arXiv subjects

Prathapasinghe Dharmawansa

Publications and source records attributed to Prathapasinghe Dharmawansa.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Sample-Deficient Analysis of the Leading Generalized Eigenvalue for the Detection of Signals in Colored Gaussian Noise

This paper investigates the signal detection problem in colored Gaussian noise with an unknown covariance matrix. To be specific, we consider a sample deficient 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 whitening operation along with the observation model induces a single spiked covariance structure on the $F$-matrix. Moreover, the sample deficiency (i.e., $m>n$) in turn makes this $F$-matrix rank deficient, thereby {\it singular}. Therefore, a simple exact statistical characterization of the leading generalized eigenvalue (l.g.e.) of a complex correlated {\it singular} $F$-matrix with a single spiked associated covariance is of paramount importance to assess the performance of the detector (i.e., the receiver operating characteristics (ROC)). To this end, we adopt the powerful orthogonal polynomial technique in random matrix theory to derive a new finite dimensional c.d.f. expression for the l.g.e. of this particular $F$-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 remarkably simple limiting profile.

eess.SP↗

arXiv2024

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.

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arXiv2023

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, we do not impose any gradient boundedness conditions for the objective functions. Furthermore, 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.

math.OC↗

arXiv2023

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.

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arXiv2022

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

Let $\mathbf{X}\in\mathbb{C}^{n\times m}$ ($m\geq n$) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ η\mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, {\color{blue}$\mathbf{u}\in\mathbb{C}^{n\times 1}$} is an arbitrary vector with unit Euclidean norm, $η\geq 0$ is a non-random parameter, and $(\cdot)^*$ represents the conjugate-transpose. This paper investigates the distribution of the random quantity $κ_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n λ_k/λ_1$, where {\color{blue}$0\le λ_1\le λ_2\le \ldots\leq λ_n<\infty$} 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., $κ_{\text{SC}}(\mathbf{X})$) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., $κ_{\text{SC}}^{-2}(\mathbf{X})$). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of $κ_{\text{SC}}^2(\mathbf{X})$ 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 $η$ scales on the order of $1/n$, $κ_{\text{SC}}^2(\mathbf{X})$ scales on the order of $n^3$. In this respect we establish simple closed-form expressions for the limiting distributions. {\color{blue}It turns out that, as $m,n\to\infty$ such that $n/m\to c\in(0,1)$, properly centered $κ_{\text{SC}}^{2}(\mathbf{X})$ fluctuates on the scale $m^{\frac{1}{3}}$}.

math.PR↗

arXiv2022

The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses

Let $\mathbf{W}\in\mathbb{C}^{n\times n}$ be a {\it single-spiked} Wishart matrix in the class $\mathbf{W}\sim \mathcal{CW}_n(m,\mathbf{I}_n+ θ\mathbf{v}\mathbf{v}^\dagger) $ with $m\geq n$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{v}\in\mathbb{C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $θ\geq 0$ is a non-random parameter, and $(\cdot)^\dagger$ represents the conjugate-transpose operator. Let $\mathbf{u}_1$ and $\mathbf{u}_n$ denote the eigenvectors corresponding to the samllest and the largest eigenvalues of $\mathbf{W}$, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity $Z_{\ell}^{(n)}=\left|\mathbf{v}^\dagger\mathbf{u}_\ell\right|^2\in(0,1)$ for $\ell=1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for $Z_{1}^{(n)}$ which is amenable to asymptotic analysis as $m,n$ diverges with $m-n$ fixed. It turns out that, in this asymptotic regime, the scaled random variable $nZ_{1}^{(n)}$ converges in distribution to $χ^2_2/2(1+θ)$, where $χ_2^2$ denotes a chi-squared random variable with two degrees of freedom. This reveals that $\mathbf{u}_1$ can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of $Z_{n}^{(n)}$ is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension $(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of $n=2,3$, and $4$, we obtain closed-form expressions.

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arXiv2020

Performance Analysis of a Two-Tile Reconfigurable Intelligent Surface Assisted $2\times 2$ MIMO System

We consider a two--tile reconfigurable intelligent surface (RIS) assisted wireless network with a two-antenna transmitter and receiver over Rayleigh fading. We show that the average received signal-to-noise-ratio (SNR) optimal transmission and combining vectors are given by the left and right singular spaces of the RIS-receiver and transmit-RIS channel matrices, respectively. Moreover, the optimal phases at the two tiles of the RIS are determined by the phases of the elements of the latter spaces. To further study the effect of phase compensation, we statistically characterize the average SNR of all possible combinations of transmission and combining directions pertaining to the latter singular spaces by deriving novel expressions for the outage probability and throughput of each of those modes. Furthermore, for comparison, we derive the corresponding expressions in the absence of RIS. Our results show an approximate SNR improvement of $2$\,dB due to the phase compensation at the RIS.

cs.IT↗

arXiv2020

Reconfigurable Intelligent Surface assisted Two-Way Communications: Performance Analysis and Optimization

In this paper, we investigate the two-way communication between two users assisted by a re-configurable intelligent surface (RIS). The scheme that two users communicate simultaneously over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at $\left(\log(ρ)/ρ\right)^L$ rate where $L$ is the number of elements, whereas the spectral efficiency increases at $\log(ρ)$ rate at large average SINR $ρ$. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solution is derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter one is solved through semidefinite programming relaxation and a proposed greedy-iterative method, which can achieve higher performance and lower computational complexity, respectively.

eess.SP↗

arXiv2019

Eigenvalue Based Detection of a Signal in Colored Noise: Finite and Asymptotic Analyses

Signal detection in colored noise with an unknown covariance matrix has a myriad of applications in diverse scientific/engineering fields. The test statistic is the largest generalized eigenvalue (l.g.e.) of the whitened sample covariance matrix, which is constructed via $m$-dimensional $p $ signal-plus-noise samples and $m$-dimensional $n $ noise-only samples. A finite dimensional characterization of this statistic under the alternative hypothesis has hitherto been an open problem. We answer this problem by deriving cumulative distribution function (c.d.f.) of this l.g.e. via the powerful orthogonal polynomial approach, exploiting the deformed Jacobi unitary ensemble (JUE). Two special cases and an asymptotic version of the c.d.f. are also derived. With this new c.d.f., we comprehensively analyze the receiver operating characteristics (ROC) of the detector. Importantly, when the noise-only covariant matrix is nearly rank deficient (i.e., $ m=n$), we show that (a) when $m$ and $p$ increase such that $m/p$ is fixed, at each fixed signal-to-noise ratio (SNR), there exists an optimal ROC profile. We also establish a tight approximation of it; and (b) asymptotically, reliable signal detection is always possible (no matter how weak the signal is) if SNR scales with $m$.

cs.IT↗

arXiv2019

Detection of a Signal in Colored Noise: A Random Matrix Theory Based Analysis

This paper investigates the classical statistical signal processing problem of detecting a signal in the presence of colored noise with an unknown covariance matrix. In particular, we consider a scenario where m-dimensional p possible signal-plus-noise samples and m-dimensional n noise-only samples are available at the detector. Then the presence of a signal can be detected using the largest generalized eigenvalue (l.g.e.) of the so called whitened sample covariance matrix. This amounts to statistically characterizing the maximum eigenvalue of the deformed Jacobi unitary ensemble (JUE). To this end, we employ the powerful orthogonal polynomial approach to determine a new finite dimensional expression for the cumulative distribution function (c.d.f.) of the l.g.e. of the deformed JUE. This new c.d.f. expression facilitates the further analysis of the receiver operating characteristics (ROC) of the detector. It turns out that, for m=n, when m and p increase such that m/p attains a fixed value, there exists an optimal ROC profile corresponding to each fixed signal-to-noise ratio (SNR). In this respect, we have established a tight approximation for the corresponding optimal ROC profile.

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arXiv2018

Outage Analysis of $2\times2 $ MIMO-MRC in Correlated Rician Fading

This paper addresses one of the classical problems in random matrix theory-- finding the distribution of the maximum eigenvalue of the correlated Wishart unitary ensemble. In particular, we derive a new exact expression for the cumulative distribution function (c.d.f.) of the maximum eigenvalue of a $2\times 2$ correlated non-central Wishart matrix with rank-$1$ mean. By using this new result, we derive an exact analytical expression for the outage probability of $2\times 2$ multiple-input multiple-output maximum-ratio-combining (MIMO-MRC) in Rician fading with transmit correlation and a strong line-of-sight (LoS) component (rank-$1$ channel mean). We also show that the outage performance is affected by the relative alignment of the eigen-spaces of the mean and correlation matrices. In general, when the LoS path aligns with the least eigenvector of the correlation matrix, in the {\it high} transmit signal-to-noise ratio (SNR) regime, the outage gradually improves with the increasing correlation. Moreover, we show that as $K$ (Rician factor) grows large, the outage event can be approximately characterized by the c.d.f. of a certain Gaussian random variable.

eess.SP↗

arXiv2017

Traffic Minimizing Caching and Latent Variable Distributions of Order Statistics

Given a statistical model for the request frequencies and sizes of data objects in a caching system, we derive the probability density of the size of the file that accounts for the largest amount of data traffic. This is equivalent to finding the required size of the cache for a caching placement that maximizes the expected byte hit ratio for given file size and popularity distributions. The file that maximizes the expected byte hit ratio is the file for which the product of its size and popularity is the highest -- thus, it is the file that incurs the greatest load on the network. We generalize this theoretical problem to cover factors and addends of arbitrary order statistics for given parent distributions. Further, we study the asymptotic behavior of these distributions. We give several factor and addend densities of widely-used distributions, and verify our results by extensive computer simulations.

cs.NI↗

arXiv2014

Roy's largest root under rank-one alternatives:The complex valued case and applications

The largest eigenvalue of a Wishart matrix, known as Roy's largest root (RLR), plays an important role in a variety of applications. Most works to date derived approximations to its distribution under various asymptotic regimes, such as degrees of freedom, dimension, or both tending to infinity. However, several applications involve finite and relative small parameters, for which the above approximations may be inaccurate. Recently, via a small noise perturbation approach with fixed dimension and degrees of freedom, Johnstone and Nadler derived simple yet accurate stochastic approximations to the distribution of Roy's largest root in the real valued case, under a rank-one alternative. In this paper, we extend their results to the complex valued case. Furthermore, we analyze the behavior of the leading eigenvector by developing new stochastic approximations. Specifically, we derive simple stochastic approximations to the distribution of the largest eigenvalue under five common complex single-matrix and double-matrix scenarios. We then apply these results to investigate several problems in signal detection and communications. In particular, we analyze the performance of RLR detector in cognitive radio spectrum sensing and constant modulus signal detection in the high signal-to-noise ratio (SNR) regime. Moreover, we address the problem of determining the optimal transmit-receive antenna configuration (here optimality is in the sense of outage minimization) for rank-one multiple-input multiple-output Rician Fading channels at high SNR.

math.ST↗

arXiv2014

Local Asymptotic Normality of the spectrum of high-dimensional spiked F-ratios

We consider two types of spiked multivariate F distributions: a scaled distribution with the scale matrix equal to a rank-one perturbation of the identity, and a distribution with trivial scale, but rank-one non-centrality. The norm of the rank-one matrix (spike) parameterizes the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for a spike located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from this value depends only on the largest eigenvalue $λ_{1}$. Furthermore, $λ_{1}$ is asymptotically normal, and the statistical experiment of observing all the eigenvalues of the F matrix converges in the Le Cam sense to a Gaussian shift experiment that depends on the asymptotic mean and variance of $λ_{1}$. In particular, the best statistical inference about a sufficiently large spike in the local asymptotic regime is based on the largest eigenvalue only. As a by-product of our analysis, we establish joint asymptotic normality of a few of the largest eigenvalues of the multi-spiked F matrix when the corresponding spikes are above the phase transition threshold.

math.ST↗

arXiv2014

Joint density of eigenvalues in spiked multivariate models

The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and linear statistics used in testing.

math.ST↗

arXiv2013

Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean

Recently, D. Wang has devised a new contour integral based method to simplify certain matrix integrals. Capitalizing on that approach, we derive a new expression for the probability density function (p.d.f.) of the joint eigenvalues of a complex non-central Wishart matrix with a rank-1 mean. The resulting functional form in turn enables us to use powerful classical orthogonal polynomial techniques in solving three problems related to the non-central Wishart matrix. To be specific, for an $n\times n$ complex non-central Wishart matrix $\mathbf{W}$ with $m$ degrees of freedom ($m\geq n$) and a rank-1 mean, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue ($λ_{\min}$). The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference $m-n$. This further facilitates the analysis of the microscopic limit for the minimum eigenvalue which takes the form of the determinant of a square matrix of size $m-n$ with the Bessel kernel. We also develop a moment generating function based approach to derive the p.d.f. of the random variable $\frac{\text{tr}(\mathbf{W})}{λ_{\min}}$, where $\text{tr}(\cdot)$ denotes the trace of a square matrix. This random quantity is of great importance in the so-called smoothed analysis of Demmel condition number. Finally, we find the average of the reciprocal of the characteristic polynomial $\det[z\mathbf{I}_n+\mathbf{W}],\; |\arg z|<π$, where $\mathbf{I}_n$ and $\det[\cdot]$ denote the identity matrix of size $n$ and the determinant, respectively.

math.ST↗

arXiv2012

Distributions of Demmel and Related Condition Numbers

Consider a random matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ ($m \geq n$) containing independent complex Gaussian entries with zero mean and unit variance, and let $0<λ_1\leq λ_{2}\leq ...\leq λ_n<\infty$ denote the eigenvalues of $\mathbf{A}^{*}\mathbf{A}$ where $(\cdot)^*$ represents conjugate-transpose. This paper investigates the distribution of the random variables $\frac{\sum_{j=1}^n λ_j}{λ_k}$, for $k = 1$ and $k = 2$. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities, and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as $n$ and $m$ tend to infinity with their difference fixed, both densities scale on the order of $n^3$. After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case $m = n$.

math.ST↗

arXiv2012

Locally Best Invariant Test for Multiple Primary User Spectrum Sensing

We consider multi-antenna cooperative spectrum sensing in cognitive radio networks, when there may be multiple primary users. A noise-uncertainty-free detector that is optimal in the low signal to noise ratio regime is analyzed in such a scenario. Specifically, we derive the exact moments of the test statistics involved, which lead to simple and accurate analytical formulae for the false alarm probability and the decision threshold. Simulations are provided to examine the accuracy of the derived results, and to compare with other detectors in realistic sensing scenarios.

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