Threshold estimation for sparse high-dimensional deconvolution

The problem of covariance estimation for a p-dimensional normal vector X ∼ N(0, Σ) observed with additional noise is studied. Only a very general non-parametric assumption is imposed on the distribution of the noise. In this semi-parametric deconvolution problem spectral thresholding estimators are constructed that adapt to sparsity in Σ. We prove that the minimax convergence rates logarithmic in log p/n with n being the sample size.