Sharp optimal recovery in the Two Gaussian Mixture Model
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In this paper, we study the non-asymptotic problem of exact recovery in the Two Gaussian mixture model when the centers are separated by some Δ>0. We present an optimal 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. Our procedure is based on a variant of Lloyd's iterations initialized by a spectral method. As a consequence of our study, we recover a sharp phase transition for the problem of exact recovery in the Gaussian mixture model. The latter phase transition happens around the critical threshold Δ^* given by Δ^*2 = σ^2n(1 + √(1+2p/nn)) .
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