We study the problem of learning a low-degree spherical polynomial of degree $\ell_0 = Θ(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network (NN) with channel attention in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\eps \in (0,1)$, a carefully designed two-layer NN with channel attention and finite width of $m \ge Θ({n^4 \log (2n/δ)}/{d^{2\ell_0}})$ trained by the vanilla gradient descent (GD) requires the lowest sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ with probability $1-δ$ for every $δ\in (0,1)$, in contrast with the representative sample complexity $Θ\pth{d^{\ell_0} \max\set{\eps^{-2},\log d}}$, where $n$ is the training daata size. Moreover, such sample complexity is not improvable since the trained network renders a sharp rate of the nonparametric regression risk of the order $Θ(d^{\ell_0}/{n})$ with probability at least $1-δ$. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $Θ(d^{\ell_0})$ is $Θ(d^{\ell_0}/{n})$, so that the rate of the nonparametric regression risk of the network trained by GD is minimax optimal. The training of the two-layer NN with channel attention consists of two stages. In Stage 1, a provable learnable channel selection algorithm identifies the ground-truth channel number $\ell_0$ from the initial $L \ge \ell_0$ channels in the first-layer activation, with high probability. This learnable selection is achieved by an efficient one-step GD update on both layers, enabling feature learning for low-degree polynomial targets. In Stage 2, the second layer is trained by standard GD using the activation function with the selected channels.

Analyzing causality in multivariate systems involves establishing how information is generated, distributed and combined, and thus requires tools that capture interactions beyond pairwise relations. Higher-order information theory provides such tools. In particular, Partial Information Decomposition (PID) allows the decomposition of the information that a set of sources provides about a target into redundant, unique, and synergistic components. Yet the mathematical connection between such higher-order information-theoretic measures and causal structure remains undeveloped. Here we establish the first theoretical correspondence between PID components and causal structure in both Bayesian networks and hypergraphs. We first show that in Bayesian networks unique information precisely characterizes direct causal neighbors, while synergy identifies collider relationships. This establishes a localist causal discovery paradigm in which the structure surrounding each variable can be recovered from its immediate informational footprint, eliminating the need for global search over graph space. Extending these results to higher-order systems, we prove that PID signatures in Bayesian hypergraphs differentiate parents, children, co-heads, and co-tails, revealing a higher-order collider effect unique to multi-tail hyperedges. We also present procedures by which our results can be used to characterize systematically the causal structure of Bayesian networks and hypergraphs. Our results position PID as a rigorous, model-agnostic foundation for inferring both pairwise and higher-order causal structure, and introduce a fundamentally local information-theoretic viewpoint on causal discovery.

This study proposes Riesz representer estimation methods based on score matching. The Riesz representer is a key component in debiased machine learning for constructing $\sqrt{n}$-consistent and efficient estimators in causal inference and structural parameter estimation. To estimate the Riesz representer, direct approaches have garnered attention, such as Riesz regression and the covariate balancing propensity score. These approaches can also be interpreted as variants of direct density ratio estimation (DRE) in several applications such as average treatment effect estimation. In DRE, it is well known that flexible models can easily overfit the observed data due to the estimand and the form of the loss function. To address this issue, recent work has proposed modeling the density ratio as a product of multiple intermediate density ratios and estimating it using score-matching techniques, which are often used in the diffusion model literature. We extend score-matching-based DRE methods to Riesz representer estimation. Our proposed method not only mitigates overfitting but also provides insights for causal inference by bridging marginal effects and average policy effects through time score functions.

In the realm of diverse high-dimensional data, images play a significant role across various processes of manufacturing systems where efficient image anomaly detection has emerged as a core technology of utmost importance. However, when applied to textured defect images, conventional anomaly detection methods have limitations including non-negligible misidentification, low robustness, and excessive reliance on large-scale and structured datasets. This paper proposes a texture basis integrated smooth decomposition (TBSD) approach, which is targeted at efficient anomaly detection in textured images with smooth backgrounds and sparse anomalies. Mathematical formulation of quasi-periodicity and its theoretical properties are investigated for image texture estimation. TBSD method consists of two principal processes: the first process learns the texture basis functions to effectively extract quasi-periodic texture patterns; the subsequent anomaly detection process utilizes that texture basis as prior knowledge to prevent texture misidentification and capture potential anomalies with high accuracy.The proposed method surpasses benchmarks with less misidentification, smaller training dataset requirement, and superior anomaly detection performance on both simulation and real-world datasets.

Statistical inference in contextual bandits is complicated by the adaptive, non-i.i.d. nature of the data. A growing body of work has shown that classical least-squares inference may fail under adaptive sampling, and that constructing valid confidence intervals for linear functionals of the model parameter typically requires paying an unavoidable inflation of order $\sqrt{d \log T}$. This phenomenon -- often referred to as the price of adaptivity -- highlights the inherent difficulty of reliable inference under general contextual bandit policies. A key structural property that circumvents this limitation is the \emph{stability} condition of Lai and Wei, which requires the empirical feature covariance to concentrate around a deterministic limit. When stability holds, the ordinary least-squares estimator satisfies a central limit theorem, and classical Wald-type confidence intervals -- designed for i.i.d. data -- become asymptotically valid even under adaptation, \emph{without} incurring the $\sqrt{d \log T}$ price of adaptivity. In this paper, we propose and analyze a penalized EXP4 algorithm for linear contextual bandits. Our first main result shows that this procedure satisfies the Lai--Wei stability condition and therefore admits valid Wald-type confidence intervals for linear functionals. Our second result establishes that the same algorithm achieves regret guarantees that are minimax optimal up to logarithmic factors, demonstrating that stability and statistical efficiency can coexist within a single contextual bandit method. Finally, we complement our theory with simulations illustrating the empirical normality of the resulting estimators and the sharpness of the corresponding confidence intervals.

Federated learning (FL) supports privacy-preserving, decentralized machine learning (ML) model training by keeping data on client devices. However, non-independent and identically distributed (non-IID) data across clients biases updates and degrades performance. To alleviate these issues, we propose Clust-PSI-PFL, a clustering-based personalized FL framework that uses the Population Stability Index (PSI) to quantify the level of non-IID data. We compute a weighted PSI metric, $WPSI^L$, which we show to be more informative than common non-IID metrics (Hellinger, Jensen-Shannon, and Earth Mover's distance). Using PSI features, we form distributionally homogeneous groups of clients via K-means++; the number of optimal clusters is chosen by a systematic silhouette-based procedure, typically yielding few clusters with modest overhead. Across six datasets (tabular, image, and text modalities), two partition protocols (Dirichlet with parameter $α$ and Similarity with parameter S), and multiple client sizes, Clust-PSI-PFL delivers up to 18% higher global accuracy than state-of-the-art baselines and markedly improves client fairness by a relative improvement of 37% under severe non-IID data. These results establish PSI-guided clustering as a principled, lightweight mechanism for robust PFL under label skew.

Survival analysis relies fundamentally on the semi-parametric Cox Proportional Hazards (CoxPH) model and the parametric Accelerated Failure Time (AFT) model. CoxPH assumes constant hazard ratios, often failing to capture real-world dynamics, while traditional AFT models are limited by rigid distributional assumptions. Although deep learning models like DeepAFT address these constraints by improving predictive accuracy and handling censoring, they inherit the significant challenge of black-box interpretability. The recent introduction of CoxKAN demonstrated the successful integration of Kolmogorov-Arnold Networks (KANs), a novel architecture that yields highly accurate and interpretable symbolic representations, within the CoxPH framework. Motivated by the interpretability gains of CoxKAN, we introduce KAN-AFT (Kolmogorov Arnold Network-based AFT), the first framework to apply KANs to the AFT model. Our primary contributions include: (i) a principled AFT-KAN formulation, (ii) robust optimization strategies for right-censored observations (e.g., Buckley-James and IPCW), and (iii) an interpretability pipeline that converts the learned spline functions into closed-form symbolic equations for survival time. Empirical results on multiple datasets confirm that KAN-AFT achieves performance comparable to or better than DeepAFT, while uniquely providing transparent, symbolic models of the survival process.

Large language models (LLMs) solve reasoning problems by first generating a rationale and then answering. We formalize reasoning as a latent variable model and derive an expectation-maximization (EM) objective for learning to reason. This view connects EM and modern reward-based optimization, and shows that the main challenge lies in designing a sampling distribution that generates rationales that justify correct answers. We instantiate and compare several sampling schemes: rejection sampling with a budget, self-taught reasoner (STaR), and prompt posterior sampling (PPS), which only keeps the rationalization stage of STaR. Our experiments on the ARC, MMLU, and OpenBookQA datasets with the Llama and Qwen models show that the sampling scheme can significantly affect the accuracy of learned reasoning models. Despite its simplicity, we observe that PPS outperforms the other sampling schemes.

We introduce two nonlinear sufficient dimension reduction methods for regressions with tensor-valued predictors. Our goal is two-fold: the first is to preserve the tensor structure when performing dimension reduction, particularly the meaning of the tensor modes, for improved interpretation; the second is to substantially reduce the number of parameters in dimension reduction, thereby achieving model parsimony and enhancing estimation accuracy. Our two tensor dimension reduction methods echo the two commonly used tensor decomposition mechanisms: one is the Tucker decomposition, which reduces a larger tensor to a smaller one; the other is the CP-decomposition, which represents an arbitrary tensor as a sequence of rank-one tensors. We developed the Fisher consistency of our methods at the population level and established their consistency and convergence rates. Both methods are easy to implement numerically: the Tucker-form can be implemented through a sequence of least-squares steps, and the CP-form can be implemented through a sequence of singular value decompositions. We investigated the finite-sample performance of our methods and showed substantial improvement in accuracy over existing methods in simulations and two data applications.

\noindent Hyper-parameter selection is a central practical problem in modern machine learning, governing regularization strength, model capacity, and robustness choices. Cross-validation is often computationally prohibitive at scale, while fully Bayesian hyper-parameter learning can be difficult due to the cost of posterior sampling. We develop a generative perspective on hyper-parameter tuning that combines two ideas: (i) optimization-based approximations to Bayesian posteriors via randomized, weighted objectives (weighted Bayesian bootstrap), and (ii) amortization of repeated optimization across many hyper-parameter settings by learning a transport map from hyper-parameters (including random weights) to the corresponding optimizer. This yields a ``generator look-up table'' for estimators, enabling rapid evaluation over grids or continuous ranges of hyper-parameters and supporting both predictive tuning objectives and approximate Bayesian uncertainty quantification. We connect this viewpoint to weighted $M$-estimation, envelope/auxiliary-variable representations that reduce non-quadratic losses to weighted least squares, and recent generative samplers for weighted $M$-estimators.