Open Access
August 2019 Sparse covariance matrix estimation in high-dimensional deconvolution
Denis Belomestny, Mathias Trabs, Alexandre B. Tsybakov
Bernoulli 25(3): 1901-1938 (August 2019). DOI: 10.3150/18-BEJ1040A


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.


Download Citation

Denis Belomestny. Mathias Trabs. Alexandre B. Tsybakov. "Sparse covariance matrix estimation in high-dimensional deconvolution." Bernoulli 25 (3) 1901 - 1938, August 2019.


Received: 1 October 2017; Revised: 1 March 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066244
MathSciNet: MR3961235
Digital Object Identifier: 10.3150/18-BEJ1040A

Keywords: Fourier methods , Minimax convergence rates , severely ill-posed inverse problem , thresholding

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
Back to Top