Gaussian Measure Research Articles

A remark on Gibbs measures with log-correlated Gaussian fields

Abstract We study Gibbs measures with log-correlated base Gaussian fields on the d-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson’s argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove nonnormalizability of the Gibbs measure. When $d = 2$ , our argument provides an alternative proof of the nonnormalizability result for the focusing $\Phi ^4_2$ -measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein’s inequality on $\mathbb R^d$ . We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) nonnormalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

Higher-order asymptotic corrections and their application to the Gamma Variance Model

We present improved methods for calculating confidence intervals and p values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can incorporate uncertainties in parameters that themselves represent uncertainties (informally, “errors on errors”) called the Gamma Variance Model. This model contains fixed parameters, generically denoted by ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, that represent the relative uncertainties in estimates of standard deviations of Gaussian distributed measurements. If the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are small, one can construct confidence intervals and p values using standard asymptotic methods. This is formally similar to the familiar situation of a large data sample, in which estimators for all adjustable parameters have Gaussian distributions. Here we address the important case where the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are not small and as a consequence the first-order asymptotic distributions do not represent a good approximation. We investigate improved test statistics based on the technology of higher-order asymptotics (modified likelihood root and Bartlett correction). The effective application of higher-order corrections removes an important computational barrier to the use of the Gamma Variance Model.

BV capacity and perimeter in abstract Wiener spaces and applications

Abstract This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity cap H ⁡ ( ⋅ ) {\operatorname{cap}_{H}(\,\cdot\,)} (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with cap H ⁡ ( ⋅ ) {\operatorname{cap}_{H}(\,\cdot\,)} is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L 1 ⁢ ( X , γ ) {L^{1}(X,\gamma)} .

Semi-tensor product-based one-bit compressed sensing

The area of one-bit compressed sensing (1-bit CS) focuses on the recovery of sparse signals from binary measurements. Over the past decade, this field has witnessed the emergence of well-developed theories. However, most of the existing literature is confined to fully random measurement matrices, like random Gaussian and random sub-Gaussian measurements. This limitation often results in high generation and storage costs. This paper aims to apply semi-tensor product-based measurements to 1-bit CS. By utilizing the semi-tensor product, this proposed method can compress high-dimensional signals using lower-dimensional measurement matrices, thereby reducing the cost of generating and storing fully random measurement matrices. We propose a regularized model for this problem that has a closed-form solution. Theoretically, we demonstrate that the solution provides an approximate estimate of the underlying signal with upper bounds on recovery error. Empirically, we conduct a series of experiments on both synthetic and real-world data to demonstrate the proposed method’s ability to utilize a lower-dimensional measurement matrix for signal compression and reconstruction with enhanced flexibility, resulting in improved recovery accuracy.

On mm-Entropy of a Banach Space with Gaussian Measure

On mm-Entropy of a Banach Space with Gaussian Measure

Multifrequency matched-field source localization based on Wasserstein metric for probability measures.

Matched-field processing (MFP) for underwater source localization serves as a generalized beamforming approach that assesses the correlation between the received array data and a dictionary of replica vectors. In this study, the processing scheme of MFP is reformulated by computing a statistical metric between two Gaussian probability measures with the cross-spectral density matrices (CSDMs). To achieve this, the Wasserstein metric, a widely used notion of metric in the space of probability measures, is employed for developing the processor to attach the intrinsic properties of CSDMs, expressing the underlying optimal value of the statistic. The Wasserstein processor uses the embedded metric structure to suppress ambiguities, resulting in the ability to distinguish between multiple sources. In this foundation, a multifrequency processor that combines the information at different frequencies is derived, providing improved localization statistics with deficient snapshots. The effectiveness and robustness of the Wasserstein processor are demonstrated using acoustic simulation and the event S5 of the SWellEx-96 experiment data, exhibiting correct localization statistics and a notable reduction in ambiguity. Additionally, this paper presents an approach to derive the averaged Bartlett processor by evaluating the Wasserstein metric between two Dirac measures, providing an innovative perspective for MFP.

Steering witnesses for unknown Gaussian quantum states

We define and fully characterize the witnesses based on second moments detecting steering in Gaussian states by means of Gaussian measurements. All such tests, which arise from linear combination of variances or second moments of canonical operators, are easily implemented in experiments. We propose also a set of linear constraints fully characterizing steering witnesses when the steered party has one bosonic mode, while in the general case the constraints restrict the set of tests detecting steering. Given an unknown quantum state we implement a semidefinite program providing the appropriate steering test with respect to the number of random measurements performed. Thus, it is a ‘repeat-until-success’ method allowing for steering detection with less measurements than in full tomography. We study the efficiency of steering detection for two-mode squeezed vacuum states, for two-mode general unknown states, and for three-mode continuous variable GHZ states. In addition, we discuss the robustness of this method to statistical errors.

On Characterization of Quantum Gaussian Measurement Channels

On Characterization of Quantum Gaussian Measurement Channels

Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces

Randomized quadratures for integrating functions in Sobolev spaces of order α ≥ 1 \alpha \ge 1 , where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O ( n − α − 1 / 2 ) O(n^{-\alpha -1/2}) is proven, where n n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.

On the planar Gaussian-Minkowski problem

The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this “uniqueness” result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.

On the L2 boundedness of pseudo-multipliers for Hermite expansions

We give various conditions for Hermite pseudo-multipliers to be bounded on L2(Rn). As a by-product we also give new results for pseudo-multipliers in the Gaussian measure setting. One of our key tools is a new integration by-parts formula for Hermite expansions.

Numerical Method for Stochastic Nonlinear Schrödinger Equation Driven by Multivariate Gaussian Measure: Algorithms and Applications

In this paper, we present a novel Galerkin spectral method for numerically solving the stochastic nonlinear Schrödinger (NLS) equation driven by multivariate Gaussian random variables, including the nonlinear term. Our approach involves deriving the tensor product of triple random orthogonal basis and random functions, which enables us to effectively handle the stochasticity and nonlinear term in the equation. We apply this newly proposed method to solve both one- and two-dimensional stochastic NLS equations, providing detailed analysis and comparing the results with Monte Carlo simulation. In addition, the proposed method is applied to the stochastic Ginzburg–Landau equation. Our method exhibits excellent performance in both spatial and random spaces, achieving spectral accuracy in the numerical solutions.

Deep learning-based parameter estimation of stochastic differential equations driven by fractional Brownian motions with measurement noise

This study proposes a general parameter estimation neural network (PENN) to jointly identify the system parameters and the noise parameters of a stochastic differential equation driven by fractional Brownian motion (FBM) from a short sample trajectory. It separately extracts deep features from the trajectory and fuses the information of sampling frequency by a two-stage neural network architecture such that the sample trajectories with variable lengths and sampling times can be properly processed. In addition, by considering additive Gaussian measurement noise in the training stage and utilizing suitable loss functions, the PENN can quantitatively estimate the level of measurement noise and reduce its negative impacts on estimating the governing parameters. Experiments on Fitzhugh–Nagumo model, Duffing oscillator and genetic toggle switch model demonstrate that the PENN can accurately estimate the system parameters, the noise intensity and Hurst exponent of the process noise as well as the signal-to-noise ratio of the measurement noise with high speed.

Scalable inference of cell differentiation networks in gene therapy clonal tracking studies of haematopoiesis.

Investigating cell differentiation under a genetic disorder offers the potential for improving current gene therapy strategies. Clonal tracking provides a basis for mathematical modelling of population stem cell dynamics that sustain the blood cell formation, a process known as haematopoiesis. However, many clonal tracking protocols rely on a subset of cell types for the characterization of the stem cell output, and the data generated are subject to measurement errors and noise. We propose a stochastic framework to infer dynamic models of cell differentiation from clonal tracking data. A state-space formulation combines a stochastic quasi-reaction network, describing cell differentiation, with a Gaussian measurement model accounting for data errors and noise. We developed an inference algorithm based on an extended Kalman filter, a nonlinear optimization, and a Rauch-Tung-Striebelsmoother. Simulations show that our proposed method outperforms the state-of-the-art and scales to complex structures of cell differentiations in terms of nodes size and network depth. The application of our method to five in vivo gene therapy studies reveals different dynamics of cell differentiation. Our tool can provide statistical support to biologists and clinicians to better understand cell differentiation and haematopoietic reconstitution after a gene therapy treatment. The equations of the state-space model can be modified to infer other dynamics besides cell differentiation. The stochastic framework is implemented in the R package Karen which is available for download at https://cran.r-project.org/package=Karen. The code that supports the findings of this study is openly available at https://github.com/delcore-luca/CellDifferentiationNetworks.

Inferring the probability distribution over strain tensors in polycrystals from diffraction based measurements

Polycrystals illuminated by high-energy X-rays or neutrons produce diffraction patterns in which the measured diffraction peaks encode the individual single crystal strain states. While state of the art X-ray and neutron diffraction approaches can be used to routinely recover per grain mean strain tensors, less work has been produced on the recovery of higher order statistics of the strain distributions across the individual grains. In the setting of small deformations, we consider the problem of estimating the crystal elastic strain tensor probability distribution from diffraction data. For the special case of multivariate Gaussian strain tensor probability distributions, we show that while the mean of the distribution is well defined from measurements, the covariance of strain has a null-space. We show that there exist exactly 6 orthogonal perturbations to this covariance matrix under which the measured strain signal is invariant. In particular, we provide analytical parametrisations of these perturbations together with the set of possible maximum-likelihood estimates for a multivariate Gaussian fit to data. The parametric description of the null-space provides insights into the strain PDF modes that cannot be accurately estimated from the diffraction data. Understanding these modes prevents erroneous conclusions from being drawn based on the data. Beyond Gaussian strain tensor probability densities, we derive an iterative radial basis regression scheme in which the strain tensor probability density is estimated by a sparse finite basis expansion. This is made possible by showing that the operator mapping the strain tensor probability density onto the measured histograms of directional strain is linear, without approximation. The utility of the proposed algorithm is demonstrated by numerical simulations in the setting of single crystal monochromatic X-ray scattering. The proposed regression methods were found to robustly reject outliers and accurately predict the strain tensor probability distributions in the presence of Gaussian measurement noise.

Continuous latent state preintegration for inertial-aided systems

Traditionally, the pose and velocity prediction of a system at time t2 given inertial measurements from a 6-DoF IMU depends on the knowledge of the system’s state at time t1. It involves a series of integration and double integration that can be computationally expensive if performed regularly, in particular in the context of inertial-aided optimisation-based state estimation. The concept of preintegration consists of creating pseudo-measurements that are independent of the system’s initial conditions (pose and velocity at t1) in order to predict the system’s state at t2. These pseudo-measurements, so-called preintegrated measurements, were originally computed numerically using the integration rectangle rule. This article presents a novel method to perform continuous preintegration using Gaussian processes (GPs) to model the system’s dynamics focusing on high accuracy. It represents the preintegrated measurement’s derivatives in a continuous latent state that is learnt/optimised according to asynchronous IMU gyroscope and accelerometer measurements. The GP models allow for analytical integration and double integration of the latent state to generate accurate preintegrated measurements called unified Gaussian preintegrated measurements (UGPMs). We show through extensive quantitative experiments that the proposed UGPMs outperform the standard preintegration method by an order of magnitude. Additionally, we demonstrate that the UGPMs can be integrated into off-the-shelf multi-modal estimation frameworks with ease based on lidar-inertial, RGBD-inertial, and visual-inertial real-world experiments.

Approximation Characteristics of Gel’fand Type in Multivariate Sobolev Spaces with Mixed Derivative Equipped with Gaussian Measure

In this paper, we study the probabilistic Gel’fand N,δ-width of multivariate Sobolev spaces MW2rTd with mixed derivative that are equipped with Gaussian measure μ in LqTd. The sharp asymptotic estimates are determined by employing the discretization method.

A Novel Complex-Valued Gaussian Measurement Matrix for Image Compressed Sensing.

The measurement matrix used influences the performance of image reconstruction in compressed sensing. To enhance the performance of image reconstruction in compressed sensing, two different Gaussian random matrices were orthogonalized via Gram-Schmidt orthogonalization, respectively. Then, one was used as the real part and the other as the imaginary part to construct a complex-valued Gaussian matrix. Furthermore, we sparsified the proposed measurement matrix to reduce the storage space and computation. The experimental results show that the complex-valued Gaussian matrix after orthogonalization has better image reconstruction performance, and the peak signal-to-noise ratio and structural similarity under different compression ratios are better than the real-valued measurement matrix. Moreover, the sparse measurement matrix can effectively reduce the amount of calculation.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Optimal numerical integration and approximation of functions on ℝd equipped with Gaussian measure

Abstract We investigate the numerical approximation of integrals over $\mathbb{R}^{d}$ equipped with the standard Gaussian measure $\gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ of mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_{q}(\mathbb{R}^{d}, \gamma )$ of the unit ball of $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ for $1 \leq q < p < \infty $ and $q=p=2$.