Sparse covariance matrix estimation in high-dimensional deconvolution

Abstract

We study the estimation of the covariance matrix Σ 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 Σ. 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.MSC 2010 subject classifications: Primary 62H12; secondary 62F12, 62G05.

Likelihood theory for the graph Ornstein-Uhlenbeck process

“…Until this point, we have assumed the availability of the adjacency matrix. However, sparse stochastic processes have become increasingly influential to handle high-dimension problems (Belomestny et al 2019;Gaïffas and Matulewicz 2019;Ma et al 2021). Regularisation through a penalty on the model parameters is an important component of this literature and we show in the next section that general Lévy-driven OU processes can be consistently transformed into GrOU processes in this context.…”

“…Stationary noise distributions are usually too simplistic to explain the intrinsic variability of the data. Volatility modulation adds a stochastic scaling factor (Belomestny et al 2019;Cai et al 2016) which follows its own dynamics to better represent exogenous source of uncertainty (Pigorsch and Stelzer 2009b;Yang et al 2020) whilst a jump component helps to model unforeseen perturbations or rare calendar events (Barndorff-Nielsen and Veraart 2012).…”

Adaptive robust estimation in sparse vector model

“…Estimators of variance or covariance matrix based on the empirical characteristic function have been studied in several papers [4,5,3,6]. The setting in [4,5,3] is different from the ours as those papers deal with the model where the non-zero components of θ are random with a smooth distribution density. The estimators in [4,5] are also quite different.…”

“…The estimators in [4,5] are also quite different. On the other hand, [3,6] consider estimators close toṽ 2 . In particular, [6] uses a similar pilot estimator for testing in the sparse vector model where it is assumed that σ ∈ [σ − , σ + ], 0 < σ − < σ + < ∞, and the estimator depends on σ + .…”

“…As explained in Section 4 below, this problem has similarities with estimation of noise level in our setting. The main difference is that instead of fixing in advance the Gaussian nominal distribution of the contamination model we assume that it belongs to a class of distributions, such as (2) or (3). Therefore, the corresponding results in Section 4 can be viewed as results on robust estimation of scale where, in contrast to the classical setting, we are interested in adaptation to the unknown nominal law.…”