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Eigendecomposition of symmetric matrices is at the heart of many computer vision algorithms. However, the derivatives of the eigenvectors tend to be numerically unstable, whether using the SVD to compute them analytically or using the Power Iteration (PI) method to approximate them. This instability arises in the presence of eigenvalues that are close to each other. This makes integrating eigendecomposition into deep networks difficult and often results in poor convergence, particularly when dealing with large matrices. While this can be mitigated by partitioning the data into small arbitrary groups, doing so has no theoretical basis and makes it impossible to exploit the full power of eigendecomposition. In previous work, we mitigated this using SVD during the forward pass and PI to compute the gradients during the backward pass. However, the iterative deflation procedure required to compute multiple eigenvectors using PI tends to accumulate errors and yield inaccurate gradients. Here, we show that the Taylor expansion of the SVD gradient is theoretically equivalent to the gradient obtained using PI without relying in practice on an iterative process and thus yields more accurate gradients. We demonstrate the benefits of this increased accuracy for image classification and style transfer., Comment: IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) PREPRINT 2021

This paper investigates robust downlink precoding for three-dimensional (3D) massive multi-input multi-output (MIMO) configuration with matrix manifold optimization. Starting with a posteriori channel model, we formulate the robust precoder design to maximize an upper bound of ergodic weighted sum-rate under a total power budget. We derive the generalized eigenvector structure for optimal precoder with matrix manifold optimization. However, since the precoding of multiple users is coupled in the structure, we maximize the objective function for each user in alternation and prove the solution of each individual problem is the generalized eigenvector corresponding to the maximum generalized eigenvalue. In accordance with this, we design an iterative algorithm and present its convergence analysis. Furthermore, we propose a Riemannian conjugate gradient (RCG) method to solve the generalized eigenvalue problem (GEP) for higher efficiency in the precoder design algorithm.

The performance of most the clustering methods hinges on the used pairwise affinity, which is usually denoted by a similarity matrix. However, the pairwise similarity is notoriously known for its vulnerability of noise contamination or the imbalance in samples or features, and thus hinders accurate clustering. To tackle this issue, we propose to use information among samples to boost the clustering performance. We proved that a simplified similarity for pairs, denoted by a fourth order tensor, equals to the Kronecker product of pairwise similarity matrices under decomposable assumption, or provide complementary information for which the pairwise similarity missed under indecomposable assumption. Then a high order similarity matrix is obtained from the tensor similarity via eigenvalue decomposition. The high order similarity capturing spatial information serves as a robust complement for the pairwise similarity. It is further integrated with the popular pairwise similarity, named by IPS2, to boost the clustering performance. Extensive experiments demonstrated that the proposed IPS2 significantly outperformed previous similarity-based methods on real-world datasets and it was capable of handling the clustering task over under-sampled and noisy datasets., 10 pages, 7 figures, 2 tables, 4 pages supplementary information appendix

Multiview subspace learning has attracted much attention due to the efficacy of exploring the information on multiview features. Most existing methods perform data reconstruction on the original feature space and thus are vulnerable to noisy data. In this article, we propose a novel multiview subspace learning method, called multiview consensus structure discovery (MvCSD). Specifically, we learn the low-dimensional subspaces corresponding to different views and simultaneously pursue the structure consensus over subspace clustering for multiple views. In such a way, latent subspaces from different views regularize each other toward a common consensus that reveals the underlying cluster structure. Compared to existing methods, MvCSD leverages the consensus structure derived from the subspaces of diverse views to better exploit the intrinsic complementary information that well reflects the essence of data. Accordingly, the proposed MvCSD is capable of producing a more robust and accurate representation structure which is crucial for multiview subspace learning. The proposed method can be optimized effectively, with theoretical convergence guarantee, by alternatively iterating the argument Lagrangian multiplier algorithm and the eigendecomposition. Extensive experiments on diverse datasets demonstrate the advantages of our method over the state-of-the-art methods.

The matrix enhancement and matrix pencil (MEMP) has been used to reduce the number of array elements in the planar array with a shaped-beam pattern. This work extends the MEMP-based synthesis method to the synthesis of pattern reconfigurable sparse planar arrays in wide angle scanning and wide-band, which exploits a necessary restriction on finding the common element positions and individual element excitations for multiple patterns. Moreover, a unitary transformation is exploited into the multiple-composite equivalent Hankel matrix, which converts the complex computations of singular value decomposition (SVD) and eigenvalue decomposition (EVD) courses into real ones and effectively reduces the computation complexity. A new tactic is also restricted on avoiding too small element spacing with the consideration of the size of array antenna element. A set of numerical experiments is provided to validate the effectiveness and advantages of the proposed method.

Heterogeneous information network (HIN) embedding aims at learning the low-dimensional representation of nodes while preserving structure and semantics in an HIN. Existing methods mainly focus on static networks, while a real HIN usually evolves over time with the addition (deletion) of multiple types of nodes and edges. Because even a tiny change can influence the whole structure and semantics, the conventional HIN embedding methods need to be retrained to get the updated embeddings, which is time-consuming and unrealistic. In this paper, we investigate the problem of dynamic HIN embedding and propose a novel Dynamic HIN Embedding model (DyHNE) with meta-path based proximity. Specifically, we introduce the meta-path based first- and second-order proximities to preserve structure and semantics in HINs. As the HIN evolves over time, we naturally capture changes with the perturbation of meta-path augmented adjacency matrices. Thereafter, we learn the node embeddings by solving generalized eigenvalue problem effectively and employ eigenvalue perturbation to derive the updated embeddings efficiently without retraining. Experiments show that DyHNE outperforms the state-of-the-arts in terms of effectiveness and efficiency.

In this paper, a novel data-driven method for weight initialization of Multilayer Perceptrons and Convolutional Neural Networks based on discriminant learning is proposed. The approach relaxes some of the limitations of competing data-driven methods, including unimodality assumptions, limitations on the architectures related to limited maximal dimensionalities of the corresponding projection spaces, as well as limitations related to high computational requirements due to the need of eigendecomposition on high-dimensional data. We also consider assumptions of the method on the data and propose a way to account for them in a form of a new normalization layer. The experiments on three large-scale image datasets show improved accuracy of the trained models compared to competing random-based and data-driven weight initialization methods, as well as better convergence properties in certain cases. publishedVersion

Detecting the number of signals (NOS) is a fundamental problem in array processing. However, most existing methods have a computational complexity of $O(M^3)$ , where M is the number of sensors. In this paper we propose a fast detection algorithm with a complexity of $O(DM^2)$ , where D is the number of signals. Different from the conventional approaches that depend on the eigenvalues of the sample covariance matrix (SCM), the proposed method is based on the Ritz values --- the eigenvalues of the projected SCM in Krylov subspace. We first present a scheme to construct the Krylov subspace via the Lanczos tridiagonalization. Then we prove that the first D Ritz values are equal to the $D$ signal eigenvalues. Furthermore, if the Krylov subspace has a dimension greater than D, the minimum Ritz value (MRV) converges to the noise power. Therefore, we can detect the NOS by comparing the MRV with a threshold determined by the noise power. Because the proposed algorithm does not perform eigenvalue decomposition, it has a simple implementation structure. Moreover, by using the Sylvester's law of inertia, the Ritz values do not need to be computed either. Simulation results show that the proposed algorithm outperforms the existing methods under the conditions of small sample size and low signal-to-noise ratio.

The paper proves the quadratic convergence of the complex HZ method for solving the positive definite generalized eigenvalue problem. The proof is made for a general cyclic pivot strategy in the case of simple eigenvalues and for any wavefront pivot strategy in the case of simple or double eigenvalues. The proof is valid for the real HZ method. The preliminary numerical tests confirm the theoretical results.

The broadening first-order sea clutters caused by the signals from different directions with various radial current velocities create severe disturbance for target detection using high-frequency surface wave radar (HFSWR). Conventional sea clutter suppression methods tend to remove the sea clutter and target signals when they are mixed in the Doppler spectrum. Based on the characteristics of the target signal and sea clutter in spatial-temporal domain, a new first-order sea clutter suppression method for HFSWR using orthogonal projection is proposed. The proposed method uses the data from multichannels and slow-time domain at the adjacent range cell to construct a covariance matrix, which can be used to obtain the sea clutter subspace by eigendecomposition. Later, original signals are projected onto the sea clutter subspace. Finally, subtract the component of the original signals in the sea clutter subspace from the original signals to achieve the suppression of sea clutter by retaining the target signals. The simulation and experimental results for a single target and multiple targets cases indicate that the proposed method can suppress the first-order sea clutter effectively, which enhances the target detection capacity in the sea clutter zone for HFSWR.

Sensor drift is one of the severe issues that gas sensors suffer from. To alleviate the sensor drift problem, a gas sensor drift compensation approach is proposed based on local discriminant subspace projection (LDSP). The proposed approach aims to find a subspace to reduce the distribution difference between two domains, i.e., the source and target domain. Similar to domain regularized component analysis (DRCA) which is a recently proposed sensor drift correction method, the mean distribution discrepancy is minimized in the common subspace in our approach. LDSP extends DRCA in two aspects, i.e., it not only takes the label information of the source data into consideration to reduce the possibility of the case that samples in the subspace with different class labels stay close to each other, but also borrows the idea of locality-preserving projection to deal with multimodal data. Specifically, inspired by local Fisher discriminant analysis (LFDA), the label information is utilized to maximize the local between-class variance of source data in the latent common subspace and simultaneously minimize the local within-class variance. The formulation of LDSP is a generalized eigenvalue problem that can be readily solved. The experimental results have shown the proposed method outperforms other gas sensor drift compensation methods in terms of classification accuracy on two public gas sensor drift datasets.

This study proposes, for the first time, an approach to remove thermal noise from the wave coherency matrix, C₂, estimated from single-look complex dual-polarization Interferometric Wide Swath mode Sentinel-1 synthetic aperture radar data. The approach is straightforward; it exploits the ThermalNoiseRemoval module, provided by the European Space Agency (ESA) in its Sentinel Application Platform (SNAP) software, to remove thermal noise from the channel intensities. Then, noise correction on the complex data is applied, in order to estimate the noise-free C₂ matrix. As a further novelty, the proposed approach can be implemented in SNAP, through the use of a processing graph that is here provided. The method is applied on a dense time series of Sentinel-1 data, collected on an agricultural area located near Seville, Spain. The impact of thermal noise on the estimation of the eigendecomposition parameters of C₂, i.e., entropy (H₂), average alpha angle (α₂), and anisotropy (A₂), is assessed for different land-cover types, namely river, rice, forest, and urban areas. Monte Carlo simulations are implemented to assess the performance of the proposed approach in estimating H₂, α₂, and A₂. Results show that the proposed noise removal method improves the estimation of these parameters for the considered land-cover classes.

For a spaceborne high-resolution and wide-swath synthetic aperture radar (HRWS-SAR) system, it usually uses the digital beamforming technology. However, in practice, because of the influences of temperature, antenna pattern, receiving antenna, and other error factors, there may exist the range synchronization time errors, amplitude errors, and phase errors among different spatial channels. These nonideal factors will significantly degrade the multichannel data reconstruction performance, resulting in a smeared SAR image. To address this issue, in this article we propose a novel channel error correction algorithm based on the orthogonal projection theory. First, the optimal weight of each Doppler ambiguity component is calculated by the orthogonal projection. Then, the cost function is constructed based on the power maximization criterion, from which the channel phase errors can be obtained. Finally, the HRWS-SAR imaging can be finely realized after performing the channel balancing. Compared with the conventional phase error estimation method, the proposed algorithm does not require to perform the matrix eigenvalue decomposition, avoiding the signal leakage phenomenon under low SNR case. The effectiveness of the proposed algorithm is validated by both airborne and space-borne real SAR data.

The finite element model updating method of dynamical systems based on results of modal tests is proposed. The purpose of updating is to change eigenspectrum. The method alters a stiffness matrix by adding an updating finite element model created on the nodes of the intial one with respect to the existing links between the linear degrees of freedom. The stiffnesses of the updating elements are utilized as the updating parameters to be defined. The objective function equals to the least square weighted sum of residuals between the target, which were determined experimentally, and current values of modal stiffnesses. The iterative solution process is carried out. At each iteration step the conjugate gradient method is applied to solve the unconstrained minimization problem. The modeshapes, which were calculated as the result of solving the generalized eigenvalue problem at the previous iteration step, are employed to calculate the current modal stiffnesses. The method does not have a limit to a size of matrices and keeps their sparsity and symmetry. It provides the model updating of selected regions of a structure and step-by-step model updating of predefined groups of eigenfrequencies. Moreover, geometrical features of a structure, such as the presence of the symmetry planes and structurally identical elements, may be taken into account. The method is implemented into a program and verified by the example of the free dynamically-scaled model of Tu-204. In order to perform the ground vibration testing, the model was suspended with a low-rigidity flexible support. The finite element model made of solid elements has been updated on the basis of the six experimentally determined sets of eigenfrequencies. The target frequencies from each set have been achieved with a high level of accuracy.

A low-complexity parametric bilinear generalized approximate message passing (PBiGAMP)-based receiver is conceived for multi-carrier faster-than-Nyquist (MFTN) signaling over frequency-selective fading channels. To mitigate the inherent ill-conditioning problem of MFTN signaling, we construct a segment-based frequency-domain received signal model in the form of a block circulant linear transition matrix, which can be efficiently calculated by applying a two dimensional fast Fourier transform. Based on the eigenvalue decomposition of the block circulant matrices, we can diagonalize the covariance matrix of the complex-valued colored noise process imposed by the associated two dimensional non-orthogonal matched filtering. Building on this model, a PBiGAMP-based parametric joint channel estimation and equalization (JCEE) algorithm is proposed for MFTN systems. In this algorithm, we introduce a pair of additive terms for characterizing the interferences arising from adjacent segments and employ the exact discrete a priori probabilities of the transmitted symbols for improving the bit error rate (BER) performance. To further enhance the system's robustness in the presence of ill-conditioned matrices, we develop a refined PBiGAMP-based JCEE algorithm by introducing a series of scaled identity matrices. Moreover, the proposed PBiGAMP-based JCEE algorithms may be readily decomposed into GAMP-based equalization algorithms, when the channel state information is perfectly known. The overall complexity of the proposed algorithms only increases logarithmically with the total number of transmitted symbols. Our simulation results demonstrate the benefits of the proposed PBiGAMP-based iterative message passing receiver conceived for MFTN signaling.

As we all know, the model mismatch, primarily when the desired signal exists in the training data, or when the sample data is used for training, will seriously affect algorithm performance. This paper combines the subspace algorithm based on direction of arrival (DOA) estimation with the adaptive beamforming. It proposes a reconstruction algorithm based on the interference plus noise covariance matrix (INCM). Firstly, the eigenvector of the desired signal is obtained according to the eigenvalue decomposition of the subspace algorithm, and the eigenvector is used as the estimated value of the desired signal steering vector (SV). Then the INCM is reconstructed according to the estimated parameters to remove the adverse effect of the desired signal component on the beamformer. Finally, the estimated desired signal SV and the reconstructed INCM are used to calculate the weight. Compared with the previous work, the proposed algorithm not only improves the performance of the adaptive beamformer but also dramatically reduces the complexity. Simulation experiment results show the effectiveness and robustness of the proposed beamforming algorithm.

Objective. Action intention understanding EEG signal classification is indispensable for investigating human-computer interactions and intention understanding mechanisms. Numerous investigations on classification tasks extract classification features by using graph theory metrics; however, the classification results are usually not good. Method. To effectively implement the task of action intention understanding EEG signal classification, we proposed a new feature extraction method by improving discriminative spatial patterns. Results. The whole frequency band and fusion band achieved satisfactory classification accuracies. Compared with other authors’ methods for action intention understanding EEG signal classification, the new method performs more satisfactorily in some aspects. Conclusions. The new feature extraction method not only effectively avoids complex values when solving the generalized eigenvalue problem but also perfectly realizes appreciable classification accuracies. Fusing the classification features of different frequency bands is a useful strategy for the classification task.

Recently proposed L2-norm linear discriminant analysis criterion via the Bhattacharyya error bound estimation (L2BLDA) is an effective improvement of linear discriminant analysis (LDA) for feature extraction. However, L2BLDA is only proposed to cope with vector input samples. When facing with two-dimensional (2D) inputs, such as images, it will lose some useful information, since it does not consider intrinsic structure of images. In this paper, we extend L2BLDA to a two-dimensional Bhattacharyya bound linear discriminant analysis (2DBLDA). 2DBLDA maximizes the matrix-based between-class distance which is measured by the weighted pairwise distances of class means and meanwhile minimizes the matrix-based within-class distance. The weighting constant between the between-class and within-class terms is determined by the involved data that makes the proposed 2DBLDA adaptive. In addition, the criterion of 2DBLDA is equivalent to optimizing an upper bound of the Bhattacharyya error. The construction of 2DBLDA makes it avoid the small sample size problem while also possess robustness, and can be solved through a simple standard eigenvalue decomposition problem. The experimental results on image recognition and face image reconstruction demonstrate the effectiveness of the proposed methods.

In this paper, the emerging orthogonal time frequency space (OTFS) modulation is firstly restructured as a precoded orthogonal frequency division multiplexing (OFDM) system, so that the well-established frequency-domain approach can be applied to perform signal in fast fading channels. Then a frequency-domain minimum mean squared error (MMSE) equalizer for OTFS is introduced and its performance is analyzed based on the eigenvalue decomposition of the channel matrix. Inspired by the frequency-domain precoding structure, an adaptive transmission scheme with frequency-domain precoding matrix composed of the eigenvectors of the channel matrix is proposed to improve the system performance under MMSE equalization, and its optimized performance is derived with simple expression. Finally, considering two extreme channel conditions, the lower and upper bounds for the diversity performance of the adaptive transmission scheme are derived. Simulation results show that the proposed adaptive transmission achieves significantly better performance for short signal frames and can work well with imperfect channel state information (CSI). The derived performance bounds can serve as benchmarks for OTFS and other precoded OFDM systems.

In this letter, the problem of 2-D direction of arrival (DOA) estimation using uniform circular array (UCA) is addressed. The performance of the beamspace transform (BT) based algorithm is degraded when shrinking the quantity of antennas because the BT raises the residual error. To overcome this problem, we propose a 2-D compensation method. Firstly, we estimate the biased 2-D DOAs via UCA-ESPRIT algorithm. Secondly, the eigenvalue decomposition (EVD) of covariance matrix without the BT is conducted to extract the noise subspace in element space. Thereafter, inspired by the relationship that the steering vector is orthogonal to the noise subspace, the first-order Taylor expansion is performed for the direction matrix which is reconstructed by the biased DOA estimates. Finally, the offsets are compensated via least squares (LS). Compared to virtual array transformation methods, the proposed method possesses better estimation performance and needs less computation cost. Multiple simulation experiments are presented to verify the efficiency of the approach.

In this paper, a new method for the cross-term-free Wigner–Ville distribution (WVD) is proposed. The proposed method is based on sliding eigenvalue decomposition (SEVD) and the WVD which has been termed as SEVD–WVD. The SEVD decomposes the multicomponent signal into a set of monocomponent signals, and then, the WVD of analytic monocomponents is added to obtain a cross-term-free time–frequency representation. We have applied the proposed technique on clean and noisy synthetic signals in order to verify its robustness. Moreover, the SEVD–WVD is applied on speech signal to show its suitability on real-world data. The proposed method has been compared with other cross-term reduction techniques for the WVD in the literature.

Semi-supervised learning is an attractive technique for using unlabeled data in classification. In this work, an efficient semi-supervised extreme learning machine (ELM) classification framework is proposed by introducing the Laplacian regularization term. It tries to build two nonparallel hyperplanes such that each hyperplane is closer to one of the two classes and farther away from the other class in ELM feature space, based on which, two new semi-supervised ELM classification algorithms are proposed. First, we formulate semi-supervised ELM as a ratio form, and reformulate it as a generalized eigenvalue problem (called Laplacian generalized eigenvalue extreme learning machine, LapGELM) to solve simply. Then we replace the form of ratio with a form of difference so that the optimization problems can be transformed into a standard eigenvalue problem (called Laplacian standard eigenvalue ELM, LapSELM) to control overfit and solve quickly. Furthermore, the proposed algorithms are implemented on various datasets with different sizes and structures. Compared with traditional methods, experiment results show the feasibility and effectiveness of the proposed algorithms due to their simplicity, rapidity, and good generalization performance.

The performance of the traditional Capon beamformer degrades sharply when the signal of interest (SOI) appears in the training data. To reduce the impact of SOI on the Capon beamformer, two methods for the interference-plus-noise covariance matrix (INCM) reconstruction are proposed in this letter. The proposed-1 method is based on the integral of the Capon spectrum without the residual noise power. In the proposed-2 method, the interference power is estimated via the orthogonality between different sparse steering vectors (SVs) to project the sample covariance matrix for the INCM reconstruction. Meanwhile, the inverse of INCM is obtained by eigenvalue decomposition and the SV of SOI is updated by the principal eigenvector of the reconstructed SOI covariance matrix (SCM). Simulation results show that the proposed methods are robust against some mismatch errors.

This paper studies the stabilizability and stabilization of continuous-time systems in the presence of stochastic multiplicative uncertainties. The authors consider multi-input, multi-output (MIMO) linear time-invariant systems subject to multiple static, structured stochastic uncertainties, and seek to derive fundamental conditions to ensure that a system can be stabilized under a mean-square criterion. In the stochastic control framework, this problem can be considered as one of optimal control under state- or input-dependent random noises, while in the networked control setting, a problem of networked feedback stabilization over lossy communication channels. The authors adopt a mean-square small gain analysis approach, and obtain necessary and sufficient conditions for a system to be mean-square stabilizable via output feedback. For single-input, single-output (SISO) systems, the condition provides an analytical bound, demonstrating explicitly how plant unstable poles, nonminimum phase zeros, and time delay may impose a limit on the uncertainty variance required for mean-square stabilization. For MIMO minimum phase systems with possible delays, the condition amounts to solving a generalized eigenvalue problem, readily solvable using linear matrix inequality optimization techniques.

For the purpose of fault monitoring of dynamic processes, this study proposes a hybrid framework of canonical variate analysis (CVA) and global-local preserving projection (GLPP) and refer to it as canonical GLPP analysis. The construction of Laplacian matrix based on the Hankel matrix provides a novel technology for preserving projection analysis. The optimal projection matrix solved by the canonical GLPP analysis framework can simultaneously preserves time series correlations and spatial distribution information in dimension reduction. This method inherits the advantages of GLPP in handling high-dimensional data with manifold learning and the strength of CVA in self-correlation analysis with state-space model. The solution to the proposed method is formulated as a generalized eigenvalue problem. The effectiveness of the canonical GLPP analysis method in fault detection and identification of dynamic processes is verified with a case study on the Tennessee Eastman process. Compared with the Fisher discriminant analysis based CVA, dynamic PCA, dynamic GLPP and the GLPP method, the proposed canonical GLPP analysis framework provides a more robust and accurate paradigm for dynamic process monitoring.

The classical one-way generalised screen propagator (GSP) and Fourier finite-difference (FFD) schemes have limitations in imaging large angles in complex media with substantial lateral variations in wave velocity. Some improvements to the classical one-way wave scheme have been proposed with optimised methods. However, the performance of these methods in imaging complex media remains unsatisfying. To overcome this issue, a new strategy for wavefield extrapolation based on the eigenvalue and eigenvector decomposition of the Helmholtz operator is presented herein. In this method, the square root operator is calculated after the decomposition of the Helmholtz operator at the product of the eigenvalues and eigenvectors. Then, Euler transformation is applied using the best polynomial approximation of the trigonometric function based on the infinite norm, and the propagator for one-way wave migration is calculated. According to this scheme, a one-way operator can be computed more accurately with a lower-order expansion. The imaging performance of this scheme was compared with that of the classical GSP, FFD and the recently developed full-wave-equation depth migration (FWDM) schemes. The impulse responses in media with arbitrary velocity inhomogeneity demonstrate that the proposed migration scheme performs better at large angles than the classical GSP scheme. The wavefronts calculated in the dipping and salt dome models illustrate that this scheme can provide a precise wavefield calculation. The applications of the Marmousi model further demonstrate that the proposed approach can achieve better-migrated results in imaging small-scale and complex structures, especially in media with steep-dipping faults.

In this paper, a two-dimensional direction-of-arrival (DOA) estimation algorithm based on nested acoustic vector-sensor (AVS) array is introduced. The minimum unit interval of the used nested AVS array is integral multiple of half-wavelength of signal. Firstly, four selection matrices are used to recombine four received vectors, by which four special cross-covariance matrices are obtained to form an extended block covariance matrix. Then, a set of parameters on elevation angles are estimated by traditional estimation of signal parameter via rotational invariance technique (ESPRIT) algorithm. Using these parameters, unambiguous elevation angles can be obtained without spectral peak search and added eigenvalue decomposition. At last, using the estimated elevation angles, the azimuth angles can be estimated by array manifold matching algorithm. The proposed algorithm can distinguish more signals than some existing ESPRIT algorithms. Many simulation results can prove the performance of the proposed algorithm in improving the accuracy of DOA estimation. Simulation results also show that the unit interval of optimal nested AVS array should be larger than half-wavelength of received signal.

There is growing awareness that errors in the model equations cannot be ignored in data assimilation methods such as four-dimensional variational assimilation (4D-Var). If allowed for, more information can be extracted from observations, longer time windows are possible, and the minimization process is easier, at least in principle. Weak constraint 4D-Var estimates the model error and minimizes a series of linear least-squares cost functions using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited memory preconditioners (LMPs) have been constructed using approximations of the eigenvalues and eigenvectors of the Hessian in the previous inner loop. If the Hessian changes signicantly in consecutive inner loops, the LMP may be of limited usefulness. To circumvent this, we propose using randomised methods for low rank eigenvalue decomposition and use these approximations to cheaply construct LMPs using information from the current inner loop. Three randomised methods are compared. Numerical experiments in idealized systems show that the resulting LMPs perform better than the existing LMPs. Using these methods may allow more efficient and robust implementations of incremental weak constraint 4D-Var.

Moment descriptors have been widely used for the analysis and representation of images. In this paper, we propose a new set of discrete orthogonal moments of fractional order, called Quaternion Cartesian Fractional Hahn Moments. The proposed QCFrHMs are based on new Fractional Hahn Polynomials and generalize the classical Quaternion Hahn Moments. First, FrHPs are proposed and defined using eigenvalue decomposition and the spectral representation of the classical Hahn polynomial matrix. Then, the proposed FrHPs are used as a kernel function to define the new Fractional Hahn Moments. Finally, based on quaternion algebra, the FrHMs for grayscale images are generalized to the QCFrHMs for color images. The proposed QCFrHMs depend on four parameters: two polynomial parameters and two fractional orders, which allow us to use them to propose a robust, blind and efficient watermarking scheme for the copyright protection of color images where the requirements of a watermarking scheme are successfully ensured thanks to the performance of the proposed QCFrHMs. Experimental results are provided to illustrate the effectiveness of the proposed color image descriptors.

This letter considers the problem of underdetermined estimation of direction-of-arrival (DOA) and polarization parameters, including the degree of polarization (DOP), polarization orientation angle (POA), and polarization ellipticity angle (PEA), for partially polarized (PP) signals with a dual-polarized coprime array. Firstly, the increased dimension due to dual-polarized sensors is reduced by adding two covariance matrices, which facilitates the coarray construction. Then, the holes in the coarray of coprime array are interpolated via utilizing the annihilating relation. At last, the DOAs and array noise variance are obtained by solving an annihilating relation based optimization problem and then used to reconstruct the coherency matrices via the weight least squares (WLS). The polarization parameters can be estimated by performing eigendecomposition on the reconstructed coherency matrices. We show the underdetermined estimation of array noise variance which is difficult for conventional methods is vital for accurate underdetermined estimation of the polarization parameters. The performances of the proposed method are verified by numerical simulations and compared with other representative methods including the Cramer-Rao bound (CRB).

Two-way wave depth migration (TWDM) can achieve a wider imaging angle in media with significant lateral velocity changes and better imaging results for complex structures with high dip angles than ...

The wide range of time scales in chemical reaction systems has become an important problem in reactive flow simulations. This work proposes an intelligent time-scale operator-splitting (OS) chemistry integration method, which is effective in reduction of numerical stiffness and model complexity. Different from most existing publications, a pretrained backpropagation neural network is used to identify the slow and fast reactions and detect the sources of model stiffness on the fly, which replaces the expensive eigendecomposition of Jacobian matrix. With the fast-slow decomposition, the chemical source term can be represented as the sum of a stiff part and a nonstiff part. A stable time-scale OS integration is performed to solve the stiff chemical ordinary differential equations, which balances the computational cost with accuracy. In the simulation, a favorable comparison of the proposed integration method with the existing ODE solvers, such as implicit Euler, explicit Euler, and Runge-Kutta, is included to show its effectiveness and merits.

A subgridding unconditionally stable finite-element time-domain method based on local eigenvalue solution (SUSL-FETD) is proposed to solve multiscale modeling. In this method, subgridding elements are used to discrete a multiscale computational domain, and the explicit central-difference method is applied for temporal discretization. Compared with traditional elements, the subgridding elements have a more complex distribution of edges and nodes. Therefore, an efficient subgridding scheme is given to form the element matrix for each subgridding element. Such system matrices assembled by all element matrices are symmetric. The subgridding FETD (S-FETD) is then further developed into the subgridding unconditionally stable FETD (SUS-FETD) by filtering spatial unstable modes. It is time-consuming to obtain the unstable modes by global eigenvalue decomposition. In this article, the local eigenvalue decomposition is proposed to get the unstable modes, which further reduces the memory storage and computing time. Numerical examples certify that the proposed SUSL-FETD has high accuracy and efficiency.

In close-talking applications, such as mobile phones, selective measurement of the near-field sound is desirable as it can be used to improve the signal-to-noise ratio. In this study, the theory of generalized radiation modes is applied to the design of microphone arrays. The generalized radiation modes are formulated as a generalized eigenvalue problem related to the specific acoustic impedances on the array surface. The real eigenvalue corresponds to the near-to-far ratio for each mode and the real eigenvector indicates the modal shape, i.e., the amplitudes and phases of the individual microphones. The microphone array is designed according to the eigenvector with the largest eigenvalue so as to maximize the near-to-far ratio. The theory is verified based on computer simulations. The proposed method is also compared with conventional gradient microphones via numerical examples.

This paper shows the different ways of using generalized Givens rotations in complex joint eigenvaluedecomposition (JEVD) problem. It presents the different schemes of generalized Givens rotation, justifies the introducedapproximations and focuses on the process of extending an algorithm developed for real JEVD to the complex JEVD.Several Joint Diagonalization problem use generalized Givens rotations to achieve the solution, many algorithmsdeveloped in the real case exist in the literature and are not generalized to the complex case. Hence, we show herein asimple and not trivial way to get the complex case from the real one. Simulation results are provided to highlight theeffectiveness and behaviour of the proposed techniques for different scenarios.

Combining forecasts from multiple temporal aggregation levels exploits information differences and mitigates model uncertainty, while reconciliation ensures a unified prediction that supports aligned decisions at different horizons. It can be challenging to estimate the full cross-covariance matrix for a temporal hierarchy, which can easily be of very large dimension, yet it is difficult to know a priori which part of the error structure is most important. To address these issues, we propose to use eigendecomposition for dimensionality reduction when reconciling forecasts to extract as much information as possible from the error structure given the data available. We evaluate the proposed estimator in a simulation study and demonstrate its usefulness through applications to short-term electricity load and financial volatility forecasting. We find that accuracy can be improved uniformly across all aggregation levels, as the estimator achieves state-of-the-art accuracy while being applicable to hierarchies of all sizes.

A Multicarrier phase-coded (MCPC) waveform design scheme with two steps for Joint radar and communication (JRC) system is developed. Firstly, an integrated MCPC waveform design method is addressed by simultaneously maximizing the Signal-to-clutter-to-noise ratio (SCNR) and the Shannon capacity, subject to both the Integrated sidelobe level ratio (ISLR) constraint and the energy constraint. This model is theoretically proved to be a convex optimization problem with respect to the absolute squares of the transmit weights corresponding to different subcarriers, of which the analytical result is also discussed. Subsequently, by further optimizing the phases of the transmit weights, minimizing the Peak to average power ratio (PAPR) for JRC system is recast as a Semidefinite programming (SDP) problem, which can be effectively solved with the Semidefinite relaxation (SDR) technique via Eigenvalue decomposition (EVD) or Complex Gaussian randomization (CGR). Numerical examples are provided to verify the effectiveness of the proposed scheme.

Standard multivariate techniques like Principal Component Analysis (PCA) are based on the eigendecomposition of a matrix and therefore require complete data sets. Recent comparative reviews of PCA algorithms for missing data showed the regularised iterative PCA algorithm (RPCA) to be effective. This paper presents two chunk-wise implementations of RPCA suitable for the imputation of “tall” data sets, that is, data sets with many observations. A “chunk” is a subset of the whole set of available observations. In particular, one implementation is suitable for distributed computation as it imputes each chunk independently. The other implementation, instead, is suitable for incremental computation, where the imputation of each new chunk is based on all the chunks analysed that far. The proposed procedures were compared to batch RPCA considering different data sets and missing data mechanisms. Experimental results showed that the distributed approach had similar performance to batch RPCA for data with entries missing completely at random. The incremental approach showed appreciable performance when the data is missing not completely at random, and the first analysed chunks contain sufficient information on the data structure.

Canonical correlation analysis (CCA) is a famous data analysis method that has been successfully used in many areas. CCA extracts meaningful information from a pair of data sets, by seeking pairs of linear combinations from two sets of variables with maximum correlation. Mathematically, CCA resorts to solving a large-scale generalized eigenvalue problem. However, as the dimension of the data sets is much larger than the number of samples, CCA may suffer from the small-sample-size (SSS) problem and the over-fitting problem. In order to overcome these difficulties, the regularized technique is often applied, but it is difficult to choose the optimal parameter in advance. In this work, we propose an Exponential Canonical Correlation Analysis (ECCA) method based on matrix exponential, which is parameter-free and can overcome the over-fitting and the SSS problems fundamentally. However, the computational overhead of the ECCA method is very high in practice. Based on the randomized singular value decomposition (RSVD), we then propose a Randomized Exponential Canonical Correlation Analysis (RECCA) method for data analysis and dimensionality reduction. Theoretical results are given to show the rationality of this randomized method, and establish the relationship between RECCA and ECCA. Numerical experiments are performed on some real-world, high-dimensional and large-sample data sets, which illustrate the superiority of the proposed algorithms over many state-of-the-art CCA algorithms.

This article studies the problem of affine transformation-based guidance of a multiquadcopter system (MQS) in an obstacle-laden environment. Such MQSs can perform a variety of cooperative tasks including information collection, inspection mapping, disinfection, and fire-fighting. The MQS affine transformation is an approach to a decentralized leader–follower coordination guided by $n+1$ leaders, where leaders are located at vertices of an $n$ -D simplex, called leading simplex , at any time $t$ . The remaining agents are followers by acquiring the desired affine transformation via local communication. Followers are contained in a rigid-size ball at any time $t$ but they can be distributed either inside or outside the leading simplex. By eigendecomposition of the affine transformation coordination, safety in a large scale MQS coordination can be ensured by constraining eigenvalues of the affine transformation. Given the initial and final configurations of the MQS, A* search is applied to optimally plan safe coordination of a large-scale MQS, minimizing the travel distance between the the initial and final configuration. The article also proposes a proximity-based communication topology for followers to assign communication weights with their in-neighbors and acquire the desired coordination with minimal computation cost.

From a brain functional connectivity (FC) matrix, we can identify the hub nodes by a new method of eigencentrality mapping, which not only counts for one node's centrality but also all other nodes' centrality values through correlation connections in an eigenvector of the FC matrix. For the resting-state functional MRI (fMRI) data (complex-valued EPI images in nature), both magnitude and phase images are useful for brain FC analysis. We herein report on brain functional hubness analysis by constructing the FC matrix from phase fMRI data and identifying the hub nodes by eigencentrality mapping. In our study, we collected a cohort of 160 complex-valued fMRI dataset (consisting of magnitude and phase in pairs), and performed independent component analysis (ICA), FC matrix calculation (in size of 50 × 50) and FC matrix eigen decomposition; thereby obtained the 50-node eigencentrality values in the eigenvector associated with the largest eigenvalue. We also compared the hub structures inferred from FC matrices under different thresholding. Alternatively, we obtained the geometric hubs among p value the 50 nodes involved in the FC matrix through the use of harmonic centrality metric. Our results showed that phase fMRI data analysis defines the resting-state brain functional hubs primarily in the central region (subcortex) and the posterior region (parieto-occipital lobes and cerebella). The brain central hubness was supported by the geometric central hubness, which, however, is distinct from the magnitude-inferred hubness in brain superior region (frontal and parietal lobes). Our findings pose a new understanding of (or a debate over) brain functional connectivity architecture.

Characterization of the rank distribution of a sparse random matrix over a finite field can be decomposed into two subproblems. The first subproblem is the characterization of the probability $p(i,n)$ that an $n$ -dimensional vector is linearly dependent of other $i$ linearly independent $n$ -dimensional vectors. The second subproblem is the characterization of the rank distribution of a sparse random matrix as a function of $p(i,n)$ . In this letter, we focus on the second subproblem and present an exact solution to it. The derivation is based on an absorbing Markov chain and the eigen decomposition of the transition matrix. Compared with the state-of-the-art expressions, the derived expression is closed-form and of lower complexity. As a necessity for the exactness of the derived expression, we prove that the derived expression is equivalent to the state-of-the-art expressions.

The stability of two-dimensional (2-D) linear systems, continuous, discrete, and mixed (hybrid) cases with real or complex coefficients, is considered in this article using a single formalism. Algebraic necessary and sufficient conditions for testing the stability of these systems are developed. The conditions are based on the characteristic polynomial to be void of zero in the stability region which depends on the case being considered. These conditions consist of the stability test of few real univariate polynomials and a real generalized eigenvalue problem. The resulting stability test requires reduced computational complexity compared to existing techniques. A numerical example is given to illustrate the merits of the proposed approach.

In this paper, a meshless Chebyshev collocation method (CCM) for the numerical solution of the eigenvalue problems of the Helmholtz equation is presented. The Chebyshev polynomials are employed for the efficient and accurate approximation of the eigenfunctions to ensure the pseudo-spectral convergence of the CCM. Two different approaches, namely the generalized eigenvalue approach and the standard eigenvalue approach, which convert the original eigenvalue problem into the generalized eigenvalue problem and the standard eigenvalue problem respectively, are implemented. Five benchmark numerical examples are presented and discussed to demonstrate the accuracy and efficiency of the proposed CCM.

The multiple signal classification (MUSIC) algorithm is a promising method for the plethora of problems related to the direction-of-arrival (DOA) estimation. Conventionally, this approach uses the whole sensor array observations to obtain the signal or noise subspace, which consequently leads to a huge computational burden. In this letter, to circumvent this problem, we make a significant modification to the traditional MUSIC algorithm. First, we compute only two sub-matrices of the sample covariance matrix (SCM) exploiting the Nystrom method avoiding its complete calculation. These matrices can be used to construct an accurate noise subspace without calculating the SCM and its eigenvalue decomposition (EVD). Furthermore, to have a uniform DOA estimation, we modify the classical ULA by displacing two antenna elements from both the ends of the array to a top and a bottom of the array axis. This unique structure improves the estimation of DOAs near and at the array end fires. Several numerical results are included to confirm the efficacy of the new method.

Accurate multi-modal predictions can vigorously support people's wise decisions. Predicting the future by leveraging eigentensor based multivariate Markov model or Z-eigenvector based multi-order Markov model has flourished in recent years. However, there is no integrated solution by combining multivariate Markov model, and multi-order Markov model with tensor-based calculation approach. Besides, the computation efficiency, and quick response of tensor-based calculation approach are seriously restricted by the curse of dimensionality arising from higher-order tensor. Therefore, this paper focuses on proposing a scalable tensor train (TT) based higher order dominant Z-eigen decomposition (HODZED) for multivariate multi-order Markov models under cloud/edge computing environments to provide quick accurate prediction. First, we propose a multivariate multi-order Markov model, and extend dominant Z-eigen decomposition to HODZED for calculating the dominant Z-eigentensor. Then, we propose two scalable TT-based HODZED (TT-HODZED), and improved TT-based HODZED (ITT-HODZED) algorithms to improve the computation efficiency. Afterwards, we present a multi-modal prediction algorithm based on the dominant Z-eigentensor. Experimental results based on the random dataset, and real-world GPS trajectory dataset demonstrate that TT-HODZED, and ITT-HODZED algorithms can significantly improve the computation efficiency, and reduce the running memory on the premise of guaranteeing the almost consistent prediction accuracy compared to the original HODZED algorithm.

In this work, a fast numerical method is considered to solve the space-fractional semilinear parabolic equations on closed surfaces. It is a challenge in that how to define the space fractional operator and corresponding semilinear parabolic equation with the energy functional defined on surface. To overcome it, using the local tangential space, we construct the spectral approximation for the space fractional operator on surfaces and apply the matrix transfer technique to avoid the difficulty of fractional nonlocality. The main advantage of the matrix transfer method is the completed diagonal representation of fractional operators from eigenvalue decomposition. Moreover, the time-discrete error estimates are presented as well as the energy stability. Various numerical examples are carried out to verify the theoretical results.

Direction of arrival (DOA) estimation using new proposed array geometries named sparse nested arrays with coprime displacement (SNACD) is discussed in this paper. The SNACD has sparse subarrays with nested relationship, and the displacement between the subarrays is coprime to the inter-element spacing of the smaller subarray. This geometry can combine the properties of nested array and coprime array, which can simultaneously achieve virtual uniform array with large aperture in co-array domain and reduce the mutual coupling influence in physical-array domain. In the co-array domain, the coprime parameter estimations are separated into the virtual direction matrix and source vector, respectively. Based on this data model, several DOA estimation methods can be applied, like the spatial smooth multiple signal classification (SS-MUSIC) and atomic norm minimization based grid-less (ANM-GL) method, which both are robust to the grid mismatch problem. To further reduce the complexity, we proceed to propose a Discrete Fourier transform with offset compensation (DFT-OC) method, which requires neither eigenvalue decomposition nor sparse recovery. The final unique DOA estimation is achieved from the intersection of the coprime estimations, which are achieved with automatically pairing. Compared to coprime array and nested array based methods, the proposed scheme obtains better DOA estimation accuracy and higher angular resolution with mutual coupling influence. Multiple simulations are presented to verify the effectiveness of our approach.

This paper presents a method for blind separation and DOA estimation of narrow-band independent signals based on eigenvalue decomposition. It can effectively improve the blind separation performance of array mixed signals under the condition of low signal-to-noise ratio. At the same time, the DOA of the separated signal is estimated. It is of great significance to improve the adaptability of array system in complex electromagnetic signal environment.

For scenarios of coherent underwater sources at low signal-to-noise ratio (SNR), a novel quaternion-based DOA algorithm without eigendecomposition is proposed using a linear vector-hydrophone array. We construct four quaternion models by judiciously arranging the received data to fully utilize the statistical information of the incident signals. To avoid the high computational complexity caused by the eigenvalue decomposition (EVD), we introduce the computationally efficient propagator method (PM) to estimate the elevation angles of the observed signals. In the quaternion algebra framework, we statistically eliminate the additive noise, which makes the PM method exhibit a robust performance in low SNR. By fully exploiting the direction information embedded in the velocity components, we achieve a high-resolution two-dimensional (2-D) DOA estimation result with a linear vector-hydrophone array. The simulations demonstrate that the proposed method offers stable estimation performance compared with the existing non-quaternion schemes without the need for any pair matching between the estimated azimuth and elevation angles.