Robustness of Covariance Estimators with Application in Activity Detection

2025-10-08

AI Summary

This paper addresses robust covariance estimation for active user detection in multi-antenna random access. We consider the scenario where the observation matrix is corrupted by noise and channel perturbations, and propose a class of structured covariance estimators parameterized by a function $g$ and a model set $mathcal{H}$. We prove that these estimators achieve arbitrarily small estimation error bounds under infinitesimal perturbations. Innovatively, we formulate activity detection as structured covariance estimation of large-scale fading coefficients and rigorously show that the relaxed maximum likelihood estimator (MLE) belongs to this robust estimator family. By integrating codebook designs satisfying the sign kernel condition, we employ nonnegative least squares and relaxed MLE for sparse recovery. Theoretically, when the number of receive antennas is sufficiently large and the number of active users satisfies $S leq lceil frac{1}{2}M^2 ceil - 1$, both estimators exactly recover the large-scale fading coefficients—establishing, for the first time, simultaneous robustness and statistical consistency guarantees for covariance-based detection in multi-antenna random access.

Technology Category

• Machine Learning: Active Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 $f{W}$ 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 $f{W}^frac{1}{2}f{Z}^{-1}f{W}^frac{1}{2}$ under the constraint that $f{Z}$ 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 $f{W}$ 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 $Sleqleftlceilfrac{1}{2}M^2 ight ceil-1$ where $S$ is the number of active users and $M$ is number of pilot symbols per user.

Problem

• Developing robust covariance estimators for perturbed observations
• Applying covariance estimation to activity detection in wireless networks
• Proposing codebooks for reliable large-scale fading coefficient recovery

Innovation

• Robust covariance estimators minimize eigenvalues under constraints
• Relaxed maximum likelihood estimator belongs to robust class
• Proposed codebooks enable recovery with sufficient antennas