Rank-one Modification Research Articles

The assignment of zero sound pressure frequencies using measured sound pressure receptances and structural receptances

In structural receptances, the zeros (antiresonances) define those frequencies at which vibrations disappear. In this paper, the zero sound pressure frequency is defined as the frequency at which the sound pressure is zero at certain locations. A method for the assignment of zero sound pressure frequencies using measured sound pressure receptances and structural receptances is proposed through two forms of structural modifications: rank-one modification and higher-rank modification. In rank-one modification, the sound pressure receptances of the modified structure are obtained by using the structural receptances with the Sherman-Morrison formula. In higher-rank modification, the modifications are determined by an analysis of the null space of a matrix consisting of structural receptances and sound pressure receptances. This method requires only a few structural receptances and sound pressure receptances, and no structural models or acoustic models are needed. Numerical examples involving a baffled plate vibrating in air and a fluid-loaded finite cylindrical shell are used to demonstrate the methods. Numerical results show that the zero sound pressure frequencies can be assigned to specific frequencies by structural modification.

Note on a rank-one modification of the singular value decomposition

In this paper, we investigate the singular value decomposition (SVD) of Σ+xyH, where Σ is an m×n real diagonal matrix, x∈Cm, and y∈Cn. We start by briefly revisiting an existing approach for determining the desired SVD by sequentially computing the eigendecomposition of two separate hermitian rank-one modifications of a real diagonal matrix. Then we introduce the notion of the rank-two secular functionwhose roots are the singular values of Σ+xyH and exploit its properties to bound each root/singular value in disjoint intervals. Once the singular values are computed, we demonstrate how to directly compute the full set of associated left/right singular vectors ultimately giving us a new method for computing the SVD of Σ+xyH in O(min(m,n)2) time.

A New Robust Subspace Recovery Algorithm (Student Abstract)

A common task in data analysis is to compute an approximate embedding of the data in a low dimensional subspace. This is used, for example, for dimensionality reduction. Robust Subspace Recovery computes the embedding by ignoring a fraction of the data considered as outliers. Its performance can be evaluated by how accurate the inliers (non-outliers) are represented. We propose a new algorithm that outperforms the current state of the art when the data is dominated by outliers. The main idea is to rank each point by evaluating the change in the global PCA error when that point is considered as an outlier. We show that this lookahead procedure can be implemented efficiently by centered rank-one modifications.

An augmented memoryless BFGS method based on a modified secant equation with application to compressed sensing

Making a rank-one modification on the classical BFGS (Broyden–Fletcher–Goldfarb–Shanno) updating formula, we develop a class of augmented BFGS methods. The suggested formula can be considered as a hybridization of the basic BFGS updating formula for Hessian with an additional rank-one term embedded to guarantee a general modified secant equation. By using the well-known Sherman–Morrison formula, the inverse of a memoryless version of the given updating formula is computed to be applied for solving large-scale problems. Convergence analysis is concisely carried out as well. At last, the practical merits of the method are investigated by numerical tests on a set of CUTEr problems as well as the well-known compressed sensing problem. Results show the computational efficiency of the given method.

Optimal Feature Manipulation Attacks Against Linear Regression

In this paper, we investigate how to manipulate the coefficients obtained via linear regression by adding carefully designed poisoning data points to the dataset or modify the original data points. Given the energy budget, we first provide the closed-form solution of the optimal poisoning data point when our target is modifying one designated regression coefficient. We then extend the analysis to the more challenging scenario where the attacker aims to change one particular regression coefficient while making others to be changed as small as possible. For this scenario, we introduce a semidefinite relaxation method to design the best attack scheme. Finally, we study a more powerful adversary who can perform a rank-one modification on the feature matrix. We propose an alternating optimization method to find the optimal rank-one modification matrix. Numerical examples are provided to illustrate the analytical results obtained in this paper.

A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications

A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications

Design of rank-one modification feedback controllers for second-order systems with time delay using frequency response methods

Abstract In this note, a novel approach to design rank-one – single input – feedback controllers for second-order systems with time delay is presented. The versatile, well-known receptance modeling is combined with classical frequency response methods of control design, and the feedback gains are computed in order to achieve a given stability margin based on sensitivity specifications. The systems under study are assumed to be stable in open-loop. A heuristic optimization technique, namely, genetic algorithm is employed to impose a minimal distance from the Nyquist plot of the controlled system to the critical point, guaranteeing thereby a predefined margin of robust stability. In addition to the well known advantages of the receptance method in obtaining experimental models, the proposed design approach brings the possibility of considering time delay without approximations, by using frequency response calculations, avoiding thereby the drawbacks inherent to the a posteriori stability verification when pole assignment techniques are used. Numerical examples help to enlighten the merits of the proposed approach.

A method for recovering Jacobi matrices with mixed spectral data

In this paper we employ the Euclidean division for polynomials to recover uniquely a Jacobi matrix in terms of the mixed spectral data consisting of its partial entries and the information given on its full spectrum together with a subset of eigenvalues of its truncated matrix obtained by deleting the last row and last column, or its rank-one modification matrix modified by adding a constant to the last element. A necessary and sufficient condition is provided for the existence of the inverse problem. A numerical algorithm and a numerical example are given.

An efficient estimation of distribution algorithm with rank-one modification and population reduction

As a type of model-based metaheuristic, estimation of distribution algorithms (EDAs) show certain advantages over other metaheuristics by using statistical learning method to estimate the distribution of promising solutions. However, the commonly-used Gaussian EDAs (GEDAs) usually suffer from premature convergence that severely limits their efficiency. In this paper, we first attempt to enhance the performance of GEDA by improving its model estimation method. The new estimation method shifts the weighted mean of high-quality solutions towards the fitness improvement direction and estimates the covariance matrix by taking the shifted mean as the center. Theoretical analyses show that the new covariance matrix is essentially a rank-one modification (R1M) of the original one. It could effectively adjust both the search scope and the search direction of GEDA, and thus improving the search efficiency. Furthermore, considering the importance of the population size tuning in GEDA, we develop a population reduction (PR) strategy which linearly reduces the population size throughout the evolution. By this means, the exploration and exploitation ability of GEDA could be balanced better in different search stages and a more proper utilization of limited computation resource can be achieved. Combining GEDA with the R1M and PR strategies, a novel EDA variant named EDA-R1M-PR is developed. The performance of EDA-R1M-PR was comprehensively evaluated and compared with that of several state-of-the-art evolutionary algorithms. Experimental results indicate that the R1M and PR strategies effectively enhance the global optimization ability of GEDA and the resultant EDA-R1M-PR significantly outperforms its competitors on a set of benchmark functions.

An upper [formula omitted]-Hessenberg reduction of a matrix through symplectic Householder transformations

In this paper, we introduce a reduction of a matrix to a condensed form, the upper J-Hessenberg form, via elementary symplectic Householder transformations, which are rank-one modification of the identity. Features of the reduction are highlighted and a general algorithm is derived. Then, we study different possibilities to specify the general algorithm in order to built better versions. We are led to two variants numerically more stables that we compare to JHESS algorithm. Also, some numerical experiments for comparing the different algorithms are given.

Recursive Slow Feature Analysis for Adaptive Monitoring of Industrial Processes

Recently, a new process monitoring and fault diagnosis method based on slow feature analysis has been developed, which enables concurrent monitoring of both operating point and process dynamics. In this paper, a recursive slow feature analysis algorithm for adaptive process monitoring is put forward to accommodate time-varying processes by updating model parameters and monitoring statistics once a new sample arrives. An important algebraic property of slow feature analysis is first established. We then show that such a property can be violated by online updating with a forgetting factor used, and a remedy is suggested. A novel algorithm based on the rank-one modification and the orthogonal iteration procedure is proposed to recursively adjust the solution to the generalized eigenvalue problem, model parameters, and associated monitoring statistics in a cost-efficient way. In addition, an improved stopping criterion for model updating is proposed based on the statistics relevant to process dynamics, which yields an intelligent maintenance mechanism of monitoring systems. The efficacy of the proposed method is finally evaluated on a real crude heating furnace system.

Adaptive Lattice Filters for Systems of Space-Time Processing of Non-Stationary Gaussian Processes

Adaptive systems protecting pulse radars from non-stationary in time (range) clutter echoes are usually tuned using training vectors composed of complex amplitudes of input signals and comprising a finite-length “sliding window” of data. From any current range gate to a subsequent one, a training sample is partially updated (or modified) by means of excluding the “old” training vectors (correspond to the current range gate) and including the “new” ones (correspond to the next range gate). As a consequence, respective estimates of adaptive system parameters are corrected according to a modified sample correlation matrix (CM), which is typically a sum of an initialCMand a modifying matrix of rank K ≥ 1. In this case it is possible to avoid re-computing these parameters based on a new training sample of full size and, instead of this, we correct them in an “economical” way employing K-rank modification of a matrix inverse to the CM estimate. This paper is devoted to comparative analysis of various (K ≥ 1)-rank modification algorithms that correct the parameters of adaptive lattice filters (ALF). Main attention is paid to synthesis as well as theoretical and experimental study of algorithms of direct (K > 1)-rank modification of the ALF parameters. These algorithms attain the said objective omitting the K-fold application of known rank-one (K = 1) modification algorithms. We also synthesize a combined algorithm (CA) of (K ≥ 1)-rank modification of the ALF parameters that is more computationally simple and more numerically robust compared to known algorithms. The ALF employing the CA can serve as an effective tool for solving various tasks of space-time adaptive signal processing in pulse radars of different purpose.

Efficient line matching with homography

In this paper, we propose a novel approach to line matching based on homography. The basic idea is to use cheaply obtainable matched points to boost the similarity between two images. Two types of homography method, which are estimated by direct linear transformation, transform images and extract their similar parts, laying a foundation for the use of optical flow tracking. The merit of the similarity is that rapid matching can be achieved by regionalizing line segments and local searching. For multiple homography estimation that can perform better than one global homography, we introduced the rank-one modification method of singular value decomposition to reduce the computation cost. The proposed approach results in point-to-point matches, which can be utilized with state-of-the-art point-match-based structures from motion (SfM) frameworks seamlessly. The outstanding performance and feasible robustness of our approach are demonstrated in this paper.

Recursive Spectral Meta-Learner for Online Combining Different Fault Classifiers

This paper considers the problem of fault classification when different fault classifiers are performed simultaneously. Based on spectral meta-learner (SML) proposed by Parisi et al. , its recursive version, i.e., recursive SML (RSML) is developed for online combining the potentially conflicting classification information. Considering different statistical properties of faults occurring at different time intervals, the binary classification information is recursively utilized. By introducing a forgetting factor, the leading eigenvector of the estimate of the time-varying covariance matrix is used as the weight vector for each classifier. Rank-one modification is then used for calculating the eigenvector in order to reduce the online computational complexity. The performance of RSML is strictly analyzed in a statistical sense, including the effect of recursive calculation and conditional dependence of different classifiers on the weight vector. Compared with majority voting and SML, a higher balanced accuracy of RSML can be verified by the benchmark Tennessee Eastman process.

Recursive transformed component statistical analysis for incipient fault detection

This paper presents a new data-driven process monitoring method called recursive transformed component statistical analysis (RTCSA) for the purpose of incipient fault detection. Without space partition, RTCSA processes data in sliding windows to obtain orthogonal transformed components (TCs) recursively using rank-one modification. The statistical information of TCs can reveal some important process features, implying that faults can be detected by monitoring the statistics of TCs. With second-order statistics, the detection index reduces to relative changes of ordered eigenvalues of the sample covariance matrix. Fault detectability is analyzed in a statistical sense, leading to the analysis of the eigenvalues of stochastic matrices, including the closed-form expressions for the probability distribution function of the arbitrary lth largest eigenvalue of a class of real uncorrelated Wishart matrices. It indicates that a scaled ordered eigenvalue is sensitive to small changes. The structure of the detection index ensures that RTCSA is sensitive to incipient faults. Compared with existing multivariate statistical process monitoring approaches such as principal component analysis (PCA) and its variants, the superior detectability of RTCSA is illustrated by a numerical example and the Tennessee Eastman process.

A Novel Noise Jamming Detection Algorithm for Radar Applications

In this letter, we devise and assess a new algorithm to detect a platform equipped with self-screening jamming systems. The latter illuminates the victim radar by means of noise-like signals leading to an increase of the constant false alarm rate threshold (at the detection stage) and a reduction of radar sensitivity. The problem is formulated in terms of a binary hypothesis test exploiting the rank-one modification of the interference covariance matrix introduced by the jammer and the generalized likelihood ratio test is derived. Performance analysis is conducted on simulated data and is aimed at highlighting the effectiveness of this new approach.

On Rényi Entropy Power Inequalities

This paper gives improved Renyi entropy power inequalities (R-EPIs). Consider a sum $S_{n} = \sum _{k=1}^{n} X_{k}$ of $n$ independent continuous random vectors taking values on $ {\mathbb {R}}^{d}$ , and let $\alpha \in [1, \infty ]$ . An R-EPI provides a lower bound on the order- $\alpha $ Renyi entropy power of $S_{n}$ that, up to a multiplicative constant (which may depend in general on $n, \alpha , d$ ), is equal to the sum of the order- $\alpha $ Renyi entropy powers of the $n$ random vectors $\{X_{k}\}_{k=1}^{n}$ . For $\alpha =1$ , the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov, which relies on the sharpened Young’s inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.

Descent Symmetrization of the Dai–Liao Conjugate Gradient Method

Symmetrizing the Dai–Liao (DL) search direction matrix by a rank-one modification, we propose a one-parameter class of nonlinear conjugate gradient (CG) methods which includes the memoryless Broyden–Fletcher–Goldfarb–Shanno (MLBFGS) quasi-Newton updating formula. Then, conducting an eigenvalue analysis, we suggest two choices for the parameter of the proposed class of CG methods which simultaneously guarantee the descent property and well-conditioning of the search direction matrix. A global convergence analysis is made for uniformly convex objective functions. Computational experiments are done on a set of unconstrained optimization test problems of the CUTEr collection. Results of numerical comparisons made by the Dolan–Moré performance profile show that proper choices for the mentioned parameter may lead to promising computational performances.

Recursive minimum component algorithms for parameter estimation of dynamic systems

Minimum component analysis is an elegant and promising approach to system identification. This paper focuses on developing related recursive algorithms. Two algorithms are presented. In the first approach, rank-one modification is used to get an accurate and quickly convergent algorithm. However, its drawback is low computational efficiency. Therefore, the second algorithm is proposed to realize faster computation. The simulation results verify the effectiveness of both algorithms.

Рекуррентные алгоритмы настройки адаптивных решетчатых фильтров

The authors analyze the algorithms intended for correction of adaptive lattice filters (ALF) parameters under K-rank (K i 1) modification of estimate correlation matrix within a “sliding” over the time (range) data window. The drawbacks of methods that correct the ALF parameters based on K-fold utilization of known algorithms of rank-one (K = 1) modification are discussed. The combined algorithm (CA) of K-rank (K i 1) modification is synthesized. Under considered conditions, the only one-fold utilization of the CA solves the task of ALF parameters correction. The paper demonstrates, that proposed CA reduces the computational complexity and enhances the numerical stability of procedure of ALF parameters correction as compared with the competing methods based on algorithms of rank-one modification.