This research aims to propose and evaluate a novel model named K-Fold Causal Bayesian Additive Regression Trees (K-Fold Causal BART) for improved estimation of Average Treatment Effects (ATE) and Conditional Average Treatment Effects (CATE). The study employs synthetic and semi-synthetic datasets, including the widely recognized Infant Health and Development Program (IHDP) benchmark dataset, to validate the model's performance. Despite promising results in synthetic scenarios, the IHDP dataset reveals that the proposed model is not state-of-the-art for ATE and CATE estimation. Nonetheless, the research provides several novel insights: 1. The ps-BART model is likely the preferred choice for CATE and ATE estimation due to better generalization compared to the other benchmark models - including the Bayesian Causal Forest (BCF) model, which is considered by many the current best model for CATE estimation, 2. The BCF model's performance deteriorates significantly with increasing treatment effect heterogeneity, while the ps-BART model remains robust, 3. Models tend to be overconfident in CATE uncertainty quantification when treatment effect heterogeneity is low, 4. A second K-Fold method is unnecessary for avoiding overfitting in CATE estimation, as it adds computational costs without improving performance, 5. Detailed analysis reveals the importance of understanding dataset characteristics and using nuanced evaluation methods, 6. The conclusion of Curth et al. (2021) that indirect strategies for CATE estimation are superior for the IHDP dataset is contradicted by the results of this research. These findings challenge existing assumptions and suggest directions for future research to enhance causal inference methodologies.

The exponential growth in data sizes and storage costs has brought considerable challenges to the data science community, requiring solutions to run learning methods on such data. While machine learning has scaled to achieve predictive accuracy in big data settings, statistical inference and uncertainty quantification tools are still lagging. Priority scientific fields collect vast data to understand phenomena typically studied with statistical methods like regression. In this setting, regression parameter estimation can benefit from efficient computational procedures, but the main challenge lies in computing error process parameters with complex covariance structures. Identifying and estimating these structures is essential for inference and often used for uncertainty quantification in machine learning with Gaussian Processes. However, estimating these structures becomes burdensome as data scales, requiring approximations that compromise the reliability of outputs. These approximations are even more unreliable when complexities like long-range dependencies or missing data are present. This work defines and proves the statistical properties of the Generalized Method of Wavelet Moments with Exogenous variables (GMWMX), a highly scalable, stable, and statistically valid method for estimating and delivering inference for linear models using stochastic processes in the presence of data complexities like latent dependence structures and missing data. Applied examples from Earth Sciences and extensive simulations highlight the advantages of the GMWMX.

Survival analysis is a branch of statistics used for modeling the time until a specific event occurs and is widely used in medicine, engineering, finance, and many other fields. When choosing survival models, there is typically a trade-off between performance and interpretability, where the highest performance is achieved by black-box models based on deep learning. This is a major problem in fields such as medicine where practitioners are reluctant to blindly trust black-box models to make important patient decisions. Kolmogorov-Arnold Networks (KANs) were recently proposed as an interpretable and accurate alternative to multi-layer perceptrons (MLPs). We introduce CoxKAN, a Cox proportional hazards Kolmogorov-Arnold Network for interpretable, high-performance survival analysis. We evaluate the proposed CoxKAN on 4 synthetic datasets and 9 real medical datasets. The synthetic experiments demonstrate that CoxKAN accurately recovers interpretable symbolic formulae for the hazard function, and effectively performs automatic feature selection. Evaluation on the 9 real datasets show that CoxKAN consistently outperforms the Cox proportional hazards model and achieves performance that is superior or comparable to that of tuned MLPs. Furthermore, we find that CoxKAN identifies complex interactions between predictor variables that would be extremely difficult to recognise using existing survival methods, and automatically finds symbolic formulae which uncover the precise effect of important biomarkers on patient risk.

Recently it has been observed that neural networks exhibit Neural Collapse (NC) during the final stage of training for the classification problem. We empirically show that multivariate regression, as employed in imitation learning and other applications, exhibits Neural Regression Collapse (NRC), a new form of neural collapse: (NRC1) The last-layer feature vectors collapse to the subspace spanned by the $n$ principal components of the feature vectors, where $n$ is the dimension of the targets (for univariate regression, $n=1$); (NRC2) The last-layer feature vectors also collapse to the subspace spanned by the last-layer weight vectors; (NRC3) The Gram matrix for the weight vectors converges to a specific functional form that depends on the covariance matrix of the targets. After empirically establishing the prevalence of (NRC1)-(NRC3) for a variety of datasets and network architectures, we provide an explanation of these phenomena by modeling the regression task in the context of the Unconstrained Feature Model (UFM), in which the last layer feature vectors are treated as free variables when minimizing the loss function. We show that when the regularization parameters in the UFM model are strictly positive, then (NRC1)-(NRC3) also emerge as solutions in the UFM optimization problem. We also show that if the regularization parameters are equal to zero, then there is no collapse. To our knowledge, this is the first empirical and theoretical study of neural collapse in the context of regression. This extension is significant not only because it broadens the applicability of neural collapse to a new category of problems but also because it suggests that the phenomena of neural collapse could be a universal behavior in deep learning.

Aqueous solubility (AS) is a key physiochemical property that plays a crucial role in drug discovery and material design. We report a novel unified approach to predict and infer chemical compounds with the desired AS based on simple deterministic graph-theoretic descriptors, multiple linear regression (MLR) and mixed integer linear programming (MILP). Selected descriptors based on a forward stepwise procedure enabled the simplest regression model, MLR, to achieve significantly good prediction accuracy compared to the existing approaches, achieving the accuracy in the range [0.7191, 0.9377] for 29 diverse datasets. By simulating these descriptors and learning models as MILPs, we inferred mathematically exact and optimal compounds with the desired AS, prescribed structures, and up to 50 non-hydrogen atoms in a reasonable time range [6, 1204] seconds. These findings indicate a strong correlation between the simple graph-theoretic descriptors and the AS of compounds, potentially leading to a deeper understanding of their AS without relying on widely used complicated chemical descriptors and complex machine learning models that are computationally expensive, and therefore difficult to use for inference. An implementation of the proposed approach is available at https://github.com/ku-dml/mol-infer/tree/master/AqSol.

This paper introduces a novel methodology for solving distributed-order fractional differential equations using a physics-informed machine learning framework. The core of this approach involves extending the support vector regression (SVR) algorithm to approximate the unknown solutions of the governing equations during the training phase. By embedding the distributed-order functional equation into the SVR framework, we incorporate physical laws directly into the learning process. To further enhance computational efficiency, Gegenbauer orthogonal polynomials are employed as the kernel function, capitalizing on their fractional differentiation properties to streamline the problem formulation. Finally, the resulting optimization problem of SVR is addressed either as a quadratic programming problem or as a positive definite system in its dual form. The effectiveness of the proposed approach is validated through a series of numerical experiments on Caputo-based distributed-order fractional differential equations, encompassing both ordinary and partial derivatives.

Time series analysis and prediction methods currently excel in quantitative analysis, offering accurate future predictions and diverse statistical indicators, but generally falling short in elucidating the underlying evolution patterns of time series. To gain a more comprehensive understanding and provide insightful explanations, we utilize symbolic regression techniques to derive explicit expressions for the non-linear dynamics in the evolution of time series variables. However, these techniques face challenges in computational efficiency and generalizability across diverse real-world time series data. To overcome these challenges, we propose \textbf{N}eural-\textbf{E}nhanced \textbf{Mo}nte-Carlo \textbf{T}ree \textbf{S}earch (NEMoTS) for time series. NEMoTS leverages the exploration-exploitation balance of Monte-Carlo Tree Search (MCTS), significantly reducing the search space in symbolic regression and improving expression quality. Furthermore, by integrating neural networks with MCTS, NEMoTS not only capitalizes on their superior fitting capabilities to concentrate on more pertinent operations post-search space reduction, but also replaces the complex and time-consuming simulation process, thereby substantially improving computational efficiency and generalizability in time series analysis. NEMoTS offers an efficient and comprehensive approach to time series analysis. Experiments with three real-world datasets demonstrate NEMoTS's significant superiority in performance, efficiency, reliability, and interpretability, making it well-suited for large-scale real-world time series data.

Emerging marketplaces for large language models and other large-scale machine learning (ML) models appear to exhibit market concentration, which has raised concerns about whether there are insurmountable barriers to entry in such markets. In this work, we study this issue from both an economic and an algorithmic point of view, focusing on a phenomenon that reduces barriers to entry. Specifically, an incumbent company risks reputational damage unless its model is sufficiently aligned with safety objectives, whereas a new company can more easily avoid reputational damage. To study this issue formally, we define a multi-objective high-dimensional regression framework that captures reputational damage, and we characterize the number of data points that a new company needs to enter the market. Our results demonstrate how multi-objective considerations can fundamentally reduce barriers to entry -- the required number of data points can be significantly smaller than the incumbent company's dataset size. En route to proving these results, we develop scaling laws for high-dimensional linear regression in multi-objective environments, showing that the scaling rate becomes slower when the dataset size is large, which could be of independent interest.

The minimum norm least squares is an estimation strategy under an over-parameterized case and, in machine learning, is known as a helpful tool for understanding a nature of deep learning. In this paper, to apply it in a context of non-parametric regression problems, we established several methods which are based on thresholding of SVD (singular value decomposition) components, wihch are referred to as SVD regression methods. We considered several methods that are singular value based thresholding, hard-thresholding with cross validation, universal thresholding and bridge thresholding. Information on output samples is not utilized in the first method while it is utilized in the other methods. We then applied them to semi-supervised learning, in which unlabeled input samples are incorporated into kernel functions in a regressor. The experimental results for real data showed that, depending on the datasets, the SVD regression methods is superior to a naive ridge regression method. Unfortunately, there were no clear advantage of the methods utilizing information on output samples. Furthermore, for depending on datasets, incorporation of unlabeled input samples into kernels is found to have certain advantages.

We introduce a general iterative procedure called risk-based calibration (RC) designed to minimize the empirical risk under the 0-1 loss (empirical error) for probabilistic classifiers. These classifiers are based on modeling probability distributions, including those constructed from the joint distribution (generative) and those based on the class conditional distribution (conditional). RC can be particularized to any probabilistic classifier provided a specific learning algorithm that computes the classifier's parameters in closed form using data statistics. RC reinforces the statistics aligned with the true class while penalizing those associated with other classes, guided by the 0-1 loss. The proposed method has been empirically tested on 30 datasets using na\"ive Bayes, quadratic discriminant analysis, and logistic regression classifiers. RC improves the empirical error of the original closed-form learning algorithms and, more notably, consistently outperforms the gradient descent approach with the three classifiers.