Projection Matrices Research Articles

Multiple Transformation Matrix-Based Adaptive Projective Vortex Formation Tracking for Multiagent Systems With a Leader of Completely Unknown Input.

This article systematically studies projective vortex formation tracking (PVFT) of linear multiagent systems (MASs) on directed graphs through multiple transformation matrices, in which the input of leader and its upper bound information are not available to any follower. First, an innovative class of distributed adaptive observer is designed using the projection matrices to capture multiple desired virtual signals of the leader. Next, a novel kind of distributed PVFT protocol based on distributed observer, local observer and coordinates coupling matrices are proposed. Two different adaptive update mechanisms and nonlinear functions are introduced in the observer and controller to override the unknown input of the leader. Then, an algorithm is given, and the protocol under the algorithm is proved from three processes of estimation, aggregation, and rotation to enable linear systems to achieve PVFT. Finally, the effects of parameters on aggregation and rotation are analyzed systematically, and several simulation examples are given to illustrate the reliability of the results.

Enabling Low Sidelobe Secret Multicast Transmission for mmWave Communication: An Integrated Oblique Projection Approach

The protection of multicast transmission with optimized secrecy in millimeter-wave (mmWave) communication is still an open problem. In this paper, we propose a low sidelobe secret multicast transmission scheme in mmWave communication based on physical layer security techniques. By utilizing oblique projection, we jointly adopt the directional modulation (DM) and the artificial noise (AN) to achieve secret multicast transmission at low computational costs, without jeopardizing the low sidelobe transmitting pattern. Specifically, considering the constraint of multibeam with the channel knowledge of all target users, a primary transmitting weight vector of the base station (BS) is designed to achieve a low sidelobe transmitting pattern. Then, in each symbol period, the transmitting weight vector of the BS is updated by performing a linear transformation on the primary weight vector with a transformation matrix, which is obtained from the oblique projection matrices of all the target users’ channel vectors. In this way, without deteriorating the primary weight vector’s low sidelobe transmitting pattern, the obtained transmitting weight vector synthesizes the expected symbols at the target users and adds randomness to the eavesdroppers at undesired directions. Finally, the simulations demonstrate that our proposed scheme achieves outstanding communication performance for the target users at low computational costs, while keeping the high symbol error rates at the undesired directions.

Multi-view robust regression for feature extraction

Recently, Multi-view Discriminant Analysis (MVDA) has been proposed and achieves good performance in multi-view recognition tasks. However, as an extension of LDA, this method still suffers from the small-class problem and has the sensitivity to outliers. In order to address these drawbacks and achieve better performance on multi-view recognition tasks, we proposed Multi-view Robust Regression (MVRR) for multi-view feature extraction. MVRR is a regression based method that imposes L2,1 norm as the metric of the loss function and the regularization term to improve robustness and obtain jointly sparse projection matrices for effective feature extraction. Moreover, we incorporate an orthogonal matrix to regress the extracted features to their scaled label to avoid the small-class problem. Therefore, MVRR guarantees the projection matrix to break through the restriction of the number of class for solving the small-class problem. We also propose an iterative algorithm to compute the optimal solution of MVRR and the convergence of MVRR is proved. Experiments are conducted on four databases to verify the performance of MVRR and the result illustrates that MVRR is robust on multi-view feature extraction.

Elementary coupling coefficients for the Wigner supermultiplet symmetry

An algorithm for evaluating Wigner coefficients of U(4) ⊃ SUS(2) ⊗ SUT(2) for the coupling [n14,n24,n34,n44]⊗[1,0,0,0]↓[m14,m24,m34,m44] with arbitrary irreducible representation [n14,n24,n34,n44] is provided based on the algebraic expressions with the expansion coefficients of the U(4)⊃SUS(2)⊗SUT(2) states in terms of those in the U(4) canonical basis and the Clebsch-Gordan coefficients of U(4) in the canonical basis. The state expansion coefficients are evaluated as the components of the null space vectors of the spin-isospin projection matrices. Program summaryProgram Title: Wigner-U4-SU2xSU2CPC Library link to program files:https://doi.org/10.17632/4b7gf2phh2.1Licensing provisions: GPLv3Programming language: Wolfram Mathematica version 9.0.1.0 and beyondNature of problem: Numerical evaluation of the elementary Wigner coefficients (EWCs) of U(4)⊃SUS(2)⊗SUT(2).Solution method: Mathematica implementation of the EWCs of U(4)⊃SUS(2)⊗SUT(2) [1] using the expansion coefficients of U(4)⊃SUS(2)⊗SUT(2) states in terms of those of the U(4) canonical basis determined as components of the null space vectors of the spin-isospin projection matrices [2].

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

An adaptive model order reduction technique for parameter-dependent modular structures

AbstractThis work is concerned with an adaptive reduced order model of modular structures assembled from parameter-dependent substructures. The substructures are reduced by proper orthogonal decomposition (POD) and connected by means of a tied contact formulation. We present a method to adapt the matrices of the substructures to parameter changes. We employ interpolation on Grassmann manifolds for the parametric adaption of the projection matrices. For the adaptation of the stiffness matrices, we use the direct empirical interpolation method (DEIM). Manifold interpolation of the reduced stiffness matrices, cannot be applied here since it would require semi-positive definiteness, which is here not fulfilled because of necessary rigid body motion modes. The novelty of this work is the application of these interpolation methods to the special problem class of POD-based tied contact model order reduction. Furthermore, we show a methodology to compute significant snapshots on the substructure level to compute a POD basis that can be used in different global structures.

Semi-supervised enhanced discriminative local constraint preserving projection for dimensionality reduction of medical hyperspectral images

Microscopic hyperspectral images has the advantage of containing rich spatial and spectral information. However, the large number of spectral bands provides a significant amount of spectral features, but also leads to data redundancy and noise, which seriously affect the recognition and classification performance of the images, as well as increasing the requirements for computation and storage. To address this issue, we propose a dimensionality reduction algorithm named enhanced discriminant local constraint preserving projection (EDLCPP). Specifically, the global spectral attention mechanism focuses on important bands, the high discriminability sample selection module measures the discriminability of samples using a modified average neighborhood margin, the graph construction module preserves the local geometric relationship and discriminant information, and the graph embedding module embeds the constructed graphs into a low-dimensional space to obtain the projection matrices. Experimental results on eight cholangiocarcinoma (CCA) hyperspectral images, Bloodcell1-3, and Bloodcell2-2 datasets have demonstrated the effectiveness of the proposed method.

Robust manifold discriminative distribution adaptation for transfer subspace learning

Transfer learning is an effective means by leveraging rich information of the labeled samples from the source domain to build an appropriate classifier for annotating the target domain. Existing transfer learning algorithms usually try to minimize the distribution difference between different domains by means of Maximum Mean Discrepancy (MMD) based metric criteria. However, the domain adaptation methods based on MMD usually have the following two problems: 1) the local geometry information of the inter-domain data is ignored, which may cause to negative transfer effect. 2) The classifier and the feature learning are not integrated into one framework, in which the global optimal solution may be not obtained. Therefore, a Robust Manifold Discriminative Distribution Adaptation (RMDDA) model is presented for transfer subspace learning. Because the difference between domains leads to the large deviation of the original features of data in the source and target domains, it is difficult to construct the manifold structure of inter-domain data. A new feature termed Histogram Feature of Neighbors (HFON) is firstly introduced to rightly capture the domain locality. Secondly, the marginal distribution, conditional distribution, and discriminative regression are integrated into a model, which can solve the problem of separating feature learning from classifier training. It is considered that the dimension of feature extraction is limited by the number of classes which is determined by discriminative regression, we introduce double projection matrices into the model. Lastly, the numerous experiments on six benchmark cross-domain image datasets prove the superiority of RMDDA algorithm compared with some state-of-the-art algorithms.

The characteristic polynomial of projections

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.

Kernel-Based KPI-Oriented Fault Detection Approach in the Presence of Outliers

Kernel partial least squares (KPLS) has poor robustness and cannot achieve effective monitoring for key performance indicators (KPI). This study investigates a new KPI-oriented robust KPLS approach to mitigate these drawbacks. In this methodology, the robust KPLS inspired by the gradient boosting principle incorporates a weighting matrix into the KPLS to mitigate the influence of outliers. Simultaneously, the associated coefficient matrix is derived in detail. Then, a decomposition approach is used to separate the process variable space into two orthogonal parts. Two strategies are discussed to obtain the unknown projection matrices based on the kernel principal component analysis and the elastic network frameworks. Finally, the performance of the proposed methods in terms of prediction, monitoring and robustness to outliers is evaluated by the Tennessee Eastman process and the three-phase flow facility. The results show that the proposed methods have good prediction accuracy and monitoring performance in the presence of outliers, demonstrating the effectiveness and advantages of the proposed approaches.

Predicting Drug-Disease Associations Through Similarity Network Fusion and Multi-View Feature Projection Representation.

Predicting drug-disease associations (DDAs) through computational methods has become a prevalent trend in drug development because of their high efficiency and low cost. Existing methods usually focus on constructing heterogeneous networks by collecting multiple data resources to improve prediction ability. However, potential association possibilities of numerous unconfirmed drug-related or disease-related pairs are not sufficiently considered. In this article, we propose a novel computational model to predict new DDAs. First, a heterogeneous network is constructed, including four types of nodes (drugs, targets, cell lines, diseases) and three types of edges (associations, association scores, similarities). Second, an updating and merging-based similarity network fusion method, termed UM-SF, is presented to fuse various similarity networks with diverse weights. Finally, an intermediate layer-mediated multi-view feature projection representation method, termed IM-FP, is proposed to calculate the predicted DDA scores. This method uses multiple association scores to construct multi-view drug features, then projects them into disease space through the intermediate layer, where an intermediate layer similarity constraint is designed to learn the projection matrices. Results of comparative experiments reveal the effectiveness of our innovations. Comparisons with other state-of-the-art models by the 10-fold cross-validation experiment indicate our model's advantage on AUROC and AUPR metrics. Moreover, our proposed model successfully predicted 107 novel high-ranked DDAs.

A Bi-Directional Two-Dimensional Deep Subspace Learning Network with Sparse Representation for Object Recognition

A principal component analysis network (PCANet), as one of the representative deep subspace learning networks, utilizes principal component analysis (PCA) to learn filters that represent the dominant structural features of objects. However, the filters used in PCANet are linear combinations of all the original variables and contain complex and redundant principal components, which hinders the interpretability of the results. To address this problem, we introduce sparse constraints into a subspace learning network and propose three sparse bi-directional two-dimensional PCANet algorithms, including sparse row 2D2PCANet (SR2D2PCANet), sparse column 2D2PCANet (SC2D2PCANet), and sparse row–column 2D2PCANet (SRC2D2PCANet). These algorithms perform sparse operations on the projection matrices in the row, column, and row–column direction, respectively. Sparsity is achieved by utilizing the elastic net to shrink the loads of the non-primary elements in the principal components to zero and to reduce the redundancy in the projection matrices, thus improving the learning efficiency of the networks. Finally, a variety of experimental results on ORL, COIL-100, NEC, and AR datasets demonstrate that the proposed algorithms learn filters with more discriminative information and outperform other subspace learning networks and traditional deep learning networks in terms of classification and run-time performance, especially for less sample learning.

Neural network representations for the inter- and intra-class common vector classifiers

Common Vector Approach (CVA) is a known linear regression-based classifier, which also enables an extension to inter-class discrimination, known as the Discriminative Common Vector Approach (DCVA). The characteristics of linear regression classifiers (LRCs) enable the possibility of a schematic implementation that is similar to the neuron model of artificial neural networks (ANNs). In this work, we explore this schematic similarity to come up with an ANN representation for both CVA and DCVA. The new representation eliminates the need for projection matrices in its implementation, hence significantly reduces the memory requirements and computational complexities of the processes. Furthermore, since the new representation is in a neural style, it is expected to provide a solid and intriguing extension of CVA (and DCVA) by further incorporating adaptation or activation processes to the already successful CVA-based classifiers.

Parametric Reduced-Order Modeling of Battery Thermal Management Systems for Varying Rates of Cooling Liquid Flow

The thermal management system in battery packs for electrical/hybrid electric vehicles plays an important role in maintaining good performance and health of battery cells. In order to optimize the thermal management system design, computational fluid dynamics (CFD) simulations are often used. The CFD is a powerful tool that can accurately simulate the thermal behavior of the system using numerical computation techniques such as finite volume method. However, the days-long computational expense means that the CFD-based simulations are still unsuitable for situations when rapid results are required, for example for online predictive control or fast-pace optimization design.A new method of reduced-order modeling of battery thermal management systems (BTMS) was presented in [1]. Given a CFD model, the algorithm based on projection on Krylov-subspace yields a reduced-order model (ROM). The ROM accurately predicts the system’s transient response to arbitrary internal heating of individual batteries and temperature of cooling liquid at inlet. Unlike conventional Krylov-subspace approaches, the algorithm does not require the system matrix of the CFD model. Therefore, the method is more suitable for use with commercial CFD software which does not allow access to such internal information. The ROM uses a drastically reduced number of degrees of freedom requiring drastically reduced simulation time.This paper extends our ROM by including the mass flow rate of cooling liquid as a model parameter. The task is accomplished by interpolating the system matrices after several local reduced-order models are constructed at discrete mass flow rates (MFR). In order to interpolate the local ROMs, we first need to transform the local ROMs onto a common subspace. The transformation matrix is obtained by applying the singular value decomposition (SVD) to the local projection matrices. In this paper, three local ROMs are built at 3 different MFRs. After these local ROMs are transformed according to the above procedure, a parametric reduced-order model (pROM) is constructed via quadratic interpolation.The constructed pROM is tested against the new MFRs different from those used for building the local ROMs. For each new MFR, the pROM produced a transient response that is almost indistinguishable from that of the CFD model (e.g., see Figure 1). The several thousand core-hour computational time needed for CFD simulations is replaced by a few tens of seconds using the pROM on a personal computer. The low computing load of the pROM is deemed adequate for on-board applications. Reference [1] Xiang L, Lee CW, Zikanov O, Hsu C-C. Efficient reduced order model for heat transfer in a battery pack of an electric vehicle. Applied Thermal Engineering. 2022; 201:117641.Figure 1 Figure 1

Mutual Domain Adaptation

To solve the label sparsity problem, domain adaptation has been well-established, suggesting various methods such as finding a common feature space of different domains using projection matrices or neural networks. Despite recent advances, domain adaptation is still limited and is not yet practical. The most pronouncing problem is that the existing approaches assume source-target relationship between domains, which implies one domain supplies label information to another domain. However, the amount of label is only marginal in real-world domains, so it is unrealistic to find source domains having sufficient labels. Motivated by this, we propose a method that allows domains to mutually share label information. The proposed method finds a projection matrix that matches the respective distributions of different domains, preserves their respective geometries, and aligns their respective class boundaries. The experiments on benchmark datasets show that the proposed method outperforms relevant baselines. In particular, the results on varying proportions of labels present that the fewer labels the better improvement.

Two-Dimensional Embedded Fuzzy Data Clustering

Fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -means (FKM) is a popular clustering method by assigning data points into respective clusters with uncertainty measured by the membership degree. Usually, FKM performs clustering according to the distance between data points in the original space, which might contain undesirable noises and redundant features; therefore, the underlying data semantic connections cannot be accurately captured. Moreover, the vectorized representation of the two-dimensional data such as image inevitably leads to the loss of structural information. In this paper, we propose a novel FKM method termed two-dimensional embedded fuzzy data clustering (2DEFC) which has two merits. First, 2DEFC directly takes 2D data as input without vectorizing them in order to retain more structural information of data. Second, the two subspace projection matrices are jointly optimized with the data membership degree for better collaborating with each other, which effectively avoids the sub-optimality limitation caused by the conventional mode of sequentially performing dimensionality reduction and clustering. An efficient algorithm is proposed to optimize the 2DEFC objective function. Besides, we provide comprehensive analysis on 2DEFC including its convergence behavior, computational complexity, and the fuzzy weighting exponent. Extensive comparative studies on benchmark 2D data sets demonstrate that the competitive performance of 2DEFC in 2D data clustering.

Joining Iso-Structured Models with Commutative Orthogonal Block Structure

In this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.

Single phase multi-view clustering using unified graph learning and spectral representation

The most widely used multi-view clustering approaches can be classified into different groups: Subspace multi-view clustering algorithms, multi-view kernel approaches, matrix factorization approaches, and spectral clustering algorithms. Most approaches rely on combining predefined individual similarity matrices from multiple views, and then estimate the common similarity matrix. Therefore, their performance can be severely affected by noisy original similarity matrices. Moreover, most of these methods integrate different spectral projection matrices of each view together, which can also affect the clustering results.To address these shortcomings, we propose a single-phase multiview clustering method with Consensus Graph Learning and Spectral Representation (MCGLSR) in this paper. Instead of directly integrating the similarity matrices of the different views, which could introduce noise, our proposed method jointly generates similarity graphs of the views and their common similarity matrix (graph matrix) using a unified global objective function. At this stage, the similarity matrices of the different views are enforced to be sufficiently similar, which eliminates the problem of noise and promotes a clearer unified data structure. Moreover, our proposed objective function is able to recover the common spectral projection and soft cluster assignments based on the common graph structure. The proposed method takes as input a kernelized representation of the views features and directly provides the individual graphs, the common graph, the common spectral representation, and the cluster assignments. Our technique allows us to obtain the final cluster assignments without requiring an external clustering algorithm. Several real-world datasets are used to evaluate our technique and demonstrate its efficacy. The code of the proposed method will be available at the following link: https://github.com/SallyHajjar/MCGLSR.git.

Accelerating unsteady aerodynamic simulations using predictive reduced-order modeling

Unsteady computational fluid dynamics (CFD) simulations are essential in aerospace engineering because they can provide high-fidelity flow fields to better understand transient physics, such as vortex shedding. However, unsteady full-order CFD simulations must repeatedly march the flow solution with a small time step and are computationally expensive. Reduced-order modeling (ROM) is a powerful approach to alleviate the above issue by decomposing the unsteady flow solutions into spatial modes and temporal coefficients, making the unsteady flow easier to simulate. Existing ROM studies mostly focused on parametric problems that use a large number of simulation samples to train an offline model (parametric ROM). Although the trained model can quickly predict any flow fields within the parameter space, the computational cost for generating the massive unsteady simulation samples is still high, especially when the number of parameters and their ranges increase. To further address the high-cost issue, we develop an efficient predictive ROM approach to accelerate individual unsteady aerodynamic simulations. We use the Galerkin projection approach to reduce the Reynolds-averaged Navier–Stokes equations, along with the discrete empirical interpolation method (DEIM) for decreasing the computation cost for nonlinear terms. In addition, we develop an efficient ROM formulation that correlates the temporal coefficients between the momentum, pressure, and turbulence equations, allowing solving fewer ROM equations at the prediction stage. The intrusive nature of the predictive ROM enables using the first portion of unsteady flow data to accelerate the rest of the simulation; no massive offline samples are needed. We use the stalled turbulent flow over the NACA0012 airfoil as the benchmark and evaluate the predictive ROM's speed and accuracy for challenging non-equilibrium scenarios caused by a large sudden change in flow conditions. The run time ratios between CFD and ROM (including the calculation of basis vectors and projection matrices) range between 13 and 45. The pressure, drag, lift, pitching moment, and flow fields simulated by the ROM approach agree reasonably well with the CFD references at various times. The proposed predictive ROM approach is, in principle, applicable to different airfoil geometries and flow conditions and can be integrated into a design optimization process for accelerating unsteady aerodynamic simulations.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.