We derive nonasymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in R d for classes of at most s-sparse vectors separated from 0 by a constant a > 0. In some cases, we get exact expressions for the nonasymptotic minimax risk as a function of d, s, a and find explicitly the minimax selectors. These results are extended to dependent or non-Gaussian observations and to the problem of crowdsourcing. Analogous conclusions are obtained for the probability of wrong recovery of the sparsity pattern. As corollaries, we derive necessary and sufficient conditions for such asymptotic properties as almost full recovery and exact recovery.Moreover, we propose data-driven selectors that provide almost full and exact recovery adaptively to the parameters of the classes.

In this paper, we study the problem of clustering in the Two component Gaussian mixture model when the centers are separated by some ∆ > 0. We present a non-asymptotic 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 non-asymptotic 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 ∆ given by ∆2 = σ 2 1 + 1 + 2p n log n log n.

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