Probability Measure Research Articles

Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2<σ<1, of the square |ζ(1/2+it)|2 of the Riemann zeta-function. We consider probability measures defined by means of Z(σ+iφ(t)), where φ(t), t⩾t0>0, is an increasing to +∞ differentiable function with monotonically decreasing derivative φ′(t) satisfying a certain normalizing estimate related to the mean square of the function Z(σ+iφ(t)). This allows us to extend the distribution laws for Z(s).

The weak mixing property on negatively curved manifolds

Given a complete manifold of negative curvature, we show that weak mixing is a generic property in the set of all probability measures invariant by the geodesic flow, as soon as the flow is topologically weakly mixing in restriction to its non-wandering set.

Wasserstein upper bounds of [formula omitted]-norms for multivariate densities in Besov spaces

While the total variation between two probability measures cannot be bounded by Wasserstein metrics in general, it is possible if they possess smooth densities. More generally, we demonstrate that Lp-distances between densities in Besov spaces can be bounded by powers of the Wasserstein metrics.

Coherent distributions on the square–extreme points and asymptotics

Abstract Let $\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1]^2$ , i.e. all those probability measures $\mu$ for which there exists a random vector $(X,Y)\sim \mu$ , a pair $(\mathcal{G},\mathcal{H})$ of $\sigma$ -fields, and an event E such that $X=\mathbb{P}(E\mid\mathcal{G})$ , $Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set $\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\mathrm{ext}(\mathcal{C})$ . We apply these results to obtain the asymptotic sharp bound $\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$ .

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

Multilevel thresholding with divergence measure and improved particle swarm optimization algorithm for crack image segmentation

Crack formation is a common phenomenon in engineering structures, which can cause serious damage to the safety and health of these structures. An important method of ensuring the safety and health of engineered structures is the prompt detection of cracks. Image threshold segmentation based on machine vision is a crucial technology for crack detection. Threshold segmentation can separate the crack area from the background, providing convenience for more accurate measurement and evaluation of the crack condition and location. The segmentation of cracks in complex scenes is a challenging task, and this goal can be achieved by means of multilevel thresholding. The arithmetic-geometric divergence combines the advantages of the arithmetic mean and the geometric mean in probability measures, enabling a more precise capture of the local features of an image in image processing. In this paper, a multilevel thresholding method for crack image segmentation based on the minimum arithmetic-geometric divergence is proposed. To address the issue of time complexity in multilevel thresholding, an enhanced particle swarm optimization algorithm with local stochastic perturbation is proposed. In crack detection, the thresholding criterion function based on the minimum arithmetic-geometric divergence can adaptively determine the thresholds according to the distribution characteristics of pixel values in the image. The proposed enhanced particle swarm optimization algorithm can increase the diversity of candidate solutions and enhance the global convergence performance of the algorithm. The proposed method for crack image segmentation is compared with seven state-of-the-art multilevel thresholding methods based on several metrics, including RMSE, PSNR, SSIM, FSIM, and computation time. The experimental results show that the proposed method outperforms several competing methods in terms of these metrics.

Global differentiable structures for the Fisher-Rao and Kantorovich-Wasserstein-Otto metrics

We develop a class of non-parametric, Banach-Sobolev manifolds of probability measures that, despite having comparatively weak topologies, support the Fisher-Rao and Kantorovich-Wasserstein-Otto (KWO) Riemannian metrics. The manifolds employ the Kaniadakis κ-deformed logarithms in their charts, and are isomorphic to the (whole) model spaces, W1,λ(μ). These are weighted Sobolev spaces with Lebesgue exponents as small as λ=4, making them suitable for approximations. The KWO metric is analysed through a bundle of Markov semigroups having generators with irregular first-order terms. Strong a-priori estimates are obtained for the associated parabolic equations. Together with a weak Poincaré/Lp-convergence theorem, these are used to establish the Hölder continuity of the “velocity field” representation of tangent vectors—a property inherited by the KWO metric. The manifolds provide a simple framework for the study of problems involving both metrics.

How many motor units is enough? An assessment of the influence of the number of motor units on firing rate calculations

The number of motor units included in calculations of mean firing rates varies widely in the literature. It is unknown how the number of decomposed motor units included in the calculation of firing rate per participant compares to the total number of active motor units in the muscle, and if this is different for males and females. Bootstrapped distributions and confidence intervals (CI) of mean motor unit firing rates decomposed from the tibialis anterior were used to represent the total number of active motor units for individual participants in trials from 20 to 100 % of maximal voluntary contraction. Bootstrapped distributions of mean firing rates were constructed using different numbers of motor units, from one to the maximum number for each participant, and compared to the CIs. A probability measure for each number of motor units involved in firing rate was calculated and then averaged across all individuals. Motor unit numbers required for similar levels of probability increased as contraction intensity increased (p < 0.001). Increased levels of probability also required higher numbers of motor units (p < 0.001). There was no effect of sex (p ≥ 0.97) for any comparison. This methodology should be repeated in other muscles, and aged populations.

An Encourage Arrival of Single-Server Queuing System with Defeat Balking and Feedback Customers

Abstract: In this paper, we develop a finite capacity Single-server Markovian queuing system with encourage arrival, defeat balking and feedback customers. The steady state solution and the system characteristic are derived for this model and various probabilistic and performance measures of the system are derived.

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

AbstractIn this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space to and also from to the space of functions of bounded mean oscillation , with respect to the probability measure on when is a multi‐index in . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure

In this paper, the worst-case probability measure over the data is introduced as a tool for characterizing the generalization capabilities of machine learning algorithms. More specifically, the worst-case probability measure is a Gibbs probability measure and the unique solution to the maximization of the expected loss under a relative entropy constraint with respect to a reference probability measure. Fundamental generalization metrics, such as the sensitivity of the expected loss, the sensitivity of the empirical risk, and the generalization gap are shown to have closed-form expressions involving the worst-case data-generating probability measure. Existing results for the Gibbs algorithm, such as characterizing the generalization gap as a sum of mutual information and lautum information, up to a constant factor, are recovered. A novel parallel is established between the worst-case data-generating probability measure and the Gibbs algorithm. Specifically, the Gibbs probability measure is identified as a fundamental commonality of the model space and the data space for machine learning algorithms.

Neural networks taking probability distributions as input: A framework for analyzing exchangeable networks

In recent years, exchangeable network structures that take datasets as input have been widely used to obtain representations of various datasets. Although they perform well, analyzing exchangeable network with a dataset as input is challenging. Given that this type of network can be viewed as a function acting on probability measures since datasets are drawn from various distributions, this paper theoretically analyzes exchangeable network structures from a probabilistic perspective. This paper proposes a probabilistic analytical framework that neural networks acting on probability measures, which is an extension of Multi-Layer Perceptrons (MLP). When only samples from each distribution are available, in this new analytical framework, neural networks acting on probability measures correspond to the traditional exchangeable structure defined on datasets. Using this new analytical framework, we can demonstrate the ability of exchangeable network structures to capture complex patterns, as it provides the universal approximation property of exchangeable network structures. Furthermore, we derive a consistency result that shows the parameter estimation of exchangeable network structures is consistent statistically under certain conditions.

Learning Ultrametric Trees for Optimal Transport Regression

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the space. However, measures supported on trees admit a closed-form optimal transport that can be computed in linear time. In this paper, we aim to find an optimal tree structure for a given discrete metric space so that the tree-Wasserstein distance approximates the optimal transport distance in the original space. One of our key ideas is to cast the problem in ultrametric spaces. This helps us optimize over the space of ultrametric trees --- a mixed-discrete and continuous optimization problem --- via projected gradient decent over the space of ultrametric matrices. During optimization, we project the parameters to the ultrametric space via a hierarchical minimum spanning tree algorithm, equivalent to the closest projection to ultrametrics under the supremum norm. Experimental results on real datasets show that our approach outperforms previous approaches (e.g. Flowtree, Quadtree) in approximating optimal transport distances. Finally, experiments on synthetic data generated on ground truth trees show that our algorithm can accurately uncover the underlying trees.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Confidence Is All You Need for MI Attacks (Student Abstract)

In this evolving era of machine learning security, membership inference attacks have emerged as a potent threat to the confidentiality of sensitive data. In this attack, adversaries aim to determine whether a particular point was used during the training of a target model. This paper proposes a new method to gauge a data point’s membership in a model’s training set. Instead of correlating loss with membership, as is traditionally done, we have leveraged the fact that training examples generally exhibit higher confidence values when classified into their actual class. During training, the model is essentially being ’fit’ to the training data and might face particular difficulties in generalization to unseen data. This asymmetry leads to the model achieving higher confidence on the training data as it exploits the specific patterns and noise present in the training data. Our proposed approach leverages the confidence values generated by the machine-learning model. These confidence values provide a probabilistic measure of the model’s certainty in its predictions and can further be used to infer the membership of a given data point. Additionally, we also introduce another variant of our method that allows us to carry out this attack without knowing the ground truth(true class) of a given data point, thus offering an edge over existing label-dependent attack methods.

Put–Call Parities, absence of arbitrage opportunities, and nonlinear pricing rules

AbstractWhen prices of assets traded in a financial market are determined by nonlinear pricing rules, different parities between call and put options have been considered. We show that, under monotonicity, parities between call and put options and discount certificates characterize ambiguity‐sensitive (Choquet and/or Šipoš) pricing rules, that is, pricing rules that can be represented via discounted expectations with respect to non‐additive probability measures. We analyze how nonadditivity relates to arbitrage opportunities and we give necessary and sufficient conditions for Choquet and Šipoš pricing rules to be arbitrage free. Finally, we identify violations of the Call‐Put Parity with the presence of bid–ask spreads.

Online as-Built Building Information Model Update for Robotic Monitoring in Construction Sites

Today, automated techniques for the update of as-built Building Information Models (BIM) make use of offline algorithms restricting the update frequency to an extent where continuous monitoring becomes nearly impossible. To address this problem, we propose a new method for robotic monitoring that updates an as-built BIM in real-time by solving a Simultaneous Localization and Mapping (SLAM) problem where the map is represented as a collection of elements from the as-planned BIM. The suggested approach is based on the Rao-Blackwellized Particle Filter (RBPF) which enables explicit injection of prior knowledge from the building’s construction schedule, i.e., from a 4D BIM, or its elements’ spatial relations. In the methods section we describe the benefits of using an exact inverse sensor model that provides a measure for the existence probability of elements while considering the entire probabilistic existence belief map. We continue by outlining robustification techniques that include both geometrical and temporal dimensions and present how we account for common pose and shape mistakes in constructed elements. Additionally, we show that our method reduces to the standard Monte Carlo Localization (MCL) in known areas. We conclude by presenting simulation results of the proposed method and comparing it to adjacent alternatives.

Sums of random polynomials with differing degrees

Let μ \mu and ν \nu be probability measures in the complex plane, and let p p and q q be independent random polynomials of degree n n , whose roots are chosen independently from μ \mu and ν \nu , respectively. Under assumptions on the measures μ \mu and ν \nu , the limiting distribution for the zeros of the sum p + q p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n → ∞ n \to \infty . In this paper, we generalize and extend this result to the case where p p and q q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of μ \mu and ν \nu , scaled by the limiting ratio of the degrees of p p and q q . Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q p + q for any pair of measures μ \mu and ν \nu , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

Ornstein[formula omitted]Uhlenbeck type processes on Wasserstein spaces

Let P2 be the space of probability measures on Rd having finite second moment, and consider the Riemannian structure on P2 induced by the intrinsic derivative on the L2-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein−Uhlenbeck (OU) type Dirichlet form on P2 whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is L2-compact. Perturbations of the OU Dirichlet form are also studied.