Matched and Mismatched Estimation of Kronecker Product of Linearly Structured Scatter Matrices Under Elliptical Distributions

Abstract

The estimation of covariance matrices is a core problem in many modern adaptive signal processing applications. For matrix-and array-valued data, e.g., MIMO communication, EEG/MEG (time versus channel), the covariance matrix of vectorized data may belong to the non-convex set of Kronecker product structure. In addition, the Kronecker factors can also exhibit an additional linear structure. Taking this prior knowledge into account during the estimation process drastically reduces the amount of unknown parameters, and then improves the estimation accuracy. On the other hand, the broad class of complex elliptically symmetric distributions, as well as the related complex angular elliptical distribution, are particularly suited to model heavy-tailed multivariate data. In this context, we first establish novel robust estimators of scatter and shape matrices (both related to a covariance matrix), having a Kronecker product structure with linearly structured Kronecker factors. Then, we conduct a theoretical analysis of their asymptotic performance (i.e., consistency, asymptotic distribution and efficiency), in matched and mismatched scenarios, i.e., when misspecifications between the true and assumed models occur. Finally, numerical results illustrate the theoretical analysis and assess the usefulness of the proposed estimators.

Unification of sparse Bayesian learning algorithms for electromagnetic brain imaging with the majorization minimization framework

“…Although this paper focuses on electromagnetic brain source imaging, Type-II methods have also been successfully developed in other fields such as direction of arrival (DoA) and channel estimation in wireless communications ( Gerstoft et al, 2016 ; Haghighatshoar and Caire, 2017 ; Khalilsarai et al, 2020 ; Prasad et al, 2015 ), Internet of Things (IoT) (Fengler et al, 2019a; 2019b), robust portfolio optimization in finance ( Feng et al, 2016 ), covariance matching and estimation ( Benfenati et al, 2020 ; Greenewald and Hero, 2015 ; Meriaux et al, 2020 ; Ollila et al, 2020 ; Ottersten et al, 1998 ; Tsiligkaridis et al, 2013 ; Werner et al, 2008 ; Zoubir et al, 2018 ), graph learning ( Kumar et al, 2020 ), and brain functional imaging ( Wei et al, 2020 ). The methods introduced in this work may also prove useful in these domains.…”

Joint Estimation of Location and Scatter in Complex Elliptically Symmetric Distributions

“…Moreover its estimation performance is way better that the SCM one in heavy-tailed data while it is almost similar in light-tailed scenarios: high gain, very small loss. These very promising results promote the use of the R-estimator to other problems, as the structured shape estimation discussed in [19,21].…”

Unification of Sparse Bayesian Learning Algorithms for Electromagnetic Brain Imaging with the Majorization Minimization Framework

“…Although this paper focuses on electromagnetic brain source imaging, Type-II methods have also been successfully developed in other fields such as direction of arrival (DoA) and channel estimation in wireless communications [18], [19], [21], [102], Internet of Things (IoT) [103], [104], robust portfolio optimization in finance [22], covariance matching and estimation [105]- [112], graph learning [113], and brain functional imaging [92]. The methods introduced in this work may also prove useful in these domains.…”