Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature.

We propose a new asymptotic expansion method for nonlinear filtering, based on a small parameter in the system noise. The conditional expectation is expanded as a power series in the noise level, with each coefficient computed by solving a system of ordinary differential equations. This approach mitigates the trade-off between computational efficiency and accuracy inherent in existing methods such as Gaussian approximations and particle filters. Moreover, by incorporating an Edgeworth-type expansion, our method captures complex features of the conditional distribution, such as multimodality, with significantly lower computational cost than conventional filtering algorithms.

PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems Tiantian Sun, Jian Zu PVD-Net: a physics-informed framework for singularly perturbed problems. Van Dyke matching principle is introduced to enhance prediction accuracy. Numerical results show our methods exceed baseline performance. Abstract Physics-informed neural networks and Physics-informed DeepONet excel in solving partial differential equations; however, they often fail to converge for singularly perturbed problems. To address this, we propose two novel frameworks, Prandtl-Van Dyke neural network (PVD-Net) and its operator learning extension Prandtl-Van Dyke Deep Operator Network (PVD-ONet), which rely solely on governing equations without data. To address varying task-specific requirements, both PVD-Net and PVD-ONet are developed in two distinct versions, tailored respectively for stability-focused and high-accuracy modeling. The leading-order PVD-Net adopts a two-network architecture combined with Prandtl's matching condition, targeting stability-prioritized scenarios. The high-order PVD-Net employs a five-network design with Van Dyke's matching principle to capture fine-scale boundary layer structures, making it ideal for high-accuracy scenarios. Numerical experiments on various models show that our proposed methods consistently outperform existing baselines under various error metrics, thereby offering a powerful new approach for multi-scale problems. Keywords: Singularly perturbed problem; Matched asymptotic expansions; Machine learning; Operator learning; Multi-scale problem1. Introduction Singular perturbation problems are prevalent in various scientific and engineering disciplines, including fluid mechanics [1, 2], aerodynamics [3, 4, 5], solid mechanics [6, 7, 8], and biological transport [9]. For example, in aerodynamics, singular perturbation methods are used to analyze rapid pressure and velocity changes around a thin airfoil.

Laplacian matrices are commonly employed in many real applications, encoding the underlying latent structural information such as graphs and manifolds. The use of the normalization terms naturally gives rise to random matrices with dependency. It is well-known that dependency is a major bottleneck of new random matrix theory (RMT) developments. To this end, in this paper, we formally introduce a class of generalized (and regularized) Laplacian matrices, which contains the Laplacian matrix and the random adjacency matrix as a specific case, and suggest the new framework of the asymptotic theory of eigenvectors for latent embeddings with generalized Laplacian matrices (ATE-GL). Our new theory is empowered by the tool of generalized quadratic vector equation for dealing with RMT under dependency, and delicate high-order asymptotic expansions of the empirical spiked eigenvectors and eigenvalues based on local laws. The asymptotic normalities established for both spiked eigenvectors and eigenvalues will enable us to conduct precise inference and uncertainty quantification for applications involving the generalized Laplacian matrices with flexibility. We discuss some applications of the suggested ATE-GL framework and showcase its validity through some numerical examples.

Motivated by recent progress in the problem of numerical K\"ahler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to K\"ahler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the $h$-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.

Solving Singularly Perturbed Differential Equations (SPDEs) presents challenges due to the rapid change of their solutions at the boundary layer. In this manuscript, We propose Asymptotic Physics-Informed Neural Networks (ASPINN), a generalization of Physics-Informed Neural Networks (PINN) and General-Kindred Physics-Informed Neural Networks (GKPINN) approaches. This is a decomposition method based on the idea of asymptotic analysis. Compared to PINN, the ASPINN method has a strong fitting ability for solving SPDEs due to the placement of exponential layers at the boundary layer. Unlike GKPINN, ASPINN lessens the number of fully connected layers, thereby reducing the training cost more effectively. Moreover, ASPINN theoretically approximates the solution at the boundary layer more accurately, which accuracy is also improved compared to GKPINN. We demonstrate the effect of ASPINN by solving diverse classes of SPDEs, which clearly shows that the ASPINN method is promising in boundary layer problems. Furthermore, we introduce Chebyshev Kolmogorov-Arnold Networks (Chebyshev-KAN) instead of MLP, achieving better performance in various experiments.

Community detection is one of the most critical problems in modern network science. Its applications can be found in various fields, from protein modeling to social network analysis. Recently, many papers appeared studying the problem of overlapping community detection, where each node of a network may belong to several communities. In this work, we consider Mixed-Membership Stochastic Block Model (MMSB) first proposed by Airoldi et al. (2008). MMSB provides quite a general setting for modeling overlapping community structure in graphs. The central question of this paper is to reconstruct relations between communities given an observed network. We compare different approaches and establish the minimax lower bound on the estimation error. Then, we propose a new estimator that matches this lower bound. Theoretical results are proved under fairly general conditions on the considered model. Finally, we illustrate the theory in a series of experiments.

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.

Abstract:: In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equation we propose an efficient inversion algorithm, AER, for inverse source problems. We prove that this simplification will not decrease the accuracy of the inversion result, especially for inverse problems with noisy data. Abstract: This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction -- diffusion -- advection equation.

Large-scale network inference with uncertainty quantification has important applications in natural, social, and medical sciences. The recent work of Fan, Fan, Han and Lv (2022) introduced a general framework of statistical inference on membership profiles in large networks (SIMPLE) for testing the sharp null hypothesis that a pair of given nodes share the same membership profiles. In real applications, there are often groups of nodes under investigation that may share similar membership profiles at the presence of relatively weaker signals than the setting considered in SIMPLE. To address these practical challenges, in this paper we propose a SIMPLE method with random coupling (SIMPLE-RC) for testing the non-sharp null hypothesis that a group of given nodes share similar (not necessarily identical) membership profiles under weaker signals. Utilizing the idea of random coupling, we construct our test as the maximum of the SIMPLE tests for subsampled node pairs from the group. Such technique reduces significantly the correlation among individual SIMPLE tests while largely maintaining the power, enabling delicate analysis on the asymptotic distributions of the SIMPLE-RC test. Our method and theory cover both the cases with and without node degree heterogeneity. These new theoretical developments are empowered by a second-order expansion of spiked eigenvectors under the $\ell_\infty$-norm, built upon our work for random matrices with weak spikes. Our theoretical results and the practical advantages of the newly suggested method are demonstrated through several simulation and real data examples.