Small Noise Research Articles

Taming numerical imprecision by adapting the KL divergence to negative probabilities

The Kullback–Leibler (KL) divergence is frequently used in data science. For discrete distributions on large state spaces, approximations of probability vectors may result in a few small negative entries, rendering the KL divergence undefined. We address this problem by introducing a parameterized family of substitute divergence measures, the shifted KL (sKL) divergence measures. Our approach is generic and does not increase the computational overhead. We show that the sKL divergence shares important theoretical properties with the KL divergence and discuss how its shift parameters should be chosen. If Gaussian noise is added to a probability vector, we prove that the average sKL divergence converges to the KL divergence for small enough noise. We also show that our method solves the problem of negative entries in an application from computational oncology, the optimization of Mutual Hazard Networks for cancer progression using tensor-train approximations.

Mechanical vibration signal compression based on speech codecs for intelligent manufacturing

ABSTRACT How to effectively compress mechanical signals so that they can support remote and real-time health monitoring is a hot issue in the context of intelligent manufacturing. Due to the characteristics of small repeatability, high noise, and high numerical accuracy of vibration signal, traditional compression algorithms cannot well balance the compression ratio, reconstruction quality, compression time indicators. Therefore, this paper presents a novel mechanical vibration signal compression scheme based on speech codecs, which can realize signal compression in many scenarios such as tool wear detection and bearing fault diagnosis. In this work, mechanical vibration signal compression is first transformed into a multi-objective optimization problem, and the feasibility of solving this problem from the perspective of speech coding is deeply analyzed. Further, this paper presents five possible compression methods with different compression ratios. The presented methods are evaluated with real-time data for tool wear detection in stone processing and public datasets for bearing fault diagnosis. The experimental results show that the proposed methods achieve high compression ratios (4.0 ~ 24.6) while ensuring low time overhead (the average time for compressing a sample point is not more than 0.0194 ms) and high reconstruction quality (the MSE index all less than 0.008), outperforming other state-of-the-art methods.

A Robust Sparse Sensor Placement Strategy Based on Indicators of Noise for Ocean Monitoring

A well-performing data-driven sparse sensor deployment strategy is critical for marine monitoring systems, as it enables the optimal reconstruction of marine physical quantities with fewer sensors. However, ocean data typically contain substantial amounts of noise, including outliers (incomplete data) and inherent measurement noise, which heightens the complexity of sensor deployment. Therefore, this study optimizes the sparse sensor placement model by establishing noise indicators, including small noise weight and large noise weight, which are measured by entropy to minimize the prediction bias. Building on this, a robust sparse sensor placement algorithm is proposed, which utilizes the block coordinate update (BCU) iteration method to solve the function. During the iterative updating process, the proposed algorithm simultaneously updates the selection matrix, reconstruction matrix, and noise matrix. This allows for effective identification and mitigation of noise in the data through evaluation. Consequently, the deployed sensors achieve superior reconstruction performance compared to other deployment methods that do not incorporate noise evaluation. Experiments are also conducted on datasets of sea surface temperature (SST) and global ocean salinity, which demonstrate that our strategy significantly outperforms several other considered methods in terms of reconstruction accuracy while enabling autonomous sensor deployment under noisy conditions.

Solving deep-learning density functional theory via variational autoencoders

In recent years, machine learning models, chiefly deep neural networks, have revealed suited to learn accurate energy-density functionals from data. However, problematic instabilities have been shown to occur in the search of ground-state density profiles via energy minimization. Indeed, any small noise can lead astray from realistic profiles, causing the failure of the learned functional and, hence, strong violations of the variational property. In this article, we employ variational autoencoders (VAEs) to build a compressed, flexible, and regular representation of the ground-state density profiles of various quantum models. Performing energy minimization in this compressed space allows us to avoid both numerical instabilities and variational biases due to excessive constraints. Our tests are performed on one-dimensional single-particle models from the literature in the field and, notably, on a three-dimensional disordered potential. In all cases, the ground-state energies are estimated with errors below the chemical accuracy and the density profiles are accurately reproduced without numerical artifacts. Furthermore, we show that it is possible to perform transfer learning, applying pre-trained VAEs to different potentials.

High-Accuracy Phase Frequency Detection Technology Based on BDS Time and Frequency Signals.

For the time and frequency signals of Beidou satellites, a high-accuracy phase frequency detection technology based on phase group synchronization is proposed. Using the Beidou receiver and satellite signals as the frequency standard and the measured signals, respectively. The Beidou receiver and the satellite signals are sent to the phase coincidence detector of the different frequencies to generate a phase coincidence point pulse, which is sent to the different frequency phase detector as a control signal to generate the phase differences between the Beidou receiver and satellite signals, and then complete the high-accuracy phase synchronization between the Beidou receiver and satellite signals. Experimental results show that when the delay resolution reaches ps level, the phase synchronization accuracy of the system can reach 10 ps, which has the characteristics of small phase noise, low development cost, simple circuit structure, and high synchronization accuracy compared with the traditional phase synchronization technologies. Therefore, it would be widely used in satellite positioning, astrometry, precision navigation, aerospace, satellite launch, power transmission, communications, radar, and other high-tech fields.

Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.

Delayed feedback active suspension control for chaos in quarter car model with jumping nonlinearity

Vehicles that operate on off-road terrain, such as rough and unpaved roads, sometimes suffer from severe vibrations. This vibration causes vehicle tires to lose contact with the supporting ground and subsequently collide with it. This jumping leads to nonlinear impact dynamics similar to those of a bouncing ball. In this study, a quarter car model with jumping nonlinearity is newly developed to represent the impact dynamics of an off-road vehicle under harsh terrain conditions. Subsequently, complicated vibrations, such as quasi-periodic and chaotic vibrations in the developed model, are identified using a bifurcation diagram and Lyapunov exponents. Additionally, delayed feedback (DF) control of the active suspension is designed for the developed model. The numerical results show that DF control is effective at eliminating chaos and reducing vibration levels. The noise robustness of the DF control is validated for sinus excitation and small random road noise. The results indicate that DF is effective even when road noise exists. Thus, this study demonstrates the feasibility of DF active suspension control for future suspension control design of the off-road vehicle.

Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

Indexing Topological Numbers on Images by Transferring Chiral Magnetic Textures

AbstractTopological analysis is widely adopted in various research fields to unveil intricate features and structural relationships implied in geometrical objects. Especially, in the fields of data analysis, exploring the topological properties of various images offers rich insights into the intrinsic geometrical information within them. In this study, a novel approach is proposed to investigate the topological properties of arbitrary grayscale images by employing a straightforward procedure used in 2D magnetism studies to calculate topological numbers. This method utilizes machine learning techniques to transfer chiral magnetic textures onto the images. Then, the topological number is then computed directly from the converted images by integrating the solid angles formed by adjacent spin vectors. The method successfully identifies the topological numbers of various grayscale images, showing stable performances against small noises. Furthermore, two applications of the method: are demonstrated topological analysis of the Modified National Institute of Standards and Technology (MNIST) dataset and the counting of blood cells in microscopic images.

Convergence analysis for minimum action methods coupled with a finite difference method

Abstract The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $\theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.

Study on the influence of void evolution of porous asphalt concrete on acoustic absorption performance under high-temperature loading

This paper investigated the void evolution in porous asphalt concrete (PAC) under high-temperature loading and the resulting decay in acoustic absorption performance. The specimen of PAC-13 with a void content of 20.2 % was designed and loaded with MMLS3 for 0–100,000 cycles at 60℃. Computed tomography techniques were employed to extract the distribution characteristics and structural parameters of voids. Subsequently, the acoustic absorption coefficients were measured at different loading stages. The results indicate that PAC experienced both rapid and gradual reduction phases in air void content under high-temperature loading. After 100,000 cycles of loading, the air void content decreased by 5.44 %, void connectivity decreased by 4.32 %, and the average number of voids in section reduced by 8.30 %. The distribution range of nearly 90 % of void sizes shifted from below 7 mm to below 6 mm after 100,000 cycles. At different loading stages, the void content, average number and average equivalent diameter of connected voids exhibited a positive correlation and similar trends with the absorption capacity of PAC for small vehicle noise. This study provides novel insights and perspectives into the acoustic performance degradation of PAC due to void evolution under high-temperature loading.

Large deviation principles and Malliavin derivative for mean reflected stochastic differential equations

In this paper, we consider a class of reflected stochastic differential equations for which the constraint is not on the paths of the solution but on its law. We establish a small noise large deviation principle, a large deviation for short time, the Malliavin derivative and the smoothness of the density. To prove large deviation principles, a sufficient condition for the weak convergence method, which is suitable for Mckean–Vlasov stochastic differential equation, plays an important role.

Least squares estimation for the Ornstein–Uhlenbeck process with small Hermite noise

Least squares estimation for the Ornstein–Uhlenbeck process with small Hermite noise

Stationary distribution and ergodicity of a stochastic multi-species model with Holling type II response function

In this paper, we consider a stochastic multi-species model with the Holling type II functional response, which includes the stochastic competition model, the stochastic mutualism model, and the stochastic hybrid model of competition and mutualism. Sufficient conditions for the stationary distribution and ergodicity of the model are established. We support our analytical findings through extensive simulation studies. We show that under relatively small noise, the model has a stationary distribution and ergodicity, while the relatively large noise leads to the non-persistent and non-stationary distribution of the model through the numerical simulation studies of three species.

Stochastic quantisation of Yang–Mills–Higgs in 3D

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}$\\end{document} is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}$\\end{document} and thus define a quotient space of “gauge orbits” O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak {O}$\\end{document}. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak {O}$\\end{document} (up to a potential finite time blow-up) associated to the stochastic YMH flow.

Finite Sample Rates for Logistic Regression with Small Noise or Few Samples

AbstractThe logistic regression estimator is known to inflate the magnitude of its coefficients if the sample size n is small, the dimension p is (moderately) large or the signal-to-noise ratio $$1/\sigma $$ 1 / σ is large (probabilities of observing a label are close to 0 or 1). With this in mind, we study the logistic regression estimator with $$p\ll n/\log n$$ p ≪ n / log n , assuming Gaussian covariates and labels generated by the Gaussian link function, with a mild optimization constraint on the estimator’s length to ensure existence. We provide finite sample guarantees for its direction, which serves as a classifier, and its Euclidean norm, which is an estimator for the signal-to-noise ratio. We distinguish between two regimes. In the low-noise/small-sample regime ($$\sigma \lesssim (p\log n)/n$$ σ ≲ ( p log n ) / n ), we show that the estimator’s direction (and consequentially the classification error) achieve the rate $$(p\log n)/n$$ ( p log n ) / n - up to the log term as if the problem was noiseless. In this case, the norm of the estimator is at least of order $$n/(p\log n)$$ n / ( p log n ) . If instead $$(p\log n)/n\lesssim \sigma \lesssim 1$$ ( p log n ) / n ≲ σ ≲ 1 , the estimator’s direction achieves the rate $$\sqrt{\sigma p\log n/n}$$ σ p log n / n , whereas its norm converges to the true norm at the rate $$\sqrt{p\log n/(n\sigma ^3)}$$ p log n / ( n σ 3 ) . As a corollary, the data are not linearly separable with high probability in this regime. In either regime, logistic regression provides a competitive classifier.

Topologically Defective Lattice Potential‐Based Gain‐Dissipative Ising Annealer with Large Noise Margin

AbstractGain‐dissipative Ising machines (GIMs) are annealers inspired by physical systems such as Ising spin glasses to solve combinatorial optimization problems. Compared to traditional quantum annealers, GIM is relatively easier to scale and can save on additional power consumption caused by low‐temperature cooling. However, traditional GIMs have a limited noise margin. Specifically, their normal operation requires ensuring that the noise intensity is lower than their saturation fixed point amplitude, which may result in increased power consumption to suppress noise‐induced spin state switching. To enhance the noise robustness of GIM, in this study a GIM based on a topologically defective lattice potential (TDLP) is proposed. Numerical simulations demonstrate that the TDLP‐based GIM can accurately simulate the bifurcation spin evolution in the Ising model. Furthermore, through the MAXCUT benchmark based on G‐set graphs, the optimal performance of TDLP‐based GIM is shown to surpass that of traditional GIMs. Additionally, the proposed TDLP‐based GIM successfully solves the MAXCUT benchmark and domain clustering dynamics benchmark based on G‐set graphs when the noise intensity exceeds its saturation fixed‐point amplitude. This indicates that the proposed system provides a promising architecture for breaking the small noise constraints required by traditional GIMs.

Uniform large deviations for stochastic Burgers–Huxley equation driven by multiplicative noise

In this paper, we establish a Freidlin–Wentzell type large deviation principle uniformly with respect to initial condition in bounded subsets of an infinite dimensional Banach space for stochastic Burgers–Huxley equation driven by multiplicative small noise. The proof is based on the weak convergence approach obtained by Salins, Budhiraja, Dupuis (Trans. Amer. Math. Soc., 2019).

Multi-mechanical frequency locking mechanism and robustness analysis of MDOF coupled resonator

Mechanical frequency locking can overcome the dependence of peak frequency on the driving amplitude and improve the frequency stability in nonlinear resonant systems. Most research focuses on the single mechanical frequency locking phenomenon, but few studies focus on multi-mechanical frequency locking phenomenon. In this study, the multi-mechanical frequency locking mechanism and robustness analysis of the peak frequency is proposed through multi-mode coupled vibration. Firstly, a multi-degree-of-freedom (MDOF) coupled resonator consisting of a low-frequency resonant element and multiple high-frequency resonant elements is designed, and the vibration energy of the resonant system can be transferred from the low-frequency resonant element driven by a harmonic excitation to the high-frequency resonant elements via the 1:3 internal resonance. Through perturbation and bifurcation analysis, the nonlinear dynamic behaviors of MDOF coupled resonator are analyzed in detail. It is found that the peak frequency of the low-frequency resonant element can remain stable in multiple drive intervals, leading to multi-mechanical frequency locking phenomenon. Then, a 3-DOF magnetic coupled resonant system is experimentally designed to demonstrate the physical conditions and influencing factors of the double mechanical frequency locking phenomenon. Typically, introducing external disturbances, the frequency locking is experimentally demonstrated to improve the robustness of peak frequency. The experimental results show that when the mechanical frequency locking phenomenon occurs, the influence of small external noise disturbance on the peak frequency can be ignored, which greatly improves the frequency stability of nonlinear resonant systems.

Complex Transitions of the Bounded Confidence Model from an Odd Number of Clusters to the Next

The bounded confidence model assumes simple continuous opinion dynamics in which agents ignore opinions which are too far from their own. The two initial variants—Hegselmann–Krause (HK) and Deffuant–Weisbuch (DW)—of the model have attracted significant attention since the early 2000s. This paper revisits the version of the HK model applied to a probability distribution, earlier studied by Jan Lorenz. It shows that the bifurcation diagram depends on the parity of the size of the discretisation and that adding a small noise to the initial conditions leads to complex transitions involving several phases.