Small Random Perturbations Research Articles

A Topological Investigation of the Cosmic Web Formation

N-body simulations of the evolution of the large-scale structure of the universe (the Cosmic Web) were run while altering attractive gravity laws and initial conditions, in order to infer which properties of the universe are revealed by its current large-scale topology. The Cosmic Web was found to develop regardless of the gravitational alteration made. Significantly altering the initial conditions from Gaussian distribution ones was found to eliminate the Cosmic Web. Thus, we require small random perturbations in otherwise uniform initial conditions in the early universe, under an attractive gravitational force, for the Cosmic Web to develop.

SIR models with vital dynamics, reinfection, and randomness to investigate the spread of infectious diseases

We investigate SIR models with vital dynamics, reinfection, and randomness at the transmission coefficient and recruitment rate. Initially, we conduct an extensive analysis of the autonomous scenario, covering aspects such as local and global well-posedness, the existence and internal structure of attractors, and the presence of gradient dynamics. Subsequently, we explore the implications of small nonautonomous random perturbations, establishing the continuity of attractors and ensuring their topological structural stability. Additionally, we study scenarios in which both the transmission coefficient and the recruitment rate exhibit time-dependent or random behavior. For each scenario, we establish the existence of attractors and delineate conditions that determine whether the disease is eradicated or reaches an endemic state. Finally, we depict numerical simulations to illustrate the theoretical results.

Effects of Small Random Perturbations in the Extended Glass–Kauffman Model of Gene Regulatory Networks

Effects of Small Random Perturbations in the Extended Glass–Kauffman Model of Gene Regulatory Networks

AAGAN: Android Malware Generation System Based on Generative Adversarial Network

With the rapid evolution of mobile malware, especially Android malware, machine learning (ML)-based Android malware detection systems have drawn massive attention. Although ML algorithms have recently led to many vital breakthroughs in malware detection, they are still particularly vulnerable to adversarial example (AE) attacks. By applying small random perturbations (e.g. simply modifying different kinds of features from the application’s manifest file), an AE attack can cause the misclassification of legitimate applications. This paper proposes AAGAN, an automated Android malware generation system based on Generative Adversarial Networks (GAN) that can successfully deceive current ML detectors. Our experiment results indicate that AEs generated by our system can flip the prediction of the state-of-the-art detection algorithms in 99% of cases using a real-world dataset. To defend against AE attacks, we improve the robustness of our detection system by alternatively retraining with these newly generated AEs. Surprisingly, after retraining five times, AAGAN can achieve an 89% success rate in bypassing our malware detection system.

Persistence of smooth manifolds for a non-autonomous coupled system under small random perturbations

We consider a non-autonomous coupled system (x,y), whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C1 invariant manifold S(t,ω)={(x,y)∈X×Y|x=σ(t,y,ω)} under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation.

Reliability and minimal detectable change of stiffness and other mechanical properties of the ankle joint in standing and walking

BackgroundAnkle joint stiffness and viscosity are fundamental mechanical descriptions that govern the movement of the body and impact an individual’s walking ability. Hence, these internal properties of a joint have been increasingly used to evaluate the effects of pathology (e.g., stroke) and in the design and control of robotic and prosthetic devices. However, the reliability of these measurements is currently unclear, which is important for translation to clinical use. Research questionCan we reliably measure the mechanical impedance parameters of the ankle while standing and walking? MethodsEighteen able-bodied individuals volunteered to be tested on two different days separated by at least 24 h. Participants received several small random ankle dorsiflexion perturbations while standing and during the stance phase of walking using a custom-designed robotic platform. Three-dimensional motion capture cameras and a 6-component force plate were used to quantify ankle joint motions and torque responses during normal and perturbed conditions. Ankle mechanical impedance was quantified by computing participant-specific ensemble averages of changes in ankle angle and torque due to perturbation and fitting a second-order parametric model consisting of stiffness, viscosity, and inertia. The test-retest reliability of each parameter was assessed using intraclass correlation coefficients (ICCs). We also computed the minimal detectable change (MDC) for each impedance parameter to establish the smallest amount of change that falls outside the measurement error of the instrument. ResultsIn standing, the reliability of stiffness, viscosity, and inertia was good to excellent (ICCs=0.67–0.91). During walking, the reliability of stiffness and viscosity was good to excellent (ICCs=0.74–0.84) while that of inertia was fair to good (ICCs=0.47–0.68). The MDC for a single subject ranged from 20%− 65% of the measurement mean but was higher (>100%) for inertia during walking. SignificanceResults indicate that dynamic measures of ankle joint impedance were generally reliable and could serve as an adjunct clinical tool for evaluating gait impairments.

Terrestrial reference frames and their internal accuracy at coordinate system level

The accuracy assessment of terrestrial reference frames (TRFs) at coordinate system level is a key task to ensure their successful use in Earth studies, satellite navigation and other geodetic positioning applications. Currently, the TRF quality specifications for the most demanding users dictate that the origin, orientation and scale should be determined at an accuracy level of 1 mm, and they should remain stable over time at a rate of 0.1 mm/yr. To evaluate the conformity of the internal accuracy of modern TRFs to such requirements, an appropriate mapping is needed to convert frame coordinate errors (and their CV matrix) in a terrestrial network to matching errors (and their CV matrix) in the realized coordinate system. Several projection schemes may be considered for this mapping problem, all of which aim at extracting the correlated part of the estimation error in TRF coordinates that is describable by small random perturbations in their coordinate system. The goal of the present paper is to investigate the inference problem of frame accuracy at coordinate system level, and to discuss not only the theoretical aspects of the required covariance projectors, but also the practical impact on the results obtained by their implementation in space geodetic solutions. For this purpose, a relevant case study is performed to evaluate the accuracy of the realized origin, orientation and scale in the ITRF frame series based on the formal CV matrices for their estimated positions and velocities in the four technique subnetworks (DORIS, SLR, VLBI, GNSS).

Partially hyperbolic random dynamics on Grassmannians

A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions, it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure, showing that with high probability, the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.

Self-supervision for Tabular Data by Learning to Predict Additive Homoskedastic Gaussian Noise as Pretext

The lack of scalability of data annotation translates to the need to decrease dependency on labels. Self-supervision offers a solution with data training themselves. However, it has received relatively less attention on tabular data, data that drive a large proportion of business and application domains. This work, which we name the Statistical Self-Supervisor (SSS), proposes a method for self-supervision on tabular data by defining a continuous perturbation as pretext. It enables a neural network to learn representations by learning to predict the level of additive isotropic Gaussian noise added to inputs. The choice of the pretext transformation is motivated by intrinsic characteristics of a neural network fundamentally performing linear fits under the widely adopted assumption of Gaussianity in its fitting error and the preservation of locality of a data example on the data manifold in the presence of small random perturbations. The transform condenses information in the generated representations, making them better employable for further task-specific prediction as evidenced by performance improvement of the downstream classifier. To evaluate the persistence of performance under low-annotation settings, SSS is evaluated against different levels of label availability to the downstream classifier (1% to 100%) and benchmarked against self- and semi-supervised methods. At the most label-constrained, 1% setting, we report a maximum increase of at least 2.5% against the next-best semi-supervised competing method. We report an increase of more than 1.5% against self-supervised state of the art. Ablation studies also reveal that increasing label availability from 0% to 1% results in a maximum increase of up to 50% on either of the five performance metrics and up to 15% thereafter, indicating diminishing returns in additional annotation.

Quenched decay of correlations for nonuniformly hyperbolic random maps with an ergodic driving system

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate transfer operators. We apply our results to obtain quenched exponential correlations decay for several non-iid random dynamical systems including small random perturbations of Lorenz maps and Axiom A attractors.

Scaling limit of small random perturbation of dynamical systems

Dans cet article, nous prouvons qu’une petite perturbation aléatoire d’un système dynamique avec plusieurs équilibres stables converge vers une chaîne de Markov dont les états sont des voisinages des équilibres stables les plus profonds, sur une échelle temporelle adaptée, à condition que la dynamique perturbée soit réversible dans le temps. Un tel résultat a été anticipé dès les années 1970, lorsque les fondements du traitement mathématique de ce problème ont été établis par Freidlin et Wentzell, mais la convergence au niveau des processus restait ouverte jusqu’à aujourd’hui. Nous résolvons ce problème en réduisant l’analyse à l’étude de la solution d’une équation de Poisson associée. De plus, nous introduisons une méthode pour effectuer cette analyse en utilisant d’une manière inédite des fonctions de test bien connues.

Topological energy of networks.

Energy is an important network indicator defined by the eigenvalues of an adjacency matrix that includes the neighbor information for each node. This article expands the definition of network energy to include higher-order information between nodes. We use resistance distances to characterize the distances between nodes and order complexes to extract higher-order information. Topological energy ( T E), defined by the resistance distance and order complex, reveals the characteristics of the network structure from multiple scales. In particular, calculations show that the topological energy can be used to distinguish graphs with the same spectrum well. In addition, topological energy is robust, and small random perturbations of edges do not significantly affect the T E values. Finally, we find that the energy curve of the real network is significantly different from that of the random graph, thus showing that T E can be used to distinguish the network structure well. This study shows that T E is an indicator that distinguishes the structure of a network and has some potential applications for real-world problems.

No evidence for synchronization of the solar cycle by a “clock”

The length of the solar activity cycle fluctuates considerably. The temporal evolution of the corresponding cycle phase, that is, the deviation of the epochs of activity minima or maxima from strict periodicity, provides relevant information concerning the physical mechanism underlying the cyclic magnetic activity. An underlying strictly periodic process (akin to a perfect “clock”), with the observer seeing a superposition of the perfect clock and a small random phase perturbation, leads to long-term phase stability in the observations. Such behavior would be expected if cycles were synchronized by tides caused by orbiting planets or by a hypothetical torsional oscillation in the solar radiative interior. Alternatively, in the absence of such synchronization, phase fluctuations accumulate and a random walk of the phase ensues, which is a typical property of randomly perturbed dynamo models. Based on the sunspot record and the reconstruction of solar cycles from cosmogenic 14C, we carried out rigorous statistical tests in order to decipher whether there exists phase synchronization or random walk. Synchronization is rejected at significance levels of between 95% (28 cycles from sunspot data) and beyond 99% (84 cycles reconstructed from 14C), while the existence of random walk in the phases is consistent with all data sets. This result strongly supports randomly perturbed dynamo models with little inter-cycle memory.

Singular quadratic eigenvalue problems: linearization and weak condition numbers

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $$\delta $$ -weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $$\delta $$ -weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

The smoothed complexity of Frank–Wolfe methods via conditioning of random matrices and polytopes

Frank–Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, a fruitful approach in many works has been the use of condition measures of polytopes. Lacoste-Julien and Jaggi introduced a condition number for polytopes and showed linear convergence for several variations of the method. The actual running time can still be exponential in the worst case (when the condition number is exponential). We study the smoothed complexity of the condition number, namely the condition number of small random perturbations of the input polytope and show that it is polynomial for any simplex and exponential for general polytopes. Our results also apply to other condition measures of polytopes that have been proposed for the analysis of Frank–Wolfe methods: vertex-facet distance (Beck and Shtern) and facial distance (Peña and Rodríguez). Our argument for polytopes is a refinement of an argument that we develop to study the conditioning of random matrices. The basic argument shows that for c>1 a d -by- n random Gaussian matrix with n \geq cd has a d -by- d submatrix with minimum singular value that is exponentially small with high probability. This also has consequences on known results about the robust uniqueness of tensor decompositions, the complexity of the simplex method and the diameter of polytopes.

Asymptotic covariance estimation by Gaussian random perturbation

In most cases, the asymptotic covariance matrix of an M-estimator is in a sandwich form. This sandwich form involves calculations of the first and second order derivatives of the loss function, which is intractable if the loss function is complex. To alleviate this problem, we propose in this article a novel method called Gaussian random perturbation. This method can be used to estimate the asymptotic covariance matrix of a general M-estimator without derivative calculations. The idea can be summarized as follows. We first generate a small random perturbation around the M-estimator. Then, we re-evaluate the loss function at the randomly perturbed M-estimator and obtain the estimators of the first and second order derivatives of the loss function via Taylor series expansion. This leads to a novel estimator for the asymptotic covariance matrix. We then rigorously show that the resulting covariance estimator is statistically consistent with two elegant characteristics. First, it involves no computation of derivatives. This makes it easier to estimate the covariance matrix of an M-estimator with a complex loss function. Second, it is convenient for parallel computing and thus attractive for massive data analysis. The consistency of the proposed asymptotic covariance estimator is demonstrated under appropriate regularity conditions. The practical usefulness of the method is further demonstrated with both simulation studies and real data analysis.

A robust technique for optimal fitting of roll-over shapes of human locomotor systems

The roll-over shape (ROS) effectively characterizes the lower limb's ability to roll forward during the single-limb support phase of human walking. ROS is modelled as an optimally fitted circular arc to the center of pressure (CoP) data transformed in the shank/leg-based local coordinate system. The commonly used method for optimal fitting of ROS is complex to implement and eliminates inherent individual variability in the ROS parameters during walking. We propose and validate a novel computerized method for optimal fitting of roll-over shapes of the lower limb during walking. Gait data of a healthy individual from Winter's book was used to generate ankle-foot and knee-ankle-foot roll-over shapes using the proposed method. The goodness of fit and form of both the roll-over shapes were validated with the literature. To test the robustness of the proposed technique, small random perturbations were introduced to the transformed CoP data and the effect of these small changes in the data on the ROS parameters was studied. The ROS parameters such as radius, arc length, subtended arc angle, and horizontal and vertical shift in the arc center did not change substantially with small changes in the CoP data. The proposed method is computationally efficient, and easy to implement for optimal fitting and characterization of ROS.

Quenched Decay of Correlations for One-dimensional Random Lorenz Maps

We study rates of mixing for small random perturbations of one-dimensional Lorenz maps. Using a random tower construction, we prove that, for Hölder observables, the random system admits exponential rates of quenched correlation decay.

Noise-induced transition in the Zeldovich–Semenov model with local and global bifurcations

The noise-induced transition of the Zeldovich–Semenov model in a continuous stirred tank reactor is investigated under small random perturbations. The deterministic model will exhibit mono- and bistable characteristics via local and global bifurcations. In the bistable zone, based on the Freidlin–Wentzell large deviation theory, the stochastic preference is explained by analyzing the required action of the fluctuational path. For the case of monostability, in the weak noise limit, the emergence of the switching line gives rise to the sudden switch of the optimal path and the sliding cycle will appear via the sliding bifurcation, which is verified by numerical methods. In addition, when there is no saddle in phase space, stochastic excitation with large-amplitude spikes is studied. On the quasi-threshold manifold, the point with the minimum quasi-potential plays the same role as the saddle, which means that the optimal path will undergo a large excursion by crossing this special point. These phenomena are verified by employing stochastic simulations.

Holstein polaron at fixed temperature: influence of the chain length

Based on the semiclassical Holstein model, the dynamics of a quantum particle in one-dimensional molecular chain with a trapping site is modeled. Numerical simulation is used to investigate the dynamics of a polaron in a chain with small random Langevin-like perturbations which imitate the thermostat. Parameter values are chosen such that the polaron energy at the trapping site is much greater than the energy of temperature fluctuations. Results of modeling demonstrate that temperature decay of a polaron depends on the chain length even at very low temperatures.