Detection of Signals in Colored Noise: Roy's Largest Root Test for Non-central $F$-matrices

This paper investigates the signal detection problem in colored noise with an unknown covariance matrix. In particular, we focus on detecting a non-random signal by capitalizing on the leading eigenvalue (a.k.a. Roy's largest root) of the whitened sample covariance matrix as the test statistic. To this end, the whitened sample covariance matrix is constructed via \(m\)-dimensional \(p \) plausible signal-bearing samples and \(m\)-dimensional \(n \) noise-only samples. Since the signal is non-random, the whitened sample covariance matrix turns out to have a {\it non-central} \(F\)-distribution with a rank-one non-centrality parameter. Therefore, the performance of the test entails the statistical characterization of the leading eigenvalue of the non-central \(F\)-matrix, which we address by deriving its cumulative distribution function (c.d.f.) in closed-form by leveraging the powerful orthogonal polynomial approach in random matrix theory. This new c.d.f. has been instrumental in analyzing the receiver operating characteristic (ROC) of the detector. We also extend our analysis into the high dimensional regime in which \(m,n\), and \(p\) diverge such that \(m/n\) and \(m/p\) remain fixed. It turns out that, when \(m=n\) and fixed, the power of the test improves if the signal-to-noise ratio (SNR) is of at least \(O(p)\), whereas the corresponding SNR in the high dimensional regime is of at least \(O(p^2)\). Nevertheless, more intriguingly, for \(m<n\) with the SNR of order \(O(p)\), the leading eigenvalue does not have power to detect {\it weak} signals in the high dimensional regime.

Signal Detection in Colored Noise Using the Condition Number of $F$-Matrices

Signal detection in colored noise with an unknown covariance matrix has numerous applications across various scientific and engineering disciplines. The analysis focuses on the square of the condition number \(\kappa^2(\cdot)\), defined as the ratio of the largest to smallest eigenvalue \((\lambda_{\text{max}}/\lambda_{\text{min}})\) of the whitened sample covariance matrix \(\bm{\widehat{\Psi}}\), constructed from \(p\) signal-plus-noise samples and \(n\) noise-only samples, both \(m\)-dimensional. This statistic is denoted as \(\kappa^2(\bm{\widehat{\Psi}})\). A finite-dimensional characterization of the false alarm probability for this statistic under the null and alternative hypotheses has been an open problem. Therefore, in this work, we address this by deriving the cumulative distribution function (c.d.f.) of \(\kappa^2(\bm{\widehat{\Psi}})\) using the powerful orthogonal polynomial approach in random matrix theory. These c.d.f. expressions have been used to statistically characterize the performance of \(\kappa^2(\bm{\widehat{\Psi}})\).

Detection of Signals in Colored Noise: Leading Eigenvalue Test for Non-central $F$-matrices

This paper investigates the signal detection problem in colored noise with an unknown covariance matrix. In particular, we focus on detecting an unknown non-random signal by capitalizing on the leading eigenvalue of the whitened sample covariance matrix as the test statistic (a.k.a. Roy's largest root test). Since the unknown signal is non-random, the whitened sample covariance matrix turns out to have a non-central $F$-distribution. This distribution assumes a singular or non-singular form depending on whether the number of observations $p\lessgtr$ the system dimensionality $m$. Therefore, we statistically characterize the leading eigenvalue of the singular and non-singular $F$-matrices by deriving their cumulative distribution functions (c.d.f.). Subsequently, they have been utilized in deriving the corresponding receiver operating characteristic (ROC) profiles. We also extend our analysis into the high dimensional domain. It turns out that, when the signal is sufficiently strong, the maximum eigenvalue can reliably detect it in this regime. Nevertheless, weak signals cannot be detected in the high dimensional regime with the leading eigenvalue.

Generalized Eigenvalue Based Detection of Signals in Colored Noise: A Sample Deficient Analysis

This paper investigates the signal detection problem in colored noise with an unknown covariance matrix. To be specific, we consider a scenario in which the number of signal bearing samples ($n$) is strictly smaller than the dimensionality of the signal space ($m$). Our test statistic is the leading generalized eigenvalue of the whitened sample covariance matrix (a.k.a. $F$-matrix) which is constructed by whitening the signal bearing sample covariance matrix with noise-only sample covariance matrix. The sample deficiency (i.e., $m>n$) in turn makes this $F$-matrix rank deficient, thereby singular. Therefore, an exact statistical characterization of the leading generalized eigenvalue (l.g.e.) of a singular $F$-matrix is of paramount importance to assess the performance of the detector (i.e., the receiver operating characteristics (ROC)). To this end, we employ the powerful orthogonal polynomial approach to derive a new finite dimensional c.d.f. expression for the l.g.e. of a singular $F$-matrix. It turns out that when the noise only sample covariance matrix is nearly rank deficient and the signal-to-noise ratio is $O(m)$, the ROC profile converges to a limit.

Flex-Net: A Graph Neural Network Approach to Resource Management in Flexible Duplex Networks

Flexible duplex networks allow users to dynamically employ uplink and downlink channels without static time scheduling, thereby utilizing the network resources efficiently. This work investigates the sum-rate maximization of flexible duplex networks. In particular, we consider a network with pairwise-fixed communication links. Corresponding combinatorial optimization is a non-deterministic polynomial (NP)-hard without a closed-form solution. In this respect, the existing heuristics entail high computational complexity, raising a scalability issue in large networks. Motivated by the recent success of Graph Neural Networks (GNNs) in solving NP-hard wireless resource management problems, we propose a novel GNN architecture, named Flex-Net, to jointly optimize the communication direction and transmission power. The proposed GNN produces near-optimal performance meanwhile maintaining a low computational complexity compared to the most commonly used techniques. Furthermore, our numerical results shed light on the advantages of using GNNs in terms of sample complexity, scalability, and generalization capability.

On the Convergence of Inexact Gradient Descent with Controlled Synchronization Steps

We develop a gradient-like algorithm to minimize a sum of peer objective functions based on coordination through a peer interconnection network. The coordination admits two stages: the first is to constitute a gradient, possibly with errors, for updating locally replicated decision variables at each peer and the second is used for error-free averaging for synchronizing local replicas. Unlike many related algorithms, the errors permitted in our algorithm can cover a wide range of inexactnesses, as long as they are bounded. Moreover, the second stage is not conducted in a periodic manner, like many related algorithms. Instead, a locally verifiable criterion is devised to dynamically trigger the peer-to-peer coordination at the second stage, so that expensive communication overhead for error-free averaging can significantly be reduced. Finally, the convergence of the algorithm is established under mild conditions.

Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

Let $\mathbf{X}\in\mathbb{C}^{m\times n}$ ($m\geq n$) be a random matrix with independent rows each distributed as complex multivariate Gaussian with zero mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ \eta \mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{u}\in\mathbb{C}^{n\times n}$ is an arbitrary vector with a unit Euclidean norm, $\eta\geq 0$ is a non-random parameter, and $(\cdot)^*$ represents conjugate-transpose. This paper investigates the distribution of the random quantity $\kappa_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n \lambda_k/\lambda_1$, where $0<\lambda_1<\lambda_2<\ldots<\lambda_n<\infty$ are the ordered eigenvalues of $\mathbf{X}^*\mathbf{X}$ (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called {\it scaled condition number} or the Demmel condition number (i.e., $\kappa_{\text{SC}}(\mathbf{X})$) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., $\kappa_{\text{SC}}^{-2}(\mathbf{X})$). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of $\kappa_{\text{SC}}^2(\mathbf{X})$ which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as $m,n\to\infty$ such that $m-n$ is fixed and when $\eta$ scales on the order of $1/n$, $\kappa_{\text{SC}}^2(\mathbf{X})$ scales on the order of $n^3$. In this respect we establish simple closed-form expressions for the limiting distributions.