Title: Sparse covariance matrix estimation in high-dimensional deconvolution,
Abstract: We study the estimation of the covariance matrix $\Sigma$ of a
$p$-dimensional normal random vector based on $n$ independent observations
corrupted by additive noise. Only a general nonparametric assumption is imposed
on the distribution of the noise without any sparsity constraint on its
covariance matrix. In this high-dimensional semiparametric deconvolution
problem, we propose spectral thresholding estimators that are adaptive to the
sparsity of $\Sigma$. We establish an oracle inequality for these estimators
under model miss-specification and derive non-asymptotic minimax convergence
rates that are shown to be logarithmic in $n/\log p$. We also discuss the
estimation of low-rank matrices based on indirect observations as well as the
generalization to elliptical distributions. The finite sample performance of
the threshold estimators is illustrated in a numerical example.