Papers

Prequential adaptation of frozen tabular foundation models under temporal drift, with each label revealed only after prediction, helps some deployments and harms others, yet current practice does not predict which. We study the sources and limits of these gains. A diagnostic anatomy attributes gains to four recurring mechanisms under a streaming protocol that removes three optimistic biases and quantifies a fourth. Within an agnostic total-variation drift class, the target conditional is only partially identified: its identified-set diameter, the wall, is irreducible from unlabeled data uniformly in sample size. A second, orthogonal $L^2$ projection wall quantifies what the frozen representation cannot express. Two canonical mechanism priors collapse the first wall. Under stated nuisance-rate conditions, the wall can be estimated from labeled historical windows at a $\sqrt N$ rate above the margin threshold $\gamma^\star=d_0/(2\alpha_s)$. At $\gamma=0$, the conditional lower-bound program depends on an open affinity estimate; the positive-margin lower branch also remains open. Semi-synthetic data illustrate the finite-sample mechanism with calibrated exponents. Stream-level proxies on eight industrial streams fall on the difficult side under a stated roughness bound, while the equality case $\gamma=\gamma^\star$ remains unresolved.

Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.

A standard spatial regression model $Y = X^\top\beta + U$, with covariates $X$ and a Gaussian random field $U$ with Matérn covariance, decouples covariate effects from spatial dependence. In several applications this is mechanistically implausible, as covariates likely drive the dynamics generating the residual covariance, and it is statistically problematic, since the generalised least-squares estimator of $\beta$ may be biased by spatial self-confounding. We propose operator-matched spatial regression, where covariates enter both pointwise and as deterministic forcings in a stochastic partial differential equation (SPDE) that generates the random effect, partitioning the apparent effect into local and operator-mediated components. The formulation arises as the steady state of a transport–relaxation equation forced by $X$, giving the parameters physical interpretations. We show that self-confounding is resolved by construction and that maximum likelihood inference for the regression coefficients, implementable through standard covariance-based methods, is calibrated when the operator parameters are known and, under expanding-domain asymptotics, when they are jointly estimated. A computationally efficient finite element implementation is introduced and justified via bounds on the discretisation error. Applications to temperature–altitude regression and air-quality monitoring show that the method matches or outperforms alternatives and produces physically meaningful results.

In certain clinical settings, patients diagnosed with a disease of interest are at risk for death as well as a serious nonfatal event, and interest lies in event-free survival (EFS), a composite endpoint defined as the time from disease onset until the earlier of the nonfatal event and death. Component-wise censoring of EFS arises when each component is subject to a different censoring mechanism. For example, the nonfatal event may be interval censored between assessments, and death is right-censored. Further, in studies where individuals enroll after disease onset (including prevalent cohort studies), EFS is left truncated. Methods to estimate EFS probability with left-truncated and right-censored data are available in the literature, but they cannot handle component-wise censoring. We propose a kernel smoothing method to non-parametrically estimate EFS in this setting. Our method can also estimate and test for differences in the restricted mean event-free survival time, and can leverage two types of supplemental data that may be available: data from participants followed for death only (not followed for the nonfatal event), and incident cohort data, which arises when there is no delay between disease onset and study enrollment. We assess the proposed method using simulations and demonstrate the method using data from the Atherosclerosis Risk in Communities (ARIC) Study to estimate dementia-free survival probability following a myocardial infarction.

Lung cancer remains a leading cause of cancer mortality because many cases are diagnosed at advanced stages. Low-dose computed tomography (LDCT) screening can reduce mortality through earlier detection. Partially observable Markov decision process (POMDP) models can personalize screening by maintaining a belief over an individual's latent cancer state. However, cancer-state transition probabilities are often generated from clinical simulations and are subject to estimation error and model misspecification. We propose a robust POMDP framework using $\ell_1$-norm ambiguity sets around the nominal transition probability. The model optimizes screening decisions against the worst-case transition probability while keeping other components fixed at nominal values. Building on the piecewise-linear and convex structure of the robust value function, we adapt point-based value iteration to compute robust screening policies. We evaluate the policies using out-of-sample simulations that perturb selected cancer-progression parameters and compare them with the nominal ENGAGE policy for representative female and male heavy-smoker cohorts at age 50. Robust POMDP policies generally outperform nominal ENGAGE in mean out-of-sample quality-adjusted life-years (QALYs), with the best performance at a moderate ambiguity radius within the tested grid. Clinical analysis shows that the robust policy reduces lung cancer deaths (LCDs) in all evaluated settings for the female cohort and in most settings for the male cohort, with additional false positives (FPs). Screening-schedule analysis shows that the robust policy recommends more LDCT screens and detects more early-stage lung cancers. These findings show that incorporating transition-model uncertainty into data-driven screening models can improve out-of-sample reliability and provide more robust decision support when clinical simulation inputs are misspecified.

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.

Generalization under distribution shift remains a core challenge in modern machine learning, yet existing learning bound theory is limited to narrow, idealized settings and is non-estimable from samples. In this paper, we bridge the gap between theory and practical applications. We first show that existing definition of concept shift breaks when the source and target supports mismatch. Leveraging entropic optimal transport, we propose a key notion: $\gamma^{*}\!$-concept shifts, and derive a general error bound unifying covariate and $\gamma^{*}\!$-concept shifts, which applies to broad loss functions, label spaces, and stochastic labeling. We further develop estimators for these shifts with concentration guarantees, and the DataShifts algorithm, which can quantify distribution shifts and estimate the error bound in most applications - a rigorous and general tool for analyzing learning error under distribution shift.

Generative artificial intelligence changes how firms reach customers, but standard marketing data do not record how often users see and notice a firm's name in generated answers. We develop Generative Marketing Mix Modeling (GMMM) to estimate the causal effects of Generative Engine Optimization (GEO) and Generative Engine Marketing (GEM). For GEO, GMMM combines repeated generated answers with question counts, shares of use across generative systems, and notice probabilities. For GEM, it combines records of sponsored placements with notice probabilities. GMMM compares expected business responses under alternative treatment sequences and establishes sufficient conditions for identifying the resulting effects. We investigate the empirical performance of the proposed method using simulated answers to product recommendation in English and Japanese.

Continuous glucose monitoring (CGM) provides high-frequency measurements of glucose dynamics and enables short-term glucose forecasting for diabetes management. Although time-series foundation models have shown strong general forecasting ability, their effectiveness for CGM prediction and the added value of multimodal dietary context remain unclear. We conduct a comprehensive empirical study using eight public CGM datasets spanning Type 1 diabetes, Type 2 diabetes, and non-diabetes populations. Under a unified protocol across multiple context lengths and prediction horizons, zero-shot foundation models did not consistently outperform strong task-specific baselines such as Elastic Net and PatchTST. In contrast, lightweight fine-tuning substantially improved forecasting performance. For example, fine-tuned Chronos-Bolt reduced RMSE by 6.5%-18.4% in the T1D cohort and by 8.6%-18.2% in the non-diabetes/T2D cohort, with comparable improvements in both in-distribution and out-of-distribution test settings. We further evaluate multimodal dietary context using CGMacros, which provides temporally aligned CGM signals, food images, and macronutrient records. A residual-based fusion framework reduced overall RMSE by approximately 3% and postprandial RMSE by approximately 15% relative to the CGM-only baseline. Moreover, Chronos-based CGM representations were more strongly correlated with observed postprandial glucose increments than representations from LSTM and CatBoost, even after those models incorporated additional dietary modalities, suggesting that pretrained temporal representations better preserve meal-induced excursion patterns. These findings show that foundation models require CGM-specific adaptation for reliable forecasting and that dietary context provides clinically meaningful signals beyond CGM alone, especially during postprandial periods.

We study the nonlocal continuity equation \[ \partial_t\mu_b =\operatorname{div}\!\left( \mu_b\nabla\log\bigl((I-b^2\Delta)^{-1}\mu_b\bigr) \right) \] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $\mu(t)$ at the sharp, uniform-in-time rate \[ \sup_{t\ge0}\|\mu_b(t)-\mu(t)\|_{L^1}\le Cb^2. \] The key estimate is the uniform dissipation of a $b$-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic $N$-particle dynamics. A weak–strong modulated energy argument gives \[ \mathbb E\!\left[ \sup_{t\ge0}W_1(\mu_b^N(t),\mu_b(t)) \right] \le C(Nb)^{-1/2} \] for iid initialization. Consequently, the choice $b\asymp N^{-1/5}$ approximates heat flow uniformly in time at rate $N^{-2/5}$.

We study reinforcement learning (RL) with transition look-ahead, where the agent may observe which states would be visited upon playing any sequence of actions before deciding its course of action. Although look-ahead can substantially improve achievable performance, [1] showed that optimal planning with multi-step transition look-ahead is NP-hard. However, this hardness was established using a discount factor close to one. It was therefore unknown whether the problem remains hard for every discount factor, and whether near-optimal planning can nevertheless be performed efficiently. We resolve both questions. First, we show that for every fixed discount factor, exact planning remains NP-hard. Second, we introduce a randomized polynomial-time approximation scheme for every fixed look-ahead depth. Third, we extend our approach to account for unknown transitions. We empirically validate the soundness of our results on the wind-farm storage-control benchmark of [2], showing that our approach, optimally accounting for -step look-ahead information, offers substantially better performance than existing algorithms. [1] Corentin Pla, Hugo Richard, Marc Abeille, Nadav Merlis, Vianney Perchet : On the Hardness of Reinforcement Learning with Transition Look-Ahead [2] Chenbei Lu, Zaiwei Chen, Tongxin Li, Chenye Wu, Adam Wierman : Reinforcement Learning with Imperfect Transition Predictions: A Bellman-Jensen Approach

We investigate mean-variance portfolio selection with an $\ell_0$-penalty to promote sparsity in asset allocations. Uncertainty in the mean return vector is incorporated through an ellipsoidal uncertainty set, yielding a robust sparse optimization framework. We characterize the structure of both local and global minimizers and exploit these properties in the risk minimization and return maximization formulations. Building on this structural insight, we develop a branch-and-bound algorithm tailored to the resulting robust sparse portfolio problems, together with a new pruning rule that can discard exponentially many candidate portfolios in a single step. Extensive computational experiments on real market data, together with comparisons against a mixed-integer second-order cone programming solver, demonstrate the effectiveness and competitiveness of the proposed approach.

Generalized latent factor models provide a flexible framework for analyzing high-dimensional non-Gaussian data, but principled estimation and uncertainty quantification under missingness remain substantially less developed. We develop a theory that connects a computationally tractable nonconvex procedure directly to statistical inference for nonlinear latent factor models with exponential-family links and partially observed entries. Our procedure combines a link-aware double-SVD initialization, a unilateral refinement that achieves rowwise consistency, and vanilla gradient descent. We show that the refined initializer enters a region of incoherence and contraction and that gradient descent remains in this region through implicit regularization, contracting rapidly down to the statistical estimation error without explicit incoherence or balancing regularization. Our central result is a uniform rowwise linear approximation for the actual output of gradient descent that isolates the leading score fluctuations from higher-order estimation and optimization errors. These expansions yield asymptotically valid individual and Gaussian multiplier-bootstrap simultaneous inference for latent factors, together with simultaneous confidence bands for missing-entry means, without requiring an additional debiasing step. The resulting estimation rate matches a restricted-class minimax lower bound up to logarithmic factors, while the theory accommodates severe missingness, weak low-rank signals, and diminishing local curvature. Simulations support the theoretical findings, and an application to large language model evaluation illustrates uncertainty-aware estimation and ranking of latent model capabilities.

In this paper, we investigate the generalization performance of distributed gradient descent algorithms in a reproducing kernel Hilbert space under a robust loss function $l_{\sigma}$. By exploiting the spectral characterization of gradient descent together with the intrinsic properties of robust loss functions, we establish optimal learning rates for the distributed kernel-based robust gradient descent (DKRGD) algorithm with an appropriately chosen scale parameter $\sigma$. The proposed parameter choice of $\sigma$ simultaneously alleviates the saturation phenomenon and guarantees statistical robustness. A key technical contribution is a novel error analysis that provides substantially sharper bounds for products of operators, thereby significantly relaxing existing restrictions on the maximum number of local machines while retaining optimal learning rates. Finally, we develop a communication-efficient strategy that further improves the convergence performance of DKRGD.

Coarse spatial resolution limits the ability of subseasonal prediction models to resolve extreme precipitation. Downscaling with either dynamical or deep generative models can overcome this issue, but the comparative performance of these models for extremes across different atmospheric regimes remains poorly understood. In this work, we evaluate the Weather Research and Forecasting (WRF) model against a diffusion-based generative model by downscaling two physically distinct, extreme precipitation events up to lead times of 3 weeks. For a fair comparison with WRF, which can downscale boundary conditions from different driving models without model-specific training, the diffusion model is trained in an unpaired fashion. Both approaches improve upon the raw European Centre for Medium-Range Weather Forecasts forecasts, in comparison to fused rain gauge-radar observations in Switzerland (CombiPrecip), but exhibit regime-dependent strengths. WRF achieves the highest probabilistic skill for a multicell, non-stationary event. Conversely, the diffusion model is more consistent across different performance metrics for the two events, outperforming WRF in a more stationary supercell event. These results demonstrate that explicit dynamical modeling can add value for specific precipitation events for subseasonal lead times, and that generative downscaling adds value more broadly in different situations.

Area-based social risk indices summarize residents' socioeconomic conditions but incompletely capture physical features of place that may affect health. We evaluated whether numerical representations of physical place produced by four geospatial foundation model families from 2022 satellite data explained residual variance in tract-level associations between the Area Deprivation Index, Social Deprivation Index, and Social Vulnerability Index with health outcomes. We used LightGBM to predict variables from the American Community Survey and 40 chronic disease and health-behavior outcomes from CDC PLACES across 82,646 census tracts in the contiguous United States, evaluating performance across 10 held-out states. Among survey variables, models were moderately predictive of some variables including housing type (R-squared up to 0.54) but weak for disability, unemployment, and income disparity. For health outcomes, models explained up to 54% of variance left unexplained by social risk indices, with the largest gains for annual checkups, arthritis, and high blood pressure. Mean total variance explained by geospatial foundation models across the 40 health-related outcomes increased from 0.31 in the smallest tract-size decile to 0.39 in the largest. Geospatial foundation models capture health-relevant features of place not represented by conventional social risk indices and may usefully augment them in epidemiological analyses.

Privacy and data-governance constraints often prevent pooling individual-level data across studies, limiting the use of conventional approaches for causal media- tion analysis in multicenter settings. We propose a federated causal meta-mediation framework for right-censored time-to-event outcomes that enables collaborative es- timation without sharing individual-level data. Our framework targets natural indirect effects in a prespecified population by combining information on mediator and outcome mechanisms across distributed data sources. A site-by-site identifi- cation strategy further allows heterogeneity across data sources to be character- ized, with a variance decomposition separating outcome-related, mediator-related, and interaction components. We develop federated one-step and targeted maxi- mum likelihood estimators that accommodate data-adaptive and machine-learning methods for nuisance-function estimation. The finite-sample performance of the proposed estimators is evaluated through numerical simulations. To illustrate the practical utility of the framework, we apply it on data from the French National Health Data System to evaluate the role of methotrexate coprescription in explain- ing the effect of TNFi versus IL-12/23 inhibitor therapy on treatment persistence among psoriatic patients.

We introduce statistical theory for the matching of finitely many objects, represented as metric measure spaces (mm-spaces). The approach is based on the second lower bound (SLB) of the Gromov-Wasserstein distance and thus is able to identify deviations in the distributions of the (pairwise) distances within each mm-space. We introduce a surrogate of the SLB barycenter which can be easily computed and expressed explicitly in terms of the distance distributions of each object. When comparing $m$ mm-spaces for $n$ randomly drawn samples in each space, the resulting statistic then can be calculated efficiently in $O(m \cdot n^2 \log(n))$ basic operations. We derive the asymptotic distribution and finite-sample bounds of the proposed test statistic, which serves as a basis for a variety of tools for statistical inference, specifically an asymptotic test for pose-invariant object discrimination and a classification method (based on the SLB barycenter) with controlled error rates. These methods are investigated in simulations and applied to the structural comparison of protein domains.

Random geometric graph detection asks whether a graph generated from $n$ independent uniform points on $S^{d-1}$, with edge probabilities depending on inner products, can be distinguished from an Erdos-Renyi graph of the same edge density using only its adjacency matrix. Mao, Wu, and Xu conjectured that the cubic trace of the standardized kernel determines the detection boundary: $n^3(\mathrm{tr}\,K^3)^2$ tending to zero or infinity should imply impossibility or strong detection, respectively. We construct counterexamples and prove a uniform result for spherical polynomial kernels of uniformly bounded degree with mean edge density bounded away from zero and one. For the integral operator $K$ of the standardized centered kernel, define \[ n_* = \min\left\{ \frac{1}{|\mathrm{tr}\,K^3|^{2/3}},\quad \frac{1}{\sqrt{\mathrm{tr}\,K^4}},\quad \frac{d}{\mathrm{tr}\,K^2} \right\}, \] with zero denominators interpreted as infinity. The total variation distance between the geometric and independent-edge models tends to zero if $n/n_* \to 0$ and to one if $n/n_* \to \infty$. The three scales correspond to triangles, four-cycles, and global geometry. The pure degree-four spherical harmonic kernel has only positive nonzero eigenvalues, yet its detection scale is $n \asymp d^5$. Throughout $d^5 \ll n \ll d^{16/3}$, the triangle and four-cycle signals and every fixed-degree standardized polynomial mean gap vanish, while the full graph remains strongly detectable. Thus the cubic-trace conjecture fails even without spectral sign cancellation. The impossibility proof combines higher-order spherical integration by parts, a graph expansion of cumulants, and a finite-order relative entropy comparison.

Multivariate time series imputation (MTSI) aims to recover missing values in temporal data composed of multiple interdependent variables. This problem is central to real-world applications such as healthcare monitoring, traffic networks, and energy systems. Recent diffusion-based approaches have shown strong potential for probabilistic imputation by learning to generate missing values through iterative denoising. However, most existing approaches perform diffusion directly in the original data space, requiring the denoising network to simultaneously capture global structure, temporal dynamics, and stochastic variability. This makes the generative task unnecessarily complex, especially when modern deterministic imputers can already provide accurate initial reconstructions. To address this limitation, we propose RDDMPI, a conditional residual diffusion framework that operates directly in residual space. Instead of modeling the full missing signal directly, we reformulate probabilistic imputation as a baseline-residual decomposition, where a pretrained model captures the dominant signal and a diffusion process models the residual uncertainty. To better exploit deterministic guidance, \model{} conditions the reverse denoising process on both the baseline-completed signal and its latent representation, while a reliability-aware conditioning mechanism adaptively controls the influence of baseline information during residual generation. This formulation simplifies the diffusion learning objective, enabling it to focus on structured correction terms rather than reconstructing the full signal. Experiments on multiple benchmark datasets demonstrate that RDDMPI consistently improves both reconstruction accuracy and uncertainty quantification.

High-dimensional two-sample mean testing is challenging when the dimension exceeds the sample size. The random projection method proposed by Lopes et al. (2011) addresses this difficulty by mapping the data to a lower dimension space where Hotelling's $T^2$ statistic can be applied, while retaining useful covariance information and gaining power when the variables exhibit non-negligible covariance structure. However, single projection tests may be sensitive to the realized projection matrix, whereas existing multiple projection procedures often rely on resampling or simulation for calibration, with limited theoretical understanding. Moreover, projection-based Hotelling tests may be less sensitive to sparse mean differences. To address these limitations, we propose a Cauchy-combined random projection test (CRPT), which applies Hotelling's $T^2$ test after multiple independent random projections and combines the projected $p$-values through the Cauchy transformation. The proposed method retains the ability of random projection methods to incorporate covariance information while reducing reliance on any single projection. Under the Gaussian assumption, we establish the null tail behavior of the proposed statistic and further investigate its asymptotic power under suitable alternatives. To improve sensitivity to sparse alternatives, we further develop a power-enhanced version of CRPT. Simulation studies and real data analysis are conducted to examine the performance and practical applicability of the proposed procedures.

A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments $P_k^i$, representing the probability that point $x^i$ belongs to class $k$. These flows have two drivers: a nonlinear reaction component that relaxes $P$ to an evolving Bayesian-like posterior, and a diffusive component that relaxes $P$ to its local average, with a diffusivity $\nu$ that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.

Identifiability of tensor decompositions is often established through linear-algebraic conditions on the factor families. For nonnegative decompositions, however, positivity provides additional information that is not captured by dimension and independence alone: nonnegative terms cannot cancel, and their supports constrain competing decompositions. We introduce a positive scattering term that quantifies this additional source of identifiability and combine it with the dimension budget underlying the Lovitz–Petrov generalization of Kruskal's theorem. For every subset of components, we obtain two sufficient conditions: a threshold of $2|S|-2$ guarantees minimality and nonnegative rank, while the stronger threshold $2|S|-1$ guarantees uniqueness among nonnegative decompositions of the same length. The key result is a positive splitting inequality for irreducible exchanges of nonnegative rank-one tensors, which combines the dimension constraint with support-induced geometric rigidity. Although the scattering term is defined through an optimization over intermediate factor spaces, we show that its mode costs are exactly $0$, $1$, or $+\infty$, yielding an exact activation characterization in terms of graph connectivity. The resulting criterion can strictly certify sparse nonnegative tensor decompositions beyond the reach of Kruskal and Lovitz–Petrov conditions, including examples for which those conditions fail even after reshaping. In the matrix case, the two criteria reduce respectively to full-rank factorization and two-sided separability.

Transportation agencies increasingly predict crash-injury severity with statistical and machine-learning models, but these models do not state how often their output contains the recorded injury level or for which groups of drivers it fails, a gap that matters most for motorcyclists and unrestrained drivers. This study develops and evaluates a certification layer that gives any fitted severity model a finite-sample, distribution-free coverage guarantee within prespecified safety strata. The layer, CHOIR (Conformal Heterogeneity-aware Ordinal Inference with Risk control), combines groupwise and weighted conformal prediction with conformal risk control to return contiguous KABCO intervals, and adds a declared sensitivity analysis for medically assessed injury and bounds on fatal omission. It is evaluated on 4.04 million Texas crashes from 2017-2023, one sampled driver per crash, with seven base models from the ordered logit to a tabular foundation model, and on held-out counties and later years. Under one pooled threshold every model reaches 0.90 coverage overall but covers motorcyclists or unrestrained drivers at 0.868 or lower, and class-balanced gradient boosting covers unrestrained drivers at only 0.374. Calibration within four safety strata places all 28 model-by-stratum estimates between 0.898 and 0.907, at the cost of sets spanning 3.6 to 4.6 of five categories for these groups, and the certified ordered logit is within 0.03 categories of the narrowest model. Injury-model coverage should therefore be certified within safety groups rather than on average, calibration rather than model complexity determines validity, and a statewide threshold should not be applied to small rural counties without local calibration data.

Modeling the joint distribution of extreme values in high-dimensional financial time series is challenging because extremes are sparse and locally extreme observations are not necessarily extreme relative to their full marginal distribution. To address this, we introduce a time-dependent network Hüsler-Reiss model in which market-informed adjacency matrices determine how strongly observations contribute to the estimation. We propose binary and weighted specifications, including the Joint Extremes Adjacency Matrix (JEAM) which combines information about individual extremeness with historical patterns of joint extreme movements. In the forecasting evaluation part, covering one-minute stock returns from three sectors of the S&P 100, JEAM achieves the best out-of-sample log scores for both tail directions; improving scores by 12.5-13.6% in the lower tail and 11.4-14.9% in the upper tail. The results show that incorporating market-informed network structures in the estimation, improves forecast evaluation of extremes across time series.

We obtain multivariate martingale central limit theorems in $p$-Wasserstein distance with respect to the $\ell_r$ norm in $\mathbb{R}^d$ for $p\geq 1$ and $r\in [1,\infty]$, which generalize the results for $p=1$ and $r=2$ in the literature. As corollaries, we obtain the Yurinskii coupling and Cramér-type moderate deviation results. We also provide an illustrative application to the stochastic gradient descent algorithm. To prove our main results, we combine Lindeberg's swapping argument with a new Gaussian convolution inequality controlling the $p$-Wasserstein distance between a Gaussian convolved with a perturbation and the Gaussian with matching mean and covariance matrix. The latter is obtained by developing the recent line of research on $p$-Wasserstein bounds.

We demonstrate explicit transformations of three independent uniforms with exact Gamma$(a,1)$, $a>0$, Poisson$(\lambda)$, $\lambda>0$, and Binomial$(N,p)$, $N\geq 1$, $0\leq p \leq 1$ laws using only elementary operations.

Many applications in engineering, including wireless broadcasting, require designs that provide performance certificates at different target outage levels. This paper studies the problem of maximizing the weighted average of such certificates in the presence of uncertainty about the true system state. The problem is shown to be equivalent to an optimization over nested prediction sets, connecting to the literature on conformal prediction and extending prior art on single-level risk-averse decision making. Furthermore, we derive a dual formulation that decouples optimization across input values. Numerical experiments on a diversity-based wireless transmission system illustrate the cost of enforcing multi-level certificates with a single shared policy and trace the Pareto trade-off between multiple reliability levels.

Score matching avoids computing the normalising constant that maximum-likelihood estimation requires. On constrained domains, its generalised variants weight the Fisher divergence so that boundary terms vanish. We derive generalised score matching on open convex subsets of $\mathbb{R}^{d}$ as the small-neighbourhood limit of minimum probability flow, in which the geometry of the neighbourhoods determines the weight. Every $C^{2}$ positive definite weight arises in this way, including those of classical score matching on $\mathbb{R}^{d}$ and of its variants for non-negative data on $\mathbb{R}_{+}^{d}$. For exponential families, we extend the standard convexity, consistency and asymptotic normality results to every such weight and show that the estimator converges to the true parameter under certain boundary conditions. For a truncated Gaussian on a polytope and a Dirichlet distribution on the simplex, proposed estimators attain the lowest median error of all methods compared, in at least 42 of 50 ground-truth configurations.