Matrix Space Research Articles

Incremental generalized multiple maximum scatter difference with applications to feature extraction

In this paper, we propose a new algorithm to implement the generalized multiple maximum scatter difference (GMMSD). Due to enhanced features of this algorithm over the original GMMSD, we named it GMMSD+. By employing a different projection from both the range of the between-class scatter matrix and the null space of the within-class scatter matrix, GMMSD+ can divide the centroid vector of each class into two components: intrinsic common component (ICC) and discriminant difference component (DCC), and then automatically discards ICC which contains little discriminative information, while keeping DCC which contains the true discriminative power. Next, we introduce a practical implementation of GMMSD+, which can accurately and efficiently update the discriminant vectors with new training samples incrementally, eliminating the complete re-computation of the training process. Our experiments demonstrate that incremental version of GMMSD+(IGMMSD+) eliminates the complete re-computation of the training process when new training samples are presented, leading to significantly reduced computational cost.

Locality preserving projection on SPD matrix Lie group: algorithm and analysis

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

Beamspace dual signal space projection (bDSSP): a method for selective detection of deep sources in MEG measurements

Objective. Magnetoencephalography (MEG) has a well-recognized weakness at detecting deeper brain activities. This paper proposes a novel algorithm for selective detection of deep sources by suppressing interference signals from superficial sources in MEG measurements. Approach. The proposed algorithm combines the beamspace preprocessing method with the dual signal space projection (DSSP) interference suppression method. A prerequisite of the proposed algorithm is prior knowledge of the location of the deep sources. The proposed algorithm first derives the basis vectors that span a local region just covering the locations of the deep sources. It then estimates the time-domain signal subspace of the superficial sources by using the projector composed of these basis vectors. Signals from the deep sources are extracted by projecting the row space of the data matrix onto the direction orthogonal to the signal subspace of the superficial sources. Main results. Compared with the previously proposed beamspace signal space separation (SSS) method, the proposed algorithm is capable of suppressing much stronger interference from superficial sources. This capability is demonstrated in our computer simulation as well as experiments using phantom data. Significance. The proposed bDSSP algorithm can be a powerful tool in studies of physiological functions of midbrain and deep brain structures.

Information Geometry for Covariance Estimation in Heterogeneous Clutter with Total Bregman Divergence.

This paper presents a covariance matrix estimation method based on information geometry in a heterogeneous clutter. In particular, the problem of covariance estimation is reformulated as the computation of geometric median for covariance matrices estimated by the secondary data set. A new class of total Bregman divergence is presented on the Riemanian manifold of Hermitian positive-definite (HPD) matrix, which is the foundation of information geometry. On the basis of this divergence, total Bregman divergence medians are derived instead of the sample covariance matrix (SCM) of the secondary data. Unlike the SCM, resorting to the knowledge of statistical characteristics of the sample data, the geometric structure of matrix space is considered in our proposed estimators, and then the performance can be improved in a heterogeneous clutter. At the analysis stage, numerical results are given to validate the detection performance of an adaptive normalized matched filter with our estimator compared with existing alternatives.

Singular-Perturbation-Based Nonlinear Hybrid Control of Redundant Parallel Robot

High-speed robots are being applied more and more widely in the scenarios of aviation manufacturing, electronic packaging, and three-dimensional printing, etc. To achieve high speed, the robots are typically designed to be lightweight, hence suffer from undesired vibration, so that the dynamic performance would deteriorate. In light of this, this paper considers the dynamic modeling and active control of an emerging redundant parallel robot. First, the recursively dynamic model is modularly established by means of Lagrangian multipliers method. For completeness, the model of permanent magnet synchronous motor is further integrated with the model of mechanical system to formulate the electromechanically coupled dynamic model (ECDM). To facilitate dynamic control, the ECDM is cleverly projected into the null space of constraint Jacobian matrix eliminating the need to compute Lagrangian multipliers explicitly. Based on singular perturbation approach and Tikhonov theorem, the ECDM is decomposed into two subsystems in different timescales. On this basis, a nonlinear hybrid control scheme is proposed for both trajectory tracking and vibration suppression. Numerical simulations suggest that the nonlinear hybrid control exhibits overwhelming performance than the conventional joint-based proportional and differential feedback control.

Nonprehensile Manipulation of an Underactuated Mechanical System With Second-Order Nonholonomic Constraints: The Robotic Hula-Hoop

A mechanical system consisting of a hoop and a pole is considered, for which the corresponding dynamic model represents an underactuated system subject to second-order nonholonomic constraints. The pursued goal is to simultaneously track a trajectory in the unactuated coordinates and to stabilize the actuated ones. For the model under consideration, the well-known noncollocated partial feedback linearization algorithm fails since the corresponding zero dynamics is unstable. In this work, we show that the actuated coordinates, i.e., the pole can be stabilized by exploiting the null space of the coupling inertia matrix without affecting the performance in the underactuated coordinates tracking. We present a formal mathematical analysis, which guarantees ultimate boundedness of all coordinates. Performed simulations bolster the proposed approach.

Quasi-Newton algorithm for optimal approximate linear regression design: Optimization in matrix space

Given a linear regression model and an experimental region for the independent variable, the problem of finding an optimal approximate design calls for minimizing a convex optimality criterion over a convex set of information matrices of feasible approximate designs. For numerical solution pure gradient methods are often used by design theorists, as vertex direction, vertex exchange, multiplicative algorithms, or combinations hereof. These methods have two major deficiencies: a slow convergence rate after a quick but rough approximation to the optimum, and often a large support of the obtained nearly optimal design. The latter feature is related to the fact that the methods optimize in the space of design measures which is usually of high or even infinite dimension, whereas the dimension of the information matrices is often small or moderate. For such situations a quasi-Newton method is revisited which was originally established by Gaffke & Heiligers (1996). In the present paper new possibilities of its application are demonstrated. The algorithm optimizes in matrix space. It shows a good global and an excellent local convergence behavior resulting in an accurate approximation of the optimum. A crucial subroutine solves convex quadratic minimization over the set of information matrices via repeated linear minimization over that set, providing thus the quasi-Newton step of the algorithm. This may also be of interest as a tool for computing an approximate design from a given information matrix and such that the support size of the design keeps Carathéodory’s bound. Illustrations are given for D- and I-optimality in particular multivariate random coefficient regression models and for T-optimal discriminating design in univariate polynomial models. Moreover, the behavior of the algorithm is tested for cases of larger dimensions: D- and I-optimal design for a third order polynomial model in several variables on a discretized cube.

L0 norm constraint based external control source allocation for the minimum cost control of directed networks

Locating a pre-given number of key nodes that are connected to external control sources so as to minimize the cost of controlling a directed network ẋ(t)=Ax(t)+Bu(t), known as the minimum cost control problem, is of critical importance. Considering a network consisting of N nodes with M external control sources, the state of art techniques employ iterative searching to determine the input matrix B that characterizes how nodes are connected to external control sources, in a matrix space RN×M. The nodes having M largest values of a defined importance index are selected as key nodes. However, such techniques may suffer from large performance penalty in some networks due to the diversity of real-life networks. To address this outstanding issue, we propose an iterative method, termed “L0-norm constraint based projected gradient method” (LPGM). We probabilistically search the input matrix in each iteration by restricting its L0 norm as a fixed value M, which implies that each control source is always only connected to a single key node during the whole searching process. Simulation results show that the solution always efficiently approaches a suboptimal key node set in a few iterations. These results provide a new point of view regarding the key nodes selection in the minimum cost control of directed networks.

Three-Dimensional Lattice Boltzmann Simulations of Single-Phase Permeability in Random Fractal Porous Media with Rough Pore–Solid Interface

Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions $$2.579 \le D_{\mathrm{m}} \le 2.893$$ characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension $$D_{\mathrm{s}}$$ and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing $$D_{\mathrm{s}}$$ from 1 to 1.1 regardless of $$D_{\mathrm{m}}$$ value, while k may slightly increase or decrease, depending on $$D_{\mathrm{m}}$$ , as $$D_{\mathrm{s}}$$ increases from 1.1 to 1.6.

Multiple Model Adaptive Attitude Control of LEO Satellite with Angular Velocity Constraints

In this paper, the multiple model adaptive control is utilized to improve the transient response of attitude control system for a rigid spacecraft. An adaptive output feedback control law is proposed for attitude control under angular velocity constraints and its almost global asymptotic stability is proved. The multiple model adaptive control approach is employed to counteract large uncertainty in parameter space of the inertia matrix. The nonlinear dynamics of a low earth orbit satellite is simulated and the proposed control algorithm is implemented. The reported results show the effectiveness of the suggested scheme.

Integrated Strategies of Backtracking Search Optimizer for Solving Reactive Power Dispatch Problem

This paper proposes backtracking search optimizer (BSO) for solving the reactive power dispatch (RPD) problem. The RPD problem is highly nonlinear, nonconvex optimization problem and is consisting of both continuous and discrete control variables. It aims to find the optimal settings of the generator voltages, tap positions of tap changing transformers, and the amount of reactive compensation that able to optimize the transmission power losses. BSO has simple structure and single control parameter. It has two new tuned mutation and crossover operators that control the amplitude of the search-direction matrix and search space boundaries. In this paper, five diversified generation strategies of mutation factor of BSO to solve the RPD. The proposed BSO strategies are applied to the IEEE standard 14-, 30-, 57-bus test systems, and a real power system at West Delta region system as a part of the unified Egyptian network. Simulation results show that the BSO strategy, which almost generates mutation factor concentrated on the mean value of the normal distribution (BSO 4) assigns multidirections through the search space and supports the diversity of populations. To show the effectiveness and superiority of proposed BSO, the obtained results are compared with different algorithms reported in the literature.

The Drazin inverse of an even-order tensor and its application to singular tensor equations

The notion of the Moore–Penrose inverses of matrices was recently extended from matrix space to even-order tensor space with Einstein product in the literature. In this paper, we further study the properties of even-order tensors with Einstein product. We define the index and characterize the invertibility of an even-order square tensor. We also extend the notion of the Drazin inverse of a square matrix to an even-order square tensor. An expression for the Drazin inverse through the core-nilpotent decomposition for a tensor of even-order is obtained. As an application, the Drazin inverse solution of the singular linear tensor equation A∗X=B will also be included.

Face recognition based on Volterra kernels direct discriminant analysis and effective feature classification

We present a novel face recognition method based on direct discriminant Volterra kernels and effective feature classification (DD-VK). One of the crucial steps involves dividing face images into patches and using the DD-VK method to extract the features of sub-image patches. DD-VK implements diagonalization to discard useless information in the null space of the inter-class scatter matrix and preserve important discriminant information in the null space of the intra-class scatter matrix. This method can simultaneously maximize inter-class distances and minimize intra-class distances in the feature space. We also introduce a novel classification scheme associated with the 2D Volterra kernel feature. Our scheme aggregates the classification information obtained from each column of the feature matrix in each image patch and uses a voting strategy to implement parent face image classification. This procedure can reduce the influence of local negative information. Experimental results show that the proposed method demonstrates good performance when dealing with conventional face recognition problems and exhibits strong robustness when dealing with block occlusion images.

Geometric target detection based on total Bregman divergence

This paper develops a geometric detection approach based upon the total Bregman divergence on the Riemannian manifold of Hermitian Positive-Definite (HPD) matrices to realize target detection in a clutter. First of all, the radar received clutter data in each range cell in one coherent processing interval is modeled and mapped into an HPD matrix space, which can be described as a complex Riemannian manifold. Each point of this manifold is an HPD matrix. Then, a class of total Bregman divergences are presented to measure the closeness between HPD matrices. Based on these divergences, the medians for a finite collection of HPD matrices are derived. Furthermore, the three divergences, namely the total square loss, the total log-determinant divergence, and the total von Neumann divergence are deduced, and their corresponding geometric detection methods are designed. The principle of detection is that if a location has enough dissimilarity from the median estimated by its neighboring locations, targets are supposed to appear at this location. Numerical experiments and real clutter data are given to demonstrate the effectiveness of the proposed geometric detection methods.

Null decomposition of trees

Let T be a tree. We show that the null space of the adjacency matrix of T has relevant information about the structure of T. We introduce the Null Decomposition of trees, which is a decomposition into two different types of trees: N-trees and S-trees. N-trees are the trees that have a unique maximum (perfect) matching. S-trees are the trees with a unique maximum independent set. We obtain formulas for the independence number and the matching number of a tree using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a tree are related to its null decomposition.

Asymptotic analysis of subspace-based data-driven residual for fault detection with uncertain reference

The local asymptotic approach is promising for vibration-based fault diagnosis when associated to a subspace-based residual function and efficient hypothesis testing tools. It has the ability of detecting small changes in some chosen system parameters. In the residual function, the left null space of the observability matrix associated to a reference model is confronted to the Hankel matrix of output covariances estimated from test data. When this left null space is not perfectly known from a model, it should be replaced by an estimate from data to avoid model errors in the residual computation. In this paper, the asymptotic distribution of the resulting data-driven residual is analyzed and its covariance is estimated, which includes also the covariance related to the reference null space estimate. The importance of including the covariance of the reference null space estimate is shown in a numerical study.

Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces

This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix ${A}{A}^T$ and the b...

Ensemble of Rotation Trees for Imbalanced Medical Datasets.

Medical datasets are often predominately composed of “normal” examples with only a small percentage of “abnormal” ones and how to correctly recognize the abnormal examples is very meaningful. However, conventional classification learning methods try to pursue high accuracy by assuming that the number of any class examples is similar to each other, which lead to the fact that the abnormal class examples are usually ignored and misclassified to normal ones. In this paper, we propose a simple but effective ensemble method called ensemble of rotation trees (ERT) to handle this problem in imbalanced medical datasets. ERT learns an ensemble through the following four stages: (1) undersampling subsets from normal class, (2) obtaining new balanced training sets through combining each subset and abnormal class, (3) inducing a rotation matrix on randomly sampling subset of each new balanced set, and in each rotation matrix space, (4) learning a decision tree on each balanced training data. Here, the rotation matrix is mainly to improve the diversity between ensemble members, and undersampling technique aims to improve the performance of learned models on abnormal class. Experimental results show that, compared with other state-of-the-art methods, ERT shows significantly better performance for imbalanced medical datasets.

Structural reanalysis via force method

We propose a novel approach for predicting forces in a structure after altering physical properties of some of its elements (new structure) using the solution of the structure before any change (parent structure). The force method for structural analysis is adopted throughout the paper. In order to obtain the forces in the new structure the method merely utilizes forces in the parent structure, diagonal/block diagonal flexibility matrix of the new structure and null space of the rectangular equilibrium matrix, which the latter only depends on structural topology and geometry. The forces in the new structure are obtained using an O(m2) scheme which performs iterative matrix to vector multiplications. A rigorous study of the convergence criterion and rate is also presented for our method. We comprehensively describe our method through three numerical examples to clarify the major steps of the process and to demonstrate its efficiency and viability.

Global FOM and GMRES algorithms for a class of complex matrix equations

In this paper, a generalized global Arnoldi process is given to produce an F -orthonormal basis of complex matrix space. Then we propose the global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method for solving a class of complex matrix equations. The new methods can be implemented by the original coefficient matrices. By using Schur complement, we give the expressions of the approximate solutions and the corresponding residuals. Some convergence results of the proposed methods are also derived. Finally, numerical examples are given to illustrate the effectiveness with comparison to other existing methods.