Papers

LLM-as-a-Judge is a flexible framework for text evaluation, which allows us to obtain scores for the quality of a given text from various perspectives by changing the prompt template. Two main challenges in using LLM-as-a-Judge are computational cost of inference using a large language model (LLM), especially when evaluating a large number of instances, and inherent bias of an LLM evaluator. To address these issues and reveal the structure of score bias caused by an LLM evaluator, we propose to apply a tensor clustering method to a given LLM-as-a-Judge score tensor, whose entries are the scores for different combinations of questions, answerers, and evaluators. Specifically, we develop a new tensor clustering method MultiwayPAM, with which we can simultaneously estimate the cluster membership and the medoids for each mode of a given data tensor. By observing the medoids obtained by MultiwayPAM, we can gain knowledge about the membership of each question/answerer/evaluator cluster. We experimentally show the effectiveness of MultiwayPAM by applying it to the score tensors for two practical datasets.

The development of credit valuation adjustment (CVA) (valuation adjustments [XVA]) [Green] has increased the importance of simple interest rate models such as the Hull-White model [Tan14] [Tsuchiya]. This is because the XVA model is an FX hybrid model, and is tractable only when the interest rate part is a simple Gaussian model. For the XVA calculation of interest rate instruments, de-correlation of the yield curve can be important even for the swap portfolio. Capturing the correlation structure in the two-factor Hull-White model is an integral element of CVA (XVA) modeling. However, the correlation structure in two-factor Hull-White model has not studied enough except for the analysis in [AndersenPiterbarg]. In this study, the correlation structure of the two-factor Hull-White model is analyzed in detail. The correlation structure of co-initial swap rates is investigated using a combination of the approximation formula and Monte-Carlo simulation. The Hull-White model captures the de-correlation of the yield curve only when the parameters (volatilities and mean reversion strength) satisfy certain relationships, making the valuation of XVA by two-factor Hull-White model effective.

As regression is a widely studied problem, many methods have been proposed to solve it, each of them often requiring setting different hyper-parameters. Therefore, selecting the proper method for a given application may be very difficult and relies on comparing their performances. Performance is usually measured using various metrics such as Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), or R-squared (R${}^2$). These metrics provide a numerical summary of predictive accuracy by quantifying the difference between predicted and actual values. However, while these metrics are widely used in the literature for summarizing model performance and useful to distinguish between models performing poorly and well, they often aggregate too much information. This article addresses these limitations by introducing a novel visualization approach that highlights key aspects of regression model performance. The proposed method builds upon three main contributions: (1) considering the residuals in a 2D space, which allows for simultaneous evaluation of errors from two models, (2) leveraging the Mahalanobis distance to account for correlations and differences in scale within the data, and (3) employing a colormap to visualize the percentile-based distribution of errors, making it easier to identify dense regions and outliers. By graphically representing the distribution of errors and their correlations, this approach provides a more detailed and comprehensive view of model performance, enabling users to uncover patterns that traditional aggregate metrics may obscure. The proposed visualization method facilitates a deeper understanding of regression model performance differences and error distributions, enhancing the evaluation and comparison process.

Classical Monte Carlo methods for pricing catastrophe insurance tail risk converge at order reciprocal root N, requiring large simulation budgets to resolve upper-tail percentiles of the loss distribution. This sample-sparsity problem can lead to AI models trained on impoverished tail data, producing poorly calibrated risk estimates where insolvency risk is greatest. Quantum Amplitude Estimation (QAE), following Montanaro, achieves convergence approaching order reciprocal N in oracle queries - a quadratic speedup that, at scale, would enable high-resolution tail estimation within practical budgets. We validate this advantage empirically using a Qiskit Aer simulator with genuine Grover amplification. A complete pipeline encodes fitted lognormal catastrophe distributions into quantum oracles via amplitude encoding, producing small readout probabilities that enable safe Grover amplification with up to k=16 iterations. Seven experiments on synthetic and real (NOAA Storm Events, 58,028 records) data yield three main findings: an oracle-model advantage, that strong classical baselines win when analytical access is available, and that discretisation, not estimation, is the current bottleneck.

AutoRegressive Conditional Heteroscedasticity (ARCH) models are standard for modeling time series exhibiting volatility, with a rich literature in univariate and multivariate settings. In recent years, these models have been extended to function spaces. However, functional ARCH and generalized ARCH (GARCH) processes established in the literature have thus far been restricted to model “pointwise” variances. In this paper, we propose a new ARCH framework for data residing in general separable Hilbert spaces that accounts for the full evolution of the conditional covariance operator. We define a general operator-level ARCH model. For a simplified Constant Conditional Correlation version of the model, we establish conditions under which such models admit strictly and weakly stationary solutions, finite moments, and weak serial dependence. Additionally, we derive consistent Yule–Walker-type estimators of the infinite-dimensional model parameters. The practical relevance of the model is illustrated through simulations and a data application to high-frequency cumulative intraday returns.

Bayesian hierarchical models are frequently used in practical data analysis contexts. One interpretation of these models is that they provide an indirect way of assigning a prior for unknown parameters, through the introduction of hyperparameters. The resulting marginal prior for the parameters (integrating over the hyperparameters) is usually dependent, so that learning one parameter provides some information about the others. In this contribution, I will demonstrate that, when the prior given the hyperparameters is a canonical distribution (a maximum entropy distribution with moment constraints), the dependent marginal prior also has a maximum entropy property, with a different constraint. This constraint is on the marginal distribution of some function of the unknown quantities. The results shed light on what information is actually being assumed when we assign a hierarchical model.

In this paper, we propose a trust-region interior-point stochastic sequential quadratic programming (TR-IP-SSQP) method for solving optimization problems with a stochastic objective and deterministic nonlinear equality and inequality constraints. In this setting, exact evaluations of the objective function and its gradient are unavailable, but their stochastic estimates can be constructed. In particular, at each iteration our method builds stochastic oracles, which estimate the objective value and gradient to satisfy proper adaptive accuracy conditions with a fixed probability. To handle inequality constraints, we adopt an interior-point method (IPM), in which the barrier parameter follows a prescribed decaying sequence. Under standard assumptions, we establish global almost-sure convergence of the proposed method to first-order stationary points. We implement the method on a subset of problems from the CUTEst test set, as well as on logistic regression problems, to demonstrate its practical performance.

We study a continuous-time diffusion approximation of policy gradient for $k$-armed stochastic bandits. We prove that with a learning rate $\eta = O(\Delta^2/\log(n))$ the regret is $O(k \log(k) \log(n) / \eta)$ where $n$ is the horizon and $\Delta$ the minimum gap. Moreover, we construct an instance with only logarithmically many arms for which the regret is linear unless $\eta = O(\Delta^2)$.

Many scientific fields compare two or more noisy time series that integrate the same underlying process but are recorded on different time scales. In paleoclimate studies, for example, proxy measurements are collected versus stratigraphic depth in a climate archive and then converted to calendar time. Synchronizing two proxy records often requires estimating an alignment that maps the depth (or preliminary age) of an input record onto the calendar–time scale of an absolutely–dated target record so that corresponding proxy signals line up. Existing alignment approaches are generally optimization–based and return a single transformation, providing limited formal uncertainty quantification. Here, we introduce BSync, a Bayesian synchronization framework that treats alignments as inference over a monotone time–mapping function to match an input to a target record. The alignment is expressed as a transformation of the input depth (or age) scale to match the target record, achieved through a link function that locally expands and compresses the input scale. The model is parameterized through interpretable local rate parameters, enabling the specification of priors on deposition times to regularize the alignment toward physically plausible deformations. BSync jointly infers the aligned chronology and provides posterior uncertainty for the time–warping function and the resulting age scale. In synthetic data experiments and a real–data case study, BSync yields well–calibrated credible intervals for the aligned time scale and achieves more accurate alignments than a state–of–the–art automated method, particularly when independent age constraints are sparse.

Recovering a tree that represents the evolutionary history of a group of species is a key task in phylogenetics. Performing this task using sequence data from multiple genetic markers poses two key challenges. The first is the discordance between the evolutionary history of individual genes and that of the species. The second challenge is computational, as contemporary studies involve thousands of species. Here we present SDSR, a scalable divide-and-conquer approach for species tree reconstruction based on spectral graph theory. The algorithm recursively partitions the species into subsets until their sizes are below a given threshold. The trees of these subsets are reconstructed by a user-chosen species tree algorithm. Finally, these subtrees are merged to form the full tree. On the theoretical front, we derive recovery guarantees for SDSR, under the multispecies coalescent (MSC) model. We also perform a runtime complexity analysis. We show that SDSR, when combined with a species tree reconstruction algorithm as a subroutine, yields substantial runtime savings as compared to applying the same algorithm on the full data. Empirically, we evaluate SDSR on synthetic benchmark datasets with incomplete lineage sorting and horizontal gene transfer. In accordance with our theoretical analysis, the simulations show that combining SDSR with common species tree methods, such as CA-ML or ASTRAL, yields up to 10-fold faster runtimes. In addition, SDSR achieves a comparable tree reconstruction accuracy to that obtained by applying these methods on the full data.

Personalized medicine aims to tailor treatments to individual patients, especially when people respond heterogeneously to therapies. A key objective is to learn individualized treatment rules that recommend optimal treatments from patient characteristics. Outcome weighted learning (OWL) is an important framework because it reformulates the task as a weighted classification problem targeting clinical benefit and using modern machine learning tools. Existing OWL theory has been focusing on specific surrogate losses and Gaussian kernels. Matern kernels, which allow adjustable smoothness and better match many real world data structures, are often more suitable and include the Gaussian kernel as a special case. This work develops a general relationship between population 0-1 risk and risks from a broad class of nonnegative surrogate losses using a constrained variational transformation. The transform simplifies for convex losses and provides simple expressions for certain nonconvex losses. A condition is established that ensures a nontrivial upper bound on the excess 0-1 risk. The paper establishes convergence rates for kernel based OWL under smoothness conditions with Matern kernels or geometric noise conditions with Gaussian kernels for both convex and nonconvex losses. It also proposes two iteratively reweighted convex optimization algorithms. Simulations and an application to ACTG 175 show strong performance.

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry–Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai–Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate – the rate attained by unregularized algorithms such as EXP3 – so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

Background: A core aspect of epidemiology is determining the impacts of potential public health interventions over time. With long follow-up periods, epidemiologists may need to consider semi-competing events, in which a terminal event, like death, precludes a non-terminal event, like hypertension. Time-varying confounding poses an additional challenge when studying time-varying interventions or actions. Existing methods do not simultaneously address semi- competing events and time-varying confounding. Methods: We propose two novel g-computation algorithms for causal effects with semi- competing events and time-varying actions. To explore performance of our novel g-computation estimators, we conducted a Monte Carlo simulation study. We then applied our estimator to investigate how cigarette smoking prevention throughout young and middle adulthood might impact prevalent hypertension using data from Waves III (aged 18-26 years) - VI (aged 39-51 years) of the National Longitudinal Study of Adolescent to Adult Health. Results: Our simulations show that the novel g-computation estimators had little bias and appropriate confidence interval coverage. They outperformed existing alternative estimators across sample sizes. In the illustrative application, the novel estimator identified a small reduction in prevalence of hypertension and risk of death in midlife had all cigarette smoking been prevented across follow-up compared to the observed smoking patterns. Conclusion: As long-running cohorts progress in age, death within the study sample will become an increasing concern for studies of aging-related outcomes, life course analyses, and investigations into chronic disease development. Our novel g-computation estimators provide a simultaneous solution.

Small area estimators that ignore the sampling design lack design consistency when the sampling mechanism is complex and may be severely biased under informative designs. Existing procedures that account for the survey weights under unit-level models typically focus on a single response variable. This paper addresses the estimation of area means for several dependent target variables under a multivariate nested error regression (MNER) model. We propose a multivariate pseudo-empirical best linear unbiased predictor that accounts for the sampling mechanism. Moreover, by aggregating the MNER model, we derive a unified predictor that can be obtained from either unit-level or area-level data. Bootstrap procedures are proposed to estimate the mean squared errors (MSEs) of the proposed predictors. Simulation experiments are conducted to examine the properties of the proposed small area estimators and the MSE estimators. Finally, an application with housing data illustrates the proposed methods.

Hyperspectral images capture vast amounts of high-dimensional spectral information about a scene, making labeling an intensive task that is resistant to out-of-the-box statistical methods. Unsupervised learning of clusters allows for automated segmentation of the scene, enabling a more rapid understanding of the image. Partitioning the spectral information contained within the data via dictionary learning in Wasserstein space has proven an effective method for unsupervised clustering. However, this approach requires balancing the spectral profiles of the data, blurring the classes, and sacrificing robustness to outliers and noise. In this paper, we suggest improving this approach by utilizing unbalanced Wasserstein barycenters to learn a lower-dimensional representation of the underlying data. The deployment of spectral clustering on the learned representation results in an effective approach for the unsupervised learning of labels.

Time-series forecasting often faces challenges from non-stationarity, particularly distributional drift, where the data distribution evolves over time. This dynamic behavior can undermine the effectiveness of adaptive optimizers, such as Adam, which are typically designed for stationary objectives. In this paper, we revisit Adam in the context of non-stationary forecasting and identify that its second-order bias correction limits responsiveness to shifting loss landscapes. To address this, we propose TS_Adam, a lightweight variant that removes the second-order correction from the learning rate computation. This simple modification improves adaptability to distributional drift while preserving the optimizer core structure and requiring no additional hyperparameters. TS_Adam integrates easily into existing models and consistently improves performance across long- and short-term forecasting tasks. On the ETT datasets with the MICN model, it achieves an average reduction of 12.8% in MSE and 5.7% in MAE compared to Adam. These results underscore the practicality and versatility of TS_Adam as an effective optimization strategy for real-world forecasting scenarios involving non-stationary data. Code is available at: https://github.com/DD-459-1/TS_Adam.

Cancer subtyping plays a crucial role in informing prognosis and guiding personalized treatment strategies. However, conventional subtyping approaches often rely on static, biopsy-derived scores that hardly capture the biological heterogeneity and temporal evolution of the disease. In this study, we propose a novel trajectory-informed clustering method for cancer subtyping that integrates multi-modal clinical data and longitudinal patient trajectories. Our method constructs a patient similarity graph using time-varying imaging-derived features, clinical covariates, and transitions among key clinical states such as therapy, surveillance, relapse, and death. This graph structure enables the identification of patient subgroups that are not only phenotypically and genotypically distinct but also aligned with patterns of disease progression. We position our approach within the landscape of existing subtyping methods and highlight its advantages in terms of temporal modeling and graph-based interpretability. Through simulation studies and application to a real world dataset of liver metastases, we demonstrate the ability of our framework to uncover clinically relevant subtypes with distinct prognostic trajectories. Our results underscore the potential of trajectory-informed clustering to enhance personalized oncology by bridging cross-sectional biomarkers with dynamic disease evolution.

We study estimation of a deterministic binary count from the shuffled outputs of a common single-message local channel. Privacy is required for every neighboring composition at fixed global parameters $(\varepsilon,\delta)$, and estimators may be biased. We obtain uniform critical Poisson and Skellam approximations and characterize the limiting minimax risk as an infimum over finite observed-mark Poisson experiments. Let $\Lambda$ be the noise intensity required by the boundary Poisson-shift profile and let $\lambda_{\min}$ be the intensity required by the full composition-maximized randomized-response profile. Binary randomized response is asymptotically minimax optimal if and only if $\Lambda=\lambda_{\min}$. When $\Lambda<\lambda_{\min}$, the minimax value lies strictly between $(\Lambda+\lambda_{\min})/2$ and $\lambda_{\min}$, and a randomized-response construction with observed rare marks strictly improves on binary randomized response. We give a finite-sample certified example. Binary randomized response is strictly suboptimal at $\varepsilon=0$ for every $0<\delta<1$, and this persists for sufficiently small positive $\varepsilon$ at each fixed target. For every fixed positive $\varepsilon$, strict suboptimality also occurs along a sequence of fixed privacy targets tending to zero. The variational characterization permits approximation by finite marked channels; it does not assert that a finite optimizer exists.

This paper proves that the Epistemic Support-Point Filter (ESPF) is the unique optimal recursive estimator within the class of epistemically admissible evidence-only filters. Where Bayesian filters minimize mean squared error and are driven toward an assumed truth, the ESPF minimizes maximum entropy and surfaces what has not been proven impossible – a fundamentally different epistemic commitment with fundamentally different failure modes. Two results locate this theorem within the broader landscape of estimation theory. The first is a unification: the ESPF's optimality criterion is the log-geometric mean of the alpha-cut volume family in the Holder mean hierarchy. The Popperian minimax bound and the Kalman MMSE criterion occupy the p=+inf and p=0 positions on the same curve. Possibility and probability are not competing frameworks: they are the same ignorance functional evaluated under different alpha-cut geometries. The Kalman filter is the Gaussian specialization of the ESPF's optimality criterion, not a separate invention. The second result is a diagnostic: numerical validation over a 2-day, 877-step Smolyak Level-3 orbital tracking run shows that possibilistic stress manifests through necessity saturation and surprisal escalation rather than MVEE sign change – a direct consequence of the Holder ordering, not an empirical observation. Three lemmas establish the result: the Possibilistic Entropy Lemma decomposes the ignorance functional; the Possibilistic Cramer-Rao Bound limits entropy reduction per measurement; the Evidence-Optimality Lemma proves minimum-q selection is the unique minimizer and that any rule incorporating prior possibility risks race-to-bottom bias.

A central question in modern deep learning is how to design optimizers whose behavior remains stable as the network width $w$ increases. We address this question by interpreting several widely used neural-network optimizers, including AdamW and Muon, as instances of steepest descent under matrix operator norms. This perspective links optimizer geometry with the Lipschitz structure of the network forward map, and enables width-independent control of both Lipschitz and smoothness constants. However, steepest-descent rules induced by standard $p \to q$ operator norms lack layerwise composability and therefore cannot provide width-independent bounds in deep architectures. We overcome this limitation by introducing a family of mean-normalized operator norms, denoted $\pmean \to \qmean$, that admit layerwise composability, yield width-independent smoothness bounds, and give rise to practical optimizers such as rescaled AdamW, row normalization, and column normalization. The resulting learning rate width-aware scaling rules recover $\mu$P scaling~\cite{yang2021tensor} as a special case and provide a principled mechanism for cross-width learning-rate transfer across a broad class of optimizers. We further show that Muon can suffer an $\mathcal{O}(\sqrt{w})$ worst-case growth in the smoothness constant, whereas a new family of row-normalized optimizers we propose achieves width-independent smoothness guarantees. Based on the observations, we propose MOGA (Matrix Operator Geometry Aware), a width-aware optimizer based only on row/column-wise normalization that enables stable learning-rate transfer across model widths. Large-scale pre-training on GPT-2 and LLaMA shows that MOGA, especially with row normalization, is competitive with Muon while being notably faster in large-token and low-loss regimes.

Adaptive enrichment trials aim to identify and recruit participants most likely to benefit from treatment based on evolving biomarker evidence, with the goal of informing individualized treatment recommendations. Bayesian methods are well suited to these designs because they allow external information to be incorporated in a principled manner. In practice, prior studies often provide only summary-level information, with subgroup-specific estimates unavailable due to design or privacy constraints. Existing dynamic borrowing approaches therefore rely on aggregate measures, such as the average treatment effect, and implicitly assume that historical information maps directly onto model parameters. In adaptive enrichment settings aimed at identifying individualized treatment effects, however, subgroup-specific treatment parameters are not identifiable when only marginal historical effects are available. To address this gap, we propose a Bayesian adaptive enrichment design that borrows information from external studies using a normalized power prior anchored on one or more summary measures, such as the average treatment effect. { To our knowledge, no existing method addresses this gap.} Interim analyses use posterior probabilities to guide early stopping for efficacy or futility, or to continue recruitment within promising biomarker-defined subgroups. Simulation studies evaluate operating characteristics across historical bias, sample size, and prior informativeness. Together with a motivating future trial in obstructive sleep apnea, the results show efficiency gains versus non-borrowing designs, including improved power, earlier stopping, and reduced expected sample size.

Surrogate modeling is an essential data-driven technique for quantifying relationships between input variables and system responses in manufacturing and engineering systems. Two major challenges limit its effectiveness: (1) large data requirements for learning complex nonlinear relationships, and (2) heterogeneous data collected from sources with varying fidelity levels. Multi-task learning (MTL) addresses the first challenge by enabling information sharing across related processes, while multi-fidelity modeling addresses the second by accounting for fidelity-dependent uncertainty. However, existing approaches typically address these challenges separately, and no unified framework simultaneously leverages inter-task similarity and fidelity-dependent data characteristics. This paper develops a novel hierarchical multi-task multi-fidelity (H-MT-MF) framework for Gaussian process-based surrogate modeling. The proposed framework decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation. The framework accommodates an arbitrary number of tasks, design points, and fidelity levels while providing predictive uncertainty quantification. We demonstrate the effectiveness of the proposed method using a 1D synthetic example and a real-world engine surface shape prediction case study. Compared to (1) a state-of-the-art MTL model that does not account for fidelity information and (2) a stochastic kriging model that learns tasks independently, the proposed approach improves prediction accuracy by up to 19% and 23%, respectively. The H-MT-MF framework provides a general and extensible solution for surrogate modeling in manufacturing systems characterized by heterogeneous data sources.

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation $\xi$ and Blest's rank correlation $\nu$ over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush–Kuhn–Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both $\xi$ and $\nu$, which provide an explicit parametrization of the exact attainable region.

Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.

We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.

Suppose $\widehat\theta_n$ is a strongly consistent estimator for $\theta_0$ in some i.i.d. situation. Let $N_\varepsilon$ and $Q_\varepsilon$ be respectively the last $n$ and the total number of $n$ for which $\widehat\theta_n$ is at least $\varepsilon$ away from $\theta_0$. The limit distributions for ${\varepsilon}^2 N_\varepsilon$ and ${\varepsilon}^2 Q_\varepsilon$ as $\varepsilon$ goes to zero are obtained under natural and weak conditions. The theory covers both parametric and nonparametric cases, multi-dimensional parameters, and general distance functions. Our results are of probabilistic interest, and, on the statistical side, suggest ways in which competing estimators can be compared. In particular several new optimality properties for the maximum likelihood estimator sequence in parametric families are established. Another use of our results is ways of constructing sequential fixed-volume or shrinking-volume confidence sets, as well as sequential tests with power 1. The paper also includes limit distribution results for the last $n$ and the number of $n$ for which the supremum distance $\|F_n-F\|\ge\varepsilon$, where $F_n$ is the empirical distribution function. Yet other results are reached for $\varepsilon^{5/2} N_\varepsilon$ and $\varepsilon^{5/2} Q_\varepsilon$ in the context of nonparametric density estimation, referring to the last time and the number of times where $|f_n(x) f(x)|\ge\varepsilon$. Finally it is shown that our results extend to several non-i.i.d. situations.

In this paper, we study the problem of detecting multiple hidden submatrices in a large Gaussian random matrix when the planted signal is inhomogeneous across entries. Under the null hypothesis, the observed matrix has independent and identically distributed standard normal entries. Under the alternative, there exist several planted submatrices whose entries deviate from the background in one of two ways: in the mean-shift model, planted entries (templates) have nonzero and possibly varying means; in the variance-shift model, planted entries have inflated and possibly varying variances. We consider two placement regimes for the planted submatrices. In the first, the row and column index sets are arbitrary. Motivated by scientific applications, in the second regime the row and column indices are restricted to be consecutive. For both alternatives and both placement regimes, we analyze the statistical limits of detection by proving information-theoretic lower bounds and by designing algorithms that match these bounds up to logarithmic factors, for a wide family of templates.

Non-negative matrix factorisation (NMF) is a widely used tool for unsupervised learning and feature extraction, with applications ranging from genomics to text analysis and signal processing. Standard formulations of NMF are typically derived under Gaussian or Poisson noise assumptions, which may be inadequate for data exhibiting overdispersion or other complex mean-variance relationships. In this paper, we develop a unified framework for both traditional and convex NMF under a broad class of distributional assumptions, including Negative Binomial and Tweedie models, where the connection between the Tweedie and the $\beta$-divergence is also highlighted. Using a Majorize-Minimisation approach, we derive multiplicative update rules for all considered models, and novel updates for convex NMF with Poisson and Negative Binomial cost functions. We provide a unified implementation of all considered models, including the first implementations of several convex NMF models. Empirical evaluations on mutational and word count data demonstrate that the choice of noise model critically affects model fit and feature recovery, and that convex NMF can provide an efficient and robust alternative to traditional NMF in scenarios where the number of classes is large. The code for our proposed updates is available in the R package nmfgenr and can be found at https://github.com/MartaPelizzola/nmfgenr.

The traditional Triangular Maximally Filtered Graph (TMFG) construction requires pre-computation and storage of a dense correlation matrix; this limits its applicability to small and medium-sized datasets. Here we identify key memory and runtime complexity challenges when using TMFG at scale. We then present the Approximate Triangular Maximally Filtered Graph (a-TMFG) algorithm. This is a novel approach to scaling the construction of artificial graphs from data inspired by TMFG. The method employs k-Nearest Neighbors Graphs (kNNG) for initial construction, and implements a memory management strategy to search and estimate missing correlations on-the-fly. This provides representations to control combinatorial explosion. The algorithm is tested for robustness to the parameters and noise, and is evaluated on datasets with millions of observations. This new method provides a parsimonious way to construct graphs for use-cases where graphs are used as input to supervised and unsupervised learning but where no natural graph exists.

Bandits with noncompliance separate the learner's recommendation from the treatment actually delivered, so the learning target itself must be chosen. A platform may care about recommendation welfare in the current mediated workflow, treatment learning for a future direct-control regime, or anytime-valid uncertainty for one of those targets. These objectives need not agree. We formalize this objective-choice problem, identify the direct-control regime in which recommendation and treatment objectives collapse, and show by example that recommendation welfare can strictly exceed every learner-measurable treatment policy when downstream actors use private information. For finite-context square-IV problems we propose BRACE, a parameter-free phase-doubling algorithm that performs IV inversion only after matrix certification and otherwise returns full-range but honest structural intervals. BRACE delivers simultaneous policy-value validity, fixed-gap identification of the operationally optimal recommendation policy, and fixed-gap identification of the structurally optimal treatment policy under contextual homogeneity and invertibility. We complement the theory with a finite-context empirical benchmark spanning direct control, mediated present-versus-future tradeoffs, weak identification, homogeneity failure, and rectangular overidentification. The experiments show that safety appears as regret on easy problems, as abstention and wide valid intervals under weak identification, as a reason to prefer recommendation welfare under homogeneity failure, and as tighter structural uncertainty when extra instruments are available. For rich contexts, we also derive an orthogonal score whose conditional bias factorizes into compliance-model and outcome-model errors, clarifying what must be stabilized for anytime-valid semiparametric IV inference.