Papers

Linear mixed-effects model (LMM) is a cornerstone of longitudinal data analysis, but is limited to adeptly make heterogeneous analyses predictable under both group-specific fixed effects and subject-specific random effects. To address this challenge, we propose a novel statistical framework by using a large model prototype: a mixed effects mixture of experts model (MEMoE). This framework integrates the divide-and-conquer paradigm of Mixture of Experts Models with classical mixed-effect modeling. In the proposed MEMoE, each expert is a full LMM dedicated to capturing the longitudinal trajectory of a specific latent subpopulation, while another model gating function learns to route subjects to the most appropriate expert in a data-driven manner based on baseline covariates. We develop a robust inferential procedure for parameter estimation based on the Laplace Expectation-Maximization algorithm, with standard errors calibrated using robust sandwich estimators to account for potential model misspecification. Extensive simulation studies and an empirical application demonstrate that MEMoE outperforms both traditional single-population LMM and conventional Mixture of Experts models in terms of parameter recovery, classification accuracy, and overall model fit.

As buildings become increasingly connected and sensor-rich, intelligent remote heating control is rapidly superseding conventional local heating control. Such control algorithms often aim at reducing energy consumption by minimizing over-heating and utilizing free solar energy, for instance. Numerous companies offering heating optimization solutions have recently emerged. After installing such a system, end-users naturally want to quantify and verify the effect of such an investment, i.e., monetary return. Methods for tracking buildings' heating efficiency are diverse, ranging from simple weather normalization to more complex modeling approaches, but lack transparency and commonly agreed best practices. The problem is further complicated by the fact that buildings constantly undergo non-control-related changes that affect their energy efficiency, making it difficult to isolate and track only control-related effects using the existing methods. In this paper, we first review and derive methods for monitoring the overall efficiency of buildings, and show their inability to isolate the control effects from other changes happening in the buildings. We then propose a model-based approach for estimating and tracking only the control-related effects. Moreover, we show how the models can decompose the total control effect into sub-components to reveal where the energy effects come from. We demonstrate the methods using real data collected over approximately 10 years from the Danfoss Leanheat Building platform. Our scope focuses on district heated buildings with substation-level (supply temperature) control, but the methodology extends to other cases as well.

This monograph provides a rigorous overview of theoretical and methodological aspects of probabilistic inference and learning with Stein's method. Recipes are provided for constructing Stein discrepancies from Stein operators and Stein sets, and properties of these discrepancies such as computability, separation, convergence detection, and convergence control are discussed. Further, the connection between Stein operators and Stein variational gradient descent is set out in detail. The main definitions and results are precisely stated, and references to all proofs are provided.

We introduce ForeComp, an R package for comparing predictive accuracy using Diebold-Mariano type tests of equal predictive ability with standard and fixed smoothing inference. The package provides a common interface for loss differential based testing and includes Plot Tradeoff, a visual diagnostic for bandwidth sensitivity and the size-power tradeoff. We illustrate the toolkit with Survey of Professional Forecasters applications and Monte Carlo evidence on finite-sample performance.

Nonparametric density estimation for compositional data supported on the simplex is examined under a missing at random mechanism. Rather than imputing missing values and estimating the density from a completed data set, we adopt a strategy based on inverse probability weighting. The proposed estimator uses an adaptive Dirichlet kernel, which ensures nonnegativity on the simplex and favorable behavior near the boundary. When the observation probabilities are unknown, they are estimated through a Nadaraya-Watson regression step. The large-sample properties of the estimator are derived, including pointwise bias and variance expansions, optimal smoothing rates, and asymptotic normality. A simulation study investigates its finite-sample performance under varying sample sizes and missing rates. Simulations show our method outperforms inverse-probability-weighted kernel density estimators based on additive and isometric log-ratio transformations of the data for certain target densities. The methodology is further illustrated through an application to leukocyte composition data from the National Health and Nutrition Examination Survey (NHANES), which allows for the identification of the modal immune profile in the sampled population.

We study the problem of state representation learning for control from partial and potentially high-dimensional observations. We approach this problem via cost-driven state representation learning, in which we learn a dynamical model in a latent state space by predicting cumulative costs. In particular, we establish finite-sample guarantees on finding a near-optimal representation function and a near-optimal controller using the learned latent model for infinite-horizon time-invariant Linear Quadratic Gaussian (LQG) control. We study two approaches to cost-driven representation learning, which differ in whether the transition function of the latent state is learned explicitly or implicitly. The first approach has also been investigated in Part I of this work, for finite-horizon time-varying LQG control. The second approach closely resembles MuZero, a recent breakthrough in empirical reinforcement learning, in that it learns latent dynamics implicitly by predicting cumulative costs. A key technical contribution of this Part II is to prove persistency of excitation for a new stochastic process that arises from the analysis of quadratic regression in our approach, and may be of independent interest.

Measurement error is prevalent across all domains of scientific research where only imprecise observations, rather than the true underlying values, can be obtained. For example, estimates of human microbiome diversity are based on small samples from a much larger, generally unobserved system and reflect both sampling error and technical variation. In high-noise settings like these, it becomes difficult to make accurate predictions and to summarize uncertainty. Methods have previously been proposed to accommodate measurement error in classic predictive models, such as linear regression. However, relatively little work has been done to address measurement error in more complex and flexible models. Bayesian additive regression trees (BART), a Bayesian nonparametric model that sums the output of many decision trees, offers robust predictions with built-in uncertainty quantification. In this work, we propose measurement error BART (meBART), a novel extension to the BART model that directly incorporates measurement error in the independent variable(s). Through simulation studies, we show that in the presence of measurement error, our model enables more accurate parameter estimation, more robust uncertainty quantification, and superior predictive performance. We illustrate the utility of our proposed approach through two biomedical applications where the predictors of interest are subject to measurement error.

Multivariate time series (MTS) modeling often implicitly imposes an artificial ordering over variables, violating the inherent exchangeability found in many real-world systems where no canonical variable axis exists. We formalize this limitation as a violation of the permutation symmetry principle and require state-space dynamics to be permutation-equivariant along the variable axis. In this work, we theoretically characterize the complete canonical form of linear variable coupling under this symmetry constraint. We prove that any permutation-equivariant linear 2D state-space system naturally decomposes into local self-dynamics and a global pooled interaction, rendering ordered recurrence not only unnecessary but structurally suboptimal. Motivated by this theoretical foundation, we introduce the Variable-Invariant Two-Dimensional State Space Model (VI 2D SSM), which realizes the canonical equivariant form via permutation-invariant aggregation. This formulation eliminates sequential dependency chains along the variable axis, reducing the dependency depth from $\mathcal{O}(C)$ to $\mathcal{O}(1)$ and simplifying stability analysis to two scalar modes. Furthermore, we propose VI 2D Mamba, a unified architecture integrating multi-scale temporal dynamics and spectral representations. Extensive experiments on forecasting, classification, and anomaly detection benchmarks demonstrate that our model achieves state-of-the-art performance with superior structural scalability, validating the theoretical necessity of symmetry-preserving 2D modeling.

Censoring high-motion volumes in fMRI is common practice to reduce effects of head motion on functional connectivity (FC). Although aggressive censoring removes more noise, it causes extensive data loss, creating a tradeoff that may ultimately improve or degrade FC accuracy. Here, we evaluate how censoring affects FC estimation and downstream brain-wide association studies (BWAS). Using extensively sampled participants from the Human Connectome Project (HCP) Retest dataset, we establish individual "ground truth" FC and assess the accuracy of FC estimated from 5-30 minute scans. We find that censoring degrades FC accuracy, with more aggressive censoring being more detrimental, particularly among participants exhibiting above-average motion. In these participants, aggressive censoring reduces FC accuracy by 30% for 30-minute scans denoised with ICA-FIX, an advanced denoising method, and by 3% for scans denoised with conventional confound regression. These effects reflect substantial data loss (34%) that outweighs comparatively modest noise reductions: 7% with ICA-FIX and 18% with confound regression. Compensating for this would require substantially longer scans (62% with confound regression; 76% with ICA-FIX), inflating data collection budgets. Introducing a repeated measures framework to separate motion trait from artifact, we find that standard QC metrics are dominated by motion trait and overstate motion bias, which is effectively mitigated with less aggressive censoring. Finally, using data from nearly 1,000 HCP participants, we demonstrate that unreliable FC substantially attenuates BWAS correlations: by ~30% under optimal conditions (longer ICA-FIX scans with no censoring) but exceeding 75% in short, aggressively censored scans. Our findings support the use of advanced denoising methods, limiting censoring, and collecting longer scans to maximize fidelity of FC and BWAS.

The success of deep generative models in scientific discovery requires not only the ability to generate novel candidates but also reliable guarantees that these candidates indeed satisfy desired properties. Recent conformal-prediction methods offer a path to such guarantees, but its application to generative modeling in drug discovery is limited by budget constraints, lack of oracle access, and distribution shift. To this end, we introduce ConfHit, a distribution-free framework that provides validity guarantees under these conditions. ConfHit formalizes two central questions: (i) Certification: whether a generated batch can be guaranteed to contain at least one hit with a user-specified confidence level, and (ii) Design: whether the generation can be refined to a compact set without weakening this guarantee. ConfHit leverages weighted exchangeability between historical and generated samples to eliminate the need for an experimental oracle, constructs multiple-sample density-ratio weighted conformal p-value to quantify statistical confidence in hits, and proposes a nested testing procedure to certify and refine candidate sets of multiple generated samples while maintaining statistical guarantees. Across representative generative molecule design tasks and a broad range of methods, ConfHit consistently delivers valid coverage guarantees at multiple confidence levels while maintaining compact certified sets, establishing a principled and reliable framework for generative modeling.

We propose DistGP: a multi-robot learning method for collaborative learning of a global function using only local experience and computation. We utilise a sparse Gaussian process (GP) model with a factorisation that mirrors the multi-robot structure of the task, and admits distributed training via Gaussian belief propagation (GBP). Our loopy model outperforms Tree-Structured GPs \cite{bui2014tree} and can be trained online and in settings with dynamic connectivity. We show that such distributed, asynchronous training can reach the same performance as a centralised, batch-trained model, albeit with slower convergence. Last, we compare to DiNNO \cite{yu2022dinno}, a distributed neural network (NN) optimiser, and find DistGP achieves superior accuracy, is more robust to sparse communication and is better able to learn continually.

Indonesia's nickel ore export ban has driven rapid expansion of smelting and hydrometallurgical processing capacity at the Indonesia Morowali Industrial Park (IMIP), now the world's largest integrated nickel processing complex, on the coast of Central Sulawesi. Whether this industrialization has degraded the adjacent marine environment remains unquantified. We apply Bayesian structural time-series (BSTS) causal inference to a multi-decadal, multi-sensor satellite ocean color record of the diffuse attenuation coefficient at 490 nm, $K_d(490)$, to test for a causal link between IMIP expansion and nearshore turbidity change. A consensus structural breakpoint, a significant posterior causal effect estimated against a Banda Sea counterfactual, and a distribution-free placebo rank test collectively establish that coastal water clarity deteriorated after the transition from initial nickel pig iron production to hyper-expansion of high-pressure acid leaching facilities for battery-grade nickel. Satellite-derived land cover analysis independently corroborates this timing, showing substantial built-area growth and concurrent tree cover loss within the IMIP footprint. The resulting euphotic zone shoaling occurs in oligotrophic waters supporting high marine biodiversity, where even moderate optical degradation may impair coral photosynthesis and compress depth-dependent reef habitat. These findings quantify a marine environmental cost absent from Indonesia's mineral downstreaming policy discourse and demonstrate a transferable, satellite-based quasi-experimental framework for causal impact assessment at coastal industrial sites in data-limited tropical settings.

We introduce a repulsive mixture model to cluster observation units represented by multivariate functional data, based on similarity of curve shapes and individual-specific covariates. We propose a repulsive prior distribution for the component-specific location parameters that depends on a B-spline curve-tailored distance, extending existent repulsive priors to the context of multivariate functional data. The proposed model favors the identification of well-differentiated clusters, avoiding the presence of redundant ones. To sample from the posterior distribution, we propose an MCMC algorithm that includes a novel split-merge step that significantly improves the chain mixing. Different features of the proposed model, including the effects of repulsion and covariates in the clustering, are evaluated through simulation. The proposed model is fitted to analyze Chronic Ankle Instability (CAI) data, focusing on identifing individuals with similar types of physical dysfunctions based on the similarity of movement patterns.

Robustness under latent distribution shift remains challenging in partially observable reinforcement learning. We formalize a focused setting where an adversary selects a hidden initial latent distribution before the episode, termed an adversarial latent-initial-state POMDP. Theoretically, we prove a latent minimax principle, characterize worst-case defender distributions, and derive approximate best-response inequalities with finite-sample concentration bounds that make the optimization and sampling terms explicit. Empirically, using a Battleship benchmark, we demonstrate that targeted exposure to shifted latent distributions reduces average robustness gaps between Spread and Uniform distributions from 10.3 to 3.1 shots at equal budget. Furthermore, iterative best-response training exhibits budget-sensitive behavior that is qualitatively consistent with the theorem-guided diagnostics once one accounts for discounted PPO surrogates and finite-sample noise. Ultimately, we show that for latent-initial-state problems, the framework yields a clean evaluation game and useful theorem-motivated diagnostics while also making clear where implementation-level surrogates and optimization limits enter.

Many statistical problems can be addressed by applying a multiple testing procedure (MTP) that controls either the Family-wise Error Rate (FWER) or False Discovery Rate (FDR) under unknown arbitrarily-interdependent $p$-values, without explicitly modeling these inter-correlations. They include the FWER-controlling Bonferroni (1936) MTP and Holm (1979) MTP; the FDR-controlling Benjamini and Yekutieli (2001) MTP; and the DP-MTP (Karabatsos, 2025), based on a Dirichlet process (DP) prior distribution supporting the entire space of MTPs that control either the FWER or FDR. For such an MTP, this study introduces a new and congenial method for Bayesian predictive power analysis, for power calculation and sample size determination for any given planned future (e.g., replication or interim) study. This novel MTP predictive power analysis method is based on a joint prior distribution defining a scale matrix mixture of asymmetric multivariate normal mean-variance mixture distributions, factorized as a general prior distribution for effect sizes (e.g., obtained from expert judgment or results of prior studies), and a uniform prior distribution for correlation matrices representing arbitrary dependencies between $p$-values of test statistics of given multiple hypothesis tests under their alternative hypotheses. The new MTP power analysis method also results in $p$-value weights which can be used to minimize the relative impacts of and assess for significance-chasing biases (e.g., publication bias, $p$-hacking, etc.) in multiple testing, without needing to assume that $p$-values (effect sizes) are independent. The new simulation-based MTP predictive power analysis method is illustrated through the analysis of $p$-values obtained by a famous study of lead exposure and re-analyzed by the previous MTP literature, using R package bnpMTP.

We prove a general result that if a Metropolis–Hastings algorithm has a proposal that is not geometrically ergodic and the acceptance rate approaches unity at a suitable rate as the state variable becomes large, then the Metropolised chain will also not be geometrically ergodic. Our conditions seem stronger than might be expected, but are shown to be necessary through a counterexample. We then turn our attention to the random walk and guided walk Metropolis algorithms. We show that if the target distribution has polynomial tails the latter converges at twice the polynomial rate of the former, but that if instead the target distribution has strictly convex potential then the random walk Metropolis behaves as a $1/2$-lazy version of the guided walk Metropolis when the state variable is large, and therefore moves at a similar (ballistic) speed.

Contingency tables are the canonical representation of multivariate categorical data. As the size of the contingency table grows exponentially with the number of variables, even a moderate number of variables, each with a moderate number of levels, results in a huge number of cells, the majority of which remains empty even with a significant amount of data. We propose efficient methods for inferring higher-order loglinear models by performing subsampling on the set of the empty cells. First, we derive the likelihood under a zero-deflated Poisson sampling scheme. This is maximized via an efficient iteratively re-weighted least squares algorithm, leading to consistent and close to efficient estimators. This method works well for moderately sized contingency tables, but runs into computational instability when the number of dimensions grows. By sacrificing some efficiency, we show that nested case-control multinomial sampling combined with a degenerate logistic regression approach is also consistent and can be applied to arbitrarily large contingency tables. We illustrate the method with an analysis of data from the General Social Survey, which consists of $15014$ observations in a $69$-dimensional contingency table with a total of $6.6\times 10^{38}$ cells.

Flow maps enable high-quality image generation in a single forward pass. However, unlike iterative diffusion models, their lack of an explicit sampling trajectory impedes incorporating external constraints for conditional generation and solving inverse problems. We put forth Variational Flow Maps, a framework for conditional sampling that shifts the perspective of conditioning from "guiding a sampling path", to that of "learning the proper initial noise". Specifically, given an observation, we seek to learn a noise adapter model that outputs a noise distribution, so that after mapping to the data space via flow map, the samples respect the observation and data prior. To this end, we develop a principled variational objective that jointly trains the noise adapter and the flow map, improving noise-data alignment, such that sampling from complex data posterior is achieved with a simple adapter. Experiments on various inverse problems show that VFMs produce well-calibrated conditional samples in a single (or few) steps. For ImageNet, VFM attains competitive fidelity while accelerating the sampling by orders of magnitude compared to alternative iterative diffusion/flow models. Code is available at https://github.com/abbasmammadov/VFM

Restricting statistical experiments via nuisance-ancillary $\sigma$-fields yields nuisance-free experiments. However, a moot point with ancillarity is that maximal ancillary $\sigma$-fields are typically not unique. There are exceptions, though, among which the limiting experiments in a locally asymptotically normal (LAN) context. Building on this, we address the maximal ancillarity uniqueness problem by adopting a Hájek-Le Cam asymptotic perspective and define the concept of sequences of locally asymptotically maximal nuisance-ancillary $\sigma$-fields. We then show that any semiparametrically efficient procedure admits versions that are measurable with respect to such $\sigma$-fields while enjoying strict finite-sample nuisance-ancillarity, hence eliminating the nuisance without the hassle of estimating it. This is in sharp contrast with classical tangent space projections, which also achieve semiparametric efficiency but only enjoy asymptotic nuisance-ancillarity – at the price, moreover, of adequately estimating the nuisance. When the nuisance is the density of some noise or innovation driving the data-generating process of a LAN experiment, we show that a sequence of locally asymptotically maximal nuisance-ancillary $\sigma$-fields is generated by the so-called center-outward residual ranks and signs based on measure transportation results. Restricting local experiments to such $\sigma$-fields yields sequences of finite-sample nuisance-free (here, distribution-free) restrictions of the original local LAN experiments that nevertheless achieve the semiparametric efficiency bounds of the original ones.

The D criterion measures the volume of the joint confidence ellipsoid for the linear model coefficients and ignores its shape, so designs with the same D value can estimate individual effects with different variances. Meanwhile, A-optimality minimizes average coefficient variance. With both criteria expressed as information values, A equals D multiplied by a sphericity index for the same ellipsoid. In a fixed coefficient basis, sphericity further factors into coefficient-variance balance and a determinant-based correlation component. In five published screening comparisons, the A-optimal design has a larger correlation component despite slightly poorer variance balance; in three it also has a smaller D value. In a seven-run family of designs that all tie under D, the two with equal coefficient variances have the lowest A value. Both sphericity components can be calculated directly from standard errors and estimate correlations available in statistical software. After whitening by a prediction moment matrix, the same determinant-sphericity factorization applies to the I-criterion.

This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n\ge2$). The objective is to recover the unknown linear and nonlinear coefficients from the associated Neumann-to-Dirichlet (NtD) map. Using a higher-order linearization approach, we establish the unique determination of both coefficients from boundary measurements. For spatial dimensions $n\ge3$, uniqueness holds under $C^\gamma(\overline{\Omega})$ regularity assumptions with $0<\gamma<1$, while in the two-dimensional case uniqueness is obtained under Sobolev regularity $W^{1,p}(\Omega)$ with $p>2$. The analysis relies on the well-posedness of the forward problem together with techniques from linear inverse problems, including Runge-type approximation arguments and Fourier analysis. In addition, we develop a numerical reconstruction framework for recovering the coefficients from boundary data. The forward problem is discretized using a finite difference scheme combined with a quasi-Newton iteration, and the inverse problem is formulated within a Bayesian inference framework. Posterior distributions of the coefficients are explored using the preconditioned Crank-Nicolson (pCN) Markov chain Monte Carlo algorithm, which provides both point estimates and uncertainty quantification. Numerical experiments demonstrate the effectiveness of the proposed reconstruction method and illustrate the theoretical uniqueness results.

Intergenerational mobility quantifies the transmission of socio-economic outcomes from parents to children. While rank-rank regression (RRR) is standard, adding covariates directly (RRRX) often yields parameters with unclear interpretation. Conditional rank-rank regression (CRRR) resolves this by using covariate-adjusted (conditional) ranks to measure within-group mobility. We improve and extend CRRR by estimating conditional ranks with a deep conditional transformation model (DCTM) and cross-fitting, enabling end-to-end conditional distribution learning with structural constraints and strong performance under nonlinearity, high-order interactions, and discrete ordered outcomes where the distributional regression used in traditional CRRR may be cumbersome or prone to misconfiguration. We further extend CRRR to discrete outcomes via an $\omega$-indexed conditional-rank definition and study sensitivity to $\omega$. For continuous outcomes, we establish an asymptotic theory for the proposed estimators and verify the validity of exchangeable bootstrap inference. Simulations across simple/complex continuous and discrete ordered designs show clear accuracy gains in challenging settings. Finally, we apply our method to two empirical studies, revealing substantial within-group persistence in U.S. income and pronounced gender differences in educational mobility in India.

Estimating changes in extremes quantiles from environmental processes non-stationary in time, from small samples is challenging since it is difficult to characterise tail non-stationarity adequately. Using annual maxima and minima of near-surface temperature (tas) from CMIP6 output for some Earth desert regions, we use generalised extreme value (GEV) regression to model changes in extreme quantiles in time. We consider candidate models with different parametric forms for the variation of GEV parameters with time, estimating parameters using Bayesian inference. We select optimal candidate models using model selection criteria, including the Akaike, Bayesian, divergence and widely-applicable information criteria. In an extreme value setting, the performance of different criteria is unreliable. We therefore undertake a simulation study using ground truth models generating data similar to our extrema, to assess performance of the criteria to minimise error in prediction of change DeltaQ in the 100-year return value of tas over (2015,2125). The Bayesian information criterion (BIC) provides best performance, out-performing the divergence and widely-applicable Information criteria. We compare BIC model selection with a stacked Bayesian model average. Using BIC-selected GEV regression, we estimate joint posterior distributions of DeltaQ, coupled over three climate scenarios, for different combinations of desert region, global climate model and climate ensemble. We find significant increases in DeltaQ for regional annual maxima under stronger forcing scenarios for all desert regions. Similar but weaker trends are observed for regional annual minima. There is evidence that extreme minima are warming more rapidly than extreme maxima in Antarctica, whereas most hot desert regions exhibit the opposite effect. An ancillary file of supplementary material is provided.

Optimizing GPU kernels manually is a challenging and time-consuming task. With the rapid development of LLMs, automated GPU kernel optimization is gradually becoming a tangible reality. However, current LLM-driven automated optimization methods narrowly focus on machine learning applications, such as PyTorch operator optimization, while overlooking broader domains like sparse matrix operations in scientific computing. Extending to these broader applications brings new challenges for the benchmark and algorithm. Therefore, developing a general-purpose automated kernel optimization method becomes our primary focus. In this paper, we address the absence of systematic evaluation for multi-scenario settings by introducing MSKernelBench, which spans multiple scenarios, including fundamental algebraic operations, common LLM kernels, sparse matrix operators, and scientific computing routines, each supporting both FP32 and BF16 precision. Building on this benchmark, we introduce CUDAMaster, a multi-agent, hardware-aware system for kernel optimization that leverages profiling information and automatically constructs the full compilation and execution toolchain. Experimental results demonstrate that CUDAMaster achieves significant speedups across most operators, outperforming Astra by about 35%. In several cases, its performance matches or surpasses that of highly optimized, closed-source libraries such as cuBLAS. A demo showcasing the original and optimized code for each operator is available at https://hanyx2021.github.io/MSKernelBenchDemo/.

Let $\mathbf{x}$ be a random vector with $n$ i.i.d.\ real-valued components in the domain attraction of an $\alpha$-stable law with $\alpha\in(0,2)$, and let $\mathbf{y}=\mathbf{x}/\|\mathbf{x}\|_2$ be the associated self-normalized vector on the unit sphere. For a (possibly random) Hermitian matrix $\mathbf{A}_n=\big(a_{ij}^{(n)}\big)$ independent of $\mathbf{y}$, we study the asymptotic law of the quadratic form $\mathbf{y}^\top \mathbf{A}_n \mathbf{y}$. Building on the sharp separation between diagonal and off-diagonal contributions in this heavy-tailed setting, we show that under a mild assumption on the Frobenius norm of the off-diagonal part of $\mathbf{A}_n$ the limiting law is solely governed by the empirical distribution of the diagonal entries and the index $\alpha$. More precisely, if $n^{-1}\sum_{i=1}^n \delta_{a^{(n)}_{ii}}$ converges weakly almost surely to a deterministic $\nu$, then $Q_n$ converges in distribution to a non-degenerate law $\mu_{\nu,\alpha}$ characterized through its Stieltjes transform. The law $\mu_{\nu,\alpha}$ is shown to be atom-free (provided that $\nu$ is non-degenerate) with an explicit density and tractable tail behavior. As an application in random matrix theory, we derive an implicit resolvent-based representation of the $\alpha$-heavy Marčenko–Pastur law $H_{\alpha,\gamma}$ for heavy-tailed sample correlation matrices and prove that $H_{\alpha,\gamma}$ has no atoms except possibly at the origin. For comparison with the light-tailed setting, we also provide a Hanson–Wright-type concentration inequality for $\mathbf{y}^\top \mathbf{A}_n \mathbf{y}$ when the components of $\mathbf{x}$ are sub-Gaussian.

In the training of neural networks, adaptive moment estimation (Adam) typically converges fast but exhibits suboptimal generalization performance. A widely accepted explanation for its defect in generalization is that it often tends to converge to sharp minima. To enhance its ability to find flat minima, we propose its new variant named inverse Adam (InvAdam). The key improvement of InvAdam lies in its parameter update mechanism, which is opposite to that of Adam. Specifically, it computes element-wise multiplication of the first-order and second-order moments, while Adam computes the element-wise division of these two moments. This modification aims to increase the step size of the parameter update when the elements in the second-order moments are large and vice versa, which helps the parameter escape sharp minima and stay at flat ones. However, InvAdam's update mechanism may face challenges in convergence. To address this challenge, we propose dual Adam (DualAdam), which integrates the update mechanisms of both Adam and InvAdam, ensuring convergence while enhancing generalization performance. Additionally, we introduce the diffusion theory to mathematically demonstrate InvAdam's ability to escape sharp minima. Extensive experiments are conducted on image classification tasks and large language model (LLM) fine-tuning. The results validate that DualAdam outperforms Adam and its state-of-the-art variants in terms of generalization performance. The code is publicly available at https://github.com/LongJin-lab/DualAdam.

Understanding how multi-scale network structure influences circadian rhythms in the suprachiasmatic nucleus (SCN) is essential for uncovering the principles of rhythmic robustness and synchronization. Previous studies using synthetic SCN networks suggested a size-dependent phenomenon, in which rhythmic activity initially strengthens with network size and then saturates, but it remains unclear whether this occurs in real SCN networks. Here, we apply geometric branch growth (GBG) and geometric renormalization (GR) to generate self-similar scaled-up and scaled-down replicas from a single-scale functional mouse SCN network. Unlike synthetic models, these SCN replicas do not exhibit size-dependent rhythms: average period, amplitude, and synchronization remain stable across scales. By increasing the average degree with network size, we reproduce size-dependent rhythms and show that they arise from network connectivity, whereas low-degree networks fragment and fail to sustain oscillations. Disrupting clustering self-similarity slightly reduces synchronization, but circadian rhythms remain robust, indicating that average degree, rather than clustering, is the dominant structural driver. These results highlight the resilience of SCN rhythms to network scaling and provide a framework for linking multi-scale network structure to biological timekeeping.

Accurate and reliable forecasting of epidemic incidences is critical for public health preparedness, yet it remains a challenging task due to complex nonlinear temporal dependencies and heterogeneous spatial interactions. Often, point forecasts generated by spatiotemporal models are unreliable in assigning uncertainty to future epidemic events. Probabilistic forecasting of epidemics is therefore crucial for providing the best or worst-case scenarios rather than a simple, often inaccurate, point estimate. We present deep spatiotemporal engression methods to generate accurate and reliable probabilistic forecasts on low-frequency epidemic datasets. The proposed methods act as distributional lenses, and out-of-sample probabilistic forecasts are generated by sampling from the trained models. Our frameworks encapsulate lightweight deep generative architectures, wherein uncertainty is quantified endogenously, driven by a pre-additive noise component during model construction. We establish geometric ergodicity and asymptotic stationarity of the spatiotemporal engression processes under mild assumptions on the network weights and pre-additive noise process. Comprehensive evaluations across six epidemiological datasets over three forecast horizons demonstrate that the proposal consistently outperforms several temporal and spatiotemporal benchmarks in both point and probabilistic forecasting. Additionally, we explore the explainability of the proposal to enhance the models' practical application for informed, timely public health interventions.

We present a unified parametric framework for modal regression applicable to continuous positive distributions, with explicit support for right-censored observations. The key contribution is a systematic analytical reparameterization of density parameters as direct functions of the conditional mode. This closed-form mapping is derived for the Gamma, Beta, Weibull, Lognormal, and Inverse Gaussian distributions, directly linking the mode to a linear predictor. Maximum likelihood estimation is performed using the censored log-likelihood, with asymptotic inference based on the observed Fisher information matrix. A Monte Carlo simulation study across multiple distributions, sample sizes, and censoring levels confirms consistent parameter recovery. Empirical bias and RMSE decrease as expected, and Wald confidence intervals achieve nominal coverage. Finally, the proposed methodology is illustrated through an application to real-world reliability data. All methodology is implemented in the open-source R package ModalCens.

In modern randomized experiments, large-scale data collection increasingly yields rich baseline covariates and auxiliary information from multiple sources. Such information offers opportunities for more precise treatment effect estimation, but it also raises the challenge of integrating heterogeneous information coherently without compromising validity. Covariate-adaptive randomization (CAR) is widely used to improve covariate balance at the design stage, but it typically balances only a small set of covariates used to form strata, making covariate adjustment at the analysis stage essential for more efficient estimation of treatment effects. Beyond standard covariate adjustment, it is often desirable to incorporate auxiliary information, including cross-stratum information, predictions from various machine learning models, and external data from historical trials or real-world sources. While this auxiliary information is widely available, existing covariate adjustment methods under CAR primarily exploit within-stratum covariates and do not provide a coherent mechanism for integrating it. We propose a unified calibration framework that integrates such information through an information proxy vector and calibration weights defined by a convex optimization problem. The resulting estimator recovers many recent covariate adjustment procedures as special cases while providing a systematic mechanism for both internal and external information borrowing within a single framework. We establish large-sample validity and a no-harm efficiency guarantee, showing that incorporating additional information sources cannot increase asymptotic variance, and we extend the theory to settings in which both the number of strata and the number of information sources grow with the sample size.