Papers

We propose GaussDetect-LiNGAM, a novel approach for bivariate causal discovery that eliminates the need for explicit Gaussianity tests by leveraging a fundamental equivalence between noise Gaussianity and residual independence in the reverse regression. Under the standard LiNGAM assumptions of linearity, acyclicity, and exogeneity, we prove that the Gaussianity of the forward-model noise is equivalent to the independence between the regressor and residual in the reverse model. This theoretical insight allows us to replace fragile and sample-sensitive Gaussianity tests with robust kernel-based independence tests. Experimental results validate the equivalence and demonstrate that GaussDetect-LiNGAM maintains high consistency across diverse noise types and sample sizes, while reducing the number of tests per decision (TPD). Our method enhances both the efficiency and practical applicability of causal inference, making LiNGAM more accessible and reliable in real-world scenarios.

Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods often require careful parameter tuning or prior knowledge of the sparsity of the signal and/or noise variance. We propose a tuning-free framework that leverages implicit regularization (IR) from overparameterization to overcome this limitation. Our approach reparameterizes the estimation matrix into factors that decouple the shared row-support from individual vector entries and applies gradient descent to a standard least-squares objective. We prove that with a sufficiently small and balanced initialization, the optimization dynamics exhibit a "momentum-like" effect where the true support grows significantly faster. Leveraging a Lyapunov-based analysis of the gradient flow, we further establish formal guarantees that the solution trajectory converges towards an idealized row-sparse solution. Empirical results demonstrate that our tuning-free approach achieves performance comparable to optimally tuned established methods. Furthermore, our framework significantly outperforms these baselines in scenarios where accurate priors are unavailable to the baselines.

Online A/B testing, the gold standard for evaluating new advertising policies, consumes substantial engineering resources and risks significant revenue loss from deploying underperforming variations. This motivates the use of Off-Policy Evaluation (OPE) for rapid, offline assessment. However, applying OPE to ad auctions is fundamentally more challenging than in domains like recommender systems, where stochastic policies are common. In online ad auctions, it is common for the highest-bidding ad to win the impression, resulting in a deterministic, winner-takes-all setting. This results in zero probability of exposure for non-winning ads, rendering standard OPE estimators inapplicable. We introduce the first principled framework for OPE in deterministic auctions by repurposing the bid landscape model to approximate the propensity score. This model allows us to derive robust approximate propensity scores, enabling the use of stable estimators like Self-Normalized Inverse Propensity Scoring (SNIPS) for counterfactual evaluation. We validate our approach on the AuctionNet simulation benchmark and against 2-weeks online A/B test from a large-scale industrial platform. Our method shows remarkable alignment with online results, achieving a 92% Mean Directional Accuracy (MDA) in CTR prediction, significantly outperforming the parametric baseline. MDA is the most critical metric for guiding deployment decisions, as it reflects the ability to correctly predict whether a new model will improve or harm performance. This work contributes the first practical and validated framework for reliable OPE in deterministic auction environments, offering an efficient alternative to costly and risky online experiments.

Federated Learning (FL) is plagued by two key challenges: high communication overhead and performance collapse on heterogeneous (non-IID) data. Analytic FL (AFL) provides a single-round, data distribution invariant solution, but is limited to linear models. Subsequent non-linear approaches, like DeepAFL, regain accuracy but sacrifice the single-round benefit. In this work, we break this trade-off. We propose SAFLe, a framework that achieves scalable non-linear expressivity by introducing a structured head of bucketed features and sparse, grouped embeddings. We prove this non-linear architecture is mathematically equivalent to a high-dimensional linear regression. This key equivalence allows SAFLe to be solved with AFL's single-shot, invariant aggregation law. Empirically, SAFLe establishes a new state-of-the-art for analytic FL, significantly outperforming both linear AFL and multi-round DeepAFL in accuracy across all benchmarks, demonstrating a highly efficient and scalable solution for federated vision.

We introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.

A major effort in modern high-dimensional statistics has been devoted to the analysis of linear predictors trained on nonlinear feature embeddings via empirical risk minimization (ERM). Gaussian equivalence theory (GET) has emerged as a powerful universality principle in this context: it states that the behavior of high-dimensional, complex features can be captured by Gaussian surrogates, which are more amenable to analysis. Despite its remarkable successes, numerical experiments show that this equivalence can fail even for simple embeddings – such as polynomial maps – under general scaling regimes. We investigate this breakdown in the setting of random feature (RF) models in the quadratic scaling regime, where both the number of features and the sample size grow quadratically with the data dimension. We show that when the target function depends on a low-dimensional projection of the data, such as generalized linear models, GET yields incorrect predictions. To capture the correct asymptotics, we introduce a Conditional Gaussian Equivalent (CGE) model, which can be viewed as appending a low-dimensional non-Gaussian component to an otherwise high-dimensional Gaussian model. This hybrid model retains the tractability of the Gaussian framework and accurately describes RF models in the quadratic scaling regime. We derive sharp asymptotics for the training and test errors in this setting, which continue to agree with numerical simulations even when GET fails. Our analysis combines general results on CLT for Wiener chaos expansions and a careful two-phase Lindeberg swapping argument. Beyond RF models and quadratic scaling, our work hints at a rich landscape of universality phenomena in high-dimensional ERM.

Partially labelled samples arise when features are observed for data, but class labels are available for only a subset. In such settings, the mechanism governing label availability may itself contain information relevant to classification, yet it is typically left unmodelled in standard semi-supervised learning procedures. The SSLfmm package implements likelihood-based Gaussian finite-mixture classification in which the label-missingness process is modelled jointly with the class distribution. It supports complete-case, missing completely at random (MCAR), entropy-based missing at random (MAR), and mixed analyses in which MCAR and MAR mechanisms may both contribute. For the mixed mechanism, the source of a missing label may be known or unknown. A common R interface is provided for model fitting, prediction, performance assessment, and simulation. We describe the statistical formulation and software implementation and demonstrate its use through reproducible simulation and a semi-synthetic Blood Transfusion application.

The compatibility constant plays an important role in evaluating the prediction error of the lasso in high-dimensional settings. However, the computation of the compatibility constant is generally difficult because it is a complicated nonconvex optimization problem. In this study, we present a numerical approach to compute the compatibility constant when the support of true regression coefficients is given. We show that the optimization problem reduces to a quadratic programming (QP) once the signs of the nonzero coefficients are specified. In this case, the compatibility constant can be obtained by solving QPs for all possible sign combinations. We also formulate a mixed-integer QP (MIQP) approach that can be applied when the number of true nonzero coefficients is large. We investigate the finite-sample behavior of the compatibility constant for simulated data under various parameter settings and compare the prediction error with its theoretical upper bound. The behavior of the compatibility constant in finite samples is also investigated through a real data analysis.

Financial markets exhibit an apparent paradox: while directional price movements remain largely unpredictable–consistent with weak-form efficiency–the magnitude of price changes displays systematic structure. Here we demonstrate that real-time order-flow entropy, computed from a 15-state Markov transition matrix at second resolution, predicts the magnitude of intraday returns without providing directional information. Analysis of 38.5 million SPY trades over 36 trading days reveals that conditioning on entropy below the 5th percentile increases subsequent 5-minute absolute returns by a factor of 2.89 (t = 12.41, p < 0.0001), while directional accuracy remains at 45.0%–statistically indistinguishable from chance (p = 0.12). This decoupling arises from a fundamental symmetry: entropy is invariant under sign permutation, detecting the presence of informed trading without revealing its direction. Walk-forward validation across five non-overlapping test periods confirms out-of-sample predictability, and label-permutation placebo tests yield z = 14.4 against the null. These findings suggest that information-theoretic measures may serve as volatility state variables in market microstructure, though the limited sample (36 days, single instrument) requires extended validation.

The emergence of large language models (LLMs) has sparked much interest in creating LLM-based digital populations that can be applied to many applications such as social simulation, crowdsourcing, marketing, and recommendation systems. A digital population can reduce the cost of recruiting human participants and alleviate many concerns related to human subject study. However, research has found that most of the existing works rely solely on LLMs and could not sufficiently capture the accuracy and diversity of a real human population. To address this limitation, we propose CrowdLLM that integrates pretrained LLMs and generative models to enhance the diversity and fidelity of the digital population. We conduct theoretical analysis of CrowdLLM regarding its great potential in creating cost-effective, sufficiently representative, scalable digital populations that can match the quality of a real crowd. Comprehensive experiments are also conducted across multiple domains (e.g., crowdsourcing, voting, user rating) and simulation studies which demonstrate that CrowdLLM achieves promising performance in both accuracy and distributional fidelity to human data.

Classification of functional data where observations are curves or trajectories poses unique challenges, particularly under severe class imbalance. Traditional Random Forest algorithms, while robust for tabular data, often fail to capture the intrinsic structure of functional observations and struggle with minority class detection. This paper introduces Functional Random Forest with Adaptive Cost-Sensitive Splitting (FRF-ACS), a novel ensemble framework designed for imbalanced functional data classification. The proposed method leverages basis expansions and Functional Principal Component Analysis (FPCA) to represent curves efficiently, enabling trees to operate on low dimensional functional features. To address imbalance, we incorporate a dynamic cost sensitive splitting criterion that adjusts class weights locally at each node, combined with a hybrid sampling strategy integrating functional SMOTE and weighted bootstrapping. Additionally, curve specific similarity metrics replace traditional Euclidean measures to preserve functional characteristics during leaf assignment. Extensive experiments on synthetic and real world datasets including biomedical signals and sensor trajectories demonstrate that FRF-ACS significantly improves minority class recall and overall predictive performance compared to existing functional classifiers and imbalance handling techniques. This work provides a scalable, interpretable solution for high dimensional functional data analysis in domains where minority class detection is critical.

We analyze a novel large-scale social-media-based measure of U.S. job satisfaction, constructed by applying a fine-tuned large language model to 2.6 billion georeferenced tweets, and link it to county-level labor market conditions (2013-2023). Logistic regressions show that rural counties consistently report lower job satisfaction sentiment than urban ones, but this gap decreases under tight labor markets. In contrast to widening rural-urban income disparities, perceived job quality converges when unemployment is low, suggesting that labor market slack, not income alone, drives spatial inequality in subjective work-related well-being.

The article by García-Donato and co-authors addresses the dual challenges of accounting for model uncertainty and missing data within the Gaussian regression frameworks from an objective Bayesian perspective. Thru the use of an imputation $g$-prior that replaces $X_\gamma^TX_\gamma$ for model $\gamma$ in the covariance of $\beta_\gamma$ with $\Sigma_{X_\gamma}$, the authors develop a coherent approach to addressing the missing data problem and model uncertainty simultaneously with random $X_\gamma$ in the missing at random (MAR) or missing completely at random (MCAR) settings, while still being computationally tractable. I discuss the connection of the imputation $g$-prior to the $g$-prior with imputed $X$, and to model selection for graphical models that provide an alternative justification for the $g$-prior for random $X$s.

We introduce a novel framework for change point detection in spherical functional autoregressive (SPHAR) processes, enabling the identification of structural breaks in spatio-temporal random fields on the sphere. Our LASSO-regularized estimator, based on penalized dynamic programming in the harmonic domain, operates without knowledge of the number or locations of change points and offers non-asymptotic theoretical guarantees. This approach provides a new tool for analyzing nonstationary phenomena on the sphere, relevant to climate science, cosmology, and beyond.

The conditional average treatment effect (CATE) is frequently estimated in clinical studies to refute a homogeneous treatment effect hypothesis. Under this regime, all patients making up the population experience identical benefit from a given treatment relative to a comparator. Uncovering heterogeneous treatment effects through inference about the CATE, however, requires that covariates truly modifying the treatment effect be reliably collected at baseline. CATE-based techniques will necessarily fail to detect violations when effect modifiers are omitted from the data due to, for example, resource constraints. Severe measurement error has a similar impact. Clinical decision makers can be misled as a result. To address these limitations, we prove that a practical homogeneous treatment effect hypothesis can be gauged through inference about contrasts of the potential outcomes' variances even when effect modifiers are missing from the data. We derive causal machine learning estimators of these contrasts and study their asymptotic properties. We establish that these estimators are doubly robust and asymptotically linear under mild conditions, permitting formal hypothesis testing about treatment effect heterogeneity. Numerical experiments demonstrate that these estimators' asymptotic guarantees are approximately achieved in finite-sample randomized and observational study data alike. These inference procedures are then used to detect heterogeneous treatment effects in the re-analysis of randomized controlled trials investigating targeted temperature management in cardiac arrest patients.

We frame novelty detection on path space as a hypothesis testing problem with signature-based test statistics. Using transportation-cost inequalities of Gasteratos and Jacquier (2023), we obtain tail bounds for false positive rates that extend beyond Gaussian measures to laws of RDE solutions with smooth bounded vector fields, yielding estimates of quantiles and p-values. Exploiting the shuffle product, we derive exact formulae for smooth surrogates of conditional value-at-risk (CVaR) in terms of expected signatures, leading to new one-class SVM algorithms optimising smooth CVaR objectives. We then establish lower bounds on type-$\mathrm{II}$ error for alternatives with finite first moment, giving general power bounds when the reference measure and the alternative are absolutely continuous with respect to each other. Finally, we evaluate numerically the type-$\mathrm{I}$ error and statistical power of signature-based test statistic, using synthetic anomalous diffusion data and real-world molecular biology data.

This paper establishes the first analytical relationship between predictive model performance and loss ratio in insurance pricing. We derive a closed-form formula connecting the Pearson correlation between predicted and actual losses to expected loss ratio. The framework proves that model improvements exhibit diminishing marginal returns, analytically confirming the actuarial intuition to prioritize poorly performing models. We introduce the Loss Ratio Error metric for quantifying business impact across frequency, severity, and pure premium models. Simulations show reliable predictions under stated assumptions, with graceful degradation under assumption violations. This framework transforms model investment decisions from qualitative intuition to quantitative cost-benefit analysis.

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

We study semiparametric factor models in high-dimensional panels where the factor loadings consist of a nonparametric component explained by observed covariates and an idiosyncratic component capturing unobserved heterogeneity. A key challenge in empirical applications is the presence of missing observations, which can distort both factor recovery and loading estimation. To address this issue, we develop a projected principal component analysis (PPCA) procedure that accommodates general missing-at-random mechanisms through inverse-probability weighting. We establish consistency and derive the asymptotic distributions of the estimated factors and loading functions, allowing the sieve dimension to diverge and permitting the time dimension to be either fixed or growing. Unlike classical PCA, PPCA achieves consistent factor estimation even when T is fixed, and the limiting distributions under missing data exhibit mixture normality with enlarged asymptotic variances. Theoretical results are supported by simulations and an empirical application. Our findings demonstrate that PPCA provides an effective and robust framework for estimating semiparametric factor models in the presence of missing data.

We introduce iterative tilting, a gradient-free method for fine-tuning diffusion models toward reward-tilted distributions. The method decomposes a large reward tilt $\exp(\lambda r)$ into $N$ sequential smaller tilts, each admitting a tractable score update via first-order Taylor expansion. This requires only forward evaluations of the reward function and avoids backpropagating through sampling chains. We validate on a two-dimensional Gaussian mixture with linear reward, where the exact tilted distribution is available in closed form.

We investigate the convergence properties of a class of iterative algorithms designed to minimize a potentially non-smooth and noisy objective function, which may be algebraically intractable and whose values may be obtained as the output of a black box. The algorithms considered can be cast under the umbrella of a generalised gradient descent recursion, where the gradient is that of a smooth approximation of the objective function. The framework we develop includes as special cases model-based and mollification methods, two classical approaches to zero-th order optimisation. The convergence results are obtained under very weak assumptions on the regularity of the objective function and involve a trade-off between the degree of smoothing and size of the steps taken in the parameter updates. As expected, additional assumptions are required in the stochastic case. We illustrate the relevance of these algorithms and our convergence results through a challenging classification example from machine learning.

We study estimation and statistical inference for reward models used in aligning large language models (LLMs). A key component of LLM alignment is reinforcement learning from human feedback (RLHF), where humans compare pairs of model-generated answers and their preferences are used to train a reward model. However, human feedback is inherently heterogeneous, creating significant challenges for reliable reward learning. To address this, we adopt a heterogeneous preference framework that jointly models the latent reward of answers and human rationality. This leads to a challenging biconvex optimization problem, which we solve via an alternating gradient descent algorithm. We establish theoretical guarantees for the resulting estimator, including its convergence and asymptotic distribution. These results enable the construction of confidence intervals for reward estimates. Leveraging these uncertainty quantification results, we conduct valid statistical comparisons between rewards and incorporate uncertainty into the best-of-$N$ (BoN) policy framework. Extensive simulations demonstrate the effectiveness of our method, and applications to real LLM data highlight the practical value of accounting for uncertainty in reward modeling for LLM alignment.

We present the winning strategy for the EVA2025 Data Challenge, which aimed to estimate the probability of extreme precipitation events. These events occurred at most once in the dataset making the challenge fundamentally one of extrapolating extreme values. Given the scarcity of extreme events, we argue that a simple, robust modeling approach is essential. We adopt univariate models instead of multivariate ones and model Peaks Over Thresholds using Extreme Value Theory. Specifically, we fit an exponential distribution to model exceedances of the target variable above a high quantile (after seasonal adjustment). The novelty of our approach lies in using martingale testing to evaluate the extrapolation power of the procedure and to agnostically select the level of the high quantile. While this method has several limitations, we believe that framing extrapolation as a game opens the door to other agnostic approaches in Extreme Value Analysis.

Shapley values, a gold standard for feature attribution in Explainable AI, face two key challenges. First, the canonical Shapley framework assumes that the worth function is additive, yet real-world payoff constructions–driven by non-Gaussian distributions, heavy tails, feature dependence, or domain-specific loss scales–often violate this assumption, leading to distorted attributions. Second, achieving sparse explanations in high-dimensional settings by computing dense Shapley values and then applying ad hoc thresholding is costly and risks inconsistency. We introduce Sparse Isotonic Shapley Regression (SISR), a unified nonlinear explanation framework. SISR simultaneously learns a monotonic transformation to restore additivity–obviating the need for a closed-form specification–and enforces an L0 sparsity constraint on the Shapley vector, enhancing computational efficiency in large feature spaces. Its optimization algorithm leverages Pool-Adjacent-Violators for efficient isotonic regression and normalized hard-thresholding for support selection, ensuring ease in implementation and global convergence guarantees. Analysis shows that SISR recovers the true transformation in a wide range of scenarios and achieves strong support recovery even in high noise. Moreover, we are the first to demonstrate that irrelevant features and inter-feature dependencies can induce a true payoff transformation that deviates substantially from linearity. Extensive experiments demonstrate that SISR stabilizes attributions across payoff schemes and correctly filters irrelevant features; in contrast, standard Shapley values suffer severe rank and sign distortions. By unifying nonlinear transformation estimation with sparsity pursuit, SISR advances the frontier of nonlinear explainability, providing a theoretically grounded and practical attribution framework.

Agentic AI systems execute a sequence of actions, such as reasoning steps or tool calls, in response to a user prompt. To evaluate the success of their trajectories, researchers have developed verifiers, such as LLM judges and process-reward models, to score the quality of each action in an agent's trajectory. Although these heuristic scores can be informative, there are no guarantees of correctness when used to decide whether an agent will yield a successful output. Here, we introduce e-valuator, a method to convert any black-box verifier score into a decision rule with provable control of false alarm rates. We frame the problem of distinguishing successful trajectories (that is, a sequence of actions that will lead to a correct response to the user's prompt) and unsuccessful trajectories as a sequential hypothesis testing problem. E-valuator builds on tools from e-processes to develop a sequential hypothesis test that remains statistically valid at every step of an agent's trajectory, enabling online monitoring of agents over arbitrarily long sequences of actions. Empirically, we demonstrate that e-valuator provides greater statistical power and better false alarm rate control than other strategies across six datasets and three agents. We additionally show that e-valuator can be used for to quickly terminate problematic trajectories and save tokens. Together, e-valuator provides a lightweight, model-agnostic framework that converts verifier heuristics into decisions rules with statistical guarantees, enabling the deployment of more reliable agentic systems.

Large language models (LLMs) produce fluent but unsupported answers - hallucinations - limiting safe deployment in high-stakes domains. We propose ECLIPSE, a framework that treats hallucination as a mismatch between a model's semantic entropy and the capacity of available evidence. We combine entropy estimation via multi-sample clustering with a novel perplexity decomposition that measures how models use retrieved evidence. We prove that under mild conditions, the resulting entropy-capacity objective is strictly convex with a unique stable optimum. We evaluate on a controlled financial question answering dataset with GPT-3.5-turbo (n=200 balanced samples with synthetic hallucinations), where ECLIPSE achieves ROC AUC of 0.89 and average precision of 0.90, substantially outperforming a semantic entropy-only baseline (AUC 0.50). A controlled ablation with Claude-3-Haiku, which lacks token-level log probabilities, shows AUC dropping to 0.59 with coefficient magnitudes decreasing by 95% - demonstrating that ECLIPSE is a logprob-native mechanism whose effectiveness depends on calibrated token-level uncertainties. The perplexity decomposition features exhibit the largest learned coefficients, confirming that evidence utilization is central to hallucination detection. We position this work as a controlled mechanism study; broader validation across domains and naturally occurring hallucinations remains future work.

Real-world measurements often comprise a dominant signal contaminated by a noisy background. Robustly estimating the dominant signal in practice has been a fundamental statistical problem. Classically, mixture models have been used to cluster the heterogeneous population into homogeneous components. Modeling such data with fully parametric models risks bias under misspecification, while fully nonparametric approaches can dissipate power and computational resources. We propose a middle path: a semiparametric method that models only the dominant component parametrically and leaves the background completely nonparametric, yet remains computationally scalable and statistically robust. So instead of outlier downweighting, traditionally done in robust statistics literature, we maximize the observed likelihood such that the noisy background is absorbed by the nonparametric component. Computationally, we propose a new approximate FFT-accelerated likelihood maximization algorithm. Empirically, this FFT plug-in achieves order-of-magnitude speedups over vanilla weighted EM while preserving statistical accuracy and large sample properties.

This paper develops new identification results for multidimensional continuous measurement-error models where all observed measurements are contaminated by potentially correlated errors and none provides an injective mapping of the latent distribution. Using third order cross moments, the paper constructs a three way tensor whose unique decomposition, guaranteed by Kruskal theorem, identifies the factor loading matrices. Starting with a linear structure, the paper recovers the full distribution of latent factors by constructing suitable measurements and applying scalar or multivariate versions of Kotlarski identity. As a result, the joint distribution of the latent vector and measurement errors is fully identified without requiring injective measurements, showing that multivariate latent structure can be recovered in broader settings than previously believed. Under injectivity, the paper also provides user-friendly testable conditions for identification. Finally, this paper provides general identification results for nonlinear models using a newly-defined generalized Kruskal rank - signal rank - of intergral operators. These results have wide applicability in empirical work involving noisy or indirect measurements, including factor models, survey data with reporting errors, mismeasured regressors in econometrics, and multidimensional latent-trait models in psychology and marketing, potentially enabling more robust estimation and interpretation when clean measurements are unavailable.

This paper is concerned with the problem of how to speed up computation for Gaussian process models trained on autocorrelated data. The Gaussian process model is a powerful tool commonly used in nonlinear regression applications. Standard regression modeling assumes random samples and an independently, identically distributed noise. Various fast approximations that speed up Gaussian process regression work under this standard setting. But for autocorrelated data, failing to account for autocorrelation leads to a phenomenon known as temporal overfitting that deteriorates model performance on new test instances. To handle autocorrelated data, existing fast Gaussian process approximations have to be modified; one such approach is to segment the originally correlated data points into blocks in which the blocked data are de-correlated. This work explains how to make some of the existing Gaussian process approximations work with blocked data. Numerical experiments across diverse application datasets demonstrate that the proposed approaches can remarkably accelerate computation for Gaussian process regression on autocorrelated data without compromising model prediction performance.

We consider additive functionals of systems of random measures whose initial configuration is given by a Poisson point process, and whose individual components evolve according to arbitrary Markovian or non-Markovian measure valued dynamics, with no structural assumptions beyond basic moment bounds. In this setting and under adequate conditions, we establish a general third moment theorem for the normalized functionals. Building on this result, we obtain the first quantitative bounds in the Wasserstein distance for a variety of moving-measure models initialized by Poisson-driven clouds of points, turning qualitative central limit theorems into explicit rates of convergence. The scope of the approach is then demonstrated through several examples, including systems driven by fractional Brownian motion, $\alpha$-stable processes, uniformly elliptic diffusions, and spectral empirical measures arising from Dyson Brownian motion, all under broad assumptions on the control measure of the initial Poisson configuration. The analysis relies on a combination of Stein's method with Mecke's formula, in the spirit of the Poisson Malliavin-Stein methodology.