Multiplicative Perturbation Research Articles

Stability of stochastic difference equations with multiplicative and additive fading noise

In this paper, the effects of multiplicative and additive fading stochastic perturbations on the solutions of stochastic difference equations are studied, in which the strength of fading stochastic perturbations is square summable. Different from the existing research, this paper mainly studies the influence of two kinds of stochastic perturbations on the stability of the system, and applies the Lyapunov functional method, the asymptotic mean square trivial sufficiency conditions are obtained for the solutions of stochastic difference equations which based on the influence of multiplicative and additive fading noises. At the same time, the conditions of stochastic perturbation fading rate are obtained. Finally, the above conditions are verified by numerical simulation examples, and the results of numerical simulation show that: (1) the additive perturbations have less influence on the system than the multiplicative perturbations; (2) the fading stochastic perturbations can make the system reach a stable state in a faster time.

Dynamics of a rank-one multiplicative perturbation of a unitary matrix

We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on [Formula: see text] that hold for a variety of unitary random matrix models.

On rank one and Weyl–von Neumann theorem for multiplicative perturbations of unitary operators

For multiplicative perturbations of unitary operators, it is presented a version of Weyl–von Neumann theorem and a sufficient conditions for generic (in the intensity parameter) singular continuous spectrum under unitary rank one perturbations.

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.

On multiplicative time-dependent perturbations of semigroups and cosine families generators

On multiplicative time-dependent perturbations of semigroups and cosine families generators

A New Treatment of Multiplicative Perturbation of Black-Scholes Operator

A New Treatment of Multiplicative Perturbation of Black-Scholes Operator

Exploring Privacy-Preserving Methods via Perturbation Data Mining Employing Diverse Noise Strategies

Knowledge discovery from data, commonly referred to as data mining. it involves the extraction of significant information, which may be previously unknown, concealed, or relevant, from extensive data sets or databases through the utilization of statistical methodologies. With the introduction of enhanced hardware technologies, there has been a proliferation in the storage and recording of personal data pertaining to individuals. Sophisticated organizations employ data mining algorithms to uncover hidden patterns or insights within data. Data mining techniques find application in diverse fields such as marketing, medical diagnosis, forecasting system, and national security. However, in scenarios where data privacy is paramount, mining certain types of data without violating the privacy of data owners presents a formidable challenge, sparking growing concerns among privacy advocates. To address these concerns, it is imperative to advance data mining procedures that are complex to individual privacy considerations. Perturbation of data plays a pivotal role in Privacy-Preserving Data Mining (PPDM). Additive data safeguard data privacy. In contrast, multiplicative data perturbation involves a series of transformations, including rotation, translation, and the addition of noise components to the perturbed data copy.

Channel estimation with Zadoff–Chu sequences in the presence of phase errors

AbstractDue to their perfect periodic autocorrelation property, Zadoff–Chu sequences are often used as stimulus signals in the measurement of radio channel responses. In this letter, the cross‐correlation of a linear shift‐invariant system's response to a Zadoff–Chu sequence is derived for the case that a multiplicative perturbation such as phase noise is present at its output. The perturbation signal is shown to incur a spurious discrete Fourier transform, thus degrading the correlation's peak to sidelobe ratio. The implications for the performance limits of channel measurement systems are discussed.

Evaluating the effects of uncertainty in interval pairwise comparison matrices

In the context of multicriteria decision analysis, Pairwise Comparison Matrices (PCMs) represent a widely-used mathematical structure to capture comparisons about criteria or alternatives. When a confidence (or uncertainty) level is associated to the provided comparison measures, PCMs are naturally extended to the so-called Interval Pairwise Comparison Matrices (IPCM). Classical approaches, based on the eigenvector associated to the largest eigenvalue of the matrix (Analytic Hierarchy Process) or optimization problems (e.g. the Logarithmic Least Squares approach), are frequently adopted in order to estimate absolute utilities from relative judgements in absence of expert's uncertainty. Conversely, in the presence of uncertain measures, computationally intensive approaches based on Monte Carlo analysis have been adopted. In this paper, starting from a given PCM, we provide an approach able to find the smallest multiplicative perturbations of its elements able to perturb the obtained ranking from the ordinal perspective. On the basis of such approach, we provide theoretical results which can be adopted also in the case of uncertain measures (e.g., IPCMs) with the aim to characterize the stability of the obtained ranking on the basis of the given uncertainty. As testified by numerical examples that conclude the paper, the proposed framework can greatly simplify the classical approaches by providing the possibility to make direct assumptions on the final outcome of Monte Carlo analysis and reducing or avoiding its required computational effort.

Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes.

We consider pairs of Gaussian orthogonal ensemble matrices which are correlated with each other and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank perturbations generate outliers in the spectrum of the matrices. We investigate the statistical correlation (i.e., the typical overlap) between the eigenvectors associated to the outlier eigenvalues of each matrix in the pair, as well as the typical overlap between the outlier eigenvector of one matrix with the eigenvectors in the bulk of the spectrum of the other matrix. We discuss implications of these results for the signal recovery problem for spiked matrices, as well as for problems of high-dimensional random landscapes.

Zabczyk Type Criteria for Asymptotic Behavior of Dynamical Systems and Applications

AbstractWe obtain new characterizations for nonuniform and uniform asymptotic behaviors of variational systems in infinite dimensional spaces, following the line of studies recently developed in Dragičević et al. (J Differ Equ 268:4786–4829, 2020, J Dyn Differ Equ 34:1107–1137, 2022, Appl Math Comput 414:1–22, 2022, J Math Anal Appl 515:1–37, 2022). First, we give complete descriptions for both nonuniform and uniform stability by means of some nonuniform conditions of convergence for series of suitable nonlinear trajectories. We apply our criteria to present a novel method of exploring the robustness of the nonuniform exponential stability under additive and multiplicative perturbations and we also deduce some consequences for the robustness of the uniform exponential stability. After that, we obtain characterizations for nonuniform and uniform instability in terms of certain nonuniform conditions imposed to the sums of some well-chosen series of nonlinear trajectories. We apply our results to provide a new technique of exploring the robustness of the nonuniform exponential instability under additive and multiplicative perturbations and discuss the consequences for uniform exponential instability. Hence, for both stability and instability we generalize the previous Zabczyk type results and extend their applicability. Next, as another class of applications, we provide nonuniform criteria of Rolewicz type for stability and instability of skew-product semiflows. Thus, we extend the Zabczyk–Rolewicz type methods in two directions: by giving local conditions for global behaviors and by employing specific nonuniform boundedness conditions. Our criteria can be applied to broad classes of discrete and continuous variational dynamical systems.

Geometric criteria for controllability and near‐controllability of driftless discrete‐time bilinear systems

AbstractGeometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete‐time bilinear systems. In this paper, controllability and near‐controllability of driftless discrete‐time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near‐controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near‐controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi‐input discrete‐time bilinear systems and to linear time‐invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.

Aesthetics and neural network image representations

We analyze the spaces of images encoded by generative neural networks of the BigGAN architecture. We find that generic multiplicative perturbations of neural network parameters away from the photo-realistic point often lead to networks generating images which appear as “artistic renditions” of the corresponding objects. This demonstrates an emergence of aesthetic properties directly from the structure of the photo-realistic visual environment as encoded in its neural network parametrization. Moreover, modifying a deep semantic part of the neural network leads to the appearance of symbolic visual representations. None of the considered networks had any access to images of human-made art.

A novel non-fragile [formula omitted] fault-tolerant course-keeping control for uncertain unmanned surface vehicles with rudder failures

This paper investigates the problem of non-fragile H∞ fault-tolerant course-keeping control (NFC) for uncertain unmanned surface vehicles (UUSV) with rudder failures. Firstly, due to the inevitable parameter perturbations in practice, the unmanned surface vehicles (USV) is modeled as a kind of uncertain Nomoto model, and an uncertain second-order under-damped system is used to describe the rudder system. Next, considering the distribution characteristic of failures, a distribution-dependent stochastic failure model is established and incorporated into the controller design. In order to investigate the fragile problems of the controller, a new non-fragile fault-tolerant scheme is given against multiplicative and additive controller gain perturbations. Finally, the effectiveness of the proposed methods is verified via two numerical examples.

A fast randomized algorithm for computing an approximate null space

Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall–skinny matrix. We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing right singular vectors using randomization by examining the quality of the solution using multiplicative perturbation theory. For an mtimes n (mge n) matrix, the algorithm runs with complexity O(mnlog n +n^3) which is faster than the standard O(mn^2) methods. In applications, numerical experiments show great speedups including a 30times speedup for the AAA algorithm and 10times speedup for the total least squares problem.

New multiplicative perturbation bounds on orthogonal projection

New multiplicative perturbation bounds for orthogonal projection in a general unitarily invariant norm, the Q-norm and the spectral norm are derived. These bounds always improve the existing bounds. An example is given to show that these new bounds are optimal.

Flocking Behaviors under Hierarchical Leadership of Thermodynamic Cucker-Smale Particles with Multiplicative White Noise and Perturbation.

The thermodynamic Cucker-Smale model (TCS model) describes dynamic consistency caused by different temperatures between multi-agent particles. This paper studies the flocking behaviors of the TCS model with multiplicative white noise under hierarchical leadership. First, we introduce the corresponding model of two particles. Then, by using mathematical induction and considering the properties of differential functions, it is proved that, under certain conditions, the group can achieve flocking. Finally, we verify the conclusion through numerical simulation results. Similarly, this paper studies the above model with perturbation functions.

High Precision Calibration for Three-Dimensional Vision-Guided Robot System

Calibration is an important premise for vision-guided robot system, especially for precise industrial operations on curved surfaces. Most of existing methods focus on one or partial union of errors including assembly error of the robot, relative pose error between the vision unit and the robot, or installation error of end effector, which is still challenging in practice. To solve this problem, this article proposes a new calibration method based on error correction matrix by which different manifestations of each error is unified. As the nonlinear equations formed by these matrices cannot be solved analytically, a numerical optimization solution based on Lie algebra is presented. Specifically, the matrices acquisition is formulated as the problem of minimizing the sum of distance deviations between actual and ideal tool center points. This problem is then solved by differentiating these matrices with a form of left multiplicative perturbation. In this way, a high precision joint calibration with multisource errors is achieved. The proposed method is verified by simulations and experiments.

Multiplicative perturbation analysis for the generalized Cholesky block downdating problem

Multiplicative perturbation analysis for the generalized Cholesky block downdating problem

On Phase Change Rate Maximization with Practical Applications

We recapitulate the notion of phase change rate maximization and demonstrate the usefulness of its solution on analyzing the robust instability of a cyclic network of multi-agent systems subject to a homogenous multiplicative perturbation. Subsequently, we apply the phase change rate maximization result to two practical applications. The first is a magnetic levitation system, while the second is a repressilator with time-delay in synthetic biology.