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We consider two problems of estimation in high-dimensional Gaussian models.The first problem is that of estimating a linear functional of the means of $n$independent $p$-dimensional Gaussian vectors, under the assumption that most ofthese means are equal to zero. We show that, up to a logarithmic factor, theminimax rate of estimation in squared Euclidean norm is between $(s^2\wedge n)+sp$ and $(s^2\wedge np)+sp$. The estimator that attains the upper bound beingcomputationally demanding, we investigate suitable versions of groupthresholding estimators that are efficiently computable even when the dimensionand the sample size are very large. An interesting new phenomenon revealed bythis investigation is that the group thresholding leads to a substantialimprovement in the rate as compared to the element-wise thresholding. Thus, therate of the group thresholding is $s^2\sqrt{p}+sp$, while the element-wisethresholding has an error of order $s^2p+sp$. To the best of our knowledge,this is the first known setting in which leveraging the group structure leadsto a polynomial improvement in the rate.  The second problem studied in this work is the estimation of the common$p$-dimensional mean of the inliers among $n$ independent Gaussian vectors. Weshow that there is a strong analogy between this problem and the first one.Exploiting it, we propose new strategies of robust estimation that arecomputationally tractable and have better rates of convergence than the othercomputationally tractable robust (with respect to the presence of the outliersin the data) estimators studied in the literature. However, this tractabilitycomes with a loss of the minimax-rate-optimality in some regimes.

Minimax estimation of a p-dimensional linear functional in sparse Gaussian models and robust estimation of the mean