Some effects in adaptive robust estimation under sparsity
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Adaptive estimation in the sparse mean model and in sparse regression exhibits some interesting effects. This paper considers estimation of a sparse target vector, of its l2-norm and of the noise variance in the sparse linear model. We establish the optimal rates of adaptive estimation when adaptation is considered with respect to the triplet "noise level-noise distribution-sparsity". These rates turn out to be different from the minimax non-adaptive rates when the tripet is known. A crucial issue is the ignorance of the noise level. Moreover, knowing or not knowing the noise distribution can also influence the rate. For example, the rates of estimation of the noise level can differ depending on whether the noise is Gaussian or sub-Gaussian without a precise knowledge of the distribution. Estimation of noise level in our setting can be viewed as an adaptive variant of robust estimation of scale in the contamination model, where instead of fixing the "nominal" distribution in advance, we assume that it belongs to some class of distributions. We also show that in the problem of estimation of a sparse vector under the l2-risk when the variance of the noise is unknown, the optimal rate depends dramatically on the design. In particular, for noise distributions with polynomial tails, the rate can range from sub-Gaussian to polynomial depending on the properties of the design.
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