Papers

We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

Individual patient data (IPD) from oncology trials are essential for reliable evidence synthesis but are rarely publicly available, necessitating reconstruction from published Kaplan-Meier (KM) curves. Existing reconstruction methods suffer from digitization errors, unrealistic uniform censoring assumptions, and the inability to recover subgroup-level IPD when only aggregate statistics are available. We developed RESOLVE-IPD, a unified computational framework that enables high-fidelity IPD reconstruction and uncertainty-aware subgroup meta-analysis to address these limitations. RESOLVE-IPD comprises two components. The first component, High-Fidelity IPD Reconstruction, integrates the VEC-KM and CEN-KM modules: VEC-KM extracts precise KM coordinates and explicit censoring marks from vectorized figures, minimizing digitization error, while CEN-KM corrects overlapping censor symbols and eliminates the uniform censoring assumption. The second component, Uncertainty-Aware Subgroup Recovery, employs the MAPLE (Marginal Assignment of Plausible Labels and Evidence Propagation) algorithm to infer patient-level subgroup labels consistent with published summary statistics (e.g., hazard ratio, median overall survival) when subgroup KM curves are unavailable. MAPLE generates ensembles of mathematically valid labelings, facilitating a propagating meta-analysis that quantifies and reflects uncertainty from subgroup reconstruction. RESOLVE-IPD was validated through a subgroup meta-analysis of four trials in advanced esophageal squamous cell carcinoma, focusing on the programmed death ligand 1 (PD-L1)-low population. RESOLVE-IPD enables accurate IPD reconstruction and robust, uncertainty-aware subgroup meta-analyses, strengthening the reliability and transparency of secondary evidence synthesis in precision oncology.

This report describes an application of artificial intelligence (AI) to the Bayesian analysis of glioblastoma survival data. It has been suggested that AI can be used to construct prior distributions for parameters in Bayesian models rather than using the difficult, unreliable, and time-consuming process of eliciting expert opinion from radiation oncologists. Here, we show how generative AI can quickly propose sensible prior distributions of the hazard ratio comparing two glioblastoma therapies, for a standard Bayesian survival model on real data. Three Chatbots generated two alternative priors each which were evaluated by a radiation oncologist and then used in a sensitivity analysis to assess posterior stability. The results suggest that, for this cancer survival analysis, priors from generative AI are a preferred alternative method to expert elicitation.

This work introduces a new class of four-dimensional variational data assimilation (4D-Var) methods grounded in data-consistent inversion (DCI) theory. The methods extend classical 4D-Var by incorporating a predictability-aware regularization term. The first method formulated is referred to as Data-Consistent 4D-Var (DC-4DVar), which is then enhanced using a Weighted Mean Error (WME) quantity-of-interest map to construct the DC-WME 4D-Var method. While the DC and DC-WME cost functions both involve a predictability-aware regularization term, the DC-WME function includes a modification to the model-data misfit, thereby improving estimation accuracy, robustness, and theoretical consistency in nonlinear and partially observed dynamical systems. Proofs are provided that establish the existence and uniqueness of the minimizer and analyze how a predictability assumption that is common within the DCI framework helps to promote solution stability. Numerical experiments are presented on benchmark dynamical systems (Lorenz-63 and Lorenz-96) as well as for the shallow water equations (SWE). In the benchmark dynamical systems, the DC-WME 4D-Var formulation is shown to consistently outperform standard 4D-Var in reducing both error and bias while maintaining robustness under high observation noise and short assimilation windows. Despite introducing modest computational overhead, DC-WME 4D-Var delivers improvements in estimation performance and forecast skill, demonstrating its potential practicality and scalability for high-dimensional data assimilation problems.

Battery safety is critical in applications ranging from consumer electronics to electric vehicles and aircraft, where undetected anomalies could trigger safety hazards or costly downtime. In this study, we present OSBAD as an open-source benchmark for anomaly detection frameworks in battery applications. By benchmarking 15 diverse algorithms encompassing statistical, distance-based, and unsupervised machine-learning methods, OSBAD enables a systematic comparison of anomaly detection methods across heterogeneous datasets. In addition, we demonstrate how a physics- and statistics-informed feature transformation workflow enhances anomaly separability by decomposing collective anomalies into point anomalies. To address a major bottleneck in unsupervised anomaly detection due to incomplete labels, we propose a Bayesian optimization pipeline that facilitates automated hyperparameter tuning based on transfer-learning and regression proxies. Through validation on datasets covering both liquid and solid-state chemistries, we further demonstrate the cross-chemistry generalization capability of OSBAD to identify irregularities across different electrochemical systems. By making benchmarking database with open-source reproducible anomaly detection workflows available to the community, OSBAD establishes a unified foundation for developing safe, scalable, and transferable anomaly detection tools in battery analytics. This research underscores the significance of physics- and statistics-informed feature engineering as well as model selection with probabilistic hyperparameter tuning, in advancing trustworthy, data-driven diagnostics for safety-critical energy systems.

We provide the first proof of learning rate transfer with width in a linear multi-layer perceptron (MLP) parametrized with $\mu$P, a neural network parameterization designed to “maximize” feature learning in the infinite-width limit. We show that under $\mu P$, the optimal learning rate converges to a non-zero constant as width goes to infinity, providing a theoretical explanation to learning rate transfer. In contrast, we show that this property fails to hold under alternative parametrizations such as Standard Parametrization (SP) and Neural Tangent Parametrization (NTP). We provide intuitive proofs and support the theoretical findings with extensive empirical results.

We introduce a framework combining geometric modeling with disease progression analysis to investigate tau deposition in Alzheimer's disease (AD) using positron emission tomography (PET) data. Focusing on the hippocampus, we construct a principal surface that captures the spatial distribution and morphological changes of tau pathology. By projecting voxels onto this surface, we quantify tau coverage, intensity, and thickness through bidirectional projection distances and interpolated standardized uptake value ratios (SUVR). This low-dimensional embedding preserves spatial specificity while mitigating multiple comparison issues. Covariate effects are analyzed using a two-stage regression model with inverse probability weighting to adjust for signal sparsity and selection bias. Using the SuStaIn model, we identify subtypes and stages of AD, revealing distinct tau dynamics: the limbic-predominant subtype shows age-related nonlinear accumulation in coverage and thickness, whereas the posterior subtype exhibits uniform SUVR increases across disease progression. Model-based predictions show that hippocampal tau deposition follows a structured spatial trajectory expanding bidirectionally with increasing thickness, while subtype differences highlight posterior hippocampal involvement consistent with whole-brain patterns. Finally, directional signal patterns on the principal surface reveal contamination from the choroid plexus, demonstrating the broader applicability of the proposed framework across modalities including amyloid PET.

Detecting multimodality in empirical distributions is a fundamental problem in statistics and data analysis, with applications ranging from clustering to the study of complex systems. In practice, however, assessing departures from unimodality in a consistent and comparable way remains challenging. Widely used methods such as Hartigan and Hartigan's Dip Test illustrate these difficulties, as the interpretation of their statistics depends strongly on sample size, requires calibration to determine significance, and, for large samples, exhibit increasing sensitivity, leading to rejection of unimodality for arbitrarily small deviations from the null. We introduce Z-Dip, a standardized measure of multimodality that addresses these limitations. By treating the Dip statistic as a random variable under the null hypothesis of unimodality and standardizing its observed value, the proposed approach yields scores that are directly comparable across datasets of different sizes. Using simulation-based calibration, we derive a universal decision threshold that closely reproduces classical Dip Test decisions without requiring sample-size-specific adjustments. Extensive validation on simulated data and on more than 88,000 empirical opinion distributions shows near-perfect agreement with the classical Dip Test while providing a more interpretable and comparable measure of modality. Finally, we propose a downsampling-based correction that mitigates residual sensitivity in extremely large samples. Open-source software and reference tables are provided to facilitate practical adoption.

Precise knowledge of causal directed acyclic graphs (DAGs) is assumed for standard approaches towards valid adjustment set selection for unbiased estimation, but in practice, the DAG is often inferred from data or expert knowledge, introducing uncertainty. We present techniques to identify valid adjustment sets despite potential errors in the estimated causal graph. Specifically, we assume that only the skeleton of the DAG is known. Under a Bayesian framework, we place a prior on graphs and wish to sample graphs and compute the posterior probability of each set being valid; however, directly doing so is inefficient as the number of sets grows exponentially with the number of nodes in the DAG. We develop theory and techniques so that a limited number of sets are tested while the probability of finding valid adjustment sets remains high. Empirical results demonstrate the effectiveness of the method.

Phase-rectified signal averaging (PRSA) is a widely used algorithm to analyze nonstationary biomedical time series. The method operates by identifying hinge points in the time series according to prescribed rules, extracting segments centered at these points (with overlap permitted), and then averaging the segments. The resulting output is intended to capture the underlying quasi-oscillatory pattern of the signal, which can subsequently serve as input for further scientific analysis. However, a theoretical analysis of PRSA is lacking. In this paper, we investigate PRSA under two settings. First, when the input consists of a superposition of two oscillatory components, $\cos(2\pi t)+A\cos(2\pi (\xi t+\phi))$, where $A>0$, $\xi\in (0,1)$ and $\phi\in [0,1)$, we show that, asymptotically when the sample size $n\to \infty$, the PRSA output takes the form $A'\sin(2\pi t)+B'\sin(2\pi \xi t)$, where $A',B'\neq 0$. Second, when the input is a stationary Gaussian random process, we establish a central limit theorem: under mild regularity conditions, the averaged vector produced by PRSA converges in distribution to a Gaussian random vector as $n\to \infty$ with mean determined by the covariance structure of the random process. These results indicate that caution is warranted when interpreting PRSA outputs for scientific applications.

Counterfactual causal inference faces significant challenges when extended to multi-category, multi-valued treatments, where complex cross-effects between heterogeneous interventions are difficult to model. Existing methodologies remain constrained to binary or single-type treatments and suffer from restrictive assumptions, limited scalability, and inadequate evaluation frameworks for complex intervention scenarios. We present XTNet, a novel network architecture for multi-category, multi-valued treatment effect estimation. Our approach introduces a cross-effect estimation module with dynamic masking mechanisms to capture treatment interactions without restrictive structural assumptions. The architecture employs a decomposition strategy separating basic effects from cross-treatment interactions, enabling efficient modeling of combinatorial treatment spaces. We also propose MCMV-AUCC, a suitable evaluation metric that accounts for treatment costs and interaction effects. Extensive experiments on synthetic and real-world datasets demonstrate that XTNet consistently outperforms state-of-the-art baselines in both ranking accuracy and effect estimation quality. The results of the real-world A/B test further confirm its effectiveness.

Bayesian neural networks (BNNs) allow rigorous uncertainty quantification in deep learning, but often come at a prohibitive computational cost. We propose three different innovative architectures of partial trace-class Bayesian neural networks (PaTraC BNNs) that enable uncertainty quantification comparable to standard BNNs but use significantly fewer Bayesian parameters. These PaTraC BNNs have computational and statistical advantages over standard Bayesian neural networks in terms of speed and memory requirements. Our proposed methodology therefore facilitates reliable, robust, and scalable uncertainty quantification in neural networks. The three architectures build on trace-class neural network priors which induce an ordering of the neural network parameters, and are thus a natural choice in our framework. In a numerical simulation study, we verify the claimed benefits, and further illustrate the performance of our proposed methodology on a real-world dataset.

Children's growth extends beyond height and weight. This paper introduces the Multidimensional Index of Child Growth (MICG), developed by the IUNS Task Force "Towards a Multidimensional Approach for Child Growth." The IUNS-MICG applies a capability- and rights-based framework covering 14 dimensions of child wellbeing, including health, care, mental wellbeing, participation, autonomy, mobility, and safety. Using data from the Young Lives Study in Ethiopia, India, Peru, and Vietnam, we tested the framework with 29 indicators. Comparisons of different weighting methods show that equal weights provide robust and policy-relevant results. MICG uncovers deprivations hidden by physical measures alone; for instance, rural girls in Peru face educational and mental wellbeing disadvantages despite similar physical growth. Further analyses show that community participation in WASH programs is linked to higher multidimensional outcomes, especially for the most deprived. We also extend MICG with a Bayesian approach to estimate children's unrealized opportunities and propose a spiderweb growth chart for visualizing multidimensional progress. MICG offers a practical, equity-focused tool to monitor, evaluate, and strengthen interventions that support the Sustainable Development Goals and ensure no child is left behind.

Toeplitz covariance estimation is a classical signal processing problem, yet the geometry of the Gaussian maximum-likelihood objective remains only partially understood. Recent Newton-type, majorization-minimization, and gradient-based methods often solve the nonconvex problem globally with enough samples, but also reveal a difficult computational landscape. We study this phenomenon through an overparameterized Carathéodory representation, which parameterizes the covariance by the frequencies and amplitudes of steering vectors. A simple example shows that, for any uniform frequency grid, optimizing only the amplitudes can leave a positive error even in the population setting, which motivates optimizing the frequencies as well. Our main result proves that this nonlinear parameterization adds no stationary points: for any sample covariance, every stationary point with a positive definite covariance gives a stationary point of the Toeplitz maximum-likelihood problem. In the population setting, every such point therefore recovers the true covariance. We also prove that gradient descent converges to the true covariance in the population setting, and to the unique maximum likelihood estimate when the sample covariance is close enough to the true covariance. These results do not require overparameterization. Numerically, overparameterization speeds up gradient descent and gives a more accurate estimate for a fixed number of iterations.

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

This paper investigates change point inference in high-dimensional time series. We begin by introducing a max-$L_2$-norm based test procedure, which demonstrates strong performance under dense alternatives. We then establish the asymptotic independence between our proposed statistic and the two max-$L_\infty$-based statistics introduced by Wang and Feng (2023). Building on this result, we develop an adaptive inference approach by applying the Cauchy combination method to integrate these tests. This combined procedure exhibits robust performance across varying levels of sparsity. Extensive simulation studies and real data analysis further confirm the superior effectiveness of our proposed methods in the high-dimensional setting.

In observational studies, the observed association between an exposure and outcome of interest may be distorted by unobserved confounding. Causal sensitivity analysis can be used to assess the robustness of observed associations to potential unobserved confounding. For time-to-event outcomes, existing sensitivity analysis methods rely on parametric assumptions on the structure of the unobserved confounders and Cox proportional hazards models for the outcome regression. If these assumptions fail to hold, it is unclear whether the conclusions of the sensitivity analysis remain valid. Additionally, causal interpretation of the hazard ratio is challenging. To address these limitations, in this paper we develop a nonparametric sensitivity analysis framework for time-to-event data. Specifically, we derive nonparametric bounds for the difference between the observed and counterfactual survival curves and propose estimators and inference for these bounds using semiparametric efficiency theory. We also provide nonparametric bounds and inference for the difference between the observed and counterfactual restricted mean survival times. We demonstrate the performance of our proposed methods using numerical studies and an analysis of the causal effect of elective neck dissection on mortality in patients with high-grade parotid carcinoma.

Cluster DAGs (C-DAGs) provide an abstraction of causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relationships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyclic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. In this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the partition admissibility constraint, thereby allowing cyclic C-DAG representations. We extend the notions of d-separation and causal calculus to this setting, significantly broadening the scope of causal reasoning across clusters and enabling the application of C-DAGs in previously intractable scenarios. Our calculus is both sound and atomically complete with respect to the do-calculus: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.

Pretrained Transformers can perform in-context learning (ICL) from a few demonstrations, but this ability can fail sharply when the test distribution differs from pretraining, a common deployment setting. We study attention temperature as a simple inference-time control for improving ICL robustness under such shifts. In a high-dimensional linear-regression framework, we analyze a Transformer with "approximate softmax" attention, which preserves softmax's normalization and temperature-dependent selectivity while remaining tractable. We derive a closed-form expression for the ICL generalization error under distribution shift, and show that it is minimized by an explicit optimal attention temperature. This characterization yields interpretable guidance by linking the best temperature to moments of the pre-softmax attention scores, and predicts when temperature adjustment can recover near Bayes-optimal performance. We validate the theory with extensive simulations, and further demonstrate gains on pretrained LLMs (GPT-2 and Llama2-7B) on question-answering benchmarks under distribution shift induced by noisy in-context demonstrations. Overall, attention temperature emerges as a principled, lightweight knob for improving the robustness of ICL in pretrained Transformers.

In response to the U.S.\ Food and Drug Administration's (FDA) Project Optimus, a paradigm shift is underway in the design of early-phase oncology trials. To accelerate drug development, seamless Phase I/II designs have gained increasing attention, along with growing interest in the efficient reuse of Phase I data. We propose a nonparametric information-borrowing method that adaptively discounts Phase I observations according to the similarity of covariate distributions between Phase I and Phase II. Similarity is quantified using a kernel-based maximum mean discrepancy (MMD) and transformed into a dose-specific weight incorporated into a power-prior framework for Phase II efficacy evaluation, such as for the objective response rate (ORR). Considering the small sample sizes typical of early-phase oncology studies, we analytically derive a confidence interval for the weight, enabling assessment of borrowing precision without resampling procedures. Simulation studies under four toxicity scenarios and five baseline-covariate settings showed that the proposed method improved the probability that the lower bound of the 95% credible interval for ORR exceeded a prespecified threshold at efficacious doses, while avoiding false threshold crossings at weakly efficacious doses. A case study based on a metastatic pancreatic ductal adenocarcinoma trial illustrates the resulting borrowing weights and posterior estimates.

The particle filter is a powerful framework for estimating hidden states in dynamic systems where uncertainty, noise, and nonlinearity dominate. This mini-book offers a clear and structured introduction to the core ideas behind particle filters-how they represent uncertainty through random samples, update beliefs using observations, and maintain robustness where linear or Gaussian assumptions fail. Starting from the limitations of the Kalman filter, the book develops the intuition that drives the particle filter: belief as a cloud of weighted hypotheses that evolve through prediction, measurement, and resampling. Step by step, it connects these ideas to their mathematical foundations, showing how probability distributions can be approximated by a finite set of particles and how Bayesian reasoning unfolds in sampled form. Illustrated examples, numerical walk-throughs, and Python code bring each concept to life, bridging the gap between theory and implementation. By the end, readers will not only understand the algorithmic flow of the particle filter but also develop an intuitive grasp of how randomness and structure together enable systems to infer, adapt, and make sense of noisy observations in real time.

Data quality is critical to Intelligent Transportation Systems (ITS), as complete and accurate traffic data underpin reliable decision-making in traffic control and management. Recent advances in low-rank tensor recovery algorithms have shown strong potential in capturing the inherent structure of high-dimensional traffic data and restoring degraded observations. However, traditional batch-based methods demand substantial computational and storage resources, which limits their scalability in the face of continuously expanding traffic data volumes. Moreover, recent online tensor recovery methods often suffer from severe performance degradation in complex real-world scenarios due to their insufficient exploitation of the intrinsic structural properties of traffic data. To address these challenges, we reformulate the traffic data recovery problem within a streaming framework, and propose a novel online robust tensor recovery algorithm that simultaneously leverages both the global spatio-temporal correlations and local consistency of traffic data, achieving high recovery accuracy and significantly improved computational efficiency in large-scale scenarios. Our method is capable of simultaneously handling missing and anomalous values in traffic data, and demonstrates strong adaptability across diverse missing patterns. Experimental results on three real-world traffic datasets demonstrate that the proposed approach achieves high recovery accuracy while significantly improving computational efficiency by up to three orders of magnitude compared to state-of-the-art batch-based methods. These findings highlight the potential of the proposed approach as a scalable and effective solution for traffic data quality enhancement in ITS.

The proliferation of saddle points, rather than poor local minima, is increasingly understood to be a primary obstacle in large-scale non-convex optimization for machine learning. Variable elimination algorithms, like Variable Projection (VarPro), have long been observed to exhibit superior convergence and robustness in practice, yet a principled understanding of why they so effectively navigate these complex energy landscapes has remained elusive. In this work, we provide a rigorous geometric explanation by comparing the optimization landscapes of the original and reduced formulations. Through a rigorous analysis based on Hessian inertia and the Schur complement, we prove that variable elimination fundamentally reshapes the critical point structure of the objective function, revealing that local maxima in the reduced landscape are created from, and correspond directly to, saddle points in the original formulation. Our findings are illustrated on the canonical problem of non-convex matrix factorization, visualized directly on two-parameter neural networks, and finally validated in training deep Residual Networks, where our approach yields dramatic improvements in stability and convergence to superior minima. This work goes beyond explaining an existing method; it establishes landscape simplification via saddle point transformation as a powerful principle that can guide the design of a new generation of more robust and efficient optimization algorithms.

We study inference on a low-dimensional functional $\beta$ in the presence of infinite-dimensional nuisance parameters. Classical inferential methods are typically based on Wald intervals, whose large-sample validity rests on asymptotic negligibility of nuisance error; for example, influence-curve based estimators (Double/Debiased Machine Learning, DML) are asymptotically Gaussian when nuisance estimators converge faster than $n^{-1/4}$. Although such negligibility can hold even in nonparametric classes, it can be restrictive. To relax this requirement, we propose Perturbed Double Machine Learning, which ensures valid inference even when nuisance estimators converge slower than $n^{-1/4}$. Our proposal is to (i) inject randomness into the nuisance estimation step to generate perturbed nuisance models, each yielding an estimate of $\beta$ and a Wald interval, and (ii) filter out perturbations whose deviations from the original DML estimate exceed a threshold. For Lasso nuisance learners, we show that, with high probability, at least one perturbation yields nuisance estimates sufficiently close to the truth, so the associated estimator of $\beta$ is close to an oracle with known nuisances. The union of retained intervals delivers valid coverage even when the DML estimator converges slower than $n^{-1/2}$. The framework extends to general machine-learning nuisance learners, and simulations show coverage when state-of-the-art methods fail.

Missing data is a fundamental challenge in data science, significantly hindering analysis and decision-making across a wide range of disciplines, including healthcare, bioinformatics, social science, e-commerce, and industrial monitoring. Despite decades of research and numerous imputation methods, the literature remains fragmented across fields, creating a critical need for a comprehensive synthesis that connects statistical foundations with modern machine learning advances. This work systematically reviews core concepts-including missingness mechanisms, single versus multiple imputation, and different imputation goals-and examines problem characteristics across various domains. It provides a thorough categorization of imputation methods, spanning classical techniques (e.g., regression, the EM algorithm) to modern approaches like low-rank and high-rank matrix completion, deep learning models (autoencoders, GANs, diffusion models, graph neural networks), and large language models. Special attention is given to methods for complex data types, such as tensors, time series, streaming data, graph-structured data, categorical data, and multimodal data. Beyond methodology, we investigate the crucial integration of imputation with downstream tasks like classification, clustering, and anomaly detection, examining both sequential pipelines and joint optimization frameworks. The review also assesses theoretical guarantees, benchmarking resources, and evaluation metrics. Finally, we identify critical challenges and future directions, emphasizing model selection and hyperparameter optimization, the growing importance of privacy-preserving imputation via federated learning, and the pursuit of generalizable models that can adapt across domains and data types, thereby outlining a roadmap for future research.

It has been shown that the chain of thought (CoT) can enhance the power of large language models (LLMs) to solve certain mathematical reasoning problems. However, the capacity of CoT is still not fully explored. As an important instance, the following basic question has not yet been answered: Does CoT expand the capability of transformers across all reasoning tasks? We demonstrate that reasoning with transformers is essentially a memorization problem for reasoning datasets. Thus, examining the power of CoT across all reasoning tasks amounts to analyzing the memorization capabilities of CoT transformers. In this paper, we give a complete description of the memorization capabilities of fixed-precision transformers with or without CoT and give a negative answer to the above-mentioned question. Precisely, we first give necessary and sufficient conditions for fixed-precision transformers with and without CoT to memorize a finite reasoning dataset and show that these two conditions do not imply each other. Then, we give lower and upper bounds for the number of parameters needed for transformers with or without CoT to memorize a finite reasoning dataset with $N$ elements, which are $\overline{\Theta}(N)$ in all cases. This implies that there exist reasoning tasks for which CoT does not enhance the reasoning power of transformers, leading to a negative answer to the above-mentioned question. Finally, we give the first results on memorizing infinite reasoning datasets by CoT transformers and show that some simple infinite datasets cannot be memorized by transformers with or without CoT.

In this short note, we characterize stability of the Kim–Milman flow map – also known as the probability flow ODE – with respect to variations in the target measure in relative Fisher information.

We investigate parameter estimation in subcritical continuous-time birth-and-death processes with multiple births. We show that the classical maximum likelihood estimators for the model parameters, based on the continuous observation of a single non-extinct trajectory, are not consistent in the usual sense: conditional on survival up to time $t$, they converge as $t \to \infty$ to the corresponding quantities in the associated $Q$-process, namely the process conditioned to survive in the distant future. We develop the first $C$-consistent estimators in this setting, which converge to the true parameter values when conditioning on survival up to time $t$, and establish their asymptotic normality. The analysis relies on spine decompositions and coupling techniques.

This paper introduces a structural equation formulation that gives rise to a new family of quasi-periodic Gaussian processes, useful to process a broad class of natural and physiological signals. The proposed formulation simplifies generation and forecasting, and provides hyperparameter estimates, which we exploit in a convergent and consistent iterative estimation algorithm. A bootstrap approach for standard error estimation and confidence intervals is also provided. We demonstrate the computational and scaling benefits of the proposed approach on a broad class of problems, including water level tidal analysis, CO\textsubscript{2} emission data, and sunspot numbers data. By leveraging the structural equations, our method reduces the cost of likelihood evaluations and predictions from $\mathcal{O}(k^2 p^2)$ to $\mathcal{O}(p^2)$, significantly improving scalability.

Medical imaging often operates under limited labeled data, especially in rare disease and low resource clinical environments. Existing multimodal and meta learning approaches improve performance in these settings but lack a theoretical explanation of why or when they succeed. This paper presents a unified theoretical framework for few shot multimodal medical imaging that jointly characterizes sample complexity, uncertainty quantification, and interpretability. Using PAC learning, VC theory, and PAC Bayesian analysis, we derive bounds that describe the minimum number of labeled samples required for reliable performance and show how complementary modalities reduce effective capacity through an information gain term. We further introduce a formal metric for explanation stability, proving that explanation variance decreases at an inverse n rate. A sequential Bayesian interpretation of Chain of Thought reasoning is also developed to show stepwise posterior contraction. To illustrate these ideas, we implement a controlled multimodal dataset and evaluate an additive CNN MLP fusion model under few shot regimes, confirming predicted multimodal gains, modality interference at larger sample sizes, and shrinking predictive uncertainty. Together, the framework provides a principled foundation for designing data efficient, uncertainty aware, and interpretable diagnostic models in low resource settings.