Robustness of Covariance Estimators with Application in Activity Detection
Hendrik Bernd Zarucha, Peter Jung, Giuseppe Caire
TL;DR
This work develops a general robustness theory for covariance estimators formed by minimizing $\sum_{m} g(\lambda_m(W^{1/2} Z^{-1} W^{1/2}))$ over a structured set $\mathcal{H}$, showing stability: small perturbations in $W$ yield small estimation errors in $Z$ when $g$ and $\mathcal{H}$ satisfy mild, or even convex, conditions. It applies this framework to activity detection with multiple antennas by examining NNLR and relaxed maximum likelihood estimators, proving that deterministic codebooks with the signed kernel condition enable robust recovery of large-scale fading coefficients as the number of antennas grows, with a precise trade-off between pilot length and $K$. The analysis leverages trace-log-det covariance estimation and Lambert $W$-based inverse functions to establish robustness for the trace-log-det estimator, and uses concentration inequalities to translate noiseless guarantees into finite-$K$ performance. A coordinate-descent algorithm is shown to converge to stationary points under the signed kernel condition, and simulations corroborate the theoretical scaling laws and robustness predictions in both noiseless and noisy settings. Overall, the paper provides a rigorous link between structured covariance estimation, deterministic codebook design, and reliable activity detection in multi-antenna random-access systems. $
Abstract
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.