Robustness of Covariance Estimators with Application in Activity Detection

The first part of this work considers a general class of covariance estimators. Each estimator of that class is generated by a real-valued function $g$ and a set of model covariance matrices $H$. If $\bf{W}$ is a potentially perturbed observation of a searched covariance matrix, then the estimator is the minimizer of the sum of $g$ applied to each eigenvalue of $\bf{W}^\frac{1}{2}\bf{Z}^{-1}\bf{W}^\frac{1}{2}$ under the constraint that $\bf{Z}$ is from $H$. It is shown that under mild conditions on $g$ and $H$ such estimators are robust, meaning the estimation error can be made arbitrarily small if the perturbation of $\bf{W}$ gets small enough. \par In the second part of this work the previous results are applied to activity detection in random access with multiple receive antennas. In activity detection recovering the large scale fading coefficients is a sparse recovery problem which can be reduced to a structured covariance estimation problem. The recovery can be done with a non-negative least squares estimator or with a relaxed maximum likelihood estimator. It is shown that under suitable assumptions on the distributions of the noise and the channel coefficients, the relaxed maximum likelihood estimator is from the general class of covariance estimators considered in the first part of this work. Then, codebooks based upon a signed kernel condition are proposed. It is shown that with the proposed codebooks both estimators can recover the large-scale fading coefficients if the number of receive antennas is high enough and $S\leq\left\lceil\frac{1}{2}M^2\right\rceil-1$ where $S$ is the number of active users and $M$ is number of pilot symbols per user.