Eigenvalue Decomposition Of Matrix Research Articles

Quantum algorithm for dynamic mode decomposition integrated with a quantum differential equation solver

We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of a dynamic mode decomposition algorithm used in diverse fields such as fluid dynamics, molecular dynamics, and epidemiology. Our quantum algorithm can also compute matrix eigenvalues and eigenvectors by analyzing the corresponding linear dynamical system. Our algorithm handles a broad range of matrices, particularly those with complex eigenvalues. The complexity of our quantum algorithm is O(polylogN) for an N-dimensional system. This is an exponential speedup over known classical algorithms with at least O(N) complexity. Thus, our quantum algorithm is expected to enable high-dimensional dynamical systems analysis and large matrix eigenvalue decomposition, intractable for classical computers. Published by the American Physical Society 2024

Eigenvector method for array pattern synthesis

AbstractA novel equation for array beam pattern synthesis is presented. The expected pattern total power is set to a fixed value, under this constraint, the difference between the main beam total power and the other part total power is maximized to achieve the array pattern synthesis. To solve the proposed pattern synthesis equation, which cannot be solved by any published traditional method, based on the Lagrange multiplier method, it is transformed into a new equation in which the array element excitation vector is the eigenvector of a matrix. Thus, the array element excitation vector can be obtained by matrix eigenvalue decomposition. Three array architectures using the method are taken as examples to show its advantages. Simulation results of the examples show that the method has better performance than other methods. The method is a noniterative approach, so it requires less computational volume to complete the pattern synthesis process. In addition, through eigenvalue decomposition, multiple eigenvectors can provide other different solutions of array elements current excitations, which can be applied in different areas.

Privacy-preserving outsourcing scheme of face recognition based on locally linear embedding

Face recognition is a biometric-based technology that identifies or verifies a person’s identity by analyzing and comparing patterns in their facial features. Among kinds of face recognition algorithms, Locally Linear Embedding (LLE) algorithm has the advantage of preserving the local structure of face data. However, the LLE algorithm requires some computationally intensive matrix calculations, which might introduce latency to the device. In this paper, we propose a privacy-preserving outsourcing scheme of face recognition based on LLE algorithm for the first time. By outsourcing matrix multiplication, linear equations, and eigenvalue decomposition of the LLE algorithm to a cloud server, we reduce the computational load on the client device. To ensure data security, we generate a novel type of sparse orthogonal matrix through square root application and random permutation of diagonal elements, used as secret keys to blind the client’s data. Meanwhile, due to the sparsity of key matrices, the complexity of client-side computation complexity is reduced from O(n3) to O(n2). The experimental results show that our proposed scheme reduces computational costs while maintaining almost identical face recognition accuracy as the original LLE-based algorithm.

Monitoring land subsidence through improved CAESAR algorithm in time-series InSAR processing

Time-series SAR interferometry (InSAR) combining permanent scatterer and distributed scatterer (DS), has been strongly developed in subsidence monitoring. It is known that the Component extrAction and sElection SAR (CAESAR) is a recently presented approach of selecting and filtering scattering mechanisms for DS. This article proposes an improved CAESAR algorithm in InSAR processing. Phase optimization is performed by eigenvalue decomposition and principal component analysis of the coherence matrix that constructed based on the identified homogeneous pixels. In addition, only the interferometric phases with low noise are used to calculate the goodness-of-fit value. The improved method has been tested for subsidence monitoring over a nonurban area located in Xiongxian, China using 25 Sentinel-1A images. The results show that the improved method can provide the high spatial density of deformation measurements with accuracy ensured. Noticeable land subsidence is revealed widely within the north of Xiongxian county, particularly in Daying, Mijiawun and Beishakou towns.

Graph Convolutional Neural Networks With Diverse Negative Samples via Decomposed Determinant Point Processes.

Graph convolutional neural networks (GCNs) have achieved great success in graph representation learning by extracting high-level features from nodes and their topology. Since GCNs generally follow a message-passing mechanism, each node aggregates information from its first-order neighbor to update its representation. As a result, the representations of nodes with edges between them should be positively correlated and thus can be considered positive samples. However, there are more non-neighbor nodes in the whole graph, which provide diverse and useful information for the representation update. Two non-adjacent nodes usually have different representations, which can be seen as negative samples. Besides the node representations, the structural information of the graph is also crucial for learning. In this article, we used quality-diversity decomposition in determinant point processes (DPPs) to obtain diverse negative samples. When defining a distribution on diverse subsets of all non-neighboring nodes, we incorporate both graph structure information and node representations. Since the DPP sampling process requires matrix eigenvalue decomposition, we propose a new shortest-path-base method to improve computational efficiency. Finally, we incorporate the obtained negative samples into the graph convolution operation. The ideas are evaluated empirically in experiments on node classification tasks. These experiments show that the newly proposed methods not only improve the overall performance of standard representation learning but also significantly alleviate over-smoothing problems.

2D-Unitary ESPRIT Based Multi-Target Joint Range and Velocity Estimation Algorithm for FMCW Radar

Millimeter-wave FMCW radar has been widely used in joint range-velocity estimation of multiple targets. However, most existing algorithms are unable to estimate the range-velocity information with high accuracy simultaneously and fail to discriminate the targets with either closely spaced ranges or closely spaced velocities in the 2D range-Doppler spectrum. In order to deal with these problems, this paper proposes a 2D-Unitary ESPRIT-based joint range and velocity estimation algorithm of multiple targets for FMCW radar. Firstly, The 1D-IF signal is constructed into a 2D virtual array signal, the virtual array signals are preprocessed by a 2D-spatial smoothing technique to generate a new matrix signal. Then, according to the 2D-Unitary ESPRIT algorithm, the 2D real-valued information of the target parameters is obtained from this matrix signal, and then a new complex-value matrix is constructed. Finally, the eigenvalue decomposition of this new complex-value matrix is performed, and the range-velocity estimates of multiple targets are, respectively, calculated from the real and imaginary parts of the eigenvalues, and paired automatically. The simulation results illustrate that the proposed algorithm not only provides highly accurate range-velocity estimates but also has high-resolution performance and achieves automatic pairing of the range-velocity estimates in multi-target scenarios, thus effectively improving the multi-target joint range and velocity estimation performance of FMCW radar.

Spectral clustering eigenvector selection of hyperspectral image based on the coincidence degree of data distribution

ABSTRACT Spectral clustering is a well-regarded subspace clustering algorithm that exhibits outstanding performance in hyperspectral image classification through eigenvalue decomposition of the Laplacian matrix. However, its classification accuracy is severely limited by the selected eigenvectors, and the commonly used eigenvectors not only fail to guarantee the inclusion of detailed discriminative information, but also have high computational complexity. To address these challenges, we proposed an intuitive eigenvector selection method based on the coincidence degree of data distribution (CDES). First, the clustering result of improved k-means, which can well reflect the spatial distribution of various types was used as the reference map. Then, the adjusted Rand index and adjusted mutual information were calculated to assess the data distribution consistency between each eigenvector and the reference map. Finally, the eigenvectors with high coincidence degrees were selected for clustering. A case study on hyperspectral mineral mapping demonstrated that the mapping accuracies of CDES are approximately 56.3%, 15.5%, and 10.5% higher than those of the commonly used top, high entropy, and high relevance eigenvectors, and CDES can save more than 99% of the eigenvector selection time. Especially, due to the unsupervised nature of k-means, CDES provides a novel solution for autonomous feature selection of hyperspectral images.

Monitoring Surface Subsidence Using Distributed Scatterer InSAR with an Improved Statistically Homogeneous Pixel Selection Method in Coalfield Fire Zones

Statistically homogeneous pixel (SHP) selection is an important process in the distributed scatterer interferometric synthetic aperture radar (DS-InSAR) approach. However, prevalent methods struggle to appropriately balance the efficiency and accuracy of selection. To solve this problem, the authors of this study improved the Hypothesis Test of Confidence Interval (HTCI) to propose an adaptive method to select the level of saliency and confidence interval for the HTCI, called Adp-HTCI. The proposed method can accurately select homogeneous pixels while inheriting the high efficiency of the HTCI. Once homogeneous pixels have been chosen, the eigenvalue decomposition of the covariance matrix is used to optimize their phase and perform temporal processing. We used the proposed method along with data on 31 scenes from the Sentinel-1 satellite from 2 June 2021 to 28 May 2022 to monitor the deformation of the surface of the fire zone in the Sikeshu coalfield. The selection results of homogeneous pixels indicate that the proposed Adp-HTCI algorithm can increase the number of selected homogeneous pixels while ensuring the accuracy of the selection results, thereby enhancing the estimation accuracy and reliability of subsequent parameter solving. The DS-InSAR results showed that the cumulative maximum subsidence in the study area within a year reached—138 mm and the point density used by the DS-InSAR approach was 17.28 times higher than that used by the StaMPS approach. The results of cross-analysis with the results of StaMPS verified the accuracy of the DS-InSAR-based approach.

Anchor-based multi-view subspace clustering with graph learning

Multi-view subspace clustering (MVSC) has been drawn wide attentions in the area of pattern recognition and data mining. However, for a multi-view dataset with n samples and V views from k clusters, MVSC commonly requires O(Vn2) memory for storing the view-specific graph matrices and O(n3) time for the eigenvalue decomposition of a shared graph matrix. Hence, most of MVSC methods are difficult to handle the large-scale multi-view data problem. To address this issue, this paper proposes an Anchor-based Multi-View Subspace Clustering with Graph Learning (AMVSCGL) method. Instead of constructing a n×n graph matrix, our method generates a shared coefficient matrix with the size of n×k based on few learned view-specific anchors. Moreover, through further merging a graph learning term, this shared coefficient matrix can simultaneously capture the global and local information among multiple views and few learned view-specific anchors for clustering. Experimental results on seven large-scale multi-view data verify our AMVSCGL’s effectiveness and superiority in comparison with some state-of-the-art methods.

Blind Estimation of Spreading Code Sequence of QPSK-DSSS Signal Based on Fast-ICA

Most of the existing estimation methods of spreading code sequence are not suitable for the QPSK-DSSS. We propose a spreading code sequence estimation method based on fast independent component analysis (Fast-ICA). It mainly includes signal preprocessing, calculations of separation matrix, and spreading code sequence. Firstly, the received signal is segmented according to the period of the spreading code sequence, and the covariance matrix can be calculated. Then, the signal subspace and corresponding eigenvalues are obtained by eigenvalue decomposition of the covariance matrix. Subsequently, the received signal matrix needs to be whitened. Finally, the Fast-ICA algorithm is used to find the separation matrix to estimate the in-phase and orthogonal spreading code sequence. The experiment result shows that the estimation of the spreading code sequence can be carried out based on Fast-ICA under a low SNR of −12 dB. Compared with the constant modulus algorithm (CMA) and the decomposition method for the real part of the self-covariance matrix (EVD-R), this method has a better performance.

Spatial-temporal trajectory anomaly detection based on an improved spectral clustering algorithm

With the development of wireless communication technology, when users use wireless networks to meet various needs, wireless networks also record a large number of users’ spatial-temporal trajectory data. In order to better pay attention to the healthy development of students and promote the information construction on campus, a spectral clustering algorithm based on the multi-scale threshold and density combined with shared nearest neighbors (MSTDSNN-SC) is proposed. Firstly, it improves the affinity distance function based on the shortest time dis-tance-shortest time distance sub-sequence (STD-STDSS) by adding location popularity and uses this model to construct the initial adjacency matrix. Then it introduces the covariance scale threshold and spatial scale threshold to perform 0–1 processing on the adjacency matrix to obtain more accurate sample similarity. Next, it constructs an eigenvector space by eigenvalue decom-position of the adjacency matrix. Finally, it uses DBSCAN clustering algorithm with shared nearest neighbors to avoid to manually determine the number of clusters. Taking Internet usage data on campus as an example, multiple clustering algorithms are used for anomaly detection and four evaluation metrics are applied to estimate the clustering results. MSTDSNN-SC algorithm reflects better clustering performance. Furthermore, the abnormal trajectories list is verified to be effective and credible.

High-Precision Time Delay Estimation Based on Closed-Form Offset Compensation

To improve the estimation accuracy, a novel time delay estimation (TDE) method based on the closed-form offset compensation is proposed. Firstly, we use the generalized cross-correlation with phase transform (GCC-PHAT) method to obtain the initial TDE. Secondly, a signal model using normalized cross spectrum is established, and the noise subspace is extracted by eigenvalue decomposition (EVD) of covariance matrix. Using the orthogonal relation between the steering vector and the noise subspace, the first-order Taylor expansion is carried out on the steering vector reconstructed by the initial TDE. Finally, the offsets are compensated via simple least squares (LS). Compared to other state-of-the-art methods, the proposed method significantly reduces the computational complexity and achieves better estimation performance. Experiments on both simulation and real-world data verify the efficiency of the proposed approach.

Multiview Clustering by Consensus Spectral Rotation Fusion.

Multiview clustering (MVC) aims to partition data into different groups by taking full advantage of the complementary information from multiple views. Most existing MVC methods fuse information of multiple views at the raw data level. They may suffer from performance degradation due to the redundant information contained in the raw data. Graph learning-based methods often heavily depend on one specific graph construction, which limits their practical applications. Moreover, they often require a computational complexity of O(n3) because of matrix inversion or eigenvalue decomposition for each iterative computation. In this paper, we propose a consensus spectral rotation fusion (CSRF) method to learn a fused affinity matrix for MVC at the spectral embedding feature level. Specifically, we first introduce a CSRF model to learn a consensus low-dimensional embedding, which explores the complementary and consistent information across multiple views. We develop an alternating iterative optimization algorithm to solve the CSRF optimization problem, where a computational complexity of O(n2) is required during each iterative computation. Then, the sparsity policy is introduced to design two different graph construction schemes, which are effectively integrated with the CSRF model. Finally, a multiview fused affinity matrix is constructed from the consensus low-dimensional embedding in spectral embedding space. We analyze the convergence of the alternating iterative optimization algorithm and provide an extension of CSRF for incomplete MVC. Extensive experiments on multiview datasets demonstrate the effectiveness and efficiency of the proposed CSRF method.

Low-Complexity Visible Light Positioning and Rotation Estimation Based on Eigenvalue Decomposition

Most existing visible light positioning (VLP) systems assume vertically placed receivers, or estimate the receiver rotations through gyroscopes. Although several recent VLP systems can deal with the arbitrary tilting of the receiver without the additional sensors, the positioning frequency, algorithm efficiency and stability are still to be improved. In this work, we propose a novel positioning and rotation estimation algorithm based on the eigenvalue decomposition of the covariance matrix of light direction vectors. Four or more light-emitting diodes (LEDs) serve as the anchors and three or more tilted photodiodes are equipped on the mobile receiver. The new algorithm's computational complexity only grows linearly with the number of LEDs, and it can distinguish and bypass the local optima by verifying if the objective function reaches its theoretical maximum. We use simulations to reveal the error patterns of the algorithm and build a low-cost prototype to verify its real performance. The measured average positioning error is 1.83 cm and the orientation estimation error is 0.04 rad. The proposed scheme can be used for positioning and navigation that require fast response, low power consumption, low cost, high accuracy and small device size.

Spectral-Compatible Transmit Beampattern Design With Minimum Peak Sidelobe for Narrowband MIMO Radar

Spectral-compatible radar emission has a paramount importance in realizing the coexistence of radar with wireless communications. In this paper, we deal with the design of spectral-compatible transmit beampattern with minimum peak sidelobe level for a narrowband multiple-input multiple-output(MIMO) radar. We also consider constraints that are imposed due to the practical high power amplifier(HPA) used in modern radars. The constraints are used to control the dynamic range on the magnitude of each signal, to control variations in the transmit power from the individual antennas and the total energy of the signals. The considered problem is challenging as it is non-convex. To this end, we propose an iterative solution that incorporates the majoriztion-minimization(MM) and alternating direction method of multipliers(ADMM) frameworks. Our solution avoids the computationally expensive operations such as matrix inverse and eigenvalue decomposition. We provide numerical results to validate the proposed solution. The results show that the method provides the intended spectral-compatible beampattern while maintaining a beampattern synthesize performance that is comparable to the state-of-the-art work.

Robotic 3-D Laser-Guided Approach for Efficient Cutting of Porcine Belly

Using autonomous meat cutting robots to cut porcine belly is a productive and more hygienic solution over traditional, labor-intensive meat cutting practices such as the transmission of the bacteria between the carcasses and manual laborers. Belly cutting is the first step of porcine meat segmentation. The robot needs to be precisely guided to cut open the porcine belly without damaging the guts inside, which facilitates subsequent processing of the guts. However, the body sizes of the pigs are different. As such, the 2-D and 3-D vision methods cannot be employed to identify the belly cutting path. In this article, a laser-guided efficient cutting path planning approach that combines the segmentation algorithm (SA) with an improved genetic algorithm (GA) is proposed. The robotic cutting system consists of an industrial robotic manipulator, customized tools, a linear laser sensor, and a PC. The salient features of the proposed scheme are as follows: 1) Spatial relationships of all the relevant units are established in Cartesian space. 2) Using the eigenvalue decomposition of the covariance matrix, the porcine belly cutting path is recognized from the 3-D laser point cloud of the carcass. 3) Aiming at efficiently optimizing the discrete path waypoints, an adaptive elite GA (AEGA) is derived by improving three operators of GA. 4) In order to alleviate the robotic cutting errors, the SA is designed to shorten the cutting path with no damage to the guts. 5) Under the limit of the peritoneal cavity, the efficient robotic cutting path is optimized by combining the SA with AEGA. Porcine belly cutting experiments are carried out in a porcine automation line under real-world conditions. Experimental results show that the proposed approach is accurate, efficient, and incurs lower costs in the long-term use of the robot.

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis.

Catalysis from single active sites is analyzed using methods developed from single-molecule kinetics. Using a stochastic Markov-state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated for model problems and for a physically realistic mechanism for an alkene hydrogenation reaction on a single-atom catalyst. The spectral analysis permits a hierarchy of timescales to be identified from the single-molecule signal, which correspond to specific relaxation modes in the network.

Multiview Spectral Clustering With Bipartite Graph.

Multi-view spectral clustering has become appealing due to its good performance in capturing the correlations among all views. However, on one hand, many existing methods usually require a quadratic or cubic complexity for graph construction or eigenvalue decomposition of Laplacian matrix; on the other hand, they are inefficient and unbearable burden to be applied to large scale data sets, which can be easily obtained in the era of big data. Moreover, the existing methods cannot encode the complementary information between adjacency matrices, i.e., similarity graphs of views and the low-rank spatial structure of adjacency matrix of each view. To address these limitations, we develop a novel multi-view spectral clustering model. Our model well encodes the complementary information by Schatten p -norm regularization on the third tensor whose lateral slices are composed of the adjacency matrices of the corresponding views. To further improve the computational efficiency, we leverage anchor graphs of views instead of full adjacency matrices of the corresponding views, and then present a fast model that encodes the complementary information embedded in anchor graphs of views by Schatten p -norm regularization on the tensor bipartite graph. Finally, an efficient alternating algorithm is derived to optimize our model. The constructed sequence was proved to converge to the stationary KKT point. Extensive experimental results indicate that our method has good performance.

A Novel Channel Errors Calibration Algorithm for Multichannel High-Resolution and Wide-Swath SAR Imaging

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.

Eigenvalue decomposition of a symmetric matrix over the symmetrized max-plus algebra

This paper discusses topics in the symmetrized max-plus algebra. In this study, it will be shown the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra. Eigenvalue decomposition is shown by using a function that corresponds to the symmetrized max-plus algebra with conventional algebra. The result obtained is the existence of eigenvalue decomposition of a symmetric matrix over symmetrized max-plus algebra and its application to determine eigenvalues and eigenvectors.