Gaussian process (GP) regression is a popular surrogate modeling tool for computer simulations in engineering and scientific domains. However, it often struggles with high computational costs and low prediction accuracy when the simulation involves too many input variables. For some simulation models, the outputs may only be significantly influenced by a small subset of the input variables, referred to as the ``active variables''. We propose an optimal kernel learning approach to identify these active variables, thereby overcoming GP model limitations and enhancing system understanding. Our method approximates the original GP model's covariance function through a convex combination of kernel functions, each utilizing low-dimensional subsets of input variables. Inspired by the Fedorov-Wynn algorithm from optimal design literature, we develop an optimal kernel learning algorithm to determine this approximation. We incorporate the effect heredity principle, a concept borrowed from the field of ``design and analysis of experiments'', to facilitate efficient computation. Through several examples, we demonstrate that the proposed method provides competitive prediction and variable selection, with favorable computational performance in the examples considered. It is an effective solution for interpreting the surrogate GP regression and simplifying the complex underlying system.
This paper analyzes realized return behavior across a broad set of crypto assets by estimating heterogeneous exposures to idiosyncratic and systematic risk. A key challenge arises from the latent nature of broader economy-wide risk sources: macro-financial proxies are unavailable at high-frequencies, while the abundance of low-frequency candidates offers limited guidance on empirical relevance. To address this, we develop a two-stage ``divide-and-conquer'' approach. The first stage estimates exposures to high-frequency idiosyncratic and market risk only, using asset-level IV regressions. The second stage identifies latent economy-wide factors by extracting the leading principal component from the model residuals and mapping it to lower-frequency macro-financial uncertainty and sentiment-based indicators via high-dimensional variable selection. Structured patterns of heterogeneity in exposures are uncovered using Mean Group estimators across asset categories. The method is applied to a broad sample of crypto assets, covering more than 80% of total market capitalization. We document short-term mean reversion and significant average exposures to idiosyncratic volatility and illiquidity. Green and DeFi assets are, on average, more exposed to market-level and economy-wide risk than their non-Green and non-DeFi counterparts. By contrast, stablecoins are less exposed to idiosyncratic, market-level, and economy-wide risk factors relative to non-stablecoins. At a conceptual level, our study develops a coherent framework for isolating distinct layers of risk in crypto markets. Empirically, it sheds light on how return sensitivities vary across digital asset categories -- insights that are important for both portfolio design and regulatory oversight.
Researchers in urban and regional studies increasingly deal with spatial data that reflects geographic location and spatial relationships. As a framework for dealing with the unique nature of spatial data, various spatial regression models have been introduced. In this article, a novel model-based gradient boosting algorithm for spatial regression models with autoregressive disturbances is proposed. Due to the modular nature, the approach provides an alternative estimation procedure which is feasible even in high-dimensional settings where established quasi-maximum likelihood or generalized method of moments estimators do not yield unique solutions. The approach additionally enables data-driven variable and model selection in low- as well as high-dimensional settings. Since the bias-variance trade-off is also controlled in the algorithm, implicit regularization is imposed which improves prediction accuracy on out-of-sample spatial data. Detailed simulation studies regarding the performance of estimation, prediction and variable selection in low- and high-dimensional settings confirm proper functionality of the proposed methodology. To illustrative the functionality of the model-based gradient boosting algorithm, a case study is presented where the life expectancy in German districts is modeled incorporating a potential spatial dependence structure.
Granger Causality (GC) offers an elegant statistical framework to study the association between multivariate time series data. Linear Vector Autoregressive models (VAR) though have nice interpretation properties but have limited practical application due to underlying assumptions on the kind of associations that can be captured by these models. Numerous attempts have already been made in the literature that exploit the functional approximation power of Deep Neural Networks (DNNs) for the task of GC estimation. These methods however treat GC as a variable selection problem. We present a novel paradigm for approaching GC. We present this idea that GC is essentially linked with prediction and if a deep learning model is used to model the time series collectively or jointly, a well regularized model may learn the true granger causal structure from the data, given that there is enough training data. We propose to uncover the learned GC structure by comparing the model uncertainty or distribution of the residuals when the past of everything is used as compared to the one where a specific time series component is dropped from the model. We also compare the effect of input layer dropout on the ability of a neural network to learn granger causality from the data. We show that a well regularized model infact can learn the true GC structure from the data without explicitly adding terms in the loss function that guide the model to select variables or perform sparse regression.
Gallstone disease is a complex, multifactorial condition with significant global health burdens. Identifying underlying risk factors and their interactions is crucial for early diagnosis, targeted prevention, and effective clinical management. Although logistic regression remains a standard tool for assessing associations between predictors and gallstone status, it often underperforms in high-dimensional settings and may fail to capture intricate relationships among variables. To address these limitations, we propose a hybrid machine learning framework that integrates robust variable selection with advanced interaction detection. Specifically, Adaptive LASSO is employed to identify a sparse and interpretable subset of influential features, followed by Bayesian Additive Regression Trees (BART) to model nonlinear effects and uncover key interactions. Selected interactions are further characterized by physiological knowledge through differential equation-informed interaction terms, grounding the model in biologically plausible mechanisms. The insights gained from these steps are then integrated into a final logistic regression model within a Bayesian framework, providing a balance between predictive accuracy and clinical interpretability. This proposed framework not only enhances prediction but also yields actionable insights, offering a valuable support tool for medical research and decision-making.
Model-X knockoff framework offers a model-free variable selection method that ensures finite sample false discovery rate (FDR) control. However, the complexity of generating knockoff variables, coupled with the model-free assumption, presents significant challenges for protecting data privacy in this context. In this paper, we propose a comprehensive framework for knockoff inference within the differential privacy paradigm. Our proposed method guarantees robust privacy protection while preserving the exact FDR control entailed by the original model-X knockoff procedure. We further conduct power analysis and establish sufficient conditions under which the noise added for privacy preservation does not asymptotically compromise power. Through various applications, we demonstrate that the differential privacy knockoff (DP-knockoff) method can be effectively utilized to safeguard privacy during variable selection with FDR control in both low and high dimensional settings.
Feature selection is a critical step in high-dimensional classification tasks, particularly under challenging conditions of double imbalance, namely settings characterized by both class imbalance in the response variable and dimensional asymmetry in the data $(n \gg p)$. In such scenarios, traditional feature selection methods applied to Random Forests (RF) often yield unstable or misleading importance rankings. This paper proposes a novel thresholding scheme for feature selection based on minimal depth, which exploits the tree topology to assess variable relevance. Extensive experiments on simulated and real-world datasets demonstrate that the proposed approach produces more parsimonious and accurate subsets of variables compared to conventional minimal depth-based selection. The method provides a practical and interpretable solution for variable selection in RF under double imbalance conditions.
The Lasso is a prominent algorithm for variable selection. However, its instability in the presence of correlated variables in the high-dimensional setting is well-documented. Although previous research has attempted to address this issue by modifying the Lasso loss function, this paper introduces an approach that simplifies the data processed by Lasso. We propose that decorrelating variables before applying the Lasso improves the stability of variable selection regardless of the direction of correlation among predictors. Furthermore, we highlight that the irrepresentable condition, which ensures consistency for the Lasso, is satisfied after variable decorrelation under two assumptions. In addition, by noting that the instability of the Lasso is not limited to high-dimensional settings, we demonstrate the effectiveness of the proposed approach for low-dimensional data. Finally, we present empirical results that indicate the efficacy of the proposed method across different variable selection techniques, highlighting its potential for broader application. The DVS R package is developed to facilitate the implementation of the methodology proposed in this paper.
Bounded continuous responses -- such as proportions -- arise frequently in diverse scientific fields including climatology, biostatistics, and finance. Beta regression is a widely adopted framework for modeling such data, due to the flexibility of the Beta distribution over the unit interval. While Bayesian extensions of Beta regression have shown promise, existing methods are limited to low-dimensional settings and lack theoretical guarantees. In this work, we propose a novel Bayesian approach for high-dimensional sparse Beta regression framework that employs a tempered posterior. Our method incorporates the Horseshoe prior for effective shrinkage and variable selection. Most notable, we propose a novel Gibbs sampling algorithm using P\'olya-Gamma augmentation for efficient inference in Beta regression model. We also provide the first theoretical results establishing posterior consistency and convergence rates for Bayesian Beta regression. Through extensive simulation studies in both low- and high-dimensional scenarios, we demonstrate that our approach outperforms existing alternatives, offering improved estimation accuracy and model interpretability. Our method is implemented in the R package ``betaregbayes" available on Github.
Cerebral aneurysms are pathological dilations of intracranial arteries that can rupture with devastating consequences, including subarachnoid hemorrhage, stroke, and death. Accumulating evidence indicates that local hemodynamic forces play a critical role in aneurysm initiation, growth, and rupture. Computational fluid dynamics (CFD) and imaging-based techniques have enabled the extraction of various hemodynamic variables to characterize these flow conditions. However, the literature is highly fragmented, with different studies adopting distinct sets of metrics such as wall shear stress (WSS), oscillatory shear index (OSI), wall shear stress gradient (WSSG), relative residence time (RRT), or endothelial cell activation potential (ECAP) making it difficult to compare results or establish standardized methodologies. This paper provides the first comprehensive catalog of hemodynamic variables used in cerebral aneurysm studies to date. By systematically identifying and organizing these parameters based on their physical basis and frequency of use, this work offers a consolidated reference to guide future research. The goal is to support consistent variable selection, enhance reproducibility, and facilitate the design of more robust studies linking vascular biomechanics to aneurysm pathophysiology. This review aims to serve as a foundational resource for researchers and clinicians seeking to incorporate hemodynamic modeling into cerebral aneurysm analysis and risk assessment.
Model-based clustering integrated with variable selection is a powerful tool for uncovering latent structures within complex data. However, its effectiveness is often hindered by challenges such as identifying relevant variables that define heterogeneous subgroups and handling data that are missing not at random, a prevalent issue in fields like transcriptomics. While several notable methods have been proposed to address these problems, they typically tackle each issue in isolation, thereby limiting their flexibility and adaptability. This paper introduces a unified framework designed to address these challenges simultaneously. Our approach incorporates a data-driven penalty matrix into penalized clustering to enable more flexible variable selection, along with a mechanism that explicitly models the relationship between missingness and latent class membership. We demonstrate that, under certain regularity conditions, the proposed framework achieves both asymptotic consistency and selection consistency, even in the presence of missing data. This unified strategy significantly enhances the capability and efficiency of model-based clustering, advancing methodologies for identifying informative variables that define homogeneous subgroups in the presence of complex missing data patterns. The performance of the framework, including its computational efficiency, is evaluated through simulations and demonstrated using both synthetic and real-world transcriptomic datasets.
Mixed-integer programming (MIP) provides a powerful framework for optimization problems, with Branch-and-Cut (B&C) being the predominant algorithm in state-of-the-art solvers. The efficiency of B&C critically depends on heuristic policies for making sequential decisions, including node selection, cut selection, and branching variable selection. While traditional solvers often employ heuristics with manually tuned parameters, recent approaches increasingly leverage machine learning, especially neural networks, to learn these policies directly from data. A key challenge is to understand the theoretical underpinnings of these learned policies, particularly their generalization performance from finite data. This paper establishes rigorous sample complexity bounds for learning B&C policies where the scoring functions guiding each decision step (node, cut, branch) have a certain piecewise polynomial structure. This structure generalizes the linear models that form the most commonly deployed policies in practice and investigated recently in a foundational series of theoretical works by Balcan et al. Such piecewise polynomial policies also cover the neural network architectures (e.g., using ReLU activations) that have been the focal point of contemporary practical studies. Consequently, our theoretical framework closely reflects the models utilized by practitioners investigating machine learning within B&C, offering a unifying perspective relevant to both established theory and modern empirical research in this area. Furthermore, our theory applies to quite general sequential decision making problems beyond B&C.
In large-scale supervised learning, penalized logistic regression (PLR) effectively addresses the overfitting problem by introducing regularization terms yet its performance still depends on efficient variable selection strategies. This paper theoretically demonstrates that label noise stemming from manual labeling, which is solely related to classification difficulty, represents a type of beneficial noise for variable selection in PLR. This benefit is reflected in a more accurate estimation of the selected non-zero coefficients when compared with the case where only truth labels are used. Under large-scale settings, the sample size for PLR can become very large, making it infeasible to store on a single machine. In such cases, distributed computing methods are required to handle PLR model with manual labeling. This paper presents a partition-insensitive parallel algorithm founded on the ADMM (alternating direction method of multipliers) algorithm to address PLR by incorporating manual labeling. The partition insensitivity of the proposed algorithm refers to the fact that the solutions obtained by the algorithm will not change with the distributed storage of data. In addition, the algorithm has global convergence and a sublinear convergence rate. Experimental results indicate that, as compared with traditional variable selection classification techniques, the PLR with manually-labeled noisy data achieves higher estimation and classification accuracy across multiple large-scale datasets.
Variable selection in ultrahigh-dimensional linear regression is challenging due to its high computational cost. Therefore, a screening step is usually conducted before variable selection to significantly reduce the dimension. Here we propose a novel and simple screening method based on ordering the absolute sample ridge partial correlations. The proposed method takes into account not only the ridge regularized estimates of the regression coefficients but also the ridge regularized partial variances of the predictor variables providing sure screening property without strong assumptions on the marginal correlations. Simulation study and a real data analysis show that the proposed method has a competitive performance compared with the existing screening procedures. A publicly available software implementing the proposed screening accompanies the article.
We propose a novel nonparametric approach for linking covariates to Continuous Time Markov Chains (CTMCs) using the mathematical framework of Reproducing Kernel Hilbert Spaces (RKHS). CTMCs provide a robust framework for modeling transitions across clinical or behavioral states, but traditional multistate models often rely on linear relationships. In contrast, we use a generalized Representer Theorem to enable tractable inference in functional space. For the Frequentist version, we apply normed square penalties, while for the Bayesian version, we explore sparsity inducing spike and slab priors. Due to the computational challenges posed by high-dimensional spaces, we successfully adapt the Expectation Maximization Variable Selection (EMVS) algorithm to efficiently identify the posterior mode. We demonstrate the effectiveness of our method through extensive simulation studies and an application to follicular cell lymphoma data. Our performance metrics include the normalized difference between estimated and true nonlinear transition functions, as well as the difference in the probability of getting absorbed in one the final states, capturing the ability of our approach to predict long-term behaviors.
Identifying relationships among stochastic processes is a key goal in disciplines that deal with complex temporal systems, such as economics. While the standard toolkit for multivariate time series analysis has many advantages, it can be difficult to capture nonlinear dynamics using linear vector autoregressive models. This difficulty has motivated the development of methods for variable selection, causal discovery, and graphical modeling for nonlinear time series, which routinely employ nonparametric tests for conditional independence. In this paper, we introduce the first framework for conditional independence testing that works with a single realization of a nonstationary nonlinear process. The key technical ingredients are time-varying nonlinear regression, time-varying covariance estimation, and a distribution-uniform strong Gaussian approximation.
Transformers have demonstrated remarkable success across various applications. However, the success of transformers have not been understood in theory. In this work, we give a case study of how transformers can be trained to learn a classic statistical model with "group sparsity", where the input variables form multiple groups, and the label only depends on the variables from one of the groups. We theoretically demonstrate that, a one-layer transformer trained by gradient descent can correctly leverage the attention mechanism to select variables, disregarding irrelevant ones and focusing on those beneficial for classification. We also demonstrate that a well-pretrained one-layer transformer can be adapted to new downstream tasks to achieve good prediction accuracy with a limited number of samples. Our study sheds light on how transformers effectively learn structured data.
Bayesian optimization (BO) is a leading method for optimizing expensive black-box optimization and has been successfully applied across various scenarios. However, BO suffers from the curse of dimensionality, making it challenging to scale to high-dimensional problems. Existing work has adopted a variable selection strategy to select and optimize only a subset of variables iteratively. Although this approach can mitigate the high-dimensional challenge in BO, it still leads to sample inefficiency. To address this issue, we introduce a novel method that identifies important variables by estimating the length scales of Gaussian process kernels. Next, we construct an effective search region consisting of multiple subspaces and optimize the acquisition function within this region, focusing on only the important variables. We demonstrate that our proposed method achieves cumulative regret with a sublinear growth rate in the worst case while maintaining computational efficiency. Experiments on high-dimensional synthetic functions and real-world problems show that our method achieves state-of-the-art performance.
An important goal in cancer research is the survival prognosis of a patient based on a minimal panel of genomic and molecular markers such as genes or proteins. Purely data-driven models without any biological knowledge can produce non-interpretable results. We propose a penalized semiparametric Bayesian Cox model with graph-structured selection priors for sparse identification of multi-omics features by making use of a biologically meaningful graph via a Markov random field (MRF) prior to capturing known relationships between multi-omics features. Since the fixed graph in the MRF prior is for the prior probability distribution, it is not a hard constraint to determine variable selection, so the proposed model can verify known information and has the potential to identify new and novel biomarkers for drawing new biological knowledge. Our simulation results show that the proposed Bayesian Cox model with graph-based prior knowledge results in more trustable and stable variable selection and non-inferior survival prediction, compared to methods modeling the covariates independently without any prior knowledge. The results also indicate that the performance of the proposed model is robust to a partially correct graph in the MRF prior, meaning that in a real setting where not all the true network information between covariates is known, the graph can still be useful. The proposed model is applied to the primary invasive breast cancer patients data in The Cancer Genome Atlas project.
Variable selection poses a significant challenge in causal modeling, particularly within the social sciences, where constructs often rely on inter-related factors such as age, socioeconomic status, gender, and race. Indeed, it has been argued that such attributes must be modeled as macro-level abstractions of lower-level manipulable features, in order to preserve the modularity assumption essential to causal inference. This paper accordingly extends the theoretical framework of Causal Feature Learning (CFL). Empirically, we apply the CFL algorithm to diverse social science datasets, evaluating how CFL-derived macrostates compare with traditional microstates in downstream modeling tasks.