This paper introduces Generalized Nonnegative Structured Kruskal Tensor Regression (NS-KTR), a novel tensor regression framework that enhances interpretability and performance through mode-specific hybrid regularization and nonnegativity constraints. Our approach accommodates both linear and logistic regression formulations for diverse response variables while addressing the structural heterogeneity inherent in multidimensional tensor data. We integrate fused LASSO, total variation, and ridge regularizers — each tailored to specific tensor modes — and develop an efficient alternating direction method of multipliers (ADMM)-based algorithm for parameter estimation. Comprehensive experiments on synthetic signals and real hyperspectral datasets demonstrate that NS-KTR consistently outperforms conventional tensor regression methods. The framework’s ability to preserve distinct structural characteristics across tensor dimensions while ensuring physical interpretability makes it especially suitable for applications in signal processing and hyperspectral image analysis. • NS-KTR: nonnegative structured Kruskal tensor regression with hybrid regularization. • Mode-specific regularization: LASSO, total variation, and ridge across tensor modes. • Unified framework supports linear and logistic regression for diverse responses. • ADMM-based optimization achieves superior accuracy with significant speedups.

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We propose a robust activity detection for grant free random access using greedy covariance-learning-based matching pursuit (RCL-MP) algorithm. The method incorporates a robust loss function into the Gaussian negative log-likelihood function, and uses matching pursuit framework for greedily selecting the indices of active users. This algorithm employs a flexible loss function effectively recovering sparse support under non-Gaussian noise conditions. Furthermore, we numerically demonstrate the robustness of RCL-MP across various conditions in massive access scenarios.

Robust hybrid beamforming for integrated sensing and communications (ISAC) system under bounded uncertainties in sensing reception is developed using algorithm unrolling technique. First, the robust hybrid beamforming design problem is formulated as an optimization problem that jointly maximizes the communication sum-rate and the worst-case sensing mutual information under the uncertainty of receive steering vector. Then, a benchmark method using projected gradient descent and ascent (PGDA) algorithm is designed to solve this optimization problem. Finally, we propose to unroll the developed PGDA algorithm using the algorithm unrolling technique. Numerical results demonstrate the advantages of the unrolled PGDA algorithm over the PGDA benchmark for addressing the newly introduced problem of robust hybrid beamforming design for ISAC.

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 ν 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 ϵ-contaminated data. The robust SBL estimators perform well for all cases and nearly identical with classical SBL for Gaussian array data.

We propose greedy Capon beamformer (GCB) 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 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 performs 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.

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.

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.

In this paper, we propose two algorithms to enhance the interpretability of the hyperbola in B-scans obtained with a Ground Penetrating Radar (GPR). These hyperbolas are the responses of buried objects or cavities. To correctly detect and classify them, a denoising is typically necessary for GPR images as the signal-to-noise ratio is low, and the various interfaces naturally present in the earth have a strong response. Both algorithms are based on a sparse convolutional coding model plus a low rank component. It is solved through an Alternating Direction Method of Multipliers (ADMM) framework. In order to take into account the presence of outliers and the artifacts caused by the acquisition, the second algorithm is based on the Huber norm instead of the classicL2-norm. These algorithms are tested on a real dataset labeled by geophysicists. The results show the denoising efficiency of this approach, and in particular the robustness of the second algorithm.

Recent investigations indicate that Sparse Bayesian Learning (SBL) is lacking in robustness. We derive a robust and sparse Direction of Arrival (DOA) estimation framework based on the assumption that the array data has a centered (zero-mean) complex elliptically symmetric (ES) distribution with finite second-order moments. In the derivation, the loss function can be quite general. We consider three specific choices: the ML-loss for the circularly symmetric complex Gaussian distribution, the ML-loss for the complex multivariate t-distribution (MVT) with ν degrees of freedom, and the loss for Huber’s M-estimator. For Gaussian loss, the method reduces to the classic SBL method. The root mean square DOA performance of the derived estimators is discussed for Gaussian, MVT, and ϵ- contaminated noise. The robust SBL estimators perform well for all cases and nearly identical with classical SBL for Gaussian noise.

This paper presents a multimodal image fusion method using a novel decomposition model based on coupled dictionary learning. The proposed method is general and can be used for a variety of imaging modalities. In particular, the images to be fused are decomposed into correlated and uncorrelated components using sparse representations with identical supports and a Pearson correlation constraint, respectively. The resulting optimization problem is solved by an alternating minimization algorithm. Contrary to other learning-based fusion methods, the proposed approach does not require any training data, and the correlated features are extracted online from the data itself. By preserving the uncorrelated components in the fused images, the proposed fusion method significantly improves on current fusion approaches in terms of maintaining the texture details and modality-specific information. The maximum-absolute-value rule is used for the fusion of correlated components only. This leads to an enhanced contrast-resolution without causing intensity attenuation or loss of important information. Experimental results show that the proposed method achieves superior performance in terms of both visual and objective evaluations compared to state-of-the-art image fusion methods.

The spatial sign covariance matrix (SSCM), also known as the normalized sample covariance matrix (NSCM), has been widely used in signal processing as a robust alternative to the sample covariance matrix (SCM). It is well-known that the SSCM does not provide consistent estimates of the eigenvalues of the shape matrix (normalized scatter matrix). To alleviate this problem, we propose BASIC (Bias Adjusted SIgn Covariance), which performs an approximate bias correction to the eigenvalues of the SSCM under the assumption that the samples are generated from zero mean unspecified complex elliptically symmetric distributions (the real-valued case is also addressed). We then use the bias correction in order to develop a robust regularized SSCM based estimator, BASIC Shrinkage estimator (BASICS), which is suitable for high dimensional problems, where the dimension can be larger than the sample size. We assess the proposed estimator with several numerical examples as well as in a linear discriminant analysis (LDA) classification problem with real data sets. The simulations show that the proposed estimator compares well to competing robust covariance matrix estimators but has the advantage of being significantly faster to compute.

The first step for any graph signal processing (GSP) procedure is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little attention has been paid to modelling such errors in the adjacency matrix, and studying their effects on GSP methods. However, modelling errors in the adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically when possible and numerically, the graph error effect on the performance of GSP methods in different types of problems such as filtering of graph signals and independent component analysis of graph signals (graph decorrelation).

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.

With a change of signal notion to graph signal, new means of performing blind source separation (BSS) appear. Particularly, existing independent component analysis (ICA) methods exploit the non-Gaussianity of the signals or other types of prior information. For graph signals, such prior information is present in a graph of dependencies in the signals. We propose BSS of graph signals which uses the prior information presented by the signal graph together with non-Gaussianity. We derive the identifiability conditions for the proposed method and compare them to the conditions when only graph or non-Gaussianity approach is used. In simulation studies, we verify that the new method can separate a broader range of graph signals and show that it is also more efficient when both approaches are useful.

A popular regularized (shrinkage) covariance estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward its grand mean. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, or t weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from a heavy-tailed distribution.

In this paper, we propose a new regularized (penalized) covariance matrix estimator which encourages grouping of the eigenvalues by penalizing large differences (gaps) between successive eigenvalues. This is referred to as fusing eigenvalues (eFusion). The proposed penalty function utilizes Tukey's biweight function that is widely used in robust statistics. The main advantage of the proposed method is that it has very small bias for sufficiently large values of penalty parameter. Hence, the method provides accurate grouping of eigenvalues. Such benefits of the proposed method are illustrated with a numerical example, where the method is shown to perform favorably compared to a state-of-art method.

Recovering a graph signal from samples is a central problem in graph signal processing. Least mean squares (LMS) method for graph signal estimation is computationally efficient adaptive method. In this paper, we introduce a technique to robustify LMS with respect to mismatches in the presumed graph topology. It builds on the fact that graph LMS converges faster when the graph topology is specified correctly. We consider two measures of convergence speed, based on which we develop randomized greedy algorithms for robust interpolation of graph signals. In simulation studies, we show that the randomized greedy robust least mean squares (RGRLMS) outperforms the regular LMS and has even more potential given a robust sampling design.

The first step in any graph signal processing (GSP) task is to learn the graph signal representation, i.e., to capture the dependence structure of the data into an adjacency matrix. Indeed, the adjacency matrix is typically not known a priori and has to be learned. However, it is learned with errors. A little, if any, attention has been paid to modeling such errors in the adjacency matrix, and studying their effects on GSP tasks. Modeling errors in adjacency matrix will enable both to study the graph error effects in GSP and to develop robust GSP algorithms. In this paper, we therefore introduce practically justifiable graph error models. We also study, both analytically and in terms of simulations, the graph error effect on the performance of GSP based on the example of independent component analysis of graph signals (graph decorrelation).

The paper considers the problem of estimating the covariance matrices of multiple classes in a low sample support condition, where the data dimensionality is comparable to, or larger than, the sample sizes of the available data sets. In such conditions' a common approach is to shrink the class sample covariance matrices (SCMs) towards the pooled SCM. The success of this approach hinges upon the ability to choose the optimal regularization parameter. Typically, a common regularization level is shared among the classes and determined via a procedure based on cross-validation. We use class-specific regularization levels since this enables the derivation of the optimal regularization parameter for each class in terms of the minimum mean squared error (MMSE). The optimal parameters depend on the true unknown class population covariances. Consistent estimators of the parameters can, however, be easily constructed under the assumption that the class populations follow (unspecified) elliptically symmetric distributions. We demonstrate the performance of the proposed method via a simulation study as well as via an application to discriminant analysis using both synthetic and real data sets.

We propose a modification of linear discriminant analysis, referred to as compressive regularized discriminant analysis (CRDA), for analysis of high-dimensional datasets. CRDA is especially designed for feature elimination purpose and can be used as gene selection method in microarray studies. CRDA lends ideas from ℓq,1 norm minimization algorithms in the multiple measurement vectors (MMV) model and utilizes joint-sparsity promoting hard thresholding for feature elimination. A regularization of the sample covariance matrix is also needed as we consider the challenging scenario where the number of features (variables) is comparable or exceeding the sample size of the training dataset. A simulation study and four examples of real life microarray datasets evaluate the performances of CRDA based classifiers. Overall, the proposed method gives fewer misclassification errors than its competitors, while at the same time achieving accurate feature selection.

In scaled lasso, the unknown regression coefficients and the scale parameter of the error distribution are estimated jointly. In lasso, the optimal penalty parameter is well-known to depend on the error scale, and it is therefore typically chosen using cross-validation. The main benefit of scaled lasso is that the penalty parameter is scale-free and can be predetermined from pure theoretical considerations. Nevertheless, scaled lasso performs poorly when there exist strong correlations between the predictors. As a remedy, we propose two different scaled elastic net (EN) formulations and derive convergent algorithms for their computation. The first formulation uses a conventional EN penalty whereas the second formulation differs from the former in that the ℓ2-loss is not squared. The former approach is referred to as the scaled EN estimator and the latter as the square-root EN estimator. We illustrate via numerical examples and simulations that the proposed methods outperform the scaled lasso, especially in the presence of high mutual coherence in the feature space.

The widely used fixed-point FastICA algorithm has been derived and motivated as being an approximate Newton-Raphson (NR) algorithm. In the original derivation, the Lagrangian multiplier is treated as a constant and an ad hoc approximation is used for Jacobian matrix in the NR update. In this letter, we provide an alternative derivation of the FastICA algorithm without approximation. We show that any solution to the FastICA algorithm is a solution to the exact NR algorithm as well. In addition, we propose a novel power iteration algorithm for FastICA which is remarkably more stable than the fixed-point algorithm, when the sample size is not orders of magnitudes larger than the dimension. Our proposed algorithm can be run on parallel computing nodes.

Independent component analysis (ICA) is a widely used technique for extracting latent (unobserved) source signals from observed multidimensional measurements. In this paper we construct a fast and robust bootstrap (FRB) method for testing hypotheses on elements of the mixing matrix in the ICA model. The FRB method can be devised for estimators which are solutions to fixed-point (FP) equations. In this paper we develop FRB test for the widely popular FastICA estimator. The developed test can be used in real-world ICA analysis of high-dimensional data sets seen e.g. in big data analysis, as it avoids the common obstacles of conventional bootstrap such as immense computational cost and lack of robustness. Moreover, instability and convergence problems of the Fast ICA algorithm when applied to bootstrap data are prevented. Simulations and examples illustrate the usefulness and validity of the developed test.

Multi-linear techniques using tensor decompositions provide a unifying framework for the high-dimensional data analysis. Sparsity in tensor decompositions clearly improves the analysis and inference of multi-dimensional data. Other than non-negative tensor factorizations, the literature on tensor estimation using sparsity is limited. In this paper, we introduce sparse regularization methods for tensor decompositions which are useful for dimensionality reduction, feature selection as well as signal recovery. One major challenge in most of the tensor decomposition algorithms is their heavy dependence on good initializations. To alleviate such a critical problem we propose a reliable method based on the ridge regression to provide good starting values taking advantage of sparsity. Combined with such initializations our sparse regularization methods show highly improved performance over the conventional methods in the demonstrated simulation studies.

The compound-Gaussian (CG) distributions have been successfully used for modelling the non-Gaussian clutter measured by high-resolution radars. Within the CG class, the complexK-distribution and the complext-distribution have been used for modelling sea clutter which is often heavy-tailed or spiky in nature. In this paper, a heavy-tailed CG model with an inverse Gaussian texture distribution is proposed and its distributional properties such as closed-form expressions for its probability density function (p.d.f.) as well as its amplitude p.d.f., amplitude cumulative distribution function and its kurtosis parameter are derived. Experimental validation of its usefulness for modelling measured real-world radar lake-clutter is provided where it is shown to yield better fits than its widely used competitors.

Recent research has revealed that circularity (or, propriety) of complex random signals can be exploited in developing optimal signal processors. In this paper, a robust estimator of circularity is pro posed. The estimate is found by solving M-estimation equations and employing a novel weighting scheme. A simple iterative algorithm for its computation is introduced. A robust circularity detector stemming from the large sample properties of the estimator is proposed. It is shown to be valid detector under the broad class of complex elliptically symmetric (CES) distributions. An illustrative example demonstrating the reliable performance of the detector in a practical signal processing application is provided.

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the mathematical foundations of complex-valued signals and tools developed for handling them are scattered in literature. There appears to be a need for a concise, unified, and rigorous treatment of such topics. In this paper such a treatment is provided. Moreover, we establish connections between seemingly unrelated objects such as real differentiability and circularity. In addition, a novel complex-valued extension of Taylor series is presented and a measure for circularity is proposed.

An important characteristic of a complex random variable z is the so-called circularity property or lack of it. We study the properties of the degree of circularity based on second-order moments, called circularity quotient, that is shown to possess an intuitive geometrical interpretation: the modulus and phase of its principal square-root are equal to the eccentricity and angle of orientation of the ellipse defined by the covariance matrix of the real and imaginary part of z. Hence, when the eccentricity approaches the minimum zero (ellipse is a circle), the circularity quotient vanishes; when the eccentricity approaches the maximum one, the circularity quotient lies on the unit complex circle. Connection with the correlation coefficient rho is established and bounds on rho given the circularity quotient (and vice versa) are derived. A generalized likelihood ratio test (GLRT) of circularity assuming complex normal sample is shown to be a function of the modulus of the circularity quotient with asymptotic chi22distribution.

For the development of future wireless systems, it is crucial to create accurate channel models. Channel sounding using antenna arrays and consequently propagation parameter estimation are key tasks in creating such models. In this paper we present an estimator for the angular distribution of the diffuse scattering component that is observed in channel sounding measurements. The angular distribution is modeled as a mixture of Von Mises distributions, which correspond to scatterer clusters. The parameters of the individual distributions as well as the mixture proportions are estimated. The large sample performance of the estimator is studied by deriving the Cramer-Rao lower bound and comparing the variance of the estimates to it. The simulations show that the the proposed estimator has asymptotically optimal performance since it attains the Cramer-Rao lower bound for relatively small sample sizes.

We will derive a stochastic maximum likelihood method for estimating spatio-temporal channel parameters. Such estimators are needed in propagation studies where extensive channel measurements and sounding are required. These are seminal tasks in the process of developing advanced channel models. The proposed method employs angular Von Mises distribution model which is appropriate for directional data typically observed in channel measurement campaigns. The signal model is stochastic. The performance of the proposed method is compared to SAGE algorithm where the signal model is deterministic. The computational complexity of the proposed method is lower and channel parameters are estimated with higher fidelity because the underlying distribution model is well-suited for directional data.

This paper addresses the problem of antenna array processing in nonGaussian noise and interference conditions. Such conditions arise due to man-made interference in indoor and outdoor mobile communication channels as well as in military communications. In this paper M-estimators of the array (pseudo-)covariance matrix based upon complex data set are introduced. Estimates of the noise and signal subspaces based on M-estimators are then used to robustify the subspace direction of arrival (DOA) estimation methods. In addition, eigenvalues based on M-estimators are used in MDL criterion, thus yielding a robust signal detection method. The reliable performance of the proposed methods are shown by simulations.