Norm Of Vector Research Articles

A modularity-maximization-based approach for detecting multi-communities in social networks

The modularity is a widely-used objective function to determine communities from a given network. The leading eigenvector method is a popular solution that applies the first eigenvector to determine the communities. The low computation cost is the major advantage of the leading eigenvector method. However, the leading eigenvector method only can split a network into two communities. To detect multiple communities, the modularity maximization is transformed to the vector partition problem (VPP). We propose an algorithm which is called as the partition at polar coordinate protocol (PPCP) to solve the VPP problem. The goal of PPCP is to find non-overlapping vertex vector sets so as to maximize the quadratic sum of the norms of community vectors. The proposed PPCP has two steps to determine the communities that are the network structure analysis and the community determination. During the network structure analysis, we obtain following issues. First, the vertex vectors belong to different communities can be separated by the distribution angles. Second, a node with a higher degree corresponds to a vertex vector with a larger norm. So, we propose three refinement functions including the noise reduction, the common-friends model and the strong connectivity hypothesis to improve the accuracy of PPCP. In our simulations, PPCP detects communities more precisely than Fine-tuned algorithm especially in the network with the weak structure. Moreover, the proposed refinement functions can capture the special properties of the network. So, PPCP with refinement functions performs much better than Fine-tuned algorithm and PPCP without refinement functions in terms of the accuracy in detecting communities.

On the Construction of Regularizing Algorithms for the Correction of Improper Convex Programming Problems

We consider convex programming problems with a possibly inconsistent constraint system. Such problems constitute an important class of improper models of convex optimization and often arise in the mathematical modeling of real-life operations research statements. Since improper problems arise rather frequently, the theory and methods of their numerical approximation (correction) should be developed, which would allow to design objective procedures that resolve inconsistent constraints, turn an improper model into a family of feasible problems, and choose an optimal correction among them. In the present paper, an approximating problem is constructed by the variation of the right-hand sides of the constraints with respect to the minimum of some vector norm. The type of the norm defines the form of a penalty function, and the minimization of the penalty function together with a stabilizing term is the core of each specific method of optimal correction of improper problems. The Euclidean norm implies the application of a quadratic penalty, whereas a piecewise linear (Chebyshev or octahedral) norm is concerned with the use of an exact penalty function. The proposed algorithms may also be interpreted as (Tikhonov) regularization methods for convex programming problems with inaccurate input information. Convergence conditions are formulated for the methods under consideration and convergence bounds are established.

Design of Linear Precoders for Correlated Sources in MIMO Multiple Access Channels

This paper focuses on distributed linear precoding when users transmit correlated information over a fading multiple-input and multiple-output multiple access channel. The precoders are optimized in order to minimize the sum-mean square error (MSE) between the source and the estimated symbols. When sources are correlated, minimizing the sum-MSE results in a non-convex optimization problem. The precoders for an arbitrary number of users and transmit and receive antennas are thus obtained via a projected steepest-descent algorithm and a low-complexity heuristic approach. For the more restrictive case of two single-antenna users, a closed-form expression for the minimum sum-MSE precoders is derived. Moreover, for the scenario with a single receive antenna and any number of users, a solution is obtained by means of a semi-definite relaxation. Finally, we also consider precoding schemes where the precoders are decomposed into complex scalars and unit norm vectors. Simulation results show a significant improvement when source correlation is exploited at precoding, especially for low signal-to-noise ratios and when the number of receive antennas is lower than the number of transmitting nodes.

Atomic Polarizations, Not Charges, Determine CH Out-of-Plane Bending Intensities of Benzene Molecules.

Infrared gas phase intensities are reported for the first time for 23 CH out-of-plane bending vibrations of eight substituted benzene molecules and naphthalene by integration of bands from the Pacific Northwest National Laboratory (PNNL) spectral library. These experimental values are found to have an rms difference of 8.7 km mol-1 with the B3LYP/6-311++G(d,p) values for intensities ranging from close to zero to 126.7 km mol-1. These intensities are found to have transferable electronic structure parameters, and their square roots are proportional to the amplitudes of the hydrogen atom displacements perpendicular to the benzene ring. Quantum Theory of Atom in Molecules (QTAIM)-Charge-Charge Transfer-Dipolar Polarization models were determined from the B3LYP/6-311++G(d,p) electronic densities. By far, the largest electronic contribution to these intensities is the dipolar polarization of the carbon atom of the displaced CH bond, 0.214 e. Smaller contributions are found for the polarizations of the displaced hydrogen atoms (-0.043 e) and nearest neighbor carbon atoms (-0.052 e), both having directions opposite to that of the carbon atom polarization of the displaced CH bond. The movements of static equilibrium hydrogen charges make the smallest contribution canceling most of the hydrogen polarization changes. In fact, the carbon atomic polarizations alone account for 96.9% of the dipole moment derivative vector norm for the CH out-of-plane bends. The polarization model is also found to be valid for seven CH out-of-plane bending vibrations of N-fused benzene ring molecules (N = 3, 4, 5).

Hyperstability of a logarithmic-type functional equation on restricted domains

Let X be a real normed vector space and f:(0,∞)→X. In this paper, we prove the hyperstability of the logarithmic functional equation fxy−yf(x)=0on Γ of Lebesgue measure zero. More precisely, we prove that if f:(0,∞)→X satisfies ‖fxy−yf(x)‖≤ϕ(x,y)for all (x,y)∈Γ+⊂{(x,y):y>α(x)}[resp.Γ−⊂{(x,y):0<y<α(x)}] of Lebesgue measure zero, where α:(0,∞)→(0,∞) is an arbitrary given function and ϕ:(0,∞)×(0,∞)→[0,∞) satisfies the condition ϕ(x,y)∕y→0 as y→∞ [resp. y→0], then f satisfies the functional equation f(xy)=yf(x) for all x>0 andy>0.

Bilateral structure based matrix regression classification for face recognition

Robust sparse representation is a well-known method in computer vision. Many sparse representation models have been proposed and perform well in face recognition. Most of them use transformed images of one dimensional vector, and such implementation ignores structural information between features. To make use of this structural information, this paper presents a novel model for face recognition. Unlike traditional sparse regression measuring differences between test sample and predicted response by vector norm, our model uses matrix norm, l2,1, to calculate residual. Instead of treating each pixel independently, the residual of a pixel is connected with all other pixels from the same line and column by means of double direction l2,1-norm. Moreover, we incorporate the specific class information into the representation model by introducing group-sparse regularizer. And then, we use the alternating direction method of multipliers approach to optimize proposed model. The extensive experiments have been conducted to evaluate the proposed methods by comparing with many sparse representation based techniques.

Combined-Step-Size Normalized Subband Adaptive Filter With a Variable-Parametric Step-Size Scaler Against Impulsive Interferences

The variable step-size normalized subband adaptive filter (NSAF) with a fixed parametric step-size scaler (SSS) results in a tradeoff between the fast convergence rate and small steady-state error, and needs to reset the algorithm to improve the tracking in abrupt change scenarios. Thus, a combined-step-size NSAF with a variable-parametric SSS (VPSSS) is proposed in this brief to address these problems. In the proposed algorithm, an adaptive parameter with respect to the smallest ${L} _{\mathbf{1}}$ -norm of the subband error vectors in a window is proposed for the VPSSS and then two different step sizes are adaptively combined by a modified sigmoidal activation function, in which this modified sigmoidal activation function is updated by using the gradient descent method to minimize the ${L} _{\mathbf{1}}$ -norm of the subband error vector. Simulation results have verified that the proposed algorithm yields improved transient behavior and tracking performance as compared to the compared algorithms.

Coefficient Groups Inducing Nonbranched Optimal Transport

In this work we consider an optimal transport problem with coefficients in a normed Abelian group $G$, and extract a purely intrinsic condition on $G$ that guarantees that the optimal transport (or the corresponding minimum filling) is not branching. The condition turns out to be equivalent to the nonbranching of minimum fillings in geodesic metric spaces. We completely characterize finitely generated normed groups and finite-dimensional normed vector spaces of coefficients that induce nonbranching optimal transport plans. We also provide a complete classification of normed groups for which the optimal transport plans, besides being nonbranching, have acyclic support. This seems to initiate a new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge-Kantorovich duality.

Generalized sensitivity indices based on vector projection for multivariate output

Analyzing the sensitivity of model outputs to inputs is important to assess risk and make decisions in engineering application. However, for model with multiple outputs, it is difficult to interpret the sensitivity index since the effect of the dimension and the correlation between multiple outputs are often ignored in the existing methods. In this paper, a new kind of sensitivity analysis method is proposed by use of vector projection and dimension normalization for multiple outputs. Through the dimension normalization, the space of multiple outputs can be unified into a dimensionless one to eliminate the effect of the dimension of the different output. After an affine coordinate system is constructed by considering the correlation of the multiple normalized outputs, a total variance vector for the multiple outputs can be composed by the individual variance of each output. Then, by projecting the variance contribution vector composed by the individual variance contribution of the input to each output on the total variance vector, the new sensitivity indices are proposed for measuring the comprehensive effect of the input on the total variance vector of multiple outputs, it is defined as the ratio of the projection of the variance contribution vector to the norm of the total variance vector. We derive that the Sobol’ indices for a scalar output and the covariance decomposition based indices for multiple outputs are special cases of the proposed vector projection based indices. Then, the mathematical properties and geometric interpretation of the proposed method are discussed. Three numerical examples and a rotating shaft model of an aircraft wing are used to validate the proposed method and show their potential benefits.

Face Hallucination with Weighted Nuclear Norm Constraint

Face hallucination via patch-pairs leaning based methods has been wildly used in the past several years. Some position-patch based face hallucination methods have been proposed to improve the representation power of image patch and obtain the optimal regressive weighted vector. The rationale behind the position-patch based face hallucination is the fact that human face is always highly structured and consequently positioned and it plays an increasingly important role in the reconstruction. However, in the existing position-patch based methods, the probe image patch is usually represented as a linear combination of the corresponding patches of some training images, and the reconstruction residual is usually measured using the vector norm such as 1-norm and 2-norm. Since the vector norms neglect two-dimensional structures inside the residual, the final reconstruction performance is not very satisfactory. To cope with this problem, we present a weighted nuclear-norm constrained sparse coding (WNCSC) model for position-patch based face hallucination. In addition, an efficient algorithm for the WNCSC is developed using the alternating direction method of multipliers (ADMM) and the method of augmented Lagrange multipliers (ALM). The advantages of the proposed model are twofold: in order to fully make use of low-rank structure information of the reconstruction residual, the weighted nuclear norm is applied to measure the residual matrix, which is able to alleviate the bias between input patches and training data, and it is more robust than the Euclidean distance (2-norm); the more flexible selection method for rank components can determine the optimal combination weights and adaptively choose the relevant and nearest hallucinated neighbors. Finally, experimental results prove that the proposed method outperforms the related state-of-the-art methods in both quantitative and visual comparisons.

The kinetic theory of the mesoscopic spacetime

At the mesoscopic scales — which interpolate between the macroscopic, classical, geometry and the microscopic, quantum, structure of spacetime — one can identify the density of states of the geometry, which arises from the existence of a zero-point length in the spacetime. This spacetime discreteness also associates an internal degree of freedom with each event, in the form of a fluctuating vector of constant norm. The equilibrium state, corresponding to the extremum of the total density of states of geometry plus matter, leads precisely to Einstein’s equations. In fact, the field equation can now be reinterpreted as a zero-heat-dissipation principle. The analysis of fluctuations around the equilibrium state (which is described by Einstein’s equations), will provide new insights about the quantum gravity.

Adaptive fuzzy decentralized control for a class of MIMO large-scale nonlinear state delay systems with unmodeled dynamics subject to unknown input saturation and infinite number of actuator failures

This paper addresses design of an adaptive fuzzy decentralized fault-tolerant controller for a class of uncertain multi-input multi-output (MIMO) large-scale nonlinear systems with unmodeled dynamics subject to unknown state time-varying delay, external disturbances, unknown input saturation and actuator faults. It is shown that the proposed fault-tolerant control (FTC) scheme can handle infinite number of actuator failures including partial and total loss of effectiveness. System uncertainties have been approximated by the fuzzy logic systems (FLSs). To cope with the unknown state time-varying delay, the Razumikhin lemma has been utilized and unmodeled dynamics has been tackled by introducing a dynamical signal. To avoid high computational burden, the dynamic surface control (DSC) technique has been used and the number of adaption laws has been reduced by updating the maximum norm of fuzzy weight vectors of each subsystem. It has been shown that all closed-loop signals are semi-globally uniformly ultimately bounded (SGUUB) and the output tracking errors converge to a small neighborhood of the origin by choosing the design parameters appropriately. Simulation results demonstrate the effectiveness of the proposed control scheme.

The fate of black hole horizons in semiclassical gravity

The presence of a horizon is the principal marker for black holes as they appear in the classical theory of gravity. In General Relativity (GR), horizons have several defining properties. First, there exists a static spherically symmetric solution to vacuum Einstein equations which possesses a horizon defined as a null-surface on which the time-like Killing vector becomes null. Second, in GR, a co-dimension two sphere of minimal area is necessarily a horizon. On a quantum level, the classical gravitational action is supplemented by the quantum effective action obtained by integrating out the quantum fields propagating on a classical background. In this note we consider the case when the quantum fields are conformal and perform a certain non-perturbative analysis of the semiclassical equations obtained by varying the complete gravitational action. We show that, for these equations, both of the above aspects do not hold. More precisely, we prove that i) a static spherically symmetric metric that would describe a horizon with a finite Hawking temperature is, generically, not a solution; ii) a minimal 2-sphere is not a horizon but a tiny throat of a wormhole. We find certain bounds on the norm of the Killing vector at the throat and show that it is, while non-zero, an exponentially small function of the Bekenstein–Hawking (BH) entropy of the classical black hole. We also find that the possible temperature of the semiclassical geometry is exponentially small for large black holes. These findings suggest that a black hole in the classical theory can be viewed as a certain (singular) limit of the semiclassical wormhole geometry. We discuss the possible implications of our results.

Calibration and Alignment of Tri-Axial Magnetometers for Attitude Determination

Ellipsoid fitting method is frequently used to calibrate tri-axial magnetometer in navigation systems, but it cannot ensure the alignment between the sensor triad and the coordinate system of the host platform. This paper presents a novel calibration method to overcome this shortcoming. When the sensor triad is rotated around a selected axis, the projection of geomagnetic vector onto this axis should be constant. On the other hand, the angle between geomagnetic vector and gravity vector is also constant. These two phenomena can be described by inner product mathematically. Either of these two constant inner products can be utilized with the constant norm of geomagnetic vector to form the proposed dual inner product method, in order to solve the 12-parameter error model of tri-axial magnetometer. Simulations and experiments prove the effectiveness and robustness of the proposed method. It can achieve reference-free calibration and alignment of tri-axial magnetometer, and help to enhance the accuracy of attitude determination based on geomagnetism.

Adaptive Neural Control of Hypersonic Vehicles with Actuator Constraints

An adaptive neural control method is proposed in this paper for the flexible air-breathing hypersonic vehicle (AHV) with constraints on actuators. This scheme firstly converts the original control problem with input constraints into a new control problem without input constraints based on the control input saturation function. Secondly, on the basis of the implicit function theorem, the radial basis function neural network (RBFNN) is introduced to approximate the uncertain items of the model. And the minimal-learning-parameter (MLP) technique is adopted to design the adaptive law for the norm of network weight vector, which significantly reduces calculations. Meanwhile, the finite-time convergence differentiator (FD) is introduced, through which the model state variables and their derivatives are accurately estimated to ensure the control effect. Finally, it is theoretically proved that the closed-loop control system is stable. And the effectiveness of the designed controller is verified by simulation.

An iterative method for optimal feedback control and generalized HJB equation

In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman &#x0028 GHJB &#x0029 equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems.

On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming

Within the context of sequential methods for solving general nonlinear programming problems, and on the grounds of a previous work of the same authors, this study deals with the theoretical reasoning behind handling the original subproblems by an augmentation strategy. We do not assume feasibility of the original problem, nor the fulfillment of any constraint qualification. The previous analysis is extended along two directions. First and foremost, the exact nature of the stationary points previously considered is alleviated under an approximate stationary perspective. Second, the current analysis has been developed using general vector norms. Therefore, despite the similarities of the obtained results with those of the prior study, the present ones have been obtained under less restrictive hypotheses, and with a more involved examination. As before, we are not concerned with the sequential method itself, nor with computational results. We focus on the features of the original problem that can be inferred from the properties of the solution of the augmented problem, with the solutions being now analyzed in an approximate sense. Examples illustrating the obtained results are included.

Architectural Optimizations for a High-Throughput Sorted QR Decomposition Circuit in MIMO Communication Systems

This paper presents the VLSI architecture of a low-latency and high-throughput sorted-QR decomposition (SQRD) engine for multiple-input multiple-output (MIMO) communication systems. In order to achieve a high processing throughput, the proposed design is architected based on a novel pipelined Givens rotation (GR) structure comprising of multi-dimension COordinate rotation DIgital computer (CORDIC) (MD-CORDIC) processing elements (PEs). Moreover, this design delivers the vector norm and conducts the sorting operation as a by-product of the vectoring operation on the execution flow of the CORDIC process. Therefore, excessive overheads for norm-calculation and sorting are excluded, and thus the latency is greatly reduced and throughput is enhanced. In addition, the proposed SQRD engine is operating directly on the complex-valued channel matrix to avoid the matrix augmentation caused by the real-valued decomposition of the channel matrix. This design has been synthesized, placed and routed, and the post-layout estimation results have shown that the processing throughput of the proposed SQRD architecture achieves an approximately 2[Formula: see text] improvement compared to the prior arts.

Command filtered robust adaptive NN control for a class of uncertain strict-feedback nonlinear systems under input saturation

This paper develops a robust adaptive neural network (NN) tracking control scheme for a class of strict-feedback nonlinear systems with unknown nonlinearities and unknown external disturbances under input saturation. The radial basis function NNs with minimal learning parameter (MLP) are employed to online approximate the uncertain system dynamics. The adaptive laws are designed to online update the upper bound of the norm of ideal NN weight vectors, and the sum of the bounds of NN approximation errors and external disturbances, respectively. An auxiliary dynamic system is constructed to generate the augmented error signals which are used to modify the adaptive laws for preventing the destructive action due to the input saturation. Moreover, the command filtering backstepping control method is utilized to overcome the shortcoming of dynamic surface control method, the tracking-differentiator-based control method, etc. Our proposed scheme is qualified for simultaneously dealing with the input saturation effect, the heavy computational burden and the “explosion of complexity” problems. Theoretical analysis illuminates that our scheme ensures the boundedness of all signals in the closed-loop systems. Simulation results on two examples verify the effectiveness of our developed control scheme.

A new Wronskian change detection model based codebook background subtraction for visual surveillance applications

Background subtraction (BS) is a popular approach for detecting moving objects in video sequences for visual surveillance applications. In this paper, a new multi-channel and multi-resolution Wronskian change detection model (MCMRWM) based codebook background subtraction is proposed for moving object detection in the presence of dynamic background conditio ns. In the prooed MCMRWM, the multi-channel information helps to reduce the false negative of the foreground object; and the multi-resolution data suppresses the background noise resulting in reduced false positives. The proposed algorithm considers the ratio between feature vectors of current frame to the background model or its reciprocal in an adaptive manner, depending on the l2 norm of the feature vector, which helps to detect the foreground object completely without any false negatives. Extensive experiments are carried out with challenging video sequences to show the efficacy of the proposed algorithm against state-of-the-art BS techniques.