We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.
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We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.
This paper addresses a new active learning strategy for regression problems. The presented Wasserstein active regression model is based on the principles of distribution-matching to measure the representativeness of the labeled dataset. The Wasserstein distance is computed using GroupSort Neural Networks. The use of such networks provides theoretical foundations giving a way to quantify errors with explicit bounds for their size and depth. This solution is combined with another uncertainty-based approach that is more outlier-tolerant to complete the query strategy. Finally, this method is compared with other classical and recent solutions. The study empirically shows the pertinence of such a representativity-uncertainty approach, which provides good estimation all along the query procedure. Moreover, the Wasserstein active regression often achieves more precise estimations and tends to improve accuracy faster than other models.
Parameters of differential equations are essential to characterize intrinsic behaviors of dynamic systems. Numerous methods for estimating parameters in dynamic systems are computationally and/or statistically inadequate, especially for complex systems with general-order differential operators, such as motion dynamics. This article presents Green's matching, a computationally tractable and statistically efficient two-step method, which only needs to approximate trajectories in dynamic systems but not their derivatives due to the inverse of differential operators by Green's function. This yields a statistically optimal guarantee for parameter estimation in general-order equations, a feature not shared by existing methods, and provides an efficient framework for broad statistical inferences in complex dynamic systems.
We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.
Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter $a$ approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form $O \left( \frac{1}{\sqrt{a}}\right)$. As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.
Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.
String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which intersects an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this paper, we characterise the bi-infinite words admitting a finite string attractor as the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.
We integrate machine learning approaches with nonlinear time series analysis, specifically utilizing recurrence measures to classify various dynamical states emerging from time series. We implement three machine learning algorithms Logistic Regression, Random Forest, and Support Vector Machine for this study. The input features are derived from the recurrence quantification of nonlinear time series and characteristic measures of the corresponding recurrence networks. For training and testing we generate synthetic data from standard nonlinear dynamical systems and evaluate the efficiency and performance of the machine learning algorithms in classifying time series into periodic, chaotic, hyper-chaotic, or noisy categories. Additionally, we explore the significance of input features in the classification scheme and find that the features quantifying the density of recurrence points are the most relevant. Furthermore, we illustrate how the trained algorithms can successfully predict the dynamical states of two variable stars, SX Her and AC Her from the data of their light curves.
Mixed methods for linear elasticity with strongly symmetric stresses of lowest order are studied in this paper. On each simplex, the stress space has piecewise linear components with respect to its Alfeld split (which connects the vertices to barycenter), generalizing the Johnson-Mercier two-dimensional element to higher dimensions. Further reductions in the stress space in the three-dimensional case (to 24 degrees of freedom per tetrahedron) are possible when the displacement space is reduced to local rigid displacements. Proofs of optimal error estimates of numerical solutions and improved error estimates via postprocessing and the duality argument are presented.
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