Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modeling

This work introduces a method to equip data-driven polynomial chaos expansion surrogate models with intervals that quantify the predictive uncertainty of the surrogate. To that end, we integrate jackknife-based conformal prediction into regression-based polynomial chaos expansions. The jackknife algorithm uses leave-one-out residuals to generate predictive intervals around the predictions of the polynomial chaos surrogate. The jackknife+ extension additionally requires leave-one-out model predictions. The key to efficient implementation is to leverage the linearity of the polynomial chaos regression model, so that leave-one-out residuals and, if necessary, leave-one-out model predictions can be computed with analytical, closed-form expressions, thus eliminating the need for repeated model re-training. In addition to the efficient computation of the predictive intervals, a significant advantage of this approach is its data efficiency, as it requires no hold-out dataset for prediction interval calibration, thus allowing the entire dataset to be used for model training. The conformalized polynomial chaos expansion method is validated on several benchmark models, where the impact of training data volume on the predictive intervals is additionally investigated.

Shrinkage to Infinity: Reducing Test Error by Inflating the Minimum Norm Interpolator in Linear Models

Hastie et al. (2022) found that ridge regularization is essential in high dimensional linear regression $y=\beta^Tx + \epsilon$ with isotropic co-variates $x\in \mathbb{R}^d$ and $n$ samples at fixed $d/n$. However, Hastie et al. (2022) also notes that when the co-variates are anisotropic and $\beta$ is aligned with the top eigenvalues of population covariance, the "situation is qualitatively different." In the present article, we make precise this observation for linear regression with highly anisotropic covariances and diverging $d/n$. We find that simply scaling up (or inflating) the minimum $\ell_2$ norm interpolator by a constant greater than one can improve the generalization error. This is in sharp contrast to traditional regularization/shrinkage prescriptions. Moreover, we use a data-splitting technique to produce consistent estimators that achieve generalization error comparable to that of the optimally inflated minimum-norm interpolator. Our proof relies on apparently novel matching upper and lower bounds for expectations of Gaussian random projections for a general class of anisotropic covariance matrices when $d/n\to \infty$.

Calibrated Principal Component Regression

We propose a new method for statistical inference in generalized linear models. In the overparameterized regime, Principal Component Regression (PCR) reduces variance by projecting high-dimensional data to a low-dimensional principal subspace before fitting. However, PCR incurs truncation bias whenever the true regression vector has mass outside the retained principal components (PC). To mitigate the bias, we propose Calibrated Principal Component Regression (CPCR), which first learns a low-variance prior in the PC subspace and then calibrates the model in the original feature space via a centered Tikhonov step. CPCR leverages cross-fitting and controls the truncation bias by softening PCR's hard cutoff. Theoretically, we calculate the out-of-sample risk in the random matrix regime, which shows that CPCR outperforms standard PCR when the regression signal has non-negligible components in low-variance directions. Empirically, CPCR consistently improves prediction across multiple overparameterized problems. The results highlight CPCR's stability and flexibility in modern overparameterized settings.

Efficient reductions from a Gaussian source with applications to statistical-computational tradeoffs

Given a single observation from a Gaussian distribution with unknown mean $\theta$, we design computationally efficient procedures that can approximately generate an observation from a different target distribution $Q_{\theta}$ uniformly for all $\theta$ in a parameter set. We leverage our technique to establish reduction-based computational lower bounds for several canonical high-dimensional statistical models under widely-believed conjectures in average-case complexity. In particular, we cover cases in which: 1. $Q_{\theta}$ is a general location model with non-Gaussian distribution, including both light-tailed examples (e.g., generalized normal distributions) and heavy-tailed ones (e.g., Student's $t$-distributions). As a consequence, we show that computational lower bounds proved for spiked tensor PCA with Gaussian noise are universal, in that they extend to other non-Gaussian noise distributions within our class. 2. $Q_{\theta}$ is a normal distribution with mean $f(\theta)$ for a general, smooth, and nonlinear link function $f:\mathbb{R} \rightarrow \mathbb{R}$. Using this reduction, we construct a reduction from symmetric mixtures of linear regressions to generalized linear models with link function $f$, and establish computational lower bounds for solving the $k$-sparse generalized linear model when $f$ is an even function. This result constitutes the first reduction-based confirmation of a $k$-to-$k^2$ statistical-to-computational gap in $k$-sparse phase retrieval, resolving a conjecture posed by Cai et al. (2016). As a second application, we construct a reduction from the sparse rank-1 submatrix model to the planted submatrix model, establishing a pointwise correspondence between the phase diagrams of the two models that faithfully preserves regions of computational hardness and tractability.

Adaptive Optimizable Gaussian Process Regression Linear Least Squares Regression Filtering Method for SEM Images

Scanning Electron Microscopy (SEM) images often suffer from noise contamination, which degrades image quality and affects further analysis. This research presents a complete approach to estimate their Signal-to-Noise Ratio (SNR) and noise variance (NV), and enhance image quality using NV-guided Wiener filter. The main idea of this study is to use a good SNR estimation technique and infuse a machine learning model to estimate NV of the SEM image, which then guides the wiener filter to remove the noise, providing a more robust and accurate SEM image filtering pipeline. First, we investigate five different SNR estimation techniques, namely Nearest Neighbourhood (NN) method, First-Order Linear Interpolation (FOL) method, Nearest Neighbourhood with First-Order Linear Interpolation (NN+FOL) method, Non-Linear Least Squares Regression (NLLSR) method, and Linear Least Squares Regression (LSR) method. It is shown that LSR method to perform better than the rest. Then, Support Vector Machines (SVM) and Gaussian Process Regression (GPR) are tested by pairing it with LSR. In this test, the Optimizable GPR model shows the highest accuracy and it stands as the most effective solution for NV estimation. Combining these results lead to the proposed Adaptive Optimizable Gaussian Process Regression Linear Least Squares Regression (AO-GPRLLSR) Filtering pipeline. The AO-GPRLLSR method generated an estimated noise variance which served as input to NV-guided Wiener filter for improving the quality of SEM images. The proposed method is shown to achieve notable success in estimating SNR and NV of SEM images and leads to lower Mean Squared Error (MSE) after the filtering process.

Learning Linear Regression with Low-Rank Tasks in-Context

In-context learning (ICL) is a key building block of modern large language models, yet its theoretical mechanisms remain poorly understood. It is particularly mysterious how ICL operates in real-world applications where tasks have a common structure. In this work, we address this problem by analyzing a linear attention model trained on low-rank regression tasks. Within this setting, we precisely characterize the distribution of predictions and the generalization error in the high-dimensional limit. Moreover, we find that statistical fluctuations in finite pre-training data induce an implicit regularization. Finally, we identify a sharp phase transition of the generalization error governed by task structure. These results provide a framework for understanding how transformers learn to learn the task structure.

Probabilistically-Safe Bipedal Navigation over Uncertain Terrain via Conformal Prediction and Contraction Analysis

We address the challenge of enabling bipedal robots to traverse rough terrain by developing probabilistically safe planning and control strategies that ensure dynamic feasibility and centroidal robustness under terrain uncertainty. Specifically, we propose a high-level Model Predictive Control (MPC) navigation framework for a bipedal robot with a specified confidence level of safety that (i) enables safe traversal toward a desired goal location across a terrain map with uncertain elevations, and (ii) formally incorporates uncertainty bounds into the centroidal dynamics of locomotion control. To model the rough terrain, we employ Gaussian Process (GP) regression to estimate elevation maps and leverage Conformal Prediction (CP) to construct calibrated confidence intervals that capture the true terrain elevation. Building on this, we formulate contraction-based reachable tubes that explicitly account for terrain uncertainty, ensuring state convergence and tube invariance. In addition, we introduce a contraction-based flywheel torque control law for the reduced-order Linear Inverted Pendulum Model (LIPM), which stabilizes the angular momentum about the center-of-mass (CoM). This formulation provides both probabilistic safety and goal reachability guarantees. For a given confidence level, we establish the forward invariance of the proposed torque control law by demonstrating exponential stabilization of the actual CoM phase-space trajectory and the desired trajectory prescribed by the high-level planner. Finally, we evaluate the effectiveness of our planning framework through physics-based simulations of the Digit bipedal robot in MuJoCo.

Multidata Causal Discovery for Statistical Hurricane Intensity Forecasting

Improving statistical forecasts of Atlantic hurricane intensity is limited by complex nonlinear interactions and difficulty in identifying relevant predictors. Conventional methods prioritize correlation or fit, often overlooking confounding variables and limiting generalizability to unseen tropical storms. To address this, we leverage a multidata causal discovery framework with a replicated dataset based on Statistical Hurricane Intensity Prediction Scheme (SHIPS) using ERA5 meteorological reanalysis. We conduct multiple experiments to identify and select predictors causally linked to hurricane intensity changes. We train multiple linear regression models to compare causal feature selection with no selection, correlation, and random forest feature importance across five forecast lead times from 1 to 5 days (24 to 120 hours). Causal feature selection consistently outperforms on unseen test cases, especially for lead times shorter than 3 days. The causal features primarily include vertical shear, mid-tropospheric potential vorticity and surface moisture conditions, which are physically significant yet often underutilized in hurricane intensity predictions. Further, we build an extended predictor set (SHIPS+) by adding selected features to the standard SHIPS predictors. SHIPS+ yields increased short-term predictive skill at lead times of 24, 48, and 72 hours. Adding nonlinearity using multilayer perceptron further extends skill to longer lead times, despite our framework being purely regional and not requiring global forecast data. Operational SHIPS tests confirm that three of the six added causally discovered predictors improve forecasts, with the largest gains at longer lead times. Our results demonstrate that causal discovery improves hurricane intensity prediction and pave the way toward more empirical forecasts.

Interpretable Machine Learning for Life Expectancy Prediction: A Comparative Study of Linear Regression, Decision Tree, and Random Forest

Life expectancy is a fundamental indicator of population health and socio-economic well-being, yet accurately forecasting it remains challenging due to the interplay of demographic, environmental, and healthcare factors. This study evaluates three machine learning models -- Linear Regression (LR), Regression Decision Tree (RDT), and Random Forest (RF), using a real-world dataset drawn from World Health Organization (WHO) and United Nations (UN) sources. After extensive preprocessing to address missing values and inconsistencies, each model's performance was assessed with $R^2$, Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE). Results show that RF achieves the highest predictive accuracy ($R^2 = 0.9423$), significantly outperforming LR and RDT. Interpretability was prioritized through p-values for LR and feature importance metrics for the tree-based models, revealing immunization rates (diphtheria, measles) and demographic attributes (HIV/AIDS, adult mortality) as critical drivers of life-expectancy predictions. These insights underscore the synergy between ensemble methods and transparency in addressing public-health challenges. Future research should explore advanced imputation strategies, alternative algorithms (e.g., neural networks), and updated data to further refine predictive accuracy and support evidence-based policymaking in global health contexts.

Risk Phase Transitions in Spiked Regression: Alignment Driven Benign and Catastrophic Overfitting

This paper analyzes the generalization error of minimum-norm interpolating solutions in linear regression using spiked covariance data models. The paper characterizes how varying spike strengths and target-spike alignments can affect risk, especially in overparameterized settings. The study presents an exact expression for the generalization error, leading to a comprehensive classification of benign, tempered, and catastrophic overfitting regimes based on spike strength, the aspect ratio $c=d/n$ (particularly as $c \to \infty$), and target alignment. Notably, in well-specified aligned problems, increasing spike strength can surprisingly induce catastrophic overfitting before achieving benign overfitting. The paper also reveals that target-spike alignment is not always advantageous, identifying specific, sometimes counterintuitive, conditions for its benefit or detriment. Alignment with the spike being detrimental is empirically demonstrated to persist in nonlinear models.