Sepsis is a leading cause of mortality, yet optimal treatment policies remain contested. Existing reinforcement learning (RL) approaches learn fixed strategies for sepsis treatment, limiting adaptability to changing clinical objectives during inference. We propose EHRMPC, a framework that decouples learning patient dynamics from optimizing treatment by training a patient digital twin in the form of a generative electronic health record (EHR) model. The digital twin predicts clinical trajectories under interventions and enables model predictive control (MPC) to optimize treatments via inference-time planning over simulations. We evaluate EHR-MPC on a multicenter ICU sepsis cohort spanning 8 hospitals in the Mass General Brigham health system using both off-policy importance sampling and on-policy simulation-based evaluation. Relative to RL baselines, EHR-MPC achieves comparable off-policy performance and improved simulation performance. Unlike RL, this work frames sepsis treatment optimization as inference-time control over learned patient dynamics, establishing a general framework for decision making with generative clinical models.

We propose methods to enhance the predictive performance of generalized additive models (GAMs) in the context of covariate extrapolation, where predictions rely on covariates beyond their observed range. When using predictive models such as GAMs, shifts in the covariate distribution between training and prediction datasets can occur. Ignoring this issue may lead to inaccurate predictions in the tail of the covariate distributions. For example, this problem is particularly critical in climate-change scenarios, where covariates simulated from future climate scenarios are likely to contain more extreme conditions. Our approach integrates GAMs for the bulk of covariate distributions with asymptotic models from multivariate extreme-value theory at high covariate values. We consider binary responses based on a latent variable assumption, and also continuous responses. For large values of the covariates, on a specific marginal scale motivated by extreme-value theory the latent variable or continuous response is assumed to depend linearly on the covariates with an additive error term, when using an appropriate link function. In an application to wildfires in Europe, we explore how the new method can improve predictions, using environmental and meteorological covariates.

From medicine to marketing to social sciences, the promise of tailoring interventions to individuals is undeniable. However, practical applications force weighing personalization's potential benefits with its possible increased cost and fragility. We introduce a statistical hypothesis test that evaluates, given historical data, evidence that a personalized intervention policy's performance will surpass deploying the best single intervention. The test maintains strict type-I error control while achieving asymptotic normality with the minimal possible variance under specified conditions. Results on diverse datasets from job training, depression treatment, education and recommendation systems demonstrate the test's versatility and its superior performance over alternatives. This test can support decision-makers throughout the intervention sciences by providing a simple and powerful quantification of the potential benefits of personalization.

We explain how important classes of Bayesian semiparametric regression fitting and inference procedures can be sped up, significantly, via the use of orthogonalized design matrices. Typically, design matrices in semiparametric regression contain predictor observations and basis functions of such data. In Bayesian semiparametric regression, loop-type approaches such as Gibbs sampling and coordinate ascent variational inference typically are required. We show that pre-loop reformulation of Bayesian semiparametric regression models involving orthogonalized design matrices lead to two orders of magnitude, with respect to column dimension, computational reduction. Our computer experiments reveal that this simple paradigm results in approximately 5- to 60-fold speed-ups.

We study the problem of recovering the correspondence between a collection of $n$ points in $\mathbb{R}^d$ and a noisy, permuted version of those points. In the high-dimensional regime $d=\omega(\log n)$, under a Gaussian model with noise variance $\sigma^2=d/(b\log n)$, prior work identifies $b=2$ as the threshold for almost exact recovery. We prove that this threshold is all-or-nothing: for every fixed $b<2$, no estimator recovers a positive fraction of the matching, and even estimating the matched point cloud in Euclidean distance is asymptotically no better than ignoring the correspondence. On the other hand, we consider a multi-view generalization of the problem where $K$ noisy, independently permuted copies of the same latent point cloud are observed. Here we show that a simple polynomial-time procedure recovers all relative matchings up to $o(n)$ errors whenever $b>K/(K-1)$. Thus multiple views can break the impossibility barrier $b=2$ for the original matching problem: in particular, for $3/2 < b < 2$, the two-view model has no nontrivial recovery, but a third view makes all latent correspondences efficiently recoverable.

The Susceptible-Exposed-Infectious-Removed (SEIR) model is a fundamental model in epidemiology. Model parameters such as the reciprocal transmission, incubation, and infectious rates are often difficult to measure directly, and they are estimated by solving an optimisation problem aiming to minimise the difference between the observed data and the model solution. However, the parameters of the standard SEIR system are not globally identifiable, causing optimisation algorithms to frequently converge to incorrect local optima and suffer from numerical stiffness. Here we show a comprehensive structural identifiability analysis of the SEIR framework, and present a globally identifiable and computationally stable reparameterisation of the model derived via an observational system approach. We fully characterise the multiple locally identifiable parameters, and by transforming the system into a globally identifiable structure, we eliminate the non-uniqueness issues in the parameter estimation approaches. Our numerical experiments demonstrate that this reformulation significantly improves convergence frequency, avoids runtime errors caused by numerical overflow, and consistently recovers the correct parameters. Furthermore, incorporating first-order sensitivity equations into the optimiser enhances the robustness and execution speed of the estimation process. Numerically well-conditioned methods for parameter identification, together with a comprehensive understanding of the identifiability of the parameters, ensure that the model yields reliable, rigorous insights for infectious disease forecasting and theoretical epidemiology.

Pairwise and network meta-analyses occupy the highest tier of evidence-based medicine and routinely inform clinical guidelines and healthcare decision-making. Current approaches typically aggregate study-level treatment effects to obtain an overall estimate. We argue that the causal estimand should come first, with the aggregation derived only afterwards: the target population and the relevant sources of between-study heterogeneity should be explicitly defined before deriving the aggregation required for identification. This shift in perspective fundamentally changes both the estimands and the methodology. We develop a unified causal framework for pairwise and network meta-analysis based on aggregate data. By defining treatment effects with respect to a clinically meaningful target population, for example, the average population represented by the contributing trials, and accounting for heterogeneity induced by treatment-effect modifiers and center effects, we show that identification naturally leads to arm-level aggregation. In the network setting, this causal formulation departs fundamentally from the conventional contrast-based paradigm: arm-level aggregation emerges from the causal formulation rather than from a modeling choice, and treatment effects are identified without relying on the treatment network itself. This perspective provides an additional conceptual argument in the long-standing contrast-based versus arm-based debate. Numerical studies show that the proposed estimators target well-defined causal effects, whereas the causal interpretation of conventional approaches remains unclear. Although both approaches often produce similar estimates, we identify settings in which they diverge, with potentially important implications for the interpretation of meta-analytic evidence.

This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.

The impact of a given training point on a statistical model is classically measured through its leave-one-out influence, which quantifies the effect of its removal from the training set on the model accuracy. While the statistics of leave-one-out influences are well understood in the low-dimensional, large sample limit $n\to \infty, d=O(1)$, they become more intricate in high dimensions, as the influence of a given sample develops non-trivial dependencies on all other training samples. For convex M-estimation under Gaussian design, in the high-dimensional limit $n\asymp d$, we show that the distribution of the influences across the training set converges to a limiting measure which we sharply characterize. Building on these results, we provide evidence that influential samples tend to lie close to the decision boundary, thereby making contact with a standard data selection heuristic in active learning.

Scientists routinely disagree not about data but about how to interpret evidence, because they implicitly operate from different epistemological frameworks without recognising it. The two dominant traditions, confirmationism and falsificationism, each capture genuine insights about scientific reasoning but face well-documented limitations. Confirmationism provides a natural account of how evidence supports hypotheses but cannot escape the problem of induction. Falsificationism provides logical rigour through deductive refutation but is undermined by the Duhem-Quine problem and offers no account of how scientists rationally accept theories and act on them. Here we argue that Bayesian epistemology provides a practical resolution to this impasse. By treating evidence probabilistically and operating over a finite, revisable set of hypotheses, the framework recovers the valid contributions of both traditions while addressing their core weaknesses. We show that confirmation and falsification emerge as special cases of Bayesian updating, that the subjectivity objection to priors is weaker than commonly supposed, and that the framework has direct practical consequences for study design, evidence synthesis, and publishing norms. Specifically, it replaces the falsifiability criterion with the more useful question of whether a hypothesis makes predictions that discriminate between competitors, and reframes the reproducibility crisis as an epistemological rather than a purely statistical problem. Adopting Bayesian epistemology, even informally as a mental model, can reduce friction between researchers, improve research efficiency, and help restore the cumulative character of scientific progress.

We study scalar-on-function linear regression when each covariate curve is observed only through finitely many noisy point evaluations. Our goal is to characterize the minimax estimation and prediction risks as joint functions of the number of trajectories $n$ and the within-trajectory resolution $m$. Working in a fixed trigonometric eigenbasis, with covariance eigenvalues decaying at rate $\alpha$ and slope function of Sobolev smoothness $s$, we derive matching minimax upper and lower bounds under two canonical sampling schemes. Under an independent random design, the minimax prediction rate is $n^{-\frac{2\alpha+2s}{2\alpha+2s+1}} + (nm)^{-\frac{2\alpha+2s}{4\alpha+2s+1}}$. The first term is the fully observed functional linear regression benchmark, while the second term captures the cost of noisy point evaluations after amplification by the inverse covariance operator. Under a common design on an equally spaced grid, the shared sampling geometry introduces additional obstructions, and the minimax prediction rate becomes $n^{-\frac{2\alpha+2s}{2\alpha+2s+1}} + (nm)^{-\frac{2\alpha+2s}{4\alpha+2s+1}} + m^{-(2\alpha+2s)} + m^{-4\alpha}$. Here the third term represents discretization error induced by the fixed grid, whereas the fourth reflects the cost of identifying unknown eigenvalues from observations on a common grid. We further construct data-driven adaptive estimators that screen the covariance scale and threshold blockwise prediction energy, attaining these rates without prior knowledge of the eigenvalue sequence or the smoothness indices. The results reveal a sharp phase transition that depends on the sampling resolution under independent design and a richer phase diagram under common design. Numerical simulations and a real data example illustrate the theoretical findings.

Relational states refer to concepts such as friendship or collaboration, in which a relationship persists over a certain amount of time. Study of relational states often involves figuring out what factors contribute to the creation or dissolution of these relationships. However, most methods available now restrict their attention to binary states, i.e., ties that are either present or absent, even though many real-world systems evolve through multiple relational states (e.g., acquaintance, friendship, close friendship). We propose a continuous-time framework for modelling and inferring relational state networks in which each edge evolves by transitioning between two or more states. In our model, transition intensities are driven by state-dependent covariates that might be decomposed into anchoring (current-state) and pulling (target-state) mechanisms, with both linear and smooth non-linear effects. We address two common sampling regimes. With full event histories, a Cox-type partial likelihood with nested case-control sampling enables efficient estimation of both parametric and smooth effects. Instead, for panel data we derive a general ODE formulation for the likelihood, which leads to a particularly efficient inference procedure for binary state model. Simulation studies confirm accurate recovery of model parameters, and an empirical application to adolescent friendship data reproduces the substantive conclusions of established modelling techniques while offering substantial computational gains. The framework preserves the interpretability of classical network effects, generalizes them to multi-state ties, and scales to larger, more complex designs under both full-history and panel sampling designs.

Flexible machine-learning methods can be sensitive to hidden confounding: they may learn associations induced by unobserved confounders rather than stable signals. Spectral deconfounding mitigates this problem by shrinking high-variance directions of the covariate matrix that, under dense confounding, carry latent confounder information. Existing work has largely focused on linear models. We develop a nonlinear spectral deconfounding framework for gradient boosting. Our approach replaces the ordinary squared-error loss by a spectral loss, which alters the boosting dynamics by slowing down learning in confounding-aligned directions. We show that deconfounding is not achieved by the spectral loss alone, but by the interaction between spectral shrinkage and regularization, especially in terms of early stopping. Moreover, we provide a mixed-model interpretation that connects LAVA-type shrinkage to random-effects adjustment and yields an empirical-Bayes procedure for tuning the spectral loss. We also extend the method to general likelihoods and nonlinear confounding using Laplace approximations and kernel random effects. Across synthetic and real-world experiments, spectrally deconfounded boosting improves estimation of the target function under hidden confounding and is substantially more scalable than existing nonlinear spectral deconfounding baselines.

Health economic evaluations are fundamentally concerned with answering causal questions by targeting estimands that contrast the costs and health consequences that would be observed under at least two different interventions. This requires the joint distribution of potential outcomes under each level of intervention, which, with appropriate causal assumptions, can in principle be identified from the joint distribution of observed health outcomes. Such data, however, are rarely available from a single source. This limitation has motivated the use of decision-analytical models to approximate the joint distribution of outcomes under each intervention directly, informed by causal parameters drawn and synthesized from multiple sources, so that the potential outcomes of interest can be approximated as an expectation over the model-implied outcome trajectories. The validity of this approach, however, depends on the credibility of the underlying assumptions. In this work, we formalize this procedure explicitly as a task of causal inference, thereby defining and decomposing decision-analytical model bias into components arising from model structure (model bias) and input parameters (target bias). Because decision-analytical models often rely on unconventional target parameters lacking straightforward observable analogues, and because bias in these parameters can propagate through the model, target bias may arise even in simple settings, a point of central focus in this work. More broadly, this work provides a unifying foundation for medical decision-analytical modelling and causal inference, making explicit the potential for decision-analytical model bias and the role of causal assumptions contributing to it. Ultimately, the resulting clinical decision is only as credible as the assumptions underlying it.

In this paper, we study several incomplete entropy measures, namely the Incomplete Weighted Cumulative Residual Entropy, the Incomplete Cumulative Residual Tsallis Entropy and its weighted version, and introduce the associated partial orderings. Their connections with certain well-known stochastic orderings are also investigated. Based on these characterizations, a class of nonparametric tests for stochastic equality against ordered alternatives is developed. The asymptotic properties of the proposed tests are derived, while their finite-sample performances are assessed through extensive Monte Carlo simulations under various alternative models and sample sizes. These tests are further compared with the test based on incomplete cumulative residual entropy proposed by Zardasht (2015).

Medical time-to-event data are frequently subject to competing risks, where the occurrence of one terminal event precludes the others and standard survival methods that treat competing events as censoring yield biased absolute-risk estimates. Correct analysis instead targets the cause-specific cumulative incidence function (CIF). This methodology has been available to applied researchers almost exclusively through R packages, forcing Python-based machine-learning workflows into a Python-to-R round trip. We present comprisk, a scikit-learn-compatible Python toolkit that consolidates the canonical competing-risks methods (a scalable competing-risks random survival forest together with Fine-Gray subdistribution-hazard regression including a penalized variant, cause-specific Cox regression, the Aalen-Johansen CIF estimator, and Gray's K-sample test) behind a single, consistent API, and adds competing-risks-aware model evaluation (inverse probability of censoring weighted time-dependent AUC and Brier score, cause-specific concordance indices with closed-form confidence intervals, and calibration curves). Every estimator is validated numerically against the established R reference implementations. The forest uses a histogram-based, numba-compiled split kernel that fits 10-22x faster than randomForestSRC at comparable discrimination on real electronic-health-record cohorts and scales to n = 10^6 on a consumer CPU. comprisk is distributed on PyPI and lets applied researchers perform correct and scalable competing-risks analysis entirely within the Python scientific stack.

In this paper, we resolve an open question of Klopp & Zadik (2026) by providing a high-probability polynomial-time, node-private algorithm which nearly matches the performance of their exponential-time node-private algorithm for exact recovery in stochastic block models. Our result involves an explicitly constructed Lipschitz surrogate for the penalized likelihood function, as well as a carefully devised accept-reject algorithm that samples community labels from the corresponding exponential mechanism in polynomial-time. We rigorously analyze the privacy, runtime, and utility of our proposed algorithm, showing that even when the number of communities K grows logarithmically with the number of nodes n, we can achieve the minimax rates for exact recovery with the privacy parameter epsilon growing as log(n), thus matching known lower bounds on the cost of privacy for this setting.

Understanding how genetic and epigenetic factors jointly influence binary health outcomes remains a major challenge in biomedical research. We propose a global test for the overall effect of interactions between DNA methylation and a set of single nucleotide polymorphisms (SNPs) on a binary phenotype. We propose a logistic functional regression model in which methylation measurements at CpG sites are transformed into smooth functional predictors interacting with discrete SNP genotypes through a localized kernel. This framework enables stable inference on region-level interactions while accounting for the spatial structure of methylation around SNPs. Extensive simulations show that the proposed test provides well-calibrated type I error and improved power over classical SNP-CpG pairwise analyses. The practical relevance of the method is illustrated using publicly available methylation and genotyping data from an obesity case-control study.

For analytic convenience, existing statistical frameworks either assume random or fixed regressors. However, it is a little awkward that they do not cover the practical case of estimating the average treatment effect in experiments with randomized treatments and non-randomized, fixed pretreatment covariates. We unify the literature by providing the theory for regressions with mixed regressors that contain both random and fixed components. Importantly, our theory allows for misspecification of the regression functions. We first establish general results for estimating equations with both random and fixed components and then use it to analyze misspecified linear regression, with applications to completely randomized experiments. We focus on the causal interpretation of the regression coefficients and standard errors even when the models are wrong. We start with the theory for independent data and then extend the discussion to clustered data.

High-dimensional interpolation is common in modern machine learning, but its tail risk is less understood than its expected prediction risk. Existing theory shows that interpolating models can perform well in expectation, yet such guarantees do not determine the probability of rare, severe errors. In operations research and stochastic decision-making applications, rare estimation errors can have disproportionate downstream effects, so tail behavior matters alongside average performance. We study the fragility of high-dimensional linear interpolators using large-deviation methods. We focus on ridgeless regression and compare it with ridge-regularized estimators. We first show that the risk of ridgeless regression can exhibit heavy-tailed behavior: although its expected risk may remain well controlled, its upper tail can decay much more slowly than that of regularized alternatives. We then quantify this phenomenon at the level of large-deviation rates. In the regime we study, ridge regularization suppresses fixed right-tail deviations at the $n^2$ scale, whereas ridgeless regression has only $n\log n$-scale decay, where $n$ is the sample size. This gap shows that interpolation can be statistically fragile even when it is accurate on average. Thus regularization affects the frequency of rare, high-impact risk events in addition to the usual bias-variance tradeoff.

Extreme value theory and compositional data analysis both study settings where relative information plays a central role. In multivariate extreme value theory, threshold exceedance limits satisfy homogeneity properties that separate the radial size of an extreme event from its relative profile. In compositional data analysis, positive vectors are analysed up to multiplicative scale, and inference is based on ratios or log-ratios between components. Consequently, both fields have developed several covariance and dependence representations of the underlying relative structure. In the Hüsler-Reiss model for extremes, these include variogram, covariance, and precision parametrizations. In compositional data analysis, analogous representations arise from pairwise log-ratios, centred log-ratios, and additive log-ratios. We establish an explicit link between the two fields that relates these different representations by a small set of simple transformations, including oblique projections, Hüsler-Reiss inverses, and the variogram map. From a methodological perspective, leveraging this algebraic connection enables the transfer of statistical approaches from one field to the other. For instance, we introduce intrinsic logistic-normal graphical models for compositional data, which are based on Hüsler-Reiss graphical models for extremes. Conversely, we explore how dimensionality reduction methods from compositional data analysis can be applied to the analysis of multivariate extremes.

Time-to-event prediction from tabular patient data is central to prognosis and biomedical decision support, but right-censored follow-up prevents direct use of ordinary regression labels. Tabular foundation models offer reusable prediction machinery for modest heterogeneous datasets, yet they generally assume fully observed outcomes. We introduce SurvFM-RMST, a censoring-aware target-interface framework that converts survival outcomes into jackknife pseudo-observation targets for restricted mean survival time, enabling multiple tabular backbones to perform horizon-specific RMST regression without survival-specific fine-tuning. In controlled simulations with known conditional RMST, SurvFM-RMST recovered restricted event-free time accurately, and pseudo-RMST targets outperformed naive restricted observed-time and event-only targets. Across 36 eligible static SurvSet datasets, SurvFM backbones were competitive with established survival and RMST-regression comparators, though relative performance varied by endpoint, horizon and practical constraints. Predicted RMST further stratified held-out patients into groups with ordered observed event-free time and event enrichment. Overall, the results support pseudo-RMST target construction as a portable interface between censored survival data and tabular foundation-model prediction.

As almost all models are wrong, a mean model's usefulness is often accepted as sufficient justification for its use. In practice, however, standard statistical theory breaks when the mean model fit is imperfect, limiting this usefulness. This tension between fit and usefulness arises from a dichotomization of model fit: the model is either right or it is wrong. Motivated by the linear regression framework, we propose an alternative viewpoint that leverages the mean model's usefulness without assuming it is right or wrong. We define a new model usefulness index and use it to share information across individual observations through the mean model. The result is an estimator of each outcome's mean that shrinks individualized means towards the shared mean model, with the degree of shrinkage governed by this usefulness index. We draw connections between our estimator and the James-Stein estimator and establish when and how our estimators of the individualized means yield more efficient inference than model-based and non-model-based alternatives. We also propose a data-dependent estimate of the usefulness index that balances statistical intuition with efficiency considerations. We illustrate our method's practical value in an analysis of personal tracker data.

We study the problem of multi-snapshot spike deconvolution, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. We adopt a variable-projection formulation that eliminates the amplitudes in closed form, reducing the task to a nonconvex least-squares problem over the spike locations alone, which we refer to as the variable-projection formulation of spike deconvolution (VarProSD). We provide an explicit characterization of the basin of convexity of the VarProSD objective in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation influence the local geometry. Within this basin, we establish that the estimator is consistent in the number of snapshots under stochastic noise, and provide a complementary, sharper error bound under adversarial noise via the local Lipschitz property of the inverse map. We further show local convergence guarantees for gradient descent when initialized within the basin. A central ingredient throughout is the use of Beurling--Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.

Many scientific and engineering applications generate responses that are not scalars or vectors, but statistical objects whose form evolves over an ordered index such as time, depth. Probability distributions are a prominent example, capturing variability and uncertainty that cannot be summarized by low-dimensional statistics. When such responses are observed sequentially, the resulting dynamic distributional trajectories pose significant challenges for regression, particularly in relating scalar predictors to both within-index variability and cross-index evolution. We propose Dynamic Fréchet Regression (DFR), a framework for modeling index-dependent trajectories of distribution-valued responses. DFR extends Global Fréchet Regression by introducing an index-aware weighting mechanism. At each index, predictions are defined as weighted Fréchet means in a metric space of distributions (e.g., Wasserstein space), preserving the intrinsic geometry of the response. The weights depend jointly on predictor similarity and index proximity, enabling index-specific prediction while borrowing strength across neighboring indices. To improve interpretability in high-dimensional settings, DFR incorporates a geometry-aware feature selection approach based on sparse metric learning, which identifies predictors driving distributional dynamics without relying on Euclidean coefficients. Simulation studies show improved predictive accuracy and feature recovery over existing methods. An application to additive manufacturing data demonstrates its ability to produce interpretable, index-specific distributional predictions.

Accurate prediction with interval-censored data is particularly challenging when censoring intervals are wide and follow-up is limited, as is common in studies of chronic diseases. Although auxiliary information from source studies may improve prediction in a target study, existing transfer learning methods typically impose restrictive assumptions on model or parameter similarity, or require access to individual-level source data. We propose a novel transfer learning method for interval-censored data that allows arbitrary source models and avoids sharing source data. Our approach transfers survival probability information from source studies through a carefully designed penalty and enables efficient computation via a simple EM algorithm. When multiple source studies are available and their informativeness is unknown, we further develop a data-adaptive aggregation procedure that is robust to negative transfer. Theoretical analysis shows that the proposed estimator attains a faster convergence rate than the target-only estimator whenever at least one source study is sufficiently informative. Extensive simulation studies and an application to data from the Alzheimer's Disease Neuroimaging Initiative demonstrate the effectiveness of our approach.

Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour, and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \cite{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theoretical results and our methodology, and model a latent-collider DAG, a protein signalling network, and a multi-fidelity heavy-ion collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability of the simulator hierarchy.

Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate payment for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative payment defines a pathwise uncertainty clock, the \emph{intrinsic time} of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact rather than upper bounds, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time, not as slack in worst-case analyses. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.

The functional organization of the brain relies on coordinated activity across spatially distributed regions, making the analysis of inter-regional dependencies fundamental. Existing connectivity measures address this predominantly through phase synchronization, which is vulnerable to volume conduction artifacts and discards amplitude-domain coupling. This study introduces the Spatial Neighboring Scattering Transform, which extends the wavelet scattering transform to the multichannel setting, yielding two descriptors that jointly capture amplitude-envelope coupling between channels and its modulation across frequency scales. SNST was evaluated on the BCI Competition IV-2a motor imagery dataset using a bias-corrected, false-discovery-rate-controlled statistical pipeline, with the validation criterion defined as spatial consistency of significant coupling across subjects. The first-order descriptor identified statistically significant amplitude coupling within a central-parietal electrode neighborhood, reproduced consistently across all subjects and both imagery conditions. The second-order descriptor revealed that this coupling is periodically gated by slow rhythms, indicating a cross-frequency amplitude-modulation structure absent from single-frequency connectivity measures. Phase lag index and weighted phase lag index, computed under an identical correction procedure and verified robust to volume conduction, identified negligible significant coupling with zero overlap with SNST findings, demonstrating that amplitude envelope coupling constitutes a largely distinct connectivity signal. These results establish SNST as a cross-channel scattering-based connectivity descriptor that recovers amplitude-envelope and cross-frequency coupling structure systematically, applicable to any multichannel EEG analysis where amplitude-domain inter-regional dependence is of interest.

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems with a convex-composite structure. In this class, both the objective and the functional inequality constraints are given by convex Lipschitz outer functions composed with smooth nonlinear inner mappings. The analysis is complicated by constraint violation in a nonconvex functional inequality system and by the lack of an a priori bound on the multipliers. To address these issues, we restrict the dual variable to an auxiliary compact set and analyze a smoothed prox-linear augmented Lagrangian method through a nonsmooth nonconvex-concave minimax reformulation. The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem. We show that, for a sufficiently large penalty parameter, all but a controlled number of iterates enter a near-feasible region. On this region, a local conic regularity condition uniformly bounds the associated prox-linear multipliers and thereby makes the artificial dual truncation inactive at the selected iterates. Building on this mechanism, we establish explicit convergence rates for the proposed method in terms of the KKT residual. With dual regularization, a global dual error bound together with a bias-balancing argument gives an $O(K^{-1/3})$ rate. In the unregularized case, under additional local structural assumptions including piecewise linearity of the outer functions, a local dual error bound yields the sharper $O(K^{-1/2})$ rate.

Large language models increasingly provide labels, evaluations, and feedback for tasks specified in natural language. When a specification admits multiple readings but the supervision channel does not reveal which is operative, additional labels reduce sampling error without resolving the resulting identification problem. We introduce Natural Language PAC (NL-PAC), a framework that uses a fixed model's thresholded decoding law to define admissible labels and candidate targets. The probability that multiple labels are admissible equals the diameter of the pointwise-admissible target class, and under target-blind supervision every learner incurs worst-case risk of at least half this diameter, at every sample size; the exact randomized minimax risk over this class is attained by a data-independent strategy. Finite-sample confidence bounds make these quantities certifiable from held-out unlabeled inputs. In a frozen Qwen~2.5--3B audit, one prespecified prompt yields a positive model-relative certificate, whereas a paraphrase and exact-rule controls yield zero. A held-out bridge audit finds that supplied candidate reading clauses fail the admissibility condition needed to transfer the certificate to coherent readings. The guarantee is specific to the audited model, prompt, threshold, and input distribution; extending it to human interpretations requires external validation.

The stochastic linear bandit, where actions are represented as vectors and rewards are linear, is a central paradigm for sequential decision making. We study a partially observed variant of this problem in which the learning agent only sees a random subset of coordinates for each action. Such partial observability arises naturally in settings like recommendation and healthcare, where full action descriptions can be expensive or even impossible to obtain. In general, this makes sublinear regret information-theoretically impossible. However, we show that this barrier can be overcome when the action vectors have low intrinsic dimension. We propose an algorithm, TOFU-POV, that estimates the latent action subspace using the masked actions, imputes current actions using an epoch-wise frozen representation, and runs OFUL in the resulting low-dimensional coordinates. Our theory shows that TOFU-POV enjoys a $\sqrt{T}$ regret that scales with the intrinsic action subspace dimension as opposed to the ambient dimension and quantifies the interaction between these quantities and the missingness, decision set size, and subspace conditioning. We also devise a rank-adaptive algorithm that does not require the knowledge of the intrinsic dimension. We complement these guarantees with a lower bound based on a novel product construction that separates usual reward-learning uncertainty from a missingness-dependent cost intrinsic to partial observation. Synthetic and real data experiments support our theory and show that TOFU-POV can substantially improve upon natural baselines in this challenging problem.

We study the active learning problem of fixed-confidence top-$k$ identification from noisy pairwise comparisons. In this problem, an algorithm sequentially chooses pairs of items to compare, observes the outcomes, and stops when it can return the set of top-$k$ items with error probability at most $\delta$. The objective is to design such a $\delta$-correct procedure that minimizes the expected number of comparisons (the sample complexity). This problem falls within the broader literature on fixed-confidence pure exploration in bandit models, where a common target is asymptotic optimality: the algorithm's expected sample complexity matches the information theoretic lower bound as $\delta \to 0$. Asymptotically optimal procedures have been developed for a range of fixed-confidence pure-exploration problems, however to the best of our knowledge, for top-$1$, or more generally top-$k$ identification from pairwise comparisons under latent utility models an asymptotically optimal algorithm has not been established. In this setting, we develop such an algorithm. We characterize the structure of the lower bound and formulate it as a saddle-point problem. This structure enables a computationally efficient primal-dual procedure that learns the asymptotically optimal comparison allocation online. We then construct an adaptive comparison-allocation algorithm that tracks the allocation learned by the primal-dual procedure and prove it is asymptotically optimal.

Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

This paper studies graph matching under the correlated $\text{Erdős-Rényi}$ (ER) graph pair model. This model first samples an $\mathrm{ER}(n,\frac{\lambda}{ns})$ base graph, whose edges are then independently subsampled twice with probability $s$ to produce two correlated $\mathrm{ER}(n,\frac{\lambda}{n})$ graphs. We propose a graph matching algorithm that has $n^{2+o(1)}$ time complexity and achieves almost exact recovery with high probability under the assumptions $\lambda=(\log n)^{\alpha+o(1)}$ for some $\alpha\in(0,1)$ and $s\in(\sqrt{C_{\mathrm{Otter}}},1]$, where $C_{\mathrm{Otter}}\approx 0.338$ is Otter's tree-counting constant. This is the first algorithm with almost quadratic time complexity in this regime of $\lambda$, while the best known result in this regime is the chandelier-counting algorithm with time complexity $O(n^{c(s)})$, where $c(s)\rightarrow \infty$ as $s$ approaches $\sqrt{C_\mathrm{Otter}}$ from above. The proposed algorithm is based on local tree correlation tests. It uses a rank-based algorithm to match the vertex pairs instead of threshold-based rules in the literature. This avoids the need of computing an explicit threshold, which is computationally difficult to obtain. To prove the almost exact recovery result, we establish a new analysis of tree correlation tests in the diverging-degree regime, where both the mean degree and the tree depth grow with $n$. Based on this new result, we establish the existence of a threshold for a threshold-based graph matching algorithm via local tree correlation tests. Finally, we couple the performance of the rank-based algorithm with the threshold-based algorithm to show almost exact recovery.

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $\epsilon > 0, \delta \in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq \epsilon$ with probability at least $1-\delta$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $\tau(\mathbf{x}; \xi)$ itself, we clip stochastic differences at the Lipschitz scale $\gamma\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - \delta$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $\epsilon$ and Lipschitz constant $\gamma \in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{\epsilon^{-5}, (1-\gamma)^{-3}\epsilon^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $\epsilon^{-3}$ nonexpansive rate (i.e., for $\gamma = 1$), and under samplewise nonexpansiveness to $\epsilon^{-2}$.

Similarity search is a primary application of embedding models trained by contrastive learning. For one of the most popular contrastive learning loss functions, InfoNCE, we show that the population risk with $k$ negative samples is $O(1/k)$ close to an expected cross-entropy which quantifies deviation between i) a softmax similarity search over unseen data using the learned embedding function, and ii) an idealised softmax search over the same data but using similarity implicitly represented in the positive sample generator. This complements existing interpretations of InfoNCE in the $k\to\infty$ limit which are phrased in terms of mutual information, and alignment versus uniformity in embeddings. To quantify generalisation performance, we introduce a new continuity bound for the InfoNCE loss, obtained via Gâteaux differentiation. The bound preserves the structure of averaging over negative samples present in the loss function and features an ``inverse temperature'' parameter which can be tuned to account for the algorithmic temperature. For embedding functions which are Lipschitz in a parameter, this yields a simple demonstration that the averaging effect of $k$ negative samples in the InfoNCE loss carries over to stabilisation of the generalisation error as $k$ grows.

This paper proposes the certainty-equivalent first-order learning (CEFOL) algorithm, a deep learning algorithm for solving discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is challenging because nonlinear certainty equivalent appears in the Bellman equation and the first-order optimality conditions but is difficult to evaluate. By introducing a separate neural network to represent the certainty equivalent, CEFOL enables the exploitation of the Bellman and model-specific first-order optimality conditions. In addition to certainty equivalent, CEFOL also uses neural networks to learn the value functions, policy functions, and Lagrange multipliers by using model-specific first-order conditions to construct residuals for minimization. By using first-order and KKT residuals to learn the policy, CEFOL directly accommodates general equality and inequality constraints on the controls, including occasionally binding constraints, without requiring penalty functions or problem-specific reformulations. We apply the algorithm to risk-sensitive and Epstein--Zin consumption-saving problems, a small-noise robust-control problem, and a DSGE model with recursive preferences and stochastic volatility. Across these applications, out-of-sample Bellman diagnostics and model-specific optimality residuals, including Euler or first-order residuals where applicable, are generally of order 1.0e-4 to 1.0e-3 over the relevant state regions, with larger values mainly near binding constraints, and the learned value and policy functions closely match VFI benchmarks when available. The CEFOL algorithm also works for dynamic programming problems with expected utility, as expected utility is a special case of recursive utility.

We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top $J$-characteristic is divisible by $2^{n-1}$. As an arithmetic consequence, any two-level design whose $J$-characteristics vanish in orders one, two, three, five, and seven but which has a nonzero odd-order $J$-characteristic must have at least $256$ runs. This settles, uniformly in the number of factors, a conjecture of Eendebak, Schoen, Vazquez, and Goos (2023) on the nonexistence of certain strength-three even--odd designs with $56$ or $64$ runs. The divisibility bound is sharp at every odd order and is attained by the even-weight half-fraction.

Within-class variance in language-model representations is commonly read as incomplete neural collapse. We argue it is allocated information storage, and that the allocation obeys a law. A one-line centering identity voids a family of simplex equiangular-tight-frame claims, including our own earlier ones; in dimensionless variance shares across 14 models, macro-category structure carries only 4-12% of representational variance and within-token context carries 79-91%, stable across a 100x parameter range. On the theory side, token-level weight decay penalizes a category in proportion to its type count, not its occurrence mass, reducing next-token prediction to an imbalanced K-class problem whose optimum orders category norms by type count. A converse floor, proved for binary categories, forces within-category dispersion to be at least proportional to the conditional mutual information I(token; context | category). The law holds: identity dispersion, not total variance, tracks this information across every tested model and partition, under a model-free estimate and even across models, where one model's information predicts another's dispersion; and over pretraining the category share overshoots, decays, and partially recovers, because the information it must carry never left.

Terminal embeddings have emerged as a powerful tool for dimension reduction. Given a set of points $P\subset \mathbb{R}^d$, a terminal embedding is a mapping $f:\mathbb{R}^d\rightarrow \mathbb{R}^t$ that preserves the pairwise distance between any pair of points $p\in P$ and $q\in \mathbb{R}^d$ up to small distortion under this mapping. Terminal embeddings have been particularly fruitful for constructing $k$-means and $k$-median coresets, where the objective is to find a typically weighted subset $\Omega$ of $P$ such that for any candidate solution, the cost of the clustering objective on $\Omega$ approximates the cost of the clustering objective on $P$ up to small distortion. Unfortunately, these techniques have not been extended to more complicated structures such as clustering time-series data under common straight-line interpolation between measurements. The main issue is that terminal embeddings, arguably the central technique in this line of research, cannot be linear and are thus not immediately suitable to preserve linear structures. In this work, we develop a generalization of terminal embeddings to affine line-segments that overcomes this issue. We showcase their applicability by using our lines-preserving terminal embeddings to obtain the first dimension-free coresets for clustering time-series under the Fréchet distance. The underlying dimension reduction uses Johnson-Lindenstrauss (JL) embeddings, and our experiments indicate that terminal embeddings perform similarly to JL and favorably against PCA for synthetic and real-world time-series, while only terminal embeddings extend pairwise distance preservation to the full ambient space.

We study log-convexity and log-concavity of densities obtained from sums and differences of two independent noncentral gamma random variables. We give a complete classification of one-sided log-convexity for noncentral gamma differences, a complete log-convexity classification for sums of two independent central gamma random variables, and sharp log-concavity criteria for central differences and for common-scale sums. As special cases, we deduce a log-convexity classification for the density of the product of two correlated normal random variables with arbitrary means and variances, and log-convexity and log-concavity classifications for the densities of the variance-gamma and McKay Type I distributions.

Authorship verification (AV) is the task of determining whether two texts were written by the same author. In a forensic context, the strength of AV evidence can be quantified using likelihood ratios. Most AV methods are score-based and deriving well-calibrated likelihood ratios from these scores requires a separate calibration model. This, in turn, requires additional amounts of case-relevant data, which is often time-consuming to obtain and prepare. This study proposes two novel normalisation techniques, the Square Root Correction and the Hapax Correction, for deriving likelihood ratios from the AV method LambdaG without the need of a calibration model (Nini et al. 2026). These corrections are designed to mitigate the overestimation of evidential strength that may result from long or highly repetitive texts. Performance is evaluated against logistic regression calibration across fifteen corpora and a range of text lengths (100-9,500 tokens), using the log-likelihood ratio cost (Cllr). The proposed methods achieve performance comparable to logistic regression calibration, with the Hapax Correction outperforming it in approximately 45% of tests (weighted by corpora). Furthermore, performance was more frequently close (within 5%) when the Hapax Correction was outperformed by logistic regression calibration, compared with the reverse comparison. Eliminating the need to train a calibration model reduces data-requirements, time and complexity, thereby increasing the accessibility and transparency of forensic text comparison. This combination of empirical performance and practical advantages supports the adoption of the proposed methods in forensic settings.

We show how an adversarial model trainer can plant backdoors in a large class of deep, feedforward neural networks. These backdoors are statistically undetectable in the white-box setting, meaning that the backdoored and honestly trained models are close in total variation distance, even given the full descriptions of the models (e.g., all of the weights). The backdoor provides access to invariance-based adversarial examples for every input, mapping distant inputs to unusually close outputs. However, without the backdoor, it is provably impossible (under standard cryptographic assumptions) to generate any such adversarial examples in polynomial time. Our theoretical and preliminary empirical findings demonstrate a fundamental power asymmetry between model trainers and model users.

This paper studies the introduction of sparse group LASSO (SGL) to the quantile regression framework. Additionally, a more flexible version, an adaptive SGL is proposed based on the adaptive idea, this is, the usage of adaptive weights in the penalization. Adaptive estimators are usually focused on the study of the oracle property under asymptotic and double asymptotic frameworks. A key step on the demonstration of this property is to consider adaptive weights based on a initial $\sqrt{n}$-consistent estimator. In practice this implies the usage of a non penalized estimator that limits the adaptive solutions to low dimensional scenarios. In this work, several solutions, based on dimension reduction techniques PCA and PLS, are studied for the calculation of these weights in high dimensional frameworks. The benefits of this proposal are studied both in synthetic and real datasets.

Estimation of signal-to-noise ratios and residual variances in high-dimensional linear models has various important applications, including heritability estimation in bioinformatics. One widely used estimator is the Gaussian random-effects maximum likelihood estimator (MLE), based on the likelihood of the homogeneous Gaussian random-effects model in which both the regression coefficients and the noise variables are assumed to be i.i.d. Gaussian. This paper studies the behavior of this likelihood estimator under model misspecification. For isotropic random designs with independent, symmetric, sub-Gaussian entries, we establish consistency and asymptotic normality of the SNR MLE for fixed dense coefficient vectors and independent, centered, heteroscedastic finite-moment noise, allowing moderately heavy-tailed errors. We also give parallel consistency and central limit results for correlated Gaussian noise as a benchmark. The asymptotic variance depends on the limiting aspect ratio, the true SNR, and a scalar noise-square fluctuation parameter. This explicit form yields feasible plug-in confidence intervals under independent noise in two cases where the fluctuation parameter can be estimated from response fourth moments: heterogeneous Gaussian noise and homogeneous non-Gaussian noise. Numerical simulations compare likelihood-based and method-of-moments confidence intervals under heterogeneous and non-Gaussian noise, and a real-data illustration demonstrates the resulting calibrations on high-dimensional text features.

Estimating probabilities of extreme events involving multiple risk factors is a critical challenge in fields such as finance and climate science. This paper proposes a parametric approach to estimate the probability that a multivariate random vector falls into an extreme failure set, based on the information in the tail pairwise dependence matrix (TPDM) only. The TPDM provides a summary of tail dependence for all pairs of components of the random vector. We propose an efficient algorithm to obtain approximate completely positive decompositions of the TPDM, enabling the construction of a max-linear model whose TPDM approximates that of the original random vector. We also provide conditions under which the approximation turns out to be exact. Based on the decompositions, we can construct max-linear random vectors to estimate failure probabilities, exploiting their computational simplicity. We apply the proposed method to estimate probabilities of extreme events for real-world datasets, including industry portfolio returns and maximal wind speeds, demonstrating its practical utility for risk assessment.

Understanding how treatment effects vary across patient characteristics is essential for personalized medicine, yet randomized controlled trials (RCTs) are often underpowered to detect heterogeneous treatment effects (HTEs). We propose a framework that improves the efficiency of conditional average treatment effect (CATE) estimation in RCTs by leveraging large observational studies (OS) while preserving RCT unbiasedness. Framing CATE estimation as a supervised learning problem, we show that estimation variance is minimized using the counterfactual mean outcome (CMO) as an augmentation function. We derive finite-sample error bounds and give conditions under which OS data improves CMO estimation, and thus CATE efficiency, even under confounding in the OS or outcome distribution shift between populations. We introduce R-OSCAR (Robust Observational Studies for CMO-Augmented RCT), a two-stage estimator that calibrates OS outcome predictions to the RCT population and corrects residual bias through regularized regression. For any OS-derived nuisance, R-OSCAR is consistent for the RCT-population CATE, and is efficient relative to RCT-only estimators when the RCT-OS outcome mean discrepancy is estimable from the RCT at lower complexity than the full RCT outcome model. A cross-fitted RCT diagnostic determines, from observable data alone, whether borrowing from a given OS is supported. Simulations show R-OSCAR can reduce the RCT sample size needed for HTE detection by up to 75%, while remaining robust to misspecification. We validate on two case studies: a semi-synthetic analysis of the Tennessee STAR study with constructed observational confounding, and the Greenlight Plus pediatric-obesity trial linked with external electronic-health-record controls, where borrowing improves control-arm estimation for small trials and the diagnostic certifies it only where the records cover the trial population.

Causal questions often arise in settings where data are hierarchical: subunits are nested within units. Consider students in schools, cells in patients, or cities in states. In these settings, unit-level variables (e.g., a school's budget) may affect subunit-level outcomes (e.g., student test scores), and subunit-level characteristics may aggregate to influence unit-level outcomes. In this paper, we show how to analyze hierarchical data for causal inference. We introduce hierarchical causal models, which extend structural causal models and graphical models by incorporating inner plates to represent nested data structures. We develop a graphical identification technique for these models that generalizes do-calculus. We show that hierarchical data can enable causal identification even when it would be impossible with non-hierarchical data--for example, when only unit-level summaries are available. We develop estimation strategies, including using hierarchical Bayesian models. We illustrate our results in simulation and through a reanalysis of the classic "eight schools" study.

This paper presents a general method that provides optimal monotone conditional error functions for confirmatory adaptive two-stage designs with conditional power based sample size recalculations. The presented method builds on a previously developed general theory for optimal adaptive two-stage designs where sample sizes are reassessed for a specific conditional power and the goal is to minimize the expected sample size. The previous theory can easily lead to a non-monotonous conditional error function, which is highly undesirable for logical reasons and, as we show, can harm type I error rate control for composite null hypotheses. We also show that type I error control is generally guaranteed with a conditional error function (CEF) that is non-increasing in the first stage p-value. We present a method that extends the existing theory by introducing an intermediate monotonising steps that can easily be implemented and provides a non-increasing conditional error function. We show mathematically that the monotonising step provides the optimal non-increasing conditional error function. We illustrate the method with several examples using optconerrf, an R package implemented for this paper.

Using Non-negative Matrix Factorization (NMF), an observed matrix is approximated by a basis matrix times a coefficient matrix. When each individual's coefficient vector is explained by covariates, the coefficient matrix factorizes into a parameter matrix and a covariate matrix -- a tri-factorization whose mean structure coincides with that of the Growth Curve Model (GCM) for longitudinal data. This correspondence has been noted but not examined. We make three contributions. First, we compare NMF with covariates and the GCM: the basis is prescribed in the GCM but optimized in NMF, and the NMF-optimized basis can be used within the GCM and may improve its fit, the two agreeing when covariate effects are non-negative. Second, the main contribution, we develop statistical inference for the parameter matrix linking covariates to basis components: conditional on the optimized basis we provide standard errors, Wald-type tests, and one-sided confidence intervals, with a simulation study confirming good calibration for covariate-effect contrasts. Third, we compare NMF with principal component analysis (PCA) and functional PCA (FPCA): its non-negative coefficients are membership probabilities giving a soft clustering directly, whereas signed PCA/FPCA scores require a downstream classifier. Illustrations use growth data and a kernel-based varying-coefficient model.

Electronic health record (EHR) systems capture a wealth of multimodal clinical data, encompassing both structured clinical codes and unstructured clinical notes. Yet, many EHR-focused studies have traditionally examined these modalities in isolation or combined them using simplistic methods, overlooking the intrinsic synergy between them. In reality, these modalities are deeply interconnected, each containing clinically relevant and complementary information that, when integrated effectively, can provide a more comprehensive understanding of patient health. Despite the success of multimodal contrastive learning in vision-language applications, its potential remains under-explored in multimodal EHR, particularly in terms of theoretical understanding. To support statistical analysis of multimodal EHR data, we propose a multimodal feature embedding generative model and design a multimodal contrastive loss to learn EHR feature representations. Our theoretical analysis demonstrates the effectiveness of multimodal learning over single-modality learning and connects the solution of the loss function to the singular value decomposition of a pointwise mutual information matrix. This connection leads to a privacy-preserving algorithm tailored for multimodal EHR representation learning. Simulation studies show that the proposed algorithm performs well under a variety of configurations. We further validate its clinical utility using real-world EHR data.

Negative Binomial regression is a staple in empirical management research, especially for the analysis of supply chain disruption risks. Its computational structure is often taken for granted: most applications omit the scoring and information equations and defer to a handful of references for details. But what if the evidence provided by those trusted sources disagrees? We reexamine results from a selection of routinely-cited work on Negative Binomial regression, especially with regard to scoring and information equations in the so-called dispersion parameter. For such parameter, we find limitations affecting each stage of the maximum likelihood estimation process, and conclude that there is no reliable expression for the corresponding element of Fisher Information Matrix. For practical relevance, we also look under the hood of an open-source software implementation in R, and show that the notation adopted has some advantages over its published counterparts. Our proposed remediation is simple: to elevate computations that are rarely made explicit. We illustrate our findings in R with the aid of a simplified numerical example that, while obfuscated due to sensitivity, is underpinned by real-world data on clinical trials supply disruptions.

This paper presents a generalized version of a U-statistics-based test for MCAR developed by Aleksić (2024). The proposed test, similar to the original, evaluates the MCAR assumption by calculating and combining the covariances between response indicators and data variables. However, unlike the preceding version, it is capable of utilizing partially observed variables, resulting in a significantly larger class of detectable alternatives. Numerical results indicate that the improved test is well-calibrated, notably outperforming the well-known MCAR test developed by Little (1988) used as a benchmark. For alternatives detectable by the original method, the improved test maintains comparable, although slightly lower, power, while consistently outperforming Little's test across all studied scenarios. For alternatives that were previously undetectable or marginally detectable, the novel test demonstrates the superior performance among the three methods. While the novel test shares the assumption of finite fourth moments of the data with Little's test, the results suggest it is more robust to this requirement, although both tests exhibit similar limitations.

Turing's estimator allows one to estimate the probabilities of outcomes that either do not appear or only rarely appear in a given random sample. We perform a simulation study to understand the finite sample performance of several related confidence intervals (CIs) and introduce an approach for selecting the appropriate CI for a given sample. We give an application to the problem of authorship attribution and apply it to a dataset comprised of tweets from users on X (Twitter). Further, we derive several theoretical results about asymptotic normality and asymptotic Poissonity of Turing's estimator for two important discrete distributions.

We present a novel representation of NBA players' shooting patterns based on Functional Data Analysis (FDA). Each player's charts of made and missed shots are treated as smooth functional data defined over a two-dimensional domain corresponding to the offensive half-court. This continuous representation enables a parsimonious multivariate functional principal components analysis (MFPCA) decomposition, producing a set of common principal component functions that capture the primary modes of variability in shooting patterns, along with player-specific scores that quantify individual deviations from the average behavior. We first interpret the principal component functions to characterize the main sources of variation in shooting tendencies. We then apply $k$-medoids clustering to the principal component scores to construct a data-driven taxonomy of players. Comparing our empirical clusters to conventional NBA position labels reveals low agreement, suggesting that our shooting-pattern representation might capture aspects of playing style not fully reflected in official designations. The application of FDA to this area introduces a flexible, interpretable, and continuous framework for analyzing player tendencies, with potential applications in coaching, scouting, and historical player or match comparisons.

Single-cell sequencing is revolutionizing biology by enabling detailed investigations of cell-state transitions. Many biological processes unfold along continuous trajectories, yet it remains challenging to extract smooth, low-dimensional representations from inherently noisy, high-dimensional single-cell data. Neighbor embedding (NE) algorithms, such as t-SNE and UMAP, are widely used to embed high-dimensional single-cell data into low dimensions. But they often introduce undesirable distortions, resulting in misleading interpretations. Existing evaluation methods for NE algorithms primarily focus on separating discrete cell types rather than capturing continuous cell-state transitions, while dynamic modeling approaches rely on strong assumptions about cellular processes and specialized data. To address these challenges, we build on the Predictability-Computability-Stability (PCS) framework for reliable and reproducible data-driven discoveries. First, we systematically evaluate popular NE algorithms through empirical analysis, simulation, and theory, and reveal their key shortcomings, such as artifacts and instability. We then introduce NESS, a principled and interpretable machine learning approach to improve NE representations by leveraging algorithmic stability and to enable robust inference of smooth biological structures. NESS offers useful concepts, quantitative stability metrics, and efficient computational workflows to uncover developmental trajectories and cell-state transitions in single-cell data. Finally, we apply NESS to six single-cell datasets, spanning pluripotent stem cell differentiation, organoid development, and multiple tissue-specific lineage trajectories. Across these diverse contexts, NESS consistently yields useful biological insights, such as identification of transitional and stable cell states and quantification of transcriptional dynamics during development.

We develop a probabilistic framework for the asymptotic analysis of a bispectrum-based estimator of primordial non-Gaussianity for isotropic random fields on the sphere in the high-resolution regime. By reformulating the estimation problem as an ordinary least squares regression, we derive the asymptotic moments of the estimator. Combining these results with Stein-Malliavin techniques on Wiener chaos yields a quantitative Gaussian approximation with an explicit convergence rate in total variation distance. The analysis relies on sharp asymptotic estimates for the deterministic weights arising from spherical harmonic coupling coefficients. Numerical experiments illustrate the predicted scaling laws and provide qualitative evidence for the asymptotic Gaussian behavior.

Loss functions define estimator optimality, yet current decision-theoretic tools say little about when different losses demand incompatible optimal procedures. This paper introduces a general framework for such incompatibilities using exclusivity regions, classes, and partitions of loss spaces relative to an abstract optimality operator. These partitions decompose a loss family into regimes such that no single estimator can be optimal across distinct regimes. We develop their basic structure, including links to conic geometry and invariance under positive scaling. The framework is formalized for quantile losses, convex margin-based classification losses, and the Huber robust-regression family on skewed models. Together with collapse results, the theory becomes a calculus of loss-design relevance: it identifies which features of a loss can affect the optimal estimator and which cannot. It yields no-free-lunch results for distinct quantile levels, robustness thresholds, and margin invariants, while also showing irrelevance results such as the fact that all classification-calibrated convex surrogates induce the same Bayes classifier. Applications include robust regression, logistic losses, elicitation theory, and model-robust loss partitions.

Several methods in statistics and machine learning target a probability distribution for which an entropy-regularised variational objective is minimised. This increased flexibility introduces a computational challenge, as one loses access to an explicit unnormalised density for the target. To mitigate this difficulty, we introduce a novel measure of suboptimality called 'gradient discrepancy', and in particular a kernel gradient discrepancy (KGD) that can be explicitly computed. In the Bayesian statistics context, KGD coincides with the kernel Stein discrepancy (KSD), and we obtain a novel characterisation of KSD as measuring the size of a variational gradient. Outside this familiar setting, KGD enables novel sampling algorithms to be developed and compared, even when unnormalised densities cannot be obtained. To illustrate this point several novel algorithms are proposed and studied, including a natural generalisation of Stein variational gradient descent, with applications to mean-field neural networks and predictively oriented posteriors presented. On the theoretical side, our principal contribution is to establish sufficient conditions for desirable properties of KGD, such as continuity and convergence control.

Spectral methods, also known as chaos expansions, are widely used in global sensitivity analysis (GSA), as they leverage orthogonal bases of L2 spaces to efficiently compute Sobol' indices, particularly in data-scarce settings. When derivatives of the model are available, a desirable property, both for modeling and GSA purposes, is for the derivatives of the basis functions to also form an orthogonal basis. We demonstrate that the only basis satisfying this property is the one associated with weighted Poincar{é} inequalities and Sturm--Liouville eigenvalue problems, which we call Poincar{é} basis. We also show that under certain conditions the Poincar{é} basis achieves the same convergence rate as the best polynomial approximation for classes of smooth functions. We then introduce a comprehensive framework for gradient-enhanced GSA that integrates recent advances both in the construction of the expansion - with gradient-enhanced regression - and in the construction of weights for derivative-based sensitivity analysis. Furthermore, the proposed methodology is applicable to a broad class of probability measures and various choices of weights. We illustrate its efficiency on a challenging flood modeling case study, where Sobol' indices are accurately estimated using limited data.

The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variates coefficients are optimized jointly to minimize the approximated variance of the resulting estimator. This approach guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated on multi-fidelity applications: estimating a proportion in an aircraft design use case and a conditional value-at-risk in an electromagnetic dataset.

There has been a growing interest in anomaly detection problems recently, whilst their focuses are mostly on anomalies taking place on the time index. In this work, we investigate a new anomaly-in-mean problem in multidimensional spatial lattice, that is, to detect the number and locations of anomaly ``spatial regions'' from the baseline. In addition to the classic minimization over the cost function with a $L_0$ penalization, we introduce an innovative penalty on the area of the minimum convex hull that covers the anomaly regions. We show that the proposed method yields a consistent estimation of the number and locations of spatial anomalies. Under the minimax framework, we characterize the optimal detection error for multidimensional spatial anomaly detection problem and reveal the trade-off between detection performance and the geometric flexibility of anomaly region shapes. Large-scale Monte Carlo simulations are carried out to examine the numeric performance of the method. The method has a wide range of applications in real-world problems. As an example, we apply it to detect the marine heatwaves using the sea surface temperature data from the European Space Agency.

This paper presents a tractable algorithm for estimating an unknown Lipschitz function from noisy observations and establishes an upper bound on its convergence rate. The approach extends max-affine methods from convex shape-restricted regression to the more general Lipschitz setting. A key component is a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, which act as universal approximators of Lipschitz functions while preserving their Lipschitz constants. Leveraging this property, the estimator attains the minimax convergence rate (up to logarithmic factors) with respect to the intrinsic dimension of the data under squared loss and subgaussian distributions in the random design setting. The algorithm integrates adaptive partitioning to capture intrinsic dimension, a penalty-based regularization mechanism that removes the need to know the true Lipschitz constant, and a two-stage optimization procedure combining a convex initialization with local refinement. The framework is also straightforward to adapt to convex shape-restricted regression. Experiments demonstrate competitive performance relative to other theoretically justified methods, including nearest-neighbor and kernel-based regressors.

We propose a statistical proxy framework for retrieval-augmented generation (RAG) that formalizes how language models balance internal predictions with retrieved evidence. We derive an optimal query-level gate, analyze hallucination via retrieval discordance, model query-memory mismatch, and validate the framework numerically on synthetic and real data.

Statistical hypothesis tests typically use prespecified sample sizes, yet data often arrive sequentially. Interim analyses invalidate classical error guarantees, while existing sequential methods require rigid testing preschedules or incur substantial losses in statistical power. We introduce a simple procedure that transforms any fixed-sample hypothesis test into an anytime-valid test while ensuring Type-I error control and near-optimal power with substantial sample savings when the null hypothesis is false. At each step, the procedure predicts the probability that a classical test would reject the null hypothesis at its fixed-sample size, treating future observations as missing data under the null hypothesis. Thresholding this probability yields an anytime-valid stopping rule. In areas such as clinical trials, stopping early and safely can ensure that subjects receive the best treatments and accelerate the development of effective therapies.

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 $O\left(\ln \frac{k}{\alpha}\right)$ for large $k$ and significance level $\alpha$. 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 $O\left(\ln \frac{k}{\alpha}\right)$ in the sparse regime (just a few non-null streams) while achieving $O\left(\frac{1}{k}\ln \frac{1}{\alpha}\right)$ under dense alternatives (many non-null steams). We validate our theory through experiments on both synthetic and real-world data.

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by $n$-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) $n$-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.

Geofencing surveillance poses a dynamic spatial sampling problem. Police agencies must select a surveillance site, choose a geofence perimeter from a set of alternatives, and identify potential suspects through reverse location warrants. At the same time, warrant magistrates must impose constraints that curtail the reach of police surveillance efforts. This sampling problem emerges because agencies commonly use fixed geofence boundaries that ignore how humans move about a chosen surveillance site (i.e., pedestrian flows or traffic patterns). This further exacerbates privacy concerns and increases the risk of selective expansion where agencies extend their data collection efforts beyond the parameters outlined in their warrant. Given the Court's recent ruling in Chatrie, there is currently a need to establish a measurable process that allows magistrates to quantify and evaluate the potential impacts of a warrant proposal. In this paper, we take the first step in introducing a set of optimal radius estimators that measure how geofence perimeters adapt to their local context. Given a surveillance site and some privacy constraint, these estimators generate surveillance perimeters whose size changes with local population densities. This allows magistrates to quantify tradeoffs between local privacy intrusions with law enforcement's surveillance needs. We discuss the properties of these estimators, their underlying assumptions, and the potential consequences of using algorithms to better protect the privacy of its citizens.

Gaussian Process (GP) models provide a flexible framework for prediction and uncertainty quantification. For most covariance functions, however, exact GP prediction with $n$ points scales as $\mathcal{O}(n^3)$, making it prohibitively expensive for large datasets or large numbers of prediction points. While nearest neighbor-based prediction can work well in certain settings, non-pathological circumstances (like measurement noise, for example) can severely restrict its efficiency. This work presents a complementary approach where one conditions on carefully designed linear combinations of data, which is particularly effective in the setting of jointly predicting many values in large connected regions of the data domain. For kernel functions that are smooth away from the origin and simple prediction domains, this method can be exponentially convergent in the number of linear combinations $r$ used for conditioning. The procedure costs $\mathcal{O}(T r^2)$ work, where $T$ is the cost of solving a linear system with the data covariance matrix, and so in many cases can be computed in linear or near-linear cost by exploiting rank structure in well-behaved covariance matrices. At the cost of $\mathcal{O}(nr^2)$ additional precomputation work, this approach can also provide predictions at arbitrary points of a designated region in $\mathcal{O}(1)$ online work, making it particularly attractive for problems where prediction points are not known in advance. After establishing favorable theoretical properties, we provide several example applications to problems in prediction and matrix approximation.

Artificial-intelligence systems are becoming ubiquitous in society, yet their predictions typically inherit biases with respect to protected attributes such as race, gender, or age. Classical fairness notions, most notably Statistical Parity (SP), demand that predictions be independent of the protected attributes, but are overly restrictive when these attributes influence mediating variables that are considered business necessities. Recent causal formulations relax SP by distinguishing allowed from not-allowed causal paths and by complementing SP with Predictive Parity (PP), requiring the predictor to replicate the legitimate influence of business-necessities. Existing path-based definitions are mainly practical when applied to categorical attributes. This paper introduces a new framework for fairness in structural causal models that is tailored to continuous protected attributes. We formalize SP and PP through path-specific partial derivatives, establish conditions under which these criteria coincide with prior causal definitions, and characterize when a fair predictor, one that satisfies SP along not-allowed paths while achieving PP along allowed paths, exists. Building on this theory, we propose a fair tuning algorithm that either constructs such a predictor or, when not possible, allows for a trade-off between SP and PP. We present experiments on simulated and real data to evaluate our proposal, compare it with previously proposed methods, and show that it performs better when PP is considered.

We propose a general robust prediction framework, termed conformity-based projective prediction (CPP), that integrates Bayesian predictive modeling with ideas from conformity-based 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 $\epsilon$-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.

A recent line of work has reframed individual decision trees as linear models on engineered features associated with their splits, opening routes for oracle inequalities and feature-importance reinterpretation, but leaving open the question of what unified geometric object a forest induces when one indexes its feature map by nodes rather than by splits. The present paper studies that object. KPP indexes the feature map by the nodes of the forest, weighted by a path metric that turns each coordinate into a component of a squared-Euclidean path-isometric embedding. KPP unifies four pillars under a single node-indexed representation whose Gram is non-diagonal and carries a metric: prediction, exact additive attribution, deterministic Lipschitz robust radius in the KPP metric, and uniform Rademacher risk bounds for regression and classification under fixed, honest, or cross-fit conditioning. All probabilistic guarantees are conditional on the representation and are stated under three explicit conditioning regimes; the robust-radius guarantee is deterministic in the KPP metric rather than in a norm on the raw input. Conjectured fast-rate refinements for both regression and classification are stated as open problems and are not claimed as theorems.

We propose a new procedure MATCH (Multiplier-Assisted Tests for Conditional Hypotheses) to test whether the non-Euclidean data match the target model, which is a general framework for significance and specification testing in Fréchet regression. MATCH covers global significance, partial significance, and the adequacy of global Fréchet regression, providing a unified way to compare unrestricted conditional Fréchet means with restricted alternatives. One of the key challenges is that the ordinary held-out loss difference is first-order degenerate under the null: the oracle losses coincide, and plug-in statistics is dominated by nuisance estimation error. MATCH uses sample splitting and independent random multipliers on held-out losses to create a nondegenerate Gaussian leading term without residuals or tangent-space coordinates. To improve data use and stability, we further develop cross-fitted tests and repeated cross-fitting with p-value merging. We establish asymptotic null validity, consistency under fixed alternatives, and local power guarantees. Simulations for distributional, symmetric positive-definite (SPD) matrix-valued, and spherical responses support the theoretical findings, and applications to county-level household income distributions and North Atlantic tropical-cyclone locations demonstrate the practical use of the proposed tests.

Local Gaussian correlation (LGC) measures dependence locally, making it a natural tool for tail dependence and financial contagion, but its estimates degrade in the joint tails, where they are most needed. Location-adaptive bandwidths have been tried for LGC and found inferior to a single global bandwidth; we explain why, and map the regime in which adaptivity does help. First, a diagnostic: across heavy-tailed data-generating processes the parametric marginal pre-transform is inert (it changes the integrated error only in the fourth decimal), while the binding constraint is the local effective sample size, with the replication dispersion following a Fisher variance floor sd ~ (1 - rho^2)/sqrt(eff_n). Second, theory: specializing the Hjort-Jones local-likelihood asymptotics to the bivariate Gaussian family that LGC fits, we derive the first location-specific AMISE-optimal bandwidth for LGC, b*(x) proportional to [(1 - rho^2)^2 / (f beta^2)]^(1/6) n^(-1/6), and validate its bias expansion directly (bias proportional to b^2 beta, R^2 approximately 0.9, slope-to-beta correlation 0.80). Third, a regime map: a Monte Carlo across dependence strengths shows the adaptive rule beats the global plug-in only at moderate dependence with curved surfaces. At weak dependence there is no curvature to exploit; at strong dependence finite-sample bias from the steep surface dominates, and adaptivity performs substantially worse, with an error that grows in the sample size. This explains the field's experience that global bandwidths are hard to beat, and locates the exception. Fourth, application: on volatility-filtered equity returns the adaptive estimator yields more stable tail-dependence surfaces under resampling. The message is cautionary: the binding constraint on tail LGC is data scarcity, not bandwidth placement, and no bandwidth, however optimal, can recover information the data do not contain.

Biological systems exhibit a hierarchical structure, characterised by directed flow from upstream regulators to downstream effects. Although this ordering provides a natural scaffold for causal inference, most causal discovery and GRN methods either ignore the tiered organisation or condition on all upstream variables, which becomes infeasible for high-dimensional omics data. We present ASCEND (Ancestral Scalable Causal discovEry via iNherited Descent), a constraint-based framework that leverages known two-tiered structure to enable genome-scale causal discovery. ASCEND introduces a divide-and-conquer strategy that maintains dynamically updated ancestral conditioning sets for each downstream variable, dramatically reducing the number of conditional independence tests required, and achieves polynomial-time complexity where traditional approaches face exponential blow-up. Through extensive simulations and real biological data, we demonstrate that ASCEND accurately recovers ancestral relationships, scales properly and much faster, and outperforms existing gene regulatory network inference methods in both causal precision and computational efficiency. The algorithm's ability to resolve directionality makes it particularly suited for integrating multi-omic data where upstream regulators (e.g., SNPs, methylation sites) and downstream responses (e.g., gene expression) are measured jointly.

We present in this paper novel accelerated fully first-order methods in \emph{Bilevel Optimization} (BLO). Firstly, for BLO under the assumption that the lower-level functions admit the typical strong convexity assumption, the \emph{(Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation} (\texttt{PRAF${}^2$BA}) algorithm leveraging \emph{fully} first-order oracles is proposed, whereas the algorithm for finding approximate first-order and second-order stationary points with state-of-the-art oracle query complexities in solving complex optimization tasks. Secondly, applying as a special case of BLO the \emph{nonconvex-strongly-convex} (NCSC) minimax optimization, \texttt{PRAF${}^2$BA} rediscovers \emph{perturbed restarted accelerated gradient descent ascent} (\texttt{PRAGDA}) that achieves the state-of-the-art complexity for finding approximate second-order stationary points. Additionally, we investigate the challenge of finding stationary points of the hyper-objective function in BLO when lower-level functions lack the typical strong convexity assumption, where we identify several regularity conditions of the lower-level problems that ensure tractability and present hardness results indicating the intractability of BLO for general convex lower-level functions. Under these regularity conditions we propose the \emph{Inexact Gradient-Free Method} (\texttt{IGFM}), utilizing the \emph{Switching Gradient Method} (\texttt{SGM}) as an efficient sub-routine to find an approximate stationary point of the hyper-objective in polynomial time. Empirical studies for real-world problems are provided to further validate the outperformance of our proposed algorithms.

This paper presents a modular approach to accelerate inference in large language models (LLMs) by adding early exit heads at intermediate transformer layers. Each head is trained in a self-supervised manner to mimic the main model's predictions, allowing computation to stop early when a calibrated confidence threshold is reached. We evaluate several confidence metrics and show that entropy provides the most reliable separation between correct and incorrect predictions. Experiments on the Pythia model suite (70M to 2.8B parameters) demonstrate that our method significantly reduces inference cost while maintaining accuracy across multiple benchmarks. We further adapt this approach to speculative decoding, introducing Dynamic Self-Speculative Decoding (DSSD), which achieves 1.66x higher token acceptance than manually-tuned LayerSkip baselines with minimal hyperparameter tuning.

A/B tests in online experiments face statistical power challenges when testing multiple candidates simultaneously, while adaptive experimental designs (AED) alone fall short in inferring experiment statistics such as the average treatment effect, especially with many metrics (e.g., revenue, safety) and heterogeneous variances. This paper proposes a fixed-budget multi-metric AED framework with a two-phase structure: an adaptive exploration phase to identify the best treatment, and a validation phase with an A/B test to verify the treatment's quality and infer statistics. We propose SHRVar, which generalizes sequential halving (SH) with a novel relative-variance-based sampling and an elimination strategy built on reward z values. It achieves a provable error probability that decreases exponentially, where the exponent H3 generalizes the complexity measure for SH and SHVar with homogeneous and heterogeneous variances, respectively. Numerical experiments demonstrate its performance and robustness.

Dimensionality reduction methods such as t-SNE and UMAP are popular methods for visualizing data with a potential (latent) clustered structure. They are known to group data points at the same time as they embed them, resulting in visualizations with well-separated clusters that preserve local information well. However, t-SNE and UMAP also tend to distort the global geometry of the underlying data. We propose a more transparent modular approach that first clusters the data, then embeds each cluster, and finally aligns the clusters to obtain a global embedding. We demonstrate this approach on several synthetic and real-world datasets and show that it is competitive with existing methods, while being much more transparent.

High quality data is needed to unlock the full potential of AI for end users. However finding new sources of such data is getting harder: most publicly-available human generated data will soon have been used. Additionally, publicly available data often is not representative of users of a particular system -- for example, a research speech dataset of contractors interacting with an AI assistant will likely be more homogeneous, well articulated and self-censored than real world commands that end users will issue. Therefore unlocking high-quality data grounded in real user interactions is of vital interest. However, the direct use of user data comes with significant privacy risks. Differential Privacy (DP) is a well established framework for reasoning about and limiting information leakage, and is a gold standard for protecting user privacy. The focus of this work, \emph{Differentially Private Synthetic data}, refers to synthetic data that preserves the overall trends of source data,, while providing strong privacy guarantees to individuals that contributed to the source dataset. DP synthetic data can unlock the value of datasets that have previously been inaccessible due to privacy concerns and can replace the use of sensitive datasets that previously have only had rudimentary protections like ad-hoc rule-based anonymization. In this paper we explore the full suite of techniques surrounding DP synthetic data, the types of privacy protections they offer and the state-of-the-art for various modalities (image, tabular, text and decentralized). We outline all the components needed in a system that generates DP synthetic data, from sensitive data handling and preparation, to tracking the use and empirical privacy testing. We hope that work will result in increased adoption of DP synthetic data, spur additional research and increase trust in DP synthetic data approaches.

We study a binary distributed hypothesis testing problem where two agents observe correlated binary vectors and communicate compressed information at the same rate to a central decision maker. In particular, we study linear compression schemes and show that simple truncation is the best linear scheme in two cases: (1) testing opposite signs of the same magnitude of correlation, and (2) testing for or against independence. We conjecture, supported by numerical evidence, that truncation is the best linear code for testing any correlations of opposite signs. Further, for testing against independence, we also compute classical random coding exponents and show that truncation, and consequently any linear code, is strictly suboptimal.

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).

We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is $1/e$-linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain $O(T^{1/2})$ static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.

Supervised detection of network attacks has always been a critical part of network intrusion detection systems (NIDS). Nowadays, in a pivotal time for artificial intelligence (AI), with even more sophisticated attacks that utilize advanced techniques, such as generative artificial intelligence (GenAI) and reinforcement learning, it has become a vital component if we wish to protect our personal data, which are scattered across the web. In this paper, we address two tasks, in the first unified multi-modal NIDS dataset, which incorporates flow-level data, packet payload information and temporal contextual features, from the reprocessed CIC-IDS-2017, CIC-IoT-2023, UNSW-NB15 and CIC-DDoS-2019, with the same feature space. In the first task we use machine learning (ML) algorithms, with stratified cross validation, in order to prevent network attacks, with stability and reliability. In the second task we use adversarial learning algorithms to generate synthetic data, compare them with the real ones and evaluate their fidelity, utility and privacy using the SDV framework, f-divergences, distinguishability and non-parametric statistical tests. The findings provide stable ML models for intrusion detection and generative models with high fidelity and utility, by combining the Synthetic Data Vault framework, the TRTS and TSTR tests, with non-parametric statistical tests and f-divergence measures.

For decades, physics-based climate models have been used to provide insights for climate decision-making. Their application is, however, constrained by significant computational and technical demands. Machine learning (ML) emulators offer a way to reduce these high computational costs; yet, it remains challenging to use ML emulators effectively in climate research. In practice, climate scientists often bypass emulators altogether, and machine learning researchers frequently develop them as methodological showcases without proving their practical utility. The reasons are diverse, ranging from limited accessibility and a lack of specialized knowledge to broader concerns about the physical grounding of ML methods. Here, we discuss limitations and introduce a framework for guiding emulator development, considering both climate science and machine learning perspectives. We argue that designing easy-to-adopt emulators that address clearly defined tasks and demonstrate their reliability is essential. This offers a promising path towards making machine-learning approaches more relevant and usable for applied climate research.

Decision-makers rely on weather forecasts to plant crops, manage wildfires, allocate water and energy, and prepare for weather extremes. Today, such forecasts enjoy unprecedented accuracy out to two weeks thanks to steady advances in physics-based dynamical models and data-driven artificial intelligence (AI) models. However, model skill drops precipitously at subseasonal timescales (2 - 6 weeks ahead), due to compounding errors, systemic model biases, and the chaotic nature of the atmosphere. To counter this degradation, we introduce probabilistic bias correction (PBC), a machine learning framework that substantially reduces systematic error by learning to correct historical probabilistic forecasts. When applied to the leading dynamical and AI models from the European Centre for Medium-Range Weather Forecasts (ECMWF), PBC doubles the modest subseasonal skill of the AI Forecasting System and improves the skill of the operationally-debiased dynamical model for 91% of pressure, 92% of temperature, and 98% of precipitation targets. We designed PBC for operational deployment, and, in ECMWF's 2025 real-time forecasting competition, its global forecasts placed first for all weather variables and lead times, outperforming the dynamical models from six operational forecasting centers, an international dynamical multi-model ensemble, ECMWF's AI Forecasting System, and the forecasting systems of 34 teams worldwide. These probabilistic skill gains translate into more accurate prediction of extreme events and have the potential to improve agricultural planning, energy management, and disaster preparedness in vulnerable communities.

Large Language Models can generate synthetic survey responses at low cost, but their accuracy varies unpredictably across questions. We study the design problem of allocating a fixed budget of human respondents across estimation tasks when cheap LLM predictions are available for every task. Our framework combines three components. First, building on Prediction-Powered Inference, we characterize a question-specific rectification difficulty that governs how quickly the estimator's variance decreases with human sample size. Second, we derive a closed-form optimal allocation rule that directs more human labels to tasks where the LLM is least reliable. Third, since rectification difficulty depends on unobserved human responses for new surveys, we propose a meta-learning approach, trained on historical data, that predicts it for entirely new tasks without pilot data. The framework extends to general M-estimation, covering regression coefficients and multinomial logit partworths for conjoint analysis. We validate the framework on two datasets spanning different domains, question types, and LLMs, showing that our approach captures 61-79% of the theoretically attainable efficiency gains, achieving 11.4% and 10.5% MSE reductions without requiring any pilot human data for the target survey.

During thDuring the last few years, the term Mechanistic Interpretability, a specific area, under the umbrella of explainable artificial intelligence (XAI), has been introduced, to explain the decisions made by complex machine learning (ML) models in critical systems like UAV intrusion detection systems (UAVIDS). In this paper, we apply best-practices for data pre-processing and examine a wide range of tree-ensembles, deep neural networks, hybrid stacking models and the latest ensemble neural networks to detect intrusions in UAV, with stratified 10-fold cross validation. With our top-performing model, XGBoost, we proceed to Shapley Additive explanations (SHAP), to analyze the global and local feature importances and understand which features, each attack targets, to mimic normal traffic and where the misclassifications occur. Furthermore a distribution analysis follows, by visually comparing violin plots and the curves of kernel density estimations. With the Westfall-Young permutation test for multiple comparisons, the Bandwidth optimization of the KDEs and the selection of Jensen-Shannon Distance for the test, we discover the true causes of false predictions, observed in Wormhole and Blackhole attacks in UAVIDS-2025. The findings provide robust, reliable and explainable models for UAV intrusion detection, along with statistical insights, which capture and clarify the masked nature of the attacks, regarding the challenge of Density Support Intersection, between these attacks, in this dataset.

Liquid-cooled exascale supercomputers dissipate heat through cooling plants organized as multiple parallel subloops, but how to allocate coolant distribution units (CDUs) across subloops and how to distribute flow among them has not been systematically addressed for facilities at this scale. This paper presents a three-layer optimization framework that jointly determines the integer partition of CDUs across subloops, the continuous flow fraction allocation, and the per-timestep co-design optimization of total flow rate and supply temperature subject to per-subloop thermal safety constraints. The Modelica simulation model is built based on the data of Frontier exascale supercomputer at Oak Ridge National Laboratory. By developing a reduced-order surrogate model, all 611 feasible partitions of 25 CDUs are evaluated across the full year operational dataset of 49,353 timesteps. Three progressively richer operational strategies are compared, ranging from flow control optimization to full three-layer co-design optimization with dynamically adjusted flow fractions. The optimal design within the surrogate optimization problem is a two-subloop plant achieving 35.48% annual cooling energy savings, only 0.18% above the current three-subloop Frontier design at 35.30%. Most of the savings are delivered by supervisory co-optimization of total flow rate and supply temperature; the distinct role of flow fraction optimization is design robustness rather than additional raw savings. Flow fraction optimization compensates for any feasible CDU-to-subloop assignment, reducing the design sensitivity by 93% and providing a low-cost software-only pathway to near-optimal performance on the existing Frontier hardware. The framework is transferable to other liquid-cooled high-performance computing this http URL is transferable to other liquid-cooled high-performance computing plants.