Group Symmetry and non-Gaussian Covariance Estimation

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments

On the convergence of Maronna's MM-estimators of scatter

In this paper, {we propose an alternative proof for the uniqueness} of Maronna's $M$-estimator of scatter (Maronna, 1976) for $N$ vector observations $\mathbf y_1,...,\mathbf y_N\in\mathbb R^m$ under a mild constraint of linear independence of any subset of $m$ of these vectors. This entails in particular almost sure uniqueness for random vectors $\mathbf y_i$ with a density as long as $N>m$. {This approach allows to establish further relations that demonstrate that a properly normalized Tyler's $M$-estimator of scatter (Tyler, 1987) can be considered as a limit of Maronna's $M$-estimator. More precisely, the contribution is to show that each $M$-estimator converges towards a particular Tyler's $M$-estimator.} These results find important implications in recent works on the large dimensional (random matrix) regime of robust $M$-estimation

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

SMARAD - Centre of Excellence in Smart Radios and Wireless Research - Activity Report 2011 - 2013

Centre of Excellence in Smart Radios and Wireless Research (SMARAD), originally established with the name Smart and Novel Radios Research Unit, is aiming at world-class research and education in Future radio and antenna systems, Cognitive radio, Millimetre wave and THz techniques, Sensors, and Materials and energy, using its expertise in RF, microwave and millimeter wave engineering, in integrated circuit design for multi-standard radios as well as in wireless communications. SMARAD has the Centre of Excellence in Research status from the Academy of Finland since 2002 (2002-2007 and 2008-2013). Currently SMARAD consists of five research groups from three departments, namely the Department of Radio Science and Engineering, Department of Micro and Nanosciences, and Department of Signal Processing and Acoustics, all within the Aalto University School of Electrical Engineering. The total number of employees within the research unit is about 100 including 8 professors, about 30 senior scientists and about 40 graduate students and several undergraduate students working on their Master thesis. The relevance of SMARAD to the Finnish society is very high considering the high national income from exports of telecommunications and electronics products. The unit conducts basic research but at the same time maintains close co-operation with industry. Novel ideas are applied in design of new communication circuits and platforms, transmission techniques and antenna structures. SMARAD has a well-established network of co-operating partners in industry, research institutes and academia worldwide. It coordinates a few EU projects. The funding sources of SMARAD are diverse including the Academy of Finland, EU, ESA, Tekes, and Finnish and foreign telecommunications and semiconductor industry. As a by-product of this research SMARAD provides highest-level education and supervision to graduate students in the areas of radio engineering, circuit design and communications through Aalto University and Finnish graduate schools. During years 2011 – 2013, 18 doctor degrees were awarded to the students of SMARAD. In the same period, the SMARAD researchers published 197 refereed journal articles and 360 conference papers