Papers

We propose a semi-structured discrete-time multi-state model to analyse mortgage delinquency transitions. This model combines an easy-to-understand structured additive predictor, which includes linear effects and smooth functions of time and covariates, with a flexible neural network component that captures complex nonlinearities and higher-order interactions. To ensure identifiability when covariates are present in both components, we orthogonalise the unstructured part relative to the structured design. For discrete-time competing transitions, we derive exact transformations that map binary logistic models to valid competing transition probabilities, avoiding the need for continuous-time approximations. In simulations, our framework effectively recovers structured baseline and covariate effects while using the neural component to detect interaction patterns. We demonstrate the method using the Freddie Mac Single-Family Loan-Level Dataset, employing an out-of-time test design. Compared with a structured generalised additive benchmark, the semi-structured model provides modest but consistent gains in discrimination across the earliest prediction spans, while maintaining similar Brier scores. Adding macroeconomic indicators provides limited incremental benefit in this out-of-time evaluation and does not materially change the estimated borrower-, loan-, or duration-driven effects. Overall, semi-structured multi-state modelling offers a practical compromise between transparent effect estimates and flexible pattern learning, with potential applications beyond credit-transition forecasting.

Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.

This article explores a general factor structure for high-dimensional nonstationary functional time series, encompassing a wide range of factor models studied in the existing literature. We investigate the asymptotic spectral behaviors of the sample covariance operator under this general data structure. A novel fundamental sufficient condition, formulated in terms of a newly introduced effective rank tailored to this setup, is established under which empirical eigen-analysis yields spurious results, rendering sample eigenvalues and eigenvectors unreliable for accurately recovering the underlying factor structure. This generalizes the results of Onatski and Wang [2021] from typical high-dimensional time series (HDTS) to the more intricate functional framework. The newly defined effective rank is rigorously analyzed through a decomposition of the effects attributable to functional factor loadings and functional factors. Contrary to the findings in the HDTS setting, empirical eigen-analysis of models with only a small number of strong non-stationary factors may still produce spurious limits in the functional framework. Therefore, additional caution is warranted when applying covariance-based statistical methods to potentially nonstationary functional data. Simulation studies are performed to determine conditions under which spurious limits occur. Real data analysis on age-specific mortality rate data from multiple locations is conducted for evidence of spurious factors induced by empirical eigen-analysis.

This study aimed to adapt the Dual-Dimensional Scale of Instrumental and Relational Dependencies on Large Language Models (LLM-D12) into Turkish and evaluate its psychometric properties among regular LLM users. A sample of 387 participants (68.5% female; mean age = 25.22 +/- 7.13) completed the translated scale, which underwent cultural-linguistic validation through forward-backward translation and expert review. Confirmatory factor analysis supported the original two-factor structure after removing one item, with strong model fit (CFI = 0.993, RMSEA = 0.073). Internal consistency was high across both subscales: Cronbach's alpha = 0.831 (instrumental), 0.876 (relational), and 0.868 (total); McDonald's omega = 0.834, 0.880, and 0.900, respectively. Test-retest reliability and item monotonicity were satisfactory. External validity was demonstrated via significant associations with ATAI, IA, and PTLLM scores. Interestingly, the lack of association with need for cognition (NFC) suggests that LLM dependency may reflect strategic cognitive offloading rather than cognitive avoidance. The Turkish version of the LLM-D12 is a valid and reliable 11-item tool for assessing both instrumental and relational dependencies on LLMs.

Contrastive learning produces coherent semantic feature embeddings by encouraging positive samples to cluster closely while separating negative samples. However, existing contrastive learning methods lack a principled construction of geometric sets in the semantic feature space with distribution-free guarantees at any user-specified coverage level. We extend conformal prediction to this setting by introducing covering sets equipped with learnable generalized hyper-ball constraints. We propose a method that constructs conformal sets guaranteeing user-specified coverage of positive samples while maximizing negative sample exclusion. We theoretically motivate volume minimization as a proxy for negative exclusion, enabling our approach to operate effectively even when negative pairs are unavailable. The positive inclusion guarantee inherits the distribution-free coverage property of conformal prediction, while negative exclusion is maximized through learned set geometry optimized on a held-out training split. Experiments on simulated and real-world image datasets demonstrate improved inclusion-exclusion trade-offs compared to standard distance-based conformal baselines.

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

Multiplex networks are a powerful framework for representing systems with multiple types of interactions among a common set of entities. Understanding their structure requires statistical tools capturing higher-order cross-layer correlations. We develop a comprehensive framework for estimating and testing dependence in exchangeable multiplex networks through motif counts. We first propose a moment-based estimation methodology that extends the multi-layer stochastic block model network histogram to arbitrary motif counts. This allows us to estimate the $2^d-1$ graphons defining a $d$-layer multiplex network. We then derive the joint asymptotic distribution of cross-layer motif counts, that is aligned motifs shared across layers. Extending existing results from the unilayer setting, we show that the limiting distribution in the motif-regular case exhibits a covariance structure involving minimum-based distances between graphons. Finally, we construct hypothesis tests to detect inter-layer similarity and dependence. This work provides a rigorous extension of motif-count asymptotics and inference procedures to the multiplex setting, providing new tools to study high-order dependencies in complex networks.

Net benefit is widely used to assess whether clinical prediction models can improve decision-making, but its interpretation is often difficult in practice. In this didactic note, we show how decision curves can be understood through two familiar quantities: positive predictive value and observed event rates in threshold-defined patient groups. We show that a model has positive net benefit compared with treating no one when, among patients classified as positive, the observed event rate exceeds the decision threshold. Similarly, a model outperforms treating everyone when, among patients classified as negative, the observed event rate is below the decision threshold. These results connect decision-curve performance directly to clinically interpretable forms of threshold-specific calibration. We also describe how positive predictive value curves can complement decision curves by showing, at each threshold, the observed event rate among patients selected for action. Together, these perspectives help explain why a prediction model performs better or worse than default strategies and may make decision curve analysis more accessible to clinical audiences.

We propose a new family of inequality indices that bridges the Hoover index and the Gini coefficient. The measure is defined as the normalized expected absolute value of a convex combination of deviations from the mean and pairwise differences, providing a continuous interpolation between these two classical indices. We establish key theoretical properties, including scale invariance, boundedness, continuity, and compliance with the Pigou-Dalton transfer principle. Analytical representations are derived, allowing explicit evaluation under gamma distributions and leading to closed-form expressions involving incomplete gamma functions. From a statistical perspective, we study the plug-in estimator, obtaining a general expression for its expectation and explicit formulas for its bias under gamma populations. Simulation results indicate good finite-sample performance, with decreasing bias and mean squared error as the sample size increases. An empirical application to GDP per capita data illustrates the practical usefulness of the proposed index as a flexible tool for inequality analysis.

Since the introduction of network psychometrics, several connections to statistical models in "classical" psychometrics (i.e., IRT, SEM, GLM) as well as to approaches from other research fields have been established. In this paper, these developments have been reviewed and synthesized and, based on an exploratory literature search, further advanced and presented in an accessible visual format. This perspective opens up promising opportunities to extend the psychometric-toolbox by incorporating and learning from statistical methodologies developed in other research domains, which often address similar or even identical problems. Highlighting these methodological commonalities may also foster collaboration across research fields that have traditionally remained largely independent. Moreover, awareness of these connections may render methodological development more systematic and goal-directed and may enable a meaningful division of labor, for example between the development of statistical methodology and its practical implementation for empirical research through software tools. Finally, these methodological advances provide new opportunities for empirical research and may contribute to a reconciliation with longstanding conceptual issues concerning psychometric constructs and, more broadly, psychological phenomena.

Efficiently aggregating spatial or temporal horizons to acquire compact representations has become a unifying principle in modern deep learning models, yet learning data-adaptive representations for long-horizon sequence data, especially continuous sequences like time series, remains an open challenge. While fixed-size patching has improved scalability and performance, discovering variable-sized, data-driven patches end-to-end often forces models to rely on soft discretization, specific backbones, or heuristic rules. In this work, we propose Reinforcement Patching (ReinPatch), the first framework to jointly optimize a sequence patching policy and its downstream sequence backbone model using reinforcement learning. By formulating patch boundary placement as a discrete decision process optimized via Group Relative Policy Gradient (GRPG), ReinPatch bypasses the need for continuous relaxations and performs dynamic patching policy optimization in a natural manner. Moreover, our method allows strict enforcement of a desired compression rate, freeing the downstream backbone to scale efficiently, and naturally supports multi-level hierarchical modeling. We evaluate ReinPatch on time-series forecasting datasets, where it demonstrates compelling performance compared to state-of-the-art data-driven patching strategies. Furthermore, our detached design allows the patching module to be extracted as a standalone foundation patcher, providing the community with visual and empirical insights into the segmentation behaviors preferred by a purely performance-driven neural patching strategy.

We study rank selection for low-rank tensor regression under random covariates design. Under a Gaussian random-design model and some mild conditions, we derive population expressions for the expected training-testing discrepancy (optimism) for both CP and Tucker decomposition. We further demonstrate that the optimism is minimized at the true tensor rank for both CP and Tucker regression. This yields a prediction-oriented rank-selection rule that aligns with cross-validation and extends naturally to tensor-model averaging. We also discuss conditions under which under- or over-ranked models may appear preferable, thereby clarifying the scope of the method. Finally, we showcase its practical utility on a real-world image regression task and extend its application to tensor-based compression of neural network, highlighting its potential for model selection in deep learning.

We study estimation and detection of high-order cumulant tensors from $n$ i.i.d. observations of a $p$-dimensional random vector, with performance measured in the tensor spectral norm. Under sub-Gaussianity, we show that the minimax rate for estimating the order-$d$ cumulant tensor is $\sqrt{p/n}\wedge 1$. In contrast to covariance estimation, the natural plug-in estimator obtained by replacing population moments in the moment–cumulant formula with sample moments, which we call the sample cumulant tensor, is generally not rate-optimal for $d\ge 3$, and we construct an estimator that attains the minimax rate. On the computational side, we study testing whether the $d$-th order cumulant tensor vanishes after whitening. We provide evidence, under a model-specific low-degree conjecture, that detection is computationally hard when $n\ll p^{d/2}$. At the same time, we identify a regime in which an efficiently computable estimator achieves error smaller than the separation at which polynomial-time tests can reliably distinguish the null from the alternative under the same conjecture. This reveals an unusual reverse detection–estimation gap: computationally efficient detection can be harder than computationally efficient estimation. The underlying reason is that the relevant loss, tensor spectral norm, is itself NP-hard to compute, creating a new form of computational–statistical gap.

A framework for probabilistic forecasting of vessel motion is developed and validated for a semisubmersible operating in long period swell. Bayesian statistical methods are applied to predictions of the heave response from a physics model using numerical wave spectra and measured motion data. Model diagnoses motivate an additional level of complexity required for the error structure in the Bayesian model, specifically to account for heteroskedasticity and time-correlated errors. The hybrid model forecasts were evaluated during periods where the heave resonance and cancellation frequencies were excited. The method is demonstrated to be effective for providing reliable quantification of uncertainty and correcting bias in the raw physics model predictions. This justifies its value for improving the efficiency and safety of offshore operations.

This book develops the theory of quasi-Banach $K_\sigma$-spaces $\mathbf{F}_\psi(\Omega)$, $\mathbf{F}_\psi^*(\Omega)$, and $D_{V,W}(\Omega)$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Sub_\varphi(\Omega)$ and $V(\varphi,\psi)$ spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach $K_\sigma$-spaces are introduced, and the fundamental properties of $\mathbf{F}_\psi(\Omega)$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from $\mathbf{F}_\psi^*(\Omega)$, and the fourth addresses approximation theory in $SF_\psi(\Omega)$. The fifth chapter examines Orlicz spaces and their connections to $\mathbf{F}_\psi(\Omega)$. Chapters six and seven treat the pre-Banach $K_\sigma$-spaces $D_{V,W}(\Omega)$, establishing their essential properties and evaluating reliability and accuracy of stochastic process models. The eighth chapter provides norm distribution estimates in $L_p(T)$ for processes from $\mathbf{F}_\psi(\Omega)$. The ninth chapter develops the Monte Carlo method for multiple integrals over $\mathbb{R}^n$ with prescribed reliability and accuracy. The final two chapters treat modeling of $Sub_\varphi(\Omega)$ processes - subclasses of $K_\sigma$-spaces - with given reliability and accuracy in $L_p(T)$ and $C(T)$ respectively. The results are substantially based on the authors' original work and that of their co-authors.

Digital twins for radiation-based imaging and therapy are most useful when they assimilate patient data, quantify predictive uncertainty, and support clinically constrained decisions. This paper presents a modular framework for actionable digital twins in radiation-based imaging and therapy and instantiates its reproducible open-data component using the \openkbpfull{} benchmark. The framework couples PatientData, Model, Solver, Calibration, and Decision modules and formalizes latent-state updating, uncertainty propagation, and chance-constrained action selection. As an initial implementation, we build a GPU-ready PyTorch/MONAI reimplementation of the \openkbp{} starter pipeline: an 11-channel, 19.2M-parameter 3D U-Net trained with a masked loss over the feasible region and equipped with Monte Carlo dropout for voxel-wise epistemic uncertainty. To emulate the update loop on a static benchmark, we introduce decoder-only proxy recalibration and illustrate uncertainty-aware virtual-therapy evaluation using DVH-based and biological utilities. A complete three-fraction loop including recalibration, Monte Carlo inference, and spatial optimization executes in 10.3~s. On the 100-patient test set, the model achieved mean dose and DVH scores of 2.65 and 1.82~Gy, respectively, with 0.58~s mean inference time per patient. The \openkbp{} case study thus serves as a reproducible test bed for dose prediction, uncertainty propagation, and proxy closed-loop adaptation, while future institutional studies will address longitudinal calibration with delivered-dose logs and repeat imaging.

In High Energy Physics, detailed calorimeter simulations and reconstructions are essential for accurate energy measurements and particle identification, but their high granularity makes them computationally expensive. Developing data-driven techniques capable of recovering fine-grained information from coarser readouts, a task known as calorimeter superresolution, offers a promising way to reduce both computational and hardware costs while preserving detector performance. This thesis investigates whether a generative model originally designed for fast simulation can be effectively applied to calorimeter superresolution. Specifically, the model proposed in arXiv:2308.11700 is re-implemented independently and trained on the CaloChallenge 2022 dataset based on the Geant4 Par04 calorimeter geometry. Finally, the model's performance is assessed through a rigorous statistical evaluation framework, following the methodology introduced in arXiv:2409.16336, to quantitatively test its ability to reproduce the reference distributions.

Physics-informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are incorporated. In this work, we present a systematic comparison of different thermodynamic structure-informed neural networks by incorporating various thermodynamics formulations, including Newtonian, Lagrangian, and Hamiltonian mechanics for conservative systems, as well as the Onsager variational principle and extended irreversible thermodynamics for dissipative systems. Through comprehensive numerical experiments on representative ordinary and partial differential equations, we quantitatively evaluate the impact of these formulations on accuracy, physical consistency, noise robustness, and interpretability. The results show that Newtonian-residual-based PINNs can reconstruct system states but fail to reliably recover key physical and thermodynamic quantities, whereas structure-preserving formulation significantly enhances parameter identification, thermodynamic consistency, and robustness. These findings provide practical guidance for principled design of thermodynamics-consistency model, and lay the groundwork for integrating more general nonequilibrium thermodynamic structures into physics-informed machine learning.

Delayed outcomes are ubiquitous in online experimentation: treatment can affect whether an outcome occurs, when it occurs, and its realized value. To accommodate staggered entry while remaining robust to environmental nonstationarity and unit-level heterogeneity, we adopt a design-based perspective and target the sample cumulative reward in each arm as a function of calendar time. Our confidence sequences allow practitioners to continuously monitor the counterfactual incremental reward, such as revenue, that would have been realized by calendar time $t$ had all entered units been assigned to treatment rather than control. The main technical challenge is the choice of design-based filtration, complicated by the presence of asynchronous potential outcome times. We show that the IPW treatment-effect estimation error is not a martingale with respect to any filtration, while each arm-specific IPW estimation error is a martingale with respect to a carefully chosen arm-specific event-time filtration. We therefore construct a confidence sequence for the treatment effect by combining two arm-level confidence sequences with a union bound, and further demonstrate that this can outperform the traditional design-based variance upper bound. Finally, we characterize the class of augmentations for which the per-arm AIPW estimation error remains a martingale.

The Global Database of Events, Language and Tone (GDELT) provides geolocated event records that can be aggregated into weekly spatiotemporal panels of event counts across regions, actors, and event types. These panels are typically sparse, bursty, and overdispersed, so calibrated probabilistic forecasting is essential for monitoring rare surges. We propose Bayesian count regression pipelines that pair deterministic deep temporal encoders with negative binomial (NB2) and zero-inflated negative binomial (ZINB2) likelihood heads. Posterior predictive simulation yields predictive quantiles and right-tail probabilities that support both forecasting and anomaly scoring. For interpretable spillover attribution, we also fit a Bayesian generalised linear model with high-dimensional lagged cross-series predictors and a two-step screen-and-refit procedure under a three-parameter beta-normal (TPBN) shrinkage prior. To connect spillovers to directional statistics, active cross-region effects are mapped to geodesic bearings on the World Geodetic System 1984 ellipsoid (WGS84) and summarised using weighted circular moments, rose diagrams, and bearing-field maps. Simulations with known spillovers and conflict-panel case studies show accurate right-tail behaviour and a practical workflow for detecting and interpreting geopolitical shocks.

The sample paths of Brownian motion are known to admit the exact Besov-type smoothness exponent 1/2 when measured in the sub-Gaussian Orlicz norm. We extend these regularity results by deriving the exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces, and we provide a rate of convergence towards this limiting value. The central technique is a new chaining bound for the Orlicz modulus of a stochastic process. The latter also applies to polyogonal partial sum processes of functional random variables and allows us to strengthen Donsker's invariance principle to all function spaces on the Besov-Orlicz scale up to the exact modulus with exponent 1/2. For the critical case, we establish the thresholded weak convergence of the Besov-Orlicz seminorm of the partial sum process. The analytical results find application in a nonparametric statistical testing problem, where Besov-Orlicz statistics are shown to detect a broader range of alternatives compared to Hölderian multiscale statistics.

Timely and accurate conflict event data are essential for real-time monitoring, forecasting, and policy response. Yet near-real-time conflict datasets such as the Armed Conflict Location & Event Data Project (ACLED) are subject to reporting delays, that is, delays between event occurrence and first inclusion in the database. Such delays can introduce bias in short-term analyses and forecasts. This study provides a statistical analysis of reporting delays for African events recorded in ACLED's weekly releases from June 30, 2024, to June 1, 2025. Treating delay as a discrete time duration, we estimate grouped proportional hazards models with additive-linear and smooth terms incorporating event-level, spatial, and country-level covariates. Our results show that more than half of events are reported within two weeks, but delays vary systematically by event type, fatalities, geographic location, and political regime. Higher-fatality events are reported more quickly, while events in more restrictive political and informational environments tend to be reported more slowly. We also find substantial between-country heterogeneity, and country-specific analyses indicate that event-level effects differ across contexts. These findings show that reporting delays are structured rather than random and that real-time conflict analysis must account for them. More broadly, they provide an empirical foundation for developing nowcasting approaches to correct short-term underreporting in conflict event data.

In this work, we investigate how to develop sharp concentration inequalities for sub-Weibull random variables, including sub-Gaussian and sub-exponential distributions. Although the random variables may not be sub-Guassian, the tail probability around the origin behaves as if they were sub-Gaussian, and the tail probability decays align with the Orlicz $\Psi_\alpha$-tail elsewhere. Specifically, for independent and identically distributed (i.i.d.) $\{X_i\}_{i=1}^n$ with finite Orlicz norm $\|X\|_{\Psi_\alpha}$, our theory unveils that there is an interesting phase transition at $\alpha = 2$ in that $\PP\l(\l|\sum_{i=1}^n X_i \r| \geq t\r)$ with $t > 0$ is upper bounded by $2\exp\l(-C\max\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $\alpha\geq 2$, and by $2\exp\l(-C\min\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $1\leq \alpha\leq 2$ with some positive constant $C$. In many scenarios, it is often necessary to distinguish the standard deviation from the Orlicz norm when the latter can exceed the former greatly. To accommodate this, we build a new theoretical analysis framework, and our sharp, flexible concentration inequalities involve the variance and a mixing of Orlicz $\Psi_\alpha$-tails through the min and max functions. Our theory yields new, improved concentration inequalities even for the cases of sub-Gaussian and sub-exponential distributions with $\alpha = 2$ and $1$, respectively. We further demonstrate our theory on martingales, random vectors, random matrices, and covariance matrix estimation. These sharp concentration inequalities can empower more precise non-asymptotic analyses across different statistical and machine learning applications.

Regression by composition provides a flexible framework for constructing conditional distributions through sequential group actions. However, when multiple flows act on the same distribution, the model becomes non-identifiable, leading to flat likelihood regions and unstable estimates. We introduce a structured regularization framework that resolves this issue by assigning flow-specific penalties. The resulting estimator is defined as a penalized maximum likelihood problem with heterogeneous regularization across flows. We establish theoretical properties, including identifiability under penalization, uniqueness of the minimizer via strict convexification, and asymptotic consistency. For the adaptive Lasso, we further prove the oracle property. An efficient proximal gradient algorithm handles non-smooth penalties. Extensive simulation studies evaluate performance under varying sample sizes, correlation structures, and signal-to-noise ratios, demonstrating that regularized methods (Lasso and Elastic Net) successfully break non-identifiability and achieve low estimation error with controlled false positive rates. An application to NHANES data on asthma and lead exposure illustrates the practical utility: the unregularized estimator yields implausible coefficients, whereas regularized estimators produce stable and interpretable models and automatically select the relevant risk transformation. The Labbe plots derived from regularized estimators indicate a protective effect of reducing lead exposure. The proposed framework bridges identifiability theory with penalized estimation and opens the door to high-dimensional and longitudinal extensions.

We study dynamic regret minimization in unconstrained adversarial linear bandit problems. In this setting, a learner must minimize the cumulative loss relative to an arbitrary sequence of comparators $\boldsymbol{u}_1,\ldots,\boldsymbol{u}_T$ in $\mathbb{R}^d$, but receives only point-evaluation feedback on each round. We provide a simple approach to combining the guarantees of several bandit algorithms, allowing us to optimally adapt to the number of switches $S_T = \sum_t\mathbb{I}\{\boldsymbol{u}_t \neq \boldsymbol{u}_{t-1}\}$ of an arbitrary comparator sequence. In particular, we provide the first algorithm for linear bandits achieving the optimal regret guarantee of order $\mathcal{O}\big(\sqrt{d(1+S_T) T}\big)$ up to poly-logarithmic terms without prior knowledge of $S_T$, thus resolving a long-standing open problem.

Tensor-on-tensor regression is an important tool for the analysis of tensor data, aiming to predict a set of response tensors from a corresponding set of predictor tensors. However, standard tensor-on-tensor regression is sensitive to outliers, which may be present in both the response and the predictor. It can be affected by casewise outliers, which are observations that deviate from the bulk of the data, as well as by cellwise outliers, which are individual anomalous cells within the tensors. The latter are particularly common due to the typically large number of cells in tensor data. This paper introduces a novel robust tensor-on-tensor regression method, named ROTOT, that can handle both types of outliers simultaneously, and can cope with missing values as well. This method uses a single loss function to reduce the influence of both casewise and cellwise outliers in the response. The outliers in the predictor are handled using a robust Multilinear Principal Component Analysis method. Graphical diagnostic tools are also proposed to visualize the different types of detected outliers. The performance of ROTOT is evaluated through extensive simulations and further illustrated using the Labeled Faces in the Wild dataset, where ROTOT is applied to predict facial attributes.

Synchronous update schemes in p-bit annealing offer a natural route to massive parallelism, but they can also induce period-2 oscillations that degrade optimization performance. In practical solvers, such oscillations matter only if they become observable within the finite runtime of the device or simulation, yet most existing analyses are formulated in terms of asymptotic stability. As a result, they do not directly address the experimentally relevant question of when oscillatory modes actually appear during finite-duration annealing. Here we develop a finite-time observability framework for synchronous tick-random p-bit dynamics. Starting from a linearized mean-field description, we derive a graph-dependent criterion that predicts whether unstable modes amplify enough within a finite observation window to produce visible signatures in quantities such as the one-step autocorrelation and the energy trace. This shifts the analysis from asymptotic instability to practical detectability and yields a principled estimate of the minimum reduction in synchrony required to suppress oscillations. We validate the framework on G-set benchmark graphs and illustrative graph families. The predicted thresholds capture both the graph dependence of oscillation onset and the finite-time conditions under which oscillations become practically observable. These results provide a graph-aware basis for selecting the update probability in synchronous p-bit annealers without relying on exhaustive instance-by-instance parameter sweeps.

Self-supervised image denoising methods have traditionally relied on architectural constraints, pseudo-pair constructions, or specialized loss functions to avoid the trivial identity mapping. Among these, approaches such as Noisier2Noise or R2R create training pairs by adding synthetic noise to noisy images. While effective, these recorruption-based approaches require precise knowledge of the noise distribution, which is often unavailable. We present Learning to Recorrupt (L2R), a self-supervised framework that does not require exact knowledge of the noise distribution. Our method introduces a learnable recorruptor jointly optimized with the denoiser through a min–max saddle-point objective. The proposed method achieves state-of-the-art performance among methods without prior knowledge of the noise distribution across unconventional and heavy-tailed noise distributions, such as log-gamma and Laplace, as well as spatially correlated noise, while obtaining competitive results on real-world denoising benchmarks characterized by signal-dependent noise.

Transformers have revolutionized machine learning by making attention a central mechanism for modeling interactions within a context. Despite the central role of attention, the theoretical capabilities of Transformers for representing contextual relations remain unclear. In this work, we address this question by developing a mathematical framework based on probability and optimal transport. We view a text as a distribution of its representations and attention as a probabilistic relation between them. This perspective reveals a connection between attention normalization and optimal transport: standard softmax normalization produces conditional relations, while Sinkhorn normalization produces joint relations with prescribed marginals. Thus, both mechanisms provide structured probabilistic relations from attention scores. Under mild conditions, we establish universal approximation results for both settings. We show that Transformer architectures with Sinkhorn normalization can approximate arbitrary contextual relations represented as joint probabilities, while standard softmax Transformers can approximate arbitrary contextual relations represented as conditional probabilities. These results provide a mathematical characterization of the expressive power of Transformers for contextual relations and show how the choice of normalization determines the probabilistic structure of the relations represented by attention.

We study the high-dimensional linear regression problem with categorical predictors that have many levels. We propose a new estimation approach, which performs model compression via two mechanisms by simultaneously encouraging (a) clustering of the regression coefficients to collapse some of the categorical levels together; and (b) sparsity of the regression coefficients. We present novel mixed integer programming formulations for our estimator, and develop a custom row generation procedure to speed up the exact off-the-shelf solvers. We also propose a fast approximate algorithm for our method that obtains high-quality feasible solutions via block coordinate descent. As the main building block of our algorithm, we develop an exact algorithm for the univariate case based on dynamic programming, which can be of independent interest. We establish new theoretical guarantees for both the prediction and the cluster recovery performance of our estimator. Our numerical experiments on synthetic and real datasets demonstrate that our proposed estimator tends to outperform the state-of-the-art.