Papers

Detecting weak signals buried in complex, non-Gaussian noise is a fundamental challenge in science and engineering, with applications ranging from radar systems and communications to industrial monitoring and gravitational wave detection. The Rao detector, a key concept in this domain, achieves asymptotically optimal performance as the number of measurements increases, but requires precise knowledge of the data's statistical properties, often relying on simplified noise models. We propose a hybrid framework that combines a lightweight neural network with the Rao detection framework to address this limitation. The neural network, trained on noise-only data, learns the optimal multivariate nonlinearity, transforming noisy data to enhance signal detectability. The newly introduced LRao detector then fully extracts the signal information, achieving asymptotically optimal performance even under challenging noise conditions. Validated on both simulated and real-world magnetic sensor data, our method significantly outperforms conventional approaches. By bridging data-driven techniques with model-based signal processing, it offers a robust and interpretable solution for signal detection across diverse applications.

Conformal prediction (CP) provides finite-sample, distribution-free marginal coverage, but standard conformal regression intervals can be inefficient under heteroscedasticity and skewness. In particular, popular constructions such as conformalized quantile regression (CQR) often inherit a fixed notion of center and enforce equal-tailed errors, which can displace the interval away from high-density regions and produce unnecessarily wide sets. We propose Co-optimization for Adaptive Conformal Prediction (CoCP), a framework that learns prediction intervals by jointly optimizing a center $m(x)$ and a radius $h(x)$.CoCP alternates between (i) learning $h(x)$ via quantile regression on the folded absolute residual around the current center, and (ii) refining $m(x)$ with a differentiable soft-coverage objective whose gradients concentrate near the current boundaries, effectively correcting mis-centering without estimating the full conditional density. Finite-sample marginal validity is guaranteed by split-conformal calibration with a normalized nonconformity score. Theory characterizes the population fixed point of the soft objective and shows that, under standard regularity conditions, CoCP asymptotically approaches the length-minimizing conditional interval at the target coverage level as the estimation error and smoothing vanish. Experiments on synthetic and real benchmarks demonstrate that CoCP yields consistently shorter intervals and achieves state-of-the-art conditional-coverage diagnostics.

We have developed and tested a spatial scan statistic for categorical, functional data (CFSS) - a data structure within which current approaches cannot identify spatial clusters. Our methodology combines an encoding scheme for categorical, functional observations with a nonparametric scan statistic. In a simulation study with three distinct scenarios, the CFSS accurately recovered the simulated spatial clusters and gave very low false positive rates, high true positive rates, and high positive predictive values. We have also used the CFSS to identify and characterize spatial clusters in French air pollution data from the winter of 2024.

Bayesian sample size calculations in clinical trials usually rely on complex Monte Carlo simulations in practice. Obtaining bounds on Bayesian notions of the false-positive rate and power often lack closed-form or approximate numerical solutions. In this paper, we focus on power and sample size calculations for Bayes factors in the two-arm binomial setting of phase II trials. We cover point-null versus composite and directional hypothesis tests, derive the corresponding Bayes factors, and discuss relevant aspects to consider when pursuing Bayesian design of experiments with the introduced approach. Based on these Bayes factors, we propose a numerical approach which allows to determine the necessary sample size to obtain prespecified bounds of Bayesian power and type-I-error rate in a computationally efficient way. Our method does not rely on Monte Carlo simulations and instead solely relies on standard numerical methods. Real-world examples of phase II trials from oncology and autoimmune diseases illustrate the advantage of the proposed calibration method. In summary, our approach allows for a Bayes-frequentist compromise by providing a Bayesian analogue to a frequentist power analysis for various Bayes factors in the two-arm binomial setting of a phase II clinical trial. The methods are implemented in our R package bfbin2arm.

We investigate the limiting behavior of discrete determinantal point processes (DPPs) towards continuous DPPs when the size of the set to sample from goes to infinity. We propose a non-asymptotic characterization of this limit in terms of the concentration of statistics associated to these processes, which we refer to as "weak coherency". This allows to translate statistical guarantees from the limiting process to the original, discrete one. Our main result describes sufficient conditions for weak coherency to hold. In particular, our study encompasses settings where both the kernel of the continuous process and its underlying space are inaccessible, or when the discrete marginal kernel is a noisy version of its continuous counterpart. We illustrate our theory on several examples. We prove that a discrete multivariate orthogonal polynomial ensemble can be used to produce coresets strictly smaller than independent sampling for the same error. We propose a process achieving repulsive sampling on an unknown manifold from a set of points sampled from an unknown density. Finally, we show that continuous DPPs can be obtained as limits on random graphs with Bernoulli edges, even when only observing the graph structure. We obtain interesting byproduct results along the way.

Electricity networks are vulnerable to weather damage, with severe events often leading to faults and power outages. Timely forecasts of fault occurrences, ranging from nowcasts to several days ahead, can enhance preparedness, support faster response, and reduce outage durations. To be operationally useful, such forecasts must quantify uncertainty, enabling risk-informed resource allocation. We present a novel probabilistic framework for forecasting fault counts that captures typical and extreme events. Non-extreme faults are modeled linearly interpolating estimates from multiple additive quantile regressions, while extreme events are described through a discrete generalized Pareto distribution. To incorporate the impact of weather fluctuations, we use ensemble numerical weather predictions, which helps to quantify uncertainty in the forecasts. This approach is designed to provide reliable fault predictions up to four days ahead. We evaluate the model through numerical experiments and apply it to historical fault data from two electricity distribution networks in Great Britain. The resulting forecasts demonstrate substantial improvements over business-as-usual and alternative modeling approaches. A practitioner trial conducted with Scottish Power Energy Networks from October 2024 to March 2025 further demonstrates the operational value of the forecasts. Engineers found them sufficiently reliable to inform decision-making, offering benefits to both network operators and electricity consumers.

As autonomous systems such as drones, become increasingly deployed in high-stakes, human-centric domains, it is critical to evaluate the ethical alignment since failure to do so imposes imminent danger to human lives, and long term bias in decision-making. Automated ethical benchmarking of these systems is understudied due to the lack of ubiquitous, well-defined metrics for evaluation, and stakeholder-specific subjectivity, which cannot be modeled analytically. To address these challenges, we propose SEED-SET, a Bayesian experimental design framework that incorporates domain-specific objective evaluations, and subjective value judgments from stakeholders. SEED-SET models both evaluation types separately with hierarchical Gaussian Processes, and uses a novel acquisition strategy to propose interesting test candidates based on learnt qualitative preferences and objectives that align with the stakeholder preferences. We validate our approach for ethical benchmarking of autonomous agents on two applications and find our method to perform the best. Our method provides an interpretable and efficient trade-off between exploration and exploitation, by generating up to $2\times$ optimal test candidates compared to baselines, with $1.25\times$ improvement in coverage of high dimensional search spaces.

Due to their efficiency and small size, decision trees and random forests are popular machine learning models used for classification on resource-constrained systems. In such systems, the available execution time for inference in a random forest might not be sufficient for a complete model execution. Ideally, the already gained prediction confidence should be retained. An anytime algorithm is designed to be able to be aborted anytime, while giving a result with an increasing quality over time. Previous approaches have realized random forests as anytime algorithms on the granularity of trees, stopping after some but not all trees of a forest have been executed. However, due to the way decision trees subdivide the sample space in every step, an increase in prediction quality is achieved with every additional step in one tree. In this paper, we realize decision trees and random forest as anytime algorithms on the granularity of single steps in trees. This approach opens a design space to define the step order in a forest, which has the potential to optimize the mean accuracy. We propose the Optimal Order, which finds a step order with a maximal mean accuracy in exponential runtime and the polynomial runtime heuristics Forward Squirrel Order and Backward Squirrel Order, which greedily maximize the accuracy for each additional step taken down and up the trees, respectively. Our evaluation shows, that the Backward Squirrel Order performs $\sim94\%$ as well as the Optimal Order and $\sim99\%$ as well as all other step orders.

We study the training dynamics of gradient descent in a softmax self-attention layer trained to perform linear regression and show that a simple first-order optimization algorithm can converge to the globally optimal self-attention parameters at a geometric rate. Our analysis proceeds in two steps. First, we show that in the infinite-data limit the regression problem solved by the self-attention layer is equivalent to a nonconvex matrix factorization problem. Second, we exploit this connection to design a novel "structure-aware" variant of gradient descent which efficiently optimizes the original finite-data regression objective. Our optimization algorithm features several innovations over standard gradient descent, including a preconditioner and regularizer which help avoid spurious stationary points, and a data-dependent spectral initialization of parameters which lie near the manifold of global minima with high probability.

We consider the optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments, including those on real-data emulators.

Non-negative matrix factorization (NMF) is widely used for parts-based representations, yet formal inference for covariate effects is rarely available when the basis is learned under non-negativity. We introduce non-negative matrix factorization with random effects (NMF-RE), a mean-structure latent-variable model $Y=X(\Theta A+U)+\mathcal{E}$ that combines covariate-driven scores with unit-specific deviations. Random effects act as a working device for modeling heterogeneity and controlling complexity; we monitor their effective degrees of freedom and enforce a df-based cap to prevent near-saturated fits. Estimation alternates closed-form ridge (BLUP-like) updates for $U$ with multiplicative non-negative updates for $X$ and $\Theta$. For inference on $\Theta$, we condition on $(\widehat X,\widehat U)$ and obtain fast uncertainty quantification via asymptotic linearization, a one-step Newton update, and a multiplier (wild) bootstrap; this avoids repeated constrained re-optimization. Simulations include a targeted stress test showing that, without df control, the random-effects penalty can collapse and inference for $\Theta$ becomes degenerate, whereas the df-cap prevents this failure mode. The non-negativity constraint induces sparse, parts-based loadings – a measurement-side variable selection – while inference on $\Theta$ identifies which covariates affect which components, providing covariate-side selection. Longitudinal, psychometric, spatial-flow, and text examples further illustrate stable, interpretable covariate-effect inference.

The conditional mean risk-sharing (CMRS) rule is an important tool for distributing aggregate losses across individual risks, but its implementation in continuous multivariate models typically requires complicated multidimensional integrals. We develop a framework to compute CMRS allocations from the joint Laplace–Stieltjes transform of the risk vector. The LSTs of the allocation measures $\nu_i(B)=\mathbb{E}[X_i\boldsymbol{1}_{\{S\in B\}}]$ are expressed as partial derivatives of the joint LST evaluated on the diagonal $t_1=\cdots=t_n$. When densities exist, this yields one-dimensional Laplace inversions for $f_S$ and $\xi_i$, and hence $h_i(s)=\xi_i(s)/f_S(s)$ on the absolutely continuous part, providing closed-form or semi-analytic solutions for a broad class of distributions. We also develop numerical inversion methods for cases where analytic inversion is unavailable. We introduce an exponential tilting procedure to stabilize numerical inversion in low-probability aggregate events. We provide several examples to illustrate the approach, including in some high-dimensional settings where existing approaches are infeasible.

Gauss's method of orbit determination (OD) and its variants are among the most popular initial state estimation techniques for astronomers and engineers alike. However, owing to its assumptions regarding the two-body problem, Gauss's method is inapplicable in the cislunar domain, where three body effects dominate. We introduce a hybrid Particle Gaussian Mixture filtering method, a purely recursive probabilistic orbit determination framework based on a combination of the Markov Chain Monte Carlo based Particle Gaussian Mixture-II (PGM-II) and Particle Gaussian Mixture-I (PGM-I) filters. This method enables us to fuse probabilistic information with angles-only observations from terrestrial telescopes for short and long-term cislunar target tracking. We demonstrate this technique on an important cislunar orbit regime.

Kernel density estimators with circular data have been studied extensively for decades, as they allow flexible estimations even when the shape of the underlying density is complex. Many recent studies have examined bias correction methods; however, these methods are limited by the order when trying to improve the convergence rate of the bias, even if the true density is sufficiently smooth. To overcome this limitation, the present study considers a new bias correction approach based on the characteristic functions of the underlying circular density. We introduce wrapped flat-top kernels, which are generated by wrapping the standard flat-top kernels defined on the real line onto the circumference of a unit circle. The asymptotic mean squared errors of the wrapped flat-top kernel density estimators are then derived. The results show that the convergence rate of these estimators is faster than that of previously introduced estimators. Furthermore, wrapped flat-top kernel density estimators achieve $\sqrt{n}$-consistency under the characteristic function of finite support, such as the circular uniform and cardioid distributions. We confirm these theoretical results in the numerical experiments. In empirical analyses, we also show that wrapped flat-top kernel density estimators effectively capture the shape of data. Therefore, such estimators are expected to allow flexible and accurate estimation in circular data analysis.

Graph Neural Networks (GNNs) face fundamental limitations in expressivity and capturing structural heterogeneity. Standard message-passing architectures are constrained by the 1-dimensional Weisfeiler-Leman (1-WL) test, unable to distinguish graphs beyond degree sequences, and aggregate information uniformly from neighbors, failing to capture how nodes occupy different structural positions within higher-order patterns. While methods exist to achieve higher expressivity, they incur prohibitive computational costs and lack unified frameworks for flexibly encoding diverse structural properties. To address these limitations, we introduce Invariant-Stratified Propagation (ISP), a framework comprising both a novel WL variant (ISP-WL) and its efficient neural network implementation (ISPGNN). ISP stratifies nodes according to graph invariants, processing them in hierarchical strata that reveal structural distinctions invisible to 1-WL. Through hierarchical structural heterogeneity encoding, ISP quantifies differences in nodes' structural positions within higher-order patterns, distinguishing interactions where participants occupy different roles from those with uniform participation. We provide formal theoretical analysis establishing enhanced expressivity beyond 1-WL, convergence guarantees, and inherent resistance to oversmoothing. Extensive experiments across graph classification, node classification, and influence estimation demonstrate consistent improvements over standard architectures and state-of-the-art expressive baselines.

Two-component mixture models are particularly useful for identifying differentially expressed genes, but their performance can deteriorate markedly when the alternative distribution departs from parametric assumptions or symmetry. We propose a semiparametric mixture model in which the null component is standard normal and the alternative follows a skew-normal scale mixture with an unspecified scale mixing distribution. This formulation accommodates skewness and heavy tails, providing a flexible and computationally tractable tool for differential gene-expression analysis without restrictive distributional assumptions. We establish identifiability and consistency of the model and develop an efficient estimation algorithm that incorporates nonparametric maximum likelihood estimation of the scale distribution. Numerical studies show notable improvements over existing parametric and nonparametric approaches for modeling the alternative distribution, and applications to colon cancer and leukemia datasets demonstrate reduced false discovery and false negative rates.

Integrated IPD-AD analysis, which combines individual participant data (IPD) with aggregate data (AD), is increasingly recognized as an effective strategy for generating more reliable and generalizable inferences from heterogeneous studies. While most existing work has focused on algorithmic approaches, this paper investigates a complementary yet underexplored question: how different forms of AD influence the efficiency of data integration. Working within a constrained maximum likelihood estimation framework, we compare commonly reported summary statistics and show that subgroup-specific summaries can substantially improve estimation efficiency. In particular, we find that AD derived from outcome-stratified subgroups (e.g., cases and controls) consistently yield greater efficiency gains than those based on covariate-stratified subgroups (e.g., age or exposure categories), especially when the outcome is continuous. Although outcome-stratified summaries are commonly reported for discrete outcomes, they are rarely provided when the outcome is continuous. Our findings therefore support the routine inclusion of outcome-stratified summaries for continuous endpoints in trial reports and public data repositories to facilitate more efficient evidence synthesis. We further extend the constrained maximum likelihood framework to accommodate dataset shift and develop a fast, non-iterative estimation procedure to improve numerical stability and scalability. We illustrate the proposed methodology with two applications: an analysis of income data under covariate shift and an analysis of housing data under prior probability shift.

Sparse plus Low-Rank $(\mathbf{S} + \mathbf{LR})$ decomposition of Large Language Models (LLMs) has emerged as a promising direction in model compression, aiming to decompose pre-trained model weights into a sum of sparse and low-rank matrices $(\mathbf{W} \approx \mathbf{S} + \mathbf{LR})$. Despite recent progress, existing methods often suffer from substantial performance degradation compared to dense models. In this work, we introduce 3BASiL-TM, an efficient one-shot post-training method for $(\mathbf{S} + \mathbf{LR})$ decomposition of LLMs that addresses this gap. Our approach first introduces a novel 3-Block Alternating Direction Method of Multipliers (ADMM) method, termed 3BASiL, to minimize the layer-wise reconstruction error with convergence guarantees. We then design an efficient transformer-matching (TM) refinement step that jointly optimizes the sparse and low-rank components across transformer layers. This step minimizes a novel memory-efficient loss that aligns outputs at the transformer level. Notably, the TM procedure is universal as it can enhance any $(\mathbf{S} + \mathbf{LR})$ decomposition, including pure sparsity. Our numerical experiments show that 3BASiL-TM reduces the WikiText2 perplexity gap relative to dense LLaMA-8B model by over 30% under a (2:4 Sparse + 64 LR) configuration, compared to prior methods. Moreover, our method achieves over 2.5x faster compression runtime on an A100 GPU compared to SOTA $(\mathbf{S} + \mathbf{LR})$ method. Our code is available at https://github.com/mazumder-lab/3BASiL.

Real-time analysis of epidemic trends and forecasts can help support public health planning and the response to seasonal respiratory disease. Here, we present two models that were used in a 2025 New Zealand winter situational assessment programme for three respiratory pathogens: SARS-CoV-2, influenza and respiratory syncytial virus (RSV). Data on SARS-CoV-2 were obtained from the national Covid-19 surveillance system; data on influenza and RSV were limited to a sentinel hospital surveillance programme. Models were run weekly from May to October 2025 on these real-time disease surveillance data and provided a quantitative representation of the current epidemic trend, along with estimates of the epidemic growth rate and 28-day ahead forecasts of case incidence. Model results and interpretation were provided in weekly reports to public health partners as part of a trans-Tasman winter programme run by the Australia–Aotearoa Consortium for Epidemic Forecasting and Analytics (ACEFA). We compare in-season results that were included in these reports to a retrospective analysis of the complete data for the season. We conclude that real-time analyses performed reasonably well, and identify some areas for improvement in future winter situational assessment programmes.

We revisit the framework of Smart PAC learning, which seeks supervised learners which compete with semi-supervised learners that are provided full knowledge of the marginal distribution on unlabeled data. Prior work has shown that such marginal-by-marginal guarantees are possible for "most" marginals, with respect to an arbitrary fixed and known measure, but not more generally. We discover that this failure can be attributed to an "indistinguishability" phenomenon: There are marginals which cannot be statistically distinguished from other marginals that require different learning approaches. In such settings, semi-supervised learning cannot certify its guarantees from unlabeled data, rendering them arguably non-actionable. We propose relatively smart learning, a new framework which demands that a supervised learner compete only with the best "certifiable" semi-supervised guarantee. We show that such modest relaxation suffices to bypass the impossibility results from prior work. In the distribution-free setting, we show that the One-Inclusion Graph learner is relatively smart up to squaring the sample complexity, and show that no supervised learning algorithm can do better. For distribution-family settings, we show that relatively smart learning can be impossible or can require idiosyncratic learning approaches, and its difficulty can be non-monotone in the inclusion order on distribution families.

We study experiments on interacting populations of humans and AI agents, where both unit types and the interaction network remain unobserved. Although causal effects propagate throughout the system, the goal is to estimate effects on humans. Examples include online platforms where human users interact alongside AI-driven accounts. We assume a human-AI prior that gives each unit a probability of being human. While humans cannot be distinguished at the unit level, the prior allows us to compute the average human composition within large subpopulations. We then model outcome dynamics through a causal message passing (CMP) framework and analyze sample-mean outcomes across subpopulations. We show that by constructing subpopulations that vary in expected human composition and treatment exposure, one can consistently recover human-specific causal effects. Our results characterize when distributional knowledge of population composition (without observing unit types or the interaction network) is sufficient for identification. We validate the approach on a simulated human-AI platform driven by behaviorally differentiated LLM agents. Together, these results provide a theoretical and practical framework for experimentation in emerging human-AI systems.

We study adaptive estimation and inference in linear inverse problems defined by conditional moment restrictions. These problems are typically ill-posed, so estimators require regularization, yet the optimal regularization strength depends on a source parameter that is almost never known. We develop a discrepancy principle that chooses this parameter from empirical losses and prescribed bounds on statistical error, without knowing the source parameter. The framework covers Regularized DeepIV (RDIV) and the Tikhonov Regularized Adversarial Estimator (TRAE). Under the stated realizability conditions and comparable fitting and calibration error bounds, both one-step procedures attain strong- and weak-error bounds that match known-source rates for source parameters up to one. We also extend the framework to iterative procedures under realizability of the population iterates, widening the range of source parameters covered by the rate comparison. Finally, we use these adaptive estimators to construct an asymptotically normal doubly robust estimator that adapts to the well-posedness of the primal and dual problems.

Uniform Manifold Approximation and Projection (UMAP) is a widely used manifold learning technique for dimensionality reduction. This paper studies UMAP, supervised UMAP, and several competing dimensionality reduction methods, including Principal Component Analysis (PCA), Kernel PCA, Sliced Inverse Regression (SIR), Kernel SIR, and t-distributed Stochastic Neighbor Embedding, through a comprehensive comparative analysis. Although UMAP has attracted substantial attention for preserving local and global structures, its supervised extensions, particularly for regression settings, remain rather underexplored. We provide a systematic evaluation of supervised UMAP for both regression and classification using simulated and real datasets, with performance assessed via predictive accuracy on low-dimensional embeddings. Our results show that supervised UMAP performs well for classification but exhibits limitations in effectively incorporating response information for regression, highlighting an important direction for future development.

When data is scarce or mistakes are costly, average-case metrics fall short. What a practitioner needs is a guarantee: with probability at least $1-\delta$, the learned policy is $\varepsilon$-close to optimal after $N$ episodes. This is the PAC promise, and between 2018 and 2025 the RL theory community made striking progress on when such promises can be kept. We survey that progress. Our organizing tool is the Coverage-Structure-Objective (CSO) framework, proposed here, which decomposes nearly every PAC sample complexity result into three factors: coverage (how data were obtained), structure (intrinsic MDP or function-class complexity), and objective (what the learner must deliver). CSO is not a theorem but an interpretive template that identifies bottlenecks and makes cross-setting comparison immediate. The technical core covers tight tabular baselines and the uniform-PAC bridge to regret; structural complexity measures (Bellman rank, witness rank, Bellman-Eluder dimension) governing learnability with function approximation; results for linear, kernel/NTK, and low-rank models; reward-free exploration as upfront coverage investment; and pessimistic offline RL where inherited coverage is the binding constraint. We provide practitioner tools: rate lookup tables indexed by CSO coordinates, Bellman residual diagnostics, coverage estimation with deployment gates, and per-episode policy certificates. A final section catalogs open problems, separating near-term targets from frontier questions where coverage, structure, and computation tangle in ways current theory cannot resolve.

We propose two nonconvex regularization methods, LogLOP-l2/l1 and AdaLOP-l2/l1, for recovering block-sparse signals with unknown block partitions. These methods address the underestimation bias of existing convex approaches by extending log-sum penalty and the Minimax Concave Penalty (MCP) to the block-sparse domain via novel variational formulations. Unlike Generalized Moreau Enhancement (GME) and Bayesian methods dependent on the squared-error data fidelity term, our proposed methods are compatible with a broad range of data fidelity terms. We develop efficient Alternating Direction Method of Multipliers (ADMM)-based algorithms for these formulations that exhibit stable empirical convergence. Numerical experiments on synthetic data, angular power spectrum estimation, and denoising of nanopore currents demonstrate that our methods outperform state-of-the-art baselines in estimation accuracy.

Large Language Models (LLMs) are pretrained on massive datasets and later instruction-tuned via supervised fine-tuning (SFT) or reinforcement learning (RL). Best practices emphasize large, diverse pretraining data, whereas post-training operates differently: SFT relies on smaller, high-quality datasets, while RL benefits more from scale, with larger amounts of feedback often outweighing label quality. Yet it remains unclear why pretraining and RL require large datasets, why SFT excels on smaller ones, and what defines high-quality SFT data. In this work, we theoretically analyze transformers trained on an in-context weight prediction task for linear regression. Our analysis reveals several key findings: $(i)$ balanced pretraining data can induce latent capabilities later activated during post-training, and $(ii)$ SFT learns best from a small set of examples challenging for the pretrained model, while excessively large SFT datasets may dilute informative pretraining signals. In contrast, RL is most effective on large-scale data that is not overly difficult for the pretrained model. We validate these theoretical insights with experiments on large nonlinear transformer architectures.

We formulate and analyze a heterogeneous random hypergraph model, and we provide an achieveability result for recovery of hyperedges from the observed projected graph. We observe a projected graph which combines random hyperedges across all degrees, where a projected edge appears if and only if both vertices appear in at least one hyperedge. Our goal is to reconstruct the original set of hyperedges of degree $d_j$ for some $j$. Our achievability result is based on the idea of selecting maximal cliques of size $d_j$ in the projected graph, and we show that this algorithm succeeds under a natural condition on the densities. This achievability condition generalizes a known threshold for $d$-uniform hypergraphs with noiseless and noisy projections. We conjecture the threshold to be optimal for recovering hyperedges with the largest degree.

Circular variables that represent directions or periodic observations arise in many fields, such as biology and environmental sciences. An important issue when dealing with circular data is how to estimate their dispersion robustly, avoiding undue effects of anomalies. This work extends three robust dispersion measures from the line to the circle. Their robustness is studied via their influence functions and relative bias curves. From these dispersion measures, robust estimators of parameters of circular distributions can be derived. This yields robust estimators for the concentration parameter of the von Mises distribution and the dispersion parameter of the wrapped normal distribution. Their breakdown values and statistical efficiencies are obtained, and they are compared in a simulation study. Building on the best performing estimator, a robust circular anomaly detection procedure is developed, and employed to visualize outliers through a circular violin plot. Three real datasets are analyzed.

Unmeasured confounding is a fundamental obstacle to causal inference from observational data. Latent-variable methods address this challenge by imputing unobserved confounders, yet many lack explicit model-based identification guarantees and are difficult to extend to richer causal structures. We propose Confounder Imputation with Stochastic Neural Networks (CI-StoNet), which parameterizes the conditional structure of a causal directed acyclic graph using a stochastic neural network and imputes latent confounders via adaptive stochastic-gradient Hamiltonian Monte Carlo. Under SUTVA and overlap, and assuming that the structural components of the data-generating process are well approximated by a capacity-controlled sparse deep neural network class, we establish model identification and consistent estimation of the mean potential outcome under a fixed intervention within this class. Although the latent confounder is identifiable only up to reparameterizations that preserve the joint treatment-outcome distribution, the causal estimand is invariant across this observationally equivalent class. We further characterize the effect of overlap on estimation accuracy. Empirical results on simulated and benchmark datasets demonstrate accurate performance, and the framework extends naturally to proxy-variable and multiple-cause settings with overlap diagnostics and bootstrap-based uncertainty quantification.

Data center cooling systems consume significant auxiliary energy, yet optimization studies rarely quantify the gap between theoretically optimal and operationally deployable control strategies. This paper develops a digital twin of the liquid cooling infrastructure at the Frontier exascale supercomputer, in which a hot-temperature water system comprises three parallel subloops, each serving dedicated coolant distribution unit clusters through plate heat exchangers and variable-speed pumps. The surrogate model is built based on Modelica and validated through one full calendar year of 10-minute operational data following ASHRAE Guideline 14. The model achieves a subloop coefficient of variation of the root mean square error below 2.7% and a normalized mean bias error within 2.5%. Using this validated surrogate model, a layered optimization framework evaluates three progressively constrained strategies: an analytical flow-only optimization achieves 20.4% total energy saving, unconstrained joint optimization of flow rate and supply temperature demonstrates 30.1% total energy saving, and ramp-constrained optimization of flow rate and supply temperature, enforcing actuator rate limits, can reach total energy saving of 27.8%. The analysis reveals that the baseline system operates at 2.9 times the minimum thermally safe flow rate, and the co-optimizing supply temperature with flow rate nearly doubles the savings achievable by flow reduction alone.