Orthogonal Matrix Research Articles

Block perturbation of symplectic matrices in Williamson’s theorem

AbstractWilliamson’s theorem states that for any $2n \times 2n$ real positive definite matrix A, there exists a $2n \times 2n$ real symplectic matrix S such that $S^TAS=D \oplus D$ , where D is an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde {S}$ diagonalizing $A+H$ in Williamson’s theorem is of the form $\tilde {S}=S Q+\mathcal {O}(\|H\|)$ , where Q is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that $\tilde {S}$ and S can be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$ . Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].

Unilateral Orthogonal Nonnegative Matrix Factorization

Unilateral Orthogonal Nonnegative Matrix Factorization

Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown View Tomography

Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown View Tomography

Torque control of a wheeled humanoid robot with dual redundant arms

Most of conventional controllers are prone to consume more computational time for controlling a wheeled humanoid robot. A nonlinear trajectory control of a wheeled humanoid robot using a model-based controller with less computational load and energy consumption is presented in this article. The upper body of the humanoid robot consists of two 6 degrees of freedom redundant arms, a three degrees of freedom torso and a two degrees of freedom neck. The nonholonomic mobile platform consists of two actuated wheels and two caster wheels. Nonlinearity of the robot dynamic model and coupling between various branches of the upper body are taken into account. The dynamic model derived using Newton–Euler approach along with decoupled Natural Orthogonal Complement matrix approach is used to derive dynamic equations of the upper body and the wheeled platform. Zero moment point based stability approach is used for verifying stable motion of the wheeled humanoid robot with minimum energy and time consumption to complete a task. The proposed computed torque controller is compared with other similar controllers developed for the same robot model to prove advantages of the proposed torque controller. Simulation results are experimentally validated with the real robot.

Optimization of plastic injection molding process parameters for cowl B (L/R) sink mark defects by using Taguchi methods and ANOVA

The plastic injection molding process on Cowl B (L/R) products that have been carried out has sink mark defects. The defects that arise occur because the composition of injection molding parameter values is not optimal in the variables of melt temperature, mold temperature, packing time, packing pressure, and cooling time. The purpose of this study is to find the optimal composition of parameter values for each variable, to minimize sink mark defects in the product. The analysis process begins with the preparation of an orthogonal array matrix to determine the design parameters to be simulated on Autodesk mold flow. These results are evaluated with a signal-to-noise ratio to determine the effect of each parameter value composition on the results of the analysis process. The Analysis of Variance (ANOVA) method is used to estimate the contribution of each independent variable to all response measurements (the dependent variable). The optimization results for sink mark defects in the sink mark index value of 1.4494%, volumetric shrinkage of 0.5053%, and sink mark estimate of 0.0608 mm are found in the composition of the parameter values of melt temperature 200°C, mold temperature 80°C, packing time 30 seconds, packing pressure 80 MPa and a cooling time of 13,365 seconds. This data is used as a reference in determining parameters before production is carried out on plastic injection molding machines so that the time and cost of testing the injection molding process are optimal.

Efficient, secure and verifiable outsourcing scheme for SVD-based collaborative filtering recommender system

With the explosive growth of e-commerce, recommender systems are widely used to generate personalized recommendation for customers. SVD-based collaborative filtering and its variants are highly accurate and scalable approaches to recommender systems. Due to the heavy computation of SVD-based collaborative filtering, outsourcing the computation is an efficient solution to reduce computational complexity. In this paper, we propose an efficient, secure and verifiable outsourcing scheme for SVD-based collaborative filtering recommender system. We use symmetric block diagonal matrices as seeds to generate secret keys, which are novel orthogonal sparse matrices to blind the target matrices of SVD. Security analysis shows that our scheme can protect the privacy of both the input and output and efficiency analysis shows that our scheme is (3m2+3n2+5mn)/(m3+n3) efficient compared to fully local algorithm. In our scheme, we also create a verification approach that is capable of detecting misbehavior from a cloud server with probability (1−12n). The experiment shows that the client achieves significant computational savings and the recommendation accuracy of the scheme is nearly as good as that of the fully local algorithm.

Idempotents in simple skew group rings and applications to skew cyclic codes

We give the form of complete sets of pairwise orthogonal primitive idempotents in a simple skew group ring. Using these idempotents, we construct some classes of skew cyclic codes. We give also some examples of MDS codes and rows orthogonal and MDS matrices. Communicated by Eric Jespers

Solar irradiation-driven synthesis of biodiesel from composite oil using biomass catalyst and process parameters optimization

Developing countries like India are highly dependent on fossil fuels, which constitute about 70% of their total energy needs. However, the availability of fossil fuel resources is constantly declining worldwide due to their abundant usage. In the process of burning fossil fuels, harmful gases are emitted into the atmosphere and pollute the air we breathe. Biodiesel is a clean and promising renewable energy source for meeting future energy demands. In this study, a solar irradiation driven transesterification setup is developed to generate biodiesel from composite oil using a waste eggshell derived catalyst. The aim is to address the techno-economic and sustainability challenges associated with biodiesel production. To perform transesterification experiments, a Taguchi L9 orthogonal design matrix is created. Experimental optimization revealed that the best process variables for producing high yield biodiesel of 90% were 2 wt% catalyst loading, 60 °C thermal state, and a 6:1 composite oil to methyl alcohol ratio in molar. The biodiesel produced underwent testing to assess its physical and chemical properties, following the guidelines outlined in the ASTM D6751 standards. The proposed solar irradiation driven transesterification setup, composite oil, and waste eggshell derived catalyst proved to be an effective and green renewable resource for producing sustainable biodiesel.

Data-Aided CSI Estimation Using Affine-Precoded Superimposed Pilots in Orthogonal Time Frequency Space Modulated MIMO Systems

An orthogonal affine-precoded superimposed pilot (AP-SIP)-based architecture is developed for the cyclic prefix (CP)-aided single input single output (SISO) and multiple input multiple output (MIMO) orthogonal time frequency space (OTFS) systems relying on arbitrary transmitter-receiver (Tx-Rx) pulse shaping. The data and pilot symbol matrices are affine-precoded and superimposed in the delay Doppler (DD)-domain followed by the development of an end-to-end DD-domain relationship for the input-output symbols. At the receiver, the decoupled pilot and data symbol are extracted by employing orthogonal precoder matrices, which eliminates the mutual interference. Furthermore, a novel pilot-aided Bayesian learning (PA-BL) technique is conceived for the channel state information (CSI) estimation of SISO OTFS systems based on the expectation-maximization (EM) technique. Subsequently, a data-aided Bayesian learning (DA-BL)-based joint CSI estimation and data detection technique is proposed, which beneficially harnesses the estimated data symbols for improved CSI estimation. In this scenario our sophisticated data detection rule also integrates the CSI uncertainty of channel estimation into our the linear minimum mean square error (LMMSE) detectors. The AP-SIP framework is also extended to MIMO OTFS systems, wherein the DD-domain input matrix is affine-precoded for each transmit antenna (TA). Then an EM algorithm-based PA-BL scheme is derived for simultaneous row-group sparse CSI estimation for this system, followed also by our data-aided DA-BL scheme that performs joint CSI estimation and data detection. Moreover, the Bayesian Cramer-Rao bounds (BCRBs) are also derived for both SISO as well as MIMO OTFS systems. Finally, simulation results are presented for characterizing the performance of the proposed CSI estimation techniques in a range of typical settings along with their bit error rate (BER) performance in comparison to an ideal system having perfect CSI.

Generalized Orthogonal Matrices for Orthogonal Transformation in Six-Phase Motors

Generalized Orthogonal Matrices for Orthogonal Transformation in Six-Phase Motors

Towards Feature Decoupling for Lightweight Oriented Object Detection in Remote Sensing Images

Recently, the improvement of detection performance always relies on deeper convolutional layers and complex convolutional structures in remote sensing images, which significantly increases the storage space and computational complexity of the detector. Although previous work has designed various novel lightweight convolutions, when these convolutional structures are applied to remote sensing detection tasks, the inconsistency between features and targets as well as between features and tasks in the detection architecture is often ignored: (1) The features extracted by convolution sliding in a fixed direction make it difficult to effectively model targets with arbitrary direction distribution, which leads to the detector needing more parameters to encode direction information and the network parameters being highly redundant; (2) The detector shares features from the backbone, but the classification task requires rotation-invariant features while the regression task requires rotation-sensitive features. This inconsistency in the task can lead to inefficient convolutional structures. Therefore, this paper proposed a detector that uses the Feature Decoupling for Lightweight Oriented Object Detection (FDLO-Det). Specifically, we constructed a rotational separable convolution that extracts rotational equivariant features while significantly compressing network parameters and computational complexity through highly shared parameters. Next, we introduced an orthogonal polarization transformation module that decomposes rotational equivariant features in both horizontal and vertical orthogonal directions, and used polarization functions to filter out the required features for classification and regression tasks, effectively improving detector performance. Extensive experiments on DOTA, HRSC2016, and UCAS-AOD show that the proposed detector can achieve the best performance and achieve an effective balance between computational complexity and detection accuracy.

Generalized Hamiltonian drift-fluid and gyrofluid reductions

We provide a procedure for deriving Hamiltonian reduced fluid models for plasmas, starting from a Hamiltonian gyrokinetic system in the approximation. The procedure generalizes, to a considerable extent, previous results. In particular, the evolution of moments with respect to the magnetic moment coordinate is also taken into account, together with background density and magnetic inhomogeneities. In the limit of vanishing Finite Larmor Radius (FLR) effects, an infinite family of reduced electron drift-fluid equations is derived, evolving all the electron moments , with and , where N and M are arbitrary non-negative integers counting the maximum order of the moments taken with respect to the parallel velocity and to the magnetic moment coordinates, respectively. An analogous result is found in the gyrofluid case and applied to the ion species. The gyrofluid result holds for , finite FLR effects and a low ratio β e between electron internal pressure and magnetic guide field pressure. In both the drift and gyrofluid case, the key for the identification of the Hamiltonian structure resides in changes of variables based on orthogonal matrices that diagonalize the Jacobi matrices associated with Hermite and Laguerre polynomials. In terms of the transformed variables, the drift and gyrofluid equations are cast in a simple form, which reduces to advection equations for Lagrangian invariants in the two-dimensional case with homogeneous background. Because the procedure requires to evolve, for a particle species s, all and only the moments , with and , not every choice for the set of moments is admissible, for fixed N and M. This might also explain the scarcity of Hamiltonian reduced fluid models obtained so far which account for anisotropic temperature fluctuations.

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.

Mechanically Prestressed Pneumatically Driven Bistable Soft Actuators

Abstract Bistable soft robots are gaining momentum for their fast speed. This study presents a novel asymmetric mechanically prestressed, pneumatically driven, bistable laminated soft actuator. Its two orthogonal stable shapes are created by prestretching two orthogonal elastomer matrix composites before bonding them to a thin core layer. Two fluidic layers with fluid channels are bonded on either side of the core layer to actuate and trigger the snap-through process of the actuator. An analytical model is proposed as follows: the actuator net energy is calculated based on polynomials with unknown coefficients, and the stable shapes of the actuator are computed as a result of pneumatic pressure and external loads with the Rayleigh–Ritz method. Bistable actuators are fabricated with different prestrains, and motion capture and tensile loading experiments are conducted for model validation. A gripper is fabricated with two bistable actuators and demonstrated to grasp a variety of objects. Sensitivity studies are performed to identify the actuator response as a function of a variety of design parameters.

Fundamental limits in structured principal component analysis and how to reach them

How do statistical dependencies in measurement noise influence high-dimensional inference? To answer this, we study the paradigmatic spiked matrix model of principal components analysis (PCA), where a rank-one matrix is corrupted by additive noise. We go beyond the usual independence assumption on the noise entries, by drawing the noise from a low-order polynomial orthogonal matrix ensemble. The resulting noise correlations make the setting relevant for applications but analytically challenging. We provide characterization of the Bayes optimal limits of inference in this model. If the spike is rotation invariant, we show that standard spectral PCA is optimal. However, for more general priors, both PCA and the existing approximate message-passing algorithm (AMP) fall short of achieving the information-theoretic limits, which we compute using the replica method from statistical physics. We thus propose an AMP, inspired by the theory of adaptive Thouless-Anderson-Palmer equations, which is empirically observed to saturate the conjectured theoretical limit. This AMP comes with a rigorous state evolution analysis tracking its performance. Although we focus on specific noise distributions, our methodology can be generalized to a wide class of trace matrix ensembles at the cost of more involved expressions. Finally, despite the seemingly strong assumption of rotation-invariant noise, our theory empirically predicts algorithmic performance on real data, pointing at strong universality properties.

Quality Analysis of Matrix Masking of Digital Audio Data

This article considered the guarantee of confidentiality task of audio information transmitted over a packet communication channel. Domain analysis showed, that in most cases cryptographical method used for guarantee of confidentiality of audio information, however, along with them, matrix methods of protective coding used, with implementation on FPGA or DSP processors. These methods, using random matrices, impose additional instrumental errors during decoding. The purpose of the study: is to assess the quality of protective coding ‒ masking of audio data by quasi-orthogonal structured matrices. The proposed method of masking digital audio information by quasi-orthogonal matrices differs from the known ones in predictable results and simplity of implementation. An alternative approach to the classical use of metrics is proposed for evaluating the result of masking. It is based on the analysis of the spectral component of the signal. It is shown that masking with a square Mersenne ‒ Walsh matrix brings digital audio information, presented in the form of a matrix, to a form close in spectrum to white noise. This reliably protects it in the communication channel from unauthorized access. The obtained results showed that the proposed method is promising and suggests the study of the influence of the choice of orthogonal matrix structures and their sizes on the results of masking.

Unsupervised neural decoding of signals recorded by thin-film electrode arrays implanted in muscles using autoencoding with a physiologically derived optimisation criterion

We present an autoencoder that learns the discharge times of individual motor units (MU) by processing multichannel electromyogram (EMG) recordings obtained by electrode arrays implanted in muscles, thereby providing a neural interface with the spinal cord. To this end, after preprocessing of the EMG via convolutive sphering, the encoder is constrained to apply an orthogonal transformation to the spatial data and the optimisation criterion enforces a temporal sparsity constraint. These constraints are based on the theoretical modelling of EMG generation. This decomposition method was evaluated on both simulated and experimental signals. For simulated signals, with a ratio of 1.5 between observations and sources, the average detection accuracy was 94% for up to 60 sources for SNR 20 dB. Moreover, the detection accuracy did not significantly decrease when decreasing the SNR to values as low as 0 dB. Experimental signals were collected in humans with thin-film invasive electrodes implanted in the tibialis anterior muscle. The results showed an accuracy on experimental data >90%. Moreover, the proposed method outperformed a state-of-the-art blind source separation approach in terms of the number of reliably detected motor units. In conclusion, we have proposed the first fully unsupervised neural network approach to the problem of neural decoding of intramuscular EMG time series by translating the available theoretical knowledge on the signal properties into an autoencoding architecture tailored to the decomposition problem.

Event‐triggered containment control for multi‐agent systems under hybrid cyber attacks

AbstractThis paper studies event‐triggered containment control problem of multi‐agent systems (MASs) under deception attacks and denial‐of‐service (DoS) attacks. First, to save limited network resources, an event‐triggered mechanism is proposed for MASs under hybrid cyber attacks. Different from the existing event‐triggered mechanisms, the negative influences of deception attacks and DoS attacks are considered in the proposed triggering function. The communication frequencies between agents are reduced. Then, based on the proposed event‐triggered mechanism, a corresponding control protocol is proposed to ensure that the followers will converge to the convex hull formed by the leaders under deception attacks and DoS attacks. Compared with the previous researches about containment control, in addition to hybrid cyber attacks being considered, the nonlinear functions related to the states of the agents are applied to describe the deception attack signals in the MAS. By orthogonal transformation of deception attack signals, the containment control problem under deception attacks and DoS attacks is reformulated as a stability problem. Then, the sufficient conditions on containment control can be obtained. Finally, a set of simulation example is used to verify the effectiveness of the proposed method.

Tire-Road Friction Coefficient Estimation for Distributed Drive Electric Vehicles Using PMSM Sensorless Control

Many tire-road friction coefficient estimation methods require the wheel angular speed sensor to provide a signal. In this paper, a tire-road friction coefficient estimation algorithm for distributed drive electric vehicles using permanent magnet synchronous motors(PMSM) sensorless control is proposed to reduce the use of wheel angular speed sensors. The wheel angular speed signal is replaced by the rotor speed signal obtained from PMSM sensorless control. First, the three degrees of freedom vehicle dynamics model and the PMSM mathematical model are established, and a strong tracking cubature Kalman filtering algorithm based on orthogonal transformation (T-STCKF) is derived to overcome the problem that the STCKF algorithm is prone to nonlocal sampling when dealing with high dimensional systems. Second, a PMSM sensorless control system based on the adaptive exponent reaching law of sliding mode control and the T-STCKF algorithm is established, and confirmed its validity. Finally, the feasibility of the proposed algorithm is demonstrated through multicase simulation and experiment and the results shows that the T-STCKF algorithm is more accurate than the STCKF algorithm.

Self-Adaptive Spherical Search With a Low-Precision Projection Matrix for Real-World Optimization.

Since the last three decades, numerous search strategies have been introduced within the framework of different evolutionary algorithms (EAs). Most of the popular search strategies operate on the hypercube (HC) search model, and search models based on other hypershapes, such as hyper-spherical (HS), are not investigated well yet. The recently developed spherical search (SS) algorithm utilizing the HS search model has been shown to perform very well for the bound-constrained and constrained optimization problems compared to several state-of-the-art algorithms. Nevertheless, the computational burdens for generating an HS locus are higher than that for an HC locus. We propose an efficient technique to construct an HS locus by approximating the orthogonal projection matrix to resolve this issue. As per our empirical experiments, this technique significantly improves the performance of the original SS with less computational effort. Moreover, to enhance SS's search capability, we put forth a self-adaptation technique for choosing the effective values of the control parameters dynamically during the optimization process. We validate the proposed algorithm's performance on a plethora of real-world and benchmark optimization problems with and without constraints. Experimental results suggest that the proposed algorithm remains better than or at least comparable to the best-known state-of-the-art algorithms on a wide spectrum of problems.