We study the discrete-to-continuum consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds under a manifold assumption. Graph convolution is defined spectrally via the graph Laplacian, whose low-frequency spectrum approximates that of the Laplace-Beltrami operator of the underlying smooth manifold, and shallow GCNNs of possibly infinite width are linear functionals on the space of measures on the parameter space. From this functional-analytic perspective, graph signals are seen as spatial discretizations of functions on the manifold, which leads to a natural notion of training data consistent across graph resolutions. To enable convergence results, the continuum parameter space is chosen as a weakly compact product of unit balls, with Sobolev regularity imposed on the output weight and bias, but not on the convolutional parameter. The corresponding discrete parameter spaces inherit the corresponding spectral decay, and are additionally restricted by a frequency cutoff adapted to the informative spectral window of the graph Laplacians. Under these assumptions, we prove $Γ$-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers, in the sense of weak convergence of the parameter measures and uniform convergence of the functions over compact sets. This provides a formalization of mesh and sample independence for the training of such networks.

We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.

Modern engineering and scientific workflows frequently require simultaneous prediction across related tasks and fidelity levels [1-6]. In such contexts, some outputs are scarce and expensive to obtain, while others are cheaper and more abundant. Multi-task Gaussian processes (MTGPs), also known as multi-output Gaussian processes, offer a principled Bayesian framework to exploit inter-task correlations, enabling knowledge sharing that improves predictive accuracy and reduces the demand for large high-fidelity datasets [7-9]. Over decades of development, MTGPs have been applied across diverse domains, including time series forecasting, multitask optimization, and multifidelity classification, demonstrating their broad utility wherever data cost asymmetries and cross-task dependencies are present [10-16]. The central motivation for MTGPs is to leverage dependencies among related tasks to enhance predictive quality when high-fidelity information is limited [17]. For example, predicting an airfoil's lift coefficient from limited, expensive high-fidelity computational fluid dynamics (CFD) simulations can benefit from correlating with sufficient low-fidelity simulations [3]. Recent work in joint multi-objective and multifidelity optimization has also utilized MT - GPs to balance exploration and exploitation across tasks, improving predictive performance and decision-making by explicitly modeling relationships among outputs and fidelities [12].

Neural networks have shown state-of-the-art performances in various classification and regression tasks. Rectified linear units (ReLU) are often used as activation functions for the hidden layers in a neural network model. In this article, we establish the connection between the Poisson hyperplane processes (PHP) and two-layer ReLU neural networks. We show that the PHP with a Gaussian prior is an alternative probabilistic representation to a two-layer ReLU neural network. In addition, we show that a two-layer neural network constructed by PHP is scalable to large-scale problems via the decomposition propositions. Finally, we propose an annealed sequential Monte Carlo algorithm for Bayesian inference. Our numerical experiments demonstrate that our proposed method outperforms the classic two-layer ReLU neural network. The implementation of our proposed model is available at https://github.com/ShufeiGe/Pois_Relu.git.

This paper establishes a rigorous theoretical foundation for the function class implicitly learned by XGBoost, bridging the gap between its empirical success and our theoretical understanding. We introduce an infinite-dimensional function class $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ that extends finite ensembles of bounded-depth regression trees, together with a complexity measure $V^{d, s}_{\infty-\text{XGB}}(\cdot)$ that generalizes the $L^1$ regularization penalty used in XGBoost. We show that every optimizer of the XGBoost objective is also an optimizer of an equivalent penalized regression problem over $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ with penalty $V^{d, s}_{\infty-\text{XGB}}(\cdot)$, providing an interpretation of XGBoost as implicitly targeting a broader function class. We also develop a smoothness-based interpretation of $\mathcal{F}^{d, s}_{\infty-\text{ST}}$ and $V^{d, s}_{\infty-\text{XGB}}(\cdot)$ in terms of Hardy--Krause variation. We prove that the least squares estimator over $\{f \in \mathcal{F}^{d, s}_{\infty-\text{ST}}: V^{d, s}_{\infty-\text{XGB}}(f) \le V\}$ achieves a nearly minimax-optimal rate of convergence $n^{-2/3} (\log n)^{4(\min(s, d) - 1)/3}$, thereby avoiding the curse of dimensionality. Our results provide the first rigorous characterization of the function space underlying XGBoost, clarify its connection to classical notions of variation, and identify an important open problem: whether the XGBoost algorithm itself achieves minimax optimality over this class.

In this brief note, we considers the problem of learning with dynamic-optimizing principal-agent setting, in which the agents are allowed to have global perspectives about the learning process, i.e., the ability to view things according to their relative importances or in their true relations based-on some aggregated information shared by the principal. Whereas, the principal, which is exerting an influence on the learning process of the agents in the aggregation, is primarily tasked to solve a high-level optimization problem posed as an empirical-likelihood estimator under conditional moment restrictions model that also accounts information about the agents' predictive performances on out-of-samples as well as a set of private datasets available only to the principal (e.g., see [1], [2], [3], [4] and [5] for further discussions on empirical likelihood methods with moment restrictions). Here, we provide a coherent mathematical argument which is necessary for characterizing the learning process behind this abstract dynamic-optimizing principal-agent learning framework. Note that, due to the inherent feedbacks behavior among the agents, the proposed learning framework remarkably offers some advantages in terms of stability and consistency, despite that both the principal and the agents do not necessarily need to have any knowledge of the sample distributions or the quality of each others datasets. Finally, it is worth remarking that such a learning framework can provide new insights in the context of collaborative learning problem with global perspectives that exploits the principal-agent setting (e.g., see [6], [7], [8] or [9] for related discussions), although we acknowledge that there are a number of conceptual and theoretical problems, such as small sample properties, still need to be addressed.

Empirical models of demand for differentiated products rely on low-dimensional product representations to capture substitution patterns. These representations are increasingly proxied by applying ML methods to high-dimensional, unstructured data, including product descriptions and images. When proxies fail to capture the true dimensions of differentiation that drive substitution, standard workflows will deliver biased counterfactuals and invalid inference. We develop a practical toolkit that corrects this bias and ensures valid inference for a broad class of counterfactuals. Our approach applies to market-level and/or individual data, requires minimal additional computation, is efficient, delivers simple formulas for standard errors, and accommodates data-dependent proxies, including embeddings from fine-tuned ML models. It can also be used with standard quantitative attributes when mismeasurement is a concern. In addition, we propose diagnostics to assess the adequacy of the proxy construction and dimension. The approach yields meaningful improvements in predicting counterfactual substitution in both simulations and an empirical application.

Canonical Polyadic (CP) tensor decomposition is a workhorse algorithm for discovering underlying low-dimensional structure in tensor data. This is accomplished in conventional CP decomposition by fitting a low-rank tensor to data with respect to the least-squares loss. Generalized CP (GCP) decompositions generalize this approach by allowing general loss functions that can be more appropriate, e.g., to model binary and count data or to improve robustness to outliers. However, GCP decompositions do not explicitly account for any symmetry in the tensors, which commonly arises in modern applications. For example, a tensor formed by stacking the adjacency matrices of a dynamic graph over time will naturally exhibit symmetry along the two modes corresponding to the graph nodes. In this paper, we develop a symmetric GCP (SymGCP) decomposition that allows for general forms of symmetry, i.e., symmetry along any subset of the modes. SymGCP accounts for symmetry by enforcing the corresponding symmetry in the decomposition. We derive gradients for SymGCP that enable its efficient computation via all-at-once optimization with existing tensor kernels. The form of the gradients also leads to various stochastic approximations that enable us to develop stochastic SymGCP algorithms that can scale to large tensors. We demonstrate the utility of the proposed SymGCP algorithms with a variety of experiments on both synthetic and real data.

Machine learning assisted state prediction of misspecified linear dynamical system via modal reduction Rohan Vittal Thorat a, Rajdip Nayek a a Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, 110016, IndiaAbstract Accurate prediction of structural dynamics is imperative for preserving digital twin fidelity throughout operational lifetimes. Parametric models with fixed nominal parameters often omit critical physical effects due to simplifications in geometry, material behavior, damping, or boundary conditions, resulting in model form errors (MFEs) that impair predictive accuracy. This work introduces a comprehensive framework for MFE estimation and correction in high-dimensional finite element (FE) based structural dynamical systems. The Gaussian Process Latent Force Model (GPLFM) represents discrepancies non-parametrically in the reduced modal domain, allowing a flexible data-driven characterization of unmodeled dynamics. A linear Bayesian filtering approach jointly estimates system states and discrepancies, incorporating epistemic and aleatoric uncertainties. To ensure computational tractability, the FE system is projected onto a reduced modal basis, and a mesh-invariant neural network maps modal states to discrepancy estimates, permitting model rectification across different FE dis-cretizations without retraining. Validation is undertaken across five MFE scenarios--including incorrect beam theory, damping misspecification, misspecified boundary condition, unmodeled material nonlinearity, and local damage --demonstrating the surrogate model's substantial reduction of displacement and rotation prediction errors under unseen excitations. The proposed methodology offers a potential means to uphold digital twin accuracy amid inherent modeling uncertainties. Keywords: Model bias, Gaussian Process, Latent Force Model, Bayesian filtering, Modal reduction, Digital twin 1. Introduction The reliable simulation of structural dynamical systems is central to engineering analysis, design, and decision-making. In practice, high-fidelity models are often impractical due to limited information, computational constraints, or simplifying assumptions in geometry, boundary conditions, damping mechanisms, and material constitutive laws. These idealizations lead to model form errors (MFEs)--systematic discrepancies between the predicted and actual system responses--which, if unaccounted for, can significantly degrade predictive accuracy. This challenge is especially critical in the context of digital twins, where model predictions directly inform monitoring and decision-making. Digital twins of structural systems integrate computational models with real-time or historical measurement data to enable continuous prediction, monitoring, and decision making [1, 2].

The use of neural networks trained on individual claims data has become increasingly popular in the actuarial reserving literature. We consider how to best input historical payment data in neural network models. Additionally, case estimates are also available in the format of a time series, and we extend our analysis to assessing their predictive power. In this paper, we compare a feed-forward neural network trained on summarised transactions to a recurrent neural network equipped to analyse a claim's entire payment history and/or case estimate development history. We draw conclusions from training and comparing the performance of the models on multiple, comparable highly complex datasets simulated from SPLICE (Avanzi, Taylor and Wang, 2023). We find evidence that case estimates will improve predictions significantly, but that equipping the neural network with memory only leads to meagre improvements. Although the case estimation process and quality will vary significantly between insurers, we provide a standardised methodology for assessing their value.