Gaussian Approximation Research Articles

Statistical‐Mechanical Theory on the Probability Distribution Function for the Net Charge of an Electrolyte Droplet

AbstractDroplets of electrolyte solutions in an insulating medium are ubiquitous in nature. The net charges of these droplets are normally nonzero, and they fluctuate. However, a theory on the probability distribution function for the net charge of droplets is lacking, so far. Thus, a statistical‐mechanical theory of a charged droplet is developed including the effect of the electrostatic energy of charging as well as the random distribution of ions. Then, the probability distribution function for the net charge of an electrolyte droplet is calculated assuming that it is generated from a macroscopic solution with the different cation and anion concentrations. Using the Gaussian approximation and Stirling's formula, the analytic results for the average and variance of the net charge of a droplet are obtained.

Memory-biomimetic deep Bayesian filtering

The widely used Bayesian filtering has a solid theoretical foundation and an efficient computational architecture, but it suffers from first-order Markovianity and the inability to learn from offline data. While recent research has utilized deep learning to overcome these limitations, issues remain with respect to theoretical rigor and scalability, resulting in the non-guaranteed optimality of filtering. In this work, we approach these problems from a biomimetic perspective. Firstly, a memory-based state evolution model is constructed inspired by the human cognitive mechanism, which enables recursive filtering while incorporating long-term state information. Secondly, a memory-biomimetic deep Bayesian filtering algorithm with good interpretability is designed, which improves the filtering process at multiple levels by naturally integrating recurrent neural networks and Bayesian filtering architecture. Thirdly, the Gaussian approximation implementation of the proposed filtering method is derived, which brings an efficient computational process with good analytical properties. Extensive experiments for airborne target tracking are performed to demonstrate the excellent estimation performance of the proposed method.

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

Field Theory for Mechanical Criticality in Disordered Fiber Networks.

Strain-controlled criticality governs the elasticity of jamming and fiber networks. While the upper critical dimension of jamming is believed to be d_{u}=2, non-mean-field exponents are observed in numerical studies of 2D and 3D fiber networks. The origins of this remains unclear. In this study we propose a minimal mean-field model for strain-controlled criticality of fiber networks. We then extend this to a phenomenological field theory, in which non-mean-field behavior emerges as a result of the disorder in the network structure. We predict that the upper critical dimension for such systems is d_{u}=4 using a Gaussian approximation. Moreover, we identify an order parameter for the phase transition, which has been lacking for fiber networks to date.

Developing animal fat substitute in low-fat meatballs: A strategy to use high internal phase emulsions stabilized by Prinsepia utilis Royle protein

In promoting healthy diet, developing animal fat substitutes for meat products has been a prominent trend in food science. In this study, Prinsepia utilis Royle protein (PuRP) with amphiphilic property was extracted from waste oil pomace. High internal phase emulsions (HIPEs) were prepared with a 75% oil phase and stabilized with 2% (w/v) PuRP due to their excellent elastic-gel property. Furthermore, the PuRP-HIPEs were used to substitute animal fat in low-fat meatballs. Below 100 mM ionic strength, the uniformly distributed PuRP-HIPEs exhibited an approximate Gaussian size distribution with an average particle size of about 100 μm. The PuRP-HIPEs exhibited good thermodynamic stability and improved the texture of meatballs. Additionally, the PuRP-HIPEs significantly increased the mobile water content in steamed meatballs, resulting in better water retention and distribution than the free-fat and lard-added meatballs. Overall, the PuRP-HIPEs could substitute 100% animal fat in meatballs and maintain their cooking characteristics.

Variational Gaussian approximation for the magnetic Schrödinger equation *

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove L 2-error bounds and observable error bounds for the approximating Gaussian wave packet.

Memory effects in colloidal motion under confinement and driving

The transport of individual particles in inhomogeneous environments is complex and exhibits non-Markovian responses. The latter may be quantified by a memory function within the framework of the linear generalised Langevin equation (GLE). Here, we exemplify the implications of steady driving on the memory function of a colloidal model system for Brownian motion in a corrugated potential landscape, specifically, for one-dimensional motion in a sinusoidal potential. To this end, we consider the overdamped limit of the GLE, which is facilitated by separating the memory function into a singular (Markovian) and a regular (non-Markovian) part. Relying on exact solutions for the investigated model, we show that the random force entering the GLE must display a bias far from equilibrium, which corroborates a recent general prediction. Based on data for the mean-square displacement (MSD) obtained from Brownian dynamics simulations, we estimate the memory function for different driving strengths and show that already moderate driving accelerates the decay of the memory function by several orders of magnitude in time. We find that the memory may persist on much longer timescales than expected from the convergence of the MSD to its long-time asymptote. Furthermore, the functional form of the memory function changes from a monotonic decay to a non-monotonic, damped oscillatory behaviour, which can be understood from a competition of confined motion and depinning. Our analysis of the simulation data further reveals a pronounced non-Gaussianity, which questions the Gaussian approximation of the random force entering the GLE.

High-dimensional multivariate analysis of variance via geometric median and bootstrapping.

The geometric median, which is applicable to high-dimensional data, can be viewed as a generalization of the univariate median used in 1-dimensional data. It can be used as a robust estimator for identifying the location of multi-dimensional data and has a wide range of applications in real-world scenarios. This paper explores the problem of high-dimensional multivariate analysis of variance (MANOVA) using the geometric median. A maximum-type statistic that relies on the differences between the geometric medians among various groups is introduced. The distribution of the new test statistic is derived under the null hypothesis using Gaussian approximations, and its consistency under the alternative hypothesis is established. To approximate the distribution of the new statistic in high dimensions, a wild bootstrap algorithm is proposed and theoretically justified. Through simulation studies conducted across a variety of dimensions, sample sizes, and data-generating models, we demonstrate the finite-sample performance of our geometric median-based MANOVA method. Additionally, we implement the proposed approach to analyze a breast cancer gene expression dataset.

Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

Abstract Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of $(X,Y)$ with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$ in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.

Quantitative Gaussian approximation of randomly initialized deep neural networks

Quantitative Gaussian approximation of randomly initialized deep neural networks

A Gaussian approximation filter for nonlinear systems with compound noises

In practical applications, measurements may contain many different noise components due to complex environments and other factors. The Gaussian approximation filter, constrained by its requirement for Gaussian characteristics in measurement noises, faces challenges in obtaining a satisfactory filtering performance under compound noise environments. In this article, an improved Gaussian approximation filter is designed for nonlinear systems with compound measurement noises. Hypothesis tests on the prior and posterior residuals are designed to identify the types of noise present in the system. Subsequently, a posterior residual-based adaptive compensation is proposed to mitigate the adverse effects of compound noise on the Gaussian approximation filter. Moreover, it is demonstrated that when the measurement acceptance rate exceeds a certain critical threshold, the estimation error remains bounded. Finally, the effectiveness of the proposed algorithm and analysis is validated through simulations in the context of target tracking.

Resilient Factor Graph-Based GNSS/IMU/Vision/Odo Integrated Navigation Scheme Enhanced by Noise Approximate Gaussian Estimation in Challenging Environments

The signal blockage and multipath effects of the Global Navigation Satellite System (GNSS) caused by urban canyon scenarios have brought great technical challenges to the positioning and navigation of autonomous vehicles. In this paper, an improved factor graph optimization algorithm enhanced by a resilient noise model is proposed. The measurement noise is resilient and adjusted based on an approximate Gaussian distribution-based estimation. In estimating and adjusting the noise parameters of the measurement model, the error covariance matrix of the multi-sensor fusion positioning system is dynamically optimized to improve the system accuracy. Firstly, according to the approximate Gaussian statistical property of the GNSS/odometer velocity residual sequence, the measured data are divided into an approximate Gaussian fitting region and an approximate Gaussian convergence region. Secondly, the interval is divided according to the measured data, and the corresponding variational Bayesian network and Gaussian mixture model are used to estimate the innovation online. Further, the noise covariance matrix of the adaptive factor graph-based model is dynamically optimized using the estimated noise parameters. Finally, based on low-cost inertial navigation equipment, GNSS, odometer, and vision, the algorithm is implemented and verified using a simulation platform and real-vehicle road test. The experimental results show that in a complex urban road environment, compared with the traditional factor graph fusion localization algorithm, the maximum improvement in accuracy of the proposed algorithm can reach 65.63%, 39.52%, and 42.95% for heading, position, and velocity, respectively.

Algorithmization and Hardware Implementation of Polar Coding for 5G Telecommunications

Abstract The article explores a recursive algorithm for determining Bhattacharyya parameters, which is a measure of the degree of channel polarization for telecommunications with polar coding. A method is given for determining the positions of information and fixed bits in a polar code. The principles of constructing a polar encoder and a polar decoder successive cancellation are considered. The Field-Programmable Gate Array implementation of the successive cancellation list decoder is considered. Experimental studies of the construction of a polar code using Gaussian approximation have been carried out. The influence of the design signal-to-noise ratio on the bit error rate of telecommunications with polar codes has been studied. The use of multi-position modulation in telecommunications with polar codes has been studied. It is expected that the results obtained will be useful in hardware and software implementation of 5G telecommunications with polar coding.

DynGMA: A robust approach for learning stochastic differential equations from data

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Real-space local self-motion of protonated and deuterated water.

We report on the self-part of the Van Hove correlation function, the correlation function describing the dynamics of a single molecule, of water and deuterated water. The correlation function is determined by transforming inelastic scattering spectra of neutrons or x rays over a wide range of momentum transfer Q and energy transfer E to space R and time t. The short-range diffusivity is estimated from the Van Hove correlation function in the framework of the Gaussian approximation. The diffusivity has been found to be different from the long-range macroscopic diffusivity, providing information about local atomic dynamics.

Equation and solution for the 4-point correlation function of galaxies in the Gaussian approximation and its parity-odd part

Equation and solution for the 4-point correlation function of galaxies in the Gaussian approximation and its parity-odd part

Nonadiabatic Dynamics with Exact Factorization: Implementation and Assessment.

In this work, we report our implementation of several independent-trajectory mixed-quantum-classical (ITMQC) nonadiabatic dynamics methods based on exact factorization (XF) in the Libra package for nonadiabatic and excited-state dynamics. Namely, the exact factorization surface hopping (SHXF), mixed quantum-classical dynamics (MQCXF), and mean-field (MFXF) are introduced. Performance of these methods is compared to that of several traditional surface hopping schemes, such as the fewest-switches surface hopping (FSSH), branching-corrected surface hopping (BCSH), and the simplified decay of mixing (SDM), as well as conventional Ehrenfest (mean-field, MF) method. Based on a comprehensive set of 1D model Hamiltonians, we find the ranking SHXF ≈ MQCXF > BCSH > SDM > FSSH ≫ MF, with the BCSH sometimes outperforming the XF methods in terms of describing coherences. Although the MFXF method can yield reasonable populations and coherences for some cases, it does not conserve the total energy and is therefore not recommended. We also find that the branching correction for auxiliary trajectories is important for the XF methods to yield accurate populations and coherences. However, the branching correction can worsen the quality of the energy conservation in the MQCXF. Finally, we find that using the time-dependent Gaussian width approximation used in the XF methods for computing decoherence correction can improve the quality of energy conservation in the MQCXF dynamics. The parameter-free scheme of Subotnik for computing the Gaussian widths is found to deliver the best performance in situations where such widths are not known a priori.

Coping with inter-observer variability when assessing AI for scoring of HER2-stained whole slides.

e12571 Background: Around 15% of breast carcinomas exhibit Human Epidermal growth factor receptor 2 (HER2) overexpression due to gene amplification. HER2 protein levels are assessed by Immunohistochemistry (IHC) with 4 scores (0, 1+, 2+, 3+). Only score 3+ (IHC) cases and score 2+ (IHC) with proven HER2 gene amplification are eligible for targeted anti-HER2 therapies; others are HER2-negative. However, tumors with score 1+ and 2+ not amplified could benefit from treatment with the new HER2-based antibody-drug conjugates, requiring an accurate HER2 scoring, which is challenged by interobserver variability. This drawback is challenging for AI-based HER2 scoring assessment, and it is thus important to understand this inter-variability and then be able to propose a probabilistic framework which quantifies and explains the discrepancy between the AI analysis and the pathologist scoring. Methods: We introduce a novel probabilistic framework designed to assign probabilities to scores generated by AI algorithms. To the best of our knowledge, this represents the pioneering effort to quantify the uncertainty surrounding AI predictive models for HER2 scores. The methodology relies on a high-dimensional multivariate Gaussian approximation of the HER2 scoring process, hinting on the precise identification of marked cells to a specified category based on membrane completeness and staining intensity. Introducing uncertainty in the categorical identification enables the quantification of uncertainty in the overall AI-scoring at the whole slide image (WSI) level. Consequently, instead of offering a singular score, we can allocate a probability of belonging to a particular category (0, 1+, 2+, or 3+) for the global score. This approach enables a comparison of the predicted likelihood with the distribution of scores among pathologists, rather than relying on a "hypothetical" ground truth, often determined by a majority-voting which can be noise-sensitive or which excludes ambiguous cases when an agreement is hard to reach. Results: For illustration, we use the Kwant HER2 algorithm developed by DiaDeep on a validation set of 68 HER2 cases, previously scored by three expert pathologists. The framework allows us to understand the discrepancy between pathologists as we introduce nuance to the AI scoring and underlines narrow probability margins in borderline cases, therefore depicting the inherent challenge for pathologists to draw a clear demarcation between closely aligned scores categories. Conclusions: The proposed framework underscores the intrinsic variability of HER2 scoring between pathologists and allows to analyze the AI predictions for HER2 scoring with more nuance, consequently enabling a more critical analysis of the results compared to a singular raw output. As we enter a new era of AI-assisted diagnosis, such a method could improve the way of assessing new AI tools and help interpret their results.

Quantum scaling for the metal–insulator transition in a two-dimensional electron system

The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase’s resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.

Constructing and Evaluating Machine-Learned Interatomic Potentials for Li-Based Disordered Rocksalts.

Lithium-based disordered rocksalts (LDRs), which are an important class of positive electrode materials that can increase the energy density of current Li-ion batteries, represent a significantly complex chemical and configurational space for conventional density functional theory (DFT)-based high-throughput screening approaches. Notably, atom-centered machine-learned interatomic potentials (MLIPs) are a promising pathway to accurately model the potential energy surface of highly disordered chemical spaces, such as LDRs, where the performance of such MLIPs has not been rigorously explored yet. Here, we represent a comprehensive evaluation of the accuracy, transferability, and ease of training of five atom-centered MLIPs, including the artificial neural network potentials developed by the atomic energy network (AENET), the Gaussian approximation potential (GAP), the spectral neighbor analysis potential (SNAP) and its quadratic extension (qSNAP), and the moment tensor potential (MTP), in modeling a 11-component LDR chemical space. Specifically, we generate a DFT-calculated data set of 10,842 configurations of disordered LiTMO2 and TMO2 compositions, where TM = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, and/or Cu. To provide a point-of-comparison on the performance of atom-centered MLIPs, we also trained the neural equivariant interatomic potential (NequIP) on a subset of our data. Importantly, we find AENET to be the best potential in terms of accuracy and transferability for energy predictions, while MTP is the best for atomic forces. While AENET is the fastest to train among the MLIPs considered at low number of epochs (300), the training time increases significantly as epochs increase (3300), with a corresponding reduction in training errors (∼60%). Note that AENET and GAP tend to overfit in small data sets, with the extent of overfitting reducing with larger data sets. Finally, we observe AENET to provide reasonable predictions of average Li-intercalation voltages in layered, single-TM LiTMO2 frameworks, compared to DFT (∼10% error on average). Our study should pave the way both for discovering novel disordered rocksalt electrodes and for modeling other configurationally complex systems, such as high-entropy ceramics and alloys.