Preconditioning is a key component of MCMC algorithms that improves sampling efficiency by facilitating exploration of geometrically complex target distributions through an invertible map. While linear preconditioners are often sufficient for moderately complex target distributions, recent work has explored nonlinear preconditioning with invertible neural networks as components of normalizing flows (NFs). However, empirical and theoretical studies show that overparameterized NF preconditioners can degrade sampling efficiency and fit quality. Moreover, existing NF-based approaches do not adapt their architectures to the target distribution. Related work outside of MCMC similarly finds that suitably parameterized NFs can achieve comparable or superior performance with substantially less training time or data. We propose a factorized preconditioning architecture that reduces NF complexity by combining a linear component with a conditional NF, improving adaptability to target geometry. The linear preconditioner is applied to dimensions that are approximately Gaussian, as estimated from warmup samples, while the conditional NF models more complex dimensions. Our method yields significantly better tail samples on two complex synthetic distributions and consistently better performance on a sparse logistic regression posterior across varying likelihood and prior strengths. It also achieves higher effective sample sizes on hierarchical Bayesian model posteriors with weak likelihoods and strong funnel geometries. This approach is particularly relevant for hierarchical Bayesian model analyses with limited data and could inform current theoretical and software strides in neural MCMC design.

The Lasso has been widely used as a method for variable selection, valued for its simplicity and empirical performance. However, Lasso's selection stability deteriorates in the presence of correlated predictors. Several approaches have been developed to mitigate this limitation. In this paper, we provide a brief review of existing approaches, highlighting their limitations. We then propose a simple technique to improve the selection stability of Lasso by integrating a weighting scheme into the Lasso penalty function, where the weights are defined as an increasing function of a correlation-adjusted ranking that reflects the predictive power of predictors. Empirical evaluations on both simulated and real-world datasets demonstrate the efficacy of the proposed method. Additional numerical results demonstrate the effectiveness of the proposed approach in stabilizing other regularization-based selection methods, indicating its potential as a general-purpose solution.

This paper studies the high-dimensional scaling limits of online stochastic gradient descent (SGD) for single-layer networks. Building on the seminal work of Saad and Solla, which analyzed the deterministic (ballistic) scaling limits of SGD corresponding to the gradient flow of the population loss, we focus on the critical scaling regime of the step size. Below this critical scale, the effective dynamics are governed by ballistic (ODE) limits, but at the critical scale, new correction term appears that changes the phase diagram. In this regime, near the fixed points, the corresponding diffusive (SDE) limits of the effective dynamics reduces to an Ornstein-Uhlenbeck process under certain conditions. These results highlight how the information exponent controls sample complexity and illustrates the limitations of deterministic scaling limit in capturing the stochastic fluctuations of high-dimensional learning dynamics.

Time-series forecasting increasingly demands not only accurate observational predictions but also causal forecasting under interventional and counterfactual queries in multivariate systems. We present DoFlow, a flow based generative model defined over a causal DAG that delivers coherent observational and interventional predictions, as well as counterfactuals through the natural encoding and decoding mechanism of continuous normalizing flows (CNFs). We also provide a supporting counterfactual recovery result under certain assumptions. Beyond forecasting, DoFlow provides explicit likelihoods of future trajectories, enabling principled anomaly detection. Experiments on synthetic datasets with various causal DAG and real world hydropower and cancer treatment time series show that DoFlow achieves accurate system-wide observational forecasting, enables causal forecasting over interventional and counterfactual queries, and effectively detects anomalies. This work contributes to the broader goal of unifying causal reasoning and generative modeling for complex dynamical systems.

Objectives: Unsupervised learning with electronic health record (EHR) data has shown promise for phenotype discovery, but approaches typically disregard existing clinical information, limiting interpretability. We operationalize a Bayesian latent class framework for phenotyping that incorporates domain-specific knowledge to improve clinical meaningfulness of EHR-derived phenotypes and illustrate its utility by identifying an asthma sub-phenotype informed by features of Type 2 (T2) inflammation. Materials and methods: We illustrate a framework for incorporating clinical knowledge into a Bayesian latent class model via informative priors to guide unsupervised clustering toward clinically relevant subgroups. This approach models missingness, accounting for potential missing-not-at-random patterns, and provides patient-level probabilities for phenotype assignment with uncertainty. Using reusable and flexible code, we applied the model to a large asthma EHR cohort, specifying informative priors for T2 inflammation-related features and weakly informative priors for other clinical variables, allowing the data to inform posterior distributions. Results and Conclusion: Using encounter data from January 2017 to February 2024 for 44,642 adult asthma patients, we found a bimodal posterior distribution of phenotype assignment, indicating clear class separation. The T2 inflammation-informed class (38.7%) was characterized by elevated eosinophil levels and allergy markers, plus high healthcare utilization and medication use, despite weakly informative priors on the latter variables. These patterns suggest an "uncontrolled T2-high" sub-phenotype. This demonstrates how our Bayesian latent class modeling approach supports hypothesis generation and cohort identification in EHR-based studies of heterogeneous diseases without well-established phenotype definitions.

We develop a Gaussian process framework for learning interaction kernels in multi-species interacting particle systems from trajectory data. Such systems provide a canonical setting for multiscale modeling, where simple microscopic interaction rules generate complex macroscopic behaviors. While our earlier work established a Gaussian process approach and convergence theory for single-species systems, and later extended to second-order models with alignment and energy-type interactions, the multi-species setting introduces new challenges: heterogeneous populations interact both within and across species, the number of unknown kernels grows, and asymmetric interactions such as predator-prey dynamics must be accommodated. We formulate the learning problem in a nonparametric Bayesian setting and establish rigorous statistical guarantees. Our analysis shows recoverability of the interaction kernels, provides quantitative error bounds, and proves statistical optimality of posterior estimators, thereby unifying and generalizing previous single-species theory. Numerical experiments confirm the theoretical predictions and demonstrate the effectiveness of the proposed approach, highlighting its advantages over existing kernel-based methods. This work contributes a complete statistical framework for data-driven inference of interaction laws in multi-species systems, advancing the broader multiscale modeling program of connecting microscopic particle dynamics with emergent macroscopic behavior.

We develop new methods to integrate experimental and observational data in causal inference. While randomized controlled trials offer strong internal validity, they are often costly and therefore limited in sample size. Observational data, though cheaper and often with larger sample sizes, are prone to biases due to unmeasured confounders. To harness their complementary strengths, we propose a systematic framework that formulates causal estimation as an empirical risk minimization (ERM) problem. A full model containing the causal parameter is obtained by minimizing a weighted combination of experimental and observational losses--capturing the causal parameter's validity and the full model's fit, respectively. The weight is chosen through cross-validation on the causal parameter across experimental folds. Our experiments on real and synthetic data show the efficacy and reliability of our method. We also provide theoretical non-asymptotic error bounds.

Ensuring algorithmic fairness remains a significant challenge in machine learning, particularly as models are increasingly applied across diverse domains. While numerous fairness criteria exist, they often lack generalizability across different machine learning problems. This paper examines the connections and differences among various sparsity measures in promoting fairness and proposes a unified sparsity-based framework for evaluating algorithmic fairness. The framework aligns with existing fairness criteria and demonstrates broad applicability to a wide range of machine learning tasks. We demonstrate the effectiveness of the proposed framework as an evaluation metric through extensive experiments on a variety of datasets and bias mitigation methods. This work provides a novel perspective to algorithmic fairness by framing it through the lens of sparsity and social equity, offering potential for broader impact on fairness research and applications.

Traditional topic models such as Latent Dirichlet Allocation (LDA) have been widely used to uncover latent structures in text corpora, but they often struggle to integrate auxiliary information such as metadata, user attributes, or document labels. These limitations restrict their expressiveness, personalization, and interpretability. To address this, we propose nnLDA, a neural-augmented probabilistic topic model that dynamically incorporates side information through a neural prior mechanism. nnLDA models each document as a mixture of latent topics, where the prior over topic proportions is generated by a neural network conditioned on auxiliary features. This design allows the model to capture complex nonlinear interactions between side information and topic distributions that static Dirichlet priors cannot represent. We develop a stochastic variational Expectation-Maximization algorithm to jointly optimize the neural and probabilistic components. Across multiple benchmark datasets, nnLDA consistently outperforms LDA and Dirichlet-Multinomial Regression in topic coherence, perplexity, and downstream classification. These results highlight the benefits of combining neural representation learning with probabilistic topic modeling in settings where side information is available.

While looped transformers (termed as Looped-Attn) often outperform standard transformers (termed as Single-Attn) on complex reasoning tasks, the theoretical basis for this advantage remains underexplored. In this paper, we explain this phenomenon through the lens of loss landscape geometry, inspired by empirical observations of their distinct dynamics at both sample and Hessian levels. To formalize this, we extend the River-Valley landscape model by distinguishing between U-shaped valleys (flat) and V-shaped valleys (steep). Based on empirical observations, we conjecture that the recursive architecture of Looped-Attn induces a landscape-level inductive bias towards River-V-Valley. Theoretical derivations based on this inductive bias guarantee a better loss convergence along the river due to valley hopping, and further encourage learning about complex patterns compared to the River-U-Valley induced by Single-Attn. Building on this insight, we propose SHIFT (Staged HIerarchical Framework for Progressive Training), a staged training framework that accelerates the training process of Looped-Attn while achieving comparable performances.