Abstract

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 at most $s$ of the means are nonzero. We show that, up to a logarithmic factor, the minimax 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 being computationally demanding, we investigate suitable versions of group thresholding estimators that are efficiently computable even when the dimension and the sample size are very large. An interesting new phenomenon revealed by this investigation is that the group thresholding leads to a substantial improvement in the rate as compared to the element-wise thresholding. Thus, the rate of the group thresholding is $s^{2}\sqrt{p}+sp$, while the element-wise thresholding has an error of order $s^{2}p+sp$. To the best of our knowledge, this is the first known setting in which leveraging the group structure leads to 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. We show that there is a strong analogy between this problem and the first one. Exploiting it, we propose new strategies of robust estimation that are computationally tractable and have better rates of convergence than the other computationally tractable robust (with respect to the presence of the outliers in the data) estimators studied in the literature. However, this tractability comes with a loss of the minimax-rate-optimality in some regimes.

Highlights

  • Linear functionals are of central interest in statistics

  • We have studied two problems: the problem of estimating a multidimensional linear functional and the one of estimating the mean of p-variate random vectors when the data is corrupted by outliers

  • A surprising outcome of our work is that exploiting the group structure of the sparsity is far more important in the problem of linear functional estimation rather than in the problem of the whole signal

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Summary

Introduction

Linear functionals are of central interest in statistics. The problems of estimating a function at given points, predicting the value of a future observation, testing the validity of a hypothesis, finding a dimension reduction subspace are all examples of statistical inference on linear functionals. The primary goal of this paper is to investigate the problem of estimation of a particular form of linear functional defined as the sum of the observed multidimensional signals. This problem is of independent interest on its own, one of our motivations for studying it is its tight relation with the problem of robust estimation. There is a vast literature on studying the problem of estimating quadratic functionals (Donoho and Nussbaum, 1990; Laurent and Massart, 2000; Cai and Low, 2006; Bickel and Ritov, 1988). The problem of estimation of nonsmooth functionals was tackled in the literature, see (Cai and Low, 2011)

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