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The Internet of Things paradigm heavily relies on a network of a massive number of machine-type devices (MTDs) that monitor changes in various phenomena. Consequently, MTDs are randomly activated at different times whenever a change occurs. This essentially results in relatively few MTDs being active simultaneously compared to the entire network, resembling targeted sampling in compressed sensing. Therefore, signal recovery in machine-type communications is addressed through joint user activity detection and channel estimation algorithms built using compressed sensing theory. However, most of these algorithms follow a two-stage procedure in which a channel is first estimated and later mapped to find active users. This approach is inefficient because the estimated channel information is subsequently discarded. To overcome this limitation, we introduce a novel covariance-learning matching pursuit algorithm that bypasses explicit channel estimation. Instead, it focuses on estimating the indices of the active users greedily. Simulation results presented in terms of probability of miss detection, exact recovery rate, and computational complexity validate the proposed technique's superior performance and efficiency., Comment: submitted to IEEE IoT journal

We propose greedy Capon beamformer (GBF) for direction finding of narrow-band sources present in the array's viewing field. After defining the grid covering the location search space, the algorithm greedily builds the interference-plus-noise covariance matrix by identifying a high-power source on the grid using Capon's principle of maximizing the signal to interference plus noise ratio (SINR) while enforcing unit gain towards the signal of interest. An estimate of the power of the detected source is derived by exploiting the unit power constraint, which subsequently allows to update the noise covariance matrix by simple rank-1 matrix addition composed of outerproduct of the selected steering matrix with itself scaled by the signal power estimate. Our numerical examples demonstrate effectiveness of the proposed GCB in direction finding where it perform favourably compared to the state-of-the-art algorithms under a broad variety of settings. Furthermore, GCB estimates of direction-of-arrivals (DOAs) are very fast to compute., Comment: Submitted for publication

In the Multiple Measurements Vector (MMV) model, measurement vectors are connected to unknown, jointly sparse signal vectors through a linear regression model employing a single known measurement matrix (or dictionary). Typically, the number of atoms (columns of the dictionary) is greater than the number measurements and the sparse signal recovery problem is generally ill-posed. In this paper, we treat the signals and measurement noise as independent Gaussian random vectors with unknown signal covariance matrix and noise variance, respectively, and derive fixed point (FP) equation for solving the likelihood equation for signal powers, thereby enabling the recovery of the sparse signal support (sources with non-zero variances). Two practical algorithms, a block coordinate descent (BCD) and a cyclic coordinate descent (CCD) algorithms, that leverage on the FP characterization of the likelihood equation are then proposed. Additionally, a greedy pursuit method, analogous to popular simultaneous orthogonal matching pursuit (OMP), is introduced. Our numerical examples demonstrate effectiveness of the proposed covariance learning (CL) algorithms both in classic sparse signal recovery as well as in direction-of-arrival (DOA) estimation problems where they perform favourably compared to the state-of-the-art algorithms under a broad variety of settings.

This chapter reviews methods for linear shrinkage of the sample covariance matrix (SCM) and matrices (SCM-s) under elliptical distributions in single and multiple populations settings, respectively. In the single sample setting a popular linear shrinkage estimator is defined as a linear combination of the sample covariance matrix (SCM) with a scaled identity matrix. The optimal shrinkage coefficients minimizing the mean squared error (MSE) under elliptical sampling are shown to be functions of few key parameters only, such as elliptical kurtosis and sphericity parameter. Similar results and estimators are derived for multiple population setting and applications of the studied shrinkage estimators are illustrated in portfolio optimization.

We propose estimating the scale parameter (mean of the eigenvalues) of the scatter matrix of an unspecified elliptically symmetric distribution using weights obtained by solving Tyler's M-estimator of the scatter matrix. The proposed Tyler's weights-based estimate (TWE) of scale is then used to construct an affine equivariant Tyler's M-estimator as a weighted sample covariance matrix using normalized Tyler's weights. We then develop a unified framework for estimating the unknown tail parameter of the elliptical distribution (such as the degrees of freedom (d.o.f.) $\nu$ of the multivariate $t$ (MVT) distribution). Using the proposed TWE of scale, a new robust estimate of the d.o.f. parameter of MVT distribution is proposed with excellent performance in heavy-tailed scenarios, outperforming other competing methods. R-package is available that implements the proposed method.

Expectation-Maximization (EM) algorithm is a widely used iterative algorithm for computing (local) maximum likelihood estimate (MLE). It can be used in an extensive range of problems, including the clustering of data based on the Gaussian mixture model (GMM). Numerical instability and convergence problems may arise in situations where the sample size is not much larger than the data dimensionality. In such low sample support (LSS) settings, the covariance matrix update in the EM-GMM algorithm may become singular or poorly conditioned, causing the algorithm to crash. On the other hand, in many signal processing problems, a priori information can be available indicating certain structures for different cluster covariance matrices. In this paper, we present a regularized EM algorithm for GMM-s that can make efficient use of such prior knowledge as well as cope with LSS situations. The method aims to maximize a penalized GMM likelihood where regularized estimation may be used to ensure positive definiteness of covariance matrix updates and shrink the estimators towards some structured target covariance matrices. We show that the theoretical guarantees of convergence hold, leading to better performing EM algorithm for structured covariance matrix models or with low sample settings., Comment: ICASSP Conference, 4 pages, 8 figures

A robust and sparse Direction of Arrival (DOA) estimator is derived for array data that follows a Complex Elliptically Symmetric (CES) distribution with zero-mean and finite second-order moments. The derivation allows to choose the loss function and four loss functions are discussed in detail: the Gauss loss which is the Maximum-Likelihood (ML) loss for the circularly symmetric complex Gaussian distribution, the ML-loss for the complex multivariate $t$-distribution (MVT) with $\nu$ degrees of freedom, as well as Huber and Tyler loss functions. For Gauss loss, the method reduces to Sparse Bayesian Learning (SBL). The root mean square DOA error of the derived estimators is discussed for Gaussian, MVT, and $\epsilon$-contaminated data. The robust SBL estimators perform well for all cases and nearly identical with classical SBL for Gaussian noise.

Graph Neural Networks (GNNs), particularly Graph Convolutional Neural Networks (GCNNs), have emerged as pivotal instruments in machine learning and signal processing for processing graph-structured data. This paper proposes an analysis framework to investigate the sensitivity of GCNNs to probabilistic graph perturbations, directly impacting the graph shift operator (GSO). Our study establishes tight expected GSO error bounds, which are explicitly linked to the error model parameters, and reveals a linear relationship between GSO perturbations and the resulting output differences at each layer of GCNNs. This linearity demonstrates that a single-layer GCNN maintains stability under graph edge perturbations, provided that the GSO errors remain bounded, regardless of the perturbation scale. For multilayer GCNNs, the dependency of system's output difference on GSO perturbations is shown to be a recursion of linearity. Finally, we exemplify the framework with the Graph Isomorphism Network (GIN) and Simple Graph Convolution Network (SGCN). Experiments validate our theoretical derivations and the effectiveness of our approach.

Covariance matrix tapers have a long history in signal processing and related fields. Examples of applications include autoregressive models (promoting a banded structure) or beamforming (widening the spectral null width associated with an interferer). In this paper, the focus is on high-dimensional setting where the dimension $p$ is high, while the data aspect ratio $n/p$ is low. We propose an estimator called Tabasco (TApered or BAnded Shrinkage COvariance matrix) that shrinks the tapered sample covariance matrix towards a scaled identity matrix. We derive optimal and estimated (data adaptive) regularization parameters that are designed to minimize the mean squared error (MSE) between the proposed shrinkage estimator and the true covariance matrix. These parameters are derived under the general assumption that the data is sampled from an unspecified elliptically symmetric distribution with finite 4th order moments (both real- and complex-valued cases are addressed). Simulation studies show that the proposed Tabasco outperforms all competing tapering covariance matrix estimators in diverse setups. A space-time adaptive processing (STAP) application also illustrates the benefit of the proposed estimator in a practical signal processing setup.

We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.