Real-valued Matrix Research Articles

Real-valued DOA estimation for a mixture of uncorrelated and coherent sources via unitary transformation

This paper presents a novel real-valued DOA estimation method to handle the scenarios where both the uncorrelated and coherent sources coexist. More specifically, an augmented matrix is constructed and then transformed into a real-valued version for the DOA estimation of uncorrelated sources by utilizing the unitary transformation, which allows an extension of the effective array aperture. Afterwards, an oblique projection operator is employed so that the contributions of the uncorrelated sources are removed. Finally, a real-valued coherent augmented matrix is constructed to estimate the remaining coherent sources. In addition, the fading coefficients are estimated by adding penalties to a constraint quadratic minimization problem, which guarantees the stability of the solution. Compared with the existing partial real-valued and complex-valued DOA estimation methods for a mixture of uncorrelated and coherent sources, the proposed method offers favorable performance in terms of both estimation accuracy and computational efficiency. Furthermore, our method makes it possible to resolve more sources than the number of sensors. Simulation results demonstrate the effectiveness and notable performance of the proposed method.

Two-dimensional discrete fractional Fourier transform-based content removal algorithm

This paper proposes a novel content removal technique for enhancing the camera identification performance. Here, very low bit rate videos with the overall noise patterns having time-varying statistics are considered. First, different two-dimensional discrete fractional Fourier transforms with different rotational angles are applied to the overall noise pattern of each frame of each video. Second, the modulus of each element of each transformed matrix is normalized to one if the rotational angles of the transforms are not equal to the integer multiples of \(\pi \). Third, the corresponding two-dimensional inverse discrete fractional Fourier transform is applied to each normalized matrix, and the corresponding real part is taken out for the further processing. Fourth, the absolute values of the elements in each normalized real-valued matrix are bounded by certain threshold values. Here, different threshold values are employed for different rotational angles. Finally, the processed matrices are averaged over all the rotational angles and all the frames of the videos of the same camera. To evaluate the performance, the correlation function is employed. Extensive computer numerical simulations are preformed. The obtained results show that the proposed method outperforms existing methods (Kang et al. in IEEE Trans Inf Forensics Secur 7(2):393–402, 2012; Li in IEEE Trans Inf Forensics Secur 5(2):280–287, 2010).

A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic–viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter Q in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter Q in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using P and S velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available.

Improved DOA estimation based on real-valued array covariance using sparse Bayesian learning

To further improve the efficiency of sparse Bayesian learning (SBL) for direction of arrival (DOA) estimation, a real-valued (unitary) formulation of covariance vector-based relevance vector machine (CV-RVM) technique is proposed in this paper. The covariance matrix of the sensor output is firstly transformed into a real-valued covariance matrix via unitary transformation, and the real-valued covariance matrix can be sparsely represented in a real-valued over-complete dictionary. Then the sparse Bayesian learning technique implemented in real domain is used to estimate the DOA. According to the property of the real-valued covariance matrix, unitary single measurement vector (USMV) CV-RVM for uncorrelated signals and unitary multiple measurement vector (UMMV) CV-RVM for correlated signals are developed, respectively. Due to the fact that the proposed methods are implemented in real domain and the snapshots are doubled via unitary transformation, the proposed methods have lower computational cost and better performance compared to the original SMV CV-RVM and MMV CV-RVM. Simulation results show the effectiveness of the proposed methods.

The Real-Valued Sparse Direction of Arrival (DOA) Estimation Based on the Khatri-Rao Product.

There is a problem that complex operation which leads to a heavy calculation burden is required when the direction of arrival (DOA) of a sparse signal is estimated by using the array covariance matrix. The solution of the multiple measurement vectors (MMV) model is difficult. In this paper, a real-valued sparse DOA estimation algorithm based on the Khatri-Rao (KR) product called the L1-RVSKR is proposed. The proposed algorithm is based on the sparse representation of the array covariance matrix. The array covariance matrix is transformed to a real-valued matrix via a unitary transformation so that a real-valued sparse model is achieved. The real-valued sparse model is vectorized for transforming to a single measurement vector (SMV) model, and a new virtual overcomplete dictionary is constructed according to the KR product’s property. Finally, the sparse DOA estimation is solved by utilizing the idea of a sparse representation of array covariance vectors (SRACV). The simulation results demonstrate the superior performance and the low computational complexity of the proposed algorithm.

Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays

In this paper, we first propose quaternion-valued neural networks (QVNNs) with unbounded time-varying delays. Some sufficient conditions on the global μ-stability in the form of both complex-valued and real-valued linear matrix inequalities (LMIs) are provided by solving two difficulties. One is decomposing the QVNN into two complex-valued systems with the plural decomposition method of quaternion, which can reduce the complexity of calculations by avoiding the non-commutativity of quaternion multiplication. The other is choosing the appropriate Lyapunov–Krasovskii functional in the form of Hermitian matrices, which is a big challenge. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.

Probabilistic low-rank matrix completion from quantized measurements

We consider the recovery of a low rank real-valued matrix M given a subset of noisy discrete (or quantized) measurements. Such problems arise in several applications such as collaborative filtering, learning and content analytics, and sensor network localization. We consider constrained maximum likelihood estimation of M, under a constraint on the entry-wise infinity-norm of M and an exact rank constraint. We provide upper bounds on the Frobenius norm of matrix estimation error under this model. Previous theoretical investigations have focused on binary (1-bit) quantizers, and been based on convex relaxation of the rank. Compared to the existing binary results, our performance upper bound has faster convergence rate with matrix dimensions when the fraction of revealed observations is fixed. We also propose a globally convergent optimization algorithm based on low rank factorization of M and validate the method on synthetic and real data, with improved performance over previous methods.

Real-valued MVDR beamforming using spherical arrays with frequency invariant characteristic

Complex-valued minimum variance distortionless response (MVDR) beamforming for wideband signals has very high computational amount. In this paper, we design a novel real-valued MVDR beamformer for spherical arrays. The dependence of the array steering matrix on source signal directions and frequencies is decoupled using spherical harmonic decomposition. Then a compensation network is designed to solve the frequency dependence of the array response and to get a new array response only determined by the spherical harmonics of the source directions. All frequency bins of wideband signals can be used together instead of being processed independently. By exploiting the property of the conjugate spherical harmonics, a unitary transform can be found to acquire a real-valued frequency invariant steering matrix (FISM). Based on the FISM, real-valued MVDR (RV-MVDR) is developed to obtain good performance with low computational amount. Simulation results demonstrate the performance of our proposed method for beamforming and direction-of-arrival (DOA) estimation by comparing with the complex-valued and real-weighted MVDR methods.

Real‐valued localisation algorithm for near‐field non‐circular sources

A novel real-valued near-field localisation algorithm for non-circular sources is proposed using the uniform linear array. On the basis of non-circularity of signals, an extended real-valued covariance matrix is constructed. Then, utilising the symmetric property of the extended array manifold, the direction of arrival and range parameters are obtained by two one-dimensional rank reduction estimators. The proposed algorithm is efficient in that it only requires second-order statistics (SOS) and real-valued computations. Moreover, it can improve the estimation accuracy and resolve more sources than the conventional SOS-based methods.

High‐resolution DOA estimation for closely spaced correlated signals using unitary sparse Bayesian learning

A novel method is proposed to effectively solve the challenging problem of direction-of-arrival (DOA) estimation for closely spaced correlated signals. A centro-Hermitian extended matrix is exploited to double the number of data samples, and then is transformed into a real-valued data matrix. An improved sparse Bayesian learning scheme is utilised to estimate DOAs by recovering the real-valued jointly row-sparse solution matrix with a reduced computational burden. The proposed method not only provides increased estimation accuracy but also has improved angular separation performance. Simulation results validate the effectiveness of the proposed method.

Adaptive multinomial matrix completion

The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. We consider a general, non-uniform, sampling scheme and give theoretical guarantees on the performance of a constrained, nuclear norm penalized maximum likelihood estimator. One important advantage of this estimator is that it does not require knowledge of the rank or an upper bound on the nuclear norm of the unknown matrix and, thus, it is adaptive. We provide lower bounds showing that our estimator is minimax optimal. An efficient algorithm based on lifted coordinate gradient descent is proposed to compute the estimator. A limited Monte-Carlo experiment, using both simulated and real data is provided to support our claims.

Reproducing electric field observations during magnetic storms by means of rigorous 3-D modelling and distortion matrix co-estimation

Electric fields induced in the conducting Earth by geomagnetic disturbances drive currents in power transmission grids, telecommunication lines or buried pipelines, which can cause service disruptions. A key step in the prediction of the hazard to technological systems during magnetic storms is the calculation of the geoelectric field. To address this issue for mid-latitude regions, we revisit a method that involves 3-D modelling of induction processes in a heterogeneous Earth and the construction of a magnetospheric source model described by low-degree spherical harmonics from observatory magnetic data. The actual electric field, however, is known to be perturbed by galvanic effects, arising from very local near-surface heterogeneities or topography, which cannot be included in the model. Galvanic effects are commonly accounted for with a real-valued time-independent distortion matrix, which linearly relates measured and modelled electric fields. Using data of six magnetic storms that occurred between 2000 and 2003, we estimate distortion matrices for observatory sites onshore and on the ocean bottom. Reliable estimates are obtained, and the modellings are found to explain up to 90% of the measurements. We further find that 3-D modelling is crucial for a correct separation of galvanic and inductive effects and a precise prediction of the shape of electric field time series during magnetic storms. Since the method relies on precomputed responses of a 3-D Earth to geomagnetic disturbances, which can be recycled for each storm, the required computational resources are negligible. Our approach is thus suitable for real-time prediction of geomagnetically induced currents by combining it with reliable forecasts of the source field.

Rotordynamic analysis using the Complex Transfer Matrix: An application to elastomer supports using the viscoelastic correspondence principle

Numerous methods are available to calculate rotordynamic whirl frequencies, including analytic methods, finite element analysis, and the transfer matrix method. The typical real-valued transfer matrix (RTM) suffers from several deficiencies, including lengthy computation times and the inability to distinguish forward and backward whirl. Though application of complex coordinates in rotordynamic analysis is not novel per se, specific advantages gained from using such coordinates in a transfer matrix analysis have yet to be elucidated. The present work employs a complex coordinate redefinition of the transfer matrix to obtain reduced forms of the elemental transfer matrices in inertial and rotating reference frames, including external stiffness and damping. Application of the complex-valued state variable redefinition results in a reduction of the 8×8 RTM to the 4×4 Complex Transfer Matrix (CTM). The CTM is advantageous in that it intrinsically separates forward and backward whirl, eases symbolic manipulation by halving the transfer matrices’ dimension, and provides significant improvement in computation time. A symbolic analysis is performed on a simple overhung rotor to demonstrate the mathematical motivation for whirl frequency separation. The CTM׳s utility is further shown by analyzing a rotordynamic system supported by viscoelastic elastomer rings. Viscoelastic elastomer ring supports can provide significant damping while reducing the cost and complexity associated with conventional components such as squeeze film dampers. The stiffness and damping of a viscoelastic damper ring are determined herein as a function of whirl frequency using the viscoelastic correspondence principle and a constitutive fractional calculus viscoelasticity model. The CTM is then employed to obtain the characteristic equation, where the whirl frequency dependent stiffness and damping of the elastomer supports are included. The Campbell diagram is shown, demonstrating the CTM׳s ability to intrinsically separate synchronous whirl direction for a non-trivial rotordynamic system. Good agreement is found between the CTM results and previously obtained analytic and experimental results for the elastomer ring supported rotordynamic system.

Research on the Gerschgorin Radii Estimator Based on Real-Valued Decomposition Technique in Picosatellites

In order to reduce the computational burden, real-valued decomposition technique is applied in picosat positioning. Through converting the correlation matrix to real-valued matrix, the real-valued decomposition technique not only reduces the computer burden of matrix decomposition, but also makes the correlation matrix more close to Toeplitz matrix through a forward-backward smoothing. Simulations confirm that the real-valued Gerschgorin Radii shows an improved performance.

Universal state transfer on graphs

A continuous-time quantum walk on a graph G is given by the unitary matrix U(t)=exp(−itA), where A is the adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any ϵ>0, there is a time t, where the (a,b)-entry of U(t) satisfies |U(t)a,b|≥1−ϵ. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for ϵ=0. In this work, we study a natural extension of this notion called universal state transfer wherein state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (i) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (each eigenvector has entries which are equal in magnitude). (ii) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, the switching automorphism group is cyclic. (iii) There is a family of complex oriented prime-length cycles which has universal pretty good state transfer. This provides a concrete example of a family of graphs with this universal property. (iv) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. Finally, we provide a spectral characterization of universal perfect state transfer for graphs switching equivalent to circulants.

Scalability of Parallel Genetic Algorithm for Two-mode Clustering

matrix having the same set of entity in the rows and cloumns is known as one-mode data matrix, and traditional one-mode clustering algorithms can be used to cluster the rows (or columns) separately. With the popularity of use of two-mode data matrices where the rows and columns have different sets of entities, the need for simultaneous clustering of rows and columns popularly known as two-mode clustering increased. Additionally, the emergence of large data sets and the prediction of Moore's law slow-down have created the challenge of clustering scalability. In this paper, we address the problem of scalability of organizing an unlabelled two- mode dataset into clusters utilizing multicore processor. We propose a parallel genetic algorithm (GA) heuristics based two-mode clustering algorithm, which is an adaptation of the classical Cuthill-McKee Matrix Bandwidth Minimization (MBM) algorithm. The classical MBM method aims at reducing the bandwidth of a sparse symmetric matrix, which we adapted to make it suitable for non-symmetric real-valued matrix. Preliminary results indicate that our algorithm is scalable on multicore processor compared to serial implementation. Future work will include more extensive experiments and evaluations of the system.

Low-Complexity Sorted QR Decomposition for MIMO Systems Based on Pairwise Column Symmetrization

QR decomposition (QRD) is a preprocessing technique for detecting symbols in multiple-input and multiple-output (MIMO) systems, but the computational complexity is prohibitively high when the systems incorporate a large number of antennas. This paper presents a low-complexity sorted QRD (SQRD) algorithm for MIMO systems. The proposed algorithm performs SQRD through orthogonalizations based on the modified Gram-Schmidt process, rearranging the column vectors of a real-valued MIMO channel matrix in such a way that the symmetry between the vectors is maintained. By using the symmetry, the computations required for orthogonalizing one of the two adjacent vectors can be eliminated effectively, which significantly reduces the computational complexity. Theoretical analyses show that the proposed algorithm reduces the computational complexity required for SQRD by 50% for any MIMO configurations, when compared to the conventional algorithm. In addition, the memory requirement to store resultant matrices is 50% of that in the conventional one.

Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds

The present research paper proposes an extension of the classical scalar Auto-Regressive Moving-Average (ARMA) model to real-valued Riemannian matrix manifolds. The resulting ARMA model on matrix manifolds is expressed as a non-linear discrete-time dynamical system in state-space form whose state evolves on the tangent bundle associated with the underlying manifold. A number of examples are discussed within the present contribution that aim at illustrating the numerical behavior of the proposed ARMA model. In order to measure the degree of temporal dependency between the state-values of the ARMA model, an extension of the classical autocorrelation function for scalar sequences is suggested on the basis of the geometrical features of the underlying real-valued matrix manifold.

Single Network Relational Transductive Learning

Relational classification on a single connected network has been of particular interest in the machine learning and data mining communities in the last decade or so. This is mainly due to the explosion in popularity of social networking sites such as Facebook, LinkedIn and Google+ amongst others. In statistical relational learning, many techniques have been developed to address this problem, where we have a connected unweighted homogeneous/heterogeneous graph that is partially labeled and the goal is to propagate the labels to the unlabeled nodes. In this paper, we provide a different perspective by enabling the effective use of graph transduction techniques for this problem. We thus exploit the strengths of this class of methods for relational learning problems. We accomplish this by providing a simple procedure for constructing a weight matrix that serves as input to a rich class of graph transduction techniques. Our procedure has multiple desirable properties. For example, the weights it assigns to edges between unlabeled nodes naturally relate to a measure of association commonly used in statistics, namely the Gamma test statistic. We further portray the efficacy of our approach on synthetic as well as real data, by comparing it with state-of-the-art relational learning algorithms, and graph transduction techniques with an adjacency matrix or a real valued weight matrix computed using available attributes as input. In these experiments we see that our approach consistently outperforms other approaches when the graph is sparsely labeled, and remains competitive with the best when the proportion of known labels increases.

Deterministic Construction of Real-Valued Ternary Sensing Matrices Using Optical Orthogonal Codes

In this letter, a new class of real-valued matrices is presented for deterministic compressed sensing. A base matrix is constructed by cyclic shifts of binary sequences in an optical orthogonal code (OOC). Then, a Hadamard matrix is used for its extension, which ultimately produces a real-valued matrix that takes the entries of 0, -1 and +1 before normalization. The new sensing matrix forms a tight frame with small coherence, which theoretically guarantees the average recovery performance of sparse signals with uniformly distributed supports. Several example sensing matrices are presented by employing a special type of OOCs obtained from modular Golomb rulers.