This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nystrom-based interpolation method.

This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the 'flat' geometry around saddle points, first-order methods can struggle to escape these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing analytic techniques do not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions, which explicitly bring out the dependence on problem dimension, conditioning of the saddle neighborhood, and more, for a class of strict-saddle functions.

Boosting methods have been introduced in the late 1980's. They were born following the theoritical aspect of PAC learning. The main idea of boosting methods is to combine weak learners to obtain a strong learner. The weak learners are obtained iteratively by an heuristic which tries to correct the mistakes of the previous weak learner. In 1995, Freund and Schapire [18] introduced AdaBoost, a boosting algorithm that is still widely used today. Since then, many views of the algorithm have been proposed to properly tame its dynamics. In this paper, we will try to cover all the views that one can have on AdaBoost. We will start with the original view of Freund and Schapire before covering the different views and unify them with the same formalism. We hope this paper will help the non-expert reader to better understand the dynamics of AdaBoost and how the different views are equivalent and related to each other.

This paper addresses challenges in robust transfer learning stemming from ambiguity in Bayes classifiers and weak transferable signals between the target and source distribution. We introduce a novel quantity called the ''ambiguity level'' that measures the discrepancy between the target and source regression functions, propose a simple transfer learning procedure, and establish a general theorem that shows how this new quantity is related to the transferability of learning in terms of risk improvements. Our proposed ''Transfer Around Boundary'' (TAB) model, with a threshold balancing the performance of target and source data, is shown to be both efficient and robust, improving classification while avoiding negative transfer. Moreover, we demonstrate the effectiveness of the TAB model on non-parametric classification and logistic regression tasks, achieving upper bounds which are optimal up to logarithmic factors. Simulation studies lend further support to the effectiveness of TAB. We also provide simple approaches to bound the excess misclassification error without the need for specialized knowledge in transfer learning.

While the test-negative design (TND), which is routinely used for monitoring seasonal flu vaccine effectiveness (VE), has recently become integral to COVID-19 vaccine surveillance, it is susceptible to selection bias due to outcome-dependent sampling. Some studies have addressed the identifiability and estimation of causal parameters under the TND, but efficiency bounds for nonparametric estimators of the target parameter under the unconfoundedness assumption have not yet been investigated. We propose a one-step doubly robust and locally efficient estimator called TNDDR (TND doubly robust), which utilizes sample splitting and can incorporate machine learning techniques to estimate the nuisance functions. We derive the efficient influence function (EIF) for the marginal expectation of the outcome under a vaccination intervention, explore the von Mises expansion, and establish the conditions for $\sqrt{n}-$consistency, asymptotic normality and double robustness of TNDDR. The proposed TNDDR is supported by both theoretical and empirical justifications, and we apply it to estimate COVID-19 VE in an administrative dataset of community-dwelling older people (aged $\geq 60$y) in the province of Qu\'ebec, Canada.

Stress testing refers to a set of methods and tools that assess the impact of an adverse scenario on a financial portfolio. An adverse scenario could, for example, be described as a downturn of macroeconomic and financial risk factors. Typically, a factor model links the risk factors with asset returns, which in turn allows to calculate the impact of the stress scenario on a portfolio. Using techniques from statistics and machine learning, we extend the universe of risk factors by aggregating existing risk factors into higher-level risk factors, such as a global risk factor, broad geographic regions or cyclical and non-cyclical industries. The methods developed also allow to evaluate the strength or weakness over time of aggregated risk factors, such as the intensity of global risk, which changes substantially over time.

The premise of identifiable and causal representation learning is to improve the current representation learning paradigm in terms of generalizability or robustness. Despite recent progress in questions of identifiability, more theoretical results demonstrating concrete advantages of these methods for downstream tasks are needed. In this paper, we consider the task of intervention extrapolation: predicting how interventions affect an outcome, even when those interventions are not observed at training time, and show that identifiable representations can provide an effective solution to this task even if the interventions affect the outcome non-linearly. Our setup includes an outcome Y, observed features X, which are generated as a non-linear transformation of latent features Z, and exogenous action variables A, which influence Z. The objective of intervention extrapolation is to predict how interventions on A that lie outside the training support of A affect Y. Here, extrapolation becomes possible if the effect of A on Z is linear and the residual when regressing Z on A has full support. As Z is latent, we combine the task of intervention extrapolation with identifiable representation learning, which we call Rep4Ex: we aim to map the observed features X into a subspace that allows for non-linear extrapolation in A. We show using Wiener's Tauberian theorem that the hidden representation is identifiable up to an affine transformation in Z-space, which is sufficient for intervention extrapolation. The identifiability is characterized by a novel constraint describing the linearity assumption of A on Z. Based on this insight, we propose a method that enforces the linear invariance constraint and can be combined with any type of autoencoder. We validate our theoretical findings through synthetic experiments and show that our approach succeeds in predicting the effects of unseen interventions.

Water resources are essential for sustaining human livelihoods and environmental well being. Accurate water quality prediction plays a pivotal role in effective resource management and pollution mitigation. In this study, we assess the effectiveness of five distinct predictive models linear regression, Random Forest, XGBoost, LightGBM, and MLP neural network, in forecasting pH values within the geographical context of Georgia, USA. Notably, LightGBM emerges as the top performing model, achieving the highest average precision. Our analysis underscores the supremacy of tree-based models in addressing regression challenges, while revealing the sensitivity of MLP neural networks to feature scaling. Intriguingly, our findings shed light on a counterintuitive discovery: machine learning models, which do not explicitly account for time dependencies and spatial considerations, outperform spatial temporal models. This unexpected superiority of machine learning models challenges conventional assumptions and highlights their potential for practical applications in water quality prediction. Our research aims to establish a robust predictive pipeline accessible to both data science experts and those without domain specific knowledge. In essence, we present a novel perspective on achieving high prediction accuracy and interpretability in data science methodologies. Through this study, we redefine the boundaries of water quality forecasting, emphasizing the significance of data driven approaches over traditional spatial temporal models. Our findings offer valuable insights into the evolving landscape of water resource management and environmental protection.

Discovering causal relationships from observational data is a fundamental yet challenging task. Invariant causal prediction (ICP, Peters et al., 2016) is a method for causal feature selection which requires data from heterogeneous settings and exploits that causal models are invariant. ICP has been extended to general additive noise models and to nonparametric settings using conditional independence tests. However, the latter often suffer from low power (or poor type I error control) and additive noise models are not suitable for applications in which the response is not measured on a continuous scale, but reflects categories or counts. Here, we develop transformation-model (TRAM) based ICP, allowing for continuous, categorical, count-type, and uninformatively censored responses (these model classes, generally, do not allow for identifiability when there is no exogenous heterogeneity). As an invariance test, we propose TRAM-GCM based on the expected conditional covariance between environments and score residuals with uniform asymptotic level guarantees. For the special case of linear shift TRAMs, we also consider TRAM-Wald, which tests invariance based on the Wald statistic. We provide an open-source R package 'tramicp' and evaluate our approach on simulated data and in a case study investigating causal features of survival in critically ill patients.

This information can be captured by an n n adjacency matrix A, where A(i, j) = 1 indicates the existence of an edge between vertices i and j, and A(i, j) = 0 indicates the absence of an edge. Traditionally, the adjacency matrix is binary and primarily used for representing unweighted graphs. Its capacity extends to handling weighted graphs, where the elements of A are assigned the corresponding edge weights. Moreover, graph representation can be used for any data via proper transformations. For instance, when vertex attributes are available, they can be transformed into a pairwise distance or kernel matrix, thus creating a general graph structure. This adaptability enables the incorporation of additional information from vertex attributes, making graph representation a powerful tool for data analysis that goes beyond traditional binary or weighted graphs. Graph embedding is a fundamental and versatile approach for analyzing and exploring graph data, encompassing a wide range of techniques such as spectral embedding [7, 8], graph convolutional neural networks [9, 10], node2vec [11, 12], among others. By projecting the vertices into a low-dimensional space while preserving the structural information of the graph, graph embedding yields vertex representations in Euclidean space that facilitate various downstream inference tasks, such as community detection [13, 14], vertex classification [15, 9], outlier detection [16, 17], etc.