in Gaussian Mixture Models without requiring spectral gap assumptions.

Findings

Spectral clustering achieves minimax optimality under specified conditions.

No spectral gap conditions are necessary for optimality.

Results apply to fixed number of clusters with high signal-to-noise ratio.

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.