Papers

Masked diffusions are popular generative models for discrete distributions. Unlike standard autoregressive sampling, they reveal several coordinates in parallel, approximating each block's joint conditional law by a product of one-coordinate conditionals. The resulting procedure, usually called tau-leaping, reduces computational cost but introduces a factorization error ($\varepsilon_\text{fact}$), even with perfectly learned predictors. We study the resulting tradeoff between generative accuracy and computational cost, focusing on how to choose a denoising schedule to minimize $\varepsilon_\text{fact}$ for a fixed sampling budget. To do so, we establish an exact integral representation of $\varepsilon_\text{fact}$ separating the schedule from the target's dependence structure, summarized by a dependence density $\rho$. This representation yields recursive stationarity equations for optimal schedules and allows us to quantify how estimation errors in $\rho$ affect schedule selection. As the dimension $N$ and sampling budget grow, we characterize the optimal schedule and quantify the cost of random block sizes relative to a deterministic planner. We highlight a fundamental dichotomy: if $\rho$ converges uniformly to a strictly positive continuous profile as $N\to\infty$, schedule optimization can only improve the leading constant of $\varepsilon_\text{fact}$, while if $\rho$ degenerates, schedule optimization can improve the asymptotic order. Examples based on stationary processes and exchangeable mixtures illustrate these regimes.

This work presents a novel probabilistic framework for the assessment of orbital debris fragment outcomes in the low lunar orbit (LLO) regime over time horizons ranging from a few hours to a hundred days after a debris generation event. The framework develops continuous distributions for the probability of sinking (lunar collision) and non-sinking over these transient time horizons, building upon NASA's Standard Breakup Model to provide insights into the variations in the likelihood of debris outcomes across the LLO regime and offering an alternative to computationally expensive Monte Carlo simulations. The effects of perturbative forces such as solar radiation pressure are used to assess the realization of debris outcomes over varying time horizons, providing an analytical framework that bounds the likelihood of each outcome. Results indicate variations in the probability of sinking over short time horizons depending on the originating location of the debris generation event, as well as a link between the physical characteristics of fragments and their likelihood of sinking over longer time horizons. The future incorporation of this framework into broader-scale orbital environment models, mission risk assessment procedures, and policy development are briefly discussed.

We study Brownian motion and Brownian bridge processes conditioned to remain in a fixed interval $[0,a]$, focusing on the conditional distribution at a single time. For a Brownian motion in $[0,a]$, conditioned on survival up to time $t$ we recall (and present in a self-contained form) the conditional density of its position at $t$ (exact extrapolation). For a Brownian bridge conditioned to remain in $[0,a]$ on $[0,T]$ we express the probability density at an interior time as a normalized product of killed transition densities (exact interpolation). These densities admit dual complementary series representations, which are linked via the Jacobi theta identity. Our main contribution is a unified exact sampling suite for both extrapolation and interpolation that (i) includes boundary endpoints and (ii) automatically switches between density representations to keep acceptance rates efficient across regimes. To that end, we derive simple proposal families which cover both small- and large-time regimes, with an automatic rule selecting the tighter envelope. Our procedures extend to exact simulation of discrete skeleton paths at multiple times via the Markov property.

We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order $h$ along the base manifold and of order one along the fibre. Its local covariance geometry leads to a Gumbel limit with an explicit intrinsic constant. Combining this limit with Gaussian approximation and cross-fitted covariance estimation yields a feasible simultaneous confidence tube for the regression field. We further discuss improved finite-sample inference with bandwidth selection and high-order bias corrections. Simulations on various manifolds support the proposed inference procedure. A randomized reconstruction of global wind data illustrates how the tube's cross-sections describe spatially varying uncertainty.

Covariate matching between treatment and control groups remains a foundational methodology in observational causal inference, designed to replicate the covariate balance of randomized controlled trials. Despite substantial methodological advancements, applied researchers continue to face an inherent trade-off between achieving rigorous balance and preserving the statistical power of the matched sample. We introduce a novel framework for a matching method named Optimal Study Interior Partitioning (OSIP). OSIP frames observational matching as a joint optimization problem governed simultaneously by a maximum cardinality constraint on the number of strata, and a strict fine balance threshold on the matching carried out in each strata. OSIP is based on two-stage optimization algorithm. In both stages it considers a projection of the (high-dimensional) data into one-dimensional space by evaluating an estimated propensity score for each unit. The method is based on enforcing a partition of the input to subsets of consecutive intervals of estimated propensity scores. The novelty arises because we consider this task as a global optimization problem. In the first stage, we employ an optimal partitioning algorithm with respect to an auxiliary (one-dimensional) goal function. In the second phase, we go back to the original distance function and apply heuristics approaches locally optimizing the solution from the first stage. We compare our framework with established methodologies from the literature across well-studied empirical benchmarks. We use the standard benchmark of Lindner cardiovascular dataset to demonstrate that OSIP provides a highly efficient sweet spot solution for the corresponding multi-criteria goal. This conclusion is verified by considering also other benchmark datasets.

As an alternative to the additive and extremal biases of average and max pooling, we introduce Geometric Mean Pooling (GMP), a signed pooling operator that combines the product of feature signs with the geometric mean of feature magnitudes. Motivated by local-to-global composition in quantum many-body physics, GMP retains both joint sign information and a characteristic multiplicative scale without introducing learnable pooling parameters. We show that non-overlapping hierarchical GMP preserves the corresponding global multiplicative statistic and evaluate it on synthetic sequence tasks, iterative coarse-graining, image classification, and molecular lipophilicity regression. On the synthetic tasks, GMP recovers product-based signals more accurately than average and max pooling and maintains predictive performance under the tested levels of multiplicative input noise. On image and molecular data, however, its effectiveness depends on the representation, target parameterization, and placement of local and global pooling. These results position GMP as a complementary, regime-dependent inductive bias for tasks in which equal-weight multiplicative composition is plausible, rather than as a universal replacement for standard pooling operators.

Frontier language models now produce professional deliverables that expert graders judge to match human work on a substantial share of economically valuable tasks, yet most enterprise GenAI initiatives fail to show a measurable business effect and a large fraction of agentic projects are expected to be cancelled. We argue that this is substantially a measurement problem: public benchmarks answer "what can the model do?", whereas a deployment decision requires "is this workflow fit, reliable, safe and worth scaling - here, on our data, under our controls?". We present EnterpriseVal, a use-case-level evaluation system that closes this gap. It comprises (i) a formal specification of the use case and of the frozen socio-technical configuration under test, model, prompts, retrieval, tools, guardrails and human oversight, with an autonomy level and consequence tier that jointly set the required evaluation intensity; (ii) a metric catalogue spanning fidelity, utility, efficiency, reliability, assurance and oversight; (iii) a grading protocol that scales blinded expert judgement with calibrated LLM-as-judge scoring through prediction-powered inference; (iv) a two-tier threshold gate, stated as an executable algorithm, that maps metric vectors with confidence bounds to REJECT/CONDITIONAL/SCALE decisions; and (v) a value-and-risk model in which the reviewer catch rate is a measured parameter. We report a pilot across three workflows in a global bank. In credit-memo drafting, human-graded citation precision reached 88% and hallucination rate 1.6% for the best model against gates of 70% and 5%; in procedure transformation, analyst refinement effort fell from an estimated 27.4 to 2.9 hours per document. We separate established results, documented pilot evidence, the proposed system and open hypotheses, and specify the experiments required for full validation

In classification-oriented adaptive sensing, posterior samples characterize uncertainty at the current measurement state and can serve two roles: they may guide the next sensing direction, while their class labels provide votes for the candidate classes and determine whether sensing should continue. We focus on the stopping layer that turns these votes into a declaration, without modifying the posterior sampler or sensing directions. A natural plug-in rule declares when the observed vote share exceeds a threshold. We show that this threshold is not itself a confidence guarantee: when the underlying vote mass equals the threshold, the plug-in rule declares about half the time. As alternatives, we calibrate a fixed-sample rule and a finite-horizon sequential rule to a prescribed false-declaration probability, and study exact curtailment, which stops a fixed-pool rule once its final verdict is forced. We then derive how one-round declaration probabilities determine posterior-sample cost and classification accuracy along a sensing path. On MNIST with DDRM and a fixed PCA-guided probe sequence, curtailment saves up to 62% of posterior samples. Among the evaluated rules at matched operating points, sequential stopping reduces the cost the most. At a high accuracy, that same sequential rule can trade more posterior samples for fewer measurements.

Simulation based inference (SBI) circumvents the challenge of intractable likelihoods by using a simulator that generates data given parameter values. For instance, neural likelihood estimation (NLE) estimates the likelihood function by training a neural network to perform conditional density estimation on simulated data given corresponding parameters. However such density estimation is only feasible for relatively low dimensional data. We extend the scalability of SBI methods to a higher dimensional problem: long sequences with a complex dependency structure. We introduce Neural Composite Likelihood Estimation (NCLE). This divides the sequence into smaller, equal-sized batches. Instead of training NLE to estimate the likelihood for an entire sequence, we estimate the likelihood for each batch separately. The product of these forms an approximate composite likelihood (CL), and we perform frequentist inference using methods from the CL literature: we get a point estimate from maximising the approximate CL and obtain confidence intervals by estimating the Godambe information matrix. We demonstrate the effectiveness of NCLE with experiments on time series models.

We develop single-loop stochastic projected damped extragradient methods for nonconvex–(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex–strongly concave and nonconvex–concave settings, respectively, where $\kappa=L/\mu$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(\kappa\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(\kappa^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

Cybersecurity risk quantification remains challenging due to limited operational data and difficulties in quantifying Loss Event Frequency (LEF). This paper introduces the Loss Event Frequency Security Analyser (LEFSA), a probabilistic framework that reformulates LEF estimation as machine-level Cyber Loss Event (CLE) prediction combined with hierarchical infrastructure-level aggregation. LEFSA estimates calibrated machine-level CLE probabilities from operational cybersecurity telemetry and aggregates them across infrastructure layers while accounting for machine-level dependencies. This provides a foundation for scalable, explainable, and operationally applicable cyber risk estimation at the level of machines, services, business processes, and the entire organization. The framework was evaluated using proprietary Managed Detection & Response telemetry from 23 organizations using Microsoft Defender for Endpoint. XGBoost achieved the strongest predictive performance, with a mean area under the receiver operating characteristic curve of 0.90 and consistently low calibration error across evaluation periods. The results demonstrate that operational cybersecurity telemetry contains substantial predictive information for future CLE occurrence, supporting probabilistic machine-level modeling and hierarchical aggregation as a promising foundation for quantitative, data-driven cyber risk management.

In this paper, we address the joint parameter estimation of drift and diffusion coefficients for heterogeneously interacting particle systems. Unlike the homogeneous setting, the interactions are governed by a graphon-weighted mean-field framework, which introduces significant analytical complexity: while homogeneous systems yield i.i.d. limits, our setting results in particles that are independent but non-identically distributed in the limit. Based on discrete observations of the system over a fixed time interval $[0, T]$, we propose a contrast function based on a pseudo-likelihood approach. We prove the consistency of the resulting estimators as the discretization step $\Delta_n \to 0$ and the number of particles $N \to \infty$. Furthermore, we establish asymptotic normality under the additional constraint $N \Delta_n \to 0$, demonstrating that the underlying graphon structure can be rigorously handled despite the lack of identical distribution in the limit.

This paper proposes a new small area estimation approach that integrates Partial Least Squares within the Fay-Herriot model to address the challenges posed by high-dimensional and highly correlated auxiliary variables sets. The resulting Partial Fay-Herriot (PFH) estimator constructs supervised components that maximize their association with the target variable, enhancing model stability and predictive efficiency. Monte Carlo simulations demonstrate that PFH estimator achieves lower mean squared error than the standard Fay-Herriot estimator and outperforms principal components-based alternatives while relying on fewer latent dimensions. The methodology is applied to the estimation of district-level per capita consumption in Mozambique, where the survey data source is complemented by large set of correlated census variables. The resulting estimates highlight pronounced geographic heterogeneity and uncover spatial clusters of deprivation. Overall, the findings show that the proposed supervised dimension-reduction approach represents an effective and easily interpretable tool for producing reliable indicators in high-dimensional contexts.

The pre-season strengths of European football clubs are usually measured by two proxies in the literature. Football Club Elo Ratings provide strictly performance-based Elo ratings from the early days of the European Cups, while Transfermarkt valuations are crowd-based estimates of squad market values. This paper compares them by evaluating their ability to forecast the results of matches played in the UEFA Champions League and the UEFA Europa League between the seasons 2020/21 and 2024/25. The two indicators yield almost identical out-of-sample accuracy when used separately. Combining the two measures leads to a modest improvement, but the best aggregation procedure is sensitive to the forecast target. Our results suggest that seeding based on Elo ratings would be (closely) optimal.

Clustering is a fundamental task in data analysis, typically addressed through centroid-based methods such as K-means. In this work, we present a general framework for multi-domain clustering via measure quantization: given samples from multiple domains, we learn a shared set of cluster prototypes by minimizing a probability metric, such as the Sinkhorn divergence or the Maximum Mean Discrepancy, between each domain's probability measure and the measure of prototypes. Data points are then assigned to clusters either via nearest centroid, or via optimal transport, a collaborative strategy that couples all samples within a domain. A mini-batch optimization strategy makes both fitting and assignment scalable, reducing memory and computational cost while preserving clustering performance. Experimental results on 5 multi-domain benchmarks spanning image, audio and sensor data show that our Sinkhorn-based method consistently outperforms classical and multi-domain clustering baselines, and that this advantage persists when scaling to hundreds of thousands of samples.

Comparing two populations at the same physical covariate value requires more than conditional means or isolated target-point decisions: researchers may need evidence about an entire conditional-distribution ordering over a continuum, even when covariate margins differ. This paper makes that common-value comparison estimable under an explicit structure–flexibility tradeoff and turns the resulting surface into simultaneous evidence for first-order stochastic dominance. Population-specific margins map the common covariate value into each group, while a fitted copula-derivative representation links conditional distributions across the region. Uniform inference propagates uncertainty from both the margins and dependence model through a one-sided statistic with unknown binding locations. Under correct specification within a finite copula class, smoothness and trimming conditions, and a uniquely best candidate family, the procedure admits uniform control and consistent calibration. Simulations show increasing rejection as alternatives become more distinguishable, alongside model-selection sensitivity and small-sample size distortion. In a descriptive PSID application, the high–low parental-education comparison satisfies the two-direction criterion after multiplicity adjustment, whereas adjacent education-group comparisons remain inconclusive. The framework therefore supports region-wide distributional comparison while making its structural and inferential boundaries explicit.

A large family of regulated release decisions accepts a production lot if and only if the pair (sample mean, sample standard deviation) of $n$ units falls in a fixed plane region. Content uniformity (USP <905>), capability release, variables sampling and percent-within-limits highway acceptance all take this form. The acceptance probability under a hypothesized unit-level law is not, so far as we are aware, evaluated exactly in current practice: for a general parent the joint finite-$n$ law of $(\bar X,s)$ is a constrained $n$-fold integral (Craig 1932 at $n=3,4$; Springer 1953 at general $n$) which admits no elementary reduction. Yet that probability is fixed by the law of the additive pair $(T_1,T_2)=(\sum_{i=1}^{n} X_i,\sum_{i=1}^{n} X_i^2)$, whose characteristic function is the $n$-th power of a one-observation transform. One two-dimensional Fourier inversion therefore delivers it for an arbitrary smooth parent - exactly, as an identity, and numerically on a grid whose dimension stays two whatever $n$ is - a deterministic alternative to both the simulation and the Normality assumption. The identity is exact; what we evaluate is a finite-grid inversion of it, whose error we report per example rather than bound. At $n=3$ the Uniform parent yields the joint law of $(\bar X,s)$, the acceptance probability and the capability law in closed form; we derive these, use them as the anchor of a validation ladder, and then work three rules end to end under non-normal parents at their prescribed sample sizes ($n=10$-$30$). Against the exact answer the Normal-theory incumbent errs by enough to change the risk a plan is designed against, and errs in both directions along the operating curve, so no single constant corrects it. In general only the full distribution suffices.

This paper addresses the problem of parameter estimation and model-based imputation for multivariate circular data lying on a p-dimensional torus in the presence of missing values. Actually, the periodic nature of the sample space invalidates conventional imputation techniques designed for Euclidean data. Then, we propose a general framework for maximum likelihood estimation under the wrapped elliptically symmetric family of distributions, with particular interest in the multivariate wrapped normal distribution, when the missing data mechanism is ignorable. The methodology leverages the conditional properties of the elliptically symmetric distributions on the unwrapped space, embedding the imputation of missing torus data into an Expectation-Maximization algorithm that treats both the wrapping coefficients and the missing entries as latent variables. Derivation of both the E and M steps is detailed and a working algorithm is discussed. Imputation methods are also taken into account. The finite-sample performance of the maximum likelihood estimator under ignorable missingness is assessed through Monte Carlo simulations under the wrapped normal specification. The methodology is also illustrated on data with artificially introduced missingness.

This paper proposes fuzzy entropic triple k-means (FE3KM), an entropy-regularized fuzzy partitioning method for three-way data arrays that simultaneously clusters objects, variables, and occasions. Unlike standard fuzzy clustering, which controls fuzziness through a nonlinear exponent m, FE3KM embeds memberships linearly within a Least-Squares (LS) objective, using row-stochastic membership matrices and entropy regularization to represent uncertainty in cluster assignments. This linear structure is what enables an exact decomposition of total deviance into within- and between-cluster components, further attributable to each mode (objects, variables, occasions) and to individual clusters within each mode. This is an interpretive property unavailable to exponent-m fuzzy models. We derive Alternating Least-Squares algorithms for FE3KM, including an accelerated variant for large arrays, and formally relate the method to tandem clustering procedures and to Tucker3/three-mode partitioning models. A simulation study assesses recovery accuracy, robustness to noise and fuzziness mis-specification, and performance against tandem and three-way clustering baselines. An application to a benchmark three-way TV ratings dataset demonstrates how FE3KM recovers interpretable object-variable-occasion structures and quantifies the contribution of each cluster and mode to overall data variability.

Introduction Risk modelling (RM) and treatment effect modelling (EM) are two approaches to build models to estimate heterogeneous treatment effects from randomized control trials (RCT). RM is a two-stage approach; it estimates a predicted baseline risk at the first stage and then includes it as the only effect modifier in a regression model in the second stage. EM is a full treatment interaction multivariable model. Both approaches have theoretical advantages and limitations, but a thorough comparison including a simulation study is missing. Methods In a theoretical review of the two approaches, we present their underlying assumptions and theoretical advantages and disadvantages. Based on our theoretical review, we design simulation scenarios for RCTs with a dichotomous outcome and evaluate the performance of both approaches with respect to the root mean square error and the bias in the predicted risk difference. Results In the theoretical part we argue that baseline risk is a treatment effect modifier in many clinical situations. RM is a dimensionality reduction approach which, however, makes strong assumptions about the role of prognostic factors modifying the treatment effect. EM's greater flexibility is a possible advantage when the sample size is large. In most simulated scenarios RM performs better than EM, even when the assumptions underlying RM are not fully met. The advantage of RM diminishes as sample size increases. Conclusion When choosing between RM and EM the available sample size and the plausibility of their underlying assumptions should be considered.

We present scentree, an open-source Python package for constructing a scenario fan and a scenario tree for multistage stochastic programming from historical data. It combines machine learning and multivariate time series models to obtain a scenario fan that captures inter-stage dependencies in the stochastic processes. This scenario fan is subsequently transformed into a scenario tree suitable for multistage stochastic optimization, providing a flexible and extensible framework for uncertainty modeling. A key contribution is the automation of the complete workflow, including model selection, parameter estimation, scenario fan generation, and scenario tree construction. Scentree does not rely on assumptions about the underlying data distribution, reducing the statistical expertise required to produce a scenario tree. Furthermore, it is agnostic to the specific multistage stochastic problem to be solved.

We revisit Breiman's observation that reducing inter-tree correlation without weakening individual trees can improve random forests. Building on this principle, we introduce two variants: Dirichlet-Multinomial Bagging Random Forest (DM) and Dirichlet-Weighted Random Forest (DW). Both modulate sample reweighting via a concentration parameter $\alpha>0$. We provide a simple theoretical criterion that clarifies when these variants behave indistinguishably from standard random forests, and we use it to guide a lightweight tuning strategy. In a controlled evaluation on public classification benchmarks, DM and DW are consistently competitive and often stronger than other random-forest (RF) baselines, with negligible additional runtime.

Some products are, by nature, designed to operate under stress conditions that repeatedly cycle between two levels, such as batteries undergoing repeated charge and discharge, or components exposed to recurring thermal or pressure cycles. When such products are tested under accelerated conditions, the applied stress is naturally elevated in the same cyclic manner rather than held constant or increased in a single step. Cyclic-stress accelerated life testing (CyALT) reflects this by alternating the applied stress between specified levels. Units are often inspected only at fixed times, so the data consist of interval failure counts rather than exact lifetimes. Hypothesis testing plays a crucial role in this setting. It can assess whether the applied stress significantly affects lifetime, or validate the parameter values used to design the experiment. Classical tests rely on the maximum likelihood estimator, which is highly sensitive to contamination. Since interval probabilities are estimated jointly across stress conditions, a single anomalous failure count can distort the entire parameter vector and severely affect the test's significance level and power. This paper develops Wald-type and Rao-type test statistics for log-normal CyALT lifetimes under interval monitoring, based on the weighted minimum density power divergence estimator. Both tests are derived for simple and composite null hypotheses. A Monte Carlo study shows that classical tests lose control of the significance level rapidly under contamination, while the robust tests stay close to nominal with only a modest loss of power under clean data. An application to air-conditioner evaporator lifetime data further illustrates the proposed tests.

Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or Lévy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.

This paper studies the practice of combining multiple outcomes into a summary index to estimate a causal effect. For common estimators and index constructions, the estimate equals a weighted sum of the estimated effects on the components, with weights that are implicit and rarely reported. The paper derives the weights and shows that, for inverse-covariance-weighted indices, they can be negative and unrestricted in magnitude, so the index effect can have the opposite sign to every component effect. The paper proposes two procedures for valid inference on the index effect: a variance estimator that accounts for the data-dependent weights, and a shifted t-test that requires no such correction. Conventional t-tests of the null of no effect remain valid. Contrary to common claims, summary indices do not generally improve power. Three published studies illustrate the results.

Real data often contain unusual observations that can exert disproportionate effects on variable selection, especially in complex predictor settings. We propose a two-stage sparse median outer product of gradients (smOPG) method for variable selection in single index models with outlier contamination. We first estimate sparse local gradients via \(\ell_1\)-penalized local median regression and then recover the active predictor set from a rank-one sparse approximation of the resulting gradient matrix using regularized singular value decomposition. The combination of median regression and local weighting provides robustness to both response outliers and leverage points. Extensive simulations across varying dimensions and contamination mechanisms demonstrate the favorable variable selection performance of smOPG relative to existing methods. Applications to air pollution and genomic data demonstrate practical utility, while theory establishes active-set recovery without requiring selection consistency of individual local regressions.

Sequential quantile inference is difficult under local differential privacy because every record is randomized before reaching the analyst and the limiting quantile variance depends on an unknown density. We develop an online procedure that combines randomized response with dynamically chained parallel stochastic gradient descent (P-SGD). The resulting Polyak–Ruppert estimator admits a strong Gaussian approximation. A cross-chain quadratic statistic, computed entirely from private iterates, consistently estimates the limiting variance without a separate online density estimator. These results yield asymptotic confidence sequences and, under polynomial chain growth, asymptotic time-uniform coverage. Arm-wise constructions support locally private quantile best-arm identification, time-uniform simple-regret bounds, and sequential A/B tests of quantile treatment effects. Simulations and salary-data analyses illustrate the finite-sample behavior and practical use of the proposed methods.

In the early stages of a pandemic, identification of a small number of infected individuals through large-scale screening is critical for pandemic control, yet remains challenging under limited reagents and testing capacity. Existing group testing methods suffer from either high computational complexity or low identification accuracy. Even worse, no available methods provide theoretically rigorous analysis for sparse identification with hard constraints caused by the sample usage constraint and the dilution effect existing ubiquitously in practical applications. In this article, we propose the Logic Screening method (LoSc), an ultra fast, accurate, and theoretically grounded framework for large-scale screening. LoSc introduces a novel decoding algorithm with a very simple selection strategy, achieving identification of all positives with only O(klogn) pooled tests. The decoding relies only on logical operations, enabling direct hardware implementation and yielding ultra fast computational implementation. Moreover, LoSc explicitly incorporates dilution and sample usage constraints into pooling designs, and establishes theoretical guarantees to guide optimal pooling configurations. Extensive simulations confirm the superior effectiveness, efficiency, and scalability. We believe LoSc offers a fast and reliable solution for automatic large-scale screening.

Modern text and image representations are often matrix-valued, with rows corresponding to tokens, patches, or other local feature vectors. Predictive information is often sparse but sample-specific, making classical sparse regression methods with a common support poorly suited to this heterogeneity. This paper formalizes an individualized sparse regression framework for matrix-valued covariates in which each observation has its own rows of interest, while the associated regression effects are shared across the population. To estimate this model, we introduce a diagonalized attention mechanism that uses query–key scores to localize sample-specific signal rows and a value matrix for downstream regression. The proposed method has a parameter dimension independent of sample size and can identify rows of interest for new observations without their responses. We establish existence theorems showing that, under suitable score-separation and concentration conditions, single-head and multi-head diagonalized attention models recover the latent rows with high probability, yielding prediction risk bounds. Our theory therefore provides a statistical explanation of how attention-based scoring localizes sample-specific signals in heterogeneous matrix-valued data. Simulations demonstrate strong prediction and localization in regression and misspecified classification across varying sample sizes, dimensions, and signal cardinalities. Real sentiment analyses show improved classification accuracy and interpretable token selection.

We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.