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We provide an a priori analysis of a certain class of numerical methods, commonly referred to as collocation methods, for solving elliptic boundary value problems. They begin with information in the form of point values of the right side f of such equations and point values of the boundary function g and utilize only this information to numerically approximate the solution u of the Partial Differential Equation (PDE). For such a method to provide an approximation to u with guaranteed error bounds, additional assumptions on f and g, called model class assumptions, are needed. We determine the best error (in the energy norm) of approximating u, in terms of the number of point samples m, under all Besov class model assumptions for the right hand side $f$ and boundary g. We then turn to the study of numerical procedures and asks whether a proposed numerical procedure (nearly) achieves the optimal recovery error. We analyze numerical methods which generate the numerical approximation to $u$ by minimizing a specified data driven loss function over a set $\Sigma$ which is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a correct data driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis a loss function $L^*$ is proposed. We also address the recent methods of Physics Informed Neural Networks (PINNs). Minimization of the new loss $L^*$ over neural network spaces $\Sigma$ is referred to as consistent PINNs (CPINNs). We prove that CPINNs provides an optimal recovery of the solution $u$, provided that the optimization problem can be numerically executed and $\Sigma$ has sufficient approximation capabilities. Finally, numerical examples illustrating the benefits of the CPINNs are given.

Estimating the sharing of genetic effects across different conditions is important to many statistical analyses of genomic data. The patterns of sharing arising from these data are often highly heterogeneous. To flexibly model these heterogeneous sharing patterns, Urbut et al. (2019) proposed the multivariate adaptive shrinkage (MASH) method to jointly analyze genetic effects across multiple conditions. However, multivariate analyses using MASH (as well as other multivariate analyses) require good estimates of the sharing patterns, and estimating these patterns efficiently and accurately remains challenging. Here we describe new empirical Bayes methods that provide improvements in speed and accuracy over existing methods. The two key ideas are: (1) adaptive regularization to improve accuracy in settings with many conditions; (2) improving the speed of the model fitting algorithms by exploiting analytical results on covariance estimation. In simulations, we show that the new methods provide better model fits, better out-of-sample performance, and improved power and accuracy in detecting the true underlying signals. In an analysis of eQTLs in 49 human tissues, our new analysis pipeline achieves better model fits and better out-of-sample performance than the existing MASH analysis pipeline. We have implemented the new methods, which we call ``Ultimate Deconvolution'', in an R package, udr, available on GitHub.

We present an information-theoretic lower bound for the problem of parameter estimation with time-uniform coverage guarantees. Via a new a reduction to sequential testing, we obtain stronger lower bounds that capture the hardness of the time-uniform setting. In the case of location model estimation, logistic regression, and exponential family models, our $\Omega(\sqrt{n^{-1}\log \log n})$ lower bound is sharp to within constant factors in typical settings.

Recently, addressing spatial confounding has become a major topic in spatial statistics. However, the literature has provided conflicting definitions, and many proposed definitions do not address the issue of confounding as it is understood in causal inference. We define spatial confounding as the existence of an unmeasured causal confounder with a spatial structure. We present a causal inference framework for nonparametric identification of the causal effect of a continuous exposure on an outcome in the presence of spatial confounding. We propose double machine learning (DML), a procedure in which flexible models are used to regress both the exposure and outcome variables on confounders to arrive at a causal estimator with favorable robustness properties and convergence rates, and we prove that this approach is consistent and asymptotically normal under spatial dependence. As far as we are aware, this is the first approach to spatial confounding that does not rely on restrictive parametric assumptions (such as linearity, effect homogeneity, or Gaussianity) for both identification and estimation. We demonstrate the advantages of the DML approach analytically and in simulations. We apply our methods and reasoning to a study of the effect of fine particulate matter exposure during pregnancy on birthweight in California.

We introduce a sparse estimation in the ordinary kriging for functional data. The functional kriging predicts a feature given as a function at a location where the data are not observed by a linear combination of data observed at other locations. To estimate the weights of the linear combination, we apply the lasso-type regularization in minimizing the expected squared error. We derive an algorithm to derive the estimator using the augmented Lagrange method. Tuning parameters included in the estimation procedure are selected by cross-validation. Since the proposed method can shrink some of the weights of the linear combination toward zeros exactly, we can investigate which locations are necessary or unnecessary to predict the feature. Simulation and real data analysis show that the proposed method appropriately provides reasonable results.

Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. Software implementations of our algorithms in Cython-wrapped C are made available at //github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development.

Genome assembly is a prominent problem studied in bioinformatics, which computes the source string using a set of its overlapping substrings. Classically, genome assembly uses assembly graphs built using this set of substrings to compute the source string efficiently, having a tradeoff between scalability and avoiding information loss. The scalable de Bruijn graphs come at the price of losing crucial overlap information. The complete overlap information is stored in overlap graphs using quadratic space. Hierarchical overlap graphs [IPL20] (HOG) overcome these limitations, avoiding information loss despite using linear space. After a series of suboptimal improvements, Khan and Park et al. simultaneously presented two optimal algorithms [CPM2021], where only the former was seemingly practical. We empirically analyze all the practical algorithms for computing HOG on real and random datasets, where the optimal algorithm [CPM2021] outperforms the previous algorithms as expected, though at the expense of extra memory. However, it uses non-intuitive approach and non-trivial data structures. We present arguably the most intuitive algorithm, using only elementary arrays, which is also optimal. Our algorithm empirically proves even better for both time and memory over all the algorithms, highlighting its significance in both theory and practice. We further explore the applications of hierarchical overlap graphs to solve various forms of suffix-prefix queries on a set of strings. Loukides et al. [CPM2023] recently presented state-of-the-art algorithms for these queries. However, these algorithms require complex black-box data structures and are seemingly impractical. Our algorithms, despite failing to match the state-of-the-art algorithms theoretically, answer different queries ranging from 0.01-100 milliseconds for a data set having around a billion characters.

In causal inference, sensitivity models assess how unmeasured confounders could alter causal analyses, but the sensitivity parameter -- which quantifies the degree of unmeasured confounding -- is often difficult to interpret. For this reason, researchers sometimes compare the sensitivity parameter to an estimate for measured confounding. This is known as calibration. Although calibration can aid interpretation, it is typically conducted post hoc, and uncertainty in the point estimate for measured confounding is rarely accounted for. To address these limitations, we propose novel calibrated sensitivity models, which directly bound the degree of unmeasured confounding by a multiple of measured confounding. The calibrated sensitivity parameter is interpretable as an intuitive unit-less ratio of unmeasured to measured confounding, and uncertainty due to estimating measured confounding can be incorporated. Incorporating this uncertainty shows causal analyses can be less or more robust to unmeasured confounding than would have been suggested by standard approaches. We develop efficient estimators and inferential methods for bounds on the average treatment effect with three calibrated sensitivity models, establishing parametric efficiency and asymptotic normality under doubly robust style nonparametric conditions. We illustrate our methods with a data analysis of the effect of mothers' smoking on infant birthweight.

This work aims to improve generalization and interpretability of dynamical systems by recovering the underlying lower-dimensional latent states and their time evolutions. Previous work on disentangled representation learning within the realm of dynamical systems focused on the latent states, possibly with linear transition approximations. As such, they cannot identify nonlinear transition dynamics, and hence fail to reliably predict complex future behavior. Inspired by the advances in nonlinear ICA, we propose a state-space modeling framework in which we can identify not just the latent states but also the unknown transition function that maps the past states to the present. We introduce a practical algorithm based on variational auto-encoders and empirically demonstrate in realistic synthetic settings that we can (i) recover latent state dynamics with high accuracy, (ii) correspondingly achieve high future prediction accuracy, and (iii) adapt fast to new environments.

We propose a meta-learning method for positive and unlabeled (PU) classification, which improves the performance of binary classifiers obtained from only PU data in unseen target tasks. PU learning is an important problem since PU data naturally arise in real-world applications such as outlier detection and information retrieval. Existing PU learning methods require many PU data, but sufficient data are often unavailable in practice. The proposed method minimizes the test classification risk after the model is adapted to PU data by using related tasks that consist of positive, negative, and unlabeled data. We formulate the adaptation as an estimation problem of the Bayes optimal classifier, which is an optimal classifier to minimize the classification risk. The proposed method embeds each instance into a task-specific space using neural networks. With the embedded PU data, the Bayes optimal classifier is estimated through density-ratio estimation of PU densities, whose solution is obtained as a closed-form solution. The closed-form solution enables us to efficiently and effectively minimize the test classification risk. We empirically show that the proposed method outperforms existing methods with one synthetic and three real-world datasets.