Sharp optimal recovery in the two component Gaussian mixture model

Abstract

In this paper, we study the problem of clustering in the Two component Gaussian mixture model when the centers are separated by some $\mathrm{\Delta }>0$. We present a nonasymptotic lower bound for the corresponding minimax Hamming risk improving on existing results. We also propose an optimal, efficient and adaptive procedure that is minimax rate optimal. The rate optimality is moreover sharp in the asymptotics when the sample size goes to infinity. Our procedure is based on a variant of Lloyd’s iterations initialized by a spectral method. As a consequence of nonasymptotic results, we find a sharp phase transition for the problem of exact recovery in the Gaussian mixture model. We prove that the phase transition occurs around the critical threshold $\overline{\mathrm{\Delta }}$ given by

$${\overline{\mathrm{\Delta }}}^2={\mathit{\sigma }}^2(1+\sqrt{1+\frac{2\mathit{p}}{\mathit{n}log\mathit{n}}})log\mathit{n}.$$