Tensor decompositions have emerged as powerful analytical tools across diverse fields including signal/image processing [1, 2, 3, 4, 5, 6], chemometrics [7], geophysics [8, 9], and neuroscience [10, 11, 12, 13]. Their effectiveness stems from their ability to approximate high-dimensional tensors with low-rank decompositions, offering efficient dimensionality reduction while preserving essential structural information. For example, tensor decomposition techniques [14, 15, 16] are applied in hyperspectral image (HSI) analysis to extract low-rank structures for dimensionality reduction [9], and used in electroencephalogram (EEG) analysis to capture latent patterns across multiple dimensions [10]. Over the past decade, tensor regression (TR) models have received attention, with numerous approaches proposed in the literature, including Tucker tensor regression [17], low-rank orthogonally decomposable tensor regression [18], Bayesian Kruskal tensor regression (KTR) [19], Bayesian low rank tensor ring completion [20], graph-regularized tensor regression [21], and tensor regression network [22].

Kolmogorov-Arnold Networks (KANs) offer a structured and interpretable framework for multivariate function approximation by composing univariate transformations through additive or multiplicative aggregation. This paper establishes theoretical convergence guarantees for KANs when the univariate components are represented by B-splines. We prove that both additive and hybrid additive-multiplicative KANs attain the minimax-optimal convergence rate $O(n^{-2r/(2r+1)})$ for functions in Sobolev spaces of smoothness $r$. We further derive guidelines for selecting the optimal number of knots in the B-splines. The theory is supported by simulation studies that confirm the predicted convergence rates. These results provide a theoretical foundation for using KANs in nonparametric regression and highlight their potential as a structured alternative to existing methods.

We address the Statistical Process Control (SPC) of high-dimensional, dynamic industrial processes from two complementary perspectives: manifold fitting and manifold learning, both of which assume data lies on an underlying nonlinear, lower dimensional space. We propose two distinct monitoring frameworks for online or 'phase II' Statistical Process Control (SPC). The first method leverages state-of-the-art techniques in manifold fitting to accurately approximate the manifold where the data resides within the ambient high-dimensional space. It then monitors deviations from this manifold using a novel scalar distribution-free control chart. In contrast, the second method adopts a more traditional approach, akin to those used in linear dimensionality reduction SPC techniques, by first embedding the data into a lower-dimensional space before monitoring the embedded observations. We prove how both methods provide a controllable Type I error probability, after which they are contrasted for their corresponding fault detection ability. Extensive numerical experiments on a synthetic process and on a replicated Tennessee Eastman Process show that the conceptually simpler manifold-fitting approach achieves performance competitive with, and sometimes superior to, the more classical lower-dimensional manifold monitoring methods. In addition, we demonstrate the practical applicability of the proposed manifold-fitting approach by successfully detecting surface anomalies in a real image dataset of electrical commutators.

A common way to estimate an unknown convex regression function $f_0: Ω\subset \mathbb{R}^d \rightarrow \mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However, this estimator is known for its tendency to overfit near the boundary of $Ω$, posing significant challenges in real-world applications. In this paper, we introduce a new estimator of $f_0$ that avoids this overfitting by minimizing a penalty on the subgradient while enforcing an upper bound $s_n$ on the sum of squared errors. The key advantage of this method is that $s_n$ can be directly estimated from the data. We establish the uniform almost sure consistency of the proposed estimator and its subgradient over $Ω$ as $n \rightarrow \infty$ and derive convergence rates. The effectiveness of our estimator is illustrated through its application to estimating waiting times in a single-server queue.

The advent of Scientific Machine Learning has heralded a transformative era in scientific discovery, driving progress across diverse domains. Central to this progress is uncovering scientific laws from experimental data through symbolic regression. However, existing approaches are dominated by heuristic algorithms or data-hungry black-box methods, which often demand low-noise settings and lack principled uncertainty quantification. Motivated by interpretable Statistical Artificial Intelligence, we develop a hierarchical Bayesian framework for symbolic regression that represents scientific laws as ensembles of tree-structured symbolic expressions endowed with a regularized tree prior. This coherent probabilistic formulation enables full posterior inference via an efficient Markov chain Monte Carlo algorithm, yielding a balance between predictive accuracy and structural parsimony. To guide symbolic model selection, we develop a marginal posterior-based criterion adhering to the Occam's window principle and further quantify structural fidelity to ground truth through a tailored expression-distance metric. On the theoretical front, we establish near-minimax rate of Bayesian posterior concentration, providing the first rigorous guarantee in context of symbolic regression. Empirical evaluation demonstrates robust performance of our proposed methodology against state-of-the-art competing modules on a simulated example, a suite of canonical Feynman equations, and single-atom catalysis dataset.

Copulas are a fundamental tool for modelling multivariate dependencies in data, forming the method of choice in diverse fields and applications. However, the adoption of existing models for multimodal and high-dimensional dependencies is hindered by restrictive assumptions and poor scaling. In this work, we present methods for modelling copulas based on the principles of diffusions and flows. We design two processes that progressively forget inter-variable dependencies while leaving dimension-wise distributions unaffected, provably defining valid copulas at all times. We show how to obtain copula models by learning to remember the forgotten dependencies from each process, theoretically recovering the true copula at optimality. The first instantiation of our framework focuses on direct density estimation, while the second specialises in expedient sampling. Empirically, we demonstrate the superior performance of our proposed methods over state-of-the-art copula approaches in modelling complex and high-dimensional dependencies from scientific datasets and images. Our work enhances the representational power of copula models, empowering applications and paving the way for their adoption on larger scales and more challenging domains. Given a collection of d continuous random variables, a simple model for their joint probability density function is the product of the corresponding d univariate densities (Peterson, 1987). Indeed, the copula uniquely and exactly represents the inter-variable dependence, unlike correlation or mutual information (Geenens, 2023), fully disentangling the marginal behaviour from the joint. This disentanglement enables a modular approach for multivariate modelling: first, model the uni-variate variables independently, and second, model their dependence with a copula.

Time series analysis has emerged as an important tool for improving patient diagnosis and management in healthcare applications. However, these applications commonly face two critical challenges: time misalignment and data sparsity. Traditional approaches address these issues through a two-step process of imputation followed by prediction. We propose MAGIC (Multi-tAsk Gaussian Process for Imputation and Classification), a novel unified framework that simultaneously performs class-informed missing value imputation and label prediction within a hierarchical multi-task Gaussian process coupled with functional logistic regression. To handle intractable likelihood components, MAGIC employs Taylor expansion approximations with bounded error analysis, and parameter estimation is performed using EM algorithm with block coordinate optimization supported by convergence analysis. We validate MAGIC through two healthcare applications: prediction of post-traumatic headache improvement following mild traumatic brain injury and prediction of in-hospital mortality within 48 hours after ICU admission. In both applications, MAGIC achieves superior predictive accuracy compared to existing methods. The ability to generate real-time and accurate predictions with limited samples facilitates early clinical assessment and treatment planning, enabling healthcare providers to make more informed treatment decisions.

Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.

We present a sensor-fusion framework for rapid, non-destructive classification and quality control of organic substances, built on a standard Arduino Mega 2560 microcontroller platform equipped with three commercial environmental and gas sensors. All data used in this study were generated in-house: sensor outputs for ten distinct classes - including fresh and expired samples of apple juice, onion, garlic, and ginger, as well as cinnamon and cardamom - were systematically collected and labeled using this hardware setup, resulting in a unique, application-specific dataset. Correlation analysis was employed as part of the preprocessing pipeline for feature selection. After preprocessing and dimensionality reduction (PCA/LDA), multiple supervised learning models - including Support Vector Machine (SVM), Decision Tree (DT), and Random Forest (RF), each with hyperparameter tuning, as well as an Artificial Neural Network (ANN) and an ensemble voting classifier - were trained and cross-validated on the collected dataset. The best-performing models, including tuned Random Forest, ensemble, and ANN, achieved test accuracies in the 93 to 94 percent range. These results demonstrate that low-cost, multisensory platforms based on the Arduino Mega 2560, combined with advanced machine learning and correlation-driven feature engineering, enable reliable identification and quality control of organic compounds.

Scaling laws have played a cornerstone role in guiding the training of large language models (LLMs). However, most existing works on scaling laws primarily focus on the final-step loss, overlooking the loss dynamics during the training process and, crucially, the impact of learning rate schedule (LRS). In this paper, we aim to bridge this gap by studying a teacher-student kernel regression setup trained via online stochastic gradient descent (SGD). Leveraging a novel intrinsic time viewpoint and stochastic differential equation (SDE) modeling of SGD, we introduce the Functional Scaling Law (FSL), which characterizes the evolution of population risk during the training process for general LRSs. Remarkably, the impact of the LRSs is captured through an explicit convolution-type functional term, making their effects fully tractable. To illustrate the utility of FSL, we analyze three widely used LRSs -- constant, exponential decay, and warmup-stable-decay (WSD) -- under both data-limited and compute-limited regimes. We provide theoretical justification for widely adopted empirical practices in LLMs pre-training such as (i) higher-capacity models are more data- and compute-efficient; (ii) learning rate decay can improve training efficiency; (iii) WSD-like schedules can outperform direct-decay schedules. Lastly, we explore the practical relevance of FSL as a surrogate model for fitting, predicting and optimizing the loss curves in LLM pre-training, with experiments conducted across model sizes ranging from 0.1B to 1B parameters. We hope our FSL framework can deepen the understanding of LLM pre-training dynamics and provide insights for improving large-scale model training.