Bayesian optimal experimental design (BOED) selects experiments to maximize information gain about model parameters. However, in decision-critical settings, reducing parameter uncertainty does not necessarily improve downstream decisions, as only specific parameter directions relevant to the objective truly matter. We propose GoBOED, a goal-driven BOED framework that directly optimizes experimental designs for a specified decision-making objective. GoBOED combines an amortized variational posterior surrogate with a differentiable convex decision layer, enabling gradient-based design optimization that is fully decision-focused. We theoretically show that GoBOED gradients are insensitive to parameter directions irrelevant to the decision objective, providing a formal justification for why goal-driven design achieves equivalent decision quality over a wider set of experimental designs than information-gain maximization. Empirically, across source localization, epidemic management, and pharmacokinetic control, GoBOED identifies designs that better align with downstream decision objectives and reveals that near-optimal design windows are substantially wider than those predicted by goal-agnostic BOED approaches.

Frontier LLMs now perform strongly across a wide range of physics evaluations, but it is hard to disentangle genuine reasoning from recall of established science. We introduce DiscoverPhysics, an interactive benchmark that asks a LLM agent to discover the laws of motion of a simulated world whose physics deliberately deviates from our own. We construct 22 worlds governed by, among others, screened and fractional-power gravity, multi-species couplings, hidden dark-matter-like particles, non-coordinate-free physics, and time-varying interactions. Each world is generated on demand by an N-body simulator, for which the agent proposes several rounds of experiments, observes raw trajectory data, and ultimately submits both a natural-language explanation of the world's physics and a Python implementation of the inferred law. Because solving a world requires the agent to design informative experiments and revise its hypotheses, the benchmark probes long-horizon reasoning over an experimental history. We evaluate submissions along two complementary axes: trajectory MSE on held-out particles and an LLM-judged explanation score following an expert-written rubric assessing conceptual understanding of each world. Across eleven frontier models, we find that the strongest agents pass only half of the worlds and consistently fail on those where latent structure must be uncovered. Open-source models lag substantially behind commercial models, both in their ability to design informative experiments and in extracting conclusions from the data. We further find that good predictive accuracy does not guarantee high explanation quality and that conceptual understanding depends on hypothesis refinement through well-chosen experiments.

This paper proposes the quantile unit-log-symmetric autoregressive moving average (QULS--ARMA) model for bounded time series on the open unit interval $(0,1)$. The model extends the unit-log-symmetric family by introducing a quantile-based reparameterization and embedding autoregressive and moving-average dynamics directly in the conditional quantile, thereby overcoming limitations of mean-based approaches and providing a coherent framework for proportion data arising from ratios of dependent positive variables. The proposed specification accommodates asymmetric behavior and heavy tails through flexible log-symmetric kernels, including the normal and Student-$t$ distributions. Parameter estimation is carried out via conditional maximum likelihood, and asymptotic properties are established. Monte Carlo simulations and an empirical application to hydroelectric energy storage proportions in Brazil assess the finite-sample performance and practical advantages of the QULS--ARMA model. The results show the good performance of the proposed estimators across a range of scenarios and kernel specifications.

Molecular signatures derived from omics data are increasingly used in epidemiological studies to characterize lifestyle exposures, either as proxies of exposure or to provide insight into disease mechanisms. These signatures are typically constructed by regressing the exposure on high-dimensional omics features. In the literature, an initial univariate screening step has sometimes been applied prior to multivariate modelling, but the causal implications of this choice have not yet been considered. Focusing on settings where the exposure causally influences molecular features (and not the reverse), we use directed acyclic graphs (DAGs) and $d$-separation arguments to show that collider bias may arise when the screening step is ignored, leading to the inclusion of non-causal features in the signature. We further demonstrate that the screening step can mitigate this bias. Our simulation studies illustrate that screening reduces the inclusion of non-causal features, albeit at the cost of lower sensitivity and reduced correlation between the exposure and the resulting signature. Overall, we recommend applying univariate screening prior to signature construction, particularly when the inclusion of non-causal features is undesirable, such as in mechanistic studies.

Stochastic gradient descent (SGD) is a foundational algorithm for large-scale statistical learning and stochastic optimization. However, statistical inference based on SGD iterates remains challenging when stochastic gradients have infinite variance, as the relevant limiting distributions depend on unknown nuisance parameters. In this paper, we develop an efficient, model-agnostic methodology for constructing confidence regions from SGD trajectories that applies in both finite- and infinite-variance regimes. The procedure is based on a joint weak convergence result for the Polyak-Ruppert averaged estimator and an empirical second-moment normalizer constructed from stochastic gradients along the SGD trajectory. This joint limit yields a self-normalized statistic in which the leading tail-dependent scaling terms cancel. We then use a subsampling calibration scheme to estimate the relevant critical values, avoiding explicit estimation of tail indices, slowly varying functions, or stable-law parameters. The resulting confidence regions are straightforward to implement and are asymptotically valid under both the finite- and infinite-second-moment regimes. Simulation studies show reliable coverage in various settings, supporting the proposed method as a practical tool for uncertainty quantification in stochastic optimization.

Benchmarks increasingly guide deployment, procurement and scientific screening, yet a score supports only the response it records, not necessarily the deployment action. We introduce deployment-complete benchmarking, which tests whether benchmark evidence determines a deployment action. A benchmark is complete for a claim exactly when the action is constant on each evidence fiber; mixed fibers expose missing deployment information, and completion curves quantify the evidence required to resolve ambiguity. In controlled response spaces, benchmark-channel conformal coverage of 94.98% transferred poorly to an unmeasured deployment channel (10.07%), whereas response-rank intervals achieved 94.91% coverage; even zero benchmark error certified only 45.4% of candidates at the largest residual size. Public audits revealed incompleteness, including 97.9% mixed Tox21 fibers and zero median certifiable fraction in main Matbench and JARVIS audits. In held-out replays, certify-then-acquire reduced false decisions from 1.19% to 0.027% in Tox21 and from 20.3% to 0.128% in JARVIS, while changing model choice and identifying deployment-relevant probes. Deployment-ready benchmarks should report evidence, supported actions, ambiguity and completion cost rather than scores alone.

Survival analysis aims to model how covariates and time jointly shape the time-to-event distribution under right censoring. Classical methods such as the Cox model and generalised additive models (GAMs) require interactions and time-varying effects to be manually specified, which is increasingly impractical on rich clinical datasets. We introduce KAPLAN-HR, a B-spline Kolmogorov-Arnold Network (KAN) for nonparametric estimation of the conditional hazard as a joint function of covariates and time. A single-layer KAPLAN-HR model recovers a GAM, while deeper architectures capture interactions and time-varying effects through composition. We establish a convergence rate for the nonparametric KAN hazard estimator that depends only on the smoothness of the underlying KAN representation and not on the covariate dimension, thereby mitigating the curse of dimensionality for KAN-representable targets. In evaluations over six clinical benchmark datasets, KAPLAN-HR matches or exceeds the predictive performance of established statistical and deep learning survival methods.

We test whether the optimal learning-rate schedule depends on bit-width during from-initialisation quantisation-aware training (QAT) for sub-100M decoder language models. A 720-run factorial grid (Phase 2) over bit-width x warmdown fraction x LR magnitude x model size x seed (FP16/INT8/INT6, 15M-100M, 5 seeds) finds the optimal warmdown is 33% at every (bit-width, size) cell. The primary hypothesis -- that INT6 QAT requires a different schedule than higher-precision training -- is falsified at FP16/INT8/INT6. A 625-run follow-up (Phase 5) probes the null along five axes: optimiser (AdamW), schedule shape (cosine), training length (up to 9x more iterations), an extended size sweep (5M-350M), and an INT4 sweep from 3M to 100M. The null is robust under all three setup changes. The INT6 penalty follows a log-linear scaling law whose fit on Phase 2 predicts the five held-out Phase 5 sizes (5M, 8M, 175M, 250M, 350M) within their 95% prediction intervals (5/5). For INT4 the picture is sharper than the higher precisions: at 50M and 100M, wd33 is decisively optimal (paired z ~ 12-15, 10/10 seeds); below 50M, across the six tested sizes from 3M to 30M, no individual size shows a statistically significant schedule preference and the per-size mean penalty oscillates within seed-level noise. The boundary is therefore a transition between a noise-dominated regime below 50M and a decisive wd33 regime at and above 50M, not a clean wd10 region. A weight-to-grid-distance probe falsifies the simplest mechanism for the FP16/INT8/INT6 null result (rapid grid-snapping): pre-warmdown, INT6-QAT weights sit at essentially the same distance from the INT6 grid as FP16 weights (ratio ~ 1.04). Practical recommendation: at sub-100M scale, tune the LR schedule once at FP16 and apply unchanged to INT8/INT6 QAT; for INT4 at 50M+ use wd33; for INT4 below 50M the schedule choice is in the noise.

The empirical distribution function assigns mass $1/n$ to each of the $n$ observations in a sample. As these are highly variable, estimation error may be reduced by replacing them with estimated observations that are asymptotically less variable. Motivated by this idea, we introduce a nonparametric estimator obtained by assigning mass $1/m$ to $m$ estimated expected order statistics, with $m$ chosen arbitrarily. The estimator enjoys several finite-sample properties and yields a rich asymptotic theory. Its estimation error relative to its population counterpart is controlled by the $L^1$ error of the empirical distribution. Moreover, every $L$-functional of the new estimator corresponds to an $L$-functional of the empirical distribution with updated weights. We establish almost sure convergence in $L^p$ norm and Wasserstein distance as $n \to \infty$, and derive weak convergence of the associated empirical quantile process in $L^p(0,1)$, for $p\in[1,\infty)$ and $m$ fixed, and for $p=1,2$ as $n,m \to \infty$. These results yield asymptotic distributions for distance-based functionals, including $L^p$ and Wasserstein metrics. Bootstrap validity is also established. Simulations show that the estimator often improves on the empirical distribution and remains competitive with kernel methods, with more stable performance across different distributional settings.

Exponential random graph models (ERGMs) are a widely used framework for network data, enabling hypothesis testing on the structural mechanisms underlying observed networks. Bayesian ERGMs provide principled uncertainty quantification and enable the incorporation of prior knowledge through fully probabilistic modelling. However, computation remains challenging because the posterior is doubly intractable, with a likelihood normalising constant that depends on unknown parameters. This paper reviews Bayesian approaches to ERGM inference, categorising inference methods into three broad classes: auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, alongside dedicated treatment of model selection. We also discuss modelling extensions for missing data, longitudinal dynamics, populations of networks, weighted networks, highlighting applications across various scientific disciplines.

The aim of this Lecture Note is to introduce the Signal Processing (SP) community to a powerful yet still under-utilised tool: the semiparametric statistics. In short, the semiparametric framework allows us to estimate or perform hypothesis testing on a finite-dimensional parameter in the presence of an infinite-dimensional nuisance parameter (i.e. a function), such as the density of the noise. Clearly, this framework is general enough to include almost every SP application. Remarkably, as the title suggests drawing on George R. R. Martin's famous book series, the greatest advantage of semiparametric statistics over parametric and non-parametric ones lies in the fact that it is able to reconcile two seemingly dichotomous concepts: statistical efficiency and robustness. Here, robustness is understood in the sense of distribution-freeness, that is the estimation performance must be robust with respect to the lack of knowledge of the functional form of the generating data distribution. To explain exactly what this means, in this Lecture Note we will focus our attention on the famous and fundamental symmetric location problem. The symmetric location problem is a fundamental problem that can be found (in various forms) in countless areas of SP: source localization, time synchronization, array signal processing, and distributed sensor networks, just to name a few. Furthermore, it is important to note that the methodology we will develop for this specific problem can be extended to much more general semiparametric estimation problems, such as the estimation of the location vector and covariance matrix in elliptical data.

We study the mean-squared error of $k$-fold cross-validation as a risk estimator, with particular emphasis on how its accuracy depends on the number of folds $k$. Despite the widespread use of cross-validation, principled guidance for choosing $k$ is largely absent, mainly due to the complex dependence between fold-wise error estimates. To obtain sharp and interpretable results, we focus on the majority algorithm in binary classification, a minimal yet nontrivial empirical risk minimization procedure. We provide a fine-grained analysis of its cross-validation behavior, showing that even this simple algorithm exhibits subtle and delicate phenomena for which existing theory provides loose and even vacuous bounds. Leveraging this analysis, we introduce a minimax framework for cross-validation risk estimation and prove that no empirical risk minimization algorithm can achieve an $O(1/n)$ minimax mean-squared error when the number of folds grows with the number of samples $n$; instead, a lower bound of order $Ω(\sqrt{k}/n)$ is unavoidable. Our results reveal fundamental limitations of cross-validation as a data-reuse strategy, clarify gaps and inaccuracies in prior theoretical work, and position the majority algorithm as a natural benchmark that any tight analysis of cross-validation should be able to explain.

We study change-point detection for high-dimensional data in regimes where inference must be performed from small batches of observations. Our primary focus is the high-dimensional, low sample size (HDLSS) regime, where the sequence length is fixed while the ambient dimension diverges. We propose a dimension-averaged angular kernel scan framework for detecting marginal distributional shifts. The statistic aggregates bounded one-dimensional angular discrepancies across coordinates, yielding a fully nonparametric, hyperparameter-free, and moment-agnostic estimator that remains well-defined without specifying, estimating, or assuming finite marginal moments, for example under heavy-tailed or contaminated distributions. For the offline single-change problem, we derive an exact population mean factorization into a universal deterministic shape function and a scalar signal factor, characterize the null covariance structure up to a scalar long-run variance factor, and establish an HDLSS multivariate central limit theorem under cross-coordinate mixing. These results lead to plug-in Gaussian calibration, asymptotic type-I error control, and power and localization guarantees, including a $d^{-1/2}$ local detection scale. We further extend the offline procedure to a fixed-window sequential monitoring procedure for high-dimensional streaming data, and obtain ARL calibration and worst-case EDD bounds. Simulation studies demonstrate that the proposed method can accurately detect and localize changes in challenging HDLSS and streaming settings where moment-based or hyperparameter-sensitive procedures may be unreliable.

We study counterfactual distribution learning for high-dimensional outcomes whose counterfactual law may concentrate near lower-dimensional structure. Standard isotropic smoothing treats all ambient directions equally, leading to unfavorable scaling and unstable local inference. We propose two diffusion-guided estimators based on semiparametric debiasing: diffusion-informed smoothing for counterfactual densities and diffusion-informed score smoothing for counterfactual scores. The estimators combine causal nuisance adjustment with geometry-adaptive localization driven by diffusion score information, removing first-order nuisance bias while aligning smoothing with local outcome geometry. We establish asymptotic expansions, risk bounds, and inference procedures for smoothed density and score-based targets, with ambient density inference obtained under additional approximation conditions. Under structural geometry conditions, the leading stochastic error is governed by an effective dimension induced by the diffusion-guided kernel, rather than by the ambient dimension. Semi-synthetic experiments based on CelebA show steeper error decay for geometry-adaptive methods, supporting the proposed effective-dimension theory.

Sketch-and-solve (SAS) is a very successful method to efficiently estimate the solution of heavily overdetermined large linear least squares problems. It uses random sketching to reduce the size of the problem, hence reducing the computational cost. Several authors have shown that averaging several solutions from SAS further improves the accuracy, which is measured by the residual associated to the approximate solution. Going further, we combine solutions from sketch-and-solve in a multilevel manner, such that the approximate solution is a combination of SAS samples obtained from small sketches and more accurate correction terms obtained from larger sketches. We first consider the variance of the estimator, which depends on the variance of the coarse samples and the correction terms. We show that the variance of the correction terms on each level follows a trend and decreases faster than the variance of the simple SAS estimator. However, we then show that the overall computational cost of our multilevel framework is slightly higher than that of the simple average estimator, so a naive application of multilevel methods appears unattractive for least squares problems.

We study a stochastic multi-armed bandit problem where an agent is granted a free exploration budget before regret accumulates, a setting not captured by the classic regret minimization or pure exploration paradigms. The goal is to design an adaptive policy that strategically explores the bandit instance in the initial free exploration phase and minimizes the cumulative regret in the subsequent phase. We formalize this regret minimization with free exploration problem and identify an interesting regime where the free exploration budget scales logarithmically with the time horizon. To quantify the amount of regret saved with high probability as a result of the availability of the free exploration phase, we introduce a novel set of policies known as $(α,β)$-probably saving policies. We propose a two-phase, probably saving algorithm, UFE-KLUCB-H, which consists of a principled free exploration policy, UFE, and a history-aware regret minimization policy KLUCB-H. Instance-dependent upper bounds on UFE-KLUCB-H are derived, showing that UFE-KLUCB-H accumulates strictly less regret than policies that do not have access to a free exploration phase. Complementarily, we derive instance-dependent lower bounds based on novel multi-instance perturbation arguments tailored to the free-exploration setting, demonstrating the near-optimality of UFE-KLUCB-H for two-valued bandits. Our upper and lower bounds reveal sharp phase transitions in the accumulated regret depending on the amount of available free exploration. Simulations are conducted to demonstrate that forced exploration and adaptivity in the algorithm lead to greater regret savings.

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of bivariate tail dependence since it evaluates the underlying copula only along the diagonal. To address this limitation, several measures of strongest manifestation of tail dependence have been proposed in the bivariate case, including a measure based on the tail copula of the underlying bivariate copula. This paper introduces and investigates the multivariate maximal tail concordance measure (MTCM) which extends the bivariate measure to the multivariate case. The MTCM quantifies the largest tail mass over lower hyperrectangles of common unit volume, while the associated maximizer identifies the direction of maximal tail probability. We establish fundamental properties of the MTCM in the multivariate case, including existence of an optimal direction. We also derive analytical representations for several important model classes. Closed-form expressions are further obtained for survival Marshall-Olkin copulas, Archimax and nested Archimedean copulas with regularly varying Archimedean generators. An application to trivariate annual sea-level maxima in England shows that the MTCM can reveal off-diagonal stress directions and substantial differences in the underlying extremal dependence not detected by likelihood- or TDC-based comparisons.

We prove that no reinforcement learning policy with confidence-gated autonomy can simultaneously achieve maximum helpfulness, optimal calibration, and full autonomy under rational oversight, whenever some tasks exceed the agent's reliable competence: the Behavioral Credibility Trilemma. The impossibility is geometric -- adding any non-affine autonomy incentive to a strictly proper scoring rule destroys strict properness, so an agent rewarded for both calibrated confidence and autonomous action systematically inflates its reported confidence on tasks below the principal's approval threshold. The Behavioral Perturbation Lemma quantifies the inflation (scaling as $w_A/(2 w_C)$ for the Brier score) and shows detection requires $Ω(1/Δ^2)$ observations. We prove the principal's optimal oversight rule is necessarily non-affine, making the impossibility unconditional and optimizer-independent across log-concave-density policy families. We formalize the Confidence-Gated Decision Problem, map existing methods onto the trilemma, and identify two constructive resolution pathways (commitment, domain separation). A 540-configuration Best-of-N experiment tests five pre-registered hypotheses, all strongly confirmed (effect sizes $d = 1.10$ to $5.32$), and adds a descriptive analysis of the achievable-$(H, C, A)$ surface geometry showing a plateau-truncated frontier consistent with the predicted inflation saturation.

In multi-modal biomedical research, integrating high-dimensional genomic data with clinical baselines is essential for precision medicine. However, standard deep neural network approaches often entangle these modalities, obscuring the specific predictive impact of genetic features and leading to possibly suboptimal predictive performance. Motivated by the landmark METABRIC cohort primary breast tumors study, we propose the Stein-Encoder, a white-box supervised framework designed to isolate the genetic signal driving clinical outcomes conditional on nuisance covariates. By leveraging Stein's method and residualization techniques, our approach constructs an interpretable single index that summarizes relevant biological heterogeneity while flexibly incorporating clinical factors and can be used to improve downstream prediction. We establish theoretical guarantees for identification, consistency and efficiency improvement. Applied to the METABRIC cohort, the Stein-Encoder outperforms unsupervised benchmarks in predictive accuracy. Crucially, it achieves structural disentanglement by revealing response-specific biological mechanisms: we find that tumor size is driven primarily by mitotic networks, whereas prognostic indices rely on a distinct proliferation-versus-immune axis. This work contributes a unified, computationally efficient framework that bridges statistical rigor with the representational power of neural networks, enabling interpretable, task-specific and efficient compression of multi-modal health data for a wide range of precision medicine applications, beyond biomarker discovery.

We derive confidence intervals and confidence sequences for causal effects in situations where the back-door or front-door criteria are applicable. Our tightest confidence intervals hold in the standard setting where the training data consists of IID observations over a system described by a given causal diagram. When interventions are allowed to depend on the past data, our confidence intervals become wider and involve a term coming from the law of the iterated logarithm, even where the number of observations is known in advance. In the sequential setting where the number of observations is not given, our confidence intervals, arranged into a confidence sequence for causal effects, involve more iterated logarithm terms and become even wider.

This paper proposes StrTransformer, a source-wise structured Transformer framework for blind source recovery and branch-wise latent modeling. Instead of using an encoder to infer latent variables, StrTransformer directly optimizes the latent source matrix together with an observation-space mixer and source-wise structural Transformer branches. The mixer enforces reconstruction consistency, while each Transformer branch imposes a differentiable structural constraint on one latent source trajectory. Specifically, each source is converted into multi-scale patch tokens, randomly masked, processed by a locality-biased Transformer, and evaluated through a masked patch reconstruction energy. This energy acts as an implicit source-wise structural prior. To encourage different latent branches to specialize into different temporal regimes, StrTransformer further introduces an ordered multi-scale controller that learns branch-specific patch-scale weights, ordered scale centers, and locality attention slopes. The resulting objective combines observation reconstruction, source-wise structural regularization, and modular auxiliary penalties for separation and scale specialization. We analyze the decoupling and coupling structure of the objective, the regularized exact-reconstruction fiber, and the reduction of permutation symmetry induced by ordered branch descriptors. A controlled case study shows that the learned branches converge to distinct temporal-scale structures and recover source-aligned latent trajectories under post-hoc evaluation.

The importance of functional data analysis has increased substantially in recent years. In machine learning, nonlinear function regression based on deep neural networks is referred to as operator learning, and many of its applications involve functional time series data. However, the theoretical understanding of nonlinear models in functional time series analysis remains limited, as most existing works focus on linear models. In this paper, we derive basic properties for analyzing adaptive learning in nonlinear functional autoregressive (NFAR) models. Specifically, we derive sufficient conditions for NFAR models to be exponentially mixing. We provide an example with a Hammerstein operator under which these conditions are satisfied. As an application of exponential mixing, we consider operator learning for NFAR models with Urysohn operators and derive convergence rates for adaptive estimators based on deep neural networks.

Single-pass uncertainty quantification (UQ) methods for classification represent uncertainty by predicting a tractable distribution over the class probability vector. While existing approaches primarily focus on enhancing the expressiveness of this distribution, they often provide limited insight into how predictive uncertainty is structured and aggregated, resulting in weak interpretability. We introduce the courtroom analogy, which conceptualizes uncertainty-aware classification as a structured debate among class-specific advocates. Each advocate forms a probabilistic opinion, and a final verdict is reached by aggregating these opinions using input-dependent plausibility weights. In this framework, each advocate's opinion is modeled as a Dirichlet distribution whose concentration parameter is decomposed into shared evidence and class-specific advocacy. This yields a structured mixture of Dirichlet distributions with semantically interpretable parameters. To instantiate this formulation, we propose Mixture of Dirichlet EXperts (MoDEX), a single-pass neural architecture that predicts the courtroom parameters, enabling efficient and expressive UQ while explicitly modeling uncertainty aggregation. We demonstrate that MoDEX enjoys strong theoretical properties and achieves state-of-the-art UQ performance across diverse benchmarks, yielding interpretable uncertainty estimates with meaningful semantics.

Classification of matches played in the last rounds of sports competitions is a well-established tool for evaluating tournament designs. Both deterministic and probabilistic approaches are available for this purpose. Our paper offers the first comparison of them by analysing the most prominent example of four-team round-robin competitions, the group stage of the FIFA World Cup. We show that both methods are highly relevant in practice: all (four) deterministic and (six) probabilistic match types occurred in the 2014 and 2018 FIFA World Cups, respectively. The probabilistic model, which accounts for the relative benefits of attacking and defending, provides deeper insights; for instance, the competitive matches from the deterministic approach can be of any of the six probabilistic types. Finally, the probabilistic framework is used to quantify and decompose the impact of the main reforms introduced for the 2026 FIFA World Cup: the expansion to 48 teams, as well as the modified qualification and tie-breaking rules.

The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.

We study optimal experimental design for multinomial logit (MNL) bandits, where an agent repeatedly selects a subset of $K$ items from a ground set of size $N$ and observes single-choice feedback. Unlike linear or generalized linear bandits, MNL bandits have a combinatorial action space, which makes classical optimal design approaches and naive optimization over all subsets computationally intractable. We propose a computationally efficient optimal design framework for MNL models that achieves both statistical efficiency and scalability through two complementary approaches: (i) an exact or certified-approximate reformulation of the design oracle as a $0$-$1$ mixed-integer linear program (MILP) with solver-certified early stopping, and (ii) a fully polynomial-time lifted design that replaces the nonlinear objective with a tractable surrogate. Using the Kiefer-Wolfowitz equivalence theorem, we establish near G-optimality guarantees and characterize the induced statistical-computational trade-offs. As an application, we develop a best assortment identification algorithm for MNL bandits with linear utilities and non-uniform revenues, and prove an instance-dependent sample complexity of $\tilde{O}\big(\frac{d \log N}{Δ^2}\big)$, where $d$ is the feature dimension, $N$ is the number of arms, and $Δ$ is the minimum revenue gap.

We study nonstationary generalized linear bandits (GLBs), where the expected reward is modeled through a nonlinear link function with an unknown time-varying parameter. This framework encompasses a broad class of reward models, including linear, Bernoulli, and binomial rewards. Existing approaches are predominantly based on maximum-likelihood estimation (MLE), using sliding-window, restart, or discounting mechanisms to handle nonstationarity. Although these methods achieve statistically efficient regret guarantees, they generally require revisiting past observations at every round, which leads to computation and memory costs that grow with time; moreover, several of them rely on a non-convex projection step. In this paper, we propose DOMD-GLB, a new algorithm for nonstationary GLBs that utilizes discounted online mirror descent (DOMD) for parameter estimation, thereby incurring only $O(1)$ computation and memory costs per round. We prove dynamic regret bounds of order $\tilde{O} \big(c_μ^{-1/2} d^{3/4} P_T^{1/4} T^{3/4}\big)$ in drifting environments and $\tilde{O}\big(c_μ^{-1/3} d^{2/3} Γ_T^{1/3} T^{2/3}\big) $in piecewise-stationary environments, where $d$ denotes the feature dimension, $T$ the time horizon, $P_T$ the path length, $Γ_T$ the number of change points, and $c_μ$ a curvature parameter associated with the link function, while substantially improving computational efficiency over prior work. To the best of our knowledge, this is the first algorithm for nonstationary GLBs with per-round computation and memory costs independent of time.

Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation--maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim σ^6$, where $n$ is the number of observations and $σ^2$ is the noise variance.

We study the geometry of determinantal point processes (DPPs) through the spectral decomposition $L=UΛU^{\top}$. The spectrum $Λ$ governs the cardinality distribution via elementary symmetric polynomials, while the eigenspace orientation $U$ governs the conditional law within each fixed-cardinality stratum. Conditioning on cardinality $k$ yields the $k$-DPP, for which the identifiability structure changes fundamentally: the spectral parameter becomes identifiable only up to a common scale, and the eigenspace rotation parameter is identifiable only through squared minors of the eigenvector matrix. We characterize the identifiability gap precisely, via three explicit invariances (scale, sign similarity, and eigenspace rotation) and a dimension-counting theorem showing the existence of additional continuous non-identifiability whenever $\binom{N}{k}<N(N+1)/2$. In contrast, for the full DPP the non-identifiability comes only from the discrete sign similarity.

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

Directed acyclic graphs provide a fundamental tool for representing directed dependence structures in multivariate network data, and are widely used to model financial and economic networks. However, accurate and interpretable estimation remains challenging under graph structural uncertainty. We propose an optimal model averaging method for directed acyclic Gaussian graphs. With a set of candidate models varying by graph structures, we average estimates from candidate models using weights that minimize a penalized negative log-likelihood criterion. In contrast to existing approaches, we not only establish the asymptotic optimality, weight consistency, and parameter consistency of the proposed method, but also explicitly characterize how different candidate models affect the convergence rate. Moreover, we prove parameter consistency even when all candidate graph models are misspecified. Results from simulation studies and a real-data analysis on the banks' international liability data show the promise of the proposed method.

The use of ordinal patterns (OPs) for analyzing the dependence structure of univariate and continuously distributed processes has gained popularity in recent years. This research goes one step further and considers the transcripts being computed from successive OPs in the time series. Transcripts constitute a kind of ``difference'' between successive OPs and thus naturally relate to two algebraic distances between OPs, the Cayley and Kendall edit distances. The original time series is transformed into a sequence of transcripts or distances, respectively, and important stochastic properties thereof are derived. It is shown that these properties differ substantially among different types of original processes. This motivates the development of various statistics based on transcripts and edit distances in order to investigate the dependence structure of the original process. In particular, the asymptotic distribution of these statistics under the null hypothesis of serial independence is derived, which is then used to implement nonparametric tests for serial dependence. A simulation study shows that these novel dependence tests have appealing power properties, often outperforming former OP-based dependence tests. A concluding real-world data example illustrates the application and interpretation of the proposed approaches in practice.

Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.

Graph Neural Networks (GNN) are currently the most popular approach for learning and prediction on graph-structured data and are deployed in various fields, from social network analysis to drug discovery. However, there is limited mathematical understanding of the performance of GNNs. We discuss the various perspectives used to study statistical generalisation in GNNs. We identify three broad frameworks. The first approach, rooted in learning theory, relies on uniform convergence bounds and the complexity of the hypothesis class of specific GNN architectures. This approach also builds on the expressivity of GNNs, typically studied through the lens of graph isomorphism tests. The second principle is to simplify the neural architecture by analysing GNNs under the asymptotics of infinitely many parameters or infinite graph size. This approach approximates GNNs using Gaussian processes, neural tangent kernels or graphon neural network operators, which allow studying the generalisation or stability of trained GNNs. The third framework studies GNNs under random graph models, often the contextual stochastic block model, and derives non-asymptotic error rates using tools from high-dimensional statistics. We highlight some key theoretical results and discuss a few limitations and open research questions for each perspective.

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{tΔ}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly defined, and we recover the classical graph-Laplacian pointwise rate $O(N^{-2/(d+6)})$ up to logarithmic factors, for diffusion times $t $ up to $O(1)$ and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of $Q_t f$ at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density $p$ is $C^3$ and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.

We consider the problem of testing mutual independence among the components of a high-dimensional random vector. Building on the rank-based max-sum framework, we introduce fixed finite-$L_q$ power-sum statistics under three general classes of rank-based correlations: simple linear rank statistics, non-degenerate rank-based U-statistics and degenerate rank-based U-statistics. The proposed statistics interpolate between the dense-alternative sensitivity of the $L_2$ statistic and the sparse-alternative sensitivity of the $L_\infty$ statistic. We establish the asymptotic independence between any fixed finite-$L_q$ block and the corresponding $L_\infty$ statistic, and combine $L_2,L_4,L_6$ and $L_\infty$ p-values through a Cauchy rule. Numerical studies show that the resulting $L_{2,4,6,\infty}$ procedure is highly robust to the sparsity of the alternative and has strong empirical power across the considered designs.

We propose a quasi maximum likelihood estimation method for Bergomi-type stochastic volatility models with parametrized kernels, focusing on the estimation of the kernel parameters from high-frequency time-series observations of option prices. We first show that the cumulative forward variance, which can be reconstructed from option prices, solves an infinite-dimensional stochastic differential equation driven by a one-dimensional Brownian motion under the Bergomi-type model. To overcome this degeneracy, we introduce a nondegenerate proxy likelihood based on the Euler-Maruyama approximation and define an estimator through the associated estimating function. We establish consistency and asymptotic mixed normality of the proposed estimator under a regular class of kernels. Simulation studies and an empirical application to SPXW option data illustrate the finite-sample performance of the method and the practical relevance of the approach.

World models for partially observed environments must imagine multiple compatible hidden futures and steer between them under counterfactual actions. Joint Embedding Predictive Architectures (JEPAs) do this in latent space, but a vector-valued latent has no internal structure for carrying the belief over hidden continuations through blind rollout. We introduce the Unitary World Model JEPA (UWM-JEPA), a JEPA world model with a density-matrix latent on a joint system-environment space and a learned unitary predictor. The construction preserves the joint-state spectrum exactly during rollout, so the predictor itself cannot dissipate the represented uncertainty. On a hidden-velocity indicator task requiring five-step forward simulation under a given action sequence with the target observation masked, UWM-JEPA reaches 0.77 accuracy and degrades monotonically as actions are perturbed; a parameter-matched LSTM-JEPA trained under the same counterfactual-target objective and action head collapses to majority-class accuracy (0.53) under every action condition. Under blind rollout, UWM-JEPA loses fewer than ten points of probe R^2 at short horizons while vector-latent baselines lose forty-one and sixty-eight; both nevertheless tie on a held-out context probe, locating the separation in the predictor rather than the encoder. Action sensitivity itself requires training against counterfactual rather than teacher-forced targets, a finding that applies beyond the unitary parameterisation. For JEPA world models to imagine under partial observability, latent geometry and predictor dynamics matter, not frozen context-encoding capacity alone.

Online experiments in ads, recommendation, and member-experience systems are often planned before the dominant interference mechanism is known. A treatment may propagate through budgets, inventory, producer exposure, graph spillovers, or temporal carryover, making the randomization design itself a statistical decision. We formulate this problem as robust design selection over uncertain exposure mechanisms. Given a finite catalog of six implementable designs, the selector compares each design by worst-case planning risk over an ambiguity set. The risk combines exposure bias, assignment-unit variance, minimum detectable effect, contamination or carryover, operational cost, and estimand mismatch. For theoretical justification, the paper develops a geometry-aware guarantee, stating that design bias is bounded by Wasserstein distance to the launch exposure distribution, and this penalty is minimax tight under Lipschitz exposure response. We also prove finite-catalog approximation and a robust selector theorem with excess-risk control, exact recovery under separation, and certified shortlists when the risk surface is flat. Empirically, the same selector gives different recommendations across samples from public datasets. It selects user-randomization on Criteo ads with dimensionless robust risk 1.295, switchbacks on Open Bandit-bts/men with risk 2.105, and cluster-randomization on KuaiRand with risk 2.240. The Open Bandit case stresses known but uneven logging support, with propensities from 0.00006 to 0.594 and a 5.17% IPS effective-sample share. Overall, the paper contributes an interference-aware experiment design framework based on mechanism-robust design decisions, where the output is either a justified design choice or an uncertainty shortlist.

While aggregate leaderboard scores drive AI development, they contain substantial measurement noise whose sources and magnitudes remain unquantified, making it unclear when rankings reflect genuine capability differences versus evaluation artifacts. We introduce a framework for measuring the latent landscape in AI benchmark ecosystems. Applying Confirmatory Factor Analysis (CFA) and Generalizability Theory to 4,000+ models from the Open LLM Leaderboard, we decompose sources of ranking variance and establish: (1) structures assumed in current reporting practice underestimate the strength of relationships between benchmarks; (2) evidence of local dependence among leaderboard items, undermining uses of benchmarks as measurement instruments under current scoring systems; (3) contributor metadata explains more rank-relevant variance ($\approx9\%$) than architecture or deployment categories in this context; (4) a manifest-score "scaling law" slope has low reliability ($R_β=0.53$); by contrast, the latent general-factor size slope is highly stable across ecosystem controls ($R_g=0.97$). We are able to provide unique insights into benchmark dynamics, such as which benchmarks are a function of LLM size and which can be oppositely impacted by post-training practices. We provide actionable diagnostics to determine how benchmark rankings can be trusted and how benchmark design can be improved.

We analyze a fixed panel of S\&P 500 stocks from 1996 to 2026 using complementary static and kinetic Ising models applied to daily binary open-to-close movements. The static pairwise model provides a long-run maximum-entropy summary of low-order dependence and reveals a sectorally organized interaction network with modest small-world structure and within-sector couplings about 2.8 times stronger than between-sector couplings, with especially coherent real estate and energy sectors. The kinetic model incorporates smooth time-varying external fields, self-memory, and directed lagged couplings to describe next-day dynamics. It reveals slow field-regime shifts around three major market-wide perturbations -- the dot-com bust, the global financial crisis, and the COVID-19 episode. Self-memory is generally weak, and the directed coupling structure is much less sector-concentrated and more asymmetric than the static network, while still reproducing the broad evolution of aggregate market movement. Taken together, the two complementary models characterize both persistent market organization and short-horizon cross-stock dynamics, providing a compact statistical physics view of interaction structure and time-varying behavior in the S\&P 500.

The boosted Frank-Wolfe algorithm accelerates the classical Frank-Wolfe algorithm by better aligning the update direction with the negative gradient. Its analysis, however, has been limited to deterministic convex problems, with step sizes that require either line search or knowledge of the Lipschitz constant of the gradient. We develop a novel step size strategy that does not depend on the Lipschitz constant of the gradient, which allows us to extend the boosted Frank-Wolfe algorithm to the stochastic setting. We prove that boosting with this step size strategy can be combined with many modern gradient estimators, including SAGA, L-SVRG, SAG, Heavy Ball momentum, and zeroth-order estimators, among others, while retaining the worst-case convergence rates of ordinary stochastic Frank-Wolfe. Our analysis also yields the first convergence rates for boosted Frank-Wolfe on nonconvex and quasar-convex objectives, results which are new even for deterministic problems. Experiments on sparse logistic regression and quantum process tomography show that stochastic boosted Frank-Wolfe achieves faster convergence per gradient oracle call (and on wall-clock) compared to the non-boosted baseline.

Human feedback often arrives as preferences rather than calibrated numeric rewards, motivating reinforcement learning from preferential feedback, also referred to as reinforcement learning from human feedback (RLHF). We present a rigorous theoretical study of preference-only learning in episodic kernel MDPs. In each episode, the learner deploys two policies from a common start state and receives a single binary label indicating which trajectory is preferred, modeled by a Bradley--Terry--Luce link on the difference of cumulative (unobserved) rewards. Under kernel-based assumptions on the reward and transition functions (one of the most general models amenable to theoretical analysis) we develop preference-based value estimation and confidence sets tailored to end-of-episode comparisons. We prove high-probability regret bounds that scale sublinearly in the number of episodes, implying that the value of the learned policy converges to that of the optimal policy.

We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in Deng et al., 2026 (arxiv:2602.04770). For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

Let $M^n$ be a compact orientable smooth Riemannian submanifold of dimension $n\geq 3$ in $\mathbb R^d$. We construct a family of deformed Hodge Laplacians $Δ_t^*$, $t>0$, acting on differential forms and defined through the extrinsic geometry of $M^n$. We prove that these operators converge uniformly, in the appropriate operator topology, to the classical Hodge Laplacian $Δ^*$ as $t\to0^+$. Given a point cloud $S_m \subset M^n$, we define empirical operators $Δ^*_{t, S_m}$ and establish their spectral convergence in probability to $Δ^*$, as $t \to 0^+$, under a suitable scaling regime $t = m ^{-\frac{1}{2n}}$. This rigorously extends the scalar Belkin--Niyogi Laplacian Eigenmaps framework to differential forms. As applications, we obtain consistent recovery procedures for the de Rham cohomology ring $H^* (M^n,\mathbf R)$, the second fundamental form of $M^n$, hence for the Riemannian curvature tensor, and consequently for the Pontryagin characteristic classes and Pontryagin numbers of $M^n$ from sampled data.

Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's dynamics. Recently, score-based diffusion models have learned global diffusion priors that effectively model (stochastic) dynamics, revealing strong potential for data assimilation. In this paper, we investigate how information from noisy observations can be incorporated to enable continuous correction and refinement of the predicted system state when using a diffusion prior. Motivated by particle filtering methods, we represent the posterior distribution using a set of particles. After receiving noisy observations, the diffusion model is guided using the observation likelihood to steer the generation process toward observation-consistent states. Nevertheless, such guidance does not guarantee sampling from the true posterior. We therefore employ a Sequential Monte Carlo approach over the diffusion trajectory, viewed as a path measure, to reweight and resample particles, thereby correcting the generation process and ensuring convergence toward the desired posterior distribution. This leads to an unbiased particle filtering method that rigorously fuses observational data with diffusion model simulations.

This work focuses on the question of learning from a large number of devices with each device holding only a single sample of data. Several real-world applications exist to this one sample per client setup up including learning from fitness trackers, data/app usage aggregators, body-worn sensing devices, and daily event monitors to name a few. When a client has only one sample, the standard federated learning paradigm breaks down as a local update based on that single point is far from being useful, especially in the earlier rounds for estimation of the model coefficients. This utility is further weakened by the privacy-inducing noise applied at every round. This work caters to this problem to enable such clients to collaboratively contribute to effectively learn a global model without leaking the privacy of their data. The proposed approach injects a single, carefully calibrated noisy perturbation to transform the sample at each client, followed by a post-processed representation which is shared with the server. These representations aggregated at the server are processed to obtain an unbiased gradient update that in expectation matches the non-private centralized gradient while preserving data privacy. This approach is different than traditional private federated learning, where the communication payloads involve model coefficients as opposed to privately transformed data samples. This method enables devices with extremely limited data to collaborate and learn accurate, privacy-preserving models without requiring large local datasets or sacrificing individual privacy.

Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a broad range of domains, including biology, medicine, astronomy, and the social and environmental sciences. Modern datasets are increasingly complex, often exhibiting rich dependency, heterogeneity, and other features that challenge traditional statistical methods. In response, high-dimensional statistics has evolved to address more sophisticated estimation and inference problems. This evolution has, in turn, fostered deep connections with and contributions to a wide range of research areas, including optimization, concentration of measure, random matrix theory, information theory, and theoretical computer science. Given the rapid pace of recent developments in high-dimensional statistics, our goal is to synthesize representative advances, highlight common themes and open problems, and point to important works that offer entry points into the field.

Offline Reinforcement Learning from Human Feedback (RLHF) pipelines such as Direct Preference Optimization (DPO) train on a pre-collected preference dataset, which makes them vulnerable to preference poisoning attack. We study label flip attacks against log-linear DPO. We first illustrate that flipping one preference label induces a parameter-independent shift in the DPO gradient. Using this key property, we can then convert the targeted poisoning problem into a structured binary sparse approximation problem. To solve this problem, we develop two attack methods: Binary-Aware Lattice Attack (BAL-A) and Binary Matching Pursuit Attack (BMP-A). BAL-A embeds the binary flip selection problem into a binary-aware lattice and applies Lenstra-Lenstra-Lovász reduction and Babai's nearest plane algorithm; we provide sufficient conditions that enforce binary coefficients and recover the minimum-flip objective. BMP-A adapts binary matching pursuit to our non-normalized gradient dictionary and yields coherence-based recovery guarantees and robustness (impossibility) certificates for $K$-flip budgets. Experiments on synthetic dictionaries and the Stanford Human Preferences dataset validate the theory and highlight how dictionary geometry governs attack success.

LLMs utilizing chain-of-thought reasoning often waste substantial compute by producing long, incorrect responses. Abstention can mitigate this by withholding outputs unlikely to be correct. While most abstention methods decide to withhold outputs before or after generation, dynamic mid-generation abstention considers early termination of unpromising reasoning traces at each token position. Prior work has explored empirical variants of this idea, but principled guidance for the abstention rule remains lacking. We present a formal analysis of dynamic abstention for LLMs, modeling abstention as an explicit action within a regularized reinforcement learning framework. An abstention reward parameter controls the trade-off between compute and information. We show that abstaining when the value function falls below this reward strictly outperforms natural baselines under general conditions. We further derive a principled and efficient method to approximate the value function. Empirical results on mathematical reasoning and toxicity avoidance tasks support our theory and demonstrate improved selective accuracy over existing methods.

The growing scale of deep learning demands distributed training frameworks that jointly reason about parallelism, memory, and network topology. Prior works often rely on heuristic or topology-agnostic search, handling communication and memory separately. Without per-device memory awareness, these methods typically ensure feasibility post hoc by sharding parameters and activations across many devices, increasing synchronization, inflating communication, and underutilizing compute-limiting scalability and efficiency on real datacenter networks. We present NEST, a network-, compute-, and memory-aware device placement framework that unifies model parallelism, topology modeling, and memory feasibility via structured dynamic programming. NEST's DP operates on operator graphs with tensor and expert parallel configurations, explicit allreduce latencies across hierarchical or arbitrary networks, and memory/compute profiles. By factoring parallelism across tensor, pipeline, data, and expert dimensions, NEST defines a principled search space for hybrid strategies while jointly optimizing co-location, network latency, and memory feasibility. Evaluations across diverse hardware and networks show NEST achieves up to 2.43 times higher throughput, better memory efficiency, and improved scalability over state-of-the-art baselines, providing a foundation for co-designing parallelization strategies and datacenter interconnects for next-generation AI infrastructure. The source code of NEST is available at: https://github.com/scai-tech/Nest

The dispersion of real data is particularly important to understand the variability of a given distribution. In addition to the central tendency, variability is of considerable interest in a wide variety of fields such as life sciences, meteorology, and economics. The modal interval (MI) describes the dispersion or spread of distribution and represents the most concentrated interval of a univariate unimodal distribution. In this study, we propose a nonlinear modal interval regression (MIR) method to smoothly estimate a conditional MI to provide a robust description of how the dispersion of a data distribution varies with the covariate. First, we use kernel density estimation (KDE) to estimate the quantile levels corresponding to the conditional MI bounds, which serve as input to the quantile loss function. Second, we fit upper and lower bound functions using the quantile loss with smoothing splines. The results of numerical experiments demonstrate that the reformulated MIR achieved higher accuracy and stability than both the conventional MIR and the KDE methods. To evaluate the effectiveness of the proposed approach, we applied the method to neonatal hormone data and identified notable rhythms in cortisol and melatonin levels during the first ten days after birth.

Many political surveys rely on post-stratification, raking, or related weighting adjustments to align respondents with the target population. But when respondents differ from nonrespondents on the outcome itself (nonignorable nonresponse), these adjustments can fail, introducing bias even into basic descriptives. We provide a practical method that corrects for nonignorable nonresponse by leveraging response-propensity proxies (e.g., interviewer-coded cooperativeness) observed among respondents to extrapolate toward nonrespondents, while directly integrating observable covariates and retaining the benefits of post-stratification with known population shares. The method generalizes the variable-response-propensity (VRP) framework of Peress (2010) from binary to ordinal outcomes, which are widely used to measure trust, satisfaction, and policy attitudes. The resulting estimator is computed by maximum likelihood and implemented in a compact R routine that handles both ordinal and binary outcomes. Using the 2024 American National Election Study (ANES), we show that accounting for nonignorable nonresponse produces substantively meaningful shifts for life satisfaction (estimated latent correlation $ρ\approx 0.53$), while yielding negligible changes for retrospective economic evaluations ($ρ\approx 0$), highlighting when nonignorable nonresponse substantively affects survey estimates.

Recent evidence suggests that analyzing the presence/absence of taxonomic features can offer a compelling alternative to differential abundance analysis in microbiome studies. However, standard approaches to differential prevalence analysis face challenges with boundary cases and multiple testing. To address these limitations, we developed DiPPER (Differential Prevalence via Probabilistic Estimation in R), a method based on Bayesian hierarchical modeling. We benchmarked our method against existing differential prevalence methods, along with two differential abundance tools, using publicly available data from 57 human gut microbiome studies. We observed considerable variation in performance across the evaluated methods. Importantly, DiPPER demonstrated high sensitivity to detect potentially differentially prevalent features while maintaining a well-calibrated family-wise error rate under the global null hypothesis. Most notably, it outperformed the alternatives in the replication of findings across independent studies. Furthermore, DiPPER provides differential prevalence estimates and uncertainty intervals that are inherently adjusted for multiple testing.

Large-scale neuroimaging studies often collect data from multiple scanners across different sites, where variations in scanners, scanning procedures, and other conditions across sites can introduce artificial site effects. These effects may bias brain connectivity measures, such as functional connectivity (FC), which quantify functional network organization derived from functional magnetic resonance imaging (fMRI). How to leverage high-dimensional network structures to effectively mitigate site effects has yet to be addressed. In this paper, we propose SLACC (Sparse LAtent Covariate-driven Connectome) factorization, a multivariate method that explicitly parameterizes covariate effects in latent subject scores corresponding to sparse rank-1 latent patterns derived from brain connectivity. The proposed method identifies localized site-driven variability within and across brain networks, enabling targeted correction. We develop a penalized Expectation-Maximization (EM) algorithm for parameter estimation, incorporating the Bayesian Information Criterion (BIC) to guide optimization. Extensive simulations validate SLACC's robustness in recovering the true parameters and underlying connectivity patterns. Applied to the Autism Brain Imaging Data Exchange (ABIDE) dataset, SLACC demonstrates its ability to reduce site effects.

Autoregressive models (ARMs) currently constitute the dominant paradigm for large language models (LLMs). Energy-based models (EBMs) represent another class of models, which have historically been less prevalent in LLM development, yet naturally characterize the optimal policy in post-training alignment. In this paper, we provide a unified view of these two model classes. Taking the chain rule of probability as a starting point, we establish an explicit bijection between ARMs and EBMs in function space, which we show to correspond to a special case of the soft Bellman equation in maximum entropy reinforcement learning. Building upon this bijection, we derive the equivalence between supervised learning of ARMs and EBMs. Furthermore, we analyze the distillation of EBMs into ARMs by providing theoretical error bounds. Our results provide insights into the ability of ARMs to plan ahead, despite being based on the next-token prediction paradigm.

This article develops a moderate-deviation limit theory for autoregressive models with jointly persistent mean and volatility dynamics. The autoregressive coefficient is allowed to drift toward unity slower than the classical 1/n rate, while the volatility persistence parameter also converges to one at an even slower, logarithmic order, so that the conditional variance process is itself nearly nonstationary and its unconditional moments may diverge. This double localization allows the variance process to be nearly nonstationary and to evolve slowly, as observed in financial data and during asset price bubble episodes. Under standard regularity conditions, we establish consistency and distributional limits for the OLS estimator of the autoregressive coefficient that remains valid in the presence of highly persistent stochastic volatility. We show that the effective normalization for least squares inference is governed by an average volatility scale, and we derive martingale limit theorems for the OLS estimator under joint drift and volatility dynamics. In a mildly stationary regime (where the autoregressive root approaches one from below), the OLS estimator is asymptotically normal. In a mildly explosive regime (where the root approaches one from above), an OLS based self normalized statistic converges to a Cauchy limit. Strikingly, in both regimes, the limiting laws of our statistics are invariant to the detailed specification of the volatility process, even though the conditional variance is itself nearly nonstationary. Overall, the results extend moderate-deviation asymptotics to settings with drifting volatility persistence, unify local to unity inference with nearly nonstationary stochastic volatility, and deliver practically usable volatility robust statistics for empirical work in settings approaching instability and exhibiting bubbles.

We study nonparametric distance-based (isotropic) local polynomial methods for estimating the boundary average treatment effect curve, a causal functional that captures treatment effect heterogeneity in boundary discontinuity designs. We establish identification, estimation, and inference results both pointwise and uniformly along the treatment assignment boundary. We show that the geometric regularity of the boundary, a one-dimensional manifold, plays a central role in determining feasible convergence rates and valid inference procedures. Our theoretical contributions are threefold. First, we derive uniform lower and upper bounds on the convergence rate of the misspecification bias of isotropic local polynomial estimators. Second, we obtain uniform distributional approximations that justify boundary-robust inference. Third, we establish minimax lower bounds for a broad class of nonparametric isotropic regression estimators. These results yield practical guidance for empirical implementation, including new bandwidth selection rules that adapt to local irregularities of the treatment-assignment boundary. We illustrate the proposed methods using simulation evidence and an empirical application, and provide companion general-purpose software.

Ecological Momentary Assessment (EMA) studies enable the collection of high-frequency self-reports of suicidal thoughts and behaviors (STBs) via smartphones. Latent stochastic differential equations (SDEs) are a promising model class for EMA data, as it is irregularly sampled, noisy, and partially observed. But SDE-based models suffer from two key limitations. (a) These models often violate domain constraints, undermining scientific validity and clinical trust of the model. (b) Training is numerically unstable without ad hoc fixes (e.g. oversimplified dynamics) that are ill-suited for high-stakes applications. Here, we develop a novel class of expressive SDEs whose solutions are provably confined to a prescribed compact polyhedral state space, matching the domains of EMA data. In this work, (1) we show why chain-rule based constructions of SDEs on compact domains fail, theoretically and empirically; (2) we derive constraints on drift and diffusion for general and stationary SDEs so their solutions remain in the desired state space; and (3), we introduce a parameterization that maps arbitrary (neural or expert-given) dynamics into constraint-satisfying SDEs. On several real EMA datasets, including a large suicide-risk study, our parameterization improves forecasts and optimization dynamics over standard latent neural SDE baselines. These contributions pave the way for principled, trustworthy continuous-time models of suicide risk and other clinical time series and extend applications of SDE-based methods (e.g. diffusion models) to domains with hard state constraints.

Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal when large noise is present. To address this issue, we propose a hybrid approach that combines Christoffel sampling with optimal experimental design. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convexity-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we further extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance.

Class imbalance remains a practical obstacle in the development of clinical prediction models for conditions such as diabetes mellitus, where the number of confirmed cases is often much smaller than the number of controls. The Synthetic Minority Over-sampling Technique (SMOTE) and its variants are widely used to address this imbalance, but they generate synthetic observations through local interpolation in feature space and do not explicitly model the joint dependence structure of the minority class. To address this challenge, our study introduces a copula-based data augmentation approach that estimates the minority-class dependence structure when generating synthetic samples and integrates with standard machine learning techniques. Specifically, we employ truncated vine copulas to represent multivariate dependence through a sequence of bivariate building blocks. We evaluate the proposed approach on three public diabetes datasets, namely the Pima Indians Diabetes dataset, the Iraqi Diabetes dataset, and the CDC BRFSS 2015 Diabetes Health Indicators dataset, which together cover a range of sample sizes, dimensionalities, and imbalance regimes. For each dataset, five resampling strategies are compared across five classifiers using a 5 by 2 cross validation protocol with Dietterich's paired t test. Our findings suggest that CopulaSMOTE can improve minority-class recovery in larger tabular diabetes datasets, particularly the CDC BRFSS dataset, but its advantages depend on the classifier and evaluation metric.

Long Short-Term Memory (LSTM) neural network models have become the cornerstone for sequential data modeling in numerous applications, ranging from natural language processing to time series forecasting. Despite their success, the problem of model selection, including hyperparameter tuning, architecture specification, and regularization choice remains largely heuristic and computationally expensive. In this paper, we propose a unified statistical framework for systematic model selection in LSTM networks. Our framework extends classical model selection ideas, such as information criteria and shrinkage estimation, to sequential neural networks. We define penalized likelihoods adapted to temporal structures, propose a generalized threshold approach for hidden state dynamics, and provide efficient estimation strategies using variational Bayes and approximate marginal likelihood methods. Several biomedical data centric examples demonstrate the flexibility and improved performance of the proposed framework.

Real-world time series are often governed by complex nonlinear dynamics. Understanding these underlying dynamics is crucial for precise future prediction. While deep learning has achieved major success in time series forecasting, many existing approaches do not explicitly model the dynamics. To bridge this gap, we introduce DeepEDM, a framework that integrates nonlinear dynamical systems modeling with deep neural networks. Inspired by empirical dynamic modeling (EDM) and rooted in Takens' theorem, DeepEDM presents a novel deep model that learns a latent space from time-delayed embeddings, and employs kernel regression to approximate the underlying dynamics, while leveraging efficient implementation of softmax attention and allowing for accurate prediction of future time steps. To evaluate our method, we conduct comprehensive experiments on synthetic data of nonlinear dynamical systems as well as real-world time series across domains. Our results show that DeepEDM is robust to input noise, and outperforms state-of-the-art methods in forecasting accuracy. Our code is available at: https://abrarmajeedi.github.io/deep_edm.

In clinical studies, the risk of the primary (terminal) event may be modified by intermediate events, resulting in semicompeting risks. To study the treatment effect on the terminal event mediated by the intermediate event, researchers wish to decompose the total effect into direct and indirect effects. In this article, we extend the randomized interventional approach to time-to-event outcomes, where both intermediate and terminal events are subject to right censoring. We envision a random draw for the intermediate event process from a reference distribution, either marginally over time-varying confounders or conditionally given the observed history. We present the identification formula for interventional effects. We also discuss some variants of the identification assumptions. We estimate the treatment effects using nonparametric maximum likelihood estimation and propose a sensitivity analysis that incorporates a latent frailty. As an illustration, we study the effect of matched unrelated donor versus haploidentical donor on death mediated by relapse in a hematopoietic cell transplantation study with graft-versus-host disease (GVHD) as the time-varying confounder. We find that matched unrelated donor transplantation is preferable in terms of survival rates under the use of post-transplant PTCy GVHD prophylaxis for lymphoma patients.

Recommender systems play a crucial role in internet economies by connecting users with relevant products. However, designing effective recommender systems faces the key challenges: the exploration-exploitation tradeoff in securing incentive to explore new products against user's self-interested preferences. While prior work addresses Bayesian Incentive Compatibility (BIC) in fixed-design linear bandits (Sellke & Slivkins, 2023), we tackle the challenge of stochastic user covariates sampled online. Unlike standard black-box reductions (Mansour et al., 2020), our two-stage framework exploits the linear reward structure to achieve sublinear regret while satisfying incentive constraints. To address it, we propose a two-stage algorithm that integrates incentivized exploration with any efficient plug-in offline learning algorithms. In the first stage, it explores products while maintaining incentive compatibility to gather optimal samples. The second stage employs inverse proportional gap sampling strategy (IPGS) integrated with any efficient learning methods to secure sublinear regret. Theoretically, we prove that algorithm RCB achieves $O(\sqrt{KdT})$ regret and simultaneously satisfies incentive constraints, and discovers the tradeoff between incentive budget and regret, validating in experiments. We demonstrate RCB's strong incentive gain, sublinear regret, and robustness through a real application on personalized warfarin dosing and simulations.

Data assimilation combines dynamical models with observations to improve state estimates. Ensemble filters sequentially assimilate observations by updating a set of samples over time, alternating between a forecast and an analysis step. Accurate and robust predictions often require preserving critical invariants such as mass, stoichiometric balance of chemical species, and electrical charge. While modern numerical solvers maintain these invariants, existing invariant-preserving analysis steps are limited to Gaussian settings. Furthermore, they can be incompatible with regularization techniques such as inflation and covariance tapering. In this work, we focus on preserving linear invariants in non-Gaussian filtering problems. Leveraging tools from measure transport theory, we introduce a novel class of nonlinear ensemble filters that preserve any desired linear invariants. Notably, we recover a constrained formulation of the Kalman filter for the special case of the Gaussian setting. We also demonstrate how to combine preserving invariants with regularization techniques in the ensemble Kalman filter. Numerical experiments illustrate the benefits of preserving linear invariants in both ensemble Kalman filters and transport-based nonlinear ensemble filters.

This paper analyzes the factorizability and geometry of transition matrices of multivariate Markov chains. Specifically, we demonstrate that the induced chains on factors of a product space can be regarded as information projections with respect to the Kullback-Leibler divergence. This perspective yields Han-Shearer type inequalities and submodularity of the entropy rate of Markov chains, as well as applications in the context of large deviations and mixing time comparison. As concrete algorithmic applications in Markov chain Monte Carlo (MCMC) and approximate inference, we provide three illustrations based on lifted MCMC, swapping algorithm and factored filtering to demonstrate projection samplers improve mixing over the original samplers. The projection sampler based on the swapping algorithm resamples the highest-temperature coordinate at stationarity at each step, and we prove that such practice accelerates the mixing time by multiplicative factors related to the number of temperatures and the dimension of the underlying state space when compared with the original swapping algorithm. Through simple numerical experiments on a bimodal target distribution, we show that the projection samplers mix effectively, in contrast to lifted MCMC and the swapping algorithm, which mix less well. In filtering, our proposed factored filtering scheme is able to scale to high dimensions with linear-in-dimension computational cost per step at the price of an approximation error that can be tracked using the distance to independence, compared with the exponential-in-dimension cost per step of the exact filter.

Bregman proximal-type algorithms (BPs), such as mirror descent, have become popular tools in machine learning and data science for exploiting problem structures through non-Euclidean geometries. In this paper, we show that BPs can get trapped near a class of non-stationary points, which we term \emph{spurious stationary points}. Such stagnation can persist for any finite number of iterations if the gradient of the Bregman kernel is not Lipschitz continuous, even in convex problems. The root cause lies in a fundamental contrast in descent behavior between Euclidean and Bregman geometries: While Euclidean gradient descent ensures sufficient decrease near any non-stationary point, BPs may exhibit arbitrarily slow decrease around spurious stationary points. As a result, commonly used Bregman-based stationarity measure, such as relative change in terms of Bregman divergence, can vanish near spurious stationary points. This may misleadingly suggest convergence, even when the iterates remain far from any true stationary point. Our analysis further reveals that spurious stationary points are not pathological, but rather occur generically in a broad class of nonconvex problems with polyhedral constraints. Taken together, our findings reveal a serious blind spot in Bregman-based optimization methods and calls for new theoretical tools and algorithmic safeguards to ensure reliable convergence.

Scaling factors in residual branches have emerged as a prevalent method for boosting neural network performance, especially in normalization-free architectures. While prior work has primarily examined scaling effects from an optimization perspective, this paper investigates their role in residual architectures through the lens of generalization theory. Specifically, we establish that wide residual networks (ResNets) with constant scaling factors become asymptotically unlearnable as depth increases. In contrast, when the scaling factor exhibits rapid depth-wise decay combined with early stopping, over-parameterized ResNets achieve minimax-optimal generalization rates. To establish this, we demonstrate that the generalization capability of wide ResNets can be approximated by kernel regression associated with the Neural Tangent Kernel (NTK). Our theoretical findings are validated through experiments on synthetic data and real-world classification tasks, including MNIST and CIFAR-100.

Incomplete multi-view clustering is an important technique to deal with real-world incomplete multi-view data. Previous works assume that all views have the same incompleteness, i.e., balanced incompleteness. However, different views often have distinct incompleteness, i.e., unbalanced incompleteness, which results in strong views (low-incompleteness views) and weak views (high-incompleteness views). The unbalanced incompleteness prevents us from directly using the previous methods for clustering. In this paper, inspired by the effective biological evolution theory, we design the novel scheme of view evolution to cluster strong and weak views. Moreover, we propose an Unbalanced Incomplete Multi-view Clustering method (UIMC), which is the first effective method based on view evolution for unbalanced incomplete multi-view clustering. Compared with previous methods, UIMC has two unique advantages: 1) it proposes weighted multi-view subspace clustering to integrate these unbalanced incomplete views, which effectively solves the unbalanced incomplete multi-view problem; 2) it designs the low-rank and robust representation to recover the data, which diminishes the impact of the incompleteness and noises. Extensive experimental results demonstrate that UIMC improves the clustering performance by up to 40% on three evaluation metrics over other state-of-the-art methods.

Transportation agencies have an opportunity to leverage increasingly-available trajectory datasets to improve their analyses and decision-making processes. However, this data is typically purchased from vendors, which means agencies must understand its potential benefits beforehand in order to properly assess its value relative to the cost of acquisition. While the literature concerned with trajectory data is rich, it is naturally fragmented and focused on technical contributions in niche areas, which makes it difficult for government agencies to assess its value across different transportation domains. To overcome this issue, the current paper explores trajectory data from the perspective of a road transportation agency interested in acquiring trajectories to enhance its analyses. The paper provides a literature review illustrating applications of trajectory data in six areas of road transportation systems analysis: demand estimation, modeling human behavior, designing public transit, traffic performance measurement and prediction, environment and safety. In addition, it visually explores 20 million GPS traces in Maryland, illustrating existing and suggesting new applications of trajectory data.

Epistemic uncertainty is often viewed as a reducible uncertainty that vanishes with increasing data. This perspective implicitly assumes parameter identifiability and equates epistemic uncertainty with predictive variability. In overparametrized neural networks, however, model parameters are typically non-identifiable due to symmetries and redundant representations. As a consequence, substantial parameter uncertainty can persist even when the underlying function is fully identified. In this work, we analyze epistemic uncertainty through the lens of non-identifiability and characterize both discrete and continuous sources of residual uncertainty. Focusing on one-hidden-layer ReLU networks, we thoroughly analyze the resulting posterior structure and validate our theoretical insights through empirical studies.

We trace a conceptual genealogy from Abraham de Moivre's derivation of the normal curve (1733) to the modern distributional approach to statistics. De Moivre's Approximatio ad Summam Terminorum Binomii gave the first systematic derivation of the Gaussian density, its normalising constant (completed by Stirling's identification of $B = \sqrt{2π}$), and its tail probabilities computed to six decimal places -- more than seventy years before Gauss. His method -- extracting information from probability laws by evaluating sums against indicator probes -- is recognisably an instance of the operational viewpoint that underlies distributional statistics. We identify a four-stage chain: coefficient extraction (De Moivre) $\to$ generating functions (Euler, Laplace) $\to$ characteristic functions (Fourier, Lévy) $\to$ distributional pairings $\langle T, φ\rangle$ (Schwartz). At each stage the probes become more flexible and the class of laws that can be studied grows wider. The distributional framework, in which a probability law is represented by a distribution--kernel pair $(T, φ) \in \mathcal{S}'(\mathbb{R}) \times \mathcal{S}(\mathbb{R})$, is the natural endpoint of this progression. We formulate and prove a distributional version of the De Moivre--Laplace theorem: the standardised binomial distribution converges to the Gaussian in $\mathcal{S}'(\mathbb{R})$, with De Moivre's original computation corresponding to the special case of indicator test functions. We also discuss the transversality framework, which provides a geometric explanation -- via infinite codimension of degeneracy strata -- for why pathologies such as moment indeterminacy, non-identifiability, and singular Fisher information are rarely encountered in parametric statistical models.

Diffusion models are increasingly used as powerful conditional generators, yet real deployments often involve multiple target distributions arising from different tasks, e.g., diverse prompt domains in text-to-image generation, or multiple environments in robotics with diffusion policies. This naturally leads to a multi-objective learning (MOL) problem. A key challenge is that achieving good Pareto trade-offs can require a generalist model class with substantially larger capacity than what suffices for solving any individual task, thereby increasing statistical cost since sample complexity typically scales with the model complexity. To reconcile this, we develop a principled MOL framework for diffusion models with limited data: a semi-supervised regime where paired (labeled) samples are scarce, but (unlabeled) condition data are abundant. We propose a two-stage training procedure that first fits lightweight specialist models from limited paired data, and then distills them into a generalist model by generating pseudo-samples. We establish generalization bounds showing that the required number of paired samples only depends on the complexity of the specialist model classes. We further extend the theory to diffusion policies for sequential decision making to account for distribution shift in on-policy rollouts. Extensive experiments on robotic control and image restoration tasks are conducted to verify our theoretical results.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

This article is the rejoinder to ``The ICML 2023 Ranking Experiment: Examining Author Self-Assessment in ML/AI Peer Review,'' to appear in the Journal of the American Statistical Association with discussion. To address the practical and theoretical points raised by the discussants, we organize our response around four core themes: (i) formulating peer review as a statistical estimation problem; (ii) mitigating equity and strategic concerns in the deployment of the Isotonic Mechanism; (iii) incorporating complementary signals such as reviewer rankings and structured metadata; and (iv) exploring a human-centered framework for peer review in the era of generative AI.

Public service programs often allocate limited resources under uncertainty about their benefits, creating a need for randomization to support credible evaluation. In practice, however, applicants commonly enter waitlists where resources are prioritized toward individuals judged to have higher need through tiered priority queues, making direct randomization difficult. Motivated by this, we develop an experimental design framework for learning treatment effects while treating those most in need where incoming applicants are randomized into priority queues based on their assessed risk scores. Treatments are then provided across queues in priority order and first-in-first-out within queue as budget becomes available. Our contributions are two-fold. First, we characterize what causal effects are identified under this priority-queue allocation. When arrivals are exogenous, treatments are conditionally randomized, and hence standard estimands are identified; when arrivals are endogenous, queue randomization instead provides an instrument for treatment, identifying local treatment effects induced by the queuing process. Second, we develop optimized queue-assignment designs that trade off statistical efficiency against prioritizing higher-need applicants. We show in the process that, despite dependence in treatment assignments induced by the design, usual iid efficiency bounds remain well-justified design objectives. We illustrate the proposed designs using data from a housing allocation program in a large U.S. county.

This paper introduces the Quantum Covariance Embedding, which embeds Positive Operator-Valued Measures into a tensor product of a Reproducing Kernel Hilbert Space and the quantum state space via a tensorized Bochner integral. This construction induces the Quantum Maximum Discrepancy that metrizes the space of quantum measurements. Applying this framework to Quantum State Tomography, we reformulate density estimation as a tensorized kernel regression, enabling optimal inference without the basis-dependent sparsity constraints that restrict existing methods. We develop a unified geometric design theory for quantum Gram superoperators, establishing that Unitary Designs are strictly E-optimal experimental designs and thus statistically superior to Pauli observables. For general structure-free estimation, we derive the exact minimax lower bound and prove that our tensorized estimators achieve this optimal rate. Furthermore, we introduce the QUAntum Regression with Kernels (QUARK) estimator to accommodate the spectral geometry of physical implementations, deriving central limit theorem and concentration inequalities. To facilitate practical estimation, we establish the exactness of trace-preserving projections and demonstrate efficient estimation under mutually unbiased bases via the fast Walsh-Hadamard transform.

We study inference-time alignment for diffusion-based generative models, aiming to steer a base model toward high-reward outputs without updating its weights. Recent Sequential Monte Carlo (SMC)-based steering methods approximate reward-tilted target distributions in a principled way, but their proposals remain largely tied to the base sampler. Since reward information is mainly used after propagation through particle reweighting and resampling, these methods can require large particle budgets and suffer from weight degeneracy and high-variance estimates. One way to reduce variance and improve particle efficiency is to iteratively learn twisting functions that provide look-ahead guidance, as in twisted SMC. However, existing learnable twisting methods are developed mainly for classical sequential inference and can be unstable when applied to diffusion-based alignment with high-dimensional state spaces and terminal, noisy, or black-box rewards. We propose Trust-Region Iterative Twisted Sequential Monte Carlo (TRI-TSMC), a trust-region framework for learning twisting functions in SMC-based inference-time alignment. Each iteration computes an exact KL-constrained update in path space, which admits a closed-form solution by tempered importance reweighting, and projects this target back to the parameterized twisted family by weighted maximum likelihood. Theoretically, we formalize the value-function interpretation of the optimal twisting function and show that it yields a zero-variance sampler. We prove that the trust-region update follows an escort path toward the target distribution, that the weighted maximum-likelihood update is a forward-KL projection, and that the path reduces residual importance-weight variance. Empirically, TRI-TSMC improves primary alignment objectives on discrete diffusion text generation and text-to-image generation under matched inference-time budgets.

Reinforcement learning algorithms are generally designed to maximize the expected return across a population. However, a policy that is optimal on average may be suboptimal for certain individuals, leading to potential safety concerns. To address this, we first formalize the notion of individual harm from a counterfactual perspective and define harm as the event in which a chosen action results in a strictly worse outcome than a baseline alternative. We then propose a general two-stage procedure for learning policies that maximize the expected return while accounting for individual harm. We further establish the finite-sample properties of the learned policy, derive an upper bound on its sub-optimality gap, and show that the harm rate remains well-controlled. Numerical experiments on both simulated and real-world datasets demonstrate the effectiveness of the proposed approach.

Integrating multimodal datasets in clinical oncology is frequently hindered by high dimensionality and blockwise missingness, where entire data sources are unavailable for specific patient subsets. Standard survival models often struggle with these gaps, leading to biased results or patient exclusion. We introduce Multimodality Stacking with Blockwise missing values (MSB), a late-fusion framework for survival analysis that independently models modality-specific features before aggregating predictions via a cross-validated stacking meta-learner. MSB was validated on the PIONeeR study (n=443 patients, 378 biomarkers across eight heterogeneous sources) to predict progression-free survival in advanced non-small cell lung cancer patients receiving immunotherapy. MSB yielded higher predictive performance (C-index) than baseline algorithms. Improvements varied by baseline strength: linear models showed a 15.9% increase (p<0.001 for the Wilcoxon signed-rank test), random survival forests gained 5.4% (p=0.002), and gradient boosting methods improved by 2.1% (p=0.030). Beyond discrimination, MSB reduced the generalization gap (train-test difference in 5 folds cross-validation repeated 3 times: 0.055 vs 0.380 for linear models). Permutation importance analysis identified routine laboratory markers, clinical features, and PD-L1 expression as primary predictive drivers. Missing block indicators showed negligible importance, suggesting the model learned from biomarker values rather than data availability patterns. MSB provides a statistically validated framework for multimodal survival prediction with blockwise missingness. By enabling systematic biomarker evaluation without requiring complete data, MSB offers a practical tool for predictive modeling in biomedical research, pending external validation. Implementation is available at https://github.com/MohamedBoussena/MSB under Inria license.

Model-assisted interval designs such as the Keyboard design are transparent and easy to implement in phase I oncology trials. However, interim decisions based solely on data from the current dose may overlook informative signals from neighbouring doses, leading to unnecessary escalation or de-escalation. We propose the shared Keyboard design, a Bayesian model-assisted design that replaces the independent beta--binomial updating scheme at each dose with a posterior induced by a Beta kernel process using kernel-weighted pseudo-counts. The design preserves the decision structure of the Keyboard design while enabling controlled borrowing across nearby doses. To prioritise overdose control, we propose an asymmetric kernel that assigns greater weight to toxicities observed at higher doses during escalation. We further extend the proposed design to accommodate adaptive dose insertion when the initial dose grid is inadequate and time-to-event outcomes when late-onset toxicities are present. Extensive simulation studies demonstrate substantial improvements in both accuracy and safety for identifying the maximum tolerated dose. In settings involving dose insertion, the proposed design identifies inserted target doses more effectively than adaptive dose modification while maintaining a comparable modification rate.

Background. The reliability paradox describes the empirical observation that cognitive tasks producing robust group-level effects often yield poor between-individual reliability. Existing approaches rely predominantly on the intraclass correlation coefficient (ICC), which captures only linear, second-moment dependence between test and retest. Methods. We introduce a normalized, information-theoretic complement to ICC, NLRΔ, defined as the difference between empirically estimated mutual information and the analytic Gaussian baseline implied by the test-retest correlation. We pair NLRΔ with ICC(2,1), bias-corrected and accelerated (BCa) bootstrap intervals, Benjamini-Hochberg false discovery rate (FDR) control, and a 24-cell multiverse over the KSG nearest-neighbour parameter, correlation method, and minimum-sample threshold. The full pipeline is governed by pre-specified claim contracts, content-addressed provenance, and SHA-256-verified raw data ingestion, and is released as the MixMind Reliability Framework. Results. Across 50 estimable primary measures from the Flanker, Stroop, Stop-Signal, Go/No-Go, and Posner task families, the median NLRΔ is -0.138 nats, with interquartile range [-0.257, -0.034]. Zero of 50 primary measures exceed the headline rule. The companion ICC(2,1) analysis recovers the classical reliability paradox pattern, and the 24-specification multiverse yields 0 of 1,200 estimable cells passing the headline rule. Conclusions. On these two public datasets, replacing or augmenting ICC with an information-theoretic reliability measure does not rescue cognitive tasks from the reliability paradox. The robust null is invariant to the analytic choices examined here. We release the full pipeline, raw-data hashes, and contracts to enable exact replication and extension to other datasets and tasks.

Random sampling is a fundamental tool in modern machine learning and numerical linear algebra for reducing the computational cost of large-scale matrix problems. Existing analyses, however, rely primarily on subspace embedding guarantees, which do not precisely characterize the statistical bias of nonlinear random oblique projections induced by sampling, which arises ubiquitously in subsampled least squares and fast low-rank approximation methods. Because (pseudo)inversion is nonlinear, these random oblique projections can be systematically biased even when the underlying sketch is unbiased, thereby introducing hidden bias into downstream least squares and low-rank approximation solutions. In this work, we develop a unified non-asymptotic theory for random oblique projections in high dimensions. We show that standard random sampling schemes generally induce a systematic statistical bias overlooked by classical subspace embedding-style analyses, and we propose a principled debiasing framework to correct it. We illustrate the power of the theory through two canonical applications. For subsampled least squares, we obtain sharp bias--variance characterizations, reveal previously unrecognized statistical suboptimality in widely used sampling schemes, and identify when debiasing yields provable improvements. For fast CUR decomposition, we develop a debiased approach with improved approximation accuracy. Numerical experiments further validate our theoretical findings.

In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler--Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE -- in KL divergence for kTULA, and in total variation for tRLMC -- valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $ M $-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $ f_i $; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $ φ$-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $ \ell_2 $-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

Quaternion neural networks are parameter-efficient and model multidimensional dependencies by representing four related features as a single entity. However, existing quaternion self-attention computes component-wise scores and applies independent softmax operations to each component, which increases the computational cost and allows attention distributions to diverge across components. We propose a shared-score quaternion self-attention mechanism that computes a single real-valued score using the quaternion inner product and applies a shared attention distribution across all components. This reduces score-computation multiplications by 75% and the number of softmax operations from four to one. We prove that, when queries and keys are produced by quaternion linear projections that induce component pre-mixing, the component-wise and shared scores lie in the same interaction subspace, indicating that independent component-wise attention primarily re-parameterizes the same interactions rather than expanding the feature interaction space. In speech enhancement, our method reduces inference time by up to 44.3% on a GPU and 58.1% on a CPU while maintaining quality, with consistent trends across vision and natural language processing.

Multiview latent-variable models provide a fundamental framework for discrete data analysis, with applications to latent structure models, topic models, and mixtures of product distributions. In the discrete setting, the joint distribution of the observed views can be represented as a nonnegative low-rank tensor, which we call a multiview density tensor. We study the problem of estimating this tensor from multinomial count data. A key challenge is that multinomial sampling induces heteroskedastic and dependent noise, so the difficulty of estimation depends not only on the ambient dimensions and rank, but also on how the probability mass is distributed across different locations of sample space. We propose a general scaling framework for density tensor estimation under multinomial sampling. This framework leads to a spectral estimator for which we prove a Frobenius-norm upper bound that directly handles heteroskedasticity and negative dependence. For the original multiview model, we obtain fiber-mass-dependent Frobenius upper bounds and minimax lower bounds showing that this dependence is unavoidable. Under $\ell_1$ loss, we develop both oracle and feasible data-driven estimators based on the same scaling principle, establish minimax lower bounds, and show near-optimality for the oracle rule at fixed rank and for slice normalization under bounded slice-to-fiber imbalance. Simulations support the theory and demonstrate the robustness of the proposed methods.

This paper studies nonparametric regression with repeated measurements when the response in the target domain is unobservable or costly to collect. We adopt a transfer learning framework that leverages a source domain with observable responses under covariate shift. The target regression function is estimated by correcting the distribution shift via the density ratio. We consider both known and unknown density ratio scenarios, which reflect different data available for nonparametric regression estimation. In both cases, we further address two settings: the uniformly bounded density ratio and the unbounded case with finite moment conditions. Under the unknown density ratio scenario, both the density ratio and the target regression function are estimated using rectified linear unit (ReLU) feedforward neural networks (FNNs), whereas under the known density ratio scenario, only the target regression function is estimated by ReLU FNNs. Theoretically, we establish non-asymptotic error bounds for the proposed estimators and prove that they achieve the minimax optimal convergence rate under the repeated measurements setting. Notably, we develop a novel approximation theory where the constants of the network parameters depend polynomially, rather than exponentially as in existing works, on the dimension, thereby mitigating the curse of dimensionality. Consequently, we derive sharper non-asymptotic bounds for the stochastic error. The finite sample performance of the proposed method is demonstrated through numerical simulations and a real data application.

Evaluating whether large language models (LLMs) capture the structure of natural language beyond local fluency remains an open challenge. Existing evaluation methods, largely based on task performance or short-context behavior, provide limited insight into the long-range statistical organization of generated text. We propose a complementary evaluation framework based on repeated subsequences. By analyzing their distribution across scales and relating it to higher-order Rényi entropies, we probe how texts reuse previously established structure under finite-length conditions. Experiments on human-written texts and length-matched GPT-generated texts show that, while power-law models can describe restricted ranges of block length, the observed entropy growth is often equally or better characterized by logarithmic--power forms. Across datasets, natural language exhibits stable entropy-growth patterns over accessible ranges, with consistent average behavior despite variability across individual texts. In contrast, GPT-generated texts show systematic and statistically significant shifts in estimated exponents with model size. These results demonstrate that repeated-subsequence entropy provides a quantitative structural diagnostic that reveals systematic differences in long-range organization, distinguishing natural language from state-of-the-art LLM outputs beyond surface-level fluency.

We analyze downstream courtroom governance in Philadelphia eviction cases using 755,004 Municipal Court landlord--tenant records filed from 1969 through 2022. Post-filing case processing is organized by repeated courtroom relationships, judge and tenant-attorney regimes, reusable agreement templates, and repeated team-property units. Among both-represented, both-attorney-named cases, 58.2% involve a plaintiff-side and tenant-side attorney pair that had appeared against one another in the prior year, and greater prior pair exposure predicts lower default, higher judgment-by-agreement, and higher served-writ rates. Judge-linked cases display statistically distinct baseline outcome, continuance, fee, and award regimes; tenant-attorney identity explains meaningful variance in both case outcomes and agreement terms. Settlement text is highly standardized: reusable templates explain strictness, waiver, lockout-trigger, payment-plan, deadline, and time-is-essence language far more strongly than raw attorney identity. Monetary burden concentrates in repeated plaintiff-attorney-property units. Assignment-cell support and balance audits indicate that judge-linked evidence reflects institutional heterogeneity rather than a clean judge lottery, and judge--triad interactions are not estimable in this docket. Eviction court emerges as a repeated institutional field that organizes bargaining, text, debt, and enforcement after cases enter the courtroom pipeline.

We introduce the Markov Distributional Conformal Prediction (MDCP) method that extends the distributional conformal prediction (previously developed for regression) to the setting of a strictly stationary Markov process. Instead of relying on a specific model structure to do prediction, the idea of distributional conformal prediction interval aligns with the Model-Free (MF) Prediction Principle. In analogy to MF prediction of Markov processes, our method exploits the probability integral transform based on estimated transition distribution functions to transform the Markov data to an i.i.d.~dataset. We show a non-asymptotic error bound of MDCPs unconditional coverage rate under a $β$-mixing condition and other standard assumptions on the kernel estimators. The asymptotic validity of the conditional prediction interval is also verified. In addition, we show that our conditional prediction interval is still asymptotically valid with Markov processes being $L^p$-$m$-approximable instead of satisfying the mixing property. Numerical simulations and real data experiments are deployed to empirically illustrate the finite-sample performance of MDCP, and compare it with the MF bootstrap prediction method.

We study the problem of detecting multiple change points in the mean vectors of an independent sequence of high-dimensional observations. We propose a family of ridge-regularized CUSUM statistics built upon the adaptable ridge-regularized Hotelling's T2 test of Li et al. (Ann. Statist. 48 (2020) 1815-1847). The proposed tests are designed for dense alternatives in the high-dimensional regime where the dimension is comparable to the sample size. By introducing ridge regularization, the procedure achieves a stable form of sample covariance normalization and attains adaptability with respect to the underlying population covariance structure. We derive the limiting distributions of the proposed statistics under mild conditions, both under the null hypothesis and under a class of local alternatives. We further develop a principled framework for selecting the regularization parameter by maximizing asymptotic power. Extensive simulation studies demonstrate that the proposed tests compare favorably with a wide range of existing methods across diverse settings. The performance of the proposed test procedure is illustrated through an application to a panel of daily log-returns from S&P 500 constituents spanning 2007-2025.

Off-dynamics offline reinforcement learning seeks to learn a target-domain policy from a large source dataset and a limited target dataset under mismatched transition dynamics. Existing approaches such as reward augmentation and data filtering are constrained to the source dataset and cannot synthesize new target behavior to improve coverage beyond the collected source trajectories. While recent model-based methods attempt to address this by learning target-aware dynamics, the generated experience is constructed only at the transition level, which leads to accumulated errors over long horizons. These limitations necessitate a shift toward trajectory-level generation for off-dynamics offline RL. We propose CEDGE, a Cross-domain Energy-guided Diffusion GEneration framework. CEDGE trains a trajectory diffusion model on source-domain trajectories and adapts the generated samples to the target domain through energy guidance. This guidance is derived by minimizing the distribution mismatch between the source and desired target-domain trajectories and is decomposed into return, domain, and behavior energy components. The resulting energy-guided trajectories are useful both for direct planning and as synthetic data for policy learning. Since target adaptation is achieved via energy guidance rather than retraining the diffusion model, CEDGE can be efficiently adapted to new target dynamics compared to previous methods. Experiments on the ODRL benchmark demonstrate that trajectory-level energy-guided generation improves diffusion planning under dynamics shifts and produces synthetic data that improves downstream target policy learning.

Reward modeling is not only a prediction problem: in KL-regularized policy optimization, the learned reward is exponentiated to define the deployed policy, so downstream value depends on errors in reward-tilted regions. We study this feedback in a Gaussian single-index model with $r^*(x) = σ^*(\langle θ^*, x\rangle)$ and $x \sim N(0, I_d)$. We analyze a two-stage neural reward model that first learns the hidden direction $θ^*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression. Exponential reward weighting changes the Hermite signal available to the first layer; for any feature-learning temperature $β_1$ above a dimension-free $O(1)$ threshold, a constant fraction of neurons recover the hidden direction, with weak-recovery complexity governed by the generative exponent. After feature recovery, we derive tilted-policy value-gap bounds for an idealized label-weighted fit with weights $e^{y/β_2}$ and a more practical surrogate-weighted fit with weights $e^{r_{a_0}(x)/β_2}$. Keeping the $β_2$-dependence explicit yields an admissible set of deployment temperatures, balancing the gain from lowering $β_2$ against the learning cost amplified by exponential weighting; in the surrogate-weighted case, proxy-dependent factors shrink this admissible set.

We study the sample complexity of robust binary hypothesis testing under three standard contamination models: $\varepsilon$-additive (Huber), $\varepsilon$-subtractive, and $\varepsilon$-total variation (TV), denoted by $n^*_{\mathrm{Hub}}(\varepsilon)$, $n^*_{\mathrm{Sub}}(\varepsilon)$, and $n^*_{\mathrm{TV}}(\varepsilon)$, respectively. For subtractive contamination, we show that least favourable distributions exist and provide explicit formulas for the same, bringing this model in line with the classical Huber and TV models. Next we show that in all three models, sample complexity may be highly unstable in the contamination parameter $\varepsilon$, increasing by polynomial factors even for $o(\varepsilon)$ perturbations. Similarly, there may be polynomial factor gaps between the sample complexities when $\varepsilon$ is known exactly versus when it is known up to $o(\varepsilon)$ error. Despite the instability of the sample complexity in all models, we show that the sample complexities across models are comparable up to constant-factor rescaling of $\varepsilon$. Specifically, for any fixed $δ_0>0$, the following hold for all distributions $p$ and $q$: (i) $n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(2\varepsilon)$, (ii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((2+δ_0)\varepsilon)$, and (iii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((1+δ_0)\varepsilon)$, and the scaling constants are tight. Finally, we extend our results to adaptive versions of the contamination models.

Identifying the most influential nodes in a network, typically using centrality measures, is a central task in applied network analysis. However, real-world networks are often constructed from noisy or incomplete data, which can distort rankings and lead to errors in identifying the true top-$k$ nodes. In this paper, we study how network noise affects the recovery of the true top-$k$ node set based on degree centrality. Specifically, we consider a noisy network observation in which edges are randomly added or removed according to a probabilistic noise model, and analyze the resulting empirical top-$k$ set. We show that top-$k$ recovery under network noise is governed by the relationship between the degree gap and the noise magnitude, which separates recoverable and unrecoverable regimes. To quantify ranking stability, we derive upper and lower bounds on the expected discrepancy between the empirical and true top-$k$ sets in a general framework and for specific network models. We also extend the analysis to eigenvector centrality, showing that similar noise-gap tradeoffs arise in spectral rankings. Simulation studies support our theoretical findings and illustrate the practical impact of network noise across a range of settings.

We establish four structural results for feature learning in wide two-layer neural networks under the Maximal Update Parametrization ($μ$P). First, we prove global existence and uniqueness of the mean-field limit of noisy gradient descent under $μ$P, identifying the maximal admissible weight $w^*$ on the moment sequence of the initialization as the reciprocal parameter-moment-growth boundary, and hence the largest weighted moment class propagated by the flow. The finite-particle approximation has uniform-in-time squared-Wasserstein rate $O(N^{-1})$. Second, we characterize identifiability of the mean-field limit: two admissible parameter measures induce the same network function in $L^2$ exactly when their active components agree modulo the finite-rank realization symmetry of the architecture. The orbit depth $D^*_{\mathrm{orb}}$ is separated from the moment-variety depth $D^*_{\mathrm{var}}$. Third, under the Barron-Hermite target condition the active support of the long-time limit measure admits a sparse-dictionary decomposition: it is supported on at most $S^*$ atoms modulo finite-rank realization symmetry, with $S^*$ bounded by an explicit coefficient-threshold number. Fourth, we derive the total feature-learning-error decomposition into statistical, optimization, propagation-of-chaos, and sparse-residual components, with a target-dependent Hermite/Barron tail replacing any initialization-only residual. The four results are tied together by an architectural identity: the triple $(w^*, D^*_{\mathrm{orb}}, S^*)$ -- the maximal admissible weight, the orbit identifiability depth, and the sparse-dictionary depth at which the target is realizable -- is the natural learning cell of the architecture-data pair $(σ, ρ)$. The proofs are self-contained except for standard results from $μ$P and mean-field Langevin theory.

Understanding cognitive processes in major depressive disorder (MDD) often relies on behavioral tasks, which are typically analyzed separately, overlooking potential correlations and shared latent structure. To address this limitation, we propose the Shared Hidden-factor Information Framework for Multiple Behavioral Tasks (SHIFT), a joint modeling approach that leverages shared information across tasks, allowing each task to benefit from information learned by the others. SHIFT introduces subject-specific latent factors that capture cross-task dependencies while accommodating individual heterogeneity in decision-making, response times (RTs), and strategy switching. To address computational challenges without requiring high-dimensional integration, we develop an expectation-maximization with variational approximation algorithm that preserves both temporal structure and between-task dependencies. Through extensive simulation studies, we demonstrate that SHIFT substantially improves estimation accuracy and efficiency relative to single-task analyses. We then apply SHIFT to a study of MDD to jointly model the Probabilistic Reward Task (PRT) and the Flanker Task (FT). Results indicate that MDD participants show lower engagement in the PRT and reduced focus in the FT compared with healthy controls. Moreover, when individuals are engaged and focused, they exhibit longer RTs. Although observed RTs do not predict treatment response, the shared parameters recovered by SHIFT showed suggestive treatment-modulation patterns, indicating their potential as exploratory behavioral markers for therapeutic outcomes.

We generalize finite-sample bounds for convex clustering to the setting where affinity weights appearing in the objective correspond to a general connected graph. These bounds and their analysis lead to a better understanding of clustering behavior under various implied connectivity structures behind the data and to new rates of convergence for centroid recovery. The new theoretical framework is based on random walks, which allow application of concentration inequalities related to random graph models, and formalizes the relationship between the clustering performance and the connectivity of the graph structures. Through the form of the bound and empirical results, we argue proper tuning of hyperparameters to convex clustering problems should also include tuning of input affinity weights.

We propose a general robust prediction framework, termed conformal-projective prediction (CPP), that integrates Bayesian predictive modeling with ideas from conformal prediction. Rather than assessing conformity through residual-based scores, the CPP criterion defines conformity distributionally: a candidate value for a future response is considered conforming to the extent that its inclusion in the data leaves the leave-one-out predictive distributions of the observed responses undisturbed. The framework requires only that the leave-one-out and swapped predictive distributions are available in closed form and that the swapped predictive mean is differentiable in the candidate value. Under these conditions, we establish a general bounded-influence proposition and a general local convexity lemma, and prove that CPP dominates any plug-in predictor with unbounded influence in asymptotic variance under $ε$-contamination models. When the posterior mean is linear in the observations, as in Gaussian linear models, basis-expansion regression, and Gaussian process regression, the swapped predictive mean is affine in the candidate value, yielding closed-form or one-dimensional optimization solutions and an efficient rank-two computational update; all general theoretical results specialize to explicit corollaries in this setting. Simulation experiments and two data analyses under the Gaussian linear model illustrate the finite-sample advantages of the proposed method, confirming the theoretical predictions across contamination levels, sample sizes, and predictor dimensions.

Low-light, long-exposure defocus deblurring remains a challenging problem due to the simultaneous presence of severe blur and complex biased noise. Existing methods typically rely on simplified noise assumptions, which limits their effectiveness under realistic imaging conditions. In this work, we propose Physen-Noise2Noise, a self-supervised deblurring framework guided by the physical model of defocus imaging, which leverages noisy multi-frame observations without requiring clean reference images. Unlike conventional Noise2Noise-based approaches that assume zero-mean noise, we derive a frequency-domain constraint inherent to the defocus imaging process and incorporate it into the learning framework via a learnable noise bias parameter. In addition, a multi-frame noisy initialization strategy is introduced to suppress complex biased noise prior to deblurring, providing a more stable starting point for reconstruction. This formulation explicitly models biased noise and enables joint bias correction and high-frequency detail recovery during training. Furthermore, we develop a pretrain-finetune variant to enhance robustness and generalization under challenging noise conditions. Extensive experiments on both simulation and real-world datasets demonstrate that the proposed method consistently outperforms state-of-the-art self-supervised approaches for defocus deblurring in the presence of complex biased noise.

This paper studies variable selection and post-selection inference for high-dimensional clustered data using marginal-model-based procedures. We show that, when covariates are heterogeneously distributed across clusters, marginal-model LASSO may use them as sparse proxies for latent cluster effects, shifting the estimation target away from the structural fixed effects and inducing false selections. To address this problem, we propose Synthetic Heterogeneous-Effects LASSO (SHEL), a fixed-effects penalized framework that incorporates cluster-level synthetic approximations to the latent heterogeneity. We establish theoretical properties of SHEL in high-dimensional settings and develop procedures for valid post-selection inference. The finite sample performance of the proposed method is investigated through extensive simulation studies. A longitudinal bulk RNA-seq dataset of enriched blood neutrophils from hospitalized COVID-19 patients is analyzed to demonstrate the method in a real application.

Uniform sampling on implicitly defined manifolds is a core primitive in motion planning, constrained simulation, and probabilistic machine learning. MASEM addresses this problem by entropy-maximizing resampling, but its resampling weights depend on a local k-nearest-neighbour density estimate whose errors can be amplified by aggressive resampling temperatures. We ask whether a polynomial-maximization moment estimator can replace the plug-in density rule without changing the surrounding MASEM architecture. The proposed PMM-MASEM module computes shell spacings from nested k-nearest-neighbour radii, estimates their standardized cumulants, and uses a gated PMM2/PMM3 estimator only when the spacing distribution departs from the flat Exp(1) regime; otherwise it falls back to the plug-in/MLE rule. This fallback is essential: on a flat homogeneous manifold the plug-in estimator is already the MLE, so PMM should not outperform it. A local Known-DGP Monte Carlo experiment confirms this gate: the selector returns MLE on flat Exp(1) spacings and reduces density MSE by 22--36% on asymmetric gamma and boundary-spacing regimes. The evidence is not uniformly positive: PMM3 worsens a platykurtic uniform spacing law, and a lightweight resampling-proxy experiment improves seven-lobes coverage but degrades the sine and swiss-roll proxies. The current evidence therefore supports an applicability-boundary result rather than a general MASEM improvement claim.

Diffusion/score-based models have emerged as powerful generative models, capable of generating high-quality samples that mimic the training data distribution. However, it has been observed that they are prone to reproducing training samples-known as "memorization"-potentially violating copyright and privacy. In this paper, we study the effect of Higher-Order Langevin Dynamics (HOLD) on this phenomenon. HOLD diffusion processes introduce auxiliary variables; if the data variable is interpreted as "position," then the auxiliary variables can be interpreted as "velocity" and "acceleration," depending on the chosen order of the model. They were originally proposed based on the intuition that they regularize the trajectories of the data variable by implicitly imposing additional dynamical constraints. Our work provides, to our knowledge, the first theoretical characterization of the regularization effect of HOLD. Specifically, we show that in HOLD, the dynamics of the data variable are governed by a low-pass-filtered version of the learned score function, with smoothness increasing with the order of HOLD. We then analyze the optimal empirical score and the possibility of distribution collapse. Together, our results explain the mitigation of memorization as the model order increases. Finally, we present an empirical study on real-world data that supports our theory and highlights this distinct advantage of HOLD over standard diffusion in practice.

This paper introduces a physics-informed generative framework that resolves the fundamental conflict between the statistical flexibility of deep learning and the rigorous theoretical constraints of fixed-income modeling. We demonstrate that standard generative models and unconstrained statistical extrapolations suffer from "manifold collapse" and severe arbitrage violations when forecasting term structures across diverse macroeconomic regimes. To overcome this, we propose a two-stage architecture. First, a Student-t Conditional Variational Autoencoder with Dynamic Level Injection (CVAEsT+LS) extracts a robust, heavy-tailed term structure manifold, effectively decoupling macroeconomic shape dynamics from absolute base rates. Second, the latent dynamic evolution is governed by a continuous-time Neural Stochastic Differential Equation (SDE) strictly penalized by a No-Arbitrage Partial Differential Equation (PDE). Empirical results across multiple sovereign currencies (USD, GBP, JPY) confirm that our synergistic approach drastically reduces out-of-sample forecasting errors -- achieving an exceptional 6.58 bps Mean Tenor RMSE -- and successfully overcomes the massive parallel drift and zero-lower-bound violations exhibited by the classical HJM model in extreme environments. Furthermore, through phase space vector field analysis, we demonstrate the model's superior capability in unsupervised macroeconomic regime detection and high-quality continuous-time scenario generation. Ultimately, this research provides a highly scalable, mathematically sound evolutionary engine for term structure modeling.

For stochastic process models, parameter inference is often severely bottlenecked by computationally expensive likelihood functions. Simulation-based inference (SBI) bypasses this restriction by constructing amortized surrogate likelihoods, but most SBI methods assume a black-box data generating process. While these surrogates are exact in the limit of infinite training data, practical scenarios force a strict tradeoff between model quality and simulation cost. In this work, we loosen the black-box assumption of SBI to improve this tradeoff for structured stochastic process models. Specifically, for neural network likelihood surrogates trained via probabilistic classification, we propose to augment the standard binary cross-entropy loss with exact score information $\nabla_θ\log p(x \mid θ)$ and adaptive weighting based on loss gradients. We evaluate our approach on case studies involving network dynamics and spatial processes, demonstrating that our method improves surrogate quality at a drastically lower computational cost than generating more training data. Notably, in some cases, our approach achieves downstream inference performance equivalent to a 10x increase in training data with less than a 1.1x increase in training time.

The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, φ)$, where $T$ is a tempered distribution and $φ$ is a rapidly decaying kernel. We develop the thesis that the kernel acts as a geometric regulariser, placing parametric statistical models in generic (transversal) position relative to degeneracy loci encoding non-identifiability, singular information, moment indeterminacy, and representation failure. Using the transversality theorems of Whitney, Thom, and Mather, we prove a finite-dimensional weak transversality theorem: for a generic kernel in any sufficiently rich family, the kernel-induced feature map avoids degeneracy strata of sufficiently high codimension. We establish verifiable conditions -- formulated as rank conditions on the Jacobian of the joint feature map -- under which the transversality hypothesis can be checked, and verify them for location families, the log-normal, Stein discrepancies, and graphical models. The present results apply to parametric models; extensions to semiparametric and nonparametric settings are discussed. The degeneracy classification includes representation degeneracy (Type 0) for models without closed-form densities and higher-order instabilities (Type IV) in non-chordal graphical models. Identifiability, robustness, moment determinacy, Fisher information regularity, Stein discrepancy, inferential separation, and the Behrens-Fisher problem all admit a unified geometric interpretation as transversality conditions on the feature map. This paper serves as a geometric companion to a series of papers developing the distributional framework.

Previous comparisons of ordinary least squares with Newey-West standard errors (OLS-NW) and Prais-Winsten (PW) regression in multiple-group interrupted time series analysis have been limited to first-order autoregressive (AR[1]) errors because PW estimation for higher-order AR[k] processes was previously unavailable. We conducted the first systematic evaluation of OLS-NW and PW under AR[2] and AR[3] error structures using Monte Carlo simulation. Simulations examined mild positive, oscillatory, and high persistent autocorrelation across varying series lengths and effect sizes. OLS-NW generally showed higher apparent power but substantially inflated Type I error and poor coverage, particularly under persistent autocorrelation, where inferential performance worsened with increasing AR order and series length. PW maintained substantially better inferential calibration across nearly all conditions. Both methods were approximately unbiased.

Asymptotically linear estimators in semiparametric models are usually studied through a von Mises expansion in which first-order inference is based on the influence-function variance. This reduction is valid only when the second-order remainder is negligible not only in probability but also in variance, a requirement not implied by the usual product-rate conditions ensuring asymptotic linearity. We study the regime in which the remainder contributes variance at order $n^{-1}$, so that the total sampling variance differs from the standard influence-function approximation by a non-vanishing first-order term. We derive a finite-sample variance decomposition separating the influence-function variance, the remainder variance, and their covariance, and characterize sandwich validity through the vanishing of scaled remainder variance: under a negligible cross term, the sandwich estimator is consistent for the total sampling variance when $n\,\mathrm{Var}(R_{\mathrm{rem}})\to 0$ and materially underestimates it in the complementary near-boundary regime $n\,\mathrm{Var}(R_{\mathrm{rem}})\to c_R>0$. We then establish asymptotic validity of two refined procedures in the near-boundary regime: the leave-one-out jackknife and the pairs bootstrap. Jackknife validity is obtained through a self-normalization argument; bootstrap validity is established directly under a Mallows--2 condition. We also extend the theory to clustered data and derive an analytic expression showing how intra-cluster correlation amplifies the sandwich gap through the remainder term. Simulations illustrate the regime and confirm the predicted coverage behaviour of the competing variance estimators.

In this paper, we revisit the notion of partial copula, originally introduced to test conditional independence, highlighting its capability to represent the dependence between two random variables after removing their dependence with a covariate. Building upon results previously presented in the literature, we show that partial copulas can be seen as a nonlinear analogue of partial correlation. Then, we prove several results showing how dependence properties of the conditional copulas constrain the form of the partial copula. Finally, a simulation study is conducted to illustrate the results and to show the potential of the partial copula as a way to describe covariate-adjusted statistical dependence. This highlights the potential of the method to be used in causal inference problems and to recover the true sign of a causal effect.

Across many risk-sensitive areas, it is critical to continuously audit machine learning systems as we receive more data to quickly determine if they are performing as designed. This auditing task can be modeled as a sequential hypothesis testing problem with $k$ data streams and a global null hypothesis that asserts the system operates as intended across all $k$ streams. Under the alternative, the standard global sequential test, which uses a Bonferroni correction, has an expected stopping time of $O\left(\ln \frac{k}α\right)$ for large $k$ and significance level $α$. In this work, we demonstrate that efficient sequential tests, relying on merging martingales via averaging and products rules, provide improved stopping times, and thus more powerful tests against the null. Using these results, we show that a balanced test can match the Bonferroni rate of $O\left(\ln \frac{k}α\right)$ in the sparse regime (just a few non-null streams) while achieving $O\left(\frac{1}{k}\ln \frac{1}α\right)$ under dense alternatives (many non-null steams). We validate our theory through experiments on both synthetic and real-world data.

We study the implicit bias of momentum-based optimizers on smooth homogeneous models. We show that \textit{momentum steepest descent} algorithms like Muon (spectral norm), MomentumGD ($\ell_2$ norm), and Signum ($\ell_\infty$ norm) are \textit{approximate} steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the $\ell_\infty$ margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.

Differential privacy provides a formal framework for releasing statistical estimators that limit how much any single observation can influence the output, by injecting calibrated random noise. We study differentially private estimation in two-stage procedures common in causal inference and individualized treatment rule (ITR) learning, in which data-dependent weights are first estimated to enforce covariate balance and a parameter of interest is then obtained by weighted empirical risk minimization. We propose Differentially Private Two-Stage Empirical Risk Minimization (DP-2ERM), which privatizes the final estimator directly through objective perturbation calibrated to the data-dependent sensitivity of the full pipeline. The analysis combines deterministic weight-perturbation bounds for several covariate-balancing methods (inverse propensity weighting, entropy balancing weighting, and maximum mean discrepancy weighting) with probabilistic sensitivity bounds for the second-stage solution. The resulting calibration is sharper than the natural stage-wise composition baseline, which the same sensitivity analysis supplies as a byproduct. Simulation studies and a benchmark application to ITR learning demonstrate the improved privacy--utility trade-off.

AI agents increasingly execute procedural workflows as sequential action traces, which obscures latent concurrency and induces repeated step-by-step reasoning. We introduce BPOP, a Bayesianframework that infers a latent dependency partial order from noisy linearized traces. BPOP models traces as stochastic linear extensions of an underlying graph and performs efficient MCMC inference via a tractable frontier-softmax likelihood that avoids #P-hard marginalization over linear extensions. We evaluate on our open-sourced Cloud-IaC-6, a suite of cloud provisioning tasks with heterogeneous LLM-generated traces, and WFCommons scientific workflows. BPOP recover dependency structure more accurately than trace-only and process-mining baselines, and the inferred graphs support a compiled executor that prunes irrelevant context, yielding substantial reductions in token usage and execution time.

We study online transfer reinforcement learning (RL) in episodic Markov decision processes, where experience from related source tasks is available during learning on a target task. A fundamental difficulty is that task similarity is typically defined in terms of rewards or transitions, whereas online RL algorithms operate on Bellman regression targets. As a result, naively reusing source Bellman updates introduces systematic bias and invalidates regret guarantees. We identify one-step Bellman alignment as the correct abstraction for transfer in online RL and propose re-weighted targeting (RWT), an operator-level correction that retargets continuation values and compensates for transition mismatch via a change of measure. RWT reduces task mismatch to a fixed one-step correction and enables statistically sound reuse of source data. This alignment yields a two-stage RWT $Q$-learning framework that separates variance reduction from bias correction. Under RKHS function approximation, we establish regret bounds that scale with the complexity of the task shift rather than the target MDP. We further show the required density ratios admit a constructive RKHS estimator with finite-sample guarantees, and empirically validate robustness to estimated and mis-specified ratios. Empirical results in both tabular and neural network settings demonstrate consistent improvements over single-task learning and naïve pooling, highlighting Bellman alignment as a model-agnostic transfer principle for online RL.

Biased gradient compression with error feedback (EF) reduces communication in federated learning (FL), but under non-IID data, the residual error can decay slowly, causing gradient mismatch and stalled progress in the early rounds. We propose step-ahead partial error feedback (SA-PEF), which integrates step-ahead (SA) correction with partial error feedback (PEF). SA-PEF recovers EF when the step-ahead coefficient $α=0$ and step-ahead EF (SAEF) when $α=1$. For non-convex objectives and $δ$-contractive compressors, we establish a second-moment bound and a residual recursion that guarantee convergence to stationarity under heterogeneous data and partial client participation. The resulting rates match standard non-convex Fed-SGD guarantees up to constant factors, achieving $O((η,η_0TR)^{-1})$ convergence to a variance/heterogeneity floor with a fixed inner step size. Our analysis reveals a step-ahead-controlled residual contraction $ρ_r$ that explains the observed acceleration in the early training phase. To balance SAEF's rapid warm-up with EF's long-term stability, we select $α$ near its theory-predicted optimum. Experiments across diverse architectures and datasets show that SA-PEF consistently reaches target accuracy faster than EF.

With grid operators confronting rising uncertainty from renewable integration and a broader push toward electrification, Demand-Side Management (DSM) -- particularly Demand Response (DR) -- has attracted significant attention as a cost-effective mechanism for balancing modern electricity systems. Unprecedented volumes of consumption data from a continuing global deployment of smart meters enable consumer segmentation based on real usage behaviours, promising to inform the design of more effective DSM and DR programs. However, existing clustering-based segmentation methods insufficiently reflect the behavioural diversity of consumers, often relying on rigid temporal alignment, and faltering in the presence of anomalies, missing data, or large-scale deployments. To address these challenges, we propose a novel two-stage clustering framework -- Clustered Representations Optimising Consumer Segmentation (CROCS). In the first stage, each consumer's daily load profiles are clustered independently to form a Representative Load Set (RLS), providing a compact summary of their typical diurnal consumption behaviours. In the second stage, consumers are clustered using the Weighted Sum of Minimum Distances (WSMD), a novel set-to-set measure that compares RLSs by accounting for both the prevalence and similarity of those behaviours. Finally, community detection on the WSMD-induced graph reveals higher-order prototypes that embody the shared diurnal behaviours defining consumer groups, enhancing the interpretability of the resulting clusters. Extensive experiments on both synthetic and real Australian smart meter datasets demonstrate that CROCS captures intra-consumer variability, uncovers both synchronous and asynchronous behavioural similarities, and remains robust to anomalies and missing data, while scaling efficiently through natural parallelisation. These results...

In case of sparse graphs, relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking transition probability matrix is considered with respect to vertex clustering. For this purpose, the random walk along the non-backtracking graph is considered, the vertices of which are the bioriented edges, and the adjacency relation depends on whether the random walk goes through the oriented edges with the rule of ``not going back in the next step''. This is encoded in the non-backtracking matrix that is the adjacency matrix of the non-backtracking graph. The structural real eigenvalues of the transition probability matrix are related to the constant multiples of the non-backtracking one, the concordance of which indicates the existence of a sparse stochastic block model behind the graph. ``Inflation--deflation'' techniques are also developed for clustering the vertices of the original graph together with real world applications.

Flow Matching (FM) has emerged as a powerful paradigm for continuous normalizing flows, yet standard FM implicitly performs an unweighted $L^2$ regression over the entire ambient space. In high dimensions, this leads to a fundamental inefficiency: the vast majority of the integration domain consists of low-density ``void'' regions where the target velocity fields are often chaotic or ill-defined. In this paper, we propose {$γ$-Flow Matching ($γ$-FM)}, a density-weighted variant that aligns the regression geometry with the underlying probability flow. While density weighting is desirable, naive implementations would require evaluating the intractable target density. We circumvent this by introducing a Dynamic Density-Weighting strategy that estimates the \emph{target} density directly from training particles. This approach allows us to dynamically downweight the regression loss in void regions without compromising the simulation-free nature of FM. Theoretically, we establish that $γ$-FM minimizes the transport cost on a statistical manifold endowed with the $γ$-Stein metric. Spectral analysis further suggests that this geometry induces an implicit Sobolev regularization, effectively damping high-frequency oscillations in void regions. Empirically, $γ$-FM significantly improves vector field smoothness and sampling efficiency on high-dimensional latent datasets, while demonstrating intrinsic robustness to outliers.

Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than nominal quantities. Under classical linear--Gaussian assumptions, we derive exact expressions for the first- and second-order moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimension-dependent contraction of the innovation covariance. The analysis is extended to NN association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum-norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation statistics cannot be preserved under nontrivial gating and association. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.

We develop Gaussian approximations for high-dimensional vectors formed by second-order $U$- and $V$-statistics whose kernels depend on sample size under independent but not identically distributed (i.n.i.d.) sampling. Our results hold irrespective of which component of the Hoeffding decomposition is dominant, thereby covering both non-degenerate and degenerate regimes as special cases. By allowing i.n.i.d.~sampling, the class of statistics we analyze includes weighted $U$- and $V$-statistics and two-sample $U$- and $V$-statistics as special cases, which cover estimators of parameters in regression models with many covariates, many-weak instruments as well as a broad class of smoothed two-sample tests and the separately exchangeable arrays, among others. In addition, we develop maximal inequalities for high-dimensional $U$-statistics with size-dependent kernels under the i.n.i.d.~setting, in a form that remains sharp across a broad range of applications, which may be of independent interest.

We introduce a density-power weighted variant for the Stein operator, called the $γ$-Stein operator. This is a novel class of operators derived from the $γ$-divergence, designed to build robust inference methods for unnormalized probability models. The operator's construction (weighting by the model density raised to a positive power $γ$ inherently down-weights the influence of outliers, providing a principled mechanism for robustness. Applying this operator yields a robust generalization of score matching that retains the crucial property of being independent of the model's normalizing constant. We extend this framework to develop two key applications: the $γ$-kernelized Stein discrepancy for robust goodness-of-fit testing, and $γ$-Stein variational gradient descent for robust Bayesian posterior approximation. Empirical results on contaminated Gaussian and quartic potential models show our methods significantly outperform standard baselines in both robustness and statistical efficiency.

Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding neural operators, enabling transferability analyses of graph neural networks (GNNs). This paper develops a unified spectral framework that brings together convergence results under different assumptions on the underlying graphon, including no regularity, global Lipschitz continuity, and piecewise-Lipschitz continuity. The framework places these results in a common operator setting, enabling direct comparison of their assumptions, convergence rates, and tradeoffs. We further illustrate the empirical tightness of these rates on synthetic and real-world graphs.

This paper introduces a robust and computationally efficient estimation framework for high-dimensional volatility models in the BEKK-ARCH class. The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings. Non-asymptotic error bounds are established for the resulting estimators under heavy-tailed regimes, and the minimax optimal convergence rate is derived. Moreover, a robust BIC and a Ridge-type estimator are introduced for selecting the model order and the number of BEKK components, respectively, with their selection consistency established under heavy-tailed settings. Simulation studies demonstrate finite-sample performance of the proposed method, and two empirical applications illustrate its practical utility. The results show that the new framework outperforms existing alternatives in both computational speed and forecasting accuracy.

Ensemble-based data assimilation (DA) methods have become increasingly popular due to their inherent ability to address nonlinear dynamic problems. However, these methods often face a trade-off between analysis accuracy and computational efficiency, as larger ensemble sizes required for higher accuracy also lead to greater computational cost. In this study, we propose a novel machine learning-based data assimilation approach that combines the traditional ensemble Kalman filter (EnKF) with a fully connected neural network (FCNN). Specifically, our method uses a relatively small ensemble size to generate preliminary yet suboptimal analysis states via EnKF. A FCNN is then employed to learn and predict correction terms for these states, thereby mitigating the performance degradation induced by the limited ensemble size. We evaluate the performance of our proposed EnKF-FCNN method through numerical experiments involving Lorenz systems and nonlinear ocean wave field simulations. The results consistently demonstrate that the new method achieves higher accuracy than traditional EnKF with the same ensemble size, while incurring negligible additional computational cost. Moreover, the EnKF-FCNN method is adaptable to diverse applications through coupling with different models and the use of alternative ensemble-based DA methods.

We numerically investigate the feasibility and limits of jointly estimating flow fields and unknown particle properties (e.g., position, size, and density) from Lagrangian particle tracking (LPT) data. LPT offers time-resolved, volumetric measurements of particle trajectories, which are markers of the carrier fluid motion. However, experimental tracks are spatially sparse and potentially noisy, and the problem of reconstructing flow fields may be further complicated by inertial particle transport, such that particle slip velocities must be determined to access the velocity field of the carrier fluid. To address this problem, we develop a data assimilation framework that couples an Eulerian representation of the flow with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. We show that flow fields and particle properties can be jointly estimated in three representative regimes: (1) In a turbulent boundary layer with noisy tracer tracks (St to 0), flow fields and true particle positions are jointly estimated, which amounts to a physics-informed particle tracking problem; (2) in homogeneous isotropic turbulence seeded with inertial particles (St ~ 1-5), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterization; and (3) in a compressible, shock-dominated flow, we report the first joint reconstructions of velocity, pressure, density, and inertial particle properties (diameter and density), highlighting both the potential and certain limits of joint estimation in supersonic regimes. A systematic sensitivity study reveals how the seeding density, noise level, and Stokes number govern reconstruction accuracy for our method.

How can we generate samples from a conditional distribution that we never fully observe? This question arises across a broad range of applications in both modern machine learning and classical statistics, including image post-processing in computer vision, approximate posterior sampling in simulation-based inference, and conditional distribution modeling in complex data settings. In such settings, compared with unconditional sampling, additional feature information can be leveraged to enable more adaptive and efficient sampling. Building on this, we introduce Conditional Generator using MMD (CGMMD), a novel framework for conditional sampling. Unlike many contemporary approaches, our method frames the training objective as a simple, adversary-free direct minimization problem. A key feature of CGMMD is its ability to produce conditional samples in a single forward pass of the generator, enabling practical one-shot sampling with low test-time complexity. We establish rigorous theoretical bounds on the loss incurred when sampling from the CGMMD sampler, and prove convergence of the estimated distribution to the true conditional distribution. In the process, we also develop a uniform concentration result for nearest-neighbor based functionals, which may be of independent interest. Finally, we show that CGMMD performs competitively on synthetic tasks involving complex conditional densities, as well as on practical applications such as image denoising and image super-resolution.

Emerging economies, particularly the MINT countries (Mexico, Indonesia, Nigeria, and Türkiye), are gaining influence in global stock markets, although they remain susceptible to the economic conditions of developed countries like the G7 (Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States). This interconnectedness and sensitivity of financial markets make understanding these relationships crucial for investors and policymakers to predict stock price movements accurately. To this end, we examined the main stock market indices of G7 and MINT countries from 2012 to 2024, using a recent graph neural network (GNN) algorithm called multivariate time series forecasting with graph neural network (MTGNN). This method allows for considering complex spatio-temporal connections in multivariate time series. In the implementations, MTGNN revealed that the US and Canada are the most influential G7 countries regarding stock indices in the forecasting process, and Indonesia and Türkiye are the most influential MINT countries. Additionally, our results showed that MTGNN outperformed traditional methods in forecasting the prices of stock market indices for MINT and G7 countries. Consequently, the study offers valuable insights into economic blocks' markets and presents a compelling empirical approach to analyzing global stock market dynamics using MTGNN.

The factor modeling for high-dimensional time series is powerful in discovering latent common components for dimension reduction and information extraction. Most available estimation methods can be divided into two categories: the covariance-based under asymptotically-identifiable assumption and the autocovariance-based with white idiosyncratic noise. This paper follows the autocovariance-based framework and develops a novel weight-calibrated method to improve the estimation performance. It adopts a linear projection to tackle high-dimensionality, and employs a reduced-rank autoregression formulation. The asymptotic theory of the proposed method is established, relaxing the assumption on white noise. Additionally, we make the first attempt in the literature by providing a systematic theoretical comparison among the covariance-based, the standard autocovariance-based, and our proposed weight-calibrated autocovariance-based methods in the presence of factors with different strengths. Extensive simulations are conducted to showcase the superior finite-sample performance of our proposed method, as well as to validate the newly established theory. The superiority of our proposal is further illustrated through the analysis of one financial and one macroeconomic data sets.

A new method of estimating population linear spectral statistics from high-dimensional data is introduced. When the dimension $d$ grows with the sample size $n$ such that $\frac{d}{n} \to c>0$, the proposed method is the first with proven convergence rate of $\mathcal{O}(n^{\varepsilon - 1})$ for any $\varepsilon > 0$ in a general nonparametric setting. For Gaussian data, a CLT for the estimation error with normalization factor $n$ is shown.

Understanding and certifying the generalization performance of machine learning algorithms -- i.e. obtaining theoretical estimates of the test error from the training error -- is a central theme of statistical learning theory. Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical VC-dimension theory. In this study, we formalize the generalization error bound by Rademacher complexity in Lean 4, building on measure-theoretic probability theory available in the Mathlib library. Our development provides a mechanically-checked pipeline from the definitions of empirical and expected Rademacher complexity, through a formal symmetrization argument and a bounded-differences analysis, to high-probability uniform deviation bounds via a formally proved McDiarmid inequality. A key technical contribution is a reusable mechanism for lifting results from countable hypothesis classes (where measurability of suprema is straightforward in Mathlib) to separable topological index sets via a reduction to a countable dense subset. As worked applications of the abstract theorem, we mechanize standard empirical Rademacher bounds for linear predictors under $\ell_2$ and $\ell_1$ regularizations, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.

In many image analysis problems, the contours of objects carry important statistical information about shape. Such contours are typically affected by deformation variables including scaling, translation, rotation, and reparametrization. Previous studies in statistical shape analysis have mainly focused on analyzing contours and shapes through discrete observations. While this approach might offer computational advantages, it overlooks the continuous nature of these objects and their underlying geometric structure. It also ignores potential dependencies between the deformation variables and their effect on the shape, which may result in a loss of statistical information and reduced interpretability. In this paper, we introduce a novel framework for analyzing shapes within the context of Functional Data Analysis (FDA). Basis expansion techniques are employed to derive analytic solutions for the estimation of deformation variables, namely scaling, translation, rotation, and reparametrization, thereby achieving curve alignment. A generative model for random contours is then developed using principal component analysis techniques. Numerical experiments on simulated data and the \textit{MPEG-7} database demonstrate that our method successfully identifies deformation parameters and captures the underlying distribution of random contours in settings where traditional FDA methods fail.

Traditional geostatistical methods assume independence between observation locations and the spatial process of interest. Violations of this independence assumption are referred to as preferential sampling (PS). Standard methods to address PS rely on estimating complex shared latent variable models and can be difficult to apply in practice. We study the use of inverse sampling intensity weighting (ISIW) for PS adjustment in model-based geostatistics. ISIW is a two-stage approach wherein we estimate the sampling intensity of the observation locations then define intensity-based weights within a weighted likelihood adjustment. Prediction follows by substituting the adjusted parameter estimates within kriging. We introduce an implementation of ISIW based on the Vecchia approximation, enabling computational gains while maintaining strong predictive accuracy. Interestingly, we found that ISIW outpredicts standard PS methods under misspecification of the sampling design, and that accurate parameter estimation had little correlation with predictive performance, raising questions about the conditions driving optimal implementation of kriging-based predictors under PS. Our work highlights the potential of ISIW to adjust for PS in an intuitive, fast, and effective manner. We illustrate these ideas on spatial prediction of lead concentrations measured through moss biomonitoring data in Galicia, Spain, and PM2.5 concentrations from the U.S. EPA Air Quality System network in California.

Constructing fast and accurate surrogate models is a key ingredient for making robust predictions in many topics. We introduce a new model, the Multiparameter Eigenvalue Problem (MEP) emulator. The new method connects emulators and can make predictions directly from observables to observables. We present that the MEP emulator can be trained with data from Eigenvector Continuation (EC) and Parametric Matrix Model (PMM) emulators. A simple simulation on a one-dimensional lattice confirms the performance of the MEP emulator. Using $^{28}$O as an example, we also demonstrate that the predictive probability distribution of the target observables can be easily obtained through the new emulator.

Time series in natural sciences, such as hydrology and climatology, and other environmental applications, often consist of continuous observations constrained to the unit interval (0,1). Traditional Gaussian-based models fail to capture these bounds, requiring more flexible approaches. This paper introduces the Matsuoka Autoregressive Moving Average (MARMA) model, extending the GARMA framework by assuming a Matsuoka-distributed random component taking values in (0,1) and an ARMA-like systematic structure allowing for random time-dependent covariates. Parameter estimation is performed via partial maximum likelihood (PMLE), for which we present the asymptotic theory. It enables statistical inference, including confidence intervals and model selection. To construct prediction intervals, we propose a novel bootstrap-based method that accounts for dependence structure uncertainty. A comprehensive Monte Carlo simulation study assesses the finite sample performance of the proposed methodologies, while an application to forecasting the useful water volume of the Guarapiranga Reservoir in Brazil showcases their practical usefulness.

We introduce a novel class of Bayesian mixtures for normal linear regression models which incorporates a further Gaussian random component for the distribution of the predictor variables. The proposed cluster-weighted model aims to encompass potential heterogeneity in the distribution of the response variable as well as in the multivariate distribution of the covariates for detecting signals relevant to the underlying latent structure. Of particular interest are potential signals originating from: (i) the linear predictor structures of the regression models and (ii) the covariance structures of the covariates. We model these two components using a lasso shrinkage prior for the regression coefficients and a graphical-lasso shrinkage prior for the covariance matrices. A fully Bayesian approach is followed for estimating the number of clusters, by treating the number of mixture components as random and implementing a trans-dimensional telescoping sampler. Alternative Bayesian approaches based on overfitting mixture models or using information criteria to select the number of components are also considered. The proposed method is compared against EM type implementation, mixtures of regressions and mixtures of experts. The method is illustrated using a set of simulation studies and a biomedical dataset.

We consider the scenario of supervised learning in Deep Learning (DL) networks, and exploit the arbitrariness of choice in the Riemannian metric relative to which the gradient descent flow can be defined (a general fact of differential geometry). In the standard approach to DL, the gradient flow on the space of parameters (weights and biases) is defined with respect to the Euclidean metric. Here instead, we choose the gradient flow with respect to the Euclidean metric in the output layer of the DL network. This naturally induces two modified versions of the gradient descent flow in the parameter space, one adapted for the overparametrized setting, and the other for the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the ${\mathcal L}^2$ cost to its global minimum at a uniform exponential convergence rate; one thereby obtains an a priori stopping time for any prescribed proximity to the global minimum. We point out relations of the latter to sub-Riemannian geometry. Moreover, we generalize the above framework to the situation in which the rank condition does not hold; in particular, we show that local equilibria can only exist if a rank loss occurs, and that generically, they are not isolated points, but elements of a critical submanifold of parameter space.

Recent work has proposed Wasserstein k-means (Wk-means) clustering as a powerful method to classify regimes in time series data, and one-dimensional asset returns in particular. In this paper, we begin by studying in detail the behaviour of the Wasserstein k-means clustering algorithm applied to synthetic one-dimensional time series data. We extend the previous work by studying, in detail, the dynamics of the clustering algorithm and how varying the hyperparameters impacts the performance over different random initialisations. We compute simple metrics that we find to be useful in identifying high-quality clusterings. We then extend the technique of Wasserstein k-means clustering to multidimensional time series data by approximating the multidimensional Wasserstein distance as a sliced Wasserstein distance, resulting in a method we call 'sliced Wasserstein k-means (sWk-means) clustering'. We apply the sWk-means clustering method to the problem of automated regime classification in multidimensional time series data, using synthetic data to demonstrate the validity and effectiveness of the approach. Finally, we show that the sWk-means method is able to identify distinct market regimes in real multidimensional financial time series, using publicly available foreign exchange spot rate data as a case study. We conclude with remarks about some limitations of our approach and potential complementary or alternative approaches.

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

We establish Chernoff-type bounds for the largest eigenvalue of sums of Hermitian random matrices generated by a time-inhomogeneous Markov chain. Our primary regime assumes a compact state space and contractivity of each Markov kernel in Wasserstein distance, i.e., positive Ollivier-Ricci curvature. This assumption is convenient to verify in inhomogeneous settings and is satisfied by several chains of practical interest, such as stochastic gradient descent on strongly convex smooth objectives. We also develop analogous bounds for noncompact state spaces under a notion of spectral gap for inhomogeneous chains introduced by Saloff-Coste and Zuniga. Finally, we illustrate the utility of our results through an analysis of the Elo rating system, a popular method for ranking players in sports analytics, under a dynamic version of the Bradley-Terry-Luce model.

Stochastic Dominance (SD) theory provides a rigorous framework for selecting superior assets tailored to the asset allocation needs of investors with varying risk preferences (i.e., risk-averse, risk-seeking, and risk-neutral). However, traditional stock clustering methods typically rely on geometric metrics such as Euclidean distance, which often fail to effectively capture the intrinsic risk dominance relationships among assets. To address this limitation, this paper proposes an innovative clustering analysis framework based on SD test statistics. Methodologically, this study deeply integrates SD theory with machine learning algorithms. Transcending the limitations of traditional reliance on geometric distance, we innovatively utilize test statistics from first-, second-, and third-order SD to construct a "Stochastic Dominance Coefficient Matrix." Building upon this matrix, we modify the classic K-means and Hierarchical Clustering algorithms. Specifically, we derive 12 distinct algorithm variants tailored to different orders of SD relationships. Simultaneously, we construct the SD-SC coefficient and the SD-DBI index as specialized validity indices to evaluate the clustering performance. Empirically, we analyze constituent stock data from a representative developed market (the US NASDAQ Index) and an emerging market (China's CSI 100 Index). The results verify the effectiveness and robustness of the proposed method. Furthermore, we apply the clustering results to the modification of the Single Index Model and the construction of Global Minimum Variance Portfolios (GMVP). The findings demonstrate that the proposed method effectively facilitates customized asset allocation for investors, holding significant theoretical value and practical implications.

Time series forecasting drives operational decisions in areas like finance, transportation, and energy. While supervised learning approaches achieve strong performance, they require domain-specific training, feature engineering, and ongoing maintenance. Large-scale foundation models have recently emerged as a zero-shot alternative, avoiding task-specific training much like LLMs. In this work, we evaluate foundation models against standard supervised approaches. Rather than focusing solely on aggregate accuracy, we analyze performance across four operational regimes: periodic human-centric systems, physically constrained processes, stochastic financial markets, and heterogeneous demand forecasting. Our results characterize optimal deployment areas. Foundation models perform well in domains with transferable periodic structures and are efficient for cold-start or long-tail scenarios. Conversely, supervised specialists maintain higher precision in systems governed by strict physical constraints. In financial domains, newer foundation models are rapidly closing the performance gap with supervised specialists. We further quantify trade-offs in inference latency, data drift adaptability, and deployment constraints. Finally, we propose a Complexity Router that assigns each series to the optimal model class using empirical features. We demonstrate that this selective routing achieves higher accuracy and significantly lower inference costs compared to deploying a universal foundation model, providing a practical framework for balancing generalization and efficiency.

Multicalibration extends classical calibration by requiring predictions to be unbiased over a rich collection of functions, encompassing both prediction slices and subpopulations. It has emerged as a powerful framework for fairness, robustness, and reliable prediction, yet the theoretical understanding of multicalibration boosting (MCBoost) remains fragmented and often relies on restrictive assumptions. In this work, we develop a unified and refined perspective on MCBoost that subsumes existing variants, including multiaccuracy, BatchGCP, and BatchMVP. We uncover several phenomena that provide new insights into its practical behavior: even highly accurate and flexible predictors can remain substantially miscalibrated; enforcing multicalibration introduces a calibration-risk trade-off; and early stopping plays a central role in controlling this trade-off. On the theoretical side, we establish a general framework for MCBoost under weaker and more realistic conditions. We show that the boosting iterates converge to a Bregman projection of the population-optimal predictor onto the cumulative span generated by the audit class, thereby explicitly characterizing the function space on which multicalibration is achieved. We further derive convergence rates under different smoothness assumptions, finite-sample guarantees, and principled stopping rules that ensure multicalibration at termination. Finally, we extend the theory of universal adaptability under covariate shift, providing more general transfer guarantees and clarifying when multicalibrated predictors generalize across domains. These results provide a more complete theoretical foundation and practical guidance for multicalibration boosting, positioning it as both a unifying framework and a reliable post-processing approach for modern predictive models.

The global push for electric vehicles (EVs) has sharply increased demand for critical minerals such as cobalt and lithium, creating a tension between rapid industrial growth and long-term sustainability. Extraction is concentrated in a few regions -- notably the Democratic Republic of Congo (DRC), Chile, and Argentina -- where it has produced serious socio-environmental harms, including ecosystem degradation, labour exploitation, and the displacement of Indigenous communities. In the DRC, cobalt mining is frequently linked to child labour and hazardous working conditions; in Chile, lithium extraction intensifies water scarcity and threatens local agriculture and biodiversity. Policy instruments such as the U.S. Inflation Reduction Act (IRA) seek to promote ethical sourcing, but an extraction-driven model continues to deepen global inequalities. This chapter examines the contested temporalities of the transition, in which the short-term economic incentives of extraction conflict with longer-term environmental and social goals. It argues for a place-based framework built on community-centred governance, sustainable mining practices, and circular-economy strategies, including recycling and material substitution, to align resource security with equity and ensure that the shift to EVs does not reproduce the injustices it aims to address.

We describe how the target trial framework can be used to plan and report analyses that attempt to answer causal questions by combining information from multiple, diverse sources. Such analyses may involve comparisons of treatments evaluated in different populations, for example when an index trial is combined with other data sources in external comparator analyses, or when extending causal inferences from a randomized trial to a new target population in generalizability and transportability analyses. When planning such analyses, the specification of the target trial supports the explicit definition of the target population with an associated sampling model. We propose this as an additional component for the target trial framework, especially relevant for analyses that combine information, because it influences the choice of eligibility criteria, the specification of the causal model, the choice of causal contrasts, and reasoning about identification strategies. Furthermore, the framework encourages careful mapping of data elements from multiple data sources to a single target trial. This mapping process can highlight potentially irreconcilable misalignments between data sources with respect to specific components of the framework -- for example, in the definitions of eligibility criteria, treatment assignment, and treatment receipt. Such misalignments can arise when attempts to specify a target trial that aligns with a specific data source introduce or worsen misalignments with other proposed data sources. The extent of such misalignments may warrant switching to other data sources, or prospectively obtaining data, to emulate the proposed target trial. We conclude that the target trial framework promotes transparent discussion about the design of and assumptions made in analyses that answer causal questions by combining information from diverse sources.

Context or prompt-level reweighting has emerged as a central algorithmic lever in Reinforcement Learning with Verified Rewards (RLVR) for improving the reasoning capability of large language models, yet the principle determining what constitutes an optimal weighting remains poorly understood. We address this gap by formulating prompt reweighting as a functional derivative of a utility functional defined in the pass-rate function space, yielding a unified optimality framework that accommodates existing schemes, including REINFORCE and GRPO. Building on this optimality framework, we propose a distribution-aware prompt reweighting approach, called CurveRL, based on a quantile coordinate transform, in which the weight assigned to each prompt depends not on the absolute value of pass rates but on its rank and density to reflect the distributional structure of the pass rates in the learning dynamics. Extensive experiments across multiple benchmarks demonstrate that our proposed CurveRL consistently outperforms GRPO and other RLVR baselines. Our study identifies context-distribution control as a principled axis for analyzing and designing prompt-reweighted RLVR algorithms. The code is released in https://github.com/zhyzmath/CurveRL.

We propose PACE-GGM, a data-adaptive differentially private method for covariance estimation that concentrates its privacy budget on the most informative entries of the empirical covariance matrix, rather than perturbing all entries. This applies in the natural setting where the modeler supplies separate bounds for each variable, so that individual entries can be measured with less noise than the full matrix. In each round, our method selects a poorly approximated entry, measures it using the Gaussian mechanism, and then reconstructs a full covariance matrix using a maximum-entropy reconstruction objective, leading to a Gaussian graphical model structure. Experiments on diverse real-world datasets demonstrate consistent improvements in estimation error with respect to the Gaussian mechanism and other baselines, particularly in high-dimensional and low-to-moderate privacy regimes.

This paper studies the joint role of long-memory dynamics,rough-volatility behavior, and persistence-based forecasting features in equity volatility modeling. We combine semiparametric long-memory estimation, rough-volatility diagnostics, and structured forecasting regressions to examine whether persistence measures contain economically meaningful forecasting information beyond conventional volatility predictors. Using a panel of 115 S&P500 constituents from November 2001 through April 2026, we document that volatility proxies exhibit substantial long-memory behavior and locally rough dynamics. The cross-sectional mean Geweke-Porter-Hudak estimate of the memory parameter is $\hat{d} = 0.226$, while the corresponding local-Whittle estimate is $\hat{d} = 0.440$, with statistical significance observed across nearly the entire panel. Rolling estimates of persistence rise substantially during the global financial crisis and the COVID period and display a positive contemporaneous association with the VIX. We then examine whether persistence-related features improve out-of-sample volatility forecasts beyond standard HAR and HAR-X benchmarks. Incorporating cross-sectional persistence aggregates, sectoral persistence measures, and persistence-by-stress interaction terms produces moderate but statistically significant forecasting improvements, particularly at longer horizons and during stress regimes. Forecast gains are strongest during periods of elevated market volatility and in volatility-managed portfolio applications. The results suggest that persistence measures may serve as useful reduced-form indicators of the duration and propagation of uncertainty in financial markets, although the paper does not claim structural identification of the economic mechanisms generating persistence.

This study presents the development and application of a scalable non-ergodic ground motion model (NGMM) for the Los Angeles area. The NGMM is trained and validated on physics-based simulated ground-motion data from a recent Statewide California Earthquake Center (SCEC) CyberShake study. The NGMM is formulated as a Gaussian Process (GP) regression model, where the prior median is defined as the ASK14 ergodic ground-motion model and the posterior median is obtained by learning the non-ergodic effects embedded in the training data. These non-ergodic effects include systematic site and path effects, which are represented in the GP using Matérn and specialized covariance kernels that explicitly characterize path vectors. Implementing the NGMM requires hyperparameter tuning and inference on large datasets (on the order of one million data points or more), posing significant computational challenges for conventional GP approaches. To address this scalability issue, this paper presents a suite of computational strategies, including sparse Cholesky inversion, parallel computing, GPU acceleration, and stochastic gradient descent minimization. Despite these advances, the full CyberShake dataset (on the order of hundreds of millions of data points) remains computationally prohibitive. Therefore, aleatory variability is modeled separately using a mixed-effects formulation to represent within-event and between-event variability. The developed NGMM has two primary applications: interpolation of partially observed ground-motion fields and predictive modeling for ground motions in unobserved earthquake scenarios. Validation results on independent datasets demonstrate accurate performance in both applications. A case study of power transmission network assessment in an Mw 6.7 Puente Hill scenario further demonstrated that the developed NGMM closely reproduces physics-based simulation results.

Input-convex neural networks (ICNNs) are widely used for log-concave density estimation, convex-potential normalizing flows, optimal transport, and transport-map inversion for high-dimensional Bayesian posteriors. These tasks share a structural constraint: the inter-layer weights of the ICNN must remain non-negative. The standard recipe, projected gradient descent (PGD) onto the non-negative cone, applies a hard, non-smooth projection -- the stiff-penalty limit of an ADMM-style constraint splitting -- and its classical convergence guarantees do not transfer to the non-smooth ICNN training landscape; the differentiable alternative, softplus reparametrization, attenuates the gradient exponentially in the weight magnitude, stalling training with dead inter-layer weights and plateaued loss. Inspired by parameter-extension lifts of PDE-constrained inverse problems, we propose the lift: instead of constraining the inter-layer weights directly, we train an unconstrained hypernetwork that emits them from a permutation-invariant summary of the input batch. This adds stochasticity to the training dynamics that softens the loss landscape, letting the iterates escape the gradient-attenuated region where direct softplus stalls. We trace this softening to three structural ingredients -- a learnable bias acting as slack, a hypernetwork body that conditions on the target batch, and a cross-covariance coupling the two through batch stochasticity -- and prove each one necessary: deleting any single ingredient collapses the cross-covariance that carries the softening. On log-concave energy-based modeling from one-dimensional toy targets to image-flavored latents, and convex-potential normalizing flows on a 21-dimensional tabular benchmark, we show that the lift reaches a lower test loss than both PGD and direct softplus, and turns a plateau-bounded training trajectory into a valley-descending one.

In 3D scene understanding, deep learning models rely on large models and extensive training to capture basic geometric structures that are present in the 3D data. However, existing methods lack explicit mechanisms to incorporate geometric information, such as learnable primitive shapes, often necessitating large models and more training data which in turn increases cost and can limit generalization. We introduce GIBLy, a lightweight geometric inductive bias layer that integrates learnable geometric priors into 3D segmentation pipelines. GIBLy enhances existing architectures -- whether MLP-based, convolution-based, or transformer-based -- by providing features aligned with simple geometric shapes (and thus human-interpretable) that improve segmentation performance with minimal computational overhead. We validate our approach across multiple 3D semantic segmentation benchmarks, demonstrating consistent performance gains, including up to +11.5% mIoU on TS40K with PTV3, while adding only 58K extra parameters. Our results highlight the benefit of explicitly encoding geometric structure to support accurate and efficient 3D scene understanding, with a lightweight add-on layer

Deploying clinical prediction models across healthcare systems often fails when key training covariates are unavailable at deployment and labeled outcomes are limited in the target domain. For example, high-performing models for out-of-hospital cardiac arrest (OHCA) rely on detailed prehospital measurements routinely collected in high-resource settings but unavailable in many international registries. Existing methods either discard missing covariates, sacrificing predictive information, or rely on untestable assumptions about their target distribution. We propose DRUM (\underline{D}istributionally \underline{R}obust \underline{U}nsupervised transfer learning with structurally \underline{M}issing covariates), a framework that transfers prediction models to target populations where certain covariates are structurally absent and outcome labels are unavailable. DRUM partitions covariates into shared components ($X$), observed across all settings, and missing components ($A$), observed only in the source. Rather than imputing missing covariates, DRUM optimizes worst-case predictive performance over the unknown target distribution of $A \mid X$ using a neural network generator, with a robustness parameter controlling allowable deviation from the source conditional. We further develop a bias correction procedure that reduces sensitivity to nuisance estimation error. Simulations show substantial improvements in both mean and worst-case prediction error under distribution shift. Applied to cross-national OHCA prediction, transferring models from a US registry to multiple Asian registries where prehospital variables are unrecorded, DRUM yields better-calibrated predictions and improved clinical classification performance across sites.

What functions can Neural Processes represent? We analyze the representational capacity of popular NP architectures: Conditional Neural Processes (CNPs), Attentive Neural Processes (ANPs), Transformer Neural Processes (TNPs), and their latent variants. We prove these architectures form a strict hierarchy. CNP-representable functions are exactly those depending on finitely many expected features of the context distribution. ANPs strictly generalize CNPs via query-dependent reweighting, enabling kernel smoothers. ConvCNPs and ANPs are incomparable; each contains functions outside the other, separated by stationarity versus translation equivariance. TNPs with $L$ self-attention layers capture $L$-hop context interactions. For latent NPs, we show finite-dimensional latents provide coherent sampling but do not circumvent encoder limitations; matching GP posterior distributions requires latent dimension scaling with context size. These results provide a theoretical foundation for architecture selection based on task structure.

This article addresses issues of model criticism and model comparison in Bayesian contexts, and focusses on the use of the so-called posterior predictive p-values (ppp values). These involve a general discrepancy or conflict measure and depend on the prior, the model, and the data. They are used in statistical practice to quantify the degree of surprise or conflict in data, and for purposes of comparing different combinations of prior and model. The distribution of such ppp values is however far from uniform, as we demonstrate for different models, making their interpretation and comparison a difficult matter. We propose a natural calibration of the ppp values, where the resulting cppp values are uniform on the unit interval under model conditions. The cppp values, which in general rely on a double simulation scheme for their computation, may then be used to assess and compare different priors and models. Our methods also make it possible to compare parametric with nonparametric model specifications, in that genuine `measures of surprise' are put on the same canonical uniform scale. Our techniques are illustrated for some applications to real data. We also present supplementing theoretical results on various properties of the ppp and cppp.

Longitudinal modified treatment policies (LMTP) are a class of interventions that allow the definition, identification, and estimation of causal effects in general settings, such as with continuous or multivariate exposures, treatment regimens that require grace periods. Targeted machine learning estimators (i.e., double/debiased) have been formulated for LMTPs that assign the exposure at time $t$ as a function of the natural value of treatment at time $t$. However, important applications such as estimating the effect of a delay in the start of a treatment require formulating LMTPs that depend not only on the natural value of treatment at time $t$ but also on the \textit{history} of the natural value of treatment prior to time $t$. This paper develops targeted learning estimators for this general case. We discuss the definition of the effects, and propose estimators that use an augmented-data version of the sequential regression form of the longitudinal g-computation formula. Our estimators are based on the efficient influence function and provide $\sqrt{n}$ inference under standard doubly robust rate assumptions on the convergence of the outcome and treatment regressions. We apply the new estimators to assess the effect of delaying a risky pain treatment by one month on 12-month incidence of opioid use disorder.

We study the generalized dynamic factor model in a long-memory setting. Unlike most recent work, which assumes a finite-dimensional factor space and short memory, our framework allows the factor space to be infinite-dimensional and the common components to exhibit long memory. We employ the two-sided estimation method of Forni, Hallin, Lippi and Reichlin (2000, Review of Economics and Statistics) to recover the common component. The long memory structure of the common component poses a challenge, as it introduces unboundedness/discontinuity in the spectral density. We address this issue by leveraging two key facts: First, the estimated operator is a projection onto the leading eigenspace and thus the eigengap provides an intrinsic scaling that partially mitigates the blow-up. Second, we perform most of our estimation in $L^p$-norm, rather than pointwise. Experimental results are presented to provide evidence supporting the theory, as well as potential improvements to it.

We study the problem of identifying dynamically distinct basins of attraction in high dimensional time-homogeneous Markov processes using only trajectory sampling. This problem is fundamental in the analysis of metastable dynamical systems, where the process rapidly mixes within basins while transitions between basins occur rarely on the timescale of interest, or even when the state space is reducible. Existing approaches typically rely on spatial discretization or spectral analysis of estimated transition operators, which can become unreliable in high dimensional settings or when the underlying basin geometry is highly nonlinear. We propose a discriminative approach to basin identification based on marginal trajectory distribution comparison. We prove a simple risk separation result: if two initial states belong to the same basin, the Bayes-optimal classifier distinguishing their marginal trajectory distributions achieves risk close to 1/2, whereas if they lie in distinct basins, the optimal risk is close to zero. This observation reduces basin detection to a two-sample discrimination problem between marginal trajectory distributions. Motivated by this principle, we develop a neural algorithm that receives a set of candidate basin representatives and iteratively merges them by estimating classification risk with a neural network that approximates the Bayes classifier. We evaluate the method on various metastable systems. These include synthetic systems constructed by embedding low-dimensional dynamics into high dimensional noisy ambient spaces. In these settings, standard spectral and clustering-based methods often fail, while our approach accurately recovers the underlying basin structure. These results display a shortcoming of existing methods and highlight trajectory discrimination as an effective tool for identifying dynamical basins in high dimensional stochastic systems.

Heritability is a central concept in the long-standing debate about nature versus nurture in biological and social sciences. However, existing notions of heritability are based on strong assumptions and do not use explicit causal models. We propose a new, counterfactual definition of heritability by adopting the potential outcomes model in causal inference. Our counterfactual heritability measures the importance of genetic inheritance by the average magnitude of difference between an individual with their hypothetical ``non-identical twin'' that is exposed to the exact same environment. We provide bounds on the counterfactual heritability that can, in principle, be computed from observational data. We then compare counterfactual heritability and its associated bounds with common notions of heritability in population-based studies, twin and sibling studies, and plant breeding experiments. Our results and comparisons highlight the importance of clarifying the causal structural assumptions and counterfactual comparisons in reasoning about heritability.

The regression of principal component scores (RPCS) on covariates is a widely used analytic approach to detect and test for associations between functional measurements and study participant characteristics. Here we show that: (1) RPCS loses power relative to Function on Scalar Regression (FoSR); (2) the amount of power loss depends on the correlation between the PCs and the true effect; (3) if not corrected for multiplicity, RPCS has inflated $α$-level; and (4) current RPCS methods do not provide valid inference for the true effect. In contrast, we show that Function on Scalar Regression (FoSR) can avoid these problems using a particular combination of modeling tools. We validate these theoretical findings through extensive simulations and illustrate their practical implications using minute-level accelerometry data from the National Health and Nutrition Examination Survey (NHANES).

Classical archetypal analysis is appealing for its interpretability, but its linear geometry can limit performance on data with strongly non-linear structure; at the same time, existing neural extensions improve flexibility while often weakening the geometric meaning of archetypes and interpolations. In this work, we develop a Riemannian version of archetypal analysis based on data-driven pullback geometry for real-valued data, with the goal of combining the interpretability of classical archetypal analysis with the expressive power of modern non-linear models. We introduce a class of deformed star distributions together with associated pullback Riemannian geometry to provide a statistical interpretation of the resulting manifold mappings, define the Riemannian archetypal mapping (RAM) as a projection onto the manifold of geodesically convex combinations of archetypes, and propose a practical optimization scheme based on convex relaxation followed by non-convex refinement. We further propose a learning scheme that yields reasonable, albeit generally suboptimal, deformed star distributions from data. Experiments on synthetic examples and MNIST show that the resulting framework produces meaningful geodesics, useful denoising projections, and geometry-aware classifications, while also clarifying where current optimization limitations remain.

Modern Artificial Intelligence achieves remarkable predictive power by optimizing statistical risk functionals over vast corpora. Yet a gap separates this from genuine intelligence: the inability to distinguish correlation from causation. This paper argues that causal inference (identifying mechanisms invariant under intervention) is AI's indispensable statistical conscience. Without causal grounding, AI systems are correlation machines: powerful in familiar domains, brittle under distribution shift, and biased in high-stakes settings. Three contributions develop this argument. First, a Statistical Necessity Theorem for Causal Generalization: any algorithm achieving out-of-distribution generalization must encode causal structure, formalizing the distinction between prediction P(Y|X) and intelligence P(Y|do(X)). Second, a unified framework connects Pearl's do-calculus, the Potential Outcomes framework, Double Machine Learning, and Invariant Risk Minimization as a family of Causal Statistical Estimators, each identifying interventional distributions under different assumptions. Third, three AI failure modes (hallucination in large language models, reward hacking in reinforcement learning from human feedback, and degradation under distribution shift) are manifestations of causal blindness, each admitting a principled statistical remedy. Trustworthy AI is, at its core, a problem of causal statistics. The statistical community is not merely equipped to solve it -- it is the only community with the foundational tools to do so rigorously.

Finite-width fully connected neural networks with Gaussian-initialized weights deviate from their infinite-width Gaussian limit, exhibiting non-vanishing higher-order cumulants. We approximate these deviations, for a neural network evaluated in a finite number of inputs, using multidimensional Edgeworth expansions of arbitrary order $4m-1$, with $m\in\mathbb{N}$. Assuming that the corresponding Gaussian limit has an invertible covariance matrix and that the activation function is polynomially bounded, we establish a bound of order $n^{-m}$ on the total variation distance between the law of the true network output and its Edgeworth approximation, with matching lower bounds. As an application, we quantify the error in Bayesian posterior distributions when the prior is replaced by its Edgeworth expansion. Our results are more general and also apply to sequences of conditionally Gaussian vectors converging to a Gaussian vector with invertible covariance.

We propose a novel kinetic Langevin sampler based on a specific splitting scheme using the exact harmonic Langevin integrator. For strongly log-concave target measures, the sampler exploits a decomposition of the strongly convex potential into a quadratic part and a convex perturbation with Lipschitz continuous gradient. For the resulting first- and second-order schemes associated with this splitting we establish convergence rates in $L^2$-Wasserstein distance as well as non-asymptotic error bounds. In particular, the contraction rate is of the same order as that of the underlying continuous dynamics. To achieve $\varepsilon$-accuracy, the required step size for the second-order scheme is comparable to that of established splitting schemes such as OBABO or UBU, which are widely used in machine learning and molecular dynamics.

Regularized Adjusted Plus-Minus (RAPM) is the standard framework for estimating individual player impact in basketball. Its application requires possession-level stint data -- records of which five players shared the court for each contiguous sequence of possessions -- a form of data the NBA did not systematically record until the late 1990s. This paper describes the construction, methodology, and validation of the first possession-level player impact database for the pre-play-by-play NBA era, covering the regular seasons from 1984--85 through 1995--96, spanning twelve published seasons. As of this writing, 2,179 regular-season games have been reconstructed across twelve published seasons, comprising 435,760 total logged possessions and 1,012 distinct player-seasons. Every game was manually reconstructed from broadcast video: lineup changes were logged at every dead-ball substitution, possessions were tallied directly from footage, and points scored by each lineup were recorded. RAPM is estimated via weighted ridge regression applied to the reconstructed stint data, using the identical mathematical framework applied to modern play-by-play records. We provide a rigorous treatment of the reconstruction protocol, the formal properties of the estimation procedure, uncertainty quantification through posterior credible intervals, a multi-criterion validation framework, and an analysis of sampling properties at partial coverage. The resulting database is the only possession-level individual impact record for this era and provides a foundation for historical analysis that has until now been technically inaccessible.

Skill libraries allow LLM agents to load task-specific instructions on demand, letting non-expert users solve domain-specific tasks through natural language without knowing which skills exist or how they work. However, performance degrades as libraries grow -- by up to 21\% when scaling from a small set of helpful skills to a 202-skill library. In this work, we formulate this performance degradation as the pass rate drop between loading a library of known-helpful skills and the full library. Moreover, we propose to decompose the pass rate drop by conditioning on the skill(s) invocation -- which skills the agent selects during a trajectory -- into two effects: \emph{skill shadowing}, where the agent selects wrong skills more often as the library expands, and \emph{context overhead}, where the enlarged context degrades execution even when selection is correct. We derive upper bounds on both effects to characterize their magnitudes of impacts to the pass rate drop. Our empirical estimates of the effects and their upper bounds both show that the \emph{skill shadowing} effect grows with library size and significantly contributes to the performance degradation, whereas the \emph{context overhead} effect remains small and indistinguishable from zero. This observed asymmetry establishes that the skill selection failure, not the enlarged context, is the primary bottleneck when expanding the skill libraries.

Remote sensing techniques have been increasingly utilised in aquatic applications in recent years. A common challenge in using optical satellite data is the presence of missing observations due to cloud cover. These data gaps can lead to missed detection of critical events, such as algal blooms, in lakes of high interest to water authorities. As a result, enhancing the completeness of optical satellite datasets is crucial for improving the monitoring and prediction of algal blooms. In this study, we compared a traditional data imputation method (i.e., linear interpolation) with deep learning models for reconstructing missing spectral bands across four lakes with historical records of algal blooms. The deep learning models adopted include CNN-based architectures (i.e., CNN, Inception Resnet, and Autoencoder) and CNN-LSTM-based architectures (i.e., CNN-LSTM, Resnet-LSTM, and Autoencoder-LSTM). Our results demonstrated that deep learning models substantially outperformed the baseline linear interpolation method in imputing spectral band values within artificially masked regions. Among these models, CNN delivered the best performance across most lakes. Furthermore, we evaluated the performance of algal bloom indices (i.e., Green/Red and NDCI) derived from the imputed imagery by comparing them with the observed data. Our results demonstrate that deep learning models are effective for imputing missing data in PlanetScope SuperDove imagery, enabling more reliable applications in water monitoring.

Robustness, domain adaptation, photometric/occlusion invariance, sensor drift, and alignment style are treated as separate literatures with separate method families. Under label-preserving deployment shift they share one geometric object: the covariance Sigma_task = Cov_{Q_n}(n) of ways inputs can change without changing the label. CORAL, adversarial training, augmentation, metric learning, Jacobian penalties, and alignment constraints are not independent tricks--they are estimators of Sigma_task. Fix that object and the Jacobian penalty is pinned by a matrix Sigma' whose range must cover range(Sigma_task)--the matching principle. We prove optimality in a linear-Gaussian model (Thm. A), necessity of range coverage for any quadratic penalty that zeros deployment drift (Thm. G), and the same dichotomy at global minima (Thm. A*_global). Wrong-direction/signal-aligned controls (Lemma C; Cor. E/E*) and seven estimators (Lemmas D1--D7), plus label-free TDI, yield a falsifiable recipe when Sigma_task must be learned. Thirteen blocks (ML through Qwen2.5-7B) test matched vs isotropic vs wrong-direction penalties on geometry and deployment drift. Twelve match theory where identifiability holds; Office-31 is a named eigengap failure. Partial passes: geometry can improve without every headline task metric moving. A pilot 7B DPO run (one epoch, 240 pairs): matched style-PMH preserves Style TDI where standard DPO degrades it. We do not claim standard training reaches global minima (assumption (O) is open), that estimated Sigma_task is always identifiable, or dominance on every leaderboard. We claim a falsifiable design recipe: estimate Sigma_task, match Sigma', run the controls, report task and geometry separately.

Estimating heterogeneous treatment effects with machine learning has attracted substantial attention in both academic research and industrial practice. However, the two communities often evaluate models under markedly different conditions. Methodological work typically relies on semi-simulated benchmarks and metrics that require counterfactual outcomes, whereas real-world applications rely on observable metrics based on ranking or test outcomes. Despite the well-known gap between methodological progress and practical deployment, the relationship between these evaluation regimes has not been examined systematically. We conduct a large-scale empirical study of treatment effect evaluation across standard semi-simulated benchmark families and real-world datasets. Our benchmark covers meta-learners paired with multiple base learners, as well as specialized causal machine learning models. We evaluate these methods using observable metrics common in application-oriented literature, alongside counterfactual metrics commonly used in methods papers. Our results reveal two complementary gaps. First, counterfactual metrics do not reliably recover the estimators preferred by observable metrics, even on the same semi-simulated benchmarks. Second, rankings obtained on semi-simulated benchmarks do not transfer to real datasets. We further find that simple meta-learners with strong base models are consistently competitive, in contrast to specialized causal models. Overall, our findings suggest that progress in treatment effect estimation research should not be assessed solely through counterfactual metrics and semi-simulated benchmarks, but it would benefit from incorporating observable metrics and real-data validation.

It is well known that, under standard regularity conditions, the maximum likelihood estimator (MLE) satisfies a central limit theorem and converges in distribution to a Gaussian random variable as the sample size grows. This paper strengthens this classical result by developing several stronger forms of asymptotic normality for the normalized MLE. With additional assumptions on the score, we first establish sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. We then prove an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. When the Fisher information of the normalized estimate is bounded, or its density has bounded first derivative, we further show that the smoothing can be removed, yielding entropic normality of the MLE itself. The proofs develop auxiliary tools that may be of independent interest, including exponential consistency bounds, high-moment estimates, and entropy-control arguments for the estimator.

History-dependent sampling can reduce long-run Monte Carlo variance by discouraging redundant revisits, but existing schemes typically encode history through empirical measure on finite state spaces, which is infeasible in high-dimensional discrete configuration spaces or ill-posed in continuous domains. We propose Score-Repellent Monte Carlo (SRMC) framework that summarizes trajectory history by a running average of score evaluations in $\mathbb{R}^d$, where $d$ is the dimension of the score and state representation. This history is converted into a surrogate target through an exponential score tilt, indexed with $α$ that represents the strength of repellence in controlling the magnitude of the history-based repulsion. The surrogate family is normalization-free in the standard MCMC sense, yielding a generic wrapper: at each iteration, any base kernel targeting $π$ can instead be run on the current surrogate $π_{θ_n}$ while the history is updated online. We analyze the coupled evolution of the history recursion and Monte Carlo estimators using stochastic approximation with controlled Markovian noise, establishing almost sure convergence and a joint central limit theorem. We further identify regimes in which the asymptotic covariance decreases as $α$ increases, with scaling $O(1/α)$, extending the near-zero-variance effect of finite-state history-dependent samplers to general state spaces with constant memory. Experiments on continuous targets and discrete energy-based models demonstrate improved estimator variance and mode coverage, while retaining $O(d)$ memory usage and modest per-iteration overhead.

Semiparametric estimators admitting a von Mises expansion often reduce inference to the influence-function variance. This reduction is justified when the second-order remainder is negligible in variance, a condition that is stronger than the usual product-rate requirement guaranteeing classical asymptotic linearity. When the remainder contributes non-negligible variance, the standard sandwich can underestimate the total sampling variance and Wald intervals can undercover; we call this the \emph{near-boundary regime}. We derive a finite-sample variance decomposition separating influence-function and remainder components, give a practical characterization of when sandwich variance can fail, and show that the leave-one-out jackknife and pairs cluster bootstrap can estimate the total variance under explicit regularity conditions. For the jackknife, consistency follows from a self-normalization argument; for the bootstrap, we work under a Mallows-2 consistency condition. An analytic expression for the amplification of the sandwich gap by intra-cluster correlation is derived for clustered data. A simulation study using a surrogate-assisted targeted learning estimator in stepped-wedge cluster-randomized trials illustrates the regime: the variance ratio $\hat{V}_{\rm JK}/\hat{V}_{\rm Sand}$ is 1.14--1.38 and persistent across cluster counts, and the refined procedures substantially improve coverage.

In this paper, we study the problem of testing whether or not a given probability measure $μ$ on $\mathbb{R}^{d}$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We call this the problem of testing the mixture model hypothesis. To tackle it, we introduce a new set of computable orthogonal invariants of $μ$, namely, the eigenvalues of the 4th moment operator $T_μ$ associated with the measure. We prove that the largest eigenvalue is always an outlier eigenvalue. Further, we show how the first and second largest eigenvalues of $T_μ$ give nonasymptotic bounds for this problem and give a complete resolution of the asymptotic version of the problem under the $L^{8}$-$L^{2}$ equivalence assumption.

We study inference for linear quantile regression with two-way clustered data. Using a separately exchangeable array framework and a projection decomposition of the quantile score, we characterize regime-dependent convergence rates and establish a self-normalized Gaussian approximation. We propose a two-way cluster-robust sandwich variance estimator with a kernel-based density ``bread'' and a projection-matched ``meat'', and prove consistency and validity of inference in Gaussian regimes. We also show an impossibility result for uniform inference in a non-Gaussian interaction regime.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

Interval-censoring frequently occurs in studies of chronic diseases where disease status is inferred from intermittently collected biomarkers. Although many methods have been developed to analyze such data, they typically assume perfect disease diagnosis, which often does not hold in practice due to the inherent imperfect clinical diagnosis of cognitive functions or measurement errors of biomarkers such as cerebrospinal fluid. In this work, we introduce a semiparametric modeling framework using the Cox proportional hazards model to address interval-censored data in the presence of inaccurate disease diagnosis. Our model incorporates sensitivity and specificity of the diagnosis to account for uncertainty in whether the interval truly contains the disease onset. Furthermore, the framework accommodates scenarios involving a terminal event and when diagnosis is accurate, such as through postmortem analysis. We propose a nonparametric maximum likelihood estimation method for inference and develop an efficient EM algorithm to ensure computational feasibility. The regression coefficient estimators are shown to be asymptotically normal, achieving semiparametric efficiency bounds. We further validate our approach through extensive simulation studies and an application assessing Alzheimer's disease (AD) risk. We find that amyloid-beta is significantly associated with AD, but Tau is predictive of both AD and mortality.

The Finite State Projection (FSP) method approximates the Chemical Master Equation (CME) by restricting the dynamics to a finite subset of the (typically infinite) state space, enabling direct numerical solution with computable error bounds. Adaptive variants update this subset in time, but multiscale systems with widely separated reaction rates remain challenging, as low-probability bottleneck states can carry essential probability flux and the dynamics alternate between fast transients and slowly evolving stiff regimes. We propose a flux-based adaptive FSP method that uses probability flux to drive both state-space pruning and time-step selection. The pruning rule protects low-probability states with large outgoing flux, preserving connectivity in bottleneck systems, while the time-step rule adapts to the instantaneous total flux to handle rate constants spanning several orders of magnitude. Numerical experiments on stiff, oscillatory, and bottleneck reaction networks show that the method maintains accuracy while using substantially smaller state spaces.

Understanding how habitats shape species distributions and abundances across river networks remains a longstanding and fundamental challenge in ecology, with direct implications for effective biodiversity management and conservation. We introduce a scalable spatial stream network (S3N) model that enables estimation, inference, and prediction with greater computational efficiency than previously possible. S3Ns extend nearest-neighbor Gaussian processes (NNGPs) to include ecologically salient stream network dependence structure. Additionally, S3Ns implement more efficient preprocessing than SSNs; while the computational cost of estimation is a function of the number of observation points and not of the number of reaches, the preprocessing is a function of both. We demonstrate that S3Ns accurately recover spatial and covariance parameters 2-3 orders of magnitude faster than existing spatial stream network models. We then apply S3Ns to estimate the population sizes and geographic distributions of 285 fish species in the entire Ohio River Basin (>4,000 river km, approximately 170,000 reaches and 9,000 observation points) on a laptop. These results indicate the promise of S3Ns for mapping freshwater variables and quantifying the influence of environmental drivers across extensive, complex river networks with many observation points.

Inferring the biophysical parameters of conductance-based models (CBMs) from experimentally accessible recordings remains a central challenge in computational neuroscience. Spike times are the most widely available data, yet they reveal little about which combinations of ion channel conductances generate the observed activity. This inverse problem is further complicated by neuronal degeneracy, where multiple distinct conductance sets yield similar spiking patterns. We introduce a method that addresses this challenge by combining deep learning with Dynamic Input Conductances (DICs), a theoretical framework that reduces complex CBMs to three interpretable feedback components governing excitability and firing patterns. Our approach first maps spike times to DIC densities at threshold using a neural network that learns a low-dimensional representation of neuronal activity. The predicted DIC values are then used to generate degenerate CBM populations via an iterative compensation algorithm, ensuring compatibility with the intermediate target DICs, and thereby reproducing the corresponding firing patterns, even in high-dimensional models. Applied to two models, this algorithmic pipeline reconstructs spiking and bursting regimes with high accuracy and robustness to variability, including spike trains generated under noisy current injection mimicking physiological stochasticity. It produces diverse degenerate populations within milliseconds on standard hardware, enabling scalable and efficient inference from spike recordings alone. Together, this work positions DICs as a practical and interpretable link between experimentally observed activity and mechanistic models. By enabling fast and scalable reconstruction of degenerate populations directly from spike times, our approach provides a powerful way to investigate how neurons exploit conductance variability to achieve reliable computation.

We demonstrate that learning procedures that rely on aggregated labels, e.g., label information distilled from noisy responses, enjoy robustness properties impossible without data cleaning. This robustness appears in several ways. In the context of risk consistency -- when one takes the standard approach in machine learning of minimizing a surrogate (typically convex) loss in place of a desired task loss (such as the zero-one mis-classification error) -- procedures using label aggregation obtain stronger consistency guarantees than those even possible using raw labels. And while classical statistical scenarios of fitting perfectly-specified models suggest that incorporating all possible information -- modeling uncertainty in labels -- is statistically efficient, consistency fails for ``standard'' approaches as soon as a loss to be minimized is even slightly mis-specified. Yet procedures leveraging aggregated information still converge to optimal classifiers, highlighting how incorporating a fuller view of the data analysis pipeline, from collection to model-fitting to prediction time, can yield a more robust methodology by refining noisy signals.

Distributed stochastic optimization intertwines (i) stochastic gradient noise, (ii) communication compression, and (iii) adaptive/normalized updates. While each factor has been studied in isolation, their joint effect under realistic assumptions remains poorly understood. In this work, we develop a unified theoretical framework for Distributed Compressed SGD (DCSGD) and its sign variant Distributed SignSGD (DSignSGD) under the recently introduced $(L_0, L_1)$-smoothness condition. From a conceptual perspective, we show that the first- and second-order modified equations from the literature do not accurately model the discrete-time step-size/stability restrictions, especially under $(L_0,L_1)$-smoothness. From a technical perspective, we propose new first-order SDEs by carefully incorporating curvature-dependent terms into their drift: This helps capture the fine-grained relationship between learning rate restrictions, gradient noise, compression, and the geometry of the loss landscape. Importantly, we do so under general gradient noise assumptions, including heavy-tailed and affine-variance regimes, which extend beyond the classical bounded-variance setting. Our results suggest that normalizing the updates of DCSGD emerges as a natural condition for stability, with the degree of normalization precisely determined by the gradient noise structure, the landscape's regularity, and the compression rate. In contrast, DSignSGD converges even under heavy-tailed noise with standard learning rate schedules. Together, these findings offer both new theoretical insights and perspectives, and practical guidance.

Adaptive experiments use preliminary analyses of the data to inform further course of action and are commonly used in many disciplines including medical and social sciences. Because the null hypothesis and experimental design are data-dependent, it has long been recognized that statistical inference for adaptive experiments is not straightforward. Most existing methods only apply to specific adaptive designs and rely on strong assumptions. In this work, we propose selective randomization inference as a general framework for analysing adaptive experiments. In a nutshell, our approach applies conditional post-selection inference to randomization tests. By using directed acyclic graphs to describe the data generating process, we derive a selective randomization p-value that controls the selective type-I error. As inference only relies on the randomness in the treatment assignment, no modelling assumptions or independent and identically distributed data are needed. We elaborate on conditions that render the proposed p-value computable and provide rejection sampling and MCMC algorithms to find a Monte Carlo approximation. Moreover, this article shows how to estimate and construct confidence intervals for a homogeneous treatment effect. Lastly, we demonstrate our method and compare it with other randomization tests using synthetic and real-world data.

In randomized controlled trials (RCTs) that focus on time-to-event outcomes, intercurrent events can arise in two ways: as semi-competing events, which modify the hazard of the primary outcome events, or as competing events, which make the definition of the primary outcome events unclear. Although five strategies have been proposed in the ICH E9 (R1) addendum to address intercurrent events in RCTs, these strategies are not easily applicable to time-to-event outcomes when aiming for causal interpretations. In this study, we show how to define, estimate, and make inferences concerning objectives that have causal interpretations within these contexts. Specifically, we derive the mathematical formulations of the causal estimands corresponding to the five strategies and clarify the data structure needed to identify these causal estimands. Furthermore, we introduce nonparametric methods for estimating and making inferences about these causal estimands, including the asymptotic variance of estimators and hypothesis tests. Finally, we illustrate our methods using data from the LEADER Trial, which aims to investigate the effect of liraglutide on cardiovascular outcomes.

NASA's Interstellar Boundary Explorer (IBEX) satellite collects data on energetic neutral atoms (ENAs) that can provide insight into the heliosphere boundary between our solar system and interstellar space. Using these data, scientists can construct maps of the ENA intensities (often, expressed in terms of flux) observed in all directions. The ENA flux observed in these maps is believed to come from at least two distinct sources: one source which manifests as a ribbon of concentrated ENA flux and one source (or possibly several) that results in a smoothly-varying globally-distributed flux. Each ENA source type and its corresponding ENA intensity map is of separate scientific interest. In this paper, we develop statistical algorithms for separating the total ENA intensity maps into two source-specific maps (ribbon and globally-distributed flux) and estimating corresponding uncertainty. Key advantages of the proposed method include enhanced model flexibility and improved propagation of estimation uncertainty. We evaluate the proposed methods on simulated data designed to mimic realistic data settings. We also propose new methods for estimating the center of the near-elliptical ribbon in the sky, which can be used in the future to study the location and variation of the local interstellar magnetic field.

We develop methods for estimating how infinitesimal policy changes affect long-term outcomes in dynamic systems. We show that dynamic marginal policy effects (MPEs) can be identified via tractable reduced-form expressions, and can be estimated under a general sequential unconfoundedness assumption. We also propose a doubly robust estimator for dynamic MPEs. Our approach does not require observing full dynamic state information (as is typically assumed for off-policy evaluation in Markov decision processes), and does not incur an exponential curse of horizon (as is typical in non-Markovian off-policy evaluation). We demonstrate practicality and robustness of our approach in a number of simulations, including one motivated by a dynamic pricing application where people use past prices to form a reference level for current prices.

Conjoint experiments randomize multidimensional profiles, offering a powerful design for recovering structural preference parameters -- including marginal rates of substitution, willingness to pay, and the distribution of preferences across a population. Yet the dominant approach in political science has focused on nonparametric causal estimands that do not leverage this potential. We propose a structural approach that embeds a deep neural network within a random utility logit model, allowing preference parameters to vary as a fully flexible function of respondent characteristics. The neural network addresses the concern that a parametric specification may not capture the true data generating process, while double/debiased machine learning provides valid inference on average preference parameters. We apply our method to three prominent conjoint studies and find rich preference heterogeneity masked by reduced-form averages: a near-zero gender effect coexists with 83% preferring female candidates, opposition to undemocratic behavior is near-universal but varies sharply in intensity, and progressive tax preferences cut across every partisan subgroup.

Conventional joint modeling approaches generally characterize the relationship between longitudinal biomarkers and discrete event occurrences within terminal, recurring or competing risk settings, thereby offering a limited representation of complex, multi-state trajectories. We propose a general multi-state joint modeling framework that unifies longitudinal biomarker dynamics with multi-state time-to-event processes defined on arbitrary directed graphs. The proposed framework also accomodates nonlinear longitudinal submodels and scalable inference via stochastic gradient descent. This formulation encompasses both Markovian and semi-Markovian transition structures, allowing recurrent cycles and terminal absorptions to be naturally represented. The longitudinal and event processes are linked through shared latent structures within nonlinear mixed-effects models, extending classical joint modeling formulations. We derive the complete likelihood, model selection criteria, and develop scalable inference procedures based on stochastic gradient descent to enable high-dimensional and large-scale applications. In addition, we formulate a dynamic prediction framework that provides individualized state-transition probabilities and personalized risk assessments along complex event trajectories. Through simulation and application to the PAQUID cohort, we demonstrate accurate parameter recovery and individualized prediction.

Motivated by the Bures distance, we introduce a new family of distances, \emph{relative translation invariant Wasserstein distances}, denoted by $RW_p$, as an extension of the classical Wasserstein distances $W_p$ for $p \in [1, +\infty)$. We establish that $RW_p$ defines a valid metric and demonstrate that this type of metric is more intrinsic than the classical Wasserstein distance. A bi-level algorithm is designed to compute the general $RW_p$ distance between arbitrary discrete distributions. Moreover, when $p = 2$, we show that the optimal coupling matrix is invariant under distributional translation in the discrete setting, and we further propose two algorithms, the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, to improve the numerical stability of computing $W_2$ distance and the optimal coupling matrix solutions. Finally, we conduct three experiments to validate our theoretical results and algorithms. The first two experiments report that the $\mathrm{RW}_2$-LP algorithm and the $\mathrm{RW}_2$-Sinkhorn algorithm, both with and without normalization, can significantly reduce the numerical errors compared to standard algorithms. The third experiment shows that $RW_p$ algorithms are computationally scalable and applicable to the retrieval of similar thunderstorm patterns in practical applications.