In many modern applications, a carefully designed primary study provides individual-level data for interpretable modeling, while summary-level external information is available through black-box, efficient, and nonparametric machine-learning predictions. Although summary-level external information has been studied in the data integration literature, there is limited methodology for leveraging external nonparametric machine-learning predictions to improve statistical inference in the primary study. We propose a general empirical-likelihood framework that incorporates external predictions through moment constraints. An advantage of nonparametric machine-learning prediction is that it induces a rich class of valid moment restrictions that remain robust to covariate shift under a mild overlap condition without requiring explicit density-ratio modeling. We focus on multinomial logistic regression as the primary model and address common data-quality issues in external sources, including coarsened outcomes, partially observed covariates, covariate shift, and heterogeneity in generating mechanisms known as concept shift. We establish large-sample properties of the resulting fused estimator, including consistency and asymptotic normality under regularity conditions. Moreover, we provide mild sufficient conditions under which incorporating external predictions delivers a strict efficiency gain relative to the primary-only estimator. Simulation studies and an application to the National Health and Nutrition Examination Survey on multiclass blood-pressure classification.

We consider the problem of conditional independence (CI) testing and adopt a kernel-based approach. Kernel-based CI tests embed variables in reproducing kernel Hilbert spaces, regress their embeddings on the conditioning variables, and test the resulting residuals for marginal independence. This approach yields tests that are sensitive to a broad range of conditional dependencies. Existing methods, however, rely heavily on kernel ridge regression, which is computationally expensive when properly tuned and yields poorly calibrated tests when left untuned, which limits their practical usefulness. We propose the Generalised Kernel Covariance Measure (GKCM), a regression-model-agnostic kernel-based CI test that accommodates a broad class of regression estimators. Building on the Generalised Hilbertian Covariance Measure framework (Lundborg et al., 2022), we characterise conditions under which GKCM satisfies uniform asymptotic level guarantees. In simulations, GKCM paired with tree-based regression models frequently outperforms state-of-the-art CI tests across a diverse range of data-generating processes, achieving better type I error control and competitive or superior power.

Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hatθ^\top x$ approximately follows the convolution of the (generally non-Gaussian) distribution of $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

Constrained sampling is an important and challenging task in computational statistics, concerned with generating samples from a distribution under certain constraints. There are numerous types of algorithm aimed at this task, ranging from general Markov chain Monte Carlo, to unadjusted Langevin methods. In this article we propose a series of new sampling algorithms based on the latter of these, specifically the kinetic Langevin dynamics. Our series of algorithms are motivated on advanced numerical methods which are splitting order schemes, which include the BU and BAO families of splitting schemes.Their advantage lies in the fact that they have favorable strong order (bias) rates and computationally efficiency. In particular we provide a number of theoretical insights which include a Wasserstein contraction and convergence results. We are able to demonstrate favorable results, such as improved complexity bounds over existing non-splitting methodologies. Our results are verified through numerical experiments on a range of models with constraints, which include a toy example and Bayesian linear regression.

Online health communities (OHCs) are vital for fostering peer support and improving health outcomes. Support groups within these platforms can provide more personalized and cohesive peer support, yet traditional support group formation methods face challenges related to scalability, static categorization, and insufficient personalization. To overcome these limitations, we propose two novel machine learning models for automated support group formation: the Group specific Dirichlet Multinomial Regression (gDMR) and the Group specific Structured Topic Model (gSTM). These models integrate user generated textual content, demographic profiles, and interaction data represented through node embeddings derived from user networks to systematically automate personalized, semantically coherent support group formation. We evaluate the models on a large scale dataset from MedHelp, comprising over 2 million user posts. Both models substantially outperform baseline methods including LDA, DMR, and STM in predictive accuracy (held out log likelihood), semantic coherence (UMass metric), and internal group consistency. The gDMR model yields group covariates that facilitate practical implementation by leveraging relational patterns from network structures and demographic data. In contrast, gSTM emphasizes sparsity constraints to generate more distinct and thematically specific groups. Qualitative analysis further validates the alignment between model generated groups and manually coded themes, showing the practical relevance of the models in informing groups that address diverse health concerns such as chronic illness management, diagnostic uncertainty, and mental health. By reducing reliance on manual curation, these frameworks provide scalable solutions that enhance peer interactions within OHCs, with implications for patient engagement, community resilience, and health outcomes.

Tensor-on-tensor (TOT) regression is an important tool for the analysis of tensor data, aiming to predict a set of response tensors from a corresponding set of predictor tensors. However, standard TOT regression is sensitive to outliers, which may be present in both the response and the predictor. It can be affected by casewise outliers, which are observations that deviate from the bulk of the data, as well as by cellwise outliers, which are individual anomalous cells within the tensors. The latter are particularly common due to the typically large number of cells in tensor data. This paper introduces a novel robust TOT regression method, named ROTOT, that can handle both types of outliers simultaneously, and can cope with missing values as well. This method uses a single loss function to reduce the influence of both casewise and cellwise outliers in the response. The outliers in the predictor are handled using a robust Multilinear Principal Component Analysis method. Graphical diagnostic tools are also proposed to identify the different types of outliers detected. The performance of ROTOT is evaluated through extensive simulations and further illustrated using the Labeled Faces in the Wild dataset, where ROTOT is applied to predict facial attributes.

We study rank selection for low-rank tensor regression under random covariates design. Under a Gaussian random-design model and some mild conditions, we derive population expressions for the expected training-testing discrepancy (optimism) for both CP and Tucker decomposition. We further demonstrate that the optimism is minimized at the true tensor rank for both CP and Tucker regression. This yields a prediction-oriented rank-selection rule that aligns with cross-validation and extends naturally to tensor-model averaging. We also discuss conditions under which under- or over-ranked models may appear preferable, thereby clarifying the scope of the method. Finally, we showcase its practical utility on a real-world image regression task and extend its application to tensor-based compression of neural network, highlighting its potential for model selection in deep learning.

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon and SGD on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and moreover Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of Muon and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.